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\section{Introduction}
There are no natural boundaries for the gravitational field analogous to the
conducting boundaries that play a major role in electromagnetism. In
principle, there is no need to introduce any. The behavior of the universe
as a whole can be posed as an initial value (Cauchy) problem. In an initial
value problem, data is given on a spacelike hypersurface ${\cal S}_0$. The
problem is to determine a solution in the future domain of dependence ${\cal
D}^{+}({\cal S}_0)$, which consists of those points whose past directed
characteristics all intersect ${\cal S}_0$. The problem is well-posed if
there exists a unique solution which depends continuously on the initial
data. The pioneering work of Y. Bruhat~\cite{bruhat} showed that the initial
value problem for the (classical) vacuum gravitational field is well-posed.
Assuming that matter fields do not spoil things, this suggests that the
global cosmological problem of treating the universe as a whole can be
solved in a physically meaningful way, i.e. in a way such that the solution
does not undergo uncontrolled variation under a perturbation of the initial
data. This is indeed the case for the presently accepted cosmological model
of an accelerating universe (positive cosmological constant) where the
conformal boundary at future null infinity ${\cal I}^+$ is spacelike. In a
conformally compactified picture, ${\cal I}^+$ acts as a spacelike cap on
the future evolution domain and no boundary condition is necessary or indeed
allowed.
In practice, of course, treating an isolated system as part of a global
cosmological spacetime is too complicated a problem without oversimplifying
assumptions such as isotropy or homogeneity. One global approach applicable
to isolated systems is to base the Cauchy problem on the analogue of a
foliation of Minkowski spacetime by the hyperboloidal hypersurfaces
\begin{equation}
t^2 - x^2 -y^2 - z^2 = T^2 , \quad t \ge T .
\end{equation} In a Penrose conformally compactified picture~\cite{penrose},
this foliation asymptotes to the light cones and extends to a foliation of
future null infinity ${\cal I}^+$. The analogue in curved spacetime is a
foliation by positive constant mean curvature hypersurfaces. Since no light
rays can enter an asymptotically flat spacetime through ${\cal I}^+$, no
boundary data are needed to evolve the interior spacetime. In addition, the
waveform and polarization of the outgoing radiation can be unambiguously
calculated at ${\cal I}^+$ in terms of the Bondi news function~\cite{bondi}.
This approach was first extensively developed by Friedrich~\cite{helconf}
who formulated a hyperbolic version of the Einstein-Bianchi system of
equations, which is manifestly regular at ${\cal I}^+$, in terms of the
conformally rescaled metric, connection and Weyl curvature. This is
potentially the basis for a very attractive numerical approach to simulate
global problems such as gravitational wave production. For reviews of
progress on the numerical implementation
see~\cite{fraurev,saschrev1,saschrev2}. There has been some success in
simulating model axisymmetric problems~\cite{frauhein}. More recently, there
have been other attempts at the hyperboloidal approach based upon the
Einstein equations for the conformal metric. Zengino{\u g}lu~\cite{zen} has
implemented a code based upon a generalized harmonic formulation in which
the gauge source terms produce a hyperbolic foliation. A mixed
hyperbolic-elliptic system proposed by Moncrief and Rinne~\cite{moncrinn}
has been implemented as an axisymmetric code~\cite{rinn} which produces long
term stable evolutions. Another hyperbolic-elliptic system based upon a
tetrad approach has been developed by Bardeen, Sarbach and
Buchman~\cite{bardsarbuch}. In spite of the attractiveness of the
hyperboloidal approach and its success with model problems, considerable
work remains to make it applicable to systems of astrophysical interest.
A different global approach is to match the Cauchy evolution inside a a
finite worldtube to an exterior characteristic evolution extending to ${\cal
I}^+$. In this approach, called Cauchy-characteristic matching, the
characteristic evolution is constructed by using the Cauchy evolution to
supply characteristic data on an inner worldtube, while the characteristic
evolution supplies the outer boundary data for the Cauchy evolution. The
success of Cauchy-characteristic matching depends upon the proper
mathematical and computational treatment of the initial-boundary value
problem (IBVP) for the Cauchy evolution. This approach has been
successfully implemented in the linearized
regime~\cite{harl} but also needs considerable additional work to apply to
astrophysical systems. See~\cite{winrev} for a review.
Instead of a global treatment, the standard approach in numerical
relativity, as in computational studies of other hyperbolic systems, is to
introduce an artificial outer boundary. Ideally, the outer boundary
treatment is designed to represent a passive external universe by allowing
radiation to cross only in the outgoing direction. This is the primary
application of the IBVP. Other possible
applications, which I will not consider, are the timelike conformal boundary
to a universe with negative cosmological constant and the membranes which
play a role in higher dimensional theories. While there are no natural
boundaries in classical gravitational theory, boundaries do play a central
role in the ideas of holographic duality introduced in higher dimensional
attempts at quantum gravity. Such applications are also beyond the scope of
this review as well as beyond my own expertise. Here, I confine my attention
to 4-dimensional spacetime, although the techniques governing a well-posed
IBVP readily extend to hyperbolic systems in any dimension.
In the IBVP, data on a timelike boundary ${\cal T}$, which meets ${\cal
S}_0$ in a surface ${\cal B}_0$, are used to further extend the solution of
the Cauchy problem to the domain of dependence ${\cal D}^{+}({\cal S}_0 \cup
{\cal T})$. In the simulation of an isolated astrophysical system containing
neutron stars and black holes, the outer boundary ${\cal T}$ is coincident
with the boundary of the computational grid and ${\cal B}_0$ is
topologically a sphere surrounding the system. However, for purposes of
treating the underlying mathematical and computational problems, it suffices
to concentrate on the local problem in the neighborhood of some point on the
intersection ${\cal B}_0$ between the Cauchy hypersurface ${\cal S}_0$ and
the boundary ${\cal T}$. For hyperbolic systems, the global solution in the
spacetime manifold ${\cal M}$ can be obtained by patching together local
solutions. This is because the finite speed of propagation allows {\it
localization} of the problem. The setting for this local problem is depicted
in Fig.~\ref{fig:bound}.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=.4]{bound}
\caption{Data on the 3-manifolds ${\cal S}_0$ and ${\cal T}$, which
intersect in the 2-surface ${\cal B}_0$, locally determine a solution
in the spacetime manifold ${\cal M}$}.
\end{center}
\label{fig:bound}
\end{figure}
The IBVP for Einstein's equations only received widespread attention after
its importance to the artificial outer boundaries used in numerical
relativity was pointed out~\cite{stewart}. The first {\it strongly}
well-posed IBVP was achieved for a tetrad version of the Einstein-Bianchi
system, expressed in first differential order form, which included the
tetrad, connection and curvature tensor as evolution fields~\cite{fn}.
Strong well-posedness guarantees the existence of a unique solution which
depends continuously on both the Cauchy data and the boundary data. A
strongly well-posed IBVP was later established for the harmonic formulation
of Einstein's equations as a system of second order quasilinear wave
equations for the metric~\cite{wpgs}. The results were further generalized
in~\cite{wpe,isol} to apply to a general quasilinear class of symmetric
hyperbolic systems whose boundary conditions have a certain hierarchical
form.
A review of the IBVP in general relativity must per force be of a different
nature than a review of the Cauchy problem. The local properties of the
Cauchy problem are now well understood. Several excellent reviews
exist~\cite{crev1,crev2,crev3,crev4}. For the IBVP, the results are not
comprehensive and are closely tied to the choice of hyperbolic reduction of
Einstein's equations. There are only a few universal features and, in
particular, there is no satisfactory treatment of the 3+1 formulation which
is extensively used in numerical relativity. For that reason, I will adopt a
presentation which differs somewhat from the standard approach with the
motivation of setting up a bare bones framework whose flexibility might be
helpful in further investigations.
My presentation is also biased by the important role of the IBVP in
numerical relativity, which treats Einstein's equations as a set of partial
differential equations (PDEs) governing the metric in some preferred
coordinate system. On the other hand, from a geometrical perspective, one of
the most fundamental and beautiful results of general relativity is that the
properties of the local Cauchy problem can be summed up in geometric terms
independent of any coordinates or explicit PDEs. This geometric formulation
only came about after the Cauchy problem was well understood from the PDE
point-of-view. The importance of the geometric approach to the numerical
relativist is that it supplies a common starting point for discussing and
comparing different formulations of Einstein's equations. Presently, the PDE
aspects of metric formulations of the IBVP are only understood in the
harmonic formulation. In order to transfer this insight into other
formulations a geometric framework can serve as an important guide. For that
reason, I will shift often between the PDE and geometric approach. When the
emphasis is on the geometric side I will use abstract indices, e.g. $v^a$ to
denote a vector field, and on the PDE side I will use coordinate indices, e.g.
$v^\mu= (v^t,v^i)$, to denote components with respect to spacetime
coordinates $x^\mu= (t,x^i)$.
The standard mathematical approach to the IBVP is to first establish the
well-posedness of the underlying Cauchy problem, and next the local
half-space problem. If these individual problems are well-posed (in a sense
to be qualified later) then the problems with more general boundaries will
also be well-posed. Thus I start my review in Sec.~\ref{sec:cauchy} by first
providing some brief background material for the Cauchy problem.
Next, in Sec.~\ref{sec:compl}, I point out the complications in going from
the Cauchy problem to the IBVP. The IBVP for Einstein's equations is not
well understood due to problems arising from the constraint equations. The
motivation for this work stems from the need for an improved understanding
and implementation of boundary conditions in the computational codes being
used to simulate binary black holes. The ability to compute the details of
the gravitational radiation produced by compact astrophysical sources, such
as coalescing black holes, is of major importance to the success of
gravitational wave astronomy. If the simulation of such systems is based upon
a well-posed Cauchy problem but not a well-posed IBVP then the
results cannot be trusted in the domain of dependence of the outer boundary.
In Sec's~\ref{sec:bare},~\ref{sec:initial} and~\ref{sec:hyperb}, I present the
underlying mathematical theory.
Early computational work in general relativity focused on the Cauchy problem
and the IBVP only received considerable attention after its importance to
stable and accurate simulations was recognized.
I discuss some history of the work on the IBVP in Sec.~\ref{sec:history},
for the purpose of pointing out some of the partial
successes and ideas which may be of future use.
In addition to the mathematical issue of an appropriate boundary condition,
the description of a binary black hole as an isolated system raises the
physical issue of the appropriate outer boundary data. In the absence of an
exterior solution, which could provide this data by matching, the standard
practice is to set this data to zero. This raises the question, discussed in
Sec.~\ref{sec:absorbc}, of how to formulate a non-reflecting outer boundary
condition in order to avoid spurious incoming radiation.
Sec's~\ref{sec:fn} and~\ref{sec:harm} describe the two strongly
well-posed formulations of the
IBVP, which are known at the present time. Neither is based upon a $3+1$
formulation and they both handle the constraints in different ways. This
prompts the discussion, in Sec.~\ref{sec:constr}, of constraint enforcement
in the $3+1$ formulations. The resolution of issues regarding
geometric uniqueness, discussed in Sec.~\ref{sec:geom}, would shed light on
the universal features of the IBVP that would perhaps guide the way to
a successful $3+1$ treatment.
There is also the computational problem of turning a well-posed IBVP into a
stable and accurate evolution code. I will not go into the details of the
large range of techniques which are necessary for the successful
implementation of a numerical relativity code. Since initiating this
review, I have learned of a separate review in progress~\cite{sarbtig} which covers
such numerical techniques in great detail. Building a numerical relativity
code is a complex undertaking. As observed by Post and Votta~\cite{postvot}
in a study of large scale computational projects, ``the peer review process
in computational science generally doesn't provide as effective a filter as
it does for experiment or theory. Many things that a referee cannot detect
could be wrong with a computational science paper\ldots The few existing
studies of error levels in scientific computer codes indicate that the
defect rate is about seven faults per 1000 lines of Fortran''. They
emphasize that ``New methods of verifying and validating complex codes are
mandatory if computational science is to fulfill its promise for science and
society''. These observations are especially pertinent for numerical
relativity where validation by agreement with experiment is not yet possible.
In that spirit, I discuss the code tests that have been proposed and carried
out for the gravitational IBVP in Sec.~\ref{sec:num}.
My aim has been to present the background material which might open new
avenues for a better understanding of the IBVP and lead to progress on some
of the important open questions posed in Sec.~\ref{sec:quest}.
\section {The Cauchy problem}
\label{sec:cauchy}
Here I summarize those aspects of the Cauchy problem which are
fundamental to the IBVP. For more detail, see~\cite{crev1,crev2,crev3,crev4}.
In contrast to Newtonian theory, which describes gravity in terms of an
elliptic Poisson equation that propagates the gravitational field
instantaneously, the retarded interactions implicit in general relativity
give rise to new features such as gravitational waves. Wave propagation
results from the mathematical property that Einstein's equations can be
reduced to a hyperbolic system of PDEs. However, the coordinate freedom in
Einstein's theory admits gauge waves which propagate with arbitrarily high
speeds, including speeds faster than light. Einstein's equations are not
a priori a hyperbolic system in which propagation speeds must be bounded and
for which an initial value problem can be posed.
This crucial step in going from Einstein's equations to a hyperbolic system
has been highlighted by Friedrich as the process of hyperbolic
reduction~\cite{Friedrich}. The first and most famous example of hyperbolic
reduction was through the introduction of harmonic coordinates, which led to
the classic result that the Cauchy problem for the harmonic formulation of
Einstein's equations is well-posed~\cite{bruhat}. Here I summarize the
hyperbolic reduction of Einstein's equations in in terms of generalized
harmonic coordinates $x^\alpha=(t,x^i)=(t,x,y,z)$, which are functionally
independent solutions of the curved space scalar wave equation
\begin{equation}
\Box x^\mu = \frac{1}{\sqrt{-g}}\partial_\alpha
(\sqrt{-g}g^{\alpha\beta}\partial_\beta x^\mu) =-\hat \Gamma^\mu ,
\end{equation}
where $\hat \Gamma^\mu$ are gauge source functions~\cite{Friedrich}. In
terms of the connection $\Gamma^\mu_{\alpha\beta}$, these harmonic
conditions are
\begin{equation}
{\cal C}^\mu :=\Gamma^\mu -\hat \Gamma^\mu =0,
\label{eq:harmcond}
\end{equation}
where
\begin{equation}
\Gamma^\mu = g^{\alpha\beta}\Gamma^\mu_{\alpha\beta}=
-\frac{1}{\sqrt {-g}}\partial_\alpha ( \sqrt{-g}g^{\alpha\mu} ).
\end{equation}
The hyperbolic reduction of the Einstein tensor results from setting
\begin{equation}
E^{\mu\nu}:= G^{\mu\nu} -\nabla^{(\mu}{\cal C}^{\nu)}
+\frac{1}{2}g^{\mu\nu}\nabla_\rho{\cal C}^\rho =0 ,
\label{eq:creduced}
\end{equation}
where $C^\nu$ is treated formally as a vector field in constructing the
``covariant'' derivatives $\nabla^{\mu}C^{\nu}$. (In generalized harmonic
formulations based upon a background connection, $C^\nu$ is a legitimate
vector field. See Sec.~\ref{sec:harm}.)
When the harmonic gauge source functions have the functional dependence
$\hat \Gamma^\nu(x,g)$, the principal part of
(\ref{eq:creduced}) reduces to the wave operator acting on the densitized
metric, i.e.
\begin{equation}
E^{\mu\nu}= \frac{1}{2\sqrt{-g}}\partial_\alpha \bigg (g^{\alpha\beta}
\partial_\beta( \sqrt{-g}g^{\mu\nu})\bigg ) + \text{lower order terms}.
\label{eq:reduced}
\end{equation}
Thus the harmonic evolution equations (\ref{eq:creduced}) are quasilinear
wave equations for the components of the densitized metric
$\sqrt{-g}g^{\mu\nu}$. The well-posedness of the Cauchy problem for the
system (\ref{eq:creduced}) then follows from known results for systems of
quasilinear wave equations. (It is important to bear in mind that such
results are local in time since there is no general theory for the global
existence of solutions to nonlinear equations.)
In turn, the well-posedness of the Cauchy problem for the harmonic Einstein
equations also follows provided that the harmonic conditions ${\cal
C}^\mu=0$ are preserved under the evolution. The proof of constraint
preservation results from applying the contracted Bianchi identity $\nabla_\mu
G^{\mu\nu} =0$ to (\ref{eq:creduced}). This leads to a homogeneous wave
equation for ${\cal C}^\mu$,
\begin{equation}
\nabla^\rho \nabla_\rho \, {\cal C}^\mu +R^\mu_\rho \,{\cal C}^\rho=0.
\label{eq:bianchi}
\end{equation}
If the initial data enforce
\begin{equation}
{\cal C}^\mu |_{{\cal S}_0}= 0
\label{eq:c0}
\end{equation}
and
\begin{equation}
\partial_t {\cal C}^\mu |_{{\cal S}_0}=0
\label{eq:ct0}
\end{equation}
then the unique solution of (\ref{eq:bianchi}) is ${\cal C}^\rho=0$. It is
easy to satisfy (\ref{eq:c0}) by algebraically determining the initial
values of $\partial_t g^{\mu t}$ in terms of the initial values of
$g^{\mu\nu}$ and their spatial derivatives. In order to see how to satisfy
(\ref{eq:ct0}) note that the reduced equations (\ref{eq:reduced}) imply
\begin{equation}
G^{\mu\nu} n_\nu=n_\nu \nabla^{(\mu}{\cal C}^{\nu)}
-\frac{1}{2}n^\mu \nabla_\rho{\cal C}^\rho,
\label{eq:Gn}
\end{equation}
where
$$ n_\nu= -\frac{1}{\sqrt{-g^{tt}}}\partial_\nu t
$$
is the unit timelike normal to the Cauchy hyperurfaces. Thus if
\begin{equation}
G^{\mu\nu} n_\nu|_{{\cal S}_0} =0,
\label{eq:hamomc}
\end{equation}
i.e. if the Hamiltonian and momentum constraints are satisfied by the
initial data, and if the reduced equations (\ref{eq:creduced}) are satisfied
then it follows that
\begin{equation}
[n_\nu \nabla^{(\mu}{\cal C}^{\nu)}
-\frac{1}{2}n^\mu \nabla_\rho{\cal C}^\rho ]|_{{\cal S}_0} =0.
\label{eq:ndc}
\end{equation}
It is easy to check that (\ref{eq:ndc}) implies that $\partial_t {\cal
C}^\mu |_{{\cal S}_0}=0$ provided ${\cal C}^\mu |_{{\cal S}_0}= 0$.
As a result, the traditional Hamiltonian and momentum constraints on the
initial data, along with the reduced evolution equations
(\ref{eq:creduced})), imply that the initial conditions (\ref{eq:c0}) and
(\ref{eq:ct0}) required for preserving the harmonic conditions are satisfied.
Conversely, if the Hamiltonian and momentum constraints are satisfied
initially, then (\ref{eq:Gn}) ensures that they will be preserved under
harmonic evolution. Thus the conditions ${\cal C}^\nu =0$ can be considered
as the constraints of the generalized harmonic formulation.
The formalism also allows constraint adjustments by which
(\ref{eq:creduced}) is modified by
\begin{equation}
E^{\mu\nu}:= G^{\mu\nu} -\nabla^{(\mu}{\cal C}^{\nu)}
+\frac{1}{2}g^{\mu\nu}\nabla_\rho{\cal C}^\rho
+A^{\mu\nu}_\sigma {\cal C}^\sigma=0,
\label{eq:creduced2}
\end{equation}
where the coefficients
$A^{\mu\nu}_\sigma$ have the dependence $A^{\mu\nu}_\sigma(x,g,\partial
g)$. Such constraint adjustments have proved to be important in
applying constraint damping~\cite{constrdamp} in the simulation of
black holes~\cite{pret1,pret2,pret3} and in suppressing long wavelength instabilities
in a shifted gauge wave test~\cite{babev} (see Sec.~\ref{sec:num}).
However, they do not change the principal part
of the reduced equations and have no effect on well-posedness.
Historically, the first Cauchy
codes were based upon the ``3+1'' or Arnowitt-Deser-Misner (ADM)
formulation of the Einstein equations~\cite{adm}.
The ADM formulation introduces a Cauchy foliation of
space-time by a time coordinate $t$ and expresses the 4-dimensional
metric as
\begin{equation}
ds^2 = -\alpha^2 dt^2 + h_{ij} \left(dx^i + \beta^i dt\right)
\left(dx^j + \beta^j dt\right) ,
\label{eq:admmet}
\end{equation}
where $h_{ij}$ is the induced 3-metric of the $t=const$ foliation,
$\alpha$ is the lapse and $\beta^i$ the shift, with the unit normal
to the foliation given by $n^\mu=(1,-\beta^i)/\alpha$.
The field equations are written in first differential form in terms
of the extrinsic curvature of the Cauchy foliation
$$
k_{ij} = \frac{1}{2}{\cal L}_n g_{ij}.
$$
This can be accomplished in many ways. In one of the earliest schemes
proposed for numerical relativity by York~\cite{york},
the requirement that the 6 spatial components of the Ricci tensor vanish, i.e
$R_{ij}=0$, yields a set of evolution equations for the 3-metric and extrinsic curvature,
\begin{eqnarray}
\partial_t g_{ij} -{\cal L}_\beta g_{ij} &=& -2\alpha k_{ij} \\
\partial_t k_{ij} -{\cal L}_\beta k_{ij} &=& -D_i D_j\alpha
+ \alpha\left({\cal R}_{ij} + k k_{ij} - 2 k_i^l k_{lj} \right) ,
\label{eq:adm}
\end{eqnarray}
where $D_i$ is the connection and ${\cal R}_{ij}$
is the Ricci tensor associated with $h_{ij}$.
The Hamiltonian and momentum constraints take the form
\begin{eqnarray}
2 G^{\mu\nu}n_\mu n_\nu ={\cal R} - k_{ij} k^{ij} + k^2 &=& 0
\label{eq:hamc} \\
G^{\mu i}n_\mu = D_j \left( k^{ij} - h^{ij} k\right) &=&0,
\label{eq:momc}
\end{eqnarray}
where ${\cal R}=h^{ij}{\cal R}_{ij}$ and $k=h^{ij}k_{ij}$.
Codes presently used for the simulation of binary black holes
apply the constraints to the initial Cauchy data
but do not enforce them during the evolution.
The choice of evolution equations may be modified by mixing in
combinations of the constraint equations. In addition, the evolution equations
must be supplemented by equations governing the lapse
and shift. There is a lot of freedom in how all this
can be done. The choices affect whether the Cauchy problem is
well-posed.
\section{Complications of the IBVP}
\label{sec:compl}
The difficulties
underlying the IBVP have recently been discussed
in~\cite{disem,hjuerg,juerg,reulsarrev}. There are several
chief complications which do not arise in the Cauchy problem.
\begin{enumerate}
\item The first complication stems from a well-known property of the
flat-space scalar wave boundary problem
\begin{equation}
(\partial_t^2 -\nabla^2)\Phi=0 \, , \quad x \ge 0, \, t\ge 0.
\label{eq:swe1}
\end{equation}
The light rays are the characteristics of the equation.
There are two characteristics associated with each
direction, e.g. the characteristics in the $\pm x$ direction. Both of
these characteristics cross the initial hypersurface $t=0$ but only one crosses
the boundary at $x=0$. As a result,
although the initial Cauchy data consist of the two pieces of information
$\Phi|_{t=0}$ and $\partial_t \Phi|_{t=0}$, only half as much boundary
data can be freely prescribed at $x=0$, e.g the Dirichlet data
$q_D=\partial_t \Phi|_{x=0}$, or the Neumann data $q_N=\partial_x \Phi|_{x=0}$ or
the Sommerfeld data $q_S=(\partial_t - \partial_x) \Phi|_{x=0} $.
Sommerfeld data is based upon
the derivative of $\Phi$ in the characteristic direction determined by
the outward normal to the boundary.
(In the first differential order formalism $(\partial_t - \partial_x) \Phi$
is an {\it ingoing variable} at the boundary.)
The choices $q_D=q_N=q_S$ do not lead to the same solution. In order
to obtain a given
physical solution, this implies that the boundary data cannot be
prescribed before the boundary condition is specified, i.e. the
boundary data for the solution
depends upon the boundary condition, unlike the situation
for the Cauchy problem. The analogue in the gravitational case is the
inability to prescribe both the metric and its normal derivative on a
timelike boundary, which implies the inability to freely prescribe both
the intrinsic 3-metric of the boundary and its extrinsic curvature. This
leads to a further complication regarding constraint enforcement at the
boundary, i.e. the Hamiltonian and momentum constraints (\ref{eq:hamc})
and (\ref{eq:momc}) cannot be enforced directly because they couple the
metric and its normal derivative.
\item For computational purposes, a Sommerfeld boundary condition is preferable
because it allows numerical noise to propagate across the boundary. Thus
discretization error can leave the numerical grid, whereas Dirichlet and Neumann
boundary conditions would reflect the error and trap it in the grid.
The Sommerfeld condition on a metric component supplies the
value of the derivative $K^\alpha \partial_\alpha g_{\mu\nu}$ in an outgoing null
direction $K^\alpha$. However, the boundary does
not pick out a unique outgoing null direction at a given point but,
instead, essentially a half cone of null directions. This complicates the geometric
formulation of a Sommerfeld boundary condition. In addition, constraint
preservation does not allow free specification of Sommerfeld data for all
components of the metric, as will be seen later in formulating the Sommerfeld
conditions (\ref{eq:hk}) - (\ref{eq:hl}).
\item The correct boundary data for the gravitational field is generally
not known except in special cases, e.g. when simulating an exact solution.
This differs from electromagnetic theory where, say, homogeneous Dirichlet
or Neumann data for the various components of the electromagnetic field correctly
describe the data for reflection from a mirror. The tacit assumption in
the simulation of an isolated system is that homogeneous Sommerfeld
data gives rise to minimal back reflection of gravitational waves from
the outer boundary. But this is an approximation which only becomes
exact in the limit of an infinite sized boundary.
\item Another major complication arises from the gauge freedom. In the
evolution of the Cauchy data it is necessary to introduce a foliation of
the spacetime by Cauchy hypersurfaces ${\cal S}_t$, with unit timelike
normal $n_a$. The evolution of the spacetime metric
\begin{equation}
g_{ab}=-n_a n_b +h_{ab}
\label{eq:gdecom}
\end{equation}
is carried out along the flow of an evolution vector field $t^a$
which is related
to the normal by the lapse $\alpha$ and shift $\beta^a$ by
\begin{equation}
t^a= \alpha n^a + \beta^a \, , \quad \beta^a n_a =0.
\label{eq:lapshif}
\end{equation}
The choice of foliation is part of the gauge freedom in the resulting solution
but does not enter into the specification of the initial data.
In the current treatments of the IBVP, the foliation is
coupled with the formulation of the boundary condition.
As a result, some gauge information enters into the
boundary condition and boundary data.
\item The partial derivative $\partial_\alpha g_{\mu\nu} $ entering into
the construction of the boundary condition for the metric has by itself
no intrinsic geometric interpretation, unless, say, a background
connection or a preferred vector field is introduced.
\item In general, the boundary moves with respect to the initial Cauchy
hypersurface in the sense that the spacelike unit outer normal $N_a$ to
${\cal T}$ is not orthogonal to the timelike unit normal $n_a$ to ${\cal
S}_0$. The initial velocity of the boundary is characterized by the
hyperbolic angle $\Theta$, where
\begin{equation}
N^a n_a =\sinh \Theta
\label{eq:hangle}
\end{equation}
Specification of $\Theta$ on the edge ${\cal B}_0$ must be included in
the data.
The coordinate specification of the location of the boundary is pure
gauge since it does not determine its location geometrically in the
sense that a curve is determined geometrically by its its acceleration,
given its initial position and velocity. Given ${\cal B}_0$ and $\Theta$,
the future location of the boundary should be determined in a
geometrically unique way. In the Friedrich-Nagy system, the motion of the
boundary is determined by specifying its mean extrinsic curvature. But
this is tantamount to a piece of Neumann data. Can this be accomplished
via a non-reflecting boundary condition of the Sommerfeld type?
\item In a reduction to first differential order form by introducing a
momentum $\Pi$, according to the example \begin{equation} n^a \partial_a
\Phi = \Pi, \label{eq:advect} \end{equation} there is a further
difficulty if $\Theta \ne 0$ at the boundary. The sign of $\Theta$
determines whether $n^a$ points outward or inward to ${\cal T}$, i.e
whether $\Phi$ is an ingoing or outgoing variable. Thus the sign of
$\Theta$ determines whether such an advection equation requires a
boundary condition. This forces a Dirichlet condition on the normal
component of the shift in some 3+1 formulations.
\item There are also compatibility conditions between the initial data
and the boundary data at the edge ${\cal B}_0$. For the example of the
scalar wave problem (\ref{eq:swe1}) with a Dirichlet boundary condition
at $x=0$, the boundary data must satisfy
\begin{equation}
\partial_t^2 \Phi |_{(t=0,x=0)}
= (\partial_x^2+\partial_y^2+\partial_z^2)\Phi |_{(t=0,x=0)},
\end{equation} where the right hand side is determined by the initial
data. An infinite sequence of such conditions follow from taking time
derivatives of the wave equation. They must be satisfied if the solution
is required to be $C^\infty$. In simple problems, this sequence of
compatibility conditions can be satisfied by choosing initial data and
boundary data with support that vanishes in a neighborhood of the edge
${\cal B}_0$. But in problems with elliptical constraints, such as occur
in general relativity, this simple approach is not possible. In numerical
relativity, these compatibility conditions are usually ignored, with the
consequence that some transient {\it junk} radiation emanating from the
edge is generated. In principle, this could be avoided by smoothly {\it gluing}
the initial data to an exterior region with Schwarzschild~\cite{corv} or
Kerr~\cite{corvsch} data. This gluing construction would avoid mathematical
difficulties but it is an implicit construction and in practice no
numerical algorithm for carrying it out has been proposed. In the
simulation of binary black holes, this edge effect combines with another
source of junk radiation which is hidden in the choice of initial data.
The tacit assumption is that the spurious radiation from these sources
is quickly flushed out of the simulation, with no significant effect after
a few crossing times. Since this issue is difficult to treat or quantify
in a useful way, I adopt the expedient assumption that all compatibility
conditions for a $C^\infty$ solution have been met.
\end{enumerate}
In order to resolve most of the above complications it appears that a
foliation ${\cal B}_t$ of the boundary ${\cal T}$ must be specified as
part of the boundary data. Such a foliation is a common ingredient of
all successful treatments to date. The foliation supplies the gauge
information which determines a unique outgoing null direction for a
Sommerfeld condition. In Sec.~\ref{sec:bare}, I specify ${\cal B}_t$
in terms of the choice of an evolution vector field on the boundary.
Most of the above complications stem from the fact the domain of
dependence determined by the boundary alone is empty. An initial value
problem for a hyperbolic system can be be consistently posed in the
absence of a boundary. But the opposite is not true for an IBVP. Without
an underlying Cauchy problem, an IBVP does not make sense. In an IBVP,
boundary data cannot determine a unique solution independently of the
initial Cauchy data and there is no domain in which the solution is
independent of the initial data. Thus a well-posed IBVP problem must be
based upon a well-posed Cauchy problem.
\bigskip
\section{The bare manifold}
\label{sec:bare}
\bigskip
In constructing an evolution code for the gravitational field, the first
step is define a spatial grid and a time update scheme. This sets up the
underlying structure necessary to store the values of the various fields.
The analogous object at the continuum level corresponds to the bare
manifold on which the gravitational field is later painted. This is the
approach I will adopt here. It provides a useful way to order the
introduction of the basic geometric quantities which enter the IBVP.
Setting up the spatial grid corresponds to the analytic specification of spatial
coordinates $x^i$ on ${\cal S}_0$. The time update algorithm corresponds to
the introduction of an evolution field $t^a$ in ${\cal M}$ which is tangent to the
boundary ${\cal T}$. (In more complicated update schemes, which I won't
consider, the boundary might move through the grid.) The evolution field
must have the property that under its flow ${\cal S}_0$ is mapped into a
foliation ${\cal S}_t$ of ${\cal M}$, and its edge ${\cal B}_0$ is mapped
into a foliation ${\cal B}_t$ of ${\cal T}$.
If a time coordinate is initiated at $t=0$ on ${\cal S}_0$ then the flow
of $t^a$ induces {\it adapted coordinates} $x^\mu=(t,x^i)$ on ${\cal M}$
by requiring
\begin{equation}
{\cal L}_t t =1
\label{eq:fol}
\end{equation}
\begin{equation}
{\cal L}_t x^i=0,
\label{eq:xi}
\end{equation}
where $ {\cal L}_t$ is the Lie derivative with respect to $t^a$. Note
that $t^a$ and the adapted coordinates $x^\mu$ are explicitly constructed
fields on ${\cal M}$ with no metric properties. They uniquely fix the
gauge freedom on ${\cal M}$ in precisely the same way that the numerical
grid and update scheme provide a unique evolution algorithm. If $x^A$ are the
coordinates on the edge ${\cal B}_0$, then the corresponding adapted
coordinates on the boundary are $(t,x^A)$, where $ {\cal L}_t x^A=0$. It
will be convenient throughout this review to let the manifold with boundary
be described by adapted coordinates
\begin{equation}
x^\mu = (t\ge 0, x \ge 0, x^A).
\label{eq:adapted}
\end{equation}
Under a diffeomorphism $\psi$ of ${\cal M}$ which maps ${\cal T}$ into
itself, $t^a \rightarrow \psi_* t^a$ where $\psi_* t^a$ can be
chosen to be any other possible evolution field. In particular, if
$\psi_* t^a =t^a$ in ${\cal M}$ and $\psi x^i =x^i$ on ${\cal S}_0$ then the
diffeomorphism must be the identity. Thus, given a coordinate gauge on
${\cal S}_0$, the choice of $t^a$ determines the remaining diffeomorphism
freedom. This allows a description of the evolution in a specific choice of
adapted coordinates without losing sight of the gauge freedom.
Note that any one-form normal to the boundary is proportional to
$\partial_a t$. However, at the bare manifold level the unit normal
cannot be specified since that involves metric information. The
projection tensor
\begin{equation}
\pi^a_b=\delta^a_b -t^a\partial_b t
\label{eq:tproj}
\end{equation}
has the properties
\begin{equation}
\pi^a_b v^b \partial_a t =0,
\end{equation}
i.e. it projects a vector field into the tangent space of ${\cal S}_t$, and
\begin{equation}
\pi^a_b w_a t^b=0,
\end{equation}
i.e. it projects a 1-form into the space orthogonal to $\partial_t$.
These are the main structures that exist in the IBVP a priori to introducing
a geometry on ${\cal M}$. There is an alternative approach in which
geometrical concepts are introduced earlier. In the Cauchy problem, the
initial data can be specified in geometrical form as tensor fields $\tilde
h_{ab}$ and $\tilde k_{ab}$ on a ``disembodied'' 3-manifold $\tilde {\cal
S}_0$ (cf. \cite{hawkel}). Only after the embedding of $\tilde {\cal S}_0$ in
${\cal M}$ is this data interpreted as the intrinsic metric $h_{ab}$ and
extrinsic curvature $k_{ab}$ of ${\cal S}_0$. The mean curvature
$k=h^{ab}k_{ab}$ can itself be interpreted as a variable determining the
location of ${\cal S}_0$. Similarly, in the IBVP the mean extrinsic
curvature of ${\cal T}$ can be interpreted as a wave equation determining
the geometric location of the boundary~\cite{fn}. However, these
interpretations assume knowledge of the spacetime geometry which is only
known after a solution is found.
This order in which the basics objects are introduced is akin to the
question: Which came first - the geometry or the manifold (or some
combination)? Here I adopt the manifold approach, which is more akin to the
spirit of numerical relativity. I assume a priori, for the given choice
of evolution field $t^a$, that ${\cal M}$ is the domain of dependence of the
initial-boundary data, i.e. it is the manifold upon which the data
determines a unique evolution. Here ``a priori'' is used in the sense of a
spacetime geometry which exists only after the solution of the IBVP is
obtained.
\bigskip
\section{Initial data}
\label{sec:initial}
\bigskip
Since Einstein's equations are second differential order in the metric, any
evolution scheme must specify $g_{\mu\nu}$ and $\partial_t g_{\mu\nu}$ on
${\cal S}_0$. The classic result of the Cauchy problem is that a
geometrically unique solution of the Cauchy problem is determined by initial
data consisting of the intrinsic metric $h_{ab}$ of ${\cal S}_0$ and its
extrinsic curvature $k_{ab}$, subject to the constraints
(\ref{eq:hamc})--(\ref{eq:momc}).
The remaining initial data necessary to specify a unique spacetime metric
consist of gauge information, i.e. gauge data that affect the resulting solution only by
a diffeomorphism. One such quantity is the lapse $\alpha$, which relates the
unit future-directed normal to the time foliation according to
\begin{equation}
n_a=-\alpha\partial_a t.
\label{eq:n}
\end{equation}
The embedding of ${\cal S}_0$ in ${\cal M}$ then gives rise to the
spacetime metric
\begin{equation}
g_{ab} =- n_a n_b +h_{ab}
\label{eq:hng}
\end{equation}
and the interpretation of $k_{ab}$ as the extrinsic curvature
through the identification
\begin{equation}
k_{ab} = h_a^c \nabla_c n_b,
\end{equation}
where $\nabla_c$ is the covariant derivative associated with $g_{ab}$.
The choice of evolution field $t^a$ supplies the remaining gauge data. It
is transverse but not in general normal to the Cauchy hypersurface so
that it determines a shift $\beta_a$ according to
\begin{equation}
\beta_a = h_{ab} t^b.
\end{equation}
This relationship supplies the metric information
$$ g_{ab} t^b =\alpha n_a +\beta_a
$$
relating $t^a$ to the unit normal $n_a$.
In the adapted coordinates, the metric has components
\begin{equation}
g_{tt}=-\alpha^2+h_{ij}\beta^i \beta^j
\end{equation}
\begin{equation}
g_{ti}=\beta_i =h_{ij}\beta^j
\end{equation}
\begin{equation}
g_{ij}=h_{ij}.
\end{equation}
The inverse metric is given by $g^{ab} = -n^a n^b +h^{ab}$, where
\begin{equation}
h^{ab}n_b=0, \quad h^{ac}h_{bc}=\delta^a_b +n^a n_b.
\end{equation}
In the adapted coordinates,
\begin{equation}
g^{tt}=-\alpha^{-2}
\end{equation}
\begin{equation}
g^{ti}=\alpha^{-2}\beta^i
\end{equation}
\begin{equation}
g^{ij}=h^{ij}\, , \quad h^{ik}h_{kj}=\delta^i_j.
\end{equation}
The implementation of the initial data into an evolution scheme depends
upon the details by which Einstein's equations are converted into a set
of PDEs governing $g_{\mu\nu}$ in the adapted
coordinates. All such schemes require specification of the initial values
of the lapse and shift, in addition to $h_{ij}$ and $k_{ij}$. Thus it can be
assumed that $g_{ab}$ is specified on ${\cal S}_0$. By Lie
transport along the streamlines of $t^a$, this then allows the
construction of a preferred stationary background metric ${\gz}_{ab}$ on
${\cal M}$ picked out by the initial data. Given the choice of evolution
field $t^a$ and the initial Cauchy data, this background metric is uniquely
and geometrically determined by
\begin{equation}
{\cal L}_t {\gz}_{ab} =0 \, ,
\quad {\gz}_{ab}|_{{\cal S}_0}=g_{ab}|_{{\cal S}_0} .
\label{eq:gz}
\end{equation}
In the adapted coordinates $ {\gz}_{\mu\nu}(t,x^i)=g_{\mu\nu}(0,x^i)$.
\bigskip
\section{Hyperbolic initial-boundary value problems}
\label{sec:hyperb}
\bigskip
There is an extensive mathematical literature on the IBVP for hyperbolic
systems. The major progress traces back to the formulation of maximally
dissipative boundary conditions for linear symmetric hyperbolic systems
due to Friedrichs~\cite{friedrichs} and Lax and Phillips~\cite{laxphil}.
There has been recent progress in obtaining results for quasilinear
systems where the boundary contains characteristics, as arises in some
formulations of Einstein's equations. Unfortunately, much of this
material is heavy on the mathematical side and not easy reading for
relativists coming from astrophysical or numerical backgrounds. In the
absence of the complications of shocks introduced by hydrodynamic
sources, relativists are content to deal with {\it smooth}, i.e.
$C^\infty$, solutions and forgo the Sobolev theory which enters a
complete discussion of the quasilinear IBVP. For relativists, the most
readable source on the theory of hyperbolic boundary problems is the
textbook by Kreiss and Lorenz~\cite{green-book}, which boasts: ``In
parts, our approach to the subject is {\it low-tech}.... Functional
analytical prerequisites are kept to a minimum. What we need in terms of
Sobolev inequalities is developed in an appendix."
Taylor's~\cite{taylor1,taylor2} treatises on partial differential
equations contain a classic treatment of pseudo-differential theory but
are less readable for relativists. Fortunately, much of the critical
formalism pertinent to Einstein's equations appears as background
material in papers on the gravitational IBVP. The material I present here
is heavily based upon those sources,
namely~\cite{fn,stewart,reulsarrev,green-book,wpgs,wpe,isol}.
There are two distinct formulations of the IBVP, depending upon whether
you consider Einstein's equations as a natural second differential order
system of wave equations or whether you reduce it to a first order
system. While the second order approach is the most economical, it is not
applicable to all formulations of Einstein's equations, particularly
those whose gauge conditions do not have the semblance of wave equations.
The first order theory has been extensively developed because of its historic
importance to the symmetric hyperbolic formulation of hydrodynamics. The IBVP
for second order systems has received less attention and some new
techniques have originated in the consideration of the Einstein problem.
There are also two distinct approaches to studying well-posedness - one
based upon energy estimates and the other based upon pseudo-differential
theory where Fourier-Laplace expansions are used to reduce the
differential operators to algebraic operators. The
pseudo-differential theory can be applied equally well to first or second
order systems. In the following, I give a brief account of the underlying
ideas in terms of some simple model problems. This will provide a
background for discussing the difficulties that arise when considering
constraint preservation in the gravitational IBVP.
The subclasses of hyperbolic systems consist of weak hyperbolicty, strong
hyperbolicity, symmetric hyperbolicity and strict hyperbolicity. These
subclasses are determined by the principal part of the system when
written as first differential order PDEs. Weakly hyperbolic systems do
not have a well-posed Cauchy problem, which turned out to be responsible
for the instabilities encountered in early attempts at numerical
relativity using naive ADM formulations with a prescribed lapse and shift.
Strong hyperbolicity is sufficient to guarantee a well-posed Cauchy
problem but not a well-posed IBVP. It is possible to base a well-posed
IBVP on either symmetric hyperbolic or strictly hyperbolic systems.
However, strictly hyperbolic systems arise very rarely and, to my
knowledge, not at all in numerical relativity. So I will limit my
discussion to the symmetric hyperbolic case.
I begin with the IBVP for a second order scalar wave equation, where the
underlying techniques are transparent rather than hidden in the
machinery of symmetric hyperbolic theory.
The generalization to systems of quasilinear wave equations,
described in Sec.~\ref{sec:swev}, can also be treated by the same techniques
as for symmetric hyperbolic systems. However, for application to the
harmonic Einstein problem, the scalar treatment suffices since the
principal part of the system consists of a common wave operator acting on
the individual components of the metric. The mathematical
analysis which is necessary for a treatment of the quasilinear
IBVP in full rigor is beyond my competence and presumably
outside the interest of someone from a more physical or
computational background. I simply state the main results and give
references when such mathematical theory must be evoked.
\bigskip
\subsection{Second order wave equations}
\bigskip
The ideas underlying the well-posedness of the IBVP are well illustrated
by the case of the quasilinear wave equation. I give some examples
which are are relevant to the harmonic formulation of Einstein's
equations and which illustrate the techniques behind both the energy
approach and the pseudo-differential approach.
\bigskip
\subsubsection{The energy method for second order wave equations}
\label{sec:enwave}
\bigskip
First consider first the linear wave equation for a scalar field,
\begin{equation}
\Phi_{tt}=\Phi_{xx}+\Phi_{yy}+\Phi_{zz}+F
\label{eq:byp1}
\end{equation}
on the half-space
\begin{equation}
x\ge 0,\quad -\infty <y<\infty,
\quad -\infty < z < \infty, \nonumber
\end{equation}
with boundary condition at $x=0$
\begin{equation}
\Phi_t-\alpha\Phi_x-\beta_2 \Phi_y-\beta_3 \Phi_z= q \, ,\quad
\alpha >0,~\beta_2^2+\beta_3^2<1 ,
\label{eq:byp2}
\end{equation}
with boundary data $q$, initial data of compact support
\begin{equation}
\Phi=f_1,\quad \Phi_t=f_2,\quad t=0
\label{eq:byp3}
\end{equation}
and forcing term $F(t,x,y,z)$.
The subscripts $(t,x,y,z)$ denote partial derivatives, e.g
$$\Phi_t={\partial \Phi \over \partial t}=\partial_t \Phi .
$$
All
coefficients and data are assumed to be real and
$\alpha >0,~\beta_2,~\beta_3$ are constants. The notation
$$
(\Phi,\Psi), \quad \|\Phi\|^2=(\Phi,\Phi);\quad (\Phi,\Psi)_B,\quad
\|\Phi\|^2_B=(\Phi,\Phi)_B,
$$
is used to denote the $L_2$-scalar product and norm over the half-space and
boundary space, respectively.
In order to
adapt the standard definition of energy estimates to second order systems, the notation
${\bf \Phi}= (\Phi,\Phi_t,\Phi_x,\Phi_y,\Phi_z)$ is used
to represent the solution and
its derivatives; and similarly ${\bf f}= (f_1,f_2,f_{1x},f_{1y},f_{1z})$ for the initial data.
For the scalar IBVP (\ref{eq:byp1})--(\ref{eq:byp3}), strong well-posedness
requires the existence of a unique solution satisfying the a priori estimate
\begin{equation}
\|{\bf \Phi}(t)\|^2+\int_0^t\|{\bf \Phi}(\tau)\|^2_B d\tau
\le K_T \left (\|{\bf f}\|^2 +\int_0^t\|F(\tau)\|^2 d\tau
+\int_0^t\|q(\tau)\|^2_B d\tau \right ),
\label{eq:swp}
\end{equation}
in any time interval $0<t<T$, where the constant $K_T$ is
independent of $F$, ${\bf f}$ and $q$.
It is important to note that (\ref{eq:swp}) estimates the derivatives of $\Phi$,
both in the interior and on the boundary, in terms of the data and the
forcing. This is referred to as ``gaining a derivative''. This property is
crucial in extending the local IBVP to global situations, e.g. where the
boundary is a sphere or where there is an interior and exterior boundary as in
a strip problem. Otherwise, reflection from the boundary could lead to the
``loss of a derivative'', which would lead to unstable behavior under
multiple reflections.
The usual procedure is to derive an energy estimate by integration by parts,
using for example $(\Phi_{yy},\Phi_z)=-(\Phi_y,\Phi_{yz})=0$. Consider first
the estimates of the derivatives of $\Phi$ in the homogeneous case $F\equiv
q\equiv 0$. Using the standard energy for a scalar field, integration by parts
gives
$$
\partial_t (\|\Phi_t\|^2+ \|\Phi_x\|^2+ \|\Phi_y\|^2 + \|\Phi_y\|^2)=
-2(\Phi_t,\Phi_x)_B.
$$
If $\beta_2=\beta_3=0$ and $\alpha >0$ in the boundary condition
(\ref{eq:byp2}) then $(\Phi_t,\Phi_x)_B \ge 0$, i.e. the boundary condition is
{\it dissipative}, and there is an energy estimate. Otherwise there is no
obvious way to estimate the boundary flux. Instead, it is possible to use a
non-standard energy $E$ for the scalar wave equation (\ref{eq:byp1}) which
does provide the key estimate if $\beta_2^2+\beta_3^2>0$.
The first step is to show that
\begin{equation}
E:=\|\Phi_t\|^2+\|\Phi_x\|^2+\|\Phi_y\|^2+\|\Phi_z\|^2-
2(\Phi_t,\beta_2 \Phi_y+\beta_3 \Phi_z)
\label{eq:byp4}
\end{equation}
is a norm for the derivatives $(\Phi_t,\Phi_x,\Phi_y,\Phi_z)$.
Since $\beta_2^2+\beta_3^2< 1$, this follows, after a rotation,
from the inequality $\Phi_t^2 + \Psi^2 -2\beta \Phi_t \Psi \ge 0$
for $\beta^2<1$.
This leads to
\medskip
{\it Lemma 1:}
The solution of (\ref{eq:byp1})--(\ref{eq:byp3}) satisfies the energy estimate
\begin{equation}
\partial_t E +\alpha \|\Phi_x\|^2_B\le E + \|F\|^2 +\frac{1}{\alpha} \|q\|^2_B.
\label{eq:byp5}
\end{equation}
\medskip
\noindent
{\it Proof.} Integration by parts gives
\begin{equation}
\partial_t \|\Phi_t\|^2=2(\Phi_t,\Phi_{tt})=
-\partial_t (\|\Phi_x\|^2+\|\Phi_y\|^2+\|\Phi_z\|^2)+
2(\Phi_t,F)-2(\Phi_t,\Phi_x)_B
\label{eq:byp6}
\end{equation}
and
\begin{equation}
2\partial_t (\Phi_t,\beta_2 \Phi_y+\beta_3 \Phi_z)
=2(\Phi_{tt}, \beta_2 \Phi_y+\beta_3 \Phi_z)
= -2(\Phi_{x},\beta_2 \Phi_y+\beta_3 \Phi_z)_B
+2 (F, \beta_2 \Phi_y+\beta_3 \Phi_z).
\label{eq:byp7}
\end{equation}
Since (\ref{eq:byp2}) implies
$$
2(\Phi_t,\Phi_x)_B=2\alpha\|\Phi_x\|^2_B+
2(\Phi_{x},\beta_2 \Phi_y+\beta_3 \Phi_z)_B+
2 (\Phi_x,q)_B,$$
subtraction of (\ref{eq:byp7}) from (\ref{eq:byp6}) leads to
\begin{eqnarray}
\partial_t E &=& 2(\Phi_t-\beta_2 \Phi_y -\beta_3 \Phi_z,F)-2\alpha\|\Phi_x\|^2_B
-2(\Phi_x,q)_B \nonumber \\
& \le& \|\Phi_t-\beta_2 \Phi_y -\beta_3 \Phi_z\|^2 + \|F\|^2
-\alpha\|\Phi_x\|^2_B +\frac{1}{\alpha}\|q\|^2_B. \nonumber
\end{eqnarray}
The identity
$$
\| \Phi_t -\beta_2 \Phi_y -\beta_3 \Phi_z\|^2 = E-\| \Phi_x\|^2 -\| \Phi_y\|^2
-\| \Phi_z\|^2 + \| \beta_2 \Phi_y+\beta_3 \Phi_z \|^2
$$
then implies (\ref{eq:byp5}) and proves the lemma.
\medskip
By integration, the lemma estimates
$$
E(T)~\hbox{and}~\int_0^T \|\Phi_x\|^2_B dt \quad \hbox{in terms of}\quad
E(0),~\int_0^T \|F\|^2dt~ \hbox{and}~
\int_0^T \|q\|^2_B dt.
$$
Strong well-posedness (\ref{eq:swp}) also requires estimates
of the boundary norms $\|\Phi_t \|_B$, $\|\Phi_y \|_B$
and $\|\Phi_z\|_B$.
First, a calculation similar to that above gives
\begin{eqnarray}
\partial_t (\Phi_x,\Phi_t)&=&(\Phi_{xt},\Phi_t)+(\Phi_x,\Phi_{tt})
\nonumber \\
&=&-\frac{1}{2}\|\Phi_t\|^2_B+(\Phi_x,\Phi_{xx})+(\Phi_x,\Phi_{yy})
+(\Phi_x,\Phi_{zz})+(\Phi_x,F) \nonumber \\
&=&-\frac{1}{2}\|\Phi_t\|^2_B-\frac{1}{2}\|\Phi_x\|^2_B
+\frac{1}{2}\|\Phi_y\|^2_B+\frac{1}{2}\|\Phi_z\|^2_B +(\Phi_x,F) .
\label{eq:byp8}
\end{eqnarray}
Estimates of $\|\Phi_t \|_B$ in terms of $\|\Phi_y \|_B$, $\|\Phi_z\|_B$,
$\|\Phi_x\|_B$ and $\|q\|_B$ can be obtained from
the boundary conditions (\ref{eq:byp2}), which give,
for any $\delta$ with $0<\delta<1$,
\begin{eqnarray}
\|\Phi_t\|^2_B&=& \|\beta_2 \Phi_y+ \beta_3 \Phi_z
+\alpha \Phi_x+q\|^2_B \nonumber \\
&\le&
\|\beta_2 \Phi_y+ \beta_3 \Phi_z\|^2_B+
2\|\beta_2 \Phi_y+ \beta_3 \Phi_z\|_B \,
\|\alpha \Phi_x+q\|_B+ \|\alpha \Phi_x+q\|^2_B \\
&\le& (1+\delta)
\|\beta_2 \Phi_y+ \beta_3 \Phi_z\|^2_B
+(1+{1\over\delta})\|\alpha \Phi_x+q\|^2_B \nonumber \\
&\le& (1+\delta)\left(\beta_2^2+\beta_3^2\right)
(\|\Phi_y\|^2_B+\|\Phi_z\|^2_B)+
(1+{1\over \delta})\|\alpha\Phi_x+q\|^2_B. \nonumber
\end{eqnarray}
Next, since $\beta_2^2+\beta_3^2<1$, $\delta$ can be chosen
such that $(1+\delta)(\beta_2^2+\beta_3^2)
\le (1-\delta).$ Therefore, by (\ref{eq:byp8}),
$$
\delta(\|\Phi_y\|^2_B+\|\Phi_z\|^2_B)\le
(1+{1\over\delta})\| \alpha \Phi_x+q\|^2_B + \|\Phi_x\|^2_B
+2\partial_t (\Phi_x,\Phi_t) -2 (\Phi_x,F).
$$
Since $(\Phi_x,\Phi_t)$ can be estimated by $E$, there follows
\medskip
{\it Lemma 2:}
\begin{eqnarray}
& \int_0^T & \left(\|\Phi_t\|^2_B+\|\Phi_x\|^2_B+\|\Phi_y\|^2_B
+\|\Phi_z\|^2_B\right)dt
\nonumber \\
&\quad & \le const (E(0)+\int_0^T \|F\|^2dt+\int_0^T \|q\|^2_B dt). \nonumber
\end{eqnarray}
\medskip
An estimate for $\Phi$ itself can easily be obtained by the change of variable
$\Phi\to e^{\mu t}\Phi$, as described in Sec.~\ref{sec:quasiw}
or in Appendix 1 of~\cite{wpe}.
Together with the results of Lemma 1 and Lemma 2, this establishes
\medskip
{\it Theorem 1:} The IBVP (\ref{eq:byp1})--(\ref{eq:byp3}) is
strongly well-posed in the sense
of (\ref{eq:swp}).
\medskip
The result can also be generalized to half-plane problems for wave
equations of the general constant coefficient form
\begin{equation}
\Phi_{tt}= 2b^i \Phi_{it}+h^{ij} \Phi_{ij},
\quad x_1\ge 0,\quad -\infty < (y,z) <\infty ,~x^i=(x,y,z)
\label{eq:byp9}
\end{equation}
where $h^{ij}$ is is a metric of $(+++)$ signature.
By coordinate transformation, (\ref{eq:byp9}) can be
transformed into (\ref{eq:byp1}) and the appropriate boundary conditions
formulated. See~\cite{wpe} for details.
This example illustrates from the PDE perspective the constructions
necessary to establish strong well-posedness. For the purpose of
establishing the strong well-posedness of the IBVP for the wave equation
on a general curved space background, it is also instructive to take
advantage of the geometric nature of the problem.
In terms of standard relativistic notation, consider the wave equation
\begin{equation}
g^{ab}\nabla_a \nabla_b\Phi = F
\label{eq:byp21}
\end{equation}
for a massless scalar field propagating on a Lorentzian spacetime ${\cal M}$
foliated by compact, $3$-dimensional time-slices ${\cal S}_t$, with boundary
${\cal T}$ foliated by ${\cal B}_t$. Here $\nabla$ denotes the covariant
derivative associated with the spacetime metric $g^{ab}$. For notational
simplicity, let $\Phi_a = \nabla_a\Phi$.
The IBVP consists in finding solutions of (\ref{eq:byp21}) subject to the
initial Cauchy data
\begin{equation}
\left. \Phi \right|_{{\cal S}_0} = f_1, \qquad
\left. n^b\Phi_b \right|_{{\cal S}_0} = f_2
\label{eq:byp21b}
\end{equation}
and the boundary condition
\begin{equation}
\left[ (T^b + a N^b) \Phi_b \right]_{\cal T} = q ,
\label{eq:byp21c}
\end{equation}
with data $q$ on ${\cal T}$. Here $n^b$ is the future-directed unit normal to
the time-slices ${\cal S}_t$ and and $N^b$ is the outward unit normal to ${\cal
T}$; $T^b$ is an arbitrary future-directed timelike vector field which is
tangent to ${\cal T}$; and $a > 0$. The motion of the boundary is described
geometrically by the hyperbolic angle $\tanh\Theta=N^b n_b$. Without loss of
generality, assume the normalization $g_{bc} T^b T^c = -1$. A Sommerfeld
boundary condition then corresponds to the choice $a=1$ for which $T^b +N^b$
points in an outgoing null direction.
In order to establish estimates, consider the energy momentum tensor
of the scalar field
$$
\Theta_b^a = \Phi_b \Phi^a -{1\over 2}\delta_b^a \Phi^c \Phi_c.
$$
The essential idea is the use of an energy associated with a timelike vector
$u^a=T^a+\delta N^a$, where $0<\delta<1$, so that $u^a$
points outward from the boundary. The corresponding energy $E(t)$ and the
energy flux $P(t)$ through ${\cal B}_t$ are
\begin{equation}
E(t) =\int_{{\cal S}_t } u^b \Theta_b^a n_a,
\label{eq:ewave}
\end{equation}
which is a covariant version of the non-standard energy (\ref{eq:byp4}),
and
\begin{equation}
P(t) =\int\limits_{{\cal B}_t} u^b \Theta_b^a N_a .
\end{equation}
It follows from the timelike property of $u^a$ that $E(t)$ is a norm for
$\Phi_a(t)$.
Energy conservation for the scalar field, i.e. integration by parts, gives
$$
\partial_t E = P
-\int\limits_{{\cal S}_t} (\Theta_{ab}\nabla^a u^b +u^a \Phi_a F) ,
$$
so that
\begin{equation}
\partial_t E \le P + const
(E+ \int\limits_{{\cal S}_t} F^2 ) .
\label{eq:byp22}
\end{equation}
The required estimates arise from an identity satisfied by the flux density
\begin{equation}
u^b \Theta_b^a N_a =
-{\delta\over 2} \left (
(N^a \Phi_a )^2 + (T^a \Phi_a )^2 + Q^{ab}\Phi_a \Phi_b \right)
+N^a \Phi_a T^b \Phi_b +\delta (N^a \Phi_a )^2
+ \delta(T^a \Phi_a )^2
\nonumber
\end{equation}
where $Q_{bc} = g_{bc} + T_b T_c - N_b N_c$ is the positive definite
2-metric in the tangent space of the boundary orthogonal to $T^a$. By
using the boundary condition to eliminate $T^a \Phi_a$ in the last group
of terms, there follows
\begin{eqnarray}
u^b \Theta_b^a N_a &=& -{\delta\over 2} \left (
(N^a \Phi_a )^2 + (T^a \Phi_a )^2
+ Q^{ab}\Phi_a \Phi_b \right) \nonumber \\
&+&\left (-a+\delta(1+a^2)\right)(N^a \Phi_a )^2
+(1-2a\delta)N^a \Phi_a q +\delta q^2 \nonumber \\
&=& -{\delta\over 2} \left (
(N^a \Phi_a )^2 + (T^a \Phi_a )^2 + Q^{ab}\Phi_a \Phi_b \right) \nonumber \\
&-&\left (a-\delta(1+a^2)\right )
\left (N^a \Phi_a -\frac{\left(1-2a\delta \right)q}
{2 \left (a-\delta(1+a^2)\right )}\right )^2
+\left (\delta+ \frac {(1-2a \delta)^2} {4\left (a-\delta(1+a^2) \right )}
\right )q^2. \nonumber
\end{eqnarray}
The choice
$$ 0< \delta < \frac{a}{1+a^2}
$$
(which also guarantees that $\delta<1$ so that $u^a$ is timelike),
gives the inequality
\begin{equation}
u^b \Theta_b^a N_a \le -{\delta\over 2} \left (
(N^a \Phi_a )^2 + (T^a \Phi_a )^2 + Q^{ab}\Phi_a \Phi_b \right)
+ const \, q^2 .
\label{eq:byp23}
\end{equation}
It now follows from (\ref{eq:byp22}) and (\ref{eq:byp23}) that
\begin{eqnarray}
\partial_t E &+& \int_{{\cal B}_t}{\delta\over 2} \bigg(
(N^a \Phi_a )^2 + (T^a \Phi_a )^2 + Q^{ab}\Phi_a \Phi_b \bigg)
\nonumber \\
&\le & const \bigg (E+ \int_{{\cal S}_t} F^2 +
\int_{{\cal B}_t} q^2 \bigg ).
\label{eq:byp24}
\end{eqnarray}
This is the required estimate of the gradient $\Phi_a$ on the boundary
(as well as usual estimate of the energy $E$) to prove that the problem
is strongly well-posed. As in the previous example, an estimate of $\Phi$
itself follows from the change of variable $\Phi\to e^{\mu t}\Phi$, which
introduces a mass term in (\ref{eq:byp21}).
\bigskip
\subsubsection{The quasilinear case}
\label{sec:quasiw}
\bigskip
The estimate (\ref{eq:byp24}) is sufficient to establish the criterion
(\ref{eq:swp}) for strong well-posedness of the IBVP for the linear
wave equation with constant metric coefficients. In order to extend the
result to the quasilinear case on a curved space background, where the
metric depends upon $\Phi$ and $\Phi_a$, it is necessary to show that the
corresponding estimates hold for arbitrarily high derivatives of
$\Phi$. In the process, this also requires stability of the system
under the addition of lower differential order terms, which arise under the
differentiation of the wave equation. These requirements
are sometimes neglected or misunderstood in the relativity literature.
More generally, local existence theorems for variable coefficient or
quasilinear equations follow by iteration of solutions of the linearized
equations with frozen coefficients. The energy estimates for the frozen
coefficient problem establish the existence of a unique solution which
depends continuously on the data. The extension of this result to the
quasilinear case first requires that the problem with variable
coefficients be strongly well-posed. For this, it already necessary to
obtain estimates for arbitrarily high derivatives of the solution to the
linearized problem.
For the purpose of illustrating the procedure, it suffices to consider
the IBVP for the 2(spatial)-dimensional wave equation with variable
coefficients
\begin{equation} \Phi_{tt}=P\Phi+R\Phi+F,\quad x\ge 0,\quad
-\infty<y<\infty, \label{eq:byp15}
\end{equation}
with smooth initial data
\begin{equation}
\Phi=f_1 \, ,\quad \Phi_t=f_2 \, , \quad t=0 ,
\label{eq:byp17}
\end{equation}
and boundary condition
\begin{equation}
\alpha(t,y) \Phi_t=\Phi_x-\mu \Phi+r(t,y) \Phi+q(t,y)\, ,\quad
\alpha(t,y) \ge const >0\, , \, \mu =const >0
\label{eq:byp16}
\end{equation}
with smooth, compatible boundary data $q$.
Here
$$ P\Phi=(a\Phi_x)_x+(b\Phi_y)_y-2\mu \Phi_t-\mu^2\Phi\, ,
\quad a>a_0=const>0, \, b>b_0=const>0\, ,
$$
is an elliptic operator which has been modified by terms which arise in
(\ref{eq:byp15}) by the transformation $\Phi \rightarrow e^{\mu t} \Phi$,
where $\mu$ introduces a mass term; and
$$
R\Phi=c_1\Phi_t+c_2\Phi_x+c_3\Phi_y+c_4\Phi
$$
are terms of lower (zeroth
and first) differential order. The coefficients $a$, $b$ and $c_i$ are smooth functions
of $(t,x,y)$.
Consider the norm for the Cauchy data
$$
E= \|\Phi_t\|^2+(\Phi_x,a\Phi_x)+(\Phi_y,b\Phi_y)+\mu^2\|\Phi\|^2.
$$
Integration by parts leads to
\begin{eqnarray}
\partial_t E& =& -4\mu\|\Phi_t\|^2+2(\Phi_t,F)+2(\Phi_t,R\Phi)-2(\Phi_t,a \Phi_x)_B
+ (a_t\Phi_x,\Phi_x)+(b_t\Phi_y,\Phi_y) \nonumber \\
&\le & const (\|F\|^2+E)-2(\Phi_t,a\Phi_x)_B .
\label{eq:byp18}
\end{eqnarray}
The boundary condition gives
\begin{eqnarray}
-(\Phi_t,a\Phi_x)_B&=& -(\Phi_t,\alpha a\Phi_t)_B
-\mu(\Phi_t, a\Phi)_B+(\Phi_t,ar\Phi+aq)_B \nonumber \\
&=&-(\Phi_t,\alpha a\Phi_t)_B-\mu(\Phi_t, a_0 \Phi)_B
-\mu\left(\Phi_t,(a-a_0)\Phi\right)_B +(\Phi_t,ar \Phi+aq)_B \\
&\le & -\frac{1}{2} \mu a_0\partial_t \|\Phi\|^2_B
-\frac{1}{2}(\Phi_t,\alpha a\Phi_t)_B+ const (\|\Phi\|^2_B+\|q\|^2_B).
\end{eqnarray}
Therefore, from (\ref{eq:byp18}),
\begin{eqnarray}
&\partial_t (E+\mu a_0\|\Phi\|^2_B )+(\Phi_t,\alpha a \Phi_t)_B \\
&\le const \left(E+\|\Phi\|^2_B+\|F\|^2+\|q\|^2_B\right),
\label{eq:byp19}
\end{eqnarray}
This establishes that the energy estimate is stable against lower order
perturbations. Now it is possible to estimate the derivatives. For the
derivatives $Y=\Phi_y$ and $T=\Phi_t$ tangential to the boundary,
differentiation of the wave equation gives
\begin{eqnarray}
Y_{tt}&=&PY+RY+R_y \Phi+(a_y \Phi_x)_x+(b_y Y)_y+F_y
\label{eq:Y} \\
T_{tt}&=&PT+RT+R_t \Phi+(a_t\Phi_x)_x+(b_t Y)_y+F_t.
\label{eq:T}
\end{eqnarray}
Here $R_y \Phi$ and $R_t \Phi$ are linear combinations of first
derivatives of $\Phi$ which have already been estimated and can be
considered part of the forcing term $F$. Also, the wave equation
(\ref{eq:byp15}) implies
$$ a \Phi_{xx}=T_t-bY_y+~ \hbox{terms that have already been estimated}, $$
so that $\Phi_{xx}$ is also lower order with respect to (\ref{eq:Y}) and
(\ref{eq:T}). Thus, except for lower order terms, $Y$ and $T$ solve the
same wave equations (\ref{eq:Y}) and (\ref{eq:T}) as $\Phi$, with the
same boundary conditions up to lower order terms. Therefore all second
derivatives $\Phi_{ay}$ and $\Phi_{at}$ can be estimated as well as
$\Phi_{xx}$. Repetition of this process gives estimates for any number of
derivatives.
In order now to show the existence of a solution to the variable
coefficient problem, one approach, which is particularly familiar to
numerical relativists, is to approximate the PDE by a stable finite
difference approximation. This approach is detailed in~\cite{green-book}
where summation by parts (SBP) is applied to the finite difference
problem in the analogous way that integration by parts is used above in
the analytic problem. The SBP approach shows that the corresponding
estimates hold independently of gridsize. Existence of a solution of the
analytic problem then follows from the limit of vanishing gridsize.
It also follows from the estimates for arbitrary derivatives of the
variable coefficient problem that Sobolev's theorems can be used to
establish similar, although local in time, estimates for quasilinear
systems. Then the same iterative methods used for first order symmetric
hyperbolic systems can be used to show that well-posedness extends
locally in time to the quasilinear case. In this way it was shown
in~\cite{wpe} that the general quasilinear wave problem
(\ref{eq:byp21})--(\ref{eq:byp21c}), where the spacetime metric now
depends upon $\Phi$ and $\Phi_a$, is strongly well-posed.
\bigskip
\subsubsection{Systems of wave equations}
\label{sec:swev}
\bigskip
The strong well-posedness of the IBVP for the quasilinear scalar wave
(\ref{eq:byp21})--(\ref{eq:byp21c}) can be generalized to a system of
coupled wave equations
\begin{equation}
g^{ab}(\Phi) \nabla_a\nabla_b\Phi^A = F^A(\Phi,\nabla\Phi),
\quad A=1,2,...N
\label{Eq:WaveSystemEq}
\end{equation}
with smooth initial data
\begin{equation}
\left. \Phi^A \right|_{{\cal S}_0} = f_1^A\; , \qquad
\left. n^b\nabla_b\Phi^A \right|_{{\cal S}_0} = f_2^A\; ,
\label{Eq:WaveSystemID}
\end{equation}
and the boundary condition
\begin{equation}
\left. \left( T^b + \alpha N^b \right )\nabla_b\Phi^A \right|_{\cal T}
= c^{a\, A}{}_B\left. \nabla_a\Phi^B \right|_{\cal T}
+ d^A{}_B\left. \Phi^B \right|_{\cal T} + q^A,
\label{Eq:WaveSystemBC}
\end{equation} where $a = a(x,\Phi) > 0$, $q^A = q^A(x)$, $c^{a\, A}{}_B
= c^{a\, A}{}_B(x,\Phi)$ and $d^A{}_B = d^A{}_B(x,\Phi)$ are smooth
functions of their arguments. All data are compatible. As before, each
time-slice ${\cal S}_t $ is spacelike, with future-directed unit normal
$n^a(x,\Phi)$, the boundary ${\cal T}$ is timelike with outward unit
normal $N^a(x,\Phi)$ and $T^a = T^a(x,\Phi)$ is an arbitrary
future-directed timelike vector field tangent to ${\cal T}$.
In~\cite{isol}, it was shown that the IBVP
(\ref{Eq:WaveSystemEq})--(\ref{Eq:WaveSystemBC}) is
strongly well posed given certain restrictions on $c^{a\, A}{}_B$, i.e.
there exists a solution locally in time which satisfies (\ref{eq:swp}) in
terms of the corresponding $L_2$ norms for $\Phi^A$ and its gradient.
One important situation in which the restrictions on $c^{a\, A}{}_B$ are
satisfied is when it can be transformed into the upper diagonal form
\begin{displaymath}
c^{a\, A}{}_B = 0,\qquad B \leq A.
\label{eq:triang}
\end{displaymath}
This has important applications to the constrained systems of wave
equations obtained by formulating Maxwell's equations in the Lorentz
gauge or Einstein's equations in the harmonic gauge~\cite{isol}, as
discussed in Sec.~\ref{sec:harm}.
\bigskip
\subsubsection {Pseudo-differential theory}
\label{sec:fourlap}
\bigskip
The previous scalar wave problems required a non-standard energy to
obtain the necessary estimates. In more general problems, an
effective choice of energy might not be obvious or even exist.
Pseudo-differential theory provides an alternative treatment of such
cases. The approach is based upon a Fourier transform for the spatial
dependence and a Laplace transform in time. It can be applied equally
well to first or second order systems. Here I illustrate how it applies
to the second order wave equation. The more general theory for a system
of equations is usually presented in first order form and is reviewed in
Sec.~\ref{sec:pseudodiff}.
As an illustrative example, consider the 2(spatial)-dimensional version
of the IBVP for the scalar wave considered in Sec.~\ref{sec:enwave},
\begin{equation}
\Phi_{tt}= \Phi_{xx}+ \Phi_{yy}+F,\quad x\ge 0,~ -\infty <y<\infty ,
\label{eq:wp1}
\end{equation}
with boundary condition at $x=0$
\begin{equation}
\Phi_t-\alpha \Phi_x-\beta \Phi_y =q, \quad \alpha>0, \, \, |\beta|<1 ,
\label{eq:wp2}
\end{equation}
with compact boundary data $q$ and initial data
\begin{equation}
\Phi(0,x,y)=f_1(x,y),\quad \Phi_t(0,x,y)=f_2(x,y) .
\label{eq:wp3}
\end{equation}
The following simple observation reveals the underlying idea. The system
cannot be well-posed if the homogeneous version of
(\ref{eq:wp1})--(\ref{eq:wp3}) with $F=q=0$ admits arbitrarily fast
growing solutions. The homogeneous system has solutions of the form
\begin{equation}
\Phi(t,x,t)=e^{st+i\omega y} \varphi(x),\quad |\varphi|_\infty <\infty ,
\label{eq:wp4}
\end{equation}
where
\begin{eqnarray}
\varphi_{xx} &-& (s^2+\omega^2)\varphi=0, \label{eq:wp5} \\
s\varphi(0)&=&\alpha \varphi_x(0)+i\beta\omega \varphi(0) .
\label{eq:wp6}
\end{eqnarray}
Here $\varphi(x)$ is a smooth bounded function, so that its maximum
norm $|\varphi|_\infty$ is finite, $\omega$ is a real constant
and $s=\eta +i\xi$ a complex constant. This poses an eigenvalue problem.
If there are solutions with $\eta=\Re\, s >0$ then
$$
\Phi_\nu =e^{s\nu t+i\nu\omega y} \varphi(\nu x)
$$
is also a solution for any $\nu >0$, so that there are solutions which
grow arbitrarily fast exponentially. Therefore, a necessary condition
for a well-posed problem is that solutions with $\Re\, s >0$ must be
ruled out by the boundary condition.
The general solution of the ordinary differential equation (\ref{eq:wp5})
is
\begin{equation}
\varphi=\sigma_1 e^{\kappa_+ x}+ \sigma_2 e^{\kappa_- x},
\label{eq:wp7}
\end{equation}
where
\begin{equation}
\kappa_{\pm}=\pm \sqrt{s^2+\omega^2}
\label{eq:eigenv}
\end{equation}
solve the characteristic equation
$$ \kappa^2-(s^2+\omega^2)=0.
$$
Here $ \Re\,\kappa_+>0$ and $ \Re \, \kappa_-<0$ for $\Re \, s>0$. By
assumption $\varphi$ is bounded which requires that $\sigma_1=0,$ i.e.
\begin{equation}
\varphi=\sigma_2 e^{\kappa_- x}.
\label{eq:wp10}
\end{equation}
Introducing (\ref{eq:wp10}) into the boundary condition (\ref{eq:wp6}) gives
\begin{equation}
(s-\alpha\kappa_- -i\beta\omega )\sigma_2=0 .
\label{eq:wp11}
\end{equation}
\smallskip\noindent
Since $\Re \, \kappa_-<0$ and by assumption $\alpha >0$, there
are no solutions with
$\Re \, s>0$.
Thus this necessary condition for a well-posed
problem is satisfied.
In order to proceed further
it is technically convenient
to assume that the initial data (\ref{eq:wp3}) vanishes. This
may always be achieved by the transformation
\begin{equation}
\Phi \rightarrow \Phi-e^{-t} f_1 - t e^{-t}(f_2+f_1),
\label{eq:homcd}
\end{equation}
so that the initial data gets swept into the forcing term $F$.
Then (\ref{eq:wp1})--(\ref{eq:wp3}) can be solved by
Fourier transform with respect to $y$ and Laplace transform with respect to $t$,
i.e. in terms of
\begin{equation}
\hat \Phi (s,x,\omega) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dy
e^{-i\omega y} \int_0^\infty dt e^{-st} \Phi(t,x,y) \, , \quad \Re \, s >0,
\end{equation}
where $\omega$ is real and $s$ is complex.
The inhomogeneous versions of (\ref{eq:wp5}) and (\ref{eq:wp6}) imply
that the coefficients satisfy
\begin{eqnarray}
\hat \Phi_{xx}&-&(s^2+\omega^2)\hat \Phi =-\hat F \nonumber \\
(s&-&i\beta \omega)\hat \Phi(s,0,\omega)-\alpha\hat \Phi_x(s,0,\omega) =\hat q(s,\omega).
\label{eq:wp12}
\end{eqnarray}
Since it has already been shown that the homogeneous system
(\ref{eq:wp5})--(\ref{eq:wp6}) has no eigenvalues for $\Re \, s>0$
and $\Re \, \kappa<0$, it follows that (\ref{eq:wp12}) has a unique solution
$\hat \Phi$. Inversion of the Fourier-Laplace transform then
gives a unique solution for $\Phi$.
The well-posedness of a variable coefficient or quasilinear wave
problem also requires estimates of the higher derivatives of $\Phi$. The
system of equations for the derivatives are obtained by differentiating the
wave equation and the boundary condition. In that process, any variable
coefficient terms in the boundary condition lead to inhomogeneous
boundary data for the derivatives. It is possible to transform
the boundary data $q$ to $0$ by a transformation analogous to
(\ref{eq:homcd}), which sweeps $q$ and its derivatives into the forcing
term $F$. If this inhomogeneous boundary data is continually subtracted out
of the boundary condition, inhomogeneous terms of higher differential order
appear in the forcing term. As a consequence, the resulting estimates would
bound lower derivatives of the solution in terms of higher derivatives of
the data, a process referred to as ``losing'' derivatives.
Instead, a different approach is necessary to establish well-posedness.
It is simple to calculate the solution for $\hat F = 0$.
Corresponding to (\ref{eq:wp10}) and (\ref{eq:wp11}), there follows
\begin{equation}
\hat \Phi(s,x,\omega)=e^{\kappa_- x}\hat \Phi(s,0,\omega)
\label{eq:wp13}
\end{equation}
where
$$
(s-\alpha\kappa_- -i\beta \omega)\hat \Phi (s,0,\omega)=\hat q(s,\omega).
$$
It is now possible to establish~\cite{wpgs}
\medskip
\noindent {\bf Boundary stability:} The solution (\ref{eq:wp13})
satisfies the estimates
\begin{eqnarray}
|\hat \Phi_x(s,0,\omega,)|&\le& K |\hat q(s,\omega)|, \nonumber \\
\sqrt{|s|^2+\omega^2}\cdot
|\hat \Phi(s,0,\omega)|&\le& K |\hat q(s,\omega)|,
\label{eq:wbstab}
\end{eqnarray}
where the constant $K$ is independent of $s$ and $\omega$.
Similar estimates hold for all the derivatives.
\medskip
The estimates (\ref{eq:wbstab}) follow from purely algebraic consequences of the eigenvalue
relations (\ref{eq:eigenv}) and (\ref{eq:wp11}).
The essential steps are to show
\begin{enumerate}
\item There is a constant $\delta_1>0$ such that$|\Re \, \kappa_- | > \delta_1 \Re \, s$
\item For all $\omega$ and $s$ with $\Re s \ge 0$,
there is a constant $\delta_2>0$ such that
$|s-\alpha\kappa_- -i\beta\omega|\ge\delta_2\sqrt{|s|^2+|\omega|^2}$.
\end{enumerate}
See~\cite{wpgs} for the details.
\medskip
Boundary stability allows the application of the theory of pseudo-differential operators
to show that the IBVP is {\em strongly well-posed in the generalized sense},
\medskip
\begin{equation}
\int_0^t \|{\bf \Phi}(\tau)\|^2d\tau+\int_0^t \|{\bf \Phi}(\tau)\|^2_B d\tau
\le K_T \left ( \int_0^t\|F(\tau)\|^2 d\tau
+\int_0^t\|q(\tau)\|^2_B d\tau \right ) , \quad 0 \le t \le T,
\label{eq:swpg}
\end{equation}
where ${\bf \Phi}=(\Phi, \Phi_a)$.
The theory is discussed in Sec.~\ref{sec:pseudodiff} in the standard
context of first order systems. But the first order theory is flexible enough to apply
to second order systems.
In particular, in Sec.~\ref{sec:pseudodiff} it is applied to show that the IBVP
for the quasilinear version of the second order wave equation with boundary
condition (\ref{eq:wp1})--(\ref{eq:wp2})
is well-posed in the generalized sense.
Strong well-posedness in the generalized sense is similar to strong
well-posedness (\ref{eq:swp}) except now the initial data has been swept into the forcing
term and the estimate for ${\bf \Phi}$ in the interior involves
a time integral. In both cases, the gradients at the boundary are estimated
by the boundary data and the forcing, i.e. a derivative is gained at the boundary.
This ensures that the well-posedness of the local halfplane problem can be
extended globally to include boundaries that lead to multiple reflections.
\bigskip
\subsubsection{Generalized eigenvalues}
\bigskip
Strong well-posedness in the generalized sense not only rules out eigenvalues
of (\ref{eq:wp6}) with $\eta=\Re(s)>0$ but also {\it generalized eigenvalues}
for which $\eta=0$. This is implicit in the estimates for boundary stability
(\ref{eq:wbstab}) in which the constant $K$ is independent of $s$.
However, generalized eigenvalues can exist in well-behaved physical systems.
A prime example is a surface wave which travels
tangential to the boundary with periodic time dependence. See~\cite{stewart}
for the treatment of such an example from Maxwell theory.
Generalized eigenvalues are ruled out by the boundary conditions
required for strong well-posedness in the generalized sense and historically
have been treated on an individual basis. However, a new approach
to this problem has recently been formulated by H-O. Kreiss~\cite{heinzpc}.
This approach splits the problem into two subproblems:
\begin{enumerate}
\item One in which the forcing vanishes, $F=0$.
\item One in which the boundary data is homogeneous, $q=0$.
\end{enumerate}
A second order wave problem is called
{\em well-posed in the generalized sense} if these subproblems satisfy
the corresponding estimates
\begin{equation}
\int_0^t \|\Phi(\tau)\|^2_B d\tau
\le K_T \int_0^t \|q(\tau)\|^2_B d\tau \, , \quad 0\le t \le T,
\label{eq:wpg1}
\end{equation}
\begin{equation}
\int_0^t \| {\bf \Phi}(\tau)\|^2d\tau \le K_T \int_0^t \|F(\tau)\|^2 d\tau \, ,
\quad 0\le t \le T.
\label{eq:wpg2}
\end{equation}
Here it is only required that $\Phi$, and not its gradient ${\bf \Phi}$,
be estimated by the boundary data $q$. Thus the solution no longer gains a
derivative at the boundary. However, no global problems arise from multiple
reflections because the estimate (\ref{eq:wpg2}) implies the gain of one
derivative in the interior.
As examples, consider the scalar wave problem (\ref{eq:wp1}) with the two
choices of boundary conditions at $x=0$,
\begin{eqnarray}
({\bf A})&\quad & \Phi_x -i\beta \Phi_y =q, \nonumber \\
({\bf B})& \quad & \Phi_x -\beta \Phi_y =q, \nonumber
\end{eqnarray}
where $\beta$ is real, with $|\beta|<1$. In case ({\bf A}), $\Phi$ is
complex. Introducing these boundary conditions into the homogeneous
system (\ref{eq:wp4})--(\ref{eq:wp6}) gives
\begin{eqnarray}
({\bf A})&\quad & \kappa =\omega \beta, \nonumber \\
({\bf B})& \quad &\kappa=-i\omega \beta, \nonumber
\end{eqnarray}
with $s^2=\kappa^2-\omega^2$.
In neither case is there a solution with $\Re s>0$ but both cases possess
generalized eigenvalues,
\begin{eqnarray}
({\bf A})& \quad& s^2=-\omega^2(1- \beta^2), \quad \Re s =0, \nonumber \\
({\bf B})& \quad &s^2=-\omega^2(1+\beta^2), \quad \Re s =0. \nonumber
\end{eqnarray}
The corresponding eigenfunctions are the {\it surface waves}
\begin{equation} ({\bf A}) \quad e^{i\omega
(\pm\sqrt{1-\beta^2}t+y)-|\omega\beta|x},
\end{equation}
and the oscillatory waves
\begin{equation} ({\bf B}) \quad e^{i\omega
(\pm\sqrt{1+\beta^2}t+\beta x+y)},
\end{equation}
which give rise to {\it glancing waves} for $\beta=0$.
By investigating the inhomogeneous problem, it can be shown that
both choices of boundary condition give rise to an IBVP which satisfies
(\ref{eq:wpg1}) and (\ref{eq:wpg2}) and is well-posed in the generalized
sense~\cite{heinzpc}.
\bigskip
\subsection{First order symmetric hyperbolic systems}
\label{sec:firstord}
\bigskip
Most of the work on the IBVP for hyperbolic systems has been directed
toward fluid dynamics, where a first order formulation is natural. Here I
describe the essentials of the two main approaches, pseudo-differential
theory and the theory of symmetric hyperbolic systems.
\bigskip
\subsubsection{Pseudo-differential theory}
\label{sec:pseudodiff}
\bigskip
In order to summarize the pseudo-differential theory for a
first order symmetric hyperbolic system consider first the
constant coefficient system
\begin{equation}
u_t={\cal P}(\partial_x)u+F,\quad
{\cal P}(\partial_x)={\cal P}^i \partial_{x_i}
=A\, \partial_{x_1}+\sum_{j=2}^m B_j \partial_{x_j}
\label{eq:A1}
\end{equation}
on the half-space
$$ t\ge 0,\quad x_1\ge 0,\quad -\infty < x_j <\infty\, , ~j=2,\ldots, m ,
$$
with initial data
\begin{equation}
u(0,x)= f(x). \label{eq:A4}
\end{equation}
Here $u(t,x)=\left(u^{(1)}(t,x),\ldots,u^{(N)}(t,x)\right)$ is a vector
valued function of the real variables $(t,x)=(t,x_1,\ldots, x_m)$ and
$A,B_j$ are constant $N\times N$ matrices. In applications to spacetime,
$m=3$ but the number of spatial dimensions does not complicate the theory.
The notation $ \langle u, v \rangle $ and $|u|^2= \langle u, u \rangle$
denotes the inner product and norm in the $N$-dimensional linear space.
All data are smooth, compatible and have compact support.
The {\it symbol} representing the principal part of the system,
\begin{equation}
{\cal P}(i\omega)=iA\omega_1+iB(\omega_-),\quad B(\omega_-)
=\sum_{j=2}^m B_j \omega_j
\, \quad |\omega |=1,
\label{eq:A2}
\end{equation}
is obtained by replacing $\partial_x$ by its Fourier representation
$i\omega=(i\omega_1, i\omega_-),~\omega_-=(\omega_2,\ldots,\omega_m)$.
Symmetric hyperbolicity requires that $A$ and $B$ be self-adjoint
matrices so that the eigenvectors of ${\cal P}$ form a complete set with
purely imaginary eigenvalues for all real $\omega$. More precisely, there
exists a symmetric, positive definite {\it symmetrizer} $H$ such that
$HA$ and $HB_j$ are self-adjoint. See~\cite{agran} for more
general applications.
Here it is also assumed that the {\it boundary matrix}
$A$ is nonsingular so that it can be transformed into the form
\begin{equation}
A=\pmatrix{-\Lambda^I & 0\cr 0& \Lambda^{II}},
\label{eq:A3}
\end{equation}
where $\Lambda^I,\Lambda^{II}$ are real positive definite diagonal
matrices acting on the $P$ dimensional subspace $u^I$ and the $(N-P)$
dimensional subspace $u^{II}$, respectively. The theory also applies to
the singular case where the boundary is uniformly characteristic, i.e.
the kernel of $A$ has constant dimension~\cite{majda}.
See Sec.~\ref{sec:md} for a treatment of the singular case by
the energy method.
The IBVP requires $P$ boundary conditions at $x_1=0$, corresponding to
the $P$ ingoing modes in the plane wave decomposition carried out below
in conjunction with (\ref{eq:A7}) and (\ref{eq:A11}). They are prescribed
in the form
\begin{equation}
u^I(t,0,x_{-})=Su^{II}(t,0,x_{-})+q(t,x_{-}),\quad x_{-}=(x_2,\ldots,x_m).
\label{eq:A5}
\end{equation}
The main ingredient of a definition of well-posedness is the estimate of
the solution in terms of the data. See Sec.~7.3 of~\cite{green-book}. By
a transformation analogous to (\ref{eq:homcd}), the IBVP (\ref{eq:A1}),
(\ref{eq:A4}), (\ref{eq:A5}) reduces to a problem with homogeneous
initial data $f=0$. The required estimate is the first order version of
the estimate (\ref{eq:swpg}) for the second order wave equation. The
first order problem is called {\em strongly well-posed in the generalized
sense} if there is a unique solution $u$ such that
\begin{equation}
\int_0^t \|u(\tau)\|^2 d\tau +
\int_0^t \|u(\tau)\|^2_B d\tau \le
K_T \bigg ( \int_0^t \|F(\tau)\|^2 d\tau +
\int_0^t \|q(\tau)\|^2_B d\tau \bigg ) \, , \quad 0\le t\le T,
\label{eq:A6}
\end{equation} where the constant $K_T$ does not depend on $F$ or $q$.
Here $ \|u\|$ and $ \|u\|_B$ are the $L_2$ norms of $|u|$ over the
half-space and the boundary, respectively.
As for the scalar wave problem considered in Sec.~\ref{sec:fourlap}, the
IBVP is not well-posed if the homogeneous system $F=q=0$ admits wave
solutions
\begin{equation}
u(t,x_1,x_-)=e^{st+i\langle \omega,x\rangle_-} \varphi(x_1),\quad
\langle \omega,x\rangle_-=\sum_{j=2}^m \omega_j x_j, \quad \Re \, s >0,
\label{eq:A7}
\end{equation}
with $|\varphi|_\infty <\infty$. The existence of such homogeneous
solutions would imply the existence of solutions which grow arbitrarily
fast exponentially.
In order to decide whether such homogeneous waves exist, introduce
(\ref{eq:A7}) into (\ref{eq:A1}) and (\ref{eq:A5}) to obtain
\begin{eqnarray}
s\varphi&=A\varphi_{x_1}+iB(\omega_-)\varphi,\quad x_1 \ge 0, \nonumber \\
\varphi^I(0)&=S\varphi^{II}(0),\quad |\varphi|_\infty <\infty.
\label{eq:A9}
\end{eqnarray}
This is an eigenvalue problem for a system of ordinary differential
equations which can be solved in the usual way. Let $\kappa$ denote the
solutions of the characteristic equation
\begin{eqnarray}
{\rm Det}|A\kappa-\left(sI-iB(\omega_-)\right)|=0,
\label{eq:A10}
\end{eqnarray}
obtained by setting $\varphi(x_1)=e^{\kappa x_1}\varphi (0)$.
\medskip
It can be shown that
\begin{enumerate}
\item There are exactly $r$ eigenvalues with $\Re \, \kappa <0$
and $n-r$ eigenvalues with $\Re \, \kappa>0.$
\item There is a constant $\delta>0$ such that, for all $s$
with real $\Re s >0$
and all $\omega_-$,
$$ |\Re \, \kappa |>\delta \Re s. $$
In particular, for $\Re \, s >0,$ there are no $\kappa$ with $\Re \, \kappa =0.$
\end{enumerate}
See~\cite{kreiss2} for the proof.
\medskip
If all eigenvalues $\kappa_j$ are distinct, the general solution of (\ref{eq:A9})
has the form
\begin{equation}
\varphi=\sum_{\Re \, \kappa_j<0} \sigma_j e^{\kappa_jx_1} h_j+
\sum_{\Re \, \kappa_j>0} \sigma_j e^{\kappa_jx_1} h_j ,
\label{eq:A11}
\end{equation}
where $h_j$ are the corresponding eigenvectors.
(If the eigenvalues are degenerate, the usual modifications apply.)
For bounded solutions, all $\sigma_j$ in the second
term are zero. Introducing $\varphi$ into the boundary conditions
(\ref{eq:A9}) at $x_1=0$ gives a linear system of $r$ equations for
the $r$ unknowns $(\sigma_1,\ldots,\sigma_r)= {\underline{\sigma}}$
of the form
\begin{equation}
C(s,\omega_-) {\underline{\sigma}} =0.
\label{eq:A12}
\end{equation}
Therefore the problem is not well-posed if for some $\omega_-$
there is an eigenvalue $s_0$ with $\Re \,s_0>0$, i.e.
${\rm Det} \,C(s_0,\omega_-)=0$. In that case, the linear system
(\ref{eq:A12}), and therefore also (\ref{eq:A9}), has a nontrivial solution.
Thus the determinant condition
\begin{equation}
{\rm Det} \,C(s_0,\omega_-)\ne 0, \quad \Re \,s_0>0 ,
\label{eq:detcond}
\end{equation}
is necessary for a well-posed problem.
Thus in order to consider a well-posed problem assume that
${\rm Det} \,C\ne 0$ for $\Re \, s >0.$ Then
the inhomogeneous IBVP (\ref{eq:A1}), (\ref{eq:A4}), (\ref{eq:A5})
can be solved by Laplace transform in time and Fourier
transform in the tangential variables. Again set $u(0,x)=f(x)= 0$. Then
\begin{eqnarray}
s\hat u&=&A\hat u_x+iB(\omega_-) \hat u+\hat F,
\label{eq:A13a} \\
\hat u^I(0)&=&S\hat u^{II}(0)+\hat q.
\label{eq:A13}
\end{eqnarray}
Since, by assumption, (\ref{eq:A9}) has only the trivial solution for $\Re \, s>0$,
(\ref{eq:A13}) has a unique solution. Inverting the Fourier and Laplace
transforms gives the solution in physical space.
As in the scalar wave case, it is simple to solve (\ref{eq:A13a}) for $\hat F = 0$.
From the inhomogeneous versions of (\ref{eq:A11}) and (\ref{eq:A12}),
$$ \hat u(s,x_1,\omega_-)=\sum_{\Re \, \kappa_j <0} \sigma_j e^{\kappa_j x_1} h_j,
$$
where the $\sigma_j$ are determined by
$$ C(s,\omega_-) {\underline{\sigma}}=\hat q.
$$
\medskip
It is now possible to establish the pseudo-differential version of boundary
stability:
\medskip
\noindent For all
$\omega,s$ with $\Re s>0,$ there is a constant $K$ independent
of $\omega,s$ and $\hat q$ such that the solutions of
(\ref{eq:A13a})--(\ref{eq:A13}) with $\hat F = 0$ satisfy
\begin{equation}
|\hat u(s,0,\omega)|\le K|\hat q(s,\omega)|.
\label{eq:A14}
\end{equation}
\medskip
Boundary stability is equivalent to the requirement
that the eigenvalue problem (\ref{eq:A9}) has
no eigenvalues for $\Re \, s\ge 0$ or that ${\rm Det} \, C(\omega_-,s)\ne 0$
for $\Re \, s\ge 0$. In particular, it rules out generalized eigenvalues.
It is essential to establish the
\medskip
\noindent {\bf Main Theorem}: If the half-space problem is boundary stable
then it is strongly well-posed in the generalized sense of (\ref{eq:A6}).
\medskip
\noindent See~\cite{kreiss2} for the proof, where boundary
stability is used to construct a
{\it symmetrizer} in the Fourier-Laplace representation
which leads to the estimate
\begin{equation}
\eta \|\hat u(s,x_1,\omega)\|^2+|\hat u(s,0,\omega)|^2\le
const\left( {1\over\eta} \|\hat F\|^2+c|\hat q|^2\right).
\label{eq:A16}
\end{equation}
Inversion of the Fourier-Laplace transform proves the theorem.
\medskip
The pseudo-differential theory
has far reaching consequences. In particular, the computational rules
for pseudo-differential operators imply:
\begin{enumerate}
\item The Laplace transform only requires that the estimates hold for
$\eta>\eta_0>0$, where $\eta_0$ is sufficiently large to allow for (controlled)
exponential growth due to lower order terms.
This is essential for extending strong well-posedness to
systems with variable coefficients.
\item Boundary stability is also valid if the symbol depends smoothly on $(t,x)$
and is not destroyed by lower order terms. Therefore the problem
can be localized and well-posedness in general domains can be
reduced to the study of the Cauchy problem and half-space problems.
\item The {\it principle of frozen coefficients} holds.
The properties of the pseudo-differential operators give rise to estimates of
derivatives in the same way as for standard partial differential equations.
Therefore strong well-posedness in the generalized sense can
be extended to linear problems
with variable coefficients and, locally in time,
to quasilinear problems.
\end{enumerate}
\medskip
Since pseudo-differential operators are much more flexible than standard
differential operators, they can be applied to second order systems as
well as first order systems. Consider, for example, the problem
(\ref{eq:wp1})--(\ref{eq:wp3}) discussed in Sec.~\ref{sec:fourlap}. After
transforming the initial data to zero, the Fourier-Laplace transform
becomes
\begin{eqnarray}
&\hat \Phi_{xx}&=(s^2+\omega^2)\hat \Phi-\hat F, \nonumber \\
& s\hat \Phi&-\alpha \hat \Phi_x- i \beta \omega \hat \Phi =\hat q.
\label{eq:A17}
\end{eqnarray}
Introduction of a new variable $\hat \Phi_x=\sqrt{|s|^2+\omega^2}\,\hat
\Psi$ gives the first order system
\begin{eqnarray}
\hat u_x &=&
\frac{1}{\sqrt{|s|^2+\omega^2}}\,
\pmatrix{ 0 & |s|^2+\omega^2\cr s^{2}+\omega^{2} & 0 \cr}
\hat u -\tilde F,\quad
\hat u=\pmatrix{\hat \Phi\cr \hat \Psi\cr}, \cr
&\cr
&{}&\frac{s}{\sqrt{|s|^2+\omega^2}}\hat \Phi-\alpha \hat \Psi
- \frac{i\beta\omega}{\sqrt{|s|^2+\omega^2}}\hat \Phi=\tilde q,
\label{eq:A18}
\end{eqnarray}
where
\begin{eqnarray}
\tilde F={1\over\sqrt{|s|^2+\omega^2}} \pmatrix{0\cr \hat F\cr},\quad
\tilde q={1\over\sqrt{|s|^2+\omega^2}} \hat q.
\label{eq:A19}
\end{eqnarray}
\medskip
The second order problem (\ref{eq:wp1})--(\ref{eq:wp3}) is strongly
well-posed in the generalized sense if the corresponding first order
problem (\ref{eq:A18}) with general data $\tilde F,\tilde q$ has this
property. The second order version of boundary stability (\ref{eq:A17})
established in Sec.~\ref{sec:fourlap}, rewritten in terms of the first
order variables, implies
\begin{eqnarray}
|\hat \Phi(s,0,\omega)|+|\hat \Psi(s,0,\omega)| \le K \, |\tilde q(s,\omega)|,
\label{eq:A20}
\end{eqnarray}
i.e. the first order order version of boundary stability (\ref{eq:A14}).
Thus the Main Theorem applies and the second order problem is
strongly well-posed in the generalized sense in the first order version
(\ref{eq:A6}), which is equivalent to the second order version
(\ref{eq:swpg}).
\bigskip
\subsubsection{The energy method for first order symmetric hyperbolic
systems}
\label{sec:md}
\bigskip
Energy estimates for first order symmetric hyperbolic systems were first
applied to Einstein's equations in harmonic coordinates by Fischer and
Marsden~\cite{fischmarsd} to give an alternative derivation of the
results of Choquet-Bruhat for the Cauchy problem. The energy method
extends to the quasilinear IBVP with {\em maximally dissipative} boundary
conditions.
Again, begin by considering the constant coefficient system
(\ref{eq:A1})
\begin{equation}
u_t={\cal P}^i\partial_i u+F,\quad {\cal P}^i\partial_i
=A\, \partial_{x_1}+\sum_{j=2}^m B_j \partial_{x_j}
\label{eq:B1}
\end{equation}
on the half-space
$$ t\ge 0,\quad x_1\ge 0,\quad -\infty < x_j <\infty\, , ~j=2,\ldots, m ,
$$
with initial data $ u(0,x)= f(x)$. As before, $A$ and $B_j$ are symmetric
$N\times N$ matrices so that the eigenvectors of $P^i$ form a complete
set with real eigenvalues. In matrix notation, there is a symmetric
positive definite symmetrizer $H_{MN}$ such that $H_{MP}A^P_N$ and
$H_{MP} B_{j}{}^{P}_N$ are symmetric. Here, the boundary matrix $A$ is
allowed to be singular, cf.~\cite{majda,rauch,secchi3}. With an
appropriate choice of symmetrizer, it can be put in the form
\begin{equation}
A=\alpha \pmatrix{-I^P & 0 &0\cr 0& O^Q& 0\cr 0&0& I^R} ,
\, \alpha>0, \quad
\quad u= \pmatrix{u^I \cr u^O \cr u^{II}}
\label{eq:B3}
\end{equation} where $I^P$and $I^R$ are identity matrices acting on the
$P$-dimensional subspace $u^I$ and the $R$-dimensional subspace $u^{II}$,
respectively,and $O^Q$ is a zero matrix acting on $Q$-dimensional kernel
$u^O$, with $N=P+Q+R$. No boundary condition can be imposed on the
components of $u^{II}$, which are the outgoing modes, or the components
of $u^O$, The components of $u^O$ satisfy
PDEs intrinsic to the boundary. They are referred to as
{\it zero velocity modes} since they have no velocity relative
to the boundary. As in the discussion of the
non-characteristic case in Sec.~\ref{sec:pseudodiff}, there are $P$
ingoing modes and the IBVP requires $P$ boundary conditions at $x_1=0$.
They are prescribed in the {\em maximal} form
\begin{equation}
u^I(t,0,x_{-})=S u^{II}(t,0,x_{-})+q(t,x_{-}),\quad
x_{-}=(x_2,\ldots,x_m) ,
\label{eq:B5}
\end{equation}
where the $P\times R$ matrix $S$ satisfies the {\em dissipative}
condition that, for homogeneous data $q=0$, the local energy flux out of
the boundary be positive,
\begin{equation}
{\cal F}: = \langle u, A u \rangle \ge 0.
\end{equation}
In the simplified form (\ref{eq:B3}), this leads to the requirement
\begin{equation}
- |Su^{II}(t,0,x_{-})|^2 + |u^{II}(t,0,x_{-})|^2 \ge 0,
\label{eq:B6}
\end{equation}
where $|u|^2 = \langle u, u \rangle$ in terms of the linear space inner
product.
The rationale for these {\em maximally dissipative} boundary conditions
results from the energy estimate for the case of homogeneous boundary
data $q=0$. Beginning with
\begin{equation}
\partial_t \langle u,u \rangle
= 2\langle u,A\, \partial_{x_1}u
+\sum_{j=2}^m B_j \partial_{x_j}u +F \rangle ,
\label{eq:B7}
\end{equation}
integration over the half-space gives
\begin{eqnarray}
\partial_t E:= \partial_t \| u \|^2
&= & 2(u,A\, \partial_{x_1}u) +2(u,F) \nonumber \\
&= & - (u,Au)_B +2(u,F), \nonumber \\
&\le &2(u,F) \le \| u \|^2+ \| F \|^2 =E + \| F \|^2 ,
\label{eq:B8}
\end{eqnarray}
where the $(u,v)$ denotes the integral of $\langle u, v\rangle $, etc.
Thus the maximally dissipative boundary conditions provide the required
energy estimate. Inhomogeneous boundary data $q$ can be transformed to
zero by the change of variable $u\rightarrow u-q$ to obtain
an analogous estimate.
More generally, if the boundary is uniformly characteristic so that the
kernel of the boundary matrix $A$ has constant dimension $Q$, then the
quasilinear IBVP problem with maximally dissipative boundary conditions
is strongly well-posed: a solution exists locally in time which satisfies
(\ref{eq:A6}). See~\cite{shuxing,ohkubo,gues,secchi2,secchi3} for
details.
The theory can also be recast in the covariant space-time form
\begin{equation}
A^\mu \partial_\mu u =F
\label{eq:fosh}
\end{equation}
where $A^\mu=(A^t,A^i)$ are symmetric matrices and $A^t$ is positive
definite. As an illustration of this and the various choices of boundary
conditions, consider the IBVP for the scalar wave equation
\begin{equation}
g^{\mu\nu}\partial_\mu \partial_\nu \Phi = 0,
\quad x\ge 0, \quad t \ge 0, \quad g^{xx}>0.
\end{equation}
This can be rewritten in the first order symmetric hyperbolic form
(\ref{eq:fosh}) for a 5-component field $u$ by introducing the auxiliary
variables
\begin{equation}
u =\left(
\begin{array}{c}
u_0 \\
u_t \\
u_i
\end{array}
\right)
= \left(
\begin{array}{c}
\Phi \\
\partial_t \Phi \\
\partial_i \Phi
\end{array}
\right) .
\end{equation}
The matrices $A^\mu$ are then given by
\begin{equation}
A^t = \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -g^{tt} & 0 \\
0 & 0 & g^{jk}
\end{array}
\right) ,
\quad
A^i = \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & -2g^{ti} & -g^{ji} \\
0 & -g^{ij} & 0
\end{array}
\right),
\end{equation}
with
\begin{equation}
F= \left(
\begin{array}{c}
u_t \\
0 \\
0
\end{array}
\right).
\end{equation}
In this first order form, the Cauchy data consist of $u|_{t=0}=f$ subject
to the constraints
\begin{equation}
{\cal C}_i:= u_i - \partial_i u_0 =0.
\label{eq:sconstr}
\end{equation}
The evolution system implies that the constraints satisfy
\begin{equation}
\partial_t {\cal C}_i=0,
\end{equation}
so that they propagate up the timelike boundary at $x=0$ and present no
complication.
The boundary matrix $(-A^x)$ (oriented in the outward normal direction)
has a 3-dimensional kernel, whose transposed basis
consists of the zero-velocity modes
\begin{equation}
(1,0,0,0,0), \quad
(0,0,-g^{xy},g^{xx}, 0), \quad \mbox{and} \quad
(0,0,-g^{xz},0, g^{xx} ) .
\end{equation}
In addition, there is one positive eigenvalue and one negative eigenvalue
\begin{equation}
\lambda_{\pm}= \pm \lambda +g^{xt},
\end{equation}
where
\begin{equation}
\lambda= \sqrt{({g^{xt}})^2+\delta_{ij}g^{xi}g^{xj}} .
\end{equation}
Thus precisely one boundary condition is required.
In terms of the normalized eigenvectors
\begin{equation}
e_{\pm} = \frac{1}{\sqrt{ \pm 2 \; \lambda \; \lambda_\pm }} \left(
\begin{array}{c}
0 \\
\pm \lambda +g^{xt} \\
+g^{xi} ,
\end{array}
\right).
\end{equation}
$u=u_+ e_ + +u_- e_- +u_O$, where
$u_O$ lies in the kernel. The boundary condition takes the form $u_+
-Su_- =q$, subject to the dissipative condition
\begin{equation}
{\cal F}=-\langle u,A^x u \rangle \ge 0.
\end{equation}
For the case of homogeneous data $q=0$, this requires that
\begin{eqnarray}
S^2 \le - {\lambda_-} \; / \; {\lambda_+} \; .
\end{eqnarray}
The limiting cases $S = \pm \sqrt{ - \; {\lambda_-} \; / \; {\lambda_+}}$
lead to reflecting boundary conditions. In the case of a standard
Minkowski metric $g^{\mu\nu}=\eta^{\mu\nu}$, these correspond to the
homogeneous Dirichlet condition and Neumann conditions
\begin{equation}
\partial_t \Phi|_B = 0 , \quad \partial_x \Phi |_B=0,
\end{equation}
and the choice $S=0$ corresponds to the Sommerfeld condition
\begin{equation}
( \partial_t -\partial_x) \Phi |_B = 0.
\end{equation}
More generally, the geometric interpretation of these boundary conditions
is obscured because the energy $E$ (\ref{eq:B8}) standardly
used in the first order theory is constructed with the linear space inner
product, as opposed to the geometrically defined energy (\ref{eq:ewave})
natural to the second order theory. It is possible to reformulate the
covariant theory of the wave equation in first order symmetric form with
boundary conditions based upon the covariant energy
(\ref{eq:ewave}). But without such a guide to begin with, a
first order symmetric hyperbolic formulation can lose touch with
the underlying geometry.
This scalar wave example also illustrates how the number of evolution
variables and constraints escalate upon reduction to first order form. As
a result, in the case of Einstein's equations, the advantages of
utilizing symmetric hyperbolic theory is counterbalanced by the increased
algebraic complexity which is further complicated by the wide freedom in
carry out a first order reduction. See Sec.~\ref{sec:firstord}.
\bigskip
\subsection{The characteristic Initial-boundary value problem}
\label{sec:civp}
\bigskip
There is another IBVP which gained prominence after Bondi's~\cite{bondi}
success in using null hypersurfaces as coordinates to describe
gravitational waves. In the null-timelike IBVP, data is given on an
initial characteristic hypersurface and on a timelike worldtube to
produce a solution in the exterior of the worldtube. The underlying
physical picture is that the worldtube data represent the outgoing
gravitational radiation emanating from interior matter sources, while
ingoing radiation incident on the system is represented by the initial
null data.
The characteristic IBVP received little attention before its
importance in general relativity was recognized. Rendall~\cite{rend}
established well-posedness of the double null problem for the quasilinear
wave equation, where data is given on a pair of intersecting
characteristic hypersurfaces. He did not treat the characteristic problem
head-on but reduced it to a standard Cauchy problem with data on a
spacelike hypersurface passing through the intersection of the
characteristic hypersurfaces so that well-posedness followed from
the result for the Cauchy problem. He extended this approach to establish
the well-posedness of the double-null formulation of the full Einstein
gravitational problem. Rendall's approach cannot be applied
to the null-timelike problem, even though the double null problem is a
limiting case.
The well-posedness of the null-timelike problem for the gravitational
case remains an outstanding problem. Only recently has it been shown
that the quasilinear problem for scalar waves is well-posed~\cite{nullt}.
The difficulty unique to this problem can be illustrated in
terms of the 1(spatial)-dimensional wave equation
\begin{equation}
( \partial_{\tilde t}^2 -\partial_{\tilde x}^2)\Phi=0,
\label{eq:tilde1d}
\end{equation}
where $(\tilde t,\tilde x)$ are standard space-time coordinates. The
conserved energy
\begin{equation}
\tilde E(\tilde t)= \frac{1}{2} \int d\tilde x \bigg( (
\partial_{\tilde t}\Phi)^2 +(\partial_{\tilde x}\Phi)^2 \bigg )
\label{eq:tildeE}
\end{equation}
leads to the well-posedness of the Cauchy problem. In characteristic
coordinates $(t=\tilde t -\tilde x, \, x=\tilde t +\tilde x)$, the wave
equation transforms into
\begin{equation}
\partial_t \partial_x \Phi =0.
\label{eq:1dphi}
\end{equation}
The conserved energy evaluated on the characteristics $t={\rm const}$,
\begin{equation}
\tilde E(t) = \int dx (\partial_ x \Phi)^2,
\end{equation}
no longer controls the derivative $\partial_t \Phi$.
The usual technique for treating the IBVP is to split the problem into a
Cauchy problem and local half-space problems and show that these
individual problems are well posed. This works for hyperbolic systems
based upon a spacelike foliation, in which case signals propagate with
finite velocity. For (\ref{eq:tilde1d}), the solutions to the Cauchy
problem with compact initial data on $\tilde t=0$ are square integrable
and well-posedness can be established using the $L_2$ energy norm
(\ref{eq:tildeE}).
However, in characteristic coordinates the 1-dimensional wave equation
(\ref{eq:1dphi}) admits signals traveling in the $+x$-direction with
infinite coordinate velocity. In particular, initial data of compact
support $\Phi(0,x)=f(x)$ on the characteristic $t=0$ admits the solution
$\Phi = g(t) +f(x)$, provided that $g(0)=0$. Here $g(t)$ represents the
profile of a wave which travels from past null infinity ($x\rightarrow
-\infty$) to future null infinity ($x\rightarrow +\infty$). Thus,
without a boundary condition at past null infinity, there is no unique
solution and the Cauchy problem is ill posed. Even with the boundary
condition $\Phi(t,-\infty)=0$, a source of compact support $S(t,x)$
added to (\ref{eq:1dphi}), i.e.
\begin{equation}
\partial_t \partial_x \Phi =S,
\label{eq:1dphis}
\end{equation}
produces waves propagating to $x=+\infty$ so that although the solution
is unique it is still not square integrable.
On the other hand, consider the modified problem obtained by setting
$\Phi=e^{ax}\Psi$,
\begin{equation}
\partial_t (\partial_x+a) \Psi=F \, , \quad \Psi(0,x)
= e^{-ax}f(x), \quad \Psi(t,-\infty)=0 \, ,\quad a>0 ,
\label{eq:1dpsi}
\end{equation}
where $F=e^{-ax} S$.
The solutions to (\ref{eq:1dpsi}) vanish at $x=+\infty$ and are square
integrable. As a result, the problem (\ref{eq:1dpsi}) is well
posed with respect to an $L_2$ norm. For the simple case where $F=0$,
multiplication of (\ref{eq:1dpsi}) by $(2a \Psi+\partial_x
\Psi+\frac{1}{2}\partial_t \Psi)$ and integration by parts gives
\begin{equation}
\frac{1}{2}\partial_t \int dx \bigg(
(\partial_ x \Psi)^2+2a^2 \Psi^2 \bigg)
=\frac{a}{2} \int dx \bigg(2(\partial_ t \Psi)\partial_ x \Psi
- (\partial_ t \Psi)^2 \bigg)
\le \frac{a}{2} \int dx (\partial_ x \Psi)^2 .
\end{equation}
The resulting inequality
\begin{equation}
\partial_t E \le {\rm const} E
\end{equation}
for the energy
\begin{equation}
E=\frac{1}{2} \int dx \bigg( (\partial_ x \Psi)^2+2a^2 \Psi^2 \bigg)
\label{eq:1denergy}
\end{equation}
provides the estimates for $\partial_x \Psi$ and $\Psi$ which are
necessary for well-posedness. Estimates for $\partial_t \Psi$, and other
higher derivatives, follow from applying this approach to the derivatives
of (\ref{eq:1dpsi}). The approach can be extended to include the source
term $F$ and other generic lower differential order terms. This allows
well-posedness to be extended to the case of variable coefficients and,
locally in time, to the quasilinear case.
Although well-posedness of the problem was established in the
modified form (\ref{eq:1dpsi}), the energy estimates can be translated back to the
original problem (\ref{eq:1dphis}). The modification in going from
(\ref{eq:1dphis}) to (\ref{eq:1dpsi}) leads to an effective modification
of the standard energy for the problem. Rewritten in terms of the
original variable $\Phi=e^{ax}\Psi$, (\ref{eq:1denergy}) corresponds to
the energy
\begin{equation}
E=\frac{1}{2} \int dx e^{-2ax} \bigg( (\partial_ x \Phi)^2
+a^2 \Phi^2 \bigg ).
\label{eq:enorm}
\end{equation}
Thus while the Cauchy problem for (\ref{eq:1dphis}) is ill posed with
respect to the standard $L_2$ norm it is well posed with respect to the
exponentially weighted norm (\ref{eq:enorm}).
This technique can be applied to a wide range of characteristic
problems. In particular, it has been applied to the quasilinear wave
equation for a scalar field $\Phi$ in an asymptotically flat curved space
background with source $S$,
\begin{equation}
g^{ab}\nabla_a \nabla_b \Phi = S (\Phi,\partial_c \Phi, x^c),
\label{eq:qw}
\end{equation}
where the metric $g^{ab}$ and its associated covariant derivative
$\nabla_a$ are explicitly prescribed functions of $(\Phi,x^a)$. In
Bondi-Sachs coordinates~\cite{bondi,sachs} based upon outgoing null
hypersurface $u={\rm const}$, the metric has the form
\begin{equation}
g_{\mu\nu}dx^\mu dx^\nu = -(e^{2\beta}W-r^{-2}h_{AB}W^A W^B)du^2
-2e^{2\beta}dudr
-2h_{AB}W^B dudx^A +r^2h_{AB}dx^A dx^B,
\label{eq:nullmet}
\end{equation}
where $x^A$ are angular coordinates, such that $(u,x^A)={\rm const}$
along the outgoing null rays, and $r$ is an areal radial coordinate. Here
the metric coefficients $(W,\beta,W^A,h_{AB})$ depend smoothly upon
$(\Phi,u,r,x^A)$ and fall off in the radial direction consistent with
asymptotic flatness. The null-timelike problem consists of determining
$\Phi$ in the exterior region given data on an initial null hypersurface
and on an inner timelike worldtube,
\begin{equation}
\Phi(0,r,x^A)=f(r,x^A)\, ,\quad \Phi(u,R,x^A) =q(u,x^A),
\quad R\le r<\infty , \quad u\ge 0.
\label{eq:ndata}
\end{equation}
\bigskip
It is shown in~\cite{nullt}:
\medskip
\noindent {\it The null-timelike IBVP
(\ref{eq:qw})--(\ref{eq:ndata}) is strongly well-posed subject to a
positivity condition that the principal part of the wave operator reduces
to an elliptic operator in the stationary case. }
\bigskip
The proof is based upon energy estimates obtained in compactified
characteristic coordinates extending to ${\cal I}^+$.
\bigskip
\section{Historical developments}
\label{sec:history}
\bigskip
Early computational work in general relativity focused on the Cauchy
problem. The IBVP only recently received serious attention
and is still a work in progress.
Although there are two formulations where strong well-posedness has been
established (see Sec's~\ref{sec:fn} and~\ref{sec:harm}), several
important questions remain. Along the way, there have been partial
successes based upon ideas of potential value in guiding future work.
\bigskip
\subsection{The Frittelli-Reula formulation}
\label{eq:fritreul}
\bigskip
The first extensive treatment of the IBVP for Einstein equations was
carried out by Stewart~\cite{stewart}, motivated at the time by the
tremendous growth in computing power which made numerical relativity a
realistic approach for applications to relativistic astrophysics. His
primary goal was to investigate how to formulate an IBVP for Einstein's
equations which would allow unconstrained numerical evolution. Stewart
focused upon a formulation of Einstein's equations due to Frittelli and
Reula~\cite{fritrel1,fritrel2}, although his approach is sufficiently
general to have application to other formulations.
The Frittelli-Reula system was chosen because it is symmetric hyperbolic
for certain choices of the free parameters. It is based upon the ADM
formalism, with metric (\ref{eq:admmet}), in which all second derivatives
are eliminated by the introduction of auxiliary variables. There is a
2-parameter freedom in the choice of first order variables. The
densitized lapse $h^p \alpha$, where $p$ is an additional adjustable
parameter, and the shift are treated as explicitly prescribed variables.
In addition to the Hamiltonian and momentum constraints
(\ref{eq:hamomc}), the integrability conditions arising from the
auxiliary variables lead to 18 additional constraints.
Another adjustable parameter controls the
freedom of mixing the constraints with the evolution system.
This net result is that the vacuum Einstein equations reduce to a first
order evolution system of the form (\ref{eq:A1}) consisting of 30
equations governing the metric variables and 22 equations governing the
constraints. This is further complicated by the lack of geometric or
tensorial properties of the evolution variables. Frittelli and Reula
analyzed the principal part of the system and showed that it is symmetric
hyperbolic for certain values of the adjustable parameters.
The well-posedness of the IBVP for such symmetric hyperbolic reductions
of Einstein's equations depends upon whether proper constraint preserving
boundary conditions can be imposed. Stewart analyzed the eigenvalues of
the boundary matrix for the linearized system using the Fourier-Laplace
method described in Sec.~\ref{sec:fourlap}. For the evolution system
governing the metric variables, he identified 6 ingoing modes, 6 outgoing
modes and 18 zero-velocity modes in the kernel which propagate tangential
to the boundary. Thus this system requires exactly 6 boundary conditions.
From the boundary matrix for the system governing constraint evolution,
he identified 3 ingoing modes, 3 outgoing
modes and 16 zero velocity modes, so that 3 boundary conditions are
required for constraint enforcement.
The Fourier-Laplace analysis of the constraint system showed that
the determinant condition (\ref{eq:detcond}) was satisfied, i.e. the
homogeneous problem had only the trivial solution, and that the linearized
system had a well-posed IBVP. The three boundary conditions
could be satisfied by requiring that the Cauchy momentum constraints
(\ref{eq:momc}) vanish on the boundary.
The analysis of the evolution system governing the metric showed that the
constraints could be enforced by a particular choice of boundary data for
the metric variables. In this way, the evolution could be freed from the
constraint system. The details are hidden in the symbolic algebra scripts
necessary to analyze the complexity of the Fourier-Laplace modes.
No discussion was given of the estimates for the derivatives which would be
necessary for the well-posedness of the nonlinear problem. There has
apparently been no further results for the Frittelli-Reula formulation
and no attempt at a numerical evolution code.
Although the results were inconclusive for the nonlinear case, Stewart's
treatment provided the first example of how to apply pseudo-differential
techniques to the IBVP for Einstein's equations
and served as the basis for much of the following work. The recognition that
constraint preservation for this system could be achieved by enforcing
the Cauchy momentum constraints on the boundary suggests a possible wider
application but whether there is any universal procedure for enforcing
the constraints in the $3+1$ formulation remains an open issue. See
Sec.~\ref{sec:constr} for a discussion.
\bigskip
\subsection{The BSSN and NOR formulations}
\bigskip
Codes based upon the Baumgarte-Shapiro-Shibata-Nakamura (BSSN)
formulation~\cite{bssn1,bssn2} have been used by the majority of
groups~\cite{mCcLpMyZ06,jBjCdCmKjvM06b,jGetal,pDfHdPeSeSrTjTjV,bssnsom}
carrying out simulations of binary black hole and neutron star systems.
The successful development of the BSSN formulation proceeded through an
interplay between educated guesses and feedback from code performance.
Only in hindsight has its success spurred mathematical analysis, which
showed that certain versions were strongly hyperbolic and thus had a
well-posed Cauchy problem~\cite{sartig,nor,gundl1}. Although significant
progress has been made in establishing some of the necessary conditions
for well-posedness and constraint preservation of the
IBVP~\cite{gundl2,horst,nunsar}, there is still no satisfactory
mathematical theory on which to base a numerical treatment of the
boundary.
Similar to the Frittelli-Reula system, the BBSN system reduces the
Einstein equations to first order form by the introduction of auxiliary
variables. In addition, there is a conformal-traceless decomposition of
the 3-geometry. The Nagy-Ortiz-Reula (NOR) system~\cite{nor} is similar
but without the conformal decomposition. Again, a number of free
parameters enter into the first order reduction and into the way that the
constraints are mixed with the evolution system. For a particular choice
of parameters the linearization off Minkowski space is symmetric
hyperbolic~\cite{nunsar} and leads to a well-posed IBVP for the
linearized problem. However, the corresponding nonlinear system is no
longer symmetric hyperbolic and there is no well-posed boundary
treatment.
Gundlach and Martin-Garcia~\cite{gundl2} studied simplified second order
versions of of the BSSN and NOR systems which were symmetric hyperbolic in
a sense they defined in~\cite{gundl3}. They were able to confirm and
generalize many of the results found in~\cite{sartig} for the first order
reduction. Of particular interest, they found that all the characteristic
modes propagated causally, in contrast to the superluminal modes present
in the first order system. The chief shortcoming of their treatment is
the incompatibility of the constraints with the dissipative boundary
conditions necessary for well-posedness.
The strong well-posedness of the IBVP for $3+1$ formulations remains an
outstanding problem. The strategy in current numerical practice for BSSN
evolution systems is to apply naive, homogeneous Sommerfeld boundary
conditions, where needed, to each evolution variable (cf.~\cite{bssnsom})
and place the boundary out far enough so that its harmful effects are
limited.
\bigskip
\subsection{Other $3+1$ studies}
\label{sec:other}
\bigskip
Many of the other early investigations on the well-posedness of the IBVP
for $3+1$ formulations centered about linearized systems~\cite{szilschbc,
calpulreulbc, calsarbc, nagysar}, spherically symmetric and 1D
spacetimes~\cite{iriond,bardbuch,calehnt, fritgombcss} or other model
problems~\cite{novak,lindbc,reulsarmod, gundl3} which simplified the
treatment. In particular, well-posedness of the linearized problem is a
necessary condition for extension to the nonlinear case.
One promising approach was based upon a generalization of the
Frittelli-Reula and Einstein-Christoffel (EC)~\cite{andy} systems, which
Kidder, Scheel and Teukolsky (KST) showed was symmetric hyperbolic for
certain values of the free parameters~\cite{kst}.
In~\cite{calpulreulbc}, energy estimates for maximally dissipative
boundary conditions were used to formulate a well posed IBVP for the
linearization of this system off Minkowski space. However, constraint
preservation limited the allowed boundary conditions to the reflecting
Dirichlet or Neumann type.
The geometric analogy between the Hamiltonian and momentum constraints
(\ref{eq:hamomc}) of the Cauchy problem and the {\it boundary constraints}
\begin{equation}
G^{ab}N_b =0,
\label{eq:bconstr}
\end{equation}
where $N_b$ is the normal to the boundary, led Frittelli and Gomez to
propose that (\ref{eq:bconstr}) be enforced as boundary
conditions~\cite{fritgombc, fritgombc2,fritgombc4}. They showed for the
EC system, with vanishing shift and certain choices of the free
parameters, that enforcing three linearly dependent combinations of the
boundary constraints would lead to preservation of the Hamiltonian and
momentum constraints. Furthermore, they showed that these linear combinations
could be used to formulate boundary conditions for three of the ingoing
metric variables. They did not study the boundary stability of the
resulting IBVP.
In~\cite{calsarbc}, the determinant condition (\ref{eq:detcond}) of the
Fourier-Laplace method was applied to the linearized (EC) system to
identify ill-posed modes arising from various choices of boundary
conditions and free parameters. Several noteworthy results were found.
For parameter choices in which the EC system was strongly but not
symmetric hyperbolic, they found that maximally dissipative boundary
conditions gave rise to ill posed modes. This is in accord with the
general theory which requires both maximally dissipative boundary
conditions and a symmetric system to guarantee a well posed IBVP. In
addition, for a range of parameters giving rise to a symmetric system,
ill-posed modes were found for boundary conditions based upon the
boundary constraints (\ref{eq:bconstr}). As in the case of the Cauchy
momentum constraints proposed by Stewart, this casts doubt on whether
such boundary constraints are universally applicable. The approach
in~\cite{calsarbc} was effective for ferreting out what doesn't work but
did not go beyond the results of~\cite{calpulreulbc} for establishing a
well-posed IBVP.
In a later study~\cite{sarbtigl}, the EC system was further generalized
to include a dynamical lapse of the Bona-Masso type~\cite{bonmass} and
fixed shift. The IBVP was analyzed in the high frequency limit, again
using the determinant condition of the Fourier-Laplace method to
determine ill-posed modes. It was found that constraint preserving
boundary conditions that were based upon the Newman-Penrose~\cite{np}
Weyl curvature component $\Psi_0$ satisfied the determinant condition
provided the evolution system was strongly hyperbolic and the constraint
propagation system was symmetric hyperbolic. Boundary conditions based
upon the Newman-Penrose $\Psi_0$ component were first introduced in the
Friedrich-Nagy system~\cite{fn}. Other $\Psi_0$ boundary conditions for
the EC system were tested
in~\cite{lindbc,caltechbc,rinne,nagysar,oRlLmS07,sarbuch,sarbuch2,
ruizhbc}. They have been improved to be highly effective absorbing
boundary conditions for gravitational waves (see Sec.~\ref{sec:absorbc})
and have performed well in numerical tests. See Sec.~\ref{sec:num}.
\bigskip
\subsection{The harmonic and Z4 formulations}
\bigskip
The IBVP for general relativity takes on one of its simplest forms in the
harmonic formulation, in which the Einstein equations reduce to 10
quasilinear wave equations. Nevertheless, progress on this problem was
not straightforward. Difficulties arose in handling the harmonic
constraints (\ref{eq:harmcond}), in which derivatives of the metric
tangential to the boundary prohibited use of standard dissipative
boundary conditions for the wave equation. An early well-posed
treatment was based upon the observation that the harmonic Cauchy problem
is well-posed~\cite{harl,mBbSjW06}. Consequently, if locally smooth
reflection symmetry were imposed across the boundary then the
well-posedness of the Cauchy problem would imply well-posedness of
resulting IBVP on either side of the boundary. The reflection symmetry
forces the troublesome tangential derivatives to vanish but it also
forces homogeneous boundary conditions of the Dirichlet or Neumann type.
Although these boundary conditions satisfy the dissipative criterion for
a well-posed IBVP, they were too restrictive for use in practical
numerical applications and did not allow
large boundary data. It took a different approach to formulate a
strongly well-posed harmonic IBVP with Sommerfeld type boundary
conditions. See Sec.~\ref{sec:harm}.
The $Z4$ formalism~\cite{z4} aims at a covariant version of hyperbolic
reduction by expressing the vacuum Einstein equations in the form
\begin{equation}
G^{\mu\nu} -\nabla^{(\mu}Z^{\nu)}
+\frac{1}{2}g^{\mu\nu}\nabla_\rho Z^\rho =0 .
\label{eq:z4reduced}
\end{equation}
When the vector field $Z^\mu$ is identified with the (generalized)
harmonic conditions ${\cal C}^\mu$, this reduces to the harmonic
formulation. However, the freedom is retained to introduce other gauge
conditions which force $Z^\mu =0$. When only 6 components of
(\ref{eq:z4reduced}) are required to vanish, it has been
shown~\cite{z4sb,constrdamp} that the $Z4$ formalism encompasses the
standard $3+1$ formulations, including the ADM, NOR, BSSN and KST
systems.
It is possible that the close analogue between the $Z4$ and harmonic
formulations might be used to shed light on the IBVP for $3+1$ systems.
Constraint preserving boundary conditions for such $Z4$ systems have been
proposed~\cite{bonabc, bonabc2,hilditch}. However, as for other $3+1$
formulations, the boundary stability necessary for a strongly
well-posed IBVP has not been established. Nevertheless, the results of
numerical tests are promising. See Sec.~\ref{sec:num}.
\bigskip
\section{Non-reflecting outer boundary conditions}
\label{sec:absorbc}
\bigskip
The correct physical description of an isolated system involves
asymptotic conditions at infinity which ensure that the radiation fields
have the proper $1/r$ falloff and that the total energy and radiative
energy loss are finite. This can be achieved by locating the outer
boundary at ${\cal I}^+$. Instead, current simulations of binary black
holes are carried out with an outer boundary at a large but finite
distance in the wave zone, i.e. many wavelengths from the source. This is
in accord with the standard practice in computational physics to impose
an artificial boundary condition (ABC) which attempts to approximate the
proper behavior of the exterior region.
At the analytic level, many ABCs are possible, even Dirichlet or Neumann
conditions, provided the proper boundary data is known to allow outgoing
radiation to pass through. However, the determination of the correct
boundary data is a global problem, which requires extending the solution
to ${\cal I}^+$ either by matching to an exterior (linearized or
nonlinear) solution obtained by some other means. The matching approach
has been reviewed elsewhere~\cite{winrev}. As shown by Gustafsson and
Kreiss, the construction of a non-reflecting boundary condition for an
isolated system in general requires knowledge of the solution in a
neighborhood of infinity~\cite{guskreis}.
Even if the outgoing radiation data for the analytic problem were known,
at the numerical level a Dirichlet or Neumann condition would reflect
waves generated by the numerical error and trap them in the grid. The
alternative approach is an ABC which is non-reflecting for homogeneous
data. Artificial boundary conditions for an isolated radiating system for
which homogeneous data is approximately valid are commonly called
absorbing boundary conditions (see e.g.~\cite{Eng77,Hig86,Tre86,
Bla88,Jia90,Ren}), or non-reflecting boundary conditions (see e.g.
\cite{Hed79,giv,kell,agh}) or radiation boundary conditions (see e.g.
\cite{Bay80,hag1}). Such boundary conditions are advantageous for
computational use. However, local ABCs are in general not perfect.
Typically they cause some partial reflection of an outgoing wave
back into the system~\cite{Lind1,Orsz,Hig86,Ren}. It is only required
that there be no spurious reflection in the limit that the boundary
approaches an infinite sphere.
A traditional ABC for the wave equation is the Sommerfeld condition. For
a scalar field $\Phi$ satisfying the Minkowski space wave equation
(\ref{eq:byp1}) with compact source, the exterior retarded field has the
form
\begin{equation}
\Phi=\frac{f(t-r,\theta,\phi)}{r} +\frac{g(t-r,\theta,\phi)}{r^2}
+\frac{h(t,r,\theta,\phi)}{r^3} ,
\label{eq:outgoing}
\end{equation}
where $f$, $g$ and $h$ are smooth bounded functions. The simplest case is
the monopole radiation field
\begin{equation}
\Phi=\frac{f(t-r)}{r}
\end{equation}
which satisfies $(\partial_t+\partial_r) (r\Phi)=0$. This motivates the
use of the Sommerfeld condition
\begin{equation}
\frac{1}{r}(\partial_t+\partial_r)(r\Phi)|_R= q(t,R,\theta,\phi)
\label{eq:somm1}
\end{equation}
on a finite boundary $r=R$. However a homogeneous Sommerfeld condition,
i.e. $q=0$, is exact only for the spherically symmetric monopole field.
The Sommerfeld boundary data $q$ for the general case (\ref{eq:outgoing})
falls off as $1/R^3$, so that a homogeneous Sommerfeld condition
introduces an error which is small only for large $R$. As an example,
\begin{equation}
q= \frac{f(t-R)\cos\theta}{R^3}
\label{eq:qdip}
\end{equation}
for the dipole solution
\begin{equation}
\Phi_{\mathrm{Dipole}}=\partial_z{\frac{f(t-r)}{r}}
=-\left( \frac{f'(t-r)}{r} + \frac{f(t-r)}{r^2} \right)
\cos\theta .
\end{equation}
A homogeneous Sommerfeld condition at $r=R$ leads to a solution $\tilde
\Phi_{Dipole}$ containing a reflected ingoing wave. For large $R$, it is
given by
\begin{equation}
\tilde \Phi_{\mathrm{Dipole}} \sim \Phi_{\mathrm{Dipole}}
+ \kappa \frac{F(t+r-2R)\cos\theta}{r},
\end{equation}
where $\partial_t f(t)=F(t)$ and the reflection coefficient has
asymptotic behavior $\kappa=O(1/R^2)$. More precisely, the Fourier mode
\begin{equation}
\tilde \Phi_{\mathrm{Dipole}}(\omega)
= \partial_z\bigg (\frac{e^{i\omega (t-r)}}{r}
+ \kappa_\omega \frac{e^{i\omega (t+r-2R)}}{r} \bigg )
\end{equation}
satisfies the homogeneous boundary condition
$(\partial_t+\partial_r)(r\tilde\Phi_{Dipole}(\omega))|_R=0$ with
reflection coefficient
\begin{equation}
\kappa(\omega) =\frac{1}{2\omega^2 R^2 +2i\omega R-1}
\sim \frac{1}{2\omega^2 R^2} .
\label{eq:skappa}
\end{equation}
Note, from (\ref{eq:qdip}) and (\ref{eq:skappa}),
\begin{equation}
\kappa \sim q R.
\label{eq:kappaq}
\end{equation}
Use of this relationship simplifies the determination of the asymptotic
falloff of the reflection coefficient by avoiding an explicit calculation
of the reflected wave. Also note, that if (\ref{eq:somm1}) is replaced by
the second order Sommerfeld condition
\begin{equation}
\frac{1}{r^3}(\partial_t+\partial_r) r^2(\partial_t+\partial_r)
(r\Phi)|_R= q_2
\label{eq:somm2}
\end{equation}
then dipole radiation has homogeneous data $q_2=0$. In this way,
the falloff rate of the reflection coefficient is reduced from
(\ref{eq:skappa}) to $\kappa_2 \sim 1/R^3$.
The exponent $n$ of the $O(1/R^n)$ falloff of the reflection coefficient
is an important measure of the accuracy of an ABC. Such reflection
coefficients can be calculated for linearized gravitational waves on a
Minkowski background, analogous to the above scalar wave example,
using either the Regge-Wheeler-Zerilli~\cite{regge,zerilli,oSmT01p}
perturbative method, as carried out in~\cite{sarbuch,sarbuch2,ruizhbc},
or by the Bergmann-Sachs~\cite{bergsachs} gravitational Hertz
potential method, as carried out in~\cite{isol}. See Sec.~\ref{sec:isol}.
The main difference in the gravitational case arises
from the gauge modes, which exist
along with the radiative degrees of freedom. In first order formulations,
this is further complicated by the modes introduced by the auxiliary
variables. The second order harmonic formulation, in which all modes
propagate on the light cone, is simplest to analyze. See
Sec.~\ref{sec:harm} for a discussion of reflection
coefficients in the harmonic case.
Local ABCs have been extensively applied to linear problems with varying
success~\cite{Lind1,Eng77,Bay80,Tre86,Hig86,Bla88,Jia90}. Some are local
approximations to exact integral representations of the solution in the
exterior of the computational domain~\cite{Eng77}, while others are based
on approximating the dispersion relation of the so-called one-way wave
equations~\cite{Lind1,Tre86}. Higdon~\cite{Hig86} showed that this last
approach is essentially equivalent to specifying a finite number of
angles of incidence for which the ABCs yield perfect transmission. Local
ABCs have also been derived for the linear wave equation by considering
the asymptotic behavior of outgoing wave solutions~\cite{Bay80}, thus
generalizing the Sommerfeld condition. Although this type of ABC is
relatively simple to implement and has a low computational cost, the final
accuracy can be limited if the assumptions about the behavior of the
waves are oversimplified. See~\cite{giv,Ren,tsy,ryab} for general
discussions.
The disadvantages of local ABCs have led to consideration of nonlocal
versions based on integral representations of the infinite domain
problem~\cite{Tin86,giv,tsy,agh}. Even when the Green
function is known, such approaches were initially dismissed as
impractical~\cite{Eng77}; however, the rapid development of computer
power and numerical techniques has made it possible to implement exact
nonlocal ABCs for the linear wave equation and Maxwell's equations in
3D~\cite{deM,kell2}. If properly implemented, this method can yield
numerical solutions to a linear problem which converge to the exact
infinite domain problem in the continuum limit, while keeping the
artificial boundary at a fixed distance. However, due to nonlocality, the
computational cost per time step usually grows at a higher power with
grid size, $\mathcal{O} (N^4)$ per time step in three spatial dimensions,
than for a local approach~\cite{giv,deM,tsy}.
The extension of ABCs to nonlinear problems is much more difficult. The
problem is normally treated by linearizing the region between the outer
boundary and infinity, using either local or nonlocal linear
ABCs~\cite{tsy,ryab}. The neglect of the nonlinear terms in this region
introduces an unavoidable error at the analytic level. However, even larger
errors are typically introduced in prescribing the outer boundary data.
The correct boundary data must correspond to the continuity of fields and
their normal derivatives when extended across the boundary into the
linearized exterior. This is a subtle global requirement for any
consistent boundary algorithm, since discontinuities in the field or its
derivatives would otherwise act as a spurious sheet source on the
boundary that would contaminate both the interior and the exterior
solutions. However, the fields and their normal derivatives constitute an
over determined set of data for the boundary problem. So it is necessary
to solve a global linearized problem, not just an exterior one, in order
to find the proper data. An expedient numerical method
to eliminate back reflection is the use of {\it sponge layers},
cf.~\cite{sponge}, in which damping terms are introduced into the
evolution equations near the outer boundary.
The designation ``exact ABC'' is given to an ABC
for a nonlinear system whose only error is due to linearization of the
exterior. An exact ABC requires the use of global techniques, such as the
boundary potential method, to eliminate back reflection at the
boundary~\cite{tsy,agh}. Furthermore, nonlinear waves intrinsically
backscatter, which makes it incorrect to try to entirely eliminate
incoming radiation from the outer region. For the nonlinear wave
equation, test results presented in~\cite{holvorc1,holvorc2} showed that
Cauchy-characteristic matching outperformed all ABC's in the existent
literature.
It is an extra challenge to apply ABCs to strongly nonlinear hydrodynamic
problems~\cite{giv}. Thompson~\cite{Tho87} generalized a previous
nonlinear ABC of Hedstrom~\cite{Hed79} to treat 1D and 2D problems in gas
dynamics. These boundary conditions performed poorly in some situations
because of difficulties in adequately modeling the field outside the
computational domain~\cite{Tho87,giv}. Hagstrom and
Hariharan~\cite{Hag88} have overcome these difficulties in 1D gas
dynamics by their use of Riemann invariants. They proposed, at the
heuristic level, a generalization of their local ABC to 3D.
In order to reduce the analytic error, an artificial boundary for a
nonlinear problem must be placed sufficiently far from the strong-field
region. This can increase the computational cost in multi-dimensional
simulations~\cite{Eng77}. There is no ABC which converges
(as the discretization is refined) to the infinite domain exact solution
of a strongly nonlinear wave problem in multi-dimensions, while keeping
the artificial boundary fixed. When the system is nonlinear and not
amenable to an exact solution, a finite outer boundary condition must
necessarily introduce spurious effects. Attempts to use
compactified Cauchy hypersurfaces which extend the domain to spatial
infinity have failed because the phase of short wavelength radiation
varies rapidly in spatial directions~\cite{Orsz}. In fact, in his pioneering
simulation of binary black holes, Pretorius~\cite{pret1,pret2} used this
effect as a numerical expedience by applying artificial dissipation to
diminish short wavelength error arising from the use of a compactified
outer boundary at spatial infinity. For a recent review of ABCs in the
computational mathematics literature, see~\cite{hagrev}.
The situation for the gravitational IBVP is not as severe as for
hydrodynamics, especially for formulations in which the gauge modes and
radiation modes propagate with the same speed. However, due to
nonlinearities, there is always some error of an analytic nature
introduced by a finite boundary which is independent of
discretization. In general, a systematic reduction of this error can only
be achieved by moving the computational boundary to larger and larger
radii. There has been recent progress in designing absorbing boundary
conditions for the gravitational field. Buchman and
Sarbach~\cite{sarbuch} have developed higher order local boundary
conditions based upon derivatives of $\Psi_0$ which are non-reflecting up
to any given multipole mode for linearized gravitational waves,
analogous to the switch from a first order
Sommerfeld condition (\ref{eq:somm1}) to the second order condition
(\ref{eq:somm2}). These boundary conditions {\it freeze} the value of
$\Psi_0$ to its initial value, i.e.
\begin{equation}
\partial_t \Psi_0 =0,
\label{eq:freeze}
\end{equation}
in order to avoid a $O$th order violation of the compatibility condition
between the initial and boundary data. They have extended this approach
to include quadrupole gravitational waves on a Schwarzschild background
and also to account for $O(M/R)$ back scatter using a nonlocal
version~\cite{sarbuch2}. For a review of this approach see~\cite{oS07}.
It has been applied to the harmonic formulation in~\cite{ruizhbc} and to
the Z4 formulation in~\cite{hilditch}. These are possibly the best
possible local boundary conditions for the gravitational radiation modes.
Lau~\cite{lau1,lau2,lau3} has formulated an exact ABC
for linearized gravitational waves on a Schwarzschild background. Based
upon the flat space work of~\cite{agh}, he reduces the calculation of the
Green function incorporating the boundary condition for the perturbed
metric to the integration of a radial ODE by using a combined spherical
harmonic and Laplace transform of the Regge-Wheeler-Zerilli equations. He
discusses the trade-off in computational cost for this nonlocal ABC
versus the larger computational domain required by a local condition.
This is similar to the trade-off between characteristic matching and the
application of local boundary conditions.
\bigskip
\section{The Friedrich-Nagy system}
\label{sec:fn}
\bigskip
Friedrich and Nagy~\cite{fn} have presented a theorem establishing the
first strongly well-posed IBVP for Einstein's equations with the
generality to handle an outgoing radiation boundary condition. The
approach uses the energy method for first order symmetric hyperbolic
systems described in Sec.~\ref{sec:md}. Their formulation is based upon
the Einstein-Bianchi system of equations with evolution variables
consisting of an orthonormal tetrad ${\bf e}_{\tilde a}$, $\tilde a =
(0,1,2,3)$, the associated connection coefficients $\Gamma^{\tilde
a}_{\tilde b \tilde c}$ and Weyl curvature tetrad components $C_{\tilde a
\tilde b \tilde c \tilde d}$. Although it differs from the metric based
formulations used in numerical relativity, the success of their treatment
suggests that many of the underlying ideas should be universally
applicable. In their words, ``There are certainly many possibilities to
discuss the initial boundary value problem and there will be as many ways
of stating boundary conditions. However, all of these should be just
modifications of the boundary conditions given in our theorem''. Perhaps
this is overstated since their formulation is 3rd differential order in
the metric, as opposed to the 2nd order $3+1$ or harmonic formulations.
Yet, all successful formulations must have the common property of
prescribing data which produces a unique solution to Einstein's
equations.
The tetrad vector ${\bf e}_0$ is chosen to be timelike and tangent to the
boundary. It is used to construct adapted coordinates
$x^\mu=(t,x^i)=(t,x,y,z)$ satisfying
\begin{equation}
{\cal L}_{{\bf e}_0} t =1\, , \quad {\cal L}_{{\bf e}_0} x^i =0,
\label{eq:fncoord}
\end{equation}
so that $e^\mu_0$ plays the role of the evolution vector field $t^\mu$
introduced in Sec.~\ref{sec:bare}. However, the evolution field
now has the metric property of being a timelike unit vector so, as a
reminder, I denote it by $T^\mu=e^\mu_0$. In accord with the notation in
Sec.~\ref{sec:bare}, let the initial hypersurface ${\cal S}_0$ be given
by $t=0$ and the boundary ${\cal T}$ be given by $x=0$, with adapted
coordinates $(t,x^A)$. The tetrad vectors are adapted to the geometry as
follows. Let $N^\mu ={e_1}^\mu \propto -\nabla^\mu x$ be the unit outer
normal to ${\cal T}$.
Extend $N^\mu$ throughout the spacetime manifold ${\cal M}$ by requiring
that it be the unit normal to the hypersurfaces ${\cal T}_c$ given by
$x=c=const>0$. On ${\cal S}_0$, the remaining tetrad vectors $e_A^\mu$,
$A=(2,3)$, are chosen to be an orthonormal dyad for the $(t=0, x=c)$
subspaces. They are then propagated throughout ${\cal M}$ by Fermi-Walker
transport along the integral curves of $T^\mu$, which lie in ${\cal
T}_c$. The connection components intrinsic to ${\cal T}_c$ are considered
to be freely specifiable gauge source functions. In addition, the mean
extrinsic curvature $K$ of ${\cal T}_c$ is also considered to be a gauge
source function. Moreover, it is shown that the equation $K=q(x^\mu)$ can
be cast as a quasilinear wave equation which determines ${\cal T}_c$
given initial data corresponding to $x=c$ and $\partial_t x=0$ at $t=0$.
By design, this choice of adapted coordinates and tetrad gauge greatly
simplify the IBVP. The evolution system consists of the gauge conditions
governing the tetrad, the equations relating the connection to the
tetrad, the vacuum equations relating the Weyl curvature to the
connection (which imply the vanishing of the Einstein tensor) and the
vacuum Bianchi identities, i.e. the tetrad version of the equations
\begin{equation}
\nabla_\mu {C^\mu}_{\nu\rho\sigma}=0.
\end{equation}
The system is over determined and subject to constraints arising from
integrability conditions. Remarkably, it can be reduced to a system with
the properties:
\begin{itemize}
\item The evolution system is symmetric hyperbolic.
\item The boundary matrix (\ref{eq:B3}) admits two ingoing variables
corresponding to combinations of the $ \Psi_0 $ and $ \Psi_4$
Newman-Penrose components of the Weyl tensor.
\item Maximally dissipative boundary conditions take the form
\begin{equation}
\Psi_0 + \alpha \Psi_4 + \beta \bar \Psi_4 = q,
\label{eq:fnpsi}
\end{equation}
where $q$ is the (complex) boundary data and $\alpha$ and $\beta$ are
coefficients subject to a dissipative condition. (Conventions here are
chosen to be consistent with Newman-Penrose conventions and differ
from~\cite{fn}.)
\item The subsidiary system governing the constraints is symmetric
hyperbolic and intrinsic to the ${\cal T}_c$ hypersurfaces, i.e. all
derivatives are tangential to ${\cal T}_c$. This gives rise to constraint
preservation without requiring any further restrictions on the boundary
conditions.
\end{itemize}
The resulting IBVP is strongly well-posed. Given
initial data on ${\cal S}$, including the hyperbolic angle $\Theta$ in
(\ref{eq:hangle}) at the edge ${\cal B}_0$, the boundary data $q$, a
choice of gauge source functions and a dissipative choice of boundary
condition, there exists a unique solution locally in time. Furthermore,
the solution depends continuously on the data.
Several important points should be noted:
\begin{itemize}
\item The specification of the mean extrinsic curvature $K$ of the
boundary geometrically determines the location of the boundary.
\item The choice of unit timelike vector $T^a$ tangent to the boundary
represents gauge freedom in the construction of the solution. It induces
a corresponding foliation of the spacetime and the boundary
according to ${\cal L}_T t=1$.
\item The geodesic curvature of the integral curves of $T^a$ constitute
gauge source functions required on the boundary. This gauge freedom feeds
into the adapted coordinates $(t,x^A)$ of the boundary. As a result, the
functional specification of $K(t,x^A)$ becomes gauge dependent. This
complication could be avoided by choosing $T^a$ to be geodesic but at the
expense of possible coordinate focusing singularities which would affect
the long term existence of the solution.
\item The outgoing null vector $K^a$ and ingoing null vector $L^a$ used
in defining $\Psi_0$ and $\Psi_4$ are determined by the choice of $T^a$
(gauge) and the boundary normal $N^a$ by
\begin{equation}
K^a=T^a+N^a\, , \quad L^a=T^a-N^a.
\label{eq:fnk}
\end{equation}
Friedrich and Nagy are careful to point out that gauge freedom prevents
any meaningful interpretation of $\Psi_0$ and $\Psi_4$ as either purely
outgoing or ingoing radiation.
\end{itemize}
\bigskip
\section{The harmonic IBVP}
\label{sec:harm}
\bigskip
The first successful treatment of the harmonic IBVP for Einstein's
equations was carried out using the pseudo-differential theory described
in Sec.~\ref{sec:fourlap}, which established strong well-posedness in a
generalized sense~\cite{wpgs}. The theory was developed for the second
order formulation of the generalized harmonic formulation
(\ref{eq:harmcond})--(\ref{eq:reduced}). The boundary conditions were
given in Sommerfeld form
in terms of the outgoing null vector $K^a=T^a+N^a$ normal to the
foliation of the boundary. Here, as in the Friedrich-Nagy approach
(\ref{eq:fnk}), $T^a$ is a future directed timelike unit vector tangent
to the boundary but now it is also chosen to be normal to its foliation
${\cal B}_t$. Recall that $n^a$, the normal to the Cauchy foliation
${\cal S}_t$, is not necessarily tangent to the boundary. The motion of
the boundary, characterized by the hyperbolic angle (\ref{eq:hangle}),
distinguishes $T^a$ from $n^a$.
In the adapted coordinates $x^\mu=(t\ge 0,x\ge 0,x^A)$ described in
(\ref{eq:adapted}), six Sommerfeld boundary conditions for the densitized
metric $\gamma^{\mu\nu} =\sqrt{-g}g^{\mu\nu}$ were given by
\begin{eqnarray}
K^\mu \partial_\mu \gamma^{AB} &=&q^{AB}(t,x^A) \label{eq:sombc1} \\
K^\mu \partial_\mu( \gamma^{tA}+\gamma^{xA} )
&=&q^{A}(t,x^A) \label{eq:sombc2} \\
K^\mu \partial_\mu ( \gamma^{tt}+2\gamma^{tx}+\gamma^{xx} )
&=&q(t,x^A), \label{eq:sombc3}
\end{eqnarray}
where the $q$'s are freely prescribed Sommerfeld data. The harmonic
constraints (\ref{eq:harmcond}) were used to supply four additional
boundary conditions, which could be expressed in the Sommerfeld form
\begin{eqnarray}
\sqrt{-g}{\cal C}^A& = &
\frac{1}{2}(\partial_t -\partial_x)( \gamma^{tA}- \gamma^{xA})
\nonumber \\
&+&\frac{1}{2}(\partial_t +\partial_x)( \gamma^{tA} +\gamma^{xA})
+\partial_B \gamma^{AB} - \sqrt{-g}\hat \Gamma^A =0
\label{eq:scbc1} \\ \nonumber \\
\sqrt{-g}( {\cal C}^t+{\cal C}^x) &=&
\frac{1}{2}(\partial_t -\partial_x)(\gamma^{tt}- \gamma^{xx})
\nonumber \\
&+& \frac{1}{2}(\partial_t +\partial_x)
(\gamma^{tt}+2\gamma^{xt}+\gamma^{xx})
+\partial_A (\gamma^{tA}+\gamma^{xA})
- \sqrt{-g}(\hat \Gamma^t+\hat \Gamma^x) =0
\label{eq:scbc2} \\
\sqrt{-g}{\cal C}^t& =&
\frac{1}{2}(\partial_t -\partial_x)(\gamma^{tt}- \gamma^{tx}) \nonumber \\
&+& \frac{1}{4}(\partial_t +\partial_x))\bigg((\gamma^{tt}+2
\gamma^{tx}+\gamma^{xx})
+(\gamma^{tt}-\gamma^{xx})\bigg)
+\partial_A \gamma^{tA} -\sqrt{-g}\hat \Gamma^t =0 .
\label{eq:scbc3}
\end{eqnarray}
Constraint preservation then follows from the homogeneous wave equation
(\ref{eq:bianchi}).
The key feature of (\ref{eq:sombc1})--(\ref{eq:scbc3}) is that they form
a sequential hierarchy of Sommerfeld boundary conditions for the metric
variables such that the source terms are given in
terms of derivatives of previous variables in the sequence. For instance,
the terms on the second line of (\ref{eq:scbc1}) are derivatives of
$\gamma^{AB}$ and $(\gamma^{tA}+\gamma^{xA})$, whose boundary
conditions are prescribed previously in (\ref{eq:sombc1}) and (\ref{eq:sombc2}).
This pattern persists for the remaining boundary conditions in the sequence,
i.e. (\ref{eq:scbc2}) and (\ref{eq:scbc3}). This structure gives rise to
a corresponding sequence of estimates for the variables in the hierarchy,
which is the key for establishing boundary stability and the strong
well-posedness of the IBVP. There is considerable freedom in the boundary
conditions provided this hierarchical structure is preserved.
The well-posedness of the harmonic IBVP was subsequently also established
using estimates for the non-standard energy (\ref{eq:ewave}) associated
with a timelike vector pointing outward from the boundary~\cite{wpe}.
The hierarchical structure of the boundary conditions corresponds to the
upper triangular property (\ref{eq:triang}), which is sufficient
condition for a well-posed IBVP for a coupled system of quasilinear wave
equations.
A more general and geometrical version of these results in
terms of a background metric $\gz_{ab}$ was presented in~\cite{isol}.
The connection $\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a$ and curvature tensor $\mbox{\em \r{R}}{}^d{}_{cab}$
associated with the background metric $\gz_{ab}$ have the same tensorial
properties as the corresponding quantities $\nabla_a$ and $R^d{}_{cab}$
associated with $g_{ab}$. In particular, the difference $\nabla_a
-\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a$ defines a tensor field $C^d_{ab}$ according to
\begin{equation}
(\nabla_a -\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a) v^d = C^d_{ab} v^b
\label{eq:cconn}
\end{equation}
for any vector field $v^b$. In terms of the (nonlinear) perturbation
\begin{equation}
f_{ab}=g_{ab}-\gz_{ab}
\label{eq:f}
\end{equation}
of the metric from the background,
\begin{equation}
C^d_{ab}
= \frac{1}{2} g^{dc}\left( \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_{a} f_{bc}
+ \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_{a} f_{bc} - \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_c f_{ab} \right).
\label{eq:chris}
\end{equation}
Since $\gz_{ab}$ is explicitly known, a solution for $ f_{ab}$ is
equivalent to a solution for $g_{ab}$. Einstein's equations are given by
\begin{equation}
E^{ab}:= G^{ab} -\nabla^{(a}{\cal C}^{b)}
+\frac{1}{2}g^{ab}\nabla_d{\cal C}^d =0
\label{eq:ceq}
\end{equation}
subject to the harmonic constraints
\begin{equation}
{\cal C}^d:= g^{ab} C^d_{ab} =0.
\label{eq:chc}
\end{equation}
In the adapted coordinates, the harmonic constraints take the form
\begin{equation}
{ \cal C}^\rho: =
g^{\mu\nu}( \Gamma^\rho_{\mu\nu}- \Gamma\hspace{-0.20cm}{}^{\mbox{\r{~}}}{}\hspace{-0.18cm}^\rho_{\mu\nu} ) =0,
\label{eq:charm}
\end{equation}
so that the background Christoffel symbols $\Gamma\hspace{-0.20cm}{}^{\mbox{\r{~}}}{}\hspace{-0.18cm}^\rho_{\mu\nu}$ appear
as harmonic gauge source functions. When the harmonic constraints are
satisfied, the reduced Einstein equations
form the desired quasilinear wave system for $f_{\mu\nu}$,
\begin{equation}
g^{\rho\sigma}\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_\rho\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_\sigma f_{\mu\nu}
= 2 g_{\lambda\tau} g^{\rho\sigma} C^\lambda{}_{\mu \rho}C^\tau{}_{\nu\sigma}
+4 C^\rho{}_{\sigma(\mu} g_{\nu)\lambda}
C^\lambda{}_{\rho \tau}g^{\sigma\tau}
- 2 g^{\rho\sigma}\mbox{\em \r{R}}^\lambda{}_{\rho\sigma(\mu} g_{\nu)\lambda} .
\label{eq:beinst}
\end{equation}
The analogue of the Sommerfeld conditions
(\ref{eq:sombc1})--(\ref{eq:scbc3}) are prescribed in terms of the
boundary decomposition of
the metric
\begin{equation}
g_{ab} =N_a N_b+H_{ab}\, , \quad H_{ab}= -T_a T_b +Q_{ab},
\label{eq:bdecom}
\end{equation}
This leads to an orthonormal tetrad $(T^a,N^a,Q^a,\bar Q^a)$ on ${\cal
T}$, where $Q^a$ is a complex null vector tangent to ${\cal B}_t$ with
normalization
\begin{equation}
Q_{ab}= Q_{(a}\bar Q _{b)}\, ,
\quad Q^a \bar Q_a=2 \, , \quad Q^a Q_a=0.
\label{eq:qnorm}
\end{equation}
(The tetrad is unique up to the spin freedom $Q^a \rightarrow e^{i\theta}
Q^a$ which does not enter in any essential way.) In terms of the
outgoing and ingoing null vector fields $K^a=T^a+N^a$ and $L^a=T^a-N^a$,
respectively, which are normal to ${\cal B}_t$, the metric has the null
tetrad decomposition
\begin{equation}
g_{ab} = - K_{(a}L _{b)}+Q_{(a}\bar Q _{b)}.
\label{eq:ntetrad}
\end{equation}
Six Sommerfeld boundary conditions which determine the components of the
outgoing null derivatives $K^a \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc}$ are then given by
\begin{eqnarray}
\frac{1}{2} K^b K^c K^a
\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc} &=& q^a K_a ,
\label{eq:ak} \\
(Q^b K^c K^a
-\frac {1}{2} K^b K^c Q^a) \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc} &=& q^a Q_a ,
\label{eq:aq}\\
(L^b K^c K^a
-\frac {1}{2} K^b K^c L^a)
\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc}&=& q^a L_a ,
\label{eq:al} \\
(\frac {1}{2}Q^b Q^c K^a
- Q^b K^c Q^a) \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc} &=& 2\sigma ,
\label{eq:sigqq}
\end{eqnarray}
in terms of boundary data $q^a$ and $\sigma$. The harmonic constraints
provide four additional boundary conditions which, in terms of the null
tetrad, can be expressed in the Sommerfeld form
\begin{eqnarray}
-2{\cal C}^a K_a =\left( Q^b\bar{Q}^c K^a+ K^b K^c L^a
- K^b \bar{Q}^c Q^a - K^b Q^c \bar{Q}^a \right)
\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc} &=& 0 ,
\label{eq:hk} \\
-2{\cal C}^a Q_a =\left( L^b Q^c K^a + K^b Q^c L^a
- K^b L^c Q^a+Q^b Q^c \bar{Q}^a \right)
\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{b c} &=& 0,
\label{eq:hq} \\
- 2{\cal C}^a L_a = \left( L^b L^c K^a+ Q^b \bar{Q}^c L^a
- \bar{Q}^b L^c Q^a- Q^b L^c \bar{Q}^a \right)
\nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc}&=& 0.
\label{eq:hl}
\end{eqnarray}
As before, constraint preservation follows from (\ref{eq:bianchi}).
Together, (\ref{eq:ak})-- (\ref{eq:hl}) provide Sommerfeld boundary
conditions for the components of $K^a \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a f_{bc}$ in the
sequential order $(KK),(QK),(LK),(QQ),(Q\bar Q),(LQ),(LL)$ in terms of
the boundary data and the derivatives of preceding components in the
sequence. This hierarchy of Sommerfeld boundary conditions satisfies the
requirements given in Sec.~\ref{sec:swev} (Theorem 1 of~\cite{isol}) for
a strongly well-posed IBVP for the quasilinear hyperbolic system
(\ref{eq:beinst}). See Sec.~\ref{sec:geom} for a geometrical interpretation
of the boundary data and Sec.~\ref{sec:num} for numerical tests.
\bigskip
\subsection{Application to an isolated system}.
\label{sec:isol}
\bigskip
The main application of the gravitational IBVP is to the spherical outer
boundary used in the simulation of an isolated system emitting radiation.
As discussed in Sec.~\ref{sec:absorbc}, in the absence of an exterior
solution, the simplest approach is the use of a Sommerfeld boundary
conditions with homogeneous data. In doing so it is important to take
advantage of the freedom in the form of the boundary conditions in order
to reduce back reflection.
Sommerfeld boundary conditions consistent with a well posed harmonic IBVP
have wide freedom regarding the addition of (i) partial derivative terms
consistent with the hierarchical structure and (ii) lower order algebraic
terms. Various choices were considered in~\cite{isol}. They were tested
by computing the resulting reflection coefficients for spherical waves in
a Minkowski space background. For this purpose the densitized metric is
approximated to linearized accuracy by \begin{equation}
\sqrt{-g}g^{\mu\nu}=\eta^{\mu\nu}+ \gamma^{\mu\nu}, \end{equation} where
$\eta^{\mu\nu}$ is the Minkowski metric. The calculation of the
reflection coefficients proceeds as for the scalar wave example in
Sec.~\ref{sec:absorbc}, as modified to deal with the gauge modes.
Linearized waves in the harmonic gauge can be constructed from the
gravitational analogue of the Hertz potential~\cite{bergsachs}, which
has the symmetries
\begin{displaymath}
H^{\mu\alpha\nu\beta}= H^{[\mu\alpha]\nu\beta}=H^{\mu\alpha[\nu\beta]}
=H^{\nu\beta\mu\alpha}
\end{displaymath}
and satisfies the flat space wave equation $\partial^\sigma
\partial_\sigma H^{\mu\alpha\nu\beta}= 0$. Then the perturbation
\begin{displaymath}
\gamma^{\mu\nu} =\partial_\alpha \partial_\beta H^{\mu\alpha\nu\beta}.
\end{displaymath}
satisfies the linearized Einstein equations $ \partial^\sigma
\partial_\sigma \gamma^{\mu\nu}=0$ in the harmonic gauge $ \partial_\mu
\gamma^{\mu\nu}=0$. Outgoing waves can be generated from the potential
\begin{equation}
H^{\mu\alpha\nu\beta}= K^{\mu\alpha\nu\beta}\frac{f(t-r)}{r}\, ,
\quad \gamma^{\mu\nu} =
K^{\mu\alpha\nu\beta} \partial_\alpha \partial_\beta \frac{f(t-r)}{r},
\label{eq:Khertz}
\end{equation}
where $ K^{\mu\alpha\nu\beta}$ is a constant tensor. (All higher
multipoles can be constructed by taking spatial derivatives.)
$K^{\mu\alpha\nu\beta}$ has 21 independent components but the choice
$K^{\mu\alpha\nu\beta}=\epsilon^{\mu\alpha\nu\beta}$ leads to
$\gamma^{\mu\nu}=0$ so there are only 20 independent waves. These can
reduced to 10 pure gauge waves for which the linearized Riemann tensor
vanishes, which correspond to the trace terms in $K^{\mu\alpha\nu\beta}$;
e.g. $K^{\mu\alpha\nu\beta}=\eta^{\alpha\nu}\eta^{\beta\mu}
-\eta^{\mu\nu}\eta^{\alpha\beta}$ leads to a monopole gauge wave. The
trace-free part gives rise to 10 independent quadrupole gravitational
waves, corresponding to spherical harmonics with $(\ell=2,-2\le m\le 2)$
in the two independent polarization states.
For a boundary at $r=R$, the Sommerfeld derivative in
the outgoing null direction is
\begin{equation}
K^\mu\partial_\mu = \partial_t +\partial_r.
\end{equation}
In formulating boundary conditions which minimize back reflection, the
property $K^\mu \partial_\mu f(t-r) =0$ is used to introduce the
appropriate powers of $r$, analogous to the scalar example
(\ref{eq:somm1}). In~\cite{isol}, the optimal choice was found to be the
Sommerfeld hierarchy
\begin{eqnarray}
\frac{1}{r^2}K_\alpha K_\beta K^\mu \partial_\mu (r^2\gamma^{\alpha\beta})
&=& q_{KK}\; ,
\label{eq:modasymKK}\\
\frac{1}{r^2} K_\alpha Q_\beta K^\mu\partial_\mu(r^2 \gamma^{\alpha\beta} )
&=& q_{KQ}\; ,
\label{eq:modasymKQ} \\
\frac{1}{r^2}Q_\alpha\bar Q_\beta
K^\mu\partial_\mu(r^2\gamma^{\alpha\beta})
-\frac{\gamma}{r} &=& q_{Q\bar Q}\; ,
\label{eq:modasymQbQ} \\
Q_\alpha Q_\beta K^\mu\partial_\mu \gamma^{\alpha\beta}
- Q_{\alpha}K_\beta Q^\mu \partial_\mu \gamma^{\alpha\beta} &=& q_{QQ}\; .
\label{eq:modasymQQ}
\end{eqnarray}
It was found that the data $q_{..}=O(1/R^4)$ for the outgoing
gravitational quadrupole waves and $q_{..} =O(1/R^3)$ for the outgoing
gauge waves. This implies, in accord with (\ref{eq:kappaq}), that
homogeneous Sommerfeld data gives rise to reflection coefficients
$\kappa=O(1/R^3)$ for the gravitational waves and $\kappa=O(1/R^2)$ for
the gauge waves. These results were confirmed using the
Regge-Wheeler-Zerilli perturbative formulation along with the metric
reconstruction method described in \cite{oSmT01p}.
The analysis of linearized waves shows that $q_{QQ}$ controls the
amplitude of the gravitational radiation passing through the boundary.
Higher order boundary conditions can be based upon replacing
(\ref{eq:modasymQQ}) by a condition on $\Psi_0$, which in the linearized
theory controls the radiation in a gauge independent manner. In this
way, the $\Psi_0$ based boundary conditions discussed in
Sec.~\ref{sec:absorbc} can be used to further increase the $1/R^n$
falloff rate of the reflection coefficients for gravitational waves.
\bigskip
\section{Constraint preservation}
\label{sec:constr}
\bigskip
The IBVP for Einstein's equations is still not well understood due to a
great extent from complications arising from the constraints. The
Hamiltonian and momentum constraints on the Cauchy data take the
universal form (\ref{eq:hamomc}) in terms of the components of the
Einstein tensor normal to the initial hypersurface. However, there is no
common way to ensure constraint preservation for the various formulations
of Einstein's equations. Even the constraints themselves take on
different forms.
The Friedrich-Nagy system (see Sec.~\ref{sec:fn}) is based upon the
Einstein-Bianchi equations which is third differential order in terms of
the metric. In that case, by a cleverly designed choice of adapted
coordinates and gauge, constraint propagation is governed by a system
tangential to the boundary. Thus there are no ingoing constraint modes
and constraint preservation is straightforward.
In the strongly well-posed harmonic system described in
Sec.~\ref{sec:harm}, the harmonic conditions ${\cal C}^a$ became
surrogates for the Hamiltonian and momentum
constraints. Because ${\cal C}^a$ satisfies a homogeneous wave
equation, there are four ingoing constraint modes. These could be
eliminated by dissipative boundary conditions with homogeneous data. In
Sec.~\ref{sec:harm}, homogeneous Dirichlet conditions on
${\cal C}^a$ were chosen. This allowed
the constraints to be enforced in terms of first differential Sommerfeld
conditions on the metric. Homogeneous Neumann or Sommerfeld conditions
on the constraints would also ensure constraint preservation but at the
expense of a more complicated coupling with the evolution system for the
metric.
The worldtube constraints which arise in the gravitational version of the
null-timelike IBVP discussed in Sec.~\ref{sec:civp} present an entirely
different aspect. In that problem, boundary data on a worldtube ${\cal
T}$ and initial data on an outgoing null hypersurface ${\cal N}_0$
determine the exterior spacetime by integration along the outgoing null
geodesics. The worldtube constraints impose conditions on the integration
constants. The Bondi-Sachs formalism~\cite{bondi,sachs} introduces
coordinates $x^\alpha=(u,r,x^A)$ based upon a family of outgoing null
hypersurfaces ${\cal N}_u$, where $u$ labels the null hypersurfaces,
$x^A$ are angular labels for the null rays and $r$ is a surface area
coordinate. The evolution system is composed of radial propagation
equations along the outgoing null rays consisting of the hypersurface
equations $G_\mu^u=G_\mu^\nu \nabla_\nu u=0$, which only contain
derivatives tangent to ${\cal N}_u$, and the evolution equations
$G_{AB} - \frac{1}{2} g_{AB} g^{CD}G_{CD}=0$.
The components of Einstein's equations independent of the hypersurface
and evolution equations are worldtube constraints (called supplementary
conditions by Bondi and Sachs),
\begin{eqnarray}
g^{AB}G_{AB}&=&0 \label{eq:triv} \\
G_A^r&=&0 \label{eq:consu}\\
G_u^r&=&0. \label{eq:consA}
\end{eqnarray}
When the hypersurface and evolution equations are satisfied, the
contracted Bianchi identity
\begin{equation}
\nabla_\nu G_\mu^\nu =0
\label{eq:bianch}
\end{equation}
implies that these equations need only be satisfied on the worldtube
${\cal T}$. The identity for $\mu=r$ reduces to $g^{AB}G_{AB}=0$ so that
(\ref{eq:triv}) becomes trivially satisfied. (Here it is necessary that
the worldtube have nonvanishing expansion so that the areal radius $r$ is
a non-singular coordinate.) The identity for $\nu=A$ then reduces to the
radial ODE
\begin{equation}
\partial_r (r^2 G_A^r) =0,
\end{equation}
so that $G_A^r$ vanishes if it vanishes on ${\cal T}$. When $G_A^r=0$,
the identity for $\nu=u$ then reduces to
\begin{equation}
\partial_r (r^2 G_u^r)=0,
\end{equation}
so that $G_u^r$ also vanishes if it vanishes on ${\cal T}$.
Thus the worldtube constraints reduce to (\ref{eq:consu}) and (
\ref{eq:consA} ), which are equivalent to the condition that the Einstein
tensor satisfy
\begin{equation}
\xi^{\mu}G_{\mu}^{\nu}N_\nu=0,
\label{eq:Gcon}
\end{equation}
where $\xi^{\mu}$ is any vector field tangent to the worldtube, whose normal
is $N_\nu$. These are the boundary analogue of the momentum
constraints for the Cauchy problem. In Stewart's treatment of the $3+1$
IBVP (see Sec.~\ref{eq:fritreul}), it was the Cauchy momentum constraints
which were enforced on the boundary. In the characteristic IBVP, it is
the worldtube constraints (\ref{eq:Gcon}) which must be enforced. They
form three components of the boundary constraints (\ref{eq:bconstr})
proposed by Frittelli and Gomez for the $3+1$ problem. (See
Sec.~\ref{sec:other}.)
This worldtube constraints (\ref{eq:Gcon}) can be interpreted as flux
conservation laws for the $\xi$-momentum contained in the
worldtube~\cite{tam},
\begin{equation}
P_\xi(u_2)-P_\xi(u_1) = \int_{u_1}^{u_2} dS_\nu \{
\nabla^\nu \nabla_\mu \xi^\mu - \nabla_\mu \nabla^{(\nu}\xi^{\mu)} \}
\end{equation}
where
\begin{equation}
P_\xi =\oint dS_{\mu\nu} \nabla^{[\nu}\xi^{\mu]}
\end{equation} and $dS_{\mu\nu}$ and $dS_\nu$ are the 2-surface and
3-volume elements on the worldtube. When $\xi^\mu$ is a Killing vector
for the intrinsic 3-metric of the world-tube, this gives rise to a strict
conservation law. For the limiting case at ${\cal I}^+$, these flux
conservation laws govern the energy-momentum, angular momentum and
supermomentum corresponding to the asymptotic symmetries~\cite{tam}. For
an asymptotic time translation, they give rise to the Bondi's
famous result~\cite{bondi} relating the mass loss to the square of the
news function.
In terms of the intrinsic metric of the worldtube
\begin{equation}
H_{\mu\nu}=g_{\mu\nu}-N_\mu N_\nu,
\end{equation}
its intrinsic covariant derivative ${\cal D}_\mu$ and its extrinsic
curvature
\begin{equation}
K_{\mu\nu}=H_\mu^\rho \nabla_\rho N_\nu,
\end{equation}
the worldtube constraints (\ref{eq:Gcon}) can be rewritten as
\begin{equation}
H_\nu^\mu
G_{\mu\rho} N^\rho ={\cal D}_\mu (K^\mu_\nu
- \delta^\mu_\nu K^\rho_\rho) =0.
\label{eq:bmom}
\end{equation}
These are the analogue of (\ref {eq:momc}) for the Cauchy problem but they
now form a symmetric hyperbolic system because of the timelike nature
of the worldtube. In terms of a dyad (\ref{eq:qnorm}) adapted to the
foliation of the worldtube, this gives rise to the {\bf Worldtube
Theorem}~\cite{josh}:
\bigskip
\noindent {\em Given $H_{ab}$, $Q^a Q^b K_{ab}$ and $K$, the worldtube
constraints constitute a well-posed initial-value problem which
determines the remaining components of the extrinsic curvature $K_{ab}$}.
\bigskip
The theorem constrains the integration constants for the
nullcone-worldtube IBVP. Similarly, they constrain the boundary data
for a $3+1$ IBVP subject to the Frittelli-Gomez conditions.
Unfortunately, for neither of these IBVPs has it been possible to combine
the boundary constraints with the evolution system in a manner consistent
with a strongly well-posed IBVP.
The enforcement of the boundary constraints is an indirect way to enforce
the Hamiltonian and momentum constraints constraints $H=G^{\mu\nu} n_\mu
n_\nu$ and $P^\mu= h^\mu_\rho G^{\rho\nu} n_\nu $, where in the $3+1$
decomposition with respect to the Cauchy hypersurfaces
$$
h_{\mu\nu}=g_{\mu\nu } + n_\mu n_\nu.
$$
The more direct
approach commonly used to investigate constraint preservation in the
$3+1$ Cauchy problem is to cast the contracted Bianchi identity
(\ref{eq:bianch}) into a hyperbolic system. The results depend upon the
particular formulation.
As a first example, consider the ADM system (\ref{eq:adm}) in which only
the 6 Einstein equations
\begin{equation}
h_\mu^\rho h_\sigma^\nu {R_\rho}^\sigma=0
\label{eq:admev}
\end{equation}
are evolved. Application of the contracted Bianchi identity gives rise to
the symmetric hyperbolic constraint propagation system
\begin{eqnarray}
n^\gamma \partial_\gamma H - \partial_j P^j
&=& B^\gamma G_{\nu \gamma }n^\nu \nonumber \\
n^\gamma \partial_\gamma P^i - h^{ij} \partial_j H
&=& B^{\mu\gamma} G_{\nu \gamma }n^\nu,
\label{eq:adnc}
\end{eqnarray}
where the coefficients $B^\gamma$ and $B^{\mu\gamma}$ arise from
Christoffel symbols and do not enter the principal part. When applied to
the IBVP, a complication arises from the component of the shift normal
to the boundary,
\begin{equation}
\beta^N=\beta^\mu N_\mu = -\alpha \sinh \Theta
\end{equation}
in terms of the lapse $\alpha$ and the hyperbolic angle $\Theta$
(\ref{eq:hangle}) governing the velocity of the boundary. Here $\beta^N
<0$ ($\beta^N >0$) for a boundary which is moving inward (outward) with
respect to the Cauchy hypersurfaces. An analysis of (\ref{eq:adnc})
shows that only one boundary condition is allowed provided $\beta^N \le
0$, i.e provided the boundary is moving inward. In that case, the theory of
symmetric hyperbolic systems guarantees that all the constraints
would be preserved if the single constraint
\begin{equation}
H+P^i N_i = G_{\mu\nu} n^\mu K^\nu =0
\label{eq:admcon}
\end{equation}
is satisfied at the boundary, where $K^\mu$ is the outgoing null vector to
the foliation of the boundary. (Additional boundary conditions are
necessary for constraint preservation if $\beta^N> 0$.) By virtue of
the evolution system (\ref{eq:admev}), the constraint (\ref{eq:admcon})
is equivalent to
\begin{equation}
G_{\mu\nu} K^\mu K^\nu =0.
\label{eq:raych}
\end{equation}
This is the Raychaudhuri equation~(cf.~\cite{wald})
\begin{equation}
K^\mu \partial _\mu \theta +\frac{1}{2}\theta^2+\sigma \bar\sigma =0,
\label{eq:drho}
\end{equation}
where $\theta$ is the expansion and $\sigma$ is the shear of the outgoing
null rays tangent to $K^\mu$. Thus, for the ADM system, constraint
preservation can be enforced by the Sommerfeld boundary condition
(\ref{eq:drho}) for $\theta$. Unfortunately, although the constraint
system has these attractive properties, the ADM evolution system is only
weakly hyperbolic and consequently leads to unstable evolution.
Next consider the BSSN evolution system, which enforces the 6 Einstein
equations
\begin{equation}
h_\mu^\rho h_\sigma^\nu {R_\rho}^\sigma
-\frac{2}{3}h_\mu^\nu H=0.
\label{eq:bssnmev}
\end{equation}
The contracted Bianchi identity now implies the constraint system
\begin{eqnarray}
n^\gamma \partial_\gamma H
-\partial_j P^j
&=& B^\gamma G_{\nu \gamma }n^\nu \\
n^\gamma \partial_\gamma P^i
+ \frac{1}{3} h^{ij} \partial_j H
&=& B^{\mu\gamma} G_{\nu \gamma }n^\nu.
\label{eq:bssnmc}
\end{eqnarray}
This is not symmetric hyperbolic and would not lead to stable constraint
preservation even for the Cauchy problem. This is remedied in the course
of introducing auxiliary variables which reduce the BSSN system to first
order form. Auxiliary constraints are mixed into the evolution system
(\ref{eq:bssnmev}) and they combine with the constraint system
(\ref{eq:bssnmc}) to form a larger symmetric hyperbolic constraint
system. There is a large freedom in the constraint-mixing parameters and
gauge conditions. For a particular choice made by N{\' u}{\~ n}ez and
Sarbach~\cite{nunsar}, the linearization off Minkowski space yields a
symmetric hyperbolic evolution system. The boundary conditions for this
system are complicated by the normal component of the shift. As discussed
in conjunction with (\ref{eq:advect}), the number of boundary conditions
required by the advection equations introduced in the first order
reduction depends upon whether $\beta^N$ is positive or negative. This
forces use of a Dirichlet condition, e.g. $\beta^N=0$, rather than a
Sommerfeld condition on the shift. Constraint preservation holds only in a
certain parameter range, $(b_1 \le1,b_2 \le 1)$ for the boundary conditions
given in equation (97) of~\cite{nunsar}. The particular choice $b_1=0$,
leads to the boundary condition~\cite{olivpc}
\begin{equation}
H-3 P^i N_i =G_{\mu\nu} n^\mu (n^\nu - 3N^\nu) = {\cal Z},
\label{eq:bssncon}
\end{equation}
where ${\cal Z}$ represents contributions from the auxiliary constraints,
or, by using the evolution system (\ref{eq:bssnmev}),
\begin{equation}
G_{\mu\nu} L^\mu L^\nu ={\cal Z},
\label{eq:inraych}
\end{equation}
where $L^\mu$ is the ingoing null vector to the boundary. It is a bizarre
feature of the $3+1$ problem that the constraint preserving boundary
condition switches from the outgoing Raychaudhuri form (\ref{eq:raych})
to the ingoing Raychaudhuri form (\ref{eq:inraych}) in going from the ADM
to the BSSN system. The Raychaudhuri equation for the outgoing null
direction cannot be imposed in the allowed range of $(b_1,b_2)$.
The widely varying nature of constraint enforcement among different
formulations does not provide any apparent insight. However, one problem
common to many first order formulations arises from the advective
derivative $n^\mu \partial_\mu$, which determines whether the auxiliary
variables are ingoing or outgoing at the boundary, depending on the sign
of $\beta^N$. This problem could be avoided by instead using the
derivative $t^\mu \partial_\mu$ determined by the evolution field, which
can always be chosen tangential to the boundary. That suggests that the
projection operator $ \pi^\mu_\nu$ associated with $t^\mu$, given in
(\ref{eq:tproj}), might be useful in separating out the evolution system
from the constraints, rather than the projection operator $h^\mu_\nu$
used in (\ref{eq:admev}) and (\ref{eq:bssnmev}). This gives rise to many
ways to obtain a symmetric hyperbolic constraint system whose boundary
treatment is independent of $\beta^N$. As a simple example, in adapted
coordinates the evolution system $G^{ij} =\lambda \delta^{ij} G^{tt}$,
with $\lambda>0$, leads via (\ref{eq:bianch}) to the symmetrizable
constraint system
\begin{eqnarray}
\partial_t G^{tt} +\partial_j G^{tj} &=& \text {lower order terms }
\nonumber \\
\partial_t G^{ti} +\lambda \partial_i G^{tt} &=&
\text {lower order terms} .
\end{eqnarray}
Independently of $\beta^N$, this system requires only 1 boundary
condition to preserve all the constraints.
\bigskip
\section{Geometric uniqueness of the IBVP}
\label{sec:geom}
\bigskip
The solution of the Cauchy problem has the important property of {\it
geometric uniqueness}, i.e. Cauchy data $(h_{ab},k_{ab})$ on ${\cal
S}_0$ determine a metric $g_{ab}$ which is unique up to
diffeomorphism. Under a diffeomorphism $\psi$, the data $(\psi^*
h_{ab},\psi^* k_{ab})$ determines an equivalent metric. As well as being
a pretty result, this has the practical application of allowing numerical
simulations with the same initial data but carried out with different
formulations and different gauge conditions to produce geometrically
equivalent spacetimes. Friedrich~\cite{hjuerg} has emphasized that this
property remains an unresolved issue for the IBVP.
There are different ways in which this property might be formulated for
the IBVP. The most demanding way would be to require that the data at a
point of the boundary be locally determined by the boundary geometry in
the neighborhood of that point. Such data might include the trace $K$ of
the extrinsic curvature of the boundary, which forms part of the data for
the Friedrich-Nagy system. However, it is clear that at least two more
pieces of data are necessary to prescribe the gravitational radiation
degrees of freedom. In the Friedrich-Nagy system, these two pieces of
data are supplied by the combination (\ref{eq:fnpsi}) of the Weyl tensor
components $\Psi_0$ and $\Psi_4$. However, the associated outgoing and
ingoing null vectors are not determined by the local geometry but
depend upon the choice of timelike evolution field $T^a$ tangent to the
boundary, according to (\ref{eq:fnk}). This could be avoided by requiring
these null vectors to satisfy the local geometric condition that they be
principle null directions of the Weyl tensor (cf.~\cite{wald}); but in a
general spacetime this would lead to four choices which would then have
to be incorporated somehow into the evolution system. An alternative,
suggested in~\cite{hjuerg}, is to base the data on the
eigenvector problem determined by the trace free part of the extrinsic
curvature of the boundary,
\begin{equation}
( K_{ab} -\frac{1}{3} H_{ab} K ) V^b = \lambda H_{ab} V^b.
\end{equation}
As in the preceding presentation, $H_{ab}$ is the intrinsic metric of the
boundary. For a spherical worldtube $r=R$ in
Minkowski space,
\begin{equation}
K_{ab} -\frac{1}{3} H_{ab} K = \frac{1} {3R}( H_{ab}
+3 \tilde T_a \tilde T_b)
\end{equation} where $\tilde T_a$ is a timelike eigenvector. This raises
the possibility of whether this eigenvector problem can be used to pick
out a locally preferred timelike direction $\tilde T_a$ in the curved space case.
Similar algebraic properties of the extrinsic curvature hold under roundness
conditions which are typically satisfied by the artificial outer boundary
of an isolated system. However, whether such an approach can be
incorporated into the evolution system and whether the two radiation
degrees of freedom can be encoded in the extrinsic curvature are not
obvious.
Neither of the two strongly well-posed formulations of the IBVP described
in Sec's~\ref{sec:fn} and~\ref{sec:harm} are based upon purely local
geometric data. In both of them, a foliation of the boundary consistent
with a choice of evolution field plays an essential nonlocal role. This
suggests that a version of geometric uniqueness based upon purely local
data might not be possible. The prescription of an evolution field
$t^a$ as part of the boundary data provides the necessary structure to pose
a version of geometric uniqueness~\cite{juerg,disem}. As explained in
Sec's~\ref{sec:bare} and~\ref{sec:initial}, the flow of the evolution
field carries the initial edge ${\cal B}_0$ into a foliation ${\cal B}_t$
of the boundary; and it carries the initial Cauchy data into a stationary
background metric $\gz_{ab}$ according to (\ref{eq:gz}). Thus the
evolution field provides the two essential structures to geometrize the
boundary data: the foliation ${\cal B}_t$ determines the outgoing null
direction $K^a$ and the preferred background metric allows the Sommerfeld
derivative to be expressed covariantly as $K^a \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a$ in terms of the
background connection. Under a diffeomorphism, the evolution field
transforms according to $t^a \rightarrow \psi_* t^a$ with the consequence
that $\gz_{ab} \rightarrow \psi^* \gz_{ab} $.
The boundary data $q^a$ and $\sigma$ for the covariant version of the
covariant Sommerfeld conditions (\ref{eq:ak})--(\ref{eq:hl}) then have
the geometric interpretation that
\begin{equation}
q^a = K^b (\nabla_b - \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_b) K^a
\label{eq:qdata}
\end{equation}
is the acceleration of the outgoing null vector $K^a$ relative to the
background acceleration, and
\begin{equation}
\sigma =
\frac{1}{2}Q^a Q^b(\nabla_a - \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_a) K_b, \\
\label{eq:shdata}
\end{equation} is the shear of $K^a$ relative to the background. The use
of the shear in posing geometrical boundary conditions for the harmonic
formulation was also suggested in~\cite{ruizhbc}. The rotation freedom
in the dyad dependence (\ref{eq:shdata}) can be removed by introducing
the rank-2 shear tensor
\begin{equation}
\sigma^{ab} =\frac{1}{2}(Q^{ac}Q^{bd}
- \frac{1}{2} Q^{ab}Q^{cd})(\nabla_c - \nabla\hspace{-0.27cm}{}^{\mbox{\r{~}}}{}\hspace{-0.22cm}_c) K_d\, , \quad
\sigma = Q_a Q_b \sigma^{ab},
\end{equation}
with $\sigma^{ab}\nabla_b t=0$.
By construction, all quantities involved in the boundary conditions map
as tensor fields under a diffeomorphism $\psi$. As a result of the
covariant form of the generalized harmonic equations (\ref{eq:ceq}) and
(\ref{eq:chc}), the solution $f_{ab}=g_{ab}-\gz_{ab}$ also maps
as a tensor field. The metric $g_{ab}$ satisfies
the generalized harmonic condition (\ref{eq:charm})
with respect to the background $\gz_{ab}$ and the mapped
metric $\psi^* g_{ab}$ satisfies the generalized harmonic condition with
respect to $\psi^* \gz_{ab}$
This can be taken one step further~\cite{disem}. As characterized
in~\cite{hawkel}, the Cauchy data $h_{ab}$ and $k_{ab}$ can be
interpreted as fields $\tilde h_{ab}$ and $\tilde k_{ab}$ on a {\it disembodied}
3-manifold $\tilde {\cal S}_0$ via its embedding ${\cal S}_0$ in the
4-dimensional spacetime manifold ${\cal M}$. A similar approach applies
to the boundary data. Let
\begin{equation}
q^a = q_N N^a + q_{\cal T}^a,
\end{equation}
so that $q_{\cal T}^a$ is tangent to the boundary ${\cal T}$. Then the
fields $q_N$, $q_{\cal T}^a$ and $ \sigma^{ab}$ are intrinsic to the
boundary. Along with the Cauchy data and the hyperbolic angle $\Theta$ at the
edge ${\cal B}_0$, they can be induced by the embedding of
a disembodied version of data. This leads via the well-posedness of the IBVP with
Sommerfeld data to a harmonic version of a
\medskip
\noindent Geometric Uniqueness Theorem:
\bigskip
\noindent Consider the 3-manifolds $\tilde {\cal T}$ and $\tilde {\cal S}_0$
meeting in an edge $\tilde {\cal B}_0$. On $\tilde {\cal S}_0$ prescribe
the symmetric tensor fields $\tilde h_{ab}$ and $\tilde k_{ab}$, subject
to the Hamiltonian and momentum constraints and the condition that
$\tilde h_{ab}$ be a Riemannian metric . On $\tilde{\cal B}_0$ prescribe
the scalar field $\tilde \Theta$. On $\tilde{\cal T}$ prescribe a smooth
foliation $\tilde{\cal B}_t$, parametrized by a scalar function $\tilde
t$, the scalar field $\tilde q_N$, the vector field $\tilde q_{\cal T}^a$
and the rank-2 tensor field $\tilde \sigma^{ab}$. Then, after embedding
$\tilde{\cal S}_0 \cup \tilde {\cal T}$ as the boundary ${\cal S}_0 \cup
{\cal T}$ of a 4-manifold ${\cal M}$ as depicted in
Fig.~\ref{fig:bound}, this provides the Sommerfeld boundary data for a
vacuum spacetime in a neighborhood of ${\cal B}_0$ which is unique up to
diffeomorphism.
\bigskip
Because the boundary data contain gauge information, this version of
geometric uniqueness is weaker than for the Cauchy problem. It is an open
question whether the boundary data can be prescribed purely in terms of
local geometric objects~\cite{hjuerg}. Note that the data contains no
information about the 3-metric of the boundary, not even that it is a
timelike 3-manifold. The geometrical interpretation of the data involves
the metric of the embedded spacetime, whose existence is in the content
of the theorem. The diffeomorphism freedom lies in the freedom in the
embedding and in the choice of evolution field $t^a$, which determines
the gauge and background geometry. See~\cite{reulsarrev} for a discussion
of these issues in the context of linearized gravitational theory.
The spatial locality of the solution can be extended, say to a boundary
with spherical topology, by patching solutions together. However, the
locality in time presents a more complicated problem regarding the
maximal development of the solution. For instance, the time foliation
might develop a gauge pathology which prematurely stops the evolution.
That makes it unclear how the maximal development for the Cauchy
problem, as constructed by Choquet-Bruhat ~\cite{gerochbr}, might be
generalized to the IBVP. A restart of the evolution at an intermediate
time in order to extend the solution would introduce a new gauge, new
initial data, a new evolution field and thus a new background metric. A
maximal development based upon the original background would have to be
based upon a maximal choice of evolution field.
The geometric nature of the Sommerfeld conditions
(\ref{eq:ak})--(\ref{eq:sigqq}) allows them to be formally applied to any metric
formulation. In a $3+1$ formulation, the metric has the decomposition
\begin{equation}
g_{\mu\nu} = -n_\mu n_\nu +\hat N_\mu \hat N_\nu +Q_{\mu\nu},
\label{eq:cdecom}
\end{equation}
where $\hat N_\mu$ is the unit normal to the boundary which lies in the
Cauchy hypersurfaces and, as before, $Q_{\mu\nu}= Q_{(\mu}\bar Q
_{\nu)}$ is the 2-metric intrinsic to its foliation ${\cal B}_t$. The
Sommerfeld boundary conditions (\ref{eq:ak})-- (\ref{eq:aq}) and
(\ref{eq:sigqq}) supply boundary data for the $\hat N^\mu \hat N^\nu
k_{\mu\nu}$, $Q^\mu \hat N^\nu k_{\mu\nu}$ and $Q^\mu Q^\nu k_{\mu\nu}$
components of the extrinsic curvature of the Cauchy foliation. However,
(\ref{eq:al}) supplies the boundary data for the normal component of the
shift, which for many $3+1$ formulations would require a Dirichlet
condition that fixes its sign. The remaining
Sommerfeld boundary conditions (\ref{eq:hk})--(\ref{eq:hl}), which
enforce the harmonic constraints, would also require modification
depending upon the particular $3+1$ gauge conditions. See~\cite{nunsar}
for a discussion relevant to the BSSN formulation. Numerical tests would
be necessary to study whether application of
(\ref{eq:ak})--(\ref{eq:aq}) and (\ref{eq:sigqq}) would improve the
performance over the present boundary treatment of $3+1$ systems.
\bigskip
\section{Numerical tests}
\label{sec:num}
\bigskip
Post and Votta~\cite{postvot} have emphasized that ``Verification and
validation establish the credibility of code predictions. Therefore it's
very important to have a written record of verification and validation
results.'' The {\em validation} of a code implies that its predictions
are in accord with observed phenomena. For the present status of
numerical relativity, in the absence of any empirical observations,
the burden falls completely on {\em verification}. Post and Votta list five
verification techniques:
\begin{enumerate}
\item ``Comparing code results with an exact answer''.
\item ``Establishing that the convergence rate of the truncation error
with changing grid spacing is consistent with expectations''.
\item ``Comparing calculated with expected results for a problem
especially manufactured to test the code''.
\item ``Monitoring conserved quantities and parameters, preservation of
symmetry properties and other easily predictable outcomes''.
\item ``Benchmarking -- that is, comparing results from those with
existing codes that can calculate similar problems''.
\end{enumerate}
The importance of the first four techniques has now been recognized by
most numerical relativity groups and their implementation in practice has
improved the integrity of the field. Individual groups cannot easily
carry out the fifth technique independently. This was the motivation
behind formation of the Apples with Apples (AwA)
Alliance~\cite{awa}.
The early attempts at developing numerical codes were primarily judged by
their ability to simulate black holes, understandably because of the
astrophysical importance of quantifying that system. When the difficulty with
numerical stability became apparent, there was increased focus on a better
mathematical and computational understanding of the analytic and
numerical algorithms. Only a few groups had based their codes upon
symmetric or strongly hyperbolic formulations of Einstein's
equations and fewer had even begun to worry about how to apply
boundary conditions. The cross fertilization between computational
mathematics and numerical relativity was entering a productive stage. At
the same time, standardized tests were developed by the AwA Alliance in
order to isolate problems, calibrate accuracy and compare code
results, \url{http://www.ApplesWithApples.org}.
Such testbeds have been historically used in computational
hydrodynamics. There are two fundamentally different types. One compares
simulations of a physically important process, such as the binary black
hole problem. The second type involve idealized situations which isolate
problems, such as the ``shock tube'' test in computational fluid
dynamics. This is the type of testbed considered by the AwA Alliance.
The first tests were designed to study evolution algorithms in the
absence of boundaries~\cite{mex1,mex2}. Five tests were based upon a
toroidal 3-manifold (equivalent to periodic boundary conditions):
\begin{itemize}
\item The {\it robust stability} test evolves random initial data in the
linearized regime. This is a pass/fail test designed as a screen to
eliminate unstable codes.
\item The {\it linearized wave} test propagates a periodic plane wave
either parallel or diagonal to an axis of the 3-torus. The test checks the
accuracy in tracking both the amplitude and phase of the wave.
\item The {\it gauge wave} test is a pure gauge version of the linearized
wave test, but with amplitude in the non-linear regime.
\item The {\it shifted gauge wave} test is based upon a gauge wave with
non-vanishing shift. Both the gauge wave and shifted gauge wave tests are
challenging because of exponentially growing modes in the analytic
problem~\cite{babev}.
\item The {\it Gowdy wave} test simulates an expanding or contracting
toroidal spacetime, which contains a plane polarized gravitational wave in a
genuinely curved, strong field context.
\end{itemize}
The wave tests provide exact solutions which allow convergence
measurements. Instabilities are monitored by the growth of the
Hamiltonian constraint. Test results were carried out for codes based
upon numerous formulations: harmonic, Friedrich-Nagy, NOR, and several
versions of BSSN and ADM. See~\cite{mex2} for the test results.
Tests of the Cauchy evolution algorithm cull out algorithms whose
boundary stability is doomed from the outset. A subsequent plan for
boundary tests was formulated by opening up one axis of the 3-torus to
form a manifold with boundary. This has the advantage that the boundaries
are smooth 2-tori, thus avoiding the complication of sharp boundary
points. This could later be extended to opening up all three axes to test
performance with a cubic boundary. For the robust stability test the
boundary data consist of random numbers. For the wave tests, the boundary
data is supplied by the exact (or linearized) solution. See~\cite{awa}
for the detailed specifications of the five AwA boundary tests.
These boundary tests were first formulated and applied in the early
development of boundary algorithms for harmonic codes. The robust
stability test~\cite{robust} was used to verify the stability of a code
based upon a well posed IBVP for linearized harmonic
gravity~\cite{szilschbc}. Subsequently, the gauge wave tests were carried
out with a harmonic code whose underlying IBVP was well-posed for
homogeneous Dirichlet and Neumann conditions~\cite{harl,mBbSjW06}. This
revealed problems in the very nonlinear regime, where the approximation
of small boundary data was violated. The shifted gauge wave test posed an
additional difficulty, beyond the unstable analytic modes that already
challenged the Cauchy evolution test~\cite{bab,babev}. The periodic time
variation of the shift produced an effective oscillation of the
boundaries which blue shifted and trapped the error resulting from the
reflecting boundary conditions. This led to unstable behavior in the
nonlinear regime.
After formulation of the strongly well-posed harmonic IBVP described in
Sec.~\ref{sec:harm}, the Sommerfeld boundary conditions were implemented
and tested in a harmonic code~\cite{harmsomm,harmexcis,seil,seilthesis}.
Numerical stability and accurate phase and amplitude tracking were
confirmed by the robust stability and linearized wave tests. An
important attribute of strong well-posedness is the estimate of boundary
values provided by the energy conservation obeyed by the principal part
of the equations. This boundary stability extends to the semi-discrete
system obtained by replacing spatial derivatives by finite differences
obeying {\it summation by parts} (the discrete counterpart of integration
by parts), so that energy conservation caries over to the semi-discrete
problem~\cite{kreissch}. Stability then extends to the fully discretized
evolution algorithm obtained with an appropriate time integrator, such as
Runge-Kutta~\cite{Kreiss-Wu}. It was found that these discrete
conservation laws were both effective and essential in controlling the
exponential analytic modes latent in the gauge wave test.
Although the analytic proof of well-posedness given
in~\cite{wpe} was based upon a scalar wave energy differing by a small
boost from the standard energy, it is interesting that these successful
code tests were based upon the standard energy. This confirms the
robustness of the underlying approach.
The oscillating boundaries in the shifted gauge wave test excite a
different type of long wavelength instability which could not be
suppressed by purely numerical techniques. Knowledge of the Sommerfeld
boundary data allows the wave to enter and leave the boundaries, but the
numerical error, although small and convergent, excites an exponential
mode of the analytic problem. However, because this instability violates
the harmonic constraints it was possible to suppress it by a harmonic
constraint adjustment of the form
(\ref{eq:creduced2})~\cite{harmsomm,babev}. This example emphasizes the
importance of understanding instabilities in the analytic problem in
order to control them in a numerical simulation.
Other wave solutions which have been used for numerical tests are
the Teukolsky waves~\cite{teuk}, which are linearized spherical waves
appropriate for testing a spherical boundary, and the nonlinear Brill
waves~\cite{brill}, which are useful for testing wave propagation during
collapse to a black hole. Teukolsky wave tests of the harmonic Sommerfeld
conditions confirm that the constraint violation error due to homogeneous
outer boundary data drops to numerical truncation error as the wave
propagates off the grid~\cite{seil}. Furthermore, when constraint damping
is applied to the interior of the grid, the error drops to machine round-off.
For Brill wave tests with the same code, homogeneous Sommerfeld
conditions lead to considerable back reflection off the boundary, as
expected from the discussion in Sec.~\ref{sec:absorbc}. In~\cite{improved},
Teukolsky waves were also used to test an implementation of the improved
higher order harmonic boundary conditions proposed by Buchman and
Sarbach~\cite{sarbuch2}.
Rinne, Lindblom and Scheel~\cite{oRlLmS07} have developed a numerical
test for comparing back reflection of waves from a spherical boundary.
First, using perturbative techniques, they construct a reference solution
for a linearized gravitational wave propagating on an exterior
Schwarzschild background. Then the full numerical code is run with a
finite spherical outer boundary. The error with respect to the reference
solution is used to compare different choices of boundary conditions.
Using the first order harmonic code described in~\cite{lLmSlKrOoR06},
they used this test to compare several boundary conditions. The
comparisons of homogeneous Sommerfeld boundary conditions were
in accord with the theoretical expectations discussed in
Sec's~\ref{sec:absorbc} and~\ref{sec:harm}. The higher order
Sommerfeld conditions and the
$\Psi_0$ freezing condition both produced less back refection than the
first order conditions (\ref{eq:modasymKK}) -(\ref{eq:modasymQQ}). The
test was also used to reveal the spurious effects arising from
a sponge boundary condition and also from the combination of
numerical dissipation and spatial compactification used by
Pretorius~\cite{pret1,pret2}. In an independent follow up of this
test~\cite{ruizhbc}, the advantages of
higher order Sommerfeld conditions were confirmed.
Although $3+1$ codes have been predominant in binary black hole
simulations, boundary tests have not been very extensive as compared to
harmonic codes. This is especially pertinent for the BSSN system.
Boundary tests based on a gravitational wave
perturbation of the Schwarzschild exterior have been carried out for the
KST system~\cite{caltechbc}. The results revealed instabilities although
they showed how improvements to the boundary conditions could reduce
constraint violation. The robust stability and Brill wave tests have been
applied to boundary conditions for a symmetric hyperbolic version of the
Einstein-Christoffel system~\cite{sarbtigl}. Although the Cauchy problem
for this system is well-posed, both tests revealed instabilities due to
the boundary algorithm. Again the tests were useful guides for
understanding the problems with the boundary treatment.
C. Bona and C. Bona-Casas~\cite{gowdy} have made the first application of
the Gowdy wave boundary test. They show how it can be applied to a
symmetric hyperbolic version of the first order Z4 formalism. Here, as in
many other studies, the Cauchy problem is well-posed but the strong
well-posedness of the IBVP depends on the boundary condition. Improper
boundary conditions can lead to instability and/or constraint violation.
They first demonstrate that the test is effective at investigating
methods for preserving the energy constraint for the Z4 system in a
strong field environment. In a subsequent work~\cite{z4cpbc}, they apply
both the Gowdy wave and robust stability boundary tests in an expanded
study of constraint violation in the Z4 framework. The robust stability
case is applied both with opening up one axis of the 3-torus and with a
fully cubic boundary. This allows testing SBP algorithms at the corners
and edges. The results show numerical stability of the proposed boundary
algorithms in the linearized regime. The Gowdy wave test extends this
study to constraint preservation in the nonlinear regime. The results
provide further evidence of numerical stability and show that
constraint violation can be kept at the level of discretization error.
\section{Open questions}
\label{sec:quest}
\bigskip
The IBVP has analytical, computational, geometrical and physical
aspects. The analytic goal is a strongly well-posed IBVP, which is the
prime necessity for the computational goal of an accurate evolution
algorithm. It is also the raison d'etre for the geometric goal of a
gauge invariant formulation of the boundary data. The prime physical
goal, at present, is the accurate simulation of binary black holes. The
binary black hole problem has taken a course of its own, which has been
remarkably successful in view of the gaps in our current understanding
of the other aspects of the IBVP.
Some important open questions which would help close those gaps
are:
\begin{itemize}
\item {\bf Question 1.} Is there a strongly well-posed IBVP based upon a
$3+1$ formulation?
\bigskip
Some insight into this question would be provided by the answer to
\item {\bf Question 2.} Can the necessary boundary data be represented by
gauge invariant, local geometric objects?
\bigskip
In the Friedrich-Nagy treatment, there are three pieces of boundary
data which are not pure gauge: the trace $K$ of the extrinsic boundary
curvature, which determines the location of the boundary, and the
Weyl curvature components encoding the two radiation degrees
of freedom. In the harmonic system, the Sommerfeld data which
encode the radiation consist of the shear, or the Weyl curvature
components for a higher order condition. This leads to
\item {\bf Question 3.} In the harmonic formulation, can the
trace $K$ be used as gauge invariant boundary data?
For the Cauchy problem, there exists a maximal
development of the solution~\cite{gerochbr}.
\item {\bf Question 4.} What is the proper formulation of the maximal
development of the IBVP?
\bigskip
\end{itemize}
The quote of Turing at the beginning of this review expresses the
relative degree of difficulty between the Cauchy problem and the IBVP.
\begin{acknowledgments}
This work was supported by NSF grant PHY-0854623 to the
University of Pittsburgh. I have benefited
from numerous discussions with H. Friedrich,
H-O. Kreiss, O. Reula, B. Schmidt and O. Sarbach.
\end{acknowledgments}
|
{
"timestamp": "2012-04-25T02:05:25",
"yymm": "1203",
"arxiv_id": "1203.2154",
"language": "en",
"url": "https://arxiv.org/abs/1203.2154"
}
|
\section{\label{sec:level1}First-level heading}
Cellular membranes are two-dimensional (2D) liquids composed of thousands of different lipids and membrane bound proteins. Though once thought of as uniform solvents for embedded proteins, a wide array of biochemical and biophysical evidence suggests that cellular membranes are quite heterogeneous (reviewed in ~\cite{Pike06,Lingwood10}). Putative membrane structures, often termed `rafts', are thought to range in size from $10-100nm$, much larger than the $a \sim 1nm$ size of the individual lipids and proteins of which they are composed. This discrepancy in scale presents a thermodynamic puzzle: na\"{\i}ve estimates predict enormous energetic costs associated with maintaining heterogeneity in a fluid membrane~\cite{Machta11}.
Parallel work in giant plasma membrane vesicles (GPMVs) isolated from living mammalian cells presents a compelling explanation for the physical basis of these proposed structures. When cooled below a transition temperature around $25^o$C, GPMVs phase separate into two 2D liquid phases~\cite{BaumgartHSHHBW07} which can be observed by conventional fluorescence microscopy. Quite surprisingly, they pass very near to a critical point in the Ising universality class at the transition temperature~\cite{Veatch08}. Near a miscibility critical point, the small free energy differences between clustered and unclustered states could allow the cell to more easily control the spatial organization of the membrane, lending energetic plausibility to the proposed structures. Although analogous critical points can be found in synthetic membranes~\cite{Veatch07,Veatch072,Honerkamp-Smith09} these systems require the careful experimental tuning of two thermodynamic parameters, as in the Ising liquid-gas transition where pressure (equivalent to the Ising magnetization) and temperature must both be tuned. Although it has been suggested that biological systems frequently tune themselves towards \textit{dynamical} and other statistical critical points~\cite{Mora11}, so far as we know membranes are the clearest example of a biological system which appears to be tuned to the proximity of a \textit{thermal} critical point.
Other plausible theoretical models have focused on 2D micro emulsions (stabilized by surfactants~\cite{Brewster09}, coupling to membrane curvature~\cite{Schick12}, or topological defects in orientational order~\cite{Korolev08}) but none have emerged from direct, quantitative experiments on membranes from living cells. It has been argued that Ising fluctuations should have vanishing contrast between the two phases~\cite{Schick12}. While this is true of macroscopic regions, a region of radius $R$ of lipids of size $a\sim 1nm$ should have contrast $\sim (R/a)^{-\beta/\nu} = (R/a)^{-1/8}$, leading to predicted composition differences of $0.7$ at the physiologically relevant $20nm$ scale, and differences of 0.5 at $R=400nm$ scale of fluorescence imaging~\cite{Veatch08}; on the length scales of interest there is plenty of contrast. Indeed, our calculations of Ising-induced forces take place at and above the critical point, where the macroscopic contrast is of course zero.
How might a cell benefit by tuning its membrane near to criticality? Presuming that functional outcomes are carried out by proteins embedded in the membrane, we focus on the effects that criticality might have on them. For embedded proteins, proximity to a critical point is distinguished by the presence of large, fluctuating entropic forces known as critical Casimir forces. Three dimensional critical Casimir forces have a rich history of theoretical study\cite{Fisher78}. In more recent experimental work~\cite{Bonn09} colloidal particles clustered and precipitated out of suspension when the surrounding medium is brought to the vicinity of the liquid-liquid miscibility critical point in their surrounding medium. Two dimensional Casimir forces like the ones studied here have been investigated for the Ising model using numerical transfer matrix techniques~\cite{Burkhardt95}, for a demixing transition using Monte-Carlo~\cite{Reynwar08} and for shape fluctuation using perturbative analytical methods~\cite{Yolcu11,Yolcu12}. Here we estimate the magnitude of composition mediated Casimir forces arising from proximity to a critical point, both in Monte-Carlo simulations on a lattice Ising model, and analytically, making use of recent developments in boundary conformal field theory(CFT)~\cite{Cardy082,Ginsparg88,DiFrancesco97}. Our motivation is biological: in a cellular membrane, these long ranged critical Casimir forces could have profound implications. More familiar electrostatic interactions are screened over around $1nm$ in the cellular environment, whereas we find the composition mediated potential can be large over tens of nanometers.
Critical Casimir forces are likely utilized by cells in the early steps of signal transduction where lipid mediated lateral heterogeneity has been shown to play vital roles. Many membrane bound proteins segregate into one of two membrane phases when biochemically extracted with detergents at low temperatures~\cite{Melkonian95}, or when proteins are localized in phase separated GPMVs~\cite{Veatch08}. Furthermore, there is evidence that some receptors change their partitioning behavior in response to ligand binding or down-stream signaling events~\cite{Holowka05}. Modeling this as a change in the coupling between the receptor protein and the Ising order parameter predicts that these bound receptors will see a change in their interaction partners. Supporting this view, ligand binding to receptor is often accompanied by spatial reorganization in which receptors and downstream molecules move into close proximity of one another~\cite{Pike06,Veatch12}, perhaps because they now share a preference for the same Ising phase. Perturbations to the lipid composition of the membrane, like cholesterol depletion~\cite{Levental09}, typically disrupt this spatial reorganization~\cite{Veatch12} and have dramatic effects on the final outcomes of signaling~\cite{Sheets99,Sil07,Gidwani03}, in our view by taking the membrane away from its critical point and interfering with the resulting long ranged forces.
We take three approaches to estimating the form of these potentials. We first consider two point-like proteins which interact with the local order parameter like local insertions of magnetic field $h_1$ and $h_2$ at $x=0$ and $x=d$. To calculate the resulting potential we write a Hamiltonian for the combined system of the Ising model with order parameter $\phi(x)$ plus proteins as $H(\left[\phi(x)\right], d) = H_{Ising}(\left[\phi(x)\right])+h_1 \phi(0)+h_2 \phi(d)$. We then write a partition function for the combined system $Z(d)=\int D\left[\phi(x)\right]e^{-\beta H((\phi(x), d))}$ and solve to lowest order in $h$ giving the potential $U_{eff}(d) =-\log(Z(d))+\log(Z(\infty))= -h_1 h_2C(d)$ with $C(d) = \langle\phi(0)\phi(d)\rangle$ the correlation function. $C(d) \sim d^{-\eta}$ when $d\ll \xi$ with the Ising model $\eta=\frac{1}{4}$ and $C(d) \sim d^{-1/2}\exp(-d/\xi)$ for $d\gg \xi$. The potential is attractive for like and repulsive for unlike insertions of field, in agreement with the scaling of the CFT result as we will show below. A protein which does not couple to the order parameter can still feel a long-ranged force if it couples to the local energy density. The energy density is also correlated with a $d^{-2}$ dependance. However, the magnitude of both of these potentials, as well as their shape at distances $d\sim r$ require the Monte-Carlo and CFT approaches described below.
Secondly, we numerically calculated potentials using Monte-Carlo on the lattice Ising model for like and unlike disk-shaped inclusions. Although absolute free energies are difficult to obtain from Monte-Carlo techniques, differences between the free energies of two ensembles, $\delta F$, conditioned on a subset of the degrees of freedom are readily available, provided the degrees of freedom in the two `macro-states' can be mapped onto each other and have substantial overlap. This information is implicitly used in a Monte-Carlo scheme where both `macro-states' are treated as members of a larger ensemble and are switched between so as to satisfy detailed balance. The Bennett method~\cite{Bennett76,Jarzynski97}, uses this information more explicitly, noting that $\exp{(-\beta\delta F)}=\left\langle e^{-\beta \delta E}\right\rangle$ can be estimated without bias from either distribution.
\begin{figure}
\includegraphics[scale=.5]{Figure2new12-21.eps}
\caption{\label{fig:Feff} Effective potentials between bound inclusions are plotted on linear (top) and log-log (bottom) graphs, for inclusions where $r_1=r_2=r$. The CFT results for both like and unlike interactions (thick dashed lines) and for potentials containing a free BC agree with the power-law scaling of the two-point function (thin black dashed line) at large lengths, but separate at small separations. We also compare to Bennett method simulations at $T_c$ as described in the text. We run simulations for each of the blocky spheres shown in (C). Each curve is plotted collapsed by using $r$ as the distance to the farthest point from its center, with no free parameters. The results of our Monte-Carlo pair potentials are all shown plotted against $d/r$ (thin solid lines with colors as in (C)) with the theory curves in dashed lines. The CFT prediction is in excellent agreement with simulation data even for very small inclusions well past the applicability of the power law prediction of the perturbative approach. The value of the potential is fit at the farthest accessible simulation point, where we add the CFT prediction.}
\end{figure}
Our `macro-states' are the location of two blocky `disks' as shown in fig~\ref{fig:Feff}C. All spins either contained in or sharing a bond with these disks are constrained to be either all up or all down. We map the degrees of freedom in one macro-state to a neighboring one by moving all of the spin values 1 lattice spacing to the right or left of the fixed spin region onto fixed spins on the other side. By integrating our measured $\beta \delta F=-\log \left<exp(-\beta \delta E )\right> $ over many sites outwards to infinity, we can in principle measure this potential to arbitrary distance. However, because the potential is long-ranged at $T_c$, we integrate it out to $50$ lattice spacings and add the CFT prediction for the potential at that distance as described below. We perform simulations using the Wolff Algorithm on $500 \times 500$ lattices under the constraint that any cluster which intersects a disk is rejected, enforcing our fixed boundary conditions. We supplement these with individual spin flips near the inclusions where almost all Wolff moves are rejected. The resulting potentials are plotted in fig.~\ref{fig:Feff}A. We collapse the Monte-Carlo curves by using the the effective radius given by the farthest point from the origin contained in the blocky lattice inclusion as the effective radius.
Finally, we use conformal field theory to make an analytical prediction for the form of these potentials. Our calculation makes extensive use of the conformal invariance of the free energy which emerges at the critical point. An element from the global conformal group can take us from the configuration in fig.~\ref{fig:ConformalMapping}A to that shown in fig.~\ref{fig:ConformalMapping}B where the two disks are concentric with spatial infinity in fig.~\ref{fig:ConformalMapping}A now lying between the two cylinders on the real axis. The radius of the outer circle $R(d,r_1,r_2)$ is now given by:
\begin{equation}
\label{eq:Rdef}
\begin{array}{clrr}
R(d,r_1,r_2)=\frac{x-2+\sqrt{(x-2)^2-4}}{2} \text{, } x=\frac{(d+2r_1)(d+2r_2)}{r_1r_2}
\end{array}
\end{equation}
The much larger local conformal group, particular to 2D, is the set of all analytic functions. We use the transformation $z' =\frac{\log(z)}{2\pi}$ gluing together the boundaries at $x=1$ and $x=0$ to give the cylinder shown in fig.~\ref{fig:ConformalMapping}C with a circumference of $1$ and length:
\begin{equation}
\label{eq:taudef}
\tau(d,r_1,r_2)=i\log(R(d,r_1,r_2))/2\pi
\end{equation}
This transformation breaks global conformal invariance and so increases the free energy by $c\log(R)/12$~\cite{Ginsparg88}, where $c=1/2$ in the Ising model. Defining a $1+1$ dimensional quantum theory on the cylinder (see ~\cite{Ginsparg88}) with `time', $t$ running down its length, our Hamiltonian for $t$ translation is $H=2\pi(L_0+\bar{L}_0-\frac{c}{12})$, where $L_0+\bar{L}_0$ is the generator of dilation in the plane.
\begin{figure}
\includegraphics[scale=0.45]{ConformalMapping.eps}
\caption{\label{fig:ConformalMapping} We consider potentials of mean force in configuration (A), with disks of radius $r_1$ and $r_2$ separated by a distance $d$ with boundary conditions $A$ and $B$. We conformally map this to configuration (B), where both disks are centered on the origin, with the first at radius $1$ and the second at radius $R(d,r_1,r_2)$. We then map this to a cylinder shown in (C) of circumference $1$ and length $-i\tau=\log(R)/2\pi$ where we associate restricted partition functions in an imaginary time $1+1$D quantum model with potentials of mean force in the original configuration.}
\end{figure}
Partition functions in this geometry are linear sums of characters of the conformal group. The representations of the conformal group particular to the Ising universality class have characters given by~\cite{Ginsparg88,Cardy08}:
\begin{equation}
\label{eq:chidef}
\begin{array}{clrr}
\chi_{0}(\tau)=\frac{1+q^2+q^3+\cdots}{q^{1/48}}=\frac{1}{2\sqrt{\eta(\tau)}}\left[\sqrt{\theta_3(q)}+\sqrt{\theta_4(q)}\right] \\
\chi_{1/16}(\tau)=\frac{1+q+q^2+2q^3+\cdots}{q^{1/48-1/16}}=\frac{1}{\sqrt{2\eta(\tau)}}\left[\sqrt{\theta_2(q)}\right] \\
\chi_{1/2}(\tau)=\frac{1+q+q^2+\cdots}{q^{1/48-1/2}}=\frac{1}{2\sqrt{\eta(\tau)}}\left[\sqrt{\theta_3(q)}-\sqrt{\theta_4(q)}\right]
\end{array}
\end{equation}
where $q=\exp{(i\pi\tau)}$, with $\eta(\tau)$ the Dedekind $\eta$ function and with $\theta(\tau)$ the Jacobi, or elliptic Theta functions.
Conformally invariant boundary conditions (BCs) can be deduced by demanding consistency between two parameterizations of the cylinder~\cite{Cardy08}. In one, time moves from one BC to the other across the cylinder with the usual Ising Hamiltonian. Alternatively, time can move around the cylinder with the BCs now entering into the Hamiltonian. There are three allowed BCs~\cite{Cardy08} which, by considering symmetry can be associated with `up', `down' and `free'. These three BCs have four non-trivial potentials between them; a repulsive `unlike' interaction between `up' and `down' BCs, an attractive `like' interaction between `ups' and `ups' or `downs' and `downs', an attractive `free-free' (Fr-Fr) interaction between two `free' BCs and a repulsive `free-fixed' (Fr-Fx) interaction between a `free' BC and either an `up' or a `down'.
The free energy in the configuration shown in figure ~\ref{fig:ConformalMapping}A can be interpreted as a potential of mean force between the bound inclusions. Choosing the convention that the potentials go to 0 as $d \rightarrow \infty$, the potential is given by $U(d)= F_{AB}(\tau)-F_{AB}(\infty)$. After undoing the mapping which changes the free energy by a central charge dependent factor so that $F_{AB}(\tau)=- \log{Z_{AB}(\tau)} + c\pi\tau/6$ (with $k_BT=1$) the potentials are given by:
\begin{equation}
\label{eq:potential}
\begin{array}{llll}
U_{\text{like}}(d,r_1,r_2)\\
\text{ }=-\log \left(\chi_{o}(2\tau)+\chi_{1/2}(2\tau)+\sqrt{2}\chi_{1/16}(2\tau)\right)+\frac{\pi\tau}{12}\\
U_{\text{unlike}}(d,r_1,r_2)\\
\text{ }=-\log \left(\chi_{o}(2\tau)+\chi_{1/2}(2\tau)-\sqrt{2}\chi_{1/16}(2\tau)\right)+\frac{\pi\tau}{12}\\
U_{\text{Fr-Fr}}(d,r_1,r_2)=-\log \left(\chi_{o}(2\tau)+\chi_{1/2}(2\tau)\right)+\frac{\pi\tau}{12}\\
U_{\text{Fr-Fx}}(d,r_1,r_2)=-\log \left(\chi_{o}(2\tau)-\chi_{1/2}(2\tau)\right)+\frac{\pi\tau}{12}\\
\end{array}
\end{equation}
with $\chi_h$ as defined in eq.~\ref{eq:chidef}, and $\tau$ as defined in eqs.~\ref{eq:Rdef} and~\ref{eq:taudef}. These potentials are plotted on regular and log-log graphs in figure~\ref{fig:Feff}. Their form is in agreement with the numerical results obtained using transfer matrix methods in~\cite{Burkhardt95}.
At large $d$, we can examine the asymptotics of the potentials using the form of each potential in eq.~\ref{eq:potential} and the series expansion of the characters as shown in eq.~\ref{eq:chidef}. For fixed BCs, the leading contribution to the potential of mean force is equal to $\pm \sqrt{2(r_1r_2)^{1/4}}d^{-\frac{1}{4}}$, with a sign which differs depending on whether the two BCs are like or unlike, in agreement with the point like approximation. For potentials that involve at least one `free' BC, similar analysis shows that the leading contribution is proportional to $d^{-2}$. All four potentials diverge at short distances like $\pm d^{-1/2}$ where in all cases the sign is positive unless both BCs are identical. We note that the origins of the two techniques leading to the curves shown in fig.~\ref{fig:Feff} are very different; arguably as different from each other as each are from a lipid bilayer. The very close agreement, even at lengths comparable to the lattice spacing speaks to the power of universality.
We also compare the form of the potential with Monte-Carlo results performed at temperatures away from the critical point where the potential has a range given roughly by $\xi$. In each case the resulting potential is a one dimensional cut through a four dimensional scaling function which could depend nontrivially on $d/r_1$,$d/r_2$,$d/\xi$ and the `polar' coordinate $h/t^{\beta\delta}$~\cite{Schofield69} describing the proximity to criticality. The dashed lines show the CFT prediction for $T=T_c$, with numerical results at $1.05$,$1.1$ and $1.2T_c$, all for the $2 \times 2$ block sphere shown at right in fig.~\ref{fig:vsT}. The repulsive potential is both deepest and sharpest at $T_c$, while the the attractive force is sharpest slightly above $T_c$, with the final potential of very similar magnitude.
We expect our results to apply, with a few caveats, to proteins embedded in real cell membranes. Proteins couple to their surrounding composition through the height of their hydrophobic regions, interactions of their membrane-proximal amino acids with their closest lipid shell and by covalent attachment to certain lipids which themselves strongly segregate into one of the two low temperature phases. In simulation our proteins couple strongly to their nearest neighbor lipids leading to potentials in excellent agreement with CFT predictions that are very different in origin. These are expected to describe any uniform boundary condition in an Ising liquid, in the limit where all lengths are large compared to the lattice spacing. When separated by lengths of order a lipid spacing (1nm) we might expect additional corrections to this form, and in particular, a weakly coupled protein may have behavior intermediate between a `free' and a `fixed' BC. In addition, a protein that couples non-uniformly around its boundary might have interesting behavior not addressed here. We note that our boundary conditions couple to two long-ranged scaling fields- the magnetization field which falls off with the a power of $-1/4$ and the energy density which falls off with a power of $-2$, both of which must be present in membranes or any other system near an Ising critical point.
It is interesting to compare this composition mediated force to other forces that could act between membrane bound proteins. Electrostatic interactions are screened over around $1nm$ in the cellular environment, making them essentially a contact interaction from the perspective of the cell. There is an analogous shape fluctuation mediated Casimir force that falls off like $d^{-6}$~\cite{Yolcu11,Yolcu12}, and is therefore also very short ranged. Membrane curvature can also mediate forces with a leading attractive term that falls off like $d^{-2}$ and a leading repulsive term that falls off like $d^{-4}$. Although they decay with a much larger power than the critical Casimir forces described above, curvature mediated potentials depend on elastic constants and are not bound to be of order $k_BT$ allowing them to become quite large at shorter distances. Using typical values~\cite{Reynwar07} the potentials are comparable at lengths $\sim 5-10nm$ to the composition mediated potential we find here~\cite{Dommersnes99}. There are numerous examples of biology using these relatively short ranged but many $k_BT$ potentials for coordinating energetically expensive and highly irreversible events like vesiculation~\cite{Reynwar07}. We propose that critical Casimir forces could mediate long ranged and reversible interactions useful for regulating a protein's binding partners. More generally, this work demonstrates that the hypothesis of criticality enables a quantitative understanding of the broad range of phenomena frequently associated with `raft' heterogeneity in cell membrane.
\begin{figure}
\includegraphics[scale=0.5]{fig3-1-2-12.eps}
\caption{\label{fig:vsT} We compare our critical results with potentials obtained from Monte-Carlo simulations away from the critical point along the temperature axis. As can be seen, the potentials are longest ranged at the critical point. The repulsive interaction is also steepest at the critical point, though the attractive one has a larger force at short distances slightly away from the critical point. }
\end{figure}
This work was supported by NIH R00GM087810, NSF DMR 1005479, and NIH T32GM008267. We thank Paul Ginsparg, Chris Henley, Markus Deserno, Cem Yolcu, Barbara Baird and David Holowka for useful discussions.
\bibliographystyle{prsty}
|
{
"timestamp": "2012-08-06T02:03:04",
"yymm": "1203",
"arxiv_id": "1203.2199",
"language": "en",
"url": "https://arxiv.org/abs/1203.2199"
}
|
\section*{Results}
\paragraph{Formulation of the problem and intuitive analysis}
We consider the hamiltonian of an integrable quantum system with an integrability-breaking perturbation $\hat{V}$: $\hat{H} = H_{0}(\hat{\vec{n}}) + \hat{V}$. Here the integrable part of the hamiltonian is a function $H_{0}$ of all the members of the complete set of the integrals of motion, where the latter are labeled as $\hat{\vec{n}} = \left(\hat{n}_{1},\,\hat{n}_{2},\,\ldots,\,\hat{n}_{d} \right)$; $d$ is the number of the degrees of freedom. The eigenstates and eigenvalues of $H_{0}(\hat{\vec{n}})$ are $|\vec{n}\rangle$ and $E_{\vec{n}}$, respectively; the corresponding quantities for $\hat{H}$ are $|\alpha\rangle$ and $E_{\alpha}$, as before. We assume that both the $E_{\vec{n}}$ and the $E_{\alpha}$ spectra are free of degeneracies.
The non-degenerate spectrum is a generic property of an integrable hamiltonian unless it has mutually non-commuting integrals of motion or commensurate frequencies \cite{landau1958,yuzbashyan2002}. However, because of the absence of level repulsion, the energy levels are often near-degenerate: they are distributed as if by a Poisson process, resulting in the exponential distribution of level spacings \cite{berry1977b}.
Now we imagine that the system is prepared in some initial nonequilibrium state $|\psi_{\mbox{\scriptsize init.}}\rangle \equiv |\psi(t\!=\!0)\rangle$, and then allowed to evolve according to the hamiltonian $\hat{H}$. Following a transient period, the expectation value of a generic observable $\hat{A}$ will, overwhelming majority of the time, fluctuate around the infinite-time average given in equation~(\ref{MAIN_A_relax_1}). Figure~\ref{f:MAIN_A_of_time} gives an example of such a process, in the regime intermediate between fully integrable and fully chaotic, showing a partial retention of the memory of the initial state.
\begin{figure}[h!]
\vspace{-1in}\mbox{}
\begin{center}
\includegraphics[scale=.7]{Olshanii_fig1.eps}
\end{center}
\caption{
\label{f:MAIN_A_of_time}
\textbf{An example of a partially retained memory of the initial state.} Shown are the $x$- and $y$-components of the hopping energy (respectively: $E_{x}$ in solid red, with initially positive value, and $E_{y}$ in dotted blue, with initially negative value), as a function of time, of a two-dimensional $33 \times 33$-site noninteracting Anderson model in the presence of an Aharonov-Bohm (A-B) flux (see the \secMethodsIINumericalIImodelsIIused{} subsection of the Methods section). A fully chaotic systems would obey the equipartition theorem, predicting that the infinite time averages of the two components should be equal, $\inftave{(E_{x}-E_{y})}=0$. However, for intermediate strengths of the integrability-breaking perturbation, a residual deviation from equipartition---strongly correlated with $\init{(E_{x}-E_{y})}$, the initial deviation---remains. The two straight horizontal lines on the right-hand side of the plot are the exact values of $\inftave{E_{x}}$ and $\inftave{E_{y}}$, computed from equation~(\ref{MAIN_A_relax_1}). In terms of the Anderson model parameters introduced in the text following the paragraph containing equation~(\ref{MAIN_central_result}), this system has $\varepsilon = 1.0$, corresponding to the Anderson's disorder parameter of $W/J = .87$. The initial state was represented by an eigenstate of the impurity-free lattice, $ |\psi_{\mbox{\scriptsize init.}} \rangle = |n_{x}\!=\!-9,\,n_{y}\!=\!+3\rangle$, whose energy was $E = -1.46 \, J$, where $J$ is the hopping constant (see the \secMethodsIINumericalIImodelsIIused{} subsection of the Methods section). At zero A-B flux, the ground state is $|n_{x}\!=\!0,\,n_{y}\!=\!0\rangle$. The A-B fluxes $\phi_{x}$ and $\phi_{y}$ through a complete loop along the $x$- and the $y$-directions, respectively, were chosen to be far from approximately commensurate; we had $\phi_{x} = (\varphi/4)\,\phi_{0}$ and $\phi_{y} = (e/10)\,\phi_{0}$, where $\phi_{0}$ is the elementary quantum of magnetic flux, $\varphi = 1.618\ldots$ is the golden ratio, and $e =2.718\ldots $ is the base of the natural logarithm. The apparent correlation between the two energies is due to the conservation of the total energy.
}
\end{figure}
Since we are concentrating on the disappearance of the integrals of motion, we take our observable of interest to be diagonal in the eigenbasis of $H_{0}(\hat{\vec{n}})$, the integrable part of the hamiltonian:
$\langle \vec{n} | \hat{A} | \vec{n}' \rangle = A_{\vec{n}} \delta_{\vec{n}\,\vec{n}'}$.
Let us choose the initial state to be an eigenstate,
$| \vec{n}_{\mbox{\scriptsize init.}} \rangle$, of $H_{0}(\hat{\vec{n}})$.
In the intuitive discussion that follows, it will help to imagine that we chose a state with a ``highly improbable'' (meaning: very different from the microcanonical average) value of the observable of interest $\hat{A}$.
According to equation~(\ref{MAIN_A_relax_1}), in the course of time evolution, the initial state $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$ gets transformed into a superposition, with essentially random phase relationships, of the eigenstates $| \alpha \rangle$ of the perturbed hamiltonian, $\hat{H}$, such that each $| \alpha \rangle$ in the superposition has a substantial overlap with $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$, i.e. a large corresponding occupation number.
Now, since the eigenstates $| \vec{n} \rangle$ of $H_{0}(\hat{\vec{n}})$ (let us call these the ``unperturbed eigenstates'') form a basis, we may expand the eigenstates $| \alpha \rangle$ of the full, perturbed hamiltonian $\hat{H}$ (the ``perturbed eigenstates'') as linear combinations of the unperturbed eigenstates. The stronger the perturbing potential $\hat{V}$, the more different the perturbed eigenstates from the unperturbed ones, and the greater the number of unperturbed eigenstates that must be linearly combined (with appreciable weights) to build any perturbed eigenstate. A standard measure of the latter number is the so-called \emphas{number of principal components}, to be defined momentarily. Among the perturbed eigenstates $| \alpha \rangle$ whose overlap with $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$ is non-negligible, a typical one will be such that when it is expanded in terms of the unperturbed eigenstates, the weight of
the state $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$ is of the order of
$\eta_{\alpha}^{\{\vec{n}\}} \equiv \sum_{\vec{n}} \left| \langle \vec{n} | \alpha \rangle \right|^4$, which is the so-called Inverse Participation Ratio (IPR) \cite{georgeot1997}. The number of principal components mentioned above is then given by $\Npcn\sim 1/\eta_{\alpha}^{\{\vec{n}\}}$; in this case, it is the number of principal unperturbed components.
As we said, we will assume that the matrix elements $V_{\vec{n}_{1}\vec{n}_{2}}$ of the integrability-breaking perturbation $\hat{V}$ obey no selection rules. We will also assume that the equi-energy surface $H_{0}(\vec{n})$ is ``sufficiently irrational,'' which is a concept that needs a bit of explaining: as is well known, energy eigenstates in an integrable system are completely characterized by their quantum numbers (which we can take to be integers) corresponding to all the conserved quantities. In a typical integrable system, for sufficiently high excited states, if one of the quantum numbers is changed by 1 while all the others are held fixed, the change in the energy of the eigenstate will, in general, be large compared to the energy level spacing; that is, there will be many eigenstates whose energies lie in between the energy of the original state and the energy of the state that differs from the original by 1 in a single quantum number. Eigenstates that are neighbors in energy generally have very different values of all quantum numbers. In fact, if one orders the eigenstates by increasing energy and looks at how their quantum numbers change from one eigenstate to the next in the sequence, these changes may appear quite random; if they do, we say that the equi-energy surface is ``sufficiently irrational.'' More precisely: let $\{(\hat{n}_{k})_{\indx}\}$ be the sequence of the values of the $k$th quantum number as one is going from one eigenstate of $H_{0}(\vec{n})$ to the next in the order of increasing energy, labeled by $\indx$. The equi-energy surface is said to be sufficiently irrational if, for every $k$, the sequence $\{(\hat{n}_{k})_{\indx}\}$ passes any simple statistical test for randomness. This property seems to be quite generic for integrable systems with incommensurable frequencies; see Supplementary Figure~\SIfIISUPPIIHzeroIIrandomization{} for an example.
These two assumptions (the absence of selection rules in the perturbing potential $\hat{V}$ and the sufficient irrationality of the equi-energy surface of $H_{0}(\vec{n})$) imply that the presence of the state $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$ as one of the components of the state $| \alpha \rangle$ does not ``bias
the selection'' of the other states $| \vec{n} \rangle$ that enter the expansion of $| \alpha \rangle$. This means that the expansion of $| \alpha \rangle$ appears as if the states (other than $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$) that enter into it were indiscriminately
chosen from a microcanonical shell around $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$. This would make---were it not for the ``systematic'' presence of the initial state in the expansion of $| \alpha \rangle$---the value of $\hat{A}$, on average, equal to the microcanonical average. The state $| \vec{n}_{\mbox{\scriptsize init.}} \rangle$ becomes \textit{the only} component of the state $| \alpha \rangle$ that ``remembers'' the initial value of $\hat{A}$. Since on average $\Npcn\sim 1/\eta$ unperturbed states enter the expansion of a perturbed state---and out of these unperturbed states that enter the expansion, usually exactly one has the initial value for the observable $\hat{A}$---we obtain the estimate that the infinite time average of $\hat{A}$ is the weighted average of the initial value and the microcanonical value of $\hat{A}$, where the weight ratio is $1:\left(\Npcn-1\right)$ in favor of the microcanonical value:
$
\inftave{A}
\approx
\eta_{\alpha}^{\{\vec{n}\}} \init{A}
+
(1 - \eta_{\alpha}^{\{\vec{n}\}}) \MC{A}
$,
where $\init{A}=\langle \vec{n}_{\mbox{\scriptsize init.}} | \hat{A} | \vec{n}_{\mbox{\scriptsize init.}} \rangle$ is the quantum expectation value of $\hat{A}$ with respect to the initial state, and $\MC{A}$ is the microcanonical average, i.e. the average over all the values $\langle \vec{n} | \hat{A} | \vec{n} \rangle$ such that the unperturbed eigenenergies of $| \vec{n} \rangle$ lie in a narrow window centered at the mean energy of the system.
\paragraph{A statistically solvable model.}
Guided by this intuitive reasoning, we have constructed a statistical model that captures the essential physics and can be solved exactly. The model has two principal ingredients. First, the unperturbed hamiltonian $\hat{H}_{0}$ is replaced by an ensemble of hamiltonians, each member of which has the same eigenenergies and eigenstates as $\hat{H}_{0}$, but which eigenstate corresponds to which eigenvalue is chosen at random. It is convenient to think of these random assignments as permuting the eigenvalues, while the eigenstates remain fixed; so each permutation $\sigma$, which sends the $N$-tuple $(1,\, 2,\, \ldots,\,N)$ to $(\sigma(1),\, \sigma(2),\, \ldots,\, \sigma(N))$, says that the $j$th largest eigenvalue corresponds to the eigenstate that in the original hamiltonian corresponded to the $\sigma^{-1}(j)$th largest eigenvalue. Second, the perturbation $\hat{V}$ is replaced by an ensemble of perturbations; the distribution of the perturbations (viewed as matrix elements between various eigenstates) is assumed to be invariant under the permutations $\sigma$ of the eigenstates. We should note that we are permuting only the unperturbed eigenstates whose unperturbed eigenvalues lie within a microcanonical energy window ${\cal W}_{\mbox{\scriptsize MC}}(E,\,\Delta E)$ centered
at the mean energy $E$ of the initial state. Also, the perturbations $\hat{V}$ are truncated so that they couple only the eigenstates within that energy window. The following result then follows (see the Supplementary Methods for the details of the derivation):
\begin{eqnarray}
\Big\langle \inftave{A} \Big\rangle_{\sigma,\,\hat{V}}
=
\frac{
\left(N-\Npca\right) \init{A}
+
N\left(\Npca-1\right) \MC{A}
}{\Npca\left(N-1\right)}
\,\,
\nonumber
\\
{}
\label{MAIN_almost_the_central_result}
\end{eqnarray}
Here $N$ gives the number of the states in the microcanonical window ${\cal W}_{\mbox{\scriptsize MC}}(E,\,\Delta E)$, and
$\Big\langle \ldots \Big\rangle_{\sigma,\,\hat{V}}$ stands for an average over the uniformly distributed permutations $\sigma$ and the permutation-invariant-distributed perturbations $\hat{V}$. It will be convenient to introduce, for any quantity $B_{\vec{n}}$ that depends on the values of the quantum numbers $\vec{n}$, the \emphb{microcanonical average} as
\[
\Thrm[B_{\vec{n}}] \equiv \frac{1}{\NinMC} \sum_{\vec{n}:\,E_{\vec{n}} \in {\cal W}_{\mbox{\scriptsize MC}}(E,\,\Delta E)} B_{\vec{n}}
\]
where $\NinMC$ is the number of unperturbed eigenstates whose eigenvalues are in the microcanonical energy window. Then the typical number of the (interacting) principal components is given by
$
\Npca \equiv 1/{ \Big\langle \etaMC(\sigma,\,\hat{V}) \Big\rangle_{\sigma,\,\hat{V}} }
$
where
$
\etaMC(\sigma,\,\hat{V})
\equiv
\Thrm\left[\etaa(\sigma,\,\hat{V})\right]
$
is the \emphb{microcanonical average of the inverse participation ratio} of the noninteracting eigenstates over the interacting ones
$\left(\mbox{in other words,}\;
\etaa \equiv \sum_{\alpha} |\langle \alpha | \vec{n} \rangle |^4\right)$,
$\init{A}=\langle \vec{n}_{\mbox{\scriptsize init.}} | \hat{A} | \vec{n}_{\mbox{\scriptsize init.}} \rangle$ is the value of $\hat{A}$ in the initial state (it is written as an expectation value even though, given our choices, the initial state is also an eigenstate of $\hat{A}$), and
$
\MC{A} \equiv \Thrm\left[A_{\vec{n}}\right]
$
is the \emphb{microcanonical average for the observable}.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Olshanii_fig2.eps}
\end{center}
\caption
{
\label{f:MAIN_ipr_and_level_statistics}
\smaller\smaller
\textbf{Testing the predictive power of equation~(\ref{MAIN_central_result}).} Our integrable system is a single particle on a two-dimensional lattice with an Aharonov-Bohm flux. Lattice parameters are the same as for Figure~\ref{f:MAIN_A_of_time}. Each row of plots corresponds to a different type of non-integrable perturbation, as follows. \IntroPanlSymb{\bfpanel{a}}\IntroPanlSymb{\bfpanel{d}} A real Gaussian random matrix perturbation acting between the eigenstates of the unperturbed lattice. \IntroPanlSymb{\bfpanel{b}}\IntroPanlSymb{\bfpanel{e}} A single impurity of fixed strength. \IntroPanlSymb{\bfpanel{c}}\IntroPanlSymb{\bfpanel{f}} An Anderson-type disorder. In plots \IntroPanlSymb{\bfpanel{a}}\IntroPanlSymb{\bfpanel{d}} and \IntroPanlSymb{\bfpanel{c}}\IntroPanlSymb{\bfpanel{f}} the data points marked with (red) squares are for the case $\varepsilon=0.25$; with (blue) circles, $\varepsilon=0.5$; and with (green) triangles, $\varepsilon=1$. In plots \IntroPanlSymb{\bfpanel{b}}\IntroPanlSymb{\bfpanel{e}} the (red) squares are also for $\varepsilon=0.25$, while the (green) triangles are for $\varepsilon=2\times 10^{5}$. Plots \bfpanel{a},\bfpanel{b},\bfpanel{c} show the infinite time average, obtained from exact time dynamics, of the ``equipartition measure'' $E_{x}-E_{y}$ versus the value this measure had in the initial state. The straight solid lines are the predictions of equation~(\ref{MAIN_central_result}). Included as insets are the representations of the lattice and its perturbation. In plots \bfpanel{d},\bfpanel{e},\bfpanel{f} we show the level-spacing histograms for the perturbed hamiltonians, which show how close the systems presented in plots \bfpanel{a},\bfpanel{b},\bfpanel{c} are to integrability or to well-developed chaos. The horizontal axis, $\Delta\mbox{e}$, is the level spacing in the unfolded spectrum. The closer the actual distribution to the curve representing the Poissonian level statistics, the more integrable the system; the closer the actual distribution to the curve representing the Gaussian Orthogonal Ensemble (GOE) or {\v S}eba or the Gaussian Unitary Ensemble (GUE), the more chaotic the system. See the main text for details.
}
\end{figure}
What we suggest now is to treat the ``real-world'' hamiltonian $\hat{H}$ as a single instance, $(\sigma = \sigma^{\mbox{\scriptsize real-world}},\,\hat{V}=\hat{V}^{\mbox{\scriptsize real-world}})$, of the ensemble of hamiltonians considered above. In this case, the ensemble average on the left-hand side of equation~(\ref{MAIN_almost_the_central_result}) will serve as the ``best predictor'' for the ``real-world'' value of the infinite-time average of the observable, $\inftave{A}$. Notice that, conversely, one of the constituents on the right-hand side of equation~(\ref{MAIN_almost_the_central_result}) will, in its turn, also have to be estimated by a ``best predictor.'' Namely, while $\init{A}$ and $\MC{A}$ are the same for all $(\sigma,\,\hat{V})$-instances and thus can be extracted from a single instance (which can be the``real-world'' instance), the quantity $\etaMC(\sigma,\,\hat{V})$ (on which $\Npca$ depends) varies from instance to instance and so its averaging over instances has a nontrivial effect. Since, however, we only have access to a single realization of $\hat{H}$, we must use the ``real-world'' value of $\etaMC(\sigma,\,\hat{V})$ as the ``best estimate'' for the ensemble average:
$
\Big\langle \etaMC(\sigma,\,\hat{V}) \Big\rangle_{\sigma,\,\hat{V}}
\approx
\etaMC(\sigma^{\mbox{\scriptsize real-world}},\,\hat{V}^{\mbox{\scriptsize real-world}})
=
\etaMC
\,,
$
where
$
\etaMC = \Thrm\left[\etaa\right]
$
is the ``real-world'' microcanonical average of $\etaMC(\sigma,\,\hat{V})$.
An undesirable feature of the expression in equation~(\ref{MAIN_almost_the_central_result}) is that it depends on the number of states in the microcanonical window, $\NinMC$. However, in the limit where $\NinMC$ greatly exceeds the number of principal components,
$
\NinMC \gg \Npca\,,
$
$\NinMC$ disappears from the expression. We finally obtain the desired relationship between the infinite-time average and the initial value of our observable of interest:
\begin{eqnarray}
\inftave{A}
=
\etaMC \init{A}
+
\left(1-\etaMC\right) \MC{A}
\label{MAIN_central_result}
\end{eqnarray}
This is almost our original estimate based on intuitive arguments, except that instead of the typical value of the IPR of the perturbed over the unperturbed states, $\eta_{\alpha}^{\{\vec{n}\}}$, we have the microcanonical average of the IPR of the unperturbed states over the perturbed states. Note the high degree of universality: the same parameter $\etaMC$ is used regardless of what observable $\hat{A}$ one is interested in (which means---since $\hat{A}$ is diagonal in the unperturbed basis---regardless of which integral of motion of $H_{0}(\hat{\vec{n}})$ one is interested in), or with which initial state one starts from among the unperturbed eigenstates.
\paragraph{Testing the analytical prediction}
The relationships in equations~(\ref{MAIN_almost_the_central_result}) and (\ref{MAIN_central_result}) are our central results; we now test them against exact time dynamics of particular physical systems.
Our integrable system will always be a $33 \times 33$-site two-dimensional lattice. We also assume that both $x$- and $y$-cycles of the lattice are threaded by an Aharonov-Bohm (A-B) solenoid each \cite{cheung1988,uski1998,heinrichs2009}. If the lattice is imagined to cover the surface of a torus, the flux is produced by two solenoids: one toroidal, contained within the torus of the lattice, and one straight, passing through the hole of the torus. The corresponding magnetic field fluxes are assumed to be highly irrational but weak (i.e. of the order of one) multiples of the elementary magnetic flux quantum; they are also presumed to be mutually irrational---see the first subsection of the Methods section, below. Their purpose is to destroy the dihedral symmetries of the square lattice while preserving the conservation of momentum in both $x$- and $y$-directions. This lifts the degeneracies in the unperturbed spectrum and randomizes the sequence of appearance of the quantum numbers (two components of the momentum vector) along the energy axis. We investigated three different types of the integrability-breaking perturbation. The first perturbation (see Figure~\ref{f:MAIN_ipr_and_level_statistics}\mfpanel{a}) completes the lattice to a \emphas{single instance} of a Deformed (\emphas{real}) Gaussian Random Matrix Model \cite{kota2001} that couples the eigenstates of the unperturbed lattice. The second perturbation (Figure~\ref{f:MAIN_ipr_and_level_statistics}\mfpanel{b}) is a single impurity of a fixed strength. This system is a lattice version of \v{S}eba billiards \cite{seba1991}, which are known to lie in between integrable and completely quantum-chaotic systems \cite{seba1991}. The third example (Figure~\ref{f:MAIN_ipr_and_level_statistics}\mfpanel{c}) is a \emphas{single instance} of the conventional Anderson disorder \cite{anderson1958}, with a rectangular distribution of the strength in each of the impurities. (We should mention that ergodicity in the context of an Anderson lattice was also studied in ref.~\citen{stotland2008}.) While the first and the second examples correspond to permutation-invariant perturbations, the third one does not. In particular, in comparison to the first two types of perturbation, in the Anderson case there are much stronger correlations between certain kinds of matrix elements. Namely, for each matrix element, consider the momentum difference between the states that the matrix element connects. If one looks at the set of matrix elements which have the same momentum difference, one finds that from one realization of the perturbation to the next, their values all change by the same factor. The observable of interest is the ``equipartition measure''---the difference between the $x$- and $y$-hopping energies, $E_{x}$ and $E_{y}$---whose value in a state of a thermal equilibrium is always zero, thanks to the $x \leftrightarrow y$ symmetry. The infinite-time average of the ``equipartition measure'' $E_{x}-E_{y}$ is shown as a function of its initial value. As the governing parameter $\varepsilon$ we have chosen the ratio between the root mean square of the off-diagonal matrix elements $V_{\vec{n}\vec{n}'}$ (the same for all the pairs $(\vec{n},\,\vec{n}')$) and the typical energy spacing at the energy of interest: $\varepsilon \equiv V_{0} \rho(E)$, where $V_{0} = \sqrt{\overline{|V_{\vec{n}\vec{n}'}|^2}}$ and $\rho(E)$ is the density of states at the energy $E$.
For each type of perturbation, we time-evolved 201 different initial states, taken to be all the eigenstates of the unperturbed lattice whose eigenvalues came from a representative microcanonical energy window; the middle eigenstate (the 101st one in the order of increasing energy within the window) had the energy of $-1.5\,J$, where $J$ is the hopping constant (see the \secMethodsIINumericalIImodelsIIused{} subsection of the Methods section). A \emphas{single instance} of the corresponding hamiltonian (which was random for the cases \bfpanel{a} and \bfpanel{c}) was used in all three cases. The different initial states have different values for $\init{(E_{x}-E_{y})}$, lying between some minimum and maximum values; we needed to group the initial states into sets with similar $\init{(E_{x}-E_{y})}$-values. Thus we partitioned the interval from the minimum to the maximum $\init{(E_{x}-E_{y})}$-value into subintervals centered at $-3 J$, $-2.5 J$, \ldots $3 J$, each of width $\Delta \init{(E_{x}-E_{y})} = .5 J$. All the initial states whose $\init{(E_{x}-E_{y})}$-values fell into the same subinterval constituted a ``group of initial states with similar $\init{(E_{x}-E_{y})}$-values.'' The points shown in plots \bfpanel{a},\bfpanel{b},\bfpanel{c} correspond to the groups: the $x$-value of a point is the center of the subinterval defining the group, while the $y$-value is the group average of $\inftave{(E_{x}-E_{y})}$. The theoretical curves, produced by equation~(\ref{MAIN_central_result}), also involved an averaging: the value of $\init{(E_{x}-E_{y})}$ which is fed into equation~(\ref{MAIN_central_result}) was, for each group, the group average of $\init{(E_{x}-E_{y})}$ rather than the center of the subinterval; this explains why the theoretical curves are not exactly straight lines. As Figures~\ref{f:MAIN_ipr_and_level_statistics}\mfpanel{a},\mfpanel{b},\mfpanel{c} show, the prediction of equation~(\ref{MAIN_central_result}) agrees very well with the numerical results.
Figures~\ref{f:MAIN_ipr_and_level_statistics}\mfpanel{d},\mfpanel{e},\mfpanel{f} are there to show where on the continuum between integrability and well-developed chaos our various systems lie. The discrete points are the plots of the actual level spacing statistics of our systems, properly unfolded \cite{guhr1998}; for the cases \bfpanel{d} and \bfpanel{f}, we also average over 16 realizations of the perturbation $\hat{V}$. For comparison, we also plot the curves corresponding to the completely integrable and (the appropriate) completely quantum-chaotic systems. The curves labeled ``Poisson'' arise for Poisson level statistics, which is characteristic for integrable systems. The most recognizable feature is the nonzero value at zero spacing, representing the absence of level repulsion in integrable systems. The Gaussian Orthogonal Ensemble (GOE) curve is valid for systems with well-developed quantum chaos in the presence of time-reversal invariance; the \v{S}eba distribution \cite{seba1991} holds for systems with singular perturbations; and the Gaussian Unitary Ensemble (GUE) is used in the cases of quantum chaos without time-reversal invariance \cite{guhr1998}. We see that when \mbox{$\varepsilon=0.25$}, the level-spacing statistics of our systems is intermediate between those of the integrable-like and the appropriate completely quantum-chaotic-like distributions. As $\varepsilon$ increases, the level spacing distributions for the real Gaussian, singular, and Anderson perturbations converge, respectively, to the GOE, \v{S}eba, and GUE predictions. (The reason for the GUE statistics in the Anderson case is the Aharonov-Bohm flux, which breaks the time-reversal invariance \cite{uski1998}. In the case of the first model the statistics remains of a GOE type, since the perturbation matrix elements were artificially fixed to real values.) The expression in equation~(\ref{MAIN_central_result}) is thus confirmed in the full range from the integrable regime all the way to the well-developed quantum chaos.
To relate our predictions to the system parameters, we further connect the inverse participation ratio $\etaMC$---otherwise emprirically irrelevant---to the governing parameter $\varepsilon$. This allows us to trace the integrability-to-chaos transition, i.e. express the memory of the initial state through the strength of the non-integrable perturbation. Figure~\ref{f:MAIN_kam} shows, for a particular group of the initial states, the values of the ``equipartition measure'' $\init{(E_{x}-E_{y})}$ as a function of the parameter $\varepsilon$. The numerical values are drawn from the set used to produce Figure~\ref{f:MAIN_ipr_and_level_statistics}. To obtain the theoretical prediction, we tabulate numerically the inverse participation ratio $\eta_{\alpha_{0}}^{\{\alpha\}}$, averaged over the unperturbed states $|\alpha_{0}\rangle$, for a Deformed (complex) Gaussian Random Matrix Model. For large values of $\varepsilon$, we also use known theoretical results \cite{fyodorov1995_R11580,frahm1995_385} for the so-called strength function, $\overline{\left| \langle \alpha_{0} | \alpha \rangle \right|^2}$, combined with the assumption \cite{flambaum1997_5144} of the Gaussian character of the fluctuation of $\langle \alpha_{0} | \alpha \rangle$:
\begin{eqnarray}
\inftave{A} - \MC{A}
\mathrel{ \mathop \approx_{\varepsilon \gtrsim 1}}
\frac{q}{2\pi^2\varepsilon^2} (\init{A}-\MC{A})
\quad,
\label{MAIN_central_result_eps_gg_1}
\end{eqnarray}
where $q=3$ for a non-integrable perturbation that belongs to the Gaussian Orthogonal class, and $q=2$ in the Gaussian
Unitary case. Note that the Aharonov-Bohm flux present in the example of the Figure~\ref{f:MAIN_kam} implies the latter. Note also that the result in equation~(\ref{MAIN_central_result_eps_gg_1}) for the Gaussian Orthogonal case can be confirmed directly \cite{frahm1995_385}, without the assumption of Gaussianity of $\langle \alpha_{0} | \alpha \rangle$. The strength function for an equidistant spectrum perturbed by a real matrix whose matrix elements have the same magnitude and random signs---the case closely related to a Poisson spectrum perturbed by a real Gaussian matrix---first appears in the classic Wigner's paper \cite{wigner1955_548}. The strength function for the latter system {\it per se} is predicted in ref. \citen{jacquod1995_3501}. (See the \secMethodsIIRelevantIItheoreticalIIresults{} subsection of the Methods section for more details.)
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Olshanii_fig3.eps}
\end{center}
\caption
{
\label{f:MAIN_kam}
\textbf{The memory of the initial state as a function of the strength of the integrability-breaking perturbation.}
For the case of Anderson disorder, we compute, for three similar initial states, $|\psi_{\mbox{\scriptsize init.}} \rangle = |n_{x}\!=\!+7,\,n_{y}\!=\!0,\pm 1\rangle$, the infinite time average of the ``equipartition measure'' $E_{x}-E_{y}$ as a function of the disorder strength $\varepsilon$. The red crosses correspond to the averages over the this group of states. The red solid line corresponds to the formula (\ref{MAIN_central_result}) combined with a numerically tabulated values of the inverse participation ratio $\etaMC$ for the case of the Deformed (complex) Gaussian Random Matrix Model. Finally, the black dotted line is given by the asymptotic analytic prediction in equation~(\ref{MAIN_central_result_eps_gg_1}). The lattice parameters are the same as for Figure~\ref{f:MAIN_A_of_time}.
}
\end{figure}
\section*{Discussion}
We have thus demonstrated that, in the case when the integrability-breaking perturbation obeys no selection rules and when the spectrum of the underlying integrable system is ``sufficiently irrational,'' it is possible to characterize the memory of the initial values of the unperturbed integrals of motion, as the strength of the perturbation is increased, by a simple and universal expression. The expression was verified for three different types of perturbations away from integrability, including a Deformed Gaussian Random Matrix Model, an isolated impurity, and the two-dimensional Anderson model; in each case, the expression works for a full range of perturbation strengths from completely integrable to completely chaotic.
Our predictions should be testable experimentally with techniques that are already available, or nearly so; likely contexts include those of the investigations of Anderson localization with cold gases in optical lattices \cite{billy2008,roati2008}, or those of quantum mirage configurations of a quantum corral \cite{manoharan2000}, modified to a \v{S}eba-type billiard. In general, memory effects should be visible as soon as the discreteness of energy levels is discernible.
Future work must try to come to terms with cases where the partial preservation of the integrals of motion is enhanced via some selection rules obeyed by the perturbing potential. The two most empirically relevant examples of selection rules come from the limitations imposed by the few-site nature of the hopping terms in typical lattice hamiltonians, and by the few-body nature of interactions in many-body systems. Encouragingly, some work relevant to these cases has already been done; ref.~\citen{brown2010} studied the role of the former type of selection rules, while the effects of the latter type on the structure of the eigenstates has been investigated in a number of works; for a review, see ref.~\citen{kota2001} and the references therein, particularly ref.~\citen{flambaum1996_5729}. More generally, one should start systematically increasing the complexity of the topology of the network of transitions.
\section*{Methods}
\paragraph{Numerical models used to verify equation~(\ref{MAIN_central_result}).}
As an example of an unperturbed hamiltonian $\hat{H}_{0}$, we use a $N_{x} \times N_{y}$ ($33 \times 33$ in the numerical examples considered) two-dimensional lattice with periodic boundary conditions and odd $N_{x}$, $N_{y}$:
\begin{eqnarray}
\hat{H}_{0}
&=&
-J \!\!\sum_{j_{x}=-\frac{N_{x}-1}{2}}^{+\frac{N_{x}-1}{2}} \sum_{j_{y}=-\frac{N_{y}-1}{2}}^{+\frac{N_{y}-1}{2}}
\!
\left(
e^{-i \frac{2\pi \Delta n_{x}}{N_{x}} } |(j_{x}+1,\,j_{y})\rangle \langle (j_{x},\,j_{y})|
+
\right.
\notag
\\
&& \qquad
\left.
e^{-i \frac{2\pi \Delta n_{y}}{N_{y}} } |(j_{x},\,j_{y}+1)\rangle \langle (j_{x},\,j_{y})|
+
h.c.
\right) ,
\label{MAIN_Integrable_hamiltonian}
\end{eqnarray}
where $|(j_{x},\,j_{y})\rangle$ are the eigenstates of position. Both $x$- and $y$-hopping constants have the same amplitude $J$. We also assume that both $x$- and $y$-cycles of the lattice are threaded by an Aharonov-Bohm (A-B) solenoid each. This leads to the hopping constants acquiring complex phase factors, with phases $-2\pi\Delta n_{x}/N_{x}$ and $-2\pi\Delta n_{y}/N_{y}$ respectively. Here $\Delta n_{x} \equiv \phi_{x}/\phi_{0}$ ($\Delta n_{y} \equiv \phi_{y}/\phi_{0}$) where $\phi_{x}$ ($\phi_{y}$)
is the magnetic flux through the $x$-cycle ($y$-cycle), $\phi_{0} = 2\pi \hbar c/q$ is the elementary quantum of magnetic flux, $c$ is the speed of light, and $q$ is the electric charge of the lattice particle. The flux is assumed to be very weak, $\Delta n_{x}(\Delta n_{y}) \sim 1$. Note also that after a suitable gauge transformation, the complex hoppings can be replaced by real ones, supplemented by twisted boundary conditions \cite{cheung1988,uski1998,heinrichs2009}. The eigenstates of the unperturbed hamiltonian are plane waves $|(n_{x},\,n_{y})\rangle$,
\begin{eqnarray*}
\langle (j_{x},\,j_{y})|(n_{x},\,n_{y})\rangle
=(N_{x} N_{y})^{-1/2}\exp\left(i \frac{2\pi n_{x}j_{x}}{N_{x}}+i \frac{2\pi n_{y}j_{y}}{N_{y}} \right)\,.
\end{eqnarray*}
The linear momentum quantum numbers $n_{x}$ and $n_{y}$ ($|n_{x}|\leq(N_{x}-1)/2$ and $|n_{y}|\leq(N_{y}-1)/2$) constitute a set of the integrals of motion. Eigenenergies of the unperturbed hamiltonian are
\begin{eqnarray*}
E_{n_{x},\,n_{y}}=-2J \left\{\cos[2\pi(n_{x}+\Delta n_{x})/N_{x}] +
\cos[2\pi(n_{y}+\Delta n_{y})/N_{y}]\right\}.
\end{eqnarray*}
The purpose of introducing the A-B flux is to generate an equi-energy surface $E_{n_{x},\,n_{y}} = E$ that is sufficiently irrational with respect to the lattice of the integer quantum numbers $n_{x}$ and $n_{y}$. To this end, the ``defects'' $\Delta n_{x}$ and $\Delta n_{y}$ must be both irrational and mutually irrational (see the first subsection of the first section of the Supplementary Methods and Supplementary Figure~\SIfIISUPPIIHzeroIIrandomization{}). In our numerical calculations, we used $\Delta n_{x} = \varphi/4$ and $\Delta n_{y} = e/10$, where $\varphi = 1.618\ldots$ is the golden ratio and $e =2.718\ldots $ is the base of the natural logarithm.
As the first example of an integrability-breaking perturbation $\hat{V}$, we considered a member of a Gaussian Orthogonal Ensemble acting between the eigenstates of the unperturbed lattice (Figure~{\ref{f:MAIN_ipr_and_level_statistics}a):
\begin{eqnarray*}
V_{\vec{n}\vec{n}'} = \sqrt{1+\delta_{\vec{n}\vec{n}'}} V_{0} \xi_{\vec{n}\vec{n}'}
\quad,
\end{eqnarray*}
where the $\xi_{\vec{n}\vec{n}'}$ are $N_{x}N_{y} (N_{x}N_{y} + 1)/2$ independent real Gaussian-distributed random variables of unit variance and zero mean, with the remaining $N_{x}N_{y} (N_{x}N_{y} - 1)/2$ matrix elements controlling the hermiticity of $\hat{V}$.
The second example (Figure~{\ref{f:MAIN_ipr_and_level_statistics}b) is a singular perturbation,
\begin{eqnarray*}
V(j_{x},\,j_{y}) = V_{0} N_{x} N_{y} \delta_{(j_{x},\,j_{y}),\,(0,\,0)}
\quad,
\end{eqnarray*}
which produces a matrix with all-equal matrix elements:
\begin{eqnarray*}
V_{\vec{n}\vec{n}'} = V_{0}
\quad.
\end{eqnarray*}
In this case, the eigenstates of the perturbed hamiltonian can be found exactly. They read
\begin{eqnarray*}
|\alpha\rangle=C_{\alpha}^\text{sing}
\sum_{n_{x}=-\frac{N_{x}-1}{2}}^{+\frac{N_{x}-1}{2}} \sum_{n_{y}=-\frac{N_{y}-1}{2}}^{+\frac{N_{y}-1}{2}}
\frac{|(n_{x},\,n_{y})\rangle}{E_{\alpha}-E_{n_{x},\,n_{y}}}
\quad,
\end{eqnarray*}
where the corresponding eigenenergies $E_{\alpha}$ are the solutions of the algebraic equation
\begin{eqnarray*}
\sum_{n_{x}=-\frac{N_{x}-1}{2}}^{+\frac{N_{x}-1}{2}} \sum_{n_{y}=-\frac{N_{y}-1}{2}}^{+\frac{N_{y}-1}{2}}
\frac{1}{E_{\alpha}-E_{n_{x},\,n_{y}}}=\frac{1}{V_{0}}
\quad,
\end{eqnarray*}
and the normalization factor is defined by
\begin{eqnarray*}
\left(C_{\alpha}^\text{sing}\right)^{-2}=
\sum_{n_{x}=-\frac{N_{x}-1}{2}}^{+\frac{N_{x}-1}{2}} \sum_{n_{y}=-\frac{N_{y}-1}{2}}^{+\frac{N_{y}-1}{2}}
\left(E_{\alpha}-E_{n_{x},\,n_{y}}\right)^{-2}
\quad.
\end{eqnarray*}
A solution of this form was at first obtained by \v{S}eba \cite{seba1990}, for a flat continuous billiard. A similar problem involving a two-dimensional lattice with periodic boundary conditions in one direction and a trapping potential in another was recently analyzed \cite{valiente2011}.
Finally, we consider an Anderson-type disorder (Figure~{\ref{f:MAIN_ipr_and_level_statistics}c):
\begin{eqnarray*}
V(j_{x},\,j_{y}) = W \zeta_{j_{x},\,j_{y}}
\quad,
\end{eqnarray*}
where
the
$\zeta_{j_{x},\,j_{y}}$ are $N_{x} N_{y}$ real independent variables, distributed uniformly between $-1$ and $+1$; $W$ is the Anderson disorder parameter. The interaction strength parameter $V_{0}$ used in the previous cases corresponds to the r.m.s.\ of the (generally complex) off-diagonal matrix elements $V_{\vec{n}\vec{n}'}$, $V_{0} \equiv \sqrt{\overline{|V_{\vec{n}\vec{n}'}|^2}} = W/(2 \sqrt{3} \sqrt{N_{x} N_{y}})$.
\paragraph{Relevant results on the IPR in random matrix models.}
To obtain the solid line in Figure~\ref{f:MAIN_kam}, we computed the inverse participation ratio $\eta_{\alpha_{0}}^{\{\alpha\}}$, averaged over all unperturbed states $| \alpha_{0} \rangle$, for a $N \times N$ Deformed Gaussian Random Matrix $\hat{h} = \hat{h}_{0} + \hat{v}$, with $N=2000$. The ``integrable'' part of the matrix, $\hat{h}_{0}$, was represented by a diagonal matrix whose $N$ diagonal entries were given by $N$ independent real random numbers uniformly distributed in the interval $[-(N/\rho)/2,\, +(N/\rho)/2]$. Here, $\rho$ is the density of states. The ``non-integrable part'', $\hat{v}$, was a random matrix drawn from the Gaussian Unitary Ensemble of random Hermitian matrices: (real) diagonal matrix elements and the real and imaginary parts of the off-diagonal matrix elements above the diagonal (those below the diagonal then being fixed by hermiticity) were given by independent, Gaussian-distributed random numbers with standard deviations $\sigma_{\mbox{\scriptsize diag.}} = V_{0}$ and $\sigma_{\mbox{\scriptsize off-diag., Re}} = \sigma_{\mbox{\scriptsize off-diag., Im}} = V_{0}/\sqrt{2}$, respectively. For completeness, the Gaussian Orthogonal case would give $\sigma_{\mbox{\scriptsize diag.}} = \sqrt{2} V_{0}$ and $\sigma_{\mbox{\scriptsize off-diag.}} = V_{0}$. In both cases, $V_{0}$ fixes the mean square of the off-diagonal matrix elements: $\overline{|v_{\alpha_{0}^{},\alpha_{0}'}|^2} = V_{0}^2$.
Equation~(\ref{MAIN_central_result_eps_gg_1}) uses the following asymptotic expression for the inverse participation ratio $\eta$:
\begin{eqnarray}
\eta
\mathrel{ \mathop \approx_{\varepsilon \gtrsim 1}}
\frac{q}{2\pi^2\varepsilon^2}
\quad,
\label{eta_eps_gg_1}
\end{eqnarray}
where $q=3$ for a non-integrable perturbation that belongs to the Gaussian Orthogonal class, and $q=2$ in the Gaussian Unitary case. In refs. \citen{fyodorov1995_R11580,frahm1995_385,wigner1955_548} and \citen{jacquod1995_3501} it was shown that for $\varepsilon \gg 1$, the strength function converges to
\begin{eqnarray}
\overline{\left| \langle \alpha_{0} | \alpha \rangle \right|^2}
\mathrel{ \mathop \approx_{\varepsilon \gtrsim 1}}
\frac{1}{\pi\rho} \frac{\hbar\Gamma/2}{ (E_{\alpha_{0}}-E_{\alpha})^2 + (\hbar\Gamma/2)^2 }
\quad,
\label{BW}
\end{eqnarray}
in both Gaussian Orthogonal and Gaussian Unitary cases. Here, $\Gamma $ is determined by the Fermi Golden Rule: $\Gamma = 2\pi V_{0}^2 \rho /\hbar$.
Also, it is known \cite{flambaum1997_5144} that in the same limit, the individual coefficients $\langle \alpha_{0} | \alpha \rangle$ behave as independent Gaussian random variables, with zero mean and with a standard deviation governed by the expression (\ref{BW}). Recall that the fourth moment of a Gaussian distribution of a variable $\xi$ is related to the second one as \cite{Sayed2008} $\overline{|\xi|^4} = q \left(\overline{|\xi|^2}\right)^2$. Also, it is important that when $\varepsilon \gg 1$, the energy width $\Gamma$ in equation~(\ref{BW}) contains many levels, and thus the sum in the definition of the inverse participation ratio, $\eta_{\alpha_{0}}^{\{\alpha\}} \equiv \sum_{\alpha} |\langle \alpha | \alpha_{0} \rangle |^4$, can be replaced by an integral. The asymptotic formula (\ref{eta_eps_gg_1}) for the inverse participation ratio immediately follows.
\paragraph{\textit{Ab initio} computations.} All numerical time evolution was computed from full exact diagonalization of the hamiltonians using the \textit{Mathematica} computer package.
\begin{addendum}
\item[\textsf{Acknowledgements}] \mbox{}\\We are grateful to F. Werner and D. Cohen for enlightening discussions on the subject.
Supported by the Office of Naval Research grants N00014-09-1-0502 (M.O. and
V.D.) and N00014-09-1-0966 (M.R.), and the National Science Foundation grants PHY-1019197 (M.O. and V.D.) and PHY-0902906 (K.J.).
\end{addendum}
|
{
"timestamp": "2012-03-12T01:00:30",
"yymm": "1203",
"arxiv_id": "1203.1972",
"language": "en",
"url": "https://arxiv.org/abs/1203.1972"
}
|
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\section{Introduction}
Quantum walks have proven to be a fruitful model to study the dynamics of particles on lattices or more general graphs. The effects of exchange statistics on multiparticle quantum walks has been studied for bosons and fermions with a notable effect of statistics. For example it has been shown that quantum walks with non-interacting bosons give rise to effective interactions which lead to Bose-Einstein condensation in graphs with spatial dimension $d<2$ \cite{Vezzani}. Quantum walks with pairs of bosons or fermions have been studied in the context of the graph isomorphism problem by relating the Green's function of the evolution to the spectrum of the graph. It was shown that these models have much more computational power when the particles are allowed to interact because they can distinguish non-isomorphic graphs that noninteracting particles cannot \cite{Coppersmith}. Recently using integrated photonics there has been experimental realization of two particle quantum walks which simulated bosonic, fermionic, and semionic (a type of Abelian anyon) statistics \cite{Sansoni}.
A natural extension of multiparticle quantum walks is to consider anyons which are particles that exist in two dimensions and have richer exchange statistics than bosons or fermions. In the case of Abelian anyons, exchanges, or braids, yield some model dependent phase $e^{i\phi}$ while for non-Abelian anyons there is a large Hilbert space of fusion degrees of freedom of the multi particle system and a braid performs a unitary transformation on this vector space. A quasi-1-dimensional model for anyonic quantum walks was introduced in Ref. \cite{Wang2010} where it was shown that Abelian anyons have quadratic variance just like standard quantum walks with trivial statistics but that non-Abelian anyons create entanglement during the walk purely due to braiding that acts to decohere the spatial degree of freedom. In Ref. \cite{isingaqw} the quantum walk on a quasi-1-dimensional ladder was studied in detail for one type of non-Abelian anyon known as the Ising model anyon. This anyon corresponds to a spin$-1/2$ irrep. of $SU(2)_{k=2}$ Chern-Simons theory, and the variance of the probability distribution of the walker was
proven to be asymptotically linear in time with coefficient one, corresponding to the classical random walk behaviour. That particular case points out that there is a fundamental difference between the dynamics of Abelian vs. non-Abelian anyons.
In this work we explore these differences for more general anyon models,
specifically for spin$-1/2$ irreps of all values of the index $k$ in $SU(2)_k$ Chern-Simons theory.
Numerical results show that for small number of time steps, the position distribution approaches the quantum walk distribution as index $k$ grows \cite{Wang2010}, and it was conjectured that the distribution looks classical
when $k\ll t$. The problem with the fully coherent anyonic quantum walk is that to evaluate the probability
distribution for $t$ iterations of the discrete evolution operator, the number of paths which contribute
to the walk grows exponentially with $t$. It turns out that evaluating each path is equivalent to computing the Jones polynomial for the link corresponding to the closed world line of that path, a computation which is itself exponentially hard (except for the cases $k=1,2,4$ \cite{Jaeger}) in the number of time steps $t$. To simplify the calculations for general $k$, we adopt here a special quantum walk protocol
which introduces loss of memory to the coin and fusion degrees of freedom. Such a protocol
allows for calculating the walk distribution for a relatively large number of time steps and to use certain approximations
to diagonalize the walk operator and obtain analytical results.
\section{Anyonic $V^2$ quantum walk}
The anyonic quantum walk is a dynamical model for a single anyon (mobile or ``walker" anyon) moving
in a lattice of fixed anyons (stationary ``background" anyons). The mobile anyon has an extra
degree of freedom called the \emph{coin} which is coupled to the direction of its movement.
Here we consider a quasi-1-dimensional lattice of background anyons where the walker has only
two directions of movement, left or right, and the coin is thus two-dimensional. Although the anyons
are arranged on a line, they exist on a two-dimensional manifold such that the mobile anyon can
pass them from above or below without coming into contact with the stationary anyons. This dynamics is accomodated by a walk on a ladder where the top(bottom) leg corresponds to mode $\ket{0}$$(\ket{1})$ of the coin, the sites correspond to the rungs and are labelled by an integer $s$, the stationary anyons are pinned in the islands of the ladder (see Fig. \ref{fig:anyonwalk}).
\begin{figure}[h]
\includegraphics[width=.6\textwidth]{aqwladder}
\caption{The anyonic quantum walk on a ladder of point contacts. The red oval is the walker anyon which
moves along the black lines, and the grey ovals are the stationary anyons which occupy the islands between the point contacts. The anyons are initialized in the state $\ket{\Phi_0}$ meaning they are created out of the vacuum in pairs as indicated by the ovals joined by strings, and half of each pair is outside the ladder not participating in the walk.
When on the upper(lower) edge the walker always to the left(right).
The occupation sites of the mobile anyon are labelled by the positions $s$ of the
point contacts where the mobile anyon can tunnel. The tunneling matrix between the legs at each rung corresponds to the coin shuffling operator $U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{F}}$.
Each lattice shift corresponds to a braid between the worldlines of the
walker anyon and a stationary anyon, represented by the generator $b_s$.}
\label{fig:anyonwalk}
\end{figure}
The arrangement in Fig. \ref{fig:anyonwalk} is a crude model of quasiparticle dynamics in
Fractional Quantum Hall samples which support anyonic quasiparticle excitations
\cite{Willett,Bonderson}. The edge modes
(horizontal in Fig. \ref{fig:anyonwalk}) carry mobile quasiparticles with a current direction determined by an externally applied voltage, and bulk quasiparticles
can be pinned at antidots within the islands.
The upper and lower edge modes are mixed by introducing point contacts controlled by front gates between the islands,
allowing tunneling from edge to edge. The coin is thus encoded as propagation to left or right
on the upper or lower edges respectively, and the scattering matrix that describes the tunneling
between the edges represents the quantum coin flip operator. The model assumes a hardcore
approximation for the anyons, such that they are always far apart, no polarization to
fusion channels occurs and the anyons interact only non-locally via their braiding statistics.
The model is also chiral: the particles on the upper edge are only allowed to move left
and on the lower edge only to the right, and all braids are therefore anti-clockwise.
When the walker shifts
between lattice sites, the wave function is multiplied by the braid generator
that corresponds to braiding the walker and the stationary anyon. The total Hilbert space
is the tensor product space of the spatial sites, fusion degrees of freedom and the coin,
$\mathcal{H}=\mathcal{H}_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{space}\otimes\mathcal{H}_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{fusion}\otimes\mathcal{H}_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{coin}$.
The position space is spanned by $\big\{\ket{s}\big\}_{s=0}^{N-1}$, the fusion space accommodates $2N$
anyons ($N-1$ stationary anyons, the walker anyon and their antiparticle pairs) with total charge
vacuum, and the coin space is spanned by $\big\{\ket{0},\ket{1}\big\}$. We assume periodic boundary
conditions, and choose the number of sites to be always larger than the length of the walk,
$N\geq4t+1$. The walker starts initially from the localized state $\ket{s_0}\bra{s_0}$. For reasons described below, rather than fully coherently evolving the system over many steps, the walk evolves coherently for two cycles followed by tracing over fusion and coin degrees of freedom. Afterward the fusion state is reset to the vacuum pair state $\ket{\Phi_0}\bra{\Phi_0}$, the coin state to $\ket{c_0}\bra{c_0}$, and the dynamics is iterated:
\begin{equation} \label{eqn:aqwevolution1}
\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}(t+1)=\mathrm{Tr}_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{f,c}}\big(\big(U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{F}}\big)^2\:
\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}(t)\otimes\ket{\Phi_0}\bra{\Phi_0}\otimes\ket{c_0}\bra{c_0}\:
\big((U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{F}})^2\big)^\dag\big)
\end{equation}
\begin{equation}
U_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{F}=I_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{space}\otimes I_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{fusion}\otimes H
\end{equation}
\begin{equation}
H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}
\end{equation}
\begin{equation} \label{eqn:aqwevolution4}
U_\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}=\sum\limits_{s=0}^{N-1}\Big(S_{s+1}^-\otimes b_{s}\otimes P_0
+S_s^+\otimes b_{s}\otimes P_1\Big)
\end{equation}
when the Hadamard matrix $H$ is chosen as the coin flip operator,
$S_s^-=\ket{s-1}\bra{s}$ and $S_s^+=\ket{s+1}\bra{s}$ are the shift operators and
$P_0=\ket{0}\bra{0},\;P_1=\ket{1}\bra{1}$ are projectors to the coin states.
The braid generators $b_s$ are the matrices associated with braiding the anyons
$s$ and $s+1$ in anticlockwise direction. Due to the periodic boundary conditions,
there is also a braid generator $b_{N-1}$ associated
with braiding the anyons 1 and $N$ which can be expressed in terms of the other generators%
\footnote{This generator can be explicitly written as
$b_{N-1}b_{N-2}\ldots b_2b_1b_2^\dag b_3^\dag\ldots b_{N-2}^\dag b_{N-1}^\dag$},
but we choose the system size so that the walker never crosses the boundary and this
generator is never applied. A string of braid generators,
$B=\prod_{s_j} b_{s_j}$ is called a \emph{braid word}. The braid matrices are determined by the anyon model
in question. In a physical setting, the properties of the underlying material supporting
the anyonic excitations determine the anyon model.
We consider a general class of non-Abelian
anyons corresponding to spin$-1/2$ irreps of the quantum group SU(2)$_k$ which is a group with a deformed version of the
SU(2) algebra. The fusion rules of the SU(2)$_k$ model satisfy the triangle inequality and integer sum condition as in SU(2):
\[ j_1\times j_2=\sum\limits_{j=|j_1-j_2|}^{j_1+j_2}j \]
but
with two restrictions on the total spin charge $j$:
\[ j\leq k/2;\quad j_1+j_2+j\leq k \]
where $j_1$ and $j_2$ are two individual charges and the parameter $k$ is the level of the theory.
The Hilbert space of SU(2)$_k$ anyons grows as $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{dim}(\mathcal{H}_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{fusion}})\propto d^N$
where $d=2\cos(\frac{\pi}{k+2})$ is the quantum dimension. Since the fusion space grows exponentially in the total number of anyons, if one is keeping track of correlations of the walker with the other anyons, the number of degrees of freedom to
keep track of grows very fast with the number of time steps and the calculations become
inefficient. To calculate the probability distribution of the walker for a given time
step, only the calculation of the trace over the fusion degrees of freedom is necessary.
Fortunately, there exists a connection between the trace over the representations
of the braid group $\bra{\Phi_0}B\ket{\Phi_0}$ and link invariants from knot theory
that simplifies calculations of the trace. The trace over the fusion degrees of
freedom can be understood in two equivalent ways: the usual matrix trace over
the bracket representation of the density matrix of the fusion state,
or the diagrammatic \emph{quantum trace} which is defined by taking the fusion diagram
representation of the state and joining the open ends of the incoming and outgoing charge indices \cite{bondersonsthesis}.
The choice of which indices are joined together determines a tracing scheme for the
braid word, and the tracing corresponds to a closure of a braid in knot theory,
such that the closed braid forms a link. The usual tracing schemes are \emph{plat tracing},
where the neighbouring charges are fused such that their total charge is vacuum,
and \emph{Markov tracing}, where each charge is connected to itself before and after time
evolution.
We choose the initial state of the anyons $\ket{\Phi_0}\bra{\Phi_0}$ such that the tracing corresponds to Markov tracing.
The anyons are initially created from vacuum pairs, after which
one member of each pair is dragged out of the system. In a braid presentation, this corresponds
to having nearest-neighbour pairs with total charge vacuum, followed by a braid word which
moves all left members of the pairs to the left.
The total number of background anyons in an $N$-site walk is $N-1$, but the total Hilbert
space accommodates $2N$ anyons (background anyons, walker anyon, and their antiparticle
pairs). Only half of the anyons participate in the quantum walk and after the two-step evolution
all anyons are fused back together with their antiparticle pairs, see Fig. \ref{fig:aqwlink}.
In this scheme, the expectation value $\bra{\Phi_0}B\ket{\Phi_0}$ corresponds diagrammatically
to the Markov trace over the braid word $B$.
\begin{figure}[h]
\includegraphics[width=.6\textwidth]{aqwlink2}
\caption{A link generated by the anyon worldlines in a quantum walk.
Particle-antiparticle pairs are initially created from the vacuum state and one member of each pair
is dragged out of the system. The walker anyon braids with the background anyons for two steps,
after which each particle is fused back with their antiparticle pairs. Such a protocol implies
that the diagrammatic tracing scheme is Markov.}
\label{fig:aqwlink}
\end{figure}
The virtue of using the state $\ket{\Phi_0}$ as the initial fusion state is that
the expectation value $\bra{\Phi_0}B\ket{\Phi_0}$ can now be expressed in terms of the link
invariant called the Kauffman bracket which is a Laurent polynomial in the variable $A$ \cite{kauffman}, denoted $\big\langle L\big\rangle(A)$,
where $L$ is a link, or a collection of tangled loops. This is written formally as
\begin{equation} \label{eqn:expectkauffman}
\bra{\Phi_0}B\ket{\Phi_0}=\frac{\big\langle L\big\rangle(A)}{d^{n-1}}
\end{equation}
where $L=(B^{{\rm Markov}})$ is the link obtained from the braid word $B$ via closure by the initial state $\ket{\Phi_0}$,
$d$ is the quantum dimension of the anyons, $n$ is the number of distinct components (strands) in the link $L$,
and the value of the parameter for SU(2)$_k$ anyons is $A=ie^{-i\pi /2(k+2)}$.
The calculation of the trace over the fusion states thus reduces to evaluation of the Kauffman
bracket of the link that is drawn when the walker anyon moves on a particular path. The evaluation of the
Kauffman bracket is not necessarily more efficient than calculating the matrix trace of the braid
representations, but if one restricts to a model where the walker evolves coherently only for a
finite amount of time steps, it turns out that for arbitrary long walks only a fixed number of links needs to be evaluated
and the quantum walk distribution can be calculated exactly for fairly large number of time steps.
In the usual quantum walk protocol, the quantum speedup occurs because the position of the walker
becomes entangled with the coin. The system evolves coherently and the correlations between the position states
and coin states preserve the memory in the system, hence the system dynamics is highly non-Markovian.
If the walk is subject to decoherence, the correlations between the position and the coin become
degraded and some of the memory in the system is lost. The loss of memory can be modeled using the
so called ``$V^n$ model" \cite{vkmodel}, where the state of the coin is erased and reset on every $n$th
step, for example by doing a measurement on the coin and preparing it over again in the initial state.
After resetting the coin, the system is in a product state and the correlations between the position
and coin are lost. Remarkably even in the $V^2$ model there is enough coherence left in the system to provide for quadratic speed up over the classical random walk. Specifically, it was shown in Ref. \cite{vkasymp} that the variance asymptotically scales like $\sigma(t)^2=K_2 t^2+K_3 t$ for some constants $K_2,K_3$.
Here the dynamics of our anyonic walk, Eq. \ref{eqn:aqwevolution1}, is the $V^2$ model where $V=U_SU_F$. We write the dynamics as a super operator on the density operator $\rho_s(t)$ for the spatial degrees of freedom of the walker:
\[
\begin{array}{rcl}
\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t+1)&=&\mathcal{E}(\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t))\\\\
&=&\sum\limits_{f,c}\Big[\big(I\otimes\bra{f}\otimes\bra{c}\big)\big(U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi S}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi F}\big)^2\Big]\;
\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t)\otimes\ket{\Phi_0}\bra{\Phi_0}\otimes\ket{c_0}\bra{c_0}\;
\Big[\big((U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi S}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi F})^2\big)^\dag\big(I\otimes\ket{f}\otimes\ket{c}\big)\Big]\\\\
&=&\sum\limits_{f,c}\Big[\big(I\otimes\bra{f}\otimes\bra{c}\big)\big(U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi S}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi F}\big)^2
\big(I\otimes\ket{\Psi_0}\otimes\ket{c_0}\big)\Big]\; \rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t)\\\\
&&\Big[\big(I\otimes\bra{\Phi_0}\otimes\bra{c_0}\big)\big((U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi S}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi F})^2\big)^\dag
\big(I\otimes\ket{f}\otimes\ket{c}\big)\Big]\\\\
&\equiv&\sum\limits_{f,c}E_{fc}\;\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t)\;E_{fc}^\dagger
\end{array}
\]
where the Kraus generators have been defined as
\begin{equation} \label{eqn:krausgens}
E_{fc}=\big(I\otimes\bra{f}\otimes\bra{c}\big)\; \big(U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi S}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi F}\big)^2\;
\big(I\otimes\ket{\Psi_0}\otimes\ket{c_0}\big).
\end{equation}
Using equations (\ref{eqn:aqwevolution1})--(\ref{eqn:aqwevolution4}) we find for the double step operator
\begin{equation} \label{eqn:doublestep}
\begin{array}{rcl}
(U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}U_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{F}})^2&=&\sum\limits_{s=0}^{N-1}\Big(\ket{s-2}\bra{s}\otimes b_{s-2}b_{s-1}\otimes P_0HP_0H
\;+\;\ket{s}\bra{s}\otimes b_{s}^2\otimes P_0HP_1H\\\\
&&+\;\ket{s}\bra{s}\otimes b_{s-1}^2\otimes P_1HP_0H\;+\;\ket{s+2}\bra{s}\otimes b_{s+1}b_{s}\otimes P_1HP_1H\Big).
\end{array}
\end{equation}
It is clear from this equation
that the double step operator shifts the coefficients of the density matrix by two rows or columns, or not at all.
By inserting this expression to the superoperator and using the completeness of the fusion and coin bases,
$\sum\limits_f\bra{f}b_s\ket{\Phi_0}\bra{\Phi_0}b_{s'}^\dagger\ket{f}=\bra{\Phi_0}b_{s'}^\dagger b_s\ket{\Phi_0}$
and $\sum\limits_c\bra{c}P_aHP_bH\ket{c_0}\bra{c_0}H^\dagger P_{a'}H^\dagger P_{b'}\ket{c}
=\delta_{a,b'}\bra{c_0}H^\dagger P_{a'}H^\dagger P_aHP_bH\ket{c_0}$, the action of the superoperator on a general element
$\ket{s}\bra{s'}$ of the spatial density matrix can be written as a sum of 7 terms:
\begin{equation} \label{eqn:superoperator}
\begin{array}{rcl}
\mathcal{E}(\ket{s}\bra{s'}) &=&\frac{1}{4}\Big[\ket{s-2} \bra{s^{\prime }-2}\;
\bra{\Phi_0}b_{s^{\prime }-1}^{\dagger}b_{s^{\prime }-2}^{\dagger }b_{s-2}b_{s-1}\ket{\Phi_0}\;
+\;\ket{s-2}\bra{s^{\prime }}\; \bra{\Phi_0}b_{s^{\prime }}^{\dagger 2}b_{s-2}b_{s-1}\ket{\Phi_0} \\\\
&&+\;\ket{s}\bra{s^{\prime }-2}\; \bra{\Phi_0}b_{s^{\prime }-1}^{\dagger }b_{s^{\prime }-2}^{\dagger}b_{s}^{2}\ket{\Phi_0}
+\;\ket{s}\bra{s^{\prime}}\; \Big(\bra{\Phi_0}b_{s^{\prime }}^{\dagger 2}b_{s}^{2}\ket{\Phi_0}\;
+\; \bra{\Phi_0}b_{s^{\prime }-1}^{\dagger 2}b_{s-1}^{2}\ket{\Phi_0}\Big) \\\\
&&-\;\ket{s}\bra{s^{\prime }+2}\; \bra{\Phi_0}b_{s'}^{\dagger}b_{s^{\prime }+1}^{\dagger}b_{s-1}^{2}\ket{\Phi_0}\;
-\;\ket{s+2}\bra{s^{\prime }}\; \bra{\Phi_0}b_{s^{\prime }-1}^{\dagger2}b_{s+1}b_{s}\ket{\Phi_0} \\\\
&&-\;\ket{s+2}\bra{s^{\prime }+2}\; \bra{\Phi_0}b_{s^{\prime }}^{\dagger }b_{s^{\prime }+1}^{\dagger}b_{s+1}b_{s}\ket{\Phi_0} \Big].
\end{array}
\end{equation}
where we have chosen $\ket{c_0}=\ket{0}$ such that $\bra{c_0}H^\dagger P_{a'}H^\dagger P_aHP_bH\ket{c_0}=\pm\frac{1}{4}$.
Equation (\ref{eqn:superoperator}) shows that for any element of the initial spatial density matrix, there are only
seven nonzero elements of the superoperator $\mathcal{E}$. Writing the density matrix in a vectorized form,
the superoperator can be written as a matrix: $\vec{\rho}\,'=\sum_{f,c}\big(E_{fc}\otimes E_{fc}^\dag\big) \vec{\rho}$,
where the superoperator matrix $\sum_{f,c}\big(E_{fc}\otimes E_{fc}^\dag\big)$ is a band matrix with seven diagonal
bands, and the rest of the elements are zero. Starting from the initial position density matrix
$\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(0)=\ket{s_0}\bra{s_0}$, the position density matrix after $t$ time steps is obtained by applying
the superoperator $t$ times:
\begin{equation} \label{eqn:spatialdm}
\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(t)=\mathcal{E}^t\big(\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s}}(0)\big).
\end{equation}
The elements of the superoperator are given by the product of the coin term
$\bra{c_0}H^\dagger P_{a'}H^\dagger P_aHP_bH\ket{c_0}=\pm\frac{1}{4}$ and the fusion term
$\bra{\Phi_0}B(s,s')\ket{\Phi_0}$. The probability distribution at time step $t$
is obtained by starting from an initially localized state $\ket{s_0}\bra{s_0}$ and applying the superoperator $t$ times.
The site probabilities are then given by the diagonal elements of the spatial density matrix. We have calculated
the variance, $\sigma(t)^2=\langle s^2\rangle-\langle s\rangle^2$, of the $V^2$ walk for various choices of the
level $k$ up to $t=100$ iterations of the superoperator, corresponding to 200 quantum walk steps.
One interesting feature of the $V^2$ anyonic walk is that the effect of braiding statistics is trivial for
Abelian anyons. If all the stationary anyons are of the same type, the walker always picks the same phase $e^{i\phi/2}$ when
braiding with them, such that
\begin{equation}
\bra{\Phi_0}B\ket{\Phi_0}=e^{-i\phi/2}e^{-i\phi/2}e^{i\phi/2}e^{i\phi/2}=1.
\end{equation}
This is similar to the fully coherent anyonic quantum walk, where the wave function always picks the phase $e^{i\phi t/2}$
during the forward time evolution and phase $e^{-i\phi t/2}$ during backward time evolution, such that the
overall effect is trivial. Another way to think about the Abelian walk is to look at the case $k=1$ in Chern-Simons theory.
There are only two charges $\{0,\frac{1}{2}\}$ in this model, the fusion channels are unambiguous:
$j_1\times j_2=j$, and the quantum dimension is $d=1$, so this model is Abelian. Substituting $A=ie^{-i\pi/6}$ to the values
of the brackets in Table \ref{table:expectvalues} in Appendix \ref{sec:brackets} gives $\bra{\Phi_0}B(s,s')\ket{\Phi_0}=1$ for all
brackets, confirming that the Abelian walk gives the same dynamics as the original $V^2$ quantum walk. The variance
of the $V^2$ walk for Abelian anyons is plotted in Fig. \ref{fig:variance} a). It shows the known fact \cite{vkmodel} that the
original $V^2$ walk propagates slower than the fully coherent quantum walk, but qualitatively the behaviour is still the same:
the variance depends quadratically on time. The best fit for the variance of the $k=1$ walk is given by $\sigma^2=0.125\,t^2+0.75\,t$.
\begin{figure}[h]
a)
\includegraphics[width=.45\textwidth]{variances1}
\quad
b)
\includegraphics[width=.45\textwidth]{variances2}
\caption{The variance $\sigma^2$ of the two-step walk as a function of time $t$ for various anyon models indexed by $k$. The scaling of the sites and time steps is chosen such that one iteration
of the superoperator $\mathcal{E}$ corresponds to two iterations of the quantum walk time evolution operator, and
at each iteration of $\mathcal{E}$ the walker shifts twice. a) Case $k=1$ corresponds to the Abelian walk, which also
coincides with the original $V^2$ quantum walk. Plotted for
comparison are the variance of the classical random walk (RW) and the ordinary quantum walk (QW) with the same initial
coin state and coin flip matrix. b) Non-Abelian SU(2)$_k$ models zoomed to the last 100 time steps. Also plotted is
the linear curve $\sigma^2=t$ which corresponds to the classical random walk.}
\label{fig:variance}
\end{figure}
In the non-Abelian case, the construction of the superoperator $\mathcal{E}$ requires evaluation of all eight expectation values
in Eq. (\ref{eqn:superoperator}) for all $s$ and $s'$. The evaluation can be done by using the relation between
the expectation values and the Kauffman bracket in Eq. (\ref{eqn:expectkauffman}). Each braid word consists of
four braid generators, so that each planar representation of the corresponding link has four crossings. A very convenient
feature of the link diagrams is that if $s$ and $s'$ are far apart, the forward and backward braid words act
on disjoint sets of strands, and the Kauffman brackets of all disjoint braids are equal
(see Fig. \ref{fig:links} in Appendix \ref{sec:brackets}). Thus, the calculation
of each expectation value involves only calculating the disjoint case and a small number of cases where $s'$ is
close to $s$. The values of the expectation values are tabulated in Appendix \ref{sec:brackets} for all unique
choices of $s'-s$.
The variances for various choices of $k$ are plotted in Fig. \ref{fig:variance} b). We observe that for $k=2$, the variance follows
exactly the linear line $\sigma^2=t$. It is interesting to note that this behaviour is exactly the same as was
shown for the fully coherent walk with $k=2$ \cite{isingaqw}. For the rest of the values of $k$, there is surprisingly small
difference in the variance, with the slope of the variance slightly increasing as a function of $k$. For $k=3$ and
$k=4$ (not shown in the figure) we find that the slope of the variance is slightly smaller than 1, $k=3$ having the
smallest slope (best fit 0.9877). The highest variance
is obtained when $k\rightarrow\infty$, which corresponds to choice of the parameters $A=i$ and $d=2$, with slope 1.0665.
It should be noted that in the fully coherent anyonic quantum walk model, taking the limit $k\rightarrow\infty$ leads
to the conventional SU(2) algebra, and the braiding generators are a non-Abelian representation of the permutation group. However when we introduced decoherence in the system, the braid words that contribute to the walk have different structure
than in the fully coherent model. In the fully coherent model, the diagonal elements of the spatial degree of freedom of the walker correspond to a sum over all paths where the walker strand and necessarily all other strands return their initial position
after forward and backward time evolution. Hence each of these paths is a trivial permutation in the fusion space and has no effect on the diagonal elements of the quantum walk.
In the $V^2$ model however, the links do not have a dedicated walker strand as can be seen from Fig. \ref{fig:links} in Appendix
\ref{sec:brackets}. The nontrivial character of the $k\rightarrow\infty$ model thus follows from the fact that the links are not identity permutations and hence
expectation values in Table \ref{table:expectvalues} are not equal to 1, as they would be in the fully coherent walk.
The key difference is that only for a highly non-Markovian environment does the $k\rightarrow \infty$ case reduce to the normal
quantum walk for only if one includes memory of all previous steps does the fusion DOF become disentangled with the spatial DOF.
\section{Approximation by circulant matrices}
\label{sec:approx}
In the previous section, we derived the exact superoperator that describes how the spatial density matrix transforms
in a single step of the walk, and obtained numerical results by applying the superoperator to the density matrix repeatedly.
We did not however obtain any expression for the density matrix after evolution by an arbitrary number of steps.
In the following, we approximate the Kraus operators $E_{fc}$ by circulant matrices which have uniform coefficients on
each diagonal, and diagonalize the Kraus operators via Fourier transform to find arbitrary powers of the superoperator
$\mathcal{E}$ in a compact form.
Calling Eqs. (\ref{eqn:krausgens}) and (\ref{eqn:doublestep}) from previous section, the Kraus generators are written explicitly as
\begin{eqnarray}
E_{fc} &=&\sum_{s}\Big[ C_{c}^{00}\;\bra{f}b_{s}b_{s+1}\ket{\Phi_0}\; \ket{s}\bra{s+2}\;
+\;\bra{f}\;(C_{c}^{01}b_{s}^{2}\;+\;C_{c}^{10}b_{s-1}^{2})\;\ket{\Phi_{0}}\; \ket{s}\bra{s} \nonumber \\
&&+\;C_{c}^{11}\;\bra{f}b_{s+1}b_{s}\ket{\Phi_{0}}\; \ket{s+2}\bra{s} \Big] \label{kraus}
\end{eqnarray}
where the coin term has been defined as $C_{c}^{ab}:=\bra{c}P_{a}HP_{b}H\ket{c_0}$.
The matrix expression of the generators is
\begin{equation}
E_{fc}=\left(
\begin{array}{cccccc}
d_{fc}(0) & & a_{fc}(0) & \cdots & b_{fc}(N-2) & \\
& d_{fc}(1) & & a_{fc}(1) & & b_{fc}(N-1) \\
b_{fc}(0) & & d_{fc}(2) & & \ddots & \\
& b_{fc}(1) & & \ddots & \ddots & a_{fc}(N-3) \\
a_{fc}(N-2) & \cdots & \ddots & & \ddots & \\
& a_{fc}(N-1) & & b_{fc}(N-3) & & d_{fc}(N-1)
\end{array}
\right) , \label{ebm}
\end{equation}
where
\begin{eqnarray}
a_{fc}(s) &=&C_{c}^{00}\bra{f}b_{s}b_{s+1}\ket{\Phi _{0}}, \label{bmatrelem} \\
b_{fc}(s) &=&C_{c}^{11}\bra{f}b_{s+1}b_{s}\ket{\Phi_{0}}, \\
d_{fc}(s) &=&C_{c}^{01}\bra{f}b_{s}^{2}\ket{\Phi_{0}}+C_{c}^{10}\bra{f}b_{s-1}^{2}\ket{\Phi_{0}}.
\end{eqnarray}
Comparing the matrix above with the general circulant matrix given in Eq. (\ref{circularm}) of the
Appendix \ref{sec:circmtx}, we see that although matrix $E_{fc}$
is not a circulant matrix, it can be turned to a circulant matrix
if the $s$-dependence of parameters $a_{fc}(s)$, $b_{fc}(s)$, $d_{fc}(s)$ is suppressed in some meaningful way, so
that the sequences of these parametrers get approximated by some respective
sequences $\widetilde{a}_{fc}$, $\widetilde{b}_{fc}$ and $\widetilde{d}_{fc}.$ This suppression of the
$s$-dependence also leads to circulant form for $E_{fc}$. The advantage of having the Kraus generators being
circulant matrices stems from the fact that the spectral decomposition problem of such matrices is solved
(see Appendix \ref{sec:circmtx}), and therefore this enables us to express the CP map of our QW and its powers in terms of the
orthogonal eigenprojections of the generators. To this end we introduce for
each $a_{fc}(s),$ $b_{fc}(s)$ and $d_{fc}(s)$ the following
approximation for each $s\in
\mathbb{Z}
_{N}$
\begin{eqnarray}
a_{fc}(s) &\approx &\widetilde{a}_{fc}(s)=\frac{1}{N}Tr(\widehat{h}^{2}E_{fc}) \label{approx} \\
d_{fc}(s) &\approx &\widetilde{d}_{fc}(s)=\frac{1}{N}Tr(E_{fc}) \\
b_{fc}(s) &\approx &\widetilde{b}_{fc}(s)=\frac{1}{N}Tr(\widehat{h}^{-2}E_{fc})\label{eqn:approx3}
\end{eqnarray}
where the elementary band matrices are defined as $\widehat{h}=\sum_{s\in\mathbb{Z}_{N}}\ket{s}\bra{s+1}$.
This is explicitly written as
\begin{eqnarray}
\widetilde{a}_{fc}&=&\frac{C_{c}^{00}}{N}\sum_{s}\bra{f}b_{s}b_{s+1}\ket{\Phi_{0}}, \label{averaging} \\
\widetilde{d}_{fc}&=&\frac{C_{c}^{01}}{N}\sum_{s}\bra{f}b_{s}^{2}\ket{\Phi_{0}}+\frac{C_{c}^{10}}{N}
\sum_{s}\bra{f}b_{s-1}^{2}\ket{\Phi_{0}} \\
\widetilde{b}_{fc}&=&\frac{C_{c}^{11}}{N}\sum_{s}\bra{f}b_{s+1}b_{s}\ket{\Phi_{0}},
\end{eqnarray}
i.e. the parameters $a_{fc}(s),$ $b_{fc}(s),$ $d_{fc}(s)$
are approximated by the arithmetic averages of the corresponding $(f,\Phi_{0})$-elements of the respective generators.
Denote $E_{fc}^w$ the matrices obtained from $E_{fc}$ by substituting the $s$-dependent terms with their averages:
\begin{eqnarray}
E_{fc}^w&=&\widetilde{a}_{fc}\sum_{s}\ket{s}\bra{s+2}+\widetilde{d}_{fc}\sum_{s}\ket{s}\bra{s}
+\widetilde{b}_{fc}\sum_{s}\ket{s+2}\bra{s} \\
&=&\widetilde{a}_{fc}\widehat{h}^2+\widetilde{d}_{fc}\widehat{I}+\widetilde{b}_{fc}\widehat{h}^{-2}.
\end{eqnarray}
The map given by the new Kraus generators $E_{fc}^w$ is not trace preserving, so we are seeking for a set of generators
$\widetilde{E}_{fc}$ which satisfy the trace preserving condition
$\sum_{f,c}\widetilde{E}_{fc}^{\dagger}\widetilde{E}_{fc}=\mathbb{I}$.
To this end we need to scale the circulant matrices $E_{fc}^{w}$
by $\sum_{fc}E_{fc}^{w\dagger}E_{fc}^{w}\equiv M$, so the new trace
preserving circulant Kraus generators are defined
$\widetilde{E}_{fc}=(\sum_{f,c}E_{fc}^{w\dagger}E_{fc}^{w})^{-1/2}\:E_{fc}^{w}\equiv\Lambda\:E_{fc}^w$ with $\Lambda=M^{-1/2}$.
A calculation gives
\begin{equation}
M=\kappa_1\:\mathbb{I}+\kappa_2\:\widehat{h}^2+\kappa_2^{\ast}\:\widehat{h}^{-2}
\end{equation}
where
\begin{equation}
\begin{array}{lll}
\kappa_1&=&\sum\limits_{f,c}\big(|\widetilde{a}_{fc}|^2+|\widetilde{d}_{fc}|^2+|\widetilde{b}_{fc}|^2\big)\\
&=& \frac{1}{4}\Big(\frac{1}{N^2}\sum_{j,j'}\bra{\Phi_0}b_{j^{\prime }+1}^{\dagger }b_{j^{\prime}}^{\dagger }b_{j}b_{j+1}\ket{\Phi_0}+\frac{2}{N^2}\sum_{j,j'}\bra{\Phi_0}b_{j^{\prime }}^{\dagger 2}b_{j}^2\ket{\Phi_0} + \frac{1}{N^2}\sum_{j,j'}\bra{\Phi_0}b_{j^{\prime }}^{\dagger }b_{j^{\prime}+1}^{\dagger }b_{j+1}b_{j}\ket{\Phi_0}\Big) \\\\
\kappa_2&=&\frac{1}{4} \Big(\frac{1}{N^2}\sum_{j,j'}\bra{\Phi_0}b_{j^{\prime }}^{\dagger}b_{j^{\prime}+1}^{\dagger}b_{j-1}^2\ket{\Phi_0}+\frac{1}{N^2}\sum_{j,j'}\bra{\Phi_0}b_{j^{\prime }}^{\dagger 2} b_{j}b_{j+1}\ket{\Phi_0}\Big).\\
\end{array}
\label{kappas}
\end{equation}
The matrix $M$ is a circulant band matrix which can be diagonalized via discrete Fourier transform $F$ (see Appendix \ref{sec:circmtx}):
\[
F^\dag MF=\sum_l\big(\kappa_1+\kappa_2\omega^{2l}+\kappa_2^{\ast}\omega^{-2l}\big)\ket{l}\bra{l}.
\]
The matrix $F^\dag MF$ is a diagonal matrix with all diagonal values nonzero, so its inverse exists. The normalization operator $\Lambda$ can now be defined as
$\Lambda=F(F^\dag MF)^{-1/2}F^\dag$, where $(F^\dag MF)^{-1/2}=\sum_l\big(\kappa_1+\kappa_2\omega^{2l}+\kappa_2^{\ast}\omega^{-2l}\big)^{-1/2}\ket{l}\bra{l}$.
A simple check verifies that $\Lambda^2M=I$ and $\Lambda=M^{-1/2}$ as desired.
Unfortunately, the normalized generators $\widetilde{E}_{fc}=\Lambda E_{fc}$ contain all even powers of the matrices $\hat{h}$ for general values of the
Chern-Simons parameter $k$ and do not admit a helpful form. However, for the special values $k=2,4$ there is a significant simplification. Notice that each of the terms in expressions for $\kappa_1,\kappa_2$ in Eq. (\ref{kappas}) is an averaged Kauffman bracket. For large $N$ most of these brackets will correspond to disjoint links, i.e. the links formed by two step walks involving the strands located at $j$ and $j^{\prime}$ do not entangle. As shown in Table \ref{table:expectvalues}, this always occurs if $|j^{\prime}-j|>3$. If we approximate the average by the value of the bracket for these disjoint links then
\begin{equation}
\begin{array}{lll}
\kappa_1&=&\frac{1}{32}\Big(6\cos(\frac{2\pi}{k+2})+4\cos(\frac{4\pi}{k+2})+2\cos(\frac{6\pi}{k+2})+5\Big)\sec^4(\frac{\pi}{k+2}) +O(1/N) \\\\
\kappa_2&=&\frac{1}{8}\Big(\cos(\frac{4\pi}{k+2})-\cos(\frac{2\pi}{k+2})+1\Big)\sec^2(\frac{\pi}{k+2})\Big)+O(1/N).
\end{array}
\end{equation}
where the error is of order $1/N$. Thus at the special values $k=2,4$ we have $\kappa_2\approx 0$ and the normalization operator becomes a
scalar multiple of the identity: $\Lambda=\kappa_1^{-1/2}\:\mathbb{I}$. Henceforth we focus on the case $k=2$ since the behaviour of $k=4$ is quite similar. The Kraus generators are given by
\begin{equation}
\widetilde{E}_{fc}=\kappa_1^{-1/2}\big(\widetilde{a}_{fc}\widehat{h}^2+\widetilde{d}_{fc}\widehat{I}+\widetilde{b}_{fc}\widehat{h}^{-2}\big).
\end{equation}
The elementary circular matrix $\widehat{h}$ can now be diagonalized as $F^\dag\widehat{h}F=\widehat{g}=\sum_{n\in\mathbb{Z}_N}\omega^n\ket{n}\bra{n}$
where $\omega=e^{2\pi i/N}$. This allows to write the Kraus generators in a diagonal form
\begin{eqnarray}
\widetilde{E}_{fc}&=&\kappa_1^{-1/2}F\big(\widetilde{a}_{fc}\widehat{g}^2+\widetilde{d}_{fc}\mathbb{I}
+\widetilde{b}_{fc}\widehat{g}^{-2}\big)F^\dag\\
&\equiv&\sum_{k\in\mathbb{Z}_N}\lambda_{fc}(k)P_{f_k}
\end{eqnarray}
with $\lambda_{fc}(k)=\kappa_1^{-1/2}\big(\widetilde{a}_{fc}\omega^{2k}+\widetilde{d}_{fc}+\widetilde{b}_{fc}\omega^{-2k}\big)$
and $P_{f_k}=F\ket{k}\bra{k}F^\dag$. The action of the superoperator $\widetilde{\mathcal{E}}$ on a spatial density matrix $\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}$ is then given by
\begin{equation}
\widetilde{\mathcal{E}}(\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}})=\sum_{f,c}\widetilde{E}_{fc}\:\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}\:\widetilde{E}_{fc}^\dag
=\sum_{f,c}\sum_{k,l\in\mathbb{Z}_N}\lambda_{fc}(k)\lambda_{fc}^*(l)P_{f_k}\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}P_{f_l}^\dag
\end{equation}
and for $t$ time steps
\begin{equation}
\widetilde{\mathcal{E}}^t(\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}})=\sum_{k,l\in\mathbb{Z}_N}\prod_{i=1}^t\sum_{f_i,c_i}
\big(\lambda_{f_ic_i}(k)\lambda_{f_ic_i}^*(l)\big)P_{f_k}\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}P_{f_l}^\dag.
\end{equation}
Let's use this compact form to calculate the diagonal probability distribution $p(s,t)$ at time $t$:
$p(s,t)=\bra{s} \widetilde{\mathcal{E}}^{t}(\rho_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{S}}(0))\ket{s}$. Let's assume that
the walker is initialized in the position eigenstate $\ket{s_0}$, that the size of the periodic lattice is $N$,
and that during the 2-step walk the coin is always reinitialized to state $\ket{c=0}$. We find for the index $k=2$:
\begin{equation} \label{eqn:distk2}
\begin{array}{rcl}
p_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{Ising}}(s,t)&=&\langle s | \widetilde{\mathcal{E}}^{t}(|s_0\rangle\langle s_0|)|s\rangle\\
&=&\frac{1}{2^tN^2}\sum_{r,l\in\mathbb{Z}_N}\omega^{(s-s_0)(r-l)}
(\omega^{2(r-l)}+\omega^{-2(r-l)})^t\\
&=&\frac{1}{2^tN^2}\sum_{m=0}^t\binom{t}{m}\sum_{r,l\in\mathbb{Z}_N}\omega^{(s-s_0)(r-l)}
\omega^{2m(r-l)}\omega^{-2(t-m)(r-l)}\\
&=&\frac{1}{2^tN^2}\sum_{m=0}^t\binom{t}{m}\sum_{r\in\mathbb{Z}_N}\omega^{r(s-s_0+4m-2t)}\sum_{l\in\mathbb{Z}_N}\omega^{-l (s-s_0+4m-2t)}\\
&=&\frac{1}{2^tN^2}\sum_{m=0}^t\binom{t}{m}(N\delta_{s-s_0+4m-2t,0})^2\\
&=&\frac{1}{2^t}\binom{t}{\frac{2t-(s-s_0)}{4}}\\
\end{array}
\end{equation}
This is the binomial distribution where the range of the sites is $s\in[-2t,2t]$ and the probabilities are nonzero only for
$s=s_0+4n,\:n\in\mathbb{Z}$, i.e. the $V^2$ model with $k=2$. Ising anyon walkers therefore have the same probability distribution as the classical random walk where every step moves two units to the right or left and the variance, scaled so that each two steps move takes place over two time intervals, is $\sigma_{\rm Ising}^2(t)=t$.
\section{Conclusions}
We have considered a lossy anyonic quantum walk protocol where the entanglement between the spatial modes of
the walker and its environment, the coin and fusion space, is lost on every second step. We calculated the time evolution
of the exact probability distributions for various values of the parameter $k$ in Chern-Simons theory.
The case $k=1$ corresponds to the Abelian anyonic quantum walk and it
was found to be equal to the standard $V^2$ quantum walk with trivial exchange statistics, the variance of which
has a leading term proportional to $t^2$. Cases $k\geq2$ are non-Abelian anyon models, for which the variance
grows linearly as a function of time for all tested values of $k$ in the time scales of up
to 100 iterations of the superoperator.
By approximating the Kraus generators
of the superoperator by circulant matrices, the generators can be diagonalized for $k=2,4$. This allows for a compact
expression for the probability distribution after arbitrary number
of iterations of the superoperator. The expression for $k=2$ is the binomial distribution which is equal to the
classical random walk distribution. Thus the $V^2$ anyonic quantum walk with Ising anyons has the same behaviour
as the fully coherent walk, which was also shown to have a linear variance with coefficient 1 \cite{isingaqw}.
In the fully coherent walk, the slowdown of the walker propagation can be explained by decoherence: the fusion Hilbert
space of non-Abelian anyons acts like an environment to the walker+coin system, degrading the quantum correlations between
the spatial and coin modes which are the origin of the quantum speedup. It was conjectured in Ref. \cite{isingaqw}
that the slowdown happens when the parameter $k$ is much smaller than the number of time steps $t$, $k\ll t$, ie. for any
finite $k$, the walk behaves classically in the large time limit. In the $V^2$ model presented here the results support that conjecture. There are two kinds of decoherence
mechanisms at work, one due to the fusion space of the anyons and the other because the entanglement between the spatial
modes and the coin is lost. The loss of quantum correlations on every second step does not change the qualitative behaviour
when the braiding statistics is Abelian, but when the additional effect of the fusion space comes into play for non-Abelian
anyons, the spreading velocity changes to diffusive for all values of $k$ for long enough times (in fact after fewer than 100 steps).
The $V^2$ model can be easily generalized to $V^n$ models where the walk evolves coherently for $n$ time steps instead of two.
Evaluation of higher number of time steps becomes increasingly hard, because the number of different Kauffman brackets that need to be computed increases as the walker is allowed to do more steps between the tracing operation.
One might expect that for large $n$, the walker propagates diffusively for $n$ steps for small $k$ and ballistically for large $k$,
before the tracing is carried out. However, as we have shown, the variance of the anyonic $V^2$ walk is linear even when $k\gg 2$,
so we expect that in the long time limit the variance is linear for all $V^n$ models, although the walker might spread ballistically
in the initial stage of the walk.
It is known that spatial randomness in the coin operator can lead to localization of the walker wave packet \cite{localization}, ie. the probability
to stay in the initial position becomes very high and the probability falls off exponentially away from the initial position.
In the anyonic quantum walk setup, spatial randomness can also occur if the occupation numbers of the background anyons
fluctuate randomly. These fluctuations might occur when thermal excitations create particle-antiparticle pairs from vacuum, for example.
It is interesting to note that if the occupation numbers of the islands are not uniform, the Abelian anyons can pick up different
phases during forward and backward time evolution, and the effect of the braiding statistics becomes nontrivial. We have calculated
evolution of the $V^2$ anyonic walk with random fillings of Abelian anyons on the islands, but observed diffusive spreading (no localization). Localization is a quantum phenomenon that requires interference of the probability amplitudes to occur. Interestingly our results show that while the memory loss in the $V^2$ model preserves enough coherence in the Abelian walk to provide for quadratic speed up without disorder, it does not provide enough to give localization with disorder. Rather the competition of the two effects yields classical behaviour.
\section*{Acknowledgements}
One of us (D.E.) is grateful to the Department of Physics and Astronomy, Macquarie University for hospitality during a sabbatical stay during which this work was initiated.
|
{
"timestamp": "2012-03-12T01:00:47",
"yymm": "1203",
"arxiv_id": "1203.1999",
"language": "en",
"url": "https://arxiv.org/abs/1203.1999"
}
|
\section{Introduction}
The ability of a material to convert heat into electricity is measured
by the dimensionless thermoelectric figure of merit $ZT$ defined by:
\begin{equation}
ZT=\frac{S^2GT}{(\kappa_\mathrm{e}+\kappa_\mathrm{l})}
\end{equation}
where $S$ denotes the Seebeck coefficient, $G$ the electrical
conductance, $T$ the temperature, $\kappa_\mathrm{e}$ the electronic
and $\kappa_\mathrm{l}$ the lattice parts of the thermal
conductance~\cite{Nolas01Book}. Due to the strong interconnection
between the parameters that control $ZT$, it has been traditionally
proved difficult to achieve values above unity, which translates to
low conversion efficiencies and limit the applications for
thermoelectricity.
The recent advancements in lithography and nanofabrication, however,
have lead to the realization of breakthrough experiments on
nanostructured thermoelectric devices that demonstrated enhanced
performance, sometimes even up to 2 orders of magnitude higher than
the corresponding bulk material values. Nanostructures provided the
possibility of independently designing the quantities that control the
$ZT$ in achieving higher values. Enhanced performance was demonstrated
for 1D nanowires (NWs)~\cite{Hochbaum08,Boukai08}, 2D thin films,
1D/2D superlattices~\cite{Venkatasubramanian01,Kim06}, as well as
materials with embedded nanostructuring~\cite{Tang10}.
Graphene, a recently discovered two-dimensional form of carbon, has
received much attention over the past few years due to its excellent
electrical, optical, and thermal
properties~\cite{Novoselov04}. Graphene, however, is not a useful
thermoelectric material. Although its electrical conductance is as
high as that of copper~\cite{Chen08}, its ability to conduct heat is
even higher~\cite{Ghosh08}, which increases the denominator of
$ZT$. To make things worse, as a zero bandgap material, pristine
graphene has a very small Seebeck coefficient~\cite{Seol10}, which
minimizes the power factor $S^2G$. Nanoengineering, however, could
provide ways to increase the Seebeck coefficient and decrease the
thermal conductivity as well.
The high thermal conductivity of graphene is mostly due to the lattice
contribution, whereas the electronic contribution to the thermal
conduction is smaller~\cite{Hone99,Balandin08}. In order to reduce the
thermal conductivity, therefore, the focus is placed on reducing phonon
conduction. Recently many theoretical studies have been performed
regarding the thermal conductivity of graphene-based
structures. Several methods, such as the introduction of vacancies,
defects, isotope doping, edge roughness and boundary scattering, can
considerably reduce thermal
conductance~\cite{Jiang11,Hu10,Sevincli10}. Importantly, in certain
instances this can be achieved without significant reduction of the
electrical conductance.
In order to improve the Seebeck coefficient graphene needs to acquire
a bandgap. This can be achieved by appropriate patterning of the
graphene sheet into nanoribbons~\cite{Han07,Zhang11}. Graphene
nanoribbons (GNRs) are thin strips of graphene, where the bandgap
depends on the chirality of the edges (armchair or zigzag) and the
width of the ribbon. Armchair GNRs (AGNRs) can be semiconductors with
a bandgap inversely proportional to their width~\cite{Han07}. Although
the acquired bandgap can increase the Seebeck coefficient, when
attempting to reduce the thermal conductivity by introducing disorder
in the nanoribbon, as described above, the electrical conductivity is
also strongly affected~\cite{Ouyang09,Areshkin07}, and the
thermoelectric performance remains low. Zigzag GNRs (ZGNRs), on the
other hand, show metallic behavior with very low Seebeck coefficient,
but as described in Ref.~\cite{Areshkin07}, the transport in ZGNRs is
nearly unaffected in the presence of line edge roughness, at least in
the first conduction plateau around their Fermi level.
In this work, by using atomistic electronic and phononic bandstructure
calculations, and quantum mechanical transport simulation, we show
that despite the zero bandgap, the thermoelectric performance of ZGNRs
can be largely enhanced. For this a series of design steps are employed:
i) Introducing extended line defects (ELDs) as described in
Ref.~\cite{Bahamon11} can break the symmetry between electrons and
holes by adding additional electronic bands. This practically provides
a sharp band edge around the Fermi level and offering a band asymmetry
which for thermoelectric purposes it practically constitutes an
``effective bandgap''. ii) Introducing background impurities enhances
the ``effective bandgap''. iii) Introducing edge roughness reduces the
lattice part of the thermal conductivity (significantly more than it
reduces the electrical conductivity). After such procedure, we
demonstrate that the figure of merit $ZT$ can be greatly enhanced and
high thermoelectric performance could be achieved.
The paper is organized as follows. In section~\ref{s:Approach} we
describe the methodology used in our calculations. In
section~\ref{s:Result} we present the results for the
electronic/phononic structure and transmission of ZGNRs for every step
of our design approach (in section~\ref{s:Transmission}), and their
influence on the thermoelectric coefficients (in
section~\ref{s:Thermoelectric}). Finally, in section~\ref{s:Summary}
we conclude.
\section{Approach}
\label{s:Approach}
In linear response regime, the transport coefficients can be evaluated
using the Landauer formula~\cite{Kim09,Jeong10,Karamitaheri11}:
\begin{equation}
G=\left (\frac{2q^2}{h}\right)I_0~~~~[1/\Omega]
\end{equation}
\vspace*{-20pt}
\begin{equation}
S=\left (-\frac{k_{\mathrm{B}}}{q}\right )\frac{I_1}{I_0}~~~~[V/K]
\label{e:seebeck}
\end{equation}
\vspace*{-20pt}
\begin{equation}
\kappa_{\mathrm{e}}=\left (\frac{2Tk_{\mathrm{B}}^2}{h}\right )\left [
I_2-\frac{I_1^2}{I_0}\right ]~~~~[W/K]
\end{equation}
Here, $h$ is the Planck constant, $k_{\mathrm{B}}$ is the Boltzmann
constant, and
\begin{equation}
I_j=\int_{-\infty}^{+\infty}\left(
\frac{E-E_{\mathrm{F}}}{k_{\mathrm{B}}T}
\right)^j{T}_{\mathrm{el}}(E)\left(- \frac{\partial f}{\partial E} \right)dE
\end{equation}
where ${T}_{\mathrm{el}}(E)$ is the electronic transmission
probability, $f(E)$ is the Fermi function and $E_{\mathrm{F}}$ is the
Fermi-level of the system. Similarly, the lattice contribution to the
thermal conductance can be given as a function of the phonon
transmission probability~\cite{Ouyang09}:
\begin{equation}
\kappa_\mathrm{l}=\frac{1}{h}\int_{0}^{+\infty}{T}_\mathrm{ph}(\omega)\hbar\omega\left(\frac{\partial
n(\omega)}{\partial T}\right)\ d(\hbar\omega)
\label{e:kp}
\end{equation}
where $n(\omega)$ denotes the Bose-Einstein distribution function and
${T}_\mathrm{ph}(\omega)$ is the phonon transmission probability~\cite{Jeong11}.
For the electronic structure, the Hamiltonian of the GNRs is described in the standard first nearest-neighbor atomistic tight-binding $p_z$ orbital approximation. The hopping parameter is set to $-2.7~\mathrm{eV}$ and the on site potential is shifted to zero so
that the Fermi level remains at $0~\mathrm{eV}$. This model has been
recently used to describe the electronic transport of ELD-ZGNR with
double-vacancies and the results are in good agreement with
first-principle calculations and experimental
studies~\cite{Bahamon11,Lahiri10}. To the best of our knowledge, only
a few first-principle calculations and experimental studies have been
conducted in structures that include ELDs~\cite{Appelhans10,Lusk10,Lahiri10}. The two main features of
the electronic structure, the asymmetry between electrons and holes,
and the metallic behavior of the ELD in the graphene ribbon channel
have been described in these studies, and are also captured by the
tight-binding model as we will demonstrate below.
For the phonon modes, the dynamic matrix is constructed using the
fourth nearest-neighbor force constant
model~\cite{Karamitaheri11}. The force constant method uses a set of
empirical fitting parameters and can be easily calibrated to
experimental measurements. We use the fitting parameters given in
Ref.~\cite{Saito98Book} for graphene-based structures. We assume that
this model is still valid under structures that include ELDs. Although verification of its validity for ELD-ZGNRs has not been demonstrated yet, i.e. using first-principle calculations, in
Ref.~\cite{Kahaly08} it was shown using DFT simulations that there is
little difference between the phonon transmission of carbon nanotube
structures with/without ELDs which could justify our model
choice. In any case, as we show below, the main influence on the
phonon transport in this work originates from edge roughness
scattering, which reduces the phonon transmission drastically. The
effect of edge roughness scattering is the dominant effect, and that
can be captured adequately by the model we employ in this work. The
influence of the ELDs on the phonon transmission is much smaller
compared to the effect of edge roughness, and therefore we still
choose to use the numerically less expensive fourth nearest-neighbor
force constant method.
In this work, the fully quantum mechanical non-equilibrium Green's
function formalism (NEGF) is used for transport calculations of
both electrons and phonons. The system geometry is defined as a set of two
semi-infinite contacts and a channel (device) with length $L$. The
device Green's function is obtained as
\begin{equation}
G_{\mathrm{el}}(E)=\left (EI-H-\Sigma_{\mathrm{s},{\mathrm{el}}}-\Sigma_{\mathrm{d},{\mathrm{el}}}
\right)^{-1}
\end{equation}
for electron calculation, where $H$ is the device Hamiltonian matrix
and $E$ is the energy. In the case of phonon transport the Green's
function is given by:
\begin{equation}
G_{\mathrm{ph}}(E)=\left (EI-D-\Sigma_{\mathrm{s},\mathrm{ph}}-\Sigma_{\mathrm{d},\mathrm{ph}}
\right)^{-1}
\end{equation}
where $D$ is the dynamic matrix and $E=\hbar \omega$~\cite{Karamitaheri12}. The contact
self-energy matrices $\Sigma_{\mathrm{s/d}}$ are calculated using the
Sancho-Rubio iterative scheme~\cite{Sancho85}. The effective
transmission probability through the channel can be achieved using the
relation:
\begin{equation}
T_{\mathrm{el/ph}}(E)=\mathrm{Trace}[\Gamma_{\mathrm{s}}G\Gamma_{\mathrm{d}}G^{\dagger}]
\end{equation}
where $\Gamma_{\mathrm{s}}$ and $\Gamma_{\mathrm{d}}$ are the
broadening functions of contacts~\cite{Datta05Book}.
This method is very effective in describing the effect of realistic
distortion in nanostructures, including all quantum mechanical
effects. In our calculation, we include long-range substrate
impurities with density of one impurity per $125~\mathrm{nm}$ and edge
distortion (roughness) up to four layers in each side of the ribbon's
edge. These are applied only on the device part and not in the contact
regions~\cite{Areshkin07}.
\section{Results and Discussion}
\label{s:Result}
An efficient thermoelectric material must be able to effectively
separate hot from cold carriers. The quantity that determines the
ability to filter carriers is the Seebeck coefficient. The Seebeck
coefficient depends on the asymmetry of the density of states around
the Fermi level. In semiconductor the Seebeck coefficient is large,
but in a metal where the density of states is more uniform in energy the
Seebeck coefficient is small. Metallic ZGNRs also have a small Seebeck
coefficient because their transmission is constant around the Fermi
level, despite the peak in the DOS at $E=0~\mathrm{eV}$ due to the edge
states. Recently, however, Bahamon et al. have investigated the
electrical properties of ZGNRs that included an ELD
(ELD-ZGNRs) along the nanoribbon's length~\cite{Bahamon11}. It was
reported that the ELD breaks the electron-hole energy symmetry in
nanoribbons, and introduces an additional electron band around the
Fermi level. In such a way an asymmetry in the density of states and
the transmission function are achieved which improves the Seebeck
coefficient as we will show further down. This particular structure
has also been recently experimentally
realized~\cite{Lahiri10}. Although the method of fabrication was
rather complicated to be able to scale for industrial applications,
nevertheless it makes studies on GNRs appropriate and interesting as
well.
\subsection{Electronic and Phononic Structure}
\label{s:Transmission}
The changes in the electronic structure of the ZGNRs after the
introduction of the ELD are demonstrated in
Fig.~\ref{fig:BandStructure}. Figure \ref{fig:BandStructure}-a shows
the atomistic geometry of the pristine ZGNR of width $W\sim
4~\mathrm{nm}$ (with 20 zigzag edge lines) and
Fig. ~\ref{fig:BandStructure}-b its electronic structure. The Fermi
level is at $E=0~\mathrm{eV}$ due to the symmetry between electron and
hole bands. Figure ~\ref{fig:BandStructure}-c shows the structure of
the ELD-ZGNR with the same width. The region in which the ELD is
introduced is shown in red color. The ELD changes the hexagons of the
GNR to pentagons and octagons after a local rearrangement of the
bonding and the introduction of two additional atoms in the unit cell.
We use a two parameter notation to describe the ELD-ZGNR structure
throughout this work as ELD-ZGNR($n_1$,$n_2$), where $n_1$ and $n_2$
are the indices of the partial-ZGNRs above and below the line defect,
respectively (i.e. the number of zigzag edge lines of atoms), although
in all cases we use $n_1=n_2$. The bandstructure of the
ELD-ZGNR(10,10) is shown in Fig.~\ref{fig:BandStructure}-d. The
thick-red line shows a new band that is introduced in the conduction
band near the Fermi energy ($E=0~\mathrm{eV}$), which corresponds to
the ELD. There are two points that result in the creation of the extra
band. Part of the physics behind this is explained by Pereira et
al. in Ref.~\cite{Pereira06}. The first point is that a defect in the
graphene system will introduce states that reside close to the Fermi
level at $E=0~\mathrm{eV}$. This is similar to the edge states of the
ribbons that tend to reside near the Fermi level. The second point
again described in Ref.~\cite{Pereira06}, is that an asymmetry in the
dispersion between electrons and holes will be created when carbon
atoms of the graphene sublattice ``A'' (or ``B'') are coupled with
atoms from ``A'' (or ``B'') again. Usually, the atomic arrangement in
graphene can be splitted in sublattices ``A'' and ``B'', where atoms
from ``A'' couple to ``B'' and vise versa. When this happens, the
dispersion is symmetric in the first-nearest neighbor tight-binding model. At a
defect side such as the ELD we consider, where ``A'' connects to ``A''
as seen in Fig.~\ref{fig:BandStructure}-c, such asymmetry can be
observed. The fact that the overall bandstructure has additional bands
compared to the pristine ribbon is also connected to the two extra
atoms in the unit cell.
Moving one step
further, in Fig.~\ref{fig:BandStructure}-e we show the geometry of a
GNR with two ELDs. We denote this structure as
2ELD-ZGNR($n_1$,$n_2$,$n_3$), where $n_1$, $n_2$, and $n_3$ denote the
the number of zigzag carbon lines above, within, and below the line
defects. Figure~\ref{fig:BandStructure}-f shows the electronic
structure of the 2ELD-ZGNR(8,4,8). In this case two additional bands
are introduced near the Fermi level as noted by the thick-red
lines. In this structure the asymmetry between electron and hole bands
around the Fermi level ($E=0~\mathrm{eV}$) is further enhanced.
$\mathbf{1^{st}}$ {\bf Design Parameter- The Effect of ELD:} Figure
\ref{fig:PristineTrans} demonstrates the increase in the asymmetry of
the bands around the Fermi level by showing how the transmission
changes when one or two ELDs are introduced in the
channel. For the pristine ZGNR, the transmission is equal to one,
indicating the existence of a single propagating band at energies
around the Fermi level (green line). With the introduction of one ELD,
the conduction band ($E>0~\mathrm{eV}$) is composed of two subbands,
whereas the valence band ($E<0~\mathrm{eV}$) is still composed of one
subband. With the introduction of two ELDs, three conduction subbands
now appear, but still only one valence subband. As it
will be shown below, this asymmetry will improve the Seebeck
coefficient. This constitutes the first design step in improving the
thermoelectric performance of ZGNRs.
There is, however, another point worth mentioning. In
Fig.~\ref{fig:Current} we show colormaps of the normalized current
spectrum at $E=0.2~\mathrm{eV}$ in the cross sections of the ELD-ZGNRs described in Fig.~\ref{fig:PristineTrans}. Figure
~\ref{fig:Current}-a shows the current spectrum of the
ELD-ZGNR(10,10). The current is zero close to the edges of the ribbon
and peaks near the center. This is demonstrated more clearly in
Fig. \ref{fig:Current}-d, which shows the current along one atomic
chain perpendicular to this channel (blue line). The black line of
Fig.~\ref{fig:Current}-d illustrates the current density on the cross
section of the pristine ZGNR channel for reference.
The current spectrum for the 2ELD-ZGNR(8,4,8) is shown in
Fig.~\ref{fig:Current}-b. The situation is now different since most of
the current is confined within the two ELDs. This, however, is the
case only when the distance between the ELDs is smaller than the
widths of the upper/lower regions. In the case where the width of the
middle region similar to the widths of the upper/lower
regions, the current is spread more uniformly in the channel as shown
in Fig.~\ref{fig:Current}-c for the 2ELD-ZGNR(7,6,7)
channel. Figure~\ref{fig:Current}-e shows again the current along one
atomic chain in the cross section of these ribbons. The current spectrum
is localized in the middle of the channel in the 2ELD-ZGNR(8,4,8)
channel (red line) compared to the pristine channel (black line). In a
2ELD-ZGNR(9,2,9) channel with a narrower middle region the current
spectrum is localized even closer around the center (blue line). A
large portion of the current is in general flowing around the ELD
regions. The design capability to localize the current spectrum in the
middle of the channel away from the edges will prove advantageous in
the presence of edge roughness since the current in this case will be
less affected. On the other hand, in the case of the 2ELD-ZGNR(7,6,7)
channel the current spectrum tends to concentrate more close to the
edges (green line).
$\mathbf{2^{nd}}$ {\bf Design Parameter- The Effect of Background
Positive Impurities:} We next illustrate the possibility of further
enhancing the asymmetry between electron and hole transport near the
Fermi level by the introduction of positively charged substrate background
impurities. The effect of background impurities is included in the
Hamiltonian in a simplified way as an effective negative long range
potential energy on the appropriate on-site Hamiltonian elements as
described in Ref.~\cite{Areshkin07}. A positive impurity in the
substrate will constitute a repulsive potential for holes (a barrier
for holes but a well for electrons) and will degrade hole transport
more effectively than electron
transport. Figure.~\ref{fig:RoughImpurity}-a shows how the
transmission of the ELD-ZGNR(10,10) channel (dashed-black line) is
affected after the introduction of positive charged impurities in the
channel (solid-blue line). Indeed, the transmission of holes below the
Fermi level ($E=0~\mathrm{eV}$) is degraded. This effect additionally
increases the asymmetry of the propagating bands and improves the Seebeck
coefficient. On the other hand, the opposite is observed when negative
impurities are introduced in the substrate. Negative impurities are a
barrier for electrons and reduce their
transmission~\cite{Neophytou06}, but do not interfere with the hole
subsystem as shown in Fig.~\ref{fig:RoughImpurity}-b. This type of
impurities will actually harm the asymmetry and needs to be avoided.
$\mathbf{3^{rd}}$ {\bf Design Parameter- The Effect of Roughness:} In
the third step of the design process we introduce the effect of edge
roughness. The inset of Fig.~\ref{fig:RoughImpurity}-c shows the
influence of edge roughness on the transmission of the ZGNR(20) of
length $125~\mathrm{nm}$. As also described in previous
studies~\cite{Areshkin07,Sevincli10}, in the first conduction plateau
the effect is negligible. In contrast to ZGNR, ELD-ZGNRs as well as
2ELD-ZGNRs are affected by edge roughness. This is because the
bandstructure of these GNRs has undergone a band folding, and
therefore, the states in the first conduction plateau have lower wave
vectors. As the long range defects can induce only small value of
momentum transfer, the momentum conservation rule indicates that, in
contrast to the ZGNR, the transport of ELD-ZGNRs and 2ELD-ZGNRs will
not remain ballistic in the presence of line edge roughness and long
range substrate impurities. This is shown in
Fig.~\ref{fig:RoughImpurity}-c, where the transmission of a roughened
$125~\mathrm{nm}$ long ELD-ZGNR(10,10) channel (solid-blue line) is reduced by
$\sim 25\%$ compared to the ballistic value (dashed-black line). Edge
roughness degrades the conductivity of holes and electrons by a
similar amount, and therefore, the level of asymmetry around the Fermi
level is retained.
Figures \ref{fig:ElTrans}-a and \ref{fig:ElTrans}-b illustrate the
influence of roughness in ELD-ZGNR channels on their transmission, for channels of different lengths and widths. In this calculation positive impurities are also
included. Figure \ref{fig:ElTrans}-a shows the transmission of edge
roughened ELD-ZGNR(10,10) versus energy for the channel lengths
$L=250$, $500$, and $2000~\mathrm{nm}$. As the channel length is increased, the
transmission drops further compared to the transmission of the ideal
channel (black-solid line). This is expected since the channel
resistance increases with increasing length. Figure
\ref{fig:ElTrans}-b illustrates the effect of the ribbon's width on the
transmission of ELD-ZGNRs with rough edges. In this case the length is
kept constant at $L=250~\mathrm{nm}$, and results for three different ribbon
with parameters (10,10), (7,7), and (5,5) are shown. As the width of
the ribbon is decreased, the effect of line edge roughness scattering
on the transmission becomes stronger because the carriers reside on
average closer to the edges.
It is worth mentioning that the effect of edge roughness on the
transmission is much stronger in AGNR than in ZGNR. Although in the
case of some AGNRs a bandgap is naturally present and the asymmetry
does not need to be created with the introduction of line defects and
impurities, the conductance is severely degraded by the roughness
which renders this type of ribbon not well suited for transport
applications~\cite{Areshkin07}. (Note that edge roughness will be
needed in order to reduce thermal conductivity as will be shown
below.)
As we mentioned above in Fig.~\ref{fig:Current}, the channel which
includes two ELDs can shift the majority of the
current spectrum in the region between the two ELDs, and thus farther
away from the edges. It is therefore expected that the 2ELD-ZGNR will
be less affected by edge roughness scattering than the ELD-ZGNR. A
comparison of the transmission of these devices with rough edges is
shown in Fig.~\ref{fig:Width}. The transmission of
ELD-ZGNR($n_1$,$n_1$), and two cases of 2ELD-ZGNR,
2ELD-ZGNR($n_2$,4,$n_2$) and the 2ELD-ZGNR($n_3$,6,$n_3$) at
$E=0.2~\mathrm{eV}$ versus their width $W$ are compared. The
parameters $n_i$ are adjusted such that the three channels have nearly
the same width $W$. The first channel belongs to the category shown in
Fig.~\ref{fig:Current}-a, the second in the category of
Fig.~\ref{fig:Current}-b, and the third in the category of
Fig.~\ref{fig:Current}-c. The third channel as shown in
Fig.~\ref{fig:Current} spreads the current spectrum more uniformly in
the channel and is expected to be affected the most from edge
roughness. All channels have the same length of $L=250~\mathrm{nm}$. For
smaller widths the effect of roughness is strong, and the
transmissions of all channels are drastically reduced. Since the
2ELD-ZGNR devices can concentrate the current spectrum around the
defect lines as shown in Fig.~\ref{fig:Current}-b and
\ref{fig:Current}-c, they effectively bring it closer to the edges and
the reduction is larger for these devices. For larger widths the
transmission of the ribbons approaches its ballistic value, which is 2
for the ELD-ZGNR devices and 3 for the 2ELD-ZGNR devices. The
transmission of the 2ELD-ZGNR($n_2$,4,$n_2$) channels increases
faster with increasing channel width, because the current spectrum is located farther from the
edges which makes it less susceptible to scattering as the width
increases. The transmission of 2ELD-ZGNR($n_3$,6,$n_3$) channel
eventually increases close to the ballistic transmission value as the
width increases, but it increases more slowly than that of the
2ELD-ZGNR($n_2$,4,$n_2$) channel.
{\bf Effect of roughness on phonon Transport:} Although the reduction
in the electronic transmission of channels with ELDs can be quite
strong when considering edge roughness, the reduction in the lattice
part of the thermal conductivity is even stronger. We take advantage of on this
effect when attempting to optimize the thermoelectric figure of
merit. The phonon transmission for the edge roughened ELD-ZGNR(10,10)
channel versus energy is shown in Fig.~\ref{fig:PhTrans}-a. Results
for channel lengths $L=10$, $100$, and $2000~\mathrm{nm}$ are shown. As
expected, the transmission decreases as the length is increased. What
is important, however, is that the decrease is much stronger than the
decrease of the electron transmission shown in
Fig.~\ref{fig:ElTrans}-a. For example, for a channel length of
$L=100~\mathrm{nm}$ the phonon transmission reduces by more than a factor of
$6X$, whereas the electronic transmission even at larger length
$L=250~\mathrm{nm}$ reduces only by $<30\%$. Interestingly, the same
order of reduction of the phonon transmission is observed for the
2ELD-ZGNRs as shown in Fig.~\ref{fig:PhTrans}-b, indicating that the
line defect does not affect phonon conduction significantly compared to the effect of edge roughness.
\subsection{Thermoelectric Coefficients}
\label{s:Thermoelectric}
The denominator of the $ZT$ figure of merit consists of the summation
of the contributions to the thermal conductivity of the electronic
system and the phononic system. In graphene the phonon part dominates
the thermal conductivity, whereas the electronic part contribution is
much smaller. The situation is different,however, in rough ELD-ZGNRs,
in which the phonon thermal conductivity is degraded more than the
electronic thermal conductivity. Figure \ref{fig:NormalizedThermal}
clearly illustrates this effect by showing the ratio of the phonon
thermal conductance to the electronic thermal conductance versus the
rough channel length. The cases of ELD-ZGNR(10,10) and
2ELD-ZGNR(8,4,8) are shown in dashed-red and dash-dot-blue lines,
respectively. For small channel lengths, where transport is
quasi-ballistic and roughness does not affect the transmission
significantly, $\kappa_{\mathrm{l}}$ is almost $5X$ larger than
$\kappa_{\mathrm{e}}$. As the length of the channel increases and the
effect of the roughness becomes significant, the phonon system is
degraded more than the electronic system, and the
$\kappa_{\mathrm{l}}$ is significantly reduced compared to
$\kappa_{\mathrm{e}}$. For lengths $L\sim 100~\mathrm{nm}$ and beyond,
$\kappa_{\mathrm{l}}$ can become even smaller than
$\kappa_{\mathrm{e}}$. The trend is the same when considering channels
with one or two ELDs. We note that from the inset of
Fig.~\ref{fig:NormalizedThermal} which shows that the ratio of the electrical
conductance $G$ over $\kappa_{\mathrm{e}}$ is almost constant, it can be
indicated that both $G$ and $\kappa_{\mathrm{e}}$ follow the same
trend, as the Wiedemann-Franz law dictates. We mention that the
$\kappa_{\mathrm{l}}$ and $\kappa_{\mathrm{e}}$ values used in
Fig.~\ref{fig:PhTrans} are extracted using the corresponding mean free
paths (MFPs) for phonons and electrons respectively, defined as
described in Ref.~\cite{Sevincli10}
\begin{equation}
T(E)=\frac{N_{\mathrm{ch}}(E)}{1+\frac{L}{\lambda(E)}}
\end{equation}
where, $T(E)$ is transmission probability, $N_{\mathrm{ch}}(E)$ is the
number of modes at energy $E$, $L$ is the given length of the channel,
and $\lambda(E)$ is the mean free path of the carriers. Alternatively,
$\kappa_{\mathrm{l}}$ and $\kappa_{\mathrm{e}}$ could be extracted
from the transmission calculations by using a statistical average over
several rough samples for each channel length. The results of both
methodologies are in good agreement for the electronic part of the
thermal conductivity. For the lattice part, the agreement is good only
for the shorter channels, below $\sim 100~\mathrm{nm}$. For larger channel
lengths, the phonon transmission is severely reduced which increases
the noise in the calculation for extracting the
$\kappa_{\mathrm{l}}$. The values extracted directly from the
integration of the phonon transmission could be as much as $2X$
larger, which could increase the
$\kappa_{\mathrm{l}}/\kappa_{\mathrm{e}}$ by a factor of $2X$ for the
longer channels. In this case the ratio
$\kappa_{\mathrm{l}}/\kappa_{\mathrm{e}}$ will be closer to unity, but
this is still a huge advantage compared to devices without roughness.
{\bf Power Factor:} Using the first design step, i.e. the effect of
ELDs, we have demonstrated that the transmission of electrons around
the Fermi level can be increased (from $T=1$ to $T=2$ and $T=3$ in the
presence of one and two ELDs, respectively). An asymmetry is thus
created between holes and electrons. This increases both the
conductivity and Seebeck coefficient of the channel as shown in
Fig.~\ref{fig:ThermoelectricP}. Figure \ref{fig:ThermoelectricP}-a
shows the conductance of the 2ELD-ZGNR(8,4,8) (blue), of the ELD-ZGNR
(10,10) (red), and of the pristine nanoribbon (green) at room
temperature $300~K$. As expected, the conductance of the channel with
two ELDs is the largest, followed by the channel with one ELD. They
are larger than the pristine channel by $\sim 3X$ and $\sim 2X$,
respectively. Figure \ref{fig:ThermoelectricP}-b shows the changes of
the Seebeck coefficient after the introduction of the ELDs in the
nanoribbon. Due to its metallic behavior and the flat transmission near the Fermi level, the pristine channel exhibits zero
Seebeck coefficient. Due to the built asymmetry after the introduction
of the ELDs, however, the Seebeck coefficient increases for both
channels. The channel with two line defects has the largest asymmetry,
and therefore the largest Seebeck coefficient (in absolute
values). Finally, the power factor in Fig.~\ref{fig:ThermoelectricP}-c
is indeed largely improved in the ELD structures, and
especially the 2ELD-ZGNR channel.
In Figure \ref{fig:ThermoelectricL} we show the same thermoelectric
coefficients for the same structures as in
Fig.~\ref{fig:ThermoelectricP}, but now edge roughness and positive
impurities are included in the calculation. The length of the channels
in this case is $2000~\mathrm{nm}$. A similar qualitative behavior is observed
as in Fig.~\ref{fig:ThermoelectricP} for both
channels. Quantitatively, however, the conductance in
Fig.~\ref{fig:ThermoelectricL}-a is now significantly reduced by a
factor of $\sim 15X$ (the dots correspond to the position of the peak
of the power factor of the devices without roughness and impurities in
Fig.~\ref{fig:ThermoelectricP}). The Seebeck coefficient in
Fig.~\ref{fig:ThermoelectricL}-b, on the other hand
increases. Finally, the peak of the power factor in
Fig.~\ref{fig:ThermoelectricL}-c reduces only slightly compared to the
peak of the power factor of the devices without edge roughness in
Fig.~\ref{fig:ThermoelectricP}-c (dots).
{\bf Thermoelectric Figure of Merit:} For the devices that include
rough edges, however, as we demonstrated in
Fig.~\ref{fig:NormalizedThermal}, the phonon thermal conductivity is
drastically reduced compared to the electronic thermal conductivity. A
large improvement is therefore expected in the $ZT$ figure of
merit. Figure \ref{fig:ZT} shows the $ZT$ figure of merit versus
energy at room temperature for the ELD-ZGNR(10,10), the
ELD-ZGNR(10,10) with impurities and roughness (red), and the
2ELD-ZGNR(8,4,8) (blue) with impurities and roughness. As indicated,
large values of $ZT$ can be achieved, especially in the case of the
device with two ELDs. The phonon lattice conductivity value used in
this calculation was extracted using the MFP method. Since as
explained above, that value could be $2X$ lower than the value
extracted from direct integration of the 'noisy' transmission, in the
inset of Fig.~\ref{fig:ZT} we show the $ZT$ versus energy using the
$\kappa_{\mathrm{l}}$ values extracted from the transmission. Indeed
the values could be reduced by a factor of $\sim 2X$, but still peak
$ZT$ values above 2 can be achieved at room temperature, which is
comparable and even better than the best thermoelectric materials to
date~\cite{Snyder08}. We note that as shown by Ref.~\cite{Sevincli10}
rough ZGNRs can have high $ZT$ values even without the presence of
ELDs. For this however, the asymmetry in the sharp edges of the higher
subbands is utilized at energies above $0.5~\mathrm{eV}$. Those
energies however, are too high and can not easily be reached. Finally
we mention here that our formalism has considered scattering only by
edge roughness and impurity scattering, whereas phonon scattering and
dephasing mechanisms are not included. However, as it is shown for 1D
NWs~\cite{Neophytou11}, the effects of impurity scattering and edge
roughness are the most important scattering effects in channels of
cross sections below $5~\mathrm{nm}$, and we expect this to hold also for GNRs
as well.
\section{Summary}
\label{s:Summary}
In this work we present a theoretical design procedure for achieving
high thermoelectric performance in zigzag graphene nanoribbon (ZGNRs)
channels, which in their pristine form have very poor performance. The
fully quantum mechanical non-equilibrium Green's function technique was
used for electron and phonon transport, and tight-binding and force
constant methods were used for the electronic and phonon bandstructure
descriptions. We show that by introducing extended line defects (ELDs)
in the length of the nanoribbon we can create an asymmetry in the
density of modes around the Fermi level, which improves the Seebeck
coefficient. ELDs increase the electronic conduction modes, which
increase the channel conductance as well. The power factor is
therefore significantly increased. In addition, we show that by
introducing edge roughness the phonon thermal conductivity
($\kappa_{\mathrm{l}}$) is drastically degraded much more than the
electronic thermal conductivity ($\kappa_{\mathrm{e}}$), or the
electronic conductance ($G$). These three effects result in large
values of the thermoelectric figure of merit, and indicate that
roughed ZGNRs with ELDs could potentially be used as efficient high
performance thermoelectric materials.
\section*{Acknowledgments}
This work, as part of the ESF EUROCORES program EuroGRAPHENE, was
partly supported by funds from FWF, contract I420-N16. This work was
also partly supported by the Austrian Climate and Energy Fund, contract
No. 825467.
|
{
"timestamp": "2012-03-12T01:01:10",
"yymm": "1203",
"arxiv_id": "1203.2032",
"language": "en",
"url": "https://arxiv.org/abs/1203.2032"
}
|
\section{Introduction}
It is widely believed that eruptions in solar atmosphere, such as flare, filament eruption and coronal mass ejection (CME), are the intermittent liberation of non-potential magnetic energy stored in coronal magnetic field. Magnetic helicity is a quantitative measure of twists, kinks, and inter-linkages of magnetic field lines (Berger \& Field 1984) and is a useful and important parameter to indicate topology and non-potentiality of a magnetic field system. Moreover, it is conserved in a closed volume, as well as in an open volume in the absence of boundary flows (Berger \& Field 1984). Since the solar corona is an open volume with the photosphere as a boundary with normal flux, the magnetic helicity can be transported from the sub-photosphere into the corona though the boundary of photosphere by emergence of helical flux and shuffling motions (Berger \& Field 1984). According to Berger (1999), the transport rate of relative helicity, $\dot{H}$, due to both processes are given by
\begin{equation}
\dot{H_{n}}=\oint2(\textbf{\textit{B}}_{t} \cdot
\textbf{\textit{A}}_{{p}}){\upsilon_{n}}dS
\end{equation}
\begin{equation}
\dot{H_{t}}=-\oint2(\mbox{\boldmath$\upsilon$}_{t}\cdot \textbf{\textit{A}}_{{p}})B_{{n}}dS
\end{equation}
where subscripts \textquotedblleft n\textquotedblright and \textquotedblleft t\textquotedblright represent the normal and tangential component, respectively. Chae (2001) has observationally determined $\dot{H_{t}}$ by regarding the horizontal motions $\mbox{\boldmath$\mu$}_{t}$ which deduced from a time series of line-of-sight magnetograms with the local correlation tracking (LCT) method (November \& Simon 1988) as the shuffling motions $\mbox{\boldmath$\upsilon$}_{t}$, and found that shuffling motions on photosphere can provide enough magnetic helicity for the CMEs, which was confirmed by a series of subsequent studies (Nindos \& Zhang 2002; Moon et al. 2002a,b; Nindos et al. 2003). It also has been proved that in some active regions (ARs), magnetic helicity transported by rotational motion is comparable with that deduced from LCT method (Zhang et al., 2008; Min et al., 2009). However, D$\acute{e}$moulin \& Berger (2003) have demonstrated that the horizontal motions $\mbox{\boldmath$\mu$}_{t}$ include both components $\mbox{\boldmath$\upsilon$}_{t}$ and $\mbox{\boldmath$\upsilon$}_{n}$ with the relation of $\mbox{\boldmath$\mu$}_{t}=\mbox{\boldmath$\upsilon$}_{t}-\frac{\upsilon_{n}}{B_{n}}\textbf{\textit{B}}_{t}$. Therefore, helicity injection calculated this way not only include the contribution of shuffling motions but also the emerging fluxes.
Emergence of twisted fluxes and shuffling motions are the popular triggers for solar eruptions and the common mechanism for magnetic helicity accumulation both in theoretical and observational works (Leka et al. 1996; Grigoryev \& Ermakova 2002; Fan \& Gibson 2004). Emergence naturally carries magnetic flux as well as helicity though the photosphere, if the emerging fluxes have magnetic helicity. While shuffling motions generate magnetic shear and then supply helicity and free energy into coronal field. But still now, we know a little about to what extent and how both processes contribute to the helicity accumulation.
It is worth noting that the presence or magnitude of flow along the field lines has no bearing on the temporal evolution of the magnetic field (as described by the ideal MHD induction equation), and also has no contribution to the helicity accumulation. So we should consider only the motion which is perpendicular to the magnetic field in the estimation of helicity accumulation. D$\acute{e}$moulin \& Berger (2003) have pointed out that there are two ways to get the normal velocity, by which we can calculate the magnetic helicity transport by flux emergence and shuffling motions separately. Firstly, since the magnetic field and velocity field are not mutually independent, we can deduce the normal velocity from known conditions under some assumptions. Secondly, if the AR locates around the center of solar disk, we can use Doppler velocity as the normal velocity. Advantages and disadvantages of both methods were discussed by many authors (D$\acute{e}$moulin \& Berger 2003; Pariat et al. 2005). So far, only the first method has been tried. Kusano et al. (2002) have deduced $\upsilon_{n}$ and $\mbox{\boldmath$\upsilon$}_{t}$ which only include component perpendicular to the magnetic field by using observations of $\textbf{\textit{B}}$ and $\mbox{\boldmath$\mu$}_{t}$ on photosphere together with the induction equation. They found that both processes have equal contribution to the helicity injection and have supplied magnetic helicity of opposite signs in AR NOAA 8210. Welsch et al. (2004) have introduced two methods to get the normal velocity. One is to deduce the $\mbox{\boldmath$\upsilon$}_{t}$ and $\mbox{\boldmath$\upsilon$}_{n}$ under an assumption of $\mbox{\boldmath$\upsilon$}\cdot\textbf{\textit{B}}=0$ and by a simple relation $\mbox{\boldmath$\upsilon$}_{t}=\mbox{\boldmath$\mu$}_{t}+\frac{\upsilon_{n}}{B_{n}}\textbf{\textit{B}}_{t}$ proposed by D$\acute{e}$moulin \& Berger (2003), and the other is to solve the same equations as Kusano et al. (2002) by different technique. Using the high spatial resolution vector magnetogram obtained by Helioseismic and Magnetic Imager (HMI) on broad the Solar Dynamics Observatory (SDO), Liu et al. calculated the velocity vector by the differential affine velocity estimator (DAVE) method, which models image motion with either the continuity equation or the convection equation (depending on a switch). And then they found that the helicity flux from shuffling term is dominant, while that from emergence term is small (private communication).
Based on a simple relation $\mbox{\boldmath$\upsilon$}_{t}=\mbox{\boldmath$\mu$}_{t}+\frac{\upsilon_{n}}{B_{n}}\textbf{\textit{B}}_{t}$ (D$\acute{e}$moulin \& Berger 2003), we find that the component of velocity which is perpendicular to the magnetic field can be deduced from the vector magnetograms and the horizontal motions (see detail in section 3). After projecting this velocity into tangential and normal component of photosphere, we can deduce the magnetic helicity transport by shuffling motions and flux emergence separately. We apply our new method to an isolated AR NOAA 10930. In order to decrease the ambiguities, we only study the day that the AR passed through the solar meridian. The paper was organized as follows. Section 2 is the description of the observations. The method of how to get the plasma velocity which is perpendicular to the magnetic field and data reduction are presented in Section 3. Section 4 are the observational results. Summary and discussion are presented in Section 5.
\section{Observations}
Figure 1 shows the morphological evolution of AR NOAA 10930 from December 08 to 13. It has a mature and stable leading sunspot, and a small and rapidly changed following sunspot. The temporal evolution of the sunspots area (thick (thin) line for leading (following) sunspot) and tilt angle are also shown in Figure 1. In order to distinguish opposite polarities in $\delta$ sunspot, only pixels whose intensity is weaker than 45\% of quiet sun are included for area estimation. The tilt angle is measured as angle between the connection of two polarities and the south direction. From Figure 1, we see that the leading sunspot was stable during its solar disk passage. The morphology of the following sunspot changed very rapidly according to emergence, cancellation and mergence, whereas its area changed a little until December 9. After that, there was a significant emergence. It slowed down very quickly at the beginning of December 10. Around 12:00 UT on December 10, the flux emergence reoccurred. Associated with the emergence, the following sunspot rotated counterclockwise around its center and moved eastward rapidly, resulting in an increase in magnetic complexity. These growing process continued till a X3.4 flare occurred on December 13.
AR NOAA 10930 passed through the solar meridian on 2006 December 11, with the location changing from W4S6 to E7S6. Meanwhile, the following sunspot area increased from $4.9\times10^{7}$ km$^{2}$ to $8.7\times10^{7}$ km$^{2}$, as a percentage of 35\% of the total area increase during its solar disk passage. The variations of tilt angle which due to the eastward motion of the following sunspot changed from 10$^{\circ}$ to 32$^{\circ}$, as a percentage of 50\% of the total increase of the tilt angle. The rotational speed of following sunspot increased up to $8^{\circ} h^{-1}$ on December 11. It maintained till a X3.4 flare occurred on December 13. And then the rotational speed slowed down to $3^{\circ} h^{-1}$ (Min et al. 2009). All the observational evidences show that December 11 was an important day for the AR evolution.
\begin{figure}
\centerline{\includegraphics[width=0.6\textwidth,clip=]{fig1_ar_evolution.eps} }
\caption{Images show AR NOAA 10930 evolution during its solar disk passage from December 8 to 14. Plots in bottom panels show temporal evolution of sunspots area and tilt angle. }
\end{figure}
\section{Method and Data Reduction}
\subsection{Deduction of Plasma Velocity Perpendicular to the Magnetic Field}
At first, we give a simple description of the relation among different velocities which are used in the present work. From observations, we can get the horizontal motions ($\mbox{\boldmath$\mu$}_{t}$) of the magnetic features by LCT method, which we indicate by purple arrow in Figure 2a. Shuffling motions of plasma on solar surface ($\mbox{\boldmath$\upsilon$}_{t}$) can be deduced from $\mbox{\boldmath$\mu$}_{t}$ by $\mbox{\boldmath$\upsilon$}_{t}=\mbox{\boldmath$\mu$}_{t}+\frac{\upsilon_{n}}{B_{n}}\textbf{\textit{B}}_{t}$. Normal velocity of plasma ($\mbox{\boldmath$\upsilon$}_{n}$) can be obtained by several ways as we introduced in previous section. Both $\mbox{\boldmath$\upsilon$}_{t}$ and $\mbox{\boldmath$\upsilon$}_{n}$
are outlined by red arrows in Figure 2a. The combined vector of them is the real motion of plasma ($\mbox{\boldmath$\upsilon$}$), which is represented by black arrow in Figure 2a. With reference to the direction of magnetic field, as shown by thick black arrow in Figure 2a, $\mbox{\boldmath$\upsilon$}$ can be divided into two components, one is parallel to the magnetic field ($\mbox{\boldmath$\upsilon$}_{\parallel}$) and the other is perpendicular to the magnetic field ($\mbox{\boldmath$\upsilon$}_{\perp}$), as we shown in Figure 2a by blue arrows. Since $\mbox{\boldmath$\upsilon$}_{\parallel}$ has no contribution to the magnetic helicity accumulation, we neglect it in the present work. Then, we get the normal velocity and shuffling motions of the plasma by dividing $\mbox{\boldmath$\upsilon$}_{\perp}$ into two components, one is tangential to the solar surface ($\mbox{\boldmath$\upsilon$}_{\perp t}$) and the other is normal to the solar surface ($\mbox{\boldmath$\upsilon$}_{\perp n}$), as we show by green arrows in Figure 2a. According to the vector relations depicted in Figure 2a, we get
\begin{equation}
\mbox{\boldmath$\upsilon$}_{\perp}=\mbox{\boldmath$\mu$}_{t}-(\mbox{\boldmath$\mu$}_{t}\cdot\textbf{\textit{b}})\textbf{\textit{b}}
\end{equation}
where \textbf{\textit{b}} is the direction vector of magnetic field. That means $\mbox{\boldmath$\upsilon$}_{\perp}$ can be deduced from $\mbox{\boldmath$\mu$}_{t}$ and $\textbf{\textit{B}}$.
\subsection{Data Reduction}
The Spectro-Polarimeter (SP) of Solar Optical Telescope (SOT) on board Hinode measures spectral profiles of full Stokes parameters of two Fe I lines at 630.15 and 630.25 nm and nearby continuum with 21.5 m$\mathring{A}$ spectral resolution (Kosugi et al. 2007; Suematsu et al. 2007; Ichimoto et al. 2007). On 2006 December 11, SP observed AR NOAA 10930 under fast map mode with the spatial resolution of 0.32$''$ and obtained 6 sets of Stokes I, Q, U and V. Vector magnetograms are derived from the inversion of the full Stokes profiles based on the assumption of Milne-Eddington (ME) atmosphere model. We use an automated ambiguity-resolution code based on the minimum energy algorithm (Leka et al. 2009) to resolve the 180$^\circ$ ambiguity of magnetograms.
The noise level is estimated by selecting one quiet region within the SP map to calculate 1$\sigma$ standard deviation of the average value of field strengths. In order to avoid ambiguity, we only analyze those pixels with $\textbf{\textit{B}}_{n}$ and $\textbf{\textit{B}}_{t}$ greater than the above deviation. Figure 2b shows a vector magnetogram on 11:10 UT. It shows that the arrows in leading sunspot are almost radial except the southwest part, as we outlined by white thick curve and labeled as A. In this region, the magnetic field is obviously left handed. Meanwhile, the following sunspot is also left handed, which corresponds to negative helicity for AR located in south hemisphere. There is so called \textquotedblleft magnetic channel\textquotedblright (Wang et al. 2008) structure along the neutral line. Magnetic field here is highly sheared, as indicated by arrows parallel to the neutral line.
The horizontal motions $\mbox{\boldmath$\mu$}_{t}$ were calculated by 2 minutes longitudinal magnetograms taken by the Narrowband Filter Imager (NFI) of SOT (Tsuneta et al. 2008) by LCT method, and is shown in Figure 2c. Our result shows the similar organized motions, such as inward motion in inner-penumbra and outward motion in outer-penumbra of the leading sunspot, and counterclockwise rotational motion and the eastward motion of the following sunspot, which are consistent with Min et al. (2009). Furthermore, horizontal motions in region A shows clockwise rotation, which correspond to positive helicity injection.
Figure 2d shows $\mbox{\boldmath$\upsilon$}_{\perp t}$ which deduced from $\mbox{\boldmath$\mu$}_{t}$ and \textit{\textbf{B}}. In this map, for the leading sunspot, the inward motion in inner-penumbra remains, while the outward motion in outer-penumbra disappears. Meanwhile, in the following sunspot, the counterclockwise rotational motion remains, while the eastward motion disappears. It means that horizontal motion which deduced by LCT method do include two components. One is the shuffling motion, which is the real horizontal motion of the plasma on the photosphere in more or less vertical magnetic field. It will drag the magnetic field associated with the plasma, and remains in the $\mbox{\boldmath$\upsilon$}_{\perp t}$ map. The other results from the flux emergence or submergence in nearly horizontal magnetic configuration and then footpoints of tubes move along the magnetic field. This motion will disappear from the $\mbox{\boldmath$\upsilon$}_{\perp t}$ map. As in our example, the outward motion in out-penumbra in leading sunspot and the eastward motion along the neutral line disappear from the $\mbox{\boldmath$\upsilon$}_{\perp t}$ map, which are motions resulting from the emergence since the sunspot is going through its growing phase. Meanwhile, the inward motion in inner-penumbra of leading sunspot and the counterclockwise rotation in following sunspot, which remain in $\mbox{\boldmath$\upsilon$}_{\perp t}$ map, are the shuffling motions of the plasma. Furthermore, in this map, the clockwise rotation in region A is more outstanding than that in $\mbox{\boldmath$\mu$}_{t}$ map.
The normal velocity, $\mbox{\boldmath$\upsilon$}_{\perp n}$, is shown in Figure 2e, where white tone represents upward motion and black one represents downward motion. It shows that the sunspot is dominated by upward motion. But at the boundary of penumbra and photosphere, the upward and downward motions are mixed, which presents a complex distribution of normal velocity. As a comparison, Doppler map, which is calculated from spectral line core fits to Fe I 630.15 nm line profile at each spatial position, is shown in Figure 2f. The pixel resolution of $\mbox{\boldmath$\upsilon$}_{\perp n}$ map is much lower than that of Doppler map, because it was deduced from $\mbox{\boldmath$\mu$}_{t}$ map. So more small features can be identified in Doppler map. However, roughly speaking both maps are consistent. Especially, there are some downward motions along the neutral line. Compared Figure 2e and 2f to 2b, we find that these down flow area are spatially corresponds to the magnetic channel.
\begin{figure}
\centerline{\includegraphics[width=0.8\textwidth,clip=]{fig2_para_map.eps} }
\caption{(a)A sketch to show the relation of different velocities which used in the present work. Purple arrow represents the horizontal motion $\mbox{\boldmath$\mu$}_{t}$. Red arrows represent the real motion of the plasma on solar surface $\mbox{\boldmath$\upsilon$}_{t}$ and the normal component of plasma motion $\mbox{\boldmath$\upsilon$}_{n}$. The combined vector of both is the real motion of the plasma ($\mbox{\boldmath$\upsilon$}$) as indicated by thin black arrow. With reference to the direction of magnetic field, as shown by thick black arrows, $\mbox{\boldmath$\upsilon$}$ can be divided into two components. One is parallel to the magnetic field ($\mbox{\boldmath$\upsilon$}_{\parallel}$) and the other is perpendicular to the magnetic field ($\mbox{\boldmath$\upsilon$}_{\perp}$), as we shown by blue arrows; (b) Vector magnetogram; (c) $\mbox{\boldmath$\mu$}_{\perp t}$; (d) $\mbox{\boldmath$\upsilon$}_{\perp t}$; (e) $\mbox{\boldmath$\upsilon$}_{\perp n}$; (f) Doppler velocity; (g) Map of helicity flow, $2(\textbf{\textit{B}}_{t} \cdot \textbf{\textit{A}}_{{p}}){\upsilon_{\perp n}}-2(\mbox{\boldmath$\upsilon$}_{\perp t}\cdot \textbf{\textit{A}}_{{p}})B_{{n}}$ (h) Map of helicity flow due to the vertical motion, $2(\textbf{\textit{B}}_{t} \cdot \textbf{\textit{A}}_{{p}}){\upsilon_{\perp n}}$; (i) Map of helicity flow due to the shuffling motion $-2(\mbox{\boldmath$\upsilon$}_{\perp t}\cdot \textbf{\textit{A}}_{{p}})B_{{n}}$
}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.4\textwidth,clip=]{fig3_helicity_ev_total.ps} }
\caption{Injection rates of helicity due to flux emergence (triangle signs), shuffling motions (star sings), and sum of both (diamond signs) for whole AR, following sunspot and leading sunspot, respectively.
}
\end{figure}
\section{Results}
Form Zhang et al. (2008) \& Park et al. (2010), we know the magnetic helicity injection in AR NOAA 10930 is predominantly negative throughout the period of its disk passage. Park et al. (2010) found that there are three time periods over which magnetic helicity injection is mainly positive for more than nine hours. The day we studied in the present work is within the second period of the positive helicity injection.
Figure 2g shows the distribution of $-2(\mbox{\boldmath$\upsilon$}_{\perp t}\cdot \textbf{\textit{A}}_{{p}})B_{{n}}dS+2(\textbf{\textit{B}}_{t} \cdot\textbf{\textit{A}}_{{p}}){\upsilon_{\perp n}}dS$ on 11:10 UT, which is a sum of local contribution by flux emergence and shuffling motions. It shows that the helicity injection in following sunspot is predominantly negative, which is consistent with the left handed twisted magnetic field and counterclockwise rotation of the sunspot. While in leading sunspot, both negative and positive helicity injection occurred. It is clearly seen that the positive helicity injection occurred at two separate regions. One is along the neutral line. The other is the region A.
The distribution of helicity injection by flux emergence and shuffling motions in Figure 2h and 2i demonstrate the origin of positive helicity injection. From both panels, we see that the positive helicity injection along the neutral line comes from $\dot{H_{n}}$, while, the other positive injection comes from $\dot{H_{t}}$ at the region A. The complex evolution properties along the neutral line was studied by several authors (Wang et al., 2008; Park et al., 2010), and was conjectured as a result of the emergence of a positive helicity system. In region A, the positive injection of the helicity in $\dot{H_{t}}$ map is consistent with the clockwise rotation of sunspot. While in $\dot{H_{n}}$ map, the negative helicity injection is consistent with the left handed twist of magnetic field.
The temporal evolution of $\dot{H}$ on December 11 was shown in Figure 3. Triangles indicate the contribution of flux emergence, stars represent the shuffling motions, and diamonds indicate the sum of both. The top panel shows that the flux emergence and shuffling motions contribute almost the same sign of magnetic helicity during whole period. The helicity injected by the flux emergence is almost negative which is consistent with the dominated left handed twist of the magnetic field for south-hemisphere AR. On the other hand, helicity injected by shuffling motion varies from time to time. It is negative until 12:00 UT and then becomes positive. The dynamical transient variation of $\dot{H}$ is consistent with $\dot{H}_{t}$, while the variation of $\dot{H}_{n}$ is a little bit stable. The variation tendency derived by us is almost the same as that by Park et al. (2010), although the exact value was not equal because of the differences of data type, observational time and calculation method.
Middle and bottom panels show the temporal evolution of $\dot{H_{t}}$ and $\dot{H_{n}}$ for following and leading sunspot, respectively. For the following sunspot, the helicity injections by the shuffling motion and flux emergence are always negative, and the helicity accumulation by both effects are $-178\times10^{40} Mx^{2}$ and $-164\times10^{40} Mx^{2}$, respectively. For the leading sunspot, the helicity injections caused by both effects are negative at first, and then associated with the flux emergence along the neutral line and the clockwise motion in the region A, it changes its sign from negative to positive. The helicity accumulation by the shuffling motion and flux emergence are $248\times10^{40} Mx^{2}$ and $46\times10^{40} Mx^{2}$, respectively. Here it is worth noting that the magnetic helicity accumulation in this active region is negative. And the period which we studied includes a positive helicity transport phase (Park et al., 2010). Even though as a whole, the magnetic helicity accumulation in this active region is $-48\times10^{40} Mx^{2}$. Our results show that the helicity accumulation mainly result from the following sunspot, but the dynamic variation mainly result from the leading sunspot. And also, even though the magnetic flux of the following sunspot is much smaller than that of the leading sunspot, it contributes most helicity accumulation because of the predominance of the negative helicity injection to the solar corona in this AR. Moreover the active region was located in the southern hemisphere and showed the left-handed (negative) helicity, violating the so-called hemispheric rule. So the characteristics of helicity evolution of the region may be specific to that category of ARs, which should be studied by further observations.
\section{Summary and Discussion}
By fully using the observational data, we have developed a new methodology by which we can estimate the magnetic helicity injection by flux emergence and shuffling motions, separately. The key point of the methodology is that we deduce the component of plasma velocity, which is perpendicular to the magnetic field, by observational data only. It provide us a useful tool to study the contribution of flux emergence and shuffling motion in energy storage and eruption initiation, especially after the launch of SDO. The derivation of the helicity injection is demonstrated for AR NOAA 10930. The observational properties of this AR during the period of our study are: a stable mature leading sunspot and a fast emerging and rapidly rotating following sunspot. Some important conclusions are obtained:
1. The sign of the helicity fluxes from flux emergence and shuffling motions are the same. The temporal variations of $\dot{H}_n$ is relatively stable, while $\dot{H}_t$ changes rapidly. The variation tendency of $\dot{H}$ is consistent with $\dot{H}_t$, implying that flux emergence provided the continuously accumulation of the helicity which may play a key role in helicity or energy storage, while the shuffling motions added some dynamic change of the helicity injection which may play a key role in eruption initiation.
2. For the following sunspot, helicity injection from flux emergence and shuffling motion are all negative. Helicity flux from flux emergence is relatively stable and compared with that from shuffling motions.
3. For the leading sunspot, helicity injection changes its sign from negative to positive for both effects, which is conjectured associated with emergence of a magnetic flux which contains opposite helicity and a clockwise rotation in southwest of the sunspot.
Usually, the rotational motion and the twisted magnetic field in sunspot occur simultaneously. We always want to know whether the observed rotation of sunspots represents the emergence of twisted magnetic flux tube, or the rotational shuffling motion twists of the magnetic flux tube. When the rotational motion represents the emergence of a twisted flux rope, one can expect that the motion results from the emergence will along the magnetic field. However in AR 10930, we find the rotational motion remains in $\mbox{\boldmath$\upsilon$}_{\perp t}$ map. So we conjecture that the rotational motion in AR NOAA 10930 is the real shuffling motion, although more observational evidence will be necessary for the final conclusion to the question stated above.
\acknowledgments
Y. Zhang wishes to address her sincere thanks to Dr. C. L. Jin for providing her code to revise the SP doppler velocity. We are grateful to the Hinode team for providing the wonderful data. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway).
|
{
"timestamp": "2012-03-12T01:01:46",
"yymm": "1203",
"arxiv_id": "1203.2096",
"language": "en",
"url": "https://arxiv.org/abs/1203.2096"
}
|
\section{Introduction\label{sec:Introduction}}
Dynamical dark matter (DDM)~\cite{DynamicalDM1,DynamicalDM2} is a new framework for
dark-matter physics in which the requirement of stability is replaced by a delicate
balancing between lifetimes and cosmological abundances across a vast ensemble of
individual dark-matter components. Due to the range of lifetimes and abundances
of these components, their collective behavior transcends that normally associated with traditional dark-matter candidates. In particular, quantities such as the total
dark-matter relic abundance, the proportional composition of the ensemble in terms
of its constituents, and the effective equation of state for the ensemble possess
a non-trivial time dependence beyond that associated with the expansion of the universe.
Indeed, from this perspective, DDM may be viewed as the most general possible framework for
dark-matter physics, and traditional dark-matter models are merely a limiting
case of the DDM framework in which the states which compose the dark sector are taken
to be relatively few in number and therefore stable.
In Ref.~\cite{DynamicalDM1}, we laid out the general theoretical features of the
DDM framework. By contrast, in Ref.~\cite{DynamicalDM2}, we presented an explicit
realization of the DDM framework: a model in which the particles which constitute the
dark-matter ensemble are the KK excitations of an axion-like field propagating in the
bulk of large extra spacetime dimensions. We demonstrated that this model has
all of the features necessary to constitute a viable realization of the general DDM
framework. In this paper, we complete our study
by performing a general analysis of all phenomenological constraints which are
relevant to this bulk-axion DDM model.
Although the analysis in this paper is primarily aimed at our specific DDM model,
many of our findings have important implications for
theories involving large extra dimensions in general. Furthermore, our analysis
can also serve as a prototype for phenomenological studies of theories in which
there exist large numbers of interacting and decaying particles.
It is important to emphasize why a general analysis of this sort is necessary,
given the existence of numerous prior studies
of the phenomenological and cosmological constraints on
axions and axion-like fields, unstable relics, and the physical properties of
miscellaneous dark-matter candidates.
As discussed in Refs.~\cite{DynamicalDM1,DynamicalDM2},
such studies are typically applicable to dark sectors involving one or only a few
fields, and are typically quoted in terms of limits on the mass, decay width,
or couplings of any individual such field. It is not at all obvious how such bounds
apply to a DDM {\it ensemble} --- a dark-matter candidate which is not
characterized by a well-defined single mass, decay width, or set of couplings.
For example, constraints on a cosmological population of unstable
particles derived from big-bang-nucleosynthesis (BBN) considerations
or bounds on distortions in the cosmic microwave background (CMB) are
generally derived under the assumption that such a population
comprises but a single particle species with a well-defined lifetime
and branching fractions. Such constraints are not directly
applicable to a DDM ensemble (in which lifetimes are balanced against
abundances), and must therefore be reexamined in this new context.
In this paper, we shall develop methods for dealing with these issues
and for properly characterizing the constraints on models in which the dark-matter
candidate is an ensemble of states rather than a single particle.
As we shall find, the presence of non-trivial mixings among the KK
excitations in our DDM model gives rise to a number of surprising effects
which are ultimately critical for its phenomenological viability. One
of these is a so-called ``decoherence''
phenomenon~\cite{DDGAxions,DynamicalDM1,DynamicalDM2} which helps to explain
how the dark matter in this model remains largely invisible to detection.
Another is a suppression, induced by this mixing, of the couplings between the
lighter particles in the dark-matter ensemble and the fields of the
Standard Model (SM). As we shall see, these effects assert themselves
in a variety of phenomenological contexts and play a crucial role in
loosening a battery of constraints which would otherwise prove extremely severe.
This paper is organized as follows.
In Sect.~\ref{sec:AxionsInED}, we briefly summarize the physics of axions in
extra dimensions and review the notational conventions
established in Ref.~\cite{DynamicalDM2}, which we once again adopt in this work.
In Sect.~\ref{sec:AbundanceConstraints}, we examine a number of processes, both
thermal and non-thermal in nature, which contribute to the generation of a
cosmological population of axions. We calculate the rates associated
with these processes and assess the relative importance of the associated production
mechanisms within different regions of model-parameter space.
In Sect.~\ref{sec:Bounds}, we then discuss the phenomenological, astrophysical, and
cosmological constraints relevant for bulk-axion DDM models and assess how
the parameter space of our model is bounded by each of these constraints. In
Sect.~\ref{sec:Combined}, we summarize the collective consequences of these
constraints on the parameter space of our bulk-axion DDM model. Finally,
in Sect.~\ref{sec:Conclusions}, we discuss the implications of our results for
future research.
\section{Generalized Axions in Extra Dimensions: A Review\label{sec:AxionsInED}}
In this section, we provide a brief review of the physics of generalized
axions in extra dimensions. (More detailed reviews can be found in
Refs.~\cite{DDGAxions,DynamicalDM2}.) By ``generalized axion,'' we mean any
pseudo-Nambu-Goldstone boson which receives its mass from instanton effects
related to a non-Abelian gauge group $G$ which confines at some scale $\Lambda_G$.
Note that the ordinary QCD axion~\cite{PecceiQuinn,WeinbergWilczekAxion}
is a special case of this, in which $G$ is
identified with $SU(3)$ color and $\Lambda_G$ is identified with
$\LambdaQCD\approx 250$~MeV. However, in this paper, we shall leave these
scales arbitrary in order to give our analysis a wider range of applicability.
We will also assume the existence of a global Abelian symmetry $U(1)_X$ which
plays the role played by the Peccei-Quinn symmetry $U(1)_{\mathrm{PQ}}$ in the
specific case of a QCD axion.
Our goal in this paper is to study the phenomenological constraints
that arise when a generalized axion is allowed to propagate in the
bulk~\cite{DDGAxions} of a theory with extra spacetime
dimensions~\cite{ADD,ADDPhenoBounds}.
In particular, we consider the case in which the axion propagates in a single,
large, flat extra dimension compactified on a $S_1/\IZ_2$ orbifold of
radius $R$. The fields of the SM are assumed to be restricted to a
brane located at $x_5=0$. We also assume that the additional
non-Abelian gauge group $G$ is restricted to the brane at $x_5=0$.
At temperatures $T \gg \Lambda_G$, the effective action for a bulk axion
in five dimensions can be written in
the form
\begin{equation}
S_{\mathrm{eff}} ~=~ \int d^4x\int_0^{2\pi R} dx_5
\left[\frac{1}{2}\partial_M a \partial^M a +
\delta(x_5)\,\big(\mathcal{L}_{\mathrm{brane}}+\mathcal{L}_{\mathrm{int}}\big)\right]~,
\end{equation}
where `$a$' denotes our five-dimensional axion field,
$\mathcal{L}_{\mathrm{brane}}$ contains the terms involving the brane fields alone,
and $\mathcal{L}_{\mathrm{int}}$ contains the interaction terms coupling the
brane-localized fields to the five-dimensional axion. The second of these terms is
given by
\begin{equation}
\mathcal{L}_{\mathrm{int}} ~=~
\frac{g_G^2\xi}{32\pi^2f_X^{3/2}} a \mathcal{G}_{\mu\nu}^a\tilde{\mathcal{G}}^{a\mu\nu}+
\sum_i\frac{c_i}{f_X^{3/2}}
(\partial_\mu a)\overline{\psi}_i\gamma^\mu\gamma^5\psi_i
+\frac{g_s^2 c_g^2}{32\pi^2f_X^{3/2}} a G_{\mu\nu}^a\tilde{G}^{a\mu\nu}
+\frac{e^2 c_{\gamma}}{32\pi^2f_X^{3/2}}
a F_{\mu\nu}\tilde{F}^{\mu\nu} +\ldots~,
\label{eq:HighT5DAxionAction}
\end{equation}
where $F_{\mu\nu}$, $G_{\mu\nu}^a$, and $\mathcal{G}_{\mu\nu}^a$ are the field strengths
respectively associated with the $U(1)_{\mathrm{EM}}$, $SU(3)$ color, and $G$ gauge
groups; $\tilde{F}_{\mu\nu}$, $\tilde{G}_{\mu\nu}^a$, and $\tilde{\mathcal{G}}^a_{\mu\nu}$ are
their respective duals; $e$, $g_s$, and $g_G$ are the respective coupling constants for
these groups; $f_X$ is the fundamental five-dimensional scale associated with the
breaking of the $U(1)_X$ symmetry; $c_\gamma$, $c_g$, and $c_i$ are coefficients
which respectively parametrize the coupling strength of the five-dimensional axion
field to the photon, gluon, and fermion fields of the SM; and $\xi$ is an $\mathcal{O}(1)$
coefficient which depends on the specifics of the axion model in question. Note that in
Eq.~(\ref{eq:HighT5DAxionAction}), we have displayed terms
involving only the light fields of the SM (\ie, the photon,
gluon, and light fermion fields), as couplings to the heavier SM fields will not
play a significant role in our phenomenological analysis.
The five-dimensional axion field can be
represented as a tower of four-dimensional KK excitations via the decomposition
\begin{equation}
a(x^\mu,x_5) ~=~ \frac{1}{\sqrt{2\pi R}}\sum_{n=0}^{\infty}
r_n a_n(x^\mu)\cos\left(\frac{nx_5}{R}\right)~,
\label{eq:AxionModeDecomp}
\end{equation}
where the factor
\begin{equation}
r_n ~\equiv~ \begin{cases}
1 &\mathrm{for~} n=0\\
\sqrt{2} &\mathrm{for~} n>0
\end{cases}
\label{eq:rmDef}
\end{equation}
ensures that the kinetic term for each mode is canonically normalized.
Substituting this expression into Eq.~(\ref{eq:HighT5DAxionAction}) and
integrating over $x_5$, we obtain
\begin{eqnarray}
S_{\mathrm{eff}} &=& \int d^4x\Bigg[\sum_{n=0}^\infty
\bigg(\frac{1}{2}\partial_\mu a_n \partial^\mu a_n
+\frac{g_G^2\xi}{32\pi^2\fhatX}
r_n a_n \mathcal{G}^a_{\mu\nu}\tilde{\mathcal{G}}^{a\mu\nu}
+\sum_i\frac{c_i}{\fhatX}
r_n(\partial_\mu a_n)\overline{\psi}_i\gamma^\mu\gamma^5\psi_i
\nonumber\\ & & ~~~~~~~~~~~
+~\frac{g_s^2c_g}{32\pi^2\fhatX}
r_n a_n G^a_{\mu\nu}\tilde{G}^{a\mu\nu}
+\frac{e^2 c_{\gamma}}{32\pi^2\fhatX}
r_n a_n F_{\mu\nu}\tilde{F}^{\mu\nu}\bigg)
-V(a)\Bigg]~,
\label{eq:HighT4DAxionAction}
\end{eqnarray}
where the axion potential is given by
\begin{equation}
V(a) ~=~ \sum_{n=0}^\infty\frac{1}{2}\frac{n^2}{R^2}a_n^2~,
\end{equation}
and where the quantity $\fhatX$, defined by the relation
\begin{equation}
\fhatX^2 ~\equiv~ 2 \pi Rf_X^3~,
\label{eq:fhatInTermsOff}
\end{equation}
represents the effective
four-dimensional $U(1)_X$-breaking scale. Note that each mode in the KK tower couples
to the SM fields with a strength inversely proportional to $\fhatX$.
Also note that at these scales, the axion mass-squared matrix
\begin{equation}
\mathcal{M}^2_{mn} ~\equiv~ \left.\frac{\partial^2 V(a)}{\partial a_m\partial a_n}
\right|_{\langle a\rangle}
\end{equation}
is purely diagonal.
At scales $T \lesssim \Lambda_G$, an additional contribution
to the effective axion potential arises due to
instanton effects. In this regime, the potential is modified to
\begin{equation}
V(a) ~=~ \sum_{n=0}^\infty\frac{1}{2}\frac{n^2}{R^2}a_n^2+
\frac{g_G^2}{32\pi^2}\Lambda_G^4
\left[1-\cos\left(\frac{\xi}{\fhatX}
\sum_{n=0}^\infty r_n a_n + \overline{\Theta}_G\right)\right]~,
\label{eq:InstantonPotential}
\end{equation}
where $\overline{\Theta}_G$ is the analogue of the QCD theta-parameter
$\overline{\Theta}$. This results in a modification of the axion
mass-squared matrix to
\begin{equation}
\mathcal{M}^2_{mn} ~=~ n^2M_c^2\delta_{mn}
+\frac{g_G^2\xi^2}{32\pi^2}\frac{\Lambda_G^4}{\fhatX^2}
r_mr_n~,
\label{eq:MassMixMatmn}
\end{equation}
where $M_c \equiv 1/R$ is the compactification scale.
This matrix above takes the form~\cite{DDGAxions}
\begin{equation}
\mathcal{M}^2 ~=~ \mX^2\left(\begin{array}{ccccc}
1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \ldots \\
\sqrt{2} & 2 + y^2 & 2 & 2 & \ldots \\
\sqrt{2} & 2 & 2+4y^2 & 2 & \ldots \\
\sqrt{2} & 2 & 2 & 2+9y^2 & \ldots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{array}\right)~,
\label{eq:AxionMassMatrixExplicit}
\end{equation}
where
\begin{equation}
y~\equiv~ \frac{M_c}{\mX}~~~~~~~ \mathrm{and}~~~~~~~
\mX^2 ~\equiv~ \frac{g_G^2\xi^2}{32\pi^2}\frac{\Lambda_G^4}{\fhatX^2}~.
\label{eq:DefsOfyandmPQ}
\end{equation}
The eigenvalues $\lambda^2$ of this mass-squared matrix are the
solutions to the transcendental equation
\begin{equation}
\frac{\pi\lambda\mX}{y}\cot\left(\frac{\pi\lambda}{\mX y}\right) ~=~ \lambda^2~.
\label{eq:TranscendentalEqForLambdas}
\end{equation}
The corresponding normalized mass eigenstates $a_\lambda$ are related to the
KK-number eigenstates $a_n$ via
\begin{equation}
a_\lambda ~=~ \sum_{n=0}^\infty U_{\lambda n} a_n
~\equiv~\sum_{n=0}^\infty\left(
\frac{r_n\widetilde{\lambda}^2}{\widetilde{\lambda}^2-n^2y^2}\right)A_\lambda a_n~,
\label{eq:DefOfalambda}
\end{equation}
where $\widetilde{\lambda}\equiv\lambda/\mX$. The dimensionless quantity $A_\lambda$ is
given by
\begin{equation}
A_\lambda ~\equiv~ \frac{\sqrt{2}}{\wtl}\left[1+ \wtl^2 + \pi^2/y^2\right]^{-1/2}~.
\label{eq:DefOfCapitalAlambda}
\end{equation}
and obeys the sum rules~\cite{DDGAxions}
\begin{equation}
\sum_\lambda A_\lambda^2 ~=~ 1~, ~~~~~~
\sum_\lambda \wtl^2A_\lambda^2 ~=~ 1~.
\label{eq:AlambdaSqdID}
\end{equation}
For $T\ll \hat{f}_X$, rewriting Eq.~(\ref{eq:HighT4DAxionAction}) in terms of the
$a_\lambda$ and expanding the axion potential given in
Eq.~(\ref{eq:InstantonPotential}) out to $\mathcal{O}(a_\lambda^6/\fhatX^6)$
yields the effective action
\begin{eqnarray}
S_{\mathrm{eff}} &=& \int d^4x
\Bigg[\sum_{\lambda}\bigg(\frac{1}{2}\partial_\mu a_\lambda
\partial^\mu a_\lambda -\frac{1}{2}\wtl^2\mX^2 a_\lambda^2
+\frac{e^2 c_{\gamma}\wtl^2A_\lambda}{32\pi^2\fhatX}
a_\lambda F_{\mu\nu}\tilde{F}^{\mu\nu}
+\frac{g_s^2 c_g\wtl^2A_\lambda}{32\pi^2\fhatX}
a_\lambda G_{\mu\nu}^a\tilde{G}^{\mu\nu a}
\nonumber\\ & & ~~~~~
+ \sum_i\frac{c_i\wtl^2A_\lambda}{\fhatX}
(\partial_\mu a_\lambda)\overline{\psi}_i\gamma^\mu\gamma^5\psi_i\bigg)
+\frac{g_G^2\xi^4\Lambda_G^4}{768\pi^2\fhatX^4}
\sum_{\lambda_i,\lambda_j,\lambda_k,\lambda_\ell}
\hspace{-0.4cm}\wtl_i^2\wtl_j^2\wtl_k^2\wtl_\ell^2
A_{\lambda_i}A_{\lambda_j}A_{\lambda_k}A_{\lambda_\ell}
a_{\lambda_i}a_{\lambda_j}a_{\lambda_k}a_{\lambda_\ell}
\Bigg]~.
\label{eq:ActionInMassEigenbasis}
\end{eqnarray}
Of course, the interaction term between the $a_\lambda$ and the gluon field
is only a useful description of the physics at temperatures above
the quark-hadron phase transition at $T \sim \LambdaQCD$. At
temperatures below this threshold, this interaction term
gives rise to an effective Lagrangian containing interactions
between the $a_\lambda$ and various hadrons, including the proton $p$,
the neutron $n$, and the charged and neutral pions $\pi^\pm$ and $\pi^0$.
This Lagrangian takes the form
\begin{eqnarray}
\mathcal{L}_{\mathrm{had}} &=& \wtl^2A_\lambda
\frac{C_{a\pi}}{f_\pi\fhatX}(\partial_\mu a_\lambda)
\Big[(\partial^\mu \pi^+)\pi^-\pi^0 + (\partial^\mu \pi^-)\pi^+\pi^0
-2(\partial^\mu \pi^0)\pi^+\pi^-\Big] ~+~
\wtl^2A_\lambda\frac{C_{an}}{\fhatX}(\partial_\mu
a_\lambda)\overline{n}\gamma^\mu\gamma^5n \nonumber\\ & & ~+~
\wtl^2A_\lambda\frac{C_{ap}}{\fhatX}(\partial_\mu
a_\lambda)\overline{p}\gamma^\mu\gamma^5p ~+~
i\wtl^2A_\lambda\frac{C_{a\pi N}}{f_\pi\fhatX}(\partial_\mu a_\lambda)
\Big[\pi^+\overline{p}\gamma^\mu n - \pi^-\overline{n}\gamma^\mu p\Big]~,
\label{eq:HadronAxionCouplings5D}
\end{eqnarray}
where the coefficients $C_{a\pi}$, $C_{an}$, \etc, depend on the details of the
theory. For example, for a ``hadronic'' QCD axion~\cite{KSVZ} (\ie, a QCD axion which
does not couple directly to the SM quarks), the coefficients $C_{ap}$ and $C_{an}$,
which determine the strength of the axion-nucleon-nucleon interactions, are
\begin{equation}
C_{ap} ~=~ 0.24 \left(\frac{z}{1+z}\right) +
0.15 \left(\frac{z-2}{1+z}\right) + 0.02~,
~~~~~~~~
C_{an} ~=~ 0.24 \left(\frac{z}{1+z}\right) +
0.15 \left(\frac{1-2z}{1+z}\right) + 0.02~,
\label{eq:CapAndCan}\\
\end{equation}
where $z=m_u/m_d\approx 0.56$ is the ratio of the up-quark and down-quark masses.
Likewise, the coefficients $C_{a\pi N}$ and $C_{a\pi}$ for such an axion are
\begin{equation}
C_{a\pi N} ~=~ \frac{1-z}{2\sqrt{2}(1+z)}~,~~~~~~~
~~~~~~C_{a\pi} ~=~ \frac{1-z}{3(1+z)}~,
\label{eq:CaNAndCapi}
\end{equation}
where $f_{\pi}\approx 93$~MeV is the pion decay constant and
$m_\pi\approx 135.0$~MeV is the neutral pion mass.
Before concluding this review,
we note that the effective coupling coefficients $c_\gamma$, $c_g$, and $c_i$
appearing in Eq.~(\ref{eq:HighT5DAxionAction}) are highly
model-dependent. They need not be $\mathcal{O}(1)$, and in many theories
any of them may vanish outright. Indeed it has been
argued~\cite{StringAxiverse} that the existence of axions and
axion-like fields which couple to electromagnetism but not to $SU(3)$
color is a generic feature of certain extensions of the SM, including
string theory. In assessing the constraints on our bulk-axion DDM model,
we shall therefore focus primarily on a ``photonic'' axion of this sort --- \ie,
a general axion with $c_g = 0$ and $c_\gamma \neq 0$. However, we shall also
discuss how such phenomenological constraints are modified in the
case of a so-called ``hadronic'' axion with non-vanishing values for both
$c_g$ and $c_\gamma$. We note that additional subtleties arise in this latter case,
due to non-trivial mixings between the $a_n$ and other pseudoscalars present in
the theory which also necessarily couple to $G_{\mu\nu}^a\widetilde{G}^{a\mu\nu}$.
These include hadrons such as $\pi^0$ and $\eta$, as well as any other axions
in the theory which play a role in
addressing the strong-CP problem~\cite{PecceiQuinn,WeinbergWilczekAxion}.
In discussing constraints on hadronic axions, we shall implicitly assume that the
full mass-squared matrix for the theory is such that the relationship between
the $a_n$ and the mass eigenstates $a_\lambda$ defined in
Eq.~(\ref{eq:DefOfalambda}) is not significantly disturbed.
Indeed, given the inherently large number of independent scales and couplings
that emerge in scenarios involving multiple axions and other pseudoscalars, this
is not an unreasonable assumption; moreover, it is straightforward to show in a
general way that these favorable conditions can always be arranged for certain
sets of axion and pseudo-scalar mixings. Such an assumption thereby enables us
to perform our phenomenological analysis in a model-independent way.
\section{Axion Production in the Early Universe\label{sec:AbundanceConstraints}}
Axions and axion-like fields can be produced via a number of different mechanisms
in the early universe. For example, these particles can be produced thermally,
via their interactions with the SM fields in the radiation bath. In
addition, a number of non-thermal mechanisms exist through which a sizable
population of axions also may be generated. These include production via
vacuum misalignment, production from the decays of cosmic strings and other
topological defects, and production from the out-of-equilibrium decays of
other, heavier fields in the theory. This last mechanism is particularly
relevant in the context of the DDM models, since, by assumption, the
dark sector in such models involves large numbers of unstable fields with
long lifetimes. Indeed, in the axion DDM model under consideration in this paper,
a non-thermal population of any $a_\lambda$ may be produced via the decays
of both heavier KK gravitons and other heavier $a_\lambda$.
In Ref.~\cite{DynamicalDM2}, we focused on misalignment production as the
primary mechanism responsible for establishing a cosmological population of
dark axions. In order for the results for the relic abundances $\Omega_\lambda$
of the $a_\lambda$ obtained there to be valid, the contributions from all
of the alternative production mechanisms mentioned above must be
subdominant for each $a_\lambda$. Therefore, in this section, we examine
each of the relevant axion-production mechanisms in turn, beginning with a
brief review of the results for misalignment production itself. Since
phenomenological constraints on scenarios involving large, flat extra dimensions
prefer that the reheating temperature $T_R$ associated with cosmic inflation be
quite low~\cite{ADDPhenoBounds}, we will hereafter operate within the context
of a low-temperature-reheating (LTR) cosmology with
$T_R \sim \mathcal{O}(\mathrm{MeV})$. Within such a cosmological context
and within the region of model-parameter space in which misalignment
production yields a total relic abundance $\Omegatot$ comparable to the observed
dark-matter relic abundance $\OmegaCDM$, we demonstrate that the contributions to each
$\Omega_\lambda$ from all other production mechanisms are indeed subdominant.
\subsection{Axion Production from Vacuum Misalignment\label{sec:MisalignmentProd}}
We begin our discussion of axion production in the early universe with
a brief review of the misalignment mechanism and its implications for
axion DDM models. (A more detailed discussion can be found in
Ref.~\cite{DynamicalDM2}.) As we shall see, this mechanism turns out
to be the dominant production mechanism for dark-matter axions in such models.
At temperatures $T\gg \Lambda_G$, the
only contributions to the axion mass-squared matrix are the
contributions from the KK masses. Since these contributions to
$\mathcal{M}^2$ are diagonal in the KK eigenbasis, no mixing occurs,
and the KK eigenstates are the mass eigenstates of the theory.
The potential for each $a_n$ with $n \neq 0$ is therefore
non-vanishing, due to the presence of the KK masses, and is
minimized at $a_n = 0$. However, the potential for the zero mode $a_0$
vanishes. In the absence of a potential for $a_0$, there is no preferred
vacuum expectation value (VEV) $\langle a_0\rangle$ which minimizes $V(a_0)$.
It therefore follows that immediately following the phase transition at
$T\sim \hat{f}_X$, the universe comprises a set of domains, each with
a different homogeneous background value for the axion field which may
be expressed in terms of a ``misalignment angle''
$\theta \equiv \langle a_0\rangle/\fhatX$.
This angle is generically expected to be $\mathcal{O}(1)$ in any particular
domain, but could also be smaller. We assume here that
$H_I \lesssim 2\pi \hat{f}_X$, where $H_I$ is the value of the Hubble parameter
during inflation, and therefore that the value of $\theta$ is
uniform over our present Hubble volume. In this case, we find that
\begin{equation}
\langle a_0\rangle = \theta\fhatX~, ~~~~~~~~~~~~~
\langle a_n\rangle = 0 \mathrm{~~~~for~~~} n\neq 0~.
\label{eq:a0initcondits}
\end{equation}
Note that the above discussion is strictly valid only in the limit in which the
Hubble volume is taken to infinity. In reality, the presence of a finite Hubble volume
limits our ability to distinguish fields with wavelengths larger than the Hubble
radius from true background values. Because of this ambiguity,
all $a_n$ for which $n/R \lesssim H_I$ can also acquire $\mathcal{O}(1)$
background values after $U(1)_X$ breaking.
In Sect.~\ref{sec:InflationScale}, we will analyze the phenomenological
consequences of this effect in detail and derive conditions under which
it can be safely neglected. As we shall demonstrate, it turns out that within our
preferred region of parameter space, these conditions involve only
mild restrictions on the cosmological context into which our model is
embedded. We will therefore assume from this point forward that the
$\langle a_n\rangle$ in our model are given by Eq.~(\ref{eq:a0initcondits}).
At temperatures down to $T \sim \Lambda_G$, the $\langle a_n\rangle$
remain fixed at these initial values. At lower temperatures, however,
the situation changes as instanton effects generate a potential for
the axion KK modes. Indeed, in the regime in which $T \ll \Lambda_G$ and
the brane mass engendered by this potential has attained the constant,
low-temperature value $\mX$ given in Eq.~(\ref{eq:DefsOfyandmPQ}), the
time-evolution of each field $a_\lambda$ is governed by an equation of the form
\begin{equation}
\ddot{a}_\lambda + \frac{\kappa}{t}\dot{a}_\lambda +
\Gamma_\lambda\dot{a}_\lambda + \lambda^2 a_\lambda ~=~ 0~,
\label{eq:TheDoubleDotEqnWithGammaLambda}
\end{equation}
where each dot denotes a time derivative, and where
\begin{equation}
\kappa ~\equiv~
\begin{cases}
3/2 &\mbox{in radiation-dominated (RD) eras}\\
2 & \mbox{in matter-dominated (MD) eras}~.
\end{cases}
\label{eq:DefOfkappaForH}
\end{equation}
When $\lambda\lesssim 3H/2$, the solution to this equation remains
approximately constant. This implies that the energy density stored in
$a_\lambda$ scales approximately like vacuum energy during this epoch. However,
at later times, when $\lambda\gtrsim 3H/2$, we see that $a_\lambda$
oscillates coherently around the minimum of its potential, with oscillations
damped by a ``friction'' term with coefficient $3H + \Gamma_\lambda$.
During this latter epoch, the energy density stored in $a_\lambda$ scales
like massive matter.
At temperatures $T \sim \Lambda_G$, the evolution of $a_\lambda$ depends more
sensitively on the explicit time-dependence of the brane mass $\mX(t)$.
In what follows, we adopt a ``rapid-turn-on'' approximation, in which
the instanton potential is assumed to turn on instantaneously at $t = t_G$,
where $t_G$ is the time at which the confining transition for the gauge
group $G$ occurs. In this approximation, $\mX(t)$ takes the form of a
Heaviside step function:
\begin{equation}
\mX(t) ~=~ \mX \Theta(t - \tG)~.
\label{eq:Heaviside}
\end{equation}
In this approximation, the $a_n$ remain
fixed at the initial values given in Eq.~(\ref{eq:a0initcondits}) so
long as $t < t_G$. At $t=t_G$, the brane mass immediately assumes its
constant, late-time value $\mX$. Since only $a_0$ is populated
immediately prior to the phase transition at $t_G$, each of the $a_\lambda$
initially acquires a background value proportional to
its overlap with $a_0$:
\begin{equation}
\langle a_\lambda(t_G)\rangle ~=~ \theta\fhatX A_\lambda~,~~~~~~
\langle \dot{a}_\lambda(t_G)\rangle ~=~ 0~.
\label{eq:alambdaInitCondits}
\end{equation}
Subsequently, after the $a_\lambda$ have been populated, each begins
oscillating at a characteristic time scale
\begin{equation}
t_\lambda ~\equiv~ \max\left\{\frac{\kappa_\lambda}{2\lambda},\tG\right\}~,
\label{eq:tlambdaInBothRegimes}
\end{equation}
where $\kappa_\lambda$ is the value of $\kappa$ corresponding to the epoch during
which this oscillation begins.
At late times $t \gg t_\lambda$, when these oscillations become rapid compared to
the rate of change of $\langle a_\lambda \rangle$ and the virial approximation is
therefore valid, one finds that the energy density $\rho_\lambda$ stored in each mode
is given by
\begin{equation}
\rho_\lambda(t) ~=~ \frac{1}{2}\theta^2\fhatX^2 \lambda^2
A_\lambda^2\left(\frac{t_\lambda}{t}\right)^{\kappa_\lambda}
e^{-\Gamma_\lambda(t-t_G)}
\label{eq:RhoOftEqnWithR}
\end{equation}
during the epoch in which the oscillation began. Computing $\rho_\lambda$ during
subsequent epochs is simply a matter of applying Eq.~(\ref{eq:RhoOftEqnWithR})
iteratively with the appropriate boundary conditions at each transition point.
Consequently, in the LTR cosmology, we have~\cite{DynamicalDM2}
\begin{equation}
\rho_\lambda^{\mathrm{LTR}}(t)~\approx~
\frac{1}{2}\theta^2\fhatX^2 \lambda^2 A_\lambda^2
e^{-\Gamma_\lambda(t-\tG)}\times\begin{cases}
\displaystyle \vspace{0.25cm}
\left(\frac{t_\lambda}{t}\right)^2~~ & t_\lambda~\lesssim~ t~\lesssim~\tRH \\
\displaystyle \vspace{0.25cm}
\left(\frac{t_\lambda^2}{\tRH^{1/2}\,t^{3/2}}\right)~~ & \tRH~\lesssim~t~\lesssim~\tMRE\\
\displaystyle
\left(\frac{t_\lambda^2\,\tMRE^{1/2}}{t^2\,\tRH^{1/2}}\right)~~ & t~\gtrsim~\tMRE~,
\end{cases}
\label{eq:RhoLambdaInLTRCosmo}
\end{equation}
where $\tRH$ denotes the reheating time --- \ie, the time at which $T = \TRH$, and the universe
transitions from an initial epoch of matter domination by the coherent oscillations of
the inflaton field to the usual radiation-dominated era.
Given the energy-density expression in
(\ref{eq:RhoLambdaInLTRCosmo}), it is straightforward to obtain the
relic abundance $\Omega_\lambda \equiv \rho_\lambda/\rhocrit$ for each
$a_\lambda$, where $\rhocrit \equiv 3M_P^2 H^2$. For the heavier modes
in the tower, for which $t_\lambda = \tG$, one finds
\begin{equation}
\Omega_\lambda^{\mathrm{LTR}} ~\approx~
3\left(\frac{\theta \fhatX \mX}{M_P}\right)^2
\tG^2
\left[1+\frac{\lambda^2}{\mX^2}+
\frac{\pi^2\mX^2}{M_c^2}\right]^{-1}
e^{-\Gamma_\lambda(t-\tG)}\times
\begin{cases}
\displaystyle\frac{1}{4} \vspace{0.25cm}~~
& 1/\lambda ~\lesssim~ t~ \lesssim~\tRH \\
\displaystyle\frac{4}{9}\left(\frac{t}{\tRH}\right)^{1/2} \vspace{0.25cm}~~
& \tRH ~\lesssim~ t ~\lesssim~ \tMRE \\
\displaystyle\frac{1}{4}\left(\frac{\tMRE}{\tRH}\right)^{1/2}~~
& t ~\gtrsim~ \tMRE~.
\end{cases}
\label{eq:OmegaLambdaOftEqnLTRtG}
\end{equation}
For the modes in the tower for which $t_\lambda > \tG$, the corresponding result is
\begin{equation}
\Omega_\lambda^{\mathrm{LTR}} ~\approx~
3\left(\frac{\theta \fhatX \mX}{M_P}\right)^2
\lambda^{-2}
\left[1+\frac{\lambda^2}{\mX^2}+
\frac{\pi^2\mX^2}{M_c^2}\right]^{-1}
e^{-\Gamma_\lambda(t-\tG)}\times
\begin{cases}
\displaystyle\frac{1}{4} \vspace{0.25cm}~~
& 1/\lambda ~\lesssim ~t ~\lesssim~ \tRH \\
\displaystyle\frac{4}{9}\left(\frac{t}{\tRH}\right)^{1/2} \vspace{0.25cm}~~
& \tRH ~\lesssim~ t ~\lesssim~ \tMRE \\
\displaystyle\frac{1}{4}\left(\frac{\tMRE}{\tRH}\right)^{1/2}~~
& t~\gtrsim ~\tMRE~.
\end{cases}
\label{eq:OmegaLambdaOftEqnLTRtlambda}
\end{equation}
The total contribution $\Omegatot$ to the dark-matter relic abundance from the
axion tower is simply the sum over these individual contributions.
While the generic behavior of $\Omegatot$ as a function of $\fhatX$, $M_c$, and
$\Lambda_G$ is somewhat complicated, simple analytical results can be obtained
in certain limiting cases of physical importance. For example, let us
consider the limit in which $t_\lambda = t_G$ for all modes in the tower and
$H_I$ is sufficiently large that none of the $a_\lambda$ which would otherwise
contribute significantly to $\Omegatot$ begin oscillating before the end of
inflation. In this limit, all of the $\Omega_\lambda$ take the form given in
Eq.~(\ref{eq:OmegaLambdaOftEqnLTRtG}), and one finds that the present-day value
of $\Omegatot$, here denoted $\Omegatotnow$, is given by the simple closed-form
expression~\cite{DynamicalDM2}
\begin{equation}
\Omegatotnow ~\approx~
\frac{3}{256\pi^2}(g_G\xi)^2\left(\frac{\theta \Lambda_G^2}{M_P}\right)^2
\tG^{3/2}\tMRE^{1/2}
\left(\frac{\tG}{\tRH}\right)^{1/2}~.
\label{eq:OmegaLambdaShelfLimit}
\end{equation}
In the opposite limit, when all of modes which contribute significantly toward $\Omegatotnow$
begin oscillating at $t_\lambda > t_G$ and have oscillation-onset times
which depend on $\lambda$ and are therefore staggered in time, $\Omega_\lambda$
is given by Eq.~(\ref{eq:OmegaLambdaOftEqnLTRtlambda}) for all $a_\lambda$. In this limit,
the expression for $\Omegatotnow$ reduces to~\cite{DynamicalDM2}
\begin{equation}
\Omegatotnow ~\approx~ \frac{3}{8} \left(\frac{\theta \fhatX}{M_P}\right)^2
\left(\frac{\tMRE}{\tRH}\right)^{1/2}~.
\label{eq:OmegaLambdaStaggeredLimit}
\end{equation}
The preferred region of parameter space from the perspective of dark-matter
phenomenology is that within which $\Omegatotnow$ represents an $\mathcal{O}(1)$
fraction of the dark-matter relic abundance inferred from WMAP data~\cite{WMAP}:
\begin{equation}
\Omega_{\mathrm{CDM}} h^2 ~=~ 0.1131 \pm 0.0034~,
\label{eq:OmegaWMAP}
\end{equation}
where $h \approx 0.72$ is the Hubble constant. From a dynamical dark-matter
perspective, it is also preferable that the full axion tower contribute
meaningfully to $\Omegatotnow$. For an $\mathcal{O}(1)$ value
of the misalignment angle $\theta$ and a reheating temperature
within the preferred range $T_R \sim 4 - 30$~MeV for theories with large
extra dimensions, one finds~\cite{DynamicalDM2} that these two conditions are realized
for $\fhatX \sim 10^{14} - 10^{15}$~GeV and $\Lambda_G \sim 10^2 - 10^5$~GeV,
provided that $M_c$ is small enough that $y \lesssim 1$.
Within this region of parameter space, the $t_\lambda$ of all $a_\lambda$
which contribute meaningfully toward $\Omegatotnow$ are staggered in time,
and therefore the lighter modes yield a proportionally greater contribution
to that total abundance. We will often focus our attention on
this particular region of parameter space when discussing constraints on
axion DDM models.
\subsection{Axion Production from Particle Decays\label{sec:IntraensembleProd}}
Another mechanism by which a non-thermal population of relic particles may
be generated in the early universe is through the decays of heavier,
unstable relics. In scenarios involving extra dimensions, these relics
include the higher KK modes of any fields which propagate within at
least some subspace of the extra-dimensional bulk. For example,
since the graviton field necessarily propagates throughout the entirety of the
bulk, a population of unstable KK gravitons
is a generic feature of all such scenarios.
In the minimal bulk-axion DDM model under consideration here, the
unstable relics whose decays can serve as a source for any given $a_\lambda$
include these KK gravitons as well as other, heavier $a_\lambda$. Moreover, since
these fields span a broad range of masses from the sub-eV to multi-TeV scale and
beyond, one would expect the population of axions produced by their collective
decays to possess by a highly non-trivial phase-space distribution.
However, as we shall demonstrate below, the total contribution
$\Gamma^{(\mathrm{IE})}$ to the decay rate of any $a_\lambda$ from
{\it intra-ensemble}\/ decays (\ie, decays to final states which include one or
more dark-sector fields in addition to any visible-sector fields that might also
be present) is far smaller than that from decays to final states involving
visible-sector fields alone.
That the total branching fraction for intra-ensemble decays is negligible
suggests that the population of axions produced by such decays will, in general,
be quite small. Thus, provided the initial abundances of the $a_\lambda$ are
set by some mechanism such as vacuum misalignment for which the $\Omega_\lambda$
of the heavier $a_\lambda$ are initially similar to or smaller than those of the
light fields, it is reasonable to assume that the contributions from intra-ensemble
decays are subleading and may therefore be safely neglected.
One class of processes which contribute to $\Gamma^{(\mathrm{IE})}$ are those which
arise due to the axion self-interactions implied by the final term in
Eq.~(\ref{eq:ActionInMassEigenbasis}). The leading such contribution comes
from three-body decay processes of the form
$a_\lambda\rightarrow a_{\lambda_1}a_{\lambda_2}a_{\lambda_3}$.
An upper bound on the total contribution $\Gamma(a_\lambda\rightarrow 3a)$
to the decay width of a given $a_\lambda$ from all kinematically allowed decays
of this form was derived in Ref.~\cite{DynamicalDM2}:
\begin{equation}
\Gamma(a_\lambda\rightarrow 3a) ~\leq~ \frac{g_G^4\xi^8}{45(4\pi)^7}
\frac{\lambda^4}{M_c^3}\left(\frac{\LambdaG}{\fhatX}\right)^8~.
\label{eq:Gammaato3a}
\end{equation}
It was also shown in Ref.~\cite{DynamicalDM2} that the partial width of the
$a_\lambda$ to a pair of photons is given by
\begin{equation}
\Gamma(a_\lambda\rightarrow \gamma\gamma) ~=~ G_\gamma (\wtl^2A_\lambda)^2
\frac{\lambda^3}{\fhatX^2}~,
\label{eq:PartialWidthToPhotons}
\end{equation}
with $G_\gamma \equiv c_\gamma^2\alpha^2/256\pi^3$, where
$\alpha \equiv e^2/4\pi$ is the fine-structure constant.
Within the preferred region of parameter space discussed above, in which
$\fhatX \sim 10^{14} - 10^{15}$~GeV and $\Lambda_G \gtrsim 10^2 - 10^5$~GeV,
we see that $\Gamma_\lambda(a\rightarrow 3a)$ is negligible compared to
$\Gamma(a_\lambda\rightarrow \gamma\gamma)$.
It then follows that $\Gamma_\lambda(a\rightarrow 3a)$
represents a vanishingly small contribution to the total decay width
$\Gamma_\lambda$ of any $a_\lambda$ in any theory with an $\mathcal{O}(1)$
value of $c_\gamma$. We therefore conclude that decays of the form
$a_\lambda\rightarrow a_{\lambda_1}a_{\lambda_2}a_{\lambda_3}$
do not play a significant role in the phenomenology of
realistic bulk-axion models of dynamical dark matter.
In addition to these decays, however, an additional set of decay channels --- those
involving lighter graviton or radion fields in the final state --- are also
open to the $a_\lambda$. In order to assess whether such decay
channels are capable of yielding a significant contribution to the relic abundance
of any of the $a_\lambda$, we begin by identifying the relevant interactions
among the modes in the KK graviton and axion towers.
Since we are considering the case of a flat extra dimension and assuming fluctuations
of the metric to be small, it is justified to work in the regime of linearized gravity.
The relevant term in the five-dimensional action is therefore
\begin{equation}
S ~=~ -\int d^4x\int^{2\pi R}_0 dy
\frac{1}{M_5^{3/2}}T_{MN}h^{MN}~,
\label{eq:Relevant5DActionTerm}
\end{equation}
where $T_{MN}$ is the stress-energy tensor, and $h_{MN}$ is the metric perturbation
defined according to the relation
\begin{equation}
g_{MN} ~=~ \eta_{MN} + \frac{2}{M_5^{3/2}}h_{MN}~.
\label{eq:LinearExpandgmunu}
\end{equation}
The piece of the stress-energy tensor which involves the five-dimensional axion field
$a$ includes both a bulk contribution and a contribution arising from terms in the
interaction Lagrangian which involve interactions of the axion with the brane-localized
fields of the SM. The bulk contribution is given by
\begin{eqnarray}
T_{MN}^{\mathrm{bulk}} ~=~
\partial_Ma\partial_Na -\frac{1}{2}\eta_{MN}(\partial_{P}a\partial^Pa)~.
\label{eq:TMNBulk}
\end{eqnarray}
Upon KK decomposition, this contribution, when coupled to
$h_{MN}$ as in Eq.~(\ref{eq:Relevant5DActionTerm}),
gives rise to three-point interactions between a KK graviton or radion
field and a pair of $a_\lambda$. These interactions lead to decays of the form
$a_\lambda\rightarrow G_{\mu\nu}^{(m)} a_{\lambda'}$, where $G_{\mu\nu}^{(m)}$
denotes a KK graviton with KK mode number $m$. In the absence of an instanton-induced
brane mass term $\mX$ for the axion field
(\ie, in the $\mX \rightarrow 0$ limit, in which
all mixing between the axion KK modes vanishes and $a_\lambda\rightarrow a_n$),
KK-momentum conservation would imply that only a single, marginal decay
channel would exist for each $a_\lambda$. Hence the contribution to $\Gamma_\lambda$
from such decays can be neglected. However, the instanton contribution to the axion
mass-squared matrix violates KK-momentum conservation, and therefore, despite the fact
that these axion-axion-graviton interactions are Planck-suppressed, they can still
potentially contribute significantly to $\Gamma_\lambda$, due to the large number
of modes into which each $a_\lambda$ can decay.
By contrast, the brane-localized contribution, which is given by
\begin{eqnarray}
T_{MN}^{\mathrm{brane}} &=& \delta(y)\delta_M^{\mu}\delta_N^\nu\bigg[
\frac{1}{2}\sum_i \frac{c_i}{f_X^{3/2}}
\Big[(\partial_\mu a) \overline{\psi}_i\gamma_\nu\gamma^5\psi_i +
(\partial_\nu a) \overline{\psi}_i\gamma_\mu\gamma^5\psi_i
-2\eta_{\mu\nu}(\partial_\rho a)\overline{\psi}_i\gamma^\rho\gamma^5\psi_i\Big]
\nonumber \\ & &
+~\frac{c_\gamma e^2}{32\pi^2 f_X^{3/2}}a \left(4\tilde{F}_{\mu\rho}F_{\nu}^{~\rho}
- \eta_{\mu\nu}\tilde{F}^{\rho\sigma}F_{\rho\sigma}\right)
+ \frac{\xi g_s^2}{32\pi^2 f_X^{3/2}}a \left(4\tilde{G}_{\mu\rho}^aG_{\nu}^{~\rho a}
- \eta_{\mu\nu}\tilde{G}^{\rho\sigma a}G_{\rho\sigma}^a\right)
\bigg]~,
\label{eq:TMNBrane}
\end{eqnarray}
leads to four-, five-, and six-point interactions between the graviton field,
the $a_\lambda$, and the various SM fields. These interactions take the same form
as those discussed in Section~\ref{sec:IntraensembleProd}, save that each vertex involves
the coupling of an additional KK graviton and is suppressed, relative to the
corresponding interaction involving the axion and SM fields alone, by an additional
factor of $M_P$. The rates for such interactions will therefore always be much
smaller than those calculated in Section~\ref{sec:IntraensembleProd}. Indeed, even the
total contribution to the decay rate of a given $a_\lambda$ from such processes, summed
over graviton KK modes, will still be suppressed by a factor of roughly $M_{5}$, where
$M_5$ denotes the five-dimensional Planck scale, relative to the contribution from decays
to SM fields alone. It is therefore sufficient, at least for our present purposes, to
neglect $T_{MN}^{\mathrm{brane}}$ and to focus solely on the interactions arising from
the bulk contribution $T_{MN}^{\mathrm{bulk}}$.
We begin our analysis of axion-axion-graviton interactions
by expanding the five-dimensional axion field, as well as the
various components $h_{\mu\nu}$, $h_{\mu 5}$, and $h_{55}$ of the metric
perturbation $h_{MN}$, in terms of KK modes. The mode expansion of the
axion field for the orbifold compactification considered here was given in
Eq.~(\ref{eq:AxionModeDecomp}); the mode expansions of
$h_{\mu\nu}$, $h_{\mu 5}$, and $h_{55}$ are analogously given by
\begin{eqnarray}
h_{\mu\nu} &=& \frac{1}{\sqrt{2\pi R}}\sum_{m=0}^\infty r_m
h_{\mu\nu}^{(m)}\cos\left(\frac{my}{R}\right)\nonumber\\
h_{\mu 5} &=& \frac{1}{\sqrt{2\pi R}}\sum_{m=1}^\infty r_m
h_{\mu 5}^{(m)}\sin\left(\frac{my}{R}\right)\nonumber\\
h_{55} &=& \frac{1}{\sqrt{2\pi R}}\sum_{m=0}^\infty r_m
h_{55}^{(m)}\cos\left(\frac{my}{R}\right)~.
\label{eq:KKModeDecompsDefs}
\end{eqnarray}
Note in particular that $h_{\mu 5}$ must be odd
with respect to the parity transformation $x_5 \rightarrow -x_5$.
Upon substituting these KK-mode decompositions into the linearized-gravity action
given in Eq.~(\ref{eq:Relevant5DActionTerm}) and integrating over
$y$, we find that the terms in the effective, four-dimensional interaction
Lagrangian which govern the interactions between the graviton and axion KK modes
consist of the following three contributions:
\begin{eqnarray}
\int_0^{2\pi R}\frac{h_{\mu\nu}T_{\mathrm{bulk}}^{\mu\nu}}{M_5^{3/2}}dy &=&
\sum_{m,n,p = 0}^\infty \frac{r_m r_n r_p}{4M_P} h^{(m)}_{\mu\nu}
\bigg[
\Big(2\partial^\mu a^{(n)}\partial^\nu a^{(p)}
-\eta^{\mu\nu} \partial^\rho a^{(n)}\partial_\rho a^{(p)}\Big)
\Delta_{mnp}^+\bigg.\nonumber\\ & & \bigg.
+\eta^{\mu\nu}\left(\frac{np}{R^2}\right)a^{(n)}a^{(p)}
\Delta_{mnp}^-\bigg]\nonumber\\
\int_0^{2\pi R}\frac{h_{55}T_{\mathrm{bulk}}^{55}}{{M_5^{3/2}}}dy &=&
\sum_{m,n,p = 0}^\infty \frac{r_mr_nr_p}{4M_P} h^{(m)}_{55}
\bigg[\partial^\rho a^{(n)}\partial_\rho a^{(p)}\Delta_{mnp}^+
+ \left(\frac{np}{R^2}\right)a^{(n)}a^{(p)}\Delta_{mnp}^-
\bigg]\nonumber\\
\int_0^{2\pi R}\frac{h_{\mu 5}T_{\mathrm{bulk}}^{\mu 5}}{M_5^{3/2}}dy &=&
\sum_{m = 1}^\infty \sum_{n,p = 0}^\infty \frac{r_mr_nr_p}{2M_P}\left(\frac{p}{R}\right)
h_{\mu 5}^{(m)}(\partial^\mu a^{(n)})a^{(p)}\Delta^-_{nmp}~,
\label{eq:IntTermsHmunu}
\end{eqnarray}
where
\begin{equation}
\Delta_{mnp}^\pm ~\equiv~ \Big[\delta_{m,n-p}+\delta_{m,p-n}\Big]\pm
\Big[\delta_{m,n+p}+\delta_{m,-n-p}\Big]~.
\end{equation}
For the purposes of computing Feynman diagrams, it is convenient to work in
the unitary gauge, in which the $h_{\mu 5}^{(m)}$ and $h_{55}^{(m)}$ fields
with $m>0$ are set to zero by the five-dimensional gauge transformations
$g_{MN}\rightarrow g_{MN} + \partial_M \epsilon_N + \partial_N \epsilon_M$,
where $\epsilon_M$ is the gauge parameter.
In this gauge, the contributions in the second and third line of
Eq.~(\ref{eq:IntTermsHmunu}) vanish (save for the interactions between
the axion KK modes and the radion field $h_{55}^{(0)}$, which will be
discussed in due time), and the physical, gauge-invariant graviton fields
\begin{equation}
G_{\mu\nu}^{(m)} ~\equiv~ h^{(m)}_{\mu\nu} + \left(\frac{R}{m}\right)
\Big[\partial_\mu h_{\nu 5}^{(m)} + \partial_\nu h^{(m)}_{\mu 5}\Big]
- \left(\frac{R^2}{m^2}\right)\partial_\mu\partial_\nu h_{55}^{(m)}
\label{eq:DefOfGmunu}
\end{equation}
reduce to $h_{\mu\nu}^{(m)}$ for all $m > 0$.
The relevant part of the effective Lagrangian consequently reduces to
\begin{equation}
\mathcal{L}_{\mathrm{int}}^{(m>0)} ~=~
-\sum_{m=1}^\infty\sum_{n,p=0}^\infty\frac{r_nr_p}{2\sqrt{2}M_P}
h_{\mu\nu}^{(m)}\bigg[
\Big(2\partial^\mu a^{(n)}\partial^\nu a^{(p)}
-\eta^{\mu\nu} \partial^\rho a^{(n)}\partial_\rho a^{(p)}\Big)
\Delta_{mnp}^+
+\eta^{\mu\nu}\left(\frac{np}{R^2}\right)a^{(n)}a^{(p)}
\Delta_{mnp}^-\bigg]~.
\label{eq:IntTermsGmunu}
\end{equation}
The expression in Eq.~(\ref{eq:IntTermsGmunu}) can be
rewritten in terms of the mass eigenstates $a_\lambda$ via the mixing
matrix $U_{\lambda n}$ in Eq.~(\ref{eq:DefOfalambda}). The result is
\begin{eqnarray}
\mathcal{L}_{\mathrm{int}}^{(m>0)} &=&
-\sum_{m=1}^\infty\sum_{n,p=0}^\infty\sum_{\lambda,\lambda'}
\frac{r_nr_p}{2\sqrt{2}M_P} h_{\mu\nu}^{(m)}U_{n\lambda}^\dagger
U_{p\lambda'}^\dagger\bigg[
\Big(2\partial^\mu a_\lambda\partial^\nu a_{\lambda'}
-\eta^{\mu\nu} \partial^\rho a_\lambda\partial_\rho a_{\lambda'}\Big)
\Delta_{mnp}^+
+\eta^{\mu\nu}\left(\frac{np}{R^2}\right)a_\lambda a_{\lambda'}
\Delta_{mnp}^-\bigg]~~\nonumber\\ &=&
-\sum_{m=1}^\infty\sum_{n=0}^\infty\sum_{\lambda,\lambda'}
\frac{r_n}{2\sqrt{2}M_P} h_{\mu\nu}^{(m)}
U_{n\lambda}^\dagger\Bigg\{\Bigg.
\Big(2\partial^\mu a_\lambda\partial^\nu a_{\lambda'}
-\eta^{\mu\nu} \partial^\rho a_\lambda\partial_\rho a_{\lambda'}\Big)
\nonumber\\ & & ~~\times~
\Big(r_{n-m}U^\dagger_{n-m,\lambda'} + r_{n+m}U^\dagger_{n+m,\lambda'}
+r_{m-n}U^\dagger_{m-n,\lambda'} + r_{-n-m}U^\dagger_{-n-m,\lambda'}\Big)
+\eta^{\mu\nu} \frac{n}{R^2} a_\lambda a_{\lambda'}\nonumber\\ & & \Bigg.
~~~~\times~
\Big[(n-m)\big(r_{n-m}U^\dagger_{n-m,\lambda'} + r_{m-n}U^\dagger_{m-n,\lambda'}\big)
+(n+m)\big(r_{n+m}U^\dagger_{n+m,\lambda'}+r_{-n-m}U^\dagger_{-n-m,\lambda'}\big)\Big]
\Bigg\}~,
\label{eq:IntLagGmunuLambdaBasis}
\end{eqnarray}
where in going from the first equality to the second we have exploited the Kronecker
deltas in $\Delta^{\pm}_{mnp}$ to evaluate the sum over $p$. It should be noted that
in the notation employed in the above expression, $U_{n\lambda}^\dagger = 0$ by
definition for $n<0$. The sum over $n$ in Eq.~(\ref{eq:IntLagGmunuLambdaBasis})
can also be performed analytically, and the resulting, final expression for the
Lagrangian in terms of the $a_\lambda$ is found to be
\begin{equation}
\mathcal{L}_{\mathrm{int}}^{(m>0)} ~=~
-\sum_{m=1}^\infty\sum_{\lambda,\lambda'}
\frac{1}{2\sqrt{2}M_P} h_{\mu\nu}^{(m)}\bigg[
\Big(2\partial^\mu a_\lambda\partial^\nu a_{\lambda'}
-\eta^{\mu\nu} \partial^\rho a_\lambda\partial_\rho a_{\lambda'}\Big)
C^{(1)}_{m\lambda\lambda'}
+\eta^{\mu\nu}M_c^2a_\lambda a_{\lambda'}
C^{(2)}_{m\lambda\lambda'}\bigg]~,
\label{eq:IntLagGmunuLambdaBasisFinal}
\end{equation}
where the coefficients $C^{(1)}_{m\lambda\lambda'}$ and $C^{(2)}_{m\lambda\lambda'}$
are given by
\begin{eqnarray}
C^{(1)}_{m\lambda\lambda'} &=&
\frac{-8m^2y^2\wtl^2\wtl'^2A_\lambda A_{\lambda'}}
{m^4y^4-2m^2y^2(\wtl^2+\wtl'^2)+(\wtl^2-\wtl'^2)^2}\nonumber\\
C^{(2)}_{m\lambda\lambda'} &=&
\frac{4\wtl^2\wtl'^2[m^2y^2(\wtl^2+\wtl'^2)-(\wtl^2-\wtl'^2)^2]
A_\lambda A_{\lambda'}}
{y^2[m^4y^4-2m^2y^2(\wtl^2+\wtl'^2)+(\wtl^2-\wtl'^2)^2]}~.
\end{eqnarray}
From the interaction Lagrangian in Eq.~(\ref{eq:IntLagGmunuLambdaBasisFinal}), it
is straightforward to obtain the Feynman rule for the graviton-axion-axion
interaction vertex in the unitary gauge:
\begin{eqnarray*}
\resizebox{1.65in}{!}{\includegraphics{FeynmanRules-ax-ax-grav-lambdabasis.eps}}
&\raisebox{1.75cm}{$\displaystyle ~=~
-\frac{i}{\sqrt{2}M_P}\bigg[
\left(k_{1\mu}k_{2\nu}+k_{1\nu}k_{2\mu}-\eta_{\mu\nu}k_1\cdot k_2\right)
C^{(1)}_{m\lambda\lambda'} -\eta_{\mu\nu}M_c^2C^{(2)}_{m\lambda\lambda'}
\bigg]$~.}
\end{eqnarray*}
Using this vertex rule along with the graviton-polarization sum
rule given in Refs.~\cite{GravFeynmanRulesGiudice,GravFeynmanRulesHan},
we find
\begin{eqnarray}
|\mathcal{M}(a_\lambda\rightarrow h_{\mu\nu}^{(m)}a_{\lambda'})|^2&=&
\frac{\big(C^{(1)}_{m\lambda\lambda'}\big)^2}{12 M_P^2 (mM_c)^4}
\Big[\lambda^4+(mM_c)^4+\lambda'^4-2(mM_c)^2\lambda^2
-2(mM_c)^2\lambda'^2-2\lambda^2\lambda'^2\Big]^2\nonumber\\ & = &
\frac{16}{3}\left(\frac{\mX^4}{M_P^2}\right)(\wtl^2A_\lambda)^2
(\wtl'^2A_{\lambda'})^2
\end{eqnarray}
for $\lambda' < \lambda$.
Consequently, the partial width of $a_\lambda$ from such a decay is
\begin{equation}
\Gamma(a_\lambda\rightarrow h_{\mu\nu}^{(m)}a_{\lambda'}) ~=~
\frac{\mX^4}{3\pi\lambda^3M_P^2}
(\wtl^2A_\lambda)^2(\wtl'^2A_{\lambda'})^2
\Big[\lambda^4+(mM_c)^4+\lambda'^4-2(mM_c)^2(\lambda^2
+\lambda'^2)-2\lambda^2\lambda'^2\Big]^{1/2}~.
\label{eq:GammaToGravAx}
\end{equation}
Once again, in order to obtain the full contribution
$\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)$ to $\Gamma_\lambda$
from decays of the form $a_\lambda\rightarrow h_{\mu\nu}^{(m)}a_{\lambda'}$,
it is necessary to sum over all combinations of
final-state graviton and axion modes which are kinematically accessible.
As before, we will approximate the mode
sums over both $m$ and $\lambda'$ as integrals. This yields the result
\begin{eqnarray}
\Gamma(a_\lambda\rightarrow h_{\mu\nu}a) &\lesssim&
\frac{4\mX^4(\wtl^2A_\lambda)^2}{3\pi\lambda^3 M_c M_P^2}
\int_{\lambda_0}^\lambda d\lambda'\int_0^{(\lambda - \lambda')/M_c}
dm (\wtl'^2A_{\lambda'})^2\nonumber \\
& & ~~~~~~\times
\Big[\lambda^4+(mM_c)^4+\lambda'^4-2(mM_c)^2(\lambda^2
+\lambda'^2)-2\lambda^2\lambda'^2)\Big]^{1/2}\nonumber \\
& = & \frac{8\mX^4(\wtl^2A_\lambda)^2}{9\pi\lambda^3 M_c^2 M_P^2}
\int_{\lambda_0}^\lambda d\lambda' (\wtl'^2A_{\lambda'})^2(\lambda+\lambda')
\Bigg[(\lambda^2 +\lambda'^2)
E\left(\frac{(\lambda-\lambda')^2}{(\lambda+\lambda')^2}\right)
-2\lambda\lambda'
K\left(\frac{(\lambda-\lambda')^2}{(\lambda+\lambda')^2}\right)
\Bigg]~,~~~~~
\label{eq:GammaaatoGaSum}
\end{eqnarray}
where $K(x)$ and $E(x)$ denote the complete elliptic integrals of the
first and second kind, respectively:
\begin{equation}
K(x) ~=~ \int_0^{\pi/2}\frac{d\theta}{\sqrt{1-x^2\sin^2\theta}}~,
~~~~~~~~
E(x) ~=~ \int_0^{\pi/2}d\theta\sqrt{1-x^2\sin^2\theta}~.
\end{equation}
\begin{figure}[b!]
\centerline{
\epsfxsize 3.0 truein \epsfbox {GravToPhotRatPlot.eps} }
\caption{The ratio
$\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)/
\Gamma(a_\lambda\rightarrow \gamma\gamma)$, shown
as a function of $\lambda$ for several different values
of $\fhatX$. Here we have set $\Lambda_G = 1$~TeV and
$\xi = g_G = 1$, and we have taken the
compactification scale to be $M_c = 10^{-11}$~GeV. It is
clear from this plot that this ratio is safely below unity
for $\fhatX$ within our preferred region
$10^{14} - 10^{15}$~GeV.
\label{fig:GravitonToPhotonWidthRatio}}
\end{figure}
In order to compare $\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)$ to the
rate for $a_\lambda$ decays to SM fields, we numerically integrate
Eq.~(\ref{eq:GammaaatoGaSum}) over $\lambda'$ and compare the
resulting expression to the decay rate
$\Gamma(a_\lambda\rightarrow \gamma\gamma)$ to photon pairs.
In Fig.~\ref{fig:GravitonToPhotonWidthRatio}, we plot the ratio
$\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)/
\Gamma(a_\lambda\rightarrow \gamma\gamma)$ as a function of
$\lambda$ for a variety of different choices of $\fhatX$. In
each case, we have set $\Lambda_G = 1$~TeV, $M_c = 10^{-11}$~GeV, and
$\xi = g_G = 1$. It is evident from this plot that only for values of
$\fhatX$ above the preferred range $\fhatX \sim 10^{14} - 10^{15}$~GeV
does the decay rate for $a_\lambda\rightarrow h_{\mu\nu}a_{\lambda'}$
become similar in magnitude to the rate for axion decays into brane fields.
Indeed, for values $\fhatX$ within this preferred range,
$\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)/\Gamma(a_\lambda\rightarrow \gamma\gamma)$
never exceeds 0.06, even for the lightest modes in the tower. Furthermore,
for values of $\fhatX$ of this magnitude, the lifetimes for all $a_\lambda$
light enough to have
$\Gamma(a_\lambda\rightarrow h_{\mu\nu}a)/\Gamma(a_\lambda\rightarrow \gamma\gamma)$
near this maximal value are parametrically larger than the present age
of the universe, even once the additional contribution to
$\Gamma_\lambda$ from $a_\lambda\rightarrow h_{\mu\nu}a_{\lambda'}$
decays is taken into account. Consequently, the decays of such
fields are not cosmologically relevant, and since the branching
fraction for all other, heavier $a_\lambda$ into final states involving
KK gravitons is utterly negligible. We therefore conclude that intra-ensemble
decays are not phenomenologically relevant for bulk-axion models of
dynamical dark matter.
Up to this point, we have focused chiefly on the effect of the tensor
KK modes of the higher-dimensional graviton field on axion production
in the early universe. However, we have yet to address the effect
of graviscalars such as the radion on axion production.
Since our minimal DDM model involves only a single
extra dimension, only a single physical graviscalar mode (proportional to
$h_{55}^{(0)}$) appears in the theory. Furthermore, while the masses
of the $h_{\mu\nu}^{(m)}$ are dictated by the compactification geometry
alone, the mass of this radion field depends on the details of the
mechanism through which the radius of the extra dimension is stabilized, and
is consequently highly model-dependent. In this paper, we assume that
the physical radion field is sufficiently heavy so as not to play a
significant role in the decay phenomenology of the light $a_\lambda$ fields
which contribute significantly to $\Omegatotnow$. Nevertheless, we note
that in scenarios which involve multiple extra dimensions of comparable size, or
scenarios in which a specific model for radius stabilization is invoked,
graviscalars may play a more significant role in the phenomenology of the dark
sector.
\subsection{Axion Production from Cosmic Strings\label{sec:CosmicStringProd}}
A population of cold axions can also be generated by the decays of topological defects.
In our axion DDM model, this includes decays of the cosmic strings associated with the
breaking of the global $U(1)_{X}$ symmetry. Such decays are relevant in situations in
which this symmetry remains unbroken until after inflation, \ie, $H_I \gtrsim 2\pi\fhatX$.
By contrast, in situations in which $H_I \lesssim 2\pi\fhatX$ and
the $U(1)_X$ is spontaneously broken prior to the inflationary epoch, cosmic
strings and other topological defects are washed out by the rapid expansion of
the universe during cosmic inflation. Consequently, in this latter case,
axion production from the decays of cosmic strings can safely be ignored.
In this paper, we are primarily interested in high values of
$\fhatX \sim 10^{14} - 10^{15}$~GeV, as these values characterize our preferred
region of parameter space. Likewise, we will primarily be interested in
relatively low values of $H_I$, which may be realized naturally in the LTR cosmology.
For this reason we shall assume that $H_I \lesssim 2\pi\fhatX$ in what follows.
We see, then, that no significant population of axions is produced by
cosmic-string decay.
\subsection{Axion Production from the Thermal Bath\label{sec:ThermalProd}}
Another mechanism through which a relic population of axions may be produced
in the early universe is a thermal one: via their interactions with the
SM fields in the radiation bath. Unlike the axion population generated by
vacuum misalignment, which is characterized by a highly non-thermal velocity
distribution (essentially that of a Bose-Einstein condensate) and is therefore
by nature cold, this population is characterized by a thermal velocity distribution.
Indeed, the properties of a thermal population of axions can differ substantially
from that of a population of axions generated via misalignment production.
A number of processes contribute to thermal axion production in the early
universe, and the processes which are the most relevant for the production of
standard axions dominate for each $a_\lambda$ in this scenario as well.
Among hadronic processes, which play an important role in axion production
when $c_g$ is non-vanishing, $q \gamma \rightarrow q a_\lambda$ and
$q g \rightarrow q a_\lambda$ dominate for $T \gtrsim \LambdaQCD$, while
pion-axion conversion off nuclei (including all processes of the form
$N \pi\rightarrow N' a_\lambda$, where $N,N' = \{n,p\}$ and $\pi$ denotes
either a charged or neutral pion) and
the purely pionic process $\pi \pi \rightarrow \pi a_\lambda$ dominate at lower
temperatures. The rate for the high-temperature process is~\cite{QGPAxionProductionRate}
\begin{equation}
\Gamma(q \gamma \rightarrow q a_\lambda)~=~\frac{g_s^2T^3\wtl^4 A_\lambda^2}
{64\pi^5\fhatX^2} \ln\left[\left(\frac{T}{m_g}\right)^2 + 0.406\right],
\end{equation}
where $m_g$ is the plasma mass
for the gluon, given in terms of the effective number of quark flavors $N_f$ at
temperature $T$ by
\begin{equation}
m_g(T) ~=~ \frac{g_s T}{3}\sqrt{3+N_f/2}.
\end{equation}
Likewise, the rates for pion-conversion off nuclei and pionic production are well
estimated by the
expressions~\cite{ChangChoiThermalNucleonRate,RaffeltPionAxionCrossSection,LTRAxionsKamionkowski}
\begin{eqnarray}
\Gamma(N \pi \rightarrow N' a_\lambda)&=&\frac{T^{7/2}m_N^{3/2}\wtl^4 A_\lambda^2e^{-m_N/T}}
{6\zeta(3)(2\pi)^{5/2}\fhatX^2f_\pi^2}
\Big[1.64\big(5C_{an}^2+5C_{ap}^2+2C_{an}C_{ap}\big)+6C_{a\pi N}^2\Big]
\int_0^\infty dx_1\frac{x_1 y_1^3}{e^{y_1}-1}\nonumber\\
\Gamma(\pi \pi \rightarrow \pi a_\lambda)&=&
\frac{3\zeta(3)T^5 C_{a\pi}^2\wtl^4 A_\lambda^2}
{1024\pi^7\fhatX^2f_\pi^2}
\int_0^\infty\int_0^\infty \frac{dx_1 dx_2 x_1^2x_2^2}{y_1y_2(e^{y_1}-1)(e^{y_2}-1)}
\int_{-1}^{1} d\mu \frac{(s-m_\pi^2)^3(5s-2m_\pi^2)}{s^2T^4}~,
\end{eqnarray}
where, once again, $\zeta(x)$ denotes the Riemann zeta function, and the effective
coupling coefficients $C_{ap}$, $C_{an}$, $C_{a\pi}$, and $C_{a\pi N}$ are given in
Eqs.~(\ref{eq:CapAndCan}) and~(\ref{eq:CaNAndCapi}). Since these processes are
mediated by strong interactions, they tend to dominate the production rate for a
hadronic axion at temperatures $T\gtrsim 100$~MeV, at which the number densities
of pions and other hadronic species are unsuppressed.
In addition to these hadronic processes, there are several process
involving the interactions between the $a_\lambda$ and the $e^\pm$ and photon fields
which contribute to the axion production rate, and indeed dominate that rate
at temperatures $T \ll \LambdaQCD$. The first of these is the inverse-decay process
$\gamma\gamma \rightarrow a_\lambda$, the rate for which is given by
\begin{equation}
\Gamma(\gamma\gamma\rightarrow a_\lambda) ~=~
2\frac{\lambda^5 G_\gamma(\wtl^2 A_\lambda)^2}{\zeta(3)\fhatX^2 T^2}
K_1\left(\frac{\lambda}{T}\right)
\label{eq:InverseDecayRate}
\end{equation}
where $K_1(x)$ and $K_2(x)$ respectively denote the Bessel function of the
first and second kind, and $G_\gamma = \alpha^2c_\gamma^2/256\pi^2$. Another
is the Primakoff process $e^\pm \gamma \rightarrow e^\pm a$. For
$T,m_e \gg \lambda$, the rate for this process
is well approximated by~\cite{QEDPlasmaAxionProductionRate}
\begin{equation}
\Gamma_{\mathrm{Prim}}(e^\pm \gamma \rightarrow e^\pm a_\lambda) ~=~
\frac{\alpha^3c_\gamma^2 n_e}{192\zeta(3)\fhatX^2}\wtl^4 A_\lambda^2
\left[\ln\left(\frac{T^2}{m_\gamma^2}\right)+0.8194\right]~,
\label{eq:ElectronPrimakoff}
\end{equation}
where the plasma mass $m_\gamma$ of the photon is given by $m_\gamma = eT/3$.
In the approximation of vanishing chemical potential,
the number density of electrons (plus positrons) $n_e$ takes the well-known form
\begin{equation}
n_e ~=~ \begin{cases}
\displaystyle \frac{3\zeta(3)}{\pi^2}T^3, ~~~~~& T \gtrsim m_e\vspace{0.25cm}\\
\displaystyle 4\left(\frac{T m_e}{2\pi}\right)^{3/2}e^{-m_e/T} ~~~~~& T \lesssim m_e~.
\end{cases}
\end{equation}
Finally, if $c_e\neq 0$ in Eq.~(\ref{eq:HighT4DAxionAction}) and the axion couples
directly to the electron field, there can be an additional contribution
to the $e^\pm \gamma \rightarrow e^\pm a$ rate from a process akin to Compton
scattering, but with an axion replacing the photon in the final state. The rate
for this process can be estimated as~\cite{RaffeltSubMeV}:
\begin{equation}
\Gamma_{\mathrm{Comp}}(e^\pm \gamma \rightarrow e^\pm a_\lambda) ~\sim~
\frac{4 \alpha c_e^2 n_e}{\fhatX^2}(\wtl^2 A_\lambda)^2 \times
\begin{cases}
\displaystyle\frac{m_e^2}{T^2} ~~& T \gtrsim m_e\vspace{0.25cm}\\
1 ~~& T \lesssim m_e~.
\end{cases}
\end{equation}
\begin{figure}[th!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {ThermalAxionRates.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {ThermalAxionRatesTot.eps}
\end{center}
\caption{A comparison of the rates associated with different axion-production
processes in the early universe. Here we have taken $M_c = 10^{-11}$~GeV,
$\fhatX =10^{15}$~GeV, $\Lambda_G = 1$~TeV, $\TRH = 5$~MeV, and
$\xi = 1$. The left panel shows the production rate for each
process for an individual axion species $a_\lambda$ with $\lambda = 1$~MeV
(\ie, a value well within the asymptotic, large-$\lambda$ regime, where the
rates are the least suppressed). The right panel shows the integrated
production rate for each process, including contributions from all modes with
$\lambda < T$. The most relevant processes for thermal axion production in
this scenario are $\pi \pi\rightarrow \pi a_\lambda$ production (red curve),
$e^\pm\gamma\rightarrow e^\pm a_\lambda$ via the Compton process
(yellow curve), $e^\pm\gamma\rightarrow e^\pm a_\lambda$ production
via the Primakoff process (orange curve), inverse decays of the form
$\gamma\gamma\rightarrow a_\lambda$ (green curve), production via the quark-gluon
process $qg\rightarrow qa$ (blue curve), and pion-production off nuclei
(purple curve). It should be noted that the Compton process requires a non-zero
electron-electron-axion coupling $c_e$, and that the curve shown here corresponds to
the case in which $c_e=1$. The value of the Hubble parameter as a function of $T$
in both the Standard (black dashed curve) and LTR (black dash-dotted curve)
cosmologies are also shown.
\label{fig:RatesVsH}}
\end{figure}
In Fig.~\ref{fig:RatesVsH}, we provide a pictorial comparison of the
rates for the axion-production processes enumerated above as functions
of temperature. The left panel shows the rates for the production of
a single axion species $a_\lambda$ with $\lambda = 1$~MeV in a scenario
with $\fhatX = 10^{15}$~GeV, $M_c = 10^{-11}$~GeV, and $\Lambda_G = 1$~TeV.
As before, we have taken $\xi=1$ and set $c_g = c_\gamma = c_e = 1$. The
red curve corresponds to the rate $\Gamma(\pi \pi \rightarrow \pi a_\lambda)$
for the pionic process; the orange curve to the rate
$\Gamma_{\mathrm{Prim}}(e^\pm \gamma \rightarrow e^\pm a_\lambda)$
for the Primakoff process; the green curve to the
rate $\Gamma(\gamma\gamma \rightarrow a)$ for the inverse-decay process;
the blue curve to the rate $\Gamma(N \pi \rightarrow N' a_\lambda)$ for
the pion-conversion process off nuclei; and the purple curve to the rate
$\Gamma(q \gamma \rightarrow q a_\lambda)$ for the quark-gluon process.
As the hadron description of the theory is valid only for $T\lesssim \LambdaQCD$, and
likewise, the quark/gluon description is only valid for $T\gtrsim \LambdaQCD$, the rates
$\Gamma(\pi \pi \rightarrow \pi a_\lambda)$, $\Gamma(N \pi \rightarrow N' a_\lambda)$, and
$\Gamma(q \gamma \rightarrow q a_\lambda)$ are only defined on one side or the other of this
scale. The yellow curve corresponds to the rate
$\Gamma_{\mathrm{Comp}}(\pi \pi \rightarrow \pi a_\lambda)$ for the Compton-like
process for $c_e = 1$. For purposes of comparison, we
also show the Hubble parameter as a function of $T$ for two different cosmologies:
the standard cosmology (black dashed curve), and an LTR cosmology with
a reheating temperature $\TRH = 5$~MeV (black dot-dashed curve).
The value of $\lambda$ we have chosen here is well within the asymptotic
regime for this choice of $M_c$ and $\fhatX$; hence the rates displayed here
represent take essentially the maximal values possible for any $a_\lambda$ in the
scenario. In the right panel of Fig.~\ref{fig:RatesVsH}, we show, for the same choice
of $M_c$ and $\fhatX$, the total contribution to the axion-production rate
obtained by summing the rates for all $a_\lambda$ for which $\lambda \leq T$ ---
\ie, those which will be kinematically accessible at a given temperature.
The most salient lesson to draw from of Fig.~\ref{fig:RatesVsH}
is that even after the contributions
from all kinetically accessible $a_\lambda$ states are included in the thermal
axion-production rate, none of the relevant processes by which a thermal population
of axions might be produced comes close to satisfying the $\Gamma \sim H$ criterion.
This implies that the $a_\lambda$, even when taken together, never attain thermal
equilibrium with the plasma after inflation ends. Furthermore, these results also
justify the claims made above, that the electron Primakoff process and inverse
decays of the form $\gamma\gamma\rightarrow a$ are the most relevant
axion-production processes for $T\lesssim \LambdaQCD$, while
hadronic processes dominate the axion-production rate for $T\gtrsim \LambdaQCD$.
Let us now estimate the contribution to $\Omegatot$ from
thermal axion production in the context of an LTR cosmology with
a reheating temperature of $\TRH=5$~MeV.
For concreteness, we focus on the case of a photonic axion with
$c_\gamma = 1$ and $c_g = c_i = 0$ for all $i$; however, the results
for other coupling assignments should not differ drastically from those
obtained here. We begin by noting that
any contribution to $\Omega_\lambda$ generated at temperatures $T \gtrsim \TRH$,
\ie, during the reheating phase, will be substantially diluted
due to entropy production from inflaton decays. It is therefore
legitimate to restrict our attention to axion production within the subsequent
RD era. For a photonic axion, the processes which contribute to thermal
axion production are inverse decays and $e^\pm\gamma\rightarrow e^\pm a$,
the latter of which, since we are assuming $c_e = 0$, is dominated by the
Primakoff process. The Boltzmann equation for the number density
$n_\lambda$ of each $a_\lambda$ is therefore effectively given by
\begin{equation}
\dot{n}_\lambda + (3H + \Gamma_\lambda) n_\lambda ~=~
C_\lambda^{\mathrm{ID}}(T) + C_\lambda^{\mathrm{Prim}}(T)
\label{eq:ThermProdBoltzEqWithn}
\end{equation}
for $T \lesssim \TRH$, where
$C_\lambda^{\mathrm{Prim}}(T)$ and $C_\lambda^{\mathrm{ID}}(T)$ are the
contact terms associated with the electron-Primakoff and inverse-decay rates given
in Eqs.~(\ref{eq:ElectronPrimakoff}) and~(\ref{eq:InverseDecayRate}), respectively.
For $T \gg \lambda,m_e$, these contact terms are well-approximated by the expressions
\begin{eqnarray}
C_\lambda^{\mathrm{Prim}}(T) & \approx &
\frac{2\alpha}{3\pi^2}G_\gamma (\wtl^2 A_\lambda)^2
\frac{T^6}{\fhatX^2}\left[\ln\left(\frac{9}{4\pi\alpha}\right) + 0.8194\right]
\nonumber \\
C_\lambda^{\mathrm{ID}}(T) & \approx & 2G_\gamma (\wtl^2 A_\lambda)^2
\frac{\lambda^5 T}{\pi^2\fhatX^2} K_1\left(\frac{\lambda}{T}\right)~,
\label{eq:ContactTermsExplicit}
\end{eqnarray}
where $K_1(x)$ denotes the Bessel function of the first kind. To obtain a
rough estimate of the relic abundance in situations in which either $m_e$ or $\lambda$
is comparable to or greater than $T$, we
modify the expression for $C_\lambda^{\mathrm{Prim}}(T)$ given in
Eq.~(\ref{eq:ContactTermsExplicit}) by including an additional exponential factor
$e^{-(\lambda + m_e)/T}$ to model the effect of Boltzmann suppression.
From Eq.~(\ref{eq:ThermProdBoltzEqWithn}) we estimate the relic abundance
of axions produced by interactions with the SM particles in the thermal bath.
To do so, we neglect the decay term and rewrite the resulting equation in
terms of the quantity $Y_\lambda \equiv n_\lambda/s$,
where $s$ is the entropy density, in order to remove the Hubble term:
\begin{equation}
s\dot{Y}_\lambda ~\approx~
C_\lambda^{\mathrm{ID}}(T) + C_\lambda^{\mathrm{Prim}}(T)e^{-(\lambda+m_e)/T}~.
\label{eq:ThermProdBoltzEqWithY}
\end{equation}
By numerically integrating this equation, we obtain an estimate of the thermal contribution
$\Omega_\lambda^{(\mathrm{therm})}$ to the abundance $\Omega_\lambda$ of each $a_\lambda$ at
present time:
\begin{equation}
\Omega_\lambda^{(\mathrm{therm})} ~\approx~ \frac{\lambda\Tnow^3\tMRE}{\rhocrit}
\int_{\Tnow}^{\TRH}
\frac{3}{\kappa(T)}\left(\frac{\TMRE}{T}\right)^{3/\kappa(T)}
\frac{g_{\ast s}(\Tnow)}{g_{\ast s}(T)}
\left[C_\lambda^{\mathrm{ID}}(T) + C_\lambda^{\mathrm{Prim}}(T)
e^{-(\lambda+m_e)/T}\right]dT~,
\label{eq:OmegaThermNow}
\end{equation}
where $g_{\ast s}(T)$ is the number of interacting degrees of freedom present in the
thermal bath at temperature $T$, and where $\kappa(T)$ is
defined in Eq.~(\ref{eq:DefOfkappaForH}).
The results of this integration are displayed in Fig.~\ref{fig:OmegaCompFromThemal}. In
this figure, we compare the contributions to the relic abundance $\Omega_\lambda$ of a
given $a_\lambda$ from misalignment production and thermal production for a
variety of different choices of the model parameters.
\begin{figure}[t!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {OmegaCompMisThermLam1TeV.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {OmegaCompMisThermLam100TeV.eps}
\end{center}
\caption{Contributions to the individual mode abundances $\Omega_\lambda$ for
a photonic axion from thermal production (dashed curves) and misalignment
production (solid curves), plotted as functions of $\lambda$
for $\fhatX = 10^{12}$~GeV (red curves),
$\fhatX = 10^{13}$~GeV (orange curves), $\fhatX = 10^{14}$~GeV (green curves),
$\fhatX = 10^{15}$~GeV (blue curves). The left panel displays the results for
$\Lambda_G = 1$~TeV, while the right panel displays the results for $\Lambda_G = 100$~TeV.
The other model parameters have been set to $M_c = 10^{-11}$~GeV, $\TRH = 5$~MeV,
$\xi = g_G = c_\gamma = 1$.
\label{fig:OmegaCompFromThemal}}
\end{figure}
It is clear from
Fig.~\ref{fig:OmegaCompFromThemal} that for these parameter assignments,
$\Omega_\lambda^{(\mathrm{therm})}$ only becomes comparable with the relic-density
contribution $\Omega_\lambda^{(\mathrm{mis})}$ from vacuum misalignment for reasonably
heavy $a_\lambda$. Neither $\Omega_\lambda^{(\mathrm{mis})}$ nor
$\Omega_\lambda^{(\mathrm{therm})}$ for such $a_\lambda$ is non-negligible
compared with the $\Omega_\lambda^{(\mathrm{mis})}$ contribution from the
lighter modes. Indeed, summing over $\lambda$ to obtain the total thermal
contribution $\Omegatot^{(\mathrm{therm})}$ to the axion relic abundance at present time
yields $3.8 \times 10^{-6} \lesssim \Omegatot^{(\mathrm{therm})} \lesssim 3.8
\times 10^{-4}$. We may therefore safely conclude that
$\Omegatot^{(\mathrm{therm})} \ll \Omegatot^{(\mathrm{mis})}$
within the preferred region of parameter space for bulk-axion models of
dynamical dark matter, and that the population of $a_\lambda$ generated by
the misalignment mechanism dominates the relic density of the DDM ensemble.
To summarize the results of this section, we have examined the primary
mechanisms through which a cosmological population of DDM axions
may be generated, including misalignment production, thermal production, and
production by decaying relics. We have shown that within the
preferred region of parameter space specified in Ref.~\cite{DynamicalDM2},
the contribution to the total present-day dark-matter relic abundance
from misalignment production $\Omegatot^{(\mathrm{mis})}$ indeed dominates over the
contributions from all other production mechanisms. This justifies the emphasis
placed on misalignment production in Ref.~\cite{DynamicalDM2}.
Still, we note that
although populations of axions produced via those other channels collectively
represent a negligible fraction of $\Omegatot$, those populations can nevertheless
play an important role in constraining bulk-axion DDM models. For example, the
thermal population of axions discussed above can still leave a significant imprint
on the diffuse X-ray spectrum despite the small size of $\Omegatot^{(\mathrm{therm})}$,
because $\Omega_\lambda^{(\mathrm{therm})} \ll \Omega_\lambda^{(\mathrm{mis})}$ when
$\lambda$ is large.
We shall return to this point in Sect.~\ref{sec:XrayGammaRay}, where we will show
that this imprint is nevertheless consistent with current observational limits.
\section{Phenomenological Constraints on Dark Axion Ensembles\label{sec:Bounds}}
In the previous section we characterized the various mechanisms
which contribute to the generation of a cosmological population of relic axions in
axion DDM models and compared the sizes of their contributions to $\Omegatotnow$.
Given that this population constitutes the dark-matter ensemble in our axion DDM model,
we now turn to examine the relevant phenomenological, astrophysical, and
cosmological constraints on that population of axions.
As we shall see, some of these constraints pertain
generically to any theory of dark matter, or to any theory containing
late-decaying relics. Others are particular to models involving light,
weakly-coupled fields. Still others pertain to theories with
large extra dimensions in general, regardless of the presence or
absence of a bulk axion field.
As we have seen in Refs.~\cite{DynamicalDM1,DynamicalDM2},
the properties of the dark-matter ensemble and its constituent
fields in our bulk-axion DDM model are determined primarily by three parameters:
the compactification scale $M_c$, the $U(1)_X$-breaking scale $\fhatX$, and the
confinement scale $\Lambda_G$ for the gauge group $G$. Because these parameters
play such a central role in characterizing the dark sector in our model,
we shall seek to phrase our phenomenological constraints in terms of restrictions
on $M_c$, $\fhatX$, and $\Lambda_G$ whenever possible.
Of course, in addition to these primary parameters, a number of other ancillary
quantities also have an impact on the phenomenology of
our model, and thus are also constrained by data.
These include the scales $H_I$ and $\TRH$ associated with cosmic
inflation, the coupling coefficients $c_g$, $c_\gamma$, and $c_i$, and so forth.
Generally speaking, these additional parameters play a subordinate role in
determining the mass spectrum and relic abundances of the $a_\lambda$,
and the values they take are typically far more model-dependent than $M_c$,
$\fhatX$, and $\Lambda_G$. Thus, while certain experimental and observational
limits serve to constrain the values these additional parameters may take, it
ultimately turns out to be possible to phrase the majority of constraints on our
model as bounds on $M_c$, $\fhatX$, and $\Lambda_G$. Indeed, as we shall see
in Sect.~\ref{sec:Combined}, most of the critical bounds can be expressed
conveniently in this manner.
We will also be interested in how these bounds constrain certain derived
quantities of physical importance, such as the quantity $y$ defined in Eq.~(\ref{eq:DefsOfyandmPQ}), which quantifies the amount of mixing that occurs
across our DDM ensemble.
\subsection{Constraints from Background Geometry\label{sec:ConstraintsLargeED}}
The first set of constraints we consider are those which apply generically
to theories with extra dimensions, independently of the presence or properties
of the bulk axion field whose KK excitations constitute the DDM ensemble in our
model. These constraints arise primarily from experimental limits on
the physical effects to which the tower of KK gravitons necessarily present
in such theories gives rise. We will primarily focus here on scenarios involving
$n$ flat extra dimensions in which the fields of the SM are localized on a 3-brane,
while gravity, as always, necessarily propagates throughout the entirety of the
$D=(4+n)$-dimensional bulk.
Perhaps the most significant and direct bound on $M_c$
in theories with extra dimensions arises due to
modifications of Newton's law at short distances as a consequence of
KK-graviton exchange. The lack of evidence for any such effect at
modified-gravity experiments to date implies constraints on the sizes and
shapes of those extra dimensions. For the case of a single large, flat extra
dimension, the current limit on the compactification scale from
such experiments is~\cite{KapnerModGrav}
\begin{equation}
M_c ~\gtrsim~ 3.9\times 10^{-12}~\mbox{GeV}~.
\label{eq:MinimumMc}
\end{equation}
This lower limit on the compactification scale is robust in the sense that
even if there exist additional compact dimensions with radii $r_i \ll 1/M_c$,
this bound is essentially unaffected. For this reason, Eq.~(\ref{eq:MinimumMc})
turns out to represent the most significant constraint on the parameter space
of bulk-axion DDM models from considerations which derive solely from
the presence of extra dimensions.
There also exist additional constraints on the compactification geometry
which arise due the relationship
between this scale, the effective four-dimensional Planck scale $M_P$, and
the fundamental scale of quantum gravity $M_D$. These constraints are generally
more sensitive to the details of the compactification scenario. In general, the
fundamental scale $M_D$ is related to $M_P$ by~\cite{ADD}
\begin{equation}
M_P^2 ~=~ V_n M_D^{2+n}~,
\label{eq:EffPlanckRel}
\end{equation}
where $V_n$ is the volume of the $n$-dimensional manifold on which the extra dimensions are
compactified. For the simple case in which this manifold is a flat, rectangular $n$-torus,
the volume $V_n$ is simply the product of $(2\pi r_i)$ for each cycle of the torus. Assuming
all radii are equal to a common radius $r$, we then have
\begin{equation}
r^{-1} ~\geq~ 2\pi M_D^{\mathrm{min}} \left(\frac{M_D^{\mathrm{min}}}{M_P}\right)^{2/n}~.
\label{eq:McBoundMDmin}
\end{equation}
Bounds on the scale $M_D$ appearing in the literature are frequently predicated on
these assumptions. However, we emphasize that in situations in which the $r_i$ are not
all equal, or in which the compactification geometry differs from that of a flat,
rectangular $n$-torus, those bounds can be considerably modified.
Under the assumption that the compactification geometry resembles that on
which Eq.~(\ref{eq:McBoundMDmin}) is predicated, one may derive constraints on $M_D$,
$r$, or combinations of the two. For example, one class of
constraints which arise in theories with extra dimensions are those
implied by the non-observation of effects related to thermal KK graviton
production in astrophysical sources such as stars~\cite{HannestadRaffeltNeutronStar}
and supernovae~\cite{KKGravitons1987A,HannestadRaffeltSupernovae}. A brief
synopsis of the most relevant bounds in this class is given
in Ref.~\cite{HannestadRaffeltNeutronStar}, all of which depend
crucially on the fundamental quantum-gravity scale $M_D$. The most stringent of
these constraints currently derives from limits on photoproduction and stellar heating
by gravitationally trapped KK gravitons in the halos of neutron stars. Indeed,
for a theory involving $n$ extra dimensions with equal radii, one finds that for
$n = 2$, the bound is $r^{-1} \geq 5.8 \times 10^{-7}$~GeV, while for $n = 3$,
one finds $r^{-1} \geq 3.8 \times 10^{-10}$~GeV~\cite{HannestadRaffeltNeutronStar}.
Collider data also place limits on $r$ and $M_D$ in theories with
extra dimensions. Searches for evidence of KK-graviton production in
the monojet (\ie, $j + \met$) channel have been performed by the
ATLAS~\cite{ATLASMonojet33pb,ATLASMonojet1fb} and CMS~\cite{CMSMonojet} collaborations.
The most recent ATLAS analysis~\cite{ATLASMonojet1fb},
conducted with $1\mathrm{~fb}^{-1}$ of integrated luminosity,
constrains $M_D \gtrsim \{3.16,2.50,2.15\}$~TeV at $95\%$~C.L.\ for $n=\{2,3,4\}$ flat
extra dimensions with equal radii. The most recent CMS analysis~\cite{CMSMonojet},
conducted at a comparable integrated luminosity, yields the slightly more stringent
constraint $M_D \gtrsim \{4.03,3.21,2.80\}$~TeV at $95\%$~C.L.\ for
the corresponding values of $n$.
Limits from searches for KK-graviton effects in the diphoton~\cite{CMSLargeEDDiphoton}
and dimuon~\cite{CMSLargeEDDimuon} channels at $36\mathrm{~pb}^{-1}$ and
$39\mathrm{~pb}^{-1}$ of integrated luminosity, respectively, have also been
derived by the CMS collaboration, but these are currently less stringent than the
constraints from the $j + \met$ channel.
It is important to realize that the aforementioned bounds on $M_D$ as a function
of the compactification geometry do {\it not}\/ necessarily translate directly into
analogous bounds on $f_X$ for a given $\fhatX$. Unlike the graviton
field, the bulk axion field in our DDM model need not necessarily propagate
throughout the entirety of the
extra-dimensional volume, but may in principle also be confined to a
$(4 + n_a)$-dimensional subspace of that volume, where $n_a < n$. When this is the case,
$\fhatX$ is related to $f_X$ by the generalization of Eq.~(\ref{eq:fhatInTermsOff}):
\begin{equation}
\fhatX^2 ~=~ V_{n_a} f_X^{2+n_a}~.
\end{equation}
Note that this relationship differs from that which exists between $M_P$ and $M_D$
because $n_a < n$.
In this paper, as in Ref.~\cite{DynamicalDM2}, we focus on the case in which
the axion field propagates in a single extra dimension of radius $R$, irrespective
of the size, shape, or number of extra dimensions which compose the totality of the
bulk. Accordingly, we define $M_c = 1/R$ to be the compactification scale
associated with this particular extra dimension, and we shall use this notation
throughout. In this paper, we are not aiming to set $M_D$ at or even near the TeV scale,
since we are not attempting to solve the hierarchy problem, but rather to address
the dark-matter problem. We will therefore
assume that the structure of any additional bulk dimensions is such that phenomenological
constraints on $M_D$ and the associated compactification geometry are satisfied.
Note, however, that the Newton's-law bound in Eq.~(\ref{eq:MinimumMc}) does apply
to $M_c$, as it applies to the compactification scale associated with any individual
extra dimension.
Another class of constraints on scenarios involving large extra dimensions
applies to ancillary variables which characterize the cosmological context in
which our model is situated.
For example, the prediction of the observed abundances of the light elements
via big-bang nucleosynthesis (BBN) is one of the greatest successes of the
standard cosmology. Consistency with these predictions requires that
effects stemming from the presence of these extra dimensions not disrupt BBN.
Successful nucleosynthesis requires that the expansion rate
of the universe during the BBN epoch, as quantified by the Hubble parameter
$H(T)$, must not deviate from its usual, four-dimensional value by more than around
10\%~\cite{ADDPhenoBounds}. In other words, there exists some temperature
$T_*\geq \TBBN \sim 1$~MeV (usually dubbed the ``normalcy temperature'' in
the literature) below which the radii of all extra dimensions are effectively fixed
and the bulk is effectively empty of energy density. A variety of different
considerations constrain $T_\ast$, most of which are related to the potentially
observable effects of KK-graviton dynamics in the early universe:
\begin{itemize}
\item Interactions between the SM fields on the brane and the bulk graviton field
result in a transfer of energy from the brane to the bulk, and a consequent cooling
of the radiation bath on the brane. Substantial energy loss via this
``evaporative cooling'' mechanism would result in a modification of the expansion
rate of the universe. At temperatures $T\lesssim T_{\mathrm{BBN}}$, such a modification
would distort the light-element abundances away from those predicted by standard BBN.
Thus, the strength of the interactions between SM particles and
excitations of the graviton field is constrained.
\item If the collective energy density associated with the graviton KK modes
is substantial, that energy density could cause the universe to become
matter-dominated too early. In extreme cases, it could even overclose
the universe.
\item Late decays of KK gravitons could result
in distortions of the abundances of light elements away from the values predicted
by BBN~\cite{ADDPhenoBounds}, which accord well with the observed values for
these abundances. Such decays could also result in significant entropy production.
\item The relationship between the Hubble parameter $H$ and the total energy density
$\rho$ of the universe is modified at early times in higher-dimensional scenarios,
even when that energy density is overwhelmingly dominated by brane-localized
states~\cite{EDHubbleCline,EDHubbleBinetruy,EDHubbleShiromizu}. Such a modification
could have a substantial effect on BBN as well.
\end{itemize}
The constraints on $T_*$ implied by these considerations have been reckoned by a
number of authors~\cite{ADDPhenoBounds,CosmoConstraintsLargeED}, and while the precise
values of the bounds so derived again depend on the number, size, and shape of the extra
dimensions, the value of $M_D$, \etc, the most stringent (which tend to come from
limits on the late decays of the excited KK modes) generally tend to restrict $T_*$
to within the rough range
$4\mbox{~MeV}\lesssim T_*\lesssim 20\mbox{~MeV}$~\cite{ADDPhenoBounds}.
One possibility for achieving such conditions is to posit that $T_*$ be identified
with the reheating temperature $\TRH$ associated with a period of cosmic inflation
initiated by an inflaton field which is localized on the same 3-brane as the SM
fields. During such an
inflationary epoch, any contributions to the energy density of the universe
from bulk states which existed prior to the inflationary epoch (save for those
which, like the contributions to $\rho_\lambda$ from vacuum misalignment, scale like
vacuum energy) are inflated away. Furthermore, if the inflaton field
decays primarily to other brane-localized states, no
substantial population of bulk states is regenerated during the subsequent
reheating phase. Thus, by adopting a LTR cosmology with a reheating temperature
$4\mbox{~MeV}\lesssim \TRH\lesssim 20\mbox{~MeV}$, we thereby ensure that
the relevant constraints related to KK-graviton production in the early universe
are satisfied. We also note that a reheating temperature of $\TRH\gtrsim 4$~MeV
is sufficient to ensure that the thermal populations of the SM fields
(and, in particular, the three neutrino species) required in standard BBN are
generated by the thermal bath after reheating~\cite{TReheatLimits,KawasakiLTR2}.
In summary, while stringent constraints exist on theories with large extra
dimensions, these constraints can be satisfied by adopting an LTR cosmology
with $4\mbox{~MeV}\lesssim \TRH\lesssim 20\mbox{~MeV}$ and a compactification
manifold for which the astrophysical bounds listed above may consistently
be satisfied for a given choice of $M_D$ and $f_X$.
Since $\Omegatotnow$ is generated via non-thermal means in our bulk-axion
model, as discussed in Sect.~\ref{sec:AbundanceConstraints}, such a low
value of $\TRH$ is not an impediment to obtaining a dark-matter relic abundance
$\Omegatotnow \approx\OmegaDM$. In fact, as shown in Ref.~\cite{DynamicalDM2},
adopting an LTR cosmology is actually an {\it asset}\/ in terms of generating a
dark-matter relic abundance of the correct magnitude. Likewise,
since the relationship between $\fhatX$ and $f_X$ need not be identical
to the relationship between $M_P$ and $M_D$,
constraints which concern the effects of KK gravitons can be
satisfied without imposing equally severe restrictions on the parameters
$\fhatX$, $M_c$, and $\Lambda_G$ which govern the properties of the dark-matter
ensemble. Indeed, the only significant model-independent constraint on these
parameters turns out to be the constraint quoted in Eq.~(\ref{eq:MinimumMc})
from tests of Newton's law at short distances.
\subsection{Axion Production with Subsequent Detection: Helioscopes and
Light Shining Through Walls\label{sec:Helioscopes}}
We now address the constraints which relate directly to the
phenomenological, astrophysical, and cosmological implications associated
with the KK tower of axion fields which constitute the DDM ensemble in our model.
We begin by discussing the limits derived from a wide variety of experiments
designed to detect axions and axion-like particles via their
interactions with the photon field. (For extensive reviews of
these experiments, see Refs.~\cite{KimReview2,JaeckelReview}.)
To date, none of these experiments have seen any conclusive evidence
for such particles, and the null results of these experiments therefore
imply constraints on the effective couplings between such axion-like particles
and the photon field.
In order to determine how the results of the experiments listed above
serve to constrain the parameter space of our bulk-axion DDM model, it
is useful to divide those experiments into several broad classes, based on
the sort of physical process each probes. One important class of experiments
comprises those in which axions are produced via their interactions with the
fields of the SM and then subsequently detected via those same
interactions. These include helioscope experiments such as CAST~\cite{CAST}
and ``light-shining-through-walls'' (LSW) experiments such as BEV, GammaeV,
and ALPS. Searches for coherent conversion of solar axions
to X-ray photons in germanium and sodium-iodide crystals via Bragg diffraction
which have been performed at experiments such as
DAMA~\cite{DAMAAxion}, TEXONO~\cite{TEXONOAxion}, SOLAX~\cite{SOLAXAxion},
and COSME~\cite{COSMEAxion} also fall into this category.
The characteristic which distinguishes experiments in this class
from others is that these experiments are affected by decoherence phenomena.
Indeed, it has been observed~\cite{DDGAxions} that in theories with bulk axions,
such phenomena result in a substantial suppression of the rate for any process
involving the production and subsequent decay of axion modes relative to
na\"{i}ve expectations.
Let us briefly review the origin of this suppression by
focusing on the interaction between the photon field and the axion KK modes given
in Eq.~(\ref{eq:HighT4DAxionAction}). (The results for the coupling between these
modes and the other SM fields are completely analogous.) We begin by defining a state
\begin{equation}
a' ~\equiv~ \frac{1}{\sqrt{N}}\sum_n^N r_n a_n~,
\end{equation}
which represents the particular linear combination of KK eigenstates $a_n$
that couples to any physics on the brane, such as $F_{\mu\nu}\widetilde{F}^{\mu\nu}$
or any pair of SM fields. Here $N \sim f_X/M_c$ denotes the number of modes in the sum.
Written in terms of $a'$, the relevant term in the interaction
Lagrangian becomes
\begin{equation}
L_{\mathrm{int}} ~\ni~
\frac{\alpha c_\gamma\sqrt{N}}{8\pi^2\fhatX}a' F_{\mu\nu}\widetilde{F}^{\mu\nu}~.
\end{equation}
In other words, $a'$ couples to the SM fields with a strength proportional to
$\sqrt{N}/\fhatX \sim 1/f_X$. Consequently, the cross-section for any
physical process which involves axions production via interactions with
the SM fields followed by subsequent detection via the same sorts of interactions
will take the form
\begin{equation}
\sigma(t) ~\propto~ \frac{N^2}{\fhatX^4} \times P(t)~,
\label{eq:CouplerToCouplerXSecBasic}
\end{equation}
where $P(t) = |\langle a'(t)|a'(t_0) \rangle|^2$ is the probability for a
state $a'$ created at time $t_0$ to be in the same state $a'$ at time $t$.
It can be shown that when $N$ is large, $P(t)$ is given by\begin{equation}
P(t) ~=~ \frac{1}{N^2} \left[\sum_\lambda \wtl^8 A_\lambda^4 +
2\sum_\lambda \sum_{\lambda'<\lambda}\wtl^4\wtl'^4A_\lambda^2A_{\lambda'}^2
\cos \left(\frac{(\lambda^2-\lambda'^2)(t-t_0)}{2p}\right) \right]~,
\label{eq:Poft}
\end{equation}
where $p$ is the initial momentum of the axion.
At very early times,
when $t\approx t_0$, the cosine factor in $P(t)$ is approximately
unity for all values of $\lambda$ and $\lambda'$. At such times, all of the
terms in the sum appearing in the second term on the right side of
Eq.~(\ref{eq:Poft}) add coherently. As a result, this term, combined together
with the first term, yields a factor on the order of $N^2$. However, as the
system evolves, the cosine terms will no longer sum coherently, and a
random-walk behavior ensues, according to which the two
terms combine to yield a factor of $\mathcal{O}(N)$ rather than of
$\mathcal{O}(N^2)$. The
time scale $\tau_D$ associated with this decoherence --- or, more precisely,
the scale at which $P(t) = 0.1 P(t_0)$ --- is found to be~\cite{DDGAxions}
\begin{equation}
\tau_D ~ \approx ~ 10^{-5} \left(\frac{2p}{\mPQ^2}\right) \frac{y^2}{N^2}
~\approx~
1.32\times 10^{-29} \left( \frac{p}{\mbox{GeV}}\right)
\left( \frac{\fhatX}{\mbox{GeV}}\right)^{-2} \mathrm{s}~,
\end{equation}
where $y$ is defined in Eq.~(\ref{eq:DefsOfyandmPQ}).
Since $\tau_D$ is clearly quite small for any combination of $p$ and $f_X$ values
of experimental relevance, any method of detecting axions which relies on their
production and subsequent detection will feel the effect of this decoherence.
By contrast, detection methods which rely on axion production without subsequent
detection (such as missing-energy signals at colliders, energy dissipation
from supernovae, \etc) or which probe for evidence of a cosmic population of
relic axions (such as microwave-cavity experiments)
will be unaffected by this phenomenon.
The consequences of axion decoherence for physical processes in the decoherence
regime are readily apparent. In this regime, as discussed above, the term in brackets
in $P(t)$ scales like $N$ rather than $N^2$; hence any cross-section
which takes the form given in Eq.~(\ref{eq:CouplerToCouplerXSecBasic})
will scale with $N\sim f_X/M_c$ according to
\begin{equation}
\sigma(t>\tau_D) ~\propto~ \frac{N}{\fhatX^4} ~\sim~ \frac{1}{N}\frac{1}{f_X^4}~.
\label{eq:CouplerToCouplerXSecSubbed}
\end{equation}
In other words, such cross-sections are suppressed by an
additional factor of $N$ relative to the na\"{i}ve expectation obtained by
setting $\fhatX \rightarrow f_X$ in Eq.~(\ref{eq:CouplerToCouplerXSecBasic}).
Thus, due to the decoherence effect,
any experimental bound on the effective coupling $G_{a\gamma\gamma}$ of
a single four-dimensional axion to the photon field which takes the form
$G^2_{a\gamma\gamma} < (G_{a\gamma\gamma}^{\mathrm{max}})^2$ translates to a
bound $G^2_{a\gamma\gamma} < (G_{a\gamma\gamma}^{\mathrm{max}})^2/\sqrt{N}$
for five-dimensional axion, rather than to
$G^2_{a\gamma\gamma} < (G_{a\gamma\gamma}^{\mathrm{max}})^2/N$.
Given the parametrization for $G_{a\gamma\gamma}$ given in
Eq.~(\ref{eq:HighT4DAxionAction}), we can phrase any such constraint as a
bound on $\fhatX$:
\begin{equation}
\fhatX ~\gtrsim~ \frac{c_\gamma\alpha}{2\pi G_{a\gamma\gamma}^{\mathrm{max}}}
\left(\frac{M_c}{f_X}\right)^{1/4}~.
\end{equation}
Using Eq.~(\ref{eq:fhatInTermsOff}), we may rewrite this constraint in the form
\begin{equation}
\fhatX ~\gtrsim~ \frac{1}{(2\pi)^{13/10}}
\left(\frac{c_\gamma\alpha}{G_{a\gamma\gamma}^{\mathrm{max}}}\right)^{6/5}
\frac{1}{M_c^{1/5}} ~=~
\big(2.50\times 10^{-4}\big)\, c_\gamma (G_{a\gamma\gamma}^{\mathrm{max}})^{-6/5}M_c^{-1/5}~.
\label{eq:HelioscopefhatLimit}
\end{equation}
The most stringent limit from the class of experiments categorized above (\ie,
those for which the phenomenon of decoherence is relevant) is currently
the $G_{a\gamma\gamma} \lesssim 8.8 \times 10^{-11} \mbox{~GeV}^{-1}$ bound
obtained by CAST~\cite{CAST}. The most stringent limit from crystalline
detectors is the
$G_{a\gamma\gamma} \lesssim 1.7 \times 10^{-9} \mathrm{~GeV}^{-1}$
bound from DAMA~\cite{DAMAAxion}, and limits on $G_{a\gamma\gamma}$ from LSW
experiments are typically roughly three orders of magnitude higher than the
CAST limit. The corresponding bound on $\fhatX$ from
Eq.~(\ref{eq:HelioscopefhatLimit}) is
\begin{equation}
\fhatX ~\gtrsim~ \left(2.92 \times 10^8\right)\, c_\gamma^{6/5} \left(\frac{M_c}{\mbox{GeV}}\right)^{-1/5}
\mbox{~GeV}~.
\label{eq:CASTConstraint}
\end{equation}
Note that even for $M_c$ at the experimental lower limit given in
Eq.~(\ref{eq:MinimumMc}), the constraint in Eq.~(\ref{eq:CASTConstraint})
is satisfied as long as $\fhatX \gtrsim 5.58 \times 10^{10}$~GeV.
\subsection{Microwave-Cavity Experiments and Direct Detection of Dark-Matter Axions}
Another class of experiments which place constraints on the couplings of
axions and axion-like fields to SM particles consists of those which
involve the direct detection of a cosmological population of axions. The most
sensitive experiments in this class are those associated with
dedicated microwave-cavity detectors such as ADMX~\cite{ADMX} and
CARRACK~\cite{CARRACK}.
Detectors of this sort are used to search for the resonant conversion of
dark-matter axions with mass $m_a$ to photons with energies
$E_\gamma \approx m_a$ in the presence of a strong magnetic field.
As a result, the observation of a signal at such a detector depends
crucially on whether the mass of the axion in question lies within the
range of photon energies probed. The axion mass range currently covered by
ADMX spans only from $1.9\times 10^{-15}$~GeV to
$3.5\times 10^{-15}$~GeV~\cite{ADMX}, and the projected future mass sensitivity
extends only as high as $10^{-13}$~GeV. Likewise, the projected sensitivity for
CARRACK extends only as high as $3.5\times 10^{-14}$~GeV.
As discussed in Ref.~\cite{DynamicalDM2}, the region of parameter space
which is the most interesting from a DDM perspective is that within which
$y \lesssim 1$ and mixing among the light axion KK modes is substantial, for
it is this region within which the full tower contributes meaningfully to
$\Omegatotnow$. Within this region of parameter space, the lightest mode in
the tower has a mass $\lambda_0 \approx M_c/2$. Taken in conjunction
with the bound on $M_c$ from modified-gravity experiments given in
Eq.~(\ref{eq:MinimumMc}), this result implies that
$\lambda_0 \gtrsim 1.5\times 10^{-12}$~GeV in highly-mixed bulk-axion scenarios.
The projected ranges for both ADMX and CARRACK lie well below this threshold
for $\lambda_0$. We therefore conclude that no meaningful constraints on
bulk-axion DDM models can be derived from the results of these experiments.
\subsection{Axion Production without Subsequent Detection:
Stars and Supernovae\label{sec:StarnAndSN}}
We now turn to examine an additional class of constraints on bulk-axion DDM
models: those related to astrophysical processes in which the $a_\lambda$
are produced through their interactions with the SM field, but never
directly detected. Among the constraints in this class are
limits on axions, moduli, and other light
scalars derived from the non-observation of their would-be effects on the
lifetimes, energy-loss rates, \etc, of various astrophysical sources such
as stars and supernovae. These effects include the following:
\begin{itemize}
\item Axions and other light fields whose interactions with the particles of the
SM are extremely weak and whose mean free paths are consequently extremely long
can dissipate energy from stars extremely efficiently. Such dissipation can
accelerate stellar cooling and result in observable alterations in stellar life
cycles, including the life cycle of our own sun.
\item Similarly, such light fields can carry away a substantial fraction of the energy
liberated by supernovae. Limits may therefore be placed on the strengths of these
interactions from the non-observation of such effects for supernova SN1987A.
\item A diffuse population of long-lived axions or KK gravitons initially produced
by stars and supernovae could decay at late times, distorting light-element abundances
and producing an observable X-ray or $\gamma$-ray signal in the keV $-$ MeV range or
higher. No evidence for such a signal has been seen by EGRET, FERMI, HEAO, Chandra,
COMPTEL, \etc
\end{itemize}
As is well known, these considerations lead to some of the most
stringent constraints on standard, four-dimensional QCD axions. We now turn to
examine how these limits constrain the parameter space of generalized bulk-axion models.
The primary distinction between processes in which the presence of the
$a_\lambda$ is ascertained by direct detection and those in which it
is only inferred from an energy deficit is that in the latter class of
processes, the $a_\lambda$ appear as particles in the asymptotic final state.
Thus, the contributions from the individual $a_\lambda$ to the
overall event rate for any such a process add not at the
amplitude level, but at the cross-section level.
The decoherence phenomena discussed in Sect.~\ref{sec:Helioscopes}
are therefore irrelevant for such processes, and the total cross-section
$\sigma^{\mathrm{prod}}_{\mathrm{tot}}$ for the production of ``missing energy'' in the
form of $a_\lambda$ fields by any given physical process is simply
the sum of the individual production cross-sections
$\sigma^{\mathrm{prod}}_\lambda$ for each axion
species. Since the effective coupling between each $a_\lambda$ and any pair of
SM fields includes a factor $\wtl^2A_\lambda/\fhatX$ from mass mixing,
as indicated in Eq.~(\ref{eq:ActionInMassEigenbasis}), each of these individual
production cross-sections scales as
\begin{equation}
\sigma^{\mathrm{prod}}_\lambda ~\propto~
\frac{1}{\fhatX^2}(\wtl^2A_\lambda)^2~.
\label{eq:SigmaProdProp}
\end{equation}
When it occurs, axion production will have a characteristic energy scale $\Ech$
determined by the surrounding environment. This energy scale may be
associated, for example, with the
temperature of a star or supernova core, or with the center-of-mass energy
$\sqrt{s}$ of a collider. Provided that $\Ech \gg M_c$ (an assumption valid
for all physical contexts of relevance in bounding the
large-extra-dimension scenarios considered here), it follows that
$\lambda \ll \Ech $ for a large number of $a_\lambda$. Such $a_\lambda$
can be considered to be effectively massless as far as production kinematics is
concerned, implying that to a very good approximation,
$\sigma^{\mathrm{prod}}_\lambda$ depends on $\lambda$ exclusively
through the coupling-modification factor appearing in
Eq.~(\ref{eq:SigmaProdProp}). (For those modes for which threshold effects
are important, such an approximation will overestimate
$\sigma^{\mathrm{prod}}_\lambda$ and result in an overly conservative bound.)
By contrast,
$\sigma^{\mathrm{prod}}_\lambda$ will be effectively zero for
those $a_\lambda$ with masses $\lambda \gg \Ech$ in any thermal environment
due to Boltzmann suppression, and will vanish outright in a non-thermal one.
Therefore, it is reasonable to evaluate
$\sigma^{\mathrm{prod}}_{\mathrm{tot}}$ by taking any additional factors in
Eq.~(\ref{eq:SigmaProdProp}) to be essentially independent of $\lambda$ and
by truncating the sum over modes at $\lambda \sim \Ech$. Thus, we find that
\begin{equation}
\sigma^{\mathrm{prod}}_{\mathrm{tot}} ~\propto~ \aleph^2(\Ech)/\fhatX^2
\label{eq:SigmaWithAlephSuppression}
\end{equation}
where the ``effective'' coupling $\aleph(\Ech)$ is given by
\begin{equation}
\aleph(\Ech) ~\equiv~
\Bigg[\sum_{\lambda=\lambda_0}^{\Ech}(\wtl^2A_\lambda)^2\Bigg]^{1/2}.
\label{eq:DefOfAleph}
\end{equation}
Since the number of modes contributing to $\sigma^{\mathrm{prod}}_{\mathrm{tot}}$
is large by assumption, and since their masses are closely spaced, it
is generally legitimate to approximate $\aleph(\Ech)$ by an integral
\begin{equation}
\aleph(\Ech) ~\approx~ \Bigg[\frac{1}{M_c}\int_{\lambda_0}^{\Ech}
(\wtl^2A_\lambda)^2 d\lambda\Bigg]^{1/2}~.
\end{equation}
The quantity $\aleph(\Ech)$ clearly plays a crucial role in the phenomenology
of bulk-axion scenarios. It is therefore worth pausing a moment to examine
in detail how $\aleph(\Ech)$ depends on the physical scales $\fhatX$, $M_c$,
and $\Lambda_G$. A straightforward calculation shows that $\aleph(\Ech)$
has the parametric scaling behaviors
\begin{equation}
\aleph(\Ech) \sim
\begin{cases}
\displaystyle
\vspace{0.25cm} \frac{\Ech^{3/2} M_c^{1/2}\fhatX^2}{\Lambda_G^4} &
~~~\displaystyle \fhatX \ll \frac{\Lambda_G^2}{M_c} \\
\displaystyle \left(\frac{\Ech}{M_c}\right)^{1/2}&
~~~\displaystyle \fhatX \gg \frac{\Lambda_G^2}{M_c}~.
\end{cases}
\label{eq:AlephCases}
\end{equation}
The first case in Eq.~(\ref{eq:AlephCases}) corresponds to $y \ll 1$, signaling
a highly-mixed axion KK tower for which $\wtl^2 A_\lambda \sim \wtl$. By contrast,
the second case corresponds to $y\gg 1$, signaling a relatively unmixed axion KK tower
for which $\wtl^2 A_\lambda \sim$~constant. These results for $\aleph(\Ech)$ are
illustrated in the left panel of Fig.~\ref{fig:AlephPanels} for
$\Ech = 30$~MeV, a value which is physically meaningful in that it corresponds
roughly to the core temperature of supernova SN1987A. Remarkably, we observe
that $\aleph(\Ech)$ experiences a {\it suppression} for $y \ll 1$. In other words,
mixing within the axion KK tower acts to suppress the magnitude of the total
production cross-section for processes in which the $a_\lambda$ appear as
missing energy. This is an important result, for it indicates that constraints on the
parameter space of our bulk-axion DDM model derived from limits on axion
production in stars, supernovae, colliders, \etc, will be considerably
weaker than one might expect from na\"{i}ve dimensional analysis. Moreover,
this result applies more generally to any theory involving KK towers of
scalar fields whose squared-mass matrix contains both brane-mass and
KK-mass terms.
\begin{figure}[t!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {AlephPlotFixedMc11.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {SMaPlotFixedLam1TeVFinal.eps}
\end{center}
\caption{The dimensionless ``effective coupling'' factor
$\aleph(\Ech)$ defined in Eq.~(\ref{eq:DefOfAleph}), shown as a
function of the relevant scales $M_c$, $\fhatX$, and $\Lambda_G$.
In the left panel, we display curves of
$\aleph(\Ech)$, each corresponding to a particular value of $\Lambda_G$
and normalized to the value $(\Ech/M_c)^{1/2}$ taken by $\aleph(\Ech)$
in the absence of mixing, as a function of $\fhatX$ with fixed
$M_c = 10^{-11}$~GeV. It is readily apparent that the net effect of mixing
within the KK axion tower
is to significantly suppress this effective coupling, thereby loosening
the corresponding production-cross-section constraints.
In the right panel, we display curves showing the overall
cross-section-suppression factor $\aleph^2(\Ech)/\fhatX^2$
as a function of $\fhatX$ for fixed $\Lambda = 1$~TeV, each corresponding
to a particular value of $M_c$. For each set of curves, we have taken
$\xi = g_G = 1$, and have chosen $\Ech = 30$~MeV, which corresponds roughly to
the core temperature $T_{\mathrm{SN}}$ of SN1987A.
\label{fig:AlephPanels}}
\end{figure}
In order to illustrate more explicitly the physical consequences of $\aleph(\Ech)$ in
our bulk-axion DDM model, we likewise display the behavior of the overall scaling
factor $\aleph^2(\Ech)/\fhatX^2$ for $\sigma^{\mathrm{prod}}_{\mathrm{tot}}$ in the right
panel of Fig.~\ref{fig:AlephPanels}. The results shown
in this panel further illustrate a significant general property of this scaling factor:
namely, that the cross-section is actually suppressed not only for large $\fhatX$, but
also for small $\fhatX$, due to the parametric behavior of $\aleph(\Ech)$ described
in Eq.~(\ref{eq:AlephCases}). Thus, for any given choice of $\Lambda_G$ and $M_c$,
there exists a maximum possible value for $\sigma^{\mathrm{prod}}_{\mathrm{tot}}$,
which is only attained at some particular value of $\fhatX$.
These results again illustrate the dramatic effect that $\aleph(\Ech)$ can have in
suppressing $\sigma^{\mathrm{prod}}_{\mathrm{tot}}$ in our bulk-axion DDM model.
Within the class of constraints from processes in which axions are produced but not
subsequently detected, use of $\aleph(\Ech)$ allows us to translate experimental
bounds on four-dimensional axion models into bounds on theories including towers
of bulk scalars.
The leading such bound is obtained from energy-loss limits from SN1987A. For
a standard four-dimensional QCD axion, this bound is roughly~\cite{RaffeltSN1987ABound}
\begin{equation}
f_a ~\gtrsim~ 4\times 10^8\mbox{~GeV}~.
\label{eq:SN1987ABoundIn4D}
\end{equation}
By contrast, in the bulk-axion scenario under consideration here, each
$a_\lambda$ light enough to be produced within the thermal environment
of SN1987A can contribute to the overall energy-dissipation rate.
Since the temperature
$T_{\mathrm{SN}}$ associated with the supernova core is roughly
$30$~MeV, the appropriate modification of Eq.~(\ref{eq:SN1987ABoundIn4D}) for
a general axion which couples to hadrons
with a coupling coefficient comparable in magnitude to that of a QCD axion is
\begin{equation}
\fhatX ~\gtrsim~ \big(4\times 10^8\mbox{~GeV}\big)\,
\aleph(T_{\mathrm{SN}})~.
\label{eq:SN1987ABoundIn5D}
\end{equation}
It then follows that in highly-mixed scenarios, this constraint can be
significantly weaker than the corresponding constraint on KK-graviton
production derived in Ref.~\cite{ADDPhenoBounds}, due to suppression by
$\aleph(T_{\mathrm{SN}})$. Indeed, the corresponding constraint on KK-graviton
production is directly obtained by replacing $\fhatX\rightarrow M_P$ and
$\aleph(T_{\mathrm{SN}})\rightarrow T_{\mathrm{SN}}/M_c$ in
Eq.~(\ref{eq:SN1987ABoundIn5D}).
While the SN1987A bound is indeed one of the most stringent constraints
on the QCD axion, it is not necessarily applicable for all general axions.
This is because the bound quoted in Eq.~(\ref{eq:SN1987ABoundIn4D}) is
predicated on the assumption that nucleon bremsstrahlung
($N + N \rightarrow N + N + a$) and other hadronic processes dominate the
rate for the production of the light scalar in question in the supernova
core. This presupposes that the light scalar couples to nuclei with a
strength comparable to that of a QCD axion. If this is not the case, however,
the constraints obtained from SN1987A energy-loss limits can differ considerably
from the standard QCD-axion bound. For example, if from among the SM particles,
the general axion couples only to the photon field, the dominant production
processes will be $e^-\gamma\rightarrow e a_\lambda$,
$p^+ \gamma\rightarrow p^+ a_\lambda$, and $p^+n\rightarrow p^+n\gamma a_\lambda$.
In this case, the considerably weaker bound~\cite{MassoALPsLongPaper}
\begin{equation}
\fhatX ~\gtrsim~ \big(2.32 \times 10^6 \mbox{~GeV}\big)\, c_\gamma
\label{eq:SN1987ABoundIn4DPhotonOnly}
\end{equation}
is obtained for a four-dimensional field. Translating this result
to the case of a KK tower of axions, as above, we find that
\begin{equation}
\fhatX ~\gtrsim~ \big(2.32 \times 10^6 \mbox{~GeV}\big)\,
c_\gamma\, \aleph(T_{\mathrm{SN}})~.
\label{eq:SN1987ABoundIn5DPhotonOnly}
\end{equation}
Furthermore, in general axion models, $c_\gamma$ may not necessarily be
of $\mathcal{O}(1)$. In other words, the SN1987A constraint is
sensitive to the $U(1)_X$ and $SU(2)\times U(1)_Y$ charges of the fields
in the model, and is thus highly model-dependent.
An analogous limit on $\fhatX$ can be derived
from observations of the lifetimes of globular-cluster (GC)
stars. The ambient temperatures $T_{\mathrm{GC}}$ of such objects are only
$\mathcal{O}(10\mathrm{~keV})$, so axion production primarily proceeds through the
Primakoff processes $\gamma +e^- \rightarrow a + e^-$ and
$\gamma + n_Z \rightarrow a + n_Z$, where $n_Z$ denotes a nucleus with atomic
number $Z$. (Note that the dominant processes in this environment differ from the
axion-nucleon-nucleon bremsstrahlung processes which dominate the axion-production
rate in supernovae.) Such a bound will therefore arise for any general
axion for which $c_\gamma \neq 1$, regardless of whether or not it couples to
the gluon field. The observation limit on axion production in
GC stars is commonly phrased as an upper bound on the effective coupling
$G_{a\gamma\gamma}$ between a standard, four-dimensional axion (or any other
similar particle) and a pair of photons, and the current bound is
$G_{a\gamma\gamma} \lesssim 1 \times 10^{-10}\mbox{~GeV}^{-1}$~\cite{PDG}.
Since $T_{\mathrm{GC}}\approx 10$~keV, the corresponding bound on $\fhatX$ is
\begin{equation}
\fhatX ~\gtrsim~ \big(1.16 \times 10^7 \mbox{~GeV} \big)\,
c_\gamma\, \aleph(T_{\mathrm{GC}})~.
\label{eq:GlobularClusterBound}
\end{equation}
Note that this constraint is independent of the SN1987A bounds, as it differs from the latter
in two significant ways. First, because the relevant production process involves
the coupling of the axion modes to photons rather than to nuclei, it depends on
$c_\gamma$ alone and not on $c_g$. Second, since $T_{\mathrm{GC}}\ll T_{\mathrm{SN}}$,
far fewer of the $a_\lambda$ will be produced with any significant frequency in GC
stars. Consequently, the enhancement factor from the sum over kinetically-accessible axion
modes for GC stars is far smaller.
Finally, bounds similar to those from SN1987A and GC stars can also be derived from the
non-observation of effects related to axion production in other astrophysical
sources, such as our own sun~\cite{SolarLifetimeBounds}. However, these bounds
are found to be subleading in comparison with the SN1987A and GC-star constraints,
essentially because they take place in far cooler environments, where the number of
kinematically accessible modes is even further suppressed by the cutoff at $\Ech$
inherent in $\aleph(\Ech)$.
\subsection{Axion Production at Colliders\label{sec:Colliders}}
We now consider the collider constraints applicable to our bulk-axion
DDM model. Due to the highly suppressed couplings between the axion and
the SM fields in standard four-dimensional axion models, collider data have
virtually no relevance in constraining the parameter space of such models.
Nevertheless, because of the huge multiplicity of light modes that arises
in theories with light bulk fields in large extra dimensions, the net
contribution to the event rates for certain processes from all of these modes
taken together can potentially yield observable signals. For example,
modes which are stable on collider time scales all appear as missing energy, and
can lead to signals in channels such as $pp\rightarrow j + \met$ and
$pp\rightarrow \gamma + \met$. In addition, the heavier, more unstable modes which
decay before exiting the detector can potentially give rise to additional
signature patterns which may include displaced vertices. Indeed, we have already
discussed in Sect.~\ref{sec:ConstraintsLargeED} how current limits from LHC data
constrain the parameter space of one such bulk field --- the higher-dimensional
graviton --- for which the monojet and monophoton channels mentioned above are of particular
importance. Since the $a_\lambda$ in our bulk-axion model couple to the fields of the
SM in much the same manner as KK gravitons, it is no surprise that the collider
phenomenology of the $a_\lambda$ turns out to be quite similar to that of KK gravitons.
We begin by discussing those signals which arise due to the combined
effect of the $a_\lambda$ which are sufficiently long-lived so as
to manifest themselves in a collider detector as missing energy.
Collider processes in which the $a_\lambda$ appear as $\met$ are yet further
examples of the class of processes discussed in the previous section in which
axions are produced but not subsequently detected. The net cross-section for any such
process is therefore likewise suppressed by axion mixing in the manner described in
Eq.~(\ref{eq:SigmaWithAlephSuppression}), with $\Ech$ given by the
center-of-mass energy $\sqrt{s}$ of the collider.
Which specific channels are relevant for the discovery of a bulk axion at hadron colliders
depends crucially on how the five-dimensional axion couples to the SM fields,
and in particular on
whether or not it couples appreciably to either light quarks or gluons. For a
field with an $\mathcal{O}(1)$ value of either $c_g$ or $c_q$ (where $q=\{u,d,s,c\}$),
the principal discovery channel at both the Tevatron and
the LHC is $pp\rightarrow j + \met$, a channel which is also one of the principal discovery
channels for KK gravitons. Thus, in order to obtain a rough estimate of the
constraints on the parameter space of our bulk-axion model from the null results
of monojet searches, we translate the bound on the fundamental scale $M_D$
established by such searches into a bound on $\fhatX$.
The cross-section for KK-graviton production in association with a single
jet at a hadron collider in a theory with $n$ large, flat
extra dimensions of equal length compactified on an $n$-torus, including
contributions from all kinematically accessible modes, is
roughly proportional to~\cite{ADDPhenoBounds}
\begin{equation}
\sigma_{\mathrm{prod}}(pp\rightarrow j + G) ~\propto ~ \left(\frac{\sqrt{s}}{2\pi}\right)^n
\frac{1}{M_{D}^{n+2}}~.
\label{eq:KKGravitonProdSigmaProp}
\end{equation}
This implies that a bound of the form $M_D > M_D^{\mathrm{min}}$ can be translated
into a rough bound on the parameter space of our bulk-axion model of the form
\begin{equation}
\frac{\aleph^2(\sqrt{s})}{\fhatX^2} ~\lesssim~ \left(\frac{\sqrt{s}}{2\pi}\right)^n
\frac{1}{(M_D^{\mathrm{min}})^{n+2}}~,
\label{eq:ColliderLimitBulkAxionTranslated}
\end{equation}
where $\aleph(\Ech)$ is defined in Eq.~(\ref{eq:DefOfAleph}).
While this approximate bound does not take into account the differences in
coupling structure between KK graviton and axion fields or the sum over
polarizations for a massive graviton, it is sufficient to obtain
parametric estimates of the resulting constraints on our three fundamental
parameters $\fhatX$, $M_c$, and $\Lambda_G$.
In Fig.~\ref{fig:LHCAExclusion}, we indicate the rough bounds on the
parameter space of our bulk-axion DDM model which can be derived in this
manner, given a chosen value of $M_D^{\mathrm{min}}$. The contours shown in this
figure correspond to constraints of the form $M_D > M_D^{\mathrm{min}}$ for
the illustrative values $M_D^{\mathrm{min}} = \{1,10,100\}$~TeV.
\begin{figure}[ht!]
\begin{center}
\epsfxsize 2.25 truein \epsfbox {LHCLimitLambda1GeV.eps}
\epsfxsize 2.25 truein \epsfbox {LHCLimitLambda1TeV.eps}
\epsfxsize 2.25 truein \epsfbox {LHCLimitLambda100TeV.eps}
\end{center}
\caption{Excluded regions of the $(\fhatX, M_c)$ parameter space of our DDM model
in which the collider constraint in Eq.~(\ref{eq:ColliderLimitBulkAxionTranslated})
is violated for $M_D^{\mathrm{min}}=1$~TeV (green); $M_D^{\mathrm{min}}=10$~TeV
(green and blue); and $M_D^{\mathrm{min}}= 100$~TeV (green, blue, and purple).
As $\Lambda_G$ increases, we see that satisfying the collider constraints becomes
increasingly easy, particularly for small $\fhatX$.
In each case, we have taken $\xi=g_c=1$ and assumed
that the axion couples to at least one light, strongly-interacting
SM particle with an $\mathcal{O}(1)$ coupling coefficient $c_g$ or $c_q$.
\label{fig:LHCAExclusion}}
\end{figure}
We now compare these results to actual constraints on $M_D$ from current experimental
data and examine the projected LHC reach for our bulk-axion DDM model.
The most stringent constraints from LHC data (which indeed come from the
$pp\rightarrow j + \met$ channel) were given in Sect.~\ref{sec:ConstraintsLargeED}.
Estimates of the future LHC reach for a theory with a single extra dimension are
$M_D^{\mathrm{min}} \approx \{14,17\}$~TeV at integrated luminosities
$\mathcal{L}_{\mathrm{int}}=\{10,100\}\mathrm{~fb}^{-1}$,
respectively~\cite{GiudiceGravitonColliders}. Likewise, Tevatron data imply a limit
$M_D^{\mathrm{min}} \approx 2.4$~TeV for a theory with a single extra
dimension~\cite{GiudiceGravitonColliders}.
Comparing these results to those in Fig.~\ref{fig:LHCAExclusion}, we see that current
collider constraints, while quite stringent, do not significantly impact the preferred
region of parameter space for our bulk-axion DDM model, even in cases in which the axion
couples to one or more strongly-interacting SM fields with an $\mathcal{O}(1)$ coupling
coefficient.
In such cases, since the most stringent current LHC limits imply a bound of roughly
$M_D^{\mathrm{min}} \approx 1$~TeV, the region of the parameter space of our model
excluded by these limits roughly corresponds to the green shaded regions shown in
Fig.~\ref{fig:LHCAExclusion}.
Since the green exclusion regions in this figure embody the most stringent such limits
applicable to our DDM model, we shall take these to represent our collider constraints
throughout the rest of this paper.
However, we note that for photonic axions and other axion species which do not couple
directly to quarks or gluons, the corresponding collider constraints (which arise from
channels such as $pp\rightarrow \gamma + \met$) are somewhat weaker.
Before concluding this section, there is one important point which deserves emphasis.
The collider processes we have been discussing thus far are those whose
event rates receive their contributions from
the low-lying modes in the tower --- \ie, those $a_\lambda$ with lifetimes
$\tau_\lambda \gtrsim 10^{-12}$~s. By contrast, those heavy
$a_\lambda$ with lifetimes $\tau_\lambda \lesssim 10^{-12}$~s
tend to decay to pairs of SM fields {\it within} the detector volume.
The decays of such states can in principle give rise to an entirely different set
of signature patterns. For example, a promptly decaying $a_\lambda$ which couples
to light quarks or gluons as well as photons would in principle contribute to event
rates in the $pp\rightarrow jjj$ and $pp\rightarrow \gamma\gamma + j$ channels.
However, since the total event rate in these channels receives contributions
from a broad spectrum of $a_\lambda$ with different $\lambda$, many
event-selection techniques which are particularly useful in standard searches
for new physics in these channels cannot be applied to a tower of decaying
bulk axions. For example, since the set of decaying axions cannot characterized
by a single, well-defined mass or cross-section, no identifiable peak can be
expected to appear in the invariant-mass distribution for the decay products of
the heavy axions. Such considerations render the results of standard searches for
new physics in these channels inapplicable to our bulk-axion model --- and indeed
to DDM models in general. Moreover, they also likely render the identification of a
conclusive signal of non-standard dark-matter physics in these channels
particularly challenging. Nevertheless, the information that could potentially
be revealed about the nature of the dark sector via such an identification is of
sufficient magnitude and importance that an analysis of the discovery potential in
these channels is an interesting topic for future study.
\subsection{Axion Decays and Distortions of the Cosmic Microwave
Background Spectrum\label{sec:CMB}}
Up to this point, we have considered those phenomenological constraints on our
DDM model which are related to the production of particles which compose our bulk-axion
ensemble, both with and without their subsequent detection. By contrast, we now
turn to discuss an entirely different set of phenomenological constraints, namely
those which arise due to the potential decays of a {\it pre-existing}\/
cosmological population of such particles. Indeed, such constraints emerge generically in
all dark-matter scenarios in which the dark sector contains unstable, long-lived
particles, and can be derived from observational limits on the physical
consequences of the late decays of those particles.
There are many considerations which can be used to place such limits on scenarios
involving decaying dark-matter particles. For example, photons produced via the decays
of such particles
can yield observable distortions in the CMB spectrum; contribute to the diffuse extragalactic
X-ray and gamma-ray backgrounds; upset BBN predictions for the primordial abundances of
light elements; and result in unacceptably large entropy production during critical epochs
in the history of the universe. Constraints on dark-matter candidates from considerations
of this sort depend not only on the decay rate of the particle species in question,
but also on the relic abundance of that species. For this reason, the constraints
applicable to single-particle models of dark-matter are generally not directly applicable
to models within the DDM framework. It is therefore necessary to revisit the observational
limits on dark-matter decays within the context of our bulk-axion model of dynamical dark
matter and assess how these limits constrain the parameter space of this model.
In this section, we begin our analysis of the constraints on the late decays of the
$a_\lambda$ in our bulk-axion DDM model by examining observational limits on the
distortions of the CMB which such decays can induce. The type of CMB distortion to
which a late-decaying particle contributes depends on the time at which that particle
decays. In the very early universe, photons produced by particle decays are brought
into thermal and kinetic equilibrium with CMB photons via a number of processes.
The dominant processes by which newly-produced photons can equilibrate {\it thermally}\/
with CMB photons are double-Compton scattering
($e^- \gamma \rightarrow e^-\gamma\gamma$) and bremsstrahlung
($e^- X^\pm\rightarrow e^- X^\pm \gamma$, where $X^\pm$ is an ion). However,
once these processes freeze out, photons produced from $a_\lambda$ decays
are unable to thermally equilibrate with the radiation bath, resulting in the
generation of a non-zero value for the pseudo-degeneracy parameter $\mu$.
The interaction rates for these processes are given by~\cite{HuAndSilkLong}
\begin{eqnarray}
\Gamma_{\mathrm{DC}}&\approx& 5.73\times 10^{-39} \left(1-\frac{Y_p}{2}\right)
(\Omega_{\mathrm{B}}h^2)
\left(\frac{T_{\mathrm{now}}}{2.7 \mbox{~K}}\right)^{3/2}
\left(\frac{\tMRE}{t}\right)^{9/4} \mbox{GeV}
\nonumber\\
\Gamma_{\mathrm{BR}}&\approx& 1.57\times 10^{-36} \left(1-\frac{Y_p}{2}\right)
(\Omega_{\mathrm{B}}h^2)^{3/2}
\left(\frac{T_{\mathrm{now}}}{2.7 \mbox{~K}}\right)^{-5/4}
\left(\frac{\tMRE}{t}\right)^{13/8} \mbox{~GeV}~,
\label{eq:GammaDCandBR}
\end{eqnarray}
where $T_{\mathrm{now}} \approx 2.725$~K is the present-day CMB temperature,
$Y_p \approx 0.23$ is the helium mass fraction, $\Omega_\mathrm{B} \approx 0.044$ is the
baryon density of the universe, and $h \approx 0.72$ is the Hubble constant.
(Note that since $z$ is quite large during the entirety of the relevant
time frame, we have here approximated $1+z\approx z$.) Once these
processes freeze out, in the sense that the rates given in Eq.~(\ref{eq:GammaDCandBR})
drop below the expansion rate $H$ of the universe, photons produced by $a_\lambda$
decay will no longer be able to attain thermal equilibrium with the CMB photons.
Even after double-Compton scattering and bremsstrahlung effectively shut
off, a number of photon-number-conserving interactions still serve to bring photons
produced at even later times into {\it kinetic}\/ equilibrium with the radiation bath.
Dominant among these processes is elastic
Compton scattering ($e^-\gamma\rightarrow e^-\gamma$),
which efficiently serves to bring photons produced by $a_\lambda$ decays into kinetic
equilibrium until a much later time $t_{\mathrm{EC}}\sim 9\times 10^9$~s, at which point
this process too effectively freezes out. However, since elastic Compton scattering
conserves photon number, it cannot similarly suffice to
bring those photons into {\it thermal}\/
equilibrium. As a result, CMB distortions in the form of a non-zero value for the
pseudo-degeneracy parameter $\mu$ can be generated by $a_\lambda$ decays during this epoch.
In addition, after elastic Compton scattering freezes out, photons produced by $a_\lambda$
decay achieve neither kinetic nor thermal equilibrium with the radiation bath.
As a result, these photons no longer contribute the generation of $\mu$, but instead
contribute to the generation of a Compton $y$ parameter (here denoted $y_C$, so as to
distinguish it from the ratio $y=M_c/\mX$). Finally, at $t\sim 10^{13}$~s,
matter and radiation decouple, and any $a_\lambda$ decays occurring after this
point not affect the CMB, but instead simply persist as a contribution to
the diffuse photon background. This last sort of contribution will be dealt
with separately, in Sect.~\ref{sec:XrayGammaRay}.
We thus see that axion decays have the potential to generate both a non-zero $\mu$
and a non-zero $y_C$. We can therefore establish constraints on our bulk-axion DDM
model by calculating the theoretical predictions for these quantities in our model
and comparing these predictions to observational data.
We begin our analysis of CMB distortions from $a_\lambda$ decays by addressing those
decays which result in the generation of the pseudo-degeneracy parameter $\mu$.
In general, provided that the additional contribution $\delta\rho_\gamma$ to the
photon energy density $\rho_\gamma$ from the decay of the $a_\lambda$ fields is small
compared to the total $\rho_\gamma$, the time-evolution of $\mu$ can
be described by the equation~\cite{HuAndSilkShort,HuAndSilkLong}
\begin{equation}
\frac{d\mu}{dt} ~=~ \frac{d\mu_a}{dt}
- \mu \left(\Gamma_{\mathrm{DC}}+\Gamma_{\mathrm{BR}}\right)~.
\label{eq:MuCMBBasic}
\end{equation}
Here $\Gamma_{\mathrm{DC}}$ and $\Gamma_{\mathrm{BR}}$ are the
interaction rates for double-Compton scattering and
bremsstrahlung, respectively, and $d\mu_a/dt$ denotes the differential
contribution to $\mu$ from axion decay. For an arbitrary $d\mu_a/dt$,
the solution to this differential equation takes the form
\begin{equation}
\mu(t) ~=~ \exp\left[
\frac{4}{5}\big(2C_{\mathrm{BR}}t^{-5/8}+C_{\mathrm{DC}}t^{-5/4}\big)\right]
\int_{t_e}^t \left[\frac{d\mu_a}{dt}(t')\right]
\exp\left[
-\frac{4}{5}\big(2C_{\mathrm{BR}}t'^{-5/8}+C_{\mathrm{DC}}t'^{-5/4}\big)\right] dt',
\label{eq:SolveDiffEqCMBmu}
\end{equation}
where $t_e \approx 1.69\times 10^3$~s is the time scale associated with
electron-positron annihilation in the early universe, and where the quantities
$C_{\mathrm{DC}}$ and $C_{\mathrm{BR}}$ are constants related
to the double-Compton-scattering and bremsstrahlung rates $\Gamma_{\mathrm{DC}}$
and $\Gamma_{\mathrm{BR}}$ in Eq.~(\ref{eq:GammaDCandBR}) by
$\Gamma_{\mathrm{DC}} \equiv C_{\mathrm{DC}} t^{-9/4}$ and
$\Gamma_{\mathrm{BR}} \equiv C_{\mathrm{BR}} t^{-13/8}$. Moreover,
the differential contribution $d\mu_a/dt$ to $\mu$ from axion
decays is given by the standard expression for
contributions due to the late injection of photons from a generic source:
\begin{equation}
\frac{d\mu_a}{dt} ~=~ \frac{1}{2.143}\left(\frac{3}{\rho_\gamma}
\frac{d\rho_\gamma}{dt}-
\frac{4}{n_\gamma}\frac{dn_\gamma}{dt}\right)~.
\label{eq:dmudtfromadecay}
\end{equation}
In general, the rate of change in the photon energy density is given by
the Boltzmann equation for the evolution of $\rho_\gamma$. In our
bulk-axion DDM model, this equation includes a source term from each
decaying state in the dark-matter ensemble. Thus, at late times, after
all of the $a_\lambda$ have already begun oscillating coherently and the
contribution to $\rho_\gamma$ from inflaton decays can safely be neglected,
we find that
\begin{equation}
\frac{d\rho_\gamma}{dt} ~=~ -4H\rho_\gamma +
\sum_\lambda \BRgamma\Gamma_\lambda\rho_\lambda~,
\label{eq:RhoLambdaAndGammaEvolEqsSGen}
\end{equation}
where $\BRgamma$ is the branching fraction of $a_\lambda$ into a pair of
photons. Note that the source term in the Boltzmann equation for $\rho_\gamma$
is simply a sum of the contributions from the various $a_\lambda$ fields.
Using Eq.~(\ref{eq:RhoLambdaAndGammaEvolEqsSGen}), along with the relations
\begin{equation}
\frac{1}{(R^4\rho_\gamma)}\frac{d(R^4\rho_\gamma)}{dt} ~=~
\frac{1}{\rho_\gamma} \left(\frac{d\rho_\gamma}{dt}+4H\rho_\gamma\right),
~~~~~~~~
\frac{1}{(R^3n_\gamma)}\frac{d(R^3n_\gamma)}{dt} ~=~
\frac{1}{n_\gamma} \left(\frac{dn_\gamma}{dt}+3Hn_\gamma\right)~,
\end{equation}
we can rewrite Eq.~(\ref{eq:MuCMBBasic}) in the form
\begin{equation}
\frac{d\mu_a}{dt} ~ = ~
\frac{1}{2.143}\bigg[\frac{3}{\rho_\gamma}
\sum_\lambda\BRgamma \Gamma_\lambda \rho_\lambda -
\frac{8}{n_\gamma} \sum_\lambda\BRgamma\Gamma_\lambda \frac{\rho_\lambda}{\lambda}
\bigg]~.
\label{eq:MuCMBFinal}
\end{equation}
For the purpose of establishing a conservative bound, we focus here on the case of
a purely photonic axion. As we saw in Sect.~\ref{sec:IntraensembleProd},
the contribution to $\Gamma_\lambda$ from intra-ensemble decays is negligible for
any $a_\lambda$ which decays on time scales relevant for the generation of CMB
distortions. It is therefore justifiable to
approximate $\Gamma_\lambda$ by the expression for $\Gamma(a\rightarrow\gamma\gamma)$
given in Eq.~(\ref{eq:PartialWidthToPhotons}) and thus to take $\BRgamma \approx 1$.
Since the energy density $\rho_\lambda$ associated with each $a_\lambda$ is given
in Eq.~(\ref{eq:RhoOftEqnWithR}), we find that in this approximation, the first source
term on the right side of Eq.~(\ref{eq:MuCMBFinal}) takes the form
\begin{equation}
\sum_\lambda \BRgamma\Gamma_\lambda \rho_\lambda ~\approx~
\frac{1}{2} \theta^2 G_\gamma m_X^4
\sum_\lambda \lambda \left(\frac{t_\lambda^2}{\tRH^{1/2}}\right)
(\wtl^2 A_\lambda)^4 e^{-
\frac{G_\gamma\lambda^3}{\fhatX^2}(\wtl^2A_\lambda)^2 (t-t_G)}
\times \begin{cases}
\vspace{0.25cm}
\tRH^{1/2} t^{-2} ~~~~ & t \lesssim \tRH \\
\vspace{0.25cm}
t^{-3/2} ~~~~ & \tRH \lesssim t \lesssim \tMRE \\
\tMRE^{1/2}t^{-2} ~~~~ & t \gtrsim \tMRE~,
\end{cases}
\label{eq:SourceTermRhoPhotExact}
\end{equation}
where we have defined $G_\gamma$ is defined below
Eq.~(\ref{eq:PartialWidthToPhotons}). The second term takes
the same form, but with one factor of $\lambda$ fewer in the summand.
In principle, one could evaluate this sum numerically at each moment in time,
and then use these results to numerically solve Eq.~(\ref{eq:MuCMBFinal}).
However, we find that by making a few additional well-motivated approximations,
we can obtain a closed-form, analytical result for $d\mu_a/dt$.
We begin by dividing the tower into sections, based on the two criteria which
determine the dependence of $\Gamma_\lambda$ and $\rho_\lambda$ on $\lambda$.
The first of these is whether the oscillation-onset time for a given
$a_\lambda$ is within the staggered regime (\ie, $t_\lambda > t_G$), or
the simultaneous turn-on regime (\ie, $t_\lambda = t_G$). In the former case,
$t_\lambda$ depends on $\lambda$ according to Eq.~(\ref{eq:tlambdaInBothRegimes});
in the latter case, $t_\lambda$ is independent of $\lambda$.
The second pertinent criterion concerns the relationship between $\lambda$ and the
quantity
\begin{equation}
\lambdatrans ~\equiv~ \pi\mX^2/M_c~,
\end{equation}
introduced in Ref.~\cite{DynamicalDM2}. This quantity corresponds roughly to
the transition point between the small-$\lambda$ regime, in which the $a_\lambda$
are highly mixed, the large-$\lambda$ regime, in which mixing is negligible.
Indeed, for $\lambda \ll \lambdatrans$, we find that
$\wtl^2 A_\lambda \approx \sqrt{2}\,\wtl/(1 + \pi^2/y^2)^{1/2}$, while for
$\lambda \gg \lambdatrans$, we find that $\wtl^2 A_\lambda \approx \sqrt{2}$. Given
these criteria,
our first approximation will be to replace $\wtl^2 A_\lambda$ with its asymptotic
large-$\lambda$ form for all $\lambda > \lambdatrans$, and with its asymptotic
small-$\lambda$ form for all $\lambda < \lambdatrans$. Our second will be to
approximate the sum over $\lambda$ by a set of source-term integrals
$I_i(m,n,\alpha,\beta,\lambda_{\mathrm{min}},\lambda_{\mathrm{max}})$, each
corresponding to a different regime in the tower of modes characterized by a
particular dependence of the integrand on $\lambda$. These
source-term integrals may be evaluated analytically by making use of
the identity
\begin{eqnarray}
I_i(m,n,\alpha,\beta,\lambda_{\mathrm{min}},\lambda_{\mathrm{max}})
&\equiv&
\alpha\int_{\mathrm{\lambda_{min}}}^{\mathrm{\lambda_{max}}} \lambda^m
e^{-\beta \lambda^n} d\lambda
\nonumber \\ &=&
\alpha\frac{1}{n} \beta^{-(m+1)/n}\Bigg[
\Gamma\left(\frac{m+1}{n},\beta \lambda_{\mathrm{min}}^n\right)-
\Gamma\left(\frac{m+1}{n},\beta \lambda_{\mathrm{max}}^n\right)\Bigg]~,
\label{eq:IntegralIdentity}
\end{eqnarray}
which is valid for $n>0$ and any real values of $m$, $\alpha$, and $\beta$.
Here $\Gamma(s,x)$ denotes the incomplete gamma function:
\begin{equation}
\Gamma(s,x) ~\equiv~ \int^\infty_x t^{s-1} e^{-t} dt~.
\label{eq:DefOfIncompleteGammaFn}
\end{equation}
Employing the approximations discussed above, we find that the first
source term on the right side of Eq.~(\ref{eq:MuCMBFinal}) reduces to
\begin{equation}
\sum_\lambda\BRgamma\Gamma_\lambda \rho_\lambda ~=~
\frac{2G_\gamma\theta^2}{M_c}
\sum_{i=1}^4
I_i\big(m_i,n_i,\alpha_i,\beta_i,\lambda_{i-1}^{\mathrm{CMB}},
\lambda_i^{\mathrm{CMB}}\big)
\times \begin{cases}
\vspace{0.25cm}
\tRH^{1/2}t^{-2} ~~ & t \lesssim \tRH \\
\vspace{0.25cm}
t^{-3/2} ~~ & \tRH \lesssim t \lesssim \tMRE \\
\tMRE^{1/2}t^{-2} ~~ & t \gtrsim \tMRE~.
\end{cases}
\label{eq:SourceTermForPhotons}
\end{equation}
Inserting this result (and the analogous result for the second source term) into Eq.~(\ref{eq:dmudtfromadecay}) and using the fact that
$\mu$-type distortions are generated by decays occurring within the RD era, we
obtain the result
\begin{equation}
\frac{d\mu_a}{dt} ~\approx~ 0.935\times
\frac{G_\gamma\theta^2}{M_c t^{3/2}}
\Bigg[\frac{3}{\rho_\gamma^{\mathrm{eq}}}\sum_{i=1}^4
I_i(m_i,n_i,\alpha_i,\beta_i,
\lambda_{i-1}^{\mathrm{CMB}},\lambda_i^{\mathrm{CMB}})
- \frac{8}{n_\gamma^{\mathrm{eq}}}\sum_{i=1}^4
I_i(m_i-1,n_i,\alpha_i,\beta_i,
\lambda_{i-1}^{\mathrm{CMB}},\lambda_i^{\mathrm{CMB}})
\Bigg]~,
\label{eq:dmuadtExplicit}
\end{equation}
where the expressions for $\alpha_i$, $\beta_i$, $m_i$, and $n_i$ valid in
each $a_\lambda$ regime are listed in Table~\ref{tab:mnbetaforCMB}.
Note that in obtaining this expression, we have assumed that the additional
contributions to $n_\gamma$ and $\rho_\gamma$ due to the injection of
photons from $a_\lambda$ are sufficiently small that these quantities
can be approximated by the equilibrium expressions
$n_\gamma^{\mathrm{eq}} = 2\zeta(3)T^3/\pi^2$ and
$\rho_\gamma^{\mathrm{eq}} ~=~ \pi^2T^4/15$.
Furthermore, we have used the fact that the time frame during which CMB
distortions to $\mu$ can arise lies entirely within the RD era.
Obtaining a final result for the magnitude of $\mu$-type
distortions to the CMB engendered by the presence of a tower of decaying DDM
axions is then simply a matter of substituting the result for $d\mu_a/dt$ in
Eq.~(\ref{eq:dmuadtExplicit}) into Eq.~(\ref{eq:SolveDiffEqCMBmu}) and
numerically evaluating the integral for a given choice of input parameters.
\begin{table}[t!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
~~~$i$~~~ & ~Oscillation regime~ &
~~Mixing regime~~ & ~~$m_i$~~ & ~~~$n_i$~~~ &
$\alpha_{i}$ & $\beta_{i}$ \\\hline
1& \multirow{2}{*}{\rule[-0.25cm]{0cm}{0.85cm}
~~$t_\lambda>t_G$~~}
& $\lambda < \lambdatrans$ \rule[-0.25cm]{0cm}{0.7cm}
& 3 & 5 & $4\tRH^{-1/2}[1+\pi^2/y^2]^{-2}$ &
~~~$2G_\gamma t(\fhatX\mX)^{-2}[1+\pi^2/y^2]^{-1}$~~~ \\
2& & $\lambda \geq \lambdatrans$ \rule[-0.25cm]{0cm}{0.7cm}
& $-1$ & 3 & $4\mX^4\tRH^{-1/2}$ & $2G_\gamma t\fhatX^{-2}$ \\\hline
3& \multirow{2}{*}{\rule[-0.25cm]{0cm}{0.85cm}
~~$t_\lambda = t_G$~~}
& $\lambda < \lambdatrans$ \rule[-0.25cm]{0cm}{0.7cm}
& 5 & 5 & ~~$t_G^{\kappa_G}\tRH^{3/2-\kappa_G}[1+\pi^2/y^2]^{-2}$~~ &
~~~$2G_\gamma t(\fhatX\mX)^{-2}[1+\pi^2/y^2]^{-1}$~~~ \\
4& & $\lambda \geq \lambdatrans$ \rule[-0.25cm]{0cm}{0.7cm}
& 1 & 3 & $\mX^4t_G^{\kappa_G}\tRH^{3/2-\kappa_G}$ & $2G_\gamma t\fhatX^{-2}$ \\\hline
\end{tabular}
\caption{Values of $m_i$, $n_i$, $\alpha_i$, and
$\beta_i$ which correspond to
different regimes, labeled by the index $i$, in a generic axion tower,
for use in Eqs.~(\ref{eq:dmuadtExplicit}) and~(\ref{eq:PhotonSpectrumSum}).
The symbol $\kappa_G$ denotes the specific value of $\kappa$, as defined in
Eq.~(\ref{eq:DefOfkappaForH}), which corresponds to $t_G$.
\label{tab:mnbetaforCMB}}
\end{center}
\end{table}
The contribution to $y_C$ from the late decays of the $a_\lambda$ may be
evaluated in much the same way as the corresponding contribution to $\mu$.
The decays which contribute to $y_C$ are those which occur during the
window $9\times 10^9 \lesssim t \lesssim 1.2 \times 10^{13}$~s, during which the
rate $\Gamma_{\mathrm{EC}}\sim H$ associated with elastic Compton
scattering can no longer bring the photons from $a_\lambda$ decay into
kinetic equilibrium even though radiation has yet to decouple from matter.
The evolution of $y_C$ is governed by the
relation~\cite{DeZotti}
\begin{equation}
\frac{dy_C}{dt} ~=~ \frac{1}{4\rho_\gamma}\frac{d\rho_\gamma}{dt}.
\end{equation}
Proceeding with the mode sum as above and adopting the same approximations
as above, we find that
\begin{equation}
\frac{dy_C}{dt} ~\approx~
\frac{2G_\gamma\theta^2}{M_c \rho_\gamma^{\mathrm{eq}}}
\sum_{i=1}^4 I_i\big(m_i,n_i,\alpha_i,\beta_i,
\lambda_{i-1}^{\mathrm{CMB}}, \lambda_{i}^{\mathrm{CMB}}\big)
\times \begin{cases}
\vspace{0.25cm} t^{-3/2} ~~~~ & t \lesssim \tMRE \\
\vspace{0.25cm} \tMRE^{1/2}t^{-2} ~~~~ & \tMRE \lesssim t \lesssim \tLS \\
0 ~~~~ & t \gtrsim \tLS~,
\end{cases}
\label{eq:dyCompdtExplicit}
\end{equation}
where $\tLS \sim 1.19 \times 10^{13}$~s is the time of last scattering and
$I_i(m,n,\alpha,\beta,\lambda_{\mathrm{min}},\lambda_{\mathrm{max}})$ is once
again given by Eq.~(\ref{eq:IntegralIdentity}). Note that since matter-radiation
equality occurs prior to last scattering, at around $\tMRE \sim 10^{11}$~s, the
epoch during which $a_\lambda$ decays can affect $y_C$ straddles both the RD and
MD eras. Numerically evaluating the expression in Eq.~(\ref{eq:dyCompdtExplicit})
from $t_{\mathrm{EC}}$ to $\tLS$, we obtain our final results for $y_C$ distortions
due to late $a_\lambda$ decay.
\begin{figure}[ht!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {CMByContourPlotLam1GeV.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {CMByContourPlotLam1TeV.eps}
\end{center}
\caption{Contours of the CMB Compton-$y$-parameter distortion $y_C$ (black lines) produced
as a result of axion decays in a bulk-axion DDM model with $\Lambda_G = 1$~GeV
(left panel) and $\Lambda_G = 1$~TeV (right panel). In each case, we have assumed a
photonic axion with $c_\gamma=1$ and have taken
$\xi = g_G = \theta = 1$, with $H_I = 1$~GeV and $\TRH=5$~MeV. Contours corresponding
to $y\equiv M_c/\mX = 1$ (solid red line) and to $y = \{0.01,0.1,10,100\}$ (dashed red lines)
are also shown. For each panel, it is evident that our bulk-axion DDM model amply
satisfies the CMB constraints in Eq.~(\protect\ref{eq:yCandmuBounds})
for all relevant values of $\fhatX$ and $M_c$, regardless of the value of $y$.
\label{fig:CMByContours}}
\end{figure}
In order to assess the CMB constraints on the parameter space of our bulk-axion
DDM model, we now compare the results obtained by numerically integrating
Eqs.~(\ref{eq:SolveDiffEqCMBmu}) and~(\ref{eq:dyCompdtExplicit}) with
observational limits on $\mu$ and $y_C$. The current limits
on these quantities are~\cite{PDG}
\begin{equation}
|\mu| < 9\times 10^{-5}~, ~~~~~~~~~
y_C < 1.2 \times 10^{-5}~.
\label{eq:yCandmuBounds}
\end{equation}
The bound on $y_C$ for a photonic axion with $c_\gamma = 1$ yields the
constraints on $\fhatX$, $M_c$, and $\Lambda_G$ shown in Fig.~\ref{fig:CMByContours}.
In this figure, we display contours of the values of $y_C$ in $(\fhatX, M_c)$ space
which arise in a bulk-axion model with $\Lambda_G = 1$~GeV (left panel) and with
$\Lambda_G = 1$~TeV (right panel). In each case, we have taken
$\xi = g_G = \theta = 1$, $H_I = 1$~GeV, and $\TRH=5$~MeV. Contours
indicating $y \equiv M_c/\mX = 1$ (solid red line) and
$y = \{0.01,0.1,10,100\}$ (dashed red lines) have also been superimposed.
For each panel, it is evident that our bulk-axion DDM model amply
satisfies the CMB constraints for all relevant values of $\fhatX$ and $M_c$, regardless
of the value of $y$.
It turns out that the constraints from the corresponding bound on $\mu$ in
Eq.~(\ref{eq:yCandmuBounds}) are even less stringent than those from the bound
on $y_C$. Thus, we conclude that both the $y_C$-type and $\mu$-type distortions
which result from $a_\lambda$ decays in our bulk-axion DDM model are well below
present experimental sensitivities. Indeed, no meaningful constraint
arises for our bulk-axion DDM model from present limits on distortions in the CMB.
As we have discussed, neither $\mu$ nor $y_C$ can be affected by
any photons which are produced by $a_\lambda$ decays at times $t \gtrsim \tLS$,
after radiation and matter decouple. Such photons do, however, contribute to
the diffuse photon background. In the next section, we will discuss the physical
effects of this diffuse photon background in detail.
\subsection{Axion Decays and Contributions to the Diffuse X-Ray and Gamma-Ray
Backgrounds\label{sec:XrayGammaRay}}
As mentioned above, the potentially observable effects of late photoproduction
from axion decays include not only distortions of the CMB, but also imprints on
the diffuse X-ray and gamma-ray backgrounds. Observational limits on such
imprints from instruments such as HEAO~\cite{HEAOdiffXRB},
COMPTEL~\cite{COMPTELdiffXRB}, XMM, and Chandra~\cite{ChandraDeepFieldData}
therefore impose additional constraints on the parameter space of our bulk-axion
DDM model.
As discussed in Sect.~\ref{sec:AbundanceConstraints}, there are two cosmological
populations of decaying $a_\lambda$ whose decays to photons can potentially leave
observable imprints on the diffuse X-ray and gamma-ray backgrounds. The first is
the population of cold axions produced by vacuum misalignment, which collectively
compose the DDM ensemble. The second is the far smaller
population of axions produced by interactions among the SM fields in the thermal
bath after inflation. While the former population provides a far
greater contribution to $\Omegatot$, the latter population contains a far
larger proportion of heavier, more unstable $a_\lambda$, as indicated in
Fig.~\ref{fig:OmegaCompFromThemal}. It is not clear {\it a priori}\/
which population yields the more stringent constraint. Thus, it is necessary to examine
the contribution to the diffuse photon background from each of these populations
in turn.
A photon produced at time $t$ with initial energy $E_\gamma(t)$ will only
contribute to the diffuse photon background if the universe remains transparent
to electromagnetic radiation over the entire range of energies through which
that photon redshifts as the universe evolves from $t$ to $\tnow$. A detailed
analysis of the time scales and photon-energy ranges for which this transparency
condition is attained is presented in Ref.~\cite{DMDecayChenKamionkowski1}.
Roughly speaking, the transparency window spans an energy range
$1 \mathrm{~keV} \lesssim E_\gamma \lesssim 10\mathrm{~TeV}$ and a time range
$10^{12}-10^{14}\mathrm{~s}\lesssim t \lesssim \tnow$, with the lower
limit depending on the particular value of $E_\gamma$. Motivated by these results,
we approximate the universe to be transparent to all photons with energies which fall
within this range at all times $t>\tLS$ and opaque to all photons otherwise. This
approximation yields a conservative bound. Moreover, we emphasize that since the
dominant contribution to the diffuse X-ray and gamma-ray flux in our model
is due to modes which decay at much later times $t\gg \tLS$, our results are
essentially insensitive to the precise contours chosen for the transparency window.
The calculation of the photon flux due to late $a_\lambda$ decays proceeds in a
manner similar to the calculation of the flux from KK-graviton decays outlined in
Ref.~\cite{CosmoConstraintsLargeED}.
The Boltzmann equation for the {\it number}\/ density $n_\gamma$ of photons
in the presence of a tower of decaying $a_\lambda$ takes the form
\begin{equation}
\dot{n}_\gamma + 3 H n_\gamma ~=~ 2\sum_\lambda
\BRgamma \Gamma_\lambda \frac{\rho_\lambda}{\lambda}~,
\end{equation}
where once again $\rho_\lambda$ is given by Eq.~(\ref{eq:RhoLambdaInLTRCosmo}).
Solving this equation for $n_\gamma$ as a function of time, we obtain
\begin{equation}
n_\gamma(t) ~=~
2 \frac{s(t)}{\sLS}
\sum_\lambda\BRgamma
\frac{\rho_\lambda(\tLS)}{\lambda}
\left[1-e^{-\Gamma_\lambda(t-\tLS)}\right]~,
\label{eq:BoltzmannSolXRayPhotons}
\end{equation}
where $s(t)$ is the entropy density of the universe at time $t$, and $\sLS$ is the
entropy density of the universe at the time of last scattering.
The present-day differential energy spectrum $dn_\gamma/dE_\gamma$ of these
photons may readily be computed from the relation
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~=~ \frac{dn_\gamma}{dt}\frac{dt}{dz}
\frac{dz}{dE_\gamma}~,
\label{eq:dndEgammaDifferentials}
\end{equation}
where $z$ is the cosmological redshift and $E_\gamma$ is the photon energy at
redshift $z$. The first of these factors may be
obtained by explicitly differentiating Eq.~(\ref{eq:BoltzmannSolXRayPhotons})
with fixed $s=\snow$, which yields a series of terms of the form
\begin{equation}
\left[\frac{dn_\gamma}{dt}\right]_\lambda ~=~
2 \left(\frac{\snow}{\sLS}\right)
\BRgamma \Gamma_\lambda
\frac{\rho_\lambda(\tLS)}{\lambda}
e^{-\Gamma_\lambda(\tnow-\tLS)}~,
\label{eq:dndtGeneralForm}
\end{equation}
one for each different value of $\lambda$.
The second factor in Eq.~(\ref{eq:dndEgammaDifferentials}) may be obtained by
noting that the relationship between time and redshift during the present,
matter-dominated era is well-approximated by $t ~=~ \tnow(1+z)^{-3/2}$.
Consequently, for each value of $\lambda$ we have
\begin{equation}
\left[\frac{dt}{dz}\right]_\lambda ~=~
-\frac{3}{2}\tnow\left(\frac{2E_\gamma}{\lambda}\right)^{5/2}
\end{equation}
during the epoch of interest. The third factor in
Eq.~(\ref{eq:dndEgammaDifferentials}) may be obtained by noting
that each of the photons produced by an axion tower state
$a_\lambda$ which decays at redshift $z$ will be monochromatic, with
energy $\lambda/2$, at the moment of decay. This implies that the present-day
energies of such photons are given by $E_\gamma(1+z) = \lambda/2$, and hence that
for each value of $\lambda$, we have
\begin{equation}
\left[\frac{dz}{dE_\gamma}\right]_\lambda ~=~
-\frac{\lambda}{2E_\gamma^2}~.
\end{equation}
Combining these expressions and summing over $\lambda$, we arrive at a general
formula for the contribution to the diffuse photon flux produced by the tower
of decaying $a_\lambda$:
\begin{equation}
\left.\frac{dn_\gamma}{dE_\gamma}\right|_{\mathrm{now}} ~=~
6\tnow\sqrt{2E_\gamma} \left(\frac{\snow}{\sLS}\right)
\sum_\lambda \BRgamma \Gamma_\lambda
\frac{\rho_\lambda(\tLS)}{\lambda^{5/2}}
e^{-\Gamma_\lambda(\tnow-\tLS)}~.
\label{eq:dndEgammaGeneralForm}
\end{equation}
Calculating the contribution to the diffuse X-ray and gamma-ray backgrounds
in our bulk-axion DDM model is then simply a matter of
applying Eq.~(\ref{eq:dndEgammaGeneralForm}) to the contribution from
the two relevant populations of decaying axions discussed above.
We begin by addressing the contribution from the population of axions
produced by vacuum misalignment --- \ie, the DDM ensemble itself.
Once again, we focus our attention on the case of a photonic axion,
for which $\Gamma_\lambda \approx \Gamma(a\rightarrow\gamma\gamma)$ and
$\BRgamma \approx 1$. In this
case, we find that the contribution to the present-day diffuse
photon background from the collective decays of the $a_\lambda$ fields is given by
\begin{equation}
\left.\frac{dn_\gamma}{dE_\gamma}\right|_{\mathrm{now}} ~=~
3 \sqrt{2E_\gamma} G_\gamma\theta^2 \mX^4
\left(\frac{\snow}{\sLS}\right)
\sum_\lambda
\left(\frac{t_\lambda^{2}\tMRE^{1/2}}{\tLS^{2}\tRH^{1/2}}\right)
\lambda^{-3/2}(\wtl^2A_\lambda)^4
e^{\frac{G_\gamma \lambda^3}{\hat{f}_G^2}
(\wtl^2A_\lambda)^2 (\tnow-t_G)}~.
\label{eq:PhotonSpectrumSum}
\end{equation}
Just as for the contributions to $\mu$ and $y_C$ in Sect.~\ref{sec:CMB},
we approximate the sum over axion modes appearing in Eq.~(\ref{eq:PhotonSpectrumSum})
as an integral over $\lambda$.
The lower limit of integration is determined by the requirement that in order
for a photon with redshifted energy $E_\gamma$ to have been produced by
the decay of the axion species $a_\lambda$ before present day, we must have
$\lambda \geq 2E_\gamma$. Likewise, photons which decay before
the processes which equilibrate them with the radiation bath freeze out
will not contribute to features in the diffuse photon background. Thus, the upper
limit of integration is set by the condition
$\lambda \lesssim 2E_\gamma (\tnow/\tLS)^{2/3}$. Furthermore, we must also require
that $\lambda$ not exceed the cutoff scale $f_G$, or be smaller than the
lightest mode in the tower. Once again, we find that the resulting integral
expressions can be written in terms of the functions
$I_i(m,n,\alpha,\beta,\lambda_{\mathrm{min}},\lambda_{\mathrm{max}})$
defined in Eq.~(\ref{eq:IntegralIdentity}):
\begin{equation}
\left.\frac{dn_\gamma}{dE_\gamma}\right|_{\mathrm{now}} ~\approx~
12G_\gamma\theta^2 \frac{\sqrt{2E_\gamma}\tnow}{M_c}
\left(\frac{\snow}{\sLS}\right)
\left(\frac{\tMRE^{1/2}}{\tLS^{2}}\right)
\sum_{i=1}^4
I_i\big(m_i-5/2,n_i,\alpha_i,\beta_i,\lambda_{i-1}^{\mathrm{XRB}},
\lambda_i^{\mathrm{XRB}}\big)~,
\label{eq:FinalXRBFormula}
\end{equation}
where the $\lambda_i^{\mathrm{XRB}}$ are analogous to the
$\lambda_i^{\mathrm{CMB}}$ appearing in Eq.~(\ref{eq:dmuadtExplicit}).
Determining the net contribution to the differential photon flux from
decays of the $a_\lambda$ for any particular choice of model parameters
is thus simply a matter of numerically evaluating Eq.~(\ref{eq:FinalXRBFormula}).
We now turn to consider the observational limits on $dn_\gamma/dE_\gamma$.
The diffuse extragalactic X-ray and gamma-ray background spectra have
been probed by a number of experiments. In the keV $-$ MeV region, the
most current data are those from HEAO, COMPTEL, XMM, and Chandra; at
energies above this, the most current data are those from EGRET and FERMI.
Over this entire energy range, the diffuse photon spectrum is well-modeled
by a set of power-law fits, and the non-observation of any discernible,
sharp features in this spectrum imposes constraints on late relic-particle
decays to photons.
For the data from the COMPTEL instrument, the best power-law fit is found to
be~\cite{COMPTELdiffXRB}
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~ = ~
10.5 \times 10^{-4} \left(\frac{E_\gamma}{5\mbox{~MeV}}\right)^{-2.4}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
~~~~~~~~ 800\mbox{~keV} \lesssim E_\gamma \lesssim 30\mbox{~MeV}~,
\label{eq:COMPTELdiffBG}
\end{equation}
while the best fit to the HEAO data is found to be~\cite{HEAOdiffXRB}
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~ = ~ \begin{cases}
\displaystyle
7.88\times 10^3 \left(\frac{E_\gamma}{\mathrm{~keV}}\right)^{-1.29}
e^{-(E_\gamma/41.13\mathrm{~keV})}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}\vspace{0.25cm}
&~~~~~~~~ 0.1\mbox{~keV}\lesssim E_\gamma \lesssim 60\mbox{~keV} \\
\displaystyle
0.43\left(\frac{E_\gamma}{\mathrm{60~keV}}\right)^{-6.5} +\vspace{0.25cm}
8.4\left(\frac{E_\gamma}{\mathrm{60~keV}}\right)^{-2.58}~~~~~~~~~~~~~~~~~~ \\
\displaystyle ~~~~~~~~~~ +~
0.38\left(\frac{E_\gamma}{\mathrm{60~keV}}\right)^{-2.05}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
~~~~~~~\vspace{0.25cm}
&~~~~~~~~ 60\mbox{~keV}\lesssim E_\gamma \lesssim 160\mbox{~keV} \\
\displaystyle
3.8\times 10^5\times \left(\frac{E_\gamma}{\mathrm{keV}}\right)^{-2.6}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
\vspace{0.25cm}~~~~~~~
&~~~~~~~~ 160\mbox{~keV}\lesssim E_\gamma \lesssim 350\mbox{~keV}\\
\displaystyle
2.0\times 10^3 \left(\frac{E_\gamma}{\mathrm{keV}}\right)^{-1.7}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}\vspace{0.25cm}
&~~~~~~~~ 350\mbox{~keV} \lesssim E_\gamma \lesssim 2\mbox{~MeV}~.
\end{cases}
\end{equation}
The Chandra satellite has improved upon these diffuse X-ray background
constraints in the $ 1\mbox{~keV}\lesssim E_\gamma \lesssim 8\mbox{~keV}$ range
by resolving a large fraction ($\sim 80$\%) of this background into point sources.
The residual spectrum in this region is well represented by the
power law~\cite{ChandraPowerLawFitHickox}
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~ = ~
2.6 \times 10^3 \left(\frac{E_\gamma}{\mbox{keV}}\right)^{-1.5}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
~~~~~~~~~~~~~ 1\mbox{~keV} \lesssim E_\gamma \lesssim 8\mbox{~keV}~.
\end{equation}
In the gamma-ray region, the most stringent current limits are those from
EGRET and FERMI. Data on the diffuse extragalactic gamma-ray background
from the former instrument~\cite{EGRETdiffGRB} are reliable
for photon energies within the range
$1.41\mbox{~GeV}\lesssim E_\gamma \lesssim 30\mbox{~MeV}$, for which we
find the best fit
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~ = ~
7.35\times 10^{-3}\left(\frac{E_\gamma}{\mathrm{MeV}}\right)^{-2.35}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
~~~~~~~~~~ 30\mbox{~MeV}\lesssim E_\gamma \lesssim 1.41\mbox{~GeV}~.
\label{eq:EGRETdiffBG}
\end{equation}
Note that data exist for
higher photon energies as well, but given that EGRET's energy resolution
is not as good at such high energies, and given that these data
have been superseded by data from FERMI, we do not use them in computing
this power-law fit. As for the FERMI data, they are well modeled by the
power law~\cite{FERMIdiffGRB}
\begin{equation}
\frac{dn_\gamma}{dE_\gamma} ~ = ~
9.59\times 10^{-3} \left(\frac{E_\gamma}{\mathrm{MeV}}\right)^{-2.41}
\mbox{~MeV}^{-1}\mbox{cm}^{-1}\mbox{s}^{-1}\mbox{str}^{-1}
~~~~~~~~~~ 274\mbox{~MeV}\lesssim E_\gamma \lesssim 70.7\mbox{~GeV}~.
\label{eq:FERMIdiffBG}
\end{equation}
\begin{figure}[b!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {XRayGraphLambda1GeV.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {XRayGraphLambda1TeV.eps} \\
\epsfxsize 3.0 truein \epsfbox {XRayGraphLambda100TeV.eps}
\end{center}
\caption{The diffuse photon-flux spectrum
$dn_\gamma/dE_\gamma$ produced from axion decays in our bulk-axion DDM model
with $\Lambda_G = 1$~GeV (upper left panel), $\Lambda_G = 1$~TeV (upper right panel),
and $\Lambda_G = 100$~TeV (lower panel).
Each solid colored curve corresponds to a different choice of $\fhatX$, within the range
$10^{6} - 10^{16}$~GeV. In all panels, we have taken
$M_c = 10^{-11}$~GeV, $H_I = 1$~GeV, $\TRH=5$~MeV, and $\xi = g_G = \theta = 1$.
By contrast, the dashed black contours represent the upper
bounds on $dn_\gamma/dE_\gamma$ derived from observational limits on the diffuse photon
flux using a number of instruments sensitive in the X-ray and gamma-ray regions.
As evident from these plots, the diffuse-photon-background contribution arising from
axion decay in our bulk-axion DDM model is consistent with all observational
limits when $\Lambda_G$ is large.
\label{fig:XRayGammaRayLimit}}
\end{figure}
In Fig.~\ref{fig:XRayGammaRayLimit}, we show a set of curves (solid colored lines)
depicting the total contribution to the diffuse gamma-ray background from the decaying
$a_\lambda$ fields, as given in Eq.~(\ref{eq:FinalXRBFormula}), for several different
values of $\fhatX$ within the range
$10^{6} - 10^{16}$~GeV. Results are shown for $\Lambda_G = 1$~GeV (upper left panel),
$\Lambda_G = 1$~TeV (upper right panel), and $\Lambda_G = 100$~TeV (lower panel).
In each case, we have taken $M_c = 10^{-11}$~GeV, $\TRH = 5$~MeV, and
$\xi = g_G = \theta = 1$. In addition, we have chosen a value for $H_I$
sufficiently large that none of the curves
shown is significantly affected by the ``inflating away'' of heavy modes which
begin oscillating before inflation ends. In addition to these curves, we
also display contours corresponding to the upper limits on the diffuse X-ray
and gamma-ray fluxes (black dashed lines) given in
Eqs.~(\ref{eq:COMPTELdiffBG}) through~(\ref{eq:FERMIdiffBG}). Any choice of model
parameters for which the differential photon flux $dn_\gamma/dE_\gamma$ exceeds
any one of these observational-limit contours for any value of $E_\gamma$ is excluded.
The results shown in Fig.~\ref{fig:XRayGammaRayLimit} indicate that while it is
not trivial to satisfy these observational limits in our bulk-axion DDM model,
the contributions to the diffuse X-ray and gamma-ray fluxes from $a_\lambda$ decay
are indeed sufficiently small that these limits are satisfied when
$\Lambda_G$ is large.
We now consider the contribution to $dn_\gamma/dE_\gamma$ from the population
of axions generated by their interactions with SM fields in the thermal bath
after inflation.
The contribution to the diffuse photon flux spectrum $dn_\gamma/dE_\gamma$
generated by such a population of axions is once again given by
Eq.~(\ref{eq:dndEgammaGeneralForm}), but with $\rho_\lambda(\tLS)$ in
Eq.~(\ref{eq:RhoLambdaInLTRCosmo}) now replaced by
\begin{equation}
\rho_\lambda(\tLS) ~\approx~ \lambda\TLS^3\tLS
\int_{\TMRE}^{\TRH}
\frac{3}{\kappa(T)}\left(\frac{\TLS}{T}\right)^{3/\kappa(T)}
\frac{g_{\ast s}(\TLS)}{g_{\ast s}(T)}
\left[C_\lambda^{\mathrm{ID}}(T) + C_\lambda^{\mathrm{Prim}}(T)
e^{-(\lambda+m_e)/T}\right]dT~,
\label{eq:RhoThermMRE}
\end{equation}
as follows from Eq.~(\ref{eq:OmegaThermNow}).
To derive an estimate for the expected contribution to
$dn_\gamma/dE_\gamma$ from the resulting equation,
we proceed in essentially the same way as we did in calculating
the contribution from axions produced via vacuum misalignment.
The results of this calculation are shown in
Fig.~\ref{fig:XRayGammaRayLimitTherm} for parameter values
within or near the preferred region of parameter space for our
bulk-axion DDM model. Specifically, we have taken $M_c = 10^{-11}$~GeV,
$\Lambda_G = 1$~TeV, $\TRH=5$~MeV, and $\xi = g_G = 1$.
The solid colored curves shown correspond to several different choices of
$\fhatX$ ranging from $\fhatX = 10^{12}$~GeV to $\fhatX= 10^{15}$~GeV. Once
again, the dashed black lines indicate the observational limits on additional
contributions to $dn_\gamma/dE_\gamma$. It is clear from
Fig.~\ref{fig:XRayGammaRayLimitTherm} that while the contribution
to the diffuse X-ray flux from thermal axions within our preferred
region of parameter space is certainly not negligible, it is also
consistent with current observational limits. We therefore conclude that even
after the contribution from thermal axions is included, our bulk-axion DDM
model is consistent with X-ray and gamma-ray data.
\begin{figure}[h!]
\begin{center}
\epsfxsize 3.0 truein \epsfbox {dndEgammaThermLam1TeV.eps} ~~~~
\epsfxsize 3.0 truein \epsfbox {dndEgammaThermLam100TeV.eps}
\end{center}
\caption{The diffuse photon flux spectrum
$dn_\gamma/dE_\gamma$ produced from the decays of a
population of $a_\lambda$ produced by interactions among the SM
particles in the thermal bath after inflation. The left
panel shows the results for $\Lambda_G = 1$~TeV, while the right
panel shows the results for $\Lambda_G = 100$~TeV. In each case,
we have taken $M_c = 10^{-11}$~GeV, $\TRH=5$~MeV, and $\xi = g_G = 1$.
The solid colored curves indicate the diffuse-photon-flux contributions
corresponding to different choices of $\fhatX$.
As in Fig.~\ref{fig:XRayGammaRayLimit}, the dashed black contours indicate the
upper bounds on $dn_\gamma/dE_\gamma$ derived from observational limits on the
diffuse X-ray and gamma-ray fluxes, and we see that our model is consistent
with these bounds.
\label{fig:XRayGammaRayLimitTherm}}
\end{figure}
\subsection{Axion Decays and Big-Bang Nucleosynthesis\label{sec:BBN}}
The accord between the primordial abundances of light nuclei inferred from
observation and the predictions for those abundances within the framework of
standard BBN has been one of the greatest triumphs of theoretical cosmology.
However, these predictions depend sensitively on the cosmological parameters
during the nucleosynthesis epoch.
For example, the presence of additional relativistic degrees of freedom in the
thermal bath during BBN can substantially distort the abundances of the light
elements away from their observed values. In addition, the decays of unstable
particles during or after the BBN epoch can also alter these abundances via the
injection of both entropy and energy into the thermal bath. We must therefore
ensure that the collective effects of $a_\lambda$ decays in our model
are sufficiently small so as not to disrupt the successful generation of
light-element abundances via standard BBN.
Limits on the abundance of a single unstable relic particle $\chi$ from
BBN are typically phrased as bounds on the number density $\hat{n}^\ast_\chi$
that $\chi$ would have at present time if it
were absolutely stable. In general, the BBN bound on $\hat{n}^\ast_\chi$ for
any given relic particle depends on the
lifetime $\tau_\chi$ of that particle. The most stringent
limits are obtained for lifetimes
$\tau_\chi \sim\mathcal{O}( 10^{9} - 10^{10}$~s),
for which the corresponding constraint is
roughly~\cite{CyburtEllisBBN,KawasakiHadronicBBN}
\begin{equation}
m_\chi\frac{\hat{n}^\ast_\chi}{n_\gamma^\ast}~\lesssim~ 10^{-13}\mbox{~GeV}~,
\end{equation}
where $n_\gamma^\ast\approx 410.5\mbox{~cm}^{-3}$ denotes the present-day number
density of photons. This limit can also be written in the form
\begin{equation}
\Omegachinodec ~\lesssim~ 1.7 \times 10^{-5}~,
\label{eq:OmegathreshChiBBNConstraint}
\end{equation}
where $\Omegachinodec$ is the relic abundance that $\chi$ would have at present
time if it were absolutely stable.
Once again, however, as with other constraints on traditional models of
decaying dark matter (such as those from the CMB and the diffuse X-ray and
gamma-ray backgrounds), these constraints are not readily applicable to models
within the DDM paradigm, since the dark-matter candidate in these models is an
ensemble with no single, well-defined mass or lifetime. Thus, we must
reexamine the derivation of the BBN constraints on decaying relic particles
in order to determine what restrictions these considerations place
on the parameter space of our bulk-axion DDM model. While a detailed calculation
of the precise limits BBN considerations impose on DDM scenarios in general
is beyond the scope of this paper, it is straightforward to demonstrate that BBN
constraints do not significantly restrict the parameter space of the particular
model which concerns us here.
\begin{figure}[b!]
\begin{center}
\epsfxsize 2.25 truein \epsfbox {OmegathreshPlotLam1GeV.eps}
\epsfxsize 2.25 truein \epsfbox {OmegathreshPlotLam1TeV.eps}
\epsfxsize 2.25 truein \epsfbox {OmegathreshPlotLam100TeV.eps}
\raisebox{0.3cm}{\large$\Omegatotnodec$}\epsfxsize 5.00 truein \epsfbox {ColorBarOmegathresh.eps}
\end{center}
\caption{Contours of the collective contribution to $\Omegatotnodec$
from the set of $a_\lambda$ with lifetimes $\tau_\lambda < \tau_\chi^{\mathrm{min}}$ for
a DDM ensemble of photonic axions with $c_\gamma=1$.
The left, center, and right panels display
the results for $\Lambda_G = 1$~GeV, $\Lambda_G = 1$~TeV, and $\Lambda_G = 100$~TeV,
respectively. In each case, we have taken $\TRH = 5$~MeV, $H_I = 100$~TeV, and
$\xi = g_G = \theta = 1$. In each panel, we see that BBN constraints
are amply satisfied throughout essentially the entire region of parameter space shown.
\label{fig:OmegaThreshPanelsLTR}}
\end{figure}
We begin by noting that in traditional, single-particle dark-matter scenarios,
an unstable dark-matter candidate $\chi$ with a relic abundance
$\Omega_\chi \sim \OmegaDM$ is generally consistent with all astrophysical
and cosmological limits on dark-matter decays, provided that
$\tau_\chi \gtrsim \tau_\chi^\mathrm{min} \sim 10^{26}$~s~\cite{DMDecayChenKamionkowski1}.
It therefore follows that any $a_\lambda$ in the DDM ensemble with a lifetime
$\tau_\lambda \gtrsim \tau_\chi^\mathrm{min}$ will have no impact on BBN within
regions of parameter space in which the WMAP constraint $\Omegatot \leq \OmegaCDM$ on
the {\it total}\/ dark-matter relic abundance is satisfied. Thus, we may safely conclude
that our bulk-axion model of dynamical dark matter is consistent with BBN constraints
within such regions of parameter space, provided that
\begin{equation}
\Omegatotnodec ~\lesssim~ 1.7 \times 10^{-5}~,
\label{eq:OmegathreshBBNConstraint}
\end{equation}
where $\Omegatotnodec$ denotes the collective contribution which
the set of $a_\lambda$ with lifetimes $\tau_\lambda < \tau_\chi^{\mathrm{min}}$
would have made to the dark-matter relic abundance at present time if they were
absolutely stable. In other words, the BBN constraint
we are imposing in Eq.~(\ref{eq:OmegathreshBBNConstraint}) effectively
rests upon the extremely conservative approach of treating all states in the
DDM ensemble whose lifetimes are less than $\tau_\chi^{\mathrm{min}}$
as if they had lifetimes $\tau_{\lambda}$ which are in the range
which is most dangerous for BBN, namely $\tau_\lambda \sim 10^{9}-10^{10}$~s.
We emphasize that while this criterion is a sufficient condition
for successful BBN, it does {\it not}\/ represent the true BBN constraint,
which is always far less stringent.
In Fig.~\ref{fig:OmegaThreshPanelsLTR}, we display contours of
$\Omegatotnodec$ for a DDM ensemble of photonic axions with $c_\gamma =1$,
as a function of $\fhatX$, $M_c$, and $\Lambda_G$.
The left panel shows the results for
$\Lambda_G = 1$~GeV, the center panel for $\Lambda_G = 1$~TeV, and the
right panel for $\Lambda_G = 100$~TeV. In each case, we have taken
$\TRH = 5$~MeV, $H_I = 100$~TeV, and $\xi = g_G = \theta = 1$.
In each panel of Fig.~\ref{fig:OmegaThreshPanelsLTR}, we see that the criterion in
Eq.~(\ref{eq:OmegathreshBBNConstraint}) is amply satisfied throughout
essentially the entire region of parameter space shown. It therefore follows
that our bulk axion model is consistent with successful BBN throughout this
region of parameter space.
\subsection{Axion Decays and Late Entropy Production\label{sec:Entropy}}
One additional physical consequence of the late decays of unstable relic
particles is the generation of entropy as those particles ``dump''
their energy density into the radiation bath. Indeed, a number of
considerations place constraints on late entropy production
from decaying particles. For example, late entropy generation can upset
the light-element predictions from standard BBN and produce observable
features in the CMB. In this section, we examine the effect of
the late decays of the $a_\lambda$ on the entropy density of the universe
in our bulk-axion DDM model as a function of time in order to verify
that no perceptible effects can arise which might serve to exclude
our model.
During any given epoch, the entropy density of the universe is dominated by
the contribution from radiation and therefore well approximated by
\begin{equation}
s ~\approx~ \sum_{i}\frac{4\rho_i(T_i)}{3T_i} ~=~
\frac{\pi^2}{30}g_{\ast s}(T)T^3~,
\end{equation}
where the index $i$ runs over all relativistic particle species,
$T_i$ is the temperature associated with any particular such species,
and $g_{\ast s}$ is the number of interacting degrees of freedom at
temperature $T$. During the early stages of the history of the universe
(prior to neutrino decoupling), all such species are characterized by a
common temperature $T_i \approx T$. During such epochs,
$g_{\ast s}(T)\approx g_{\ast}(T)$, and the entropy
density is therefore directly proportional to the total energy density
$\rhorad$ of radiation. Indeed, even during subsequent epochs,
$g_{\ast s}$ and $g_\ast$ remain roughly similar, and $\rhorad$ remains a
good indicator of the entropy density. Thus, by evaluating the contribution
to $\rhorad$ from $a_\lambda$ decays in our bulk axion DDM model, we can
assess the effect of these decays on both the energy and entropy densities
of the universe.
In the LTR cosmology, as in the standard cosmology, $\rhorad$ evolves
according to an equation similar to
Eq.~(\ref{eq:RhoLambdaAndGammaEvolEqsSGen}):
\begin{equation}
\frac{d\rhorad}{dt} ~=~ -4H\rhorad + \Gamma_\phi \rho_\phi+
\sum_\lambda \BRrad\Gamma_\lambda\rho_\lambda~.
\label{eq:RhoRadEvolEqGen}
\end{equation}
This equation assumes
the presence of a tower of decaying $a_\lambda$, where $\BRrad$ is the
total branching fraction of $a_\lambda$ into relativistic particles. Note,
however, that since we are working within the context of LTR cosmology,
the effects of inflaton decays on the energy and entropy densities of the
universe remain relevant until very late times $t\sim \tRH$.
Thus we have explicitly included an additional source term
$\Gamma_\phi \rho_\phi$ in Eq.~(\ref{eq:RhoRadEvolEqGen}) to account for
the effect of such inflaton decays, where $\Gamma_\phi$ and $\rho_\phi$ respectively
denote the decay rate and energy density of the inflaton field $\phi$.
The contribution to $\rhorad$ from inflaton decays can readily be calculated
from standard results pertaining to the LTR cosmology (for a review, see, \eg, Ref.~\cite{LTRAxionsKamionkowski}).
As the universe exits the inflationary epoch at a time $t_I \approx 2/(3H_I)$,
the energy density stored in the inflaton field is initially
$\rho_{\phi}=\rhocrit =3H_I^2 M_P^2$. During subsequent epochs, the inflaton
source term for radiation is approximately given by
\begin{equation}
\Gamma_\phi \rho_\phi ~\approx~ \frac{3H_I^2 M_P^2}{2\tRH}
\left(\frac{t_I}{t}\right)^\kappa e^{-t/2\tRH}~,
\label{eq:InflatonRadSourceTerm}
\end{equation}
where $\kappa$ is defined as in Eq.~(\ref{eq:DefOfkappaForH}), and
we have used the fact that the inflaton-decay rate is related to the
reheating time by $\Gamma_\phi \approx 1/(2\tRH)$. Note that
this source term is negligible at times $t\gg\tRH$, when by definition
$\rho_\phi \ll \rhorad$, and hence can safely be neglected at such times.
By contrast, at early times $t\lesssim \tRH$, the inflaton source term
is expected to dominate in Eq.~(\ref{eq:RhoRadEvolEqGen}), in the sense
that
\begin{equation}
\Gamma_\phi \rho_\phi ~\gg~ \sum_\lambda \BRrad\Gamma_\lambda \rho_\lambda~.
\end{equation}
Whenever this condition is satisfied,
the contribution to $\rhorad$ from $a_\lambda$ decays is inconsequential
compared to that from inflaton decays, and the axion source term can therefore
safely be neglected.
In assessing the contribution from $a_\lambda$ decay, we once again
choose to focus on the case of a photonic axion with $c_\gamma = 1$; this
implies that the decay mode $a_\lambda \rightarrow \gamma \gamma$ dominates
the contribution to $\rhorad$. In this
case, the source term for radiation due to $a_\lambda$ decay is just the
source term for photons given in Eq.~(\ref{eq:SourceTermForPhotons}). In
this case, solving Eq.~(\ref{eq:RhoRadEvolEqGen}) for $\rhorad$, we find that
\begin{equation}
\rhorad(t) ~=~ \rhoradbar(t)
+ \int_{t_G}^{t}\left(\frac{t'}{t}\right)^{4\kappa/3} \sum_\lambda
\BRgamma \Gamma_\lambda \rho_\lambda(t')dt'~,
\end{equation}
where $\rhoradbar(t)$ is the solution for $\rhorad(t)$
in the absence of any additional contribution from $a_\lambda$ decays.
Once again making use of the integral functions
$I_i\big(m,n,\alpha,\beta,\lambda_{\mathrm{min}},\lambda_{\mathrm{max}}\big)$
defined in Eq.~(\ref{eq:IntegralIdentity}) to approximate the sum over modes,
we obtain
\begin{equation}
\rhorad(t) ~=~ \rhoradbar(t)
+ \frac{2G_\gamma\theta^2}{M_c} \int_{t_G}^t dt'
\frac{t'^{\kappa/3}}{t^{4\kappa/3}}
\sum_{i=1}^4
I_i\big(m_i,n_i,\alpha_i,\beta_i,\lambda_{i-1}^{\mathrm{CMB}},
\lambda_i^{\mathrm{CMB}}\big)
\times \begin{cases}
\vspace{0.25cm}
\tRH^{1/2} ~~ & t \lesssim \tRH \\
\vspace{0.25cm}
1 ~~ & \tRH \lesssim t \lesssim \tMRE \\
\tMRE^{1/2} ~~ & t \gtrsim \tMRE~.
\end{cases}
\end{equation}
\begin{figure}[b!]
\begin{center}
\epsfxsize 2.25 truein \epsfbox {RhoRadEvolPlotLambda1GeVHI100TeV.eps}
\epsfxsize 2.25 truein \epsfbox {RhoRadEvolPlotLambda1TeVHI100TeV.eps}
\epsfxsize 2.25 truein \epsfbox {RhoRadEvolPlotLambda100TeVHI100TeV.eps}
\end{center}
\caption{The total contribution to the radiation
energy density $\rhorad$ from photonic $a_\lambda$ decays in our
bulk-axion DDM model (solid lines), plotted as functions of time
for a variety of different choices of $\fhatX$.
The left panel shows the results for $\Lambda_G = 1$~GeV, the center panel
for $\Lambda_G = 1$~TeV, and the right panel for $\Lambda_G = 100$~TeV.
In each case, we have assumed a photonic axion with $\xi=g_G=\theta=1$, and
we have taken $M_c = 10^{-11}$~GeV, $\TRH = 5$~MeV, and $H_I = 100$~TeV.
Also shown in each panel is the total value of $\rhorad$ as a function of time
in the LTR cosmology (black dashed line), which includes the contribution from
inflaton decay. In all cases, the collective contribution to
$\rhorad$ from $a_\lambda$ decays at all times $t < \tnow$ remains negligible
compared to the primordial contribution generated via inflaton decays during
reheating. Thus our bulk-axion DDM model does not lead to overproduction of either
radiation-energy density or entropy during any prior cosmological epoch.
\label{fig:RhoRadVsTime}}
\end{figure}
In Fig.~\ref{fig:RhoRadVsTime}, we show how the contribution to $\rhorad$ from
$a_\lambda$ decays in our bulk-axion DDM model evolves with time for a variety
of different choices of model parameters. The left panel
shows results for $\Lambda_G = 1$~GeV, the center panel for $\Lambda_G = 1$~TeV,
and the right panel for $\Lambda_G = 100$~TeV. The solid colored curves in each panel
correspond to different choices of $\fhatX$ within the range $10^{10} - 10^{16}$~GeV.
For all curves shown, we have assumed a photonic axion with $c_\gamma=1$, and
we have taken $M_c = 10^{-11}$~GeV, $\TRH = 5$~MeV, $H_I = 100$~TeV, and
$\xi=g_G=\theta=1$. The black dashed curve represents the total value of
$\rhorad$, which includes the standard contribution from inflaton
decays during the reheating epoch.
Since such inflaton decays constitute the dominant source for
radiation prior to the end of reheating, the range of times shown in
each panel extends from $\tRH$ to present time. The value of $H_I$
has been chosen here to be sufficiently large that the effect of heavier
$a_\lambda$ with $\lambda \gtrsim 3H_I/2$ being inflated away is unimportant.
Note, however, that for significantly smaller values of $H_I$, the contribution
to $\rhorad$ from axion decays can be further suppressed by this effect.
The differences among the curves shown in Fig.~\ref{fig:RhoRadVsTime} for different
choices of $\fhatX$ and $\Lambda_G$ ultimately stem from the effects of axion mixing
on the abundances $\rho_\lambda$ and decay widths $\Gamma_\lambda$ of the
individual axion modes.
The results shown in the left panel correspond to the case in which
$\Lambda_G$ is sufficiently small that $y\gg 1$ for all choices of $\fhatX$
shown. In this small-mixing regime, $\lambda \gtrsim \lambdatrans$ for all but
the lowest-lying mode in the axion KK tower, and
Eqs.~(\ref{eq:DefsOfyandmPQ}) and~(\ref{eq:RhoLambdaInLTRCosmo}) imply that
$\rho_\lambda \propto\fhatX^{-2}$ and $\Gamma_\lambda \propto\fhatX^{-2}$.
It therefore follows that the photon source term
$\BRgamma\Gamma_\lambda\rho_\lambda$ associated with each $a_\lambda$
within this regime decreases uniformly and substantially with increasing $\fhatX$,
as indicated. By contrast, as $\Lambda_G$ is increased,
several competing effects play an increasingly important role in determining the
magnitude of $\BRgamma\Gamma_\lambda\rho_\lambda$ for certain $\lambda$.
This is because $\lambdatrans$ increases with increasing $\Lambda_G$; hence
for large $\Lambda_G$ a greater number of the
$a_\lambda$ are brought into the $\lambda \lesssim \lambdatrans$ regime, in which
$\rho_\lambda \propto\fhatX^2$ and $\Gamma_\lambda\propto \fhatX^2$.
Increasing $\fhatX$ therefore has the effect of increasing the initial magnitude of
the photon source terms associated with the $a_\lambda$ in this regime. However,
the lifetimes of these modes also increase with increasing $\fhatX$, and hence
the transfer of their energy density to radiation is deferred until later times,
when $\rhorad$ is smaller and the contribution from $a_\lambda$ decays
can have a proportionally greater impact. The interplay between these effects
results in the behavior shown in the right two panels of Fig.~\ref{fig:RhoRadVsTime}.
Note that the curves for the total energy density shown in
Fig.~\ref{fig:RhoRadVsTime}, which are dominated by the
contribution from inflaton dynamics, drop more rapidly as a function of time than
the contributions from axion dynamics. This reflects the continuing generation
of new radiation energy density from the ongoing decays of the individual $a_\lambda$
within our DDM ensemble. In all cases, however, the collective contribution to
$\rhorad$ from $a_\lambda$ decays at all times $t < \tnow$ remains negligible
compared to the primordial contribution generated via inflaton decays during
reheating. Thus our bulk-axion DDM model does not lead to overproduction of either
radiation-energy density or entropy during any prior cosmological epoch.
\subsection{Vacuum Energy and Overclosure\label{sec:Overclosure}}
In traditional dark-matter scenarios involving a single, stable dark-matter
candidate $\chi$, the dark-matter relic abundance $\Omega_\chi$ increases
monotonically up to and beyond the present time. As a result, verifying
that $\Omega_\chi$ satisfies WMAP constraints at the present time is sufficient to
guarantee that $\chi$ does not overclose (or prematurely matter-dominate)
the universe at all previous times as well.
However, one of the hallmarks of the DDM scenario is that this is no longer true:
although $\Omegatot$ likewise experiences a Hubble-driven growth during the earliest
phases of the evolution of the universe, this quantity can nevertheless drop
during later epochs. This is possible within the DDM framework because
the single, stable dark-matter candidate $\chi$ characteristic of most traditional
dark-matter scenarios is replaced by a complex, multi-component
dark-matter ensemble whose constituents can have a broad spectrum of lifetimes
and abundances. As a result, the decays of certain dark-matter components within the
ensemble can cause $\Omegatot$ to decline --- even prior to the present day.
Indeed, such behavior for $\Omegatot$ can be quite dramatic, and is illustrated
in Fig.~6 of Ref.~\cite{DynamicalDM1} for the special case in which the DDM ensemble
consists of a KK tower of decaying dark fields. Thus, within the DDM framework,
it is no longer sufficient to verify that $\Omegatot$ satisfies overclosure constraints
at the present time; we must also verify that it has satisfied such overclosure
constraints (and constraints from premature matter- or vacuum-energy domination)
at all prior moments during the history of the universe.
It turns out, however, that this is not a problem in our bulk-axion DDM model. Since our
model already satisfies WMAP constraints at present time within our preferred region of
parameter space~\cite{DynamicalDM2}, it can run afoul of overclosure constraints in the
past only if the negative rate of change of $\Omegatot$ is sufficiently great that
$\Omegatot$ might have exceeded unity within the past history of the universe. However,
as discussed in Refs.~\cite{DynamicalDM1,DynamicalDM2}, this rate of change is described
by an effective equation-of-state parameter $w_{\mathrm{eff}}$, and two things are
already known about the value of this parameter in our model: first, it is extremely
small at the present day, \ie,
$10^{-23} \lesssim w_{\mathrm{eff}} \lesssim 10^{-12}$~\cite{DynamicalDM2}, and second,
it was even smaller in the past. Indeed, this latter assertion follows from the
generic behavior of $w_{\mathrm{eff}}$ shown in Fig.~8 of Ref.~\cite{DynamicalDM1}:
for a generic KK tower, $w_{\mathrm{eff}}$ reaches its maximum at the present day and
is exponentially smaller prior to this time. Thus, working backwards from the present
epoch, and given the finite age of the universe, we see that it is not possible for
$\Omegatot$ to have violated overclosure bounds at any point during the history of the
universe.
One related concern which arises in our bulk-axion DDM model, due to our reliance on
the misalignment mechanism for the generation of the primordial relic abundances of
the $a_\lambda$ is the risk of premature vacuum domination. Indeed, any $a_\lambda$
for which
$t_\lambda > t_G$ will contribute to the total dark-energy abundance $\Omegavac$ during
the period when $t_G \lesssim t \lesssim t_\lambda$, within which its energy density
$\rho_\lambda$ is non-vanishing but before which it begins oscillating.
Since $\rho_\lambda$ remains constant during this period, the contribution
to $\Omegavac$ scales like $\Omega_\lambda \propto t^2$ during any MD or RD epoch. Since
this represents a rate of increase far faster than that associated with matter or
radiation, the threat of premature vacuum domination from fields which remain as
vacuum energy for a long duration is of particular concern. Indeed, in extreme
cases, such fields could potentially give rise to an additional period of inflation,
leading to gross inconsistencies with the predictions of BBN, CMB data, and so forth.
In our bulk-axion model, however, it is straightforward to demonstrate that
no such inconsistencies with observational data arise.
The masses of all of the $a_\lambda$, with the sole exception of the zero mode $a_0$,
are bounded from below by the Newton's-law-modification constraint in
Eq.~(\ref{eq:MinimumMc}), since $\lambda_i \geq M_c/2$ for $i>0$. For all such modes
with $t_\lambda > t_G$, this constraint on $\lambda$ implies a bound
$t_\lambda > 6.75\times 10^{-14}$~s on the oscillation-onset time of the mode.
(The remaining modes, for which $t_G = t_\lambda$, never contribute to $\Omegavac$.)
This time scale is sufficiently early that the collective vacuum-energy contribution
from these $a_\lambda$ poses no threat of overclosure or premature
vacuum-domination. The $\Omega_\lambda$ contributions from these fields simply do
not have time to grow to a problematic size.
This leaves only the contribution from $a_0$,whose oscillation time scale
can be substantially longer than the upper limit quoted above for the higher
modes in situations in which $y \gg 1$. Since $A_{\lambda_{0}} \approx 1$ in this limit,
Eq.~(\ref{eq:RhoLambdaInLTRCosmo}) implies that prior to the time $t_{\lambda_0}$ at
which it begins oscillating, the relic abundance of $a_0$ is given by
\begin{equation}
\Omega_{\lambda_0} ~\approx~ \frac{3}{2}\frac{\mX^2\fhatX^2}{M_P^2}
\left(\frac{t}{\kappa}\right)^2~.
\end{equation}
Therefore, one finds that by the time of oscillation, which is given by
$t_{\lambda_0} \approx \kappa_{\lambda_{0}}/2\mX$
in this limit, $\Omega_{\lambda_0}$ will have grown to
\begin{equation}
\Omega_{\lambda_0}(t_{\lambda_0}) ~\approx~ \frac{3}{8} \frac{\fhatX^2}{M_P^2}~.
\end{equation}
This result is independent of $\mX$, and implies that the contribution of the
$a_0$ to $\Omegavac$ is not a cause for concern for sub-Planckian values of $\fhatX$.
Indeed, this is to be expected: in this regime, $a_0$ functions effectively
like a four-dimensional axion. Early vacuum-energy domination is known not to be
a problem for light axions and axion-like particles (see Ref.~\cite{JaeckelReview} and
references therein) in purely four-dimensional theories.
\subsection{Misalignment Production and Isocurvature
Perturbations\label{sec:isocurvature}}
In an inflationary cosmology, fluctuations in the energy density of any
population of particles produced thermally, \ie, via rapid interactions
in the radiation bath during the reheating phase, stem from the primordial
perturbations in the energy density of the inflaton field. Consequently,
such fluctuations are of the so-called adiabatic type --- that is, they
represent spatial variations in the {\it total} energy density, but not in the
relative contributions of individual particle species to that total density.
Such variations, in turn, imply fluctuations in the local spacetime curvature
and are therefore sometimes also referred to as curvature perturbations.
By contrast, fluctuations in the energy density of any population of particles
produced via means uncorrelated with the inflaton field (and therefore non-thermal)
can also give rise to fluctuations of the isocurvature
type --- \ie, perturbations in the relative contributions of different species
to the total energy density, with that total energy density held fixed.
Recent WMAP observations of the CMB power spectrum,
taken in combination with baryon acoustic oscillation (BAO) measurements and
supernova data, place a stringent bound~\cite{WMAP} on any deviations from
adiabaticity in primordial energy-density fluctuations. This bound is typically
expressed in terms of the fractional contribution $\alpha_0$ to the
CMB power spectrum from axion isocurvature perturbations:
\begin{equation}
\alpha_0 ~\equiv~ \frac{\langle (\delta T/T)^2_{\mathrm{iso}}\rangle}
{\langle(\delta T/T)^2_{\mathrm{tot}}\rangle} < 0.072~,
\label{eq:WMAPAlphaIsoBound}
\end{equation}
where $\langle (\delta T/T)^2_{\mathrm{tot}}\rangle$ and
$\langle (\delta T/T)^2_{\mathrm{iso}}\rangle$ respectively denote the
total average root-mean-squared fluctuation in the CMB temperature, and the
average root-mean-squared temperature fluctuation due to isocurvature
perturbations alone. Since the $a_\lambda$ fields which compose our dynamical
dark-matter ensemble are presumed to be produced non-thermally, via the
misalignment mechanism, it is necessary to investigate the implications of this
bound for our model.
Our discussion of isocurvature perturbations in our bulk-axion DDM model
in large part parallels the discussion of such
perturbations in traditional QCD axion models presented in
Ref.~\cite{HertzbergAxionCosmology}, to which we refer the reader for a
more complete introduction and discussion of the formalism and methodologies used.
It turns out to be convenient to express the fluctuations of any given $a_\lambda$ in
terms of the fractional change $S_\lambda$ in the ratio of
its number density $n_\lambda$ to the entropy density $s$
of the universe. This quantity can be written in the form
\begin{equation}
S_\lambda ~\equiv~ \frac{\delta(n_\lambda/s)}{(n_\lambda/s)}
~=~ \frac{\delta n_\lambda}{n_\lambda} - 3 \frac{\delta T}{T}~.
\label{eq:DefOfSlambda}
\end{equation}
We assume that the production of all other particle species $\psi_i$
(\ie, the SM fields) ultimately results from inflaton decay, and that the
density fluctuations for these species are purely adiabatic, with $S_i = 0$.
Since, by definition, the fluctuation $\delta\rho$ in the total energy density
vanishes for isocurvature fluctuations, it therefore follows that the sum of
the fluctuations in the energy densities of the various particle species
obeys a constraint which may be written in the form
\begin{equation}
\sum_\lambda \rho_\lambda \left(S_\lambda + 3 \frac{\delta T}{T}\right) +
3\sum_i \rho_i \frac{\delta T}{T} + 4\rho_{\mathrm{rad}}\frac{\delta T}{T} ~=~ 0~,
\label{eq:IsocurvatureNRGConservationEq}
\end{equation}
where the $\rho_i$ denote the energy densities associated with massive species other
than the $a_\lambda$, and $\rhorad$ once again denotes the total energy density of
radiation.
In our bulk-axion DDM model, the abundances of
all of the $a_\lambda$ are determined by a single misalignment angle
$\theta$. As discussed in Ref.~\cite{DynamicalDM1}, this reflects the
ultimate five-dimensional nature of the axion field.
This in turn implies that the density fluctuations
$\delta n_\lambda$ for all of these fields are determined by the
fluctuations $\delta\theta$ in this misalignment angle generated by
quantum fluctuations during inflation.
The fact that the fluctuations $\delta n_\lambda$ are all
determined by $\delta\theta$ implies that the $S_\lambda \equiv S$ are
essentially equal for all $a_\lambda$; hence
Eq.~(\ref{eq:IsocurvatureNRGConservationEq}) simplifies to
\begin{equation}
\Omegatot S ~=~ -3\left(\Omega_{\mathrm{mat}} +
\frac{4}{3}\Omega_{\mathrm{rad}}\right) \frac{\delta T}{T}~,
\label{eq:SlambdaEq}
\end{equation}
where $\Omega_{\mathrm{mat}}$ denotes the total abundance of matter
in the universe, including the contributions from baryonic matter, the
ensemble of dark axions, and any other particles which might contribute to
the dark-matter relic abundance, and $\Omega_{\mathrm{rad}}$ is
the relic-abundance contribution from radiation. This expression is
identical to that which describes the isocurvature perturbations associated
with a single, four-dimensional
axion. Therefore, assuming that the fluctuations in $\theta$ are Gaussian, it
follows that in our axion DDM model, $\alpha_0$ is given by the
standard expression~\cite{HertzbergAxionCosmology}
\begin{equation}
\alpha_0 ~=~\frac{8}{25} \left(\frac{\Omegatot^\ast}{\Omega_{\mathrm{mat}}^\ast}\right)^2
\frac{1}{\langle(\delta T/T)^2_{\mathrm{tot}}\rangle}\,
\frac{\sigma_\theta^2(2\theta^2 + \sigma_\theta^2)}
{(\theta^2+\sigma_\theta^2)^2}~,
\label{eq:IsocurvatureAlpha1}
\end{equation}
where $\Omega_{\mathrm{mat}}^\ast$ denotes the present-day value of
$\Omega_{\mathrm{mat}}$, and where $\sigma_\theta^2\equiv \langle(\delta\theta)^2\rangle$
denotes the variance associated with fluctuations in $\theta$.
This result makes intuitive sense. Although our DDM model has essentially partitioned
the total dark-matter abundance amongst a large number of different KK axion fields,
the underlying five-dimensional nature of the KK tower has correlated the individual
fluctuations of these fields so that they are governed by the fluctuation of a single
misalignment angle $\theta$. It is therefore not a surprise that the expected magnitude
for isocurvature fluctuations in our model turns out to be no greater than it is
standard, four-dimensional axion models.
All that remains, then, for us to do in order to determine the value of
$\alpha_0$ in our bulk-axion DDM model, is to assess the magnitude of
$\sigma^2_\theta$. Assuming again that the fluctuations in $\theta$ are
Gaussian, this quantity is given by
\begin{equation}
\sigma_\theta^2 ~=~
\frac{H_I^2}{4\pi^2\fhatX^2}~.
\end{equation}
Since we are operating within the context of an LTR cosmology with
$\TRH \sim \mathcal{O}(\mathrm{MeV})$, as discussed above, it is by no means
problematic (and in fact quite natural) for $H_I \ll \fhatX$. Therefore,
as long as $\theta \sim \mathcal{O}(1)$, as might be expected from naturalness
considerations, it can safely be assumed that $\theta \gg \sigma_\theta$.
Substituting into Eq.~(\ref{eq:IsocurvatureAlpha1}) the experimentally
observed~\cite{WMAP} values
$\langle (\delta T/T)^2_{\mathrm{tot}}\rangle \approx (1.1\times 10^{-5})^2$
and $\Omega_{\mathrm{mat}}^\ast\approx 0.262$ we find that $\alpha_0$ is well
approximated by
\begin{equation}
\alpha_0 ~\approx~ 1.95\times 10^{9}
\left(\frac{H_I\Omegatotnow}{\fhatX\theta}\right)^2
\label{eq:IsocurvatureAlpha2}
\end{equation}
in our bulk-axion model. Combining this result with the upper bound on $\alpha_0$
quoted in Eq.~(\ref{eq:WMAPAlphaIsoBound}) yields the constraint
\begin{equation}
H_I ~\lesssim~ 6.07 \times 10^{-6}
\left(\frac{\theta\fhatX}{\Omegatot^\ast}\right)~.
\label{eq:HIConstraintIsocurvature}
\end{equation}
We consider the case in which $\Omegatotnow \approx \OmegaDM$ and in which the axion
ensemble is responsible for essentially the entirety of the observed dark-matter
relic abundance. This ocrresponds to $\fhatX \approx 10^{14} - 10^{15}$~GeV.
We then find that for $\theta \sim \mathcal{O}(1)$,
the resulting constraint $H_I \lesssim 10^9 - 10^{10}$~GeV on the
Hubble parameter during inflation is
relatively mild. Indeed, there is no difficulty in satisfying this constraint in
either the standard or the LTR cosmology. We thus conclude that isocurvature
perturbations do not present any problem for our bulk-axion model of dynamical
dark matter. Moreover, a low scale for $H_I$ can be regarded as natural in the context
of an LTR cosmology.
It is worth remarking, however, that the above results have implications
for the detection of primordial gravitational waves.
Limits on primordial gravitational waves from observations of the CMB can
be conveniently parametrized in terms of the scalar-to-tensor ratio
$r$. For example, consider single-field models of inflation, in which
$r = 16\epsilon$, where $\epsilon = M_P^2(V'/V)^2/(4\pi)$ is
the inflaton slow-roll parameter, with $V$ and $V'$ denoting
the inflaton potential and its first derivative with respect to the inflaton field,
respectively~\cite{LiddleAndLyth}. In the context of our bulk-axion DDM model, the standard
relation (see, \eg, Ref.~\cite{WMAP}) between $r$ and $\alpha_0$ takes the form
\begin{equation}
r ~=~ \frac{2\theta^2\fhatX^2}{M_P^2}
\left(\frac{\OmegaDM}{\Omegatotnow}\right)^2
\frac{\alpha_0}{1-\alpha_0}~.
\end{equation}
As discussed above, consistency with the bounds in Eqs.~(\ref{eq:WMAPAlphaIsoBound}) and~(\ref{eq:HIConstraintIsocurvature}) requires that $H_I \ll 2\pi f_X\theta$
and $\alpha_0 \ll 1$. In this regime, one finds that the
expected tensor-to-scalar ratio is essentially independent of $\Omegatotnow$
and well approximated by
\begin{equation}
r ~\approx~ 2.7 \times 10^{8}
\left(\frac{H_I}{M_P}\right)^2~.
\label{eq:TensorToScalarTheoretical}
\end{equation}
Current WMAP observations, again in conjunction from BAO and supernova data,
place an upper bound $r<0.22$ on the tensor-to-scalar ratio~\cite{WMAP}.
Thus, Eq.~(\ref{eq:TensorToScalarTheoretical}) results in a constraint
$H_I \lesssim 6.7 \times 10^{13}$~GeV on the Hubble scale during
inflation --- a constraint which Eq.~(\ref{eq:HIConstraintIsocurvature})
implies is already automatically satisfied, even for $\mathcal{O}(1)$
values of the misalignment angle $\theta$. The upshot is therefore that
while there is no conflict between current limits on isocurvature
perturbations and the predictions of our bulk-axion DDM model, the
requirement that $H_I$ be relatively small in this model suggests that
$r$ should likewise be quite small --- at least in the simplest of
inflationary scenarios. Constraints on the spectral index $n_s$ from
WMAP~\cite{WMAP} can simultaneously be satisfied for small $r$ without
difficulty, for example in negative-curvature models of
inflation, which tend to predict small $r$~\cite{WMAPInflation}.
In summary, we conclude that current constraints on isocurvature
perturbations can be satisfied in our bulk-axion DDM model without
too much difficulty. However, we note that any conclusive measurement
of $r$ within the sensitivity range of the Planck satellite would have
severe ramifications for this model.
\subsection{Axion Abundances and Quantum Fluctuations During
Inflation\label{sec:InflationScale}}
Thus far in this paper, we have disregarded the effects of the quantum fluctuations
that naturally arise for any massless or nearly massless field during the
inflationary epoch. In particular, the low-momentum modes of any $a_\lambda$
in our model with a mass $\lambda \lesssim H_I$ have wavelengths which exceed the
Hubble length during inflation; excitations of such low-momentum modes are
therefore indistinguishable from a VEV and consequently do not
inflate away. These excitations necessarily yield a primordial
energy-density contribution in our bulk-axion DDM model which cannot be
avoided in any inflationary cosmology. Consistency with the relic-abundance
predictions discussed in Sect.~\ref{sec:MisalignmentProd} therefore requires
that this primordial energy density be small compared to that which results
from misalignment production.
In particular, it is possible to formulate a condition that
ensures that these quantum fluctuations not invalidate our previous analysis.
Clearly, one criterion that any such condition must enforce is that
such fluctuations not have a significant effect on the total relic
abundance of the ensemble. We may formulate this
constraint as a requirement that the difference between the full present-day
relic abundance $\Omegatotjitternow$, which incorporates the effect of these
fluctuations, and the result $\Omegatotnow$ obtained in the absence of such
corrections be negligible --- \ie, that
\begin{equation}
\big|\,\Omegatotjitternow - \Omegatotnow\big|
~\ll~ \Omegatotnow~.
\label{eq:HubbleJitterCondit1}
\end{equation}
While the condition in Eq.~(\ref{eq:HubbleJitterCondit1}) is certainly a
necessary one, it is not by itself sufficient to ensure that
vacuum fluctuations during inflation do not lead to phenomenological
difficulties for our model. This is because within DDM framework,
dark-matter stability is not a requirement, and consistency
with observational constraints is arranged by balancing decay widths
against abundances across the entire dark-matter ensemble. Indeed,
as we have demonstrated, misalignment production
provides precisely the right relationship between the $\Omega_\lambda$ and
$\Gamma_\lambda$ to mitigate the deleterious effects of the heavier, more
unstable states in our ensemble and render our model phenomenologically
viable. We must therefore ensure that this delicate balance is not
disrupted by the effects of vacuum fluctuations during inflation.
Within the preferred region of parameter space of our bulk-axion DDM model,
as discussed in Sect.~\ref{sec:MisalignmentProd}, the oscillation-onset times
for the lighter $a_\lambda$ in the tower are staggered in time. As a result,
these lighter modes collectively dominate in $\Omegatot$. It therefore follows
that whether or not the total-relic-abundance constraint in
Eq.~(\ref{eq:HubbleJitterCondit1}) is satisfied depends primarily on how
vacuum fluctuations affect the abundances of these most abundant modes alone.
By contrast, the balancing of lifetimes against abundances depends on the
properties of the full KK tower, and not merely on the attributes of the
lighter modes which dominate $\Omegatot$. The corresponding condition we
impose on our model therefore represents an even stronger constraint than the
one appearing in Eq.~(\ref{eq:HubbleJitterCondit1}) and indeed subsumes
it. To wit, we require
that the full relic abundance $\widetilde{\Omega}_\lambda$ of {\it each
axion mode} not differ significantly from the corresponding abundance
$\Omega_\lambda$ obtained in the absence of corrections due to
vacuum fluctuations during inflation --- \ie, that
\begin{equation}
\big|\,\widetilde{\Omega}_\lambda - \Omega_\lambda\big|
~\ll~ \Omega_\lambda ~~~~~ \mbox{for~all}~\lambda~.
\label{eq:HubbleJitterCondit2}
\end{equation}
We emphasize that this is an overly conservative constraint, and that
consistency with observational data is certainly possible
even if vacuum fluctuations do have a significant effect on the
abundances of certain $a_\lambda$. However, as we shall demonstrate,
the restriction that this overly conservative constraint imposes on our model
(which primarily turns out to take the form of an upper bound on $H_I$ for
any allowed choice of $\fhatX$, $M_c$, and $\Lambda_G$)
is not terribly severe.
In order to determine how this condition restricts
the parameter space of our model, we must first assess what effect
vacuum fluctuations during inflation have on the individual energy densities
$\rho_\lambda$ and relic abundances $\Omega_\lambda$ of the constituent fields
in our dark-matter ensemble. We begin by noting a generic result in
inflationary cosmologies (for a review, see Ref.~\cite{LindeInflationReview}),
namely that the variance $\langle \phi^2\rangle$ in the amplitude of any light scalar
$\phi$ with a mass $m_\phi \lesssim H_I$ induced by vacuum fluctuations during inflation
is given by
\begin{equation}
\langle\phi^2\rangle ~\sim~ \frac{H_I^3\delta t_I}{4\pi^2}~,
\label{eq:GenericFlucInPhi}
\end{equation}
where $\delta t_I$ denotes the duration of inflation. A fluctuation of this order
will therefore be induced in the amplitude of any axion in our dark-matter ensemble
with a mass smaller than $H_I$. Moreover, we note that the
relationship between $\delta t_I$ and $H_I$ is constrained by the fact
that successful resolution of the smoothness and flatness problems
requires $N_e \approx H_I \delta t_I \gtrsim 60$, where $N_e$ denotes the
number of $e$-foldings of inflation. In typical scenarios, $N_e$ lies only
slightly above this lower bound; hence $\delta t_I$ is typically expected to
be such that $H_I \delta t_I \sim \mathcal{O}(100)$. We will frequently
express our results in terms of $N_e$ in what follows.
We begin our discussion the effect of these fluctuations on the abundances
of the constituent particles in our dark-matter ensemble by examining
the simple case in which $t_G \lesssim t_I$. In this case, the axion
mass-squared matrix attains its
asymptotic, late-time form before inflation ends, and the $a_\lambda$ are
consequently already the axion mass eigenstates during the inflationary epoch.
Thus, we find that the total energy density associated with each $a_\lambda$
with $\lambda \lesssim H_I$ at the end of inflation is given by
\begin{equation}
\rho_\lambda(t_I) ~\approx~ \frac{1}{2}\lambda^2
\left(\theta A_\lambda \fhatX +
\eta_\lambda\frac{H_I\sqrt{N_e}}{2\pi}\right)^2~,
\label{eq:RhoLambdaInftIlesssimtG}
\end{equation}
where
\begin{equation}
\eta_\lambda ~\sim~
\begin{cases}
\mathcal{O}(1) & \lambda \lesssim H_I \\
0 & \lambda \gtrsim H_I
\end{cases}
\label{eq:DefOfEtaLambdaFluctuation}
\end{equation}
is a random coefficient of which parametrizes the fluctuation in
the field $a_\lambda$.
Before proceeding further, we remark that the above results depend critically
on the assumption that $t_G \lesssim t_I$. In other words, we have assumed that
the instanton dynamics associated with the gauge group $G$ has already occurred
and made its contributions to the KK masses prior to the onset of the quantum
fluctuations that arise due to inflation. By contrast, if $t_G \gtrsim t_I$,
the quantum fluctuations will occur first, when the axion mass matrix is still
diagonal and when the KK momentum modes and mass eigenstates coincide.
In such cases, these are the modes which develop quantum fluctuations,
and the mode-mixing induced by the instanton dynamics occurs only later.
This distinction is important, because the resulting energy
density for each $a_\lambda$ takes a somewhat different form
when $t_G\gtrsim t_I$:
\begin{equation}
\rho_\lambda ~=~
\frac{1}{2} \lambda^2 \,\left[
\sum_{n=0}^\infty U_{\lambda n} \left(
\theta \hat f_X \delta_{n,0} + \eta_n
\frac{H_I \sqrt{N_e}}{2\pi}\right)\right]^2~.
\label{eq:RhoLambdaInftIgtrsimtG}
\end{equation}
In this expression, $U_{\lambda n}$ is the mixing matrix in
Eq.~(\ref{eq:DefOfalambda}) and $\eta_n$ is the analogue of $\eta_\lambda$
discussed above, with $\eta_n$ taking non-zero values only when $n\lesssim H_I/M_c$.
{\it A priori}\/, this expression results in a different value for $\rho_\lambda$
than that in Eq.~(\ref{eq:RhoLambdaInftIlesssimtG}). However, it turns out that
the eventual constraints associated with Eq.~(\ref{eq:RhoLambdaInftIgtrsimtG}) are
no more stringent than those which we shall eventually calculate for
Eq.~(\ref{eq:RhoLambdaInftIlesssimtG}). In order to understand why this is the
case, let us consider an even more dramatic situation in which $\eta_n$ actually
takes a fixed, positive value $\overline{\eta}$ for all $n$ --- even values of $n$
beyond the inflationary cutoff $H_I/M_c$. In this case, we can make use of
the identity
\begin{equation}
\sum_{n=0}^\infty\, U_{\lambda n} ~=~ f(\wtl)\, A_\lambda~,
\end{equation}
where $f(\wtl)\equiv (\wtl^2 + \sqrt{2}-1)/\sqrt{2}$,
in order to rewrite Eq.~(\ref{eq:RhoLambdaInftIgtrsimtG}) in the form
\begin{equation}
\rho_\lambda ~=~ \frac{1}{2} \lambda^2 \,\left[
\theta A_\lambda \hat f_X + f(\wtl)\, {\overline{\eta}}\,
\frac{H_I\sqrt{N_e}}{2\pi}\right]^2~.
\end{equation}
Remarkably, this is essentially the same expression as we would have obtained
from Eq.~(\ref{eq:RhoLambdaInftIlesssimtG}) when $\eta_\lambda =\overline{\eta}$
for all $\lambda$, except that the fluctuation contribution now comes multiplied
by an extra ``scaling'' factor $f(\wtl)$. It is easy to verify that
$f(\wtl)\to 1$ as $\wtl\rightarrow\infty$, whereas for small $\wtl$
we find that $f(\wtl)\ll 1$. This indicates that the effects of the
inflation-related quantum fluctuations are actually {\it suppressed}\/ for the
lighter modes, relative to what occurs in the case with $t_G\lesssim t_I$.
The magnitude of this suppression depends on $y$, and is more severe when
$y\ll 1$ ({\it i.e.}\/, when the axion modes are more fully mixed). We thus
conclude that the contributions from the quantum fluctuations that arise during
inflation are greater when they occur {\it after}\/ the instanton dynamics turns
on (and after the KK mode-mixing), rather than before. We shall therefore
concentrate on the $t_G\lsim t_I$ case in what follows.
Given the result in Eq.~(\ref{eq:RhoLambdaInftIlesssimtG}), we see that
the effect of vacuum fluctuations on the $\rho_\lambda$ will be small
for values of $\lambda$ which satisfy the condition
\begin{equation}
\theta A_\lambda \fhatX ~\gtrsim~ \frac{H_I\sqrt{N_e}}{2\pi}~.
\label{eq:VacMisSurpassesHubbleFluc}
\end{equation}
Since $A_\lambda$ is a monotonically decreasing function of $\lambda$, it follows
that within any given tower of $a_\lambda$, there exists a critical mass value
\begin{equation}
\lambdafluc ~\equiv~ \frac{\mX}{\sqrt{2}} \left[
\sqrt{\left(1+\frac{\pi^2}{y^2}\right)^2
+\frac{32\pi^2\theta^2\fhatX^2}{N_e H_I^2} }
-\left(1+\frac{\pi^2}{y^2}\right)\right]^{1/2}
\label{eq:Lambdafluc}
\end{equation}
below which the effect of vacuum fluctuations on the corresponding energy density
$\rho_\lambda$ is negligible. These $\rho_\lambda$ are therefore well approximated
by Eq.~(\ref{eq:RhoLambdaInLTRCosmo}), and the corresponding abundances
$\widetilde{\Omega}_\lambda$ are given by Eq.~(\ref{eq:OmegaLambdaOftEqnLTRtG})
or Eq.~(\ref{eq:OmegaLambdaOftEqnLTRtlambda}), depending on the value of $t_\lambda$.
By contrast, for $\lambdafluc \lesssim \lambda \lesssim H_I$, the effect of vacuum
fluctuations overwhelms the effect of vacuum misalignment.
The initial energy density of each $a_\lambda$ in this regime is therefore
effectively set at $t_I$ and is approximately given by
\begin{equation}
\rho_\lambda(t_I) ~\approx~ \frac{N_e}{8\pi^2} \lambda^2 H_I^2~.
\end{equation}
Since the Newton's-law-modification bound on $M_c$ in Eq.~(\ref{eq:MinimumMc})
implies that $t_\lambda \lesssim \tRH$ for each such field, it therefore follows
that at all subsequent times, the corresponding relic abundance is given by
\begin{equation}
\widetilde{\Omega}_\lambda ~\approx~
\frac{3N_e H_I^2}{4\pi^2 M_P^2}
e^{-\Gamma_\lambda(t-t_I)}\times
\begin{cases}
\displaystyle\frac{1}{4} \vspace{0.25cm}~~
& 1/\lambda ~\lesssim ~t ~\lesssim~ \tRH \\
\displaystyle\frac{4}{9}\left(\frac{t}{\tRH}\right)^{1/2} \vspace{0.25cm}~~
& \tRH ~\lesssim~ t ~\lesssim~ \tMRE \\
\displaystyle\frac{1}{4}\left(\frac{\tMRE}{\tRH}\right)^{1/2}~~
& t~\gtrsim ~\tMRE~.
\end{cases}
\label{eq:OmegaLambdaInf}
\end{equation}
\begin{figure}[t!]
\begin{center}
\epsfxsize 2.25 truein \epsfbox {HIminFlucLifePlotOut_Lambda1GeV.eps}
\epsfxsize 2.25 truein \epsfbox {HIminFlucLifePlotOut_Lambda1TeV.eps}
\epsfxsize 2.25 truein \epsfbox {HIminFlucLifePlotOut_Lambda100TeV.eps}
\raisebox{0.3cm}{\large$H_I^{\rm crit}$}
\epsfxsize 5.00 truein \epsfbox {ColorBarHIminFlucLife.eps}
\end{center}
\caption{Contours of the critical value $H_I^{\rm crit}$ in $(\fhatX, M_c)$
parameter space. As discussed in the text, choosing $H_I \ll H_I^{\rm crit}$
guarantees that misalignment production dominates over
vacuum fluctuations in determining the relic abundance
$\widetilde{\Omega}_\lambda$ of all $a_\lambda$ in our DDM ensemble,
as desired. Here, we have taken $\TRH = 5$~MeV,
$N_e = 100$, and $\xi = g_G = \theta = 1$ and assumed a photonic axion
with $c_\gamma=1$. The left, center, and right panels display
the results for $\Lambda_G = 1$~GeV, $\Lambda_G = 1$~TeV, and
$\Lambda_G = 100$~TeV, respectively.
\label{fig:HIMaxFlucPanelsLTR}}
\end{figure}
To summarize, we see that the axion KK tower separates into three distinct regimes
within each of which different physics plays a principal role in determining
$\widetilde{\Omega}_\lambda$. In the $\lambda \lesssim \lambdafluc$ regime,
the effect of vacuum fluctuations on
$\widetilde{\Omega}_\lambda$ is negligible and the results in
Sect.~\ref{sec:MisalignmentProd} continue to hold. In the
$\lambdafluc \lesssim \lambda \lesssim H_I$ regime, the opposite is true: vacuum
fluctuations dominate and the abundances of the $a_\lambda$ are given by
Eq.~(\ref{eq:OmegaLambdaInf}). Finally, in the
$\lambda \gtrsim H_I$ regime, the wavelengths of even the lowest-lying momentum
modes of each $a_\lambda$ fall short of the Hubble length during the
inflationary epoch. Such modes therefore behave unambiguously
like particles, and are consequently inflated away.
We are now ready to address the constraint
we have imposed on the individual abundances $\widetilde{\Omega}_\lambda$
in Eq.~(\ref{eq:HubbleJitterCondit2}). Since the effect of vacuum fluctuations
is negligible both for $\lambda \gtrsim H_I$ and for $\lambda \lesssim \lambdafluc$,
it follows that this constraint will be satisfied whenever $H_I \ll \lambdafluc$.
Moreover, since $\lambdafluc$ itself decreases with increasing $H_I$, as indicated
in Eq.~(\ref{eq:Lambdafluc}), we find that our constraint may be expressed in
the form $H_I \ll H_I^{\rm crit}$, where $H_I^{\rm crit}$ is the value of the
Hubble parameter during inflation for which $H_I = \lambdafluc$.
In Fig.~\ref{fig:HIMaxFlucPanelsLTR}, we display contours of
$H_I^{\rm crit}$ as a function of the model parameters $\fhatX$, $M_c$, and $\Lambda_G$.
For the large values of $\Lambda_G$ characteristic of our preferred region of
parameter space, we observe that the constraint in Eq.~(\ref{eq:HubbleJitterCondit2})
is satisfied for
$H_I \ll H_I^{\rm crit} \sim \mathcal{O}(10 - 100\mathrm{~GeV})$. For smaller
values of $\Lambda_G$, although the constraint is certainly more severe, we
nevertheless observe that the bound can be satisfied for
$H_I \ll H_I^{\rm crit} \sim \mathcal{O}(10 - 100\mathrm{~keV})$.
This condition on $H_I$ has non-trivial implications for the construction of
explicit inflationary models, since values of $H_I$ of this magnitude tend to be
rather non-generic~\cite{TensorToScalarNonGeneric} among typical classes of
inflationary potentials. However, as discussed in Ref.~\cite{DynamicalDM2},
such a scale for $H_I$ is certainly not excluded
(see, \eg, Refs.~\cite{LTRAxionsKamionkowski,lowscaleinflation}). Moreover, a
small value for $H_I$ fits naturally within the context of the LTR cosmology.
\subsection{Other Astrophysical Constraints on Light Axions\label{sec:AdditionalConstraints}}
In addition to the constraints we have discussed above, there exist
a number of additional astrophysical and cosmological bounds on theories
involving light axions and axion-like particles.
Indeed, particles of this sort can give rise to a number of
potentially observable effects~\cite{StringAxiverse}, such as a rotation of the CMB
polarization, modifications of the matter power spectrum, and the enhanced spindown
of rotating black holes. However, in order to give rise to
observable effects of this sort, the particle in question must be exceedingly light,
with a mass $m\lesssim 10^{-10}$~eV. In the extra-dimensional scenario we are
discussing here, the Newton's-law-modification constraint on the
compactification scale $M_c$ stated in Eq.~(\ref{eq:MinimumMc}) implies
that all $a_\lambda$ in the tower have masses $\lambda\gtrsim 10^{-3}$~eV in
any scenario in which $y\lesssim 1$, \ie, in which the full tower of $a_\lambda$
contributes significantly to $\Omegatot$.
Consequently, the additional constraints on ultra-light axions and axion-like fields
discussed in Ref.~\cite{StringAxiverse} are not relevant for our bulk-axion DDM model.
\section{Synthesis:~ Combined Phenomenological Constraints
on Axion Models of Dynamical Dark Matter\label{sec:Combined}}
In the previous section, we enumerated the individual astrophysical, phenomenological,
and cosmological considerations which potentially constrain our bulk-axion DDM
model, and we evaluated the restrictions that each placed
on the parameter space of this model. In this section, we summarize how these
individual results, taken together, serve to constrain that parameter space.
Our particular interest concerns the preferred
region of parameter space outlined in Ref.~\cite{DynamicalDM2}, namely
$\fhatX \sim 10^{14} - 10^{15}$~GeV, $\Lambda_G \sim 10^2 - 10^5$~GeV, and $M_c$
chosen sufficiently small that $y \lesssim 1$. Indeed, this is the region within
which the full KK tower contributes non-trivially to the total dark-matter
relic abundance.
\begin{figure}[b!]
\begin{center}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionLam1GeV.eps}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionLam1TeV.eps}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionLam100TeV.eps}
\end{center}
\caption{Exclusion regions associated with all applicable phenomenological constraints
discussed in this paper
for our bulk-axion DDM model with $\Lambda_G = 1$~GeV (left panel),
$\Lambda_G = 1$~TeV (center panel), and $\Lambda_G = 100$~TeV (right panel).
In each case, we have taken $\xi=g_G=1$, $\TRH = 5$~MeV, and $H_I = 10^{-3}$~GeV,
and we have assumed that the axion only couples to the photon field with
$c_\gamma=1$. The shaded regions are respectively excluded by
data from helioscope measurements (red), collider considerations (magenta),
tests of Newton's-law modifications via E\"{o}tv\"{o}s-type experiments (purple),
measurements of the diffuse extragalactic X-ray and gamma-ray spectra (orange),
observations of the lifetimes of globular-cluster stars (yellow), energy-loss
limits from supernova SN1987A (cyan), the model-consistency requirement
$\Lambda_G < f_X$ (gray), overproduction of thermal axions (green),
and the upper bound on the dark-matter relic abundance from WMAP (brown).
The dashed black line corresponds to $y=\pi$; smaller values of $y$ correspond
to the region below and to the left of this line.
\label{fig:MasterConstraintPlotPhotonic}}
\begin{center}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionHadronicLam1GeV.eps}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionHadronicLam1TeV.eps}
\epsfxsize 2.25 truein \epsfbox {FinalExclusionHadronicLam100TeV.eps}
\end{center}
\caption{Same as Fig.~\protect\ref{fig:MasterConstraintPlotPhotonic}, but for a
``hadronic'' axion --- \ie, an axion which couples to both photons and gluons
(and hence to pions, nucleons, and other hadrons), but not directly to SM quarks
or leptons. For these panels, we have taken $c_g = c_\gamma = 1$.
\label{fig:MasterConstraintPlotHadronic}}
\end{figure}
In Fig.~\ref{fig:MasterConstraintPlotPhotonic}, we show the combined exclusion
regions for a purely photonic axion with $c_\gamma = 1$ for
$\Lambda_G = 1$~GeV (left panel), $\Lambda_G = 1$~TeV (center panel), and
$\Lambda_G = 100$~TeV (right panel). The shaded regions displayed in each of the
plots are excluded by the various considerations discussed in Sect.~\ref{sec:Bounds}.
Specifically, the exclusion regions appearing in these panels are those associated with
helioscope limits on solar axion production (red), collider considerations (magenta),
tests of Newton's-law modifications via E\"{o}tv\"{o}s-type experiments (purple),
measurements of the diffuse extragalactic X-ray and gamma-ray spectra (orange),
observations of the lifetimes of globular-cluster stars (yellow), energy-loss
limits from supernova SN1987A (cyan), the model-consistency requirement
$\Lambda_G < f_X$ discussed in Ref.~\cite{DynamicalDM2} (gray), and the $3\sigma$
upper bound on the dark-matter relic abundance from WMAP (brown).
The additional requirement that the relic abundance be primarily determined by
the misalignment mechanism (as envisioned in our DDM model) excludes the
green-shaded region, within which a substantial population of $a_\lambda$ is
generated via interactions with SM particles in the thermal bath.
The remaining unshaded regions of parameter space are the regions
within which our DDM model is consistent with all of these constraints.
The dashed black line indicates the contour $y = \pi$; smaller values of $y$
correspond to the region below and to the left of this line. As discussed in
Ref.~\cite{DynamicalDM2}, we are particularly
interested in the unshaded region of parameter space which falls below and to the
left of this line, since this is the region within which not only are all of the
aforementioned constraints satisfied, but also the full tower of $a_\lambda$
contributes non-trivially to $\Omegatot$.
As we see in Fig.~\ref{fig:MasterConstraintPlotPhotonic},
for small $\Lambda_G$ the most stringent constraint on the parameter space of our
model is the one derived from energy-loss limits from SN1987A. The constraint from
globular-cluster stars is also reasonably stringent, and the constraint derived
from missing-energy processes such as $pp\rightarrow \gamma + \met$ at the LHC is
estimated to be of roughly the same order. However, as the $y=\pi$ contour
superimposed over each panel in Fig.~\ref{fig:MasterConstraintPlotPhotonic}
indicates, the full tower of $a_\lambda$ contributes significantly to $\Omegatot$
for all $\Lambda_G \gtrsim 100$~GeV. Indeed, this is precisely the $\Lambda_G$
regime associated with the preferred region of parameter space for our model.
We therefore conclude that within this region of parameter space,
a photonic bulk-axion DDM ensemble constitutes a viable dark-matter
candidate.
In Fig.~\ref{fig:MasterConstraintPlotHadronic}, we consider all of the same
constraints as in Fig.~\ref{fig:MasterConstraintPlotPhotonic}, but for the
case of a hadronic axion with $c_g = c_\gamma=1$. In this case,
since the $a_\lambda$ couple to hadrons, the constraints from SN1987A and
from axion production via interactions among the SM particles in the radiation
bath both become even more stringent. Again, as in the photonic-axion case,
we find that the leading constraint for small $\Lambda_G$ is that from
SN1987A, and that as $\Lambda_G$ increases, the model-consistency constraint
becomes increasingly stringent. However, as in the photonic-axion case, we
see that within the preferred region of parameter space for our model, a hadronic
bulk axion is also consistent with experimental and observational limits. Thus
a hadronic bulk-axion DDM ensemble is a viable dark-matter candidate as well.
We also observe that the exclusion contours in
Figs.~\ref{fig:MasterConstraintPlotPhotonic}
and~\ref{fig:MasterConstraintPlotHadronic} associated
with SN1987A energy-loss limits, globular-cluster-star evolution, collider constraints,
and axion overproduction from SM particles in the radiation bath have the same slope.
This is because all of these constraints involve the production of light
axions which are never directly detected, and thus involve physical processes whose
amplitudes include a single coupling factor between the $a_\lambda$ and a
pair of SM fields. By contrast, the slopes of the constraint contours associated
with other classes of physical processes can be quite different. The
helioscope-constraint contour, for example, is related to processes in which
axions are both produced and subsequently detected via their interactions with
SM fields. Likewise, the contour associated with
limits on features in the diffuse X-ray and gamma-ray backgrounds is due to
processes involving the decays of a preexisting cosmological population of axions,
and therefore depends not only on the couplings of the $a_\lambda$ to SM fields,
but to their relative abundances as well. The slopes of these constraint
contours consequently differ from those which characterize the contours
associated with SN1987A energy-loss limits, globular-cluster-star evolution, and
so forth.
\section{Discussion and Conclusions\label{sec:Conclusions}}
In Ref.~\cite{DynamicalDM1}, we proposed a new framework for dark-matter physics
which we call ``dynamical dark matter'' (DDM). The fundamental idea underpinning
DDM is that the requirement of stability is replaced by a delicate balancing between
lifetimes and cosmological abundances across a vast ensemble of individual
dark-matter components. If Ref.~\cite{DynamicalDM1}, we developed the general
theoretical features of this new framework. By contrast, in Ref.~\cite{DynamicalDM2},
we presented a ``proof of concept,'' namely an explicit realization of the DDM
framework in which the DDM ensemble is realized as the infinite tower of KK
excitations of an axion-like field propagating in the bulk of large extra spacetime
dimensions.
In this paper, we have completed this study by systematically investigating all of
the experimental, astrophysical, and cosmological constraints which apply to this
DDM model. Some of these constraints pertain to theories with large extra dimensions
in general, while others pertain specifically to our model. Among the bounds
we have considered are constraints from limits on $a_\lambda$ production by
astrophysical sources such as stars and supernovae; constraints related to
the effects of late relic-axion decays on BBN, the CMB, and the diffuse
X-ray and gamma-ray backgrounds;
collider constraints on missing-energy processes such as $pp\rightarrow j + \met$ and
$pp\rightarrow \gamma + \met$; constraints on isocurvature perturbations generated
as a consequence of misalignment production; constraints on the production of
relativistic axions due to interactions in the thermal bath after inflation; and
constraints on the direct detection of dark axions by microwave-cavity detectors and
other, similar instruments. We have verified that all of these constraints are
satisfied within the preferred region of parameter space for our model --- namely,
that in which the bulk-axion DDM ensemble accounts for the observed dark-matter
relic abundance, while at the same time the full tower of axion modes contributes
meaningfully to that abundance. We therefore conclude that this bulk-axion DDM
model is indeed phenomenologically viable, and that the overall DDM framework
is a self-consistent alternative to traditional approaches to the dark-matter
problem.
While the focus of this paper has been on the specific bulk-axion
DDM model presented in Ref.~\cite{DynamicalDM2}, we note that many of our
results, and in many places our entire methodology, have a far
wider range of applicability.
For example, much of the formalism developed in Sect.~\ref{sec:Bounds}
for evaluating the cosmological constraints on decaying dark matter in
our bulk-axion DDM model is applicable to any model in which the dark
sector comprises a large number of fields. This is true for issues as diverse
as BBN, diffuse photon backgrounds, or stellar cooling.
Likewise, irrespective of issues pertaining to dark-matter physics,
many of our results and techniques may have applicability to theories
with large numbers of axions, such as the recently discussed ``axiverse''
theories~\cite{StringAxiverse,StringAxiversePheno}.
Thus, we believe that the methods developed and employed in this paper can
serve as a prototype for future phenomenological studies of not only the
DDM framework, but also, more generally, any theories in which there exist
large numbers of interacting and decaying particles.
\section*{Acknowledgments}
We would like to thank K.~Abazajian, Z.~Chacko, M.~Drees, J.~Feng, J.~Kumar,
R.~Mohapatra, M.~Ramsey-Musolf, S.~Su, T.~Tait, S.-H.~H.~Tye,
X.~Tata, and N. Weiner for discussions.
KRD is supported in part by the U.S. Department of Energy
under Grant DE-FG02-04ER-41298 and by the National Science Foundation through
its employee IR/D program. BT is supported in part by DOE grant
DE-FG02-04ER41291. The opinions and conclusions expressed here are those of
the authors, and do not represent either the Department of Energy or
the National Science Foundation.
|
{
"timestamp": "2012-03-12T01:00:06",
"yymm": "1203",
"arxiv_id": "1203.1923",
"language": "en",
"url": "https://arxiv.org/abs/1203.1923"
}
|
\section{Introduction}
The one-dimensional (1d) Hubbard model is the simplest model of strongly interacting dynamics of spinful fermions on a lattice within a single-band approximation.
Its Hamiltonian reads
\begin{equation}
H=-t\sum_{i=1}^{L-1}\sum_{s\in\{\uparrow,\downarrow\}}(c_{i,s}^\dagger c_{i+1,s}+{\rm h.c.})+U\sum_{i=1}^L{n_{i\uparrow}n_{i\downarrow}},
\label{eq:hubbard}
\end{equation}
where the operators $c_{i,s}$, $c^\dagger_{i,s}$, for spin $s\in\{\uparrow,\downarrow\}$ and site $i\in\{1,\ldots,L\}$ are standard fermionic annihilation and creation operators and $n_{i,s}:=c^\dagger_{i,s} c_{i,s}$ are the density operators. The model is solvable by a coordinate Bethe ansatz~\cite{lieb} and possesses an infinite number of conservation laws~\cite{shastry:88}. While stationary properties of the 1d Hubbard model are well understood, see the monograph~\cite{book}, much less is known about its dynamics, for instance about the transport behavior. In thermodynamically the most interesting regime, at half-filled band and zero magnetization $\sum_{i=1}^L \ave{n_{i,s}} = L/2$, studied in the present paper, the model is gapped for charge excitations and gapless for spin excitations, for any $U\neq 0$. At zero temperature it is therefore an example of a Mott (charge) insulator and ballistic (ideal) spin conductor. Transport can be qualitatively characterized by a Drude weight -- a linear response indicator of ballistic transport, defined as the weight of zero-frequency singular term $\delta(\omega)$ in the real part of conductivity $\sigma(\omega)$. Spin and charge Drude weights at zero temperature have been calculated in Ref.~\cite{shastry:90}, with finite-size corrections given in Ref.~\cite{finiteL}, while a regular part of $\sigma(\omega)$ is studied in Ref.~\cite{finiteom}. At nonzero temperature on the other hand no rigorous result is known and there is no consensus between numerical and Bethe ansatz based results. Thermodynamic Bethe ansatz suggests \cite{fujimoto:98} that, even at half-filling, the charge Drude weight is finite, so the model was predicted to exhibit ideal charge transport; for a similar conclusion see also Quantum Monte Carlo calculation in Ref.~\cite{kirchner:99}. Analytical calculations at large $U$ on the other hand support vanishing charge Drude weight~\cite{peres:00}, for a study of low-energy excitations see also Ref.~\cite{gu:07}. Exact numerical simulations of small systems, again at half-filling and at high/infinite temperature, suggest~\cite{prelovsek}
$\sim 1/L$ scaling of the charge Drude weight. For temperatures much smaller than the gap semiclassical arguments together with field-theoretical scattering rate predicts diffusive transport~\cite{sachdev}. Vanishing finite-temperature Drude weight in thermodynamic limit (TL) $L\to\infty$ offers the possibility of an insulating or diffusive behavior.
However, at high or infinite temperatures, non-equilibrium transport properties of 1d Hubbard model in either charge or spin sector are not known as there has been up to date no analytical or numerical method capable of reliably treating this regime. In this paper we employ non-equilibrium steady state simulations~\cite{pz:09} using an efficient matrix product ansatz~\cite{vidal} for the time-dependent density matrix and provide a clear evidence of diffusive transport for both attractive $U<0$ and repulsive $U>0$ cases at infinite temperature. Namely, we show $1/L$ scaling of charge as well as of spin current and clear linear density profiles.
\section{Boundary driven Hubbard chain}
Using the Jordan-Wigner transformation we can map the 1d Hubbard model (\ref{eq:hubbard}) to a spin-$1/2$ ladder system. Namely, writing $c_{i\uparrow}=P^{(\sigma)}_{i-1} \sigma_i^-
where $P^{(\sigma)}_i=\sigma_1^{\rm z} \cdots \sigma_i^{\rm z}$ for spin-up fermions,
and $c_{i\downarrow}=P^{(\sigma)}_L P^{(\tau)}_{i-1} \tau_i^-$
where $P^{(\tau)}_i=\tau_1^{\rm z} \cdots \tau_i^{\rm z}$ for spin-down fermions, one can verify that fermionic operators $c_{i,s},c^\dagger_{i,s}$ satisfy canonical anticommutation relations
provided $\sigma^\alpha_i$ and $\tau^\alpha_i$ are two sets of Pauli matrices (and $\sigma^\pm_j:=(\sigma^{\rm x}_j \pm{\rm i} \sigma^{\rm y}_j)/2$, $\tau^\pm_j:=(\tau^{\rm x}_j \pm{\rm i} \tau^{\rm y}_j)/2$).
Writing the Hubbard Hamiltonian (\ref{eq:hubbard}) in spin-ladder form one obtains
\begin{eqnarray}
H=&-&\frac{t}{2}\sum_{i=1}^{L-1} (\sigma_i^{\rm x} \sigma_{i+1}^{\rm x}+\sigma_i^{\rm y} \sigma_{i+1}^{\rm y}+\tau_i^{\rm x} \tau_{i+1}^{\rm x}+\tau_i^{\rm y} \tau_{i+1}^{\rm y})+\nonumber \\
&+&\frac{U}{4}\sum_{i=1}^{L}(\sigma_i^{\rm z}+1)(\tau_i^{\rm z}+1).
\label{eq:ladder}
\end{eqnarray}
The spin-1/2 ladder system consists of two $XX$ chains in two legs and a $Z-Z$ type interchain coupling along the rungs. For numerical simulations of the Hubbard model we shall use this ladder formulation (\ref{eq:ladder}).
To induce a nonequilibrium situation two legs are coupled to independent reservoirs. Their action is decribed in an effective way via the Lindblad equation~\cite{lin} for the density matrix $\rho$ of the ladder system,
\begin{equation}
\frac{{\rm d}}{{\rm d}t}{\rho}={\rm i} [ \rho,H ]+ {\cal L}^{\rm dis}(\rho),
\label{eq:Lindblad}
\end{equation}
where the dissipative term is expressed in terms of Lindblad operators $L_k$, as
\begin{equation}
{\cal L}^{\rm dis}(\rho)=\sum_k \left( [ L_k \rho,L_k^\dagger ]+[ L_k,\rho L_k^{\dagger} ] \right).
\end{equation}
We use eight Lindblad operators acting locally on the first and last sites of each leg, injecting or absorbing fermions (spinons) with certain probability:
\begin{equation}
L_{1,2}=\sqrt{\Gamma(1\mp \mu)}\,\sigma^{\pm}_1, \quad L_{3,4}=\sqrt{\Gamma(1\pm \mu)}\,\sigma^{\pm}_L,
\end{equation}
for the first, and
\begin{equation}
L_{5,6}=\sqrt{\Gamma(1\mp \mu)}\,\tau^{\pm}_1, \quad L_{7,8}=\sqrt{\Gamma(1\pm \mu)}\,\tau^{\pm}_L,
\label{eq:chargedriv}
\end{equation}
for the second leg.
$\Gamma$ is the strength of the coupling to the baths while $\mu$ is a driving strength playing the role of a chemical potential bias. As demonstrated in previous studies of 1d spin chains~\cite{pz:09} the precise form of Lindblad operators does not influence the bulk properties. Because of dissipative terms the time-dependent solution $\rho(t)$ of the Lindblad equation converges after a long time to a time-independent state called a nonequilibrium steady state (NESS), $\rho_\infty = \lim_{t\to\infty}\rho(t)$, which is unique \cite{unique}.
Once the steady state is reached, expressed in terms of a matrix product operator of a given bond dimension (following the method described in detail in Ref.~\cite{pz:09} straightforwardly adapted for the spin ladder), expectation values of arbitrary observables in the NESS can be efficiently evaluated. All expectation values considered in this paper are taken with respect to the NESS, that is $\langle A \rangle = \,{\rm tr}\,{(\rho_\infty A)}$, which we will -- when it is clear from the context, and to simplify notation -- denote just by $A$. In each NESS calculation we have carefully checked that the convergence is reached, i.e., we evolve the Lindblad equation (\ref{eq:Lindblad}) until a time-independent state is obtained, and that the results are stable with respect to increasing bond dimension \cite{technical}. For $\mu=0$, i.e., no driving, one has an equilibrium setting, resulting in a trivial NESS $\rho_\infty \propto \mathbbm{1}$. This means that for small driving $\mu$ we are studying nonequilibrium behavior at an infinite temperature. Note that such an infinite temperature state is separable in the operator space. Since the efficiency of the numerical method crucially depends on the entanglement infinite-temperature nonequilibrium states are the easiest
ones to calculate because the entanglement is expected to be smaller than at finite temperatures.
Expectation values of fermionic observables are obtained from the corresponding ones in the ladder formulation, for instance, particle densities are $n_{i\uparrow}=(\sigma^{\rm z}_i+1)/2$ and $n_{i\downarrow}=(\tau^{\rm z}_i+1)/2$. Magnetization currents of the two spin species, defined through the continuity equations ${\rm d}(\sigma^{\rm z}_i/2)/{\rm d} t=j^{(\sigma)}_{i}-j^{(\sigma)}_{i-1}$, ${\rm d}(\tau^{\rm z}_i/2)/{\rm d} t=j^{(\tau)}_{i}-j^{(\tau)}_{i-1}$, are $j^{(\sigma)}_i=-\frac{t}{2}(\sigma_i^{\rm x} \sigma_{i+1}^{\rm y}-\sigma_i^{\rm y} \sigma_{i+1}^{\rm x})$, $j^{(\tau)}_i=-\frac{t}{2}(\tau_i^{\rm x} \tau_{i+1}^{\rm y}-\tau_i^{\rm y} \tau_{i+1}^{\rm x})$. In fermionic picture the particle (charge) current is $j^{(\rm c)}_i=j^{(\sigma)}_i+j^{(\tau)}_i$, while the spin current is $j^{(\rm S)}_i=(j^{(\sigma)}_i-j^{(\tau)}_i)/2$. Particle density is $n_i=n_{i\uparrow}+n_{i\downarrow}$, while spin density is $s_i=(n_{i\uparrow}-n_{i\downarrow})/2$.
Because of the same driving at both ladder legs the currents $j^{(\sigma)}$ and $j^{(\tau)}$ are the same. Therefore, NESS is such that it has a nonzero charge current and zero spin current. We have also performed simulations with Lindblad operators on the $\tau$-chain driving transport in the opposite direction, that is with
\begin{equation}
L_{5,6}=\sqrt{\Gamma(1\pm \mu)}\,\tau^{\pm}_1, \quad
L_{7,8}=\sqrt{\Gamma(1\mp \mu)}\,\tau^{\pm}_L.
\label{eq:spindriv}
\end{equation}
In such a case of spin driving the NESS has a nonzero spin current and zero charge current because $j^{(\tau)}=-j^{(\sigma)}$ holds. Furthermore, we stress that spin and charge transport are interchanged under the particle-hole transformation for the {\em down} spin fermions only and simultaneously chaging the sign of $U$. Namely, taking $R:=\prod_{i=1}^L \tau^{\rm x}_i=R^\dagger$ one finds $R j^{({\rm S})}_i R^\dagger = j^{({\rm c})}_i/2$ and $R H(U) R^\dagger = H(-U)$, provided one takes a symmetric interaction term $(n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2})$ in (\ref{eq:hubbard}) or, equivalently, adds a chemical term $-U N/2$ to $H$ with $N= \sum_{i,s} n_{i,s}$. Even though our master equation evolution (\ref{eq:Lindblad}) does not strictly conserve $N$, we have checked explicitly that the results based on Hamiltonians $H$ and $H-U N/2$ are identical.
\section{Results}
\subsection{Evidence of diffusion: density profiles and scaling of currents}
We set $t=1$, $\Gamma=1$ and $\mu=0.2$, except in Fig.~\ref{fig:profilmu1} where $\mu=1$.
Driving $\mu=0.2$ corresponds to equilibrium density in the reservoirs of $n_{{\rm L},s}=0.4$ at the left end and $n_{{\rm R},s}=0.6$ at the right end. The average filling ratio is therefore $n=1/2$, $\sum_{i=1}^L n_{i\uparrow}=\sum_{i=1}^L n_{i\downarrow}=L/2$. The value $\mu=0.2$ is at the upper end of a linear response regime. For large drivings $\mu \gtrsim 0.6$ one gets a negative differential conductance effect \cite{benenti:09}, where the current decreases with increasing driving. The main goal of this paper is to classify spin and charge transport, whether it is {\em ballistic}, {\em diffusive} or {\em anomalous}. For NESS, different transport regimes are reflected in the scaling of the current on the system size. Fixing the driving strength $\mu$, in a ballistic conductor the current is independent of the system length, $j \sim L^0$, for a diffusive conductor it scales as $j \sim 1/L$, whereas in the anomalous case the current is proportional to a fractional power of $L$. We therefore calculated NESSs for different sizes $L$. Typical density profile is shown in Fig.~\ref{fig:profilU1}.
One can see that the densities of spin-up and spin-down fermions are linear in the bulk. Jumps in the density at the boundaries are due to over-simplified Lindblad operators that are not ``matched'' to the bulk dynamics, i.e., there are boundary resistances.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.47\textwidth]{profilU1.eps}}
\caption{(Color online)~Density profile $n_{i,s}$ along the chain for $L=100$, $U=1$. Apart from jumps at the boundary, density is linear which is typical for diffusive conductors. Solid black line, overlapping with the numerical points, is a best-fitting linear function.}
\label{fig:profilU1}
\end{figure}
\begin{figure}[h!]
\centerline{\includegraphics[width=0.49\textwidth]{jodn.eps}}
\caption{(Color online)~Scaling of charge current $j^{({\rm c})}$ divided by the extrapolated density drop $L\nabla n$ with the system size $L$ for different interactions $U$. Thick full (red) line, overlapping with $U=1.0$ data, is $\sim 30.4/L$, indicating a diffusive transport. Two dashed lines also suggest $\sim 1/L$ scaling.}
\label{fig:jodn}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=0.53\textwidth,angle=-90]{profilU2all.eps}
\caption{(Color online)~(a) Density profiles $n_{i,s}$ at $U=2$, and (b) the corresponding connected density-density correlation function $\ave{ n_{i\uparrow} n_{i\downarrow}}_{\rm c}$. Data are shown for $L=16$ (blue squares), $L=32$ (green circles) and $L=64$ (red triangles), all for charge driving. Inset in (a): scaling of the jump between the reservoir and the 1st particle and between the 1st and 2nd particles, with size $L$. Black dashed lines indicate $\sim 1/L$ and $\sim 1/L^{0.7}$. Note that at $L=64$ boundary jumps still account for around $25\%$ of the total density difference between the chain ends.}
\label{fig:U2}
\end{figure}
Because these jumps are rather large we have fitted a linear function to the density profile in the bulk, thereby obtaining the density gradient $\nabla n_{i\uparrow}=\nabla n_{i\downarrow}=\nabla n_i/2$. In Fig.~\ref{fig:jodn} we then plot the scaling of the charge current (which is in the NESS independent of the site) with the gradient of the charge density. At interaction strength $U=1$ one can see a nice scaling $j^{({\rm c)}} \sim 1/L$. Together with a linear density profiles this is a clear indication of diffusive charge transport. As mentioned, for spin transport virtually the same behavior is obtained (data not shown).
For $U=2$ the scaling is not quite as good. It seems that for shorter chains $j^{({\rm c})}$ decreases with $L$ slower than $1/L$, however, for two largest sizes that we managed to calculate, a crossover to $\sim 1/L$ scaling is clearly suggested. For smaller interaction $U=0.5$ the convergence seems better, but contrary to the $U=2$ case, the current approaches the asymptotic scaling ($\sim 1/L$) from above, i.e. it decays a bit faster for short chains. It is not clear whether $U=1$ corresponds to a crossover point between the two behaviors since details of convergence in the thermodynamic limit might depend on a particular choice of boundary Lindblad operators. In Fig.~\ref{fig:profilU0.5all}(a) we show density profiles for $U=0.5$ which, apart from considerable boundary jumps, again look linear.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.47\textwidth,angle=-90]{profilU0.5all.eps}}
\caption{(Color online)~Density profiles (a) and connected correlations (b) for $U=0.5$. Other parameters are the same as in Fig.~\ref{fig:U2}.}
\label{fig:profilU0.5all}
\end{figure}
Looking at the density profiles at $U=0.5,1,2$ which are linear in the bulk already for rather small sizes $L$, it seems natural to conjecture that the transport is diffusive in TL $L \to \infty$ for all nonzero finite values of $U$. Different transient scaling of the current with $L$ for shorter chains is likely due to rather strong boundary effects. That the boundary effects are notable can also be seen in the inset of Fig.~\ref{fig:U2}(a), where we show the jump in the density between the reservoir and the first site $n_{1\uparrow}-n_{{\rm L}\uparrow}$, as well as between the first two sites in the system $n_{2\uparrow}-n_{1\uparrow}$. While the boundary effects show a tendency to disappear in TL, at $U=2$ and largest length $L=64$ they are still non-negligible. One can try to optimize the coupling constant $\Gamma$ in order to minimize the boundary effects, however we found that $\Gamma \approx 1$ is usually close to the optimal value which does not seem to depend on $L$.
For $U=0.5$ the boundary effects are larger than for larger $U$ cases studied, $U=1$ and $U=2$, see Fig.~\ref{fig:profilU0.5all}(a).
This is probably due to a smaller bulk resistivity which makes the effects of the contacts (contact resistivity) relatively larger.
\subsection{Density-density correlations}
In Figs.~\ref{fig:U2}(b),~\ref{fig:profilU0.5all}(b), and also ~\ref{fig:corelU1}(b), we show the connected spin-up spin-down correlation function $\langle n_{i\uparrow}n_{i\downarrow}\rangle_{\rm c}=\langle n_{i\uparrow}n_{i\downarrow} \rangle - \langle n_{i\uparrow} \rangle \langle n_{i\downarrow} \rangle$, that gives on-site correlations between two fermion species. If we extrapolate our finite-$L$ data to TL, we find \begin{equation}
\langle n_{i\uparrow}n_{i\downarrow}\rangle_{\rm c} \propto (\mu^2-(2\ave{n_{i\uparrow}}-1)(2\ave{n_{i\downarrow}}-1)),
\end{equation}
with a proportionality prefactor depending on $U$ only, yielding a {\em parabolic correlation profile} for our {\em linear} density profiles. Interestingly, in the middle of the chain the connected correlations become independent of the system size, while they are going to zero at the boundaries\cite{footnotea}.
In Fig.~\ref{fig:corelU1} we in addition show also, for $U=1$ case, the non-connected correlations $\ave{n_{i\uparrow}n_{i\downarrow}}$ (top frame (a), red squares), or centered non-connected correlations $\ave{(n_{i\uparrow}-1/2)(n_{i\downarrow}-1/2)}$ (bottom frame (b), red dotted line). We can see that the non-connected correlations have similar shapes as density profiles.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.47\textwidth,angle=-90]{corelU1.eps}}
\caption{(Color online)~Density-density correlation functions for the case of charge driving (red squares) and for spin driving (blue circles). Frame (b) shows the connected correlations (symbols), frame (a) the non-connected ones. In (b) we also show $\langle (n_{i\uparrow}-\frac{1}{2})(n_{i\downarrow}-\frac{1}{2}) \rangle$ (dashed curves). All is for $U=1$, $L=32$.}
\label{fig:corelU1}
\end{figure}
Considering spin driving (\ref{eq:spindriv}), we find numerically the same diffusion constant as for charge driving (\ref{eq:chargedriv}), while the connected correlations $\langle n_{i\uparrow}n_{i\downarrow}\rangle_{\rm c} $ change sign (see Fig.~\ref{fig:corelU1}(b), blue circles). Note that this implies that conductivities at infinite temperature do not depend on the sign of interaction $U$.
We have also checked explicitly, by comparing data for $U=-1$ with $U=1$, that density profiles, currents and correlations are insensitive to the sign of $U$.
One can see that in the presence of spin current without charge current, non-connected correlations, shown in Fig.~\ref{fig:corelU1}(a) with blue circles, are practically independent of the site and are slightly smaller than $1/4$.
Note that in both cases, of charge and spin driving, the connected correlations scale as $\sim \mu^2$ and are of purely nonequilibrium origin, i.e. they vanish in the equilibrium limit ($\mu=0$).
\subsection{Large interaction $U$}
In the limit $U \to \infty$ the low energy excitations of the half-filled Hubbard model can be effectively described by the 1d isotropic Heisenberg model. In our open system formulation this
mapping cannot be strictly implemented, due to the presence of high-temperature baths which drive the system locally away from half-filling. It is therefore an interesting question whether the transport properties of the Hubbard model in the limit of large $U$ are qualitatively the same as for the Heisenberg model.
In Fig.~\ref{fig:fixedn} we plot density profiles in our open Hubbard model for increasing values of $U$, keeping $L$ fixed, and find
increasingly $\arcsin{}$-like shape, similar as in the isotropic Heisenberg model which displays an anomalous transport~\cite{znidaric:11} with the magnetization current scaling as $\sim 1/\sqrt{L}$.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.47\textwidth]{fixedn.eps}}
\caption{(Color online)~As one increases $U$ at fixed length $L=32$ the rescaled density profiles $(2n_{i,s}-1)/\mu$, shown for $U=1$ and $U=5$, become similar to $\frac{2}{\pi}\arcsin{[\frac{2i-1}{L}-1]}$ (full red curve), found in the isotropic Heisenberg model \cite{znidaric:11}.}
\label{fig:fixedn}
\end{figure}
This perhaps explains slower decay of the current with $L$ in the Hubbard model for small $L$'s and larger $U$, seen for instance in Fig.~\ref{fig:jodn} at $U=2$. Note that the limits $U \to \infty$ and $L \to \infty$ do not commute. In order to recover the Heisenberg behavior in TL one has to first let $U \to \infty$ and only then $L \to \infty$.
\subsection{Strong driving, $\mu=1$}
In previous subsections we have shown that the weakly driven Hubbard model displays diffusive behavior. Here we show that for strong driving, where the system is far away from equilibrium, the behavior
of physical observables can be dramatically different. For example, we briefly discuss the case of maximal driving $\mu=1$ and find that the current scales sub-diffusively as $j^{\rm c} \propto 1/L^2$.
The corresponding density profile is shown in Fig.~\ref{fig:profilmu1}. We can see that the profile is in TL given by a simple cosine shape $n_{i,s} = \sin^2(\pi (2i-1)/4L)$, exactly the same as has been found analytically in the isotropic Heisenberg model at strong driving~\cite{prosen:11}. This suggests that
similar exact solution for NESS at maximum driving $\mu=1$ as for the Heisenberg spin chain is also achievable for the Hubbard model and points to wider applicability of the algebraic method proposed in
Refs.~\cite{prosen:11,prosen:11b}.
\begin{figure}[t!]
\centerline{\includegraphics[width=0.47\textwidth]{profilmu1.eps}}
\caption{(Color online)~At maximal driving, $\mu=1$, density profiles have a cosine shape (red squares, for $L=50$, $U=1$) while the current scales as $\sim 1/L^2$ (data not shown), exactly as in the Heisenberg model at maximal driving \cite{prosen:11}. In the inset we show convergence of $\Delta n=n_{i,s}-\sin^2{(\pi x/2)}$ with $L$.}
\label{fig:profilmu1}
\end{figure}
\begin{figure}[h!]
\centerline{\includegraphics[width=0.47\textwidth,angle=-90]{nonhalf.eps}}
\caption{(Color online)~Density profiles (a) and currents (b) for a non-half-filled Hubbard system. We show data for $L=16$ (squares) and $L=32$ (circles), all for $U=1$. Density profiles in the bulk are flat and currents do not depend on $L$, indicating ballistic transport. In both frames upper two sets of symbols are for one fermion species, lower two for the other.}
\label{fig:nonhalf}
\end{figure}
\subsection{Non-half-filled case}
So far, all the results shown were for symmetric driving, producing on average half-filled bands. However, if the filling of the two fermion species is not $1/2$ we expect the transport to be ballistic at high temperatures. This follows from an existence of a nontrivial constant of motion, which in the non-half filled case possesses nonvanishing overlap with the spin/charge currents~\cite{Zotos:97}. In order to numerically verify the consistency of our non-equilibrium setup with this expectation, we choose a nonsymmetric driving with $L_{1,2}=\sqrt{\Gamma(1\mp \mu_{\rm L\uparrow})}\,\sigma^{\pm}_1$ and $L_{3,4}=\sqrt{\Gamma(1\pm \mu_{\rm R \uparrow})}\,\sigma^{\pm}_L$ for the first chain, where $\mu_{\rm L\uparrow}=0.5$ and $\mu_{\rm R\uparrow}=0.9$, while $L_{5,6}=\sqrt{\Gamma(1\mp \mu_{\rm L \downarrow})}\,\tau^{\pm}_1$ and $L_{7,8}=\sqrt{\Gamma(1\pm \mu_{\rm R\downarrow})}\,\tau^{\pm}_L$ with $\mu_{\rm L\downarrow}=0.4$ and $\mu_{\rm R\downarrow}=0.8$ for the 2nd chain. Density profiles can be seen in Fig.~\ref{fig:nonhalf}. One can see that the gradient is very small (or zero); what is more, currents are almost independent of system size $L$. Namely, in Fig.~\ref{fig:nonhalf}b currents are within numerical errors the same for $L=16$ and $L=32$ (small inhomogeneities visible in the Figure are due to truncation errors). If the transport were diffusive, as is the case for the half-filled system, the current for $L=32$ should be half as large as for $L=16$. We therefore confirm that the non-equilibrium transport is clearly ballistic for a non-half-filled Hubbard model.
\section{Conclusion}
Summarizing our findings about the transport in half-filled zero-magnetization 1d Hubbard model at an infinite temperature, we have shown that at finite interaction $U$ both charge and spin transport are diffusive. This conclusion is based on the scaling of the currents with the system size for up to $100$ particles, as well as on perfectly linear density profiles away from the boundaries. This complements ballistic spin transport at $U=0$ and arbitrary temperature and anomalous transport at high temperature and $U=\infty$. We acknowledge support by the grants P1-0044 and J1-2208 of Slovenian Research Agency (ARRS).
|
{
"timestamp": "2012-07-03T02:03:30",
"yymm": "1203",
"arxiv_id": "1203.1727",
"language": "en",
"url": "https://arxiv.org/abs/1203.1727"
}
|
\section{Introduction}
The activity of star formation (SF) in a present-day galaxy is
strongly related to the local galaxy density and stellar mass
\citep[][]{kwh+04}. Massive early-type galaxies lie in
higher density environments \citep[][]{Dressler80} and are dominated by
redder, older stars than less massive ones. The specific star
formation rate (i.e., SFR per unit stellar mass) of galaxies tends
to be lower in denser environments \citep[][]{kwh+04}, pointing to a picture where
more massive galaxies form stars at a lower
rate per unit mass than less massive ones. Therefore, the
bulk of stars in present-day massive galaxies must have formed at earlier
epochs than stars in less massive galaxies \citep[e.g.,][]{khw+03b,tmb+05}. The standard models of galaxy
formation have difficulty reproducing these red and dead massive
galaxies, unless feedback mechanisms (e.g., by active galactic nuclei-AGN) are introduced that prevent the gas
from cooling and forming stars. The star formation history
of massive galaxies is not yet fully understood.
The Brightest Cluster Galaxies (BCGs) are at the most luminous and
massive end of galaxy population. They are usually located at or
close to the centres of dense clusters of galaxies
\citep[e.g.,][]{jf84,sks+05}. Most of them are dominated by old
stars without prominent ongoing star formation. It has been shown
that BCGs are different from other massive galaxies (non-BCGs) in
the surface brightness profiles and some basic scaling relations
\citep[e.g.,][]{mms64,Oemler73,Oemler76,Schombert86,Schombert87,Schombert88,glc+96,pmp+06,lfr+07,bhs+07,vbk+07,dqm+07,lxm+08},
which may indicate a distinct formation mechanism.
Recent studies from numerical simulations and semi-analytic models
in the cold dark matter hierarchical structure formation framework
indicate that a large part of stellar mass in BCGs may have formed
before redshift three, and later dry (dissipationless) mergers
play an important role in their stellar mass assembly
\citep{glp+04,db07}. This picture is largely consistent with
observations. For examples, many examples of dry mergers involving
central galaxies in groups and clusters at z$<$1 have been
reported \citep[e.g.,][]{lauer88,vff+99,mlf+06,jml+07,tvf+05,tmg+08,rfv07,mgh+08,lmd+09},
although some studies of BCGs in the more distant universe disagree with
this scenario \citep[e.g.,][]{wad+08,scb+11}.
The inclusion of AGN feedback in \citet{db07} can efficiently
truncate the initial starburst and ensure that the progenitor of BCGs experiences virtually no star formation
in any evolution. However, some recent studies from the ultraviolet luminosities, infrared emission or line emission
show increasing evidence for ongoing star formation and post-starbursts in some BCGs
\citep[e.g.,][]{allen95,cga98,cae+99,edge01,hm05,mrb+06,emr+06,wes06,ehb+07,obp+08,cdv+08,pkb+09,oqo+10,lwh+12}.
The existence of blue cores and UV excess in some BCGs are also interpreted as evidence for ongoing
star formation \citep[e.g.,][]{bhb+08,pkb+09,wok+10,hmd10}. Although active star formation in these BCGs
is compelling, the starbursts may have very short timescales (shorter than 200 Myr) and only contribute a small mass fraction (less than 1 percent, \citealt{pkb+09}).
The BCGs with ongoing SF studied in previous works are mostly selected
from X-ray cluster samples, and usually reside in cooling flow clusters.
It has been shown that star formation in these BCGs is correlated with the cooling timescale ($\rm t_{cool}$) of the gas \citep[][]{rmn08}, which is a strong
indicator of their connections. However, previous studies are based on small samples and are biased
toward X-ray luminous clusters, which may not be a representative of this population.
It has also been shown nearby optically-selected local BCGs (e.g., z$<$0.1) have little indication
for enhanced active star formation \citep{ehb+07,vbk+07,wok+10}. In this study, we search for BCGs with ongoing
SF in clusters at higher redshift. We select a sample of 120 early-type BCGs at $0.1<z<0.4$ from two large
optically-selected cluster catalogues of SDSS-WHL \citep[][]{WHL09} and GMBCG \citep[][]{hmk+10}.
This sample is roughly an order of magnitude larger than previous ones.
They are selected with strong emission lines in their optical spectra, with both {\rm H}$\alpha$~and [O II]${\lambda}$3727
line emission, which indicates significant ongoing star formation.
We investigate their statistical properties and make a comparison with
a control sample selected from X-ray luminous clusters. For the first time, we probe the dependence
of SF activities in these BCGs on their stellar masses and cluster environments.
We also reconstruct their star formation history using stellar population synthesis models,
and discuss their physical connections with the cooling flows and galactic cannibalism.
The structure of the paper is as follows. We describe our sample selection and data analysis in \S2 and \S3, and
present our results in \S4. A summary and discussion are given in \S5.
Throughout this paper we adopt a cosmology with a matter density parameter
$\Omega_{\rm m}=0.3$, a cosmological constant
$\Omega_{\rm\Lambda}= 0.7$, and a Hubble constant of $H_0=70\,{\rm
km \, s^{-1} Mpc^{-1}}$, i.e., $h=H_0/(100\,{\rm km \, s^{-1} Mpc^{-1}})=0.7$.
\section{Sample Selection\label{sec:sample}}
We identify early-type BCGs with significant star formation from two large optically-selected
cluster catalogues of SDSS-WHL \citep[][]{WHL09} and GMBCG \citep[][]{hmk+10}.
The SDSS-WHL cluster catalogue was constructed from the SDSS DR6 photometric galaxy catalogue,
which includes 39,668 clusters in the redshift
range $0.05<z<\sim0.6$ with more than eight luminous ($M_r\leq-21$) member galaxies
within a radius of $0.5$\,Mpc and a photometric redshift interval $z\pm0.04(1+z)$.
The GMBCG catalogue was constructed from the SDSS DR7 photometric catalogue by identifying the red sequence plus BCG feature,
which includes 55,424 clusters in the redshift range $0.1<z<0.55$. We use the data at $0.1<z<0.4$
because both catalogues are relatively complete out to $z\sim0.4$. There is another cluster catalogue by \citet[][]{spd+11}.
we do not use this sample since it is not yet public at the time of writing. These three cluster samples overlap
but also differ in their lists of clusters. SDSS-WHL and GMBCG samples give fully consistent results (see below) concerning SF activities, and thus our conclusions should not be much affected by which catalogue we use.
In our selection, we require BCGs to have spectroscopic observations and their spectra have been parameterised by the MPA/JHU
team\footnote{http://www.mpa-garching.mpg.de/SDSS/DR7/}. We discard the BCGs with concentration index $C=R_{90}/R_{50}<2.5$ in the $i-$band (to select
early-type objects, \citealt[][]{bsa+03}) and those with a median signal-to-noise ratio (S/N) per pixel of the whole spectrum smaller than 3.
As a result, we obtain 10,996 and 15,181 early-type BCGs at $0.1<z<0.4$ from the SDSS-WHL and GMBCG catalogues, respectively.
\begin{figure*}
\centering
\includegraphics[angle=0,width=0.7\textwidth]{f1.eps}
\caption{The diagnostic diagrams of [NII]$\lambda$6584/{{\rm H}$\alpha$} versus [OIII]$\lambda$5007/{{\rm H}$\beta$} for
emission-line early-type BCGs in the SDSS-WHL catalogue (left panel) and GMBCG catalogue (right panel), respectively.
The purely star-forming galaxies, composites and AGNs are shown with solid circles, open circles and crosses, respectively.
The sources in X-ray luminous clusters are shown with red symbols.
The solid line is from \citet{kgk+06}, and the dashed line is from \citet{kht+03}.
} \label{fig:diagnose}
\end{figure*}
\citet{bpt81} proposed a suite of three diagnostic diagrams to classify the dominant energy source in emission-line galaxies,
which are commonly known as the Baldwin-Phillips-Terlevich (BPT) diagrams and are based on four emission line ratios,
[OIII]$\lambda$5007/{\rm H}$\beta$, [NII]$\lambda$6584/{\rm H}$\alpha$, [SII]$\lambda$6716+6731/{{\rm H}$\alpha$} and [OI]$\lambda$6300/{\rm H}$\alpha$.
Here, we only use the emission line diagnostic diagram of [OIII]$\lambda$5007/{\rm H}$\beta$~versus [NII]$\lambda$6584/{{\rm H}$\alpha$}
because the other lines ([SII]$\lambda$6716,6731 and [OI]$\lambda$6300)
are usually very weak in early-type galaxies \citep{hg09}.
We select the emission-line BCGs, requiring
(1) the lines of [NII]$\lambda$6584, H$\alpha$, [OIII]$\lambda$5007, H$\beta$ and [O II]$\lambda$3727 are detected as
emission lines and have the S/N$>$3.
(2) the equivalent widths (EWs) of both H$\alpha$~and [O II]$\lambda$3727 lines are greater than 3$\AA$.
It should be noted that some previous studies (e.g., Crawford et al. 1999; Donahue et al. 2010) have shown that
the line emission in BCGs with large equivalent width of H$\alpha$~is dominated by star formation. The cut of
H$\alpha$~EW $>3\AA$ is thus strongest in our criteria to select sources with SF activities. The cut of [O II] EW $>3\AA$
only reject $\sim1\%$ of sources with large H$\alpha$~EW, which does not affect our statistical result.
However, the inclusion of the [O II]$\lambda$3727 line here allows us to investigate the origin of [O II]$\lambda$3727 line emission in
star-forming BCGs (see \S3.1).
Our criteria inevitably reject weak emission-line BCGs. It is acceptable since the ability to distingush their types by the BPT
diagram will be poor \citep[e.g.,][]{hst+05}.
In total, 159 and 201 objects satisfy our criteria in the selected SDSS-WHL and GMBCG early-type BCGs respectively.
The fraction is $\sim1.4\%$ (159/10,996) for SDSS-WHL sample, $\sim1.3\%$ (201/15,181) for GMBCG sample, respectively.
The diagnostic diagram mentioned above then classifies these emission-line BCGs into different types, which are shown in Figure~\ref{fig:diagnose}.
We here detect 3 purely star-forming objects, 59 composites (SF $+$ AGN) and 98 AGNs in the SDSS-WHL objects (see the left panel of Figure~\ref{fig:diagnose}),
and 7 purely star-formings objects, 71 composites and 123 AGNs in the GMBCG objects (right panel).
We select those classified as purely star-forming objects or composites as our targets, which constitute a very rare population relative
to the whole sample ($62/10996\sim0.56\%$ for SDSS-WHL objects, $78/15181\sim0.5\%$ for GMBCG objects). Notice that 20 of these targets
overlap in these two catalogues. Thus we finally obtain a total of 120 early-type BCGs with significant ongoing star formation
(9 purely star-forming objects and 111 composites), which are listed in Table 1.
We also identify a control BCG sample from X-ray luminous clusters. The {\it ROSAT} All Sky Survey detected 18,806
bright sources \citep{vab+99} and 105,924 faint sources \citep{vab+00} in the
0.1--2.4 keV band, of which 378 extended sources in the northern hemisphere
and 447 extended sources in the southern hemisphere have been identified as clusters of galaxies \citep{bvh+00,bsg+04}.
However, many objects in recent catalogues of SDSS clusters may also be unidentified X-ray clusters \citep[][]{WHL09} since
SDSS have detected many new clusters of galaxies. We follow \citet[][]{WHL09} to cross-identify the {\it ROSAT}
X-ray bright and faint sources with two spectroscopic catalogues to construct a new sample of X-ray cluster candidates.
We first select X-ray sources with a projected separation of $r_p<0.3$ Mpc from the BCGs and hardness ratios of 0--1 as our targets \citep[][]{WHL09}.
In total, we obtain 112 targets in the {\it ROSAT} bright source catalog and 194 targets in the {\it ROSAT} faint source catalog for SDSS-WHL clusters, respectively.
We also obtain 125 targets in the {\it ROSAT} bright source catalog and 236 targets in the {\it ROSAT} faint source catalog for GMBCG clusters, respectively.
\begin{figure}
\centering
\includegraphics[angle=0,width=0.47\textwidth]{f2.eps}
\caption{Top panels: the incidence rates (fractions) of identified emission-line BCGs (dots connected with dashed line)
and BCGs with SF (dots connected with solid line) as a function of cluster richness for SDSS-WHL clusters (left) and GMBCG clusters (right), respectively.
Bottom panels: the normalised distributions of cluster richness of optically-selected samples (dots connected with dashed line)
and X-ray luminous sample (dots connected with solid line) for SDSS-WHL clusters (left) and GMBCG clusters (right), respectively.
Poisson errors are shown.
} \label{fig:fraction}
\end{figure}
We first cross-correlate selected emission-line BCGs with these X-ray candidates.
There are 24 SDSS-WHL emission-line BCGs and 25 GMBCG emission-line BCGs in {\it ROSAT} bright source catalog respectively.
Only 3 emission-line BCGs and 5 GMBCG emission-line BCGs are found in the {\it ROSAT} faint source catalog.
The fractions of emission-line BCGs relative to the X-ray luminous samples ($24/112\sim21\%$ for SDSS-WHL objects, $25/125 \sim 20\%$ for GMBCG objects)
are almost one order of magnitude higher than that in optically-selected sample ($\sim1.4\%$).
The derived fraction in X-ray luminous sample ($\sim20\%$) is slightly lower than previous
results (e.g., $27\%$ from Crawford et al. 1999; $22\%$ from Donahue et al. 2010). The difference may be a result of
different selection criteria or sample sizes. \citet[][]{dbw+10} analysed a small sample (with 32 objects), and the detected emission-line BCGs
by H$\alpha$~EW $>1\AA$ showed the typical forbidden line emission as well. Our incidence rate is close to their result.
\citet[][]{cae+99} used a relatively large statistical sample (with 256 central dominant galaxies) and selected emission-line galaxies based on H$\alpha$~line emission only.
If we select emission-line BCGs by H$\alpha$~emission line only and apply the same H$\alpha$~luminosity detection limits as \citet[][]{cae+99},
namely extrapolating their slope to our redshift range according to the expected $L \propto z^2$ relation (see Crawford et al. 1999 for details)
and correcting the difference in cosmological parameters. As a result, We obtain 29 SDSS-WHL H$\alpha$ emitters and 35 GMBCG H$\alpha$ emitters
in X-ray luminous clusters. The fractions ($29/112\sim26\%$ for SDSS-WHL objects, $35/125 \sim 28\%$ for GMBCG objects) are almost the same
with that of \citet[][]{cae+99}.
We then cross-correlate those BCGs with SF with these X-ray candidates. There are 11 SDSS-WHL BCGs with SF and 10 GMBCG BCGs with SF
in {\it ROSAT} bright source catalog. Notice that 8 sources with SF overlap in these two X-ray samples. Therefore 13 out of a total of 120
early-type BCGs with SF are likely to be in X-ray luminous clusters. In fact, 11 of these 13 sources are known
X-ray luminous clusters according to the NASA/IPAC Extragalactic Database (NED), which have been indicated in Table 1.
The fractions of BCGs with SF relative to the whole X-ray luminous
samples ($11/112\sim9.8\%$ for SDSS-WHL objects, $10/125 \sim 8.0\%$ for GMBCG objects) are also one order of magnitude higher than
that in optically-selected sample ($\sim 0.5\%$).
We show the fractions of these BCGs with SF (solid line) and selected emission-line BCGs (dashed line) as
a function of cluster richenss for the SDSS-WHL objects (top-left panel) and GMBCG objects (top-right panel) in Figure~\ref{fig:fraction} respectively.
Notice that the relations with cluster richness for sources in these two catalogues
are shown separately because the cluster richness in these catalogues is estimated with different algorithms.
It can be seen that the more massive clusters tend to habour higher fractions of emission-line BCGs and SF BCGs.
The fractions are usually the highest in the richest clusters. It indicates that the incidence rates of emission-line BCGs and SF BCGs
in a cluster sample may be much higher above some minimum cluster richness (mass). It can thus be understood that the incidence rates of
emission-line BCGs and BCGs with SF are higher in X-ray luminous clusters than optically-selected ones since X-ray selected clusters
are usually more massive (see bottom panels of Figure~\ref{fig:fraction}).
\input{table1}
\section{Data Analysis}
\subsection{SFR estimates \label{sec:assembly}}
It has been known that {\rm H}$\alpha$~emission is sensitive to the most recent star formation. The SFR based on {\rm H}$\alpha$~emission is an
indicator of the nearly instantaneous SFR since it is produced by ionization by the hottest and youngest stars.
We derive the SFRs of our target BCGs by the {\rm H}$\alpha$~line following \citet[][]{kennicutt98}
\begin{equation}
\mathrm{SFR(H_{\alpha})}=7.9\times10^{-42} L_\mathrm{H_{\alpha}}\,M_{\odot}\rm{yr^{-1}}
,
\end{equation}
where $L_\mathrm{H_{\alpha}}$ is the extinction-corrected luminosity of {\rm H}$\alpha$~emission in units of
$10^{42}\ \rm{ergs\ s^{-1}}$.
The derived SFRs({\rm H}$\alpha$) range from 0.16 $M_\odot/\mathrm{yr}$ to 129.9 $M_\odot/\mathrm{yr}$,
with an average value of 7.7 $M_\odot/\mathrm{yr}$.
\begin{figure}
\includegraphics[angle=0,width=0.47\textwidth]{f3.eps}
\caption{ The line ratio [O II]/[O III] versus the
luminosity of [O III]. The solid line shows the least-square
regression for type I (narrow line) AGNs given by
\citet[][]{khi06}. The symbols are the same as in Figure~\ref{fig:diagnose}.
Our targets clearly show enhanced [O II]/[O III] line ratios relative to the \citet[][]{khi06} line. } \label{fig:kim}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.45\textwidth]{f4.eps}
\caption{
The comparison between the estimated SFR({\rm H}$\alpha$) from the {\rm H}$\alpha$~line and SFR([O II]) from the [O II] line.
The symbols are the same as in Figure~\ref{fig:diagnose}.
It shows that these two estimates are roughly consistent with each other.
} \label{fig:sfr-contrast}
\end{figure}
We also estimate their SFRs by the [OII]$\lambda$3727 line and make a comparison with SFRs({\rm H}$\alpha$).
In order to estimate the contributions by AGN on the [O II] emission, we
follow \citet[][]{ww08} to investigate the luminosity of [O III]
as a function of the line ratio of [O II]/[O III] for our BCGs (see
Figure~\ref{fig:kim}). The symbols are the same as in Figure~\ref{fig:diagnose} except
that BCGs in X-ray luminous clusters are shown with red symbols.
Notice that hereafter we do not show the overlapping objects in the GMBCG catalog in the total sample.
It is clear that our composite BCGs show enhanced [O II]/[O III] ratios just like
purely star-forming BCGs as these two types of objects reside in the same region.
We can estimate the [O II] luminosities emitted from the HII regions by assuming the
enhanced [O II]/[O III] ratios are caused by star formations \citep[][]{khi06}.
The mean value of the [O II]/[O III] ratios is $\sim$ 7.0 for our BCGs.
Given that the average [O III] luminosity $L_\mathrm{[O III]}$
$\sim$ 7.8 $\times$ 10$^{40}$ ergs s$^{-1}$, the [O II]/[O III]
ratio predicted by the regression line (see Figure~\ref{fig:kim})
given by \citet[][]{khi06} is only $\sim$ 0.43. It means that on average
$\sim$ 92\% of the [O II] emission for our BCGs can be attributed to star formation.
We use the recent calibration of \citet[][]{kgj04}
\begin{equation}
\mathrm{SFR(O~II)}=7.9\times\frac{C_1{\times}L_{\mathrm{[O~II],42}}}{16.73-1.75[\log(\mathrm{O/H})+12]}M_{\odot}\,\rm{yr^{-1}}
\end{equation}
to estimate the SFR(O II), where $L_\mathrm{[O II],42}$ is the
extinction-corrected luminosity of [O II] emission in units of
$10^{42}\ \rm{ergs\ s^{-1}}$. $C_1$ is the correction factor due to the enhanced [O II]/[O III] ratio. The metallicity is fixed to be $\log
(\rm{O/H})+12=8.9$, corresponding to the solar value. The
derived SFRs(O II) ranges from 0.12 $M_\odot/\mathrm{yr}$ to
196.9 $M_\odot/\mathrm{yr}$, with an average value of 10.9
$M_\odot/\mathrm{yr}$.
\begin{figure*}
\includegraphics[angle=0,width=0.75\textwidth]{f5.eps}
\caption{
The spectral synthesis of an example BCG, WHLJ150407.5-024816 (RXC J1504.1-0248).
The observed spectrum $O_\lambda$ (green), model spectrum $M_\lambda$ (red) and
error spectrum (blue) of $O_\lambda$
are shown in the top left panel, respectively. The residual spectrum $E_\lambda$ (purple) is shown in the bottom left panel.
The flux intensities in the left two panels are both normalised at 4020{\rm \AA}\, by $4.5\times10^{-16}\, {\rm ergs\ s^{-1}\ cm^{-2}}$.
The light and mass weighted stellar population fractions, $x_{\rm j}$ and $\mu_{\rm j}$, are shown in the top right and bottom right panels,
respectively. Several derived quantities (see text for details) from the fitting are shown at the top right corner.
}
\label{fig:fitting}
\end{figure*}
The comparison between the SFR({\rm H}$\alpha$) and SFR([O II]) for our BCGs is shown in Figure~\ref{fig:sfr-contrast},
which shows that these two estimates are consistent with each other and most of the [OII]$\lambda$3727 line emission
in star-forming BCGs can be attributed to star formation.
It should be noted that our SFR estimates are from the SDSS spectra within 3$''$ fiber diameter, which corresponds to
an average size of $\sim$ 10 kpc for our BCGs at $0.1<z<0.4$. It has been known that the majority of the line emissions in BCGs
may be contained within this aperture \citep[e.g.,][]{hcf07}. We thus do not correct the aperture effect for our BCGs,
as in previous studies \citep[e.g.,][]{cae+99,obp+08}.
\subsection{Spectral synthesis}
\begin{figure*}
\includegraphics[angle=0,width=0.95\textwidth]{f6.eps}
\caption{ The derived total SFR and specific SFR (SSFR) by the {\rm H}$\alpha$~line versus the total stellar mass ($\rm log {M_{\rm \ast,tot}}$)
and cluster richness, respectively. The symbols are the same as in Figure~\ref{fig:diagnose}, except that 8 BCGs in known cooling flow clusters
are shown as red boxes. The size of each box is inversely proportional to its cooling time ($\rm t_{cool}$) (see Sec. 4.4).
At the top right corner of each panel, we show
the correlation coefficient and corresponding significance level for the null
hypothesis of no correlation as given by the Spearman-Rank order test.
}
\label{fig:environment2}
\end{figure*}
We use the spectral synthesis code \textsc{starlight}\ \citep{cms+05} to derive stellar populations of our BCGs.
\textsc{starlight}\ fits the observed spectrum $O_\lambda$ with a model spectrum $M_\lambda$,
which is made up of a pre-defined set of base spectra. It carries out the fitting with a simulated annealing plus
Metropolis scheme to yield the minimum $\chi^2 = \sum_\lambda [(O_\lambda - M_\lambda) w_\lambda]^2$, where $w_\lambda^{-1}$ is
the error in $O_\lambda$ at each wavelength. It models $M_\lambda$ by a combination
\begin{equation}
\label{Mlambda}
M_\lambda = M_\lambda(\vec{x}, A_V, v_\star, \sigma_\star) =
\sum_{j=1}^{N_\star} x_j \gamma_{j,\lambda} r_\lambda
\end{equation}
where $\gamma_{j,\lambda} \equiv b_{\lambda,j} \otimes G(v_\star, \sigma_\star)$,
$b_{\lambda,j} \equiv {B_{\lambda,j}\overwithdelims () B_{\lambda_{0},j}}$ is the normalised flux of the $j^{th}$ spectrum,
$B_{\lambda,j}$ is the $j^{th}$ component of base spectrum,
$B_{\lambda_{0},j}$ is the value of the $j^{th}$ base spectrum at the normalisation
wavelength $\lambda_0$, $G(v_\star, \sigma_\star)$ is the Gaussian distribution
centred at velocity $v_\star$ and $\sigma_\star$ is the line-of-sight velocity dispersion,
$x_j$ is the fraction of flux due to component {\it j} at $\lambda_0$,
$r_\lambda \equiv 10^{-0.4(A_\lambda-A_V)}$ is the global extinction term represented by $A_V$. The residual spectrum $E_\lambda$
including emission lines can be obtained by subtracting the model spectrum from the observed one as $E_\lambda = O_\lambda - M_\lambda$.
In this work, we take simple stellar populations (SSPs) from the
BC03 evolutionary synthesis models \citep{bc03} as our base spectra.
We adopt the spectral templates with $N_\ast$$=42$ SSPs -- 3 metallicities (Z=0.2, 1 and 2.5 $Z_\odot$)
and 14 ages (3, 5, 10, 25, 40, 100, 280, 500, 900 Myr and 1.4, 2.5, 5, 10, 13 Gyr),
computed with the ``Padova 1994" evolutionary tracks \citep{abb+93,bfb+93,fbb+94a,fbb+94b,gbc+96}
and the \citet{Chabrier03} initial mass function (IMF). We
follow \citet{mwg+10} to model the extinction using the
dust extinction law given by Calzetti et al. (1994, 2000) and Calzetti (1997). We show a typical example of
spectral fitting for our BCGs in Figure~\ref{fig:fitting}. The top left panel
shows the observed spectrum $O_\lambda$ (green) and the model $M_\lambda$ (red).
The bottom left panel gives the residual spectrum $E_\lambda =O_\lambda - M_\lambda$ (purple).
The light-weighted stellar population fractions $x_{\rm j}$ are shown in the top right panel.
The mass-weighted population fractions $\mu_{\rm j}$ are shown in the bottom right panel.
\texttt{STARLIGHT} presents the current stellar mass and the
fraction of each stellar component.
Following \citet{cms+05}, we can derive the mean ages of the stellar population weighted by the flux and stellar mass, respectively.
\begin{equation}
\label{logtl}
\langle {\rm log} t_\star \rangle_L = \sum _{j=1}^{N_\star} x_j \, {\rm log} \, t_j
\end{equation}
\noindent where $x_j$ is the fraction of flux contributed by certain SSP and
\begin{equation}
\label{logtm}
\langle {\rm log} t_\star \rangle_M = \sum _{j=1}^{N_\star} \mu_j \,{\rm log}\, t_j
\end{equation}
\noindent where $\mu_j$ is the fraction of stellar mass contributed by each SSP.
The results of the population synthesis are presented in \S4.2.
\section{Results} \label{sec:results}
\subsection{Dependence on stellar mass and environment} \label{sec:env}
\begin{table*}
\footnotesize
\setlength{\tabcolsep}{0.025in}
\caption{ The normalised percentage of flux and mass contributed on average by
stellar populations of different age for the whole optically-selected 120 BCGs with SF,
13 BCGs with SF in X-ray luminous clusters, 200 optically-selected quiescent BCGs,
and 116 quiescent (without distinct emission-lines) BCGs in X-ray luminous clusters,
respectively. The 1\,$\sigma$ (68.3\%) confidence interval for each average value is given in the parentheses.
}
\centering
\begin{tabular}{c|c|c|c|c}
\hline \multicolumn{1}{c|}{SSP} &
\multicolumn{1}{c|}{SF (optical)} &
\multicolumn{1}{c|}{SF (X-ray)} &
\multicolumn{1}{c}{quiescent(optical)} &
\multicolumn{1}{c}{quiescent(X-ray)}
\\
\hline
$f_{\rm burst}$ $\rm (t<0.1Gyr)$ & 7.63 (0.00,13.65) & 18.37 (1.01,32.11) & 1.38 (0.00,5.40) & 3.16 (0.00,7.12)\\
$f_{\rm young}$ $\rm (t <0.5Gyr)$ & 19.92 (3.91,39.10) & 39.77 (5.35,61.20) & 10.68 (0.00,17.34) & 9.24 (0.00,17.96)\\
$f_{\rm middle}$$\rm (0.5Gyr< t <2.5Gyr)$ & 24.31 (1.25,44.73) & 24.80 (6.12,50.04) & 20.53 (0.00,42.35) & 22.44 (0.00,36.54)\\
$f_{\rm old}$$\rm (t>2.5Gyr)$ & 55.77 (22.75,77.05) & 35.44 (17.22,58.56) & 68.79 (42.92,81.18) & 68.32 (48.23,88.91) \\
\hline
$m_{\rm burst}$ $\rm (t<0.1Gyr)$ & 0.04 (0.00,0.06) & 0.17 (0.00,0.23) & 0.002 (0.00,0.005) & 0.012 (0.00,0.016)\\
$m_{\rm young}$ $\rm (t <0.5Gyr)$ & 0.51 (0.00,1.87) & 1.68 (0.00,2.97) & 0.21 (0.00,0.59) & 0.13 (0.00,0.34) \\
$m_{\rm middle}$$\rm (0.5Gyr< t <2.5Gyr)$ & 11.73 (0.00,28.36) & 19.07 (2.00,40.12) & 4.65(0.00,9.60) & 5.72 (0.00,12.21)\\
$m_{\rm old}$$\rm (t>2.5Gyr)$ & 87.76 (57.35,93.06) & 79.25 (43.23,97.25) & 95.14(88.20,97.60) & 94.14 (84.06,97.01)\\
\hline
\end{tabular}
\label{Table2}
\end{table*}
The stellar mass for each BCG inside the fiber aperture has been estimated from
our spectral synthesis, which are roughly consistent
with that obtained by fits on the photometry by the MPA/JHU team.
The specific SFR (inside the fiber aperture) for our BCGs can thus be derived.
The total stellar mass can be obtained by multiplying the factor, $C_2 \equiv 10^{-0.4(m_{\rm petro}-m_{\rm fiber})}$,
between the fiber magnitudes and total (Petrosian) magnitudes.
We follow the method of MPA/JHU team to take the correction factor averaged over the five SDSS bands, weighted by $1/{\rm \Delta {C_2}^2}$ , where $\rm \Delta {C_2}$
is the error in the correction factor $C_2$ (estimated from the errors in the photometry).
The estimated SFR and specific SFR by the {\rm H}$\alpha$~line versus the galaxy total stellar mass ($\rm \log\, {M_{\rm \ast,tot}}$) and the cluster richness
for our BCGs are shown in Figure~\ref{fig:environment2} (the relations are similar if the SFR estimated by the [O II] line is used), respectively.
The symbols are the same as in Figure~\ref{fig:diagnose}, except that 8 BCGs in known cooling flow clusters
are shown as red boxes. The size of each box is inversely proportional to its cooling time ($\rm t_{cool}$).
Notice that the rest 5 BCGs in X-ray luminous sample are also likely to be in cooling flow clusters (see \S4.4 for discussions).
We perform the Spearman-Rank order correlation test for each relation. The corresponding correlation coefficient
and the significance level of the null hypothesis that there is no correlation are given in each panel of Figure~\ref{fig:environment2}.
It can be seen that there is an obvious trend that more massive BCGs with SF and those in richer clusters tend to have higher SFR and specific SFR, but with large scatters.
BCGs with SF in X-ray luminous clusters are often located in the densest environment and have the highest SFR and specific SFR,
which shows they appear to be forming stars at a higher rate.
BCGs with SF in cooling flow clusters (red boxes) usually have the most active star formation (also see \S4.4).
\subsection{Star formation history} \label{sec:sfh}
We investigate the star formation history of these BCGs with SF through spectral synthesis method.
We combine 42 SSPs into three ages: young-, middle- and old-age stellar populations.
The young-age stellar population includes the SSPs with age less than 0.5 Gyr,
the old-age population is SSPs with age larger than 2.5 Gyr,
and the intermediate-age population is the SSPs between them. We also show a burst population,
defined as SSPs with age less than 0.1 Gyr, connected with the most recent star formation activity.
We make a direct comparison between these BCGs with SF and a randomly-selected sample
of 200 quiescent BCGs (without emission lines of both {\rm H}$\alpha$~and [O II], i.e. sources
with the EWs of {\rm H}$\alpha$~and [O II] $\leq 0$ excluded at the 95.4\% confidence level)
in MPA/JHU catalogues.
These two samples are matched in total stellar mass. We have shown that the majority ($\sim80\%$) of BCGs in two X-ray luminous samples are not classified
as emission-line BCGs. They have also no significant emission lines (`quiescent').
We make another control sample with 116 objects.
The normalised fractions of their stellar populations are listed in Table 2, which provide a coarse star formation history of these BCGs.
\begin{figure}
\includegraphics[angle=0,width=0.44\textwidth]{f7.eps}
\caption{Top panel: the Balmer line absorption index, ${\rm H}\delta_A$ plotted
against the 4000\AA~break strength, $D_n(4000)$ for the BCGs with SF and quiescent BCGs (blue dots),
respectively.
Bottom panel: specific SFR (SSFR) versus the 4000\AA~break strength, $D_n(4000)$ for the BCGs with SF.
The symbols are the same as in Figure~\ref{fig:diagnose}. The BCGs in X-ray luminous clusters are shown with red symbols, in particular, 8 BCGs in known cooling flow clusters are shown as red boxes. The sizes of boxes
are inversely proportional to their cooling times ($\rm t_{cool}$).
} \label{fig:hdelta-d4000}
\end{figure}
The flux-weighted average population fraction is sensitive
to the star formation activity. Nearly 20\% flux are from the young stellar population
for the whole sample. This fraction can increase to $\sim$40\% for BCGs
with SF in X-ray luminous clusters. A large fraction ($\sim$40\%)
of the young stellar population is from the recent burst (with age $\rm t <0.1Gyr$).
BCGs with SF and quiescent BCGs have comparable fraction of intermediate-age stellar population, which
indicates that SF activities in BCGs contribute mainly to the fraction of young stellar population and
the timescale of SF activity is short. However, the fractions of stellar mass
in different age bins are significantly different from the flux-weighted ones.
The majority of stellar mass of BCGs with SF is contributed by the old population.
The stellar mass of the young stellar population is small ($\sim 0.5\%$ on average),
which is consistent with the result of \citet[][]{pkb+09} obtained from a smaller sample.
It shows that the stellar population will not have significant differences with that of normal BCGs when their
star formation is quenched. The stellar population of BCGs with SF are still predominantly old, not very different with the quiescent (normal) BCGs.
Notice that about $\sim 12\%$ of stars formed within the last 2.5 Gyr or so for our BCGs with SF (the first column in Table 2). The derived stellar mass inside the fiber aperture is $8.2\times10^{10}$$\rm M_\odot$ on average. If the typical SFR ($\sim 7.7 M_\odot\,{\rm yr}^{-1}$) in bursts is the same as the current one, then the total star formation duration will be 1.3 Gyr. This is consistent with a scenario that the rejuvenation SF activities in these BCGs may be sporadic \citep[e.g.,][]{sts+07}.
In any case, it should be emphasized that the scatters in these estimates are quite high (see Table 2).
It has been shown by \citet[][]{khw+03a} that the plane
defined by the 4000\AA~break strength, $D_n(4000)$, and Balmer
line absorption index, ${\rm H}\delta_A$, is also a powerful diagnostic for
the star formation history of galaxies. We show the ${\rm H}\delta_A$ absorption index as
a function of $D_n(4000)$ for these BCGs with SF in the top panel
of Figure~\ref{fig:hdelta-d4000}, and compare with quiescent
BCGs (blue dots). We also show the relation of the specific SFR versus $D_n(4000)$
for our BCGs with SF in the bottom panel of Figure~\ref{fig:hdelta-d4000}.
The values of $D_n(4000)$ and ${\rm H}\delta_A$ are taken from the MPA/JHU catalog.
As can be seen, BCGs with more active SF activities (higher specific SFR) tend to have lower $D_n(4000)$
values and stronger ${\rm H}\delta_A$ absorption than quiescent BCGs.
It means they have a higher fraction of young stars, and are more likely
to be experiencing sporadic star formation events at the present day \citep[][]{khw+03a}.
This analysis is consistent with our results obtained through spectral synthesis method.
\subsection{SF activity \& cooling flow}
It has been shown that active star formation in BCGs may be connected with the cooling flow of intracluster medium (ICM) in many X-ray
clusters \citep[e.g.,][and reference therein]{rmn08}. There are 11 known BCGs in our 13 targets in X-ray luminous clusters.
We collect their information from the literature, and find that 8 out of 11 ($\sim73\%$) have been identified to be
in cooling flow clusters \citep[e.g.,][]{df08,rmn08}.
The rest-frame optical spectra and colour images of BCGs in these eight clusters (RXC J1504.1-0248, Zw 3146, A1835,
RX J1532.8+3021, MS 1455.0+2232, RX J1720.2+2637, RX J2129.6+0005, A1204) are shown
in Figure~\ref{fig:cooling}, ordered with increasing cooling time of ICM. Their cooling time $\rm t_{cool}$
are obtained from \citet[][]{rmn08} and/or \citet[][]{bfs+05}, which are marked in each panel of Figure~\ref{fig:cooling}.
Although not all objects have measured cooling time and the derived values are somewhat different for the same source from the two studies,
it is still apparent in Figure~\ref{fig:cooling} that the four BCGs with shortest cooling times have bluer colours,
which is consistent with the result of \citet[][]{rmn08} that bluer BCGs reside in clusters with shorter cooling times.
These eight BCGs in cooling flow clusters are shown with red boxes in the plot of specific SFR versus $D_n(4000)$ (the bottom panel of Figure~\ref{fig:hdelta-d4000}),
with the size of each box inversely proportional to the cooling time. It can be seen that BCGs in cooling flow clusters
usually have more active SF activities (higher specific SFR): the four BCGs (RXC J1504.1-0248, Zw 3146, A1835, RX J1532.8+3021)
with the shortest $\rm t_{cool}$ have the most active SF activities. In fact, it has been shown two (Zw 3146 and A1835) of them can even
be classified as luminous infrared galaxies (LIRGs) \citep[][]{emr+06}. This is a strong indicator that the cooling
flow and star formation in these BCGs are connected. Figure~\ref{fig:cooling} also shows that BCGs in cooling flow clusters
(in particular the four with the shortest $\rm t_{cool}$) usually have very flat optical spectra (smaller 4000\AA~break strength, see the bottom panel of Figure~\ref{fig:hdelta-d4000}).
It should be noted that we do not know whether or not the remaining 5 targets in our X-ray luminous sample are also
in cooling flow clusters. Their rest-frame spectra and colour images are shown in Figure~\ref{fig:restXray}.
It can be seen that their images and optical spectra are very similar to those of 8 BCGs in known cooling flow clusters,
particularly for two GMBCG objects (GMBCG J221.85842+08.47364, J355.27875+00.30927).
These two objects have extremely blue images and very flat optical spectra, with extremely high SFR \& SSFR
(two red circles on the top of right panels of Figure~\ref{fig:environment2}). In any case,
the majority (even 100\%) of BCGs with SF in X-ray luminous sample are likely to be in cooling flow clusters.
\begin{figure*}
\includegraphics[angle=0,width=0.9\textwidth]{f8.eps}
\caption{ Spectra and colour images of the eight known early-type BCGs with significant ongoing SF in cooling flow clusters.
Each spectrum is shifted to the rest-frame wavelength, corrected for the Galactic extinction, and smoothed
using a 15 \AA~box. The size of each colour image corresponds to 200\,kpc by 200\,kpc. The objects are ordered with increasing
cooling time ($\rm t_{cool}$) in units of $10^8 yr$, taken from \citet[][R08]{rmn08} and/or \citet[][B05]{bfs+05}.
}
\label{fig:cooling}
\end{figure*}
\begin{figure*}
\includegraphics[angle=0,width=0.9\textwidth]{f9.eps}
\caption{
Spectra and colour images of the other 5 early-type BCGs with ongoing SF in 13 X-ray luminous clusters.
Each spectrum is shifted to the rest-frame wavelength, corrected for the Galactic extinction, and smoothed
using a 15\,\AA~box. The size of each colour image corresponds to 200\,kpc by 200\,kpc.
}
\label{fig:restXray}
\end{figure*}
\section{Summary \& Discussion} \label{sec:discussion}
Only a few BCGs have been reported with ongoing star formation in previous studies and the majority of them are identified
from X-ray cluster samples. In this paper, we identify a large sample of 120 early-type BCGs at $0.1<z<0.4$ from two large
optically-selected cluster catalogues of SDSS-WHL \citep[][]{WHL09} and GMBCG \citep[][]{hmk+10}. Their optical spectra
show strong emission lines of both [O II]${\lambda}$3727 and {\rm H}$\alpha$, indicating significant ongoing star formation.
This sample is not biased toward X-ray luminous clusters, and is thus more representative of this population.
We investigate their statistical properties and make a comparison with a control sample selected from X-ray luminous clusters.
We also investigate their star formation history using stellar population synthesis models. The main results can be summarised as follows.
\begin{enumerate}
\item
The incidence rates of emission-line BCGs and BCGs with SF in X-ray luminous clusters
are almost one order of magnitude higher than those in optically-selected clusters.
\item
More massive BCGs with SF in richer clusters tend to have higher SFR and specific SFR,
which shows they appear to be forming stars at a higher rate. BCGs with SF in X-ray luminous clusters
usually have more active SF activities.
\item
The star formation history of BCGs with SF can be well described by a recent minor and short starburst superimposed
on an old stellar component, with the recent episode of star formation contributing $<$ 1 percent of the total stellar mass ($\sim 0.5\%$ on average).
The star formation history may be episodic, lasting a substantial fraction of the time in the last 2.5 Gyr (see Table 2 and section 4.2).
\item
BCGs with SF in cooling flow clusters usually have very flat optical spectrum and the most active SF activities.
Star formation in these BCGs and cooling flow are correlated.
\end{enumerate}
\begin{figure*}
\includegraphics[angle=0,width=0.95\textwidth]{f10.eps}
\caption{ Colour images of 30 sample early-type BCGs with distinct merger features in non-X-ray luminous clusters.
The size of each colour image corresponds to 200\,kpc by 200\,kpc.
}
\label{fig:minor-merger}
\end{figure*}
Although the short SF activity appears to be a rather common phenomenon during the evolution of BCGs, the source of the cold gas
required to fuel the star formation is unclear. The correlation between SF activities in BCGs in cooling flow clusters
and the gas cooling timescale \citep[][]{rmn08} suggests a clear (cooling) origin of the cold gas in these systems.
However, the majority of our star-forming BCGs have less active SF activities. They either lie in non-cooling flow (non-X-ray luminous) clusters
or may be below the ROSAT detection limit.
Recent theoretical studies often assume a very efficient form of AGN feedback, which may suppress the star formation completely.
On the other hand, \citet[][]{bhb+08} suggest that AGN feedback may not fully compensate the energy lost via radiative cooling,
allowing the gas to cool at a reduced rate. It remains to be seen how universal this mode operates in BCGs and what kind of SFR it can sustain.
Another possible and attractive mechanism is through the known galactic cannibalism that appears
frequently in cluster environments due to dynamical friction. We indeed find a large fraction of sample BCGs in non-X-ray luminous clusters with distinct
merger features (see 30 examples\footnote{Colour images and corresponding spectra for all 120 target BCGs are
available at http:///www.jb.man.ac.uk/\~\,smao/liu.tar.gz} in Figure~\ref{fig:minor-merger}).
If there is some remaining cold gas in the captured satellite galaxy,
then the merger may supply fresh cold gas ($\sim$ few $10^8M_\odot$) that can trigger a new episode of star formation \citep[][]{pfb+06}.
It will be interesting to explore this issue further using our sample in a future study.
\section*{Acknowledgments}
We thank X. Y. Xia, Z. G. Deng, Z. L. Wen, Cheng Li, Lin Yan, J. Wang for useful discussions and comments,
We acknowledge the anonymous referee for a constructive report that much improved the paper.
This project is supported by the NSF of China 11103013. SM and XMM acknowledge the Chinese Academy of Sciences for financial support.
Funding for the creation and distribution of the SDSS Archive has
been provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Aeronautics and Space Administration,
the National Science Foundation, the U.S. Department of Energy,
the Japanese Monbukagakusho, and the Max Planck Society. The SDSS
Web site is http://www.sdss.org/. The SDSS is managed by the
Astrophysical Research Consortium (ARC) for the Participating
Institutions. The Participating Institutions are The University of
Chicago, Fermilab, the Institute for Advanced Study, the Japan
Participation Group, The Johns Hopkins University, the Korean
Scientist Group, Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the
Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, University of Pittsburgh, Princeton University, the
United States Naval Observatory, and the University of Washington.
|
{
"timestamp": "2012-03-09T02:03:37",
"yymm": "1203",
"arxiv_id": "1203.1840",
"language": "en",
"url": "https://arxiv.org/abs/1203.1840"
}
|
\section{Observational Setup\label{setup}}
\subsection{Requirements for AO observations near the limb}
To successfully operate an AO system, its wave front sensor (WFS) needs an object image with high contrast. If the images have insufficient
contrast, the wavefront can neither be measured nor corrected accurately. On the solar
disk, the contrast of the granulation pattern reaches about 20\,\% percent
\citep[][]{2010ApJ...723L.154H}, but near the limb the granulation contrast
reduces strongly. One thus needs to find a suitable solar target other than granulation with a high contrast near the limb. Two types of solar features can be used: pores or
sunspots with a local intensity reduction, and faculae with an enhanced intensity. Whereas sunspots or pores are not necessarily found
where needed, suitable faculae can nearly always be encountered. A minor, but
still important issue is that the WFS also should be equipped with a
device for adjusting its light level. The light level near the limb can be
reduced by up to a factor of two relative to the center of the disk. Using a wavelength in the core of a chromospheric line for the WFS might provide a solution to obtaining a high-contrast image beyond the limb, but its feasibility is doubtful: the light level off the limb drops by an order of magnitude to about 1\,--\,3\,\% of the continuum intensity \citep{2011A&A...531A.173B}.
Now, even if features such as facula provide a high-contrast lock point for a
successful operation of the AO, they also add a requirement: the AO lock point
cannot be changed during the observation. This poses no problem for observations
with 2D spectrometers \citep[e.g.,][]{2010A&A...520A.115B}, but complicates things for observations with slit spectrographs that sequentially scan the solar surface. One possible solution for this problem is achieved in the optical layout of the German VTT sketched in the left panel of Fig.~\ref{vtt_layout}. The AO WFS and the spatial scanning are decoupled here by the fact that the beam splitter (BS), which feeds the AO WFS, is the scanning device at the same time. The AO WFS is fed with the beam that is transmitted through the BS, whereas the post-focus instruments receive the reflected beam. Tilting the BS thus moves the FOV of the science instruments without changing the lock point of the AO WFS. One drawback is that the optimum AO correction is only obtained in a limited radius around the lock point which will then not be centered inside the FOV.
\begin{figure}
\plotone{beck_fig1.eps}
\caption{Left: Schematical layout of the VTT. Middle: side view of the instrumentation in first and ground floor. Right: top view of the first floor. The dashed arrow denotes the configuration in 2011 where only POLIS was used.\label{vtt_layout}}
\end{figure}
\subsection{Instrumental setup}
For the simultaneous observations of the four chromospheric spectral lines in
different wavelength regimes, we used two different combinations of the
post-focus instruments at the VTT at that time: the main spectrograph
(\ion{He}{i} 1083\,nm, \ion{Ca}{ii} IR), the Triple Etalon SOlar Spectrometer
\citep[TESOS, H$\alpha$;][]{1998A&A...340..569K,2002SoPh..211...17T}, and the
POlarimetric LIttrow Spectrograph \citep[POLIS, \ion{Ca}{ii} H and
H$\epsilon$, H$\alpha$;][]{2005A&A...437.1159B}. In the following, we will
describe the two different setups used in 2010 and 2011, respectively. POLIS
has been decommissioned by the KIS at the end of 2010, so that only TESOS
remains for observations in the visible range now.
The middle and right panels of Fig.~\ref{vtt_layout} show a sketch of the
light distribution in the two observation runs. Behind the exit of the
adaptive optics' light path, we inserted a dichroic infrared-visible beam
splitter (IR/VIS BS) that splits the incoming light at 800\,nm. For the setup
used in 2010, it reflected the VIS part of the light towards TESOS while
transmitting the IR part towards the main spectrograph. On the main
spectrograph, the Tenerife Infrared Polarimeter
\citep[TIP,][]{2007ASPC..368..611C} was mounted to record spectropolarimetric
data in the wavelength region near 1083\,nm. Next to the cryostat of TIP, we added a PCO 4000 camera to observe the \ion{Ca}{ii} IR line at 854\,nm.
The VIS part of the light was split once more into a blue ($<$450\,nm) and red
fraction using a dichroic pentaprism close to the entrance of TESOS. The blue
part was folded across the observing room in an one-to-one imaging of the focal plane in front of TESOS onto the focal plane inside POLIS (right panel of Fig.~1). POLIS was not fully operational at that time because of software problems on its camera PCs. We thus mounted a small pick-up mirror behind its grating that reflected the spectrum sidewards out of the instrument towards a second PCO 4000. We used a broad-band interference filter (1\,nm FWHM) centered at 396.85\,nm placed directly in front of the CCD camera to select the order of the spectrum.
In the setup used in 2011, we reflected the light from the IR/VIS BS directly towards POLIS. Because the POLIS \ion{Ca}{ii} H camera was operational again, we
now used the pick-up mirror inside POLIS to reflect the wavelength range of H$\alpha$ towards a PCO 4000 camera. The slit width corresponded to 0\farcs5 for POLIS and 0\farcs36 for the main spectrograph.
The data shown here were taken on 12 Apr 2011 between UT 07:49 and 08:14 at the solar position $(x,y)= (-933'',172'')$. The spatial scanning was done in 150 steps of 0\farcs36 with an integration time of 8 seconds per slit position. The spectral coverage for each line, and the spectral and spatial sampling per pixel are listed in Table \ref{tab_spec}. The data were corrected for stray light with an approach similar to that described in \citet{2011A&A...535A.129B}. A measurement of the instrumental PSF was also obtained for an eventual spatial deconvolution to improve the spatial resolution.
\begin{table}
\caption{Wavelength ranges in nm and spectral\,/\,spatial sampling per pixel.\label{tab_spec}}
\centerline{\begin{tabular}{cccc}\noalign{\smallskip}
\tableline
\noalign{\smallskip}
\ion{He}{i} & \ion{Ca}{ii} IR & H$\alpha$ & \ion{Ca}{ii} H \cr
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
1082.34\,--\,1083.45& 853.45\,--\,855.09& 655.12\,--657.04& 396.34\,--\,396.95 \cr
1.0 pm\, /\, 0\farcs18 & 0.8 pm\,/\,0\farcs18 & 2.0 pm\,/\,0\farcs22 & 1.9 pm\,/\,0\farcs30 \cr
\noalign{\smallskip}
\tableline
\end{tabular}}
\end{table}
\begin{figure}
\plotone{beck_fig2.eps}
\caption{Overview of the observed FOV. Counter-clockwise, starting left bottom: continuum intensity, blue wing, line core, and red wing. The limb region is displayed in a logarithmic intensity scale at the top of each panel. Left to right in each panel: \ion{Ca}{ii} H, H$\alpha$, \ion{Ca}{ii} IR, \ion{He}{i} 1083 nm.\label{fig_overview}}
\end{figure}
\section{Intensity and velocity structuring \label{general}}
Figure \ref{fig_overview} shows an overview of the FOV as seen in \ion{Ca}{ii} H, H$\alpha$, \ion{Ca}{ii} IR 854.2\,nm, and \ion{He}{i} 1083\,nm. A sunspot located about 5\,Mm from the limb was the AO lock point. The alignment of the FOV in the different wavelengths was only done preliminary without taking into account the temporal variation of the displacements caused by differential refraction \citep[e.g.,][]{2008A&A...479..213B}. The maps of the line-core intensity show various regular spicules near the limb, and a few macrospicules \citep[see, e.g.,][and references therein]{2011A&A...535A..58M}: an isolated macrospicule near the left half of the FOV and a cluster of (macro)spicules near the middle of the FOV.
The fine-scale structure of the features is more pronounced in the intensity
maps to the blue and red of the line cores taken at wavelengths corresponding to a Doppler shift of about $\pm 15\,$kms$^{-1}$. Especially in H$\alpha$ and \ion{Ca}{ii} IR, individual spatially extended, coherent features in the
line-core map decompose into multiple strands in the images to the blue and
red of the line core (e.g., at $x,y = 18$\,Mm, 35\,Mm). This behavior affects both the regular spicules near the limb and the macrospicules, but especially the largest macrospicule marked in Fig.~\ref{fig_overview} shows a pronounced difference. The blue-wing images show mainly the left half of the macrospicule, whereas in the red-wing images the right half is brighter. A similar variation can also be seen still on the disk comparing the three different wavelengths in H$\alpha$. In the line-wing images, bundles of clustered elongated dark streaks appear everywhere in the FOV, whereas the line-core map shows no matching structuring at the same locations. The dark streaks end in locations with enhanced emission in the \ion{Ca}{ii} H line (compare for instance the red line-wing images of H$\alpha$ and \ion{Ca}{ii} H).
\begin{figure}
\centerline{\resizebox{10.cm}{!}{\plotone{beck_fig3.eps}}}
\caption{Intensity difference between red and blue wing.\label{dopplergram}}
\end{figure}
Figure \ref{dopplergram} shows the difference image obtained by subtracting the image to the red of the line core from that to the blue. Especially in \ion{Ca}{ii} IR, the characteristic spatial scale parallel to the limb is smaller in the difference image than in, e.g., the core-intensity map in Fig.~\ref{fig_overview}. The reason why the effects are most pronounced for this line are the line width and the steepness of the profile near the line core. Small Doppler shifts suffice already for a large intensity difference in images taken at a fixed wavelength. The variation of the Doppler shifts along the macrospicule is also very prominent for all lines but \ion{Ca}{ii} H, whose complex line formation makes it less suitable for such an approach to estimate velocities.
\section{Properties of a macrospicule\label{macrospic}}
The pattern of Doppler shifts in the largest macrospicule is visualized in
more detail in Fig.~\ref{macspic} which shows the intensity spectra
along the central axis of the macrospicule. All four spectral lines clearly
exhibit a smooth transition from blue shifts of about -20 kms$^{-1}$ near the
limb to redshifts at a height of 10 Mm. The location of the absorption
cores in average profiles was used as zero velocity reference. The polarimetric observations in \ion{He}{i} 1083 nm with TIP also provided us with the polarization signal of the macrospicule (Fig.~\ref{polspic}). We point out that the interesting fact is already that a polarization signal is seen at a spatial sampling of (0\farcs36)$^2$ with an 8-sec integration time. The macrospicule shows significant linear polarization almost along its full extent. In Stokes $V$, the polarity of the signal seems to flip twice along the length, with an iterative sequence of negative (black) and positive (white) polarization amplitude. This could imply a twisted flux tube, with additional either a constant rotation as required by the Doppler shifts, or a helical wave pattern running along the magnetic field lines.
\begin{figure}
\centerline{\resizebox{12.cm}{!}{\plotone{beck_fig4.eps}}}
\caption{Intensity profiles along the axis of the macrospicule. Left to right: \ion{Ca}{ii} H, H$\alpha$, \ion{Ca}{ii} IR, \ion{He}{i} 1083 nm. \label{macspic}}
\end{figure}
\begin{figure}
\centerline{\resizebox{9.cm}{!}{\plotone{beck_fig5.eps}}}
\caption{Polarization profiles in \ion{He}{i} 1083 nm along the axis of the macrospicule. Left to right: Stokes I, total linear polarization $L=\sqrt{Q^2+U^2}$, Stokes $V$.\label{polspic}}
\end{figure}
\section{Summary and discussion\label{summary}}
From the successful simultaneous observation at the VTT of four of the
strongest chromospheric spectral lines near the solar limb with real-time
correction by adaptive optics, we can derive the following list of requirements for similar observations:
\begin{itemize}
\item Adjustable light level in the AO WFS
\item Independence of AO lock point and scanning procedure
\item Availability of suitable instrumentation
\item Proper light distribution by wavelengths (both setups used had a 100\,\% efficiency for each wavelength range)
\item Possibility of image rotation for compensation of differential refraction effects.
\end{itemize}
These technical requirements have to be taken into account already in the design phase of a telescope and its instrumentation, because some of them are very difficult (or impossible) to implement afterwards.
For the analysis of the data, we can only outline the prospects of the data at
this stage, using individual co-spatial profiles along the central axis of the
macrospicule as an example (Fig.~\ref{last_fig}). The simultaneous information
of emission amplitude and line width of various lines from the same chemical
element (Ca) gives restrictions on densities and temperature. These
restrictions are even more refined by the other two lines originating from
different chemical elements. The velocity structure can already be
investigated by any of the lines alone, because they all show similar
patterns. The \ion{He}{i} triplet at 1083\,nm alone provides information on
the thermodynamical structure of the atmosphere by the strength of its two
components, but they also change by coronal EUV radiation
\citep{2008ApJ...677..742C}. Adding the information on the thermodynamics from
the other lines then presents the possibility to determine the coronal
radiation instead. The spectra in Fig.~\ref{last_fig} also provide some information about the optical thickness of the macrospicule: the Ca lines and H$\alpha$ show either a double reversal in the line core (suggestive of self-absorption) or have a flat-topped emission up to a height of about 8 Mm above the limb. This suggests that the macrospicule is optically thick along most of its length.
\begin{figure}
\centerline{\resizebox{11.25cm}{!}{\plotone{beck_fig6.eps}}}
\caption{Individual spectra along the axis of the macrospicule. Clockwise, starting left top: \ion{Ca}{ii} H, H$\alpha$, \ion{He}{i} 1083\,nm, \ion{Ca}{ii} IR 854\,nm. Subsequent spectra were displaced by an arbitrary amount for better visibility. \label{last_fig}}
\end{figure}
The macrospicule shows a clear variation of Doppler shifts along its length with a change by about 30 kms$^{-1}$ on about 12\,Mm length. Together with the change of polarity in Stokes $V$ of \ion{He}{i} at 1083\,nm this suggest a configuration of a twisted flux tube, or a helical wave running along the surface of a flux tube (see \citet{2011A&A...535A..58M} for a different possible scenario).
\section{Conclusion \label{concl}}
For the future planned 4 m-class telescopes, the current data are very
promising. Polarization signals in the off-limb chromosphere can be detected on
(0\farcs36)$^2$ pixels with 1 m-class telescopes. If one thus does not aim for
diffraction-limited observations, the 4 m-class telescopes should provide a
much better S/N ratio at an improved spatial resolution compared with
the current one.
\acknowledgements The VTT is operated by the Kiepenheuer-Institut f\"ur Sonnenphysik (KIS) at the Spanish Observatorio del Teide of the Instituto de Astrof\'{\i}sica de Canarias (IAC). The POLIS instrument has been a joint development of the High Altitude Observatory (Boulder, USA) and the KIS. C.B~acknowledges partial support from the Spanish Ministerio de Ciencia e Innovaci\'on through project AYA 2010-18029.
|
{
"timestamp": "2012-03-12T01:01:58",
"yymm": "1203",
"arxiv_id": "1203.2114",
"language": "en",
"url": "https://arxiv.org/abs/1203.2114"
}
|
\section{Introduction}
The blue horizontal branch (BHB) and RR Lyrae stars are well-established
tracers of the oldest stars in the galactic halo although their galactic
distribution may not coincide with other halo tracers such
as the turn-off stars (Bell et al. 2010). This paper continues those on
other surveys for BHB and RR Lyrae stars (Kinman et al. 2007b and
Kinman \& Brown 2011) and extends previous work in the Anticentre direction
(Kinman et al. 1994). Our BHB and RR Lyrae stars are in the range
10$<V<$17 and so range in distance from those in the solar
neighbourhood to those distant enough to be included
in the SDSS DR 7 survey (Abazajian et al. 2009).
This allows our selection methods for BHB stars to be compared with other
methods for identifying halo SDSS stars ( Smith et al. 2010
and Ruhland et al. 2011).
The globular-cluster halo is thought to consist of an Old Halo and a Young Halo
in which the clusters have different horizontal branch (HB) morphologies that
can be interpreted as an age difference (Zinn 1993). The shape of the field
star halo changes with galactocentric distance (Schmidt 1956; Kinman et al. 1966)
which suggests that it may not be homogeneous.
It is now usual to postulate that the field stars belong to an ``inner halo"
that was formed $in situ$ and an ``outer halo" that has largely been
accreted.
Earlier work on the field star halo has been summarized by Helmi (2008).
Since then, there have been several more investigations of halo structure
using various tracers. Surveys using BHB stars have been made by Smith et al.
(2010), Xue et al. (2011), Ruhland et al. (2011) and Deason et al. (2011).
Surveys using RR Lyrae stars have been made by Watkins et al. (2009), and
Sesar et al. (2010). The stars in these surveys are mostly too distant to
have good proper motions from which space motions could be derived. They do,
however, allow space densities to be determined as a function of
galactocentric distance. The surveys of Deason et al. (2011) for BHB stars
and of Sesar et al. (2011) for turn-off stars both show breaks in the slopes
of their density distributions at 28 kpc from the galactic centre; this
suggests that the halo may have two components.
Recent surveys for stars that are close enough for existing proper motions to
allow a kinematic analysis include the survey for subdwarfs by Smith et al.
(2009), but this survey does not extend far enough in galactocentric distance
to sample the outer halo. Carollo et al. (2007, 2010) have analyzed the
space motions of a large number of stars within 4 kpc and find that a
two-component halo is needed to account for the galactic rotation of these
stars. They find that there is an outer halo that is more metal-poor and that
has a more retrograde rotation than the inner halo. It should be noted that
two halos overlap spatially and the outer halo is less centrally concentrated
than the inner. The difference in metallicity between the two halos
has been confirmed by De Jong et al. (2010). Recently
Carollo et al. (2012) have shown that the fraction of carbon-enhanced
metal-poor stars is twice as great in the outer halo as in the inner halo.
Sch\"{o}nrich, Asplund \& Casagrande (2011) re-analysed the Carollo et al.
(2010) data and failed to find any reliable evidence for an outer
counter-rotating halo. Inter-alia they critized the luminosity classification
of the turn-off stars. In a rebuttal, Beers et al. (2012) re-analyzed their
data (re-classifying the turn-off stars and criticizing the main-sequence
luminosity relation used by Sch\"{o}nrich et al.); they substantiate their
original conclusion that the inner halo is nearly non-rotating
while the outer halo has ``a retrograde signature" with a transition
at 15 to 20 kpc from the Sun.
Recent simulations of galaxy formation support the idea that the halos of
galaxies like the Milky Way have a dual origin and have been formed both
in-situ and by accretion (Zolotov et al., 2009, 2010, 2011), (Oser et al.,
2010). (Font et al., 2011), (McCarthy et al., 2012).
Zolotov (2011) discusses a dual halo in which
the role of accretion increases outwards from the Galactic centre and the
halo is formed solely by accretion if R$_{gal}$ $>$ 20 kpc.
In these simulations, the fraction of the halo that is accreted depends
upon the mass of the galaxy. The average of the 400 simulations given by
Font et al. has the inner ``in situ" component dropping to
20\% and the accreted component rising to 80\% at a galactocentric distance
of 20 kpc. McCarthy et al. (2012) find that the ``in situ" component has a
flattened distribution and a rotation that is intermediate between that of
the disc and the ``outer halo".
In this paper we examine halo stars in the Anticentre direction
because this is the best direction in which to study the transition from the
``in situ" to the accreted halo and the kinematic properties of the outer halo.
We do not have sufficient data to discuss the abundance differences between the
two halos.
Sec. 2 identifies the sources and their galactic distributions from
which our BHB candidates and our RR Lyrae stars are taken. In Sec. 3 we give
the photometric data for the BHB stars and in Appendix A we describe the
techniques used to identify these stars and the methods used to estimate their
distances. Sec. 4 gives the photometric data and periods for the RR Lyrae stars
and shows that they mostly belong to Oosterhoff type 1; in Appendix B we give
the ephemerides for the new RR Lyrae stars and the methods used to estimate
their distances. Sec. 5 gives the adopted distances, proper motions and radial
velocities of our program stars together with their galactic space motons.
It shown in Sec. 5 that the galactic rotation velocity (V) becomes more
retrograde with increasing galactocentric distance. Appendix C gives details of
the sources of the proper motions. Sec.6 introduces the angular momenta
L$_{z}$ and L$_{\perp}$ and their relation to the galactic rotation (V) and
the maximum height above the plane of the star's orbit (z$_{max}$). It is
shown the halo becomes more spherical with increasing galactocentric distance.
Appendix D discusses the location of thick disc in the L$_{z}$ and L$_{\perp}$
plot. Appendix E gives the angular momenta of the RR Lyrae and BHB stars near
the North Galactic Pole, those of the local BHB stars and those of the
galactic globular clusters within 10 kpc. Appendix F discusses the properties
of the kinematic groups H99 and K07. The results of the paper are summarized
in Sec. 7.
\section{Target Selection}\label{s:tsel}
Our study of the Anticentre halo began with a search for RR Lyrae stars in
the fields
RR VI ($l$ = 180$^{\circ}$, $b$ = +26$^{\circ}$) and RR VII
($l$ = 183$^{\circ}$, $b$ = +37$^{\circ}$); each field covers an area of 30
deg$^{2}$ (Kinman et al. 1982). Later Sanduleak provided BHB star candidates
for the RR VII field and these were discussed by Kinman et al. (1994).
These samples of BHB and RR Lyrae stars have been enlarged for the present
paper. New BHB candidates were taken from the objective prism surveys of
Pesch \& Sanduleak (1989; Case A-F stars) and of Beers et al.(1996; BPS BS stars).
We are indebted to Dr Peter Pesch (1996, private communication) for
sending us unpublished candidate stars from the Case survey.
These are given the prefix P in column 2 of Table \ref{t:bhb}
unless there is a previous identification in the literature.
Our methods of
selecting BHB stars from these candidates and the calculation of their
distances are described in Appendix A. Apart from the RR Lyrae stars in
fields RR VI and RR VII, the additional RR Lyrae stars have mostly been
found among BHB candidates that were found to be variable.
Seven of these RR Lyrae stars have not been previously identified and
their light curves and ephemerides are given in Appendix B.
All our program stars are listed in Tables \ref{t:bhb} and \ref{t:rrl} for the BHB
and RR Lyrae stars respectively. These tables give positions and photometric
data for the BHB stars and, also, metallicities and periods for the RR
Lyrae stars; sources of these data are given in notes to these tables.
The galactic distributions of
our program stars are shown separately for the BHB and RR Lyrae stars in
Fig. \ref{f:bl}. Not only do our BHB and RR Lyrae stars cover somewhat different
areas of the Anticentre sky, but they also cover different magnitude ranges
so that the volumes of space that they occupy only partially overlap.
Also, our selection of RR Lyrae stars favours the bluer (Bailey type $c$)
and may miss some of the redder (Bailey type $ab$)
RR Lyrae variables and so our sample may not be complete. This must be
taken into account in comparing the properties of our two samples.
The period-amplitude distribution of our RR Lyrae sample is shown
in Fig. \ref{f:pa}
for the variables with galactic latitudes less than 50$^{\circ}$.
The solid and dotted curves show the loci of the Oosterhoff type I and II variables
respectively (these were taken from Cacciari et al. 2005).
Most of our RR Lyrae stars lie close and to the left of the Oo I curve; this suggests
that the majority are Oo I variables.
The four stars that are most likely to be Oo II variables are
indicated by their numbers in Fig. \ref{f:pa}. This preponderance of Oo I variables in
the Anticenter is compatible with the discovery bu Miceli et al. (2008) that
the Oo II variables are more concentrated towards the Galactic centre than
the Oo I variables. The ratio of Oo I to Oo II variables should therefore
increase with galactocentric distance and so the Oo I variables should
predominate towards the Anticentre.
\section{The BHB Stars.}
Our BHB stars were chosen from {\it candidates} in the sources given in
Section \ref{s:tsel}.
Table \ref{t:bhb} gives the equatorial and galactic coordinates of these stars,
together with photometric data (using the system used in Kinman
et al. 1994)). Details of the photometric observing are given in
Kinman et al. (1994) for photoelectric observations and Kinman \& Brown
(2011) for CCD observations.
Table \ref{t:bhb} also gives the $GALEX$ $NUV$ magnitude
(effective wavelength 2267 \AA) that was taken from $MAST$.\footnote{The Multimission
Archive at the STSci, http://archive.stsci.edu/.}
We assumed that stars with $V<$12.5 had saturated $GALEX$ $NUV$ magnitudes
and should not be used. We also give the $2MASS$ $K$ magnitude that was
taken from the $2MASS$ Point Source Catalog using the $Vizier$ access tool.
\begin{figure*}
\includegraphics[width=13.0cm, bb=17 208 490 610, clip=true]{f1.eps}
\caption{The distribution in galactic coordinates (expressed in degrees) of
(left) the BHB
candidates from Table \ref{t:bhb} and, (right) our program RR Lyrae stars
(Table \ref{t:rrl}).
Seven RR Lyrae stars with b $>$ 50$^{\circ}$ are omitted from this Figure.
\label{f:bl}
}
\end{figure*}
The selection of BHB stars from the candidates is described in Appendix A.
We show there how a weight W (Table \ref{t:bhb}, column 13) was assigned to each
star that depends on the probability that it is a BHB star. The stars
were given a type (Table \ref{t:bhb}, column 15) that depended on this weight.
Stars with a high probability of being BHB stars were classified as $BHB$,
those with a high probability of not being BHB were classified as $A$ while
intermediate types were classified as $bhb$. A comparison of our
classifications with those obtained by methods based on SDSS photometry
suggests that stars with both $BHB$ and $bhb$ classifications have a high
probability of being BHB stars.
Appendix A also contains a discussion of five different methods of getting
the absolute magnitudes (and hence distances) of BHB stars. These distances
and the adopted distances are given in Table \ref{t:A1}.
The reddenings of both BHB stars and RR Lyrae stars were taken from
Schlegel et al. (1998).
\begin{figure}
\includegraphics[width=8.5cm, bb=95 210 490 610, clip=true]{f2.eps}
\caption{The $V$ amplitude {\it vs.} $\log$ Period plot for our RR Lyrae
sample with galactic latitudes less than 50$^{\circ}$.
Filled circles indicate stars for which a $V$ amplitude was available.
Crosses indicate stars where the $V$ amplitude was obtained by dividing the
$B$ amplitude by 1.31. Encircled filled circles and crosses
show Bailey type $c$ variables. The solid and dotted curves show the
expected loci for Oo type I and Oo II variables of Bailey type $ab$
respectively and were
taken from Cacciari et al. (2005). The four likely Oo II variables are
given their numbers in Table \ref{t:rrl}.
\label{f:pa}
}
\end{figure}
\section{The RR Lyrae Stars.}
Most of the RR Lyrae stars in Table \ref{t:rrl} are listed in the General
Catalogue of Variable Stars (GCVS, Kholopov et al., 1985) and subsequent
Name Lists and have the traditional identification by constellation;
those without GCVS names are taken from Pier et al. (2003), Kinman et al.
(2004) and Kinman \& Brown (2010). Seven of the stars in Table \ref{t:rrl} have not
been previously identified as RR Lyrae stars; their light curves and
ephemerides are given in Appendix B.
The mean $K$ magnitudes ($\langle K \rangle$) in Table \ref{t:rrl} were derived from
the 2MASS $K$ magnitudes using the method given in Feast et al. (2008).
We follow the methods given in Kinman et al. (2007b) to derive distances for
the RR Lyraes. These are briefly restated in Appendix B where we give
distances by three separate methods together with adopted distances (D) that
are adjusted to be on the same scale as those adopted for the BHB stars
(Appendix A).
\subsection{Oosterhoff types of the RR Lyrae stars.}
The globular clusters with Oosterhoff type I RR Lyrae stars are known to
have different kinematics (more retrograde orbits) than those containing
Oosterhoff type II RR Lyrae stars (Lee \& Carney, 1999; van den Bergh
1993).\footnote{The simple division into two Oosterhoff types does not cover all
cases (Smith et al. 2011) but is a good approximation for our own Galaxy.}
The Oosterhoff type is determined from the period-amplitude diagram
and this is shown in Fig. \ref{f:pa} for our RR Lyrae sample with
b$\leq$50$^{\circ}$. In this figure the loci for the
Oosterhoff type I and II variables are shown by solid and dotted curves
respectively (these curves were taken from Cacciari et al. 2005).
Most of our RR Lyrae stars lie close and to the left of the Oo I curve.
Type $ab$ stars that lie to the left of the Oo I curve may either be
metal-rich or have smaller mean amplitudes because of Blazhko effect.
This suggests that the majority of our stars
are Oo I variables. Four stars that are most likely to be Oo II variables
are indicated by their numbers in Fig. \ref{f:pa}.
Stars 29, 30 and 46 are type $c$ variables while 28 is type $ab$.
These four stars have $\langle$Z$\rangle$ = 3.9~kpc and
$\langle$R$_{gal}$$\rangle$ = 14.3~kpc compared with $\langle$Z$\rangle$ =
6.6~kpc and $\langle$R$_{gal}$$\rangle$ = 19.6~kpc for the whole sample.
The preponderance of Oo I variables in our Anticenter fields and the smaller
$\langle$R$_{gal}$$\rangle$ of our Oo II variables is to be expected if
the Oo II variables are more concentrated towards the Galactic centre than
the Oo I variables (Miceli et al. 2008). The smaller $\langle$Z$\rangle$ of
our Oo II variables also agrees with the preponderance of Oo I variables
at high Z that was found by De Lee (2008). This suggests that
the Oo II variables are not only more concentrated to the Galactic
centre but also form a more flattened system than the Oo I variables.
\section{The Galactic Space Motions of the Program Stars.}
Tables \ref{t:bhb2} and \ref{t:rrl2} give the parallaxes, proper motions,
radial velocities and the
Galactic Space Motions (U,V,W) with respect to the Local Standard of Rest
(LSR) for the BHB and bhb stars and the RR Lyrae stars respectively. The
parallaxes are derived from the adopted distances (D) given
in Tables \ref{t:A1} and
\ref{t:B2} in Appendices A and B respectively.
The error in the parallax is derived from the rms scatter given in the last
columns of Tables \ref{t:A1} and \ref{t:B2} and does not include any systematic
error in the case of the BHB stars. In the case of the RR Lyrae stars, a small
distance-dependent error has been added in quadrature to the rms scatter to
derive the error of the parallax as explained in the appendix Sec. B2.
Heliocentric space-velocity components U, V, and W were derived from
the data listed in these tables. We used the program by
Johnson \& Soderblom (1987) (updated for the J2000
reference frame and further updated with the transformation
matrix derived from the Vol. 1 of the Hipparcos data catalogue).
This program gives a right-handed system for U, V and W in which these vectors
are positive towards the directions of the Galactic centre, but we here use
the left-handed system so as to be comparable with most other recent work.
These heliocentric velocities were then corrected to velocities relative to
the LSR using the solar motion relative to the LSR
${\rm (U,V,W)_\odot=(10.0,\,5.25,\, 7.17)}$~km\,s$^{-1}$
(Dehnen \& Binney 1998).
\subsection{Radial velocities and Proper Motions}
The sources of our radial velocities are given in column (8) (S$_{RV}$) of
Tables \ref{t:bhb2} and \ref{t:rrl2}.
The Bologna velocities were derived from spectra taken with the 3.5m-LRS (TNG)
spectrograph. The Kitt Peak velocities were derived from spectra taken with
RC spectrograph on the 4m Kitt Peak telescope and kindly made available to us
by Nick Suntzeff (1997, private communication).
The remaining velocities were taken from the literature
as given in the notes to Tables \ref{t:bhb2} and \ref{t:rrl2}.
The absolute proper motions given in this paper come primarily from
astrometric data assembled from the Second Guide Star Catalog (GSC-II; Lasker
et al. 2008), and the Seventh Data Release of the Sloan Digital Sky Survey
(SDSS DR7; Abazajian et al. 2009; Yanny et al. 2009). Details are given in
Appendix C. In the case of a few of the brighter stars ($V$ $<$ 12.3) we have
chosen the proper motions given in the NOMAD Catalog (Zacharias et al.
2004). In the case of th BHB star P 30-38, we chose the proper motion
given by the SDSS DR7 because the
GSC-II---SDSS proper motion has unusually large errors. The SDSS DR7
proper motions have also been used for the stars (mostly BHB stars)
for which GSC-II---SDSS proper motions were not available.
We have only used stars that have radial velocities
to compute U, V and W, and since only 4 of the BHB stars that
only have SDSS DR7 proper motions also have radial velocities,
possible systematic differences between the SDSS DR7 and the GSC-II---SDSS
proper motions should have little effect on our overall results\footnote{ In
our discussion of halo stars at the North Galactic Pole (Kinman et al. 2007b),
the BHB and RR Lyrae stars were both closely grouped near the NGP and it
seemed reasonable to adopt a zero radial velocity for stars whose radial
velocity was not known in computing their galactic velocity V. This assumption
has not been made for the more widely spread stars in the Anticentre.}.
A comparison of the SDSS DR7 and the
GSC-II---SDSS proper motions (Appendix C) shows good agreement at the
1 mas y$^{-1}$ level; this corresponds to a tangential velocity of 40
km s$^{-1}$ at a distance of 8.5 kpc.
\begin{figure*}
\includegraphics[width=18.0cm, bb=30 270 545 517, clip=true]{f3.eps}
\caption{In (a) the ordinate is $\langle$V$_{lsr}$$\rangle$ in km s$^{-1}$.
In (b) the ordinate is $\langle$U$_{lsr}$$\rangle$ in km s$^{-1}$.
In (c) the ordinate is $\langle$W$_{lsr}$$\rangle$ in km s$^{-1}$. The
abscissa is the Galactocentric distance in kpc. NGP BHB stars (blue open circles
); NGP RR Lyrae stars (red open circles); Anticentre BHB stars (blue full circles);
Anticentre RR Lyrae stars (red full circles); Local BHB stars (blue full triangles);
Local RR Lyrae (red full triangles). It is seen that
$\langle$V$_{lsr}$$\rangle$ becomes more retrograde while
$\langle$U$_{lsr}$$\rangle$ and $\langle$W$_{lsr}$$\rangle$ remain unchanged
with increasing Galactocentric distance.
}
\label{f:3}
\end{figure*}
\begin{center}
\begin{table*}
\caption{Space Motions U, V \& W with respect to LSR and their dispersions
$\sigma_{u}$,$\sigma_{v}$ \& $\sigma_{w}$ in km s$^{-1}$. \label{t:uvw}}
\begin{tabular}{@{}lccccccccccc@{}}
\noalign{\smallskip} \hline
Sample& N& U & $\sigma_{u}$ & V & $\sigma_{v}$ & W& $\sigma_{w}$ &$\langle$Z$
\rangle$ & $\langle$ D $\rangle$ & $\langle$ R$_{gal}$ $\rangle$& Notes \\
& &km s$^{-1}$&km s$^{-1}$&km s$^{-1}$ & km s$^{-1}$ & km s$^{-1}$ & km s$^{-1}$ & kpc & kpc & kpc & \\
\hline \hline
bhb & 8 &--73$\pm$41 &115$\pm$29 &--192$\pm$45 & 121$\pm$30& --55$\pm$48 &
134$\pm$34 & 3.8 & 6.2 &13.4& 1 \\
BHB (A) & 28& +19$\pm$26 & 131$\pm$18 &--329$\pm$21& 99$\pm$13 & -14$\pm$20 &
99$\pm$13 & 3.7 & 6.1 &13.2 & 2 \\
RR (A ) & 15& +3$\pm$40 & 152$\pm$28 &--228$\pm$26& 95$\pm$17 & +18$\pm$28 &
108$\pm$20 & 2.8 & 4.5 &11.7 & 3 \\
BHB (B) & 3&--177$\pm$69 & 89$\pm$36 &--181$\pm$95&107$\pm$44 & +86$\pm$98 &
118$\pm$48 & 6.0 & 9.8 &16.9 & 4 \\
RR (B) & 16& +31$\pm$29 & 97$\pm$17 &--263$\pm$40&121$\pm$21 & +6$\pm$28 &
78$\pm$14 & 6.8 & 13.4 &20.7 & 5 \\
LOCAL BHB & 27 & +9$\pm$25 &129$\pm$18 &--205$\pm$15 & 79$\pm$ 11 &--7$\pm$20 &
101$\pm$ 14 & ... & ... & 8.0 & 6 \\
LOCAL RR & 32 &--12$\pm$29 &160$\pm$20 &--204$\pm$19 & 107$\pm$13 &+11$\pm$20 & 113$\pm$14 & ... & ... & 8.0 & 7 \\
\hline
\end{tabular}
Notes to Table: \\
\bf{(1)} Stars classified as bhb (possible BHB stars) ~~~~~~
\bf{(2)} Halo BHB stars with D $<$ 8.5 kpc. ~~~~~~
\bf{(3)} Halo RR Lyrae stars with D $<$ 8.5 kpc. ~~~~~~
\bf{(4)} Halo BHB stars with D $>$ 8.5 kpc. ~~~~~~
\bf{(5)} Halo RR Lyrae stars with 8.5 $<$ D $<$ 17 kpc. ~~~~~~
\bf{(6)} Local sample of BHB stars (Kinman et al. (2007b). ~~~~~~~
\bf{(7)} Local sample of Halo RR Lyrae stars within 1 kpc from Maintz \&
de Boer (2005). Velocity dispersions are upper limits. For further
details see text. \\
\end{table*}
\end{center}
\subsection{Discussion of the space motions U, V \& W}
Table \ref{t:uvw} gives the space motions for various subgroups of our program stars;
all velocities are with respect to the LSR. The possible
BHB stars (type bhb) have a V motion and velocity dispersions ($\sigma_{u}$,
$\sigma_{v}$ and $\sigma_{w}$) that are similar to those of the local BHB
stars. This suggests that the majority are BHB stars but, conservatively,
we have not included them in any of the other samples.
We have used the angular momenta (L$_{\perp}$ and (L$_{z}$) to distinguish
between {\it disc} and {\it halo} stars. These quantities are defined in the
Appendix of Kepley et al. (2007) and are given for our program stars in
columns 15 and 16 of Tables \ref{t:bhb2} and \ref{t:rrl2}.
Following the discussion given in our
Appendix D2, we identify the BHB star RR7 064 and the RR Lyrae stars
TW Lyn and P 82 06 as probable {\it disc} stars and have excluded them
from further discussion.
We assume that $\langle$V$\rangle$ = --V$_{LSR}$ = --220 km s$^{-1}$
for zero halo rotation although higher values are possible\footnote{e.g.
$-$236$\pm$15 km s$^{-1}$(Reid \& Brunthaler 2004);
$-$246$\pm$7 km s$^{-1}$ (Brunthaler et al. 2011).}.
We divide our program stars into samples according to distance: (A) those
nearer than 8.5 kpc, and (B) those with distances between 8.5 and 17.0 kpc.
At 17 kpc, an error of 1 mas in the proper motion will give an error of 80
km s$^{-1}$ in the transverse velocity. With our relatively small samples,
the inclusion of more distant stars would not add useful information.
Our results are given in Table \ref{t:uvw}.
The samples that contain an adequate number (N) of stars, namely
BHB(A), RR (A) and RR (B), have mean
U and W velocities that are essentially zero; this suggests that
the systematic errors in our proper motions are not having a significant
effect on the results for these samples.
The velocity dispersions in Table \ref{t:uvw} were corrected following
Jones \& Walker (1988); if the observed dispersion in U is Disp(U), and
$\xi_{i}$ is the error in U of star $i$, then the corrected dispersion
$\sigma_{u}$ is given by:
\begin{eqnarray}
\sigma_u^2 = (Disp(U))^2 - \frac{1}{n} \sum_{i=1}^{n} \xi_{i}^{2} \nonumber
\end{eqnarray}
The corrected dispersions of U, V \& W in our Anticentre samples are also
comparable (within their errors) with those of the local samples.
Fig.s 3(a), 3(b)And 3(c) are plots of the galactic velocity components
$\langle$V$_{lsr}$$\rangle$,
$\langle$U$_{lsr}$$\rangle$ and
$\langle$W$_{lsr}$$\rangle$ respectively against
galactocentric distance (R$_{gal}$) for both our Anticentre stars and
those at the North Galactic Pole (Kinman et al. 2007b). Although there is
considerable scatter between the different samples, there is clearly a trend
in $\langle$V$_{lsr}$$\rangle$ from zero
galactic rotation in the solar neighbourhood to a strong retrograde
rotation for R$_{gal}$ greater than 12.5 kpc. On the other hand,
both $\langle$U$_{lsr}$$\rangle$ and
$\langle$W$_{lsr}$$\rangle$ are essentially zero at all Galactocentric
distances. This supports our conclusion that the trend of
$\langle$V$_{lsr}$$\rangle$ with galactocentric distance
is real and not produced by systematic errors in the proper motions.
If the outer halo has a significantly retrograde rotation, as originally found by
Carollo et. al (2007, 2010), and confirmed by Beers et al. (2012), this suggests that
the outer halo dominates beyond R$_{gal}$=12.5~kpc.
\begin{figure*}
\includegraphics[width=15.0cm]{f4.eps}
\caption{A plot of L$_{\perp}$ against L$_{z}$ for (a) stars in the Anticentre,
and (b) stars at the North Galactic Pole. The black dotted curve is the outer
contour of the majority of stars studied by Morrison et al. (2009); the black
full curve is the flattened distribution that they discovered. The black
and green rectangles are the locations of the groups discovered by Helmi et
al. (1999) and Kepley et al. (2007) respectively. The magenta box shows the
location of the Thick disc. BHB and RR Lyrae stars are shown by blue and red
filled circles respectively. Selected outliers from Kepley et al. are shown
by green triangles and subdwarf outliers from Smith et al. (2009) by yellow
open circles.
}
\label{f:4}
\end{figure*}
\begin{figure*}
\includegraphics[width=15.0cm]{f5.eps}
\caption{A plot of L$_{\perp}$ against L$_{z}$ for (a) stars within 1 kpc and
(b) those with distances from between one and two kpc. The red filled circles
are RR Lyrae stars taken from the catalogue of Maintz \& de Boer (2005). Local
BHB stars are shown by blue open circles and the outliers from Kepley et al.
(2007) are shown by green filled triangles. The black contours and black, green and
magenta boxes are described in Fig. \ref{f:4}.
}
\label{f:5}
\end{figure*}
\begin{figure*}
\includegraphics[width=13.0cm, bb=95 270 490 500, clip=true]{f6.eps}
\caption{A plot of L$_{\perp}$ against L$_{z}$ for globular clusters that are
within 10 kpc. The two outlying clusters NGC 3201 and NGC 6205 (M13) are
indicated by their NGC numbers. Globular clusters with [Fe/H] $>$ -1.0 are
shown by black filled triangles and those with [Fe/H] $<$ -1.0 by black open
circles. The black curves and black, green and
magenta boxes are described in Fig. \ref{f:4}.
}
\label{f:6}
\end{figure*}
\begin{figure*}
\includegraphics[width=15.0cm, bb=50 260 545 490, clip=true]{f7.eps}
\caption{ Plots showing the correlation between (a) the galactic rotation V
and L$_{z}$, and (b) the maximum height of the orbit above the plane
(z$_{max}$) and L$_{\perp}$.
The V and z$_{max}$ are from Maintz \& de Boer (2005)
for 56 halo RR Lyrae stars.
}
\label{f:7}
\end{figure*}
\section{Structure in the motions of our halo stars.}
Plots of the angular momenta L$_{\perp}$ and L$_{z}$ can be used to
demonstrate kinematic structure among halo stars (e.g. Helmi et al. 1999).
We give a L$_{\perp}$ {\it vs.} L$_{z}$ plot for our Anticentre BHB and
RR Lyrae stars in Fig \ref{f:4}(a) and in Fig \ref{f:4}(b) for our North
Galactic Pole sample of these stars (Kinman et al. 2007b).
Similar plots are shown in Fig \ref{f:5} for local RR Lyrae and BHB stars
and in Fig \ref{f:6} for the globular clusters within 10 kpc.
Definitions of L$_{\perp}$ and L$_{z}$ are given by Kepley et al. (2007).
We calculated these quantities and their errors with a program
that was kindly made available by Heather Morrison and modified for our
use by Carla Cacciari. The values of L$_{\perp}$ and L$_{z}$ for the
North Galactic Pole RR Lyrae stars and BHB stars, the local BHB stars and the
globular clusters within 10 kpc are tabulated in Appendix E, where we also
compare our L$_{\perp}$ and L$_{z}$ with those calculated by Re Fiorentin et al.
(2005) and Morrison et al. (2009) for a small sample of halo
stars\footnote{The data for the local RR Lyraes in
Fig. \ref{f:5} were either taken from Morrison et al. (2009) or calculated
from the data given by Maintz \& de Boer (2005); in this latter case no errors
are given since Maintz \& de Boer do not give errors for their data.
The L$_{\perp}$ and L$_{z}$ for the globular clusters were calculated
from data given in Table 3 of Vande Putte and Cropper (2009).}.
L$_{z}$ correlates with galactic rotation: in the left-handed system of
coordinates, objects with positive L$_{z}$ are prograde (the Sun has
L$_{z}$$\sim$1760 kpc km s$^{-1}$) and those with negative L$_{z}$ are retrograde.
L$_{\perp}$ correlates with the maximum height of the orbit above the plane
(Fig. \ref{f:7}).
Morrison et al. (2009) investigated the L$_{\perp}$ {\it vs.} L$_{z}$ plot for
246 local metal-poor stars. The majority ($\sim$90\%) of their sample,
which we will call the {\it main concentration}, are in the location
bounded by the black dotted line in Figs. \ref{f:4}, \ref{f:5} \& \ref{f:6}.
This is taken from
the outer contour of their Fig. 3. They also discovered a flattened component
whose location is shown by the full black contour in our Figs. \ref{f:4},
\ref{f:5} \& \ref{f:6}. The remaining 10\% of their sample lie outside the
black dotted contour and, following Kepley et al. (2007) and Smith et al.
(2009), we call them {\it outliers}.
About a third of these {\it outliers} in the Morrison et al.
(2009) sample belong to the prograde group (H99) discovered by Helmi et al.
(1999) and further investigated by Re Fiorentin et al. (2005); its location
of the majority of stars in this group is
shown by the black rectangle in Figs \ref{f:4}, \ref{f:5} \& \ref{f:6}.
The green rectangle shows
the location of another group suggested by Kepley et al. (2007). The
magenta box in Figs. \ref{f:4}, \ref{f:5} \& \ref{f:6} shows the location
of stars in the {\it main
concentration} that have a high probability belonging to the Thick Disc;
this is discussed in Appendix D.
The review by Klement (2010) lists sixteen halo ``streams" that have been
identifed among stars in the solar neighbourhood.
All except the H99 and Kapteyn Group lie within the
{\it main concentration} in the L$_{\perp}$ {\it vs.} L$_{z}$ plot. An
example of structure within the {\it main concentration} is shown by the RR
Lyrae stars at distances between 1 and 2 kpc (Fig \ref{f:5}b) which are less evenly
distributed than those at distances less than 1 kpc (Fig. \ref{f:5}a). In general, the
identification of structure in this {\it main concentration} is only possible
for stars with relatively large proper motions and well determined distances.
In discussing our program stars, we shall therefore largely confine ourselves to
discussing the {\it outliers} and the ratio of the number of
{\it outliers} to the the number in the {\it main concentration}.
\begin{figure*}
\includegraphics[width=15.0cm, bb=30 214 550 600, clip=true]{f8.eps}
\caption{ (a) The percentage of outliers in each sample of BHB stars (blue filled
circles) and RR Lyrae stars (red filled circles) as a function of
Galactocentric distance in kpc. The samples are described in Table \ref{t:out-mc}.
(b) The angular momentum L$_{\perp}$ (in kpc km s$^{-1}$) for the BHB stars
(blue filled circles) and RR Lyrae stars (red filled circles) for the NGP and
Anticentre samples within 8.5 kpc as a function of Galactocentric distance.
{\it outliers} have a more spherical distribution and more retrograde orbits
than those in the {\it main concentration} and they consitute a larger fraction
of the halo with increasing galactocentric distance.
}
\label{f:8}
\end{figure*}
\begin{center}
\begin{table*}
\caption{Numbers of stars that are $outliers$ and in the $Main~Concentration$
in the Anticentre and North Galactic Pole Fields. \label{t:out-mc}
}
\begin{tabular}{@{}lccccc@{}}
\noalign{\smallskip} \hline
Field$^{\dagger}$ & No. in & No. of & Per cent. &$\langle$R$_{gal}$$\rangle$ & $\langle$
R$_{gal}$$\rangle$ \\
& Main & Outliers& of & $M. C.$ & $outliers$ \\
& Conc.& & Outliers &(kpc) & (kpc) \\
\hline \hline
ANTICENTRE BHB & 15 & 14&48\%&12.5$\pm$0.4 & 14.1$\pm$0.4 \\
ANTICENTRE RR & 9 & 6&40\%&11.2$\pm$0.6 & 12.5$\pm$0.7 \\
ANTICENTRE BHB + RR & 24 & 20&45\%&12.0$\pm$0.4 & 13.6$\pm$0.5 \\
NGP BHB & 51 & 9&15\%& 9.7$\pm$0.2 & 11.4$\pm$0.4 \\
NGP RR & 22 & 9&29\%&10.2$\pm$0.3 & 11.2$\pm$0.6 \\
NGP BHB + RR & 73 & 18&20\%&9.8$\pm$0.2 & 11.3$\pm$0.3 \\
\hline
\end{tabular}
Notes to Table: \\
$^{\dagger}$~~~ The Anticentre fields are those described in this paper and the
Fields at the NGP are those described in Kinman et al. (2007b). \\
\end{table*}
\end{center}
\begin{center}
\begin{table*}
\caption{Candidates for membership of the K07 group. \label{t:k07}}
\begin{tabular}{@{}lccccccc@{}}
\noalign{\smallskip} \hline
Star& Type$^{\dagger}$& L$_{\perp}$ & L$_{z}$ &$\langle$$V$$\rangle$ & D & [Fe/H] & Period \\
& &kpc km s$^{-1}$&kpc km s$^{-1}$ & mag. & kpc & & days \\
\hline \hline
HD 214925 & RG$^{a}$ & 1322$\pm$119 &--2177$\pm$205& 9.30 & 2.15 &--2.15 & ... \\
AT VIR & RR$^{a}$ & 1576$\pm$93 &--1712$\pm$149& 11.34& 1.30 &--1.60 & 0.5257 \\
RV CAP & RR$^{a}$ & 1464$\pm$146 &--2090$\pm$239& 11.04& 1.06 &--1.72 & 0.4477 \\
& & & & & & & \\
RR7-066 & BHB$^{b}$ & 1960$\pm$1028 &--1533$\pm$645& 15.31& 8.47 & ... & ... \\
CHSS 608 & BHB$^{b}$ & 1742$\pm$452 &--2152$\pm$657& 14.81& 6.76 & ... & ... \\
& & & & & & & \\
AF-115 & BHB$^{c}$ & 1571$\pm$457 &--1484$\pm$494 & 15.42& 8.13 & ... & ... \\
SA57-032 & BHB$^{c}$ & 1621$\pm$604 &--1514$\pm$670 & 15.13& 7.81 & ... & ... \\
AF-041 & BHB$^{c}$ & 1668$\pm$511 &--1928$\pm$635 & 15.02& 7.46 & ... & ... \\
AF-053 & BHB$^{c}$ & 2091$\pm$768 &--1609$\pm$828 & 15.19& 6.94 & ... & ... \\
SA57-001 & BHB$^{c}$ & 1430$\pm$342 &--1742$\pm$503 & 14.43& 5.68 & ... & ... \\
AF-108 & BHB$^{c}$ & 1279$\pm$224 &--2185$\pm$370 & 13.86& 3.82 & ... & ... \\
& & & & & & & \\
IP COM & RR$^{c}$ & 1878$\pm$780 &--1860$\pm$780 & 14.85& 7.25 &--1.48 & 0.6406 \\
EO COM & RR$^{c}$ & 1743$\pm$483 &--1889$\pm$528 & 14.74& 6.94 &--1.67 & 0.6320 \\
MQ COM & RR$^{c}$ & 1572$\pm$393 &--1895$\pm$501 & 14.23& 5.40 & ... & 0.6224 \\
IS COM & RR$^{c}$ & 1472$\pm$177 &--1642$\pm$338 & 13.80& 4.44 & ... & 0.3146 \\
\hline
\end{tabular}
Notes to Table: \\
$^{\dagger}$~~~ RG = red giant; BHB = blue horizontal branch star; RR = RR
Lyrae star. The superscripts a, b \& c indicate that the star belongs to
the solar neighbourhood, the Anticentre fields of the present paper or
the NGP fields of Kinman et al. (2007b) respectively. \\
\end{table*}
\end{center}
\subsection{Ratio of the number of $outliers$ to
the number in the {\it main concentration}.}
The ratio of the number of $outliers$ to the number in the {\it main
concentration} is a simple measure of the spread of halo stars in the
L$_{\perp}$ $vs$ L$_{z}$ plot. Our sample of
globular clusters within 10 kpc contains 5 (with [Fe/H] $>$ --1.0)
that belong to the disc or bulge. Among the remainder, 24 belong to the main
concentration and 2 (or 8\%) are {\it outliers}. The two {\it outliers}
are NGC 3201 and NGC 6205 (M13). Although these two clusters are listed as
``young" by Mar\'{i}n-Franch et al. (2009), Dotter et al. (2010)
give ages of 12.00$\pm$0.75 and 13.0$\pm$0.50 Gyr for NGC 3201 and
NGC 6205 respectively, which does not support this description.
NGC 6205 (together with NGC 5466, NGC 6934 and NGC 7089) is one of a group
of four globular clusters with similar L$_{\perp}$ and L$_{z}$ that are
discussed by Smith et al. (2009) in connection with an overdensity in the
subdwarfs that they studied. Smith et al. list the properties of 12
{\it outlier} subdwarfs that lie at heliocentric distances up to 5 kpc.
They are shown by yellow open circles in Figs. \ref{f:4}(a) and (b).
They show some tendency to occur in groups among themselves in their
L$_{\perp}$ {\it vs} L$_{z}$ plot
(as discussed by Smith et al.) but their locations in our plot
(Figs. \ref{f:4}(a) and (b)) show
little in common with those of our RR Lyrae and BHB {\it outliers}.
The local BHB stars within 1 kpc all belong to the {\it main concentration}
and have no {\it outliers}. The RR Lyrae stars within 1 kpc have 4
{\it outliers} that belong to the H99 group (RZ CEP, XZ CYG, CS ERI and TT
LYN); MT TEL is a possible retrograde {\it outlier} that lies just outside
the {\it main concentration}. The RR Lyrae stars at distances between 1 and 2
kpc have 2 {\it outliers} that belong to the H99 group (TT CNC and AR SER),
2 that belong to the Kepley retrograde group (AT VIR and RV CAP), and one
prograde {\it outlier} (U CAE) besides a number that are on the edge of the
{\it main concentration}. Kepley et (2007) found that XZ CYG belongs to the
H99 group and CS ERI is also likely to be a member of this group.
The H99 and K07 {\it outlier} groups are discussed further in Appendix F.
Of the 188 RR Lyrae stars within 2 kpc
for which we have data, 41 are likely to belong to the thick disc.
Of the remaining 147 halo stars, there are 5 in Fig. 5a and 10 in Fig.
5b that formally lie outside the {\it main concentration} and so would
formally be considered {\it outliers}. Five of those in Fig. 5b,
however, lie so close the boundary of the {\it main concentration} that
(with reasonable assumptions as to their error bars) it seems likely
that most belong to the {\it main concentration}. The 10 certain
{\it outliers} comprise 7$\pm$2\% of the total. If we include the 5 that
lie close to the {\it main concentration} boundary, there are 15
{\it outliers} or 10$\pm$3\% of the total. These percentages are
comparable with those found (10\%) by both Helmi et al. (1999) and
Morrison et al. (2009) among their local samples of metal-poor halo stars.
Table \ref{t:out-mc} gives the number of stars in the {\it main concentration} and the
number of {\it outliers} for the BHB and RR Lyrae stars within 8.5 kpc in
both the Anticenter and NGP (Kinman et al. 2007) fields. The percentage of
{\it outliers} and the L$_{\perp}$ of the stars in these
fields is shown plotted against galactocentric distance in Figs. \ref{f:8}(a)
and \ref{f:8}(b) respectively. It can be seen
from Fig. \ref{f:4} and Fig \ref{f:8}(b) that the majority of {\it outliers} have
greater L$_{\perp}$ and more negative L$_{z}$ than the stars in the {\it main
concentration}. This shows, according to the correlations shown in Fig. 7,
that the orbits of the {\it outliers} tend to be more retrograde and reach
larger $\mid$z$_{max}$$\mid$ than the stars in the {\it main concentration}.
The increase in the percentage of {\it outliers} with galactocentric
distance shown in Fig. \ref{f:8}(a) therefore implies that
{\it as the galactocentric distance increases, the halo has
an increasing contribution from stars that have more retrograde orbits and
a more spherical distribution than the stars in the {\it main concentration}
that predominate in the solar neighbourhood.} This result is in general
agreement with the observational results of Carollo et al. (2007,2010) and Beers et
al. (2012) and the simulations of Oser et al. (2010), Font et al. (2011) and
McCarthy et al. (2012). Our observational support for the duality of the halo is
important because (as the simulations have shown) dual halos are a general property
of the stellar spheroids of disk galaxies whose masses are comparable with that
of the Milky Way. We are grateful to the referee for asking us to emphasize
this point.
We note that a simulation of a ``smooth halo" with a Gaussian distribution of
velocities (e.g. the right-handed L$_{\perp}$ $vs.$ L$_{z}$ plot of Fig. 5 in
Smith et al. 2009) gives a {\it main concentration} that is similar in shape
but smoother than that shown by the observations. In this connection we note
that Hattori \& Yoshii (2011) conclude that violent relaxation has been effective
for stars within a scale radius of 10 kpc from the Galactic centre.
{\it We suggest that the stars of the {\it main concentration} are those
where this relaxation has been most effective}.
\section{Summary and Conclusions}
Fifty one BHB stars and 12 possible BHB stars are identified in the
Anticentre. Our selection criteria for these stars give results
that agree with those used by Smith et al. (2010) and
Ruhland et al. (2011). Fifty eight RR Lyrae stars are identified in the
Anticentre; 7 of these are new and their light curves are given in
Appendix B. Photometric data for the BHB and RR Lyrae stars are given in
Tables \ref{t:bhb} and \ref{t:rrl} respectively. Five methods are used to get distances for the
BHB stars and three methods for the RR Lyrae stars; these are compared and
combined to give distances on a uniform scale. Absolute proper motions
(largely derived from the GSCII and SDSS DR7 databases) are given for
all these stars and also radial velocities for 31 of the BHB and 37 of
the RR Lyrae stars (Tables \ref{t:bhb2} and \ref{t:rrl2}).
Our conclusions are itemized below:
\begin{enumerate}
\item
All but 4 of the 58 RR Lyrae stars in the Anticentre fields are of
Oosterhoff type I; this agrees with the Oo II stars being more centrally
concentrated in the Galaxy than those of Oo type I (Miceli et al. 2008).
Oo I globular clusters tend to have retrograde orbits
(Lee \& Carney, 1999; van den Bergh, 1993); the field RR Lyrae stars in the
Anticentre tend to have retrograde orbits.
\item
We combined the kinematic data of our Anticentre stars with
those of the stars in the North Galactic Pole fields (Kinman et al. 2007b).
In the combined data, the Galactic V motion (Fig. \ref{f:3}) is significantly
retrograde for {\it both} BHB and RR Lyrae stars with R$_{gal}$ $>$ 10 kpc.
This agrees with the findings of Carollo et al. (2007), Carollo et al. (2010)
and Beers et al. (2012) that the {\it outer halo}
shows retrograde rotation compared with the rotation of the stars in the
solar neighbourhood where the {\it inner halo} predominates. The lack of any
similar trend in the Galactic U motion makes it unlikely that the trend in
the V motion is caused by a systematic error in the proper motions.
\item
Angular momenta plots (L$_{\perp}$ $vs.$ L$_{z}$) for the BHB and RR Lyrae stars
in the Anticentre fields and the North Galactic Pole fields are compared with
similar plots for these stars in the solar neighbourhood and for the globular
clusters nearer than 10 kpc. We suggest that halo stars belong to either of
two groups --- either the
{\it main concentration} or the {\it outliers} --- according to whether they lie
inside or outside a contour in this plot which encloses the majority of
metal-poor stars in the solar neighbourhood (as defined by Morrison et al.
2009). We suggest that the stars in the {\it main concentration} are those for
which violent relaxation has been most effective (Hattori \& Yoshii, 2011).
The ratio of {\it outliers} to {\it main concentration} stars increases with
galactocentric distance (Fig. \ref{f:8}). The {\it outliers} primarily have retrograde
orbits. Since L$_{\perp}$ correlates with z$_{max}$ (the orbit's maximum height
above the galactic plane), this also implies that the halo becomes more
spherical with increasing galactocentric distance
(c.f. Schmidt, 1956; Kinman et al., 1966, Miceli et al., 2008 Table 2).
It also agrees with the simulations (McCarthy et al., 2012) that predict
that the inner halo should be more flattened than the outer halo.
\item
A review of the RR Lyrae stars in the H99 group of {\it outliers} (Helmi
et al. 1999) shows that there are six RR Lyrae stars that are likely
members (all probably of Oo type I) and that their mean [Fe/H] is --1.59.
Their mean {\it rms} scatter in [Fe/H] is 0.16 which is comparable with the
likely errors in these metallicities. These RR Lyrae stars therefore form a
more homogeneous set than the later-type stars in H99 (Roederer et al. 2010)
and they could have originated from a single globular cluster.
Another grouping with similar L$_{\perp}$ and L$_{z}$) (which we
call K07) contains 15 BHB and RR Lyrae stars at distances in the range 1.1
to 8.5 kpc. K07 contains two pairs of RR Lyrae stars (AT VIR \& RV CAP and
IP COM \& EO COM); the stars in each pair have similar properties. Better data
are needed to verify membership of the other stars in K07.
\end{enumerate}
\section*{Acknowledgments}
We thank D.R. Soderblom for kindly making available the
program to calculate the UVW space motions and Heather Morrison for allowing us
to use her program for computing L$_{\perp}$ and L$_{z}$ and the referee
for comments which helped improve the paper.
This research has made use of 2MASS data provided by the NASA/IPAC
Infrared Science Archive, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology, under contract with the National
Aeronautics and Space Administration.\\
The GSCII is a joint project of the Space Telescope Science
Institute (STScI) and the INAF-Osservatorio Astronomico di Torino
(INAF-OATo). \\
This work is partly based on observations made with the Italian Telescopio
Nazionale Galileo (TNG) operated on the island of La Palma by the Fundacion
Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de
Canarias.\\
This work has been partly supported by the MIUR (Mi\-ni\-ste\-ro
dell'Istruzione, dell'Universit\`a e della Ricerca) under
PRIN-2001-1028897 and PRIN-2005-1060802.
\begin{center}
\begin{landscape}
\topmargin 45mm
\thispagestyle{empty}
\begin{table*}
\caption{Positions and Photometry for the BHB candidate stars. The equatorial
coordinates are for J2000. The magnitudes and colours $V$,$B$,$(u-B)_{K}$,
$NUV$ and $K$ are defined in the text.
W is a weight and its relation to the Type (column 14) is given in
Appendix A. \label{t:bhb}
}
\label{t:bhb1}
\begin{tabular}{@{}clccccccccccccc@{}}
\hline
No &ID& RA&DEC&l & b&$V$&$B-V$&$(u-B)_{K}$&E$(B-V)$& $NUV$&$K$ & W & Type& Note\\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15) \\
\hline
1&BS 17438-0126&08 04 07.0 &+38 10 30 &182.5&+30.00&13.55$\pm$0.01&
$+$0.215$\pm$0.006&1.985$\pm$0.010&0.046&16.146$\pm$0.004&12.800$\pm$0.024&
$-$6 & A & \\
2&P 54-32.5 &08 05 30.9 &+41 08 02 &179.2&+30.9&15.38$\pm$0.01&
$+$0.137$\pm$0.013&2.026$\pm$0.013&0.064&17.904$\pm$0.033&14.813$\pm$0.094
&$+$4 & bhb & \\
3&AF 186 &08 06 11.1 &+40 15 01 &180.3&+30.8&15.66$\pm$0.01&
$+$0.155$\pm$0.013&2.121$\pm$0.013&0.055&18.238$\pm$0.014&15.281$\pm$0.120
&$+$8 & BHB & \\
4&AF 189 &08 09 21.8 &+38 18 00 &182.6&+31.0&15.20$\pm$0.01&
$+$0.085$\pm$0.016&2.137$\pm$0.043&0.048&17.616$\pm$0.029&15.108$\pm$0.133
&$+$8 & BHB & \\
5&P 54-111 &08 12 06.5 &+38 50 53 &182.1&+31.7&14.58$\pm$0.01&
$+$0.136$\pm$0.009&2.119$\pm$0.013&0.039&17.232$\pm$0.008&14.053$\pm$0.053
&$+$8 & BHB & \\
6&P 54-122 &08 16 00.3 &+40 10 00 &180.7&+32.6&15.17$\pm$0.01&
$+$0.036$\pm$0.008&1.965$\pm$0.013&0.045& ..... &15.152$\pm$0.140
&$+$4 &bhb & \\
7&P 54-120 &08 16 02.9 &+39 25 11 &181.6&+32.5&12.49$\pm$0.01&
$+$0.216$\pm$0.002&1.984$\pm$0.005&0.039&15.400$\pm$0.010&11.940$\pm$0.022
&$-$3 & A & \\
8&P 54-119 &08 17 41.6 &+39 04 29 &182.1&+32.8&14.26$\pm$0.01&
$+$0.214$\pm$0.007&2.087$\pm$0.011&0.038&17.254$\pm$0.023&13.532$\pm$0.039
&$+$6 &BHB & \\
9&BS 17444-0025&08 21 33.5 &+42 31 36 &178.2&+34.0&10.11$\pm$0.01&
$+$0.136$\pm$0.005&2.138$\pm$0.005&0.052& ..... &09.684$\pm$0.013
&$+$4 &bhb & 1\\
10&AF 209 &08 21 48.5 &+42 27 36 &178.2&+34.1&16.41$\pm$0.02&
$+$0.056$\pm$0.022&2.042$\pm$0.032&0.051&18.889$\pm$0.031&$>$15.43
&$+$6 &BHB & \\
11&AF 210 &08 21 59.7 &+42 18 55 &178.4&+34.1&15.42$\pm$0.01&
$+$0.189$\pm$0.012& ..... &0.051&18.090$\pm$0.020&15.339$\pm$0.189
&$+$4 &bhb & \\
12&AF 214 &08 22 51.5 &+36 18 04 &185.6&+33.3&15.66$\pm$0.01&
$+$0.226$\pm$0.014&2.088$\pm$0.020&0.058&18.751$\pm$0.049&15.126$\pm$0.126
&$+$8 &BHB & \\
13&RR7 002 &08 22 00.6 &+37 09 40 &184.5&+33.3&15.13$\pm$0.01&
$+$0.201$\pm$0.010&2.167$\pm$0.020&0.058&18.119$\pm$0.031&14.412$\pm$0.086
&$+$8 &BHB &2 \\
14&P 81-42 &08 23 47.9 &+44 32 44 &175.8&+34.7&14.43$\pm$0.01&
$+$0.050$\pm$0.004&2.039$\pm$0.011&0.045&16.622$\pm$0.010&14.142$\pm$0.059
&$+$8 &BHB & \\
15&RR7 008 &08 24 09.6 &+41 43 47 &179.2&+34.4&15.14$\pm$0.01&
$+$0.118$\pm$0.010&2.120$\pm$0.020&0.043&17.601$\pm$0.010&14.510$\pm$0.081
&$+$8 &BHB &3 \\
16&RR7 015 &08 26 30.4 &+38 10 16 &183.5&+34.3&11.75$\pm$0.01&
$+$0.208$\pm$0.010&2.081$\pm$0.020&0.042& ..... &11.007$\pm$0.020
&$+$10&BHB &4 \\
17&P 81-39 &08 27 16.7 &+45 18 10 &174.9&+35.3&15.69$\pm$0.01&
$+$0.035$\pm$0.008&2.002$\pm$0.011&0.031&17.902$\pm$0.023&15.200$\pm$0.176
&$+$8 &BHB & \\
18&RR7 021 &08 28 02.3 &+40 21 57 &181.0&+34.9&15.33$\pm$0.01&
$+$0.104$\pm$0.010&2.118$\pm$0.020&0.042&18.088$\pm$0.032&14.885$\pm$0.131
&$+$1 &bhb & \\
19&P 81-72 &08 28 14.1 &+44 40 46 &175.7&+35.5&11.79$\pm$0.01&
$+$0.202$\pm$0.001&2.024$\pm$0.003&0.028& ..... &11.281$\pm$0.019
&$+$1 &bhb & 5\\
20&P 81-79 &08 28 19.9 &+45 26 09 &174.8&+35.5&13.37$\pm$0.01&
$+$0.243$\pm$0.002&1.999$\pm$0.005&0.026&16.445$\pm$0.004&12.732$\pm$0.025
&$-$6 & A & \\
21&RR7 023 &08 28 29.6 &+40 27 47 &180.9&+35.0&12.64$\pm$0.01&
$+$0.059$\pm$0.010&2.111$\pm$0.020&0.044&14.926$\pm$0.007&12.347$\pm$0.022
&$+$9 &BHB & 6\\
22&RR7 036 &08 32 26.5 &+39 27 25 &182.2&+35.7&15.25$\pm$0.01&
$+$0.165$\pm$0.010&2.137$\pm$0.020&0.044&18.000$\pm$0.046&14.765$\pm$0.102
&$+$12&BHB & 7\\
23&P 81-101 &08 34 24.5 &+46 00 23 &174.2&+36.6&15.43$\pm$0.01&
$+$0.154$\pm$0.015&2.116$\pm$0.020&0.029&18.146$\pm$0.029&14.620$\pm$0.070
&$+$5 &bhb & \\
24&P 81-121 &08 35 25.7 &+44 00 41 &176.7&+36.7&15.65$\pm$0.01&
$+$0.045$\pm$0.012&1.975$\pm$0.014&0.030&17.676$\pm$0.011&15.316$\pm$0.149
&$+$8 &BHB & \\
25&RR7 043 &08 35 29.7 &+42 01 19 &179.1&+36.5&16.62$\pm$0.01&
$+$0.036$\pm$0.015&1.986$\pm$0.020&0.027&18.721$\pm$0.034&15.853$\pm$0.238
&$+$8 &BHB & 8\\
26&P 28-045 &08 37 42.5 &+36 07 28 &186.5&+36.2&14.315$\pm$0.01&
$+$0.135$\pm$0.008&2.133$\pm$0.021&0.035& ..... &13.809$\pm$0.042
&$+$4 &bhb & \\
27&RR7 053 &08 38 52.8 &+40 54 32 &180.6&+37.0&15.05$\pm$0.01&
$+$0.160$\pm$0.010&2.175$\pm$0.020&0.041&17.782$\pm$0.033&14.4:
&$+$8 &BHB & 9 \\
28&P 81-162 &08 39 57.3 &+46 26 45 &173.7&+37.6&16.05$\pm$0.01&
$+$0.071$\pm$0.021&2.047$\pm$0.015&0.027&18.341$\pm$0.010&15.819$\pm$0.228
&$+$8 &BHB &10 \\
29&RR7 058 &08 40 47.5 &+38 13 49 &184.0&+37.1&15.43$\pm$0.01&
$+$0.064$\pm$0.010&2.020$\pm$0.020&0.041&17.641$\pm$0.026&15.055$\pm$0.124
&$+$8 &BHB &11 \\
30&RR7 060 &08 42 16.8 &+36 45 35 &185.9&+37.2&14.49$\pm$0.01&
$+$0.160$\pm$0.010&2.102$\pm$0.020&0.035&17.164$\pm$0.023&13.836$\pm$0.055
&$+$11&BHB & 12 \\
31&P 82-04 &08 42 41.3 &+42 47 07 &178.3&+37.9&15.84$\pm$0.02&
$+$0.061$\pm$0.014&2.069$\pm$0.029&0.033&18.123$\pm$0.027&15.544$\pm$0.208
&$+$8 &BHB &13 \\
32&RR7 064 &08 43 04.9 &+41 39 05 &179.7&+37.9&11.22$\pm$0.01&
$+$0.115$\pm$0.010&2.100$\pm$0.010&0.032& ..... &10.820$\pm$0.018
&$+$8 &BHB & 14 \\
33&RR7 066 &08 43 40.9 &+39 49 20 &182.1&+37.8&15.31$\pm$0.01&
$+$0.130$\pm$0.010&2.087$\pm$0.020&0.035&17.901$\pm$0.030&14.771$\pm$0.108
&$+$8 & BHB& 15 \\
34&P 81-167 &08 45 20.2 &+46 41 34 &173.4&+38.5&14.44$\pm$0.01&
$+$0.109$\pm$0.008&2.119$\pm$0.014&0.035&16.957$\pm$0.027&13.959$\pm$0.049
&$+$8 &BHB & \\
35&P 11419-01 &08 46 48.8 &+29 49 04 &194.6&+36.8&12.79$\pm$0.01&
$+$0.123$\pm$0.006&2.079$\pm$0.024&0.043&15.376$\pm$0.010&12.348$\pm$0.022
&$+$9 &BHB &16 \\
36&RR7 082 &08 47 09.9 &+42 16 04 &179.0&+38.7&10.69$\pm$0.01&
$+$0.069$\pm$0.002&1.972$\pm$0.003&0.028& ..... &10.475$\pm$0.018
&$-$3 & A &17 \\
37&AF 293 &08 47 41.0 &+36 10 50 &186.8&+38.2&16.24$\pm$0.01&
$+$0.193$\pm$0.013&2.055$\pm$0.036&0.031&18.904$\pm$0.015&15.435$\pm$0.189
&$+$8 &BHB & \\
38&P 11419-04 &08 47 59.8 &+31 31 06 &192.6&+37.4&14.88$\pm$0.01&
$+$0.044$\pm$0.004&2.017$\pm$0.017&0.038&17.028$\pm$0.022&14.739$\pm$0.087
&$+$8 &BHB & \\
39&RR7 084 &08 48 15.0 &+40 28 45 &181.3&+38.8&15.76$\pm$0.01&
$+$0.141$\pm$0.010&2.097$\pm$0.020&0.028&18.476$\pm$0.024&15.148$\pm$0.143
&$+$8 &BHB &18 \\
40&RR7 091 &08 49 10.7 &+39 40 06 &182.4&+38.9&14.16$\pm$0.01&
$+$0.042$\pm$0.010&2.046$\pm$0.020&0.026&16.343$\pm$0.010&13.906$\pm$0.043
&$+$12&BHB &19 \\
\hline
\end{tabular}
\end{table*}
\end{landscape}
\end{center}
\begin{center}
\begin{landscape}
\topmargin 50mm
\addtocounter{table}{-1}
\thispagestyle{empty}
\begin{table*}
\caption{Continued.
}
\label{t:bhb1}
\begin{tabular}{@{}clccccccccccccc@{}}
\hline
No &ID& RA&DEC&l & b&$V$&$B-V$&$(u-B)_{K}$&E$(B-V)$& $NUV$&$K$ & W & Type& Note\\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15) \\
\hline
41&RR7 090 &08 49 23.3 &+40 23 40 &181.5&+39.0&15.65$\pm$0.01&
$+$0.089$\pm$0.010&2.090$\pm$0.020&0.024&18.028$\pm$0.019&15.285$\pm$0.125
&$+$8 &BHB &20 \\
42&P 82-49 &08 53 25.8 &+44 26 21 &176.3&+39.9&15.48$\pm$0.01&
$+$0.108$\pm$0.008&2.099$\pm$0.017&0.029&17.905$\pm$0.032&14.977$\pm$0.103
&$+$8 &BHB & 21 \\
43&BS 16473-0090&08 57 34.5 &+43 28 21 &177.6&+40.7&10.91$\pm$0.01&
$+$0.215$\pm$0.010&1.991$\pm$0.010&0.023& ..... &10.414$\pm$0.020
&$ $0 & A & \\
44&BS 16473-0102&08 58 27.1 &+47 04 08 &172.8&+40.8&13.93$\pm$0.01&
$+$0.103$\pm$0.005&2.092$\pm$0.017&0.021&16.338$\pm$0.014&13.504$\pm$0.033
&$+$8 &BHB & \\
45&BS 17139-0069&09 06 14.4 &+30 58 33 &194.3&+41.2&14.45$\pm$0.01&
$+$0.146$\pm$0.004&2.073$\pm$0.015&0.025&17.052$\pm$0.044&13.879$\pm$0.046
&$+$8 &BHB & \\
46&TON 384 &09 06 56.8 &+30 04 20 &195.5&+41.1&15.25$\pm$0.01&
$+$0.056$\pm$0.011&2.126$\pm$0.061&0.028&17.721$\pm$0.035&14.759$\pm$0.078
&$+$4 &bhb & 22 \\
47&BS 16468-0026&09 07 13.9 &+40 23 48 &181.7&+42.4&14.72$\pm$0.01&
$+$0.077$\pm$0.005&2.089$\pm$0.054&0.020&17.101$\pm$0.018&14.186$\pm$0.045
&$+$8 &BHB & \\
48&AF 379 &09 10 37.1 &+38 55 10 &183.8&+43.0&15.26$\pm$0.02&
$+$0.064$\pm$0.010&2.018$\pm$0.023&0.019&17.547$\pm$0.026&15.149$\pm$0.139
&$+$8 &BHB & \\
49&AF 386 &09 13 30.0 &+36 49 22 &186.7&+43.4&15.01$\pm$0.01&
$+$0.088$\pm$0.005&2.005$\pm$0.032&0.020&17.476$\pm$0.370&14.679$\pm$0.089
&$+$6 &BHB & \\
50&AF 390 &09 15 35.7 &+38 36 12 &184.3&+43.9&15.37$\pm$0.01&
$+$0.042$\pm$0.025&2.091$\pm$0.020&0.019&17.497$\pm$0.022&15.117$\pm$0.149
&$+$8 &BHB & \\
51&BS 16468-0078&09 16 19.4 &+40 16 02 &181.9&+44.1&11.67$\pm$0.01&
$+$0.024$\pm$0.001&1.998$\pm$0.010&0.014& ..... &11.446$\pm$0.017
&$+$7 &BHB &23\\
52&P 30-16 &09 16 53.1 &+35 52 17 &188.1&+44.0&14.40$\pm$0.01&
$+$0.123$\pm$0.006&2.058$\pm$0.030&0.019& ..... &13.879$\pm$0.055
&$+$4 &bhb & \\
53&BS 16468-0080&09 17 13.4 &+41 20 43 &180.5&+44.2&14.09$\pm$0.01&
$+$0.023$\pm$0.009&1.900$\pm$0.017&0.021&15.915$\pm$0.012&14.009$\pm$0.057
&$+$4 &bhb & \\
54&P 30-28 &09 17 44.4 &+33 20 52 &191.6&+43.9&10.28$\pm$0.01&
$+$0.119$\pm$0.004&2.083$\pm$0.003&0.018& ..... &09.757$\pm$0.016
&$+$7 &BHB & 24 \\
55&BS 16468-0090&09 18 25.8 &+39 29 59 &183.0&+44.5&14.10$\pm$0.01&
$+$0.132$\pm$0.011&2.075$\pm$0.004&0.016&16.674$\pm$0.017&13.676$\pm$0.045
&$+$8 &BHB & \\
56&CHSS 608 &09 18 59.0 &+29 40 46 &196.7&+43.6&14.81$\pm$0.01&
$+$0.089$\pm$0.011&2.045$\pm$0.030&0.021&17.276$\pm$0.023&14.353$\pm$0.071
&$+$11&BHB & \\
57&P 11424-28 &09 20 23.3 &+31 17 11 &194.5&+44.2&14.50$\pm$0.01&
$+$0.104$\pm$0.007&2.049$\pm$0.024&0.023&17.052$\pm$0.022&14.008$\pm$0.053
&$+$8 &BHB & \\
58&P 30-38 &09 21 27.6 &+35 24 14 &188.8&+44.9&14.39$\pm$0.01&
$+$0.142$\pm$0.015&2.046$\pm$0.050&0.018&17.029$\pm$0.016&13.929$\pm$0.057
&$+$8 &BHB & \\
59& 57-121 &09 25 48.5 &+39 04 30 &183.7&+45.9&15.25$\pm$0.02&
$+$0.005$\pm$0.020&2.040$\pm$0.033&0.014&17.310$\pm$0.025&15.396$\pm$0.209
&$+$6 &BHB & \\
60& AF 419 &09 25 56.3 &+28 50 07 &198.2&+45.0&15.19$\pm$0.02&
$+$0.113$\pm$0.016&2.100$\pm$0.043&0.019&17.636$\pm$0.029&14.611$\pm$0.093
&$+$9 &BHB & 25 \\
61&P 11424-70 &09 30 02.7 &+31 54 14 &194.1&+46.3&14.57$\pm$0.01&
$+$0.072$\pm$0.019&2.072$\pm$0.032&0.020&16.939$\pm$0.019&14.236$\pm$0.073
&$+$8 &BHB & \\
62&BS 16927-22 &09 32 43.6 &+39 28 31 &183.1&+47.3&11.18$\pm$0.01&
$+$0.103$\pm$0.005&2.084$\pm$0.011&0.017& ..... &10.710$\pm$0.019
&$+$4 &bhb & \\
63&CHSS 663 &09 33 48.8 &+29 07 14 &198.2&+46.7&15.18$\pm$0.01&
$+$0.056$\pm$0.011&2.077$\pm$0.035&0.018&17.282$\pm$0.024&14.959$\pm$0.112
&$+$12&BHB & \\
64&P 11424-82 &09 34 23.5 &+29 48 10 &197.3&+46.9&14.99$\pm$0.01&
$+$0.136$\pm$0.017&2.115$\pm$0.048&0.019&17.554$\pm$0.026&14.379$\pm$0.068
&$+$8 &BHB & \\
65&BS 16940-45 &09 37 07.2 &+36 09 48 &188.0&+48.1&13.55$\pm$0.01&
$+$0.021$\pm$0.011&2.009$\pm$0.008&0.014&15.602$\pm$0.015&13.336$\pm$0.030
&$+$8 &BHB & \\
66&BS 16927-55 &09 40 31.5 &+41 48 32 &179.5&+48.6&14.53$\pm$0.01&
$+$0.003$\pm$0.005&1.993$\pm$0.046&0.012&16.592$\pm$0.023&14.276$\pm$0.064
&$+$6 &BHB & \\
67&BS 16940-0070&09 42 36.7 &+34 38 18 &190.4&+49.1&14.98$\pm$0.01&
$+$0.016$\pm$0.008&2.002$\pm$0.083&0.011&17.201$\pm$0.031&14.815$\pm$0.077
&$+$4 &bhb & \\
68&BS 16940-0072&09 43 10.3 &+33 57 10 &191.4&+49.2&13.99$\pm$0.01&
$+$0.107$\pm$0.008&2.032$\pm$0.004&0.014&16.453$\pm$0.015&13.524$\pm$0.027
&$+$8 &BHB & \\
\hline
\end{tabular}
\noindent Notes to table: \\
\bf{(1)} BD +42 1850;~
\bf{(2)} AF 211;~
\bf{(3)} AF 217;~
\bf{(4)} Str\"{o}mgren $\beta$ = 2.758; CHSS (class 3);~
\bf{(5)} Str\"{o}mgren $\beta$ = 2.855;~ \\
\bf{(6)} Str\"{o}mgren $\beta$ = 2.879; CHSS (class 3);~
\bf{(7)} CHSS (class 4);~
\bf{(8)} AF 241;~
\bf{(9)} AF 256;~
\bf{(10)} US 1430;~ \\
\bf{(11)} AF 262.~
\bf{(12)} CHSS (class 3);~
\bf{(13)} US 1513;~
\bf{(14)} Str\"{o}mgren $\beta$ = 2.856; CHSS (class 3);~
\bf{(15)} AF 271;~ \\
\bf{(16)} Str\"{o}mgren $\beta$ = 2.801;~
\bf{(17)} Str\"{o}mgren $\beta$ = 2.892;~
\bf{(18)} AF 297;~
\bf{(19)} CHSS (class 4);~
\bf{(20)} AF 307;~ \\
\bf{(21)} BD +42 1926; US 1862;~
\bf{(22)} AF 368;~
\bf{(23)} Str\"{o}mgren $\beta$ = 2.858; \\
\bf{(24)} Str\"{o}mgren $\beta$ = 2.827; BD +33 1834;~
\bf{(25)} CHSS 632 (class 1). \\
\noindent References to Notes: \\
AF nnn (Pesch \& Sanduleak, 1989); CHSS (Brown et al., 2003); \\
US nnnn (Usher \& Mitchell, 1982). \\
\end{table*}
\end{landscape}
\end{center}
\begin{center}
\topmargin 45mm
\thispagestyle{empty}
\begin{table*}
\caption{Positions, Photometry and Abundances for the RR Lyrae stars. The
equatorial coordinates are for J2000. The magnitudes and colours $V$
and $K$ are defined in the text. Sources are given in the Notes. \label{t:rrl}
}
\label{t:rr1}
\begin{tabular}{@{}clcccccccccccc@{}}
\hline
No &ID& RA&DEC&l & b&Type&$\log$P& [Fe/H] & $\langle V \rangle$ &$V_{amp}$ &
$\langle K \rangle$ & E$(B-V)$ & Notes \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14) \\
\hline
1 & V385 Aur &07 25 56.0&+38 12 59&180.3&+22.8&$ab$&$-$0.266 &... &
(17.41) &(0.59) & ... &0.053& \\
2 & V386 Aur &07 26 13.2&+40 52 50&177.6&+23.6&$ c$&$-$0.516 &$-$1.75&
(16.75) &(0.50) & ... &0.063 & 5 \\
3 & V387 Aur &07 27 01.0&+36 38 46&182.0&+22.5&$ab$&$-$0.308 &$-$1.32&
(16.92) &(1.05) & ... &0.056 & 5 \\
4 & V389 Aur &07 30 10.8&+38 21 54&180.4&+23.6&$ab$&$-$0.249 &... &
(17.53) &(0.95) & ... &0.057 & \\
5 & VX Lyn &07 31 51.9&+39 07 47&179.7&+24.1&$ab$&$-$0.257 &$-$1.58&
17.01 &(0.82) & ... &0.057 & 4 \\
6 & VY Lyn &07 32 26.0&+38 50.05&180.1&+24.2&$c$&$-$0.451 &$-$1.57&
15.75 &(0.37) & 14.61$\pm$0.12&0.062& 4 \\
7 & VZ Lyn &07 32 40.8&+41 37 38&177.1&+25.0&$c$&$-$0.487 &$-$1.48&
16.20 &(0.42) & 15.43$\pm$0.19&0.054 & 4 \\
8 & WX Lyn &07 35 38.5&+39 15 27&179.8&+24.9&$ab$&$-$0.257 &$-$1.72&
16.84 &(0.69) & ... &0.049 & 4 \\
9 & AS Lyn &07 40 32.9&+41 11 37&178.0&+26.3&$ab$&$-$0.298 &$-$1.2:&
(18.35) &1.05) & ... &0.049 & 1 \\
10 & WZ Lyn &07 40 45.7&+39 18 51&180.1&+25.8&$ab$&$-$0.207 &$-$1.89&
(14.25) &(0.95) & 13.11$\pm$0.03&0.049 & 2 \\
11 & XZ Lyn &07 44 48.4&+40 12 44&179.3&+26.8&$c$&$-$0.549 &... &
(16.32) &(0.50) & ... &0.050 & \\
12 & TW Lyn &07 45 06.3&+43 06 42&176.1&+27.5&$ab$&$-$0.317 &$-$0.43&
11.99 &1.00 & 10.78$\pm$0.02&0.046 & 3,8 \\
13 & YY Lyn &07 45 30.1&+37 22 59&182.4&+26.2&$c$&$-$0.476 &$-$1.87&
14.98 &(0.46) & 14.08$\pm$0.11&0.065 & 4 \\
14 & YZ Lyn &07 45 40.9&+40 22 32&179.2&+27.0&$ab$&$-$0.304 &$-$0.6:&
(17.47) &(0.80) & ... &0.052 & 1 \\
15 & AU Lyn &07 49 35.3&+41 42 57&177.9&+28.0&$ab$&$-$0.197 &$-$1.8:&
(17.82) &(0.76) & ... &0.048 & 1 \\
16 & ZZ Lyn &07 50 21.8&+37 42 00&182.3&+27.3&$ab$&$-$0.313 &$-$1.42&
15.80 &(1.03) & 15.08$\pm$0.14&0.048 & 4 \\
17 & RW Lyn &07 50 39.2&+38 27 15&181.5&+27.5&$ab$&$-$0.302 &$-$1.53&
12.90 &(1.20) & 11.66$\pm$0.02&0.040 & 4,9 \\
18 & AV Lyn &07 54 09.6&+42 49 04&176.9&+29.1&$ab$&$-$0.233 &$-$1.7:&
(16.62) &(0.76) & ... &0.048 & 1 \\
19 & AC Lyn &07 54 42.1&+38 54 20&181.2&+28.4&$ab$&$-$0.256 &$-$1.50&
16.38 &(0.85) & ... &0.047 & 4 \\
20 & AD Lyn &07 56 23.0&+39 22 58&180.8&+28.8&$c$&$-$0.450 &$-$1.46&
15.85 &(0.49) & 15.13$\pm$0.15&0.061 & 4 \\
21 & AW Lyn &07 57 24.5&+43 12 29&176.5&+29.8&$ab$&$-$0.333 &$-$1.6:&
(16.13) &(0.92) & 15.00$\pm$0.11&0.036 & 1 \\
22 & AX Lyn &07 59 46.4&+39 16 30&181.1&+29.4&$ab$&$-$0.331 &... &
(18.54) &(0.76) & ... &0.048 & \\
23 & AY Lyn &08 00 29.9&+40 39 24&179.6&+29.8&$c$&$-$0.503 &... &
(16.88) &(0.47) & ... &0.045 & \\
24 &P 54-13 &08 01 56.2&+41 01 18&179.2&+30.2&$ab$&$-$0.226 &... &
15.20 &0.90 & 13.52$\pm$0.04&0.058 & 6 \\
25 & AZ Lyn &08 03 39.8&+42 30 45&177.6&+30.7&$ab$&$-$0.324 &$-$2.24&
16.47 &(0.84) & ... &0.046 & 5 \\
26 & BB Lyn &08 04 36.2&+42 29 01&177.6&+30.9&$ab$&$-$0.253 &$-$1.36&
(16.86) &(0.92) & ... &0.048 & 5 \\
27 & BC Lyn &08 09 37.4&+42 33 31&177.7&+31.9&$ab$&$-$0.281 &$-$1.6:&
(17.00) &(1.07) & ... &0.048 & 1 \\
28 & AF 194 &08 12 00.6&+40 39 20&180.0&+32.0&$ab$&$-$0.075 &... &
15.84 &0.45 & 14.37$\pm$0.08&0.048 & 7 \\
29 & AF 197 &08 13 46.4&+38 03 02&183.1&+31.8&$c$&$-$0.411 &... &
15.50 &0.40 & 14.35$\pm$0.07&0.038 & 7 \\
30 & DQ Lyn &08 23 41.0&+37 28 11&184.2&+33.6&$c$&$-$0.306 &... &
11.41 &0.37 & 10.44$\pm$0.02&0.044 & 6 \\
31 & RR7 032 &08 30 41.8&+40 24 24&181.0&+35.4&$ab$&$-$0.201 &... &
14.58 &0.65 & 13.22$\pm$0.03&0.047 & 6 \\
32 & RR7 034 &08 31 52.2&+38 32 14&183.3&+35.4&$c$&$-$0.539 &... &
15.32 &0.29 & 14.64$\pm$0.09&0.039 & 6 \\
33 &P 81 129 &08 32 49.6&+43 16 02&177.5&+36.2&$c$&$-$0.510 &... &
14.46 &0.52 & 13.67$\pm$0.04&0.022 & 6 \\
34 &AF Lyn &08 35 57.4&+41 01 11&180.4&+36.5&$ab$&$-$0.237 &$-$1.56 &
16.12 &(0.76) & 14.87$\pm$0.09&0.039 & 4 \\
35 &P 82 06 &08 43 56.7&+43 22 13&177.6&+38.2&$c$&$-$0.548 &... &
14.15 &0.30 & 13.45$\pm$0.03&0.024 & 6 \\
36 & AI Lyn &08 44 02.6&+38 54 48&183.2&+37.8&$ab$&$-$0.250 &... &
(17.10) &(0.92) & ... &0.032 & \\
37 & AK Lyn &08 45 55.1&+39 14 55&182.8&+38.2&$ab$&$-$0.329 &$-$1.56&
16.00 &(1.07) & 14.86$\pm$0.11&0.030 & 4 \\
38 & EN Lyn &08 46 07.0&+38 02 53&184.4&+38.1&$ab$&$-$0.204 &... &
13.53 &0.52 & 12.18$\pm$0.02&0.035 & 6 \\
39 & RR7-086 &08 48 26.2&+36 20 08&186.6&+38.4&$c$&$-$0.451 &... &
16.14 &0.63 & 15.51$\pm$0.15&0.029 & 7 \\
40 & AL Lyn &08 49 13.1&+38 49 31&183.5&+38.8&$ab $&$-$0.293 &$-$1.90 &
16.52 &(0.99) & 15.18$\pm$0.11&0.035 & 4 \\
41 & AM Lyn &08 49 50.2&+36 56 00&185.9&+38.7&$ab$&$-$0.294 &... &
(17.30)&(1.31) & ... & 0.033 & \\
42 &P 82-32 &08 50 39.5&+43 40 03&177.3&+39.4&$ab$&$-$0.304 &... &
15.07&1.16 & 14.24$\pm$0.06 & 0.031 & 6 \\
43 & AF 316 &08 50.46.3&+41 18 54&180.3&+39.3&$c$&$-$0.462 &... &
16.13 &0.46 & 15.09$\pm$0.12 & 0.027 & 7 \\
44 & RR7-101 &08 51 40.2&+40 17 11&181.6&+39.4&$c$&$-$0.482 &... &
16.15 &0.60 & 15.14$\pm$0.11 & 0.022 & 7\\
45 & TT Lyn &09 03 07.8&+44 35 08&176.1&+41.7&$ab$&$-$0.224 &$-$1.35&
09.85& 0.70 & 08.61$\pm$0.02 & 0.018 & 10,11 \\
46 &AF 400 &09 18 17.0&+31 58 49&193.5&+43.9&$c$&$-$0.403 &... &
14.10& 0.40 & 13.37$\pm$0.03 & 0.018 & 7 \\
47 &AF 430 &09 30 23.3&+33 53 11&191.2&+46.6&$c$&$-$0.515 &... &
14.90& 0.40 & 14.21$\pm$0.04 & 0.016 & 7 \\
48&BS 16927-123&09 44 36.4&+41 08 39&180.4&+49.4&$c$&$-$0.445 &... &
13.18 &0.46 & 12.36$\pm$0.02 & 0.017 & 6 \\
49 & X LMi &10 06 06.7&+39 21 28&182.5&+53.7&$ab$&$-$0.165 &$-$1.41&
12.35 &1.02 & 11.06$\pm$0.01 & 0.018 & 12,13 \\
50 &AG UMa &10 48 56.3&+42 40 14&172.9&+60.7&$ab$&$-$0.335 &... &
(15.42) &1.71 & 14.53$\pm$0.10 & 0.012 & 14 \\
\hline
\end{tabular}
\end{table*}
\end{center}
\begin{center}
\topmargin 50mm
\addtocounter{table}{-1}
\thispagestyle{empty}
\begin{table*}
\caption{Continued.
}
\label{t:bhb1}
\begin{tabular}{@{}clcccccccccccc@{}}
\hline
No &ID& RA&DEC&l & b&Type&$\log$P& [Fe/H] & $\langle V \rangle$ & $V_{amp}$ &
$\langle K \rangle$ & E$(B-V)$ & Notes \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14) \\
\hline
51 &BK UMa &10 50 18.9&+42 34 08&172.9&+61.0&$ab$&$-$0.197 &$-$1.29&
12.91& 0.54 & 11.50$\pm$0.02 & 0.012& 12,15 \\
52 &AK UMa &10 53 13.2&+41 19 02&174.9&+61.9&$c$&$-$0.309 &... &
(16.08)& (0.46) & 15.00$\pm$0.12 & 0.012 & 16 \\
53 &AO UMa &11 07 39.8&+40 33 58&174.1&+64.7&$ab$&$-$0.251 &... &
(15.54)& (1.22) & 14.62$\pm$0.10 & 0.015 & \\
54 &BN UMa &11 16 22.9&+41 14 02&170.9&+65.9&$d$&$-$0.398 &... &
13.50 & 0.50 & 12.58$\pm$0.03 & 0.014 & 6,17 \\
55 &CK UMa &12 01 36.4&+31 54 12&186.2&+78.2&$ab$&$-$0.214 &... &
14.08 & 0.53 & 12.72$\pm$0.03 & 0.024 & 6 \\
\hline
\end{tabular}
Notes to table: \\
\bf{(1)} [Fe/H] from Saha \& Oke (1984).~~
\bf{(2)} [Fe/H] from private communication from Suntzeff (1990).~~
\bf{(3)} [Fe/H] from Jurcsik et al. (2006).~~
\bf{(4)} [Fe/H] and $\langle V \rangle$ from Pier, Saha \& Kinman (2003).~~
\bf{(5)} [Fe/H] and $\langle V \rangle$ from Kinman, Saha \& Pier (2004).~~
\bf{(6)} $\langle V \rangle$ from Kinman \& Brown (2010).~~
\bf{(7)} $\langle V \rangle$ from this paper (appendix).~~
\bf{(8)} $\langle V \rangle$ from Schmidt, Chab \& Reiswig (1995).~~
\bf{(9)} $\langle V \rangle$ from Schmidt \& Seth (1996).~~
\bf{(10)} $\langle V \rangle$ from Liu \& Janes (1990).~~
\bf{(11} [Fe/H] from Sodor, Jurcsik \& Szeidl (2009).~~
\bf{(12)} $\langle V \rangle$ from Schmidt (2002).~~
\bf{(13} [Fe/H] from Jurcsik \& Kovacs (1996).~~
\bf{(14)} $\langle V \rangle$ from Kinemuchi et al. (2006).~~
\bf{(15)} [Fe/H] from Kemper (1982).~~
\bf{(16)} Bailey type and period uncertain.~~
\bf{(17)} McClusky (2008) showed that this star is an RRd. The period given
is that of the first overtone. \\
\end{table*}
\end{center}
\begin{center}
\begin{landscape}
\topmargin 45mm
\thispagestyle{empty}
\begin{table*}
\caption{Parallaxes, Proper Motions, Radial Velocities, Galactic Distances and
Galactic Space Velocities for the BHB stars. \label{t:bhb2}
}
\begin{tabular}{@{}clcccccccccccccc@{}}
\hline
No &ID&$\Pi$ &$\mu_{\alpha}$ & $\mu_{\delta}$&S$_{\mu}$&RV&S$_{RV}$&D &Z &R$_{gal}$ &U &V & W &L$_{\perp}$ & L$_{z}$ \\
& & (mas) & (mas y$^{-1}$) & (mas y$^{-1}$)& &(km s$^{-1}$)& &(kpc)&(kpc)&(kpc)&(km s$^{-1}$)&(km s$^{-1}$)&(km s$^{-1}$)&(kpc km s$^{-1})$ &(kpc km s$^{-1}$) \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15)&(16) \\
\hline
2&P 54-32.5 & 0.119$\pm$ 0.002&--1.3$\pm$ 0.6&--0.2$\pm$ 1.6&1&--017.4$\pm$ 4& 1& 8.4& 4.3 &15.8& +002$\pm$015 & +007$\pm$061 &--047$\pm$024 & + 1299$\pm$264 &+3445$\pm$929 \\
3&AF 186 & 0.104$\pm$0.002& 0.0$\pm$3.0&--10.0$\pm$3.0&3& ... &..& 9.6& 4.9 &17.0& ... & ... & ... & ... & ... \\
4&AF 189 & 0.135$\pm$0.004& +4.0$\pm$3.0&--5.0$\pm$3.0&3& ... &..& 7.4& 3.8 &14.8& ... & ... & ... & ... & ... \\
5&P 54-111 & 0.165$\pm$0.003& +4.1$\pm$1.4&--19.6$\pm$2.4&1&--054.5$\pm$ 4& 1& 6.1& 3.2 &13.5& --081$\pm$022&--570$\pm$067 &--016$\pm$036 &+1258$\pm$255&--4615$\pm$884 \\
6&P 54-122 & 0.144$\pm$0.004& +3.9$\pm$0.5&--7.1$\pm$0.8&1&--128.7$\pm$ 4& 1& 6.9& 3.7 &14.3& --168$\pm$009&--244$\pm$027 &+011$\pm$014 &+0808$\pm$225&--0348$\pm$374\\
8&P 54-119 & 0.185$\pm$0.003& +1.7$\pm$1.2&--20.5$\pm$0.9&1&--194.2$\pm$ 4& 1& 5.4& 2.9 &12.9& --166$\pm$017&--513$\pm$025 &--139$\pm$026 &+1554$\pm$297 &--3697$\pm$320\\
9&BS 17444-0025 & 1.348$\pm$0.031& +5.9$\pm$0.6&--27.0$\pm$ 0.6&2& ...&..& 0.7& 0.4 & 8.6& ... & ... & ... & ... & ... \\
10&AF 209 & 0.075$\pm$0.002& +2.5$\pm$3.0&--4.0$\pm$3.0&3& ... & ..&13.3& 7.5 &20.5& ... & ... & ... & ... & ... \\
11&AF 210 & 0.111$\pm$0.002& +9.0$\pm$3.8& 0.8$\pm$1.3&1&--084.9$\pm$ 4& 1& 9.0& 5.0 &16.3& --295$\pm$091& --015$\pm$060 &+277$\pm$135 & +5938$\pm$2401&+3204$\pm$925 \\
12&AF 214 & 0.100$\pm$0.002&--1.0$\pm$2.0&--1.0$\pm$ 2.0&3& ... &..&10.0& 5.5 &17.2& ... & ... & ... & ... & ... \\
13&RR7 002 & 0.128$\pm$0.002&--5.6$\pm$2.2&--8.9$\pm$1.7&1&+249.0$\pm$ 4& 1& 7.8& 4.3 &15.1& +322$\pm$047&--308$\pm$066 &--081$\pm$069 & +2655$\pm$1139&--1091$\pm$957\\
14&P 81-42 & 0.191$\pm$0.003&--3.0$\pm$3.0&--10.0$\pm$3.0&3& ... &..& 5.2& 3.0 &12.7& ... & ... & ... & ... & \\
15&RR7 008 & 0.129$\pm$0.002&--1.2$\pm$0.7&--13.6$\pm$ 1.0&1&--005.8$\pm$ 4& 1& 7.8& 4.4 &15.0& +054$\pm$016&--487$\pm$038 &--085$\pm$023 &+1905$\pm$329&--3838$\pm$548\\
16&RR7 015 & 0.592$\pm$0.009&--20.2$\pm$1.0&--35.0$\pm$0.7&2&+238.4$\pm$ 4& 1& 1.7& 1.0 & 9.4& +287$\pm$006&--263$\pm$006 &--028$\pm$007 &+0544$\pm$075 &--0380$\pm$066\\
17&P 81-39 & 0.106$\pm$0.002&--2.0$\pm$3.0&--2.0$\pm$3.0&3& ...& ..& 9.4& 5.5 &16.6& ... & ... & ... & ... & ... \\
18&RR7 021 & 0.120$\pm$0.002&--1.9$\pm$0.9&--8.3$\pm$1.2&1&+092.9$\pm$ 4& 1& 8.3& 4.8 &15.6& +129$\pm$019 &--310$\pm$046 &--037$\pm$027 & +1287$\pm$454&--1318$\pm$686\\
19&P 81-72 & 0.575$\pm$0.016&--2.5$\pm$0.9&--2.8$\pm$0.6&2& ...& ..& 1.7& 1.0 & 9.5& ... & ... & ... & ... & ... \\
21&RR7 023 & 0.436$\pm$0.007& +7.7$\pm$1.6&--22.6$\pm$0.7&2&--059.3$\pm$ 4&1& 2.3& 1.3 &10.0& --091$\pm$010 &--248$\pm$009 & +014$\pm$014 & +0267$\pm$140&--0281$\pm$087\\
22&RR7 036 & 0.121$\pm$0.002&--1.5$\pm$3.0&--7.5$\pm$3.0&3&+160.0$\pm$ 40& 3& 8.3& 4.8 &15.5& +171$\pm$074 &--284$\pm$120 & +017$\pm$099 & +1650$\pm$1048&--894$\pm$1765\\
23&P 81-101 & 0.107$\pm$0.002& +1.0$\pm$3.0&--2.0$\pm$3.0&3& ...& ..& 9.3& 5.6 &16.5& ... & ... & ... & ... & ... \\
24&P 81-121 & 0.100$\pm$0.006& 0.0$\pm$3.0&--8.0$\pm$3.0&3& ...& ..&10.0& 6.0 &17.1& ... & ... & ... & ... & ... \\
25&RR7 043 & 0.067$\pm$0.002& +1.0$\pm$3.0&--3.0$\pm$3.0&3& ...& ..&14.9& 8.9 &21.9& ... & ... & ... & ... & ... \\
26&P 28-045 & 0.186$\pm$0.003&--5.0$\pm$1.6&--12.2$\pm$0.8&1&--356.6$\pm$ 4& 1& 5.4& 3.2 &12.7& --225$\pm$025 &--257$\pm$021 & --345$\pm$034 & +3528$\pm$498&--0568$\pm$265\\
27&RR7 053 & 0.132$\pm$0.002&--2.9$\pm$2.0&--16.0$\pm$2.5&1&--241.0$\pm$ 5& 2& 7.6& 4.6 &14.8& --111$\pm$43 &--555$\pm$090 & --266$\pm$057 & +3626$\pm$916&--4700$\pm$1258\\
28&P 81-162 & 0.086$\pm$0.001& +1.0$\pm$3.0&--1.0$\pm$3.0&3& ... & ..&11.6& 7.1 &18.6& ... & ... & ... & ... & ... \\
29&RR7 058 & 0.120$\pm$0.002& +1.0$\pm$0.9&--7.1$\pm$1.5&1&+030.0$\pm$ 5& 2& 8.3& 5.0 &15.5&--006$\pm$023 &--279$\pm$060 & +027$\pm$029 & +0709$\pm$374&--0866$\pm$871\\
30&RR7 060 & 0.169$\pm$0.003& +0.3$\pm$1.5&--6.1$\pm$1.1&1&+063.2$\pm$ 4& 1& 5.9& 3.6 &13.2& +033$\pm$025 &--168$\pm$031 & +033$\pm$032 & +0534$\pm$314&+0668$\pm$391\\
31&P 82-04 & 0.097$\pm$0.001& +2.0$\pm$3.0&--3.0$\pm$3.0&3& ... &..&10.3& 6.3 &17.3& ... & ... & ... &... & ... \\
32&RR7 064 & 0.784$\pm$0.016&--5.1$\pm$0.7&--5.4$\pm$0.7&2&+034.8$\pm$ 4& 1& 1.3& 0.8 & 9.0& +038$\pm$004 & --024$\pm$004 &+002$\pm$004 &+0159$\pm$007&+1758$\pm$038\\
33&RR7 066 & 0.118$\pm$0.002&--1.9$\pm$2.1&--8.5$\pm$1.1&1&--055.0$\pm$ 5& 2& 8.5& 5.2 &15.6& +003$\pm$054&--325$\pm$044 & --114$\pm$069 & +1960$\pm$1028&--1533$\pm$645\\
34&P 81-167 & 0.178$\pm$0.003&--10.0$\pm$3.0&--2.0$\pm$3.0&3& ...& ..& 5.6& 3.5 &12.9& ... & ... & ... & ... & ... \\
35&P 11419-01 & 0.386$\pm$0.007& +6.1$\pm$2.2&--18.3$\pm$0.9&1&+287.1$\pm$ 4& 1& 2.6& 1.6 &10.1& +141$\pm$017&--277$\pm$012 & +196$\pm$021 & +1734$\pm$238& --0494$\pm$124\\
37&AF 293 & 0.074$\pm$0.001&--5.0$\pm$3.0&--6.0$\pm$3.0&3& ... &..&13.5& 8.4 &20.4& ... & ... & ... & ... & ... \\
38&P 11419-04 & 0.157$\pm$0.003&--0.8$\pm$1.2&--7.1$\pm$0.3&1&+170.7$\pm$ 4& 1& 6.4& 3.9 &13.5& +116$\pm$023&--233$\pm$010 & +059$\pm$028 & +0444$\pm$319&--0040$\pm$139\\
39&RR7 084 & 0.094$\pm$0.001& 0.0$\pm$3.0&--7.0$\pm$3.0&3&--065.0$\pm$40& 3&10.6& 6.7 &17.6&--057$\pm$099&--344$\pm$151 & --050$\pm$120 & +2533$\pm$1425&--2029$\pm$2446 \\
40&RR7 091 & 0.211$\pm$0.003&--0.7$\pm$1.4&--15.3$\pm$.9&1&--041.0$\pm$ 4& 1& 4.7& 3.0 &12.1&--027$\pm$020&--335$\pm$020 & --054$\pm$024 & +0696$\pm$256&--1349$\pm$235\\
\hline
\end{tabular}
\end{table*}
\end{landscape}
\end{center}
\begin{center}
\begin{landscape}
\topmargin 50mm
\addtocounter{table}{-1}
\thispagestyle{empty}
\begin{table*}
\caption{Continued.\label{t:bhb2}
}
\begin{tabular}{@{}clcccccccccccccc@{}}
\hline
No &ID&$\Pi$ &$\mu_{\alpha}$ & $\mu_{\delta}$&S$_{\mu}$&RV&S$_{RV}$&D &Z &R$_{gal}$ &U &V & W &L$_{\perp}$ & L$_{z}$ \\
& & (mas) & (mas y$^{-1}$) & (mas y$^{-1}$)& &(km s$^{-1}$)& &(kpc)&(kpc)&(kpc)&(km s$^{-1}$)&(km s$^{-1}$)&(km s$^{-1}$)&(kpc km s$^{-1}$)&(kpc km s$^{-1}$) \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15)&(16) \\
\hline
41&RR7 090 & 0.101$\pm$ 0.002& +3.0$\pm$ 3.0&--4.0$\pm$ 3.0&3&--112.0$\pm$40& 3& 9.9& 6.2 &16.9&--180$\pm$095&--185$\pm$143 & +032$\pm$111 & +2525$\pm$1525& +0502$\pm$2242 \\
42&P 82-49 & 0.111$\pm$0.002&--2.0$\pm$3.0&--4.0$\pm$ 3.0&3& ...& ..& 9.0 & 5.8 &16.0& ... & ... & ... & ... & ... \\
44&BS 16473-0102 & 0.220$\pm$0.004& +8.0$\pm$3.0&--12.0$\pm$ 3.0&3& ...& ..& 4.5& 3.0 &11.8& ... & ... & ... & ... & ... \\
45&BS 17139-69 & 0.170$\pm$0.003&--3.3$\pm$0.7&--9.2$\pm$ 1.0&1&+105.4$\pm$ 4& 1& 5.9& 3.9 &12.9& +096$\pm$014&--265$\pm$027 & --027$\pm$015 & +0740$\pm$231& --0446$\pm$331\\
46&TON 384 & 0.124$\pm$0.002&--0.7$\pm$1.3&--3.6$\pm$ 1.1&1&--174.7$\pm$ 4& 1& 8.1& 5.3 &14.9& --140$\pm$035&--094$\pm$044 & --146$\pm$039 &+1415$\pm$582&+1510$\pm$610\\
47&BS 16468-0026 & 0.154$\pm$0.003&--1.8$\pm$1.2&--13.4$\pm$ 1.0&1&+211.2$\pm$ 4& 1& 6.5& 4.4 &13.5& +181$\pm$024&--412$\pm$032 & +098$\pm$027 &+1031$\pm$231&--2418$\pm$418 \\
48&AF 379 & 0.123$\pm$0.003& +3.5$\pm$3.0&--9.5$\pm$ 3.0&3& ...& ..& 8.1& 5.5 &15.0& ... & ... & ... & ... & ... \\
49&AF 386 & 0.136$\pm$0.002& +2.2$\pm$0.9&--8.4$\pm$ 1.2&1&+009.6$\pm$ 4& 1& 7.4& 5.1 &14.2&--075$\pm$022&--286$\pm$042 & +053$\pm$023 & +1164$\pm$379&--0925$\pm$557 \\
50&AF 390 & 0.118$\pm$0.002&--2.0$\pm$3.0&--5.0$\pm$ 3.0&3& ...& ..& 8.5& 5.9 &15.3& ... & ... & ... & ... & ... \\
51&BS 16468-0078 & 0.654$\pm$0.010&--19.8$\pm$0.6&--29.1$\pm$ 0.7&2& ...& ..& 1.5& 1.1 & 9.2& ... & ... & ... & ... & ... \\
52&P 30-16 & 0.173$\pm$0.003& +1.0$\pm$3.0&--4.7$\pm$ 3.0&3& ...& ..& 5.8& 4.0 &12.8& ... & ... & ... & ... & ... \\
53&BS 16468-0080 & 0.224$\pm$0.007& +1.4$\pm$1.2&--17.9$\pm$ 1.6&1&--001.7$\pm$ 4& 1& 4.5& 3.1 &11.6&--037$\pm$017&--372$\pm$036 & +030$\pm$018 & +0691$\pm$184&--1700$\pm$405 \\
54&P 30-28 & 1.161$\pm$0.017& +9.5$\pm$0.7&--20.6$\pm$ 0.7&2& ...& ..& 0.9& 0.6 & 8.6& ... & ... & ... & ... & ... \\
55&BS 16468-0090 & 0.200$\pm$0.004&--0.4$\pm$1.6&--8.0$\pm$ 1.6&1&+221.6$\pm$ 4&..& 5.0& 3.5 &12.1& +145$\pm$027&--193$\pm$039 & +155$\pm$027 & +1287$\pm$404& +0345$\pm$446 \\
56&CHSS 608 & 0.148$\pm$0.002&--3.4$\pm$1.2&--12.5$\pm$ 1.6&1&+029.0$\pm$ 4&..& 6.8& 4.7 &13.6& +023$\pm$029&--393$\pm$051 & --104$\pm$029 & +1742$\pm$452&--2152$\pm$657\\
57&P 11424-28 & 0.169$\pm$0.003& +1.3$\pm$1.4&--7.9$\pm$ 1.6&1&+011.6$\pm$ 4& 1& 5.9& 4.1 &12.8&--061$\pm$030&--217$\pm$043 & +017$\pm$030 & +0589$\pm$363&--0030$\pm$527 \\
58&P 30-38 & 0.175$\pm$0.003& +2.0$\pm$3.0&--11.0$\pm$ 3.0&3&--179.3$\pm$ 4& 1& 5.7& 4.0 &12.7& --204$\pm$059&--269$\pm$079 &--096$\pm$059 & +0911$\pm$568&--0716$\pm$946\\
59& 57-121 & 0.125$\pm$0.005&--1.8$\pm$1.2&--8.6$\pm$ 1.4&1&+066.1$\pm$ 5& 2& 8.0& 5.7 &14.7& +066$\pm$032&--329$\pm$056 & +005$\pm$031 & +0912$\pm$391&--1451$\pm$760\\
60& AF 419 & 0.121$\pm$0.003& +3.3$\pm$1.0&--6.2$\pm$ 2.2&1&+072.7$\pm$ 4& 1& 8.3& 5.8 &14.9& --098$\pm$033&--242$\pm$086 & +116$\pm$030 & +2213$\pm$539& --0479$\pm$1186 \\
61&P 11424-70 & 0.168$\pm$0.003& +3.3$\pm$1.2&--13.8$\pm$ 2.9&1&+152.7$\pm$ 4& 1& 6.0& 4.3 &12.8&--035$\pm$028&--397$\pm$080 & +148$\pm$024 & +2057$\pm$372&--2154$\pm$970\\
62&BS 16927-22 & 0.773$\pm$0.012& +6.8$\pm$0.8&--29.8$\pm$ 0.6&2&+082.2$\pm$ 4& 1& 1.3& 1.0 & 8.9& +003$\pm$004&--177$\pm$005 & +098$\pm$004 & +0872$\pm$039&+0378$\pm$041 \\
63&CHSS 663 & 0.130$\pm$0.003&--9.0$\pm$2.0&--11.0$\pm$ 2.0&3& ...& ..& 7.7& 5.6 &14.3& ... & ... & ... & ... & ... \\
64&P 11424-82 & 0.132$\pm$0.002&--6.6$\pm$1.4&--14.7$\pm$ 1.3&1&+030.7$\pm$ 4& 1& 7.6& 5.5 &14.2& +087$\pm$038&--533$\pm$047 &--187$\pm$035 & +3559$\pm$605&--3896$\pm$620 \\
65&BS 16940-45 & 0.276$\pm$0.004&--3.7$\pm$1.8&--16.7$\pm$ 1.5&1&--101.0$\pm$ 4& 1& 3.6& 2.7 &10.7&--0063$\pm$023&--275$\pm$025 &--116$\pm$021 & +1050$\pm$278 &--0594$\pm$258\\
66&BS 16927-55 & 0.181$\pm$0.006&--2.5$\pm$0.9&--12.3$\pm$ 1.8&1&+043.1$\pm$ 4& 1& 5.5& 4.1 &12.4& +050$\pm$018&--323$\pm$047 & +015$\pm$016 & +0513$\pm$193&--1204$\pm$548 \\
67&BS 16940-0070 & 0.144$\pm$0.002& +2.6$\pm$1.2&--3.5$\pm$ 1.3&1&--082.3$\pm$ 4& 1& 6.9& 5.3 &13.6& --146$\pm$030& --091$\pm$043 & --001$\pm$025 & +1080$\pm$374&+1473$\pm$538 \\
68&BS 16940-0072 & 0.210$\pm$0.003& +5.2$\pm$1.2&--11.5$\pm$ 1.0&1&--135.1$\pm$ 4& 1& 4.8& 3.6 &11.6& --225$\pm$020&--223$\pm$024 & --029$\pm$018 & +0525$\pm$246 &--0175$\pm$265 \\
\hline
\end{tabular}
Notes to table: \\
\bf{(1)} Sources of proper motions (S$_{\mu}$) (1) GSCII-SDSS (2) Nomad
(3) SDSS (DR 7) \\
\bf{(2)} Sources of Radial Velocities (S$_{RV}$): (1) Bologna (2) Kitt Peak
4-m (3) Kinman et al. (1994) \\
\bf{(3)} Distances: (D) Heliocentric distance; (Z) Height above Plane; R$_{gal}$
Galactocentric distance assuming Solar Galactocenttric Distance = 8.0 kpc \\
\end{table*}
\end{landscape}
\end{center}
\begin{center}
\begin{landscape}
\topmargin 45mm
\thispagestyle{empty}
\begin{table*}
\caption{Parallaxes, Proper Motions, Radial Velocities, Galactic Distances and
Galactic Space Velocities for the RR Lyrae stars. \label{t:rrl2}
}
\begin{tabular}{@{}clcccccccccccccc@{}}
\hline
No &ID&$\Pi$ &$\mu_{\alpha}$ & $\mu_{\delta}$&S$_{\mu}$&RV&S$_{RV}$&D &Z &R$_{gal}$ &U &V & W &L$_{\perp}$ & L$_{z}$ \\
& & (mas) & (mas y$^{-1}$) & (mas y$^{-1}$)& &(km s$^{-1}$)& &(kpc)&(kpc)&(kpc)&(km s$^{-1}$)&(km s$^{-1}$)&(km s$^{-1}$)&(kpc km s$^{-1}$)&(kpc km s$^{-1}$ ) \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15)&16 \\
\hline
1 & V385 Aur & 0.044$\pm$0.002&--3.4$\pm$3.3& +0.6$\pm$3.2& 1 & ... & ..&22.7 & 8.8& 30.3 & ... & ... & ... & ...& ... \\
2 & V386 Aur & 0.060$\pm$0.003& +2.1$\pm$2.1&--5.0$\pm$1.7& 1 & +116.0$\pm$30.0& 3 &16.7 & 6.7& 24.2 & +091$\pm$071 &--412$\pm$135 & +106$\pm$152 & +3982$\pm$2474&--4534$\pm$3191\\
3 & V387 Aur & 0.057$\pm$0.003& +1.4$\pm$1.2&--0.6$\pm$0.9& 1 &--003.0$\pm$30.0& 3 &17.5 & 6.7& 25.1 &--050$\pm$046 &--074$\pm$080 & +092$\pm$087 & +3223$\pm$1794&+3486$\pm$1954 \\
4 & V389 Aur & 0.042$\pm$0.002&--1.2$\pm$3.0& 3.3$\pm$0.8& 1 & ... & ..&23.8 & 9.5& 31.3 & ... & ... & ... & ... & ... \\
5 & VX Lyn & 0.053$\pm$0.003&--1.8$\pm$1.2&--4.1$\pm$1.9& 1 & +001.3$\pm$15.0& 1 &18.9 & 7.7& 26.4 & +099$\pm$049 &--296$\pm$168 &--228$\pm$103 & +6730$\pm$2927&--1919$\pm$4259\\
6 & VY Lyn & 0.098$\pm$0.003& +2.2$\pm$1.0&--1.6$\pm$2.1& 1 & +114.5$\pm$15.0& 1 &10.2 & 4.2& 17.8 & +061$\pm$026 &--095$\pm$101 &129$\pm$050 & +2097$\pm$931&+2160$\pm$1743 \\
7 & VZ Lyn & 0.078$\pm$0.004&--1.9$\pm$1.4&--2.2$\pm$2.2& 1 &--182.1$\pm$15.0& 1 &12.8 & 5.4 &20.3 &--109$\pm$044 &--095$\pm$129 &--200$\pm$086 & +3604$\pm$1684&+2498$\pm$2548 \\
8 & WX Lyn & 0.056$\pm$0.003&--2.1$\pm$1.1&--3.3$\pm$1.0& 1 & +026.3$\pm$15.0& 1 &17.9 & 7.5 &25.3 & +121$\pm$042 &--214$\pm$087 &--211$\pm$086 & +6093$\pm$2401&+148$\pm$2131 \\
9 & AS Lyn & 0.030$\pm$0.002& +1.1$\pm$0.7&--3.1$\pm$2.0& 1 &--179.0$\pm$46.0& 3 &33.3 &14.8 &40.7 &--175$\pm$078 &--502$\pm$309 &--026$\pm$119 & +7604$\pm$3882 &--10531$\pm$11848\\
10 & WZ Lyn & 0.180$\pm$0.004& +1.3$\pm$0.6&--13.0$\pm$1.2& 1 & +197.0$\pm$20.0& 4 & 5.6 & 2.4 &13.2 & +193$\pm$020 &--337$\pm$031 &+043$\pm$018 & +371$\pm$107&--1511$\pm$411 \\
11 & XZ Lyn & 0.072$\pm$0.004& +2.1$\pm$1.6&--1.5$\pm$1.0& 1 &--032.0$\pm$20.0& 4 &13.9 & 6.3 &21.3 &--086$\pm$051 &--125$\pm$073 &+087$\pm$096 & +2885$\pm$1689&+1926$\pm$1482 \\
12 & TW Lyn & 0.592$\pm$0.011& +0.6$\pm$3.8& 2.6$\pm$2.9& 1 &--039.0$\pm$05.0& 5 & 1.7 & 0.8 & 9.5 &--050$\pm$015 &+021$\pm$022 & --002$\pm$027 & +307$\pm$121&+2294$\pm$213 \\
13 & YY Lyn & 0.133$\pm$0.002& +5.7$\pm$0.7&--5.9$\pm$1.4& 1 &--093.1$\pm$15.0& 1 & 7.5& 3.3 &15.1 &--165$\pm$018 & --249$\pm$049& +092$\pm$026 & +1932$\pm$412&--478$\pm$723 \\
14 & YZ Lyn & 0.047$\pm$0.002&--0.5$\pm$0.7&--3.7$\pm$1.3& 1 & +033.0$\pm$40.0& 3 &21.3 & 9.7 &28.6 & +087$\pm$051 &--349$\pm$133 &--098$\pm$070 & +4038$\pm$2027&--3482$\pm$3594 \\
15 & AU Lyn & 0.035$\pm$0.002&--0.8$\pm$1.7&--0.8$\pm$2.1& 1 & +096.0$\pm$45.0& 3 &28.6 &13.4 &35.8 & +139$\pm$120 & --056$\pm$283 &--060$\pm$209 & +8978$\pm$5151&+5307$\pm$9477 \\
16 & ZZ Lyn & 0.089$\pm$0.001&--1.4$\pm$1.4&--1.4$\pm$0.9& 1 & +147.2$\pm$15.0& 1 &11.2 & 5.1 &18.7 & +160$\pm$034 &--054$\pm$048 & --006$\pm$061 & +1636$\pm$775 &+3045$\pm$872 \\
17 & RW Lyn & 0.358$\pm$0.016& +7.3$\pm$1.7&--15.7$\pm$2.5& 1 &--149.8$\pm$15.0& 1 & 2.8 & 1.3 &10.6 &--169$\pm$016 &--216$\pm$034 & --023 $\pm$022 & +197$\pm$139&+32$\pm$354 \\
18 & AV Lyn & 0.062$\pm$0.003& +0.5$\pm$0.8&--4.9$\pm$1.1& 1 &--126.0$\pm$30.0& 3 &16.1 & 7.8 &23.4 &--081$\pm$041 &--373$\pm$084 & --87$\pm$056 & +2114$\pm$1030 &--3311$\pm$1887 \\
19 & AC Lyn & 0.071$\pm$0.004& +1.1$\pm$0.9&--5.9$\pm$1.2& 1 &--025.8$\pm$15.0& 1 &14.1 & 6.7 &21.5 &--030$\pm$031 &--395$\pm$085 & --024$\pm$054 & +1733$\pm$766 &--3581$\pm$1761 \\
20 & AD Lyn & 0.088$\pm$0.006& +1.2$\pm$1.1&--8.0$\pm$2.6& 1 & +123.6$\pm$15.0& 1 &11.4 & 5.5 &18.8 &+108$\pm$033 &--435$\pm$143 & +037$\pm$058 & +1724$\pm$805&--3855$\pm$2606\\
21 & AW Lyn & 0.080$\pm$0.003&--1.2$\pm$1.1&--4.2$\pm$1.9& 1 & +93.0$\pm$30.0 & 3 &12.5 & 6.2 &19.8 &+142$\pm$044 &--217$\pm$111& --047$\pm$060 & +2048$\pm$1087&--41$\pm$2092 \\
22 & AX Lyn & 0.026$\pm$0.001& +3.4$\pm$1.4&--2.0$\pm$2.8& 1 & ... & ..&38.5 &18.9 &45.6 & ... & ... & ... & ... & ... \\
23 & AY Lyn & 0.055$\pm$0.003& +1.2$\pm$1.4&--6.0$\pm$1.2& 1 & ... & ..&18.2 & 9.0 &25.4 & ... & ... & ... & ... & ... \\
24 &P 54-13 & 0.130$\pm$0.011& +2.6$\pm$1.0&--10.1$\pm$1.1& 1 & +069.0$\pm$10.0&2 & 7.7 & 3.9 &15.2 &+044$\pm$021 &--374$\pm$051 &+061$\pm$032 & +1046$\pm$404&--2273$\pm$801 \\
25 & AZ Lyn & 0.063$\pm$0.003& +3.3$\pm$1.0&--4.6$\pm$1.5& 1 & +087.0$\pm$33.0& 3 &15.9 & 8.1 &23.1 &--012$\pm$050 &--374$\pm$114 & 207$\pm$070 & +4933$\pm$1673&--3328$\pm$2492 \\
26 & BB Lyn & 0.058$\pm$0.003&--0.6$\pm$0.7&--6.5$\pm$1.4& 1 & +058.0$\pm$33.0& 3 &17.2 & 8.9 &24.4 & +136$\pm$044 &--505$\pm$118 &--083$\pm$054 & +4183$\pm$1429&--6557$\pm$2752 \\
27 & BC Lyn & 0.053$\pm$0.003& +3.3$\pm$1.2&--3.1$\pm$0.7& 1 & +239.0$\pm$45.0& 3 &18.9 &10.0 &26.0 & +075$\pm$068 &--308$\pm$067 &+345$\pm$091 & +7628$\pm$2718 &--2159$\pm$1627 \\
28 & AF 194 & 0.088$\pm$0.002& +0.5$\pm$0.8&--4.7$\pm$1.3& 1 & ... & ..&11.4 & 6.0 &18.6 & ... & ... & ... & ... & ... \\
29 & AF 197 & 0.105$\pm$0.003&--1.0$\pm$3.9&--8.2$\pm$1.8& 1 & ... & ..& 9.5 & 5.0 &16.9 & ... & ... & ... & ...& ... \\
30 & DQ Lyn & 0.649$\pm$0.043&--1.9$\pm$0.8&--28.7$\pm$1.0& 2 & +053.0$\pm$10.0& 2 & 1.5 & 0.9 & 9.3 & +047$\pm$009 &--202$\pm$016 &--006$\pm$008 & +106$\pm$61&+167$\pm$144 \\
31 & RR7 032 & 0.163$\pm$0.004&--1.4$\pm$1.1&--11.9$\pm$1.1& 1 & +42.0$\pm$10.0 & 2 & 6.1 & 3.6 &13.5 & +069$\pm$020 &--334$\pm$033 & --039$\pm$026 & +902$\pm$330 &--1470$\pm$432 \\
32 & RR7 034 & 0.112$\pm$0.002& +0.1$\pm$0.7&--2.9$\pm$0.7& 1 & +316.0$\pm$10.0& 2 & 8.9 & 5.2 &16.1 & +249$\pm$019 &--130$\pm$030 &+178$\pm$025 & +1519$\pm$427&+1474$\pm$459 \\
33 &P 81 129 & 0.164$\pm$0.003&--0.5$\pm$2.0&--7.3$\pm$0.9& 1 & +003.0$\pm$10.0& 2 & 6.1 & 3.6 &13.4 & +022$\pm$035 &--204$\pm$026 & --019$\pm$047 & +648$\pm$477 &+198$\pm$336 \\
34 &AF Lyn & 0.079$\pm$0.002& +1.2$\pm$0.9&--5.9$\pm$1.0& 1 &--121.7$\pm$15.0& 1 &12.7 & 7.5 &19.7 &--132$\pm$034 &--354$\pm$061 & --036$\pm$044 & +1462$\pm$557&--2444$\pm$1102 \\
35 &P 82 06 & 0.190$\pm$0.003& +4.0$\pm$3.0&--3.0$\pm$3.0& 3 &--211.0$\pm$10.0& 2 & 5.3 & 3.3 &12.6 &--234$\pm$047 &--84$\pm$075 & --045$\pm$059 & +919$\pm$470&+1684$\pm$903 \\
36 & AI Lyn & 0.049$\pm$0.003&--1.5$\pm$1.2&--0.2$\pm$2.0& 1 & ... &.. &20.4 &12.5 &27.2 & ... & ... & ... & ... & ... \\
37 & AK Lyn & 0.084$\pm$0.004& +0.8$\pm$1.2&--8.7$\pm$1.1& 1 & +237.6$\pm$15.0& 1 &11.9 & 7.4 &18.8 & +157$\pm$044 &--495$\pm$067 & 148$\pm$054 & +2656$\pm$783&--4686$\pm$1203 \\
38 & RR7-079 & 0.260$\pm$0.007& +1.0$\pm$1.1&--13.5$\pm$1.2& 1 &--032.0$\pm$10.0& 2 & 3.8 & 2.4 &11.3 &--047$\pm$015 &--238$\pm$023 & --021$\pm$017 & +209$\pm$137 &--212$\pm$253 \\
39 & RR7-086 & 0.073$\pm$0.005&--0.7$\pm$1.1&--4.0$\pm$0.7& 1 & ... & ..&13.7 & 8.5 &20.6 & ... & ... & ... & ... & ... \\
40 & AL Lyn & 0.066$\pm$0.005& +1.0$\pm$0.8&--5.6$\pm$1.2& 1 &--065.8$\pm$15.0& 1 &15.2 & 9.5 &22.0 &--105$\pm$038 &--398$\pm$087 & --011$\pm$046 & +2273$\pm$932&--3595$\pm$1789 \\
\hline
\end{tabular}
\end{table*}
\end{landscape}
\end{center}
\begin{center}
\begin{landscape}
\topmargin 50mm
\addtocounter{table}{-1}
\thispagestyle{empty}
\begin{table*}
\caption{Continued. \label{t:rrl2}
}
\begin{tabular}{@{}clcccccccccccccc@{}}
\hline
No &ID&$\Pi$ &$\mu_{\alpha}$ & $\mu_{\delta}$&S$_{\mu}$&RV&S$_{RV}$&D &Z &R$_{gal}$ &U &V & W &L$_{\perp}$ & L$_{z}$ \\
& & (mas) & (mas y$^{-1}$) & (mas y$^{-1}$)& &(km s$^{-1}$)& &(kpc)&(kpc)&(kpc)&(km s$^{-1}$)&(km s$^{-1}$)&(km s$^{-1}$)&(kpc km s$^{-1}$) &(kpc km s$^{-1}$ ) \\
(1) &(2)&(3) &(4) &(5) &(6) &(7) &(8) &(9) &(10) &(11) &(12)&(13)&(14)&(15)&16 \\
\hline
41 & AM Lyn & 0.045$\pm$0.002& +2.7$\pm$1.0&--3.7$\pm$1.3& 1 & ... & ..&22.2 &13.9 &28.9 & ... & ... & ...& ... & ... \\
42 &P 82-32 & 0.122$\pm$0.004& +5.9$\pm$0.7&--9.5$\pm$1.6& 1 & +060.0$\pm$10.0& 2 & 8.2 & 5.2 &15.2 &--084$\pm$019 &--377$\pm$065 & +213$\pm$022 & +3615$\pm$414 &--2219$\pm$931\\
43 & AF 316 & 0.078$\pm$0.002& +1.9$\pm$1.1&--4.8$\pm$1.5& 1 & ... & ..&12.8 & 8.1 &19.7 & ... & ... & ... & ... & ... \\
44 & RR7-101 & 0.077$\pm$0.003& +0.5$\pm$1.0&--4.8$\pm$1.2& 1 & ... & ..&13.0 & 8.2 &19.8 & ... & ... & ... & ... & ... \\
45 & TT Lyn & 1.408$\pm$0.029&--81.9$\pm$1.5&--41.8$\pm$0.9&2 &--065.0$\pm$05.0& 5 & 0.7 & 0.5 & 8.5 &+132$\pm$006 &--131$\pm$004 &--240$\pm$007 & +2108$\pm$62&+751$\pm$36 \\
46 &AF 400 & 0.184$\pm$0.011& +0.2$\pm$1.3&--8.5$\pm$1.4& 1 & ... & ..& 5.4 & 3.8 &12.4 & ... & ... & ... & ... & ... \\
47 &AF 430 & 0.132$\pm$0.003&--1.4$\pm$1.1&--8.2$\pm$0.8& 1 & ... & ..& 7.6 & 5.5 &14.3 & ... & ... & ... & ... & ... \\
48&BS 16927-123 & 0.289$\pm$0.007&--15.6$\pm$2.4&--4.6$\pm$1.3& 1 & +070.0$\pm$10.0& 2 & 3.5 & 2.6 &10.6 &+223$\pm$029 &--094$\pm$022 &--100$\pm$026 & +1654$\pm$327&+1295$\pm$220\\
49 & X LMi & 0.437$\pm$0.007& +7.8$\pm$1.3&--17.3$\pm$0.7& 2 &--082$\pm$20.0 & 6 & 2.3 & 1.8 & 9.5 &--152$\pm$016 &--165$\pm$008 &+004$\pm$018 & +345$\pm$144 &+507$\pm$78 \\
50 &AG UMa & 0.104$\pm$0.002&--1.6$\pm$1.0& --8.4$\pm$1.9&1 & ... & ..& 9.6 & 8.4 &15.2 & ... & ... & ... & ... & ... \\
51 &BK UMa & 0.348$\pm$0.013&--13.0$\pm$1.5&--16.9$\pm$2.2&1 & +171.4$\pm$05.0& 7 & 2.9 & 2.5 & 9.7 &+174$\pm$019 &--253$\pm$031 & +120$\pm$012 & +692$\pm$159&--341$\pm$292\\
52 &AK UMa & 0.075$\pm$0.002&--3.7$\pm$6.1&--3.3$\pm$6.3 &1 & ... & ..&13.3 &11.8 &18.5 & ... & ... & ... & ... & ... \\
53 &AO UMa & 0.096$\pm$0.003&--2.8$\pm$1.3&--6.3$\pm$1.3 &1 & ... & ..&10.4 & 9.4 &15.6 & ... & ... & ... & ... & ... \\
54 &BN UMa & 0.248$\pm$0.006&+12.5$\pm$1.3&--17.3$\pm$1.7 &1 & +019.0$\pm$10.0& 2 & 4.0 & 3.7 &10.3 &--306$\pm$024 &--223$\pm$033 & +175$\pm$014 & +2810$\pm$223&+45$\pm$316 \\
55& CK UMa & 0.201$\pm$0.003&--3.8$\pm$1.3& +0.8$\pm$2.4 &1 & +016.0$\pm$10.0& 2 & 5.0 & 4.9 &10.2 & +080$\pm$037 & --018$\pm$053 &+004$\pm$013 & +1073$\pm$285&+1817$\pm$479\\
\hline
\end{tabular}
Notes to table: \\
\bf{(1)} Sources of proper motions (S$_{\mu}$) (1) GSCII-SDSS (2) Nomad
(3) SDSS (DR 7) \\
\bf{(2)} Sources of Radial Velocities (S$_{RV}$):
(1) Pier, Saha, Kinman (2003); (2) Kinman, Brown (2010);
(3) Saha, Oke (1984); (4) Pier (unpublished); (5) Fernley \& Barnes (1997);
(6) Layden (1994); (7) Jeffery et al. (2007). \\
\bf{(3)} Distances: (D) Heliocentric distance; (Z) Height above Plane; R$_{gal}$
Galactocentric distance assuming Solar Galactocenttric Distance = 8.0 kpc \\
\end{table*}
\end{landscape}
\end{center}
\clearpage
|
{
"timestamp": "2012-03-12T01:02:13",
"yymm": "1203",
"arxiv_id": "1203.2146",
"language": "en",
"url": "https://arxiv.org/abs/1203.2146"
}
|
\section{Introduction: Motels and Bells}
In chapter 6 of his book ``Time Travel and other Mathematical
Bewilderments'' \cite{Gard}, Martin Gardner discusses the
following problem, a special case of D. H. Lehmer's ``motel
problem'' from \cite{Lehmer}.
\bigskip
{\sl Mr.\ Smith manages a motel.
It consists of $n$ rooms in a straight row.
There is no vacancy.
Smith is a psychologist who plans to study the effects
of rearranging his guests in all possible ways.
Every morning he gives them a permutation.
The weather is miserable, raining almost daily.
To minimize his guests' discomfort, each daily rearrangement
is made by exchanging only the occupants of two adjoining rooms.
Is there a simple algorithm that will run through all
possible rearrangements by switching
only one pair of adjacent occupants
at each step}?
\bigskip
For the purposes of this article, we refer to this problem as the
{\it motel problem}. Before discussing a solution we consider
another problem, which at first might seem unrelated. This
problem concerns the change-ringing of bells, so we first provide
a brief introduction to this topic and an explanation of how
permutations arise in the ringing of bells.
Around the year 1600 in England it was discovered that by altering
the fittings around each bell in a bell tower, it was possible for
each ringer to maintain precise control of when his (there were no
female ringers then) bell sounded. This enabled the ringers to
ring the bells in any particular order, and either maintain that
order or change the order in a precise way.
Suppose there are $n$ bells being rung, numbered $1, 2, 3, \ldots
,n$ in order of pitch, number 1 being the highest. When the bells
are rung in order of descending pitch $1, 2, 3, \ldots ,n$, we say
they are being rung in {\it rounds}. A change in the order of the
bells, such as rounds $1 \ 2 \ 3 \ 4 \ 5$ being changed to $2\ 1\
4\ 3\ 5$, can be considered as a permutation in the symmetric
group on five objects. In the years 1600--1650 a new craze emerged
where the ringers would continuously change the order of the bells
for as long as possible, while not repeating any particular order,
and return to rounds at the end. This game evolved into a
challenge to ring the bells in every possible order, without any
repeats, and return to rounds. We will give a precise statement of
this challenge shortly. However, the reader can now see why
bell-ringers were working with permutations. It has been pointed
out before \cite{WhiteSted} that permutations were first studied
in the 1600's in the context of the ringing of bells in a certain
order, and not in the 1770's by Lagrange, in the context of roots
of polynomials. And, by the way, the craze continues to this day.
\bigskip
A solution to the motel problem is a sequence of all the
arrangements (orderings) of $n$ objects, with the property that
each arrangement is obtained from its predecessor by a single
interchange of two adjacent objects. Algorithms for generating the
$n!$ arrangements subject to this condition can be found in S. M.
Johnson \cite{Johnson} and H. F. Trotter \cite{Trotter}; see also
the papers by D. H. Lehmer \cite{Lehmer} and M. Hall Jr.--D. Knuth
\cite{HallK}. Johnson and Trotter discovered the same algorithm,
and we describe their algorithm below. Gardner \cite{Gard} credits
Steinhaus \cite{Steinhaus} as being the first to discover this
method. Such algorithms are similar to what bell-ringers are
trying to find, as we shall now explain.
In fact, the Johnson-Steinhaus-Trotter algorithm was discovered
in the 17th century by bell-ringers.
We elaborate on this in section \ref{jst}.
Let us refer to the $n$ bells listed in a particular order or
arrangement as a {\it row}. For example, when $n=5$, rounds is
the row $1 \ 2 \ 3 \ 4 \ 5$. We will use cycle notation for
permutations. Permutations act on the positions of the bells, and
not on the bells themselves; for example, the permutation $(1\
2)(3\ 4)$ changes the row $5\ 4\ 3\ 2\ 1$ into the row $4\ 5\ 2\
3\ 1$. It is important not to confuse the numbers standing for the
bells with the numbers in the cycles, which refer to the positions
of the bells. Permutations act on the right, i.e., $XY$ means
first do the permutation $X$, then do $Y$. As usual, $S_n$ and
$A_n$ denote the symmetric group and alternating group on $n$
objects.
The task enjoyed by bell-ringers, which we refer to as the {\it
ringers problem}, is to make a list of rows subject to the
following rules:
\bigskip
(1) The first and last rows must be rounds.
\bigskip
(2) No row may be repeated (apart from rounds which appears twice,
first and last).
\bigskip
(3) Each bell may only change place by one position when moving
from one row to the subsequent row.
\bigskip
(4R) No bell occupies the same position for more than two
successive rows.
\bigskip
A {\it method} is any set of rows that obeys these rules. The
origins of rules (1) and (2) have been explained in the
introduction. Rule (3) is there for physical reasons. The purpose
of rule (4R) is mainly to make ringing interesting for the
ringers. For more details see \cite{Fletch} or \cite{WhiteRing}.
Clearly, $n!+1$ is the longest possible length that a list of rows
satisfying rules (1), (2), (3), and (4R) could have, and a list of
this length is called an {\it extent}. The ringers problem is
essentially to construct methods of various lengths, and of
particular interest is the construction of extents.
The motel problem is not identical to the ringers problem, for if
we replace rule (4R) by the following:
\bigskip
(4M) Only two bells may change place when moving from one row to
the subsequent row
\bigskip then a list of $n!+1$ rows satisfying rules (1), (2), (3),
and (4M) is precisely a solution to the motel problem.
We can now see that there is certainly a similarity between the
motel problem and the ringers problem. Some previous {\it Monthly}
articles have considered the motel problem \cite{HallK},
\cite{Lehmer}, and other {\it Monthly} articles have considered
the ringers problem, \cite{Dick}, \cite{Fletch}, \cite{WhiteRing},
\cite{WhiteSted}.
We should mention that in practice ringers do have some other
rules, but rules (1), (2), (3), (4R) are the most important. We
ignore the other rules, and some other ringing matters, for the
purposes of this article. For more details see \cite{Fletch} or
\cite{WhiteRing}.
In the following sections we discuss these two problems and some
solutions. In section \ref{jst} we present the first solution to
the motel problem, which was discovered by ringers in the 1600's.
Section \ref{plainhunt} discusses the simplest solution to the
ringers problem, and section \ref{plainbob} gives a more
complicated solution. We shall see how ringers were manipulating
cosets of a dihedral subgroup of the symmetric group in an effort
to meet their challenge. Section \ref{ccc} proposes a new method.
In section \ref{groupform} we present a general group-theoretic
framework for the problems. We discuss approaches to finding
extents in section \ref{leads}, and give some proofs of
impossibility. In section \ref{proofs} we also present a proof
of a remarkable 1948 result of R. A. Rankin which provides a
beautiful solution to a particular problem dating from 1741. The
first solution to this problem was given in 1886 by W.\ H.\
Thompson. This solution is more subtle than the proofs in section
\ref{leads}. In section \ref{newthm} we present a small
result concerning $S_4$.
No knowledge of change-ringing is assumed for this
article. We keep the bell-ringing terminology to a minimum,
although a little is necessary. The mathematics involved is
elementary group theory.
\bigskip
\section{The Johnson-Steinhaus-Trotter solution} \label{jst}
\bigskip
The Johnson-Steinhaus-Trotter solution of the motel problem
described below satisfies rules (1), (2), (3), (4M), but not rule
(4R) for $n>3$, and is therefore not a solution to the ringers
problem for $n>3$.
\bigskip
The solution to the motel problem in Johnson \cite{Johnson},
Steinhaus \cite{Steinhaus} and Trotter \cite{Trotter} (see also
\cite{Gard}, \cite{HallK}, \cite{Lehmer}) is a beautifully simple
idea. We summarise the idea as follows. Construct all arrangements
of $n+1$ objects inductively, using all arrangements of $n$
objects. The induction begins because 1 is a solution for $n=1$,
or $12, 21$ is a solution for $n=2$. Expand the list of all
arrangements of $n$ objects by replacing each arrangement by $n+1$
copies of itself. Place the new $(n+1)$th object $M$ on the left
of the first arrangement. To obtain the next arrangement, we
interchange $M$ with the object to the right of $M$. After doing
this $n$ times, we have $M$ on the extreme right. Then we leave
$M$ here for one step, as the old $n$ objects undergo their first
rearrangement. Next $M$ moves to the left, one place at a time,
and then $M$ stays at the extreme left for one step after arriving
there, as the old $n$ objects undergo their second rearrangement.
We continue this process until we reach the end of the list.
For example, to obtain the six arrangements of three objects from
the two arrangements of two objects, we first expand and obtain
23, 23, 23, 32, 32, 32. Then we weave 1 through these as
instructed in the algorithm: 123, 213, 231, 321, 312, 132.
To get the solution for four objects, we write the solution for
three objects as 234, 324, 342, 432, 423, 243. Then, with 1 as the
new object $M$, we obtain the list of rows shown below. The reader
should check the list, and ignore the $A$'s, $B$'s and $C$'s for
the moment.
\bigskip
\begin{tabular}{rlllrl}
1\ 2\ 3\ 4 & &\\ & $A$ & & $A$ & & $A$ \\ 2\ 1\ 3\ 4 & & 3\ 1\
4\ 2 & & 4\ 1\ 2\ 3\\ & $B$ & & $B$ & & $B$ \\ 2\ 3\ 1\
4 & & 3\ 4\ 1\ 2 & & 4\ 2\ 1\ 3\\ & $C$ & & $C$ & & $C$
\\ 2\ 3\ 4\ 1 & & 3\ 4\ 2\ 1 & & 4\ 2\ 3\ 1\\ & $A$ & & $A$
& & $A$ \\ 3\ 2\ 4\ 1 & & 4\ 3\ 2\ 1 & & 2\ 4\ 3\ 1\\ & $C$
& & $C$ & & $C$ \\ 3\ 2\ 1\ 4 & & 4\ 3\ 1\ 2 & & 2\ 4\ 1\ 3\\
& $B$ & & $B$ & & $B$ \\ 3\ 1\ 2\ 4 & & 4\ 1\ 3\ 2
& & 2\ 1\ 4\ 3\\ & $A$ & & $A$ & & $A$ \\ 1\ 3\ 2\ 4 & & 1\ 4\
3\ 2 & & 1\ 2\ 4\ 3\\ & $C$ & & $C$ & & $C$ \\ 1\ 3\ 4\ 2 & & 1\
4\ 2\ 3 & & 1\ 2\ 3\ 4
\end{tabular}
\bigskip
Now we explain the $A, B, C$ notation. The $A$ stands for the
permutation $(1\ 2)$, $B$ for $(2\ 3)$ and $C$ for $(3\ 4)$,
acting on the positions. Between two rows we have listed the
permutation used to get from one to the other. A shorter way to
write this is simply to list the sequence of $A$'s, $B$'s and
$C$'s,
$$A,B,C,A,C,B,A,C,A,B,C,A,C,B,A,C,A,B,C,A,C,B,A,C,$$
and we will sometimes do this in the sequel.
\bigskip
We may claim that the motel problem was actually solved in the
1600's by bell-ringers. In the early days ringers did not have
rule (4R), and all early methods of changing the order of the
bells involved only two adjacent bells switching place at any one
time. Such changes were called ``plain changes.'' And so, in fact,
rule (4M) was being used instead of rule (4R). Thus, the ringers
problem in those days was identical with the motel problem. An
exact copy of the solution given above to the motel problem for
$n=4$ can be seen in a book dating from c.\ 1621. The full 120
plain changes on five bells were being rung by the mid seventeenth
century.
Fabian Stedman's Campanalogia \cite{Stedman} published in 1677
gives the Johnson-Steinhaus-Trotter solution of the motel problem
for $n=3, 4, 5$ and $6$.
\bigskip
Moreover, the general pattern of the algorithm has been noted, as
is evidenced by the following (paraphrased) quote concerning the
$6!$ plain changes on 6 bells.
\bigskip
{\sl The method of the seven hundred and twenty, has an absolute
dependency upon the method of the sixscore changes on five bells;
for five of the notes are to make the sixscore changes, and the
sixth note hunts continually through them, and every time it leads
or lies behind them, one of the sixscore changes must then be
made. The method of the seven hundred and twenty is in effect the
same as that of the sixscore; for as the sixscore comprehended the
twenty-four changes on four, and the six on three; so likewise the
seven hundred and twenty comprehend the sixscore changes on five,
the twenty-four changes on four, and the six changes on three.} --
F.\ Stedman, 1677
\bigskip
Although the general pattern had been observed, plain changes on
more than 6 bells are difficult to ring and bell-ringers
progressed to other ways of ringing. There is an earlier book than
Stedman's, entitled {\it Tinntinnalogia}, written by Richard
Duckworth and published in 1668, which also discusses plain
changes on 3, 4, 5 and 6 bells. This book presumably gives the
Johnson-Steinhaus-Trotter solution to the motel problem, but we
have not been able to confirm this. Nevertheless, the evidence
from Stedman's book would appear to show that their algorithm had
been discovered three hundred years before. Knuth \cite{Knuth}
states that there is a document written by Mundy dating from 1653
which gives the algorithm.
\bigskip
The following observation will be important in the next sections.
The solution to the motel problem given above for $n=4$ is
equivalent to a way of writing down the elements of $S_4$ in an
ordered list, with the property that each element of the list is
obtained from the preceding element by multiplication on the right
by one of $A, B$ or $C$. The list would begin $A, AB, ABC, ABCA,
ABCAC, \ldots $.
\bigskip
\section{Plain Hunt} \label{plainhunt}
\bigskip
The simplest method for bell-ringers is called Plain Hunt. We will
describe Plain Hunt on 5 bells, and then the generalisation to $n$
bells. But note that not all methods generalise easily to $n$
bells.
Plain Hunt on 5 bells uses two permutations applied
alternately to rounds, until rounds comes back again.
The permutations are $X=(1\ 2)(3\ 4)$ and
$Y=(2\ 3)(4\ 5)$. We write $X$ or $Y$ between two
rows to indicate which permutation has been used
to get from one to the other.
After each element $X$ or $Y$ we list the product of all
the elements so far.
\bigskip
\begin{tabular}{rll}
& & \\
1\ 2\ 3\ 4\ 5 & & \\
& $X,\ (1\ 2)(3\ 4)$\\
2\ 1\ 4\ 3\ 5 & &\\
& $Y,\ \ XY=(1\ 3\ 5\ 4\ 2)$\\
2\ 4\ 1\ 5\ 3 & &\\
& $X,\ \ XYX=(1\ 4)(3\ 5)$\\
4\ 2\ 5\ 1\ 3 & &\\
& $Y,\ (1\ 5\ 2\ 3\ 4)$\\
4\ 5\ 2\ 3\ 1 & &\\
& $X,(1\ 5)(2\ 4)$\\
5\ 4\ 3\ 2\ 1 & &\\
& $Y,(1\ 4\ 3\ 2\ 5)$\\
5\ 3\ 4\ 1\ 2 & &\\
& $X,(1\ 3)(2\ 5)$\\
3\ 5\ 1\ 4\ 2 & &\\
& $Y,(1\ 2\ 4\ 5\ 3)$\\
3\ 1\ 5\ 2\ 4 & &\\
& $X,(2\ 3)(4\ 5)$\\
1\ 3\ 2\ 5\ 4 & &\\
& $Y$,identity\\
1\ 2\ 3\ 4\ 5 & &
\end{tabular}
\bigskip
Using the above shorthand, we could write this as
$$ X, Y, X, Y, X, Y, X, Y, X, Y.$$
After 10 permutations we return to rounds. In other words, there
is a total of 11 rows in Plain Hunt on 5, if we include both
rounds at beginning and end. We could have predicted this; since
$X$ and $Y$ generate a group of order 10 it follows that there can
be no more than 10 permutations in Plain Hunt on 5, and indeed in
any method only using $X$ and $Y$. However, we should note that
given any two permutations $A$ and $B$, if they generate a group
of order $m$ it does not follow that we can ring a method with
$m+1$ rows using $A$ and $B$. This is because of Rule (3).
In Plain Hunt on 6 bells, the permutations used are
$X=(1\ 2)(3\ 4)(5\ 6)$ and $Y=(2\ 3)(4\ 5)$.
On 7 bells we use
$X=(1\ 2)(3\ 4)(5\ 6)$ and $Y=(2\ 3)(4\ 5)(6\ 7)$.
On 8 bells we use
$X=(1\ 2)(3\ 4)(5\ 6)(7\ 8)$ and $Y=(2\ 3)(4\ 5)(6\ 7)$.
The generalisation to $n$ bells is now clear.
Note above that $X$ and $Y$ are products of disjoint
transpositions of consecutive numbers. This is
demanded by rule (3).
The permutations $X$ and $Y$ generate a subgroup $H_{2n}$ of $S_n$
of order $2n$, which we will call the {\it hunting subgroup} (even
though it is better known as the dihedral group $D_{2n}$). Note
that if we take a list of the elements after the comma above, we
obtain a list of the elements of the subgroup $H_{10}$.
\bigskip
{\bf Remark 1}.
Here is the general idea for solving the ringers problem:
to devise a method with more than $2n$ rows,
we must throw another permutation into the mix.
We wish to use as few permutations as possible
in order to keep the method as simple as possible,
while obtaining a method as long as possible,
hopefully with all $n!+1$ rows.
The first solutions employed by ringers to the ringers problem
involved cosets of the hunting/dihedral subgroup $H_{2n}$ of $S_n$
generated by $X=(1\ 2)(3\ 4)\ldots (n-1,\ n)$ and $Y=(2\ 3)(4\
5)\ldots (n-2,\ n-1)$ when $n$ is even, and $X=(1\ 2)(3\ 4)\ldots
(n-2,\ n-1), Y=(2\ 3)(4\ 5)\ldots (n-1,\ n)$ when $n$ is odd.
These particular involutions are used for the transitions between
successive rows because rules (3) and (4R) will be obeyed.
As long as we do not apply $X$ or $Y$ twice in succession, rule
(2) will be obeyed.
\bigskip
{\bf Remark 2}. Let us mention here another rule, which roughly
states that each bell follows the same path. We will not go into
any further detail on this. One can see that this is indeed the
case in Plain Hunt, and that it will also hold in a coset of the
hunting subgroup. This is why we shall assume for this article
that all methods are a union of cosets of the hunting subgroup.
Using cosets keeps the method simple, one of the goals from Remark
1. How to choose the cosets such as to obey rules (1)-(3) and (4R)
is the real question.
\bigskip
\section{Plain Bob}\label{plainbob}
\bigskip
Probably the next simplest method after Plain Hunt is called
Plain Bob.
This method dates from about 1650.
We will describe Plain Bob on 4 bells,
and then 6 bells.
The idea of Plain Bob is to combine Plain Hunt with some
particular cosets.
We do this because Plain Hunt is not yet a solution to the ringers
problem of finding an extent of all $n!+1$ rows. We obtain a
solution to the ringers problem for $n=4$ using the hunting
subgroup $H_8$, a group of order 8, and two left cosets, which are
$(2\ 4\ 3)H_8$ and $(2\ 3\ 4)H_8$. The advantage of using cosets
of a subgroup is that distinct cosets are disjoint and therefore
rule (2) is automatically satisfied. This fact was surely known
to, and utilised by, the early composers. Here is the full
solution, which uses one other permutation, namely $Z=(3\ 4)$,
apart from $X$ and $Y$. This $Z$ is used in order to switch into
the cosets.
We spell this out in detail. First consider Plain Hunt on 4
bells:
\bigskip
\begin{tabular}{rll}
& & \\ 1\ 2\ 3\ 4 & & \\ & $X,(1\ 2)(3\ 4)$\\ 2\ 1\
4\ 3 & &\\ & $Y,(1\ 3\ 4\ 2)$\\ 2\ 4\ 1\ 3 & &\\ & $X,(1\ 4)$\\
4\ 2\ 3\ 1 & &\\ & $Y,(1\ 4)(2\ 3)$\\ 4\ 3\ 2\ 1 & &\\ & $X,(1\
3)(2\ 4)$\\ 3\ 4\ 1\ 2 & &\\ & $Y,(1\ 2\ 4\ 3)$\\ 3\ 1\ 4\ 2 &
&\\ & $X,(2\ 3)$\\ 1\ 3\ 2\ 4 & &\\ & $Y$, identity\\ 1\ 2\ 3\
4 & &
\end{tabular}
\bigskip
where $X=(1\ 2)(3\ 4)$ and $Y=(2\ 3)$. The sequence of elements
after the commas consists of the elements of $H_8$.
In Plain Bob on 4 bells, instead of doing the final $Y$ which
takes us back to rounds, we do $Z=(3\ 4)$ instead. We then
continue with $X$ and $Y$ alternately. This has the effect of
taking us into the left coset $(Y^{-1}Z)H_8$ of the hunting
subgroup $H_8$, i.e., after the comma we will be listing the
elements of $(Y^{-1}Z)H_8$. Here is what happens.
\begin{tabular}{rll}
& & \\
3\ 1\ 4\ 2 & &\\
& $X,\ XYXYXYX=(2\ 3)= Y^{-1}$\\
1\ 3\ 2\ 4 & &\\
& $Z,\ Y^{-1}Z=(2\ 3)(3\ 4)=(2\ 4\ 3)$\\
1\ 3\ 4\ 2 & &\\
& $X,\ Y^{-1}ZX$=(1\ 2\ 3) \\
3\ 1\ 2\ 4 & &\\
& $Y,\ Y^{-1}ZXY$=(1\ 3) \\
3\ 2\ 1\ 4 & &\\
\hfil\vdots & \vdots & \hfill \vdots
\end{tabular}
\bigskip
At the same point in this coset when we have reached
$Y^{-1}ZXYXYXYX=$ \break $Y^{-1}ZY^{-1}$, instead of doing the
final $Y$ next (which would cause us to repeat $1\ 3\ 4\ 2$,
disobeying rule (2)) we do $Z$ again, which takes us into the
coset $(Y^{-1}Z)^2H_8$. Again we alternate between $X$ and $Y$ to
take us through this coset, and at the same point again we do $Z$,
which takes us back to rounds. Here is the full set of rows.
\bigskip
\begin{tabular}{rlllrl}
1\ 2\ 3\ 4 & &\\
& $X,(1\ 2)(3\ 4)$ & & $X,(1\ 2\ 3)$ & & $X,(1\ 2\ 4)$ \\
2\ 1\ 4\ 3 & & 3\ 1\ 2\ 4 & & 4\ 1\ 3\ 2\\
& $Y,(1\ 3\ 4\ 2)$ & & $Y,(1\ 3)$ & & $Y,(1\ 3\ 2\ 4)$ \\
2\ 4\ 1\ 3 & & 3\ 2\ 1\ 4 & & 4\ 3\ 1\ 2 \\
& $X,(1\ 4)$ & & $X,(1\ 4\ 3\ 2)$ & & $X,(1\ 4\ 2\ 3)$ \\
4\ 2\ 3\ 1 & & 2\ 3\ 4\ 1 & & 3\ 4\ 2\ 1\\
& $Y,(1\ 4)(2\ 3)$ & & $Y,(1\ 4\ 2)$ & & $Y,(1\ 4\ 3)$ \\
4\ 3\ 2\ 1 & & 2\ 4\ 3\ 1 & & 3\ 2\ 4\ 1\\
& $X,(1\ 3)(2\ 4)$ & & $X,(1\ 3\ 4)$ & & $X,(1\ 3\ 2)$ \\
3\ 4\ 1\ 2 & & 4\ 2\ 1\ 3 & & 2\ 3\ 1\ 4\\
& $Y,(1\ 2\ 4\ 3)$ & & $Y,(1\ 2\ 3\ 4)$ & & $Y,(1\ 2)$ \\
3\ 1\ 4\ 2 & & 4\ 1\ 2\ 3 & & 2\ 1\ 3\ 4\\
& $X,(2\ 3)$ & & $X,(2\ 4)$ & & $X,(3\ 4)$ \\
1\ 3\ 2\ 4 & & 1\ 4\ 3\ 2 & & 1\ 2\ 4\ 3\\
& $Z,(2\ 4\ 3)$ & & $Z,(2\ 3\ 4)$ & & $Z$,identity \\
1\ 3\ 4\ 2 & & 1\ 4\ 2\ 3 & & 1\ 2\ 3\ 4
\end{tabular}
\bigskip
This solution could be represented by the sequence of permutations
$$X,Y,X,Y,X,Y,X,Z,X,Y,X,Y,X,Y,X,Z,X,Y,X,Y,X,Y,X,Z.$$
There are a few important observations to make here. Firstly, by
adding cosets of $H_8$ to Plain Hunt, we have increased the number
of rows in our set of changes. This is the basic idea of all the
methods considered in this paper, as we said in Remark 2.
Secondly, we have listed (after the comma)
the permutations in the order
$$H_8\backslash \{\hbox{identity}\}, \ (Y^{-1}Z)H_8, \ (Y^{-1}Z)^2H_8,
\ \hbox{identity}$$
where within each coset we use the order from Plain Hunt.
Thirdly, we note that we obtained 3 cosets in total
because $Y^{-1}Z=(2\ 4\ 3)$ has order 3.
We also note that the union of the 3 cosets is all
of $S_4$, so in this case we obtained the maximum
number of permutations possible, an extent.
This does not happen in general.
\bigskip
Next we summarise Plain Bob on 6 bells. Recall that the idea is
to add cosets of $H_{12}$ to Plain Hunt on 6, so that we obtain
more rows. Of course, this must be done without disobeying any of
rules (1), (2), (3), (4R). We recall that the generators of the
hunting subgroup are $X=(1\ 2)(3\ 4)(5\ 6)$ and $Y=(2\ 3)(4\ 5)$.
In addition we use $Z=(3\ 4)(5\ 6)$, in the same way as in Plain
Bob on 4 bells. Since $Y^{-1}Z=(2\ 3)(4\ 5)(3\ 4)(5\ 6)=(2\ 4\ 6\
5\ 3)$ has order 5, we obtain 5 cosets of $H_{12}$ for a total of
$60$ permutations (61 rows including both rounds).
Let us check that rules (1)--(3) and (4R) are satisfied. Because
$X, Y, Z$ are all products of disjoint transpositions of
consecutive numbers and they are the only permutations used to get
from one row to the next, rule (3) is satisfied. Rule (1) is also
satisfied because of the construction, and rule (2) is satisfied
because the cosets are distinct, and therefore disjoint. Rule (4R)
is satisfied because it is satisfied for Plain Hunt, and Plain Bob
is a union of cosets of Plain Hunt. Here we see that group theory
provides us with a construction of a method, and a proof that it
obeys the rules, without writing out all the rows.
\bigskip
One can also ring Plain Bob on an odd number of bells. For any
$n$, Plain Bob on $n$ bells uses $n-1$ cosets of $H_{2n}$ and so
has $2n(n-1)$ permutations. Only when $n=4$ does $2n(n-1)=n!$.
This solution to the ringers problem dates from the 17th century,
and can be found in Stedman's 1677 book \cite{Stedman}.
Note that bell 1 here behaves in the same way as in the section
\ref{jst} solution to the motel problem. The other bells behave
differently however. Unfortunately, this solution to the ringers
problem does not generalise to arbitrary $n$ in an obvious way,
unlike the motel problem. The task of combining cosets of $H_{2n}$
to obtain a solution to the ringers problem for $n>4$ is highly
nontrivial, and solutions involve many clever ideas. See
\cite{WhiteRing} for more details.
\section{A New Method}\label{ccc}
We now introduce a "new" method on 5 bells which is a union of
cosets of $H_{10}$. We mimic the construction of Plain Bob on 5
bells except that instead of using $Z=(3\ 4)$ we will use $Z= (1\
2)$. This results in 6 cosets of $H_{10}$ because $Y^{-1}Z=(2\
3)(4\ 5)(1\ 2)=(1\ 2\ 3)(4\ 5)$ has order 6. Thus we obtain a
method with 61 rows (including both rounds).
This method is not listed in the collection of known methods,
so we propose to call it Christ Church
Dublin Differential Doubles.
This is now recognised, see the web site \cite{Smith}
under differentials.
We outline a bob for this method in
section \ref{leads}.
\bigskip
\section{A Group-Theoretic Formulation}\label{groupform}
\bigskip
As explained at the end of section \ref{jst}, the solution to the
ringers problem given above is equivalent to a way of writing down
the elements of $S_4$ in a list, with the property that each
element of the list is obtained from the preceding element by
multiplying by one of $X, Y$ or $Z$. This idea motivates the
following definition.
\bigskip
\subsection{Unicursal Generation}
\bigskip
{\bf Definition}. Let $G$ be a finite group of order $n$, and let
$T$ be a subset of $G$. We say that $T$ generates $G$
$\underline{unicursally}$ if the elements of $G$ can be ordered
$g_1, g_2, \ldots ,g_n$ so that for each integer $i$, there exists
$t_i\in T$ such that $g_{i+1}=g_it_i$. (Here subscripts are
considered modulo $n$.)
In this framework, the motel problem can be restated as follows:
is $S_n$ generated unicursally by $T=\{(1\ 2), (2\ 3), (3\ 4),
\ldots ,(n-1,\ n)\}$ ?
The ringers problem can be restated as follows: is $S_n$ generated
unicursally by a subset $T$ satisfying the following conditions:
\begin{enumerate}
\item each element of $T$ is a product of disjoint transpositions
of consecutive numbers (this is rule 3) \item $t_i$ and $t_{i+1}$
have no common fixed point for any $i$ (this is rule 4R).
\end{enumerate}
\bigskip
{\bf Example 1.} Let $G=S_3$, and let $T=\{(1\ 2),(2\ 3)\}$. Then
$T$ generates $G$ unicursally. The reader will find it useful to
verify this small example.
\bigskip
{\bf Example 2.} Let $G$ be the $D_4$ from section
\ref{plainhunt}, and let $T=\{(1\ 2)(3\ 4), (2\ 3)\}$. Then $T$
generates $G$ unicursally as shown in section \ref{plainhunt}.
\bigskip
{\bf Example 3.} Let $G=S_4$, and let $T=\{(1\ 2)(3\ 4), (2\ 3),
(3\ 4)\}$ as in section \ref{plainbob} in the solution to the
ringers problem for $n=4$.
\bigskip
{\bf Example 4.} Let $G=S_n$. The solution to the the motel
problem described in section \ref{jst} shows that $S_n$ is
generated unicursally by the $n-1$ elements of $T=\{(1\ 2), (2\
3), (3\ 4), \ldots ,$ $(n-1,\ n)\}$.
\bigskip
{\bf Example 5.} Let $G$ be any finite group and let $T=G$. Then
$G$ is generated unicursally by $T$, and any ordering of the
elements of $G$ can be used, since $g(g^{-1}h)=h$ for any $g, h\in
G$.
\bigskip
{\bf Example 6.} Let $G=S_n$ and let $$T=\{(1\ 2), (1\ 2)(3\ 4)(5\
6)\ldots , (2\ 3)(4\ 5)(6\ 7)\ldots \}.$$ It is shown in
\cite{Rapaport} (see also \cite{WhiteRing}) that $T$ generates $G$
unicursally. Knuth \cite{Knuth} states that Rapaport's result has
been generalised by Savage.
\bigskip
{\bf Remarks}. In order for a subset $T$ to generate $G$
unicursally it is necessary that $T$ generates $G$. This condition
is not sufficient, as examples 7, 8 and 9 below show.
We are usually interested in the case when $T$ is a small, and
often minimal, set of generators. The general question of whether
a given $G$ is generated unicursally by a given $T$ seems very
difficult. This problem may be related to word problems in the
group $G$.
Given a generating set $T$ for a finite group $G$, the Cayley
colour graph $C_T(G)$ is the graph with the elements of $G$ for
vertices, and all directed edges $(x,xt)$ where $t\in T$. Each
directed edge is coloured by the generator $t$. If every element
of $T$ has order 2, then the graph may be considered undirected.
Usually assumptions are made to ensure that $C_T(G)$ has no loops
or multiple edges. The group $G$ acts regularly and transitively
on the vertices of $C_T(G)$, and is the automorphism group of
$C_T(G)$. The following theorem is clear.
\begin{theorem}
A group $G$ is generated unicursally by $T$ if and only if the
Cayley colour graph defined by $G$ and $T$ is Hamiltonian.
\end{theorem}
This is the point of view of Rapaport \cite{Rapaport} and White
\cite{WhiteCamb}, \cite{WhiteRing}, \cite{WhiteSted}. White has
written several papers on bells and topological graph theory.
\bigskip
The motel problem and the ringers problem are concerned with
specific types of subset $T$ of the symmetric group $S_n$. For the
motel problem, as we have said above, $T$ is the set $\{(1\ 2),
(2\ 3), (3\ 4), \ldots (n-1,\ n)\}$. For the ringers problem,
elements of $T$ will be products of disjoint transpositions of
consecutive numbers (because of rule 3), and one must ensure that
$t_i$ and $t_{i+1}$ have no common fixed point (because of rule
4R). Which particular $T$ is chosen depends on the method.
Two methods will be discussed in this paper, Plain Bob
and Grandsire.
Let us generalise and consider the unicursal generation of
$S_n$ by elements other than products of disjoint transpositions.
First we make a few simple observations about arbitrary groups.
The classification of groups generated by $T$ of size 1 is
straightforward.
\begin{theorem}
A group $G$ is generated unicursally by a subset
of size 1 if and only if $G$ is cyclic.
\end{theorem}
Proof. Suppose $G$ is generated unicursally by $T=\{x\}$. Assuming
$g_1=1$ (w.l.o.g.) then $g_2=x$, and then $g_3=x^2$, and
$g_{i+1}=g_ix=x^i$ for any $i$. The other implication is clear.
\bigskip
The classification of groups generated by $T$ of size 2 is
nontrivial. Clearly if $G$ is isomorphic to a direct product of
two cyclic groups then $G$ is unicursally generated by a subset of
size 2. The following theorem is a remarkable result of R.\ A.\
Rankin on this case.
The result in Rankin's paper \cite{Rankin} is more general than
the version we state here, and is somewhat based on ideas of
Thompson \cite{Thompson}. The end result is very simply stated in
group theoretic language, even though the problem and the proof
are somewhat combinatorial. We give a proof in section
\ref{proofs}.
Let $\langle g \rangle$ denote the subgroup generated by $g$.
\begin{theorem} \label{rankthm}
(Rankin, 1948) Let $G$ be a finite group. Suppose that $G$
is generated by $T=\{x,y\}$, and that $\langle x^{-1}y \rangle$
has odd order. If $G$ is generated unicursally by $T$, then
$\langle x \rangle$ and $\langle y \rangle$ have odd index in $G$.
\end{theorem}
\bigskip
{\bf Example 7.} It is easily checked that $G=A_4$ is generated
by $T=\{A, B\}$ where $A=(1\ 2\ 3), B=(1\ 2\ 4)$. However, $A_4$
is not generated unicursally by $T$ by Rankin's theorem, because
$A^{-1}B=(1\ 3\ 4)$ has odd order but $\langle A \rangle$ has
index 4 in $A_4$.
\bigskip
{\bf Example 8.} It is easily checked that $A_5$ is generated by
$T=\{A, B\}$, where $A=(1\ 3\ 5\ 4\ 2)$ and $B=(3\ 5\ 4)$. Here
$A^{-1}B$ has order 3, so Rankin's theorem applies. Since $\langle
B \rangle$ has even index in $A_5$ we conclude that $A_5$ is not
generated unicursally by $T$.
\bigskip
{\bf Example 9.} It is not hard to show that $S_n$ is generated by
the transposition $A=(n-1,\ n)$ and the $(n-1)$-cycle $B=(1\ 2\ 3\
\ldots n-1)$. We may well ask whether $S_n$ is generated
unicursally by $A$ and $B$. Suppose $n$ is odd and $n>3$. Then
$S_n$ is not generated unicursally by $T=\{A, B\}$ by Rankin's
theorem, because $A^{-1}B=(1\ 2\ \ldots n)$ has odd order but
$\langle A \rangle$ has even index in $S_n$ for $n>3$.
\bigskip
{\bf Example 10.} It is not hard to show that $S_n$ is generated
by $\sigma=(1\ 2\ \ldots n)$ and $\tau = (1\ 2)$.
We (of course) ask if $S_n$ is generated unicursally by these
elements.
Rankin's theorem shows that the answer is negative if $n\geq 4$ is
even, since $\tau^{-1} \sigma$ is an $(n-1)$-cycle.
We will mention this example again soon.
\bigskip
From the discussion in the previous sections, the following is now
obvious (and has been observed before, see \cite{HallK} for
example). Let $t_i$ be as above.
\begin{theorem}
\begin{enumerate}
\item The existence of an extent on $n$ bells
satisfying rules (1)-(3), where the allowed permutations between
rows are $X_1, \ldots ,X_k$, is equivalent to $T=\{X_1, \ldots
,X_k\}$ generating $S_n$ unicursally. \item The existence of an
extent on $n$ bells satisfying rules (1),(2),(3),(4R), where the
allowed permutations between rows are $X_1, \ldots ,X_k$, is
equivalent to $T=\{X_1, \ldots ,X_k\}$ generating $S_n$
unicursally with the additional property that $t_i$ and $t_{i+1}$
have no common fixed point for any $i$.
\end{enumerate}
\end{theorem}
A permutation $X_i$ being an allowed transition between rows is
equivalent to $X_i$ being a product of disjoint transpositions of
consecutive numbers, by rule (3). No bell staying in the same
place for more than two rows (rule (4R)) is equivalent to no two
consecutive transitions having a common fixed point.
As we mentioned in Remark 2 at the end of section 3, we assume
that the methods in this article are cosets of $H_{2n}$.
Therefore, included in $T$ will be $X$ and $Y$, the generators for
the hunting subgroup $H_{2n}$. But note that even if $S_n$ is
generated unicursally by $T$, it does not follow that it is
generated unicursally as cosets of $H_{2n}$.
\bigskip
{\bf Remark 3}. In the ringing methods we discuss in this paper,
one can divide up an extent on $n$ bells into groups of $2n$ rows
called {\it leads}, roughly (but not exactly) corresponding to
cosets of $H_{2n}$. A method composed of cosets of $H_{2n}$ is
also a method made up of a succession of leads. In this paper we
use only two types of leads, and we consider methods and extents
made from a sequence of these leads. In these cases then, one can
show that the existence of an extent on $n$ bells of this type is
equivalent to the alternating group $A_{n-1}$ being generated
unicursally by $T'$, where $T'$ is a set of generators related to
$T$. For more details on this, see section 6.
The general question of whether
a given $G$ is generated unicursally by a given $T$
seems very difficult.
\bigskip
\subsection{The famous old question}
\bigskip
The famous old question mentioned in the introduction concerns the
three permutations $X=(1\ 2)(3\ 4)(5\ 6), Y=(2\ 3)(4\ 5)(6\ 7)$
and $Z=(1\ 2)(4\ 5)(6\ 7)$ in $S_7$, and whether $S_7$ is
generated unicursally by $X, Y, Z$ in a particular way. We will
explain this in detail in section \ref{leads}.
This was asked in 1741 by a bell-ringer John Holt, who was able to
construct a method of $4998$ permutations, but could not obtain a
method of $7!=5040$ permutations. He then (naturally!) queried
the existence of such an extent. As in Remark 3, it can be shown
(see \cite{Rankin}, or section \ref{leads}) that this question is
equivalent to:
\bigskip
{\bf Question A}. Is $A_6$ (acting on $\{2,3,4,5,6,7\}$) generated
unicursally by the two permutations $(3\ 4\ 6\ 7\ 5)$ and $(2\ 4\
7)(3\ 6\ 5)$ ?
\bigskip
The first proof that the answer is no is due to Thompson
\footnote{Thompson was a civil servant in India at the time,
and a Cambridge mathematics gradute.} (1886)
\cite{Thompson}, with some case-by-case analysis. An insightful
proof was given by Rankin (1948) \cite{Rankin}, where he came up
with theorem \ref{rankthm} based somewhat on Thompson's ideas.
See also \cite{Rankin2} and a proof by Swan \cite{Swan}.
We present Rankin's proof in section \ref{proofs}, in our special
case only. Rankin's result is more general. Most of the ideas of
the proof can be found in our proof of the special case in section
\ref{proofs}. We also show in section \ref{proofs} that $4998$ is
best possible.
\bigskip
\subsection{Open Questions}
\bigskip
The first concerns example 9. The argument there works when $n$
is odd, so it is natural to inquire as to what happens when $n$ is
even. Thus, we wonder whether $S_n$ is generated unicursally by
$A=(n-1,\ n)$ and $B=(1\ 2\ 3\ \ldots n-1)$. Rankin's theorem does
not apply directly. However, in the $n=4$ case, by modifying the
argument in the proof of Rankin's theorem, we will show (see
section \ref{newthm}) that $S_4$ is not generated unicursally by
$A=(1\ 2\ 3)$ and $B=(3\ 4)$. The question for even $n\geq 6$
remains open, as far as we are aware.
\bigskip
{\bf Problem 1}: Let $n\geq 6$ be even. Is $S_n$ generated
unicursally by $T=\{A,B\}$ where $A=(n-1,\ n)$ and $B=(1\ 2\ 3\
\ldots n-1)$ ?
\bigskip
According to Knuth \cite{Knuth} a similar question
was asked in 1975 by Nijenhuis and Wilf in their book
\emph{Combinatorial Algorithms}. They asked if $S_n$ is generated
unicursally by $\sigma=(1\ 2\ \ldots n)$ and $\tau = (1\ 2)$.
Rankin's theorem shows that the answer is negative if $n\geq 4$ is
even (see example 10).
Recently, Ruskey-Jiang-Weston \cite{rjw} did a computer
search for $n=5$ and did find that $S_5$ IS unicursally generated
by $\sigma$ and $\tau$. Thus this question is different to problem
1.
\bigskip
This example is not relevant to bells because the
generators are not products of disjoint transpositions.
However, as the answer to problem 1 may well be negative,
the discussion raises the natural question of whether
$S_n$ is generated unicursally by any two of its elements.
(Example 4 shows that $S_n$ is generated unicursally by
three of its elements.)
We have found that $S_4$ IS generated unicursally by
$A=(1\ 2\ 3)$ and $B=(1\ 2\ 3\ 4)$.
Here is one listing which does the trick:
$$B,A,B,A,A,B,B,B,A,B,B,A,A,B,B,B,A,A,B,A,B,B,B,A.$$
The question for $n\geq 5$ remains open.
\bigskip
{\bf Problem 2:} Is $S_n$ generated
unicursally by some two elements for all $n$?
\bigskip
By the above comments, the $n=5$ case has been done.
{\bf Remarks}. A similar eighteenth-century problem to our famous
old question, concerning another method called Stedman, remained
unsolved until 1995. We may discuss this in a future article.
Right transversals of $PSL(2,5)$ in $S_7$ are used to construct
extents in this problem, see \cite{WhiteRing}.
Readers interested only in the proof of the answer to question A
should skip ahead to section \ref{proofs}.
\bigskip
\section{Leads}\label{leads}
\bigskip
In this section we shall explain in detail the two types of leads
mentioned in Remark 3. We only deal in this section with the
methods of Plain Bob on 4 and 6 bells, and Grandsire on 5 and 7
bells, although our remarks have wider application. Then we shall
explain the origin of the famous old question.
\subsection{Plain Bob}
Consider the 25 rows in Plain Bob on 4 bells in section 4; the
first one is rounds, and then the rows can be divided into three
sets of eight. Each of these sets of eight is called a {\it lead}.
In each of these leads, note that each bell is twice in the first
position. Also note that bell number 1 is always in the first
position in the last two rows of each lead. The second of these
two rows, which is the last row of the lead, is called the {\it
lead head}. This holds in general, for Plain Bob on $n$ bells,
where leads have $2n$ rows. In Plain Bob on 4 bells, the lead
heads are $1\ 3\ 4\ 2$, $1\ 4\ 2\ 3$, and $1\ 2\ 3\ 4$. The
following are simple observations:
\begin{enumerate}
\item The first, second and third lead heads are the result of
$P=Y^{-1}Z=(2\ 4\ 3)$, $P^2$ and $P^3$ respectively acting on
rounds. \item When considering only lead heads, we may drop the 1
in the first position. \item Plain Bob can then be described by
elements of $S_{n-1}$ acting on lead heads.
\end{enumerate}
\bigskip
Now we can fully describe Plain Bob on 6 bells by its lead heads.
Here is the first lead (with initial rounds included as well):
\begin{tabular}{rll}
& & \\
1\ 2\ 3\ 4\ 5\ 6& & \\
& $X, (1\ 2)(3\ 4)(5\ 6)$\\
2\ 1\ 4\ 3\ 6\ 5 & &\\
& $Y,(1\ 3\ 5\ 6\ 4\ 2)$\\
2\ 4\ 1\ 6\ 3\ 5 & &\\
& $X,(1\ 4)(3\ 6)$\\
4\ 2\ 6\ 1\ 5\ 3 & &\\
& $Y, (1\ 5\ 4)(2\ 3\ 6)$\\
4\ 6\ 2\ 5\ 1\ 3 & &\\
& $X,(1\ 6)(2\ 4)(3\ 5)$\\
6\ 4\ 5\ 2\ 3\ 1 & &\\
& $Y,(1\ 6)(2\ 5)(3\ 4)$\\
6\ 5\ 4\ 3\ 2\ 1 & &\\
& $X,(1\ 5)(2\ 6)(3\ 4)$\\
5\ 6\ 3\ 4\ 1\ 2 & &\\
& $Y,(1\ 4\ 5)(2\ 6\ 3)$\\
5\ 3\ 6\ 1\ 4\ 2 & &\\
& $X,(1\ 3)(2\ 5)(4\ 6)$\\
3\ 5\ 1\ 6\ 2\ 4 & &\\
& $Y,(1\ 2\ 4\ 6\ 5\ 3)$\\
3\ 1\ 5\ 2\ 6\ 4 & &\\
& $X,(2\ 3)(4\ 5)$\\
1\ 3\ 2\ 5\ 4\ 6 & &\\
& $Z, (2\ 4\ 6\ 5\ 3)$\\
1\ 3\ 5\ 2\ 6\ 4 & &
\end{tabular}
\bigskip
In this case $P=Y^{-1}Z=(2\ 4\ 6\ 5\ 3)$ and the lead heads are
$3\ 5\ 2\ 6\ 4$, $5\ 6\ 3\ 4\ 2$, $6\ 4\ 5\ 2\ 3$, $4\ 2\ 6\ 3\ 5$
and $2\ 3\ 4\ 5\ 6$, corresponding to $P, P^2, P^3, P^4$ and $P^5$
respectively, acting on $2\ 3\ 4\ 5\ 6$. Each of these leads is
called a {\it plain} lead. Each lead head is obtained from the
previous lead head by applying $P$. There are five leads because
$P$ has order 5. This sequence of five plain leads is called a
{\it plain course}.
\bigskip
As we said in Remark 3, there are only two types
of lead considered in this paper.
Let us now describe the other type of lead, at least as
far as Plain Bob on 6 bells goes.
Mathematically there is no reason to have a method
made of only two or three
types of lead, but this is usually what is done
in practice for simplicity and historical reasons
(recall Remark 1).
The complete method is made up of a succession of
leads.
Any plain lead may be described by the sequence of permutations
$$X, Y, X, Y, X, Y, X, Y, X, Y, X, Z.$$
Alternatively, considering only lead heads, we describe a plain
lead by $P$, and the plain course (which has 60 permutations) by
the sequence of five plain leads
$$P, P, P, P, P.$$
The other type of lead is called a {\it bob} lead and
may be described by the
sequence of permutations
$$X, Y, X, Y, X, Y, X, Y, X, Y, X, W$$
where $W=(2\ 3)(5\ 6)$.
If this were done from rounds we would get
\begin{tabular}{rll}
& & \\
1\ 2\ 3\ 4\ 5\ 6& & \\
& $X,\ (1\ 2)(3\ 4)(5\ 6)$\\
2\ 1\ 4\ 3\ 6\ 5 & &\\
& $Y,(1\ 3\ 5\ 6\ 4\ 2)$\\
2\ 4\ 1\ 6\ 3\ 5 & &\\
& $X,(1\ 4)(3\ 6)$\\
4\ 2\ 6\ 1\ 5\ 3 & &\\
\ \ \ \ \vdots & \vdots & \\
& & \\
3\ 5\ 1\ 6\ 2\ 4 & &\\
& $Y,(1\ 2\ 4\ 6\ 5\ 3)$\\
3\ 1\ 5\ 2\ 6\ 4 & &\\
& $X,(2\ 3)(4\ 5)$\\
1\ 3\ 2\ 5\ 4\ 6 & &\\
& $W,(4\ 6\ 5)$\\
1\ 2\ 3\ 5\ 6\ 4 & &
\end{tabular}
\bigskip
This is a bob lead.
As we used $P$ to denote a plain lead we shall use
$B$ to denote a bob lead.
We can now construct longer methods using
a combination of plain and bob leads.
Here is one such method:
$$P, P, P, P, B, P, P, P, P, B, P, P, P, P, B.$$
This corresponds to doing the first 59 of the 60 permutations in
the plain course. The 60th permutation that would be performed in
a plain course is $Z$, which would bring us back to rounds.
Instead of this last $Z$, we do $W=(2\ 3)(5\ 6)$. This has the
effect of putting us into another coset, namely $BH_{12}$ where
$B=Z^{-1}W=(3\ 4)(5\ 6)(2\ 3)(5\ 6)=(2\ 3\ 4)$. We then repeat the
same 59 permutations, then do $W$ again, then the 59 and then $W$
again, which returns us to rounds since $B$ has order 3. We finish
up with a method of 180 permutations. By a similar argument as in
section \ref{plainbob}, rules (1),(2),(3),(4R), are obeyed.
\bigskip
We have still not succeeded in getting an extent
of $6!=720$ permutations.
It is possible that some other
sequence of plain and bob leads will give us an extent.
The following result ends all hope of this.
\begin{theorem} \label{pbminor}
There does not exist an extent of Plain Bob on 6 bells using
plain and bob leads. The longest possible method using plain and
bob leads has $360$ permutations, and there does exist such a
method.
\end{theorem}
Proof: The key to the proof is to observe that $P, B, Z$ and $W$
are all even permutations. The fact that $P$ and $B$ are even
implies that any lead head will be an even permutation of $2\ 3\
4\ 5\ 6$. Also, the row before a lead head is the result of
applying either $Z^{-1}$ or $W^{-1}$ to the lead head. Since $Z$
and $W$ are even, we see that in any method of plain and bob leads
all rows with bell 1 in the first position are followed by an even
permutation of $2\ 3\ 4\ 5\ 6$. The result follows, because if we
did obtain an extent we would get all possible permutations of $2\
3\ 4\ 5\ 6$ following 1.
This argument also shows that any method using plain
and bob leads has at most $5!/2=30$ leads, since each lead
has two rows with 1 in the first position, and these
rows must be followed by an even
permutation of $2\ 3\ 4\ 5\ 6$.
Each lead has 12 rows, so a method
with plain and bob leads has at most
$12\times 30 = 360$ permutations.
To show that $360$ is possible we give an ordering:
$$B, P, P, P, B, B, P, P, P, P,
B, P, P, P, B, B, P, P, P, P, B, P, P, P, B, B, P, P, P, P.$$ The
reader may check that this sequence of plain and bob leads obeys
rules (1),(2),(3),(4R).
\bigskip
{\bf Remarks}.
The use of plain and bob leads applies to Plain Bob
on $n$ bells (and other methods).
The number of leads in an extent on $n$ bells is
$n!/(2n)=(n-1)!/2$, which is the cardinality of $A_{n-1}$.
If $P$ and $B$ generate $A_{n-1}$ unicursally, then
we can construct an extent made up of plain and bob leads.
On 6 bells it is true that $P$ and $B$ generate $A_5$, but example
4 and theorem \ref{pbminor} both show that they do not generate
$A_5$ unicursally. Example 4 used Rankin's theorem, but theorem
\ref{pbminor} gives a different and shorter proof. The argument in
theorem \ref{pbminor} is shorter because parity can be used to
answer the question. The famous old question is an analogous
question about $A_6$ requiring a more delicate argument since it
is nearly possible to acheive an extent.
In the language of section \ref{proofs}, the longest chain
generated by $P$ and $B$ has length 30.
This proof gives the idea of how to construct an extent: use odd
permutations for $Z$ and $W$ but even permutations for $P$ and
$B$. Any method with these properties has a chance of working.
This idea leads to results of Saddleton (see theorems 4.8 and 4.11
of \cite{WhiteRing}).
In practice another type of lead, called a single lead,
is used to obtain an extent.
\bigskip
\subsection{Grandsire}
\bigskip
We now consider another method, the last of this article.
The method named Grandsire
(pronounced grand-sir)
is rung on an odd number of bells.
It was developed in the 1650's by Robert Roan on 5 bells,
and extensions to 7 and more bells took place in the
late 1600's or later.
The problem we referred to in
the abstract is on 7 bells, but first we explain
Grandsire on 5 bells.
The hunting subgroup $H_{10}$ is generated by $X=(1\ 2)(3\ 4)$ and
$Y=(2\ 3)(4\ 5)$ as usual. We introduce $Z=(1\ 2)(4\ 5)$, but the
first difference in Grandsire from Plain Bob is that we do $Z$ at
the very start. This is irrelevant from a mathematical point of
view. Then we do $Y$, and then alternate $X$ and $Y$ until we have
run through the coset $ZH_{10}$. The last permutation done will be
$Y$, and in total we will have done $ZYXYXYXYXY=ZX$. Then we
repeat the permutations, i.e., do $Z, Y, X, Y, \ldots , X, Y$
until we have run through the coset $(ZXZ)H_{10}$, and then we
repeat the permutations again, running through $(ZXZXZ)H_{10}$,
and then we are back at rounds. Here are the rows of 3 plain
leads, a plain course. Neglecting the first row which is the
initial rounds, each column is a plain lead.
\bigskip
\begin{tabular}{rlllrl}
1\ 2\ 3\ 4\ 5 & &\\
& $Z$ & & $Z$ & & $Z$ \\
2\ 1\ 3\ 5\ 4 & & 2\ 1\ 5\ 4\ 3 & & 2\ 1\ 4\ 3\ 5\\
& $Y$ & & $Y$ & & $Y$ \\
2\ 3\ 1\ 4\ 5 & & 2\ 5\ 1\ 3\ 4 & & 2\ 4\ 1\ 5\ 3 \\
& $X$ & & $X$ & & $X$ \\
3\ 2\ 4\ 1\ 5 & & 5\ 2\ 3\ 1\ 4 & & 4\ 2\ 5\ 1\ 3\\
& $Y$ & & $Y$ & & $Y$ \\
3\ 4\ 2\ 5\ 1 & & 5\ 3\ 2\ 4\ 1 & & 4\ 5\ 2\ 3\ 1\\
& $X$ & & $X$ & & $X$ \\
4\ 3\ 5\ 2\ 1 & & 3\ 5\ 4\ 2\ 1 & & 5\ 4\ 3\ 2\ 1\\
& $Y$ & & $Y$ & & $Y$ \\
4\ 5\ 3\ 1\ 2 & & 3\ 4\ 5\ 1\ 2 & & 5\ 3\ 4\ 1\ 2\\
& $X$ & & $X$ & & $X$ \\
5\ 4\ 1\ 3\ 2 & & 4\ 3\ 1\ 5\ 2 & & 3\ 5\ 1\ 4\ 2\\
& $Y$ & & $Y$ & & $Y$ \\
5\ 1\ 4\ 2\ 3 & & 4\ 1\ 3\ 2\ 5 & & 3\ 1\ 5\ 2\ 4\\
& $X$ & & $X$ & & $X$ \\
1\ 5\ 2\ 4\ 3 & & 1\ 4\ 2\ 3\ 5 & & 1\ 3\ 2\ 5\ 4\\
& $Y$ & & $Y$ & & $Y$ \\
1\ 2\ 5\ 3\ 4 & & 1\ 2\ 4\ 5\ 3 & & 1\ 2\ 3\ 4\ 5
\end{tabular}
\bigskip
It is because the first lead head is the result of $ZX=(1\ 2)(4\
5)(1\ 2)(3\ 4)=(3\ 4\ 5)$ which has order 3, that we get back to
rounds after $3\times 10=30$ permutations, and a plain course has
3 plain leads.
\bigskip
As with the Plain Bob method, we
sometimes add bob leads to the above plain course
to obtain a longer method.
The bob lead uses the permutation $Z$
applied instead of the last $X$ in a plain lead,
which would be
{\it two places before} the next appearance
of $Z$ in a plain course.
\begin{tabular}{rll}
1\ 2\ 3\ 4\ 5 \\ & $Z$ & \\ 2\ 1\ 3\ 5\ 4 \\
& $Y$ & \\ 2\ 3\ 1\ 4\ 5 \\ & $X$ & \\ 3\ 2\ 4\
1\ 5 \\ & $Y$ & \\ 3\ 4\ 2\ 5\ 1 \\ & $X$ & \\ 4\ 3\ 5\
2\ 1 \\ & $Y$ & \\ 4\ 5\ 3\ 1\ 2 \\ & $X$ & \\
5\ 4\ 1\ 3\ 2 \\ & $Y$ & \\ 5\ 1\ 4\ 2\ 3 \\ & $Z$
& \\ 1\ 5\ 4\ 3\ 2 \\ & $Y$ & \\ 1\ 4\ 5\ 2\ 3 \\ & $Z$&
\end{tabular}
\bigskip
If we use the bob before the first lead head, as shown above, the
first lead head will be $1\ 4\ 5\ 2\ 3$ instead of $1\ 2\ 5\ 3\
4$. This is the result of $(2\ 4)(3\ 5)$ applied to rounds. Since
this permutation has order 2, we will return to rounds after using
the bob twice in that place. In other words, the method consisting
of leads $B, P, P, B, P, P$ increases the number of permutations
from $30$ to $60$.
It is reasonable to ask, as usual,
if we could obtain a larger set
of permutations by using plain and bob leads in a
different arrangement.
The answer is no. To see this, simply note that each
of $X, Y, Z$ is even.
Therefore the largest possible number of permutations
they can generate is 60 (the order of $A_5$), which
in fact is the case as we have shown.
\begin{theorem} \label{grandoubles}
There does not exist an extent of Grandsire on 5 bells
using plain and bob leads.
The longest possible method using plain and bob leads has $60$
permutations, and there does exist such a method.
\end{theorem}
To obtain the maximum number
of permutations on 5 bells
we would need to use an odd permutation,
which involves another type of lead called a
single lead. We do not discuss single leads in
this article.
\bigskip
Next we consider Grandsire on 7 bells, or Grandsire Triples as it
is known to ringers, which is more interesting mathematically than
Grandsire on 5 bells. In this case $X=(1\ 2)(3\ 4)(5\ 6)$ and
$Y=(2\ 3)(4\ 5)(6\ 7)$ generate the hunting subgroup $H_{14}$, and
Grandsire Triples uses $Z=(1\ 2)(4\ 5)(6\ 7)$. As on 5 bells, we
do $Z$ first and then alternate $Y$ and $X$, and repeat. The first
lead head is the result of $ZX=(1\ 2)(4\ 5)(6\ 7)(1\ 2)(3\ 4)(5\
6)=(3\ 4\ 6\ 7\ 5)$ on rounds, which is $1\ 2\ 5\ 3\ 7\ 4\ 6$.
Since $ZX$ has order 5 there are 5 plain leads and 70 permutations
in a plain course of Grandsire on 7 bells.
To extend the method we use bob leads. The bob lead uses the
permutation $Z$ applied instead of the last $X$ in a plain lead,
as in Grandsire on 5 bells. If we do a bob lead on the earliest
possible occasion, the first lead head would become $1\ 7\ 5\ 2\
6\ 3\ 4$. This is the result of $B=(2\ 4\ 7)(3\ 6\ 5)$ acting on
rounds. Since $B$ has order 3, we obtain a total of 210
permutations if we use $B, P, P, P, P, B, P, P, P, P, B, P, P, P,
P$.
It is possible to obtain larger sets of permutations by using
plain and bob leads in different sequences. We may then reasonably
wonder as to the largest set we can get. On 5 bells we used that
fact that $X, Y, Z$ are even to obtain an upper bound (which was
60). This argument will not work here, since $X$ and $Y$ are odd.
It is, in fact, conceivable that we could achieve an extent of all
$7!=5040$ permutations. As before, it is enough to consider the
action of $P$ and $B$ on lead heads. In 1741 John Holt came very
close to an extent and obtained $4998$ permutations. This gave
rise to the famous old question mentioned in the introduction and
section \ref{groupform}:
\bigskip
{\bf Famous Old Question}. Is it possible to ring all the $5040$
permutations on seven bells using the Grandsire method and plain
and bob leads only? In other words, does there exist an extent of
Grandsire on 7 bells using plain and bob leads?
\bigskip
Considering only lead heads, first note that there would be
$5040/14=360$ lead heads. Next check that $P=(3\ 4\ 6\ 7\ 5)$ and
$B=(2\ 4\ 7)(3\ 6\ 5)$ generate $A_6$. If $P$ and $B$ generate
$A_6$ unicursally then we would obtain the extent we are looking
for. We therefore arrive at the following question in order to
answer the Famous Old Question:
\bigskip
{\bf Question A}. Is $A_6$ (acting on $\{2,3,4,5,6,7\}$) generated
unicursally by $P=(3\ 4\ 6\ 7\ 5)$ and $B=(2\ 4\ 7)(3\ 6\ 5)$ ?
\subsection{A bob for the new method}
We must define a bob lead for the "new" method proposed in section
\ref{ccc}. We propose the permutation $W=(3\ 4)$ instead of the
last $Z$ in a lead. Thus a plain lead has the permutations
\[
X,Y,X,Y,X,Y,X,Y,X,Z
\]
and a bob lead has the permutations
\[
X,Y,X,Y,X,Y,X,Y,X,W.
\]
The sequence of leads $P,P,P,B,P,P,P,B,P,P,P,B$ yields an extent
of all 120 permutations.
\bigskip
\section{Proofs} \label{proofs}
\bigskip
We now proceed to give Rankin's proof that question A has a
negative answer. As we explained in section \ref{leads}, this
implies that the answer to the Famous Old Question is no. The
proof is by contradiction.
\medskip
We begin by supposing that $A_6$ is
generated unicursally by
$P=(3\ 4\ 6\ 7\ 5)$ and $B=(2\ 4\ 7)(3\ 6\ 5)$.
Assume there exists an ordering $g_1, g_2, \ldots ,g_{360}$
of the elements of $A_6$,
with the property that for each $i$,
$g_{i+1}=g_{i}t_i$ for some $t_i\in T=\{P, B\}$
(with subscripts modulo 360).
We will call any sequence of elements
$g_1, \ldots , g_m$ a {\it chain} of length $m$
if it has the property that for each $i$,
$g_{i+1}=g_{i}t_i$ for some $t_i\in T=\{P, B\}$.
With subscripts modulo $m$,
we consider a chain to be an infinite cyclic sequence.
If $g_{i+1}=g_{i}P$ then we say that $g_i$ is
{\it acted on} by $P$, and similarly for $B$.
Each element of $A_6$ is acted on by exactly
one of $P$ and $B$, by assumption.
The key is to consider left cosets of the cyclic
subgroup $C$ of order 5 generated by
$$\gamma= BP^{-1}=(2\ 4\ 7)(3\ 6\ 5)(3\ 5\ 7\ 6\ 4)=
(2\ 3\ 4\ 6\ 7).$$
This was noted by Thompson, who called these
cosets ``Q-sets".
We present Rankin's argument in a series of
observations, each one a lemma.
The idea of the proof is to show that, under a certain
transformation, the parity of the
number of chains remains constant.
\begin{lemma}\label{lemma1}
Every element of a coset $xC$ of $C$ is acted on by the same
element.
\end{lemma}
Proof: Suppose $x\gamma^i$ is acted on by $P$ (a similar argument
holds for $B$). Then the next element in the chain is
$x\gamma^iP$. But
$x\gamma^iP=x\gamma^{i-1}(BP^{-1})P=x\gamma^{i-1}B$. To avoid
repetition therefore, $B$ cannot act on $x\gamma^{i-1}$, so $P$
must act on $x\gamma^{i-1}$. This argument is valid for any $i$.
\bigskip
We now consider where the elements of a coset $xC$ appear in the
chain. For each $i$ between 1 and 5, we define a positive integer
$k_i$ between 1 and 5, by letting $x\gamma^{k_i}$ be the next
element of $xC$ in the chain after $x\gamma^{i}$. This defines a
permutation in $S_5$, which in two-line notation is
$$\sigma (x)=\pmatrix{1&2&3&4&5\cr
k_1&k_2&k_3&k_4&k_5\cr}$$ for each coset $xC$.
\begin{lemma}\label{lemma2}
The permutation $\sigma (x)$ is a 5-cycle.
\end{lemma}
Proof: Our assumption on the existence of a chain of length 360
implies that $\sigma (x)$ is a 5-cycle.
\bigskip
We will now rearrange the chain. It is possible that
the `length 360 chain'
property may be destroyed during the rearrangement,
i.e., the single chain of length 360
may become several (disjoint) chains of smaller length.
Here is how the rearrangement is done. Let $xC$ be a coset acted
on by $B$ (if there is no such coset then every element of $A_6$
is acted on by $P$, which is impossible if there is only one
chain). Then the next element in the chain after $x\gamma^i$ is
$x\gamma^iB$. The chain can be divided up into 5 segments with
respect to $xC$, each segment being a sequence beginning with
$x\gamma^iB$ and ending with $x\gamma^{k_i}$. By definition of
$k_i$, there are no elements of $xC$ in the chain from
$x\gamma^iB$ to the element immediately preceding $x\gamma^{k_i}$.
In other words, in these segments with respect to $xC$, the only
element in a segment that is in $xC$ is the last element.
$$ \cdots x\gamma^i][x\gamma^iB \cdots x\gamma^{k_i}][x\gamma^{k_i}B \cdots $$
We permute these segments, so that the segment after $x\gamma^i$
now begins with $x\gamma^{i-1}B$ (and ends with
$x\gamma^{k_{i-1}}$).
$$ \cdots x\gamma^i][x\gamma^{i-1}B \cdots x\gamma^{k_{i-1}}]
[x\gamma^{k_{i-1}-1}B \cdots $$
\begin{lemma}\label{lemma3}
After the rearrangement, the coset $xC$ is acted on by $P$. All
other cosets are unaffected, in terms of whether they are acted on
by $P$ or $B$.
\end{lemma}
Proof: Note that
$x\gamma^{i-1}B=x\gamma^{i-1}BP^{-1}P=x\gamma^iP$, so the element
after $x\gamma^i$ is $x\gamma^iP$. This implies that $x\gamma^i$,
and therefore the coset $xC$, is now acted on by $P$ in the
rearrangement. The proof of the second part is clear from the
construction.
\begin{lemma}\label{lemma4}
After the rearrangement there may be more than one chain, but
every chain contains an element of $xC$.
\end{lemma}
Proof: The rearrangement may alter the number of chains because,
for example, we may have $k_1=3$ and $k_2=2$. In this case, after
rearranging, we would have
$$[x\gamma B\cdots x\gamma^3][x\gamma^2B\cdots x\gamma^2]$$
which is a chain, and the other segments would form at least
one other chain.
If there were a chain not containing any element of $xC$ after the
rearrangement, this chain would not be affected by the
rearranging, and would therefore have existed before the
rearrangement. But there was only one chain before the
rearrangement!
\bigskip
The next element of $xC$ after $x\gamma^i$ in the new arrangement
is $x\gamma^{k_{i-1}}$, so we define a permutation $\tau(x)$ in a
similar manner to $\sigma(x)$:
$$\tau (x)=\pmatrix{1&2&3&4&5\cr
k_5&k_1&k_2&k_3&k_4\cr}.$$
In the following, ``cycles'' means disjoint cycles including
1-cycles, as usual.
\begin{lemma}\label{lemma5}
The number of cycles in $\tau(x)$ is equal to the number of chains
after the rearrangement.
\end{lemma}
Proof: This follows from Lemma \ref{lemma4}.
\begin{lemma}\label{lemma6}
$(1\ 2\ 3\ 4\ 5)\tau(x)=\sigma(x)$.
\end{lemma}
Proof: This is straightforward.
\begin{lemma}\label{lemma7}
The number of cycles in $\tau(x)$ is odd.
\end{lemma}
Proof: Suppose $\tau (x)$ has $k$ cycles. The number of cycles in
$(1\ 5)\tau(x)$ is $k+1$ if 1 and 5 are in the same cycle in
$\tau(x)$, and $k-1$ otherwise. Hence the number of cycles in
$(1\ 4)(1\ 5)\tau(x)$ has the same parity as the number of cycles
in $\tau(x)$. Similarly the number of cycles in $(1\ 2)(1\ 3)(1\
4)(1\ 5)\tau(x)$ has the same parity as the number of cycles in
$\tau(x)$. Lemmas \ref{lemma2} and \ref{lemma6} now give the
result.
\bigskip
We point out that this proof works since 5 is odd, so any odd
number could be used. If an even number is used instead of 5 then
the proof shows that the parity changes. This will be used in the
proof of theorem \ref{newthms4}.
Let $r_x$ denote the number of
chains after the rearrangement with respect to $xC$.
The following fact is all-important.
\begin{lemma}\label{lemma8}
$r_x$ is odd.
\end{lemma}
Proof: Combine Lemmas \ref{lemma5} and \ref{lemma7}.
We now rearrange again with respect to another coset $yC$ that is
acted on by $B$. (If there is no such coset, skip to the Famous
Old Theorem below.) Let $r_y$ denote the number of chains after
the rearrangement with respect to $yC$. Again we define
$k_i=k_i(y)$ and $\sigma (y)$ in a similar manner. Lemma
\ref{lemma2} becomes
\begin{lemma}\label{lemma2'}
The number of chains (before rearrangement with respect to $yC$)
containing elements of $yC$ is equal to the number of cycles in
$\sigma (y)$.
\end{lemma}
Lemma \ref{lemma3} shows that after rearranging, $yC$ is acted on
by $P$. Lemma \ref{lemma4} becomes
\begin{lemma}\label{lemma4'}
The number of chains not containing elements of $yC$ remains
constant.
\end{lemma}
Proof: Shown in proof of Lemma \ref{lemma4}.
We define $\tau(y)$ similarly to $\tau(x)$. Lemma \ref{lemma5}
becomes
\begin{lemma}\label{lemma5'}
$r_x - (\hbox{number of cycles in }\sigma (y))=
r_y - (\hbox{number of cycles in }\tau(y))$.
\end{lemma}
Proof: The lefthand side is the number of chains before
rearrangement not containing elements of $yC$, and the righthand
side is the number of such chains after rearrangement.
Lemma \ref{lemma6} remains the same, with $y$ in place of $x$, and
Lemma \ref{lemma7} becomes
\begin{lemma} \label{lemma7'}
$(\hbox{number of cycles in }\tau(y) ) \equiv
(\hbox{number of cycles in }\tau(x)) \ \hbox{ mod } \ 2$.
\end{lemma}
Proof: Same as Lemma \ref{lemma7}.
Again, the following will be important.
\begin{lemma} \label{lemma8'}
$r_y$ is odd.
\end{lemma}
Proof: By Lemma \ref{lemma8} and Lemmas \ref{lemma5'} and
\ref{lemma7'}.
\bigskip
The series of lemmas shows that after this second
rearrangement with respect to $yC$, the number of chains
is still odd.
Repeat the rearrangement with respect to
every coset acted on by $B$
until all cosets are acted on by $P$.
The point is that after each rearrangement with respect to a coset
acted on by $B$, the coset is now acted on by $P$, and
also we can apply the lemmas and conclude that the
number of chains remains odd.
\begin{theorem} $($Thompson$)$
\begin{enumerate} \item $A_6$
(acting on $\{2,3,4,5,6,7\}$) is not generated unicursally by
$P=(3\ 4\ 6\ 7\ 5)$ and $B=(2\ 4\ 7)(3\ 6\ 5)$. \item It is not
possible to ring all the $5040$ permutations on seven bells using
the Grandsire method and plain and bob leads. In other words,
there does not exist an extent of Grandsire on 7 bells using plain
and bob leads. \end{enumerate}
\end{theorem}
Proof: (Rankin) We showed in section \ref{leads} that (2) is
implied by (1). To prove (1), by the above lemmas we may assume
that all cosets are acted on by $P$. Then every element of $A_6$
is acted on by $P$. The chains we have must therefore be the
cosets of the subgroup $M$ generated by $P$. By Lemma
\ref{lemma8'}, the number of chains is odd. However, since $P$ has
order 5, there are $|A_6:M|=360/5=72$ cosets, which is an even
number. This contradiction proves the theorem.
\bigskip
We remark that the roles of $P$ and $B$ could be interchanged in
the above proof, since the subgroup generated by $B$ also has even
index.
The following is shown in \cite{Dick}.
\begin{corollary} The largest number of permutations that can be
rung on seven bells, using the Grandsire method and plain and bob
leads, is $4998$.
\end{corollary}
Proof: We know that one chain of length 360 (in $A_6$) is not
possible. The shortest possible chain has length 3 since $B$ has
order 3, so the longest possible chain has length $\leq 357$. A
chain with 357 leads has $357\times 14 =4998$ permutations.
As we said earlier, there does exist a method with 4998
permutations due to Holt, so 4998 is best possible.
\bigskip
\section{A New Result}\label{newthm}
\bigskip
We next prove a small result concerning problem 1 of section
\ref{groupform}. Observe that the result is not trivial since
there are $2^{24}$ possible orderings of $A$ and $B$.
\begin{theorem} \label{newthms4}
$S_4$ is not unicursally generated by $A=(3\ 4)$
and $B=(1\ 2\ 3)$.
\end{theorem}
Proof: Suppose to the contrary that $S_4$ is unicursally generated
by $A$ and $B$. As in the sequence of lemmas, we rearrange with
respect to left cosets of $C=\langle A^{-1}B\rangle$ that are
acted on by $A$. After the rearrangement, such a coset is acted on
by $B$.
Since $C$ has order 4 which is even, the proof of Lemma
\ref{lemma7} shows that after the rearrangement with respect to
$xC$, the number of cycles in $\tau(x)$ is even, and in general
that the parity of the number of cycles changes after each
rearrangement with respect to a coset of $C$.
Suppose we perform a total of $m$ rearrangements to get all cosets
acted on by $B$. By Lemma \ref{lemma3} second sentence, the number
of cosets acted on initially by $A$ is $m$. Note that $m\leq 6$
as $|C|=4$. After all $m$ rearrangements, the number of chains has
the opposite parity to $m$, by the previous paragraph.
By assumption, we start with one chain of length 24.
If $m$ is even then
the number of chains (after all rearrangements)
is odd. The proof of Rankin's theorem above shows that
$\langle B \rangle$ has odd index, a contradiction.
The only alternative is that $m$ is odd, and we have 1, 3, or 5
cosets of $C$ acted on by $A$ initially. If 1 coset was acted on
by $A$, then 5 cosets were acted on by $B$. Therefore 20 elements
of $S_4$ were acted on by $B$, which implies there are three
consecutive $B$'s. This is not possible as $B$ has order 3. If 5
cosets were acted on by $A$, then 1 coset was acted on by $B$, and
a similar argument gives a contradiction.
The final possibility is that 3 cosets were acted
on by $A$ and 3 by $B$.
Then 12 elements of $S_4$ are acted on by each
of $A$ and $B$.
Since $A$ has order 2, we must have $A$ and $B$
alternating in the chain.
But $AB$ and $BA$ have order 4, so neither of these
work.
\bigskip
|
{
"timestamp": "2012-03-09T02:03:32",
"yymm": "1203",
"arxiv_id": "1203.1835",
"language": "en",
"url": "https://arxiv.org/abs/1203.1835"
}
|
\section{Introduction}
The lower central series $\lbrace{L_i(A),i\ge 1\rbrace}$, of a noncommutative algebra $A$ is defined inductively by $L_1(A)=A$, and $L_{i+1}(A)=[A,L_i(A)]$. In other words, $L_i(A)$ is spanned by "$i$-commutators"
$[a_1,[a_2,...[a_{i-1}, a_i]...]]$, where $a_1,...,a_i\in A$. It is
interesting to consider the successive quotients
$B_i(A)=L_i(A)/L_{i+1}(A)$, i.e., $i$-commutators modulo
$i+1$-commutators. The role of the $B_i(A)$ is, roughly, that they
characterize, step by step, the deviation of $A$ from being
commutative; this is somewhat similar to the quasiclassical
perturbation expansion in quantum mechanics.
The quotients $B_i(A)$ can be put together in a direct sum
$B(A)=\oplus_{i\ge 1}B_i(A)$. This is a graded Lie algebra: we have $[B_i(A),B_j(A)]\subset B_{i+j}(A)$. It turns out that this
Lie algebra has a big central part (i.e., a part commuting with
everything) in degree 1, namely the image $I$ of $AL_3(A)$
in~$B_1$. Denote $B_1(A)/I$ by $\bar B_1(A)$. Then we have a graded
Lie algebra $\bar B(A)=\bar B_1(A)\bigoplus\oplus_{i\ge 2}B_i(A)$,
generated by degree 1 (i.e., $\bar B_1(A)$).
The lower central series of the free algebra $A_n=A_n(\mathbb{Q})$ over the
field of rational numbers $\mathbb{Q}$ was studied in a number of
papers \cite{DE}, \cite{DKM}, \cite{AJ}, \cite{BJ}, following the
pioneering paper \cite{FS}. In this paper, Feigin and Shoikhet
observed a surprising fact: the graded spaces $\bar{B}_1(A_n)$ and
$B_i(A_n)$, $i\ge 2$, have polynomial, rather than exponential growth,
even though the dimensions of the homogeneous
components of $A_n$ itself grow exponentially. This stems from the
fact that, even though $A_n$ is "the most noncommutative" of all
algebras, the spaces $\bar{B}_1(A_n)$ and $B_i(A_n)$, $i\ge 2$, can be
described in terms of usual, ``commutative" geometry, namely in terms
of tensor fields on the $n$-dimensional space. This allows one to completely describe $\bar
B_1(A_n)$ and $B_2(A_n)$ in terms of differential forms \cite{FS}
and $B_3(A_n)$ as well as $B_i(A_n)$ for some $n$ and $i>3$ in terms
of more complicated tensor fields \cite{DE}, \cite{DKM}, \cite{AJ},
\cite{BJ}. However, the problem of describing of $B_i(A_n(\Bbb Q))$
for all $i,n$, and in particular of computation of the Hilbert series
of these spaces still remains open.
This paper is dedicated to the study of $\bar B_1(A_n)$ and $B_i(A_n)$ over the integers $\Bbb Z$ and over a finite field $\Bbb F_p$. In this case, rich new structures emerge. Namely, in the case of $\Bbb Z$, the groups $\bar B_1(A_n)$, $B_i(A_n)$ develop torsion, and it is interesting to study the pattern of this torsion. In the case of $\Bbb F_p$ the dimensions of certain homogeneous subspaces of $\bar B_1(A_n)$, $B_i(A_n)$ differ from those over $\Bbb Q$, and the patterns of these discrepancies are of interest.
Our main results are as follows. First of all, we give a complete description of $\bar{B}_1(A_n(\Bbb Z))$ in terms of differential forms, and of the torsion in $\bar B_1(A_n(\mathbb{Z}))$ in terms of the De Rham cohomology of the n-dimensional space over $\mathbb{Z}$. Since $\bar B_1(A_n(\Bbb F_p))=\bar B_1(A_n(\Bbb Z))\otimes \Bbb F_p$, this completely describes $\bar B_1(A_n(\Bbb F_p))$ in terms of differential forms. Since the Lie algebra
$\bar B$ is generated by $\bar B_1$,
this description implies a uniform bound for dimensions of
the homogeneous subspaces, which depends only on $i$ and $n$ (``polynomial growth''
for $B_i(A_n(\Bbb F_p))$).
Also, we give a description
of the torsion in $B_2(A_n(\Bbb Z))$, in terms of the De Rham cohomology over the integers
(conjectural for $2^r$-torsion). Since
$B_2(A_n(\Bbb F_p))=B_2(A_n(\Bbb Z))\otimes \Bbb F_p$, this describes
$B_2(A_n(\Bbb F_p))$ in terms of differential forms. Further, we give an explicit formula for some torsion elements in $B_2(A_3(\Bbb Z))$, and show that they are a basis of torsion.
We also present some theoretical and computer results (using the MAGMA package, \cite{M}) for torsion in $B_i(A_n(\Bbb Z))$ for $i>2$, and
propose a number of conjectures based on them.
In particular, we conjecture that the degree of any torsion element with respect to each variable
is divisible by the order of the torsion.
Finally, we provide some computer data and theoretical results for
torsion in $B_i$ in the supercase, i.e. for
free algebras $A_{n,k}$ in $n+k$ generators, where
the first $n$ generators are even and the last $k$ generators are odd, and
formulate a number of open questions based on this data. However, the pattern of torsion in higher $B_i$ remains mysterious even at the conjectural level, and will be the subject of further investigation.
The organization of the paper is as follows.
In Section 2, we discuss preliminaries.
In Section 3, we describe the structure of
$\bar B_1(A_n)$. In Section 4, we describe the structure of $B_2(A_n)$.
In Section 5, we discuss the experimental data and conjectures
for the structure of $B_i(A_n(\Bbb Z))$ for $i>2$.
In Section 6, we discuss the theoretical results, experimental data and open problems in the supercase.
Finally, in Section 7, we outline some directions of further research.
{\bf Acknowledgments.} The work of P.E. was partially supported
by the NSF grant DMS-1000113. The work of D.J. was supported by the NSF grant DMS-1103778. S.B., W.K., and J.L.
are grateful to the PRIMES program at the MIT Mathematics Department, where this research was done.
\section{Preliminaries}
\subsection{The lower central series}
Let $A$ be an algebra over a commutative ring $R$.
In this paper, we will consider algebras over $R = \mathbb{Q},\mathbb{F}_p,\mathbb{Z}$.
\begin{defn}
The lower central series filtration of $A$ is defined as $L_1(A):=A, L_{k+1}(A):= [A,L_{k}(A)]$ for $k \in \mathbb{N}$.
\end{defn}
In other words, we have \[ L_k = \text{Span}_R \{[a_1, [a_2, \ldots, [a_{k-1}, a_k]] \ldots] | a_1,\ldots, a_k\in A\}. \]
\begin{defn}
The associated graded Lie algebra of $A$ is \[ B(A):= \displaystyle\bigoplus_{k \ge 1}{} B_k(A) \] where $B_k(A):=L_k/L_{k+1}$ for all $k \in \mathbb{N}$.
\end{defn}
\begin{defn}
The two-sided ideals $M_k(A)$ are defined as $M_k(A) := AL_k$ for $k \in \mathbb{N}$.
\end{defn}
\begin{lem} \label{lem:FSLemma1}
$[A,M_3] \subset L_3$.
\end{lem}
\begin{proof}
Lemma 2.2.1 from \cite{FS} asserts this for algebras over $\mathbb{C}$. However, the formulas in the proof only involve $\pm 1$ coefficients, so the proof applies for an arbitrary ring $R$.
\end{proof}
\begin{defn}
The abelian group $\bar{B}_1$ is the quotient $\bar{B}_1:=A/(L_2+M_3)$.
\end{defn}
Thus, $\bar{B}_1$ is the quotient of $B_1$ by the image $I$ of $M_3$ in $B_1$, which by Lemma ~\ref{lem:FSLemma1} is central in $B(A)$.
\begin{cor}
The graded module
$$
\bar{B}(A) := \bar{B}_1 \oplus \displaystyle \bigoplus_{k \ge 2} B_k
$$
is a graded Lie algebra generated in degree 1.
\end{cor}
\begin{defn}
The free algebra $A_n(R)$ of rank $n$ over $R$ is the free $R$-module generated by all finite words in the letters $x_1,\ldots, x_n$; the multiplication is given by concatenation of words.
\end{defn}
\subsection{De Rham cohomology over the integers}
We will see that torsion in $\bar B_1(A_n(\Bbb Z))$ and $B_2(A_n(\Bbb Z))$ is related to the De Rham cohomology
of the $n$-dimensional space over the integers. So let us discuss how to compute this cohomology.
Denote by $\Omega(R^n)$ the module of differential forms in $n$ variables with coefficients in $R$.
We have a rank decomposition $\Omega(R^n)=\oplus_{i=0}^n \Omega^i(R^n)$,
and the De Rham differential $d: \Omega^i(R^n)\to \Omega^{i+1}(R^n)$, which defines the
De Rham complex of the $n$-dimensional space over $R$. We will use the notation $\Omega_{cl}$ for closed forms, $\Omega_{ex}$ for exact forms,
$\Omega^{ev}$ for even forms, $\Omega^{odd}$ for odd forms, $\Omega^+$ for positive rank
forms, $\Omega^{\ge k}$ for forms of rank $\ge k$, etc.
If $R$ is a $\Bbb Q$-algebra
then by the Poincar\'e lemma, the De Rham complex is acyclic in degrees $i>0$,
and its zeroth cohomology $H^0$ consists of constants (i.e., equals $R$).
However, this fails for general rings. In this section we compute the cohomology
of this complex for $R=\Bbb Z$.
\begin{lem}\label{omega1} We have:
$$H^k(\Omega(\mathbb{Z}))[m] \cong \left\{\begin{array}{ll} \mathbb{Z},& k=0, \, m=0\\ \mathbb{Z}/m, & k=1, \, m > 0, \\0,& \mathrm{otherwise} \end{array}\right.$$
(Here $[m]$ denotes the degree $m$ part).
\end{lem}
\begin{proof} Straightforward computation.\end{proof}
\begin{cor}\label{kunnethcor} Let $H^r_n := H^r (\Omega (\mathbb{Z}^n))$. We have a (noncanonically) split short exact sequence in cohomology,
\[ 0 \to H^{r}_{n-1} \oplus H^{r-1}_{n-1}
\otimes H^1_1 \to H^r_n \to \mathrm{Tor}(H^r_{n-1}, H^1_1) \to 0. \]
Moreoever, this is a short exact sequence of graded abelian groups, with respect to the natural $\mathbb{Z}^n$-grading on each term.
\end{cor}
\begin{proof}
Apply the
K\"unneth Theorem and Lemma \ref{omega1} to the decomposition
$$
\Omega(\mathbb{Z}^n)=\Omega(\mathbb{Z}^{n-1})\otimes \Omega(\mathbb{Z}).
$$
\end{proof}
\begin{cor}\label{Hev-description} (see also \cite{FR}) Suppose that all $m_i>0$. Then we have an isomorphism:
$$\text{H}^{i}(\Omega(\Bbb Z^n))[m_1, \ldots , m_n] \cong (\mathbb{Z}/gcd(m_1,\ldots,m_n))^{{n-1}\choose{i-1}},
$$
where $[m_1,...,m_n]$ denotes the part of multidegree $m_1,...,m_n$.
\end{cor}
\begin{rem}
Note that this describes the cohomology completely, since
the case when $m_i=0$ for some $i$ reduces to a smaller number of variables.
\end{rem}
\begin{proof}
We choose a splitting of the short exact sequence of Corollary \ref{kunnethcor}, and apply Lemma \ref{omega1} to obtain:
$$H^r_n \cong H^{r}_{n-1} \oplus \bigoplus_{m\geq 1} \left(H^{r-1}_{n-1}\otimes \mathbb{Z}/m \oplus \mathrm{Tor}(H^r_{n-1}, \mathbb{Z}/m)\right).$$
Note that each $\mathbb{Z}/m$ in the sum above is in multi-degree $(0,\ldots,0,m)$. Thus the $(m_1,\ldots,m_n)$-graded part of the above short exact sequence reads:
$$H_n^i[m_1,...,m_n]\cong H_{n-1}^i[m_1,...,m_{n-1}]/m_n\oplus H_{n-1}^{i-1}[m_1,...,m_{n-1}]/m_n.$$
The formula now follows by induction on $n$ and $i$, and the identity ${{r+1}\choose{i+1}}={{r}\choose{i}} + {{r}\choose{i+1}}$.
\end{proof}
\subsection{Universal coefficient formulas for $\bar{B}_1$ and $B_2$}
Let $p$ be a prime and $\mathbb{Z}^N \supset A \supset B$ be abelian groups.
For an abelian group $A$, let ${\rm tor}_p(A)$ denote the $p$-torsion of $A$.
\begin{lem}\label{lem:Universal}
Let $A_p$ and $B_p$ be the images of $A$ and $B$ in ${\Bbb F}_p^N$.
If ${\rm tor}_p(\mathbb{Z}^N / A)=0$ (in particular, if $\mathbb{Z}^N / A$ is free)
then the natural map \[ A_p / B_p \rightarrow (A/B) \otimes \mathbb{F}_p \] is an isomorphism.
\end{lem}
\begin{proof}
We have $\mathbb{Z}^N/A = \dfrac{\mathbb{Z}^N / B}{A/B}$. Since ${\rm tor}_p(\mathbb{Z}^N/A)=0$, this implies that we have an isomorphism of $p$-local parts
$(\mathbb{Z}^N /B)\otimes {\Bbb Z}_{(p)} = (A/B \oplus \mathbb{Z}^N / A)\otimes \Bbb Z_{(p)}$,
where $\Bbb Z_{(p)}$ is the ring of rational numbers whose denominator is coprime to $p$.
When we tensor the equality with $\mathbb{F}_p$, we see that $(A/B) \otimes \mathbb{F}_p$ is the kernel of the projection map $(\mathbb{Z}^N / B) \otimes \mathbb{F}_p \rightarrow (\mathbb{Z}^N / A) \otimes \mathbb{F}_p$, which is $A_p / B_p$, as desired.
\end{proof}
\begin{cor}\label{unco}
We have
\begin{enumerate}
\item[\begin{small}1)\end{small}] $\bar{B}_1(A_n(\mathbb{F}_p)) = \bar{B}_1 (A_n(\mathbb{Z})) \otimes \mathbb{F}_p$;
\item[\begin{small}2)\end{small}] $B_2(A_n(\mathbb{F}_p)) = B_2(A_n(\mathbb{Z})) \otimes \mathbb{F}_p$.
\end{enumerate}
\end{cor}
\begin{proof}
\
\begin{enumerate}
\item[\begin{small}1)\end{small}]Apply Lemma ~\ref{lem:Universal} for $\mathbb{Z}^N = A = A_n[m]$ and $B = (L_2 + M_3)[m]$, where $[m]$ means total degree $m$.
\item[\begin{small}2)\end{small}] Apply Lemma ~\ref{lem:Universal} for $\mathbb{Z}^N = A_n[m]$, $A = L_2(A_n)[m]$, and $B = L_3(A_n)[m]$. $\mathbb{Z}^N / A = B_1[m]$ is freely spanned by cyclic words of length $m$, so it is free, and the lemma applies.
\end{enumerate}
\end{proof}
\begin{rem}
This corollary is false for higher $B_i$.
For instance, one can show that
$$
\dim (B_4(A_3(\Bbb Z))[2,2,2]\otimes \Bbb F_2)\ne \dim B_4(A_3(\Bbb F_2))[2,2,2].
$$
(see Subsection \ref{discr}).
\end{rem}
\section{The Structure of $\bar{B}_1(A_n)$}
It will be useful to adapt the presentation of $A_n/M_3$ given in \cite{FS} to general base ring $R$.
This is accomplished by the following proposition.
\begin{prop}\label{relaa}
The algebra $A_n/M_3$ is the algebra generated by
$x_1, \ldots x_n$, and $u_{ij}$ for $i, j \in [1 \ldots n]$ and $i \neq j$, with the following relations:
\begin{enumerate}
\item[\begin{small}1)\end{small}] $u_{ij} = [x_i, x_j]$, and so $u_{ij} + u_{ji} = 0$;
\item[\begin{small}2)\end{small}] $[x_i, u_{jk}] = 0$ for all $i, j, k$;
\item[\begin{small}3)\end{small}] $u_{ij}$ commute with each other: $[u_{ij}, u_{kl}] = 0$;
\item[\begin{small}4)\end{small}] $u_{ij}u_{kl} = 0$ if any of the $i, j, k, l$ contain repetitions;
\item[\begin{small}5)\end{small}] $u_{ij}u_{kl}=-u_{ik}u_{jl}$ if $i, j, k, l$ are all distinct.
\end{enumerate}
\end{prop}
\begin{proof}
Relation (1) is the definition of $u_{ij}$,
and relations (2),(3) are obvious, so we will prove
only relations (4),(5).
4) We want to show that $[x,y][x,z]$ is in $M_3$.
But,
$$[x,y][x,z]=[[x,y]x,z]-[[x,y],z]x=[[x,yx],z]-[[x,y],z]x,
$$
which is in $M_3$.
5) We want to show that $[x,y][z,t]+[x,z][y,t]$ is in $M_3$.
Modulo $M_3$, this is
$$
[x,y[z,t]+z[y,t]]=[x,[yz,t]]+[x,z[y,t]-[y,t]z],
$$
which is obviously in $L_3$ (not just $M_3$).
Also, it is easy to show using relations (1)-(5) that
in the quotient of the algebra $A_n$ by these relations,
we have $[[a,b],c]=0$ for all $a,b,c$. Indeed,
using the Jacobi identity and the Leibniz rule, it suffices to show
that $[x_i,[x_j,c]]=0$. This is straightforward, substituting
$c=x_{i_1}...x_{i_m}$. The proposition is proved.
\end{proof}
For $I\subset \{1,\ldots, n\}$, of even cardinality $k$, $I=\lbrace{i_1,...,i_k\rbrace}$,
$i_1<...<i_k$, define $u_I:=u_{i_1,i_2}\cdots u_{i_{k-1},i_k}$.
\begin{prop}\label{anm3} $A_n/M_3$ is a free abelian group with basis
$x_1^{k_1}\cdots x_n^{k_n} u_I$, for $k_i\geq 0$, and $I$ as above.
\end{prop}
\begin{proof}
It is clear that the given elements are a spanning set.
Let us show that they are a basis. It is enough to show it
over $R=\Bbb Z$, hence over $R=\Bbb Q$, which is known from
\cite{FS}.
\end{proof}
Here is a coordinate-free description of $A/M_3$.
Let $V$ be a finitely generated free $R$-module,
and $A=T_RV$ be the tensor algebra of $V$.
Define the algebra $\widetilde\Omega(V)$
to be the quotient of the tensor algebra $T_R(V\oplus V)$
with generators $x_v,y_v$, $v\in V$ ($R$-linearly depending on $v$)
by the relations
$$
[x_v,x_u]=y_vy_u, [x_v,y_u]=0, v,u\in V.
$$
Note that this implies that $y_vy_u=-y_uy_v$ and $y_v^2=0$.
We have a decomposition $\widetilde{\Omega}(V)=\widetilde{\Omega}^{odd}(V)\oplus \widetilde{\Omega}^{ev}(V)$
into parts of odd and even degree with respect to the $y$-grading.
Also we have a differential on $\widetilde{\Omega}(V)$ given by $dx_v=y_v$.
Also, let $\Omega(V)$ be the algebra of differential forms
on $V^*$, generated by $x_v$ and $y_v$ with relations
$$
[x_v,x_u]=0,\ [x_v,y_u]=0,\ y_vy_u=-y_uy_v,\ y_v^2=0
$$
and differential $dx_v=y_v$.
If $R$ contains $1/2$, we can define the Fedosov
$*$-product on $\Omega(V)$ by
$a*b=ab+\frac{1}{2}da\wedge db$.
Denote $\Omega(V)$ equipped with this product by $\Omega(V)_*$.
\begin{prop}\label{phimap}
(i) If $R$ contains $1/2$ then there is a unique differential graded algebra homomorphism
$\phi: \widetilde{\Omega}(V)\to \Omega(V)_*$ such that $\phi(x_v)=x_v$,
and it is an isomorphism.
(ii) Over any $R$, one has ${\rm gr}(\widetilde{\Omega}(V))=\Omega(V)$,
where the associated graded is taken with respect to the
descending filtration by degree in the $y$-variables.
(iii) There exists a unique algebra homomorphism $\zeta: A/M_3\to \widetilde{\Omega}^{ev}(V)$,
such that $\phi(v)=x_v$. Moreover, $\phi$ is an isomorphism.
\end{prop}
\begin{proof}
(i) It is easy to check that the relations $[x_v,x_u]=y_vy_u$
and $[x_v,y_u]=0$ are satisfied in $\Omega(V)_*$, so, there is a unique
homomorphism $\phi$, which is surjective.
Also, it is clear that the Hilbert series
of $\widetilde{\Omega}(V)$ is dominated by the Hilbert series of $\Omega(V)$,
so $\phi$ is injective, hence an isomorphism.
(ii) It is easy to see that there is a surjective homomorphism
$\theta: \Omega(V)\to {\rm gr}\widetilde{\Omega}(V)$.
So it suffices to show that it is an isomorphism over $\Bbb Q$,
which follows from (i) by comparison of Hilbert series.
(iii) It is shown similarly to the proof of Proposition \ref{relaa}
that in the algebra $\widetilde{\Omega}(V)$,
one has $[[[a,b],c]=0$, so that $\zeta$ exists and is surjective.
But it follows from (ii) and Proposition \ref{relaa}
that the Hilbert series of the two algebras are the same, so
$\zeta$ is injective, i.e. an isomorphism.
\end{proof}
Thus, $\widetilde{\Omega}(V)$ is a quantization of
the DG algebra of differential forms $\Omega(V)$.
Nevertheless, over a ring $R$ not containing $1/2$, in particular
$R=\Bbb F_2$, they are not isomorphic even
as graded $GL(V)$-modules. Namely, the degree
2 component of $\widetilde{\Omega}(V)$ is $V\otimes V$
(the corresponding isomorphism is defined by $v\otimes u\mapsto x_vx_u$,
$v,u\in V$), while for $\Omega(V)$ it is $S^2V\oplus \wedge^2V$, which is not the same thing as $V\otimes V$
in characteristic $2$. However, we will need to work with rings not containing $1/2$ (such as $\Bbb Z$ and $\Bbb F_2$).
For this reason we will use a poor man's version of $\phi$, the map $\varphi$, introduced in the following proposition.
Note that it is not an algebra map and is not $GL(n)$-invariant.
\begin {prop}\label{coro} There exists a unique isomorphism of $R$-modules $\varphi:A_n/M_3\to\Omega^{ev}$
such that \[ \varphi(x_1^{k_1}\cdots x_n^{k_n}u_I) = x_1^{k_1}\cdots x_n^{k_n} dx_I, \]
where $dx_I = d_{x_{i_1}} \wedge d_{x_{i_2}} \wedge \ldots \wedge d_{x_{i_{k-1}}} \wedge d_{x_{i_k}}$.
Moreover, one has $\varphi([a,x_i])=d\varphi(a)\wedge dx_i$.
\end{prop}
\begin{proof}
The first statement is a direct consequence of Proposition \ref{anm3}, and the second one follows by an easy
direct computation.
\end{proof}
\begin{thm}\label{thm:Thm1}
The map $\varphi$ induces an isomorphism $\varphi: L_2(A/M_3) \rightarrow \Omega^{ev}_{ex}$.
\end{thm}
\begin{proof} We will need the following lemma.
\begin{lem}\label{degg1} We have $L_2(A_n)=\sum_{i=1}^n [A_n,x_i]$.
\end{lem}
\begin{proof}
This follows from the identity $[a,bc]+[b,ca]+[c,ab]=0$.
\end{proof}
By this lemma, $L_2(A_n)$ is spanned by $[a,x_i]$, $a\in A_n$.
But by Proposition \ref{coro},
$\varphi([a,x_i])=d\varphi(a)\wedge dx_i$.
Thus, $\varphi(L_2)$ is contained in $\Omega^{ev}_{ex}$.
On the other hand, $\Omega^{ev}_{ex}$ is spanned by elements of the form
$\omega=df\wedge dx_{i_1}\wedge...\wedge dx_{i_{2r+1}}$. So if $a=\varphi^{-1}(fdx_{i_1}\wedge...\wedge dx_{i_{2r}})$,
then $\varphi([a,x_{i_{2r+1}}])=\omega$, implying the opposite inclusion.
\end{proof}
\begin{cor}\label{deco}
Over any base ring $R$, one has
$$
\bar{B}_1(A_n) =\Omega^{ev}(R^n)/\Omega^{ev}_{ex}(R^n).
$$
In particular, for $R=\Bbb Z$, one has
$$
\bar{B}_1(A_n)=H^{ev,+}(\Omega(\mathbb{Z}^n))\oplus\Omega^{ev}(\mathbb{Z}^n)/\Omega^{ev,+}_{cl}(\mathbb{Z}^n),$$
where $H^{ev,+}:=H^2\oplus H^4\oplus...$ is the even De Rham cohomology of $\Bbb Z^n$.
\end{cor}
\begin{proof}
It is clear that
for any algebra $A$,
$\bar{B}_1(A) \cong B_1(A/M_3)$.
Therefore, by Theorem \ref{thm:Thm1},
$\bar{B}_1(A_n) =\Omega^{ev}/\Omega^{ev}_{ex}$.
So we have a short exact sequence
$$
0\to H^{ev,+}(\Omega(\mathbb{Z}^n))\to \bar{B}_1(A_n) \to \Omega^{ev}(\mathbb{Z}^n)/\Omega^{ev,+}_{cl}(\mathbb{Z}^n)\to 0,
$$
But it is easy to see that the quotient is a free group
(indeed, if $m\omega$ is a closed form for an integer $m$, then $\omega$ is closed as well). This implies
the statement.
\end{proof}
Now, the Poincare lemma over $\mathbb{Q}$ implies that $H^{ev,+}(\Omega(\mathbb{Z}^n))$ is torsion.
Thus, we have
\begin{thm} \label{barB1thm} The torsion in $\bar B_1(A_n\mathbb{Z}))$ is given by the equality
$$
{\mathrm{tor}}\bar{B}_1(A_n(\mathbb{Z}))= H^{ev,+}(\Omega(\mathbb{Z}^n)) = H^2(\Omega(\mathbb{Z}^n)) \oplus H^4(\Omega(\mathbb{Z}^n)) \oplus \ldots = \bigoplus^{\lfloor\frac{n}{2}\rfloor}_{k=1} H^{2k} (\Omega(\mathbb{Z}^n)).$$
\end{thm}
Combining with Corollary \ref{Hev-description}, we have a complete description of the torsion part of $\bar{B}_1(A_n(\mathbb{Z}))$.
\begin{prop}\label{b1an} If $m_1,...,m_n>0$ then the torsion in $\bar B_1(A_n(\Bbb Z))[m_1,...,m_n]$ is isomorphic to
$(\Bbb Z/{\rm gcd}(m_1,...,m_n))^{2^{n-2}}$.
\end{prop}
\begin{cor}\label{b1a2}
For $q,r>0$, the torsion in $\bar B_1(A_2(\mathbb{Z}))[q,r]$ is isomorphic to $\mathbb{Z}/{\rm gcd}(q,r)$ and
spanned by the element $x^{q-1}y^{r-1}[x,y]$.
\end{cor}
Proposition \ref{b1an} allows us to explicitly compute $\bar{B}_1(A_n(\mathbb{F}_p))$ for all $p$.
\begin{cor}\label{b1anp}
If all $m_i$ are positive, then $\dim \bar B_1(A_n(\Bbb F_p))[m_1,...,m_n]$
is $2^{n-2}$ if there exists $i$ such that $m_i$ is not divisible by $p$, and $2^{n-1}$
otherwise (i.e., if all $m_i$ are divisible by $p$).
\end{cor}
\begin{proof}
By Corollary \ref{unco}, $\bar B_1(A_n(\Bbb F_p))=\bar B_1(A_n(\Bbb Z))\otimes \Bbb F_p$.
It follows from the results of \cite{FS} that the rank of the free part
of $\bar B_1(A_n(\Bbb Z))[\bold m]$ is $2^{n-2}$. So the result follows from
Corollary \ref{b1an}.
\end{proof}
\section{The Structure of $B_2(A_n)$}
\subsection{Torsion elements in $B_2(A_n(\Bbb Z))$}
In this section we study the torsion in $B_2(A_n(\Bbb Z))$.
We will show below that for $n=2$ there is no torsion,
so the first interesting case is $n=3$.
In this subsection we describe the torsion in the case $n=3$.
Later we will give a more general result
which applies to any $n$; however, first we explicitly
work out the cases $n=3$ and $n=2$ for the reader's convenience.
Let us denote the generators of $A_3(\Bbb Z)$ by $x,y,z$.
Let $s, q, r$ be positive integers, and $T(s, q, r) = [z, z^{s-1}x^{q-1}y^{r-1}[x,y]] \in B_2(A_3(\mathbb{Z}))$.
\begin{thm} \label{TorsionTheorem}
\
\begin{enumerate}
\item[\begin{small}1) \end{small}] The element $T(s, q, r)$ is torsion of order dividing $\gcd(s, q, r)$.
\item[\begin{small}2) \end{small}] If $\gcd(s, q, r) = 2$ or $3$, the order of $T(s, q, r)$ is exactly equal to $\gcd(s, q, r)$.
\end{enumerate}
\end{thm}
\begin{proof}
\
\begin{enumerate}
\item[\begin{small}1) \end{small}]
We start with the identity
\begin{equation}\label{ide}
[z,w[x,y]] = [[w,y],xz] -[z,[y,wx]] + [x,[w,zy]] + x[z,w]y +[w,z]yx
\end{equation}
which is checked by a direct calculation.
Setting in this identity $w=z^{s-1}$, we get that $[z,z^{s-1}[x,y]] \in L_3(A_3(\mathbb{Z}))$. Now replacing $y$ with $x^{q-1}y^r$, we get that $[z,z^{s-1}[x,x^{q-1}y^r]] \in L_3(A_3(\mathbb{Z}))$. Setting $u:=[z,z^{s-1}x^{q-1}y^{r-1}[xy]]$, and using Lemma \ref{lem:FSLemma1}, we get that $ru \in L_3(A_3(\mathbb{Z}))$. Similarly, $[z,z^{s-1}[y,x^qy^{r-1}]] \in L_3(A_3(\mathbb{Z}))$, and using Lemma \ref{lem:FSLemma1}, $qu \in L_3(A_3(\mathbb{Z}))$.
It remains to show that $su \in L_3(A_3(\mathbb{Z}))$. To this end, we set $m=q-1$, $k=r-1$, and write using Lemma \ref{lem:FSLemma1} (mod $L_3(A_3(\mathbb{Z}))$):
\begin{align*}
[z,x^my^k[xy]]&=[z,x^m[y^kx,y]] = [x^m,z[y^kx,y]] =[x^m,[zy]y^kx] \\
&=[x^m,[zy^k,y]x] =-[x^m,x[y,zy^k]] =-[x,x^m[y,zy^k]],
\end{align*}
which is in $L_3(A_3(\mathbb{Z}))$ by identity (\ref{ide}).
So, putting $z^s$ instead of $z$ we get that $su=0$, as desired.
\item[\begin{small}2) \end{small}] We consider first the case when $\gcd(s, q, r) = 2$. Let $i = s-1, j = q-1, k = r -1$, and consider the element $F(x,y,z):=[x,x^iy^jz^k[yz]] = T(s, q, r)$. It suffices to show that this element is nontrivial in $B_2(A_3(\mathbb{F}_2))$.
Let $t_x, t_y, t_z$ be commutative variables (i.e. they commute with $x,y,z$ and each other). Consider $F(x+t_x,y+t_y,z+t_z)$, and take the coefficient of tridegree \linebreak $(i-1,j-1,k-1)$ in $t_x,t_y,t_z$. Clearly, it is $ijk[x,xyz[yz]]$. But we know, from computer calculations in MAGMA, that this is nonzero in $B_2(A_3(\mathbb{F}_2))$. We see that for all odd $i,j,k$, $F(x,y,z)$ is nonzero in $B_2(A_3(\mathbb{F}_2))$.
A similar procedure works in the case where $\gcd(s,q,r)=3$, with $i,j,k$ of the form \ $3m-1$. The relevant coefficient (of degree $i-2,j-2,k-2$) would be $\binom{i}{2}\binom{j}{2}\binom{k}{2}[x,x^2y^2z^2[yz]]$, and the element $[x,x^2y^2z^2[yz]]$ is nonzero in $B_2(A_3(\mathbb{F}_3))$ by a computer calculation.
\end{enumerate}
\end{proof}
\begin{rem}
1. Actually, the argument in the proof for part 2 would work for any $\gcd(s,q,r) = m$
if we knew that $[x,x^{m-1}y^{m-1}z^{m-1}[yz]]$ has order exactly
$m$ in $B_2(A_3(\mathbb{Z}))$.
2. Below we will give another proof that the order of $T(s,q,r)$ is exactly ${\rm gcd}(s,q,r)$, which works
when ${\rm gcd}(s,q,r)$ is odd.
\end{rem}
\subsection{Torsion in $B_2(A_2(\Bbb Z))$}
\begin{thm} \label{noTorsionB2}
$B_2(A_2(\mathbb{Z}))$ has no torsion.
\end{thm}
\begin{proof} We make use of the following:
\begin{lem} \label{torLemma1}
We have $B_2(A_n(\mathbb{Z})) = \sum_{i=1}^n [x_i,\bar{B}_1(A_n(\mathbb{Z}))]$.
\end{lem}
\begin{proof}
The statement follows from Lemma \ref{degg1}.
\end{proof}
Let us denote the generators of $A_2$ by $x,y$.
\begin{lem} \label{torLemma2}
If $T$ is a torsion element of $\bar{B}_1(A_2(\mathbb{Z}))$, then $[x,T]=[y,T]=0$ in $B_2(A_2(\mathbb{Z}))$.
\end{lem}
\begin{proof}
By Corollary \ref{b1a2}, torsion elements are linear combinations of
elements of the form $T=x^{q-1}y^{r-1}[xy]$ (corresponding to the
2-form $x^{q-1}dx \wedge y^{r-1}dy$). Now,
$[x,T]=[x,x^{q-1}y^{r-1}[xy]]$. This is the specialization of
$T(1,q,r)$ under setting $z=x$ (where by ``specialization", we mean
that we apply the homomorphism $A_3 \rightarrow A_2$ such that $x,y,z$
go to $x,y,x$, respectively). Since $T(1,q,r)=0$ in $B_2$, we
conclude that the specialization is zero as well, i.e., $[x,T]=0$.
Similarly, $[y,T]=0$.
\end{proof}
We showed in Theorem \ref{thm:Thm1}, that \[ \bar{B}_1(A_2) = \Omega^0\oplus\Omega^{2}/\Omega^{2}_{ex}. \]
Because the last summand is all torsion, by Lemma \ref{torLemma2}, we
can strengthen the formula for $B_2$ in Lemma \ref{torLemma1} to say
that $B_2=[x,\Omega^0]+[y,\Omega^0]$. Let us denote the $\mathbb{Z}$-span of $x$ and $y$ by $V$. Since $x,y$ is a basis of $V$,
we see that $B_2=[V,\Omega_0]$, i.e. $B_2$ is a quotient of $V\otimes
\Omega_0$.
Note that we have an identification $V\otimes \Omega^0\cong \Omega^1$, via $x\otimes f+y\otimes g\mapsto fdx+gdy$. Let us show that closed 1-forms map to zero in $B_2$.
Let $f,g \in \Omega^0$ be such that $fdx+gdy$ is a closed form, i.e. $f_y=g_x$. We may assume that this form is homogeneous of bidegree $(q, r)$. Then we can set $f=qx^{q-1}y^r/d, g=rx^qy^{r-1}/d$, where $d=\gcd(q, r)$.
Define lifts $\hat f$ and $\hat g$ of $f,g$ to $A_2$
by the formulas
$$
\hat f=\sum_{i=0}^{\frac{q}{d}-1}x^iy^{\frac{r}{d}}
(x^{\frac{q}{d}}y^{\frac{r}{d}})^{d-1}x^{\frac{q}{d}-1-i},
$$
$$
\hat g=\sum_{i=0}^{\frac{r}{d}-1}y^i
(x^{\frac{q}{d}}y^{\frac{r}{d}})^{d-1}x^{\frac{q}{d}}y^{\frac{r}{d}-1-i}.
$$
Then it is easy to see that
\[
[x,\hat f]+[y,\hat g]=0.
\]
in $A_2$. This implies that the closed 1-form $fdx+gdy$ is killed, as desired.
Thus, we see that $B_2(A_2(\mathbb{Z}))$ is a quotient of the
free group $\Omega^1/\Omega^1_{cl}$. But by the Feigin-Shoikhet theorem \cite{FS},
this free group already has the same Hilbert series as $B_2(A_2(\mathbb{Q}))$. So, there is no torsion.
\end{proof}
\subsection{Torsion in $B_2(A_n(\Bbb Z))$}
\begin{conj}\label{torb2}
The graded abelian group $B_2(A_n(\Bbb Z))$
is isomorphic to
$$
\oplus_{i\ge 1}\Omega^{2i+1}(\Bbb Z^n)/\Omega^{2i+1}_{ex}(\Bbb Z^n).
$$
Thus, the torsion in $B_2(A_n(\mathbb{Z}))$ is isomorphic to
\[ H^3(\Omega(\mathbb{Z}^n)) \oplus H^5(\Omega(\mathbb{Z}^n)) \ldots =
\bigoplus^{\lfloor{\frac{n-1}{2}\rfloor}} _{k=1} H^{2k+1}(\Omega(\mathbb{Z}^n)). \]
In particular, the torsion of $B_2(A_3(\mathbb{Z}))[q,r,s]$ is spanned by the element $T(s, q, r)$, which has order exactly $\gcd(s, q, r)$.
\end{conj}
This conjecture implies the following conjecture in characteristic $p$:
\begin{conj}\label{b2charp}
If all $m_i$ are positive, then $\dim B_2(A_n(\Bbb F_p))[m_1,...,m_n]$
is $2^{n-2}$ if there exists $i$ such that $m_i$ is not divisible by $p$, and $2^{n-1}-1$
otherwise (i.e., if all $m_i$ are divisible by $p$).
\end{conj}
The following theorem shows that Conjectures \ref{torb2} and \ref{b2charp} hold at least as upper bounds.
\begin{thm}\label{upbo}
The torsion in $B_2(A_n(\mathbb{Z}))$ is a quotient of
\[ H^3(\Omega(\mathbb{Z}^n)) \oplus H^5(\Omega(\mathbb{Z}^n)) \ldots = \bigoplus^{\lfloor{\frac{n-1}{2}\rfloor}} _{k=1} H^{2k+1}(\Omega(\mathbb{Z}^n)). \]
In particular, the torsion of $B_2(A_3(\mathbb{Z}))[q,r,p]$ is spanned by the element $T(p, q, r)$.
Also, the numbers in Conjecture \ref{b2charp} are upper bounds for
$\dim B_2(A_n(\Bbb F_p))[m_1,...,m_n]$.
\end{thm}
\begin{proof}
Let $V$ be the $\Bbb Z$-span of the generators $x_1,...,x_n$.
By Lemma \ref{torLemma1} and Corollary \ref{deco}, we have a surjective
homomorphism $\xi: V\otimes \Omega^{ev}(\Bbb Z^n)/\Omega^{ev}_{ex}(\Bbb Z^n)
\to B_2(A_n(\Bbb Z))$.
Moreover, for any $a,b,c,z\in A_n$, we have
$$
[a,[b,c]z]+[b,[a,c]z]=[a[b,c],z]-[[b,c],za]+[b[a,c],z]-[[a,c],zb]=
$$
$$
=[[ab,c],z]-[[b,c],za]+[[b,[a,c]],z]-[[a,c],zb]
\in L_3,
$$
which shows that the image of $[a,[b,c]z]$ in $B_2$
is skewsymmetric in $a,b,c$. This implies that
$\xi$ is a composition of a homomorphism
$$
\eta: \Omega^1(\Bbb Z^n)\oplus \Omega^{odd,\ge 3}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n)\to B_2(A_n(\Bbb Z))
$$
and the map
$v\otimes \omega\mapsto dv\wedge \omega$.
So the theorem follows from the following lemma.
\begin{lem}\label{le}
The image $\tilde I$
of $\Omega^1_{cl}(\Bbb Z^n)\oplus \Omega^{odd,\ge 3}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n)$ under $\eta$
coincides with the image $I$ of $\Omega^{odd,\ge 3}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n)$
under $\eta$.
\end{lem}
Indeed, by the results of \cite{FS}, the natural map
$\gamma: \Omega^1(\mathbb{Z}^n)/\Omega^1_{cl}(\mathbb{Z}^n)\to B_2(A_n(\Bbb Z))/\tilde I$ is an isomorphism
after tensoring with $\Bbb Q$. Since the group
$\Omega^1(\mathbb{Z}^n)/\Omega^1_{cl}(\mathbb{Z}^n)$ is free, $\gamma$ is injective. Hence,
Lemma \ref{le} shows that we have an exact sequence
$$
0\to K\to \Omega^{odd,\ge 3}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n)\to B_2(A_n(\Bbb Z))\to
\Omega^1(\Bbb Z^n)/\Omega^1_{cl}(\Bbb Z^n)\to 0,
$$
where the last nontrivial map is induced by $\gamma^{-1}$.
So, since $\Omega^1/\Omega^1_{cl}$ is a free group, we have
$$
B_2(A_n(\Bbb Z))\cong \Omega^1(\Bbb Z^n)/\Omega^1_{cl}(\Bbb Z^n)\oplus
(\Omega^{odd,\ge 3}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n))/K.
$$
Moreover, as follows from the Feigin-Shoikhet results \cite{FS}, this isomorphism
holds without $K$ after tensoring with $\Bbb Q$, which implies that
$K$ is a torsion group. So we get
$$
{\rm tor}_2B_2(A_n(\Bbb Z))\cong
(\Omega^{odd,\ge 3}_{cl}(\Bbb Z^n)/\Omega^{odd,\ge 3}_{ex}(\Bbb Z^n))/K=
(H^3\oplus H^5\oplus...)/K,
$$
as desired.
\begin{proof} (of Lemma \ref{le}) The proof is similar to the argument at the end of
the proof of Theorem \ref{noTorsionB2}.
Namely, let $f_1dx_1+...+f_ndx_n$ be a closed 1-form, which is homogeneous of multidegree $m_1,...,m_n$.
We may assume that $m_i>0$, otherwise we are reduced to smaller $n$.
Let $d={\rm gcd}(m_1,...,m_n)$.
Then we can set $f_i=\frac{m_i}{d}x_i^{-1}\prod_j x_j^{m_j}$.
Define lifts $\hat f_i$ of $f_i$ to elements in $A_n$ by the formulas
$$
\hat f_i=\sum_{j=0}^{\frac{m_i}{d}-1}x_i^jx_{i+1}^{\frac{m_{i+1}}{d}}\dots x_n^{\frac{m_n}{d}}
(x_1^{\frac{m_1}{d}}\dots x_n^{\frac{m_n}{d}})^{d-1}
x_1^{\frac{m_1}{d}}\dots x_{i-1}^{\frac{m_{i-1}}{d}}x_i^{\frac{m_i}{d}-j-1}.
$$
It is easy to see that $\sum_i [x_i,f_i]=0$.
This implies that $\eta(\sum_i f_idx_i)\in I$,
as desired.
\end{proof}
\end{proof}
\begin{rem}
Theorem \ref{upbo} implies that Conjecture \ref{b2charp} holds
for $p=2,3$ and $n\le 4$. The proof is analogous to the proof of Theorem
\ref{TorsionTheorem}(2).
\end{rem}
\subsection{Proof of 2-localized versions of Conjectures \ref{torb2} and \ref{b2charp}}
\subsubsection{The statement}
\begin{thm}\label{maint}
(i) Conjecture \ref{torb2} holds over $\Bbb Z[1/2]$.
So if ${\rm gcd}(q,r,s)=2^\ell(2k+1)$ then
the order of $T(q,r,s)$ is divisible by $2k+1$.
(ii) Conjecture \ref{b2charp} holds for $p>2$.
\end{thm}
The rest of the section is the proof of Theorem \ref{maint}.
In view of Corollary \ref{unco} (2), it suffices to prove (i).
We will apply the theory from the appendix to \cite{DKM}, namely Theorem 7.2.
\subsubsection{First cyclic homology of the free algebra over the integers}
Let $A=A_n(\Bbb Z)$. First of all we compute
the first cyclic homology group $HC_1(A)$.
This computation is known (due to Loday and Quillen), but we will give it
here for the convenience of the reader.
Recall that $HC_1(A)$ is the quotient of the kernel of the commutator map
$[,]:\wedge^2A\to A$ by the elements $ab\wedge c+bc\wedge a+ca\wedge b$,
$a,b,c\in A$. Also recall that if $a=x_{i_1}...x_{i_N}$ is a cyclic
word in $A/[A,A]$ (i.e., a word up to cyclic permutation)
then we can define its noncommutative partial derivative $\partial_i a\in A$
by the formula
$$
\partial_ia:=\sum_{k: i_k=i}x_{i_{k+1}}...x_{i_N}
x_{i_1}...x_{i_{k-1}}
$$
Finally, note that for any cyclic word $a$ and any positive
integer $m$, the partial derivative $\partial_i(a^m/m)$,
has integer coefficients, even though $a^m/m$ does not.
\begin{thm}\label{cychom} (\cite{LQ}, Proposition 5.4)
If $a$ is a non-power word (i.e., a nontrivial
word that is not a power of another word)
and $m>1$ is a positive integer, then the noncommutative differential
$$
d(a^m/m):=\sum_i \partial_i(a^m/m)\otimes x_i
$$
defines
an element of $HC_1(A)$ of order $m$. Moreover,
$HC_1(A)$ is the direct sum of the cyclic
subgroups $\Bbb Z/m$ spanned by these elements.
\end{thm}
\begin{proof}
We have the Connes-Loday-Quillen exact sequence (\cite{LQ}, Theorem 1.6)
$$
HC_0(A)=A/[A,A]\to HH_1(A)\to HC_1(A)\to 0.
$$
Now, since $A=TV$ is a free algebra,
$HH_1(A)$ is the kernel of the commutator map $A\otimes V\to A$,
and the map $d: HC_0(A)\to HH_1(A)$ is
the noncommutative differential.
In characteristic zero, the kernel of the map $d$ is just constants,
and $HC_1=0$. This implies that over $\Bbb Z$, $HC_1$ is a torsion group,
and in each multidegree $\bold m$, one has
$$
HC_1(A)[\bold m]=\hat J_{\bold m}/J_{\bold m},
$$
where $J_{\bold m}:={\rm Im}(d)[\bold m]$, and
$\hat J_{\bold m}$ is the saturation of $J_{\bold m}$
inside $HH_1(A)$. Now, $\hat J_{\bold m}/J_{\bold m}$
is clearly a direct sum of groups $\Bbb Z/m$ generated by
elements $d(a^m/m)$, where $a$ is a non-power word,
which implies the statement.
\end{proof}
\subsubsection{Conclusion of proof of Theorem \ref{maint}}
Now we are ready to prove Theorem \ref{maint}.
We will work over $\Bbb Z[1/2]$ and for brevity will not
explicitly show it in the notation.
Recall from the appendix to \cite{DKM}, proof
of Theorem 7.2, that we have an exact sequence
\begin{equation}\label{exase}
HC_1(A)\to \wedge^2(A/(L_2+M_3))/W\to B_2(A)\to 0,
\end{equation}
where $W$ is spanned by the images of the
elements $ab\wedge c+bc\wedge a+ca\wedge b$,
$a,b,c\in A$.
Now recall that $\Omega^{ev}$ is equipped with
the Fedosov product $a*b=ab+\frac{1}{2}da\wedge db$,
and analogously to \cite{FS}, by Proposition \ref{phimap}
we have an algebra isomorphism
$\phi: A/M_3\to \Omega^{ev}_*$,
which maps $A/(L_2+M_3)=\bar B_1(A)$ isomorphically onto
$\Omega^{ev}/\Omega^{ev}_{ex}$.
So, the middle term of the sequence (\ref{exase}) is
$\wedge^2(\Omega^{ev}/\Omega^{ev}_{ex})/W$,
where $W$ is now spanned by the elements $ab\wedge c+bc\wedge a+ca\wedge b$,
$a,b,c\in \Omega^{ev}/\Omega^{ev}_{ex}$ (note that it is not important
whether $ab$ is the usual or Fedosov product, as they differ by an exact form,
and we are working modulo exact forms).
Now, it is shown as in \cite{DKM}, Theorem 7.1,
that the algebra of differential forms is pseudoregular in the sense of
\cite{DKM}, so as a result the map
$$
\theta: \wedge^2(\Omega^{ev}/\Omega^{ev}_{ex})/W\to \Omega^{odd}/\Omega^{odd}_{ex}
$$
given by $\theta(a\otimes b)=a\wedge db$ is an isomorphism.
Thus, we have an exact sequence
$$
HC_1(A)\to \Omega^{odd}/\Omega^{odd}_{ex}\to B_2(A)\to 0,
$$
where the first map is defined by the formula
$$
T=\sum_i f_i\otimes x_i\mapsto \sum_i \phi(f_i)\wedge dx_i.
$$
Our job is to show that this map lands in $\Omega^1/\Omega^1_{ex}$.
By Theorem \ref{cychom}, for this it suffices to show that
for every cyclic word $a$ of multidegree $(m_1,...,m_n)$,
$$
\sum_i \phi(\partial_i(a^m/m))\wedge dx_i\in \Omega^1+\Omega^{odd,\ge 3}_{ex}.
$$
This is equivalent to saying that the form
$$
\omega(a,m):=\sum_i (\phi(\partial_i(a^m/m))-\partial_i(a^m/m))\wedge dx_i
$$
belongs to $ \Omega^{odd,\ge 3}_{ex}$.
Since this form is in $\Omega^{\ge 3}$ and is clearly closed,
it suffices to show that it represents the trivial class in the
De Rham cohomology. But by Corollary \ref{Hev-description},
for this it suffices to show that $\omega(a,m)$ is divisible by $mD$, where
$D={\rm gcd}(m_1,...,m_n)$.
To this end, let us compute $\omega(a,m)$ explicitly.
Suppose that $w$ be a cyclic word. By a shuffle subword of $w$ we will mean
a cyclic word obtained by crossing out some letters from $w$.
Let $1\le i_1<i_2<...<i_{2k+1}\le n$.
Let $N_{i_1,...,i_{2k+1}}(w)$ be the number of
shuffle subwords of $w$ which are even permutations of
$x_{i_1},...,x_{i_{2k+1}}$ minus the number of shuffle subwords
which are odd permutations. Note that since the length of the shuffle subword is odd,
it makes sense to talk about its parity, as it does not change under cyclic permutation.
\begin{lem}\label{form}
One has
$$
\sum_i \phi(\partial_iw)\wedge dx_i=\sum_{k\ge 0}\frac{1}{2^k}\sum_{i_1<...<i_{2k+1}}N_{i_1,...,i_{2k+1}}(w)dx_{i_1}\wedge...\wedge dx_{i_{2k+1}}.
$$
\end{lem}
\begin{proof}
This follows by direct calculation using the formula for the Fedosov product.
\end{proof}
\begin{cor}
One has
$$
\omega(a,m)=\frac{1}{m}\sum_{k\ge 1}\frac{1}{2^k}\sum_{i_1<...<i_{2k+1}}
N_{i_1,...,i_{2k+1}}(a^m)dx_{i_1}\wedge...\wedge dx_{i_{2k+1}}.
$$
\end{cor}
Therefore, the theorem follows from
the following combinatorial lemma.
\begin{lem}\label{comb}
Let $a$ be a cyclic word
of degrees $m_i$ with respect to $x_i$, and $D={\rm gcd}(m_1,...,m_n)$.
Then for $k\ge 1$, the number
$N_{i_1,...,i_{2k+1}}(a^m)$ is divisible by $m^{k+1}D$.
\end{lem}
\begin{proof}
Let $y_i$, $i=1,...,n$
be anticommuting variables (i.e., $y_iy_j=-y_jy_i$ and $y_i^2=0$).
Suppose that $a=x_{j_1}...x_{j_M}$ (where $M=\sum m_i$),
and consider the product $Y(a,m)=((1+y_{j_1})...(1+y_{j_M}))^m$.
It is easy to see that $N_{i_1,...,i_{2k+1}}(a^m)$
is the coefficient of $y_{i_1}...y_{i_{2k+1}}$ in $Y(a,m)$.
However, it is easily shown by induction that
$$
(1+y_{j_1})...(1+y_{j_M})=(1+m_1y_1+...+m_ny_n)\prod_{1\le r<s\le n}(1+y_{j_r}y_{j_s}).
$$
This implies that
$$
Y(a,m)=(1+mm_1y_1+...+mm_ny_n)\prod_{1\le r<s\le n}(1+my_{j_r}y_{j_s})
$$
and the statement follows.
\end{proof}
The theorem is proved.
\section{Experimental data and conjectures}
\subsection{Experimental data}
\label{exp-data-I}
In this subsection we summarize the experimental data obtained by direct computation in MAGMA.
In the tables below, multi-degree components are greater than or equal to $1$ since cases with one degree being $0$ reduce to a smaller number of variables. Due to the $S_n$ action permuting generators, we list only multi-degrees with weakly descending entries. Note that we are asserting the torsion subgroups are trivial for omitted degrees in each specified range.
\begin{table}[htpb]
\caption{Torsion in $B_\ell(A_2(\mathbb{Z}))$ in degrees $(i,j)$, with $i\geq j$, such that $2 \le \ell \le i+j \le 12$.}
\begin{center}
\begin{tabular}{ | l | | l | l | l | l |}
\hline
$(i,j) \backslash \ell$: & 5 & 6 & 7 & 9\\ \hline
(4,4) &$\mathbb{Z}/2$&&&\\ \hline
(6,4) &$\mathbb{Z}/2$&&$\mathbb{Z}/2$&\\ \hline
(6,6) &$\mathbb{Z}/2$&$\mathbb{Z}/3 $ & $(\mathbb{Z}/2)^3$& $(\mathbb{Z}/2)^3$\\ \hline
(8,4) &$\mathbb{Z}/2$ && $\mathbb{Z}/2$&$(\mathbb{Z}/2)^2$\\ \hline
\end{tabular}
\end{center}
\label{torsiontwo}
\end{table}
\begin{rem}
Note that we did not discover torsion in $B_2(A_2(\mathbb{Z}))$. This lack of torsion is proved in Theorem \ref{noTorsionB2}.
\end{rem}
\begin{table}[htpb]
\caption{Torsion in $B_\ell(A_3(\mathbb{Z}))$ in degrees $(i,j,k)$, with $i\geq j \geq k$, such that $2 \le \ell \le i+j+k$ and either $i,j,k \le 3$ or $j,k \le 4$, $i \le 2$.}
\begin{center}
\begin{tabular}{ | l | | l | l | l | l | l |}
\hline
$(i,j,k) \backslash \ell$: & 2 & 3 & 5 & 7 \\ \hline
(2,2,2) & $\mathbb{Z}/2$&&& \\ \hline
(3,3,3) &$\mathbb{Z}/3 $ & $\mathbb{Z}/3$&&\\ \hline
(4,2,2) &$\mathbb{Z}/2$ && $(\mathbb{Z}/2)^2$& \\ \hline
(4,4,2) &$ \mathbb{Z}/2$&&$(\mathbb{Z}/2)^5$&$(\mathbb{Z}/2)^5$ \\ \hline
\end{tabular}
\end{center}
\label{torsionthree}
\end{table}
\begin{table}[htpb]
\caption{Torsion in $B_\ell(A_4(\mathbb{Z}))$ in degrees $(i,j,k,l)$, with $i\geq j \geq k \geq l$, such that $2 \le \ell \le i+j+k+l$ and $i,j,k,l \le 2$.}
\begin{center}
\begin{tabular}{ | l | | l | l |}
\hline
$(i,j,k,l) \backslash \ell$ & 2 &5 \\ \hline
(2,2,2,2) & $(\mathbb{Z}/2)^3$ & $(\mathbb{Z}/2)^5$ \\ \hline
\end{tabular}
\end{center}
\label{torsionfour}
\end{table}
\subsubsection{Discrepancies between ranks of $B_\ell$ in characteristic zero and positive characteristic}\label{discr}
The difference
$D=\dim B_\ell(A_3(\mathbb{\Bbb F}_2))-\dim B_\ell(A_3(\mathbb{Q}))$ in degrees $(i,j,k)$
was calculated using MAGMA in degrees up to $(3,3,3)$ and $(4,2,2)$
and was found to be nonzero in the following cases:
\[
\begin{array}{|c|c|c|} \hline
\ell & (i,j,k) & D \\ \hline \hline
2 & (2, 2, 2) &1 \\ \hline
2 & (4, 2, 2) &1 \\ \hline
4 & (2, 2, 2) &-1 \\ \hline
4 & (4, 2, 2) &-1 \\ \hline
5 & (4, 2, 2) &2 \\ \hline
6 & (4, 2, 2) &-2 \\ \hline
\end{array}
\]
The difference
$D=\dim B_\ell(A_3(\mathbb{\Bbb F}_3))-\dim B_\ell(A_3(\mathbb{Q}))$ in degrees $(i,j,k)$
calculated using MAGMA in degrees up to $(3,3,3)$ and $(4,2,2)$
and was found to be nonzero in the following cases:
\[
\begin{array}{|c|c|c|} \hline
\ell & (i,j,k) & D \\ \hline \hline
2 & (3, 3, 3)& 1 \\ \hline
5 & (3, 3, 3) &-1 \\ \hline
\end{array}
\]
\subsection{Conjectures}
The theoretical results of this paper and the computational results in the previous subsection motivate the following conjectures.
\begin{conj}
If a torsion element $T$ in $B_\ell(A_n(\mathbb{Z}))$ has degree $m$ with respect to some generator $x_j$, then the order of $T$ divides $m$.
\end{conj}
\begin{conj}
There is no torsion in $B_\ell(A_n(\mathbb{Z}))[m_1, \ldots , m_n]$ unless $\ell \le m_1+ \cdots + m_n-3$.
\end{conj}
\begin{rem}\label{weakerst}
One can prove a weaker statement that there is no torsion unless $ \ell \le m_1+ \cdots + m_n-2$. Indeed, it is clear that $B_\ell[m_1,...,m_n]$ has no torsion if $m_1+...+m_n=\ell$ because $L_{\ell+1}[m_1,...,m_n]=0$ in this case. Also, in this case $L_\ell[m_1,...,m_n]$ is a saturated subgroup in the free algebra $A_n(\mathbb{Z})$ (i.e., if $jx\in L_\ell$ for a positive integer $j$, then $x\in L_\ell$) since this is a graded component of the free Lie algebra in $n$ generators over $\mathbb{Z}$ inside its universal enveloping algebra, which is $A_n(\mathbb{Z})$. This means that $B_{\ell-1}[m_1,...,m_n]$ is also free in this case.
\end{rem}
Also we propose the following questions:
{\bf Question 1.} Suppose that $\ell>2$ and $n$ are fixed.
Can there exist $p$-torsion
in $B_\ell(A_n(\Bbb Z))$ for arbitrarily large $p$?
What if $\ell,n$ are allowed to vary?
(In the computer search, we only found $2$-torsion
and $3$-torsion).
{\bf Question 2.}
Does $2$-torsion in $B_\ell(A_2(\mathbb{Z}))$ where $\ell>2$ occur only for odd $\ell$?
\section{The supercase}\label{supercase}
In this section we consider the super-extension of the preceding results, in the
style of the paper \cite{BJ}. Namely, let $A_{n,k}(R)$
be the free algebra generated over a commutative ring $R$
by $n$ even generators $x_1,...,x_n$ and $k$ odd generators, $y_1,...,y_k$.
This is really the same algebra as $A_{n+k}(R)$; the only thing that changes is the notion of
the commutator. Namely, for two words $a,b\in A_{n,k}$,
we define $[a,b]=ab-(-1)^{d_ad_b}ba$, where $d_a$ is the number of odd
generators $y_j$ in the word $a$. Then we define the modules
$L_i$ and $B_i$ in the same way as in the usual case.
The structure of $\bar B_1(A_{n,k})$ and $B_i(A_{n,k})$ over $\Bbb Q$ was studied in \cite{BJ}.
In particular, the structure of $\bar B_1$, $B_2$, and $B_3$ was completely computed.
Here we provide some results and data and formulate a number of questions regarding the structure of
$\bar B_1(A_{n,k})$ and $B_i(A_{n,k})$ over $\Bbb Z$.
\subsection{Experimental data}
Tables 4-8 list the torsion subgroup of $B_\ell(A_{m,n})(\mathbb{Z})$ for small values of $\ell,m,n$. Note that, as in Section \ref{exp-data-I}, we are asserting these subgroups to be trivial for all omitted degrees in the described range.
\begin{table}[h!]
\caption{Torsion in $B_\ell(A_{1,1}(\mathbb Z))$ in degrees $(r,s)$, where $r$ is the degree of the even generator and $s$ is the degree of the odd generator, such that $2 \leq \ell \leq r+s \leq 11$, $r,s\leq 9$, excepting $(r,s)=(8,3)$.}
$ \begin{array}{ |c||c|c|c|c|c|c|c|c | } \hline
(r,s) \backslash \ell &2&3&4&5&6&7&8&9 \\ \hline
(2,2)& \mathbb{Z}/ 2 &&&&&&& \\ \hline
(3,3)&&&\mathbb{Z}/2&&&&& \\ \hline
(4,2)&\mathbb{Z}/2&&\mathbb{Z}/2&&&&& \\ \hline
(3,4)&&&\mathbb{Z}/2&\mathbb{Z}/2&&&& \\ \hline
(4,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&&&& \\ \hline
(2,6)&\mathbb{Z}/2&&\mathbb{Z}/2&&\mathbb{Z}/2&&&\\ \hline
(3,5)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^2&&& \\ \hline
(4,4)&\mathbb{Z}/2&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^3&&& \\ \hline
(5,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^2&&& \\ \hline
(6,2)&\mathbb{Z}/2&&\mathbb{Z}/2&&\mathbb{Z}/2&&& \\ \hline
(3,6)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^2&&\\ \hline
(4,5)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^4&& \\ \hline
(5,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^4&& \\ \hline
(6,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^2&\mathbb{Z}/2&& \\ \hline
(3,7)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3& \\ \hline
(4,6)&\mathbb{Z}/2&&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^8 &\\ \hline
(5,5)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{11}&(\mathbb{Z}/2)^{13}&(\mathbb{Z}/2)^9& \\ \hline
(6,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^6& \\ \hline
(7,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^3&\\ \hline
(8,2)&\mathbb{Z}/2&&\mathbb{Z}/2&&\mathbb{Z}/2&&\mathbb{Z}/2 &\\ \hline
(3,8)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^5&(\mathbb{Z}/2)^3\\ \hline
(4,7)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^{11}&(\mathbb{Z}/2)^{19}&(\mathbb{Z}/2)^9\\ \hline
(5,6)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{12}&(\mathbb{Z}/2)^{20}&(\mathbb{Z}/2)^{31}&(\mathbb{Z}/2)^{14}\\ \hline
(6,5)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{11}&(\mathbb{Z}/2)^{18}&(\mathbb{Z}/2)^{27}&(\mathbb{Z}/2)^{13}\\ \hline
(7,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^{13}&(\mathbb{Z}/2)^8\\\hline
\end{array}$
\end{table}
\begin{table}[h!]
\caption{Torsion in $B_\ell(A_{0,2}(\mathbb Z))$ in degrees $(s_1,s_2)$, $s_1\ge s_2$, such that $2 \leq \ell \leq s_1+s_2 \leq 11$, $2\leq s_1,s_2\leq 9$.}
$ \begin{array}{ |c||c|c|c|c|c|c|c|c| } \hline
(s_1,s_2) \backslash \ell &2&3&4&5&6&7&8&9 \\ \hline
(3,3)&&&\mathbb{Z}/2&&&&& \\ \hline
(4,2)&\mathbb{Z}/2&&\mathbb{Z}/2&&&&& \\ \hline
(4,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&&&& \\ \hline
(4,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^3&&& \\ \hline
(5,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^2&&& \\ \hline
(5,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^4&& \\ \hline
(6,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^2&& \\ \hline
(5,5)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{12}&(\mathbb{Z}/2)^{14}&(\mathbb{Z}/2)^{10}& \\ \hline
(6,4)&\mathbb{Z}/2&&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^8& \\ \hline
(7,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3& \\ \hline
(8,2)&\mathbb{Z}/2 &&\mathbb{Z}/2 && \mathbb{Z}/2 && \mathbb{Z}/2&\\ \hline
(6,5)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^{12}&(\mathbb{Z}/2)^{21}& (\mathbb{Z}/2)^{33}&(\mathbb{Z}/2)^{15}\\ \hline
(7,4)&&&\mathbb{Z}/2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^{11}&(\mathbb{Z}/2)^{19}&(\mathbb{Z}/2)^9\\ \hline
(8,3)&&&\mathbb{Z}/2&\mathbb{Z}/2&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^5&(\mathbb{Z}/2)^3\\ \hline
\end{array}
$\end{table}
\begin{table}[h!]
\caption{Torsion in $B_\ell(A_{2,1}(\mathbb Z))$ in degrees $(r_1,r_2,s)$, $r_1\ge r_2$, such that either $r_1,r_2,s\leq 3$ or $r_1\leq 4, r_2\leq 3, s\leq 2$.}
$\begin{array}{|c||c|c|c|c|c|c|} \hline
(r_1,r_2,s) \backslash \ell &2&3&4&5&6&7 \\ \hline
(2,1,3)&&&\mathbb{Z}/2&&& \\ \hline
(2,2,2)&(\mathbb{Z}/2)^2&&(\mathbb{Z}/2)^2&&& \\ \hline
(3,1,2)&&&\mathbb{Z}/2&&& \\ \hline
(2,2,3)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^2&& \\ \hline
(3,1,3)&&&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^2&& \\ \hline
(3,2,2)&&&(\mathbb{Z}/2)^2&\mathbb{Z}/2&& \\ \hline
(4,1,2)&&&\mathbb{Z}/2&\mathbb{Z}/2&& \\ \hline
(3,2,3)&&&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^{10}& \\ \hline
(3,3,2)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^5& \\ \hline
(4,2,2)&(\mathbb{Z}/2)^2&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^5& \\ \hline
(3,3,3)&&\mathbb{Z}/3&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^{18}&(\mathbb{Z}/2)^{40}&(\mathbb{Z}/2)^{20}+\mathbb{Z}/3 \\ \hline
(4,3,2)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^{12}&(\mathbb{Z}/2)^7 \\ \hline
\end{array}$
\end{table}
\begin{table}[h!]
\caption{torsion in $B_\ell(A_{1,2}(\mathbb Z))$ in degrees $(r,s_1,s_2)$, $s_1\ge s_2$, such that either $r,s_1,s_2\leq 3$ or $r \leq 4, s_1\leq 3, s_2 \leq 2$.}
$\begin{array}{|c||c|c|c|c|c|c|} \hline
(r,s_1,s_2) \backslash \ell &2&3&4&5&6&7 \\ \hline
(1,2,2)&&\mathbb{Z}/3&&&& \\ \hline
(1,3,2)&&\mathbb{Z}/3&\mathbb{Z}/2+\mathbb{Z}/3&&& \\ \hline
(2,2,2)&\mathbb{Z}/2&\mathbb{Z}/3&\mathbb{Z}/2+\mathbb{Z}/3&&& \\ \hline
(2,3,1)&&&\mathbb{Z}/2&&& \\ \hline
(3,2,1)&&&\mathbb{Z}/2&&& \\ \hline
(1,3,3)&&\mathbb{Z}/3&(\mathbb{Z}/2)^3+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^2+\mathbb{Z}/3&& \\ \hline
(2,3,2)&&\mathbb{Z}/3&(\mathbb{Z}/2)^4+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^2+\mathbb{Z}/3&& \\ \hline
(3,2,2)&&\mathbb{Z}/3&(\mathbb{Z}/2)^3+\mathbb{Z}/3&(\mathbb{Z}/2)^2+\mathbb{Z}/3&& \\ \hline
(3,3,1)&&&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^2&& \\ \hline
(4,2,1)&&&\mathbb{Z}/2&\mathbb{Z}/2&& \\ \hline
(2,3,3)&&\mathbb{Z}/3&(\mathbb{Z}/2)^8+(\mathbb{Z}/3)^3&(\mathbb{Z}/2)^{11}+(\mathbb{Z}/3)^3&(\mathbb{Z}/2)^{12}+(\mathbb{Z}/3)^3& \\ \hline
(3,3,2)&&\mathbb{Z}/3&(\mathbb{Z}/2)^7+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^{10}+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^{11}+(\mathbb{Z}/3)^2& \\ \hline
(4,2,2)&\mathbb{Z}/2&\mathbb{Z}/3&(\mathbb{Z}/2)^3+\mathbb{Z}/3&(\mathbb{Z}/2)^5+\mathbb{Z}/3&(\mathbb{Z}/2)^6+(\mathbb{Z}/3)^2& \\ \hline
(4,3,1)&&&(\mathbb{Z}/2)^3&(\mathbb{Z}/2)^5&(\mathbb{Z}/2)^6& \\ \hline
(3,3,3)&&&(\mathbb{Z}/2)^{11}+(\mathbb{Z}/3)^3&(\mathbb{Z}/2)^{25}+(\mathbb{Z}/3)^4&(\mathbb{Z}/2)^{60}+(\mathbb{Z}/3)^7&(\mathbb{Z}/2)^{28}+(\mathbb{Z}/3)^3 \\ \hline
(4,3,2)&&\mathbb{Z}/3&(\mathbb{Z}/2)^7+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^{15}+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^{38}+(\mathbb{Z}/3)^4&(\mathbb{Z}/2)^{19}+(\mathbb{Z}/3)^2 \\ \hline
\end{array}$
\end{table}
\begin{table}[h!]
\caption{Torsion in $B_\ell(A_{0,3}(\mathbb Z))$ in degrees $\mathbf{s}=(s_1,s_2,s_3)$, , such that either $2 \leq \ell \leq s_1+s_2+s_3\leq 10$ and $s_1,s_2,s_3\leq 8$, or $\mathbf{s}=(7,3,1)$.}
{\tiny$
\begin{array}{|c||c|c|c|c|c|c|c|c|} \hline
\mathbf{s} \backslash \ell &2&3&4&5&6&7&8&9 \\ \hline
(2,2,1)&&\mathbb{Z}/3&&&&&& \\ \hline
(2,2,2)&(\mathbb{Z}/2)^2&(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^2+(\mathbb{Z}/3)^2&&&&& \\ \hline
(3,2,1)&&\mathbb{Z}/3&\mathbb{Z}/2+\mathbb{Z}/3&&&&& \\ \hline
(3,2,2)&&\mathbb{Z}/3&(\mathbb{Z}/2)^4+(\mathbb{Z}/3)^4&(\mathbb{Z}/2)^2+(\mathbb{Z}/3)^2&&&& \\ \hline
(3,3,1)&&\mathbb{Z}/3&(\mathbb{Z}/2)^3+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^2+\mathbb{Z}/3&&&& \\ \hline
(4,2,1)&&&(\mathbb{Z}/2)^2+\mathbb{Z}/3&\mathbb{Z}/2&&&& \\ \hline
(3,3,2)&&&(\mathbb{Z}/2)^8+(\mathbb{Z}/3)^4&(\mathbb{Z}/2)^{10}+(\mathbb{Z}/3)^6&(\mathbb{Z}/2)^{11}+(\mathbb{Z}/3)^4&&& \\ \hline
(4,2,2)&\mathbb{Z}/2&&(\mathbb{Z}/2)^6+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^7+(\mathbb{Z}/3)^3&(\mathbb{Z}/2)^8+(\mathbb{Z}/3)^2&&& \\ \hline
(4,3,1)&&&(\mathbb{Z}/2)^4+\mathbb{Z}/3&(\mathbb{Z}/2)^6+\mathbb{Z}/3&(\mathbb{Z}/2)^7+\mathbb{Z}/3&&& \\ \hline
(5,2,1)&&&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^3&&&\\ \hline
(3,3,3)&&\mathbb{Z}/3&(\mathbb{Z}/2)^{12}+\mathbb{Z}/3&(\mathbb{Z}/2)^{25}+(\mathbb{Z}/3)^{10}&(\mathbb{Z}/2)^{64}+(\mathbb{Z}/3)^{18}&(\mathbb{Z}/2)^{28}+(\mathbb{Z}/3)^6&& \\ \hline
(4,3,2)&&&(\mathbb{Z}/2)^9+\mathbb{Z}/3&(\mathbb{Z}/2)^{19}+(\mathbb{Z}/3)^5&(\mathbb{Z}/2)^{50}+(\mathbb{Z}/3)^{10}&(\mathbb{Z}/2)^{22}+(\mathbb{Z}/3)^3&&\\ \hline
(4,4,1)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^{10}+\mathbb{Z}/3&(\mathbb{Z}/2)^{29}+(\mathbb{Z}/3)^2&(\mathbb{Z}/2)^{14}+\mathbb{Z}/3&&\\ \hline
(5,2,2)&&&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^{10}+\mathbb{Z}/3&(\mathbb{Z}/2)^{28}+(\mathbb{Z}/3)^3&(\mathbb{Z}/2)^{12}&&\\ \hline
(5,3,1)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^{21}+\mathbb{Z}/3&(\mathbb{Z}/2)^{10}&&\\ \hline
(6,2,1)&&&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^3&&\\ \hline
(5,3,2)&&&(\mathbb{Z}/2)^9&(\mathbb{Z}/2)^{21}+\mathbb{Z}/3&(\mathbb{Z}/2)^{90}+(\mathbb{Z}/3)^7&(\mathbb{Z}/2)^{94}+(\mathbb{Z}/3)^4&(\mathbb{Z}/2)^{62}+(\mathbb{Z}/3)^3&\\ \hline
(5,4,1)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^{48}+\mathbb{Z}/3&(\mathbb{Z}/2)^{55}+\mathbb{Z}/3&(\mathbb{Z}/2)^{36}+\mathbb{Z}/3&\\ \hline
(6,2,2)&(\mathbb{Z}/2)^2&&(\mathbb{Z}/2)^8&(\mathbb{Z}/2)^{10}&(\mathbb{Z}/2)^{41}+\mathbb{Z}/3&(\mathbb{Z}/2)^{39}&(\mathbb{Z}/2)^{30}&\\ \hline
(6,3,1)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^{27}&(\mathbb{Z}/2)^{29}&(\mathbb{Z}/2)^{21} &\\ \hline
(7,2,1)&&&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^2&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^6&(\mathbb{Z}/2)^6&\\ \hline
(7,3,1)&&&(\mathbb{Z}/2)^4&(\mathbb{Z}/2)^7&(\mathbb{Z}/2)^{27}&(\mathbb{Z}/2)^{36}&(\mathbb{Z}/2)^{63}& (\mathbb{Z}/2)^{27}\\\hline
\end{array}
$}
\end{table}
\subsection{General statements and questions}
\subsubsection{Torsion in $\bar B_1(A_{n,k}(\Bbb Z))$}
\begin{thm}\label{b1barsup}
Let $k\ge 1$, $n+k\ge 2$, and $m_1,...,m_{n+k}>0$. Then
the torsion in $\bar B_1(A_{n,k}(\Bbb Z))[\bold m]$
is $(\Bbb Z/2)^{2^{n+k-2}}$ if all $m_i$ are even, and zero
otherwise.
\end{thm}
\begin{rem}
We see that the behavior of torsion is different from the case $k=0$, when there is torsion of all orders.
\end{rem}
\begin{proof}
Let $x_1,...,x_n,y_1,...,y_k$ be the even and odd generators, respectively,
and let $\Omega(\Bbb Z^{n|k})$ be the algebra of differential forms on these variables
over $\Bbb Z$ (so the elements $x_i$ and $dy_j$ are even, while $dx_i$ and $y_j$ are odd).
Then similarly to Section 3, we have a linear isomorphism
$\varphi: A_{n,k}(\Bbb Z)/M_3\to \Omega(\Bbb Z^{n|k})^{ev}$, which maps
$L_2$ into exact forms of positive degree. However, the image
is not the entire set of exact positive forms; for instance,
if $y$ is an odd variable, then $(dy)^{2r}=d(y(dy)^{2r-1})$ is an exact form but
is not in the image of $L_2$. On the other hand, $2(dy)^{2r}$ is
$\varphi([y,y^{2r-1}])=\varphi(2y^r)$. More generally, it is easy to show
as in Section 3 that if $\omega$ is a positive exact form then $2\omega\in \varphi(L_2)$.
This shows that we have an exact sequence
$$
0\to K\to \bar B_1(A_{n,k}(\Bbb Z))\to \Omega^{ev}/\Omega^{ev}_{ex}\to 0,
$$
where $K$ is a vector space over $\Bbb F_2$.
Now, as in the even case, we have
$$
(\Omega^{ev}/\Omega^{ev}_{ex})[m_1,...,m_{n+k}]=
(\Omega^{ev}/\Omega^{ev,+}_{cl})[m_1,...,m_{n+k}]\oplus H^{ev,+}(\Bbb Z^{n|k})[m_1,...,m_{n+k}],
$$
where the first summand is a free group and the second summand
is torsion. Furthermore, it is easy to show that for positive $m$,
$H^i(\Bbb Z^{0|1})[m]=0$ (as the Poincar\'e lemma for odd variables holds over $\Bbb Z$),
so using the K\"unneth formula as in the even case,
we see that for $k>0$ we have $H^{ev,+}(\Bbb Z^{n|k})[m_1,...,m_{n+k}]=0$.
This means that the group $(\Omega^{ev}/\Omega^{ev}_{ex})[m_1,...,m_{n+k}]$ is free, and hence
the above exact sequence splits.
It remains to compute the dimension of $K$. To do so, note that
$\bar B_1(A_{n,k}(\Bbb Z))\otimes \Bbb F_2=\bar B_1(A_{n,k}(\Bbb F_2))$.
On the other hand, $\bar B_1(A_{n,k}(\Bbb F_2))=\bar B_1(A_{n+k}(\Bbb F_2))$
(the super-structure does not matter in characteristic $2$),
so the dimension of the right hand side in multidegree $(m_1,...,m_{n+k})$
is known from Section 3. Finally, the rank of the free part $\bar B_1(A_{n,k}(\Bbb Z))[\bold m]$ is known from
\cite{BJ}. Subtracting the two, we get the statement.
\end{proof}
\subsubsection{Torsion in $B_1(A_{n,k}(\Bbb Z))$}
In the case $k>0$, unlike the purely even case $k=0$, the group
$B_1(A_{n,k}(\Bbb Z))$ has torsion. Namely, recall that
$B_1(A_n(\Bbb Z))$ has a basis consisting of cyclic words, i.e., words in the generators
up to cyclic permutation. This remains true for $A_{n,k}$, except that
some words turn out to be equal to minus themselves, thus being 2-torsion.
It is clear that such words are exactly even powers of odd words.
This implies the following result.
\begin{prop}\label{torb1}
The group ${\rm tor}B_1(A_{n,k}(\Bbb Z))$ is a vector space over $\Bbb F_2$, whose multivariable Hilbert series
is given by the formula
$$
h_{n,k}(t_1,...,t_n,u_1,...,u_k)=
$$
$$
\sum_{\bold m: m_{n+1}+...+m_{n+k}\text{ is odd}} a_{m_1,...,m_{n+k}}
\frac{t_1^{2m_1}...t_n^{2m_n}u_1^{2m_{n+1}}...u_k^{2m_{n+k}}}
{1-t_1^{2m_1}...t_n^{2m_n}u_1^{2m_{n+1}}...u_k^{2m_{n+k}}},
$$
where $f(z):=\sum a_{m_1,...,m_{n+k}} z_1^{m_1}...z_{n+k}^{m_{n+k}}$
is the Hilbert series of the free Lie algebra on $n+k$ generators:
$$
f(z)=\sum_{d\ge 1}\frac{\mu(d)}{d}\log(1-z_1^d-...-z_{n+k}^d),
$$
where $\mu(d)$ is the M\"obius function.
In particular, torsion in multidegree $\bold m$ is nonzero if and only if
all $m_i$ are even, and \linebreak $(m_{n+1}+...+m_{n+k})/{\rm gcd}(\bold m)$ is odd.
\end{prop}
\begin{ex}\label{twovar} Let $\nu(m)$ be the maximal power of $2$ dividing $m$.
For $n=0$, $k=2$, the torsion in $B_1$ in bidegree $(m_1,m_2)$ is nonzero
iff $m_1$ and $m_2$ are even, and $\nu(m_1)\ne \nu(m_2)$. For $n=1$ and $k=1$,
the torsion in $B_1$ in bidegree $(m_1,m_2)$ is nonzero
iff $m_1$ and $m_2$ are even, and $\nu(m_1)\ge \nu(m_2)$.
\end{ex}
\begin{proof}
It is clear that any nontrivial word is a power of a non-power word (i.e. a word which is not a power of another word)
in a unique way, and words generating 2-torsion are exactly the even powers of non-power words of odd
total degree with respect to the odd generators (if $z$ is such a non-power word, then
$2z^{2l}=[z,z^{2l-1}]$).
Also, it is well known that the generating function for non-power words modulo cyclic permutation
is the same as the Hilbert series of the free Lie algebra.
Indeed, if the generating function for non-power words modulo cyclic permutation
is $f(v)=\sum a_{\bold m}v^{\bold m}$, where $v^{\bold m}=v_1^{m_1}...v_{n+k}^{m_{n+k}}$,
then
$$
\sum_{\bold m} a_{\bold m}(\sum m_j)\frac{v^{\bold m}}{1-v^{\bold m}}=\frac{\sum v_j}{1-\sum v_j},
$$
since any nontrivial word is uniquely a power of a non-power word.
(The factor $\sum m_j$ appears because non-power words are considered up to cyclic permutations,
which act freely on them).
Applying the inverse of the Euler operator, $E^{-1}$, which divides terms of total degree $N$ by $N$, we get
$$
\sum_{\bold m} a_{\bold m}\log (1-v^{\bold m})=\log(1-\sum v_j),
$$
or
$$
\prod_{\bold m} (1-v^{\bold m})^{a_{\bold m}}=1-\sum v_j,
$$
which implies that $a_{\bold m}$ are the dimensions of the components of the free Lie algebra.
These two statements imply the proposition.
\end{proof}
\subsubsection{Torsion in $B_2(A_{n,k}(\Bbb Z))$}
\begin{prop}\label{no4tor}
If $k\ge 1$ and $m_i>0$ then torsion in $B_2(A_{n,k}(\Bbb Z))[m_1,...,m_{n+k}]$
is a vector space over $\Bbb F_2$, which is a quotient of
$\Omega^{odd}_{ex}(\Bbb Z^{n|k})[\bold m]\otimes {\Bbb F}_2$. In particular, there is no 4-torsion,
and $B_2(A_{n,k}(\Bbb Z[1/2]))[m_1,...,m_{n+k}]$
is a free abelian group.
\end{prop}
\begin{proof}
It is shown in the proof of Theorem \ref{b1barsup} that
$\bar B_1(A_{n,k}(\Bbb Z))[\bold m]$ is a quotient of
$\Omega^{ev}(\Bbb Z^{n|k})[\bold m]/2\Omega^{ev}_{ex}(\Bbb Z^{n|k})[\bold m]$.
Therefore, analogously to the proof of Theorem \ref{upbo} one shows that
$B_2(A_{n,k}(\Bbb Z))[\bold m]$ is a quotient of
$\Omega^{odd}(\Bbb Z^{n|k})[\bold m]/2\Omega^{odd}_{ex}(\Bbb Z^{n|k})[\bold m]$.
But as explained in the proof of Theorem \ref{b1barsup},
the De Rham cohomology groups vanish in positive multidegrees.
Hence, the subgroup $\Omega^{odd}_{ex}(\Bbb Z^{n|k})[\bold m]$
is saturated. So the torsion in $B_2(A_{n,k}(\Bbb Z))[\bold m]$ is a quotient of
$\Omega^{odd}_{ex}(\Bbb Z^{n|k})[\bold m]/2$, as desired.
\end{proof}
Let us now discuss the 2-torsion in $B_2(A_{n,k}(\Bbb Z))$ (which by Proposition \ref{no4tor}
is all the torsion in positive degrees for $k>0$).
Arguing as in Lemma \ref{lem:Universal}, we see that for any abelian groups
$\Bbb Z^N\supset A\supset B$,
we have an exact sequence
$$
{\rm tor}_p(\Bbb Z^N/A)\to (A/B)\otimes \Bbb F_p\to A_p/B_p\to 0,
$$
which implies that
$$
\dim(A_p/B_p)\le \dim ((A/B)\otimes \Bbb F_p)\le \dim(A_p/B_p)+\dim {\rm tor}_p(\Bbb Z^N/A).
$$
Setting $p=2$, $\Bbb Z^N=L_1[\bold m]$, $A=L_2[\bold m]$, and $B=L_3[\bold m]$,
this becomes
$$
\dim B_2(A_{n+k}(\Bbb F_2))[\bold m]\le \dim B_2(A_{n,k}(\Bbb Z))[\bold m]\otimes \Bbb F_2\le \dim B_2(A_{n+k}(\Bbb F_2))[\bold m]+\dim {\rm tor}_2(B_1(A_{n,k}(\Bbb Z))[\bold m].
$$
Subtracting $\dim B_2(A_{n+k}(\Bbb Q))$, we get
\begin{prop}\label{ineq}
$$
\dim {\rm tor}_2B_2(A_{n+k}(\Bbb Z))[\bold m]\le \dim {\rm tor}_2(B_2(A_{n,k}(\Bbb Z)))[\bold m]\le
$$
$$
\le\dim {\rm tor}_2B_2(A_{n+k}(\Bbb Z))[\bold m]+
\dim {\rm tor}_2(B_1(A_{n,k}(\Bbb Z))[\bold m].
$$
\end{prop}
\begin{rem}
1. Both inequalities in Proposition \ref{ineq} can be strict.
For instance, for $B_2(A_{2,1}(\Bbb Z))[2,2,2]$, they look like $1\le 2\le 3$.
2. The upper bound of Proposition \ref{ineq} is asymptotically bad -- it grows exponentially with degrees,
while the bound of Proposition \ref{no4tor} is uniform in degrees. However, in particular degrees
the bound of Proposition \ref{ineq} may be better. For example, for $n=k=1$ and $\bold m=(2,4)$,
then there is a unique odd exact form up to scaling, namely $xdx(dy)^4=d(xydx(dy)^3)$, so
Proposition \ref{no4tor} gives the bound of $1$, while Proposition \ref{ineq} implies that
${\rm tor}B_2(A_{1,1}(\Bbb Z))[2,4]=0$.
\end{rem}
Proposition \ref{ineq} together with Proposition \ref{torb1}
implies the following corollary.
\begin{cor}
If the group $B_2(A_{n,k}(\Bbb Z))[\bold m]$ has nontrivial 2-torsion,
then all $m_i$ are even. Moreover, unless
$(m_{n+1}+...+m_{n+k})/{\rm gcd}(\bold m)$ is odd, this 2-torsion
depends only on $n+k$ (i.e., is the same as the 2-torsion in $B_2(A_{n+k}(\Bbb Z))$).
\end{cor}
\begin{cor}
If $n+k=2$ then necessary conditions
for a nontrivial 2-torsion are the conditions of Example
\ref{twovar}.
\end{cor}
Note that this agrees with the results of computer calculations.
\subsubsection{Results and questions on torsion in higher $B_\ell$}
\begin{prop}\label{nontor}
Nontrivial torsion in $B_\ell(A_{n,k}(\Bbb Z[1/2]))[m_1,...,m_{n+k}]$ implies \linebreak $\ell \le m_1 + \ldots + m_{n+k} - 2$.
\end{prop}
\begin{proof}
Same as in Remark \ref{weakerst} (but using the free Lie superalgebra).
\end{proof}
\begin{rem}
1. Proposition \ref{nontor} seems to be true over $\Bbb Z$ but it seems that the proof would need to be modified, since
Lie superalgebras don't behave well in characteristic $2$.
2. We see from the tables that there {\it can} be torsion
if $\ell=m_1 + \ldots + m_{n+k}-2$ (this does not seem to occur
in the even case).
\end{rem}
\begin{prop}
If $n+k=2$, there is no torsion in $B_\ell[m_1,m_2]$ if either $m_1$ or $m_2$ equals $1$.
\end{prop}
\begin{proof} Denote the generators of $A_{n,k}(\Bbb Z)$ by $x,y$, and consider
$A_{n,k}(\Bbb Z)[m,1]$. A basis of this group is the collection of elements
$x^iyx^j$, where $i+j=m$.
Consider first the case when $x$ is even.
Let us identify $A_{n,k}(\Bbb Z)[m,1]$ with the space of polynomials of a variable $z$
with integer coefficients and degree $\le m$, by $x^iyx^j\mapsto z^i$.
In this case, the operation of bracketing with $x^l$ acting from $A_{n,k}(\Bbb Z)[m-l,1]$ to
$A_{n,k}(\Bbb Z)[m,1]$ corresponds to the operator $P\mapsto (z^l-1)P$ on polynomials.
This implies that $L_{i+1}(A_{n,k}(\Bbb Z))[m,1]$ is identified with
polynomials of degree $\le m$ which are divisible by $(z-1)^i$. So we see that
$B_{i+1}(A_{n,k}(\Bbb Z))[m,1]=\Bbb Z$ for $i=0,1.,,,m$, and zero for larger $i$,
so there is no torsion.
If $x$ is odd, the argument is similar. Namely, if $y$ is even,
we should use the assignment $x^iyx^j\mapsto (-1)^{ij}z^i$,
while if $y$ is odd, we should use the assignment
$x^iyx^j\mapsto (-1)^{i(j+1)}z^i$.
\end{proof}
\begin{thm}
The dimensions of $B_i(A_{n,k}(\mathbb{F}_p))[\mathbf{m}]$ for $i\ge 2$ are bounded from above by a constant $K_i$
independent of $ \mathbf{m}$ and $p$ (but depending on $n,k$).
\end{thm}
\begin{proof}
Since $[ab,c]+[bc,a]+[ca,b]=0$, we have
$$
[ab,[c,d]]+[bc,[a,d]]+[ca,[b,d]]=[c,[ab,d]]+[a,[bc,d]]+[b,[ca,d]].
$$
Let $A=A_{n,k}(\Bbb F_p)$. Suppose we know that $L_i=[A_{\le r},L_{i-1}]$, where
$A_{\le r}$ is the part of $A$ of degree $\le r$. Let $u$ be a word of degree $\ge 2r+2$.
Then we can write $u$ as $ab$, where the degrees of $a$ and $b$ are $\ge r+1$.
Since $L_i$ is the sum of the spaces $[c,L_{i-1}]$, where $c$ has degree $\le r$,
we find that $[u,L_i]$ is contained in the sum of spaces $[v,L_i]$, where $v$ has degree
$\le 2r+1$. This implies that $L_{i+1}=[A_{\le 2r+1},L_i]$.
Since $L_2=[A_{\le 1},L_1]$ by Lemma \ref{degg1}, we obtain by induction in $i$ that
$L_{i+1}=[A_{\le 2^i-1},L_i]$ and hence $B_{i+1}=[A_{\le 2^i-1},B_i]$. This implies that if $K_i$ is defined, then
one can take $K_{i+1}=K_i\dim A_{\le 2^i-1}$.
It remains show that $K_2$ is defined. To show this, recall that $B_2=\sum [x_i,\bar B_1]$.
On the other hand, by the results of \cite{BJ} and Theorem \ref{b1barsup},
the dimensions of the homogeneous components of $\bar B_1$ are bounded by some constant
$K$. So we can take $K_2=(n+k)K$, and we are done.
\end{proof}
\begin{rem}
The bound $K_i$ given by this proof is very poor
(it grows like $2^{i^2/2}$). For
$p\ge 3$, one can get a much better
bound, which grows exponentially with $i$.
Namely, since $L_\ell(A_{n,k}(\Bbb Z[1/2]))[\ell]$ is a saturated subgroup of
$A_{n,k}(\Bbb Z[1/2])[\ell]$, it follows from Lemma 3.1 in \cite{AJ}
that over any ring $R$ containing $1/2$, one has
$$
[(ab+ba)c,B_i]\subset [Q,B_i],
$$
where $Q$ is the span of at most quadratic monomials in $a,b,c$
(where $a,b,c\in A_{n,k}$). This implies that if $E$ is a complement to the
image in $\bar B_1$ of $1$ and all elements $(ab+ba)c$ where $a,b,c$ are of positive degree, then
$B_{i+1}=[E,B_i]$. But for $E$ we can take the span of all words of degrees $1,2$
in the generators. The number of such words is $\frac{(n+k)^2}{2}+\frac{3n+k}{2}$.
Thus, for $p\ge 3$ we can take
$$
K_i=(\frac{(n+k)^2}{2}+\frac{3n+k}{2})^{i-2}(n+k)K.
$$
for $i\ge 2$.
We expect that by a suitable strengthening of Lemma 3.1 in \cite{AJ},
this argument can be adapted to get an exponentially growing bound
$K_i$ also in the case $p=2$.
\end{rem}
Let us now state some questions that are motivated by the data in the tables.
{\bf Question 3.} Suppose $k\ge 1,n+k\ge 2$. Can $B_i(A_{n,k}(\Bbb Z))$
have $p$-torsion with $p>n+k$? For example, if $n+k=2$ (i.e., $(n,k)=(0,2)$ or $(1,1)$), can
$B_i$ have $p$-torsion for odd $p$? Is the torsion in $B_i$ in this case a vector space over $\Bbb F_2$?
{\bf Question 4.} Is there torsion in $B_3(A_{0,2}(\Bbb Z))$?
{\bf Question 5.} If $n+k=3$, can there be torsion in $B_3(A_{n,k}(\Bbb Z))[m_1,m_2,m_3]$
($m_1,m_2,m_3> 0$) other than 3-torsion?
\begin{rem} We note that, as follows from the above tables,
it is not true in the supercase that if a torsion element $T$ in $B_i$ has degree $m$
with respect to some generator, then the order of $T$ divides $m$.
For example, for $n=1,k=2$, we have a 3-torsion element in multidegree $(1,2,2)$.
\end{rem}
\subsection{Lower central series and modular representations}
Let us discuss the connection of the theory of lower central series with modular
representation theory. It is clear that the algebraic supergroup scheme $GL(n|k)(\Bbb Z)$ acts
on each $B_i(A_{n,k}(\Bbb Z))[m]$ (elements of total degree $m$),
with decomposition into multidegrees being the weight decomposition.
Moreover, for each prime $p$, the $p$-torsion in this representation is a subrepresentation, which actually
factors through $GL(n|k)(\Bbb F_p)$. Moreover, the tensor product of this representation with the algebraic closure
$\overline{\Bbb F_p}$ naturally extends to a representation of $GL(n|k)(\overline{\Bbb F_p})$.
In the purely even case, from computer data it appears that this modular representation
is always trivial at the Lie superalgebra level. However, in the supercase we see nontrivial
Lie superalgebra representations.
For instance, for $n=1$ and $k=2$, we have a 3-torsion element in $B_3$ in multidegree $(1,2,2)$,
which is unique up to scaling in degree $5$. Its degrees are not divisible by $3$, so
it defines a 1-dimensional representation of $GL(1|2)$ in characteristic $3$
which is nontrivial at the Lie superalgebra level.
It is easy to see that at the Lie superalgebra level,
this is in fact the well-known Berezinian representation
with weight $(1,-1,-1)$ (which is the same as $(1,2,2)$ in characteristic $3$).
Now consider the 3-torsion in $B_3$ for $n=0$ and $k=3$, which gives representations of $GL(3)$
(see the last table in Subsection 6.1). In this case,
in degree $5$ we have three independent torsion elements,
in multidegrees $(1,2,2)$, $(2,1,2)$ and $(2,2,1)$, respectively.
Clearly, they generate a copy of the representation $V^*\otimes (\wedge^3V)^{\otimes 2}$, where
$V$ is the 3-dimensional vector representation spanned by the generators
of $A_{0,3}$.
In degree $6$, there is an $8$-dimensional representation,
with weights being all the permutations of $(1,2,3)$, as well as $(2,2,2)$ (with multiplicity $2$).
This representation has the same character as ${\mathfrak{sl}}(V)\otimes (\wedge^3V)^{\otimes 2}$.
Note that we cannot claim without additional argument that they are isomorphic,
since ${\mathfrak{sl}}(V)$ is reducible ($1\in {\mathfrak{sl}}(V)$ in characteristic $3$ as $V$ is $3$-dimensional).
All we can say is that its composition series consists of
a $7$-dimensional irreducible representation and the
1-dimensional representation $(\wedge^3V)^{\otimes 2}$.
In degree $7$, we have a $6$-dimensional representation (whose weights are
permutations of $(1,3,3)$ and $(2,2,3)$, with 1-dimensional weight spaces),
which is clearly $S^2V^*\otimes (\wedge^2V)^{\otimes 3}$.
Finally, we have a representation $(\wedge^3V)^{\otimes 3}$ (trivial at the Lie algebra level)
sitting in degree $9$.
Actually, by analogy with \cite{FS}, for $p>2$
the $p$-torsion in $B_i$ is a representation of the
Lie superalgebra $W_{n|k}$ of supervector fields in characteristic $p$
(this action is compatible with the action of $GL(n|k)$).
In particular, we have an action of the Lie superalgebra
${\mathfrak{pgl}}(n+1|k)$ which sits inside $W(n|k)$ as the Lie algebra of
the supergroup of projective transformations. Looking at the action of this subalgebra,
we get more information.
For instance, consider the case $n=0$, $k=3$.
In this case, the 3-torsion in $B_3$
tensored with the algebraic closure $\overline{\Bbb F}_p$
carries an action of the Harish-Chandra pair $({\mathfrak{sl}}(1|3),GL(3))$
(which is equivalent to an action of the supergroup $SL(1|3)$).
Note that the trivial 1-dimensional representation of ${\mathfrak{sl}}(1|3)$ has
a family of lifts $\chi^{\otimes r}$ to $SL(1|3)$, parametrized by integers $r$;
namely $\chi|_{GL(3)}=(\wedge^3V)^{\otimes 3}$.
Now consider the 3-torsion $X$ in degrees 5,6,7 in $B_3(A_{0,3}(\Bbb Z))$
tensored with the algebraic closure (so $\dim(X)=17$).
Let us describe the composition series of $X$ as a representation of $SL(1|3)$.
It is easy to see that the irreducible module for $SL(1|3)$ with lowest degree component
$V^*\otimes (\wedge^3V)^{\otimes 2}$ is $\tilde V^*\otimes \chi$,
where $\tilde V$ is the 4-dimensional defining representation of $SL(1,3)$.
As a $GL(3)$-module, this representation is $V^*\otimes (\wedge^2V)^{\otimes 2} \oplus (\wedge^3V)^{\otimes 2}$.
Thus, we see that $\tilde V^*\otimes \chi$
is contained in the composition series of $X$.
The remaining $GL(3)$-modules are the 6-dimensional and 7-dimensional
irreducible representations of $GL(3)$, and they have to comprise an irreducible $SL(1|3)$-module,
since any $SL(1|3)$ module with the trivial action of the odd generators of the Lie algebra
must be trivial for the whole Lie algebra. Thus, we see that the composition series of $X$ has length 2 and consist of a 4-dimensional and a
13-dimensional irreducible representations, living n degrees 5,6 and 6,7, respectively.
They have Hilbert series $3t^5+t^6$ and $7t^6+6t^7$.
In a similar way one should be able to study the 3-torsion
in $B_i(A_{n,k})$ for $i>3$ (as well as higher $p$-torsion).
\section{Further work}
Besides attempting to answer the questions stated above and demystify the enigmatic pattern of torsion in
$B_i(A_n)$ and more generally $B_i(A_{n,k})$, we can point out the following interesting directions of future research.
1. Over $\mathbb{Z}[1/2]$, the Lie algebra $W_n$ of vector fields on the $n$-dimensional space
acts on $\bar B_1(A_n)$ and $B_i(A_n)$ for $i\ge 2$, similarly to \cite{FS}
(in characteristic 2, this can be generalized but a
modification is needed).
It would be interesting to study this action and to apply representation theory of $W_n$ to
the study of the torsion in $B_i(A_n)$. In particular, we conjecture that
the Lie algebra $W_n$ acts trivially
on the torsion in $B_i(A_n)$
(by analogy with the fact that vector fields
on a manifold act trivially on its cohomology).
Note that this conjecture implies the conjecture that the order
of a torsion element divides its degree with respect
to each variable $x_i$, by considering the vector field $x_i\frac{\partial}{\partial x_i}$.
We expect that in this study over $\mathbb{F}_p$, the restricted structure
of the Lie algebra $W_n$ will play a role.
Also it would be interesting to extend this approach to the supercase, where
there is an action of the Lie superalgebra $W_{n|k}$.
2. It would be interesting to generalize the results
of this paper from the quotients $B_i$ to the quotients
$N_i:=M_i/M_{i+1}$. (We note that a search in low
degrees found no torsion in these quotients).
3. It would be interesting to extend the results of this paper to algebras with relations, along the lines
of the appendix to \cite{DKM} and of \cite{BB}.
\pagestyle{empty}\singlespace
|
{
"timestamp": "2012-03-09T02:04:31",
"yymm": "1203",
"arxiv_id": "1203.1893",
"language": "en",
"url": "https://arxiv.org/abs/1203.1893"
}
|
\section{Introduction}
In this article we derive an equation for the cusp anomalous dimension for all angles
and for all values of the 't Hooft coupling $\lambda =g_{YM}^2 N$ in the planar limit
of ${\cal N} =4$ super Yang Mills. We obtain a system of non-linear integral equations of the form
of a Thermodynamic Bethe Anstaz (TBA) system. The value of the cusp anomalous dimension can
be obtained from a solution of the TBA system. This is also equal to the quark/anti-quark potential
on the three sphere, see figure \ref{CuspDiagram}.
\begin{figure}[h]
\centering
\def16cm{12cm}
\input{fig1.pdf_tex}
\caption{({\bf a}) A Wilson line with a cusp angle $\phi$. ({\bf b}) Under the plane to cylinder map
the two half lines in (a) are
mapped to a quark anti-quark pair sitting at two points on $S^3$ at a relative angle of $\pi -\phi$. The
quark anti-quark lines are extended along the time direction. }
\label{CuspDiagram}
\end{figure}
The cusp anomalous dimension is associated with the logarithmic divergence arising from
a Wilson loop with a cusped contour \cite{PolyakovCusp}
\begin{equation} \la{cuspdef}
\langle W \rangle \sim e^{ - \Gamma_{ {\rm cusp}}(\phi,\lambda) \log { L_{\rm IR} \over \epsilon_{\rm UV}} }\,,
\end{equation}
where $L_{\rm IR}$ and $\epsilon_{\rm UV}$ are IR and UV cutoffs respectively.
The locally supersymmetric Wilson loop in ${\cal N}=4$ super Yang Mills also includes a coupling
to the scalar fields specified by a direction in the internal space $\vec n$ (with $\vec n^2 =1$)
\begin{equation} \la{wildef}
W \sim {\rm Tr}\left[ P e^{ i \oint A\cdot dx + \oint |dx| \vec n\cdot \vec \Phi } \right]\,.
\end{equation}
Instead of considering the same vector $\vec n$ on the two lines that make the cusp, we can
take two vectors $\vec n$ and $\vec n'$. This introduces a second angle $\cos \theta = \vec n\cdot\vec n'$. Thus we have the generalized cusp anomalous dimension
$\Gamma_{ {\rm cusp}}(\phi, \theta , \lambda)$
\cite{Drukker:1999zq}.
$\Gamma_{ {\rm cusp}}(\phi, \theta)$
can be computed in terms of a solution of the TBA system of equations presented in this article. We can
also consider the continuation
$\phi = i \varphi$, where $\varphi$ is a boost angle in Lorentzian signature.
Before describing the computation, let us make some general remarks.
\subsection{Remarks on the cusp anomalous dimension }
$\Gamma_{ {\rm cusp}}$ is related to a variety of physical observables:
\begin{itemize}
\item
It characterizes the IR divergences that arise when we scatter massive colored particles. Here
$\varphi$ is the boost angle between two external massive particle lines. For each consecutive pair of lines in the color ordered diagram we get a factor of the form \nref{cuspdef}, where
$L_{IR}$ is the IR cutoff and $\epsilon^2_{\rm UV}$ is the given by the square of the
sum of the momenta of the two consecutive particles. More explicitly, the angle is given by
$ \cosh \varphi =- { p_1 . p_2 \over \sqrt{ p_1^2 p_2^2 } } $. This relation is general for
any conformal gauge theory. See \cite{Korchemsky:1991zp,Becher:2009kw} and references therein.
In ${\cal N}=4$ super Yang Mills the massive particles can be
obtained by setting some Higgs vevs to be non-zero $\vec \Phi$. Then the angle $\theta$ is
the angle between the Higgs vevs associated to consecutive massive particles \cite{LadderPaper}.
\item
The IR divergences of massless particles are characterized by $\Gamma^{\infty}_{ {\rm cusp}}$ which
is the coefficient of the large $\varphi$ behavior of the cusp anomalous dimension, $\Gamma_{ {\rm cusp}} \propto \varphi \Gamma_{ {\rm cusp}}^\infty $. $\Gamma^{\infty}_{ {\rm cusp}}$ was computed in the seminal paper
\cite{BES}. Note that $\Gamma^{\infty}_{ {\rm cusp}}$ is also sometimes called the ``cusp
anomalous dimension'' though it is a particular limit of the general, angle dependent ``cusp
anomalous dimension'' defined in \nref{cuspdef} .
\item
By the plane to cylinder map this quantity is identical with the energy of a static quark and
anti-quark sitting on a spatial three sphere at an angle $ \pi - \phi$.
\begin{equation}
\Gamma_{ {\rm cusp}}(\phi,\theta) = V( \phi , \theta )\,.
\end{equation}
See figure \ref{CuspDiagram} . This potential depends on the angle $\phi$ as well as on the internal
orientations of the quark and anti-quark, which define the second angle $\theta$.
\item
In particular, in the small $\delta = \pi -\phi $ limit we get the same answer as the quark-anti-quark
potential in flat space\footnote{This limit does not commute with the perturbative expansion in $\lambda$. So \nref{qqpot} is correct if $ \delta \ll \lambda $. If we expand first in $\lambda$ and then
take the $\delta \to 0$ limit we get a different answer due to IR divergences that arise
in the naive perturbative expansion. These also arise in QCD, The origin of these logs are discussed in \cite{Pineda,LadderPaper}. }
\begin{equation} \label{qqpot}
\Gamma_{ {\rm cusp}}( \phi , \lambda ) \sim { v( \theta, \lambda ) \over \delta }\,,~~~~~~{\rm when }~~~~~\delta = \pi - \phi \to 0 \,,
\end{equation}
where $v(\lambda)$ is the coefficient of the quark-anti-quark potential, $V = { v(\theta,\lambda) \over r }$, for a quark and an anti-quark at distance $r$ in flat space and
couplings to the Higgs fields which are rotated by a relative angle $\theta$.
\item
In the small $\phi$ limit the cusp anomalous dimension goes as $\phi^2$ and one can
define a Bremsstrahlung function $B$ by
\begin{equation}
\Gamma_{ {\rm cusp}} \sim - (\phi^2 - \theta^2 ) B(\lambda) ~~~~~~~~~ \phi , \theta \ll 1\,.
\end{equation}
This function $B$ can be computed exactly using localization, see \cite{Correa:2012at} and \cite{Fiol:2012sg}.
Here we will derive a set of integral equations that also determines $B$. In this way
we can link the localization and integrability exact solutions. This function
$B$ is also related to a variety of observables, see \cite{Correa:2012at,Fiol:2012sg}
for further discussion.
\end{itemize}
Another motivation to study the cusp anomalous dimension is the study of amplitudes.
Amplitudes are also functions of the angles between particles. Here we get a very simple function
of one angle which has a structure very similar to amplitudes, since it is related to
amplitudes of massive particles. Thus, obtaining exact results for this quantity is useful
to learn about the general structure of the amplitude problem.
\subsection{Method }
The method to obtain the equation is a bit indirect and we need several preliminary results
that are interesting in their own right. Just for orientation we will outline the main
idea and method for its derivation.
The method consists of the following steps
\begin{itemize}
\item
We first consider the problem of computing the spectrum of local operators
on a Wilson line. We consider the particular case of operators with a large charge,
i.e. operators containing a large number, $L$, of the complex scalar field $Z$ insertions. These
insertions create a BMN vacuum \cite{BMN}.
\item
In the large $L$ limit the problem can be solved using an asymptotic Bethe Ansatz that involves the propagation of certain ``magnons''. These equations describe magnons moving on a long strip of length $L$ with two boundaries associated to the Wilson loop on each of the ``sides'' of the operator, see
\cite{DrukkerKawamoto}. The propagation of the magnons in the bulk is the usual one \cite{Review}.
The new feature is the existence of a boundary. The magnons are reflected at the boundary and one
needs the boundary reflection matrix. This is fixed in two steps.
\item
We determine the matrix structure of the reflection matrix from group theory, as in
\cite{BeisertDynamic,HMopen,CY,CRY}. This reflection matrix is such that it obeys the
boundary Yang Baxter equation \cite{Ghoshal:1993tm}. This is evidence that the boundary
condition preserves integrability.
\item
We derive a crossing equation for the reflection phase and we find a solution.
\item
Doing a time/space flip, so that now we have the mirror theory between two
boundary states separated by a mirror ``time'' $L$. See figure \ref{TBAfigure}.
We can apply a symmetry generator that
rotates one boundary relative to the other, so that we introduce the two angles.
\item
We compute this overlap using TBA equations for any $L$,
focusing on the ground state
energy, which is extracted by taking
the large $T$ limit of the computation in figure \ref{TBAfigure}. These boundary
TBA equations
can be derived following a method similar to the relativistic case \cite{LeClair:1995uf}.
\item
We set $L=0$ we get the cusp anomalous dimension.
\end{itemize}
Let us discuss these steps in a bit more detail. First we should note that the exact integrability methods, as currently understood, work best to compute energies of states. Thus, we should phrase the computation of the cusp anomalous dimension as the computation of an energy. This is very simple. Under the usual plane to cylinder map, the cusp on the plane maps into two static quark and anti-quark
lines on $S^3 \times R$. The quark and anti-quark lines are extended along the time direction, and
they are separated by an angle $\pi -\phi$ on the $S^3$, see figure \ref{CuspDiagram}. The case $\phi=0$, which is the
straight line in the plane, is mapped to a quark-anti-quark pair at opposite points on the sphere.
If $\theta=0$, this is a BPS configuration and the cusp anomalous dimension vanishes exactly for all $\lambda$. In fact,
for $\theta = \pm \phi$ we continue to have a BPS configuration \cite{Zarembo:2002an} and the cusp anomalous dimension
continues to vanish. In general, the cusp anomalous dimension is the energy of this quark-anti-quark configuration, as a function of
the two angles, $\phi$ and $\theta$.
The configuration with $\theta = \phi =0$ preserves 16 supercharges which, together with the bosonic symmetries,
give rise to a $OSp(4^*|4)$ symmetry group. This is important to determine the boundary reflection matrix.
Let us begin by considering an apparently unrelated problem which is the problem of computing the
anomalous dimension of operators inserted along a Wilson loop. First we consider a straight
Wilson loop and we insert an operator at the point $t=0$.
For example, we can consider an insertion of a complex scalar field $Z$ on the contour
\begin{equation}
\la{simplein}
P e^{i \int_{-\infty}^0 ( A + i \Phi_4)} Z(0) e^{i \int_0^\infty( A_t + i \Phi_4) } = B_l Z(0) B_r\,.
\end{equation}
These are operators that live on the loop and should not be confused with closed string
operators. We denote these operators as $ B_l Z B_r $, where $B_{l,r}$ stands for the usual
path ordered exponentials of the gauge field. The operator considered above is BPS if $Z$ is constructed out of
scalars that do not appear in $B_{l,r}$ \nref{simplein}.
To be definite, we consider $Z = \Phi_5 + i \Phi_6$.
We can similarly consider operators of the form $B_l Z^L B_r$ which continue to be BPS. The straight Wilson loop
is invariant under dilatations, so we can characterize the operators by their dimension
under dilatations. These operators have dimension $\Delta = L$.
Determining the scaling dimension of operators of this type, but with more general insertions, is easier in the large $L$ limit. Then, we can solve this problem by considering impurities propagating along a long chain of $Z$'s. The impurities are the same as the ones that were used to solve the closed string problem in a similar regime \cite{BeisertDynamic,BeisertStaudacher}. The new aspect is that the impurities can be reflected from the boundaries at the end of the chain. This picture was discussed at the 1-loop order in the weak coupling limit in \cite{DrukkerKawamoto}. To proceed, we need to determine the boundary reflection matrix to all orders in the coupling. The matrix structure can be determined by group theory, as in \cite{BeisertDynamic}. The phase factor is more subtle. We write a crossing equation for it and we solve it following the strategy outlined in \cite{Volin,VieiraVolin}. At this stage we have completely solved the problem for operators with large $L$. Up to corrections of order $e^{- ({\rm const}) L }$, we can find the energy of any open string state by solving the appropriate Asymptotic Bethe equations.
After we have found the boundary reflection matrix we can then consider the possibility of rotating the half Wilson line that is associated with it. This rotation will simply act on the indices of the reflection
matrix via a global transformation. Now we can consider states of the form $B_l Z^L B_r(\theta,\phi)$, where we have rotated one of the sides of the Wilson line. This operator is no longer BPS but its energy is very small when $L$ is very large, i.e. it has zero energy up to $e^{- ({\rm const}) L }$ corrections. These are called Luscher (or wrapping) corrections.
\begin{figure}[h]
\centering
\def16cm{10cm}
\input{TBAfigure.pdf_tex}
\caption{The BTBA trick. The same partition function can be viewed in two ways \nref{openclosed}.
In the open string channel it is a trace over all states in the open string Hilbert space. In this case
Euclidean time runs along the $T$ arrow. Alternatively we can view it as the propagation of a closed
string along the $L$ arrow. The closed string has length $T$ and propagates over a Euclidean time $L$. The
two boundary conditions, now lead to two boundary states that create the closed strings that propagate
along the closed string channel. }
\label{TBAfigure}
\end{figure}
Before writing down the Thermodynamic Bethe Ansatz that describes the most general finite $L$ state,
we will make some checks on the phase that has been obtained. As a non-trivial
check one can get the first corrections to the ground state energy for large $L$.
Namely, we are interested in the anomalous dimension of the operator of the form
$B_l Z^L B_r(\theta,\phi)$. This correction is given by a Luscher-type formula.
This formula can be most simply understood by considering the problem in the mirror picture.
Namely, we exchange space and time in the open string picture.
In other words, we have the equivalence
\begin{equation}
\la{openclosed}
Z_{B_l,B_r}^{\rm open} = {\rm Tr}_{\rm open} [ e^{ - T H^{\rm open}_{B_l ,B_r}} ] = \langle B_l | e^{ - L H_{\rm closed} }|B_r \rangle\,,
\end{equation}
where $H^{\rm open}_{B_l ,B_r}$ is the open chain Hamiltonian on a strip of length $L$ and $H_{\rm closed}$ is
the closed chain Hamiltonian of the mirror theory on a circle of size $T$. So now we have a closed string exchanged between two boundary states. The analytic continuation of the boundary reflection matrix gives us the probability of emitting a pair of particles from the boundary state. It turns out that this continued reflection matrix has a pole at zero mirror momentum which implies that we can create
single particles \cite{Ghoshal:1993tm}. The coefficient of the pole in the reflection
matrix at zero mirror momentum determines the prefactor of the Luscher correction \cite{Bajnok:2004tq}.
We compute this at strong coupling and we find agreement with a direct string theory computation.
Furthermore, the {\it leading order } correction at weak coupling, going like $g^2$, also comes from this Luscher type term. In this way we match the leading corrections at weak and strong coupling. This constitutes a test of the boundary reflection matrix. In particular the very existence of the pole at zero mirror momentum is due to the phase factor of the matrix,
which we derived by solving the crossing equation.
Finally, one can write down a Thermodynamic Bethe Ansatz equation that describes the finite
$L$ situation. This follows the standard route for getting the energies of states of an integrable field theory with a boundary. The derivation of these equations is very similar to the derivation of the equations for closed string states. The new element is that instead of a thermodynamic partition function we have the overlap between two boundary states, as in
\nref{openclosed}. The derivation of TBA equations for integrable systems with a boundary
was considered in \cite{LeClair:1995uf}.
The boundary states are given in terms of the analytic continuation of the boundary reflection matrix.
The TBA system of equations arises from evaluating this exact overlap between the two boundary states in an approximate way by
giving the densities. Most of the TBA equations come from the entropy terms, which are the
same in our case. Thus the boundary TBA equations are very similar in structure to the
closed ones. We obtain
\begin{equation}
\log Y_A = \log (\kappa^l_A \kappa^r_A) - 2 L E_{m,A} + K_{AB} * \log ( 1 + Y_B )\,.
\end{equation}
The cusp anomalous dimension, or quark/anti-quark potential
is given schematically by
\begin{equation}
{\cal E} = - { 1 \over 2 \pi } \sum_A \int\limits_0^\infty d q_A \log(1 + Y_A )\,.
\end{equation}
Here $E_{m,A}$ and $q_A$ are the energies and momenta of the excitations in the mirror
theory. The equations will be given below in their full detail, \nref{1stTBA}-\nref{lastTBA}.
The information about the boundary
is contained in $\kappa_A$ which comes from the reflection phase of the theory and depends
on the boundary state.
{\bf Note:} We were informed that similar ideas were pursued in \cite{DrukkerTBA}.
\section{ Spectrum of operators on a Wilson line }\la{sec2}
Let us first discuss the symmetries preserved by a straight Wilson line. Let us
start with the bosonic symmetries. It preserves an $SL(2) \times SU(2) \times SO(5)$ symmetry
group. The $SO(5)$ is the subset of $SO(6)$ that leaves $\Phi^4$ invariant, where
$\Phi^4$ is the scalar that couples to the Wilson line.
The $SU(2)$ factor corresponds to the spatial rotations around the loop. The $SL(2)$ factor
contains time translations,
dilatations and special conformal transformations along the time direction.
In addition, we preserve half of the supercharges. The full supergroup is $OSp(4^*|4)$. The
star means it is the real form of $SO(4)$ such that $SO(4^*) \sim SL(2) \times SU(2)$.
Now we can consider the insertion of an operator of the form $Z^L$ on the Wilson loop,
we can denote this as $B_l Z^L B_r$. Here we choose $Z$ to be $Z = \Phi^5 + i \Phi^6$.
The operator $Z$ inserted at the origin
preserves an $SU(2|2)^2$ subgroup of the full symmetry group of the
theory.
The Wilson loop, together with the $Z$ insertions at the origin preserve an
$SU(2|2)_D $ subgroup of all the symmetry groups we mentioned. This is a diagonal
combination of the two $SU(2|2)$ factors preserved by $Z$.
This common preserved symmetry is very useful for analyzing this problem.
These operators are BPS, and they have protected anomalous dimension,
${\cal E} \equiv \Delta - J_{56} =0$.
Note that on $S^3$ we have a flux tube that goes between the quark and the anti-quark. These operators
inserted on the Wilson loop are mapped to
to various excitations of the flux tube.
\subsection{The boundary reflection matrix }
Recall that the bulk excitations are in a fundamental representation of each of the two
$\widetilde{su}(2|2)$ factors of the $\widetilde{su}(2|2)^2$ symmetry of the $Z$-vacuum.
The tilde means that we are considering the momentum dependent
central extensions discussed in \cite{BeisertDynamic,Beisertnonlin}.
In other words, we can think of them as particles with two indices
$\Psi_{A,\dot B}$,
where $A$ labels the fundamental of the first $\widetilde{su}(2|2)$ and $\dot B$ labels the {fundamental} of the second $\widetilde{su}(2|2)$ factor of the $\widetilde{su}(2|2)^2$ symmetry of the infinite chain. This central extension determines
the dispersion relation for the excitations
\begin{eqnarray} \la{constraint}
{ i \over g } & = & x^+ + { 1 \over x^+} - x^- - { 1 \over x^-}~,
\\
e^{ i p } &= & { x^+ \over x^- } ~,~~~~~~~~~~
\epsilon = i g \({ 1 \over x^+ }- { 1 \over x^-} -x^++x^-\)
= \sqrt{ 1 + 16 g^2 \sin^2{\tfrac{p}{2}} }\,, \label{enfromx}
\end{eqnarray}
Throughout this paper we define $g$ as\footnote{Note that $g \not = g_{YM}$.}
\begin{equation} \la{defofg}
g \equiv {\sqrt{ \lambda} \over 4 \pi } = { \sqrt{ g^2_{YM} N } \over 4 \pi }\,.
\end{equation}
The scattering matrix between two particles has the form
$S_{A \dot A , B \dot B}^{C \dot C D \dot D} = S_0^2 \hat S_{A B}^{ CD} \hat S_{\dot A \dot B}^{\dot C \dot D } $\cite{BeisertDynamic}.
Namely, it is the product of a phase factor $S_0^2$ and two identical matrices, one for each $\widetilde{su}(2|2)$ factor. These matrices are fixed (up to an overall factor) by the $\widetilde{su}(2|2)$ symmetry of the theory \cite{BeisertDynamic,Beisertnonlin}. These matrices depend on the
two momenta, $p_1$ and $p_2$, of the scattered variables. The phase factor $S_0(p_1,p_2)$ was guessed in \cite{BHL,BES}, and
a nice derivation was given in \cite{Volin,VieiraVolin}.
\begin{figure}[h]
\centering
\def16cm{12cm}
\input{refunfolded2.pdf_tex}
\caption{Unfolding of $R(p)$ into $S(p,-p)$. There is a non-trivial map between dotted and checked indices. See appendix \ref{Rmatrix} for details.}
\label{unfolding}
\end{figure}
In our problem we need to fix a reflection matrix of the form $R_{A \dot B}^{C \dot D }(p) $.
Let us consider first the reflection from the right boundary, see figure \ref{unfolding}.
This matrix depends on only one momentum $p$, the momentum of the incident magnon.
The boundary is invariant under an $\widetilde{su}(2|2)_{\rm D} $ symmetry group, which is diagonally
embedded in the $\widetilde{su}(2 |2)^2$ symmetry group of the bulk of the spin chain (see appendix \ref{Rmatrix}).
A similar problem was studied in \cite{CRY} and the matrix part of the reflection is the same.
Thus the symmetries constraining the reflection matrix are exactly the same as those
constraining the bulk scattering matrix for each of the $\widetilde{su}(2|2)$ factors.
From this argument we expect that the matrix structure should be completely fixed. In fact,
the matrix structure should be essentially the same as what we encounter
in the matrix $\hat S_{AB}^{CD}(p,-p)$, or $
R_{A \dot A}^{C \dot C }(p) \propto \hat S_{A \dot A}^{C \dot C }(p,-p) $.
One is tempted to say that the scattering phase factor would be $S_0(p,-p)$. However, this is not fixed by the
symmetries, and will not be true as we discuss below. In the presence of a boundary, we can do a kind of ``unfolding''
of the spin chain. Here each bulk magnon is viewed as a pair of magnons of $\widetilde{su}(2|2)_{\rm D}$, one with momentum $p$ to the left
of the boundary and one with momentum $-p$ to the right of the boundary. See figure \ref{unfolding} .
This completely solves the problem of fixing the matrix structure of the reflection matrix.
The full reflection matrix, in complete detail, is given in appendix \ref{Rmatrix}. One can
also check that it obeys the boundary Yang Baxter equation. But this is clear from the
``unfolded'' picture in terms of a single chain. We should emphasize that we have assumed
that there are no boundary degrees of freedom.
We do not see any evidence of any boundary degrees
of freedom at either weak or strong coupling, so this is a reasonable assumption.
Before we determine the phase, let us make a side remark. There is a variety
of problems that give rise to a spin chain with boundaries and preserve the same symmetries,
$OSp(4^*|4)$. We can consider an open string ending on a D5 brane that wraps $AdS_4 \times S^2$, or
$AdS_2 \times S^4$. In fact, there is a whole family of BPS branes of this kind that
arises by adding flux for the $U(1)$ gauge field on the brane worldvolume on the $S^2$ or $AdS_2$.
In fact, in the limit of large electric flux on the $AdS_2 \times S^4$ brane we get a boundary condition
like the Wilson loop one. In fact the $AdS_2 \times S^4$ branes can be interpreted as Wilson loops in
the $k$-fold antisymmetric representation of $U(N)$ \cite{Yamaguchi:2006tq}.
In all these cases one can choose the BMN vacuum (or choose the field $Z$) in such a way
that we preserve the $\widetilde{su}(2|2)_{\rm D}$ of the spin chain.
Therefore, we would get the same matrix structure for the reflection matrix, again assuming that there
are no boundary degrees of freedom. However, they would differ in the choice of a phase factor.
Below we get a phase factor which has all the right properties to correspond to the one of the Wilson loop.
It would be interesting to fix the phase factor also for these other cases, but we leave this to the future.
In order to fix the phase factor we write a crossing equation. We derive this by writing
the identity state of \cite{BeisertDynamic}, scattering it through the boundary and
demanding that the full phase is equal to one. Denoting the phase factor as $R_0$,
defined more precisely in appendix \ref{Rmatrix}, we obtain the crossing equation
\begin{equation}
\la{bdycross}
R_0(p) R_0(\bar p) = \sigma(p, - \bar p)^2\,,
\end{equation}
where the bar indicates the action of the crossing transformation. Here $\sigma(p_1,p_2)$ is the
bulk dressing phase, discussed in \cite{BES,VieiraVolin}. We are going to $\bar p$ along the
the same contour in
momentum space that we choose in the formulation of the bulk crossing equation.
In addition, we also should impose the unitarity condition
\begin{equation}
R_0(p) R_0(-p) =1\,.
\end{equation}
We now write the ansatz
\begin{equation} \la{fullphase}
R_0(p) = { 1 \over \sigma_B(p) \sigma(p,-p)} \left( { 1 + { 1 \over (x^-)^2 } } \over 1 + { 1 \over (x^+)^2 } \right)\,.
\end{equation}
Here $\sigma$ is the bulk dressing phase. This would be our naive choice for a phase
factor. The explicit factors of $x^\pm$ have been chosen only to simplify the final formula.
We have an unknown factor $\sigma_B(p)$.
Now \nref{bdycross} becomes
\begin{equation}\la{crossingequation}
\sigma_B(p) \sigma_B(\bar p) = { x^- + { 1 \over x^-} \over x^+ + { 1 \over x^+} }\,.
\end{equation}
We can now solve this equation using the method proposed in \cite{Volin,VieiraVolin}.
We give the details in appendix \ref{crossingsolution}. We obtain
\begin{eqnarray} \la{phasefa}
\sigma_B & = & e^{ i \chi(x^+) - i \chi(x^-) }\,,
\\ \la{chiintegral}
i \chi(x) & = & i \Phi(x) = \oint\limits_{|z|=1} { d z \over 2 \pi i } { 1 \over x - z } \log \left\{ \sinh[ 2 \pi g ( z + { 1 \over z} ) ] \over 2 \pi g ( z + { 1 \over z } ) \right\}\,,~~~~~~|x|>1\,.
\end{eqnarray}
This expression is valid when $|x|>1$. The value for $\chi$ in other regions
is given by analytic continuation. We have also introduced the function $\Phi(x)$ which is
given by the integral for all values of $x$. When $|x|<1$ these two functions differ by
\begin{equation} \la{phasefacon}
i \chi(x) = i \Phi(x) + \log \left\{ \sinh[ 2 \pi g ( x + { 1 \over x} ) ] \over 2 \pi g ( x + { 1 \over x } ) \right\}\,, ~~~~~~~~|x|< 1
\end{equation}
The ambiguities in the choice of branch cuts for the logarithm cancel out when we compute
$\sigma_B$ in \nref{phasefa}. Note that $\chi(x) = \chi(-x)$.
So far, we have found {\it a particular}
solution of the boundary crossing equation. Still, the true dressing phase
might require the inclusion of further CDD factors.
In order to make a conjecture for the exact
boundary dressing phase, we need to compare against some explicit computations.
Before doing so, let us observe that, given $\sigma_B(p)$, we can define an infinite family of solutions by taking
\begin{equation}
\sigma_B^{(s)}(p) = \left(\frac{x^-+\frac{1}{x^-}}{x^++\frac{1}{x^+}}\right)^s\left[\sigma_B(p)\right]^{1-2s}\,.
\label{othersols}
\end{equation}
By computing the dressing phase in the physical regime we will be able to show that $s=0$ is
the solution we want.
The proposal for the phase factor for the reflection matrix, given in \nref{fullphase}, \nref{phasefa} is one of the important results of this paper.
We will perform various checks on its validity.
\subsection{Checks of the boundary reflection phase in the physical region }
\begin{figure}[h]
\centering
\def16cm{12cm}
\input{soltitonscatterin3.pdf_tex}
\caption{Computation of the reflection phase at strong coupling. We have a soliton at
the boundary, which is at rest at $\sigma=0$. There is also an image soliton coming from
the right. Then the soliton with momentum $p$ scatters through the soliton at rest and the
one with momentum $-p$, leading to a certain time delay. From the time delay we can
compute the derivative of the reflection phase with respect to the energy. }
\label{SolitonScattering}
\end{figure}
Let us describe how
to compute the boundary dressing phase at strong coupling.
We have to consider the open string solution that corresponds to a 1/2 BPS Wilson line carrying a large
$J_{56}$ charge given in \cite{DrukkerKawamoto}. This solution describes the transition from the boundary
Wilson line to an infinite BMN vacuum.
It is convenient to understand this solution in the conformal gauge, when we set the
stress tensor on the $S^5$ equal to one, and the stress tensor of the $AdS_5$ to minus one.
In these variables, the problem only involves an $AdS_2 \times S^2$ subspace and we can
perform the Pohlmeyer reduction in each factor. The $S^2$ part gives rise to a sine gordon
theory and the solution is just half of a soliton at rest. More precisely, the center of
the sine gordon soliton sits at the boundary. In the $AdS_2$ part we have a sinh gordon
theory, and the solution is a sinh-gordon ``soliton''. This is a singular solution which is
the direct analytic continuation of the sine gordon soliton. The singularity reflects the
fact that the string goes to the $AdS$ boundary. If we compute the energy, there is a divergent
part and the finite part is zero. The setup is explained in more detail in \cite{DrukkerKawamoto}. The fact that the finite part of the energy is zero is consistent with
the absence of a boundary impurity transforming non-trivially under $\widetilde{su}(2|2)$.
A bulk magnon is a sine gordon soliton, and leaves the $AdS$ part of the solution unperturbed.
In the presence of a boundary, we need to put also the ``image'' of this soliton and the
configuration looks as in figure \ref{SolitonScattering}.
The reflection involves the scattering of the soliton
with the image soliton as well as the scattering with the soliton at rest. These
soliton scattering phases were computed in \cite{HM,HMopen}. So the strong coupling limit of the right
boundary scattering phase $R_0(p) = e^{i\delta_{\rm R}(p)}$ is given by
\begin{equation}
\delta_{\rm R}(p) = -8g\cos\tfrac{p}{2}\log\cos\tfrac{p}{2} -4g\cos\tfrac{p}{2}\log\left(\frac{1-\sin\tfrac{p}{2}}{1+\sin\tfrac{p}{2}}\right)\,.
\label{deltaR}
\end{equation}
The first term in (\ref{deltaR}) is exactly what one gets from the strong coupling limit of the factor $1/\sigma(p,-p)$ \cite{HM}.
We will see that the second term
corresponds to $\sigma_B(p)^{-1}$. At strong coupling can expand \nref{chiintegral} as
\begin{equation} \la{chistr}
i \chi(x ) \sim 4 g \left[ - 1 + ( x + { 1 \over x } ) { 1 \over 2 i } \log {( x + i ) \over (x-i)}
\right] + {\cal O}(1)\,,
\end{equation}
which leads, for physical excitations, to
\begin{equation} \la{sigstrong}
{1\over i}\log \sigma_B(p) = 4 g \cos \tfrac{p}{2} \log \left( {1 - \sin { p \over 2} \over 1 + \sin { p \over 2} } \right)\,.
\end{equation}
This indicates that we must pick the case $s=0$ from the family of solutions (\ref{othersols}).
Finally, let us discuss the behavior at weak coupling. The bulk dressing phase $\sigma$
has its first contribution at order $g^6$, leading to four loop corrections to anomalous dimensions.
On the other hand, the boundary dressing phase, $\sigma_B$,
receives its first contribution at $g^4$, so that it will start modifying anomalous
dimensions of operators inserted on the Wilson loop (dual to open string states) at three loops.
\subsection{ Reflection matrix for a Wilson line at general angles }
\label{Rwithangles}
We will need the boundary reflection matrix for a Wilson line sitting at general angles,
$\phi$ and $\theta$. In particular, we want the left and right boundaries
of the open chain to be rotated by relative angles.
We can obtain the boundary state of the Wilson line at
a different position on the $S^3$, or the $S^5$, by applying a symmetry transformation on $B_r$.
This should be a
symmetry that is broken by $B_r$. So for example, we can apply an $SU(2)_L$ rotation on
the $S^3$ which is in one of the $SU(2)$ factors in the $SO(4)$ group of rotations of the 3-sphere. If we
apply an $SU(2)_L$ rotation with an angle $2 \phi$, we will get that the quark is rotated by
an angle $\phi$ on the $S^3$, away from the south pole. See figure \ref{CuspDiagram}(b). Note that the
$SU(2)_L$ we are considering is a symmetry of the $Z$ vacuum.
We have a similar feature on the $S^5$. We can also apply a rotation in an $SU(2)_{L'}$ factor inside
$SO(4)\subset SO(6)$ (this $SO(4)$ leaves the $Z$ vacuum invariant).
The reflection matrix is very easy to obtain. We pick these two $SU(2)$ generators so that they sit in
the bosonic part of one of the $\widetilde{ su}(2|2) $ factors of the $\widetilde{ su}(2|2)^2 $ symmetry of the bulk. Then they will simply introduce some
phases of the form $e^{ i \phi}$ or $e^{ i \theta}$ when a state is reflected from the boundary and
its $SU(2)$ quantum number changes. The $SU(2)_L$ or $SU(2)_{L'}$ quantum numbers can change because they are not symmetries of the boundary state. More explicitly, the reflection matrix from a boundary state at angles $\phi,\theta$ is
given by
\begin{equation}
R^{B \dot B }_{A \dot A } (\theta, \phi) = ( m^{-1}) ^{B}_{D} m_{A}^{C} R^{D \dot B }_{C \dot A }(0,0)
~,~~~~~~~{\rm with}~~~~~ \la{mone}
m = { \rm diag}(e^{i \theta } , e^{ - i \theta} , e^{ i \phi} , e^{ - i \phi } )\,.
\end{equation}
Note that the matrix $m$ acts only on the undotted indices since we did a rotation inside only one of the
$ \widetilde{ su}(2|2)$ factors.
\subsection{ Luscher computations and checks in the mirror region }
\la{Luschersub}
In this subsection we start considering the problem with two boundaries. In other
words the operator $B_l Z^L B_r(\phi, \theta)$. Here $\phi, ~\theta$ are the relative orientations of
the two boundaries. On the plane, this corresponds to a cusp, plus an operator of the form $Z^L$ at
the tip.
In the limit $L \gg 1$ we get the naive superposition of the two boundaries and the
energy of the state is zero (${\cal E} = \Delta - L =0$), regardless of the orientation of the two boundaries.
The leading correction is of the form $e^{ - ({\rm constant}) L }$.
These corrections come from the exchange of particles
along the ``mirror'' channel. The boundary sources
particles, which then travel to the other boundary. These corrections sometimes go under
the name of Luscher corrections. Of course the familiar Yukawa potential is a simple example where
the leading correction comes from the exchange of a single massive particle.
In order to derive the precise correction formula it is convenient to describe in more detail the
mirror theory. In the bulk of the worldsheet the
mirror theory was discussed in various papers, see \cite{ArutyunovFrolov}
for example. This theory is defined by exchanging the space and time directions of the spin chain we have
been considering so far. Thus, instead of
\nref{enfromx} we define $ q = i \epsilon $ and $E_m = i p $, and use the same formulas
as in \nref{enfromx}. Here $q$ is the mirror momentum and $E_m$ is the mirror energy.
In order for these to be real we will need to pick a solution of \nref{constraint} with
$|x^+|>1$ and $|x^-|<1$. From the expression for $q$, we can write
\begin{eqnarray}
z^{[\pm a]} &=& \frac1{4g}\left(\sqrt{1+\frac{16g^2}{a^2+q^2}}\pm1\right)\left(q+ia\right)\,,
\label{zpm}
\\
E_m &= &2 { \rm{ arcsinh }} { \sqrt{ q^2 + a^2} \over 4 g }\,. \la{mirrdis}
\end{eqnarray}
Here $z^\pm$ just denote the values of $x^\pm$ in the mirror region. We have also written
the dispersion relation in the mirror region, for an arbitrary
bound state. The elementary mirror magnon has $a=1$.
When we have a boundary, this time/space flip turns the boundary into a boundary state, see figure
\ref{TBAfigure}.
Then a suitable analytic continuation of the boundary reflection matrix characterizes the
boundary state. The boundary state creates a supersposition of many particles. The total
mirror momentum should be zero since it is translational invariant. So, schematically the
state has the form
\begin{equation} \la{BdyState}
| B \rangle = |0\rangle + \int\limits_0^\infty { d q \over 2 \pi } K^{A \dot A , B\dot B } (q) a^{ \dagger}_{-q \, A \dot A} a^\dagger_{q \, B \dot B } | 0 \rangle + \cdots
\end{equation}
with
\begin{equation} \la{kfromr}
K^{A \dot A , B\dot B }(q) = \left[ R^{-1}(z^+,z^-)\right]^{A \dot A }_{ D\dot D } {\cal C}^{D \dot D ,B\dot B}\,,
\end{equation}
where we put the mirror values \nref{zpm}.
Here $R$ is the right reflection matrix, with $z^\pm$ continued
to the mirror region \nref{zpm}. This amounts to an analytic
continuation of the reflection matrix. Here $a^\dagger_{q \, A \dot A}$ is the creation operator of
a magnon with momentum $q$. ${\cal C}$ is a charge conjugation matrix.
In the case of a relativistic model with a single particle \nref{kfromr} reads $K(\theta) = { 1/ R(\theta - i { \pi \over 2 } ) } $, \cite{Ghoshal:1993tm}. The formula \nref{kfromr} can be obtained by performing a
$\pi/2$ rotation of the boundary condition.
Due to the independence of reflection events from a boundary, we
can exponentiate \nref{kfromr} to get the full boundary state \cite{Ghoshal:1993tm,LeClair:1995uf}.
Similarly, we can form a future boundary state.
This is a
boundary state that annihilates the particles. It is given by
\begin{equation} \la{BdyStatetwo}
\langle B | = \langle 0 | + \langle 0 | \int\limits_0^\infty { d q \over 2 \pi }
\bar K_{A \dot A , B\dot B } (q) a ^{A \dot A} _q a_{-q}^{B \dot B } + \cdots
\end{equation}
with
\begin{equation} \la{barkdef}
\bar K_{A \dot A , B\dot B }(q) = \left[ R^{-1}\left(-{ 1\over z^-},-{ 1 \over z^+}\right)\right]^{D\dot D }_ { B\dot B }
{\cal C}_{D \dot D A \dot A}\,.
\end{equation}
In the relativist case \nref{barkdef} would be $\bar K(\theta) = { 1 \over R(- i { \pi \over 2} -
\theta ) } $.
When $L$ is very large the leading $L$-dependent contribution comes from the exchange of
this pair of particles and we can write the corresponding contribution as
\begin{equation} \la{luschn}
\delta {\cal E}= - \int\limits_0^\infty {dq \over 2 \pi } e^{ - 2 L E_m(q) } t(q) ~,~~~~~~ t(q) = {\rm Tr}[ K(q) \bar K( q ) ]\,.
\end{equation}
This formula is correct whenever the integral is finite.
In our case, the phase factor $\sigma_B$ has a pole at $q=0$. In the physical region $\sigma_B(p)$ was
perfectly finite. This pole in the mirror region is
crucial for obtaining the correct answer. But first we need
to generalize \nref{luschn} to the situation when we have a pole at
$q=0$. The physical interpretation of this pole at $q=0$ is that the boundary state is
sourcing single particles states in the mirror theory \cite{Ghoshal:1993tm}.
For a similar case in the AdS/CFT context see \cite{CYLuscher}.
Obviously such source has to contain only zero momentum particles.
A careful analysis leads to the formula \cite{Bajnok:2004tq}
\begin{equation} \label{luscherfull}
{\cal E} \sim
- \int\limits_0^\infty { d q \over 2 \pi } \log \left\{
1 + e^{ - 2 L E_m(q) } {\rm Tr}[ K(q) \bar K( q ) ] \right\}
\sim - { 1 \over 2 }
e^{ - L E_m(0)} \sqrt{ q^2 {\rm Tr}[ K(q) \bar K( q ) ] |_{q=0} }\,.
\end{equation}
In the last equality we extracted the leading term in the integral, which comes only
from the coefficient of the pole. Notice that the $L$
dependence is precisely what we expect from the exchange of a single particle.
We should sum over all the particles that can be exchanged. The mirror theory
contains bound states indexed by an integer $a=1,2, \cdots$, and we should sum over them.
In appendix \ref{ComputationOft} we show that we can evaluate $t(q)$ for a fundamental mirror particle and
we obtain
\begin{equation} \la{texpr}
t(q) = \sigma_B(z^+,z^- )
\sigma_B\left(- { 1\over z^-} ,- { 1 \over z^+} \right) \left( { z^- \over z^+ } \right)^2
\left( {\rm Tr}[ (-1)^F] \right)^2\,,
\end{equation}
where the trace is over the four states of a single $\widetilde { su} (2|2)$ magnon.
Let us now give a simple explanation for this formula, for more details see appendix \ref{ComputationOft}. We can write the reflection
matrices that appear in $K$ and $\bar K$ \nref{kfromr} \nref{barkdef}
in terms of bulk S-matrices for the
unfolded theory, namely in terms of bulk $S$ matrices for a single $\widetilde{su}(2|2)$ factor.
The matrix in $K$ is essentially ${\cal S}(-p,p)$ and the one in $\bar K$ is
${\cal S}( \bar p , -\bar p)$. When we multiply these matrices we can use the bulk
crossing equation to get the identity. Here we should use the full bulk matrix, including
the bulk $\sigma$ factor. This is the reason that the bulk $\sigma$ factor disappears
from the final formula \nref{texpr}, but the boundary one remains. The factor of $z^-/z^+$ arises
from the factor in parenthesis in
\nref{fullphase}. Finally, the $(-1)^F$ is related to the
fact that we have fermions. Here $F$ is the fermion number.
When we perform the TBA trick, we get periodic fermions in
Euclidean time if we started with periodic fermions in the spatial directions. Of course
a periodic fermion in Euclidean time is the same as the trace with a $(-1)^F$ inserted.
The operations that lead to \nref{texpr} can be understood graphically as in figure \ref{Unfolding}.
\begin{figure}[h]
\centering
\def16cm{14cm}
\input{Unfolding5.pdf_tex}
\caption{({\bf a}) We have a strip with pairs of particles being exchanged. The two colors represent
the two types of indices. In ({\bf b}) we unfolded this into a cylinder computation. The $K$ matrices
became $S$ matrices for a single $\widetilde{su}(2|2)$. ({\bf c}) Using crossing we have moved the
lines. The red circles indicates the action of the matrix $m$. }
\label{Unfolding}
\end{figure}
Of course, for a fundamental magnon ${\rm Tr}[ (-1)^F] =0$. This is good, since it is saying
that the correction vanishes in the BPS situation.
If we rotate one boundary relative to the other then we need to perform the replacement
\begin{equation} \la{texprmone}
{\rm Tr}[ (-1)^F] \longrightarrow {\rm Tr}[ (-1)^F m ] = - 2 ( \cos \phi - \cos \theta )\,.
\end{equation}
where $m$ is given in \nref{mone}. Again, we see that it vanishes in the BPS case
$\phi = \pm \theta$.
To write down the full Luscher formula we need to compute $t(q)$ also for the bound states
of the mirror theory. One can first use the standard fusion procedure to get the bound
state reflection matrix. Then one can use the same argument as above to eliminate the
bulk $S$ matrices, as in figure \ref{Unfolding}.
The final formula is
\begin{eqnarray} \la{texprall}
t_a(q) &=& \sigma_B(z^{[+a]},z^{[-a]} )
\sigma_B\left(- { 1\over z^{[-a]}} ,- { 1 \over z^{[+a]}} \right) \left( { z^{[-a]} \over z^{[+a]} } \right)^2 \left( {\rm Tr}[ (-1)^F m_a ] \right)^2\,,~~
\\ \la{tracealla}
&~&~~~~~~~~ {\rm Tr}[ (-1)^F m_a ] = (-1)^a 2 ( \cos \phi - \cos \theta){ \sin a \phi \over \sin \phi }\,,`
\end{eqnarray}
where now the trace is over all the states of a magnon boundstate in a single copy of $\widetilde{su}(2|2)$, see equation \nref{mfora} in appendix \ref{ComputationOft}.
As anticipated, an important property of $\sigma_B$ is that it has a pole at $q=0$.
More precisely the combination of $\sigma_B$ in \nref{texprall} becomes
\begin{eqnarray} \la{chifact}
e^{ i \chi(z^{[+a]}) - i \chi(z^{[-a]}) + i \chi (1/z^{[-a]}) -i \chi(1/z^{[+a]})}
\!\!&=&\!\!
\frac{2\pi g(z^{[-a]}+\tfrac{1}{z^{[-a]}})}{\sinh[2\pi g(z^{[-a]}+\tfrac{1}{z^{[-a]}})]}
\frac{2\pi g(z^{[+a]}+\tfrac{1}{z^{[+a]}})}{\sinh[2\pi g(z^{[+a]}+\tfrac{1}{z^{[+a]}})]}{\nonumber}\\
&& \times e^{i(\Phi(z^{[+a]})-\Phi(z^{[-a]})+\Phi(1/z^{[-a]})-\Phi(1/z^{[+a]} ) ) }\,.
\end{eqnarray}
Here we used that $z$ is in the mirror kinematics and we used \nref{phasefacon} to evaluate
$\chi(x)$ when $|x|<1$. We have also used that $\chi(-x) = \chi(x)$.
Each of the sinh factors leads to a pole at $q=0$. Namely, using
\nref{zpm}
we get
\begin{equation}
2 \pi g (z^{[\pm a] } + { 1 \over z^{ [\pm a ]} }) =\pm i \pi a + \pi q \sqrt{ 1 + { 16 g^2 \over a^2 } } + {\cal O}(q^3)\,.
\end{equation}
for small $q$. We then can write the pole part of \nref{chifact} as
\begin{eqnarray}
e^{ i \chi(z^{[+a]}) - i \chi(z^{[-a]}) + i \chi (1/z^{[-a]}) -i \chi(1/z^{[+a]})}
&\sim & { 1 \over q^2} { a^4 \over ( a^2 + 16 g^2 ) } F(a,g) ^2 + {\cal O}(1)\,,
\\
\la{ffun}
{\rm with}~~~~~~~F(a,g)^2 &\equiv & e^{i(\Phi(z^{[+a]})-\Phi(z^{[-a]})+\Phi(1/z^{[-a]})-\Phi(1/z^{[+a]}))} |_{q=0}\,,
\end{eqnarray}
where the last factor is evaluated at $q=0$.
Then we find the coefficient of the double pole of $t$ as
\begin{equation} \la{smallqt}
\lim_{q\to 0 } q^2\, t_a (q) = 4 { (\cos\phi-\cos\theta )^2 \over
\sin^2 \phi } \sin^2(a\,\phi) { a^4 \over (a^2 + 16 g^2 ) } \left( -a + \sqrt{ a^2 + 16 g^2} \over
a + \sqrt{a^2 + 16 g^2} \right)^2 F(a,g) ^2\,.
\end{equation}
The factor in parenthesis is $(z^{[-a]}/z^{[a]} )^2 $.
Finally, inserting this into the expression for the energy \nref{luscherfull}, we find
\begin{equation}
\Delta {\mathcal E} \sim - { (\cos\phi-\cos\theta ) \over
\sin\phi } \sum_{a=1}^\infty (-1)^a \left( -1 + \sqrt{ 1 + 16 g^2/a^2} \over
1 + \sqrt{1 + 16 g^2/a^2} \right)^{ 1 + L }\!\!\!\!\! \sin(a\,\phi) { a \over \sqrt{ 1 + 16 g^2/a^2 } } F(a,g)\,.
\label{luschergen}
\end{equation}
The factor in parenthesis is just $e^{ - E_m(a) (L+1) }$, representing the
exchange of a bound state in the mirror channel.
The sign $(-1)^a$ is a bit subtle and has to do with the correct sign we should pick for
the square root in \nref{luscherfull}. The correct sign is easier to understand for
an angle of the form $\phi = \pi - \delta$, for small $\delta$. In this case we have a
quark antiquark configuration and it is clear that we should get a negative contribution
to the energy. In fact, we can think of the overlap of the two boundary states as computing a
kind of norm or inner product. We see that in terms of $\delta$ the expression has the
expected sign. In other words, for small $\delta$ we get the positive sign of the square root
in \nref{luscherfull}.
Of course, once we get the expression for small $\delta$ we can write it in
terms of $\phi$, or even analytically continue $\phi = - i \varphi$.
\subsubsection{Leading Luscher correction at weak coupling}
The expression \nref{luschergen} gives the leading Luscher correction at all values of the coupling
for large $L$. Let us now examine it at weak coupling. Then the factor in parenthesis in \nref{luschergen}
is of order $g^2$. So, at leading order, we get a term of the form $g^{2 + 2 L }$. This has the
interesting implication that this leading ``wrapping'' correction appears at $L+1$ loops. In particular for
$L=0$, the one loop contribution comes from such a term!. In fact, expanding \nref{luschergen} to leading order
in $g^2$, and setting $L=0$, we can set $F =1$ to this order and obtain
\begin{eqnarray}
\Gamma_{ {\rm cusp}} &=&-4 g^2 { (\cos\phi-\cos\theta ) \over
\sin \phi } \sum_{a=1}^\infty (-1)^a { \sin a \phi \over a }
\\
&=& 2 g^2 { (\cos\phi-\cos\theta ) \over
\sin \phi } \phi\,,
\end{eqnarray}
which coincides exactly with the leading 1-loop contribution to $\Gamma_{ {\rm cusp}} (\phi,\theta)$
computed in \cite{Drukker:1999zq}.
We can also do the computation of the leading order term for any $L$, we get
\begin{equation}
{\cal E} = - g^ { 2 + 2 L} { (\cos\phi-\cos\theta ) \over
\sin \phi } { (-1)^L ( 4 \pi )^{ 1 + 2 L } \over (1 + 2 L)! }
B_{1 + 2 L} \left( { \pi - \phi \over 2 \pi } \right) + {\cal O}( g^{ 4 + 2 L } )
\end{equation}
where $B_n(x)$ is the Bernoulli polynomial, which is a polynomial of degree $2L+1$. In \cite{LadderPaper} a
particular class of diagrams was identified which produced the same expression.
\subsubsection{Leading Luscher correction at strong coupling}
We can also compute the leading large $L$ correction at strong coupling. We simply evaluate
the large $g$ limit of \nref{luschergen}. First we note that
\begin{equation}
\left( { z^{[-a]} \over z^{[+a]} } \right)^{L+1} \sim e^{ - { a \over 2 g } L } = e^{ - L E_m(q=0) }\,.
\end{equation}
This implies that to leading order in $e^{ - L}$ we only need to consider the case $a=1$.
The expansion of the function $F$ is done in appendix \ref{funfexp} eqn. \nref{ffunres}.
Putting everything together we find that the leading strong coupling
correction goes as
\begin{equation} \la{luschstrong}
{ \cal E } = ( \cos \phi - \cos \theta ) { 16 g \over e^2 } e^{ - { L \over 2 g } }\,.
\end{equation}
This agrees precisely with the result computed directly from classical string theory
in appendix \ref{luschapp}, see \nref{finres}. This constitutes a nontrivial check of the
reflection phase. Notice, the funny factor of $e^{-2}$ which is correctly matched.
\section{The open Asymptotic Bethe Ansatz equations}
We will now write down the asymptotic Bethe ansatz (ABA) equations that
describe the spectrum of operators with large $L$ inserted on the Wilson loop.
These give rise to a spin chain with two boundaries, which are separated by a large distance $L$.
Moreover, the ABA equations are used to derive the BTBA system by embedding them into the closed equations,
as we do in appendix \ref{embedingderivation}.
In order to obtain the ABA equations we have to diagonalize the way the bulk and boundary scattering matrices act.
This can be done by formulating a nested Bethe ansatz, which defines impurities at different levels
of nesting. Here we just sketch the computation, which is a straightforward generalization of the case
with periodic boundary conditions studied in \cite{BeisertDynamic}.
Consider an asymptotic state with $N^{\rm I}$ bulk magnons, or level I excitations, on the half-line
with a right boundary. We will introduce a second boundary and relative angles later, when writing down
the Bethe equations. In particular, we can consider a state whose level I impurities all carry the
same $SU(2|2)_{\rm D}$ index\footnote{The choice of index is arbitrary.}.
Say, for example, in the unfolded notation,
\begin{equation}
|\Psi_3(p_1)\cdots \Psi_3(p_{N^{\rm I}})\Psi_{\check 3}(-p_{N^{\rm I}})\cdots \Psi_{\check 3}(-p_1)\rangle \equiv |0\rangle^{\rm II}\,,
\la{levelIIvac}
\end{equation}
which is regarded as the level II vacuum state. Of course, we could also consider states
where $N^{\rm II}$ out of the $N^{\rm I}$ level I impurities have different indices.
Those should be understood as $N^{\rm II}$ impurities in the level II vacuum state. In total,
such states will contain $N^{\rm I}$ level I impurities and $N^{\rm II}$ level II impurities.
In general, we have $|\Psi_{a_1}(y_1)\cdots \Psi_{a_{N^{\rm II}}}(y_{N^{\rm II}})\rangle^{\rm II}$ for $a_k=1,2$, where $y_k$ are auxiliary parameters associated with the level II impurities.
Similarly, a third level of nesting can be defined. If all the level II excitations carry the same index,
for instance $|\Psi_1(y_1)\cdots \Psi_1(y_{N^{\rm II}})\rangle^{\rm II}\equiv|0\rangle^{\rm III}$,
we can define a level III vacuum state. Then, magnons $\Psi_2$ will be treated as level III impurities
propagating in $|0\rangle^{\rm III}$. For the kind of $SU(2|2)$ spin chain we are considering, this level III
is the final level of nesting\footnote{$\Psi_4$ are not considered as elementary but as double excitations.}.
Then, to formulate a coordinate Bethe ansatz, bulk and boundary scattering factors among excitations of different levels
have to be introduced to write the nested wavefunctions. Those can be determined by imposing certain compatibility conditions.
Namely, that the action of the bulk and boundary scattering matrices on wavefunctions with higher level impurities
just pulls out the same factor as when acting on the level II vacuum state. Naturally, the bulk scattering factors are exactly the same as the ones obtained in the periodic case \cite{BeisertDynamic},
\begin{eqnarray}
S^{\rm I,I}(x_1^\pm,x_2^\pm) \!\!& = &\!\! - S_0(p_1,p_2)\,,
\\
S^{\rm I,II}(x^\pm,y) \!\!& = &\!\! 1/S^{\rm II,I}(y,x^\pm) = -\frac{y-x^{-}}{y-x^{+}}\,,
\\
S^{\rm III,II}(w,y)\!\!& = &\!\! \frac{w-y-{1\over y}+\frac{i}{2g}}{w-y-{1\over y}-\frac{i}{2g}} = \frac{w-v+\frac{i}{2g}}{w-v-\frac{i}{2g}}\,,
\\
S^{\rm III,III}(w_1,w_2) \!\!& = &\!\! \frac{w_1-w_2-\frac{i}{g}}{w_1-w_2+\frac{i}{g}}\,,
\end{eqnarray}
where
\begin{equation}
S_0(p_1,p_2)^2 = \frac{(x_1^+-x_2^-)(1-\frac{1}{x_1^-x_2^+})}{(x_1^--x_2^+)(1-\frac{1}{x_1^+x_2^-})} {1\over\sigma(p_1,p_2)^2}
\end{equation}
is the bulk dressing factor and $v = y +{1\over y}$. All other bulk scattering factors are trivial.
The reflection factors can be derived in the same way. The level II vacuum \nref{levelIIvac} containing
$N^{\rm I}$ magnons becomes a lattice with $2N^{\rm I}$ sites. Consider a single level II
impurity propagating in this vacuum from left, i.e. propagating along the left (undotted) indices
of the bulk magnons. Undotted and dotted indices can only mix by the reflection of the rightmost
bulk magnon. That could make us think there exists a defect in the middle of the level II vacuum lattice which
separates the 3 and $\check 3$ indices of the rightmost level I impurity. In principle, the level
II impurity could be reflected and transmitted across such defect, see figure \ref{levelIIdefect}.
\begin{figure}[h]
\centering
\def16cm{14cm}
\input{level2ref.pdf_tex}
\caption{Propagation of a single level II impurity across the defect.}
\label{levelIIdefect}
\end{figure}
However, and because the boundary scattering matrix $R(p)\propto S(p,-p)$, the compatibility condition we obtain
from the reflection of the rightmost level I impurity is analogous to the ones we obtain from the scattering of
two level I impurities. In this way, the compatibility conditions imply that level II impurities are purely transmitted.
In other words $\tilde R^{\rm II}=0$ and $R^{\rm II}=1$. Analogously, the reflection of level III impurities
is determined. In summary, we have
\begin{equation}
R^{\rm I}(x^\pm) = R_0(p)\,, \quad R^{\rm II}(y) = 1\,, \quad R^{\rm III}(w) = 1\,,
\end{equation}
where $R_0(p)$ is the boundary phase factor \nref{fullphase}.
Let us now put the system in a finite strip by introducing another boundary. We will then
have certain quantization conditions on the rapidities for all kind of excitations,
namely the Bethe ansatz equations. We will introduce the left boundary with relative angles
with respect to the right one, by using the rotation discussed in section \ref{Rwithangles}.
To understand how this rotation affects the factors $R^{\rm I}$, $R^{\rm II}$ and $R^{\rm III}$
it is enough to consider the action of $m$, defined in \nref{mone}, on the following key
components of the reflection matrix
\begin{eqnarray}
R_{3\check 3}^{3\check 3} \mapsto R_{3\check 3}^{3\check 3}\,,
&\Rightarrow & R^{\rm I} \mapsto R^{\rm I}\,,{\nonumber}\\
R_{1\check 3}^{3\check 1} \mapsto e^{i\theta-i\phi} R_{1\check 3}^{3\check 1}\,,
&\Rightarrow & R^{\rm II} \mapsto e^{i\theta-i\phi} R^{\rm II}\,,\\
R_{2\check 1}^{1\check 2} \mapsto e^{-2i\theta}R_{2\check 1}^{1\check 2}\,,
&\Rightarrow & R^{\rm III} \mapsto e^{-2i\theta} R^{\rm III}\,,{\nonumber}
\end{eqnarray}
Let us finally write down the nested Bethe ansatz equations. They are obtained
by picking an impurity of any level of nesting and moving it through all the
other impurities twice and reflecting it from both boundaries as it is
shown in the left picture of figure \ref{openbethe}.
\begin{figure}[h]
\centering
\def16cm{17cm}
\input{openbethe2.pdf_tex}
\caption{ Bethe equation for the open chain. ({\bf a}) The original picture with boundaries. The particle goes to one boundary, then the other, and finally back to the origina position. ({\bf b}) The unfolded picture. We
have a closed circle. The leftmost solid line is identified the rightmost one.
The motion that leads to the Bethe equations involves moving the magnon with momentum $p$ around
the closed circle and at the same time we also move its partner which has momentum $-p$ around
the circle in the opposite direction. }
\label{openbethe}
\end{figure}
If we go to the unfolded picture what we have is periodic chain of length $2L$, where for every level I excitation of momentum $p_k$ there exists a mirrored one of momentum $-p_k$, figure \ref{openbethe}. Such duplication does not occur for higher levels of nesting, for which the excitations do not necessarily come in pairs. When moving around the level I excitations to derive the Bethe equations,
we have to recall that their duplication is an artifact of the unfolding. Every pair represents a single magnon in the original picture. When we move the original magnon, it looks like moving the pair simultaneously in the unfolded picture. Then, for level I impurities we pick up the factors that correspond to simultaneously moving around the pair with momentum $p_k$ and $-p_k$ in opposite directions.
For level II impurities, we have to collect the factors corresponding to going through all the level I pairs and all the level III impurities (scattering between level II particles is trivial). Finally, for level III impurities we get the factors of going through all the level II impurities and all the other level III impurities (scattering between level III and level I particles is trivial).
The resulting set of open Bethe ansatz equations is the following
\begin{align}
1 &= \left(\frac{x_{k}^{+}}{x_{k}^{-}}\right)^{2L}
\left( { 1 + { 1 \over (x^-)^2 } } \over 1 + { 1 \over (x^+)^2 } \right)^2 { 1\over \sigma_B(p_k)^2 \sigma(p_k,-p_k)^2}
\prod_{l=1}^{N^{\rm II}}\frac{y_{l}-x_{k}^{-}}{y_{l}-x_{k}^{+}}\frac{y_{l}+x_{k}^{-}}{y_{l}+x_{k}^{+}}, \label{BI}
\\
& \quad \prod_{l\neq k}^{N^{{\rm I}}}
\frac{(x_k^+ - x_l^-)(1-\frac{1}{x_k^-x_l^+})}{(x_k^--x_l^+)(1-\frac{1}{x_k^+ x_l^-})}
\frac{(x_l^+ + x_k^+)(1+\frac{1}{x_l^-x_k^-})}{(x_l^-+x_k^-)(1+\frac{1}{x_l^+ x_k^+})}
{1\over \sigma(p_k,p_l)^2 \sigma(p_l,-p_k)^2}{\nonumber}
\\
1 & = e^{i\theta-i\phi} \prod_{l=1}^{N^{{\rm I}}}\frac{y_{k}-x_{l}^{+}}{y_{k}-x_{l}^{-}}\frac{y_{k}+x_{l}^{-}}{y_{k}+x_{l}^{+}}
\prod_{l=1}^{N^{\rm III}}\frac{w_{l}-v_{k}-\frac{i}{g}}{w_{l}-v_{k}+\frac{i}{g}}
\label{BII}\\
1 &= e^{-2i\theta} \prod_{l=1}^{N^{\rm II}}\frac{w_{k}-v_{l}+\frac{i}{g}}{w_{k}-v_{l}-\frac{i}{g}}
\prod_{l\neq k}^{N^{\rm III}}\frac{w_{k}-w_{l}-\frac{2i}{g}}{w_{k}-w_{l}+\frac{2i}{g}}
\label{BIII}.
\end{align}
Eq. \nref{BI} can be re-written, including $l=k$ in the second product, as
\begin{align}
1 &= -\left(\frac{x_{k}^{+}}{x_{k}^{-}}\right)^{2L}{x^-+{1\over x^-}\over x^++{1\over x^+}}
{ 1\over \sigma_B(p_k)^2 }
\prod_{l=1}^{N^{\rm II}}\frac{y_{l}-x_{k}^{-}}{y_{l}-x_{k}^{+}}\frac{y_{l}+x_{k}^{-}}{y_{l}+x_{k}^{+}}, \label{BI2}
\\
& \quad \prod_{l=1}^{N^{{\rm I}}}
\frac{(x_k^+ - x_l^-)(1-\frac{1}{x_k^-x_l^+})}{(x_k^--x_l^+)(1-\frac{1}{x_k^+ x_l^-})}
\frac{(x_l^+ + x_k^+)(1+\frac{1}{x_l^-x_k^-})}{(x_l^-+x_k^-)(1+\frac{1}{x_l^+ x_k^+})}
{1\over \sigma(p_k,p_l)^2 \sigma(p_l,-p_k)^2}{\nonumber}
\end{align}
As usual, the energy is given by
\begin{equation}
{\cal E} = \sum_{k=1}^{N^I } \epsilon( p_k)
\end{equation}
\section{The boundary TBA equations}
\la{BdyTBA}
The Bethe equations (\ref{BI})-(\ref{BIII}) presented in the previous section are the correct description of the spectrum for large chains, $L\gg 1$. As $L$ becomes small,
wrapping effects come into play and the Bethe equations are no longer valid. Moreover, in this paper, we are mainly interested in $L=0$.
A description of the spectrum that is valid for any $L$
is the Boundary Thermodynamic Bethe Ansatz (BTBA) equations. These are a set of integral equations that govern the dynamics in the mirror channel. That is, the dynamics of excitations after exchanging the two dimensional space and time directions \cite{Zamolodchikov:1989cf,Ghoshal:1993tm}, see figure \ref{TBAfigure}.
The TBA equations can be derived from the knowledge of the spectrum of states and bound
states in the mirror channel. This spectrum was derived in \cite{Arutyunov:2009zu}.
The derivation of the TBA equations then follows the standard route given in
\cite{Yang:1968rm,Zamolodchikov:1989cf,Bombardelli:2009ns,Arutyunov:2009ur,Gromov:2009bc}. In the case that we have a boundary we can
follow essentially the same route. We use the boundary state defined in
section \ref{sec2}, and the untangling of boundary reflection matrices described
in figure \ref{Unfolding}. Then we
get a TBA which looks very similar to what we would obtain for a closed chain
of twice the length $L$, except for the fact that for each particle
of momentum $q$ we get one of momentum $-q$,
since the boundary state creates such a pair of particles. The consequence of this is
that the $Y$ functions obey a reflection property
\begin{equation} \la{FLIP}
Y_{a,s} (u) = Y_{a,-s}(-u)
\end{equation}
The set of $Y_{a,s}$ functions is the same as the one we have for
the closed string problem \cite{Bombardelli:2009ns,Arutyunov:2009ur,Gromov:2009bc}.
However, due to \nref{FLIP} we can restrict our attention to the ones with $s\geq 0$.
The boundary data appears as chemical potentials which depend
on the angles, $\theta,~\phi$,
as well as a $u$ dependent chemical potential given by the boundary
dressing phase $\sigma_B$. The precise form of the equations is derived in appendix
\nref{TBAderivation}.
\begin{figure}[h]
\centering
\def16cm{16cm}
\input{Ysystem.pdf_tex}
\caption{
(a) Set of $Y_{a,s}$ functions for the closed string problem. Here we have the same set but
the additional condition \nref{FLIP} implies that we can restrict to the set in (b). }\label{Ysystem}
\end{figure}
Let us summarize the final equations
\begin{eqnarray}
\label{1stTBA}
\log{Y_{1, 1}\over{\bf Y}_{1,1}}\!\!&=&\!\! K_{m-1}*\log{1+{{\overline Y}_{1, m}}\over1+\overline{\bf Y}_{1, m}}{1+{\bf Y}_{m,1}\over1+Y_{m,1}}+{\cal R}^{(01)}_{1\,a}* \log(1+Y_{a,0}) \\
\label{2ndTBA}
\log{{\overline Y}_{2, 2}\over\overline{\bf Y}_{2,2}}\!\!&=&\!\!\ \ K_{m-1}*\log{1+{{\overline Y}_{1, m}}\over1+\overline{\bf Y}_{1, m}}{1+{\bf Y}_{m,1}\over1+Y_{m,1}}+{\cal B}^{(01)}_{1\,a}* \log(1+Y_{a,0})
\\
\label{3rdTBA}
\log{{\overline Y}_{1, s}\over\overline{\bf Y}_{1, s}}\!\!&=&\!\!- K_{s-1,t-1}*\log{1+{\overline Y}_{1, t}\over1+\overline{\bf Y}_{1, t}}-K_{s-1}\hat *\log{1+Y_{1,1}\over1+{\overline Y}_{2,2}}
\\
\log{Y_{a,1}\over{\bf Y}_{a,1}}\!\!&=&\!\!- K_{a-1,b-1}*\log{1+Y_{b,1}\over1+{\bf Y}_{b,1}}-K_{a-1}\hat*\log{1+Y_{1,1}\over 1+{\overline Y}_{2,2}}{\nonumber}
\\
&&\qquad\qquad\qquad+\[{\cal R}^{(01)}_{ab}+{\cal B}^{(01)}_{a-2,b}\]*\log(1+Y_{b,0})
\label{4thTBA}
\\
\label{lastTBA}
\log{Y_{a,0}\over{\bf Y}_{a,0}}\!\!&=&\!\!\[2{\cal S}_{a\,b}-{\cal R}_{a\,b}^{(11)}+{\cal B}_{a\,b}^{(11)}\]*\log(1+Y_{b,0})+2\[{\cal R}_{a\,b}^{(1\,0)}+{\cal B}_{a,b-2}^{(1\,0)}\]\s*\log{1+Y_{b,1}\over1+{\bf Y}_{b,1}}{\nonumber}
\\
&&+2{\cal R}_{a\,1}^{(1\,0)}\hs*\log{1+Y_{1, 1}\over1+{\bf Y}_{1, 1}} -2{\cal B}_{a\,1}^{(1\,0)}\hs*\log{1+{\overline Y}_{2,2}\over1+\overline{\bf Y}_{2,2}}
\end{eqnarray}
where we used the conventions of \cite{GKV,GKKV} for the kernels and integration contours\footnote{
The convolutions of terms depending on $Y_{1,1}$ or ${\overline Y}_{2,2}$ are over a finite range $|u| \leq 2 g$. We use $\hat*$ as a reminder of that.}.
We have also defined the barred $Y$'s as ${\overline Y}_{a,s}^{(\text{here})}=1/Y_{a,s}^{(\text{there})}$, (see appendix \ref{kernels} for a summary). Here, the momentum carrying $Y_{a,0}$ functions
are defined as symmetric functions $Y_{a,0}(-u)=Y_{a,0}(u)$ and $\s*f(v)=[*f(v)+*f(-v)]/2$ is a symmetric convolution\footnote{For the ground state, we expect all functions to be symmetric, $Y_{a,s} (u) = Y_{a,s}(-u)$. But for excited states \nref{FLIP} only requires
the $Y_{a,0}$ functions to be symmetric.
The equation for excited states could in principle be obtained by analytic continuation from these equations \cite{Dorey:1996re,Bazhanov:1996aq,GKV,GKKV}. }.
There are implicit sums over one of the
indices of the kernels\footnote{ The indices of $Y_{1,m}$ or $Y_{m,1}$ run over $m\geq 2 $. For $Y_{b,0}$ they run over $b\geq 1 $. The same as in \cite{GKKV}. }.
The bold face ${\bf Y}$'s represent the asymptotic large $L$ solution. This is the solution
we obtain when the convolutions with the momentum carrying $Y_{a,0}$'s are dropped. These asymptotic solutions are the only place
where the angles and the boundary dressing phase enter. They are given by
\begin{eqnarray}\la{asymptotic1}
{\bf Y}_{1,1}\!\!&=&\!\!-\frac{\cos\theta}{\cos\phi}\ ,\qquad\quad\overline{\bf Y}_{1,s}=\frac{\sin^2\theta}{\sin[(s+1)\theta]\sin[(s-1)\theta]}
\\
\overline{\bf Y}_{2,2}\!\!&=&\!\!-\frac{\cos\theta}{\cos\phi}\ ,\qquad\quad{\bf Y}_{a,1}=\frac{\sin^2\phi}{\sin[(a+1)\phi]\sin[(a-1)\phi]}\\
\la{asymptotic2}
{\bf Y}_{a,0}&=&4
{ e^{i\chi(z^{[+a]}) + i \chi(1/z^{[-a]})} \over e^{ i \chi(z^{[-a]}) + i \chi(1/z^{[+a]})} } \left(\frac{z^{[-a]}}{z^{[+a]}}\right)^{2L+2}\!\!\!\!\!\!\!\! (\cos\phi-\cos\theta)^2\,\frac{\sin^2 a\, \phi}{\sin^2\phi}\,.\la{asymptotic3}
\end{eqnarray}
where $\chi$ is the function defining the boundary dressing phase (\ref{phasefa}).
Notice that the length $L$ appears only in \nref{asymptotic2}.
Here $z^{[\pm a]}$ are the solutions of
\begin{equation} \la{uExpression}
u = g \left( z^{[+a]} + { 1 \over z^{[+a]}} \right) - i { a \over 2 } = g \left( z^{[-a]} + { 1 \over z^{[-a]}}
\right) + i { a \over 2 } = { q \over 2} \sqrt{ 1 + { 16 g^2 \over a^2 + q^2 } }
\end{equation}
in the mirror region with $ |z^{[+ a]}|>1$ and $|z^{[- a]}|<1$.
Once we solve this system of equations, we can compute the ground state energy
as
\begin{equation}
{\cal E} =
-\sum_{a=1}^{\infty}
\int\limits_0^{\infty}\frac{dq}{2\pi} \log(1+Y_{a,0})\,,
\label{exactE}
\end{equation}
where $q$ is the mirror
momentum of each magnon bound state
\begin{equation}
q = g \left[ z^{[+ a]} - z^{[- a]}- { 1 \over z^{[+ a]} } + { 1 \over
z^{[- a]} } \right]
\end{equation}
\subsection{Recovering the Luscher result }
\la{reclu}
As a simple check of these equations let us rederive the results of section
\ref{Luschersub}. In the large $L$ limit we see that
the factor $ \left(\frac{z^{[-a]}}{z^{[+a]}}\right)^{2L+2} = e^{ - E_m 2 (L+1) }$ is very small.
This implies that the ${\bf Y}_{a,0}$ in \nref{asymptotic2} are very small. So we expect
that the $Y_{a,0}$ are also small and that we can
set them to zero in all the convolution terms of the TBA equations. In this limit,
the energy is given by inserting the asymptotic form ${\bf Y}_{a,0}$, \nref{asymptotic2}, in
the expression for the energy \nref{exactE}. One would be tempted to expand
the logarithm in \nref{exactE}, since ${\bf Y}_{a,0}$ is very small. However,
${\bf Y}_{a,0}$ has a double pole a $u=0$, or $q=0$, coming from the boundary
dressing phase. In other words, it
behaves as
\begin{equation} \la{smallq}
{\bf Y}_{a,0} \sim { G_a^2 \over q^2 } + {\cal O}(1)
\end{equation}
for small $q$.
We can then write the integrals in \nref{exactE} as
\begin{eqnarray}
\int\limits_0^{\infty}\frac{dq}{2\pi} \log(1+{\bf Y}_{a,0}) & = &
\int\limits_0^{\infty}\frac{dq}{2\pi} \log\left(1+{ G_a^2 \over q^2 } \right) +
\int\limits_0^{\infty}\frac{dq}{2\pi} \log{ (1+{\bf Y}_{a,0}) \over
\left(1+{
G_a^2 \over q^2 } \right)}
\end{eqnarray}
In the second term we can certainly expand to first order in ${\bf Y}_{a,0}$ and $G_a^2$,
which produces a result which is of order $e^{ - 2 E_m ( L +1)}$. The first term, however, gives $G_a/2 \sim e^{ - E_m (L+1)}$, which is bigger. So we get
\begin{equation} \la{enget}
{\cal E} \sim - { 1 \over 2} \sum_{a=1}^\infty { G_a }
\end{equation}
But this is precisely the same as what we got in section \ref{sec2}. Namely,
\nref{luscherfull} is the same as \nref{enget} after we realize that $G_a$ defined
in \nref{smallq} is essentially
the same as \nref{smallqt}, using \nref{asymptotic2}. This is not too surprising
since \cite{Bajnok:2004tq} derived \nref{luscherfull} by appealing to
TBA equations. In summary,
\nref{enget} agrees precisely with \nref{luschergen}.
In the next section we will perform a weak coupling check of the equations.
We will derive a simplified
set of equations that describe the small angle limit $\theta, ~ \phi \sim 0 $ and
we will expand and solve the resulting equations up to order $g^6$.
\section{The near BPS limit}
\label{smallphi}
When $\phi=\theta$ the Wilson loop is BPS and the energy vanishes.
As we deform the angles away from this supersymmetric configuration, the energy behaves as
\begin{equation} \la{DevBPS}
\Gamma_{ {\rm cusp}}(\phi,\theta) = - ( \phi^2 -\theta^2 ) { 1 \over 1 - { \phi^2 \over \pi^2 } } B( \tilde \lambda ) + {\cal O}((\phi^2 -\theta^2)^2) ~,~~~~~~~~~ \tilde \lambda = \lambda ( 1 - { \phi^2 \over \pi^2 } )\,.
\end{equation}
The function $B$, also known as the ``Bremsstrahlung function", is related to a variety of physical quantities
\cite{Correa:2012at,Fiol:2012sg}. It was computed exactly in \cite{Correa:2012at,Fiol:2012sg}
using localization. In the planar limit we get
\begin{equation}
B={ 1 \over 4 \pi^2 } { \sqrt{\tilde\lambda } I_2( \sqrt{\tilde\lambda} ) \over I_1( \sqrt{\tilde\lambda} ) } + {\cal O}( 1/N^2 ) \la{bplanar}
\end{equation}
On the one hand, this allows us to test the BTBA equation to high loop orders by penetrating deep into almost all parts of the equation. On the other hand, the simplicity of \nref{bplanar}
suggests that, in the near BPS limit, the BTBA equations can be drastically simplified.
The equations we will find in this limit are not that simple.
We hope that understanding how to simplify them will teach us how to simplify TBA equation in general.
In this section we will study the BTBA equations in this limit. We will show that the BTBA equations can be reduced to a simplified set of equations. We will then solve them to 3-loop order.
Here we restrict the discussion to $\theta =0$\footnote{The general near BPS case, with $\theta \not =0$,
has a similar degree of complexity. In fact, we have explicitly expanded the equations up to second order in $\lambda$ and verified the corresponding expansion in \nref{bplanar}. But we will not give the details here.}, so that
$\tilde\lambda=\lambda$ and $\Gamma_{ {\rm cusp}}(\phi,\theta) = -\phi^2 B(\lambda ) + {\cal O}(\phi^4)$.
We also set $L=0$ to extract the cusp anomalous dimension. It is important to note that now $\phi$ is the smallest parameter. In particular, it is smaller than $\lambda$.
In this small angle limit, the momentum carrying Y-functions are of order $Y_{a,0}={\cal O}(\phi^4)$ and therefore very small. This limit reminds us of the large $L$ asymptotic limit where the momentum carrying $Y_{a,0}$'s are exponentially suppressed. However, as opposed to the asymptotic limit, in the small angle limit,
we cannot drop the convolutions with the momentum carrying $Y_{a,0}$'s. Instead, we remain with a simplified set of non linear equations. The reason is that the large value of $\log Y_{a,0}$ is not due to
the sources in the BTBA equations. Instead, it is due to the fact that
the fermionic $Y$-functions ($Y_{1,1}$ and $Y_{2,2}$) approach $-1$ and lead to a big contribution through the $\log(1+Y_{1,1})$ and $\log(1+Y_{2,2})$ terms in the convolutions.
\subsection{The simplified equations at small angles}
As the momentum carrying Y-functions are small, they only contribute to $B(\lambda)$ through their double pole. We define $\mathbb C_a$ as the coefficient of the double pole at $u=0$,
\begin{equation}
\lim_{\substack{q\to0\\ \phi\to0}} Y_{a,0}
=\[- {\phi^2\over 2 u}{\mathbb C}_{a}\]^2
\la{defC}
\end{equation}
The energy, which is dominated by the value of
$Y_{a,0}$ at the double pole, reduces to
\begin{equation}
{\cal E}=\frac{\phi^2}{2}\sum_{a=1}^\infty{{\mathbb C}_a\over\sqrt{1+{16g^2/ a^2}}}\,,
\label{EfromC}
\end{equation}
where square root factor comes from the $q\to0$ limit of $(q/2u)$, see \nref{uExpression}.
In this small $\phi$-limit, the other $Y$-functions can be expanded as
\begin{eqnarray}\la{expansion}
Y_{1,1}&=&-1-\phi^2\,\Psi+{\cal O}(\phi^4)\,,\qquad\quad
Y_{m,1}={\cal Y}_m\[1+\phi^2(\Omega_m-{\cal X}_m)/2\] + {\cal O}(\phi^4)\,,\\
\overline{Y}_{2,2}&=&-1-\phi^2\,\Phi+{\cal O}(\phi^4)\,,\qquad\quad
\overline{Y}_{1,m}={\cal Y}_m\[1+\phi^2(\Omega_m+{\cal X}_m)/2\] +{\cal O}(\phi^4)\,.{\nonumber}
\end{eqnarray}
where we assumed that to leading order $Y_{1,1}=Y_{2,2}=-1$ and $Y_{m,1}=Y_{1,m}$. It is not difficult to see that this assumption is consistent with the BTBA equations. Moreover, we find that the functions $\Omega_m$ drop out of the equations.
We find that the BTBA equations (\ref{1stTBA})-(\ref{asymptotic3}) reduce to
\begin{eqnarray}
\Psi\!\!\!&=&\!\!\!{1\over2}+K_{m-1}*\[{\cal X}_m{{\cal Y}_m\over 1+{\cal Y}_m}+{1\over3}\]-\pi\,{\mathbb C}_{a}\,{\cal R}^{(01)}_{1\,a}(u,0)
\la{wlanyc1}
\\
\la{wlanyc2}\\
\Phi\!\!\!&=&\!\!\!{1\over2}+K_{m-1}*\[{\cal X}_m{{\cal Y}_m\over 1+{\cal Y}_m}+{1\over3}\]-\pi\,{\mathbb C}_{a}\,{\cal B}^{(01)}_{1\,a}(u,0)
\\
\log {\cal Y}_m\!\!\!&=&\!\!\!-K_{m-1,n-1}*\log\(1+{\cal Y}_n\)-K_{m-1}\hat*\log{\Psi\over\Phi}
\la{wlanyc3}\\
{\cal X}_m\!\!\!&=&\!\!\!-{m^2\over3}-K_{m-1,n-1}*\[{\cal X}_n{{\cal Y}_n\over 1+{\cal Y}_n}+{1\over3}\]+\pi\,{\mathbb C}_{n}\[{\cal R}^{(01)}_{mn}+{\cal B}^{(01)}_{m-2,n}\](u,0)
\la{wlanyc4}
\\
\Delta_\text{conv} \!\!\!&=&\!\!\! \left. \left\{ {\cal R}_{a\,1}^{(1\,0)}\hat*\log({\Psi\over 1/2})
-{\cal B}_{a\,1}^{(1\,0)}\hat*\log({\Phi\over 1/2})+\[{\cal R}_{a\,b}^{(1\,0)}
+{\cal B}_{a,b-2}^{(1\,0)}\]*\log\({1+{\cal Y}_{b}\over1+{1\over b^2-1}}\) \right\}\right|_{u=0} \la{deltaconv}
\\
{\mathbb C}_a \!\!\!&=&\!\!\! (-1)^a a^2 F(a,g){z_0^{[-a]}\over z_0^{[+a]}} e^{ \Delta_\text{conv} } \la{defc}
\end{eqnarray}
where $z_0^{[\pm a ]}$ denote the values of $z^{[\pm a ]}$ at $q=0$ \nref{zpm}. In \nref{deltaconv} we
are evaluating the non-convoluted variable of the kernels at $u=0$. The hat on $\hat *$ is a convolution
over the range $|u| \leq 2 g $. $F(a,g)$ is given in \nref{ffun}.
These equations are derived by implementing the expansion of Y-functions \nref{expansion} in the TBA system of equations \nref{1stTBA}-\nref{lastTBA}.
Let us make a couple of comments. First, the factors of $1/2$, $1/3$, $m^2/3$
stand for the subtraction of the asymptotic solutions. These read
\begin{equation}
\underline\Psi=\underline\Phi= {1\over2}\,, \qquad
\underline{{\cal Y}_{m}}={1\over m^2-1}\,, \qquad
\underline{{\cal X}_m} = -{m^2\over3 }\end{equation}
Second, note that in the BPS vacuum where $\phi=0$, the TBA equations are not well defined and need a regularization. A regulator commonly used is a twist for the fermions \cite{Frolov:2009in}.
Here, the angle $\phi$ can be viewed as a physical regulator.
As opposed to other regulators,
the leading order solution ${\cal Y}_m$ is a non trivial function of the coupling.
\subsection{Weak coupling expansion of the small $\phi$ TBA}
To test the BTBA equations, we have solved the small angle simplified equations, \nref{wlanyc1}-\nref{defc},
up to three loops. In this section
we will present the results. The derivation is given in appendix \ref{smallphisolution}.
The small $\phi$ TBA equations, \nref{wlanyc1}-\nref{defc}, are certainly simpler than the general TBA equations \nref{1stTBA}-\nref{lastTBA}, but they continue to be non-linear.
However, if we make a weak coupling expansion we obtain a linear system of integral equations order by order.
To solve these linear equations we find it useful to first simplify the TBA equations as in \cite{Arutyunov:2009ux,Arutyunov:2009ax}.
To simplify \nref{wlanyc1} and
\nref{wlanyc2}, we take a convolution of the equations with ${\mathpzc s}*{\mathpzc s}^{-1}$ where
\begin{equation}
{\mathpzc s}(u) = \frac{1}{2\cosh(\pi u)}
\end{equation}
The other equations can also be simplified as shown in the appendix \ref{smallphisolution}.
Then \nref{wlanyc1}-\nref{defc} become
\begin{eqnarray}
\Phi-\Psi\!\!\! &=&\!\!\!\pi\; {\mathbb C}_a \hat K_{y,a}(u,0)\,,
\label{ferdiftba}\\
\Phi + \Psi\!\!\! &=&\!\!\!
- 2 {\mathpzc s}*{{\cal X}_{2}\over1+{\cal Y}_{2}}
+ 2 \pi {\mathpzc s}*{\cal R}^{(01)}_{2\,n}(u,0){\mathbb C}_n
-\pi\,{\mathbb C}_{a}\,K_a(u,0)\,,
\label{feraditba}
\\
\log{\cal Y}_m\!\!\! &=&\!\!\!{\mathpzc s}* I_{m,n}\log{{\cal Y}_n \over1+{\cal Y}_n }+\delta_{m,2}\, {\mathpzc s}\hat*\log{\Phi\over\Psi}\,,\label{yosimtba}\\
{\cal X}_m\!\!\! &=&\!\!\!{\mathpzc s}*I_{m,n}{{\cal X}_n \over 1+{\cal Y}_n }+\pi {\mathpzc s}\ {\mathbb C}_m
+\delta_{m,2}\,{\mathpzc s}\hat*(\Phi-\Psi)\,,
\label{xosimtba}
\\
\Delta_\text{conv} \!\!\!&=&\!\!\! \left. \left\{ {\cal R}_{a\,1}^{(1\,0)}\hat*\log({\Psi\over1/2})
-{\cal B}_{a\,1}^{(1\,0)}\hat*\log({\Phi\over 1/2})+\[{\cal R}_{a\,b}^{(1\,0)}+{\cal B}_{a,b-2}^{(1\,0)}\]*\log\({1+{\cal Y}_{b}\over1+{1\over b^2-1}}\)\right\}\right|_{u=0} \la{deltaconvnew}
\\
{\mathbb C}_a \!\!\!&=&\!\!\! (-1)^a a^2 F(a,g){z_0^{[-a]}\over z_0^{[+a]}} e^{ \Delta_\text{conv} } \la{defcnew} \end{eqnarray}
where $I_{m,n}=\delta_{m+1,n}+\delta_{m-1,n}$ and $\hat K_{y,a}$ is defined in appendix \ref{smallphisolution}.
Now expanding the functions $\Psi$, $\Phi$, ${\cal Y}_n$ and ${\cal X}_n$ in powers of $g^2$,
we can obtain them order by order by solving a linear system of equations. Up to three loops
(see appendix \ref{smallphisolution} for details) we find that
\begin{equation}
{\mathbb C}_a = 4(-1)^a g^2 + 8(-1)^a \[\pi^2-{4\over a^2}\]g^4 +16(-1)^a \[{\pi^4\over 3}-{4\pi^2\over a^2}+{20\over a^4}\]g^6 +{\cal O}(g^8)\,,
\end{equation}
Finally, the relation \nref{EfromC}, we obtain the expression for energy up to 3-loop order\footnote{
We encounter the sum $\sum_{a=1}^{\infty} (-1)^a = - { 1 \over 2}$. This can be understood by regularizing it
as $ \lim_{ \phi \to 0 } \left[ \sum_{a=1}^{\infty} (-1)^a { \sin a \phi \over a \phi } \right] = - { 1 \over 2 }$. }
\begin{equation}
{\cal E}=-\phi^2 \[g^2 -g^4\frac{2 \pi ^2 }{3}+g^6\frac{2 \pi ^4 }{3}+ O(g^8)\]=-\phi^2 \[\frac{\lambda }{16 \pi ^2}-\frac{\lambda ^2}{384 \pi ^2}+\frac{\lambda^3}{6144 \pi ^2}
+{\cal O}(\lambda^4)\]\,,
\end{equation}
In perfect agreement with the expansion of (\ref{bplanar}).
\section{Conclusions and discussion}
In this paper we have considered the problem of computing the
quark anti-quark potential on the 3-sphere in ${\cal N}=4 $ super Yang
Mills in the planar approximation. Since the planar theory
is integrable \cite{Review}, we expected to be able to derive an
exact expression.
Indeed, we found a system of boundary TBA equations \nref{1stTBA}-\nref{lastTBA}
which determines the potential
as a function of three parameters: the planar coupling $\lambda$,
the geometric angle $\phi$, which sets the angular separation on the 3-sphere and
an internal angle $\theta$ which is the relative orientation of the coupling to
the scalar field for the quark and the anti-quark.
This quark and anti-quark configuration gives rise to an integrable
system with a boundary. This is most clearly seen in the string theory picture
where we have a string going between the two lines on the boundary. One might be surprised
that we have a boundary since the string is infinitely long. However, note that the local
geometry of the string near the boundary is $AdS_2$, which indeed has a boundary.
The energy is then the
ground state energy , or Casimir energy, on the strip and it is given in terms of the solution of the
TBA equations \nref{exactE}.
This is the energy of the flux tube connecting the quark and anti-quark. These TBA equations
should also enable one to compute the energies of excitations of the flux tube. These correspond to
operators that are inserted on the Wilson loop.
The quark anti-quark potential on $S^3$ is the same as the cusp anomalous dimension as a function of
the angles, $\Gamma_{cusp}(\phi,\theta,\lambda)$.
The derivation of the boundary TBA equations is similar to the one in other
integrable models with boundary \cite{LeClair:1995uf}.
A crucial step is the determination of
the boundary reflection matrix. The matrix part is fixed by the symmetries and
the dressing phase was found by solving the boundary
crossing equation and the final answer is in \nref{phasefa},
\nref{chiintegral}.
Since there is always a certain amount of guesswork in determining the dressing
phase, we have checked it at strong coupling and we have seen that it
gives the right value both in the physical and mirror regions.
A crucial feature of the dressing phase is that it contains a pole at zero
mirror momentum. This is crucial for the proposed phase to work at weak coupling.
Note that the boundary dressing phase is responsible for the
leading order contribution in the mirror picture, while it
only starts contributing at three loops for
anomalous dimensions in the physical picture.
The pole simply means that the boundary
is sourcing single particle states.
The BTBA equations were written in \nref{1stTBA}-\nref{lastTBA}. They look very similar to the bulk
TBA equations \cite{Gromov:2009bc,Arutyunov:2009ux,Bombardelli:2009ns},
except that the boundary conditions for large $u$ are different. They now
depend on the angles. In addition, for the momentum carrying nodes, the $Y_{a,0}$,
there is an extra source term involving the boundary dressing phase.
We have obtained a simplified set of equations, \nref{ferdiftba}-\nref{defcnew},
which describes the small
angle region, $\phi , ~\theta \ll 1 $. In this region, the simplest way to solve the
problem is through supersymmetric localization, as explained in \cite{Correa:2012at}.
The planar answer is
\begin{equation} \la{brem}
\Gamma_{cusp}(\phi, \theta=0, \lambda) = - \phi^2 B + {\cal O}(\phi^4) ~,~~~~~~~~~~~~~~~
B = { 1 \over 4 \pi^2 } { \sqrt{\lambda } I_2( \sqrt{\lambda} ) \over I_1( \sqrt{\lambda} ) }
\end{equation}
So, we know the answer by independent means. Thus, these simplified BTBA equations
should reproduce \nref{brem}. Indeed, directly expanding these
simplified equations up to third order in the coupling we reproduced the expansion of \nref{brem}.
However, these ``simplified'' equations are vastly
more complex than the simple Bessel functions in \nref{brem}!.
Thus, there should be a way to simplify these equations much further and directly get the
simple answer \nref{brem}. Hopefully, the methods used to simplify the equation will also
be useful in order to simplify the full BTBA equations for general angles.
Note that in \cite{Gromov:2011cx} the TBA system for closed strings was reduced to a set of
equations involving a finite number of functions. It is very
likely that the same method works in
our case.
Notice that the simplified small angle equations
connect the integrability and the localization exact solutions.
In particular, computing the function $B$ by both methods would enable us to see whether
the coupling constant $\lambda$ that
appears in both approaches is the same or not. Of course, we expect them to be the same for ${\cal N}=4$
super Yang Mills.
However, if one could generalize the discussion in this paper to Wilson loops in ABJM theory \cite{Aharony:2008ug}, then
this small angle region could enable us to compute the undetermined function $h(\lambda)$ that appears
in the integrability approach to the ABJM theory \cite{Gromov:2008qe}.
In principle, one might wonder whether the Wilson loop leads to an integrable boundary condition.
We have found that the reflection matrix obeys the boundary Yang Baxter equation. The TBA equations were
derived assuming integrability. So all the checks we performed on them are further evidence that the
Wilson loop boundary condition is indeed integrable.
There are further checks of the equations that one should be able to do.
In particular,
one would like to reproduce the BES equation \cite{BES} for $\varphi \to \infty$.
It would also be nice to take the small $\delta = \pi -\phi$ limit. In this limit the answer should
go like $1/\delta $ and probably one can obtain again a simplified equation for the coefficient. This
determines the quark anti-quark potential in the flat space limit.
One should also be able to take the strong coupling limit of the equations and reproduce the
result derived from classical strings in $AdS_5 \times S^5$ in \cite{Forini:2010ek,arXiv:1105.5144}.
It is likely that the ideas in \cite{Gromov:2009tq,Gromov:2009at}
would enable this.
Though solving the TBA equation analytically looks difficult, it should be possible
to solve the equations numerically. The problem should be very similar to the one solved in
\cite{Gromov:2009zb}.
It would also be nice to study the problem of determining the open string spectrum on
the $AdS_4 \times S^2$ or $AdS_2 \times S^4$ D-branes which also preserve the same amount of
symmetry. The only difference with the current paper should be a different choice for the
boundary dressing phase. For this reason, the TBA equations would be the same, except
for the choice of the boundary dressing phase.
The study of perturbative amplitudes at weak coupling has found remarkably
simple underlying structures. It would be interesting to study these structures in the
context of the cusp anomalous dimension, where we have a function of a single angle $\phi$.
In particular, it would be nice to see how to connect those structures with the TBA approach
described here. This would most probably lead to both a simplification of this TBA approach as well
as some hints on the exact structure underlying the amplitude problem.
Throughout this paper we have considered the locally BPS Wilson loop which contains the
coupling to the scalar, as in \nref{wildef}. Of course, one can also consider the
Wilson loop which does not couple to the scalars, $W = tr P e^{ i \oint A } $. It would be interesting
to see whether this leads to an integrable boundary condition. At strong coupling this loop leads to
a Neumann boundary condition on the $S^5$ \cite{Alday:2007he}, which is classically integrable.\footnote{See \cite{Dekel:2011ja} for a systematic study of classically integrable boundary conditions.}
{\bf Note:} We were informed that similar ideas were pursued in \cite{DrukkerTBA}.
{\bf Acknowledgements }
We would like to thank N. Arkani-Hamed, B. Baso,
S. Caron-Huot, N. Drukker, D. Gaiotto, N. Gromov, I. Klebanov, P. Vieira and A. Zamolodchikov for discussions.
A. S. would like to thank Nordita for warm hospitality. This work was supported in part by U.S.~Department of Energy grant \#DE-FG02-90ER40542.
Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. The research of A.S.
has been supported in part by the Province of Ontario through ERA grant ER 06-02-293.
D.C would like to thanks IAS for hospitality. The research of D.C has been supported in part by a CONICET-Fulbright fellowship and grant PICT 2010-0724.
|
{
"timestamp": "2012-08-02T02:07:38",
"yymm": "1203",
"arxiv_id": "1203.1913",
"language": "en",
"url": "https://arxiv.org/abs/1203.1913"
}
|
\section{Introduction}\label{sec1}
Graphene after experimental discovery of its high-quality freestanding samples, \cite{novo}
has attracted a strong attention.\cite{cast}
Charge carriers in a single-layer graphene possess a gapless, linear spectrum close to the
$K$ and $K'$ points \cite{novo,cast,wallace} and manifest behavior of chiral massless particles with a
"light speed" equal to the Fermi velocity, $v_{F}$.
Graphene shows a lot of unusual effects, e.g.: the Klein paradox\cite{cast,klein,kat}, i.e., the perfect
transmission through arbitrarily high and wide barriers upon normal
incidence (as far as a Dirac-type Hamiltonian is valid), a half-integer quantum Hall effect (QHE) \cite{cast,brey,aba,gus0}, and a zitterbewegung, \cite{cast,zit1,zit2}
i.e., effect induced by a lateral confinement of Dirac fermions. Properties of the latter effect are essentially
modified by a strong magnetic field. Extra Dirac points in the energy spectrum for superlattices in graphene
have been obtained if the amplitude of periodic potential is sufficiently large while its period is small enough.
\cite{park2009,brey2009,barbier2010} In particular, this leads to new properties of the QHE.\cite{park2009}
Graphene's edges have also been studied considerably, \cite{cast,brey,aba,gus0,gus,milt}
in particular, in connection with the QHE \cite{cast,brey,aba,gus0}; for some phenomena it matters
a type, the armchair or zigzag, of edges. \cite{cast,brey,aba,gus0}
Edge magnetoplasmons (EMPs) in graphene have been studied only recently; \cite{balev2011}
it is shown that in the $\nu=2$ QHE regime at the armchair edge, and in the presence of a smooth steplike electrostatic
lateral confining potential, the chirality, spectrum, spatial structure, and number of the fundamental EMPs depend strongly
on the position of the Fermi level $E_{F}$.
In the case (i) of Ref. \onlinecite{balev2011}, when $E_{F}$ intesects (see Fig. 1, cf. with Fig. 1(a)
of Ref. \onlinecite{balev2011}) four degenerate states of the zero LL
at one location and two degenerate states of this LL at a different location,
two fundamental EMPs are present: counterpropagating and with essential spatial overlap.
This is in contrast with EMPs in conventional two-dimensional electron systems (2DES) which
give only one fundamental EMP at the $\nu=2$ QHE regime, with negligible spin-splitting;
for conventional 2DES different types of EMPs have been studied theoretically\cite{volkov91,aleiner94,wen91,stone92,bal,bal2000,bal99}
and experimentally. \cite{ashoori92,ernst96,kukushkin09}
Above two counterpropagating EMPs can be on resonance if a strong coupling of the EMPs holds at the ends of the segment
$L_{x}^{em} \leq L_{x}$, where $L_{x}$ is the length of graphene channel.\cite{balev2011}
In present study for the case qualitatively outlined in Fig. 1 (i.e., it is the case (i) of Ref. \onlinecite{balev2011})
we explore theoretically effect of a weak and smooth superlattice potential
$V_{s}(x)=V_{s}\cos(Gx)$ with $G=2\pi/a_{0}$, upon EMPs. Here, in agreement with speculations of \cite{balev2011}
that a strong Bragg coupling is possible due to a weak superlattice along
the edge (with period $L_{x}^{em}$, if $L_{x}/L_{x}^{em} \gg 1$), we show that
$V_{s}(x)$ can have a strong effect on two fundamental EMPs leading to manifestation of
resonance effects; in particular, referred to in the abstract.
Present EMPs in graphene with the superlattice are very different from the EMPs treated previously for
conventional 2DES with a superlattice. \cite{bal2000b}
In Sec. II A we obtain the wave functions and the spectra of LLs in an infinitely
large graphene flake in the presence of a perpendicular magnetic field and of a smooth
electrostatic confining potential, along the $y$ direction, as without
$V_{s}(x)$ so in its presence.
In Sec. II B we study the combined effect of a smooth, step-like electrostatic confining potential
and of armchair graphene edges, at $y=\pm L_{y}/2$, and of the superlattice potential
$V_{s}(x)$ on the local Hall conductivity
in the $\nu=2$ QHE regime. In Sec. III we obtain strong renormalization of the EMPs in graphene by a weak superlattice
potential. We make concluding remarks in Sec. IV.
\begin{figure}[ht]
\vspace*{-0.5cm
\includegraphics [height=10cm, width=8cm]{Fig1empgap}
\vspace*{-3.0cm
\caption{(Color online) Energy spectrum of the $n=0, \pm 1$ LLs
as a function of the quantum number $y_{0}$, at the right half of symmetric graphene
channel with armchair edges and the
smooth electrostatic potential, Eq. (\ref{eq17}),
for the Fermi level $E_{F}=V_{0}/2$.
Spatially separated edge states
are created at $y_{r}^{u}$ and at $y_{r}^{d}$ (marked by an upward arrow)
as the branches of the $n=0$ LL cross $E_{F}$.
The $\nu=2$ QHE is manifested in dc magnetotransport.
}
\end{figure}
\section{Graphene Channel and local Hall conductivity}\label{sec2}
\subsection{Effect of a smooth potential and of a weak periodic potential on the LLs}
We consider a long and a wide flat graphene flake of length $L_{x}$ and width $L_{y}$, with armchair edges at $y=\pm L_{y}/2$
in the presence of a perpendicular magnetic field ${\bf B}=B\hat{z}$, of a smooth confining potential $V_y=V(y)$ along the
$y$ direction of electrostatic origin, and of a one-dimensional (1D) periodic potential $V_{s}(x)$ along the $x$ direction.
We assume that $V_{s}(x)=V_{s}\cos(Gx)$ is a weak 1D modulation potential of period $a_{0}$.
For definiteness we assume that the potential $V_y$ is symmetric.
If it is not otherwise stated, we consider solutions with energy and wave vector close to the K point;
we present pertinent results for the energies and wave vectors
close to the K$^\prime$ point (valley) as well. In the nearest-neighbor, tight-binding model the one-electron Dirac Hamiltonian, for massless electrons, is $\mathcal{H} = \mathcal{H}_{0}+ \mathds{1} V_{s}(x)$ where
$\mathcal{H}_{0} = v_F \vec{\sigma}\cdot \hat{\vec{p}} + \mathds{1} V_y$, with $\mathds{1}$ the $2\times 2$ unit matrix.
Explicitly $\mathcal{H}_{0} $ is given by ($e > 0$)
\begin{equation}
\mathcal{H}_{0} = v_F\matt{V_y/v_F}{p_x-i p_y-eBy}{p_x+i p_y-eBy}{V_y/v_F},
\label{eq1}
\end{equation}
where $p_x$ and $p_y$ are components of the momentum operator ${\bf p}$ and $v_F \approx 10^6 m/s$ the Fermi velocity. The vector potential is taken in the Landau gauge, ${\bf A}=(-By,0,0)$.
\subsubsection{Landau levels for a smooth potential $V(y)$}
First, we present properties of the LLs in the absence of periodic potential, when only
a smooth potential and armchair termination are assumed \cite{balev2011}.
The equation $(\mathcal{H}_{0} - E)\psi = 0$ admits solutions of the form
\begin{equation}
\psi^{(0)}({\bf r})= e^{i k_{x \alpha} x}\Phi(y)/\sqrt{L_x},\quad \Phi(y) = \kvec{A\Phi_A(y)}{B\Phi_B(y)} ,
\label{eq2}
\end{equation}
where the components $\Phi_A(y)$ and $\Phi_B(y)$ correspond to the two sublattices and the coefficients A and B satisfy the relation $|A|^2+|B|^2=1$; ${\bf r}=\{x,y\}$.
Introducing the magnetic length $\ell_0=(\hbar/eB)^{1/2}$, $y_0=\ell_{0}^2k_{x \alpha}$, the variable $\xi=(y-y_0)/\ell_0$,
and assuming that $V_y$ is a smooth function of $y$, with a characteristic scale $\Delta y \gg \ell_{0}$,
it follows \cite{balev2011} at $r=\ell_0 a/\hbar v_F \ll 1$ for the $n=0$ LL that
the energy $E_{0,k_{x \alpha}}^{(0)}=V(y_{0})$ and
\begin{eqnarray}
&&\hspace*{-0.9cm}\Phi_{A\kappa }^{0}(\xi)=\Phi_{B\kappa }^{0}(\xi)=1/(\pi \ell_{0}^{2} )^{1/4}
e^{-\xi^2/2}, \nonumber \\*
&&A_\kappa^{0}=(1/r)[1-\kappa(1-r^2)^{1/2}]B_\kappa^{0} ,
\label{eq3}
\end{eqnarray}
where $a=\partial V/ \partial \xi|_{\xi=0}$, and $\kappa=+(-)$ corresponds to the K (K$^\prime$) valley.
here $B_{+}^{0}=A_{-}^{0}\approx 1$ and $A_{+}^{0}=B_{-}^{0}\approx r/2 \ll 1$.
Finally, without periodic potential for any $n=0, \pm 1,\pm 2,...$ LL and $y_{0}$ not too close to the
graphene lattice termination at $y=\pm L_{y}/2$ (see Fig. 1), the
eigenvalues $E_{n, k_{x \alpha}}^{(0),\kappa}=E_{n, y_0}^{(0),\kappa}$ can be written as
\begin{equation}
E_{n, k_{x \alpha}}^{(0),\kappa}=sgn(n) \frac{\hbar v_F}{\ell_0}\sqrt{2|n|} +V(y_0),\quad n=0,\pm 1,...
\label{eq4}
\end{equation}
where the sign function $sgn(n)=1$ and $-1$ for $n>0$ and $n<0$, respectively.
Notice that each $n \neq 0$ LL is twice degenerate with respect to the valley quantum number $\kappa$.
However, the latter index is kept in the left hand side of Eq. (\ref{eq4}) as for $y_{0}(k_{x \alpha})$ close to
the armchair edge these eigenvalues, and especially strongly $E_{0, k_{x \alpha}}^{(0),\kappa}$, become dependent on $\kappa$;
notice, for these conditions $\kappa$ can not be related only to one valley \cite{brey,aba,gus0}.
Accordingly, for $n \neq 0$ LL and $y_{0}$ not too close to the graphene lattice termination, see Fig. 1, the
eigenvalues (\ref{eq4}) are four times degenerate. Wave functions pertinent to the eigenvalues Eq. (\ref{eq4})
we will denote also as $\psi^{(0),\kappa}_{n,k_{x \alpha}}({\bf r})$.
\subsubsection{Effect of potential $V(y)$ and of a weak smooth periodic potential $V_{s}(x)$ on the LLs}
Assuming $\hbar v_{F}/\ell_{0} \gg \hbar v_{g}(k_{x \alpha}) G \gg V_{s}/2$ and $G\ell_{0} \ll 1$ where the group velocity
$v_{g}(k_{x \alpha})=\hbar^{-1} dE^{(0)}_{0,k_{x \alpha}}/d k_{x \alpha}$, now we will study
how a weak periodic potential $V_{s}(x)$ modifies energies and
wave functions of the LLs for a smooth potential $V(y)$; in particular, for conditions pertinent to Fig. 1.
We calculate the eigenvalues and eigenfunctions corresponding to the
Hamiltonian $\mathcal{H} = \mathcal{H}_{0}+ \mathds{1} V_{s}(x)$ using the perturbation theory \cite{landaulif}.
Because further there will be important only the eigenstates localized, along $y$, near a right edge of the channel,
we will write formulas assuming, in particular, such eigenstates.
Similar with Ref. \cite{bal2000b}, we can neglect by a small ``nonresonance'' contributions, $n_{\beta} \neq n_{\alpha}$.
Then keeping only the ``resonance'' contributions, $n_{\beta}=n_{\alpha}$, and the terms of the first order over
$V_{s}$, e.g., for the eigenfunctions of the $n=0$ LL we obtain
\begin{equation}
\psi^{\kappa}_{0,k_{x \alpha}}({\bf r})=\psi^{(0),\kappa}_{0,k_{x \alpha}}
+\frac{V_{s}}{2\hbar v_{g}(k_{x \alpha}) \; G}
\Big[\psi^{(0),\kappa}_{0,k_{x \alpha}-G}
-\psi^{(0),\kappa}_{0,k_{x \alpha}+G}\Big] .
\label{eq5}
\end{equation}
Further, the eigenvalues are well approximated by the zero-order terms, i.e., $E_{n, k_{x \alpha}}^{\kappa} \approx E_{n, k_{x \alpha}}^{(0),\kappa}$,
where, for $y_{0}$ not too close to the graphene lattice termination, $E_{n, k_{x \alpha}}^{(0),\kappa}$ are given by Eq. (\ref{eq4}).
Indeed, the first order corrections are exactly nullified and the second order ones
are very small, e.g.: $\sim (V_{s}/2V_{0})^{2} \ll 1$ for the model potential Eq. (\ref{eq17}).
\subsection{Local Hall conductivity in the $\protect\nu=2$ QHE regime}
Extending magnetotransport formulas for the local Hall conductivity $\sigma_{yx}({\bf r})$ of a standard 2DES in the channel,
in the presence of a smooth lateral potential \cite{bee,thouless93,bal96}, we obtain, for linear responses
and in strong magnetic fields, $\sigma_{yx}({\bf r})$ in the form \cite{zhe}
\begin{equation}
\sigma_{yx}({\bf r})= n({\bf r}) e/B ,
\label{eq14}
\end{equation}
where the local electron density $n({\bf r})$ is smooth, on the characteristic scale $\ell_{0}$, as along
$y$, mainly monotonic, so along $x$, with a weak periodic modulation. It is given by
\begin{equation}
n({\bf r})= \sum_{\alpha\kappa}f_{\alpha\kappa}\langle\alpha\kappa| \mathds{1}\delta({\bf r}-\hat{{\bf r}})|\alpha\kappa\rangle ,
\label{eq15}
\end{equation}
with $\alpha=\{n,k_{x \alpha}\}$; $\sigma_{yx}(y)=-\sigma_{xy}(y)$. So far in Eqs. (\ref{eq14})-(\ref{eq15})
only the electrons from the conduction band LLs are assumed. However, if the valence band LLs
can essentially contribute to $\sigma_{yx}({\bf r})$, as for Fig. 1, the local hole density $p({\bf r})$
will contribute to the right hand part of Eq. (\ref{eq14}) by changing $n({\bf r})$ on $[n({\bf r})-p({\bf r})]$.
Then, for conditions relevant to Fig. 1, when only $(n=0, \kappa=\pm)$ LLs can essentially contribute to $\sigma_{yx}({\bf r})$
or a diagonal component of the local conductivity tensor,
equation (\ref{eq14}) is rewritten as
\begin{eqnarray}
\nonumber
\sigma_{yx}(x,y)&=&\frac{2e^{2}L_{x}}{h}
\sum_{\kappa=\pm} \int_{-\infty}^{\infty} dy_{0} [f_{0,y_{0},\kappa}- \delta_{\kappa,-}] \\
&&\times \left\langle \psi^{\kappa}_{0,k_{x \alpha}}({\bf r})\left\|\right.\psi^{\kappa}_{0,k_{x \alpha}}({\bf r})\right\rangle,
\label{eq16}
\end{eqnarray}
where $f_{n,y_{0},\kappa}$ is the Fermi function, the two-component column spinor wave function
$\left|\psi^{\kappa}_{0,k_{x \alpha}}({\bf r})\right\rangle$ is given by Eq. (\ref{eq5});
the factor $2$ accounts for spin degeneracy.
In Eqs. (\ref{eq15}),(\ref{eq16}) $\kappa=\pm$ is understood as the pseudospin
quantum number \cite{balev2011}; e.g., for $y_{0}>0$ only at $(L_y/2-y_0)/\ell_0 \gg 1$ it can be well approximated as the valley index.
A strong splitting between the electron, $\kappa=+$, and the hole, $\kappa=-$, branches
of the $n=0$ LL \cite{cast,milt,brey}, due to
hybridization of the valley states
take place nearby the armchair edge, at $|L_y/2-y_0| \leq \ell_0$.
The eigenvalues of the $n=0$ LL for
$\kappa=+(-)$ increase (decrease) with increasing $y_{0}$.
However, for the $n \geq 1$ LLs the $\kappa=\pm$ branches at the armchair edge
have a small splitting, due to hybridization of the valley states, as their eigenvalues increase with increasing $y_{0}$;
these branches are attributed to the electron band.
Notice, the electron, $(n=0, \kappa=+)$, LL stems from the conduction band and the hole, $(n=0, \kappa=-)$, LL
arises from the valence band.
We now consider the situations depicted
in Fig. 1 for a wide symmetric armchair graphene ribbon $L_{y}>2y_r \gg \Delta y \gg \ell_0$. For definiteness
the smooth lateral potential is assumed as follows
\begin{equation}
\hspace*{-0.2cm}V(y)=
(V_0/2)\Big[2+\Phi ((y-y_r)/\Delta y) +\Phi ((y+y_r)/\Delta y)\Big],
\label{eq17}
\end{equation}
where $\Phi (x)$
is the probability integral. In Fig. 1 we have
$L_{y}=18 \Delta y$, $V_{0}=\hbar v_{F}/\sqrt{2} \ell_0$, $y_{r}=5 \Delta y$, and
$\Delta y=10\ell_{0}$. When the Fermi level $E_F$ is between the bottoms of the $n=0$ and $n=1$ LLs,
at $y_{0}=0$, and the condition $V_0\gg 2k_BT$ holds, the occupation of the $n \geq 1$ LLs is negligible; the same holds for
the $n=0$ LL in the regions of $y_{0}$ that are well above $E_F$,
see Fig. 1. In addition to the smoothness of the potential Eq.(\ref{eq17}), we assume armchair edges
of the graphene sheet at $y=\pm L_y/2$,
which cause the bending of the LLs, \cite{cast,milt,brey}
and $L_y/2-y_r\geq \Delta y$.
For conditions of Fig. 1 and qualitatively similar, the dc magnetotransport measurements will manifest the $\nu=2$ QHE.
Further, for conditions qualitatively similar with those of Fig. 1, in agreement with speculations of Ref. \cite{balev2011} we will show that
a weak periodic potential $V_{s}(x)$ can have a strong effect on two fundamental EMPs leading to manifestation of the
resonance effects. First, for convenience of a reader, we will present expressions for the local Hall conductivity,
obtained in Ref. \cite{balev2011}, that are pertinent to $V_{s}(x) \to 0$.
\subsubsection{Effect of a smooth potential, an armchair edge and a weak periodic potential
on local Hall conductivity in the $\protect\nu=2$ QHE regime}
Now, for different regions of the graphene channel we will present expressions as for, obtained in Ref. \cite{balev2011},
the unperturbed
local Hall conductivity, $\sigma_{yx}^{(0)}(y)$,
so for the main contribution induced by a finite $V_{s}(x)$, $\sigma_{yx}^{(1)}(x,y) \propto V_{s}$.
The superscript in $\sigma_{yx}^{(1)}(x,y)$ indicates that it is of the first order over $V_{s}$.
Correspondingly, this contribution in Eq. (\ref{eq16}) stems
from the first order contributions to the wave function Eq. (\ref{eq5});
to calculate $\sigma_{yx}^{(1)}(x,y)$ we will need also assume that $V_{s}/k_{B}T \ll 1$.
Point out that $\sigma_{yx}^{(0)}(y)$ is given by the right hand side of Eq. (\ref{eq16}) if to substitute
$\left|\psi^{\kappa}_{0,k_{x \alpha}}({\bf r})\right\rangle$, calculated by taking into account the first order
corrections, by the zero order wave function $\left|\psi^{(0),\kappa}_{0,k_{x \alpha}}({\bf r})\right\rangle$.
In case (i), for $y_{0}>0$ and $(y_{r}^{d}-y_{0})/\ell_{0} \gg 1$, from
Eqs. (\ref{eq3})-(\ref{eq4}), (\ref{eq16})-(\ref{eq17}) it follows \cite{balev2011}
\begin{equation}
\sigma_{yx}^{(0)}(y)=\frac{2e^2}{h} \tanh\left(\frac{V(y_r^u)-V(y)}{2k_{B} T}\right) ,
\label{eq18}
\end{equation}
where $V(y)$ is so smooth on the scale of $\ell_{0}$ that $\ell_{0} dV(y_{r}^{u})/dy \ll k_{B} T$;
the factor $4$ accounts for spin and pseudospin degeneracy.
Introducing the characteristic length $\ell_{T}=\ell_0(k_BT\ell_0/\hbar v_g^{u})$,
this condition of smoothness can be rewritten as $\ell_{0} \ll \ell_{T}$,
where $v_{g}^{u}=\ell_{0}^{2}\, \hbar^{-1} dV(y_{r}^{u})/dy$ is the group velocity at the edge $y_{r}^{u}$.
Notice, for conditions of Fig. 1 it follows that $v_{g}^{u}/v_{F}=(\ell_{0}/\sqrt{2 \pi} \Delta y) \lll 1$,
due to $\ell_{0}/\Delta y \ll 1$. Point out, Eq. (\ref{eq18}) shows that at $y=y_{r}^{u}$ the Hall conductivity
changes its sign from the electron type of charge carriers to the hole one.
For $(\ell_T/\Delta y)^2\ll 1$, Eq. (\ref{eq18}) can be rewritten as\cite{balev2011}
\begin{equation}
\sigma_{yx}^{(0)}(y)={2e^2\over h} \tanh\left(\frac{y_r^u-y}{2\ell_{T}}\right).
\label{eq21}
\end{equation}
From Eq. (\ref{eq21})
it follows \cite{bal2000,balev2011}
\begin{equation}
\frac{d\sigma_{yx}^{(0)}(y)}{dy}=
-{4e^2\over h} \left[\frac{1}{4\ell_T} cosh^{-2}\left(\frac{y-y_{r}^{u}}{2\ell_{T}}\right)\right].
\label{eq22}
\end{equation}
Further, for $L_{y}/2 \geq y \geq L_{y}/2-5\ell_{0}$ pertinent
numerical results \cite{brey,milt,gus,aba} for $\nu=2$ we approximate by the same analitical expression for
$\left[n(y)-p(y)\right]$ as in Ref. \onlinecite{balev2011}. That gives
\begin{equation}
\hspace*{-0.2cm}
\sigma_{yx}^{(0)}(y)=\frac{2e^{2}}{h} \int_{-\infty}^{\infty} \frac{dy_0}{\sqrt{\pi} \ell_{0}}
\,e^{-(y-y_0)^2/ \ell_0^2} \left[f_{0,y_0,-}-1\right] ,
\label{eq23}
\end{equation}
where it is used that $E_{0,y_0,-}$
is a sharply decreasing function at $y_{0} \approx y_{r}^{d}$ such that
the Fermi function in Eq. (\ref{eq23}) is very fastly growing at $y_{0} \approx y_{r}^{d}$ on a scale
$\ell_{d} \ll \ell_{0}$. Here appears a new characteristic scale $\ell_{d}=
k_{B}T \ell_{0}^{2}/\hbar |v_{g}^{d}|$ as
for a change of $y_{0}$ on $\ell_{d}$, at
$y_{0} \approx y_{r}^{d}$, the value of $E_{0,y_0,-}$ will change on $k_{B}T$;
point out that $|v_{g}^{d}| \ll v_{F}$ is implicit in Fig. 1.
From Eq. (\ref{eq23}) it follows \cite{balev2011}
\begin{equation}
d\sigma_{yx}^{(0)}(y)/dy=(2e^2 /h\sqrt{\pi}\ell_0)\, e^{-(y-y_r^d)^2/\ell_0^2} ,
\label{eq24}
\end{equation}
by changing the derivatives over $y$ to those
over $y_{0}$ and integrating by parts.
In a similar manner, for case (i) and $y>0$, we obtain
from Eqs. (\ref{eq3})-(\ref{eq5}), (\ref{eq16}) that
$\sigma_{yx}^{(1)}({\bf r})=\sigma_{yx}^{(1)}(y)\cos \left(G x \right)$, where
\begin{eqnarray}
\nonumber
\sigma_{yx}^{(1)}(y)&=&-\frac{e^{2}}{h} \frac{V_{s}}{k_{B} T}
\left[ \cosh^{-2}\left(\frac{y-y_{r}^{u}}{2\ell_{T}}\right) \right. \\
&&\left.+\frac{2 k_{B} T \ell_{0}}{\sqrt{\pi} \hbar |v_{g}^{d}|} e^{-\left(y-y_{r}^{d}\right)^{2}/\ell_{0}^{2}} \right],
\label{eq25}
\end{eqnarray}
here it is taken into account that $v_{g}^{d}<0$. Due to $k_{B}T \ell_{0}/\hbar |v_{g}^{d}| \ll 1$,
the amplitude of the
second term in the square brackets of Eq. (\ref{eq25}) is less than of the first one.
In addition, Eq. (\ref{eq25}) shows that a small parameter $V_{s}/k_{B}T \ll 1$ warrants
that $|\sigma_{yx}^{(1)}(x,y) \ll |\sigma_{yx}^{(0)}(y)|$; in particular, at $y \approx y_{r}^{u}$.
\section{Strong renormalization of the EMPs in graphene by a weak superlattice potential}
Now we will study effect of the superlattice potential, $V_{s}(x)$, on the fundamental EMPs for case (i),
i.e., for conditions like in Fig. 1; dissipation is neglected.
For $V_{s}(x) \equiv 0$, our fundamental EMPs
will coincide with ones obtained in Ref. \onlinecite{balev2011}.
Similar with Ref. \onlinecite{balev2011}, we expect that the charge excitation due to EMPs at
the right part of channel will be strongly localized
at $y_{r}^{u}$ ( $\rho^{ru}(t,{\bf r})$) and $y_{r}^{d}$ ( $\rho^{rd}(t,{\bf r})$).
Then the components of the current density ${\bf j}(\omega,{\bf r})$ in the
low-frequency limit $\omega\ll v_F/\ell_0$ are given, cf. with,\cite{balev2011} as
\begin{eqnarray}
\nonumber
\hspace*{-0.25cm}j_x(\omega,{\bf r})&=&-[\sigma_{yx}^{(0)}(y)+\sigma_{yx}^{(1)}({\bf r})]E_y(\omega,{\bf r}) \\*
&+&v_g^u \rho^{ru}(\omega,{\bf r})+v_g^d \rho^{rd}(\omega,{\bf r}) ,
\label{eq26}
\end{eqnarray}
\begin{equation}
j_y(\omega,{\bf r})=[\sigma_{yx}^{(0)}(y)+\sigma_{yx}^{(1)}({\bf r})]E_x(\omega,{\bf r}),
\label{eq27}
\end{equation}
where we have suppressed the factor $\exp(-i\omega t)$ common to all terms in Eqs. (\ref{eq26}) and (\ref{eq27}).
From Eqs. (\ref{eq26})-(\ref{eq27}), Poisson's equation, the linearized continuity equation
and using that for EMPs ${\bf E}(\omega,{\bf r})=-{\bf \nabla} \varphi(\omega,{\bf r})$), we obtain
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}&&-i\omega(\rho^{ru}(\omega, {\bf r})+\rho^{rd}(\omega, {\bf r}))+
[v_g^u \partial_{x}\rho^{ru}(\omega, {\bf r}) \\*
\nonumber
&&+v_g^d \partial_{x} \rho^{rd}(\omega,{\bf r})]+
[\partial_{x} \sigma_{yx}^{(1)}({\bf r})] \partial_{y} \varphi(\omega,{\bf r}) \\*
&&-[\partial_{y} \sigma_{yx}^{(0)}(y)+\partial_{y} \sigma_{yx}^{(1)}({\bf r})]
\partial_{x} \varphi(\omega,{\bf r})=0 ,
\label{18d}
\end{eqnarray}
where $\partial_{x}=\partial/\partial x$. Point out, in Eq. (\ref{18d}) the coefficients are invariant
for translations along $x$ on any distance integral of the period $a_{0}$.
Then in Eq. (\ref{18d}) we assume that
\begin{eqnarray}
\nonumber
&&\rho^{ru}(\omega, {\bf r})=\sum_{\ell=-1}^{1} \rho^{ru}_{\ell}(\omega,k_{x},y) e^{ik_{x}^{(\ell)} x} ,
\\*
\nonumber
&&\rho^{rd}(\omega, {\bf r})=\sum_{\ell=-1}^{1} \rho^{rd}_{\ell}(\omega,k_{x},y) e^{ik_{x}^{(\ell)} x} , \\*
&&\varphi(\omega,{\bf r})=\sum_{\ell=-1}^{1} \varphi_{\ell}(\omega,k_{x},y) e^{ik_{x}^{(\ell)} x} ,
\label{19d}
\end{eqnarray}
where $\ell=-1,0, 1$, $k_{x}^{(\ell)}=k_{x}+2\pi \ell/a_{0}$, and
$k_{x} \equiv k_{x}^{(0)}$.
In Eq. (\ref{19d}), for a metallic gate at a distance $d$ from the 2DES
(e.g., it can be a heavily doped Si separated from the graphene sheet by a SiO$_{2}$ layer of thickness $d=300$ nm),
is given by
\begin{eqnarray}
\nonumber
&&\varphi_{\ell}(\omega,k_{x},y)=\frac{2}{\epsilon}
\int_{-\infty}^{\infty} dy^{\prime}
R_{g}(|y-y^{\prime}|,k_{x}^{(\ell)};d) \\*
&& \times \left[\rho^{ru}_{\ell}(\omega, k_x,y^{\prime})+\rho^{rd}_{\ell}(\omega, k_x,y^{\prime})\right] ,
\label{20d}
\end{eqnarray}
where $R_g(...)$ is given by
\begin{eqnarray}
\hspace*{-0.89cm} R_{g}(|y-y^{\prime}|,k_{x}^{(\ell)};d)&=&K_{0}(|k_{x}^{(\ell)}||y-y^{\prime }|) \nonumber \\
&-& K_{0}(|k_{x}^{(\ell)}|\sqrt{(y-y^{\prime })^{2}+4d^{2}}),
\label{21d}
\end{eqnarray}
where $K_0(x)$ is the modified Bessel function. Without of a metallic gate, $d \to \infty$, the
dielectric constant $\epsilon $ is spatially homogeneous if not stated otherwise.
Multiplying Eq. (\ref{18d}) by $ i \exp(-ik_{x} x)$ and then integrating over $x$, $\int_{0}^{L_{x}} dx$, we
obtain
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&(\omega-k_{x} v_g^u) \rho^{ru}_{0}(\omega,k_{x},y)+
(\omega-k_{x} v_g^d) \rho^{rd}_{0}(\omega,k_{x},y) \\*
\nonumber
&&-\frac{\pi}{a_{0}} \sigma_{yx}^{(1)}(y)
\left[\partial_{y} \varphi_{-1}(\omega,k_{x},y)-\partial_{y} \varphi_{1}(\omega,k_{x},y) \right]\\*
\nonumber
&&+[\partial_{y} \sigma_{yx}^{(0)}(y)] k_{x} \varphi_{0}(\omega,k_{x},y)+\frac{1}{2} [\partial_{y} \sigma_{yx}^{(1)}(y)] \\*
&&\times \left[k_{x}^{(-1)}\varphi_{-1}(\omega,k_{x},y)+k_{x}^{(1)}\varphi_{1}(\omega,k_{x},y) \right]=0.
\label{22d}
\end{eqnarray}
Further, multiplying Eq. (\ref{18d}) by $ i \exp(-ik_{x}^{(-1)} x)$ and then integrating
over $x$, $\int_{0}^{L_{x}} dx$, we obtain
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&(\omega-k_{x}^{(-1)} v_g^u) \rho^{ru}_{-1}(\omega,k_{x},y)+
(\omega-k_{x}^{(-1)} v_g^d) \rho^{rd}_{-1}(\omega,k_{x},y) \\*
\nonumber
&&+\frac{\pi}{a_{0}} \sigma_{yx}^{(1)}(y)\partial_{y} \varphi_{0}(\omega,k_{x},y)+
[\partial_{y} \sigma_{yx}^{(0)}(y)] k_{x}^{(-1)} \varphi_{-1}(\omega,k_{x},y) \\*
&&+\frac{1}{2} k_{x} [\partial_{y} \sigma_{yx}^{(1)}(y)] \varphi_{0}(\omega,k_{x},y)=0.
\label{23d}
\end{eqnarray}
In addition, multiplying Eq. (\ref{18d}) by $ i \exp(-ik_{x}^{(1)} x)$ and then integrating
over $x$, $\int_{0}^{L_{x}} dx$, we obtain
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&(\omega-k_{x}^{(1)} v_g^u) \rho^{ru}_{1}(\omega,k_{x},y)+
(\omega-k_{x}^{(1)} v_g^d) \rho^{rd}_{1}(\omega,k_{x},y) \\*
\nonumber
&&-\frac{\pi}{a_{0}} \sigma_{yx}^{(1)}(y)\partial_{y} \varphi_{0}(\omega,k_{x},y)+
[\partial_{y} \sigma_{yx}^{(0)}(y)] k_{x}^{(1)} \varphi_{1}(\omega,k_{x},y) \\*
&&+\frac{1}{2} k_{x} [\partial_{y} \sigma_{yx}^{(1)}(y)] \varphi_{0}(\omega,k_{x},y)=0.
\label{24d}
\end{eqnarray}
For $y_r^d - y_r^u\gg \ell_T$, from Eqs. (\ref{22d})-(\ref{24d})
it follows that $\rho^{ru}_{\ell}(\omega, k_x,y)$ and $\rho^{rd}_{\ell}(\omega, k_x,y)$
can be well approximated (cf. Ref. \onlinecite{balev2011}) as
\begin{eqnarray}
\nonumber
&&\rho^{ru}_{\ell}(\omega, k_x,y)=
\Big[4\ell_T\cosh^2({y-y_r^u\over 2\ell_T})\Big]^{-1}\,\rho^{ru}_{\ell}(\omega, k_x) , \\*
&&\rho^{rd}_{\ell}(\omega, k_x,y)= (1/\sqrt{\pi}\ell_0) e^{-(y-y_r^d)^2/\ell_0^2} \,\rho^{rd}_{\ell}(\omega, k_x).
\label{25d}
\end{eqnarray}
In addition, we can neglect by overlap between $\rho^{ru}(\omega, k_x,y)$ and $\rho^{rd}(\omega, k_x,y)$
in Eqs. (\ref{22d})-(\ref{24d}). Then, by integration of Eqs. (\ref{22d})-(\ref{24d}) over $y$
within separate regions around $y_{r}^{u}$ and $y_{r}^{d}$, we obtain by straightforward calculations
six coupled linear homogeneous equations for six unknown functions:
$\rho^{ru}_{\ell}(\omega, k_x)$ and
$\rho^{rd}_{\ell}(\omega, k_x)$, where $\ell=-1, 0, 1$. They read, with $y_{r}^{du}\equiv y_{r}^{d}-y_{r}^{u}>0$,
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{+,0}^{(i)}(k_{x};d)] \rho^{ru}_{0}(\omega,k_{x})-
2c_{h}k_{x}R_{g}(y_{r}^{du},k_{x};d) \\*
\nonumber
&&\times \rho^{rd}_{0}(\omega,k_{x}) +c_{h} k_{x} \left(\frac{V_{s}}{k_{B}T} \ell_{T}\right)
\left[R_{g}^{\prime}(y_{r}^{du},k_{x}^{(-1)};d) \right. \\*
&&\times \left.\rho^{rd}_{-1}(\omega,k_{x}) +R_{g}^{\prime}(y_{r}^{du},k_{x}^{(1)};d) \rho^{rd}_{1}(\omega,k_{x})
\right]=0,
\label{26d}
\end{eqnarray}
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{-,0}^{(i)}(k_{x};d)] \rho^{rd}_{0}(\omega,k_{x})+
c_{h}k_{x}R_{g}(y_{r}^{du},k_{x};d) \\*
\nonumber
&&\times \rho^{ru}_{0}(\omega,k_{x})-c_{h} k_{x} \left(\frac{V_{s}}{k_{B}T} \ell_{T} \frac{v_{g}^{u}}{2|v_{g}^{d}|}\right)
\left[R_{g}^{\prime}(y_{r}^{du},k_{x}^{(-1)};d)\right. \\*
&&\left. \times \rho^{ru}_{-1}(\omega,k_{x}) +R_{g}^{\prime}(y_{r}^{du},k_{x}^{(1)};d) \rho^{ru}_{1}(\omega,k_{x})
\right]=0,
\label{27d}
\end{eqnarray}
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{+,0}^{(i)}(k_{x}^{(-1)};d)] \rho^{ru}_{-1}(\omega,k_{x})-
2c_{h}k_{x}^{(-1)} \\*
\nonumber
&&\times R_{g}(y_{r}^{du},k_{x}^{(-1)};d) \rho^{rd}_{-1}(\omega,k_{x}) +c_{h} k_{x}^{(-1)} \left(\frac{V_{s}}{k_{B}T} \ell_{T}\right)
\\*
&&\times R_{g}^{\prime}(y_{r}^{du},k_{x};d) \rho^{rd}_{0}(\omega,k_{x})=0,
\label{28d}
\end{eqnarray}
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{-,0}^{(i)}(k_{x}^{(-1)};d)] \rho^{rd}_{-1}(\omega,k_{x})+
c_{h}k_{x}^{(-1)} \\*
\nonumber
&&\times R_{g}(y_{r}^{du},k_{x}^{(-1)};d) \rho^{ru}_{-1}(\omega,k_{x}) -c_{h} k_{x}^{(-1)} \left(\frac{V_{s}}{k_{B}T} \right.
\\*
&&\left. \times \ell_{T} \frac{v_{g}^{u}}{2|v_{g}^{d}|}\right)
R_{g}^{\prime}(y_{r}^{du},k_{x};d) \rho^{ru}_{0}(\omega,k_{x})=0,
\label{29d}
\end{eqnarray}
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{+,0}^{(i)}(k_{x}^{(1)};d)] \rho^{ru}_{1}(\omega,k_{x})-
2c_{h}k_{x}^{(1)} \\*
\nonumber
&&\times R_{g}(y_{r}^{du},k_{x}^{(1)};d) \rho^{rd}_{1}(\omega,k_{x}) +c_{h} k_{x}^{(1)} \left(\frac{V_{s}}{k_{B}T} \ell_{T}\right) \\*
&&\times R_{g}^{\prime}(y_{r}^{du},k_{x};d) \rho^{rd}_{0}(\omega,k_{x})=0,
\label{30d}
\end{eqnarray}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig2empgap}
\vspace*{-0.7cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d\to \infty)$, without the gate, of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=500$\,cm$^{-1}$, $V_{s}/k_{B}T=0.3$,
$y_{r}^{du}=10 \ell_{0}$, $\Delta y=10 \ell_{0}$, $B=9$T, $T=77$K, $\ell_{T}/\ell_{0}=2$, $v_{g}^{u}=4 \times 10^{6}$ cm/s,
$v_{g}^{d}=-3 \times 10^{7}$ cm/s, $\epsilon=2$.
Panel (a) presents the dispersion relations within the first Brillouin zone,
$\pi/a_{0}> k_{x} \geq -\pi/a_{0}$, by the curves 1 (solid), 2 (dashed), 3 and 4 (dotted),
5 and 6 (dash-dotted).
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -165$\,cm$^{-1}$ and $\omega \approx -1.36 \times 10^{11}$\,s$^{-1}$,
with the gap $\approx 0.98 \times 10^{9}$\,s$^{-1}$; here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Panel (d) presents a zoom of the anticrossing for the branches 3 and 5.
}
\end{figure}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig3empgap}
\vspace*{-0.7cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=3000$nm) of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=500$\,cm$^{-1}$ and other parameters of Fig. 2 except $d=3000$nm.
Panel (a) presents the dispersion relations within the first Brillouin zone by the curves 1 (solid), 2 (dashed), 3 and 4 (dotted),
5 and 6 (dash-dotted).
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -189$\,cm$^{-1}$ and $\omega \approx -1.17 \times 10^{11}$\,s$^{-1}$,
with the gap $\approx 0.90 \times 10^{9}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$.
Panel (d) presents a zoom of the anticrossing for the branches 3 and 5.
}
\end{figure}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig4empgap}
\vspace*{-2.5cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=300$nm) of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=500$\,cm$^{-1}$ and other parameters of Fig. 2 except $d=300$nm.
Panel (a) presents the dispersion relations within the first Brillouin zone by the curves 1 (solid), 2 (dashed), 3 and 4 (dotted),
5 and 6 (dash-dotted).
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -222$\,cm$^{-1}$ and $\omega \approx -0.90 \times 10^{11}$\,s$^{-1}$,
with the gap $\approx 0.73 \times 10^{9}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$.}
\end{figure}
\begin{eqnarray}
\nonumber
\hspace*{-0.99cm}
&&[\omega-\omega_{-,0}^{(i)}(k_{x}^{(1)};d)] \rho^{rd}_{1}(\omega,k_{x})+
c_{h}k_{x}^{(1)} \\*
\nonumber
&&\times R_{g}(y_{r}^{du},k_{x}^{(1)};d) \rho^{ru}_{1}(\omega,k_{x}) -c_{h} k_{x}^{(1)}
\left(\frac{V_{s}}{k_{B}T} \right. \\*
&&\times \left. \ell_{T} \frac{v_{g}^{u}}{2|v_{g}^{d}|}\right) R_{g}^{\prime}(y_{r}^{du},k_{x};d) \rho^{ru}_{0}(\omega,k_{x})=0,
\label{31d}
\end{eqnarray}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig5empgap}
\vspace*{-0.7cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d\to \infty)$, without the gate, of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=5000$\,cm$^{-1}$, the rest of parameters are the same as in Fig. 2.
Panel (a) presents the dispersion relations within the first Brillouin zone,
$\pi/a_{0}> k_{x} \geq -\pi/a_{0}$, by the curves 1 (solid), 2 (dashed), 3 and 4 (dotted),
5 and 6 (dash-dotted).
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -1860$\,cm$^{-1}$ and $\omega
\approx -1.15 \times 10^{12}$\,s$^{-1}$,
with the gap $\approx 9.0 \times 10^{9}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Panel (d) presents a zoom of the anticrossing for the branches 3 and 5.
}
\end{figure}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig6empgap}
\vspace*{-0.7cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=3000$nm) of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for the gate at $d=3000$nm; as $2\pi/a_{0}=5000$\,cm$^{-1}$ so the other
parameters are the same as in Fig. 5.
Panel (a) presents the dispersion relations within the first Brillouin zone.
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -1920$\,cm$^{-1}$ and $\omega \approx -1.13
\times 10^{12}$\,s$^{-1}$, with the gap
with the gap $\approx 8.8 \times 10^{9}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Panel (d) presents a zoom of the anticrossing for the branches 3 and 5.}
\end{figure}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig7empgap}
\vspace*{-2.5cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=300$nm) of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=5000$\,cm$^{-1}$ and other parameters of Fig. 5 except $d=300$nm.
Panel (a) presents the dispersion relations within the first Brillouin zone.
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -2220$\,cm$^{-1}$ and $\omega \approx -8.95 \times 10^{11}$\,s$^{-1}$,
with the gap $\approx 7.3 \times 10^{9}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.}
\end{figure}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig8empgap}
\vspace*{-2.5cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d\to \infty)$, without the gate, of six EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d})
for $2\pi/a_{0}=50000$\,cm$^{-1}$, the rest of parameters are the same as in Fig. 2.
Panel (a) presents the dispersion relations within the first Brillouin zone,
$\pi/a_{0}> k_{x} \geq -\pi/a_{0}$, by the curves 1 (solid), 2 (dashed), 3 and 4 (dotted),
5 and 6 (dash-dotted).
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4, at
$k_{x} \approx -22150$\,cm$^{-1}$ and $\omega \approx -8.6 \times 10^{12}$\,s$^{-1}$,
with the gap $\approx 6.9 \times 10^{10}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; a finite gap is absent.}
\end{figure}
where, by using the same notations as in Ref. \onlinecite{balev2011}, we have $c_{h}=4e^{2}/h \epsilon$,
\begin{equation}
\omega_{+,0}^{(i)}(k_x,d)=k_x v_g^u+ 2c_{h}k_x a_{p}(k_{x};d) ,
\label{32d}
\end{equation}
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig9empgap}
\vspace*{-2.5cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=3000$nm)
of the EMPs calculated from Eqs. (\ref{26d})-(\ref{31d})
for
parameters of Fig. 8, except $d=3000$nm.
Panel (a) presents the dispersion relations within the first Brillouin zone.
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
with the gap $\approx 6.9 \times 10^{10}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; a finite gap is absent.
}
\end{figure}
\begin{equation}
\omega_{-,0}^{(i)}(k_x,d)=-k_x|v_g^d|-c_{h}k_x \, a_{m}(k_{x};d) ,
\label{33d}
\end{equation}
with the matrix elements
\begin{equation}
\hspace*{-0.3cm}a_{p}(k_{x};d)=\frac{1}{16} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
\frac{dx dt \; R_{g}(\ell_{T}|x-t|,k_{x};d)}{\cosh^{2}(x/2)\cosh^{2}(t/2)} ,
\label{34d}
\end{equation}
\begin{equation}
a_{m}(k_{x};d)=\frac{1}{\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{dx dt}{e^{x^2+t^2}}
R_{g}(\ell_{0}|x-t|,k_{x};d) .
\label{35d}
\end{equation}
Point out, if the graphene is located between two dielectric media with dielectric constants: $\epsilon_{1}$ for the halfspace
below of the graphene, and $\epsilon_{2}$ for the dielectric layer between the graphene and a metallic gate, - then
in all above expressions $\epsilon=(\epsilon_{1}+\epsilon_{2})/2$.
In addition, in Eqs. (\ref{26d})-(\ref{31d})
\begin{eqnarray}
\hspace*{-0.89cm} && R_{g}^{\prime}(y_{r}^{d}-y_{r}^{u},k_{x};d)=|k_{x}| \left[K_{1}(|k_{x}|y_{r}^{du}) \right.
\nonumber \\*
&&\left.-\frac{y_{r}^{du}}{\sqrt{(y_{r}^{du})^{2}+4d^2}}
K_{1}(|k_{x}|\sqrt{(y_{r}^{du})^{2}+4d^{2}})\right] .
\label{36d}
\end{eqnarray}
Notice that in the long-wavelength limit, $k_{x} \ell_{T} \ll 1$, and for large $d$, such that the effect of
gate, $\propto \exp(-2|k_{x}| d) \ll 1$, can be neglected, from Eqs. (\ref{21d}),
(\ref{32d})-(\ref{36d}) we obtain that: $R_{g}(|y-y^{\prime}|,k_{x};d) \approx \ln(2/|k_x(y-y')|)-\gamma$,
where $\gamma$ is the Euler constant,
$a_{p}(k_{x};d) \approx \left[\ln(1/|k_x|\ell_T)-0.145\right]$,
$a_{m}(k_{x};d) \approx \left[\ln(1/|k_x|\ell_0)+3/4\right]$,
and $R_{g}^{\prime}(y_{r}^{du},k_{x};d) \approx 1/y_{r}^{du}$.
For $\nu=2$ and case (i), in Fig. 2 we plot the dispersion relations $\omega(k_x,d\to \infty)$ of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $2\pi/a_{0}=500$\,cm$^{-1}$, $V_{s}/k_{B}T=0.3$,
$y_{r}^{du}=\Delta y=10 \ell_{0}$, $B=9$T, $T=77$K, $\ell_{T}/\ell_{0}=2$, $v_{g}^{u}=4 \times 10^{6}$ cm/s,
$v_{g}^{d}=-3 \times 10^{7}$ cm/s, $\epsilon=2$, and $\ell_{0}\approx 8.5$ nm.
Fig. 2(a) presents the dispersion relations of these EMPs within the first Brillouin zone
by the curves 1-6. Notice, Fig. 2(a) shows that the branches 3 and 4 have
$\omega \approx \pm 2.0 \times 10^{11}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 2(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -165$\,cm$^{-1}$ and $\omega \approx -1.36 \times 10^{11}$\,s$^{-1}$, with the gap
$\approx 0.98 \times 10^{9}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 2(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 2(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Panel (d) presents a zoom of the anticrossing for the branches 3 and 5
at $k_{x} \approx -159.4$\,cm$^{-1}$ and $\omega \approx 2.611 \times 10^{11}$\,s$^{-1}$, with the gap
$\approx 1.37 \times 10^{6}$\,s$^{-1}$.
In Fig. 2 and below, it is used that the exact dispersion relation $\omega(k_x,d)$
of any EMP mode can be presented in the form periodic in the reciprocal space, i.e, $\omega(k_x,d)=\omega(k_x \pm 2\pi/a_{0},d)$,
and continuous across the borders of the Brillouin zone, $\omega(\pi/a_{0}-0,d)=\omega(\pi/a_{0}+0,d)$. The latter, in particular, does not allow
an infinite group velocity for the EMP. Point out that the dispersion curves 1 (solid), 2 (dashed), 3 and 4 (dotted) have correct periodic and continuous form
in the reciprocal space, $k_{x}$, and both qualitatively and quantitavely well describe dispersion of pertinent EMP modes in graphene with the superlattice:
1 and 2 are the main fundamental EMPs, 3 and 4 are the first excited fundamental EMPs.
However, an approximate dispersion curves 5 and 6 (dash-dotted) qualitatively
correctly represent pertinent exact dependencies only nearby the anticrossings of 5 with 3 and of 6 with 4.
So the curves 5 and 6 are shown only within a small part of the first Brillouin zone in Fig. 2(a). In addition, as the second
order contributions over the periodic
potential are neglected (as well as an additional contributions in Eqs. (19) with the $\ell=\pm 2$) the Fig. 2(d) gives
only rough approximation for this anticrossing and, in particular, for its gap.
In Fig. 3 we plot the dispersion relations $\omega(k_x,d=3000$nm) of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $d=3000nm$, the rest of parameters are the same as in Fig. 2.
Fig. 3(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.9 \times 10^{11}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 3(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -189$\,cm$^{-1}$ and $\omega \approx -1.17 \times 10^{11}$\,s$^{-1}$, with the gap
$\approx 0.90 \times 10^{9}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 3(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 3(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Fig. 3(d) presents a zoom of the anticrossing for the branches 3 and 5
at $k_{x} \approx -121$\,cm$^{-1}$ and $\omega \approx 2.34 \times 10^{11}$\,s$^{-1}$, with the gap
$\sim 5 \times 10^{5}$\,s$^{-1}$.
\begin{figure}[ht]
\vspace*{-0.5cm}
\includegraphics [height=11cm, width=8cm]{Fig10empgap}
\vspace*{-2.5cm}
\caption{(Color online) The dispersion relations $\omega(k_x, d=300$nm)
of the EMPs calculated from Eqs. (\ref{26d})-(\ref{31d})
for
parameters of Fig. 8, except $d=300$nm.
Panel (a) presents the dispersion relations within the first Brillouin zone.
Panel (b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -22900$\,cm$^{-1}$ and $\omega \approx -8.2 \times 10^{12}$\,s$^{-1}$, with the gap
$\approx 6.7 \times 10^{10}$\,s$^{-1}$.
Panel (c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; a finite gap is absent.
}
\end{figure}
In Fig. 4 we plot the dispersion relations $\omega(k_x,d=300$nm) of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $d=300nm$, the rest of parameters are the same as in Fig. 2.
Fig. 4(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.6 \times 10^{11}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 4(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -222$\,cm$^{-1}$ and $\omega \approx -0.90 \times 10^{11}$\,s$^{-1}$, with the gap
$\approx 7.3 \times 10^{8}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 4(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 4(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
The anticrossing for the branches 3 and 5
holds at $k_{x} \approx -56.7$\,cm$^{-1}$ and $\omega \approx 1.79 \times 10^{11}$\,s$^{-1}$.
In Fig. 5 we plot the dispersion relations $\omega(k_x,d\to \infty)$
of the EMPs calculated from Eqs. (\ref{26d})-(\ref{31d}) for
$2\pi/a_{0}=5000$\,cm$^{-1}$ by the curves 1-6. Fig. 5(a) presents the dispersion
relations of these EMPs within the first Brillouin zone. Notice, Fig. 5(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.787 \times 10^{12}$\,s$^{-1}$ at $k_{x}
\approx 0$. Fig. 5(b) presents a zoom of the anticrossing for the
branches 2 and 4, at $k_{x} \approx -1860$\,cm$^{-1}$ and $\omega
\approx -1.15 \times 10^{12}$\,s$^{-1}$, with the gap $\approx 9.0
\times 10^{9}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value
of group velocity for pertinent $k_{x}$. A panel for the
anticrossing of the branches 1 and 3 it follows from the Fig. 5(b)
by changing $k_{x}$ on $-k_{x}$ and $\omega$ on $-\omega$. Fig. 5(c)
presents a zoom of the branches 1 and 2 at $k_{x} \approx 0$ and
$\omega \approx 0$; here a finite gap is absent. Panel (d) presents
a zoom of the anticrossing for the branches 3 and 5 at $k_{x}
\approx -1112$\,cm$^{-1}$ and $\omega \approx 2.16 \times
10^{12}$\,s$^{-1}$, with the gap $\sim 5 \times 10^{6}$\,s$^{-1}$.
In Fig. 6 we plot the dispersion relations $\omega(k_x,d=3000$nm) of
the EMPs calculated from Eqs. (\ref{26d})-(\ref{31d}) for
$2\pi/a_{0}=5000$\,cm$^{-1}$. Fig. 6(a) presents the dispersion
relations of these EMPs within the first Brillouin zone by the
curves 1-6. Notice, Fig. 6(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.783 \times 10^{12}$\,s$^{-1}$ at
$k_{x} \approx 0$. Fig. 6(b) presents a zoom of the anticrossing
for the branches 2 and 4, at $k_{x} \approx
-1920$\,cm$^{-1}$ and $\omega \approx -1.13
\times 10^{12}$\,s$^{-1}$, with the gap $\approx 8.8
\times 10^{9}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value
of group velocity for pertinent $k_{x}$. A panel for the
anticrossing of the branches 1 and 3 it follows from the Fig. 6(b)
by changing $k_{x}$ on $-k_{x}$ and $\omega$ on $-\omega$. Fig. 6(c)
presents a zoom of the branches 1 and 2 at $k_{x} \approx 0$ and
$\omega \approx 0$; here a finite gap is absent. Fig. 6(d) presents
a zoom of the anticrossing for the branches 3 and 5 at $k_{x}
\approx -1081$\,cm$^{-1}$ and $\omega \approx
2.143 \times 10^{12}$\,s$^{-1}$, with the gap $\sim
3.7 \times 10^{6}$\,s$^{-1}$.
In Fig. 7 we plot the dispersion relations $\omega(k_x,d=300$nm) of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $d=300nm$, the rest of parameters are the same as in Fig. 5.
Fig. 7(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.60 \times 10^{12}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 7(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -2220$\,cm$^{-1}$ and $\omega \approx -8.95 \times 10^{11}$\,s$^{-1}$, with the gap
$\approx 7.3 \times 10^{9}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 7(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 7(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
In Fig. 8 we plot the dispersion relations $\omega(k_x,d\to \infty)$ of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $2\pi/a_{0}=50000$\,cm$^{-1}$.
Fig. 8(a) presents the dispersion relations of these EMPs within the first Brillouin zone
by the curves 1-6. Notice, Fig. 8(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.42 \times 10^{13}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 8(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -22150$\,cm$^{-1}$ and $\omega \approx -8.6 \times 10^{12}$\,s$^{-1}$, with the gap
$\approx 6.9 \times 10^{10}$\,s$^{-1}$: here the EMPs 2 and 4 have a zero value of group velocity for pertinent $k_{x}$.
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 5(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 8(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
In Fig. 9 we plot the dispersion relations $\omega(k_x,d=3000$nm) of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $d=3000$nm; in addition, $2\pi/a_{0}=50000$\,cm$^{-1}$
and other parameters coincide with those of Fig. 8.
Fig. 9(a) presents the dispersion relations of these EMPs within the first Brillouin zone
by the curves 1-6. Notice, Fig. 9(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.42 \times 10^{13}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 9(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -22150$\,cm$^{-1}$ and $\omega \approx -8.6 \times 10^{12}$\,s$^{-1}$, with the gap
$\approx 6.9 \times 10^{10}$\,s$^{-1}$: notice, the parameters of this anticrossing are very close to the ones
of similar anticrossing in Fig. 8(b).
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 9(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 9(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
In Fig. 10 we plot the dispersion relations $\omega(k_x,d=300$nm) of the EMPs
calculated from Eqs. (\ref{26d})-(\ref{31d}) for $d=300$nm; in addition, $2\pi/a_{0}=50000$\,cm$^{-1}$
and other parameters coincide with those of Fig. 8.
Fig. 10(a) presents the dispersion relations of these EMPs within the first Brillouin zone
by pertinent curves, 1-6. Notice, Fig. 10(a) shows that the branches 3 and 4 have
$\omega \approx \pm 1.41 \times 10^{13}$\,s$^{-1}$ at $k_{x} \approx 0$.
Fig. 10(b) presents a zoom of the anticrossing for the branches 2 and 4,
at $k_{x} \approx -22900$\,cm$^{-1}$ and $\omega \approx -8.2 \times 10^{12}$\,s$^{-1}$, with the gap
$\approx 6.7 \times 10^{10}$\,s$^{-1}$. Notice, the parameters of this anticrossing are close to the ones
in Fig. 8(b), in particular, in Fig. 10(b) the gap is only $3\%$ smaller than in Fig. 8(b).
A panel for the anticrossing of the branches 1 and 3 it follows from the Fig. 10(b) by changing $k_{x}$ on $-k_{x}$
and $\omega$ on $-\omega$.
Fig. 10(c) presents a zoom of the branches 1 and 2
at $k_{x} \approx 0$ and $\omega \approx 0$; here a finite gap is absent.
Notice, Fig. 10(a) shows that the anticrossing of the curves 3 and 5 takes place
at $k_{x} \approx 1730$\,cm$^{-1}$ and $\omega \approx 1.45 \times 10^{13}$\,s$^{-1}$.
It is natural to call in Figs. 2-10: the fundamental EMPs 1, 2 as the main fundamental EMPs, and
the fundamental EMPs 3, 4 as the first excited fundamental EMPs.
Figs. 2-10 show that for case (i), outlined in Fig.1, a strong Bragg coupling is possible due to a weak superlattice along
the edge, with the period $a_{0}$, if $L_{x}/a_{0} \gg 1$. In particular, for
frequencies in the THz range: cf. Figs. 8-10. We expect that for the frequency that corresponds to zero group velocity
of pertinent main fundamental EMP branch, or pertinent first excited fundamental EMP branch, and its vicinity the response of
the system will have a strong resonance.
\section{ Concluding Remarks }
At the edge of a wide armchair graphene ribbon in the $\nu=2$ QHE regime and with a smooth monotonic electrostatic potential, we
investigated the appearance of EMPs that show zero group velocity, for characteristic frequencies,
and finite frequency gaps due to effect of a weak superlattice potential. The superlattice potential
from two original fundamental EMPs (present here without superlattice \cite{balev2011}), due to a strong Bragg alike
coupling of them, gives correctly within as a whole first Brillouin zone two main fundamental EMPs
(branches 1 and 2 on Figs. 2-10) and two first excited fundamental EMPs (branches 3 and 4 on Figs. 2-10).
In addition, for the wave vector within the center of the first Brillouin zone, i.e., $k_{x} \to 0$,
only the frequencies of the main fundamental EMPs, $1$ and $2$, tend to zero as the frequencies, e.g., of the first excited
fundamental EMPs, $3$ and $4$, tend to finite values.
As at the frequency that corresponds to zero group velocity
of the main fundamental EMPs so at one for the first excited fundamental EMPs the response of
the system should have a strong resonance, e.g., in the THz range; see Figs. 8-10.
Next we list and discuss the approximations used.
Point out that in Fig. 1 (as well as in Fig. 1 of\cite{balev2011}) it is implicit that $(v_{g}(k_{x \alpha})/v_{F})^{2} \ll 1$
for any shown $y_{0}=\ell_{0}^{2}k_{x \alpha}$. Then extra Dirac points in the energy spectrum\cite{park2009,brey2009,barbier2010}
due to present smooth, weak superlattices will not appear as here it follows that $\hbar v_{F} G \gg V_{s}/2$, which is opposite to
the key condition for extra Dirac points.\cite{park2009,brey2009}
Here for the EMPs in the $\nu=2$ QHE
regime dissipation is neglected, which is well justified as here the EMP damping can be
related only with inelastic scattering processes within narrow
temperature belts, of width $k_BT$, of each edge state
that are much weaker than scattering processes
due to a static disorder.\cite{balev2011}
The latter makes a dominant contribution to the transport scattering time in a 2DES of
graphene \cite{novo,cast,vasko09} for $B=0$.
Needless to say that
for a more accurate account of the EMPs studied here, dissipation must be included in the treatment.
We have neglected by nonlocal effects that usually have minor effect on
fundamental EMPs \cite{bal99}.
We emphasize that our study of the
fundamental EMPs for the armchair termination of a graphene channel
cannot be directly extended to zigzag termination as some important properties of the wave functions and the
energy levels are different than those of the armchair termination, cf. \cite{cast,brey,aba,gus0,gus}.
We relegate the study of EMPs along zigzag edges to a future work.
It is used a simple analytical model of a smooth, lateral confining potential Eq. (\ref{eq17}), however,
our main results are robust to modifications of its form and parameters if the
qualitative conditions of Fig. 1 are realized in a graphene channel in the $\nu=2$ QHE regime.
In Figs. 2-10 it is used that the exact dispersion relation $\omega(k_x,d)$
of any EMP mode can be presented in the form periodic in the reciprocal space, i.e, $\omega(k_x,d)=\omega(k_x \pm 2\pi/a_{0},d)$,
and continuous across the borders of the Brillouin zone, $\omega(\pi/a_{0}-0,d)=\omega(\pi/a_{0}+0,d)$.
The latter, in particular, does not allow an infinite group velocity for the EMP. Point out that the dispersion
curves of first four EMP modes, 1 - 4 , have correct periodic and continuous form
in the reciprocal space, $k_{x}$, and both qualitatively and quantitavely well describe dispersion of
these EMP modes in graphene with the superlattice. However, an approximate dispersion curves 5 and 6 (dash-dotted) in Figs. 2-10
qualitatively correctly represent pertinent exact dependencies only nearby the anticrossings of 5 with 3 and of 6 with 4.
So the curves 5 and 6 are shown only within a small part of the first Brillouin zone in Figs. 2(a)-10(a).
In addition, as the second order contributions over the periodic
potential are neglected (as well as an additional contributions in Eqs. (\ref{19d}) with the $\ell=\pm 2$) the
Figs. 2(d), 3(d), 5(d), 6(d) give only rough approximation for this anticrossing and, in particular, for its gap.
\begin{acknowledgments}
O. G. B. acknowledges support by Brazilian FAPEAM
(Funda\c{c}\~{a}o de Amparo \`{a} Pesquisa do Estado do Amazonas)
Grant and by the Brazilian Council for Research (CNPq)
APV Grant No. 452849/2009-8. A. C. A. Ramos thanks the FUNCAP (Funda\c{c}\~{a}o Cearense de Apoio
ao Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico) for financial
support.
\end{acknowledgments}
|
{
"timestamp": "2012-03-13T01:01:32",
"yymm": "1203",
"arxiv_id": "1203.2284",
"language": "en",
"url": "https://arxiv.org/abs/1203.2284"
}
|
\chapter*{Statements of Contributions}
\textbf{Declaration by author}
This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.
I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award.
I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the General Award Rules of The University of Queensland, immediately made available for research and study in accordance with the \textit{Copyright Act 1968}.
I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material.
\textbf{Statement of Contributions to Jointly Authored Works Contained in this Thesis}
This thesis is partly by publication. It contains the following publications, which I have co-authored as the first author, directly as Chapters 2,3 and 4.
\citep{Singh10c} - Incorporated as \textit{Chapter 2}. The details of the SU(2) symmetric iTEBD algorithm were developed by myself and Prof. Guifre Vidal. The implementation of the algorithm in MATLAB and numerical simulations was performed by myself. Most of the manuscript was prepared by Prof. Guifre Vidal. Section 5, which describes the details of the symmetric iTEBD algorithm, and Figs. 3 and 4 were prepared by myself. Prof. Huan-Zhou Qiang was responsible for clarifying several results from the representation theory of SU(2) that were crucial to the development of the algorithm. He also pointed out the exact results that were used in Fig.~4 to draw a comparison with the numerical results.
\citep{Singh10b} - Incorporated as \textit{Chapter 3}. The theoretical formalism presented in this paper was developed mostly by myself and Prof. Guifre Vidal. Robert~N.~C.~Pfeifer joined the effort at an advanced stage of the project and contributed ideas that were important for the final presentation of the formalism. The manuscript was mostly written by Prof. Guifre Vidal, in close correspondence with myself and Robert~N.~C.~Pfeifer. Implementation in MATLAB and numerical simulations were performed by me. However, these results were not included in the final draft of the manuscript.
\citep{Singh11} - Incorporated as \textit{Chapter 4}. The theoretical ideas presented in this paper were developed mostly by myself and Prof. Guifre Vidal. The MATLAB implementation and numerical results reported in the manuscript were performed by myself. Robert~N.~C.~Pfeifer also independently and simultaneously coded an implementation in MATLAB that was an important support for the validity of the ideas presented in the paper. Myself and Robert~N.~C.~Pfeifer jointly proposed a precomputation scheme that was used in both the implementations. The main structure and content of the paper, including all the figures, was prepared by myself in close supervision of Prof. Guifre Vidal. The final draft was critically revised by Prof. Guifre Vidal. Robert. N.C. Pfeifer also contributed to the corrections of the figures and the manuscript at a pre-submission stage, and assisted in writing the Appendix.
\textbf{Statement of Contributions by Others to the Thesis as a Whole}
The overall motivation and research direction of this thesis was provided by Prof. Guifre Vidal.
\newpage
\textbf{Statement of Parts of the Thesis Submitted to Qualify for the Award of Another Degree}
None.
\textbf{Published Works by the Author Incorporated into the Thesis}
\citep{Singh10c} - Incorporated as Chapter 2.
\citep{Singh10b} - Incorporated as Chapter 3.
\citep{Singh11} - Incorporated as Chapter 4.
\textbf{Additional Published Works by the Author Relevant to the Thesis but not Forming Part of it}
None.
\chapter*{Acknowledgments}
The completion of this thesis would have been impossible without the tireless supervision and wise guidance of Guifre Vidal. From him I have learnt numerous practical aspects of both research and life. For that my gratitude remains well beyond words.
I thank all my fellow research group members: Philippe Corboz, Glen Evenbly, Andy Ferris, Jacob Jordan, Ian McCulloch, Roman Orus, Robert Pfeifer and Luca Tagliacozzo, for providing me with an intellectually rich and competitive research environment. I also acknowledge the support of Huan-Qiang Zhou, whose research group I visited twice during the completion of this work. I thank him for always putting pulpable enthusiam into physics and physics into words.
My personal perseverance was fueled by my parents who have made me what I am today, and by Meru, my dear friend, who has been like my shadow in the last few years, especially during the strenuous times of this project.
Finally, I would like to make a special mention of the blessings of my grandsire figure, Sant Baba Pritam Das Ji, who forever nourishes my spirit.
\
\chapter*{Abstract}
Understanding and classifying phases of matter is a vast and important area of research in modern physics. Of special interest are phases at low temperatures where quantum effects are dominant. Theoretical progress is thwarted by a general lack of analytical solutions for quantum many-body systems. Moreover, perturbation theory is often inadequate in the strongly interacting regime. As a result, numerical approaches have become an indispensable tool to address such problems. In recent times, numerical approaches based on tensor networks have caught widespread attention. Tensor network algorithms draw on insights from Quantum Information theory to take advantage of special entanglement properties of low energy quantum many-body states of lattice models. Examples of popular tensor networks include Matrix Product States, Tree tensor Network, Multi-scale Entanglement Renormalization Ansatz and Projected Entangled Pair States. The main impediment of these methods comes from the fact that they can only represent states with a limited amount of entanglement. On the other hand, exploitation of symmetries, a powerful asset for numerical methods, has remained largely unexplored for a broad class of tensor networks algorithms.
In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically \textit{protect} the symmetry and \textit{exploit} it for computational gain in tensor network simulations. Our formalism is generic. It can readily be adapted to specific tensor network representations and to a wide spectrum of physical symmetries. The latter includes conservation of total particle number (U(1) symmetry) and of total angular momentum (SU(2) symmetry), and also more exotic symmetries (anyonic systems). The generality of the formalism is due to the fact that the symmetry constraints are imposed at the level of individual tensors, in a way that is independent of the details of the tensor network. As a result, we are led to a framework of symmetric tensors. Such tensors are then used as building blocks for tensor network representations of quantum-many states in the presence of symmetry.
For a long time several physical problems of immense interest have remained elusive to numerical methods mostly owing to extremely high simulation costs. These include systems of frustrated magnets and interacting fermions that are relevant in the context of quantum magnetism and high temperature superconductivity. With symmetry now as a potent ally, tensor network algorithms may finally be used to draw positive insights about such systems.
\newpage
\textbf{Keywords:} tensor networks, symmetry, U(1), SU(2), MERA, MPS, spin networks, anyons
\textbf{Australian and New Zealand Standard Research Classifications (ANZSRC):}
\\\p{0}020401 Condensed Matter Characterisation Technique Development (50\%)
\\\p{0}020603 Quantum Information, Computation and Communication (50\%).
\afterpreface
\chapter{Introduction\label{sec:introduction}}
The study of quantum many-body phenomena is of pivotal interest in modern physics. Important areas of research, such as the characterization of exotic phases of quantum matter and of quantum phase transitions\citep{Sachdev}, or even the possible realization\citep{Kitaev03, Nayak08} of a quantum computer, rely on our understanding of collective phenomena in quantum many-body systems. Theoretical progress in these research areas is hindered both by a general lack of analytical results as well as the inadequacy of perturbation theory in the strongly interacting regime where the interesting physics often lies. As a result, development of numerical approaches to probe such systems has become a flourishing research industry. However, numerical methods are limited by staggering computational costs.
The number of parameters required to describe a generic quantum many-body wavefunction on a lattice grows exponentially with the number of sites in the lattice. An immediate consequence is that exact diagonalization can only be applied to small systems. In particular, thermodynamic properties often remain inaccessible by this method. Conventional alternatives include Quantum Monte Carlo sampling techniques. These are well established numerical methods that have been used extensively in several areas of Mathematics and Physics. Of interest here is that these techniques have been applied\citep{Prokofev98, Evertz03, Syljuasen02, Sandvik05} successfully to several quantum lattice models. On the other hand, Quantum Monte Carlo techniques suffer from the notorious sign problem\citep{Loh90, Henelius00} that hinders their application to certain systems of immense interest. Notable examples include systems of frustrated magnets and of interacting fermions that are relevant in the context of quantum magnetism and high temperature superconductivity\citep{Anderson87}.
In recent years, new approaches based on tensor networks have caught widespread attention. Such approaches can be regarded as generalizations of the density matrix renormalization group (DMRG) method\citep{White92, White93, Schollwock05, McCulloch08}, which is highly successful for one-dimensional systems. The potential of tensor network algorithms relies on the fact that, as DMRG, they can address systems of frustrated spins and interacting fermions but, unlike DMRG, they can also be applied to two dimensional systems, both of large size and of infinite size. The main impediment of such methods comes from the fact that simulation costs increase rapidly with the amount of entanglement in the system. Consequently, tensor networks can only represent states with a limited amount of entanglement. On the other hand, exploitation of symmetries has remained largely unexplored for a broad class of tensor networks algorithms.
Symmetries, of fundamental importance in physics, require special treatment in numerical studies. Unless explicitly preserved at the algorithmic level, they are bound to be destroyed by the accumulation of small errors, in which case significant features of the system might be concealed. On the other hand, when properly handled, the presence of a symmetry can be exploited to reduce simulation costs.
The goal of this thesis is to extend the tensor network formalism to the presence of symmetries. We develop a generic framework that can be applied to adapt any given tensor network representation and algorithm to both numerically protect symmetries and exploit them for computational gain.
\section{Tensor network states and algorithms}
Tensor networks are an efficient parameterization of low energy quantum many-body states of lattice models. The degrees of freedom of the model are arranged on a lattice $\mathcal{L}$ made of $L$ sites where each site is described by a Hilbert space of dimension $d$. As a result, the Hilbert space dimension of $\mathcal{L}$ grows exponentially with the number of sites $L$. Thus, a generic quantum many-body state on the lattice is parameterized by exponentially many parameters. On the other hand, the dynamics of the system are typically governed by a \textit{local} Hamiltonian $\hat{H}$, that is, $\hat{H}$ decomposes as the sum of terms involving only a small number of sites, and whose strength decays with the distance between the sites. The locality of the dynamics often implies that only a relatively small amount of entanglement is present in the ground state. In such circumstances, tensor networks offer a good description of the ground state. Moreover, the description is efficient, in that the total number of parameters encoded into the tensor networks grows roughly linearly with $L$.
Examples of tensor network states for one dimensional systems include the matrix product state\citep{Fannes92,Ostlund95,Perez-Garcia07} (MPS), which results naturally from both Wilson's numerical renormalization group\citep{Wilson75} and White's DMRG and is also used as a basis for simulation of time evolution, e.g. with the time evolving block decimation (TEBD)\citep{Vidal03,Vidal04,Vidal07} algorithm and variations thereof, often collectively referred to as time-dependent DMRG\citep{Vidal03,Vidal04,Daley04,White04,Schollwock05b,Vidal07}; the tree tensor network\citep{Shi06} (TTN), which follows from coarse-graining schemes where the spins are blocked hierarchically; and the multi-scale entanglement renormalization ansatz\citep{Vidal07b, Vidal08, Evenbly09, Giovannetti08, Pfeifer09, Vidal10} (MERA), which results from a renormalization group procedure known as entanglement renormalization\citep{Vidal07b,Vidal10}. For two dimensional lattices there are generalizations of these three tensor network states, namely projected entangled pair states\citep{Verstraete04, Sierra98, Nishino98, Nishio04, Murg07, Jordan08, Gu08, Jiang08, Xie09, Murg09} (PEPS), 2D TTN,\citep{Tagliacozzo09, Murg10} and 2D MERA\citep{Evenbly10, Evenbly10b, Aguado08, Cincio08, Evenbly09b, Konig09} respectively. As variational ans\"atze, PEPS and 2D MERA are particularly interesting since they can be used to address large two-dimensional lattices, including systems of frustrated spins\citep{Murg09, Evenbly10} and interacting fermions,\citep{Corboz09, Kraus10, Pineda10, Corboz10, Barthel09, Shi09, Corboz10b, Pizorn10, Gu10} where Monte Carlo techniques fail due to the sign problem.
Some popular tensor networks are summarized in table \ref{table:tn}.
\begin{table}
\centering
\begin{tabular}{|c| c|}
\hline
\textit{One dimensional} & \textit{Two dimensional} \\ [0.5ex]
\hline\hline
MPS & PEPS \\ \hline
1D TTN & 2D TTN \\ \hline
1D MERA & 2D MERA \\
[1ex
\hline
\end{tabular}
\caption{
Popular tensor networks in one and two spatial dimensions.\label{table:tn}
}
\end{table}
\section{Symmetries}
The presence of symmetries is a universal trait of physical theories. Symmetry has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. The importance of symmetries in physical theories was firmly grounded by the famous \textit{Noether's theorem}, a rigorous result that links the presence of a symmetry to the conservation of a physical quantity.
In this thesis we will be concerned with symmetries exhibited in quantum lattice models. The many-body Hamiltonian $\hat H$ may be invariant under certain transformations, which form a group $\mathcal{G}$ of symmetries\citep{Cornwell97}. Under the action of the symmetry transformation, the Hilbert space of the theory is divided into symmetry sectors labeled by quantum numbers or conserved charges. The symmetry group $\mathcal{G}$ may be \textit{Abelian} or \textit{non-Abelian}, depending on whether or not the total effect of applying two symmetry transformations depends on the order in which the transformations are applied. The symmetry sectors associated with an Abelian symmetry correspond to one-dimensional invariant subspaces. In contrast, the dimension of the symmetry sectors associated with a non-Abelian symmetry may be larger than one.
On the lattice, one can distinguish between \textit{space} symmetries, which correspond to some permutation of the sites of the lattice, and \textit{internal} symmetries, which act on the vector space of each site. An example of space symmetry is invariance under translations by some unit cell, which leads to conservation of quasi-momentum. An example of internal symmetry is SU(2) invariance, e.g. spin isotropy in a quantum spin model. An internal symmetry can in turn be \textit{global}, if it transforms the space of each of the lattice sites according to the same transformation (e.g. a spin independent rotation); or \textit{local}, if each lattice site is transformed according to a different transformation (e.g. a spin-dependent rotation), as it is in the case of lattice gauge models. A global internal SU(2) symmetry gives rise to conservation of total spin. Table \ref{table:symmetry} lists some examples of Abelian and non-Abelian physical symmetries.
\begin{table}
\centering
\begin{tabular}{|c| c| c|}
\hline
\textit{Conserved physical quantity} & \textit{Symmetry group} & \textit{Abelian/non-Abelian}\\ [0.5ex]
\hline\hline
Parity of particle number & $Z_2$ & Abelian\\ \hline
Particle number, spin projection & U(1) & Abelian\\ \hline
Total angular momentum, total spin & SU(2)& non-Abelian\\ \hline
Particle number and spin & U(1) $\times$ SU(2)& non-Abelian \\ \hline
Spin and isospin & SU(2) $\times$ SU(2)& non-Abelian \\ \hline
Total anyonic charge & e.g. $\mbox{SU(2)}_k$ & e.g.,non-Abelian\\
[1ex
\hline
\end{tabular}
\caption{
Examples of global internal physical symmetries.\label{table:symmetry}
}
\end{table}
By targeting a specific symmetry sector during a calculation, computational costs can often be significantly reduced while explicitly preserving the symmetry. It is therefore not surprising that symmetries play an important role in numerical approaches.
\section{Incorporating symmetries into tensor network algorithms}
Exploiting symmetries has been of great interest in numerical approaches, since it allows selection of a specific charge sector within the kinematic Hilbert space, and leads to significant reduction of computational costs.
In the context of tensor network algorithms, benefits of exploiting the symmetry have been extensively demonstrated especially in the context of MPS. Both \textit{space} and \textit{internal} symmetries, Abelian and non-Abelian, have been thoroughly incorporated into DMRG code and have been exploited to obtain computational gains\citep{Ostlund95,White92,Schollwock05b,Ramasesha96,Sierra97,Tatsuaki00,McCulloch02,Bergkvist06,Pittel06,McCulloch07,Perez-Garcia08,Sanz09}.
Symmetries have also been used in more recent proposals to simulate time evolution with MPS\citep{Vidal04,Daley04,White04,Schollwock05b,Vidal07,Daley05,Danshita07,Muth10,Mishmash09,Singh10c,Cai10}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{mpscompare}
\end{center}
\caption{ Computational gain obtained by exploiting the symmetry in an MPS algorithm. Computation time (in seconds) for one iteration of the infinite TEBD algorithm, as a function of the MPS bond dimension $\chi$ is shown. Here $\chi$ is a refinement parameter, a larger $\chi$ leads to a better accuracy of the method. For sufficiently large $\chi$, exploiting symmetry leads to reductions in computation time. The horizontal line on this graph shows that this reduction in computation time equates to the ability to evaluate MPSs with a higher bond dimension $\chi$: For the same cost per iteration incurred when optimizing a regular MPS in MATLAB with bond dimension $\chi=220$, one may choose instead to optimize a U(1)-symmetric MPS with $\chi=380$ or an SU(2)-symmetric MPS with $\chi=1300$. \label{fig:mpscompare}}
\end{figure}
Figure~\ref{fig:mpscompare} is demonstrative of the colossal computational gain that has been obtained by exploiting the symmetry in the context of the MPS. (In Fig.~\ref{fig:meracompare} we show an analogous comparison for exploiting symmetries in the context of the MERA.)
However, when considering symmetries, it is important to notice that an MPS is a trivalent tensor network. That is, in an MPS each tensor has at most three indices. The Clebsch-Gordan coefficients\citep{Cornwell97} (or coupling coefficients) of a symmetry group are also trivalent, and this makes incorporating the symmetry into an MPS by considering symmetric tensors particularly simple. In contrast, tensor network states with a more elaborate network of tensors, such as MERA or PEPS, consist of tensors having a larger number of indices. In this case a more general formalism is required in order to exploit the symmetry.
In this thesis we will describe how to incorporate a global internal symmetry, given by a compact and reducible group $\mathcal{G}$, into tensor network algorithms. We will develop a generic strategy that is independent of the details of the underlying tensor network. We will do this by imposing the symmetry constraints at the level of individual tensors that constitute the tensor network. We will then also describe how symmetric tensors are manipulated such that the symmetry is both \textit{preserved} and \textit{exploited} for computational gain. Having built a framework of symmetric tensors, we will adapt an arbitrary tensor network to the presence of symmetry by using symmetric tensors as building blocks for the tensor network. The resulting tensor network represents a class of quantum many-body wavefunctions that are invariant (or more generally covariant) under the symmetry transformation. Algorithms based on such symmetric tensor networks will also be adapted to the presence of symmetry. This will be achieved by expressing each step of an algorithm in terms of symmetric manipulations of the tensors.
As a concrete illustration, we will extensively describe the implementation of U(1) and SU(2) symmetries into the MPS and MERA. With these implementations at hand, we will demonstrate the colossal benefits of incorporating the symmetry into tensor network algorithms. These include addressing specific symmetry sectors of the Hilbert space, compactification of the tensor network representation and computational speedup in numerical simulations. For example, in a lattice spin model endowed with spin isotropy the ground state is constrained to the spin zero or singlet sector of the Hilbert space. Therefore, in a numerical probing for the state, it is sufficient to restrict attention to the singlet subspace. This can, in turn, potentially result in a substantial reduction of computational costs.
\section{Plan of the thesis}
This thesis is comprised of three published papers corresponding to chapters 2,3 and 4 and additional chapters 5 and 6. The material in chapters 5 and 6 was under preparation for publication at the time of submitting this thesis. The following is a brief summary of all the chapters.
In Chapter 2 we dive straight into the core of the problem. We describe the implementation of a non-Abelian symmetry for the case of the simplest tensor network: the MPS. This will serve to illustrate the key points that are required to be considered when implementing symmetries into tensor network algorithms. In addition, this chapter also demonstrates the benefits of exploiting symmetries in the case of MPS algorithms. We adapt the infinite time evolving block decimation (iTEBD) algorithm to the presence of a global SU(2) symmetry. This is of interest in its own right since the iTEBD algorithm has been immensely successful in simulations of infinite 1D quantum many-body systems. This is also the first implementation of a non-Abelian symmetry into the iTEBD algorithm and has resulted in a significant enhancement of this algorithm.
In Chapter 3 we go beyond the class of MPS algorithms. We describe the general strategy to incorporate a wide spectrum of symmetries into more complex tensor network states and algorithms. We consider tensor networks made of symmetric tensors, that is, tensors that are invariant under the action of the symmetry. We develop a formalism to characterize and manipulate symmetric tensors.
In Chapter 4 we implement the general formalism for the case of an Abelian symmetry. The implementation of an Abelian symmetry is simplified by the fact that symmetric tensors are easier to characterize. In a basis labeled by the charges of the symmetry, a symmetric tensor has a sparse block structure. We explain how this block structure can be exploited for computational gain in a practical implementation of the symmetry. The benefits of exploiting the symmetry are numerically demonstrated by exploiting U(1) symmetry in the context of the MERA.
In Chapters 5 and 6 we describe the implementation of a non-Abelian symmetry. The details of implementing the symmetry are more involved, since the structural tensors are highly non-trivial. However, the computational gain that results from exploiting a non-Abelian symmetry is significantly larger than that obtained by exploiting an Abelian symmetry. Moreover, the practical scheme presented to implement a non-Abelian symmetry can be readily extended to incorporate more exotic symmetry constraints such as those corresponding to the presence of anyonic degrees of freedom. The benefits of exploiting a non-Abelian symmetry are numerically demonstrated by means of our implementation of SU(2) symmetry in the context of the MERA.
Finally, in Chapter 7 we draw conclusions and discuss potential applications and future directions of this work.
\textbf{Note on References:} In addition to the references listed at the end of the thesis, chapter wise references appear at the end of chapters 2, 3 and 4 that correspond to published papers.
\chapter{Exploiting symmetries in MPS algorithms: An Example\label{sec:ch2}}
We kick start the main discussion of the thesis by describing how to incorporate an SU(2) symmetry into a specific MPS algorithm: the iTEBD algorithm. The iTEBD algorithm has been immensely successful in simulations of infinite 1D quantum many-body systems.
We follow a straightforward implementation of the symmetry into the algorithm. We consider an MPS that is made of trivalent SU(2) invariant tensors. A trivalent SU(2) invariant tensor decomposes into two pieces. One piece contains the degrees of freedom whereas the other corresponds to the Clebsch-Gordan coefficients of SU(2). We describe how the iTEBD algorithm can be enhanced by exploiting this decomposition of the MPS tensors. The resulting symmetric algorithm is obtained to be 300 times faster (See Fig.~\ref{fig:mpscompare}). We use the symmetric algorithm for a numerical study of a critical quantum spin chain.
This chapter serves to illustrate the main conceptual ingredients that are required to incorporate symmetries into tensor network algorithms. In the next chapter, we will generalize these ingredients by going beyond the specific details of the SU(2) symmetry group and the MPS representation.
\includepdf[pages={2-13}]{su2mps.pdf}
\chapter{Tensor networks and symmetries: Theoretical formalism\label{sec:ch3}}
In this chapter we develop a generic theoretical formalism to incorporate a symmetry into tensor network algorithms. We consider a wide class of symmetries that are described by a compact and reducible group $\mathcal{G}$ that is multiplicity free, that is, the tensor product of two charges of the group does not contain multiple copies of a charge. Our strategy revolves around tensors that are invariant under the action of the symmetry. As a result, we formulate a framework of symmetric tensors.
A symmetric tensor transforms covariantly (or remains invariant) under the action of the symmetry. In a basis labeled by the symmetry charges, the tensor decomposes into a set of \textit{degeneracy tensors} and \textit{structural tensors}. While the degeneracy tensors contain the degrees of freedom, the components of the structural tensors are generalizations of the coupling coefficients of the group, and are determined completely by the symmetry. Moreover, any symmetry preserving manipulation of the tensor can be performed in parts. For instance, a permutation of the indices of a symmetric tensor breaks into the permutation of the corresponding degeneracy indices and the permutation of the corresponding structural indices. Therefore, this \textit{canonical decomposition} of a symmetric tensor allows for both a compact description of the tensor and a computational speedup in numerical manipulations of it.
We also point out a numerical connection to the formalism of \textit{spin networks}\citep{Penrose71, Major99}. A spin network is a mathematical object that appears, for example, in Loop Quantum Gravity\citep{Rovelli98}, where it is used\citep{Rovelli95} to facilitate a description of quantum spacetime. In our formalism, a tensor network made of symmetric tensors decomposes into a linear superposition of spin networks. Also, manipulating a symmetric tensor network requires \textit{evaluating} a spin network. Thus, our work highlights the importance of spin networks in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.
\includepdf[pages={1-4}]{symTNpaper.pdf}
\section{Errata}
The following equations appear erroneously in the publication. They are to be corrected as follows.
The tensors $P$ and $Q$ that appear in Eqs.11 and 12 do not carry degeneracy indices and spin indices corresponding to the coupled charges $e$ and $f$, since these indices are summed over in the description.
In Eq.~11 the components of $P$ and $Q$ read as $(P^{abcd}_e)_{\alpha_a \beta_b \gamma_c \delta_d}$ and $(Q^{abcd}_e)_{m_a n_b o_c p_d}$ respectively and the sum is only over different values of charge $e$. The corrected equation reads,
\begin{equation}
(T)_{ijkl} = \sum_e (P^{abcd}_e)_{\alpha_a \beta_b \gamma_c \delta_d} (Q^{abcd}_e)_{m_a n_b o_c p_d},\nonumber ~~~~~~~~~~~~~~~~~~(11)
\end{equation}
In Eq.~12 the components of $P$ and $Q$ read as $(\tilde{P}^{abcd}_f)_{\alpha_a \beta_b \gamma_c \delta_d}$ and $(\tilde{Q}^{abcd}_f)_{m_a n_b o_c p_d}$ respectively and the sum is only over different values of charge $f$. The corrected equation reads,
\begin{equation}
(T)_{ijkl} = \sum_f (\tilde{P}^{abcd}_f)_{\alpha_a \beta_b \gamma_c \delta_d} (\tilde{Q}^{abcd}_f)_{m_a n_b o_c p_d},\nonumber ~~~~~~~~~~~~~~~~~~(12)
\end{equation}
A similar correction holds for Eq.~15 which is a generalization of Eqs.11 and 12. The sum is only over different values of the intermediate charges $e_1 \ldots e_{t'}$. The corrected equation reads,
\begin{equation}
(T)_{i_1i_2 \ldots i_t} = \sum_{e_1 \ldots e_{t'}}(P^{a_1 \ldots a_t}_{e_1 \ldots e_{t'}})_{\alpha_{a_1} \ldots \alpha_{a_{t'}}} (Q^{a_1 \ldots a_t}_{e_1 \ldots e_{t'}})_{m{a_1} \ldots m_{a_{t'}}},\nonumber ~~~~~~~~~(15)
\end{equation}
\chapter{Implementation of Abelian symmetries\label{sec:ch4}}
In this chapter we specialize the general formalism to the case of Abelian symmetries. Abelian symmetries appear frequently in the context of lattice models with particles (bosons or fermions) as well as those with spins. In the former, they include particle number conservation and parity conservation, whereas in the latter they appear as conservation of spin projection.
The analysis of exploiting an Abelian symmetry is made simple by the fact that the structural tensors in this case are trivial. On the other hand, an implementation of an Abelian symmetry serves to expose the practical difficulties that are encountered when incorporating symmetries into complicated tensor networks. We pay special attention to such implementation level concerns. Certain operations in the algorithm depend only on the symmetry and not on the components of the tensors involved. We exploit this fact to \textit{precompute} the output of such operations and store their result in memory. This is particularly advantageous in an \textit{iterative} algorithm where tensor components are updated or optimized by repeating a set of computations. The runtime cost of the iterative algorithm can be significantly reduced by reusing the precomputed results from memory. By making use of precomputation we obtained a substantial computational gain from exploiting the symmetry in our \textsc{MATLAB} implementation. However, this was achieved at the expense of storing potentially large amounts of precomputed data.
The discussion is conducted in the specific context of U(1) symmetry associated, for example, with conservation of particle number or of spin projection. We describe how to implement elementary tensor manipulations such as permutation and reshape of indices in a U(1) symmetric way. We also present a concrete implementation of the U(1) symmetry in the context of the MERA. We consider a MERA that is made of U(1) symmetric tensors. Then using the U(1) MERA we demonstrate the benefits of including symmetries into tensor networks.
\includepdf[pages={1-22}]{u1.pdf}
\chapter{Implementation of non-Abelian symmetries \Rmnum{1} \label{sec:ch5}}
In this and the following chapter we will address the implementation of global non-Abelian symmetries into tensor network algorithms. We consider the specific context of an internal SU(2) symmetry, that gives rise to spin isotropy. This is an extremely important symmetry that appears amply in lattice spin models.
In this chapter we will focus on the conceptual aspects of incorporating the symmetry. The theoretic formalism developed in Chapter 3 will be adapted to the specific case of SU(2) symmetry. We consider tensors that are invariant under the action of SU(2). The structural tensors, that are part of the canonical decomposition, are highly non-trivial (and are given in terms of the Clebsch-Gordan coefficients). However, the key advantage of the canonical decomposition is that it allows tensor manipulations, such as reshape or permutation of indices, to be broken into an independent manipulation of degeneracy tensors and of structural tensors.
In the context of numerical simulations the canonical decomposition leads to a computational gain. Computational cost is incurred only when manipulating degeneracy tensors. One the other hand, structural tensors are manipulated \textit{algebraically} by exploiting properties of the Clebsch-Gordan coefficients. Manipulations of the structural tensors reduce to evaluating a spin network. This process involves integrating over the degrees of freedom associated with spin projection, which are therefore suppressed in the outcome of the tensor manipulation, as is expected in a spin isotropic description.
In Chapter 6 we will describe a specific implementation scheme of elementary manipulations of SU(2)-invariant tensors. We will address several concerns that are of importance in the practical implementations of the symmetry.
\section{Preparatory Review: Tensor network formalism \label{sec:tensornetwork}}
In this section we review the basic formalism of tensors and tensor networks. Even though we do not make any explicit reference to symmetry here, our formalism is directed towards SU(2)-invariant tensors.
We begin by recalling the basic notion of a tensor. A \textit{tensor} $\hat{T}$ is a multidimensional array of complex numbers $\hat{T}_{i_1i_2 \ldots i_k}$. The \textit{rank} of a tensor is the number $k$ of indices. The \textit{size} of an index $i$, denoted $|i|$, is the number of values that the index takes, $i \in {1, 2, \ldots, |i|}$. The \textit{size} of a tensor $\hat{T}$, denoted $|\hat{T}|$, is the number of complex numbers it contains, namely, $|\hat{T}|~=~|i_1|~\times~|i_2|~\times~\ldots~\times~|i_k|$.
\subsection{Tensors as linear maps \label{sec:tensornetwork:tensors}}
For the purpose of this thesis we regard a rank-$k$ tensor as a linear map. To this end, we first equip each index $i_l,~l=1,2,\ldots,k$, of the tensor with a direction: `in' or `out', that is, either incoming into the tensor or outgoing from the tensor respectively. We denote by $\vec{D}$ the directions associated with the indices of tensor $\hat{T}$, namely, $\vec{D}(l) =$ `in' if $i_l$ is an incoming index and $\vec{D}(l) =$ `out' if $i_l$ is outgoing.
Let us also use index $i$ of the tensor to label a basis $\ket{i}$ of a complex vector space $\mathbb{V}^{[i]} \cong \mathbb{C}^{|i|}$ of dimension $|i|$. Then a rank-one ($k = 1$) tensor with an outgoing index $i$ represents a vector in $\mathbb{V}^{[i]}$, a rank-two ($k = 2$) tensor $\hat{T}$ with one incoming index $a$ and one outgoing index $b$ represents a matrix and so on.
A tensor can be unambiguously regarded as a linear map from a vector space to complex numbers $\mathbb{C}$. For instance, a vector can be regarded as a linear map from $\mathbb{V}^{[i]}$ to $\mathbb{C}$, a matrix $\hat{T}$ can be regarded as a linear map from $(\mathbb{V}^{[a]})^* \otimes \mathbb{V}^{[b]}$ to $\mathbb{C}$ where $(\mathbb{V}^{[a]})^*$ is the dual of vector space $\mathbb{V}^{[a]}$ etc. More generally, we can use a rank-$k$ tensor $\hat{T}$ to define a linear map from the tensor product of $k$ vector spaces to $\mathbb{C}$ in the following way. Define a set $\mathbb{W}^{[i_l]},~l=1,2,\ldots,k$, of $k$ spaces where
\begin{equation}
\mathbb{W}^{[i_l]} = \left\{
\begin{array}{cc} \mathbb{V}^{[i_l]} &\mbox{ if } \vec{D}(l)=\mbox{`out'},\\
(\mathbb{V}^{[i_l]})^* &\mbox{if } \vec{D}(l)=\mbox{`in'},
\end{array} \right.
\end{equation}
where the $(\mathbb{V}^{[i_l]})^*$ is the dual of vector space $\mathbb{V}^{[i_l]}$. Then tensor $\hat{T}$ can be regarded as a linear map from the product space $\bigotimes_l \mathbb{W}^{[i_l]}$ to $\mathbb{C}$,
\begin{equation}
\hat{T} : \bigotimes_l \mathbb{W}^{[i_l]} \rightarrow \mathbb{C}. \label{eq:tensormap}
\end{equation}
We will find this viewpoint of tensors useful for subsequent generalization to SU(2)-invariant tensors.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{tensor}
\end{center}
\caption{ Graphical representation of tensors by means of a shape e.g. circle or a ``blob'' with emanating lines corresponding to indices of the tensor. Indices may be incoming or outgoing as indicated by arrows. (i) Graphical representations of tensors with rank $0,1$ and $2$, corresponding to a complex number $c \in \mathbb{C}$, a vector $\ket{v} \in \mathbb{C}^{[i]}$ and a matrix $\hat{M} \in \mathbb{C}^{|a|\times|b|}$. (ii) Graphical representation of a tensor $\hat{T}$ with components $\hat{T}_{abcd}$ with directions $\vec{D} = \{\mbox{`in', `out', `in', `out'}\}$. Notice that the indices emerge in a counterclockwise order, perpendicular to the boundary of the blob; the first index is identified by the presence of by a dot within the blob close to the index. All arrows in a tensor diagram point downwards, that may be interpreted as choosing a metaphorical ``time'' that flows downwards.\label{fig:tensor}}
\end{figure}
It is convenient to use a graphical representation of tensors, as illustrated in Fig. 1, where a tensor is depicted as a ``blob'' (or by a shape e.g., circle, square etc.) and each of its indices is represented by a line emerging \textit{perpendicular} from the boundary of the blob. In order to specify which index corresponds to which emerging line, we follow the prescription that the lines corresponding to indices $\{i_1, i_2, \ldots , i_k\}$ emerge in counterclockwise order. The first index corresponds to the line emerging closest to a mark (black dot) inside the boundary of the blob (or the first line encountered while proceeding counterclockwise from nine o'clock in case the tensor is depicted as a circle without a mark). The direction of an index is depicted by attaching an arrow to the line corresponding to the index. We follow a convention that all arrows in a diagram point downwards.
\subsection{Elementary manipulations of a tensor\label{sec:tensor:manipulations}}
A tensor can be transformed into another tensor in several elementary ways. These include, \textit{reversing} the direction of one or several of its indices, \textit{permuting} its indices, and/or \textit{reshaping} its indices.
\textit{Reversing} the direction of an index corresponds to mapping the vector space that is associated with the index to its dual. For example, in
\begin{equation}
(\hat{T}')_{\overline{a}b} = \hat{T}_{ab},
\label{eq:bend}
\end{equation}
if index $a$ is associated to a vector space $\mathbb{V}^{[a]}$, then index $\overline{a}$ that is obtained by reversing the direction of $a$ is associated with the dual space $(\mathbb{V}^{[a]})^*$. Since all arrows in a diagram point downwards, reversing the direction of an index $i$ is depicted [Fig.~\ref{fig:tensorman}(i)] by `bending' the line corresponding to $i$ upwards if it is an outgoing index or downwards if it is an incoming index. Since tensor $\hat{T}'$ is components wise equal to tensor $\hat{T}$ arrows appear to be irrelevant in the absence of the symmetry. However, arrows will play an important role when we consider SU(2)-invariant tensors since they specify how the group acts on each index of a given tensor.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{tensorman}
\end{center}
\caption{Transformations of a tensor: (i) Reversing direction of an index, Eq.~(\ref{eq:bend}). $(ii)$ Permutation of indices, Eq.~(\ref{eq:permute}). (ii) Fusion of indices $a$ and $b$ into $d = a \times b$, Eq.~(\ref{eq:fuse}); splitting of index $d=a \times b$ into $a$ and $b$, Eq.~(\ref{eq:split}).\label{fig:tensorman}}
\end{figure}
A \textit{permutation} of indices corresponds to creating a new tensor $\hat{T}'$ from $\hat{T}$ by simply changing the order in which the indices appear, e.g.
\begin{equation}
(\hat{T}')_{bac} = \hat{T}_{abc}.
\label{eq:permute}
\end{equation}
Permutation of indices is depicted by intercrossing indices, as illustrated in Fig.~\ref{fig:tensorman}(ii). Note that when the permutation involves the first index of the tensor the mark, that indicates the first index, is also shifted to a new location within the blob. It is useful to note that an arbitrary permutation of the indices can be broken into a sequence of \textit{swaps} of adjacent indices wherein the position of two indices are interchanged at a time.
Last but not the least, a tensor $\hat{T}$ can be \textit{reshaped} into a new tensor $\hat{T}'$ by `fusing' two adjacent indices into a single index and/or `splitting' an index into two indices. For instance, in
\begin{eqnarray}
(\hat{T}')_{dc} = \hat{T}_{abc},~~~~~~~d = a\times b,
\label{eq:fuse}
\end{eqnarray}
tensor $\hat{T}'$ is obtained from tensor $\hat{T}$ by fusing indices $a \in \left\{1, \cdots, |a|\right\}$ and $b \in \left\{1, \cdots, |b|\right\}$ together into a single index $d$ of size $|d| = |a| \cdot |b|$ that runs over all pairs of values of $a$ and $b$, i.e. $ d \in \left\{ (1,1), (1,2), \cdots, (|a|, |b|-1), (|a|,|b|) \right\}$, whereas in
\begin{eqnarray}
\hat{T}_{abc} = (\hat{T}')_{dc},~~~~~~~d = a\times b,
\label{eq:split}
\end{eqnarray}
tensor $\hat{T}$ is recovered from $\hat{T}'$ by splitting index $d$ of $\hat{T}'$ back into indices $a$ and $b$. Reshape of the indices is depicted as shown in Fig.~\ref{fig:tensorman}(iii).
\subsection{Multiplication of two tensors\label{sec:tensor:multiply}}
Given two matrices $\hat{R}$ and $\hat{S}$ with components $\hat{R}_{ab}$ and $\hat{S}_{bc}$, we can multiply them together to obtain a new matrix $\hat{T}$, $\hat{T} = \hat{R}\cdot \hat{S}$, with components
\begin{equation}
\hat{T}_{ac} = \sum_{b} \hat{R}_{ab}\hat{S}_{bc},
\label{eq:Mmultiply}
\end{equation}
by summing over or \textit{contracting} index $b$. The multiplication of matrices $\hat{R}$ and $\hat{S}$ is represented graphically by connecting together the emerging lines of $\hat{R}$ and $\hat{S}$ corresponding to the contracted index, as shown in Fig.~\ref{fig:multiply1}(i).
Matrix multiplication can be generalized to tensors, such that, an incoming index of one tensor is identified and contracted with an outgoing index of another. For instance, given tensor $\hat{R}$ with components $\hat{R}_{abcde}$ and directions $\{\mbox{`in', `out', `in', `out', `out'}\}$, and tensor $\hat{S}$ with components $\hat{S}_{cdfbg}$ and directions $\{\mbox{`out', `in', `in', `in', `out'}\}$, we can define a tensor $\hat{T}$ with components $\hat{T}_{gafe}$ that are given by,
\begin{equation}
\hat{T}_{afge} = \sum_{bcd} \hat{R}_{abcde}\hat{S}_{cdfbg}.
\label{eq:tensormult}
\end{equation}
Note that each of the indices $b, c$ and $d$, that are contracted, is incoming into one tensor and outgoing from the other. The multiplication is represented graphically by connecting together the lines emerging from $\hat{R}$ and $\hat{S}$ corresponding to each of these indices, as shown in Fig.~\ref{fig:multiply1}(ii).
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{multiply1}
\end{center}
\caption{(i) Graphical representation of the matrix multiplication of two matrices $\hat{R}$ and $\hat{S}$ into a new matrix $\hat{T}$, Eq.~(\ref{eq:Mmultiply}). (ii) Graphical representation of an example of the multiplication of two tensors $\hat{R}$ and $\hat{S}$ into a new tensor $\hat{T}$, Eq.~(\ref{eq:tensormult}). \label{fig:multiply1}}
\end{figure}
Multiplication of two tensors can be broken down into a sequence of elementary steps by transforming the tensors into matrices, multiplying the matrices together, and then transforming the resulting matrix back into a tensor. Next we describe these steps for the contraction given in Eq.~(\ref{eq:tensormult}). They are illustrated in Fig.~\ref{fig:multiply2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{multiply2}
\end{center}
\caption{Graphical representations of the five elementary steps 1-5 into which one can decompose the multiplication of the tensors of Eq.~(\ref{eq:tensormult}).\label{fig:multiply2}}
\end{figure}
\begin{enumerate}
\item \textit{Reverse} and \textit{Permute} the indices of tensor $\hat{R}$ in such a way that the indices $b, c$ and $d$ that are contracted appear in the last positions as \textit{outgoing} indices and in a given order, e.g. $b\overline{c}d$, and the remaining indices $a$ and $e$ appear in the first positions as \textit{incoming} indices; similarly reverse and permute the indices of $\hat{S}$ so that the indices $b, c$ and $d$ appear in the first positions as \textit{incoming} indices and in the same order, $\overline{b}c\overline{d}$, and the remaining indices $\overline{f}$ and $g$ appear in the last positions as \textit{outgoing} indices,
\begin{align}
(\hat{R}')_{a\overline{e} ~b\overline{c}d} &= (\hat{R})_{abcde}, \nonumber \\
(\hat{S}')_{\overline{b}c\overline{d} ~\overline{f}g} &= (\hat{S})_{cdfbg}. \label{eq:multi1}
\end{align}
\item \textit{Reshape} tensor $\hat{R}'$ into a matrix $\hat{R}''$ by fusing into a single index $u$ all the indices that are not contracted, $u = a\times \overline{e}$, and into a single index $y$ all indices that are contracted, $y = b \times \overline{c} \times d$; similarly reshape tensor $\hat{S}'$ into a matrix $\hat{S}''$ with indices $\overline{y} = \overline{b} \times \overline{c} \times d$ and $w = \overline{f}\times g$ (indices $b, c$ and $d$ are required to be fused according to the \textit{same} fusion sequence in the two tensors. A possible fusion sequence may involve, for example, first fusing $b$ and $c$ and then fusing the resulting index with $d$),
\begin{align}
(\hat{R}'')_{uy} &= (\hat{R}')_{a\overline{e}b\overline{c}d}, \nonumber \\
(\hat{S}'')_{yw} &= (\hat{S}')_{b\overline{c}d\overline{f}g}. \label{eq:multi2}
\end{align}
\item \textit{Multiply} matrices $\hat{R}''$ and $\hat{S}''$ to obtain a matrix $\hat{T}'$ with components
\begin{equation}
(\hat{T}'')_{uw} = \sum_{y} (\hat{R}'')_{uy} ~~(\hat{S}'')_{y w}. \label{eq:multi3}
\end{equation}
\item \textit{Reshape} matrix $\hat{T}'$ into a tensor $\hat{T}$ by splitting indices $u = a\times \overline{e}$ and $w = \overline{f}\times g$,
\begin{equation}
(\hat{T'})_{aefg} = (\hat{T}'')_{uw}. \label{eq:multi4}
\end{equation}
\item \textit{Reverse} and \textit{Permute} indices of tensor $\hat{T}'$ in the order in which they appear in $\hat{T}$,
\begin{equation}
\hat{T}_{afge} = (\hat{T}')_{a\overline{e}\overline{f}g}. \label{eq:multi5}
\end{equation}
\end{enumerate}
The contraction of Eq.~(\ref{eq:tensormult}) can be implemented at once, without breaking the multiplication down into elementary steps. However, it is often more convenient to compose the above elementary steps since, for instance, in this way one can use existing linear algebra libraries for matrix multiplication. In addition, it can be seen that the leading computational cost in multiplying two large tensors is not changed when decomposing the contraction in the above steps.
\subsection{Factorization of a tensor\label{sec:tensor:factorize}}
A matrix $\hat{T}$ can be factorized into the product of two (or more) matrices in one of several canonical forms. For instance, the \textit{singular value decomposition}
\begin{equation}
\hat{T}_{ab} = \sum_{c,d} \hat{U}_{ac}\hat{S}_{cd}\hat{V}_{db}
= \sum_{c} \hat{U}_{ac}s_{c}\hat{V}_{cb}
\label{eq:singular}
\end{equation}
factorizes $\hat{T}$ into the product of two unitary matrices $\hat{U}$ and $\hat{V}$, and a diagonal matrix $\hat{S}$ with non-negative diagonal elements $s_c = \hat{S}_{cc}$ known as the \textit{singular values} of $\hat{T}$ [Fig.~\ref{fig:decompose}(i)].
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{decompose}
\end{center}
\caption{(i) Factorization of a matrix $\hat{T}$ according to a singular value decomposition, Eq.~(\ref{eq:singular}). (ii) Factorization of a rank-4 tensor $\hat{T}$ according to one of several possible singular value decompositions. \label{fig:decompose}}
\end{figure}
On the other hand, the \textit{eigenvalue} or \textit{spectral decomposition} of a square matrix $\hat{T}$ is of the form
\begin{equation}
\hat{T}_{ab} = \sum_{c,d} \hat{M}_{ac}D_{cd}(\hat{M}^{-1})_{db}
= \sum_{c} \hat{M}_{ac}\lambda_{c}(\hat{M}^{-1})_{cb}
\label{eq:spectral}
\end{equation}
where $\hat{M}$ is an invertible matrix whose columns encode the eigenvectors $\ket{\lambda_c}$ of $\hat{T}$,
\begin{equation}
\hat{T} \ket{\lambda_{c}} = \lambda_c \ket{\lambda_c},
\end{equation}
$\hat{M}^{-1}$ is the inverse of $\hat{M}$, and $\hat{D}$ is a diagonal matrix, with the eigenvalues $\lambda_c=\hat{D}_{cc}$ on its diagonal. Other useful factorizations include the LU decomposition, the QR decomposition, etc. We refer to any such decomposition generically as a \textit{matrix factorization}.
A tensor $\hat{T}$ with more than two indices can be converted into a matrix in several ways by specifying how to join its indices into two subsets. After specifying how tensor $\hat{T}$ is to be regarded as a matrix, we can factorize $\hat{T}$ according to any of the above matrix factorizations, as illustrated in Fig.~\ref{fig:decompose}(ii) for a singular value decomposition. This requires first reversing directions, permuting and reshaping the indices of $\hat{T}$ to form a matrix, then decomposing the latter, and finally restoring the open indices of the resulting matrices into their original form by undoing the reshapes, permutations and reversal of directions.
\subsection{Tensor networks and their manipulation\label{sec:tensor:TN}}
A \textit{tensor network} $\mathcal{N}$ is a set of tensors whose indices are connected according to a network pattern, e.g. Fig.~\ref{fig:TN}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{tn}
\end{center}
\caption{(i) Example of a tensor network $\mathcal{N}$. (ii) Tensor $\hat{T}$ of which the tensor network $\mathcal{N}$ could be a representation. (iii) Tensor $\hat{T}$ can be obtained from $\mathcal{N}$ through a sequence of contractions of pairs of tensors. Shading indicates the two tensors to be multiplied together at each step. The product tensor is depicted by a blob that covers the two tensors that are multiplied.\label{fig:TN}}
\end{figure}
Given a tensor network $\mathcal{N}$, a single tensor $\hat{T}$ can be obtained by contracting all the indices that connect the tensors in $\mathcal{N}$ [\fref{fig:TN}(ii)]. Here, the indices of tensor $\hat{T}$ correspond to the open indices of the tensor network $\mathcal{N}$. We then say that the tensor network $\mathcal{N}$ is a tensor network decomposition of $\hat{T}$. One way to obtain $\hat{T}$ from $\mathcal{N}$ is through a sequence of contractions involving two tensors at a time [Fig.~\ref{fig:TN}(iii)]. Notice how a tensor that is obtained by contracting a region of a tensor network is conveniently depicted by a blob or shape that covers that region.
From a tensor network decomposition $\mathcal{N}$ for a tensor $\hat{T}$, another tensor network decomposition for the same tensor $\hat{T}$ can be obtained in many ways. One possibility is to replace two tensors in $\mathcal{N}$ with the tensor resulting from contracting them together, as is done in each step of Fig.~\ref{fig:TN}(iii). Another way is to replace a tensor in $\mathcal{N}$ with a decomposition of that tensor (e.g. with a singular value decomposition). In this thesis, we will be concerned with manipulations of a tensor network that, as in the case of multiplying two tensors or decomposing a tensor, can be broken down into a sequence of operations from the following list:
\begin{enumerate}
\item Reversal of direction of indices of a tensor, Eq.~(\ref{eq:bend}).
\item Permutation of the indices of a tensor, Eq.~(\ref{eq:permute}).
\item Reshape of the indices of a tensor, Eqs.~(\ref{eq:fuse})-(\ref{eq:split}).
\item Multiplication of two matrices, Eq.~(\ref{eq:Mmultiply}).
\item Factorization of a matrix (e.g. singular value decomposition Eq.~(\ref{eq:singular}) or spectral decomposition Eq.~(\ref{eq:spectral}).
\end{enumerate}
These operations constitute a set $\mathcal{P}$ of \textit{primitive} operations for tensor network manipulations (or, at least, for the type of manipulations we will be concerned with). In Sec.~\ref{sec:blockmoves} (and again in Chapter 6) we discuss how this set $\mathcal{P}$ of primitive operations can be generalized to tensors that are invariant under the action of the group SU(2).
Next we review basic background material concerning the representation theory of the group SU(2) without reference to tensor network states and algorithms. This review is distributed over Sections \ref{sec:symmetry}, \ref{sec:su2lattice} and \ref{sec:fusiontree}. We refer the reader to \citep{Cornwell97} and Chapters 3 and 4 of \citep{Sakurai} for additional supporting material for these sections.
\section {Representations of the group SU(2)}
\label{sec:symmetry}
In this section we consider the action of SU(2) on a vector space that is an irreducible representation of the group, and then more generally on a vector space $\mathbb{V}$ that is a reducible representation, namely, $\mathbb{V}$ decomposes as a direct sum of (possibly degenerate) irreducible representations. We also characterize vectors belonging to $\mathbb{V}$ and linear operators acting on $\mathbb{V}$ that are invariant under the action of SU(2).
Let $\mathbb{V}$ be a finite dimensional vector space on which SU(2) acts \textit{unitarily} by means of transformations $\hat{W}_{\textbf{r}}~:~\mathbb{V}~\rightarrow~\mathbb{V}$,
\begin{equation}
\hat{W}_{\textbf{r}} \equiv e^{i\textbf{r}\cdot \textbf{J}} = e^{i(r_x\hat{J}_x+r_y\hat{J}_y+r_z\hat{J}_z)}.
\label{eq:exp}
\end{equation}
Here $\textbf{r} \equiv (r_x,r_y,r_z) \in \mathbb{R}^3$ parameterizes the group elements and $\textbf{J} \equiv (\hat{J}_x, \hat{J}_y, \hat{J}_z)$; $\hat{J}_x, \hat{J}_y$ and $\hat{J}_z$ are traceless hermitian operators that are said to \textit{generate} the representation $\hat{W}_{\textbf{r}}$ of SU(2). These operators close the lie algebra su(2), namely,
\begin{equation}
[\hat{J}_{\alpha}, \hat{J}_{\beta}] = i\sum_{\gamma=x,y,z}\epsilon_{\alpha\beta\gamma}\hat{J}_{\gamma}, ~~~~~~~ \alpha,\beta = x,y,z,
\label{eq:algebra}
\end{equation}
where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita symbol. The operators $\hat{J}_x, \hat{J}_y$ and $\hat{J}_z$ are associated, for example, with the projection of angular momentum or spin along the three spatial directions $x, y$ and $z$, respectively.
It follows that,
\begin{equation}
[\textbf{J}^2, \hat{J}_{\alpha}] = 0, ~~~~~~ \textbf{J}^2 =\sum_{\alpha=x,y,z}\hat{J}_{\alpha}^2.\label{eq:casimir}
\end{equation}
\subsection{Irreducible representations\label{sec:symmetry:irreps}}
Let vector space $\mathbb{V}$ transform as an irreducible representation (or irrep) of SU(2) with spin $j$. Here $j$ can take values $0,\frac{1}{2},1,\frac{3}{2},2,\ldots$ and $\mathbb{V}$ has dimension $2j+1$. We choose an orthonormal basis $\ket{jm_j}$, the \textit{spin basis}, in $\mathbb{V}$ that is a simultaneous eigenbasis of the operators $\textbf{J}^2$ and $\hat{J}_{z}$,
\begin{align}
\textbf{J}^{2}\ket{j m_j} &= j(j+1)\ket{j m_j}, \nonumber \\
\hat{J}_{z} \ket{jm_j} &= m_j \ket{jm_j}.
\label{eq:irrepz}
\end{align}
Here $m_j$ is the magnitude of the spin projection along the $z$ direction and can assume values in the range $\{-j, -j\!+\!1,\ldots, j\}$. In this basis, the action of the operators $\hat{J}_{x}$ and $\hat{J}_{y}$ on the space $\mathbb{V}$ is conveniently described in terms of the raising operator $\hat{J}_{+} = \hat{J}_{x} + i\hat{J}_{y}$ and the lowering operator $\hat{J}_{-} = \hat{J}_{x} - i\hat{J}_{y}$ as
\begin{equation}
\hat{J}_{\pm} \ket{jm_j} = \sqrt{j(j+1) - m_j (m_j \pm 1)} \ket{j, (m_j \pm 1)}.
\label{eq:irrepladder}
\end{equation}
The operator $\textbf{J}^2$ can be written as
\begin{equation}
\textbf{J}^{2} = j(j+1)\hat{I}_{2j+1},
\end{equation}
where $\hat{I}_{2j+1}$ is the Identity acting on the irrep $j$.
\textbf{Example 1: } Consider that vector space $\mathbb{V}$ is a spin $j=0$ irrep of SU(2). Then $\mathbb{V}$ has dimension one, $\mathbb{V}~\cong~\mathbb{C}$. The operators $\hat{J}_{\alpha}$ are trivial, $\hat{J}_x~=~\hat{J}_y~=~\hat{J}_z~=~(0)$.\markend
\textbf{Example 2: }Consider a two-dimensional vector space $\mathbb{V}$ that transforms as an irrep $j=\frac{1}{2}$. Then the orthogonal vectors (in column vector notation)
\begin{align}
\begin{pmatrix} 1 \\ 0 \end{pmatrix} \equiv \; \ket{j\!=\!\frac{1}{2}, m_{\frac{1}{2}}\!=\!-\frac{1}{2}},
\begin{pmatrix} 0 \\ 1 \end{pmatrix} \equiv \; \ket{j\!=\!\frac{1}{2}, m_{\frac{1}{2}}\!=\!\frac{1}{2}},
\label{eq:basiseg1}
\end{align}
form a basis of $\mathbb{V}$. In this basis the operators $\hat{J}_{x}, \hat{J}_{y}, \hat{J}_{z}$ and $\textbf{J}^2$ read as
\begin{align}
\hat{J}_{x} \equiv \; \begin{pmatrix} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \end{pmatrix},~
\hat{J}_{y} \equiv \; \begin{pmatrix} 0 & -\frac{i}{2} \\ \frac{i}{2} & 0 \end{pmatrix},~
\hat{J}_{z} \equiv \; \begin{pmatrix} -\frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix},~\textbf{J}^2 \equiv \; \begin{pmatrix} \frac{3}{4} & 0 \\ 0 & \frac{3}{4} \end{pmatrix}. \label{eq:eg2c1}
\end{align}
In terms of Pauli matrices $\hat{\sigma}_{\alpha}$ we have
\begin{equation}
\hat{J}_{\alpha} = \frac{\hat{\sigma}_{\alpha}}{2}, ~~~ \alpha = x,y,z.\markend
\end{equation}
\textbf{Example 3: } Also consider a three-dimensional vector space $\mathbb{V}$ that transforms as an irrep $j=1$. The orthogonal vectors
\begin{align}
\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix} \equiv \; \ket{j\!=\!1, m_{1}\!=\!-1}, ~~
\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \equiv \; \ket{j\!=\!1, m_{1}\!=\!0}, ~~
\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \equiv \; \ket{j\!=\!1, m_{1}\!=\!1},
\label{eq:basiseg2}
\end{align}
form a basis of $\mathbb{V}$. In this basis, operators $\hat{J}_x, \hat{J}_y, \hat{J}_z$ and $\textbf{J}^2$ read as
\begin{align}
\hat{J}_x &\equiv \; \begin{pmatrix} 0 & \frac{1}{\sqrt{2}} & 0\\\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}\\0 & \frac{1}{\sqrt{2}} & 0 \end{pmatrix},
\hat{J}_y \equiv \; \begin{pmatrix} 0 & \frac{i}{\sqrt{2}} & 0\\-\frac{i}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}}\\0 & -\frac{i}{\sqrt{2}} & 0 \end{pmatrix}, \nonumber \\
\hat{J}_z &\equiv \; \begin{pmatrix} -1 & 0 & 0\\0 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}, ~~~~~\textbf{J}^2 \equiv \begin{pmatrix} 2 & 0 & 0\\0 & 2 & 0\\ 0 & 0 & 2 \end{pmatrix}.\markend
\label{eq:eg2c2}
\end{align}
\subsection{Reducible representations \label{sec:symmetry:rreps}}
More generally, SU(2) can act on the vector space $\mathbb{V}$ reducibly, in that, $\mathbb{V}$ may decompose as the direct sum of irreps of SU(2),
\begin{equation}
\boxed{
\begin{split}\mathbb{V} &\cong \bigoplus_j d_j\mathbb{V}_{j} \\
&\cong \bigoplus_j\left(\mathbb{D}_j \otimes \mathbb{V}_j\right).
\end{split}
}
\label{eq:decoV}
\end{equation}
Here space $\mathbb{V}_{j}$ accommodates a spin $j$ irrep of SU(2) and $d_j$ is the number of times $\mathbb{V}_{j}$ occurs in $\mathbb{V}$. The decomposition can also be re-written in terms of a $d_j-$dimensional space $\mathbb{D}_j$. We say that irrep $j$ is $d_j$-fold degenerate and that $\mathbb{D}_j$ is the degeneracy space. The total dimension of space $\mathbb{V}$ is given by $\sum_j d_j (2j+1)$.
Let $t_j=1,2,\ldots,d_j$ label an orthonormal basis $\ket{jt_j}$ in the space $\mathbb{D}_j$. Then a natural choice of basis of the space $\mathbb{V}$ is the set of orthonormal vectors $\ket{jt_jm_j} \equiv \ket{jt_j}\otimes\ket{jm_j}$, where $j$ assumes various values that occur in the direct sum decomposition, Eq.~(\ref{eq:decoV}).
In this basis the action of SU(2) on $\mathbb{V}$ is given by
\begin{equation}
\hat{W}_{\textbf{r}} \equiv \bigoplus_j \left(\hat{I}_{d_j} \otimes \hat{W}_{\textbf{r},j}\right),
\label{eq:decow}
\end{equation}
as generated by the operators
\begin{equation}
\hat{J}_{\alpha} \equiv \bigoplus_j \left(\hat{I}_{d_j} \otimes \hat{J}_{\alpha, j}\right),~~~\alpha=x,y,z.
\label{eq:decoS}
\end{equation}
Here $\hat{I}_{d_j}$ is a $d_j \times d_j$ Identity and operators $\hat{J}_{\alpha, j}$ generate the irreducible represention $\hat{W}_{\textbf{r},j}$ on space $\mathbb{V}_j$.
The operator $\textbf{J}^2$ takes the form
\begin{equation}
\textbf{J}^2 \equiv \bigoplus_j j(j+1)\left(\hat{I}_{d_j}\otimes\hat{I}_{2j+1}\right).\label{eq:decoJJ}
\end{equation}
\textbf{Example 4:} Let vector space $\mathbb{V}$ transform as an irrep $j=\frac{1}{2}$ with a finite degeneracy $d_{\frac{1}{2}} = 3$. The space $\mathbb{V}$ decomposes as $\mathbb{V}~\cong~\mathbb{D}_{\frac{1}{2}}~\otimes~\mathbb{V}_{\frac{1}{2}}$ where $\mathbb{D}_{\frac{1}{2}}$ is a three-dimensional degeneracy space and $\mathbb{V}_{\frac{1}{2}}$ corresponds to the space of Example 1.
The total dimension of space $\mathbb{V}$ is $d_{\frac{1}{2}}(2j+1) = 6$. The vectors
\begin{align}
\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \equiv \ket{j=\!\frac{1}{2}, t_0\!= 1},~~
\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \equiv \ket{j=\!\frac{1}{2}, t_0\!= 2},~~
\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \equiv \ket{j=\!\frac{1}{2}, t_1\!= 3}.
\label{eq:basiseg21}
\end{align}
form a basis of $\mathbb{D}_{\frac{1}{2}}$. A basis of $\mathbb{V}$ can be obtained as the product of the the basis (\ref{eq:basiseg21}) of $\mathbb{D}_{\frac{1}{2}}$ and the basis (\ref{eq:basiseg1}) of $\mathbb{V}_{\frac{1}{2}}$. In this basis of the operators $\hat{J}_{\alpha}$ take the form of Eq.~(\ref{eq:decoS}). For instance,
\begin{equation}
\hat{J}_x \equiv \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \otimes \begin{pmatrix} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \end{pmatrix} =
\begin{pmatrix} 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \end{pmatrix}
\label{eq:eg2}
\end{equation}
Similarly, operators $\hat{J}_y$ and $\hat{J}_z$ read as
\begin{align}
\hat{J}_y &\equiv
\begin{pmatrix} 0 & \frac{i}{2} & 0 & 0 & 0 & 0 \\ -\frac{i}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{2} & 0 & 0 \\ 0 & 0 & -\frac{i}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{i}{2} \\ 0 & 0 & 0 & 0 & -\frac{i}{2} & 0 \end{pmatrix}, \nonumber \\
\hat{J}_z &\equiv
\begin{pmatrix} -\frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \end{pmatrix}.
\end{align}
The operator $\textbf{J}^2$ reads
\begin{equation}
\textbf{J}^2 \equiv \frac{1}{2}(\frac{1}{2}+1)\hat{I}_{3}\otimes\hat{I}_{2} \equiv
\begin{pmatrix} \frac{3}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{3}{4} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{3}{4} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{3}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{3}{4} \end{pmatrix}.\markend
\label{eq:eg2s2}
\end{equation}
\textbf{Example 5:} Consider a five-dimensional Hilbert space $\mathbb{V}$ that decomposes into two different irreps $j=0$ and $j=1$ with degeneracy dimensions $d_0=2$ and $d_1=1$ so that irrep $j=0$ is two-fold degenerate. The space $\mathbb{V}$ decomposes as $\mathbb{V}~\cong~(\mathbb{D}_0~\otimes~ \mathbb{V}_{0})~\oplus~(\mathbb{D}_1~\otimes~\mathbb{V}_{1})$, where $\mathbb{D}_0$ is the two-dimensional degeneracy space of irrep $j=0$ and $\mathbb{D}_1$ is the one-dimensional degeneracy space of irrep $j=1$.
The orthogonal vectors
\begin{align}
\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} &\equiv \; \ket{j=0, t_0 = 1, m_0=0},\nonumber \\
\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} &\equiv \; \ket{j=0, t_0 = 2, m_0=0}, \nonumber \\
\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} &\equiv \; \ket{j=1, t_1 = 1, m_1=-1},\nonumber \\
\begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} &\equiv \; \ket{j=1, t_1 = 1, m_1=0},\nonumber \\
\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} &\equiv \; \ket{j=1, t_1 = 1, m_1=1}.
\label{eq:basiseg3}
\end{align}
form a basis of $\mathbb{V}$. In this basis, the operators $\hat{J}_{\alpha}$ take the form
\begin{equation}
\hat{J}_{\alpha} = (\hat{I}_{d_0} \otimes \hat{J}_{\alpha, 0}) \oplus (\hat{I}_{d_1} \otimes \hat{J}_{\alpha, 1}),~~~\alpha=x,y,z,
\end{equation}
where $\hat{J}_{\alpha, 0}$ and $\hat{J}_{\alpha, 1}$ are the generators of irrep $j=0$ (Examples 1) and irrep $j=1$ (Examples 3) respectively. Operators $\hat{J}_{\alpha}$ and $\textbf{J}^2$ read as
\begin{align}
\hat{J}_x &\equiv \; \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \end{pmatrix},\nonumber \\
\hat{J}_y &\equiv \; \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 \\0 & 0 & -\frac{i}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\0 & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 \end{pmatrix}, \nonumber \\
\hat{J}_z &\equiv \; \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 \end{pmatrix}, \nonumber \\
\textbf{J}^2 &\equiv \; \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\0 & 0 & 0 & 2 & 0 \\0 & 0 & 0 & 0 & 2 \end{pmatrix}.\markend
\label{eq:eg3}
\end{align}
\subsection{SU(2)-invariant states and operators \label{sec:symmetry:states}}
We are interested in states and operators that have a simple transformation rule under the action of SU(2).
A pure state $\ket{\Psi} \in \mathbb{V}$ with a well defined spin $j$ belongs to the subspace $\mathbb{D}_j \otimes \mathbb{V}_j$. In the spin basis $\ket{jt_jm_j}$ the state $\ket{\Psi}$ can be expanded as
\begin{equation}
\ket{\Psi} = \sum_{t_jm_j} (\Psi_j)_{t_jm_j} \ket{jt_jm_j},
\label{eq:welldefj}
\end{equation}
Under the action of SU(2), $\ket{\Psi}$ transforms to another pure state $\ket{\Psi'}$, $\ket{\Psi'}~=~\hat{W}_{\textbf{r}}\ket{\Psi}$, within the same subspace $\mathbb{D}_j~\otimes~\mathbb{V}_j$. The components $(\Psi'_j)_{t'_jm'_j}$ of $\ket{\Psi'}$ are related to those of $\ket{\Psi}$ as
\begin{equation}
(\Psi'_j)_{t'_jm'_j} = \sum_{t_jm_j} (W_{r,j})_{t'_jm'_jt_jm_j}(\Psi_j)_{t_jm_j}.
\label{eq:welldefj1}
\end{equation}
In case of vanishing spin $j=0$, the state $\ket{\Psi}$ transforms trivially under the action of SU(2), $\hat{W}_{r,j=0} \equiv 1$. In this case Eq.~(\ref{eq:welldefj1}) reduces to,
\begin{equation}
\hat{W}_{\textbf{r}} \ket{\Psi} = \ket{\Psi}, ~~~~~ \forall \textbf{r} \in \mathbb{R}^3.
\label{eq:invPsi1}
\end{equation}
That is, $\ket{\Psi}$ remains \textit{invariant} under the action of SU(2). Equivalently, it is annihilated by the action of generators,
\begin{equation}
\hat{J}_{\alpha}\ket{\Psi} = 0, ~~~~~~~~~~~\alpha=x,y,z.
\label{eq:invPsi2}
\end{equation}
An SU(2)-invariant state $\ket{\Psi}$ can be expanded in the basis $\ket{j~=~0,~t_0~,m_0~=~0}$ of the spin $j=0$ subspace, $\mathbb{D}_{0} \otimes \mathbb{V}_0$,
\begin{equation}
\boxed{\ket{\Psi} = \sum_{t_0} (\Psi_0)_{t_0} \ket{j=0,t_0,m_0=0},}
\label{eq:nPsi3}
\end{equation}
where we have used $(\Psi_0)_{t_0}$ as a shorthand for $(\Psi_{j=0})_{t_0,m_0=0}$.
A linear operator $\hat{T}: \mathbb{V} \rightarrow \mathbb{V}$ is SU(2)-invariant if it commutes with the action of the group,
\begin{equation}
[\hat{T}, \hat{W}_{\textbf{r}}] = 0, ~~~~~~~~~\forall \textbf{r} \in \mathbb{R}^3,
\label{eq:invOp2}
\end{equation}
or equivalently, if it commutes with the generators $\hat{J}_{\alpha}$,
\begin{equation}
[\hat{T}, \hat{J}_{\alpha}] = 0, ~~~~~~~~~~~\alpha=x,y,z.
\label{eq:invOp1}
\end{equation}
Notice that the operator $\textbf{J}^2$ is SU(2)-invariant, Eq.~(\ref{eq:casimir}).
An SU(2)-invariant operator $\hat{T}$ decomposes as
\begin{align}
&\boxed{\hat{T} = \bigoplus_{j} \left(\hat{T}_{j} \otimes \hat{I}_{2j+1}\right),} \nonumber \\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{(Schur's Lemma)} \label{eq:Schur}
\end{align}
where $\hat{T}_{j}$ is a $d_j\times d_j$ matrix that acts on the degeneracy space $\mathbb{D}_j$. This decomposition implies, for instance, that operator $\hat{T}$ transforms states with a well defined spin $j$ [such as $\ket{\Psi}$ of Eq.~(\ref{eq:welldefj})] into states with the same spin $j$. Thus, SU(2)-invariant operators \textit{conserve} spin.
\textbf{Example 2 revisited:} A generic state $\ket{\Psi} \in \mathbb{V}_{\frac{1}{2}}$ has the form,
\begin{equation}
\ket{\Psi} = \begin{pmatrix} (\Psi_{j=\frac{1}{2}})_{m_{\frac{1}{2}}=-\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{m_{\frac{1}{2}}=\frac{1}{2}} \end{pmatrix}~~ \in \mathbb{C}^2.
\end{equation}
and is an eigenstate of $\textbf{J}^2_{\frac{1}{2}}$ with eigenvalue $\displaystyle \frac{3}{4}$.
According to Schur's lemma, an SU(2)-invariant operator $\hat{T}$ acting on $\mathbb{V}_{\frac{1}{2}}$ must be proportional to the Identity,
\begin{equation}
\hat{T} = \; (T_{j=\frac{1}{2}}) \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix},~T_{j=\frac{1}{2}} \in \mathbb{C}. \label{eq:sparseT1}\markend
\end{equation}
\textbf{Example 4 revisited:} A generic state $\ket{\Psi}$ in the vector space $\mathbb{V} \cong 3\mathbb{V}_{\frac{1}{2}}$ of Example 3 has the form
\begin{equation}
\ket{\Psi} = \begin{pmatrix} (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=1,m_{\frac{1}{2}}=-\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=1,m_{\frac{1}{2}}=\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=2,m_{\frac{1}{2}}=-\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=2,m_{\frac{1}{2}}=\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=3,m_{\frac{1}{2}}=-\frac{1}{2}} \\ (\Psi_{j=\frac{1}{2}})_{t_{\frac{1}{2}}=3,m_{\frac{1}{2}}=\frac{1}{2}} \end{pmatrix}~~ \in \mathbb{C}^6.
\end{equation}
Similar to the previous example, $\ket{\Psi}$ is an eigenstate of $\textbf{J}^2_{\frac{1}{2}}$ with eigenvalue $\displaystyle \frac{3}{4}$..
An SU(2)-invariant operator $\hat{T} : \mathbb{V} \rightarrow \mathbb{V}$ must be of the form
\begin{align}
\hat{T} &= \; \begin{pmatrix} T_{11} & T_{12} & T_{13}\\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \nonumber \\
&= \begin{pmatrix} T_{11} & 0 & T_{12} & 0 & T_{13} & 0\\ 0 & T_{11} & 0 & T_{12} & 0 & T_{13} \\ T_{21} & 0 & T_{22} & 0 & T_{23} & 0 \\ 0 & T_{21} & 0 & T_{22} & 0 & T_{23} \\ T_{31} & 0 & T_{32} & 0 & T_{33} & 0 \\ 0 & T_{31} & 0 & T_{32} & 0 & T_{33} \end{pmatrix},\label{eq:sparseT2}
\end{align}
where we have used $T_{ij}$ as a shorthand for $(T_{\frac{1}{2}})_{ij}~\in~\mathbb{C}$. Notice that $\textbf{J}^2$ in Eq.~(\ref{eq:eg2s2}) has this form.\markend
\textbf{Example 5 revisited:} A generic state $\ket{\Psi} \in \mathbb{V}, \mathbb{V} \cong 2\mathbb{V}_0 \oplus \mathbb{V}_1$ has the form
\begin{equation}
\ket{\Psi} = \begin{pmatrix} (\Psi_0)_{1,0} \\ (\Psi_0)_{2,0} \\ ~~(\Psi_1)_{1,-1} \\ (\Psi_1)_{1,0} \\ (\Psi_1)_{1,1} \end{pmatrix}~~ \in \mathbb{C}^5.
\label{eq:ex2rev}
\end{equation}
(For simplicity we have omitted explicit labels in the subscripts.) In contrast to the previous two examples, a generic state $\ket{\Psi} \in \mathbb{V}$ is not an eigenstate of $\textbf{J}^2$, that is, $\ket{\Psi}$ is generally not a state with a well defined spin $j$.\\
An SU(2)-invariant vector $\ket{\Psi}$ has the form
\begin{equation}
\ket{\Psi} = \begin{pmatrix} (\Psi_0)_{1,0} \\ (\Psi_0)_{2,0} \\ 0 \\ 0 \\ 0 \end{pmatrix}, ~~~~~~~~~~~~~
\label{eq:ex3rev}
\end{equation}
with non-trivial components only in the spin $j=0$ subspace. Notice that this state is annihilated by the action of the operators $\hat{J}_{\alpha}$ [Eq.~(\ref{eq:eg3})] in accordance with Eq.~(\ref{eq:invPsi2}).
A state with a well defined spin $j=1$ must be of the form
\begin{equation}
\ket{\Psi_1} = \begin{pmatrix} 0 \\ 0 \\ ~~(\Psi_1)_{1,-1} \\ (\Psi_1)_{1,0} \\ (\Psi_1)_{1,1} \end{pmatrix}~~ \in \mathbb{C}^5,
\label{eq:ex2rev1}
\end{equation}
with non-trivial components only in the spin $j=1$ subspace.
An SU(2)-invariant operator $\hat{T}$ e.g. $\textbf{J}^2$ in Eq.~(\ref{eq:eg3}) has the form
\begin{align}
\hat{T} &= \; \begin{pmatrix} \left(T_0\right)_{11} & \left(T_0\right)_{12}\\ \left(T_0\right)_{21} & \left(T_0\right)_{22}\end{pmatrix} \otimes (1) \oplus \begin{pmatrix} \left(T_1\right)_{11} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \nonumber \\
&= \begin{pmatrix} \left(T_0\right)_{11} & \left(T_0\right)_{12} & 0 & 0 & 0\\ \left(T_0\right)_{21} & \left(T_0\right)_{22} & 0 & 0 & 0 \\ 0 & 0 & \left(T_1\right)_{11} & 0 & 0 \\ 0 & 0 & 0 & \left(T_1\right)_{11} & 0 \\ 0 & 0 & 0 & 0 & \left(T_1\right)_{11} \end{pmatrix},
\label{eq:sparseT3}
\end{align}
where $\left(T_0\right)_{11},~\left(T_0\right)_{12},~\left(T_0\right)_{21},~\left(T_0\right)_{22},~\left(T_1\right)_{11}~\in~\mathbb{C}$.\markend
Notice that the SU(2)-invariant vector in Eq.~(\ref{eq:ex3rev}) and the SU(2)-invariant matrices in Eqs.~(\ref{eq:sparseT1}),(\ref{eq:sparseT2}) and (\ref{eq:sparseT3}) have a \textit{sparse} structure. In particular, the non-trivial components of an SU(2)-invariant matrix $\hat{T}$ are organized into blocks $\hat{T}_j$. This block structure can be exploited for computational gain. An SU(2)-invariant matrix can be stored compactly by storing the degeneracy blocks $\hat{T}_j$, while matrix multiplication and matrix factorizations can be performed block-wise [Sec. \ref{sec:blockmoves}] resulting in a significant speedup [Fig.~\ref{fig:multsvdcompare}] for these operations.
\section{Tensor product of representations \label{sec:su2lattice}}
So far we have described the action of SU(2) on a single vector space. Let us now consider the action of SU(2) on a space $\mathbb{V}$ that is a tensor product of $L$ vector spaces,
\begin{equation}
\mathbb{V} \equiv \bigotimes_{l=1}^{L} \mathbb{V}^{(l)}, \label{eq:latticespace}
\end{equation}
where each vector space $\mathbb{V}^{(l)}, l=1,2,\ldots,L,$ transforms as a finite dimensional representation of SU(2) as generated by spin operators $\hat{J}^{(l)}_{\alpha}, \alpha=x,y,z$. We consider the action of SU(2) on the space $\mathbb{V}$ that is generated by the \textit{total} spin operators,
\begin{equation}
\boxed{\hat{J}_{\alpha} \equiv \sum_{l=1}^{L} \hat{J}^{(l)}_{\alpha}, ~~~ \alpha = x,y,z,} \label{eq:totspinops}
\end{equation}
(each term in the sum acts as $\hat{J}^{(l)}_{\alpha}$ on site $l$ and the Identity on the remaining sites) and which corresponds to the unitary transformations,
\begin{equation}
\hat{W}_{\textbf{r}} \equiv e^{\mathrm{i}\textbf{r}\cdot \textbf{J}} = \bigotimes_{l=1}^L e^{\mathrm{i}\textbf{r}\cdot \textbf{J}^{(l)}} = \bigotimes_{l=1}^L \hat{W}_{\textbf{r}}^{(l)}.\label{eq:latticerep}
\end{equation}
As a example consider two vector spaces $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ on which the action of SU(2) is generated by spin operators $\hat{J}^{(A)}_{\alpha}$ and $\hat{J}^{(B)}_{\alpha}$ respectively. We can then consider the action of the group on the product space $\mathbb{V}^{(AB)} \cong \mathbb{V}^{(A)} \otimes \mathbb{V}^{(B)}$ as generated by the total spin operators $\hat{J}^{(AB)}_{\alpha},~\alpha=x,y,z,$ that are given by
\begin{equation}
\hat{J}^{(AB)}_{\alpha} \equiv \hat{J}^{(A)}_{\alpha}\otimes \hat{I} + \hat{I} \otimes \hat{J}^{(B)}_{\alpha}.
\end{equation}
Similarly, we can consider the action of SU(2) on the product of three vector spaces, $\mathbb{V}^{(A)}, \mathbb{V}^{(B)}$ and $\mathbb{V}^{(C)}$, that is generated by spin operators $\hat{J}^{(ABC)}_{\alpha},~\alpha=x,y,z$,
\begin{equation}
\hat{J}^{(ABC)}_{\alpha} \equiv \hat{J}^{(A)}_{\alpha}\otimes \hat{I}\otimes \hat{I} + \hat{I} \otimes \hat{J}^{(B)}_{\alpha}\otimes \hat{I}+\hat{I}\otimes \hat{I}\otimes \hat{J}^{(C)}_{\alpha}, \label{eq:genprodthree}
\end{equation}
where $\hat{J}^{(A)}_{\alpha}, \hat{J}^{(B)}_{\alpha}$ and $\hat{J}^{(C)}_{\alpha}$ are the spin operators that act on the three vector spaces.
A basis of $\mathbb{V}$ can be obtained in terms of the spin basis of each vector space in the product, Eq.~\ref{eq:latticespace}. However, it is convenient to introduce a \textit{coupled} basis: the simultaneous eigenbasis of the \textit{total} spin operators $\hat{J}_z$ and $\textbf{J}^{2}$. In the coupled basis SU(2)-invariant states $\ket{\Psi} \in \mathbb{V}$,
\begin{equation}
\ket{\Psi}=\hat{W}_{\textbf{r}}\ket{\Psi},~~~\forall\textbf{r}\in\mathbb{R}^3, \label{eq:latticeinv}
\end{equation}
and SU(2)-invariant operators $\hat{T}: \mathbb{V} \rightarrow \mathbb{V}$,
\begin{equation}
[\hat{T}, \hat{W}_{\textbf{r}}] = \hat{T},~~~\forall\textbf{r}\in\mathbb{R}^3,
\end{equation}
have a sparse structure, namely, $\ket{\Psi}$ has non-trivial components only in the spin zero sector of $\mathbb{V}$ while $\hat{T}$ is block-diagonal [Eq.~(\ref{eq:Schur})].
In the remainder of the section we focus on the tensor product of \textit{two} representations. We first discuss the case where the two sites transform as irreducible representations and then the more general case of reducible representations. The tensor product of several representations can then be analyzed by considering a sequence of pairwise products.
\subsection{Tensor product of two irreducible representations\label{sec:symmetry:tp:irrep}}
Let vector spaces $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ transform as irreps $j_A$ and $j_B$ respectively. The space $\mathbb{V}^{(AB)}$ is, in general, reducible and decomposes as
\begin{equation}
\mathbb{V}^{(AB)} \cong \bigoplus_{j_{AB}} \mathbb{V}_{j_{AB}}^{(AB)},
\label{eq:tensorirrep}
\end{equation}
where the total spin $j_{AB}$ assumes all values in the range
\begin{equation}
\boxed{
\begin{split}
&j_{AB}:\{|j_A-j_B|, |j_A-j_B|+1, \ldots, j_A+j_B\}. \\
&\mbox{(Fusion rules)}
\end{split}
\label{eq:fusionrules}
}
\end{equation}
Let $\ket{j_Am_A} \in \mathbb{V}^{(A)}$ and $\ket{j_Bm_B} \in \mathbb{V}^{(B)}$ denote the spin basis of the respective vector spaces. Then the vectors
\begin{equation}
\ket{j_{A}m_{j_A};j_B m_{j_B}} \equiv \ket{j_{A}m_{j_A}} \otimes \ket{j_B m_{j_B}} \label{eq:prod2}
\end{equation}
form a basis of $\mathbb{V}^{(AB)}$. Introduce a coupled basis $\ket{j_{AB}m_{AB}} \in \mathbb{V}^{(AB)}$ that fulfills
\begin{align}
{\textbf{J}^2}^{(AB)}\ket{j_{AB} m_{j_{AB}}} &= j_{AB}(j_{AB}+1) \ket{j_{AB} m_{j_{AB}}}, \nonumber \\
\hat{J}_z^{(AB)}~\ket{j_{AB} m_{j_{AB}}}
&= m_{j_{AB}}~\ket{j_{AB} m_{j_{AB}}}.
\end{align}
The coupled basis is related to the product basis (\ref{eq:prod2}) by means of the transformation
\begin{equation}
\boxed{
\begin{split}
\ket{j_{AB} m_{j_{AB}}} = \sum_{m_{j_{A}} m_{j_{B}}} \cfusespin{A}{B}{AB}
\ket{j_{A} m_{j_{A}} ;j_{B} m_{j_{B}}}.
\end{split}
\label{eq:cg}
}
\end{equation}
Here
\begin{equation}
\cfusespin{A}{B}{AB} \equiv \braket{j_{A}m_{j_A};j_{B}m_{j_B}}{j_{AB}m_{j_{AB}}}
\end{equation}
are the \textit{Clebsch-Gordan coefficients}, which vanish unless $j_A, j_B$ and $j_{AB}$ are compatible, that is, $j_A, j_B$ and $j_{AB}$ fulfill
\begin{equation}
|j_A-j_B| \leq j_{AB} \leq j_A+j_B, \label{eq:compatiblej}
\end{equation}
and $m_{j_A}, m_{j_B}$ and $m_{j_{AB}}$ fulfill
\begin{equation}
m_{j_{AB}} = m_{j_A} + m_{j_B}.\label{eq:compatiblem}
\end{equation}
The product basis can be expressed in terms of the coupled basis as
\begin{equation}
\boxed{
\begin{split}
\ket{j_{A} m_{j_{A}};j_{B} m_{j_{B}}} = \sum_{m_{j_{AB}}} \csplitspin{AB}{A}{B} \ket{j_{AB}m_{j_{AB}}},\label{eq:revcg}
\end{split}
}
\end{equation}
where
\begin{equation}
\csplitspin{AB}{A}{B} \equiv \cfusespin{A}{B}{AB}.
\label{eq:splitcg}
\end{equation}
We graphically represent tensors $C^{\mbox{\tiny \,fuse}}$ and $C^{\mbox{\tiny \,split}}$ differently from usual tensors, as shown in Fig.~\ref{fig:cg}(i).
Tensor $\cfusespin{A}{B}{AB}$ is depicted by means of two incoming lines and one outgoing line that emerge from a point. The outgoing line corresponds to the spin index $(j_{AB}, m_{j_{AB}})$. The incoming lines that are encountered first and second when proceeding \textit{clockwise} from the outgoing line correspond to the spin indices $(j_{A}, m_{j_{A}})$ and $(j_{B}, m_{j_{B}})$ respectively.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{cg}
\end{center}
\caption{(i) The graphical representation of the Clebsch-Gordan tensors: $C^{\mbox{\tiny \,fuse}}$ and $C^{\mbox{\tiny \,split}}$. (ii) Tensors $C^{\mbox{\tiny \,fuse}}$ and $C^{\mbox{\tiny \,split}}$ are isometries and thus yield the Identity when contracted pairwise either as (i) or (ii), Eq.~(\ref{eq:cgunitary}). \label{fig:cg}}
\end{figure}
Analogously, tensor $\csplitspin{AB}{A}{B}$ is depicted by means of one incoming line and two outgoing lines that emerge from a point. The spin index $(j_{AB}, m_{j_{AB}})$ corresponds to the incoming line in this case while spin indices $(j_{A}, m_{j_{A}})$ and $(j_{B}, m_{j_{B}})$ correspond to the outgoing lines in the order in which they are encountered when proceeding \textit{counterclockwise} from the incoming line.
Tensor $C^{\mbox{\tiny \,fuse}}$ and tensor $C^{\mbox{\tiny \,split}}$ fulfill the orthogonality identities,
\begin{align}
\sum_{m_{j_A}m_{j_B}}\csplitt{j_{AB}m_{j_{AB}}}{j_{A}m_{j_{A}}}{j_{B}m_{j_{B}}}\cdot\cfuse{j_{A}m_{j_{A}}}{j_{B}m_{j_{B}}}{j'_{AB}m_{j'_{AB}}}
&=\delta_{j_{AB}j'_{AB}}\delta_{m_{j_{AB}}m_{j'_{AB}}} \nonumber \\
\sum_{j_{AB}m_{j_{AB}}}\cfuse{j_{A}m_{j_{A}}}{j_{B}m_{j_{B}}}{j_{AB}m_{j_{AB}}}\cdot\csplitt{j_{AB}m_{j_{AB}}}{j'_{A}m_{j'_{A}}}{j'_{B}m_{j'_{B}}}
&=\delta_{j_{A}j'_{A}}\delta_{m_{j_{A}}m_{j'_{A}}} \delta_{j_{B}j'_{B}}\delta_{m_{j_{B}}m_{j'_{B}}}
\label{eq:cgunitary}
\end{align}
The special graphical representations for these tensors allows one to depict the above identities in an intuitive way, as shown in Fig.~\ref{fig:cg}.(ii)-(iii).
\textbf{Example 6: } Let both vector spaces $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ transform as a spin $\frac{1}{2}$ irrep, $j_A=j_B=\frac{1}{2}$ (Example 2). The space $\mathbb{V}^{(AB)}$ decomposes into a direct sum of irreps,
\begin{align}
\mathbb{V}^{(AB)} &\cong \mathbb{V}^{(A)} \otimes \mathbb{V}^{(B)} \cong \mathbb{V}^{(AB)}_{0} \oplus \mathbb{V}^{(AB)}_{1}. \label{eq:eg40}
\end{align}
A basis can be introduced in $\mathbb{V}^{(AB)}$ in terms of the spin basis [Eq.~(\ref{eq:basiseg1})] of $\mathbb{V}^{(A)}$ and of $\mathbb{V}^{(B)}$. The coupled basis of $\mathbb{V}^{(AB)}$,
\begin{align}
&\ket{j_{AB}=0, m_{j_{AB}}=0},\nonumber \\
&\ket{j_{AB}=1, m_{j_{AB}}=-1},~\ket{j_{AB}=1, m_{j_{AB}}=0},~\ket{j_{AB}=1, m_{j_{AB}}=1}, \nonumber
\end{align}
is related to the product basis by means of the Clebsch-Gordan coefficients,
\begin{align}
\ket{j_{AB}=0, m_{AB}=0}~~\textbf{=}&~~\cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\half}{00} \ket{j_A=\frac{1}{2}, m_A=\frac{-1}{2};j_B=\frac{1}{2}, m_B= \frac{1}{2}}\nonumber \\
&+\cfuse{\frac{1}{2}\half}{\frac{1}{2}\frac{-1}{2}}{00}\ket{j_A=\frac{1}{2}, m_A=\frac{1}{2};j_B=\frac{1}{2}, m_B=\frac{-1}{2}} \label{eq:halfsinglet}\\
\ket{j_{AB}=1, m_{AB}=-1}~~\textbf{=}&~~\cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\frac{-1}{2}}{1-1}\ket{j_A=\frac{1}{2}, m_A=\frac{-1}{2};j_B=\frac{1}{2}, m_B=\frac{-1}{2}}\\
\ket{j_{AB}=1, m_{AB}=0}~~\textbf{=}&~~\cfuse{\frac{1}{2}\half}{\frac{1}{2}\frac{-1}{2}}{10} \ket{j_A=\frac{1}{2}, m_A=\frac{1}{2};j_B=\frac{1}{2}, m_B=\frac{-1}{2}}\nonumber\\
&+\cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\half}{10}\ket{j_A=\frac{1}{2}, m_A=\frac{-1}{2};j_B=\frac{1}{2}, m_B=\frac{1}{2}} \\
\ket{j_{AB}=1, m_{AB}=1}~~\textbf{=}&~~\cfuse{\frac{1}{2}\half}{\frac{1}{2}\half}{11}\ket{j_A=\frac{1}{2}, m_A=\frac{1}{2};j_B=\frac{1}{2}, m_B=\frac{1}{2}}.\label{eq:eg41}
\end{align}
For completeness, we list below the numerical value of the Clebsch-Gordan coefficients that appear in Eqs.~(\ref{eq:halfsinglet})-(\ref{eq:eg41}),
\begin{align}
\cfuse{\frac{1}{2}\half}{\frac{1}{2}\frac{-1}{2}}{00} &= \frac{1}{\sqrt{2}}, ~~~ \cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\half}{00} = \frac{-1}{\sqrt{2}},~~~\cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\frac{-1}{2}}{1-1} = 1, \nonumber \\
\cfuse{\frac{1}{2}\half}{\frac{1}{2}\frac{-1}{2}}{10} &= \frac{1}{\sqrt{2}},~~~\cfuse{\frac{1}{2}\frac{-1}{2}}{\frac{1}{2}\half}{10} = \frac{1}{\sqrt{2}}, ~~~ \cfuse{\frac{1}{2}\half}{\frac{1}{2}\half}{11} = 1. \nonumber\markend
\end{align}
\subsection{Tensor product of two reducible representations\label{sec:symmetry:tp:general}}
Let us now consider that vector spaces $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ transform reducibly under the action of SU(2). We have
\begin{equation}
\mathbb{V}^{(A)} \cong \bigoplus_{j_{A}}\left(\mathbb{D}^{(A)}_{j_A} \otimes \mathbb{V}^{(A)}_{j_A}\right), ~~~\mathbb{V}^{(B)} \cong \bigoplus_{j_{B}}\left(\mathbb{D}^{(B)}_{j_B} \otimes \mathbb{V}^{(B)}_{j_B}\right).
\label{eq:AandB}
\end{equation}
The space $\mathbb{V}^{(AB)}$ decomposes into a direct sum of irreps,
\begin{equation}
\mathbb{V}^{(AB)} \cong \bigoplus_{j_{AB}} d_{j_{AB}} \mathbb{V}^{(AB)}_{j_{AB}} \cong \bigoplus_{j_{AB}}\left(\mathbb{D}^{(AB)}_{j_{AB}} \otimes \mathbb{V}^{(AB)}_{j_{AB}}\right),
\label{eq:decoVAB}
\end{equation}
where the total spin $j_{AB}$ takes all values that are compatible with \textit{any} pair of irreps $j_A$ and $j_B$.
Let $\ket{j_At_{j_A}m_{j_A}}$ and $\ket{j_Bt_{j_B}m_{j_B}}$ denote the spin basis of $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ respectively. We introduce a coupled basis $\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}} \in \mathbb{V}^{(AB)}$ that fulfills,
\begin{align}
{\textbf{J}^2}^{(AB)}\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}} &= j_{AB}(j_{AB}+1) \ket{j_{AB} t_{j_{AB}} m_{j_{AB}}}, \nonumber \\
\hat{J}_z^{(AB)}~\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}}
&= m_{j_{AB}}~\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}}.
\end{align}
and which is related to the product basis,
\begin{equation}
\ket{j_{A}t_{j_{A}}m_{j_A};j_B t_{j_{B}} m_{j_B}} \equiv \ket{j_{A}t_{j_{A}}m_{j_A}} \otimes \ket{j_B t_{j_{B}}m_{j_B}}, \nonumber
\end{equation}
by means of a transformation,
\begin{equation}
\boxed{
\begin{split}
\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}} =
\sum_{t_{j_{A}} t_{j_{B}}}\sum_{m_{j_{A}} m_{j_{B}}} \fusespin{A}{B}{AB}
\ket{j_{A} t_{j_{A}}m_{j_{A}};j_{B} t_{j_{B}} m_{j_{B}}}.
\end{split}
\label{eq:basischange}
}
\end{equation}
The components $\fusespin{A}{B}{AB}$ can be expressed in terms of the Clebsch-Gordan coefficients as
\begin{equation}
\boxed{
\begin{split}
\fusespin{A}{B}{AB} = \tfusespin{A}{B}{AB} \cfusespin{A}{B}{AB}.
\end{split}
\label{eq:basischange1}
}
\end{equation}
Let us explain how this expression is obtained. From the definition, Eq.~(\ref{eq:basischange}), we have
\begin{align}
\fusespin{A}{B}{AB}\equiv
\braket{j_{AB} t_{j_{AB}} m_{j_{AB}}}{j_{A} t_{j_{A}}m_{j_{A}};j_{B} t_{j_{B}} m_{j_{B}}}.\label{eq:fusedef}
\end{align}
According to the direct sum decomposition, Eq.~(\ref{eq:decoVAB}), each vector $\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}}$ belongs to the subspace $\mathbb{D}^{(AB)}_{j_{AB}} \otimes \mathbb{V}^{(AB)}_{j_{AB}}$ where it factorizes as
\begin{equation}
\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}} = \ket{j_{AB} t_{j_{AB}}} \otimes \ket{j_{AB} m_{j_{AB}}}.
\end{equation}
Similarly, we can factorize vectors $\ket{j_{A} t_{j_{A}} m_{j_{A}}}$ and $\ket{j_{B} t_{j_{B}} m_{j_{B}}}$. The expression Eq.~(\ref{eq:basischange1}) can then be obtained by substituting these factorizations into Eq.~(\ref{eq:fusedef}) and re-arranging the terms as shown below,
\begin{align}
\fusespin{A}{B}{AB}&=\braket{j_{AB}t_{j_{AB}}}{j_{A}t_{j_A};j_{B}t_{j_B}} \braket{j_{AB}m_{j_{AB}}}{j_{A}m_{j_A};j_{B}m_{j_B}},\nonumber \\
&= \tfusespin{A}{B}{AB} \cfusespin{A}{B}{AB},\nonumber
\end{align}
where $\ket{j_{A}t_{j_A};j_{B}t_{j_B}} \equiv \ket{j_{A}t_{j_A}} \otimes \ket{j_{B}t_{j_B}}$.
Here $X^{\mbox{\tiny \,fuse}}$ is a one to one map that relates vectors $\ket{j_{AB} t_{j_{AB}}} \in \mathbb{D}^{(AB)}_{j_{AB}}$ to the vectors $\ket{j_{A}t_{j_A};j_B t_{j_B}} \in \mathbb{D}^{(AB)}_{j_{AB}}$. It can be regarded as a rank-$3$ tensor such that each component of $X^{\mbox{\tiny \,fuse}}$ is either a zero or a one. We have,
\begin{equation}
\tfusespin{A}{B}{AB} \!\!\!\!\!= \left\{
\begin{array}{cc} 1&\mbox{ if } \ket{j_{AB}t_{j_{AB}}} = \ket{j_{A}t_{j_A};j_{B}t_{j_B}},\\
0&\mbox{otherwise }.
\end{array} \right.
\end{equation}
The product basis can, in turn, be expressed in terms of the coupled basis,
\begin{equation}
\boxed{
\begin{split}
\ket{j_{A} t_{j_{A}}m_{j_{A}};j_{B} t_{j_{B}} m_{j_{B}}} =
\sum_{t_{j_{AB}}}\sum_{m_{j_{AB}}}\splitspin{AB}{A}{B}\ket{j_{AB} t_{j_{AB}} m_{j_{AB}}},\label{eq:revbasischange}
\end{split}
}
\end{equation}
where
\begin{equation}
\boxed{
\begin{split}
\splitspin{AB}{A}{B} =\tsplitspin{AB}{A}{B}\cdot\csplitspin{AB}{A}{B},
\label{eq:u1split1}
\end{split}
}
\end{equation}
and
\begin{equation}
\tsplitspin{AB}{A}{B} \equiv \tfusespin{A}{B}{AB}.
\label{eq:u1split2}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{su2fuse}
\end{center}
\caption{The graphical representations of (i) the fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ and (ii) the splitting tensor. For fixed values of $j_A, j_B$ and $j_{AB}$ each of these tensors factorizes into a $X$ and a $C$ tensor.\label{fig:su2fuse}}
\end{figure}
We refer to the tensors $\Upsilon^{\mbox{\tiny \,fuse}}$ and $\Upsilon^{\mbox{\tiny \,split}}$ as the \textit{fusing} tensor and the \textit{splitting} tensor respectively, since they will play an instrumental role in fusing and splitting indices of an SU(2)-invariant tensor. The special graphical representation of these tensors and their decomposition into $X$ and $C$ tensors is shown in Fig.~\ref{fig:su2fuse}. Tensors $\tfusespin{A}{B}{AB}$ and $\tsplitspin{AB}{A}{B}$ are graphically represented by means of a circle enclosing an arrow head and three lines emerging from the circle corresponding to the three indices of the tensors. The three lines in the diagrams of $X^{\mbox{\tiny \,fuse}}$ and $X^{\mbox{\tiny \,split}}$ correspond to the degeneracy indices $(j_A, t_{j_A}), (j_B, t_{j_B})$ and $(j_{AB}, t_{j_{AB}})$ by using the same assignment rules that were introduced for tensors $C^{\mbox{\tiny \,fuse}}$ and $C^{\mbox{\tiny \,split}}$ respectively. Other features of the graphical representation include an arrow head that is placed within the circle to indicate the direction of the fusion and a small rectangle, placed on the line carrying the coupled spins, that represents a permutation of basis elements.
We notice that tensor $X^{\mbox{\tiny \,fuse}}$ can be decomposed into two pieces. The first piece (depicted as the circle enclosing an arrow head) expresses a basis $\{\ket{j_{A}t_{j_{A}}; j_{B}t_{j_{B}} \equiv \ket{j_{A}t_{j_A}} \otimes \ket{j_{B}t_{j_B}}}\}$ of $\mathbb{D}^{(AB)}$ as the direct product of the basis $\{\ket{j_{A}t_{j_A}}\}$ of $\mathbb{D}^{(A)}$ and the basis $\{\ket{j_{B}t_{j_B}}\}$ of $\mathbb{D}^{(B)}$. Note that this procedure does not always lead to the set $\{\ket{j_{A}t_{j_{A}}; j_{B}t_{j_{B}}}\}$ being ordered such that states corresponding to the same total spin $j_{AB}$ are adjacent to each other within the set. However, we require that the basis associated to an index be maintained as such (this ensures, for example, that an SU(2)-invariant matrix is block diagonal when expressed in such a basis). This ordering is achieved by means of the second piece (depicted as the small rectangle): a permutation of basis states $\{\ket{j_{A}t_{j_{A}}; j_{B}t_{j_{B}}}\}$ that reorganizes them according to their total spin $j_{AB}$, so that they are identified in an one-to-one correspondence with the coupled states $\{\ket{j_{AB}t_{j_{AB}}}\}$. In particular, this description of the tensors $X^{\mbox{\tiny \,fuse}}$ and $X^{\mbox{\tiny \,split}}$ can be exploited to multiply together several such tensors, such as in Fig.~\ref{fig:fmove1}(iv), in a fast way.
By construction, a resolution of Identity can be obtained in terms of tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ and tensor $\Upsilon^{\mbox{\tiny \,split}}$, as shown in Fig.~\ref{fig:su2fuse}(ii)-(iii).
\textbf{Example 7: } Let vector spaces $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ correspond to the vector space of Example 4, that is,
\begin{align}
\mathbb{V}^{(A)} &\cong 3\mathbb{V}^{(A)}_{\frac{1}{2}} \cong \mathbb{D}_{\frac{1}{2}}^{(A)}\otimes\mathbb{V}^{(A)}_{\frac{1}{2}}, \nonumber \\
\mathbb{V}^{(B)} &\cong 3\mathbb{V}^{(B)}_{\frac{1}{2}} \cong \mathbb{D}_{\frac{1}{2}}^{(B)} \otimes \mathbb{V}^{(B)}_{\frac{1}{2}}.
\end{align}
The space $\mathbb{V}^{(AB)}$ decomposes as
\begin{align}
\mathbb{V}^{(AB)} \cong \left(\mathbb{D}^{(AB)}_{0} \otimes \mathbb{V}^{(AB)}_{0}\right) \oplus \left(\mathbb{D}^{(AB)}_{1} \otimes \mathbb{V}^{(AB)}_{1}\right),
\label{eq:eg50}
\end{align}
where,
\begin{align}
\mathbb{V}_{\frac{1}{2}}^{(A)} \otimes \mathbb{V}_{\frac{1}{2}}^{(B)} \cong (\mathbb{V}^{(AB)}_{0} \oplus \mathbb{V}^{(AB)}_{1}), \label{eq:eg71}
\end{align}
and
\begin{align}
\mathbb{D}^{(AB)}_{0} &\cong \mathbb{D}_{\frac{1}{2}}^{(A)} \otimes \mathbb{D}_{\frac{1}{2}}^{(B)}, \label{eq:degbasis1}\\
\mathbb{D}^{(AB)}_{1} &\cong \mathbb{D}_{\frac{1}{2}}^{(A)} \otimes \mathbb{D}_{\frac{1}{2}}^{(B)}.\label{eq:eg72}
\end{align}
Recall that the basis of the l.h.s. and r.h.s. of Eq.~(\ref{eq:eg71}) are related by the transformations Eqs.~(\ref{eq:halfsinglet})-(\ref{eq:eg41}). Let us now consider how the basis of the l.h.s. and r.h.s. of Eq.~(\ref{eq:degbasis1}) and of Eq.~(\ref{eq:eg72}) are related. In Eq.~(\ref{eq:degbasis1}), for instance, the vectors
\begin{equation}
\ket{j_{AB}=0, t_{j_{AB}}} \in \mathbb{D}^{(AB)}_{0},~~t_{j_{AB}}=1,2,\ldots,9, \nonumber
\end{equation}
are related to the vectors
\begin{equation}
\ket{j_A = \frac{1}{2}, t_{j_A}; j_B = \frac{1}{2}, t_{j_B}}, \in \mathbb{D}^{(AB)}_{0} ~~t_{j_A}, t_{j_B} = 1,2,3, \nonumber
\end{equation}
in straightforward way by associating the vectors in a one to one fashion in the order in which they appear in the respective basis. For example, the change of basis maps the vector
\begin{equation}
\ket{j_{AB}=0, t_{j_{AB}}=1} \mbox{ to } \ket{j_A = \frac{1}{2}, t_{j_A}=1; j_B = \frac{1}{2}, t_{j_B}=1} \nonumber
\end{equation}
and the vector
\begin{equation}
\ket{j_{AB}=0, t_{j_{AB}}=2} \mbox{ to } \ket{j_A = \frac{1}{2}, t_{j_A}=2; j_B = \frac{1}{2}, t_{j_B}=1} \nonumber
\end{equation}
and so on. The basis of the l.h.s. and r.h.s. of Eq.~(\ref{eq:eg72}) are related in a similar way. This one to one mapping can be encoded into $\tfusespin{A}{B}{AB}$ by setting the numerical value of the following components equal to one,
\begin{align}
\tfuse{\frac{1}{2} 1}{\frac{1}{2} 1}{01} &,~~ \tfuse{\frac{1}{2} 1}{\frac{1}{2} 2}{02} ,~~ \tfuse{\frac{1}{2} 1}{\frac{1}{2} 3}{03}, \nonumber \\
\tfuse{\frac{1}{2} 2}{\frac{1}{2} 1}{01} &,~~ \tfuse{\frac{1}{2} 2}{\frac{1}{2} 2}{02} ,~~ \tfuse{\frac{1}{2} 2}{\frac{1}{2} 3}{03}, \nonumber\\
\tfuse{\frac{1}{2} 3}{\frac{1}{2} 1}{01} &,~~ \tfuse{\frac{1}{2} 3}{\frac{1}{2} 2}{02} ,~~ \tfuse{\frac{1}{2} 3}{\frac{1}{2} 3}{03},\nonumber
\end{align}
and
\begin{align}
\tfuse{\frac{1}{2} 1}{\frac{1}{2} 1}{11} &,~~ \tfuse{\frac{1}{2} 1}{\frac{1}{2} 2}{12} ,~~ \tfuse{\frac{1}{2} 1}{\frac{1}{2} 3}{13}, \nonumber\\
\tfuse{\frac{1}{2} 2}{\frac{1}{2} 1}{11} &,~~ \tfuse{\frac{1}{2} 2}{\frac{1}{2} 2}{12} ,~~ \tfuse{\frac{1}{2} 2}{\frac{1}{2} 3}{13}, \nonumber \\
\tfuse{\frac{1}{2} 3}{\frac{1}{2} 1}{11} &,~~ \tfuse{\frac{1}{2} 3}{\frac{1}{2} 2}{12} ,~~ \tfuse{\frac{1}{2} 3}{\frac{1}{2} 3}{13}.\nonumber\markend
\end{align}
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{su2fuse1}
\end{center}
\caption{Tensors $\Upsilon^{\mbox{\tiny \,fuse}}$ and $\Upsilon^{\mbox{\tiny \,split}}$ are unitary and thus yield the Identity when contracted pairwise either as (i) or (ii).
\label{fig:su2fuse1}}
\end{figure}
\textbf{Example 8: } As another example of the change of basis $X^{\mbox{\tiny \,fuse}}$, consider that $\mathbb{V}^{(A)}$ and $\mathbb{V}^{(B)}$ correspond to the vector spaces of Example 3 and Example 5 respectively. That is,
\begin{align}
\mathbb{V}^{(A)} &\cong \mathbb{V}^{(A)}_{1} \cong (\mathbb{D}^{(A)}_{1} \otimes \mathbb{V}^{(A)}_{1}), \nonumber \\
\mathbb{V}^{(B)} &\cong 2\mathbb{V}^{(B)}_{0} \oplus \mathbb{V}^{(B)}_{1} \cong (\mathbb{D}_{0}^{(B)}\otimes\mathbb{V}^{(B)}_{0}) \oplus (\mathbb{D}_{1}^{(B)}\otimes\mathbb{V}^{(B)}_{1}).
\end{align}
The space $\mathbb{V}^{(AB)}$ decomposes as
\begin{align}
\mathbb{V}^{(AB)} \cong (\mathbb{D}_{0}^{(AB)} \otimes \mathbb{V}^{(AB)}_{0}) \oplus(\mathbb{D}_{1}^{(AB)} \otimes \mathbb{V}^{(AB)}_{1})\oplus (\mathbb{D}_{2}^{(AB)} \otimes\mathbb{V}^{(AB)}_{2}),
\label{eq:eg60}
\end{align}
where
\begin{align}
\mathbb{D}_{0}^{(AB)} &\cong \mathbb{D}_{1}^{(A)} \otimes \mathbb{D}_{1}^{(B)} \label{eq:x1} \\
\mathbb{D}_{1}^{(AB)} &\cong (\mathbb{D}_{0}^{(A)} \otimes \mathbb{D}_{1}^{(B)}) \oplus (\mathbb{D}_{1}^{(A)} \otimes \mathbb{D}_{0}^{(B)}) \label{eq:x2} \\
\mathbb{D}_{2}^{(AB)} &\cong \mathbb{D}_{1}^{(A)} \otimes \mathbb{D}_{1}^{(B)}. \label{eq:x3}
\end{align}
The transformation that relates the bases of l.h.s. and r.h.s. of Eq.~(\ref{eq:x1}) and of Eq.~\ref{eq:x3}) is straightforward, we set
\begin{align}
\tfuse{0 1}{1 1}{0 1} = \tfuse{1 1}{1 1}{2 1} = 1. \nonumber
\end{align}
The basis of the l.h.s. and r.h.s. of Eq.~(\ref{eq:x2}) can be related by mapping the three vectors
\begin{equation}
\ket{j_{AB}=1, t_{j_{AB}}=1,2,3} \in \mathbb{D}_{1}^{(AB)},\nonumber
\end{equation}
in a one to one manner, to the two vectors
\begin{equation}
\ket{j_A = 1, t_{j_A}=1; j_B = 0, t_{j_B}=1,2} \in (\mathbb{D}_{0}^{(A)} \otimes \mathbb{D}_{1}^{(B)}),\nonumber
\end{equation}
and the vector
\begin{equation}
\ket{j_A = 0, t_{j_A}=1; j_B = 1, t_{j_B}=1} \in (\mathbb{D}_{1}^{(A)} \otimes \mathbb{D}_{0}^{(B)}). \nonumber
\end{equation}
This is encoded into $X^{\mbox{\tiny \,fuse}}$ by setting
\begin{align}
\tfuse{0 2}{1 1}{1 2} = \tfuse{1 1}{1 1}{0 1} = \tfuse{1 1}{1 1}{1 3} = 1.\markend \nonumber
\end{align}
\section{Review: Fusion trees \label{sec:fusiontree}}
When considering the tensor product of more than two representations one can obtain several coupled bases of the product space. These correspond to taking the product of the vector spaces according to different sequences of pairwise products. The spaces are linearly ordered in a given way and we only consider a pairwise product of `adjacent' spaces in this linear ordering. For example, when considering the tensor product of three representations,
\begin{equation}
\mathbb{V}^{(ABC)} \cong \mathbb{V}^{(A)} \otimes \mathbb{V}^{(B)} \otimes \mathbb{V}^{(C)},
\end{equation}
one can consider either the pairwise products
\begin{align}
\mathbb{V}^{(D)} &\cong \mathbb{V}^{(A)} \otimes \mathbb{V}^{(B)}, \nonumber \\
\mathbb{V}^{(ABC)} &\cong \mathbb{V}^{(D)} \otimes \mathbb{V}^{(C)}, \label{eq:order1}
\end{align}
or the pairwise products
\begin{align}
\mathbb{V}^{(E)} &\cong \mathbb{V}^{(B)} \otimes \mathbb{V}^{(C)}, \nonumber \\
\mathbb{V}^{(ABC)} &\cong \mathbb{V}^{(A)} \otimes \mathbb{V}^{(E)}. \label{eq:order2}
\end{align}
Considering the tensor product one way or the other leads to two different coupled bases in $\mathbb{V}^{(ABC)}$.
More generally, there exist several choices of a coupled basis in the tensor product of $L$ representations, Eqs.~(\ref{eq:latticespace})-(\ref{eq:latticerep}). A useful way to specify a particular sequence of pairwise products is by means of a \textit{fusion tree}.
A fusion tree, denoted $\boldsymbol{\tau}$, is a \textit{directed} trivalent tree such that each node of the tree represents the tensor product of two incoming spaces into the outgoing space. The tree has a total of $L+1$ open links which correspond to the $L$ vector spaces $\mathbb{V}^{(1)}, \mathbb{V}^{(2)}, \ldots, \mathbb{V}^{(L)}$ and the product space $\mathbb{V}$. The internal links correspond to the intermediate product spaces that appear in a sequence of pairwise products. Figure~\ref{fig:fusion} illustrates two different fusions trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$ that correspond to two different ways of considering the tensor product of three and of four representations. The sequence of fusions proceeds from top to bottom.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{fusion}
\end{center}
\caption{Examples of two fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$ that describe two different ways of considering the tensor product of (i) three vector spaces $\mathbb{V}^{(A)}, \mathbb{V}^{(B)}$ and $\mathbb{V}^{(C)}$ and (ii) four vector spaces $\mathbb{V}^{(A)}, \mathbb{V}^{(B)}, \mathbb{V}^{(C)}$ and $\mathbb{V}^{(D)}$. Fusion proceeds from top to bottom.
\label{fig:fusion}}
\end{figure}
A fusion tree can also be specified as a list of fusions. For example, the fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$ depicted in Fig.~\ref{fig:fusion}(i) can be specified as
\begin{align}
\boldsymbol{\tau} &\equiv \{A,B\rightarrow D; D,C\rightarrow (ABC) \}, \nonumber \\
\boldsymbol{\tau}' &\equiv \{B,C\rightarrow E; A,E\rightarrow (ABC) \}. \nonumber
\end{align}
Fusion trees play an important role in our discussion. In the present context, a fusion tree characterizes a coupled basis of a tensor product space. In Sec.~\ref{sec:symTensor} fusion trees are also used to characterize different canonical decompositions of an SU(2)-invariant tensor.
In the remainder of the section we consider coupled bases that are labeled by different fusion trees. We define the unitary transformation that relates two such coupled bases. This transformation is important since it also relates two different canonical decompositions of an SU(2)-invariant tensor, as discussed in Sec.~\ref{sec:symTensor}. For purpose of illustration, we first characterize the coupled basis in the simple case of the tensor product of three representations before proceeding to the generic case of $L$ representations.
\subsection{Tensor product of three irreps}
Let vector spaces $\mathbb{V}^{(A)}, \mathbb{V}^{(B)}$ and $\mathbb{V}^{(C)}$ transform as irreps $j_A, j_B$ and $j_C$ respectively. The space $\mathbb{V}^{(ABC)}$ is in general reducible, and may contain several copies of an irrep $j_{ABC}$. Let us first consider the sequence (\ref{eq:order1}) of tensor products corresponding to a fusion tree $\boldsymbol{\tau}$. The vector spaces $\mathbb{V}^{(D)}$ and $\mathbb{V}^{(ABC)}$ decompose as
\begin{equation}
\mathbb{V}^{(D)} \cong \bigoplus_{j_D} \mathbb{V}^{(D)}_{j_D},\label{eq:intermediate1}
\end{equation}
and
\begin{equation}
\mathbb{V}^{(ABC)} \cong \bigoplus_{j_{ABC}, j_D} \mathbb{V}_{j_{ABC}, j_D}^{(ABC)}. \label{eq:tensorirrep3}
\end{equation}
Notice that we can use the values of $j_D$ that appear on the r.h.s. of Eq.~(\ref{eq:intermediate1}) to label different copies of $j_{ABC}$ that appear on the r.h.s. of Eq.~(\ref{eq:tensorirrep3}). Thus, a coupled basis of $\mathbb{V}^{(ABC)}$ can be labeled as $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau};j_D}$.
Let $(\hat{Q}^{j_D}_{j_A j_B j_C j_{ABC}})_{m_{j_A} m_{j_B} m_{j_C} m_{j_{ABC}}}$ denote the transformation from the product basis\\ $\ket{j_Am_{j_A}}\otimes\ket{j_Bm_{j_B}}\otimes\ket{j_Cm_{j_C}}$ to the coupled basis $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau}; j_D}$. This change of basis can be expressed in terms of Clebsch-Gordan coefficients as
\begin{align}
&(\hat{Q}^{j_D}_{j_A j_B j_C j_{ABC}})_{m_{j_A} m_{j_B} m_{j_C} m_{j_{ABC}}}\equiv
\sum_{m_{j_D}}\cfusespin{A}{B}{D} \cdot \cfusespin{D}{C}{ABC},
\label{eq:cob1}
\end{align}
where $\cfusespin{A}{B}{D}$ relates the basis $\ket{j_A, m_{j_A}}~\otimes~\ket{j_B,m_{j_B}}$ to the intermediate basis $\ket{j_Dm_{j_D}}$, and $\cfusespin{D}{C}{ABC}$ relates this intermediate basis to the coupled basis $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau};j_D}$.
Alternatively, we can first consider the tensor product $\mathbb{V}^{(B)} \otimes \mathbb{V}^{(C)}$ (corresponding to the sequence (\ref{eq:order2}) of tensor products characterized by another fusion tree $\boldsymbol{\tau}'$),
\begin{equation}
\mathbb{V}^{(E)} \cong \mathbb{V}^{(B)} \otimes \mathbb{V}^{(C)} \cong \bigoplus_{j_E} \mathbb{V}^{(E)}_{j_E}, \label{eq:intermediate2}
\end{equation}
and use irrep $j_E$ to label another coupled basis $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau}';j_E}$ of $\mathbb{V}^{(ABC)}$. Denote by $(\hat{Q}'^{j_E}_{j_A j_B j_Cj_{ABC}})_{m_{j_A} m_{j_B} m_{j_C} m_{j_{ABC}}}$ the change of basis to this new coupled basis. In terms of Clebsch-Gordan coefficients we have
\begin{align}
(\hat{Q}'^{j_E}_{j_A j_B j_C j_{ABC}})_{m_{j_A} m_{j_B} m_{j_C} m_{j_{ABC}}} \equiv
\sum_{m_{j_E}}\cfusespin{B}{C}{E}\cdot\cfusespin{A}{E}{ABC}.
\label{eq:cob2}
\end{align}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{fmove}
\end{center}
\caption{(i) In the tensor product of three irreps, two coupled bases that are labeled by fusion trees $\boldsymbol{\tau}$ (\ref{eq:order1}) and $\boldsymbol{\tau}'$ (\ref{eq:order2}) are related to one another by means of the recoupling coefficients $\hat{F}^{j_D j_E}_{j_A j_B j_C j_{ABC}}$. (ii) A recoupling coefficient is given as the `scalar product' of $\hat{Q}$ and $\hat{Q}'$.
\label{fig:fmove}}
\end{figure}
The two coupled bases $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau};j_D}$ and $\ket{j_{ABC}m_{j_{ABC}};\boldsymbol{\tau}';j_E}$ are related by a transformation that is given by a rank-$6$ tensor $\hat{F}$ with components $\hat{F}^{j_Dj_E}_{j_Aj_Bj_Cj_{ABC}}$,
\begin{equation}
\boxed{\hat{Q}'^{j_E}_{j_A j_B j_C j_{ABC}} = \sum_{j_D}\hat{F}^{j_Dj_E}_{j_Aj_Bj_Cj_{ABC}} \hat{Q}^{j_D}_{j_A j_B j_C j_{ABC}}.}
\label{eq:fmove}
\end{equation}
Here $\hat{F}^{j_Dj_E}_{j_Aj_Bj_Cj_{ABC}}$ are the \textit{recoupling coefficients}\citep{Cornwell97} of SU(2). By using Eqs.~(\ref{eq:cob1}) and (\ref{eq:cob2}), the recoupling coefficients can be explicitly expressed in terms of Clebsch-Gordan coefficients,
\begin{align}
\hat{F}^{j_Dj_E}_{j_Aj_Bj_Cj_{ABC}} &\equiv \frac{1}{2j_{ABC}+1} \times \nonumber \\
&\sum_{\textbf{m}} \left(\cfusespin{A}{B}{D} \cfusespin{D}{C}{ABC}\right. \nonumber\\
&~~~\left.\cfusespin{B}{C}{E} \cfusespin{D}{C}{ABC}\right),\label{eq:f}
\end{align}
where $\textbf{m} \equiv \{m_{j_D},m_{j_E},m_{j_A},m_{j_B},m_{j_C},m_{j_{ABC}}\}$. Notice that, since the $m$'s are summed over, the recoupling coefficients depend only on the $j$'s. Also recall that the recoupling coefficients are proportional to the 6-j symbols of the group,
\begin{equation}
\hat{F}^{j_Dj_E}_{j_Aj_Bj_Cj_{ABC}} = \alpha \left\{\begin{array}{ccc} j_A&j_B&j_D\\j_C&j_{ABC}&j_E \end{array}\right\},
\label{eq:sixj}
\end{equation}
where
\begin{equation}
\alpha \equiv (-1)^{(j_A+j_B+j_C+j_{ABC})}\sqrt{(2j_D+1)(2j_E+1)}.
\end{equation}
\subsection{Tensor product of three reducible representations}
Consider the action of SU(2) on the space $\mathbb{V}^{(ABC)}$,
\begin{equation}
\mathbb{V}^{(ABC)} \cong \mathbb{V}^{(A)}\otimes \mathbb{V}^{(B)} \otimes \mathbb{V}^{(C)},\nonumber
\end{equation}
where $\mathbb{V}^{(A)}, \mathbb{V}^{(B)}$ and $\mathbb{V}^{(C)}$ are reducible representations of SU(2). It induces a decomposition
\begin{equation}
\mathbb{V}^{(ABC)} \cong \bigoplus_{j_{ABC}} \left(\mathbb{D}^{(ABC)}_{j_{ABC}} \otimes \mathbb{V}^{(ABC)}_{j_{ABC}}\right),
\label{eq:decoVABC}
\end{equation}
where $j_{ABC}$ takes all values that are compatible with any $j_A, j_B$ and $j_C$.
Extending the argument for irreps, we can relate the coupled basis of $\mathbb{V}^{(ABC)}$ to the product basis by first considering the sequence (\ref{eq:order1}) of tensor products and using two $\Upsilon^{\mbox{\tiny \,fuse}}$ tensors
\begin{equation}
\fuse{A}{B}{D},~~~\fuse{D}{C}{(ABC)}
\end{equation}
to relate at each step the coupled basis with the product basis. Alternatively, we can consider the sequence (\ref{eq:order2}) of tensor products and use the different set of fusing tensors
\begin{equation}
\fuse{B}{C}{E},~~~\fuse{A}{E}{(ABC)}
\end{equation}
to relate the product basis to the coupled basis at each step. The respective change of basis transformation for the two cases is depicted in Fig.~\ref{fig:fmove1}(i).
The two coupled bases, so obtained, are related by means of a matrix $\hat{\Gamma}$ that decomposes, according to Schur's Lemma [Eq.~(\ref{eq:Schur})] as
\begin{equation}
\boxed{\hat{\Gamma} = \bigoplus_{j_{ABC}} (\hat{D}_{j_{ABC}} \otimes \hat{I}_{j_{ABC}}),} \label{eq:gammasplit}
\end{equation}
where the components of $\hat{D}_{j_{ABC}}$ can be expressed in terms of recoupling coefficients. This decomposition can be derived as follows.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{fmove1}
\end{center}
\caption{(i) The change of basis from the product basis to two different coupled bases in the fusion of three reducible representations is given in terms of two fusing tensors. (ii) The two coupled basis are related by means of the matrix $\hat{\Gamma}$ [Eq.~(\ref{eq:gammasplit})] that is obtained by contracting a tensor network made of tensors $\Upsilon^{\mbox{\tiny \,fuse}}$ and tensors $\Upsilon^{\mbox{\tiny \,split}}$. (iii) The components of $\hat{\Gamma}$ are given in terms of recoupling coefficients. This can be seen by performing the contraction piecewise. For fixed values of spins $j_A, j_B, j_C, j_{ABC}, j_D$ and $j_E$ the tensor network decomposes into a tensor network made of $X$ tensors and a spin network. The former is contracted to obtain a matrix $\hat{D}'$ whereas the latter can be replaced by the Identity $\hat{I}_{2j_{ABC}+1}$ and a recoupling coefficient [Fig.~\ref{fig:fmove}(ii)].
\label{fig:fmove1}}
\end{figure}
The matrix $\hat{\Gamma}$ is obtained by contracting the tensor network made of tensors $\Upsilon^{\mbox{\tiny \,fuse}}$ and tensors $\Upsilon^{\mbox{\tiny \,split}}$ that is shown in Fig.~\ref{fig:fmove1}(ii). This contraction can be performed piecewise [Fig.~\ref{fig:fmove1}(iii)]. For fixed values of $j$'s on all links the tensor network factorizes into two pieces since each constituent tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ and tensor $\Upsilon^{\mbox{\tiny \,split}}$ factorizes into a $X$ and a $C$ tensor. The tensor network made of $C$ tensors equates [Fig~\ref{fig:fmove}(ii)] the Identity times the recoupling coefficient $\hat{F}^{j_E}_{j_A j_B j_C j_{ABC}}$. The matrix $\hat{D}_{j_{ABC}}$ in Eq.~(\ref{eq:gammasplit}) is then defined as
\begin{equation}
\hat{D}_{j_{ABC}} \equiv \sum_{j_A j_B j_C j_D j_E} \hat{F}^{j_E}_{j_A j_B j_C j_{ABC}} \hat{D'}_{j_A j_B j_C j_{ABC}}^{j_E}, \label{eq:D}
\end{equation}
where $\hat{D'}_{j_A j_B j_C j_{ABC}}^{j_E}$ denotes the matrix that is obtained by contracting together the $X$ tensors. Here the sum is over all values of $j_A, j_B, j_C, j_D,$ and $j_E$ that are compatible with a given value of $j_{ABC}$.
\subsection{Tensor product of $L$ irreps \label{sec:symmetry:tpL}}
In a similar way, we can consider the tensor product of four irreps; different choices of a coupled basis, corresponding to different fusion trees, are related by the $9-j$ symbols and so on.
More generally, let us consider the tensor product of $L$ representations, Eq.~(\ref{eq:latticespace}), where each space $\mathbb{V}^{(l)}$ ($l=1,2,\ldots, L$) transforms as an irrep $j_{l}$. A coupled basis can be labeled by a fusion tree $\boldsymbol{\tau}$ and the set of intermediate irreps $j_{e_1}, j_{e_2}, \ldots, j_{e_Z}$ that are assigned to the internal links of $\boldsymbol{\tau}$. We denote by
\begin{equation}
\ket{jm_{j}; \boldsymbol{\tau}; j_{e_1} \ldots j_{e_{Z}}}~\in~\mathbb{V} \label{eq:couplebasis}
\end{equation}
such a basis. By attaching the appropriate Clebsch-Gordan tensor $C^{\mbox{\tiny \,fuse}}$ to each node of $\boldsymbol{\tau}$ and contracting the resulting tree tensor network we can obtain tensors
\begin{equation}
\hat{Q}^{j_{e_1} \ldots j_{e_{Z}}}_{j_{1} \ldots j_{L}}(\boldsymbol{\tau}), \label{eq:Q}
\end{equation}
that mediate the change from the product basis to this coupled basis.
\begin{figure}[t]
\begin{center}
\includegraphics[width=14cm]{spin}
\end{center}
\caption{The spin network that relates two coupled bases labeled by fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$. The spin network is proportional to the Identity, the proportionality factor is the coefficient $\hat{S}_{j_A j_B j_C j_D j}^{j_E j_F j_G j_H}$, which can be shown to be the product of two recoupling coefficients.
\label{fig:spin}}
\end{figure}
Another coupled basis $\ket{jm_{j}; \boldsymbol{\tau}'; j_{f_1} \ldots j_{f_{z}}}$ corresponding to a different fusion tree $\boldsymbol{\tau}'$ is related to the basis (\ref{eq:couplebasis}) by the transformation
\begin{equation}
\boxed{\hat{Q}^{j_{f_1} \ldots j_{f_{Z}}}_{j_{1} \ldots j_{L}}(\boldsymbol{\tau}') = \sum_{j_{f_1} \ldots j_{f_{Z}}} \hat{S}_{j_1 \ldots j_L}^{j_{e_1} \ldots j_{e_{Z}} j_{f_1} \ldots j_{f_{Z}}}(\boldsymbol{\tau}, \boldsymbol{\tau}') \hat{Q}^{j_{e_1} \ldots j_{e_{Z}}}_{j_{1} \ldots j_{L}}(\boldsymbol{\tau}),} \label{eq:genrecoup}
\end{equation}
where the coefficients $\hat{S}_{j_1 \ldots j_k}^{j_{e_1} \ldots j_{e_z} j_{f_1} \ldots j_{f_z}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ can be expressed in terms of the recoupling coefficients [Eq.~(\ref{eq:fmove})].
As an example, consider two different ways of coupling four spins $j_A, j_B, j_C$ and $j_D$ according to the fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$ that are shown in Fig.~\ref{fig:fusion}(ii). The two coupled bases are related by coefficients $\hat{S}_{j_A j_B j_C j_D j}^{j_{E} j_{F} j_{G} j_{H}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ that are defined according to the equality depicted in Fig.~\ref{fig:spin}. Note that the tensor network made of Clebsch-Gordan tensors, shown in Fig.~\ref{fig:spin}, is an instance of a \textit{spin network}. In this case, the spin network has two open links and can therefore be regarded as an SU(2)-invariant operator. The equality in the figure then simply depicts that the spin network is proportional to the Identity. The numerical value of the coefficient $\hat{S}_{j_A j_B j_C j_D j}^{j_{E} j_{F} j_{G} j_{H}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ can be calculated without contracting the spin network, but by instead following a procedure called \textit{evaluating} a spin network. Section~\ref{sec:sn} illustrates with simple examples the procedure to evaluate a spin network corresponding to a generic coefficient $\hat{S}_{j_1 \ldots j_k j}^{j_{e_1} \ldots j_{e_z} j_{f_1} \ldots j_{f_z}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$. For instance, it is shown that $\hat{S}_{j_A j_B j_C j_D j}^{j_{E} j_{F} j_{G} j_{H}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ can be expressed in terms of two recoupling coefficients,
\begin{equation}
\hat{S}_{j_A j_B j_C j_D j}^{j_{E} j_{F} j_{G} j_{H}}(\boldsymbol{\tau}, \boldsymbol{\tau}') = \hat{F}_{j_A j_B j_C j_{F}}^{j_{E} j_{G}} \hat{F}_{j_{E} j_C j_D j}^{j_{F} j_{H}}.
\end{equation}
\subsection{Tensor product of $L$ reducible representations}
Finally, consider the tensor product of $L$ reducible representations. A coupled basis, labeled by a given fusion tree, is related to the product basis by means of a transformation that is obtained by attaching a tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ to each node of the fusion tree, and contracting the resulting tree tensor network.
Two different choices of a coupled basis, corresponding to two different fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$, are related by a matrix $\hat{\Gamma}(\boldsymbol{\tau}, \boldsymbol{\tau}')$. The matrix $\hat{\Gamma}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ is a generalization of the matrix with the same name that appears Eq.~(\ref{eq:gammasplit}). This matrix is obtained by contracting a tensor network [e.g. Fig.~\ref{fig:spin1}] made of tensors $\Upsilon^{\mbox{\tiny \,fuse}}$ and tensors $\Upsilon^{\mbox{\tiny \,split}}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{spin1}
\end{center}
\caption{The transformation $\hat{\Gamma}$ that relates two different coupled bases, labeled by fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$, when fusing four reducible representations. Matrix $\hat{\Gamma}$ decomposes into a degeneracy matrix $\hat{D}_j$ and the Identity.
\label{fig:spin1}}
\end{figure}
For a fixed value of the total spin $j$, matrix $\hat{\Gamma}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ decomposes in terms of degeneracy matrices $\hat{D}_j$. The components of the latter can be expressed in terms of recoupling coefficients by generalizing Eq.~(\ref{eq:D}). We obtain,
\begin{equation}
\hat{D}_j = \sum \hat{S}_{j_1 \ldots j_k}^{j_{e_1} \ldots j_{e_{z}, j} j'_{e_1} \ldots j'_{e_{z}}} \hat{D}'^{j_{e_1}, \ldots, j_{e_z}}_{j_1, \ldots, j_k, j}, \label{eq:genD}
\end{equation}
where the sum runs over all spin labels but excluding $j$.
\section{Block structure of SU(2)-invariant tensors\label{sec:symTensor}}
In this section we consider tensors that are invariant under the action of the symmetry. We explain how such tensors decompose into a compact canonical form which exploits their symmetry. The canonical form can be understood as a block structure in the tensor components. In Sec.~\ref{sec:blockmoves} we then adapt the set $\mathcal{P}$ of primitive tensor network manipulations to work in this form. With the formalism of SU(2)-invariant tensors at hand we then consider tensor network decompositions made of SU(2)-invariant tensors in Sec.~\ref{sec:symTN}.
\subsection{SU(2)-invariant tensors}
Consider a rank-$k$ tensor $\hat{T}$ with indices $\{i_1, i_2, \ldots, i_k\}$ and directions $\vec{D}$. Each index $i_l$ is associated with a vector space $\mathbb{V}^{(l)}$ on which SU(2) acts by means of transformations $\hat{W}_{\textbf{r}}^{(l)}$.
Also consider the action of SU(2) on the space $\mathbb{V}~\equiv~\bigotimes_{l=1}^{k}~\mathbb{V}^{(l)}$ given by
\begin{equation}
\hat{Y}^{(1)}_{\textbf{r}}\otimes \hat{Y}^{(2)}_{\textbf{r}}\otimes \ldots \otimes \hat{Y}^{(k)}_{\textbf{r}},
\label{eq:Xtrans}
\end{equation}
where
\begin{equation}
\hat{Y}^{(l)}_{\textbf{r}} = \left\{
\begin{array}{cc} \hat{W}^{(l)~*}_{\textbf{r}}& ~~~\mbox{ if } \vec{D}(l) = \mbox{ `in' } ,\\
\hat{W}^{(l)}_{\textbf{r}}& ~~~~\mbox{ if } \vec{D}(l) = \mbox{ `out' }.
\end{array} \right.
\label{eq:y}
\end{equation}
($\hat{W}_{\textbf{r}}^{(l)~*}$ denotes the complex conjugate of $\hat{W}_{\textbf{r}}^{(l)}$.) That is, $\hat{Y}^{(l)}_{\textbf{r}}$ acts differently depending on whether index $i_l$ is an incoming or outgoing index. We then say that tensor $\hat{T}$ is SU(2) \textit{invariant} if it is invariant under the transformation of Eq.~(\ref{eq:Xtrans}). In components we have
\begin{equation}
\boxed{
\begin{split}
\sum_{i_1, i_2, \ldots, i_k} \left(\hat{Y}^{(1)}_{\textbf{r}}\right)_{i_1'i_1} \left(\hat{Y}^{(2)}_{\textbf{r}} \right)_{i_2' i_2} \ldots \left( \hat{Y}^{(k)}_{\textbf{r}} \right)_{i_k' i_k} \hat{T}_{i_1 i_2 \ldots i_k}=\hat{T}_{i_1' i_2' \ldots i_k'}&,
\label{eq:Tinv}
\end{split}
}
\end{equation}
for all $\textbf{r} \in \mathbb{R}^3$.
In the remainder of this section we explore the consequences of the constraints in Eq.~(\ref{eq:Tinv}). The main result is as follows. By writing each index $i_l$ of the tensor in a spin basis, $i_{l} = (j_l, t_{j_l}, m_{j_l})$, the tensor is revealed to have a block structure, namely, the non-trivial components are organized into blocks that are supported on orthogonal subspaces. For a given value of spin $j_l$, the index $i_l$ splits into a \textit{degeneracy index} $(j_l, t_{j_l})$ and a \textit{spin index} $(j_l, m_{j_l})$. An SU(2)-invariant tensor $\hat{T}$ decomposes into a set of \textit{degeneracy tensors}, denoted by $\hat{P}$ and carrying all the degeneracy indices, and a set of \textit{structural} tensors, denoted $\hat{Q}$, carrying all the spin indices. The degeneracy tensors contain all the degrees of freedom and correspond to the `blocks' alluded above. On the other hand, the structural tensors are completely determined by the symmetry since they can be factorized into a trivalent tree tensor network made of Clebsch-Gordan coefficients. Examples of structural tensors include Eq.~(\ref{eq:Q}), however, a structural tensor may not generally decompose according to a fusion tree. We refer to the decomposition $(\hat{P}, \hat{Q})$ as the \textit{canonical decomposition} or the \textit{canonical form} of tensor $\hat{T}$. The main benefit of the canonical form lies in the fact that $\hat{T}$ can be specified compactly by means of only the degeneracy tensors.
In the ensuing discussion we describe the canonical decomposition of SU(2)-invariant tensors on a case by case basis. We explicitly describe the canonical form of SU(2)-invariant tensors with one to three indices. The canonical form in these cases is unique up to overall numerical factors. On the other hand, an SU(2)-invariant tensor with four or more indices can be decomposed in several equivalent ways. We illustrate this with examples without resorting to a complete theoretical characterization of the canonical form in all cases. A more rigorous characterization is developed Chapter 6 where we consider a special canonical form of SU(2)-invariant tensors, namely, tree decompositions. A tree decomposition corresponds to decomposing \textit{both} the degeneracy tensors and the structural tensors according to a \textit{fusion} tree. We find this decomposition more convenient from an implementation point of view. In Chapter 6, we also describe how to construct a tree decomposition for any SU(2)-invariant tensor, how two different tree decompositions of the same tensor are related to one another, and how primitive tensor manipulations are adapted to tree decompositions.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{one}
\end{center}
\caption{Implication of the symmetry constraints fulfilled by a rank-$1$ SU(2)-invariant tensor with (i) an outgoing index, and (ii) an incoming index. The only allowed spin on the one index is $j=0$.\label{fig:one}}
\end{figure}
\subsection{One index\label{sec:symTensor:inv:one}}
An SU(2)-invariant tensor $\hat{T}$ with an outgoing index $a$ fulfills the constraint [Fig.~\ref{fig:one}(i)]
\begin{equation}
(\Psi)_{a'} = \sum_{a} (\hat{W}_{\textbf{r}})_{a'a}(\Psi)_{a},\label{eq:inv11}
\end{equation}
where $\hat{W}_{\textbf{r}}$ is the representation of SU(2) on the vector space associated to index $a$.
Let us now write index $a$ in the spin basis $a~=~(j,~t_{j},~m_{j})~=~(0,t_0,0)$. Then we have
\begin{equation}
\hat{T}_a = (\hat{P})_{t_0},
\label{eq:canonone}
\end{equation}
where $(\hat{P})_{t_0}$, shorthand for $(\hat{T}_{j_a=0})_{t_0, m_{0}=0}$, encodes the non-trivial components of $\hat{T}$. Since the only relevant irrep on the one index is $j=0$ the structural tensors are trivial. Therefore, tensor $\hat{T}$ can be stored compactly as $\hat{P}$.
On the other hand, an SU(2)-invariant tensor $\hat{T}$ with an incoming index $a$ fulfills
\begin{equation}
(\Psi)_{a'} = \sum_{a} (\hat{W}^*_{\textbf{r}})_{a'a}(\Psi)_{a},\label{eq:inv12}
\end{equation}
or equivalently
\begin{equation}
(\Psi)_{a'} = \sum_{a} (\Psi)_{a}(\hat{W}^{\dagger}_{\textbf{r}})_{a'a},\label{eq:inv12}
\end{equation}
where $\hat{W}_{\textbf{r}}^*$ and $\hat{W}_{\textbf{r}}^{\dagger}$ are the complex conjugate and adjoint of $\hat{W}_{\textbf{r}}$ respectively. The canonical form of $\hat{T}$ is the same as that stated as Eq.~(\ref{eq:canonone}).
\subsection{Two indices\label{sec:symTensor:inv:two}}
An SU(2)-invariant matrix $\hat{T}$, possibly rectangular, with indices $a$ and $b$ fulfills [Fig.~\ref{fig:two}(i)]
\begin{align}
\hat{T}_{a'b'} &= \sum_{ab}\left(\hat{W}^{(A)*}_{\textbf{r}}\right)_{a'a}\left(\hat{W}_{\textbf{r}}^{(B)}\right)_{b'b}\hat{T}_{ab},\nonumber \\
&=\sum_{ab}\left(\hat{W}_{\textbf{r}}^{(B)}\right)_{b'b}\hat{T}_{ab}\left(\hat{W}^{(A)\dagger}_{\textbf{r}}\right)_{aa'},
\label{eq:twoinv}
\end{align}
where $\hat{W}_{\textbf{r}}^{(A)}$ and $\hat{W}_{\textbf{r}}^{(B)}$ are the representations of SU(2) on the vector space associated to index $a=(j_a,m_{j_a},t_{j_a})$ and $b=(j_b,m_{j_b},t_{j_b})$ respectively. Schur's Lemma establishes that the matrix $\hat{T}$ decomposes as
\begin{align}
(\hat{T})_{ab} = (\hat{P}_{j_aj_b})_{t_{j_a} t_{j_b}} \delta_{j_aj_b} \delta_{m_{j_a} m_{j_b}},
\label{eq:canon:two1}
\end{align}
which can also be written in a block-diagonal form,
\begin{align}
\hat{T} &= \bigoplus_j \hat{T}_j, \nonumber \\
&=\bigoplus_j (\hat{P}_j \otimes \hat{I}_j).\label{eq:canon:two2}
\end{align}
Here the sum is over all values $j$ of spin $j_a$ that are equal to a value of spin $j_b$.
A rank-$2$ SU(2)-invariant tensor $\hat{T}$ with both incoming indices $a$ and $b$ is associated with fusing spins $j_a$ and $j_b$ into a total spin 0. It fulfills [Fig.~\ref{fig:two}(ii)]
\begin{align}
\hat{T}_{a'b'} &= \sum_{ab}\left(\hat{W}^{(A)*}_{\textbf{r}}\right)_{a'a}\left(\hat{W}_{\textbf{r}}^{(B)*}\right)_{b'b}\hat{T}_{ab}, \nonumber \\
&= \sum_{ab}\hat{T}_{ab}\left(\hat{W}^{(A)\dagger}_{\textbf{r}}\right)_{aa'}\left(\hat{W}_{\textbf{r}}^{(B)\dagger}\right)_{bb'}
\label{eq:twoinv}
\end{align}
and decomposes as
\begin{align}
(\hat{T})_{ab} = (\hat{P}_{j_aj_b})_{t_{j_a} t_{j_b}} \cfuse{j_am_{j_a}}{j_b m_{j_b}}{00}.
\label{eq:canon:two2}
\end{align}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{two}
\end{center}
\caption{Implication of the symmetry constraints fulfilled by rank-$2$ SU(2)-invariant tensors as resulting in the decomposition of the tensors into degeneracy tensors $\hat{P}$ and structural tensors.
\label{fig:two}}
\end{figure}
Similarly, a rank-$2$ SU(2)-invariant tensor with both outgoing indices $a$ and $b$ fulfills [Fig.~\ref{fig:two}(iii)]
\begin{align}
\hat{T}_{a'b'} = \sum_{ab}\left(\hat{W}^{(A)}_{\textbf{r}}\right)_{a'a}\left(\hat{W}_{\textbf{r}}^{(B)}\right)_{b'b}\hat{T}_{ab},
\label{eq:twoinv}
\end{align}
and decomposes as
\begin{align}
(\hat{T})_{ab} = (\hat{P}_{j_aj_b})_{t_{j_a} t_{j_b}} \csplitt{00}{j_am_{j_a}}{j_b m_{j_b}}.
\label{eq:canon:two3}
\end{align}
Both incoming (or both outgoing) spins $j_a$ and $j_b$ are compatible with the total spin 0 only for values $j$ of spin $j_a$ such that $j_a~=~j_b~=~j$ and for values $m$ of $m_a$ such that $m_{j_a}~=~-m_{j_b}~=~m$. Therefore, we can recast the canonical decompositions of Eqs.~(\ref{eq:canon:two2})-(\ref{eq:canon:two3}) in a block-diagonal form,
\begin{align}
\hat{T} &= \bigoplus_j \hat{T}_j, \nonumber \\
&=\bigoplus_{j} \left(\hat{P}_{j} \otimes \hat{\omega}_{j}\right),
\label{eq:canonblock2}
\end{align}
where $\hat{\omega}_j$ is a $(2j+1) \times (2j+1)$ reverse diagonal matrix with diagonal components
\begin{equation}
(\hat{\omega}_j)_{m,-m} \equiv \cfuse{jm}{j~-m}{00} = \csplitt{00}{jm}{j~-m} = \frac{(-1)^{j-m}}{\sqrt{2j+1}}. \label{eq:singletomega}
\end{equation}
To summarize, the canonical form of a rank-$2$ SU(2)-invariant tensor reads
\begin{equation}
\hat{T} = \bigoplus_{j} \left(\hat{P}_{j} \otimes \hat{Q}_{j}\right).
\label{eq:canonblock2}
\end{equation}
Here $\hat{P}_{j}$ contains the degrees of freedom of $\hat{T}$ that are not fixed by the symmetry, namely, $\hat{P}_{j}$ transforms trivially under the action of the SU(2), Eq.~(\ref{eq:decow}). On the other hand $\hat{Q}_j$ is determined by the symmetry according to the directions $\vec{D}$ of indices $a$ and $b$,
\begin{align}
\hat{Q}_j &= \hat{I}_{j}~~~~~~\mbox{ if } \vec{D} = \{\mbox{`in', `out'}\} \mbox{ or } \{\mbox{`out', `in'}\},\label{eq:blkmat}\\
&= \hat{\omega}_{j}~~~~~\mbox{ if } \vec{D} = \{\mbox{`in', `in'}\} \mbox{ or } \{\mbox{`out', `out'}\},
\end{align}
Thus, a rank-$2$ SU(2)-invariant tensor $\hat{T}$ can be stored compactly as
\begin{equation}
\{\{a=(j_a,t_{j_a}, m_{j_a}), b=(j_b,t_{j_b}, m_{j_b})\}, \vec{D}, \{\hat{P}_j\}\}.
\end{equation}
\textbf{Example 10:} Consider a rank-$2$ SU(2)-invariant tensor $\hat{T}$ with both outgoing indices and with each index associated to the vector space $\mathbb{V}$ of Example 8, $\mathbb{V}~\equiv~\mathbb{V}_0~\oplus~3\mathbb{V}_1~\oplus~\mathbb{V}_2$. Tensor $\hat{T}$ has the canonical form
\begin{equation}
\hat{T} \equiv (\hat{P}_0 \otimes \hat{\omega}_0) \oplus (\hat{P}_1 \otimes \hat{\omega}_1) \oplus (\hat{P}_2 \otimes \hat{\omega}_2), \label{eq:eg10}
\end{equation}
where
\begin{align}\label{eq:eg101}
\hat{\omega}_0 &\equiv 1, \nonumber \\
\hat{\omega}_1 &\equiv \begin{pmatrix} 0 & 0 & \frac{1}{\sqrt{3}} \\ 0 & -\frac{1}{\sqrt{3}} & 0 \\ \frac{1}{\sqrt{3}} & 0 & 0\end{pmatrix}, \nonumber \\
\hat{\omega}_2 &\equiv \begin{pmatrix} 0 & 0 & 0 & 0 & \frac{1}{\sqrt{5}} \\ 0 & 0 & 0 & -\frac{1}{\sqrt{5}} & 0 \\ 0 & 0 & \frac{1}{\sqrt{5}} & 0 & 0 \\ 0 & -\frac{1}{\sqrt{5}} & 0 & 0 & 0 \\ \frac{1}{\sqrt{5}} & 0 & 0 & 0 & 0\end{pmatrix}.
\end{align}
The total number of complex coefficients contained in tensor $\hat{T}$ is $|\hat{T}| = 15 \times 15 = 225$. However, the tensor can be stored compactly as
\begin{equation}
\{\{a, b\}, \{\mbox{`out', `out'}\}, \{\hat{P}_0, \hat{P}_1, \hat{P}_2\}\}, \nonumber
\end{equation}
where the total number of complex coefficients that are contained in tensors $\hat{P}_0, \hat{P}_1$ and $\hat{P}_2$ is
\begin{equation}
|\hat{P_0}|+|\hat{P_1}|+|\hat{P_2}| = 1\times 1 + 3\times 3 + 1\times 1 = 11.
\end{equation}
Therefore, by exploiting the symmetry the number of coefficients that need to be stored is twenty times smaller.\markend
\subsection{Three indices\label{sec:symTensor:inv:three}}
Consider a rank-$3$ SU(2)-invariant tensor $\hat{T}$ with incoming indices $a$ and $b$ and outgoing index $c$. It fulfills [Fig.~\ref{fig:three}(i)]
\begin{align}
\hat{T}_{a'b'c'} &=\sum_{abc}\left(\hat{W}^{(A)*}_{\textbf{r}}\right)_{a'a}\left(\hat{W}_{\textbf{r}}^{(B)*}\right)_{b'b}\left(\hat{W}_{\textbf{r}}^{(C)}\right)_{c'c}\hat{T}_{abc},\nonumber\\
&=\sum_{abc}\left(\hat{W}_{\textbf{r}}^{(C)}\right)_{c'c}\hat{T}_{abc}\left(\hat{W}^{(A)\dagger}_{\textbf{r}}\right)_{aa'}\left(\hat{W}^{(B)\dagger}_{\textbf{r}}\right)_{bb'},
\label{eq:invthree2}
\end{align}
where $\hat{W}_{\textbf{r}}^{(A)}, \hat{W}_{\textbf{r}}^{(B)}$ and $\hat{W}_{\textbf{r}}^{(C)}$ are the representations of SU(2) on indices $a=(j_a,m_{j_a},t_{j_a}), b=(j_b,m_{j_b},t_{j_b})$ and $c=(j_c,m_{j_c},t_{j_c})$ respectively. The \textit{Wigner-Eckart theorem} establishes that $\hat{T}$ decomposes as
\begin{align}
(\hat{T})_{abc} = (\hat{P}_{j_aj_bj_c})_{t_{j_a} t_{j_b} t_{j_c}} \cfusespin{a}{b}{c}.
\label{eq:three22}
\end{align}
That is, for compatible values of the spins $j_a, j_b$ and $j_c$, tensor $\hat{T}$ factorizes into tensor $\hat{P}_{j_aj_bj_c}$ containing degrees of freedom and a Clebsch-Gordan tensor that mediates the fusion of spins $j_a$ and $j_b$ into spin $j_c$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{three}
\end{center}
\caption{Examples of the constraints fulfilled by rank-$3$ SU(2)-invariant tensors and their implication as resulting in the decomposition of the tensors into degeneracy tensors $\hat{P}$ and a Clebsch-Gordan tensors.
\label{fig:three}}
\end{figure}
An SU(2)-invariant tensor $\hat{T}$ with another combination of incoming and outgoing indices has a canonical decomposition that differs in the Clebsch-Gordan coefficients. For example, if $\hat{T}$ is an SU(2)-invariant tensor with incoming indices $b$ and outgoing indices $a$ and $c$ then it fulfills
\begin{align}
\hat{T}_{a'b'c'} &=\sum_{abc}\left(\hat{W}^{(A)}_{\textbf{r}}\right)_{a'a}\left(\hat{W}_{\textbf{r}}^{(B)*}\right)_{b'b}\left(\hat{W}_{\textbf{r}}^{(C)}\right)_{c'c}\hat{T}_{abc},\nonumber\\
&=\sum_{abc}\left(\hat{W}_{\textbf{r}}^{(A)}\right)_{a'a}\left(\hat{W}^{(C)}_{\textbf{r}}\right)_{cc'}\hat{T}_{abc}\left(\hat{W}^{(B)\dagger}_{\textbf{r}}\right)_{bb'},
\label{eq:invthree2}
\end{align}
and decomposes as
\begin{align}
(\hat{T})_{abc} = (\hat{P}_{j_aj_bj_c})_{t_{j_a} t_{j_b} t_{j_c}} \csplitspin{b}{a}{c}.\label{eq:three2}
\end{align}
More generally, a rank-$3$ SU(2)-invariant tensor with any combination of incoming and outgoing indices decomposes as
\begin{align}
&(\hat{T})_{abc} = (\hat{P}_{j_aj_bj_c})_{t_{j_a} t_{j_b} t_{j_c}} (\hat{Q}_{j_aj_bj_c})_{m_{j_a} m_{j_b}m_{j_c}}. \nonumber \\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{(Wigner Eckart Theorem)}\label{eq:wigner}
\end{align}
The block structure can be made more explicit by recasting Eq.~(\ref{eq:wigner}) as
\begin{align}
\hat{T} &\equiv \bigoplus_{j_aj_bj_c} \hat{T}_{j_aj_bj_c}, \nonumber \\
&\equiv \bigoplus_{j_aj_bj_c} \left(\hat{P}_{j_aj_bj_c} \otimes \hat{Q}_{j_aj_bj_c}\right),\label{eq:threetensor}
\end{align}
where we use the direct sum symbol $\bigoplus$ to denote that the different tensors $\hat{T}_{j_aj_bj_c}$ are supported on orthonormal subspaces of the tensor product of the spaces associated with indices $a,b$ and $c$, and where the direct sum runs over all compatible values of $j_a, j_b$ and $j_c$. The components $(\hat{Q}_{j_aj_bj_c})_{m_{j_a}m_{j_b}m_{j_c}}$ are determined by the directions $\vec{D}$ of the indices,
\begin{align}
\cfusespin{a}{b}{c}&~~~~\mbox{if }\vec{D} = \{\mbox{`in', `in', `out'}\},\\
\cfusespin{a}{c}{b}&~~~~\mbox{if }\vec{D} = \{\mbox{`in', `out', `in'}\},\\
\cfusespin{a}{c}{b}&~~~~\mbox{if }\vec{D} = \{\mbox{`out', `in', `in'}\}, \\
\csplitspin{b}{a}{b}&~~~~\mbox{if }\vec{D} = \{\mbox{`out', `in', `out'}\},\\
\csplitspin{c}{a}{b}&~~~~\mbox{if }\vec{D} = \{\mbox{`out', `out', `in'}\},\\
\csplitspin{a}{b}{c}&~~~~\mbox{if }\vec{D} = \{\mbox{`in', `out', `out'}\},\\
\beta\csplitspin{a}{b}{c}&~~~~\mbox{if }\vec{D} = \{\mbox{`out', `out', `out'}\},\label{eq:case1}\\
\beta\cfusespin{a}{b}{c}&~~~~\mbox{if }\vec{D} = \{\mbox{`in', `in', `in'}\},
\label{eq:threeall}
\end{align}
where $\beta = (-1)^{j_a-j_b+m_c}\sqrt{2j_c+1}$.
To summarize, a rank-$3$ SU(2)-invariant tensor $\hat{T}$ can be stored in the most compact way as
\begin{equation}
\{\{a, b, c\}, \vec{D}, \{\hat{P}_{j_aj_bj_c}\}\},
\end{equation}
where the indices $a, b$ and $c$ are specified in the spin basis,
\begin{equation}
a = (j_a,t_{j_a},m_{j_a}),~~b = (j_b,t_{j_b},m_{j_b}),~~c = (j_c,t_{j_c},m_{j_c}).
\end{equation}
\textbf{Example 11:} Consider a rank-$3$ SU(2)-invariant tensor $\hat{T}$ such that each index, $a, b$ and $c$, is associated to the vector space $\mathbb{V}$ of Example 8.
Tensor $\hat{T}$ can be stored by storing the degeneracy tensors,
\begin{align}
&\hat{P}_{0,0,0},~~\hat{P}_{0,1,1},~~\hat{P}_{0,2,2},~~\hat{P}_{1,0,1},~~\hat{P}_{1,1,0},~~\hat{P}_{1,1,1},~~\hat{P}_{1,1,2}, \nonumber \\
&\hat{P}_{1,2,1},~~\hat{P}_{1,2,2},~~\hat{P}_{2,0,2},~~\hat{P}_{2,1,1},~~\hat{P}_{2,2,0},~~ \hat{P}_{2,2,1},~~\hat{P}_{2,2,2}, \nonumber
\end{align}
corresponding to all compatible values of $j_a, j_b$ and $j_c$.
The total number of complex coefficients that are contained in the degeneracy tensors is $\displaystyle \sum_{j_a j_b j_c}|\hat{P}_{j_aj_bj_c}| = 45$, whereas $|\hat{T}| = |a|\times|b|\times|c| = 15^3 = 3375$ components; the reduction in the number of coefficients is seventy-five times, much greater than that computed for rank-$2$ tensors in Example 10. In general, the sparsity of SU(2)-invariant tensors increases with increasing number of indices.\markend
\subsection{Four indices\label{sec:symTensor:inv:four}}
A rank-$4$ SU(2)-invariant tensor may be decomposed in several ways in terms of degeneracy tensors and structural tensors in correspondence with the existence of different fusion trees for four spins.
Consider a rank-$4$ SU(2)-invariant tensor $\hat{T}$ with incoming indices $a=(j_a,t_{j_a},m_{j_a}), b=(j_b,t_{j_b},m_{j_b})$ and $c=(j_c,t_{j_c},m_{j_c})$ and outgoing index $d=(j_d,t_{j_d},m_{j_d})$. It fulfills
\begin{align}
\hat{T}_{a'b'c'd'}
=\sum_{abcd}\left(\hat{W}_{\textbf{r}}^{(D)}\right)_{d'd}\hat{T}_{abcd} \left(\hat{W}^{(A)}_{\textbf{r}}\right)^{\dagger}_{aa'}
\left(\hat{W}_{\textbf{r}}^{(B)}\right)^{\dagger}_{bb'}
\left(\hat{W}_{\textbf{r}}^{(C)}\right)^{\dagger}_{cc'}
\label{eq:invfour}
\end{align}
where $\hat{W}^{(A)}_{\textbf{r}}, \hat{W}^{(B)}_{\textbf{r}}, \hat{W}^{(C)}_{\textbf{r}}$ and $\hat{W}^{(D)}_{\textbf{r}}$ are the representations of SU(2) on the indices $a, b, c$ and $d$ respectively.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{four}
\end{center}
\caption{Two equivalent canonical decompositions of a rank-$4$ SU(2)-invariant tensor corresponding to two different ways of fusing the three incoming indices into the outgoing index.
\label{fig:four}}
\end{figure}
Tensor $\hat{T}$ decomposes as
\begin{align}
(T)_{abcd} = \sum_{j_e}&(\hat{P}^{j_e}_{j_aj_bj_cj_d})_{t_{j_a}t_{j_b}t_{j_c}t_{j_d}} \cdot(\hat{Q}^{j_e}_{j_a j_b j_c j_d})_{m_{j_a} m_{j_b} m_{j_c} m_{j_d}},
\label{eq:deco41}
\end{align}
where the sum is over all values of the intermediate spin $j_e$.\\
The coefficients $(\hat{Q}^{j_e}_{j_a j_b j_c j_d})_{m_{j_a} m_{j_b} m_{j_c} m_{j_d}}$ [Eq.~(\ref{eq:cob1})] mediate the fusion of the spins $j_a, j_b$ and $j_c$ into a total spin $j_d$ according to a fusion tree, for example, first fusing $j_a$ and $j_b$ and then fusing the resulting spin with $j_c$.
Alternatively, tensor $\hat{T}$ can be decomposed as
\begin{align}
(T)_{abcd} = \sum_{j_f}&(\hat{P}'^{j_f}_{j_aj_bj_cj_d})_{t_{j_a}t_{j_b}t_{j_c}t_{j_d}} \cdot(\hat{Q}'^{j_f}_{j_a j_b j_c j_d})_{m_{j_a} m_{j_b} m_{j_c} m_{j_d}},
\label{eq:deco42}
\end{align}
in terms of different structural coefficients $(\hat{Q}'^{j_f}_{j_a j_b j_c j_d})_{m_{j_a} m_{j_b} m_{j_c} m_{j_d}}$ [Eq.~(\ref{eq:cob2})] that are associated with fusing the spins according to a different fusion tree, namely, fusing spin $j_a$ with the spin obtained by first fusing $j_b$ and $j_c$.
Since Eqs.~(\ref{eq:deco41}) and (\ref{eq:deco42}) represent the same tensor $\hat{T}$, the tensors $\hat{P}$ and $\hat{P}'$ are related by
\begin{equation}
\hat{P}'^{j_f}_{j_aj_bj_cj_d} = \sum_{j_e}\hat{F}^{j_ej_f}_{j_aj_bj_cj_d} \hat{P}^{j_e}_{j_aj_bj_cj_d},
\label{eq:fmove2}
\end{equation}
where $\hat{F}^{j_ej_f}_{j_aj_bj_cj_d}$ are the recoupling coefficients [Eq.~\ref{eq:fmove}].
\subsection{$k$ indices\label{sec:symTensor:inv:k}}
Finally, consider a rank-$k$ SU(2)-invariant tensor $\hat{T}$ with all \textit{outgoing} indices and which fulfills Eq.~(\ref{eq:Tinv}). By writing each index in a spin basis, $i_l = (j_l, t_{j_l}, m_{j_l})$, tensor $\hat{T}$ can be decomposed as
\begin{align}
(\hat{T})_{i_1 \ldots i_k} \equiv \!\!\!\!\!\sum_{j_{e_1} \ldots j_{e_l}}\left(\hat{P}^{j_{e_1}\ldots j_{e_l}}_{j_{1} \ldots j_{k}}\right)_{t_{j_1} \ldots t_{j_k}}\!\!\!\!\!\cdot\left(\hat{Q}^{j_{e_1} \ldots j_{e_l}}_{j_{1} \ldots j_{k}}\right)_{m_{j_1}\ldots m_{j_k}}.
\label{eq:Tcanon}
\end{align}
Here tensor $\hat{Q}^{j_{e_1}\ldots j_{e_l}}_{j_{1} \ldots j_{k}}$ [Eq.~(\ref{eq:Q})] is a transformation characterized by a fusion tree $\boldsymbol{\tau}$ whose internal links are decorated by the spins $\{j_{e_1},\ldots, j_{e_l}\}$. Another canonical form of the tensor $\hat{T}$,
\begin{align}
(\hat{T})_{i_1 \ldots i_k} \equiv \!\!\!\!\!\sum_{j'_{e_1} \ldots j'_{e_l}}\!\!\!\left(\hat{P}'^{j'_{e_1}\ldots j'_{e_l}}_{j_{1} \ldots j_{k}}\right)_{t_{j_1} \ldots t_{j_k}}\!\!\!\!\!\cdot\left(\hat{Q}'^{j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}}\right)_{m_{j_1}\ldots m_{j_k}},
\label{eq:Tcanon1}
\end{align}
comprises of different degeneracy tensors $\hat{P}'^{j'_{e_1}\ldots j'_{e_l}}_{j_{1} \ldots j_{k}}$ and different structural tensors $\hat{Q}'^{j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}}$ where the latter is a transformation characterized by another fusion tree $\boldsymbol{\tau}'$.
The two canonical forms, Eq.~(\ref{eq:Tcanon}) and (\ref{eq:Tcanon1}), are related as
\begin{equation}
\hat{P}'^{j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}} = \sum_{j_{e_1} \ldots j_{e_l}} \hat{S}^{j_{e_1} \ldots j_{e_l}j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}}(\boldsymbol{\tau}, \boldsymbol{\tau}') \hat{P}^{j_{e_1} \ldots j_{e_l}}_{j_{1} \ldots j_{k}},\label{eq:canonmap}
\end{equation}
where the coefficients $\hat{S}^{j_{e_1} \ldots j_{e_l}j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ are those which appear in Eq.~(\ref{eq:genrecoup}).
Thus tensor $\hat{T}$ can be compactly stored as
\begin{equation}
\{\{i_1, i_2, \ldots, i_k\}, \boldsymbol{\tau}, \vec{D}, \{\hat{P}^{j_{e_1} \ldots j_{e_l}}_{j_{1} \ldots j_{k}}\}\}. \label{eq:datak}
\end{equation}
For an arbitrary combination of incoming and outgoing indices of the tensor, the canonical decomposition is characterized by intermediate spins $j_{e_1} \ldots j_{e_l}$ that are assigned to the links of a trivalent tree that is more general than the fusion tree. Furthermore, two canonical decompositions are related by means of more generic spin networks than those considered (See Section~\ref{sec:sn}) for evaluating the coefficients $\hat{S}^{j_{e_1} \ldots j_{e_l}j'_{e_1} \ldots j'_{e_l}}_{j_{1} \ldots j_{k}}(\boldsymbol{\tau}, \boldsymbol{\tau}')$. A rigorous result is presented in Chapter 6 where we describe the generic transformation that relates two tree decompositions of an SU(2)-invariant tensor.
\section{Manipulations of SU(2)-invariant tensors \label{sec:blockmoves}}
In this section we consider manipulations of SU(2)-invariant tensors that belong to the set $\mathcal{P}$ [Sec.~\ref{sec:tensor:TN}] of primitives: reversal of indices, permutation of indices, reshaping of indices and matrix operations (matrix multiplication and matrix factorizations). We will adapt these manipulations to the presence of the symmetry by implementing them in such a way that the canonical form is maintained.
Our approach will be to describe the basic transformations that are instrumental in implementing the symmetric version of these manipulations and demonstrate their use with simple examples. A more rigorous treatment of adapting the primitive tensor manipulations for SU(2)-invariant tensors is presented in Chapter 6. The basic transformations are symmetry preserving and can be described by means of specical SU(2)-invariant tensors. Consequently, a symmetric manipulation decomposes into the manipulation of the degeneracy tensors and the manipulation of the structural tensors. Computational cost is incurred only by the manipulation of degeneracy tensors. On the other hand, the manipulation of structural tensors can be performed \textit{algebraically} by applying relevant properties of Clebsch-Gordan coefficients. This fact is responsible for obtaining computational speedup from exploiting the symmetry.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{cupcap}
\end{center}
\caption{(i) Graphical representation of the cup tensor. For simplicity, we choose the same graphical representation for a cup tensor applied to an index that carries a single spin $j$ as well as for one that is applied on an index $i$ that carries several spins $j$ (possibly with degeneracy $d_j$). ($c = 2j+1$ is a normalization constant.) (ii) The analogous graphical representation of the cap tensor. (iii) Multiplying together a cup tensor and a cap tensor yields the Identity.}
\label{fig:cupcap}
\end{figure}
\subsection{Reversal of indices \label{sec:blockmoves:symbend}}
An index $i=(j, t_j, m_j)$ of an SU(2)-invariant tensor can be reversed by means of the `cup' and `cap' transformations. The cup transformation is given by a rank-$2$ SU(2)-invariant tensor $\hat{\Omega}^{\mbox{\tiny \,cup}}$ with both \textit{incoming} indices. It can be used to reverse an \textit{outgoing} index. Analogously, the cap transformation is given by a rank-$2$ SU(2)-invariant tensor $\hat{\Omega}^{\mbox{\tiny \,cap}}$ with both \textit{outgoing} indices and can be used to reverse an \textit{incoming} index.
In the canonical form, the cup and cap tensors read as block-diagonal matrices,
\begin{align}
\hat{\Omega}^{\mbox{\tiny \,cup}} &\equiv \bigoplus_j (\hat{I}_{d_j} \otimes \hat{\Omega}^{\mbox{\tiny \,cup}}_j),\label{eq:cup} \\
\hat{\Omega}^{\mbox{\tiny \,cap}} &\equiv \bigoplus_j (\hat{I}_{d_j} \otimes \hat{\Omega}^{\mbox{\tiny \,cap}}_j),\label{eq:cap}
\end{align}
where $d_j = |t_j|$ and where
\begin{align}
\hat{\Omega}^{\mbox{\tiny \,cup}}_j &\equiv \sqrt{2j+1}\hat{\omega}_j, \label{eq:cupj} \\
\hat{\Omega}^{\mbox{\tiny \,cap}}_j &\equiv (-1)^{2j} \sqrt{2j+1}\hat{\omega}_j. \label{eq:capj}
\end{align}
Here $\hat{\omega}_j$ is the tensor defined in Eq.~(\ref{eq:singletomega}). By definition, the cup transformation inverts the action of the cap transformation and vice-versa,
\begin{align}
\hat{\Omega}^{\mbox{\tiny \,cup}}_j~\hat{\Omega}^{\mbox{\tiny \,cap}}_j = \hat{\Omega}^{\mbox{\tiny \,cap}}_j~\hat{\Omega}^{\mbox{\tiny \,cup}}_j = \hat{I}_{2j+1}. \label{eq:wiggle}
\end{align}
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{bend}
\end{center}
\caption{Reversing an index of an SU(2)-invariant tensor by means of a cup tensor. (i) Reversing an index of an SU(2)-invariant tensor $\hat{T}$ with two outgoing indices to obtain a matrix $\hat{T}'$. (ii) The reversal in $(i)$ as performed on the canonical form of $\hat{T}$. The degeneracy index can be reversed without affecting the components of the degeneracy tensor $\hat{P}$. The reversal of the spin index equates to replacing the shaded region with a Clebsch-Gordan tensor and a numerical factor $\frac{1}{2j+1}$, which is absorbed into $\hat{P}$ to obtain $\hat{P}'$. (iii) Reversing an index of a rank-$3$ SU(2)-invariant tensor. (iv) The reversal in $(iii)$ as performed on the canonical form of $\hat{T}$. The reversal of the spin index equates to replacing the shaded region with a Clebsch-Gordan tensor and a recoupling coefficient $\hat{F}^{j_aj_b}_{0j_bj_aj_c}$. The recoupling coefficient appears as a result of introducing a resolution of Identity [Fig.~\ref{fig:cg}(iii)] on the spins $j_b$ and $j_a$ and then simplifying the resulting diagram by applying the equality in Fig.~\ref{fig:fmove}(ii).
\label{fig:bend}}
\end{figure}
Reversal of index $i$ of tensor $\hat{T}$ can be decomposed into the reversal of the degeneracy index $(j, t_j)$ of the degeneracy tensors and reversal of the spin index $(j, m_j)$ of the structural tensors. Reversal of the degeneracy index is trivial since the cup and cap transformations act as the Identity $\hat{I}_{d_j}$ on it whereas the reversal of the spin index is mediated by transformations $\hat{\Omega}^{\mbox{\tiny \,cup}}_j$ and $\hat{\Omega}^{\mbox{\tiny \,cap}}_j$.
Figure~\ref{fig:cupcap}(i)-(ii) introduces a graphical representation of the cup and cap tensors. The cup tensor is depicted as a small circle with two incoming lines (forming a `cup') whereas a cap tensor is depicted as a small circle with two outgoing lines (forming a `cap') \citep{Baez, Biamonte10}.
Next, we illustrate how an outgoing index of a tensor can be reversed by means of the cup transformation. A cap transformation can be used to reverse an incoming index in an analogous way.
\textbf{Example 12:} Consider a rank-$2$ SU(2)-invariant tensor $\hat{T}$ with outgoing indices\\$a~=~(j_a, t_{j_a}, m_{j_a})$ and $b~=~(j_b, t_{j_b}, m_{j_b})$ and which is given in the canonical form,
\begin{equation}
\{\{a, b\}, \{\mbox{`out', `out'}\}, \{\hat{P}_{j}\}\},\label{eq:rev1}
\end{equation}
where $j$ assumes all values of $j_a$ that are equal to a value of $j_b$. Consider reversing index $a$ of $\hat{T}$ as shown in Fig.~\ref{fig:bend}(i). The resulting tensor (or matrix) $\hat{T}'$ is obtained by multiplying tensor $\hat{T}$ with a cup by contracting $a$. We follow the convention that multiplying with a cup corresponds to bending index $a$ upwards from the \textit{left} in the graphical representation. The same index can be bent upwards from the right by multiplying with the \textit{transpose} of the cup.
The resulting matrix $\hat{T}'$ has the canonical form
\begin{equation}
\{\{a, b\}, \{\mbox{`in', `out'}\}, \{\hat{P}'_{j}\}\},
\end{equation}
where
\begin{equation}
\boxed{
\hat{P}'_{j} = \frac{\hat{P}_j}{\sqrt{2j+1}}.\label{eq:degrev}
}
\end{equation}
In order to explain this expression consider Fig.~\ref{fig:bend}(ii) where the reversal is depicted as it is performed on the canonical form of $\hat{T}$. Reversal of the spin index equates to replacing the shaded region by a straight line. This corresponds to applying the following algebraic identity
\begin{equation}
\hat{\Omega}^{\mbox{\tiny \,cup}}_j~\hat{\omega}_{j} = \frac{\hat{I}_{2j+1}}{\sqrt{2j+1}}.
\end{equation}
The factor $\frac{1}{\sqrt{2j+1}}$ is absorbed into the degeneracy tensor $\hat{P}_j$, Eq.~(\ref{eq:degrev}), to obtain the final canonical form.\markend
\textbf{Example 13:} Consider a rank-$3$ SU(2)-invariant tensor $\hat{T}$ which is given in the canonical form,
\begin{equation}
\{\{a, b, c\}, \{\mbox{`in', `out', `out'}\}, \{\hat{P}_{j_aj_bj_c}\}\}. \label{eq:rev2}
\end{equation}
Consider reversing index $b$ of tensor $\hat{T}$ as shown in Fig.~\ref{fig:bend}(iii) by multiplying $\hat{T}$ with a cup such that $b$ is contracted. The canonical form of $\hat{T}'$ reads
\begin{equation}
\{\{a, b, c\}, \{\mbox{`in', `in', `out'}\}, \hat{P}'_{j_aj_bj_c}\},\label{eq:rev22}
\end{equation}
where
\begin{equation}
\boxed{
\hat{P}'_{j_aj_bj_c} = \hat{F}^{j_a j_b}_{0 j_b j_a j_c} \hat{P}_{j_aj_bj_c}.
}
\end{equation}
The recoupling coefficient $\hat{F}^{j_a j_b}_{0 j_b j_a j_c}$ appears due to the reversal of the spin index $(j_b, m_{j_b})$, as shown in Fig.~\ref{fig:bend}(iv). The Clebsch-Gordan tensor and the cup within the shaded region are replaced with another Clebsch-Gordan tensor and a recoupling coefficient. This is achieved by applying a resolution of Identity on spins $j_a$ and $j_b$ and simplifying the resulting diagram by applying the equality shown in Fig.~\ref{fig:fmove}(ii).\markend
The procedure of reversing the spin index illustrated in Example 13 can be applied to reverse a spin index of a generic rank-$k$ SU(2)-invariant tensor. Recall that a structural tensor is maintained as a trivalent tree of Clebsch-Gordan tensors. Reversal of the spin index corresponds to multiplying a cup with a Clebsch-Gordan tensor within this tree. Then, as in Example 13, we proceed by replacing the Clebsch-Gordan tensor and the cup by another Clebsch-Gordan tensor and a recoupling coefficient. The recoupling coefficient is absorbed into the degeneracy tensor to obtain the canonical form of the resulting tensor. In this way we can reverse an index of a generic rank-$k$ SU(2)-invariant tensor.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{permutecanon}
\end{center}
\caption{Permutation of indices [Fig.~\ref{fig:tensorman}(ii)] as performed on the canonical form of a rank-$3$ SU(2)-invariant tensor. The permutation decomposes into permutation of the degeneracy indices and permutation of the spin indices. The latter equates to replacing the Clebsch-Gordan tensor and a `cross' with a Clebsch-Gordan tensor and a numerical factor $\braid{a}{b}{c}$.\label{fig:permutecanon}}
\end{figure}
\subsection{Permutation of indices\label{sec:blockmoves:permute}}
Let us focus on the swap of two adjacent indices of an SU(2)-invariant tensor. As mentioned in Sec.~\ref{sec:tensor:manipulations} an arbitrary permutation of indices can be applied as a sequence of a number of such swaps.
Consider the swap e.g. Eq.~(\ref{eq:permute}) of two adjacent indices of a rank-$3$ SU(2)-invariant tensor $\hat{T}$ that is given in the canonical form,
\begin{equation}
\{\{a,b,c\}, \{\mbox{`in', `in', `out'}\}, \{\hat{P}_{j_aj_bj_c}\}\}. \nonumber
\end{equation}
Then tensor $\hat{T}'$ that is obtained as result of swapping indices $a$ and $b$ has the canonical form
\begin{equation}
\{\{b,a,c\}, \{\mbox{`in', `in', `out'}\}, \{\hat{P}'_{j_b j_a j_c}\}\}, \nonumber
\end{equation}
where
\begin{equation}
\boxed{
\hat{P}_{j_b j_a j_c}' = \braid{a}{b}{c} \hat{P}_{j_a j_b j_c}.\label{eq:perm11}
}
\end{equation}
Here $\hat{R}^{\mbox{\tiny \,swap}}$ is a rank-$3$ SU(2)-invariant tensor with components $\braid{a}{b}{c}$,
\begin{equation}
\braid{a}{b}{c} \equiv (-1)^{j_a+j_b-j_c}, \label{eq:braid3}
\end{equation}
which mediates the swap of the spin indices $(j_a, m_{j_a})$ and $(j_b, j_{m_b})$ that fuse into index $(j_c, j_{m_c})$, see Fig.~\ref{fig:permutecanon}. That is,
\begin{align}
\cfusespin{b}{a}{c} = \braid{a}{b}{c} \cfusespin{a}{b}{c}.
\end{align}
(The same tensor $\braid{a}{b}{c}$ also relates, in a similar way, tensor $\csplitspin{c}{b}{a}$ and tensor $\csplitspin{c}{a}{b}$.)
When swapping two adjacent indices of a generic rank-$k$ tensor the degeneracy tensors $\hat{P}$ and $\hat{P}'$ are also related directly by the swap tensor $\hat{R}^{\mbox{\tiny \,swap}}$, such as in Eq.~(\ref{eq:perm11}), if we work in a canonical form in which the indices that are swapped belong to the \textit{same} node of the trivalent tree that characterizes the canonical form.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5cm]{bendpermute}
\end{center}
\caption{Illustration of the property that reversal of indices ``commutes '' with a permutation of them.\label{fig:bendpermute}}
\end{figure}
Notice how the canonical form of an SU(2)-invariant tensor facilitates a computational speedup for permutation of indices: permuting indices of the tensor is reduced to permuting indices of the much smaller degeneracy tensors. Figure~\ref{fig:permutereshapecompare} illustrates the computational speedup corresponding to a permutation of indices performed using our reference implementation MATLAB. In this implementation permutation of several indices is performed without necessarily breaking the permutation into swaps, see Sec.~\ref{sec:symTN:permute} in Chapter 6.
One can also consider manipulations that involve both reversing indices and permuting them. In this context it is useful to note that these manipulations ``commute'' with one another, as illustrated in Fig.~\ref{fig:bendpermute}.
\subsection{Reshape of indices\label{sec:blockmoves:reshape}}
The transformation that implements the reshape of indices of an SU(2)-invariant tensor depends on the directions of the indices. We analyze three distinct cases. First, we consider fusion of two outgoing indices into an outgoing index and splitting of an outgoing index into two outgoing indices. Second, we consider the analogous reshape of incoming indices. And third, we consider the fusion of an incoming index with an outgoing index.
Let us consider the fusion e.g. Eq.~(\ref{eq:fuse}) of two outgoing indices of an SU(2)-invariant tensor $\hat{T}$. In order to obtain the reshaped tensor $\hat{T}'$ in a canonical form it is required that the fused index be maintained in the spin basis. However, the direct product of indices $d = a \times b$ may result in an index that does not label a spin basis. Therefore, we fuse indices by multiplying $\hat{T}$ with the fusing tensor $\fuse{a}{b}{d}$ such that indices $a$ and $b$ are contracted [Fig.~(\ref{fig:reshapecanon}(i))],
\begin{equation}
\hat{T}'_{dc} \equiv \sum_{ab} \hat{T}_{abc} \fuse{a}{b}{d},
\label{eq:su2fuseEx}
\end{equation}
or in the canonical form [Fig.~\ref{fig:reshapecanon}(ii)] ,
\begin{equation}
\boxed{
\hat{P}'_{j_dj_c} = \sum_{j_aj_b}\sum_{t_{j_a}t_{j_b}} \tfusespin{a}{b}{d} \hat{P}_{j_aj_bj_c}.\label{eq:su2fusecanon}
}
\end{equation}
Notice that the fusion of the spin indices, here, is straightforward. We proceed by multiplying with tensor $C^{\mbox{\tiny \,fuse}}$ and replacing the resulting `loop' in the figure with a straight line [Fig.~\ref{fig:cg}(ii)]. The fusion of two adjacent indices of a generic rank-$k$ SU(2)-invariant tensor follows a straightforward generalization of Eq.~(\ref{eq:su2fusecanon}). By working in a canonical form that is characterized by a trivalent tree in which the two indices belong to the same node, the fusion of the spin indices involves a simple loop elimination, similar to the one illustrated in Fig.~\ref{fig:reshapecanon}(ii).
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{reshapecanon}
\end{center}
\caption{(i) Fusion of two outgoing indices of an SU(2)-invariant tensor by means of the fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$. (ii) Fusion in the canonical form of the tensor decomposes into the fusion of the degeneracy indices using tensor $X^{\mbox{\tiny \,fuse}}$, and the fusion of the spin indices using tensor $C^{\mbox{\tiny \,fuse}}$. The latter can be performed for free since the loop can directly be replaced with a straight line [Fig.~\ref{fig:cg}(ii)]. (iii) Splitting of an outgoing index into two indices by means of the splitting tensor $\Upsilon^{\mbox{\tiny \,split}}$. (iv) Splitting the index in the canonical form of the tensor. \label{fig:reshapecanon}}
\end{figure}
The original tensor $\hat{T}$ can be recovered from $\hat{T}'$ by splitting the index $d$ back into indices $a$ and $b$. This is achieved by multiplying tensor $\hat{T}'$ with the splitting tensor $\splitt{d}{a}{b}$ such that $d$ is contracted [Fig.~\ref{fig:reshapecanon}(iii)],
\begin{equation}
\hat{T}_{abc} \equiv \sum_{d} \hat{T}'_{dc} \splitt{d}{a}{b},\label{eq:su2splitEx}
\end{equation}
or in the canonical form [Fig.~\ref{fig:reshapecanon}(iv)],
\begin{equation}
\boxed{
\hat{P}'_{j_dj_c} = \sum_{j_aj_b}\sum_{t_{j_a}t_{j_b}} \tfusespin{a}{b}{d} \hat{P}_{j_aj_bj_c}.\label{eq:su2fusecanon}
}
\end{equation}
Notice that the sum in Eq.~(\ref{eq:su2fusecanon}) implies that a reshaped tensor $\hat{P}'_{j_aj_d}$ involves a linear combination of several tensors $\hat{P}_{j_aj_bj_c}$. Thus, performing the fusion in the canonical form requires more work than reshaping regular indices which is a simple rearrangement of the tensor components. As a result, fusing indices of SU(2)-invariant tensors can be more expensive than fusing indices of regular tensors, as illustrated in Fig.~\ref{fig:permutereshapecompare} for a reshaping done in MATLAB.
Next, let us consider fusing two incoming indices into a single incoming index. This can be done in one of two equivalent ways. The first involves multiplying the tensor with a \textit{splitting tensor} by contracting the two incoming indices. Equivalently, if one prefers to use the \textit{fusing tensor}, one can reverse the two indices, multiply with the fusing tensor, and finally reverse the fused index. The two approaches are depicted in Fig.~\ref{fig:reshapein}(i). The fused index can be split back into the original indices by reverting the fusion. In the first approach this is done by multiplying with a fusing tensor while in the second approach this is done by multiplying with a splitting tensor and then reversing the two indices [Fig.~\ref{fig:reshapein}(ii)].
Finally, consider the fusion of an incoming index with an outgoing index to produce, say, an outgoing index. This can be achieved by reversing the incoming index and then fusing the indices by means of a fusing tensor. The fused index should be split in a consistent manner by reverting this fusion procedure.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{reshapein}
\end{center}
\caption{(i) Fusion of two incoming indices into an incoming index, and (ii) splitting an incoming index into two incoming indices can be performed in one of two way.\label{fig:reshapein}}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{permutereshapecompare}
\end{center}
\caption{Computation times (in seconds) required to permute indices of a rank-four tensor $\hat{T}$, as a function of the size of the indices. All four indices of $\hat{T}$ have the same size $9d$, and therefore the tensor contains $|\hat{T}| = 9^4d^4$ coefficients. The figures compare the time required to perform these operations using a regular tensor and an SU(2)-invariant tensor, where in the second case each index contains three different values of spin $j=0,1,2$, each with degeneracy $d$, and the canonical form of Eq.~(\ref{eq:deco41}) is used. The upper figure shows the time required to permute two indices: For large $d$, exploiting the symmetry of an SU(2)-invariant tensor by using the canonical form results in shorter computation times. The lower figure shows the time required to fuse two adjacent indices. In this case, maintaining the canonical form requires more computation time. Notice that in both figures the asymptotic cost scales as $O(d^4)$, or the size of $\hat{T}$, since this is the number of coefficients which need to be rearranged. We note that the fixed-cost overheads associated with symmetric manipulations could potentially vary substantially with choice of programming language, compiler, and machine architecture. The results given here show the performance of our MATLAB implementation of SU(2) symmetry.}
\label{fig:permutereshapecompare}
\end{figure}
\subsection{Multiplication of two matrices\label{sec:symTN:multiply}}
Let $\hat{M}$ and $\hat{N}$ be two SU(2)-invariant matrices given in the canonical form
\begin{align}
\hat{M} = \bigoplus_j (\hat{M}_j \otimes \hat{I}_{2j+1}), ~~~ \hat{N} = \bigoplus_j (\hat{N}_j \otimes \hat{I}_{2j+1}). \label{eq:matmult11}
\end{align}
Then the SU(2)-invariant matrix $\hat{T} = \hat{M} \hat{N}$ obtained by multiplying together matrices $\hat{M}$ and $\hat{N}$ has the canonical form
\begin{equation}
\hat{T} = \bigoplus_j (\hat{T}_j \otimes \hat{I}_{2j+1}),\label{eq:matmult4}
\end{equation}
where $\hat{T}_j$ is obtained by multiplying matrices $\hat{M}_j$ and $\hat{N}_j$,
\begin{equation}
\hat{T}_j = \hat{M}_j \hat{N}_j.\label{eq:blockmult}
\end{equation}
Clearly, computational gain is obtained as a result of performing the multiplication $\hat{T} = \hat{R}\hat{S}$ block-wise. This is illustrated by the following example.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{multsvdcompare}
\end{center}
\caption{Computation times (in seconds) required to multiply two matrices (upper panel) and to perform a singular value decomposition (lower panel), as a function of the size of the indices. Matrices of size $9d \times 9d$ are considered. The figures compare the time required to perform these operations using regular matrices and SU(2)-invariant matrices, where for the SU(2) matrices each index contains three different values of the spin $j=0,1,2$, each with degeneracy $d$, and the canonical form of Eq.~(\ref{eq:canonblock2}) is used. That is, each matrix decomposes into three blocks of size $d \times d$. For large $d$, exploiting the block diagonal form of SU(2)-invariant matrices results in shorter computation time for both multiplication and singular value decomposition. The asymptotic cost scales with $d$ as $O(d^3)$, while the size of the matrices grows as $O(d^2)$.(For matrix multiplication, a tighter bound of $O(d^{2.5})$ for the scaling of computational cost with $d$ is seen in this example.) We note that the fixed-cost overheads associated with symmetric manipulations could potentially vary substantially with choice of programming language, compiler, and machine architecture. The results given here show the performance of our MATLAB implementation of SU(2) symmetry.
\label{fig:multsvdcompare}}
\end{figure}
\textbf{Example 13 : (Computational gain from blockwise multiplication)} Consider vector space $\mathbb{V}$ that decomposes as $\mathbb{V} \cong d_j\mathbb{V}_j$ where $j$ assumes values $1,\cdots,q$ and let $d_j=d, \forall j$. The dimension of the space $\mathbb{V}$ is $dp$ where $p=\sum_{j=1}^{q}(2j+1)=q^2+q$.
Consider an SU(2)-invariant matrix $\hat{T}: \mathbb{V} \rightarrow \mathbb{V}$. Since there are $q$ blocks $\hat{T}_j$ and each block has size $d\times d$, the SU(2)-invariant matrix $\hat{T}$ contains $qd^2$ coefficients. For comparison, a regular matrix of the same size contains $d^2p^2$ coefficients, a number greater by a factor of $O(q^3)$.
Let us now consider multiplying two such matrices. We use an algorithm that requires $O(l^3)$ computational time to multiply two matrices of size $l\times l$. The cost of performing $q$ multiplications of $d\times d$ blocks in Eq.~\ref{eq:blockmult} scales as $O(qd^3)$. In contrast the cost of multiplying two regular matrices of the same size scales as $O(d^3p^3)$, requiring $O(q^5)$ times more computational time. Figure \ref{fig:multsvdcompare} shows a comparison of the computation times when multiplying two matrices for both SU(2)-invariant and regular matrices. \markend
\subsection{Factorization of a matrix\label{sec:symTN:factorize}}
The factorization of an SU(2)-invariant matrix $\hat{T}$ can also benefit from the block-diagonal structure. Consider, for instance, the singular value decomposition (SVD), $\hat{T} = \hat{U}\hat{S}\hat{V}$, where $\hat{U}$ and $\hat{V}$ are unitary matrices and $\hat{S}$ is a diagonal matrix with non-negative components. If $\hat{T}$ has the canonical form
\begin{equation}
\hat{T} = \bigoplus_j (\hat{T}_j \otimes \hat{I}_{2j+1}),
\end{equation}
we can obtain the SU(2)-invariant matrices
\begin{align}
\hat{U} = \bigoplus_j (\hat{U}_j \otimes \hat{I}_{2j+1}),\nonumber \\
~\hat{S} = \bigoplus_j (\hat{S}_j \otimes \hat{I}_{2j+1}),\nonumber \\
~\hat{V} = \bigoplus_j (\hat{V}_j \otimes \hat{I}_{2j+1}),\nonumber
\label{eq:svd0}
\end{align}
by performing SVD of each degeneracy matrix $\hat{T}_j$ independently,
\begin{equation}
\hat{T}_j = \hat{U}_j \hat{S}_j \hat{V}_j.
\label{eq:svd1}
\end{equation}
A different factorization of $\hat{T}$, such as spectral decomposition or polar decomposition, can be obtained by the analogous factorization of the blocks $\hat{T}_j$.
The computational savings are analogous to those described in Example 13 for the multiplication of matrices. Figure \ref{fig:multsvdcompare} shows a comparison of computation times required to perform a singular value decomposition on SU(2)-invariant and regular matrices using MATLAB.
\section{Supplement: Examples of evaluating a spin network \label{sec:sn}}
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{evalspin1}
\end{center}
\caption{Illustration of evaluating a spin network.}
\label{fig:evalspin1}
\end{figure}
Let us consider a spin network $\mathcal{S}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ that is constructed by means of two fusion trees $\boldsymbol{\tau}$ and $\boldsymbol{\tau}'$ in the following way. First obtain a tree tensor network $\mathcal{T}$ by attaching a tensor $C^{\mbox{\tiny \,fuse}}$ to each node the fusion tree $\boldsymbol{\tau}$. A tensor $C^{\mbox{\tiny \,fuse}}$ mediates the fusion of the incoming spins into the outgoing spin. Next, obtain the \textit{splitting tree} that is dual to $\boldsymbol{\tau}'$. A splitting tree is obtained by reversing the direction of all links of a fusion tree. In the graphical representation this corresponds to a horizontal reflection of the fusion tree. Then obtain a tree tensor network $\mathcal{T}'$ by attaching to each node of the splitting tree a tensor $C^{\mbox{\tiny \,split}}$ that mediates the splitting of the incoming spin into outgoing spins. The spin network $\mathcal{S}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ is obtained by connecting the open links of the two tree tensor networks: $\mathcal{T}$ and $\mathcal{T}'$.
Since the spin network $\mathcal{S}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ has two open links it can be contracted to obtain an SU(2)-invariant matrix, which according to Schur's lemma is proportional to the Identity. An important property of $\mathcal{S}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ is that this proportionality factor can be \textit{evaluated} algebraically without contracting the spin network. This property can be exploited to suppress the potentially high cost of contracting spin networks in numerical simulations.
The spin network $\mathcal{S}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ can be evaluated in terms of the values of basic spin networks that are shown in Fig.~\ref{fig:cg}(ii) and Fig.~\ref{fig:fmove}(ii). The first step of the evaluation procedure generally involves expressing the spin network as a composition of these basic spin networks. This can be achieved by applying, possibly several times, a resolution of Identity, Fig.~\ref{fig:cg}(i), on appropriate links of the spin network. Then one proceeds by recursively applying the equalities in Fig.~\ref{fig:cg}(ii) and Fig.~\ref{fig:fmove}(ii) to regions of the spin network, eventually replacing the spin network with a straight line and an overall numerical factor. Figure~\ref{fig:evalspin1}(i) illustrates these steps for the simple case of evaluating the spin network of Fig.~\ref{fig:spin} in terms of two recoupling coefficients.
We can also consider spin networks that have intercrossing lines such as those which appear when applying permuting indices of an SU(2)-invariant tensor. Such a spin network can be evaluated in terms of recoupling coefficients \textit{and} swap factors, as illustrated in Fig.~\ref{fig:evalspin1}(ii).
\chapter{Implementation of non-Abelian symmetries \Rmnum{2}}
In this chapter we will describe a specific scheme for implementing non-Abelian symmetries. Our implementation is based on a special canonical form, the \textit{tree decomposition}, of symmetric tensors.
A tree decomposition of a symmetric tensor corresponds to representing and storing the tensor as a tree tensor network made of two parts: (i) a symmetric \textit{vector} and (ii) possibly several \textit{splitting tensors} $\Upsilon^{\mbox{\tiny \,split}}$. We describe how to implement the tensor manipulations within the set $\mathcal{P}$ of primitive operations based on tree decompositions. In a tree decomposition, reshape and permutation of indices take a very simple form. In order to obtain the vector of the output tensor, one needs to simply multiply the vector of the initial tree decomposition with a matrix $\hat{\Gamma}$ [Fig~\ref{fig:spin}(ii)] that depends only on the permutation or reshape.
This approach offers several advantages. Reshapes and permutations can be performed without breaking them into pairwise fusions and swaps, as was described in the previous chapter. More importantly, one can precompute (that is, compute before running the algorithm) the matrices $\hat{\Gamma}$ since these do not depend on the actual components of the tensor being manipulated. This is of special advantage in the case of iterative algorithms, where by pre-computing these matrices one also eliminates the cost of evaluating spin networks at runtime, thus substantially reducing the computational costs.
\subsection{Tree decompositions of SU(2)-invariant tensors\label{sec:vectorform:tree}}
Consider a rank-$k$ SU(2)-invariant tensor $\hat{T}$ with indices $\{i_1, i_2, \ldots, i_k\}$ and directions $\vec{D}$. Let us apply the following transformations on tensor $\hat{T}$ to obtain a vector. First reverse all incoming indices of $\hat{T}$ to obtain another tensor $\hat{T}'$. Then fuse the indices of $\hat{T}'$ according to a given fusion tree $\boldsymbol{\tau}$ to obtain an SU(2)-invariant vector $\hat{v}$. This gives rise to a decomposition of tensor $\hat{T}$ in terms of the vector $\hat{v}$, a set of splitting tensors that revert the fusion sequence $\boldsymbol{\tau}$, and a set of cup tensors that reverse the split indices that are identified with the incoming indices of $\hat{T}$. We refer to such a decomposition as a \textit{tree decomposition} of $\hat{T}$ and denote it as $\mathcal{D}(\hat{T})$. It is completely specified by the following list of elements:
\begin{equation}
\mathcal{D}(\hat{T}) \equiv (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}, \hat{v}). \label{eq:datastruct}
\end{equation}
Here the fusion tree $\boldsymbol{\tau}$ determines the splitting tensors that are part of the decomposition while the directions $\vec{D}$ indicate the presence or absence of a cup tensor on the open indices of the tree decomposition.
A tree decomposition of a rank-$6$ SU(2)-invariant tensor is shown in Fig.~\ref{fig:treeDeco1}. The tree decomposition in the diagram can be specified as,
\begin{align}
\mathcal{D}(\hat{T}) \equiv (\{i_1, i_2, i_3, i_4, i_5, i_6\},
\{\mbox{`in', `out', `out', `in-R', `out', `out'}\}, \boldsymbol{\tau}, \hat{v}),\label{eq:tree6}
\end{align}
where
\begin{align}
\boldsymbol{\tau} : \{i_4, i_5 \rightarrow i_e;~~i_2, i_3 \rightarrow i_c;~~i_e, i_6 \rightarrow i_c;~~&i_1, i_d \rightarrow i_b;
i_b, i_c \rightarrow i_a\}. \nonumber
\end{align}
In the graphical representation of a tree decomposition $\mathcal{D}(\hat{T})$ the vector $\hat{v}$ appears at the top of the tree, the `body' of the tree comprises of splitting tensors that are connected according to the fusion tree $\boldsymbol{\tau}$ and the indices $\{i_1, i_2, \ldots, i_6\}$ are associated, from left to right, to the open lines at the bottom of the tree. Some open indices are bent upwards by attaching cup tensors. A value $\vec{D}(l) = \mbox{`in'}$ indicates a cup tensor is attached to index $i_l$ while $\vec{D}(l) = \mbox{`out'}$ indicates its absence. We additionally denote by $\vec{D}(l) = \mbox{`in-R'}$ that the transpose of a cup tensor is attached to index $i_l$. In the graphical representation, the values $\vec{D}(l) = \mbox{`in'}$ or $\vec{D}(l) = \mbox{`in-R'}$ correspond to bending the index $i_l$ upwards from the left or from the right respectively.
The vector $\hat{v}$ is obtained by applying an resolution of Identity, denoted $\mathcal{I}(\boldsymbol{\tau})$, on tensor $\hat{T}'$, as shown on the r.h.s. of Fig.~\ref{fig:treeDeco1}. The resolution of Identity $\mathcal{I}(\boldsymbol{\tau})$ is given by a tensor network made of a set of fusing tensors, that fuse the indices of $\hat{T}'$ according to the fusion tree $\boldsymbol{\tau}$, and the corresponding set of splitting tensors that inverts the fusion. The vector $\hat{v}$ is obtained by contracting $\hat{T}'$ with the fusing tensors.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{treedeco1}
\end{center}
\caption{ A tree decomposition $\mathcal{D}(\hat{T})$ of a rank-$6$ SU(2)-invariant tensor $\hat{T}$ with indices $\{i_1, i_2, \ldots, i_6\}$. (All internal indices are summed over.) The tree decomposition consists of an SU(2)-invariant vector $\hat{v}$, a set of splitting tensors and cup tensors that are attached to the incoming indices of $\hat{T}$. The tree decomposition is obtained by applying a resolution of Identity $\mathcal{I}(\boldsymbol{\tau})$ on the tensor. The fusing and splitting tensors that constitute $\mathcal{I}(\boldsymbol{\tau})$ are connected together according to the fusion tree $\boldsymbol{\tau}$. \label{fig:treeDeco1}}
\end{figure}
We emphasize that the cup tensors are stored \textit{as part} of the tree decomposition without consuming them into the tree. This is done to simplify reshape and permutation of indices of a tree decomposition since these operations can be performed without noticing the cup tensors. For instance, in order to permute the open indices of a tree decomposition one may proceed by detaching any cup tensors from the tree, permuting the indices and re-attaching the cup tensors to the updated tree, a direct application of the commutation property depicted in Fig.~\ref{fig:bendpermute}. On the other hand, manipulations that involve summing over an index that is attached to a cup tensor are an exception. For example, when multiplying two SU(2)-invariant matrices, each given as a tree decomposition, the cup tensor has to be properly considered to obtain the resultant matrix. This is discussed in Sec.~\ref{sec:symTN:matrixops}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{treeDeco4}
\end{center}
\caption{ (i) Two different, but equivalent, tree decompositions $\mathcal{D}^{X}(\hat{T})$ and $\mathcal{D}^{Y}(\hat{T})$ of a rank-$4$ SU(2)-invariant tensor $\hat{T}$. The two decompositions are characterized by different fusion trees $\boldsymbol{\tau}^{X}$ and $\boldsymbol{\tau}^{Y}$. (ii) Tree decompositions $\mathcal{D}^{X}(\hat{T})$ and $\mathcal{D}^{Y}(\hat{T})$ are obtained by applying the resolutions of Identity $\mathcal{I}(\boldsymbol{\tau}^{X})$ and $\mathcal{I}(\boldsymbol{\tau}^{Y})$ on $\hat{T}$.
\label{fig:treeDeco4}}
\end{figure}
\subsection{Mapping between tree decompositions\label{sec:symTN:tree:gamma}}
The same tensor $\hat{T}$ may be expressed in different tree decompositions corresponding to different choices of the fusion tree. Two different fusion trees $\boldsymbol{\tau}^{X}$ and $\boldsymbol{\tau}^{Y}$ lead to two different tree decompositions $\mathcal{D}^{X}(\hat{T})$ and $\mathcal{D}^{Y}(\hat{T})$ of the same tensor $\hat{T}$. As an example we show two different but equivalent tree decompositions of a rank-$4$ tensor in Fig. \ref{fig:treeDeco4}(i). The two decompositions
\begin{equation}
(\{i_1, i_2, i_3, i_4\}, \vec{D}, \boldsymbol{\tau}^{X}, \hat{v}^{X}) \mbox{ and } (\{i_1, i_2, i_3, i_4\}, \vec{D}, \boldsymbol{\tau}^{Y}, \hat{v}^{Y}), \nonumber
\end{equation}
are obtained by applying on the tensor the resolutions of Identity $\mathcal{I}(\boldsymbol{\tau}^{X})$ and $\mathcal{I}(\boldsymbol{\tau}^{Y})$ respectively that are separately depicted in Fig.~\ref{fig:treeDeco4}(ii).
Suppose now that we have a tensor $\hat T$ in a tree decomposition $\mathcal{D}^{X}(\hat{T})$ and we wish to transform it into another tree decomposition $\mathcal{D}^{Y}(\hat{T})$. We find it convenient to obtain the vector $\hat{v}^{Y} \in \mathcal{D}^{Y}(\hat{T})$ in steps, as shown in Fig.~\ref{fig:MtoM}. First detach all cup tensors from $\mathcal{D}^{X}(\hat{T})$ and apply the resolution of Identity $\mathcal{I}(\boldsymbol{\tau}^{Y})$ on the open indices of the tree. Then contract the splitting tensors in $\mathcal{D}^{X}(\hat{T})$ and the fusing tensors in $\mathcal{D}^{Y}(\hat{T})$ to obtain a matrix $\hat{\Gamma}(\boldsymbol{\tau}^{X}, \boldsymbol{\tau}^{Y})$. The new vector $\hat{v}^{Y}$ can be obtained by multiplying $\hat{v}^{X}$ with the matrix $\hat{\Gamma}(\boldsymbol{\tau}^{X}, \boldsymbol{\tau}^{Y})$,
\begin{equation}
\hat{v}^{Y} = \hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y)\hat{v}^X.
\label{eq:Gamma}
\end{equation}
Thus, the matrix $\hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y)$ can be used to map from one tree decomposition of an SU(2)-invariant tensor into another tree decomposition of the same tensor. Recall that the components of $\hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y)$ can be expressed in terms of recoupling coefficients [Fig.~\ref{fig:spin1}, Eq.~(\ref{eq:genD})].
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{MtoM}
\end{center}
\caption{ Mapping a tree decomposition $\mathcal{D}^{X}(\hat{T})$ of an SU(2)-invariant tensor $\hat{T}$ to another tree decomposition $\mathcal{D}^{Y}(\hat{T})$ by applying a resolution of Identity $\mathcal{I}(\boldsymbol{\tau}^Y)$ to $\mathcal{D}^{X}(\hat{T})$. We obtain an intermediate matrix $\hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y)$ by contracting together the splitting tensors in $\mathcal{D}^{X}(\hat{T})$ and the fusing tensors in $\mathcal{I}(\boldsymbol{\tau}^Y)$ and then multiply it with vector $\hat{v}^X$ to obtain vector $\hat{v}^Y$. \label{fig:MtoM}}
\end{figure}
To summarize the above procedure we define a template routine NEWTREE that takes as input a tree decomposition $\mathcal{D}^X(\hat{T})$ and a fusion tree $\boldsymbol{\tau}^Y$ and returns the tree decomposition $\mathcal{D}^Y(\hat{T})$ of the tensor. The routine reads
\begin{align}
&\textbf{NEWTREE} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Input:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}^X(\hat{T}) := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}^X, \hat{v}^X) \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boldsymbol{\tau}^Y \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Output:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}^Y(\hat{T}) := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}^Y, \hat{v}^Y) \nonumber \\
&\mbox{--------} \nonumber \\
&\mbox{Compute } \hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y) \nonumber \\
&\hat{v}^Y = \hat{\Gamma}(\boldsymbol{\tau}^X, \boldsymbol{\tau}^Y) \hat{v}^X \nonumber \\
&\mathcal{D}^Y(\hat{T}) := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}^Y, \hat{v}^Y) \nonumber \\
&\textbf{return}(\mathcal{D}^Y(\hat{T})). \label{algo:newtree}
\end{align}
(where $:=$ denotes `stored as'.)
Recall that the matrix $\hat{\Gamma}$ is sparse [Eq.~(\ref{eq:D})]. It can be shown that the matrix-vector multiplication, Eq.~(\ref{eq:Gamma}), can be performed with a cost that is $O(|\hat{v}|)$ by means of sparse multiplication.
Next we describe how manipulations in the set $\mathcal{P}$ of primitive tensor manipulations [Sec.~\ref{sec:tensor:TN}] are performed on tree decompositions. Consider an SU(2)-invariant tensor $\hat{T}$ that has been given as a tree decomposition $\mathcal{D}(\hat{T})$,
\begin{equation}
\mathcal{D}(\hat{T}) \equiv (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}, \hat{v}).\nonumber
\end{equation}
Let $\hat{T}'$ denote the SU(2)-invariant tensor that is obtained from tensor $\hat{T}$ as a result of a manipulation in $\mathcal{P}$. Also, let $\mathcal{D}(\hat{T}')$ denote a tree decomposition of $\hat{T}'$,
\begin{equation}
\mathcal{D}(\hat{T}') \equiv (\{i'_1, i'_2, \ldots, i'_m\}, \vec{D}', \boldsymbol{\tau}', \hat{v}').\nonumber
\end{equation}
We will describe how the components of vector $\hat{v}'$ are determined systematically in terms of components of the vector $\hat{v}$.
\subsection{Reversal of indices\label{sec:symTN:reverse}}
Reversal of an index of a tree decomposition is trivial since the cup tensors are stored as part of the tree decomposition. It corresponds to attaching a cup (or its transpose) to the corresponding open index of the tree in case the index is outgoing or detaching the cup from the index in case it is incoming. This simple procedure is summarized in the following template routine which describes reversal of possibly several indices of tensor $\hat{T}$ according to new directions $\vec{D}'$ provided as input. No computation is involved in the procedure. Only information pertaining to the directions of indices is updated,
\begin{align}
&\textbf{REVERSE} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Input:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}) := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vec{D}' \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Output:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}') := (\{i_1, i_2, \ldots, i_k\}, \vec{D}', \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\mbox{-----} \nonumber \\
& \mathcal{D}(\hat{T}') := (\{i_1, i_2, \ldots, i_k\}, \vec{D}', \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\textbf{return}(\mathcal{D}(\hat{T}')) \label{algo:reverse}
\end{align}
\subsection{Permutation of indices\label{sec:symTN:permute}}
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{permute1}
\end{center}
\caption{ Permuting indices of a rank-$4$ SU(2)-invariant tensor $\hat{T}$ given in a tree decomposition. The ``crossings'' in the diagram can be absorbed into the tree by applying a resolution of Identity $\mathcal{I}(\boldsymbol{\tau}')$ (for a given fusion tree $\boldsymbol{\tau}'$). An intermediate matrix $\hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}')$ is obtained by contracting together the splitting tensors in $\mathcal{D}(\hat{T})$, the permutation \textbf{p} and the fusing tensors in $\mathcal{I}(\boldsymbol{\tau}')$. The matrix is multiplied with the vector $\hat{v}$ to determine the updated vector $\hat{v}'$.}
\label{fig:permute1}
\end{figure}
The procedure to permute indices of a tree decomposition is illustrated in Fig.~\ref{fig:permute1}(i). We consider a rank-$4$ SU(2)-invariant tensor with indices $\{a,b,c,d\}$ and apply a permutation \textbf{p},
\begin{equation}
\{a, c, b, d\} = \textbf{p}(\{a, b, c, d\}). \nonumber
\end{equation}
The permutation is depicted by intercrossing index $a$ and index $c$ of the tree. This crossing can be `absorbed' into the tree by applying the resolution of Identity $\mathcal{I}(\boldsymbol{\tau}')$ on the tree. In order to determine the vector $\hat{v}'$ we contract the splitting tensors in $\mathcal{D}(\hat{T})$ and the fusing tensors in $\mathcal{I}(\boldsymbol{\tau}')$ to obtain a matrix $\hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}')$, which is then multiplied with the initial vector $\hat{v}$,
\begin{equation}
\hat{v}' = \hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}') \hat{v}. \label{eq:permsu2}
\end{equation}
Clearly, the above procedure can be employed to apply any permutation \textbf{p} on the indices of a rank-$k$ SU(2)-invariant tensor. Matrix $\hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}')$ is a generalization of matrix $\hat{\Gamma}(\boldsymbol{\tau}, \boldsymbol{\tau}')$ [e.g. Fig.~\ref{fig:spin1}] in that it additionally includes a permutation of indices. The latter can then be seen as a special instance of $\hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}')$ with a trivial permutation of indices. The components of the degeneracy part $\hat{D}$ of matrix $\hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}')$ are given by Eq.~(\ref{eq:genD}) where coefficients $\hat{S}_{j_1 \ldots j_k}^{j_{e_1} \ldots j_{e_{z}, j} j'_{e_1} \ldots j'_{e_{z}}}$ can be expressed in terms of recoupling coefficients \textit{and} swap factors (see Sec.~\ref{sec:sn}).
Finally, note that the cup tensors do not play any role in the permutation since reversal of indices commutes with a permutation of them [Fig.~\ref{fig:bendpermute}]. In practice, all cup tensors can be detached from the tree before applying the permutation and then re-attached to the updated tree.
The procedure to permute indices of a tree decomposition is summarized in the following template routine:
\begin{align}
&\textbf{PERMUTE} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Input:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\hat{T} := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boldsymbol{\tau}' \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textbf{p} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Output:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}') := (\{i'_1, i'_2, \ldots, i'_k\}, \vec{D}', \boldsymbol{\tau}', \hat{v}') \nonumber \\
&\mbox{-----} \nonumber \\
&\mbox{Compute } \hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}') \nonumber \\
&\hat{v}' = \hat{\Gamma}(\boldsymbol{\tau}, \textbf{p}, \boldsymbol{\tau}') \hat{v} \nonumber \\
&\{i'_1, i'_2, \ldots, i'_k\} = \textbf{p}(\{i_1, i_2, \ldots, i_k\}) \nonumber \\
&\hat{T}' := (\{i'_1, i'_2, \ldots, i'_k\}, \vec{D}', \boldsymbol{\tau}', \hat{v}') \nonumber \\
&\textbf{return}(\mathcal{D}(\hat{T}')) \label{algo:permute}
\end{align}
\subsection{Reshape of indices\label{sec:symTN:reshape}}
Consider fusing a pair of adjacent indices $i_l$ and $i_{l+1}$ of the tree decomposition $\mathcal{D}(\hat{T})$. Let us suppose that indices $i_l$ and $i_{l+1}$ do not carry cup tensors and also that they belong to the same node in $\mathcal{D}(\hat{T})$. Indices $i_l$ and $i_{l+1}$ can be fused into an index $i$ by applying the tensor $\fuse{i_l}{i_{l+1}}{i}$ and using the equality shown in Fig.~\ref{fig:su2fuse1}(i) to immediately obtain the tree decomposition $\mathcal{D}(\hat{T}')$. This is illustrated in Fig.~\ref{fig:reshape}(i). Note that the final vector $\hat{v}'$ is the same as the initial vector $\hat{v}$. The updated fusion tree $\boldsymbol{\tau}'$ can be obtained from $\boldsymbol{\tau}$ by deleting the node $\{i_l, i_{l+1} \rightarrow i\}$ from $\boldsymbol{\tau}$. We denote this as,
\begin{equation}
\boldsymbol{\tau}' = \boldsymbol{\tau} - \{i_l, i_{l+1} \rightarrow i_m\}. \nonumber
\end{equation}
The original tree decomposition may be recovered from $\mathcal{D}(\hat{T}')$ by splitting index $i$ back into indices $i_l$ and $i_{l+1}$. This operation is again straightforward since it does not involve a computation of vector components, as illustrated in Fig.\ref{fig:reshape}(ii). The original fusion tree $\boldsymbol{\tau}$ is recovered by concatenating a node to $\boldsymbol{\tau}'$,
\begin{equation}
\boldsymbol{\tau} \equiv \boldsymbol{\tau}' \cup \{i\rightarrow i_l, i_{l+1}\}.
\end{equation}
Now let us consider fusing indices $i_l$ and $i_{l+1}$ that \textit{do not} belong to the same node of $\mathcal{D}(\hat{T})$. In this case one can first map $\mathcal{D}(\hat{T})$ into another tree decomposition $\tilde{\mathcal{D}}(\hat{T})$ in which indices $i_l$ and $i_{l+1}$ belong to the same node and then proceed with the fusion on the tree $\mathcal{D}(\hat{T})$ as described above. This can be done by applying the procedure NEWTREE (\ref{algo:newtree}) with inputs $\tilde{\mathcal{D}}(\hat{T})$ and the desired fusion tree.
Consider the template routine FUSE that fuses indices according to a set of \textit{disjoint} fusion trees $\boldsymbol{\tau}_1, \boldsymbol{\tau}_2, \ldots$ where each fusion tree specifies fusion of a subset of adjacent indices $\{i_m, i_{m+1}\}, \{i_n,i_{n+1},i_{n+2}\}, \ldots$:
\begin{align}
&\textbf{FUSE} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Input args:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}) := (\{i_1, i_2, \ldots, i_k\}, \vec{D}, \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\{\boldsymbol{\tau}_1, \boldsymbol{\tau}_2, \ldots\} \nonumber\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{Final indices: } \{i'_1, i'_2, \ldots, i'_{l}\} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{Fusion tree of final tensor: }\boldsymbol{\tau}' \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Output args:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}') := (\{i'_1, i'_2, \ldots, i'_{l}\}, \vec{D}, \boldsymbol{\tau}', \hat{v}')\nonumber \\
&\mbox{--------} \nonumber\\
&\boldsymbol{\tau}'' = \boldsymbol{\tau}' \cup \boldsymbol{\tau}_1 \cup \boldsymbol{\tau}_2 \cup \ldots \nonumber \\
&\mathcal{D}''(\hat{T}) = \mbox{NEWTREE}(\mathcal{D}(\hat{T}), \boldsymbol{\tau}'') \nonumber \\
&\hat{v}' = \hat{v}'' \nonumber \\
&\mathcal{D}(\hat{T}') := (\{i'_1, i'_2, \ldots, i'_{l}\}, \boldsymbol{\tau}', \hat{v}') \nonumber \\
&\textbf{return}(\mathcal{D}(\hat{T}'))
\end{align}
Notice that here we essentially apply the total fusion at once by concatenating the input fusion trees $\boldsymbol{\tau}_1, \boldsymbol{\tau}_2, \ldots$ into a single tree and then applying the procedure NEWTREE. Consequently, the fusion is carried out by means of a single matrix-vector multiplication. Furthermore, the computational cost incurred by the procedure is dominated by the cost of this step. As mentioned previously, this cost is $O(|\hat{v}|)$.
Also consider the following routine to split indices $\{i'_1, i'_2, \ldots\}$ of a tree decomposition $\mathcal{D}(\hat{T}')$ (typically the output of FUSE) by reversing the fusion sequence encoded in fusion trees $\boldsymbol{\tau}_1, \boldsymbol{\tau}_2, \ldots$:
\begin{align}
&\textbf{SPLIT} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Input args:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}') := (\{i'_1, i'_2, \ldots, i'_l\}, \vec{D}, \boldsymbol{\tau}', \hat{v}') \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{Final indices: } \{i_1, i_2, \ldots, i_{k}\} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\boldsymbol{\tau}_1, \boldsymbol{\tau}_2, \ldots \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\textbf{Output args:} \nonumber \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{D}(\hat{T}) := (\{i_1, i_2, \ldots, i_{k}\}, \vec{D}, \boldsymbol{\tau}, \hat{v})\nonumber\\
&\mbox{--------} \nonumber \\
&\boldsymbol{\tau} = \boldsymbol{\tau}' \cup \boldsymbol{\tau}_1 \cup \boldsymbol{\tau}_2 \cup \ldots \nonumber \\
&\hat{v} = \hat{v}' \nonumber \\
&\hat{T} := (\{i_1, i_2, \ldots, i_{k}\}, \boldsymbol{\tau}, \hat{v}) \nonumber \\
&\textbf{return}(\hat{T})
\end{align}
Note that no computation of vector components is involved in this procedure.
Let us now describe how to reshape indices that may carry cup tensors. First consider the fusion of indices each of which carries a cup tensor. We proceed by detaching the cup tensors from the indices, applying the procedure FUSE and finally attaching a cup tensor to each of the fused indices. Analogously, an index that carries a cup tensor may be split into two indices, by detaching the cup tensor, applying the procedure SPLIT and attaching a cup tensor to each of two indices so obtained.
Finally, consider the fusion of an index that carries a cup tensor with an index that does not carry a cup tensor. The fusion proceeds by detaching the cup tensor and then applying the procedure FUSE on the indices. The fusion is to be reversed in a consistent manner by first applying the procedure SPLIT on the fused index and then attaching a cup tensor to the originally incoming index.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{reshape}
\end{center}
\caption{ (i) Fusion of two adjacent indices that belong to the same node of the tree decomposition proceeds by simply deleting the node to obtain the updated tree decomposition. (ii) Splitting an index corresponds to concatenating a new node to the tree decomposition.}
\label{fig:reshape}
\end{figure}
\subsection{Matrix multiplication and factorizations \label{sec:symTN:matrixops}}
Let us consider how matrix operations are performed on tree decompositions. Two SU(2)-invariant matrices, each given as a tree decomposition, may be multiplied together by first obtaining the matrices in a block-diagonal form (from the respective tree decompositions), performing a block-wise multiplication (Sec.~\ref{sec:symTN:multiply}) and recasting the resulting block-diagonal matrix into a tree decomposition.
An SU(2)-invariant matrix may be factorized e.g. singular value decomposed in a similar way. One proceeds by obtaining the matrix in a block-diagonal form, performing block-wise factorization (Sec.~\ref{sec:symTN:factorize}), and recasting each of the factor block-diagonal matrices into a tree decomposition.
In the remainder of the section we explain how a block-diagonal form is obtained from a tree decomposition and vice-versa.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{recovermat}
\end{center}
\caption{ The block-diagonal form of an SU(2)-invariant matrix is obtained from its tree decomposition by performing two multiplications. At each step the tensors within the shaded region are multiplied together. The same steps are also shown in the canonical form.}
\label{fig:recovermat}
\end{figure}
\subsubsection{Block-diagonal matrix from the tree decomposition}
Consider a tree decomposition $\mathcal{D}(\hat{T})$ of an SU(2)-invariant matrix $\hat{T}$. The decomposition $\mathcal{D}(\hat{T})$ comprises of a vector $\hat{v}$, a splitting tensor $\Upsilon^{\mbox{\tiny \,split}}$ and a cup tensor $\hat{\Omega}^{\mbox{\tiny \,cup}}$. We wish to obtain, from $\mathcal{D}(\hat{T})$, the corresponding block diagonal matrix,
\begin{equation}
\hat{T} = \bigoplus_j (\hat{T}_j \otimes \hat{I}_{2j+1}).\label{eq:blockform}
\end{equation}
This can be achieved by multiplying together the vector, the splitting tensor and the cup tensor. We perform this multiplication in two simple steps as shown in Fig.~\ref{fig:recovermat}. We first multiply $\hat{v}$ with $\Upsilon^{\mbox{\tiny \,split}}$ to obtain an intermediate SU(2)-invariant tensor $\hat{T}'$ that takes the canonical form,
\begin{equation}
\hat{T}'_j = \bigoplus_j (\hat{T}'_j \otimes \hat{\omega}_j),
\end{equation}
where the components $(\hat{T}'_j)_{t't''} $ of $\hat{T}'_j$ are given by
\begin{equation}
(\hat{T}'_j)_{t't''} = \sum_{t} \hat{v}_t~\tsplitt{0t}{jt'}{jt''}.\label{eq:absorbcup1}
\end{equation}
We then multiply (algebraically) tensor $\hat{T}'$ with the cup tensor to obtain the block-diagonal matrix $\hat{T}$ to obtain
\begin{equation}
\hat{T}_j = \frac{1}{2j+1} \hat{T}'_j.\label{eq:absorbcup2}
\end{equation}
\subsubsection{Tree decomposition from the block-diagonal matrix}
The tree decomposition $\mathcal{D}(\hat{T})$ can be obtained from the block-diagonal form (\ref{eq:blockform}) in a straightforward manner by reverting the previous procedure. We first multiply $\hat{T}$ with a cap tensor to obtain tensor $\hat{T}'$. Once again, the outcome of this multiplication follows algebraically. We obtain
\begin{equation}
(\hat{T}'_j)_{t't''} = (2j+1)(\hat{T}_j)_{t't''}.\label{eq:conv2tree1}
\end{equation}
We then fuse the indices of $\hat{T}'$ to obtain a vector $\hat{v}$,
\begin{equation}
\hat{v}_t = \sum_j \sum_{t't''} (\hat{T}'_j)_{t't''} \tfuse{jt'}{jt''}{0t}.\label{eq:conv2tree2}
\end{equation}
The tree decomposition $\mathcal{D}(\hat{T})$ comprises of vector $\hat{v}$, the splitting tensor $X^{\mbox{\tiny \,split}}$ that reverts the fusion in Eq.~(\ref{eq:conv2tree2}) and a cup tensor.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{tensormult}
\end{center}
\caption{ The five steps of tensor multiplication [Fig.~\ref{fig:multiply2}] as adapted to the presence of the symmetry. For simplicity, the tree decompositions are not explicitly shown; the circle representing a tensor masks the tree decomposition of the tensor.}
\label{fig:tensormult}
\end{figure}
\subsection{Multiplication of two tensors\label{sec:symTN:multTens}}
We can now consider the multiplication of two SU(2)-invariant tensors by breaking it into a sequence of five elementary steps consisting of reversals, permutes, reshapes and matrix multiplication, as was exemplified in Sec.~\ref{sec:tensor:multiply} and Fig.~\ref{fig:multiply2}. Here the elementary steps are performed on the tree decompositions of tensors. Figure~\ref{fig:tensormult} illustrates the five steps of the tensor multiplication, Eq.~(\ref{eq:tensormult}), as adapted to tree decompositions. For simplicity, we have not shown the tree decompositions in the figure and the circle that depicts a tensor can be imagined to mask the tree decomposition of the tensor.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{sn1}
\end{center}
\caption{ The number of spin networks to be evaluated when permuting or reshaping an SU(2)-invariant tensor, as a function of the number of indices of the tensor and the number of different spins $j$ assigned to each index of the tensor. The plot shows the increase in the number of spin networks that are evaluated when reshaping tensors with increasing number of indices. It also illustrates the corresponding increase that results from increasing the number of different values of spin $j$ that are assigned to the indices of the tensor. (Note that the number of spin networks does not depend on the degeneracy dimension of a spin $j$.) Consequently, the cost of reshaping tensors with a large number of indices may potentially become significant.}
\label{fig:sn1}
\end{figure}
\subsection{Discussion on computational performance\label{sec:precompute}}
The core of obtaining computational gain from exploiting the symmetry lies in block-wise matrix operations [Fig.~\ref{fig:multsvdcompare}] while permutation and reshape of indices are applied mainly to obtain block-diagonal matrices from tensors. As has been illustrated in Fig.~\ref{fig:permutereshapecompare} the cost of reshaping, for instance, SU(2)-invariant tensors can be significantly larger than that incurred in reshaping regular tensors, and in some case can lead to a severe degradation of the overall gain obtained by exploiting the symmetry. Let us analyze the cost associated to reshape and permutation of indices of an SU(2)-invariant tensor.
We have described that by working on tree decompositions reshape and permutation of indices equates to multiplying a matrix $\hat{\Gamma}$ with a vector.
The computation of $\hat{\Gamma}$ may be costly since it generally involve evaluating many spin networks, see Fig.~\ref{fig:sn1}. Consequently, the cost of reshaping or permuting tensors with a large number of indices may become significant. This is more so the case for \textit{iterative} algorithms where a fixed set of manipulations repeat many times. For example, one may optimize tensors iteratively in a variational algorithm such that the components of the tensor are updated in the current iteration and used as an input to the subsequent iteration. Note that each iteration involves evaluating a large number of spin networks, albeit the \textit{same} spin networks are evaluated in each iteration. This fact can be exploited to \textit{pre-compute} the transformations $\hat{\Gamma}$ once, say in the first iteration of the algorithm, and storing them in memory for \textit{reuse} in subsequent iterations. By precomputation of these matrices the cost of evaluating many spin networks is suppressed from the runtime costs.
In our MATLAB implementation the use of such a precomputation scheme resulted in a significant speed-up of simulations at the cost of storing additional amounts of precomputed data. In the passing we also remark that since all computations have been reduced to matrix operations, computational performance can also be potentially enhanced by parallelizing and vectorizing the underlying matrix operations.
\section{Tensor networks with SU(2) symmetry: A practical demonstration\label{sec:symTN}}
Consider a lattice $\mathcal{L}$ made of $L$ sites where each site $l$ is described by a vector space $\mathbb{V}^{(l)}$ that transforms as a finite dimensional representation of SU(2). The vector space $\mathbb{V}^{(\mathcal{L})}$ of the lattice is given as
\begin{equation}
\mathbb{V}^{(\mathcal{L})} \equiv \bigotimes_l \mathbb{V}^{(l)}.
\end{equation}
Consider a state $\ket{\Psi} \in \mathbb{V}^{(\mathcal{L})}$ that is invariant, Eq.~\ref{eq:latticeinv}, under the action of SU(2) on the vector space $\mathbb{V}^{(\mathcal{L})}$. \textit{We describe $\ket{\Psi}$ by means of a tensor network made of SU(2)-invariant tensors.}
It readily follows that the tensor obtained by contracting such a tensor network is SU(2)-invariant, as illustrated in Fig. \ref{fig:symTN}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{symTN}
\end{center}
\caption{ A tensor network $\mathcal{N}$ made of SU(2)-invariant tensors represents an SU(2)-invariant tensor $\hat{T}$. This is seen by means of two equalities. The first equality is obtained by inserting resolutions of the Identity $\hat{I} = \hat{W}_{\textbf{r}} \hat{W}^{\dagger}_{\textbf{r}}$ on each index connecting two tensors in $\mathcal{N}$. The second equality follows from the fact that each tensor in $\mathcal{N}$ is SU(2)-invariant. }
\label{fig:symTN}
\end{figure}
On the one hand, by storing each constituent tensor of the tensor network in a canonical form we can ensure a compact tensor network description of $\ket{\Psi}$. On the other, computational speedup can be obtained by exploiting the sparse canonical form of the tensors when performing manipulations of individual tensors in a tensor network algorithm.
In the remainder of the section we illustrate the implementation of SU(2) symmetry in tensor network algorithms with practical examples. We do so in the context of the Multi-scale Entanglement Renormalization Ansatz, or MERA, and present numerical results from our reference implementation of SU(2) symmetry in MATLAB.
\subsection{Multi-scale entanglement renormalization ansatz\label{sec:MERA:ansatz}}
Figure \ref{fig:mpsmera} shows a MERA that represent states $\ket{\Psi} \in \mathbb{V}^{(\mathcal{L})}$ of a lattice $\mathcal{L}$ made of $L=18$ sites. Recall that the MERA is made of layers of isometric tensors, known as disentanglers $\hat{u}$ and isometries $\hat{w}$, that implement a coarse-graining transformation. In this particular scheme, isometries map three sites into one and the coarse-graining transformation reduces the $L=18$ sites of $\mathcal{L}$ into two sites using two layers of tensors. A collection of states on these two sites is then encoded in a top tensor $\hat{t}$, whose upper index $a=1,2,\cdots, \chi_{\tiny \mbox{top}}$ is used to label $\chi_{\tiny \mbox{top}}$ states $\ket{\Psi_a} \in \mathbb{V}^{(\mathcal{L})}$. This particular arrangement of tensors corresponds to the 3:1 MERA described in \citep{Evenbly09}. We will consider a MERA analogous to that of Fig.~\ref{fig:mpsmera} but with $Q$ layers of disentanglers and isometries, which we will use to describe states on a lattice $\mathcal{L}$ made of $2\times 3^{Q}$ sites.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{mpsmera}
\end{center}
\caption{ MERA for a system of $L=2\times 3^{2}= 18$ sites, made of two layers of disentanglers $\hat{u}$ and isometries $\hat{w}$, and a top tensor $\hat{t}$.}
\label{fig:mpsmera}
\end{figure}
We will use the MERA as a variational ansatz for ground states and excited states of quantum spin models described by a local Hamiltonian $\hat{H}$. In order to find an approximation to the ground state of $\hat{H}$, we set $\chi_{\tiny\mbox{top}}=1$ and optimize the tensors in the MERA so as to minimize the expectation value
\begin{equation}
\bra{\Psi} \hat{H} \ket{\Psi}
\end{equation}
where $\ket{\Psi}\in \mathbb{V}^{(\mathcal{L})}$ is the pure state represented by the MERA. In order to find an approximation to the $\chi_{\tiny\mbox{top}}>1$ eigenstates of $\hat{H}$ with lowest energies, we optimize the tensors in the MERA so as to minimize the expectation value
\begin{equation}
\sum_{a=1}^{\chi_{\tiny\mbox{top}}}\bra{\Psi_a} \hat{H} \ket{\Psi_a},~~~~\braket{\Psi_a}{\Psi_{a'}} = \delta_{aa'}.
\end{equation}
The optimization is carried out using the MERA algorithm described in \citep{Evenbly09}, which requires contracting tensor networks (by sequentially multiplying pairs of tensors) and performing singular value decompositions.
\subsection{MERA with SU(2) symmetry\label{sec:MERA:ansatz}}
An SU(2)-invariant version of the MERA, or SU(2) MERA for short, is obtained by simply considering SU(2)-invariant versions of all of the isometric tensors, namely the disentanglers $\hat{u}$, isometries $\hat{w}$, and the top tensor $\hat{t}$. This requires assigning a spin operator to each index of the MERA. We can characterize the spin operator by two vectors, $\vec{j}$ and $\vec{d}$: a list of the different values the spin takes and the degeneracy associated with each such spin, respectively. For instance, an index characterized by $\vec{j} = \{0, 1\}$ and $\vec{d} = \{2, 1\}$ is associated to a vector space $\mathbb{V}$ that decomposes as $\mathbb{V} \cong d_0\mathbb{V}_0 \oplus d_1 \mathbb{V}_1$ with $d_0 = 2$ and $d_1 = 1$.
Let us explain how a spin operator is assigned to each link of the MERA. Each open index of the first layer of disentanglers corresponds to one site of $\mathcal{L}$. The spin operator on any such index is therefore given by the quantum spin model under consideration. For example, a lattice with a spin-$\frac{1}{2}$ associated to each site corresponds to assigning spin-$\frac{1}{2}$ operators [Eq.~(\ref{eq:eg2c1})] to each of the open indices.
For the open index of the tensor $\hat{t}$ at the very top the MERA, the assignment of spins will depend on spin sector $J$ that one is interested in. For instance, in order to find an approximation to the ground state and first seven excited states of the quantum spin model within the spin sector $J$, we choose $\vec{j} = \{J\}$ and $\vec{d} = \{8\}$.
For each of the remaining indices of the MERA, the assignment of the pair $(\vec{j}, \vec{d})$ needs careful consideration and a final choice may only be possible after numerically testing several options and selecting the one which produces the lowest expectation value of the energy.
For demonstrative purposes, we will use the SU(2) MERA as a variational ansatz to obtain the ground state and excited states of the spin-$\frac{1}{2}$ antiferromagnetic quantum Heisenberg chain that is given by,
\begin{align}
\hat{H} = \sum_{s=1}^L \hat{h}^{(s, s+1)}, \label{eq:heisenberg}
\end{align}
where
\begin{align}
\hat{h}^{(s, s+1)} &= 4\left(\hat{J}_x^{(s)}\hat{J}_x^{(s+1)} + \hat{J}_y^{(s)}\hat{J}_y^{(s+1)} + \hat{J}_z^{(s)}\hat{J}_z^{(s+1)}\right),
\end{align}
$\hat{J}_{x}, \hat{J}_y$ and $\hat{J}_z$ are the spin-$\frac{1}{2}$ operators [Eq.~(\ref{eq:eg2c1})]. The model has a global SU(2) symmetry, since the Hamiltonian commutes with the spin operators acting on the lattice $\mathcal{L}$. This follows from the fact that each local term $\hat{h}^{(s, s+1)}$ in the Hamiltonian commutes with the two site spin operators,
\begin{equation}
[\hat{h}^{(s, s+1)}, \hat{J}^{(s)}_{\alpha} + \hat{J}^{(s+1)}_{\alpha}] = 0,~~~\alpha={x,y,x}.
\end{equation}
Each spin-$\frac{1}{2}$ degree of freedom of the Heisenberg chain is described by a vector space $\mathbb{V} \cong \mathbb{V}_{\frac{1}{2}}$ that is spanned by two orthonormal states [Eq.~(\ref{eq:basiseg1})],
\begin{equation}
\ket{j=\frac{1}{2}, m=-\frac{1}{2}} \mbox{ and } \ket{j=\frac{1}{2}, m=\frac{1}{2}}. \nonumber
\end{equation}
For computational convenience, we will consider a lattice $\mathcal{L}$ where each site contains two spins. Therefore each site of $\mathcal{L}$ is described by a space $\mathbb{V} \cong \mathbb{V}_0 \oplus \mathbb{V}_1$, where $d_0=1$ and $d_1=1$, also discussed in Example 6. This corresponds to the assignment $\vec{j} = \{0, 1\}$ and $\vec{d} = \{1, 1\}$ at the open legs at the bottom of the MERA. Thus, a lattice $\mathcal{L}$ made of $L$ sites corresponds to a chain of $2L$ spins.
Table \ref{table:degdist} lists some of the spin and degeneracy dimensions assignment (for the internal links of the MERA) that we have used in the numerical computations for $L=54$ (or 108 spins). For a given value of $\vec{j}$ and $\vec{d}$ the corresponding dimension $\chi$ can be obtained as,
\begin{equation}
\chi = \sum_{j \in \vec{j}} (2j+1)\times d_j.
\end{equation}
Figure~\ref{fig:gserror} shows the error in the ground state energy of the Heisenberg chain as a function of the bond dimension $\chi$, for the assignments of $\vec{j}$ and $\vec{d}$ that are listed in Table \ref{table:degdist}. For the choice of spin assignments listed in the table the error is seen to decay polynomially with $\chi$, indicating increasingly accurate approximations to the ground state.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{gserror}
\end{center}
\caption{Error in ground state energy $\Delta E$ as a function of $\chi$ for the Heisenberg model with $2L=108$ spins and periodic boundary conditions, in the singlet sector, $J=0$. The error is calculated with respect to the exact solutions, and is seen to decay polynomially with $\chi$ for the particular choice of spins listed in Table \ref{table:degdist}. \label{fig:gserror}}
\end{figure}
\begin{table}
\centering
\begin{tabular}{c| c| c}
\hline\hline
Total bond dimension, $\chi$ & Spins $\vec{j}$ & Degeneracies $\vec{d}$ \\ [0.5ex]
\hline
4 & $\{0,1\}$ & $\{1,1\}$ \\
8 & $\left\{0,1\right\}$ & $\left\{2,2\right\}$ \\
17 & $\left\{0,1,2\right\}$ & $\left\{3,3,1\right\}$ \\
21 & $\left\{0,1,2\right\}$ & $\left\{4,4,1\right\}$ \\
30 & $\left\{0,1,2\right\}$ & $\left\{5,5,2\right\}$ \\
39 & $\left\{0, 1, 2\right\}$ & $\left\{6,6,3\right\}$ \\
43 & $\left\{0, 1, 2\right\}$ & $\left\{7,7,3\right\}$ \\
52 & $\left\{0, 1, 2\right\}$ & $\left\{8,8,4\right\}$ \\
75 & $\left\{0, 1, 2, 3\right\}$ & $\left\{9,9,5,2\right\}$ \\
[1ex
\hline
\hline
\end{tabular}
\caption{
Example of spin assignment in an SU(2) MERA for the anti-ferromagnetic spin chain with $L = 54$ sites (or $108$ spins).\label{table:degdist}
}
\end{table}
\subsection{Advantages of exploiting the symmetry}
We now discuss some of the advantages of using the SU(2) MERA.
\subsubsection{Selection of spin sector}
An important advantage of the SU(2) MERA is that it exactly preserves the SU(2) symmetry. In other words, the states resulting from a numerical optimization are exact eigenvectors of the total spin operator $\textbf{J}^2 : \mathbb{V}^{(\mathcal{L})} \rightarrow \mathbb{V}^{(\mathcal{L})}$. In addition, the total spin $J$ can be pre-selected at the onset of optimization by specifying it in the open index of the top tensor $\hat{t}$.
Figure~\ref{fig:spec} shows the low energy spectrum of the Heisenberg model $\hat{H}$ for a periodic system of $L=54$ sites (or $108$ spins), including the ground state and several excited states in the spin sectors $J=0, 1, 2$. The states have been organized according to spin projection $m_J$. We see that states with different spin projections $m_J$ (for a given $J$) are obtained to be exactly degenerate, as implied by the symmetry.
Similar computations can be performed with the regular MERA. However, the regular MERA cannot guarantee that the states obtained in this way are exact eigenvectors of $\textbf{J}^2$. Instead the resulting states are likely to have total spin fluctuations. This is shown in inset of Fig.~\ref{fig:spec}, which corresponds to the zoom in of the region in the plot that is enclosed within the box. The inset shows (black asterix points) the corresponding energies obtained for the enclosed two-fold degenerate $J=1$ states using the regular MERA. We see that the states corresponding to different values of $m_J$ are obtained with different energies.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{spec}
\end{center}
\caption{ Low energy spectrum of $\hat{H}$ with $L=54$ sites (=108 spins). Depicted states have spin $J$ of zero ($\times$, blue loops), one (+, red loops), or two ($\circ$, green loop). The superscript $^2$ close to the boundary of a loop indicates that the loop encloses two-fold degenerate states e.g., the second, third and fourth spin-1 triplets are twofold degenerate. The inset shows a zoom in of the region enclosed within the box. It compares the energies of the two-fold degenerate spin-one states within the box with those obtained using the regular MERA (black asterix points). Since the symmetry is not protected, the states obtained with the regular MERA corresponding to different $m_J$ do not have the same energies.
\label{fig:spec}}
\end{figure}
Also note that by using the SU(2) MERA, the three sectors $J=0,1$ and $2$ can be addressed with independent computations. This implies, for instance, that finding the gap between the first singlet ($J=0$) and the first $J=2$ state, can be addressed with two independent computations by respectively setting $(J=0, \chi_{\tiny\mbox{top}}=1)$ and $(J=2, \chi_{\tiny\mbox{top}}=1)$ on the open index of the top tensor $\hat{t}$. However, in order to capture the first $J=2$ state using the regular MERA, we would need to consider at least $\chi_{\tiny\mbox{top}} = 20$ (at a larger computational cost and possibly lower accuracy), since this state has only the $20$\textsuperscript{th} lowest energy overall.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{memcompare}
\end{center}
\caption{ Memory cost (in number of components) for storing the MERA as a function of the bond dimension $\chi$. The horizontal line on this graph shows that this reduction in memory cost equates to the ability to store MERAs with a higher bond dimension $\chi$: For the same amount of memory required to store a MERA with bond dimension $\chi=15$, one may choose instead to store a U(1)-symmetric MERA with $\chi=26$ or an SU(2)-symmetric MERA with $\chi=39$. \label{fig:memcompare}}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{meracompare}
\end{center}
\caption{ Computation time (in seconds) for one iteration of the MERA energy minimization algorithm, as a function of the bond dimension $\chi$. For sufficiently large $\chi$, exploiting the SU(2) symmetry leads to reductions in computation time. The horizontal line on this graph shows that this reduction in computation time equates to the ability to evaluate MERAs with a higher bond dimension $\chi$: For the same cost per iteration incurred when optimizing a regular MERA in MATLAB with bond dimension $\chi=18$, one may choose instead to optimize a U(1)-symmetric MERA with $\chi=26$ or an SU(2)-symmetric MERA $\chi=33$.\label{fig:meracompare}}
\end{figure}
\subsubsection{Reduction in memory and computational costs}
The use of SU(2)-invariant tensors in the MERA also results in a reduction of computational costs. We compared the memory and computational costs associated with using the regular MERA and the SU(2) MERA. We also found it instructive to compare the analogous costs associated with a MERA that is made of tensors that remain invariant under only a subgroup U(1) of the symmetry group. This entails introducing the spin projection operators $\hat{J}_z$ on the links of the MERA and imposing the invariance of constituent tensors under the action of these operators. For such a U(1)-MERA, imposing such constraints corresponds to conservation of the total spin projection $m_J$, while the total spin may fluctuate. (The explicit construction of the U(1)-MERA was discussed in Chapter 4).
Figure \ref{fig:memcompare} shows a comparison of the total number of complex coefficients that are required to be stored for $L=54$ sites (corresponding to 108 spins) in the three cases: regular MERA, U(1) MERA and the SU(2) MERA. U(1)-invariant tensors (see Chapter 4) have a block structure in the eigenbasis of $\hat{J}_z$ operators on each index of the tensor, and therefore they incur a smaller memory cost in comparison to regular tensors. For example, it can be seen that for the same memory required to store a regular MERA with $\chi=15$, one can instead consider storing a U(1)-MERA with $\chi=21$. On the other hand, SU(2)-invariant tensors are substantially more sparse. When written in the canonical form, SU(2)-invariant tensors are not only block-sparse but each block, in turn, decomposes into a degeneracy part and a structural part such that the structural part need not be stored in memory. With the same amount of memory that is required to store, for example, a $\chi=15$ regular MERA, one can already store a $\chi=39$ SU(2) MERA.
In Fig.~\ref{fig:meracompare} we show an analogous comparison of the computational performance in the three cases. We plot the computational time required for one iteration of the energy minimization algorithm of \citep{Evenbly09} (during which all tensors in the MERA are updated once), as a function of the total bond dimension $\chi$ for the cases of regular MERA, U(1) MERA and SU(2) MERA. We see that for sufficiently large $\chi$, using SU(2)-invariant tensors leads to a shorter time per iteration of the optimization algorithm. In the case of symmetric tensors we considered pre-computation of repeated operations, see Sec.~\ref{sec:precompute}.
\chapter{Conclusions and Outlook\label{sec:conclusion}}
In this thesis we have described how to incorporate global internal symmetries into tensor network states and algorithms.
On the theoretical side we developed a framework to characterize and manipulate symmetric tensors. Any given tensor network can be adapted to the presence of a symmetry by imposing the constituent tensors to be symmetric. Symmetric tensors are very sparse objects. Their judicious use and careful manipulation can lead to an enormous computational gain in numerical simulations. This has been extensively demonstrated in this thesis by means of our reference MATLAB implementation.
On the implementation side, we have described a practical scheme for protecting and exploiting the symmetry in numerical simulations. We proposed the use of \textit{tree decompositions} of a symmetric tensor. Several advantages of this scheme were discussed, not excluding the overall simplicity and elegance of the method. We hope to have provided a suitable implementation framework for researchers who are familiar with the theoretical aspects of incorporating the symmetries into tensor network algorithm but nonetheless find the practical implementation challenging.
In implementing symmetries we have gone beyond the case of MPS, which being a trivalent tensor network is simpler to handle. We described the construction of the U(1) and SU(2) symmetric versions of the MERA. Our Abelian implementation led to computational gains measuring up to an increase of ten to twenty times. The analogous gain from the non-Abelian implementation was much larger, measuring up to an increase of forty to fifty times. These gains may be used either to reduce overall computation time or to permit substantial increases in the MERA bond dimension $\chi$, and consequently in the accuracy of the results obtained. Therefore the exploitation of symmetries, especially non-Abelian symmetries, can be an invaluable tool for numerically challenging systems. This is more so the case in two dimensional lattice models where simulation costs are much more severe. An example of a potential application is to a system of interacting fermions that appears in the context of high temperature conductivity. Here even though symmetries are present in the model, they have not been throughly exploited in the context of tensor network algorithms.
Although we have given special attention to specific symmetry groups, U(1) and SU(2), the formalism presented in this thesis may equally well be applied to any reducible compact non-Abelian group that is multiplicity free. In particular, one can consider composite symmetries such as SU(2)$\times$U(1), corresponding to spin isotropy and particle number conservation and SU(2)$\times$SU(2) corresponding to conservation of spin and isospin etc. Such a symmetry is characterized by a set of charges $(a_1, a_2, a_3,\ldots)$; when fusing two such sets of charges $(a_1,a_2,a_3,\ldots)$ and $(a'_1,a'_2,a'_3,\ldots)$, each charge $a_i$ is combined with its counterpart $a'_i$ according to the relevant fusion rule. Once again, this behaviour may be encoded into a single fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$.
Our implementation scheme can also be readily extended to incorporate more general symmetry constraints such as those associated with conservation of total fermionic and anyonic charge. One proceeds by defining the following tensors for the relevant charges,
\begin{itemize}
\item the fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ that encodes the fusion of two charges,
\item the recoupling coefficients $\hat{F}$ that relate various ways of fusing three charges, and
\item the swap tensor $\hat{R}^{\mbox{\tiny \,swap}}$ that encodes the swap behaviour of two charges.
\end{itemize}
Note that within our specific implementation framework, one may instead just define the linear maps $\hat{\Gamma}$ that mediate tensor manipulations for the respective charges.
As an example, consider fermionic constraints where the relevant charge, $p$, is the parity of fermion particle number. Charge $p$ takes two values, $p=0$ and $p=1$ corresponding to even or odd number of fermions. The fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$ encodes the fusion rules that specify how charges $p$ and $p'$ fuse together to obtain a charge $p''$. These correspond to the fusion rules for the group $Z_2$, given as,
\begin{align}
(p=0) \times (p'=0) &\rightarrow (p''=0), \nonumber \\
(p=0)\times (p'=1) &\rightarrow (p''=1), \nonumber \\
(p=1) \times (p'=0) &\rightarrow (p''=1), \nonumber \\
(p=1) \times (p'=1) &\rightarrow (p''=0). \nonumber
\end{align}
The recoupling coefficients $\hat{F}_{p_a p_b p_c p_d}^{p_e p_f}$, associated with the fusion of three charges $p_a, p_b$ and $p_c$ are simple in this case owing to the Abelian fusion rules. They take a value $\hat{F}_{p_a p_b p_c p_d}^{p_e p_f} = 1$ for all values of intermediate charges $p_e$ and $p_f$ that appear when fusing the three charges one way or the other.
The final ingredient is the tensor $\hat{R}^{\mbox{\tiny \,swap}}$, which in this case is defined as,
\begin{align}
\hat{R}^{\mbox{\tiny \,swap}}_{p=0, p'=0 \rightarrow p''=0} &= 1,~~ \hat{R}^{\mbox{\tiny \,swap}}_{p=0, p'=1 \rightarrow p''=1} = 1, \nonumber \\
\hat{R}^{\mbox{\tiny \,swap}}_{p=1, p'=0 \rightarrow p''=1} &= 1,~~ \hat{R}^{\mbox{\tiny \,swap}}_{p=1, p'=1 \rightarrow p''=0} = -1.\nonumber
\end{align}
In a similar way, one can encode the corresponding fusion rules for anyonic charges into the fusing tensor $\Upsilon^{\mbox{\tiny \,fuse}}$. In the case of anyons, the recoupling coefficients $\hat{F}$ are obtained as solutions to the \textit{pentagon equations} whereas the tensors $\hat{R}^{\mbox{\tiny \,swap}}$ are replaced with the anyonic braid operators that are obtained as solutions to the \textit{hexagon equations}, see \citep{Trebst08, Feiguin07, Pfeifer10}. Thus, having defined these tensors for the relevant charges, the formalism and the implementation framework presented in this thesis can be readily adapted to incorporate the constraints corresponding to the presence of fermionic or anyonic charges.
In a different context, preservation of symmetry can be crucial even without demanding a computational gain. Recently, a novel classification of symmetric
phases in 1D gapped spin systems was undertaken in \citep{Chen11}. In the absence of any symmetry, in this classification all states are equivalent to trivial product states. However, by preserving certain symmetries many phases were reported to exist with a different \textit{symmetry protected topological order}. As an alternative, such a classification could potentially be addressed with the symmetric version of the MERA, since the MERA is adept at characterizing fixed points of the renormalization group flow which correspond to different phases.
Symmetries are a fundamental aspect of nature. Nearly all physical phenomenon can be explained by the presence or the absence of a symmetry. In numerical methods, the preservation of symmetries may well be a necessary requirement for simulating certain aspects of the system. As a computational aid, symmetries will play a crucial role in pushing forward the frontiers of tensor network algorithms in the coming years.
\bibliographystyle{plainnat_dotfill}
|
{
"timestamp": "2012-03-16T01:00:47",
"yymm": "1203",
"arxiv_id": "1203.2222",
"language": "en",
"url": "https://arxiv.org/abs/1203.2222"
}
|
\section{Introduction}
Up to a certain energy level standard model(SM) has an impressive success explaining experimental data so far.
Nowadays LHC is running with one of the main motivation of discovering SM Higgs boson and if SM Higgs boson to be discovered,
the leading problem to focus will be stabilizing its mass against quadratically divergent radiative corrections,
namely the hierarchy problem. Eventually the hierarchy problem is not the only problem that SM faces.
SM also does not correctly account for neutrino oscillations, thus nonzero masses of neutrinos, it can not explain dark matter phenomena,
it has no explanation to strong CP problem, and many more. So it is obligatory to go beyond SM.
So far with this purpose many theories and models have been introduced such as grand unified thories(GUT)\cite{GUT}, supersymmetric models(SUSY)\cite{SUSY},
models with extra dimensions\cite{ED}, $331$ models\cite{331mod}, left-right symmetric models\cite{LR}, (B-L) extended SM models\cite{BL}, doublet and triplet Higgs models\cite{DoubletM,TripletM}, Little Higgs(LH) models\cite{lh1,lh2,lhSimple,lhProduct} and many more.
Each of these models have motivations to solve one or more of the problems that SM encounters.
Among these approaches Little Higgs models deserve attention due to their elegant solution to hierarchy problem.
Little Higgs model was first proposed by Arkani-Hamed\cite{lh1}, and following the original idea several
variations of LH models were introduced. Little Higgs models solve hierarchy problem by first enlarging the symmetry group of SM,
and then by using collective symmetry mechanism to cancel out divergences of Higgs mass. The LH models differ in their assumed
symmetry group and in the representation of scalar multiplets and they can be classified into two groups as simple group and
product group LH models\cite{lh1,lhSimple,lhProduct}. Among product group LH models, the most economical one is the Littlest Higgs Model\cite{lh1}
which has a global $SU(5)$ symmetry containing weakly gauged $(SU2\otimes U1)^2$ subgroup. Littlest Higgs model as a consequence of its
enlarged symmetry contains new heavy gauge bosons and a new heavy scalar sector arising from a complex $SU(2)$ group
containing two neutral scalars, a charged scalar and a double charged scalar. The importance of this scalar sector is
that especially charged scalars have very distinct collider signatures\cite{ays1,ays2,ays3,akeyord}.
The main problem the original Littlest Higgs model faced was satisfying electroweak precision data(EWPD) and to be
consistent with the recent bounds on the lightest heavy scalar mass arising from searchs at Tevatron\cite{B2csaki,csaki1,perelstein2ew,B1rizzo,Bdawson,Bkilian,Bdias,Tevatron}. The
free parameters of the model were strictly constrained meaning a severe fine tunning has to be done. In order to overcome
this problem T parity were introduced which like R parity of SUSY introduces a discrete $Z2$ symmetry to the model\cite{Tparity}.
As a consequence of implementing T parity interactions between SM particles and new particles are restricted, thus parameters
were relaxed. Another consequence of this restriction is that the lightest new heavy gauge boson become a perfect candidate
for dark matter\cite{Tparitydarkmatter}. But in order to account for non zero neutrino masses T parity is broken. Introducing
T parity is not the only way of saving littlest Higgs model from strict constraints. Another method, which is also used in this work,
is to charge fermions under both $SU(2)$ gauge groups\cite{B2csaki}.
As mentioned, one of the problems of SM is that it can not account for nonzero masses of neutrinos. The existence of complex $SU(2)$ scalar group
in the littlest Higgs model allows neutrinos to gain their masses by implementing Majorana like mass term in the Yukawa Lagrangian without need of right handed neutrinos. The interactions of lepton doublets and complex $SU(2)$ scalars in Yukawa Lagrangian predicts lepton flavor and number violation by unit two, directly from decays of charged scalars,
and this is an interesting and distinguishing feature of littlest Higgs model\cite{thanlept1,gaurlept1,cinlept_L2yue,gaurlept2}. Search for lepton flavor violating signals is one of the most interesting
topics in collider physics, and in these searches one channel under investigation is the lepton flavor violation
resulting from decays of charged and double charged scalars. Thus the models containing scalar , with hypercharge
two are expected to give the most promising results. In litarature there are several models containing a scalar triplet
and lepton flavor violating signals in most of these models have been examined\cite{331}. Thus it will also be
interesting to know the possible productions of charged scalars in littlest Higgs model and their lepton flavor violating
signals at LHC.
In this work, we study the main production channels of heavy charged scalars and their lepton flavor violating collider signatures at LHC. In literature
there are several works estimating sizable production rates of heavy charged scalars which are either model independent or arising from other models rather than littlest Higgs via
$pp \to \phi^{++}\phi^{-}$,
$pp \to \phi^+\phi^-$ and $pp \to \phi^{++}\phi^{--}$ at hadron colliders\cite{EXmodelindependent}. In these searches they are basically investigating the possibility of a
charged scalar satisfying the experimental bounds such as $M_\phi\geq150GeV$\cite{mphi}. In the littlest Higgs model due to the restrictions on the symmetry breaking scale, the heavy scalars cannot have a mass lower than $0.5TeV$. Thus these production channels including their lepton flavor and number violating final states need more investigation in the context of the littlest Higgs model.
In proton collisions the charged scalars of the
littlest Higgs model can be produced exculisively or in pairs. The exculisive production of the single charged scalar with a gauge boson mainly come from exchange of gauge bosons or neutral heavy scalars in s-channel.
In the model couplings of the boson exchanges are dependent on $v'$(vacuum expectation value of the scalar triplet), thus when $v'$ is small enough to allow lepton flavor violation they are not observable. For the production of double charged scalars
the exculisive channels $\bar q q' \rightarrow W^-_{L(H)} \rightarrow W^+_L \phi^{--}$, $q q \rightarrow W^-_{L(H)} W^-_{L(H)} q'q' \rightarrow \phi^{--} q'q'$,
$\bar q q' \rightarrow W^-_{L(H)} \rightarrow W^+_H \phi^{--}$ and the pair production process $\bar q q' \rightarrow W^-_{L(H)} \rightarrow \phi^+ \phi^{--}$ are analyzed in Ref. \cite{ref1Charged}. As also mentioned in Ref. \cite{ref1Charged}, the first three production processes are supressed by a factor of $v^{\prime 2}$, hence only the process $\bar q q' \rightarrow W^-_{L(H)} \rightarrow \phi^+ \phi^{--}$ can have
significant production rates at LHC. In addition to
these production processes, heavy charged scalars of the littlest Higgs model can be produced in pairs at LHC.
In this work, the pair production processes: $pp \to \phi^{++}\phi^{-}$,
$pp \to \phi^+\phi^-$ and $pp \to \phi^{++}\phi^{--}$ via proton collisions at LHC are examined. In section two, we beriefly reviewed littlest Higgs model mostly following Ref. \cite{thanrev}, and we present the necessary formulation used
in this work. In section three we present the production cross sections of production channels investigated in this work. We also present the final state analysis
considering possible lepton flavor and number violating signals with our numerical estmates for LHC in section three. Finally at
section four we concluded that
through the all possible final states, three of them can be observed as lepton flavor or number violation at LHC.
\section{Theoretical Framework}
The littlest Higgs model, as the most economical of the product group little Higgs models, have a global symmetry $SU(5)$ with a locally gauged
subgroup $\left(SU(2)\otimes U(1)\right)^2$. At a scale $\Lambda\sim 4 \pi f$, the global symmetry $SU(5)$ is broken spontaneously to $SO(5)$ with a chosen vacuum
condensate. At low energies the dynamics of the symmetry breaking mechanism can be described by Nambu Goldstone boson (NGB) degrees of freedom,
to each broken generator there exist a NGB. It is possible to represent the group structure as $1_0\oplus3_0\oplus 2_{1/2}\oplus 3_1$, where subscribes denote the hypercharge of the group.
In the collective symmetry breaking mechanism, breaking of a global symmetry also triggers the symmetry breaking of the gauged subgroup of $\left(SU(2)\otimes U(1)\right)^2$
to $SU(2)\otimes U(1)$ of the standard model. During this symmetry breaking four of the NGBs are eaten by gauge bosons acquiring them their masses, while
in the representationps a doublet $H$ and a triplet $\Phi$ remain physical. For these NGBs a Coleman Weinberg potential can be generated at one loop level.
The generated scalar potential triggers usual electroweak symmetry breaking(EWSB) at chosen vacua of $v$ and $v'$ respectively for $H$ and $\Phi$. After
EWSB, there are four new heavy scalars, $\phi^0$, $\phi^P$, $\phi^+$ and $\phi^{++}$ which remain in the scalar sector in addition to a light scalar $H$ which is identified as the SM Higgs boson.
All scalars excluding $H$ are degenerate in mass:
\begin{eqnarray}
M_\phi =\frac{\sqrt{2} f}{v\sqrt{1-(\frac{4 v'
f}{v^2})^2}}M_H,
\end{eqnarray}
where $M_H$ is the mass of the Higgs boson, $f$ is the higher symmetry breaking scale of the littlest Higgs model, $v=246 GeV$ and $v'\leq \frac{v^2}{4f}$ are the vacuum expectation values (VEVs) of the Higgs field
and the scalar triplet respectively which are bounded by electroweak precision data.
The final particle content and properties of the gauge sector is dependent on the mixings of $U(1)$ and $SU(2)$ subgroups during the spontaneous breaking of $\left(SU(2)\otimes U(1)\right)^2$ to $SU(2)\otimes U(1)$. The mixing angles between the $SU(2)$
subgroups and between the $U(1)$ subgroups are defined respectively
as:
\begin{equation}\label{ssp}
s=\frac{g_2}{\sqrt{g_{1}^2 + g_{2}^2 }}~~,~~~~~~~~ s^\prime=\frac{g'_2}{\sqrt{g_{1}^{\prime 2} + g_{2}^{\prime
2}
}}~~,
\end{equation}
where $g_{i}$ and
$g_{i}'$ are the corresponding couplings of $SU(2)_{i}$and
$U(1)_{i}$. After EWSB, gauge sector get additional mixings and mass terms resulting the final spectrum of gauge bosons. In the littlest Higgs model, gauge sector consists of heavy new gauge bosons
$W_H^{\pm}$, $Z_H$, $A_H$ and light gauge bosons identified as SM gauge bosons; $W_L^{\pm}$, $Z_L$ and one massless boson $A_H$ identified as photon.
The final masses of gauge bosons to the order of
$\frac{v^2}{f^2}$ are expressed as\cite{thanrev}:
\begin{eqnarray}\label{massesvectors}
M_{W_L^{\pm}}^2 &=& m_w^2 \left[
1 - \frac{v^2}{f^2} \left( \frac{1}{6}
+ \frac{1}{4} (c^2-s^2)^2
\right) + 4 \frac{v^{\prime 2}}{v^2} \right], \nonumber \\
M_{W_H^{\pm}}^2 &=& \frac{f^2g^2}{4s^2c^2}
- \frac{1}{4} g^2v^2
+ \mathcal{O}(v^4/f^2)= m_w^2\left( \frac{f^2}{s^2c^2v^2}-1\right)
,\nonumber \\
M_{A_L}^2 &=& 0 ,\nonumber \\
M_{Z_L}^2 &=& m_z^2
\left[ 1 - \frac{v^2}{f^2} \left( \frac{1}{6}
+ \frac{1}{4} (c^2-s^2)^2
+ \frac{5}{4} (c^{\prime 2}-s^{\prime 2})^2 \right)
+ 8 \frac{v^{\prime 2}}{v^2} \right],
\nonumber \\
M_{A_H}^2 &=&
\frac{f^2 g^{\prime 2}}{20 s^{\prime 2} c^{\prime 2}}
- \frac{1}{4} g^{\prime 2} v^2 + g^2 v^2 \frac{x_H}{4s^2c^2}
= m_z^2 s_{{\rm w}}^2 \left(
\frac{ f^2 }{5 s^{\prime 2} c^{\prime 2}v^2}
- 1 + \frac{x_H c_{{\rm w}}^2}{4s^2c^2 s_{{\rm w}}^2} \right),
\nonumber \\
M_{Z_H}^2 &=& \frac{f^2g^2}{4s^2c^2}
- \frac{1}{4} g^2 v^2
- g^{\prime 2} v^2 \frac{x_H}{4s^{\prime 2}c^{\prime 2}}
= m_w^2 \left( \frac{f^2}{s^2c^2 v^2}
- 1 - \frac{x_H s_{{\rm w}}^2}{s^{\prime 2}c^{\prime 2}c_{{\rm w}}^2}\right) ,
\end{eqnarray}
where $m_w\equiv gv/2$, $m_z\equiv {gv}/(2c_{{\rm w}})$ and $x_H =
\frac{5}{2} g g^{\prime}
\frac{scs^{\prime}c^{\prime} (c^2s^{\prime 2} + s^2c^{\prime 2})}
{(5g^2 s^{\prime 2} c^{\prime 2} - g^{\prime 2} s^2 c^2)}$. In these
equations $s_{{\rm w}}$ and $c_{{\rm w}}$ are the usual weak mixing angles.
In littlest Higgs model, the free parameters are the symmetry braking scale $f$ and mixing angles $s$ and $s'$ and they are constrained by observables\cite{B2csaki,csaki1,perelstein2ew,B1rizzo,Bdawson,Bkilian,Bdias}.
The data from Tevatron and LEPII constrain the mass of the lightest heavy scalar as $M_{A_H}\gtrsim900GeV$\cite{Bdawson,Tevatron}. In the original formulation of the littlest Higgs model, these data imposes strong constraints on symmetry breaking scale($f>3.5 - 4 TeV$). But in this work by gauging fermions in both $U(1)$ subgroups, fermion boson couplings are modified as done in\cite{B2csaki}.
With this modification the symmetry breaking scale can be lowered to $f=0.75TeV(M_\phi\simeq0.5TeV)$ while mixing angles are restricted to be $s=0.8$ and $s'=0.6$,
which allows the mass of the $A_H$ to be at the order of few $GeV$s with a large decay width. For larger values of $f$, the mixing agles are less restricted.
Finally, the fermions of the littlest Higgs model gain their masses through EWSB due to the Yukawa Lagrangian with an extended scalar sector. The additional scalar triplet of the model enables to
implement a Majorano type mass term in Yukawa Lagrangian\cite{thanlept1,gaurlept1,cinlept_L2yue,gaurlept2}, such as:
\begin{equation}\label{lepviol1}
{\cal L}_{LFV} = iY_{ij} L_i^T \ \phi \, C^{-1} L_j + {\rm h.c.},
\end{equation}
where $L_i$ are the lepton doublets $\left(
\begin{array}{cc}
l &\nu_l \\
\end{array}
\right)$,
and $Y_{ij}$ is the Yukawa coupling with $Y_{ii}=Y$ and $Y_{ij(i\neq
j)}=Y'$. Due to this term neutrinos gain mass without need of right handed neutrinos and also the lepton flavor violation arise from the decays of heavy scalars
up to number of two. The values of Yukawa couplings $Y$ and $Y'$ are restricted
by the current constraints on the neutrino
masses\cite{neutrinomass}, given as; $M_{ij}=Y_{ij}v'\simeq
10^{-10}GeV$. Since the vacuum expectation value $v'$ has only an
upper bound; $v'<\frac{v^2}{4f}$, $Y_{ij}$ can be taken up to order of unity
without making $v'$ unnaturally small. In this work the values of
the Yukawa mixings are taken to be $10^{-10}\leq Y\leq 1$,
$Y'\sim 10^{-10}$, and the vacuum expectation value $1GeV\geq v'\geq10^{-10}
GeV$.
While studying the production rates and final collider signals of littlest Higgs model scalars, their decay modes including the lepton flavor violating decays which are studied in T.Han et al\cite{thanlept1} are required. Due to their lepton flavor
violating modes, the total widths of the charged scalars will depend on the Yukawa
couplings $Y_{ii}=Y$ and $Y_{ij(i\neq j)}=Y'$. The decay modes and width of
$\phi^{++}$ are given as\cite{thanlept1}:
\begin{eqnarray}\label{dwp2}
\nonumber \Gamma_{\phi^{++}}&=&\Gamma (W^+_L W^+_L)+3 \Gamma ( \ell^+_i \ell^+_i) +3 \Gamma (\ell^+_i \ell^+_j )\\
&\approx& \frac{v^{\prime 2} M_{\phi}^3}{2 \pi v^4}+\frac{3}{8\pi } |Y|^2 M_\phi+\frac{3}{4\pi } |Y'|^2 M_\phi
\end{eqnarray}
For the single charged scalar, the decay modes and width are given
by\cite{thanlept1}:
\begin{eqnarray}\label{dwp1}
\nonumber \Gamma_{\phi^+}&=&3 \Gamma ( \ell^+_i \bar\nu_i)+6 \Gamma
(\ell^+_i \bar\nu_j)+\Gamma ( W_L^+ H)+\Gamma
(W_L^+ Z_L)+\Gamma (t \bar{b})+\Gamma (T \bar{b})\\
&\approx & \frac{N_c M^2_t M_\phi }{32 \pi f^2}+\frac{v^{\prime
2}M^3_\phi }{2\pi v^4} +\frac{3}{8\pi
}
|Y|^2 M_\phi+\frac{3}{4\pi
}
|Y'|^2 M_\phi.
\end{eqnarray}
In this final expression, the decay of single charged scalar to $T\bar{b}$ is neglected since in the parameter space considered in this work, $M_\phi \sim M_T$, hence this decay is suppressed. It is seen from the decay widths of the scalars that lepton number
violation is proportional to $|Y|^2$ if the final state leptons are
from the same family and to $|Y'|^2$ for final state leptons are
from different generations.
\section{The results and discussions}
The scattering amplitudes of the processes $pp \to \phi^{++}\phi^{-}$,
$pp \to \phi^+\phi^-$ and $pp \to \phi^{++}\phi^{--}$ depend on the parameters $s$, $s'$ and $f$, as well as center of mass(cms) energy $\sqrt{S}$ of LHC. While
dependence on the mixing angles is weak, the dependence on the symmetry breaking scale $f$ and so on the mass of the new scalars is significant. In this work, we first examined the dependence
of the production cross sections of the charged pairs to $M_\phi$ for different values of $\sqrt{S}$. In our calculations we have chosen $s=0.8$ and $s'=0.6$ allowed by the precision data. The symmetry breaking
$f$ is taken in the range $0.75TeV$ to $3TeV$, thus the corresponding values for the mass of the heavy scalars vary in between $0.5TeV$ to $2TeV$. In our numerical calculations we have taken
$M_H=120\pm3 GeV$, $M_{Z_L}=91.188\pm0.002 GeV$, $M_{W_L}=80.40\pm0.02GeV$ and $s_W=0.47$\cite{pdg}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{Plot_allx2.eps}
\end{center}
\caption{The production cross sections of the scalar pairs with respect to $M_\phi$ for $\sqrt{S}=2TeV$, $\sqrt{S}=7TeV$ and $\sqrt{S}=14TeV$ when $s=0.8$ and $s'=0.6$.}
\label{fig::1}
\end{figure}
In figure \ref{fig::1}, we have plotted the production cross sections of the scalar pairs with respect to $M_\phi$ for $\sqrt{S}=2TeV$, $\sqrt{S}=7TeV$ and $\sqrt{S}=14TeV$ when $s=0.8$ and $s'=0.6$. It is seen from Fig.\ref{fig::1} that
the productions of heavy scalar pairs are not observable at $\sqrt{S}=2TeV$, since their rates are at the order of $10^{-9}pb$. At $\sqrt{S}=7TeV$, the production rates of the scalar pairs are in the order of $10^{-4}pb$, which is in the reach for LHC. For the case
$M_{\phi}=0.5TeV$ and $\sqrt{S}=14TeV$ scattering amplitudes for the processes $pp \to \phi^{++}\phi^{-}$,
$pp \to \phi^+\phi^-$ and $pp \to \phi^{++}\phi^{--}$ reach to values $2.9\times10^{-3}pb$, $0.5\times10^{-3}pb$ and $1.2\times10^{-3}pb$ respectively. Thus if LHC reaches to an integrated luminosity of $100fb^{-1}$, which is planned to be achieved within two years\cite{LHClum}, up to hundreds of $\phi^{++}\phi^{--}$ and
$\phi^{++}\phi^{-}$ pairs can be produced. On the other hand number of $\phi^{+}\phi^{-}$ production can not exceed 50 events even in maximal conditions. Due to these
predictions, for the final state analysis we concentrate on the production channels $pp \to \phi^{++}\phi^{-}$ and $pp \to \phi^{++}\phi^{--}$.
At LHC, heavy charged scalars of the littlest Higgs model will be identified from their lepton flavor violating decay modes\cite{akeyord,thanlept1,gaurlept1,thanrev,gaurlept2,aysbook}. As stated in Eqs. \ref{dwp2} and \ref{dwp1}, the decay modes of the charged scalars are strongly dependent on
Yukawa couplings $Y$ and $Y'$, thus to the VEV of the scalar triplet, $v'$. By choosing $v'$ in the range $1GeV>v'>10^{-10}GeV$, the Yukawa couplings are chosen as $1>Y>10^{-10}$ and $Y'\simeq 10^{-10}$. For this range, we plotted the
final decay modes of charged pairs $\phi^{++}\phi^{--}$ and $\phi^{++}\phi^{-}$ in figure \ref{fig::decays}.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=7cm]{BR_Plot_22.eps}\hskip0.2cm\includegraphics[width=7cm]{BR_Plot_12.eps}
\vskip2mm
\hskip1cm (a)\hskip5cm (b)
\end{center}
\caption{Production rates of final decay modes of $\phi^{++}\phi^{--}$(a) and $\phi^{++}\phi^{-}$(b) vs $Y$.}
\label{fig::decays}
\end{figure}
For $\phi^{++}\phi^{--}$ pair, it is seen from Fig. \ref{fig::decays}(a) that for $Y<10^{-4}$ both of the double charged scalars decay into SM particles. The final state in this case will be
two $W_L$ pairs. Due to low production rates and huge background at LHC, this case has no significance. For $Y>10^{-3}$ both scalars will
decay into leptons violating lepton flavor. In this case $2/3$ of the final stay will be $l_il_i \bar{l}_j \bar{l}_j$ where $i$ and $j$ stand for different family of leptons, and the lepton flavor is explicitly violated by four giving a collider signal of four leptons consist of two same sign leptons from same family. This channel is free from any SM backgrounds.
Also for $Y>10^{-3}$, $1/3$ of the double charged pairs wil decay into $l_il_i\bar{l}_i\bar{l}_i$. Even if these decays happen via lepton flavor violating decays of
double charged scalars, the final signal will be indistinguishable from four leptons coming from SM backgrounds. For the range $10^{-3}>Y>10^{-4}$, one of the double charge scalar decay
into SM bosons, and the other one decay into same sign leptons of same family, thus the final signal will be
$W^-_L W^-_L \bar{l}_i \bar{l}_i$. In the view of the collider observables, this scenario is the most interesting because
it will yield lepton number violation by two. Since $W_L$ decay into jets with a branching ratio of $0.6$, $36\%$
of the final states resulting from $W^-_L W^-_L \bar{l}_i \bar{l}_i$ will be two leptons of same sign
and same family acompanied by jets, violating lepton number by two and free from any backgrounds at LHC.
For $\phi^{++}\phi^{-}$, we plotted production rates of the final signals depending on $Y$ in Fig. \ref{fig::decays}(b). For
$Y<10^{-3}$, both scalars will decay into SM particles and the final mode will be $W_L^+W_L^+\bar{t}b$. Due
to low production rates of charged scalars and huge SM background, this case is not promising at LHC.
For $Y$ close to unity ($Y\simeq1$),both scalars decay into leptons. In this case $2/3$ of final states will be
$\bar{l}_i\bar{l}_i l_j v_j$ and $1/3$ of the final states $\bar{l}_i\bar{l}_i l_i v_i$. In this scenario,
$\bar{l}_i\bar{l}_i l_j v_j$ final states will be violating lepton flavor, which is a significant observable at LHC free from any SM backgrounds. For $\bar{l}_i\bar{l}_i l_i v_i$, since all of the leptons are from same family, the final states cannot be distinguished from SM backgrounds.
For $\phi^{++}\phi^{-}$ the most interesting final state can be observed when $10^{-3}<Y<1$. In this case the double charged scalar will decay into two leptons of same sign and family and the single charged scalar will decay into jets. In this case the final collider signal will be $\bar{l}_i \bar{l}_i \bar{t}b$,
two same sign and family leptons plus jets, violating lepton number by two and free from any backgrounds.
After investigating the production rates of $\phi^{++}\phi^{--}$ and $\phi^{++}\phi^{-}$ pairs and analyzing the final states of the processes, we combine the results to find possible lepton flavor and number violating final events at LHC. The final states are free from SM backgrounds and they are listed as:
\begin{itemize}
\item $l_il_i\bar{l}_j\bar{l}_j$, four lepton final states: These final states are coming from the decays of $\phi^{++}\phi^{--}$. For $Y>10^{-3}$, $2/3$ of double charged pairs will decay in four leptons, with lepton flavor violation by four. In this final state double charged scalars can be reconstructed from same sign and same family lepton pairs.
\item $\bar{l}_i\bar{l}_i l_j v_j$, three leptons plus missing energy: For $Y\simeq1$, these final states are coming from the decays of $\phi^{++}\phi^{-}$. $2/3$ of the scalar pairs will decay into two same sign same family leptons plus one additional lepton with opposite sign from another family and the missing energy of the neutrino. In this final state the observed lepton flavor violation will be two since the family of the neutrino can not be identified.
\item $\bar{l}_i\bar{l}_i \bar{t} b$, two leptons plus jets: This is the most interesting final state arising from the decays of scalar pairs because the resulting signal violates lepton number by two. This final state can be observed from the final decays of $\phi^{++}\phi^{-}$ when $10^{-1}>Y>10^{-3}$, and also from the semileptonic decays of $\phi^{++}\phi^{--}$ if two of the final state $W_L$'s decay into jets when $10^{-3}>Y>10^{-4}$.
\end{itemize}
\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{noevents7x.eps}
\end{center}
\caption{Dependence of total number of lepton number and lepton flavor violating final states from decays of scalar pairs at LHC when $\sqrt{S}=7TeV$ and $M_\phi=0.5TeV$ for an integrated luminosity of $100fb^{-1}$ on Yukawa coupling $Y$.}
\label{fig::No7}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{noevents14x.eps}
\end{center}
\caption{Dependence of total number of lepton number and lepton flavor violating final states from decays of scalar pairs at LHC when $\sqrt{S}=14TeV$ and $M_\phi=0.5TeV$ for an integrated luminosity of $100fb^{-1}$ on Yukawa coupling $Y$.}
\label{fig::No14}
\end{figure}
Finally, we estimated the dependence of total number of lepton number and lepton flavor violating final states on $Y$
for $\sqrt{S}=7TeV$, $\sqrt{S}=14TeV$ when $M_\phi=0.5TeV$ in figures \ref{fig::No7} and \ref{fig::No14} respectively, and for
$\sqrt{S}=14TeV$ when $M_\phi=0.75TeV$ in figure \ref{fig::No14x2}. In this estimated we assumed an integrated luminosity of
$100fb^{-1}$ for LHC. It is seen from Fig. \ref{fig::No7} that for $M_\phi=0.5TeV$ and $\sqrt{S}=7TeV$, there can be
about 20 lepton number violating two lepton plus jet signals observable at LHC for $10^{-3}<Y<10^{-1}$ free from any backgrounds. For $\sqrt{S}=7TeV$, three lepton and four lepton channels have lower event numbers and they are not promising.
Our final analysis for $M_\phi=0.5TeV$ and $\sqrt{S}=14TeV$ plotted in figure \ref{fig::No14} shows that in this case the number of lepton flavor and number violating final states are more promising. For $10^{-3}<Y<10^{-1}$ there can be 80 four lepton signals violating lepton flavor by four, and also few hundreds of two leptons plus jets final states violating lepton number by two. If $Y\sim 1$, this case is dominated by hundreds of three lepton plus missing energy signals with an observed lepton flavor violation by two. For $\sqrt{S}=14TeV$ and $M_\phi=0.75TeV$, i.e. $f\sim1TeV$ the final number of lepton flavor and number violating events are plotted in figure \ref{fig::No14x2}. For higher values of $M_\phi$, the number of events are reduced almost $60\%$, but still in the reach since they are all free from any SM backgrounds. In this case about $30$ to $40$ signals of two leptons plus jets violating lepton number by two can be expected if $10^{-3}<Y<10^{-1}$. The expected number of events in four lepton or three lepton channels are at the order of $10$, if $10^{-3}<Y<10^{-1}$ or $Y\sim1$ respectively.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=9cm]{noevents14_2x.eps}
\end{center}
\caption{Dependence of total number of lepton number and lepton flavor violating final states from decays of scalar pairs at LHC when $\sqrt{S}=14TeV$ and $M_\phi=0.75TeV$ for an integrated luminosity of $100fb^{-1}$ on Yukawa coupling $Y$.}
\label{fig::No14x2}
\end{figure}
\section{Conclusion}
In conclusion, the charged scalars of the littlest Higgs model can be produced via $pp \to \phi^{++}\phi^{-}$,
$pp \to \phi^+\phi^-$ and $pp \to \phi^{++}\phi^{--}$ processes at LHC. Since the production rates are low, the detection of these scalar productions
can only be done from SM background free final states. For the littlest Higgs model, these signals are the lepton flavor and number violating
final states arising from the structure of its scalar sector containing a complex $SU(2)$ group with hypercharge two which can interact with SM particles at tree level.
In this work we found that depending on the Yukawa coupling $Y$, lepton flavor and number violating final states can be detected at
LHC. For $Y\sim1$, lepton flavor violation by four in the channel $l_il_i\bar{l}_j\bar{l}_j$ and lepton flavor violation by two
in the channel $\bar{l}_i\bar{l}_i l_j v_j$ can be observed. If the Yukawa coupling is in the range $10^{-3}<Y<0.1$, lepton number violation by two in the channel
$\bar{l}_i\bar{l}_i \bar{t} b$ can be observed. In both of these final states the littlest Higgs heavy scalar can be reconstructed from invariant mass distributions of
same sign and same family lepton pairs. To identify such a signal coming from littlest Higgs model, the mass of the scalar should satisfy $M_\phi\geq 0.5TeV$ due to constraints coming
from experimental data. If such final states are achieved at LHC, this will be a discriminating discovery for littlest Higgs model among other
little Higgs models and models containing doubly charged scalars.
|
{
"timestamp": "2012-03-13T01:00:54",
"yymm": "1203",
"arxiv_id": "1203.2232",
"language": "en",
"url": "https://arxiv.org/abs/1203.2232"
}
|
\section{Introduction}
The problem of vacuum energy is probably one of the most interesting puzzles of modern physics. The observable energy density of the vacuum is at least 60 orders of magnitude smaller {than the value expected from the Standard Model}. {There is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.}
These problem can be explained through the anthropic principle. In a well-known article \cite{Weinberg}, Weinberg estimated the upper limit for the effective cosmological constant as
$$
\Lambda_{max}<5000\Lambda_{0},
$$
where $\Lambda_{0}$ is an observable value. Higher values suppress hierarchical structure formation in the universe, and, therefore, lead to cosmologies completely devoid of life as we know it.
A further development in the application of anthropic argument was the inclusion of selection rules such as a self-sampling assumption or "mediocrity principle", namely {the notion that there is nothing very unusual about our civilization.} Acceptance of mediocrity principle, grounded on statistical approach, explicitly implies the existence of a multiverse that serves a role of a statistical ensemble. {In multiverse, we can estimate the probabilities of observing any given event $j$.} Such a probability is factorized as
\begin{equation}
P_j\sim {\bar{P}}_jf_j,
\label{factor}
\end{equation}
where ${\bar{P}}_j$ is the concentration of j-type bubbles and $f_j$ is the anthropic factor proportional to a total amount of observers residing inside the j-type bubble.
At first glance it may seem that the approach based on (\ref{factor}), could solve the problem of cosmological constant. Indeed, the anthropic factor decreases with increasing $\Lambda$. However, there is another difficulty. As we shall see, when $\Lambda \rightarrow 0$ the number of observers within a bubble increases and tends to infinity. This phenomenon may be called the "infrared divergence" \cite{Infra-1}, \cite{Infra}. Therefore the value $\Lambda=0$ is preferred from the anthropic point of view.
The main plane of article is the following. The next section is devoted to calculations of probabilities in multiverse according to (\ref{factor}). The existence of the "infrared divergence" is demonstrated. In the third section we use another method to calculate the probabilities that the divergence doesn't appear. In conclusion possible objections are considered.
\section{Infrared divergence at $\Lambda\rightarrow 0$}
The probability $P_{\Lambda}$ to find oneself in a universe with a given value of vacuum energy can be determined via
\begin{equation}
P_{\Lambda}\sim N(\Lambda)\bar{P}(\Lambda),
\label{1}
\end{equation}
where $\bar{P}(\Lambda)$ is an a priori probability distribution (the relative abundance of different values of $\Lambda$ associated with the different types of bubbles in the multiverse), and $N(\Lambda)$ is an anthropic factor proportional to the total number of observers in a given region of a multiverse. The aforementioned number is evidently related to the star formation rate \cite{Vilenkin-0}, which can be estimated from the astrophysical data:
\begin{equation}
N(\Lambda)\sim \int^{t_{c}}_{0} \dot{n}(t, \Lambda)V_{c}(t)dt,
\label{2}
\end{equation}
where $\dot{n}(t, \Lambda)$ is the star formation rate in a comoving volume $V_{c}$ and $t_{c}$ defines a time of collapse. Obviously, for those universes whose expansion has a de Sitter-type asymptote $t_{c}=\infty$.
If the universe has zero curvature and the radiation is small enough, the Friedman equations become integrable and the scale factor may be written as
\begin{equation}
a=T^{2/3}\sinh^{2/3}{\tau}
\label{eq:8}
\end{equation}
where $T = 2/\sqrt{3 \Lambda}$ and $\tau$ is a dimensionless time $\tau=t/T$.
In order to calculate the comoving volume, $V_{c}$, we'll use the causal patch cut off. The causal patch is the region within the cosmological horizon \cite{Susskind-1}, \cite{Susskind-2}. Such a choice will result in
\begin{equation}
V_{c}(t)=\frac{4\pi}{3} \left(\int^{\infty}_{t}\frac{dt}{a(t)}\right)^{3}=\frac{4\pi T}{3} \left(\int^{\infty}_{\tau}\frac{d\tau}{\sinh^{2/3}(\tau)}\right)^{3}
\label{Vc}
\end{equation}
For $t\ll t_{\Lambda}=\sqrt{3/\Lambda}$ the scale factor changes according to a power law
$$
a(t)\approx t^{2/3}.
$$
When $\Lambda\rightarrow 0$, $V_{c}(t)$ diverges. The anthropic factor is inversely related to the cosmological constant. The resulting $P(\Lambda)$ will be determined by the previously chosen function $\bar{P}(\Lambda)$. The natural choice is a flat prior distribution i.e. $dP(\Lambda)/d\Lambda=0$, because the interval of anthropically acceptable values of $\Lambda$ is small in comparison with Planck scale. {As we shall see, for such a distribution the probability to find oneself in a universe with smaller values of cosmological constant is larger.} Therefore the case $\Lambda=0$ is preferred.
\textbf{Remark}. One note that for $\Lambda=0$ Eq. (\ref{Vc}) is not applicable. In this case the infinity does not follow by formally taking the limit as $\Lambda\rightarrow0$. But for Friedmann universe without vacuum energy there is no cosmological horizon and therefore the comoving volume is infinite.
Let`s consider this fact in detail. The behavior of $\dot{n}(t)$ for our universe depends on the particular model of star formation \cite{Star-1},\cite{Star-2},\cite{Star-3}, \cite{Star-4}. However, the difference in $\dot{n}(t)$ using different star formation models
is not large, since all these models predict that $\dot{n}(t)$ reaches a maximum after a couple of billions of years and then is subject to a relatively fast decrease. The height and the width of the maximum depend on the cosmological constant.
Bousso and Leichenauer (hereafter BL), developed a semi-analytic model \cite{Star-5} of the star formation rate as a function of time, studying in particular how spatial curvature, amplitude of primordial density perturbations, and cosmological constant influence the SFR. Differently from previous papers (e.g., Hernquist \& Springel 2003, hereafter HS) their model is principally interested in how large changes in the studied parameters affect the SFR. HS model, for example is no longer valid when one studies SFR under large variations of the quoted parameters.
In order to understand the shape of the SFR, we have to recall two things:
a) structure formation originates from tiny perturbations already present in the early Universe. They expanded with Universe and collapsed before recombination giving rise to dark matter haloes that formed the gravitational wells in which baryons felt after recombination;
b) there was no star formation until structure formed with $T_{} > 10^4$ K. After structure formed, the SFR started to rise and reached a maximum.
Apparently there are significant differences between the results of BL model and HS and his calculations. One should point out that BL star formation model predicts that the rate at the present day is much smaller than in the other models. Also the other models seem to fit the data better at these late times. This is natural, since these models are adjusted to fit the data and BL model is not. This is also why the other two models closely agree with each other close to the present epoch.
A similar question arise concerning the height and position of peaks in these models. The top amplitude of SFR differs significantly: the biggest in BL model, the smallest for the fossil model \cite{Star-2}. In reality, the data about the peak of the star formation rate is much less certain \cite{Star-6}. One can see that the other models do not agree with each other concerning the epoch when the peak formed and its width. According to BL (private communication) the data, for large redshift, near the peak, was not as reliable as the data for small redshift and so it's much less clear which model is closer to reality.
The detailed calculations in \cite{Star-5} show that for sufficiently wide range of $\Lambda$, SFR varies only slightly (all other parameters remain fixed to the observed values; see especially figs. 3 and 4 in \cite{Star-5}).
One question at this point is why for large $\Lambda$ values, the SFR weakly depends on $\Lambda$ (e.g., $\Lambda=10\Lambda_{0}$), while for small $\Lambda$ ($0<\Lambda<\Lambda_{0}$), we have the same SFR, and the same total stellar mass production per unit of comoving volume as in our universe.
A brief explanation of this issue is that in universe, star formation peaked around 3 Gyr and dropped off since then for a number of reasons. At 3 Gyr the cosmological constant was dynamically unimportant and we might as well set it to zero.
Vacuum energy became important only several Gyr later, when star formation was already lower. Without a cosmological constant, it is true that there would be more mergers in the future but these will mostly be halos that are too massive to cool efficiently. So while $\Lambda$ does suppress hierarchical structure formation in the future, it has little effect on star formation, and decreasing $\Lambda$ would not significantly increase star formation in the future.
Therefore, when one is interested in numerical estimations, one can assume (for example in the range $0<\Lambda<10\Lambda_{0}$) that:
$$
\dot{n}(t, \Lambda)\approx \dot{n}(t, \Lambda_{0}),\quad 0<\Lambda<10\Lambda_{0}
$$
The accuracy of these estimates, in any case, is limited by our knowledge of star formation mechanisms. In the following, we consider the analytical fit for star formation rate in our universe given in \cite{Star-1}. The dependence of SFR from redshift $z$ is
\begin{equation}
\dot{n}(z)=\dot{n}_{m}
\frac{b\exp\left[a(z-z_m)\right]}{b-a+a\exp\left[b(z-z_m)\right]},
\label{eqnoldfit}
\end{equation}
with $a= 3/5$, $b=14/15$, $z_m=5.4$, and $\dot{n}_{0}= 0.15\,{\rm
M}_\odot{\rm yr}^{-1}{\rm Mpc}^{-3}$ is the maximal star formation rate at $z=z_{m}$. {We plot this expression in Fig. 1(a), where the star formation rate reaches a peak at a redshift $z_{m}=5.4$, declining roughly exponentially towards both low and high redshift.} Using the well-known relation between $t$ and redshift $z$
$$
H_{0}dt=\frac{-dz}{((1+\Omega_{m}z)(1+z)^{2}-(z+2)z\Omega_{\Lambda})^{1/2}},
$$
where $H_{0}$ is the Hubble constant in our time and $\Omega_{m, \Lambda}=\rho_{m,\Lambda}/(\rho_{m}+\rho_{\Lambda})$, we can derive
the time dependence of $\dot{n}(t)$ {depicted in Fig. 1(b)}. We used $\Omega_\Lambda = 0.72\pm 0.04$ from the
WMAP results of \cite{WMAP} and $H_0 =72\pm 8$ km/s/Mpc from the Hubble Space Telescope key project \cite{HST}. {From Fig. 1b, one can see that star formation rate reaches maximum at $t_{m} \approx 1.5$ Gyr and then steadily declines.}
\begin{figure}
\begin{center}
\includegraphics{sfr.eps}\\
(a)\\
\includegraphics{sfr-1.eps}\\
(b)
\\
\caption{Star formation rate ($M_{\odot}yr^{-1}Mpc^{-3}$) as function of redshift (a) and time (b).}
\end{center}
\end{figure}
Therefore we can estimate the anthropic factor for $0<\Lambda<10\Lambda_{0}$ using the time dependence of SFR in fig. 1(b). From (\ref{2}),(\ref{Vc}) it follows that
\begin{equation}
N(\Lambda)\sim \Lambda^{-1/2}\int_{0}^{t_{c}}d t \dot{n}(t)\left(\int_\tau^{\infty}\frac{d\tau}{\sinh^{2/3}(\tau)}\right)^{3}.
\label{antr}
\end{equation}
In Eq. (\ref{antr}) integration is over dimensionless time which is linked to our dimensionless time $\tau_{0}$ by the relation $\tau=\tau_{0}\sqrt{\Lambda/\Lambda_{0}}$. For simplicity, we assume that $\dot{n}(t)=0$ after $t_{f}\sim 14$ Gyr. For our universe, $\tau_{f}=\tanh^{-1}\Omega_{\Lambda}$. This assumption is in a good agreement with VIMOS VLT Deep Survey data \cite{Star-7}. According to VVDS, SFR declines steadily by a factor 4 from $z=1.2$ to $z=0.05$, since in this phase both giant and intermediate galaxy populations decline. The most luminous sources ceased to efficiently produce new stars 12 Gyrs ago (at $z\sim4$), while intermediate luminosity sources
continued to produce stars till 2.5 Gyrs ago (at $z\sim0.2$).
It is convenient to define the relative anthropic factor as:
\begin{equation}
N_{rel}(\alpha)=\frac{N(\Lambda)}{N(\Lambda_{0})}\sim \alpha^{-1/2}\int_{0}^{t_{f}}d t \dot{n}(t)\left(\int_{t/T}^{\infty}d\tau\ f(\tau)\right)^{3}\times
\label{antr1}
\end{equation}
$$
\times\left[\int_{0}^{t_{f}}d t \dot{n}(t)\left(\int_{t/T_{0}}^{\infty}d\tau f(\tau)\right)^{3}\right]^{-1},
$$
where $\alpha=\Lambda/\Lambda_{0}$ and $f(\tau)=\sinh^{-2/3}(\tau)$.
Following \cite{Bousso}, one can assume that observers appeared in universe after a "delay time" of the order of some billions of years. One can re-write Eq. (\ref{2}) as
\begin{equation}
N(\Lambda)\sim \int^{t_{c}}_{\Delta t} \dot{n}(t-\Delta t, \Lambda)V_{c}(t)dt\sim \int^{t_{c}+\Delta t}_{0} \dot{n}(t, \Lambda)V_{c}(t+\Delta t)dt,
\label{22}
\end{equation}
where $\Delta t$ is a time delay. In this case in (\ref{antr1}) one need just to change the limit of the inner integral: $t\rightarrow (t+\Delta t)$.
The results of calculations for $\Delta t=0, \quad 5, \quad 10$ Gyr are given in figs. 2 - 4, respectively. {In the quoted figures, data of numerical calculations are marked by crosses, while solid lines are analytical fits.} {The result of the quoted figures is that the relative anthropic factor (and therefore the probability to find oneself in a universe with a given $\Lambda$) increases with decreasing $\Lambda$. The rapidity of this increasing is smallest for $\Delta t=0$ and largest for $\Delta t=10$ Gyr.}
\begin{figure}
\begin{center}
\includegraphics{1.eps}
\includegraphics{1-1.eps}
\\
\caption{The relative anthropic factor for $\Delta t=0$. As shown in the upper panel, we use a logarithmic scale for small values of $\Lambda$.}
\end{center}
\end{figure}
We found the following analytical fit for numerical estimations plotted in Figs. 2-4:
\begin{equation}
N_{rel}(\alpha) \approx \left\{\begin{array}{ll} \label{Nrel}
\alpha^{-\gamma_{1}}\exp(-\alpha/\beta),\quad 0.1\Lambda_{0}\leq\Lambda\leq10\Lambda_{0}\\
\alpha^{-\gamma_{2}(\alpha)},\quad 0<\Lambda\leq 0.1\Lambda_{0}\end{array}\right.
\end{equation}
The parameters $\gamma_{1}$ and $\beta$ depend on $\Delta t$ (see table). Parameter $\gamma_{2}$ slowly decreases with decreasing $\alpha$. When $\Lambda\rightarrow 0$ the relative anthropic factor tends to
$$
N_{rel}(\alpha)\rightarrow\frac{C}{\alpha^{1/2}},\quad \alpha\rightarrow 0,
$$
where
$$
C=\int_{0}^{t
_{f}}d t \dot{n}(t)\left(\int_{0}^{\infty}d\tau\ f(\tau)\right)^{3}\left(\int_{0}^{t_{f}}d t \dot{n}(t)\left(\int_{t/T_{0}}^{\infty}d\tau f(\tau)\right)^{3}\right)^{-1}
$$
is a constant. At $\Lambda=0$, our calculations lose all meaning because in this case relative anthropic factor becomes infinitely large. Hence the probability that randomly selected observer measure a value $\Lambda=0$ is exactly equal to 1.
\begin{center}\begin{tabular}{|c|c|c|}
\hline
$\Delta t$, Gyr & $\gamma_{1}$ & $\beta$ \\
\hline
0 & 0.79 & 30 \\
5 & 1.08 & 10 \\
7.5 & 1.21 & 10 \\
10 & 1.33 & 10\\
12.5 & 1.45 & 10 \\
\hline
\end{tabular}
\end{center}
\begin{figure}
\begin{center}
\includegraphics{2.eps}
\includegraphics{2-1.eps}
\caption{The relative anthropic factor for $\Delta t=5$ Gyr.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics{3.eps}
\includegraphics{3-1.eps}
\\
\caption{The relative anthropic factor for $\Delta t=10$ Gyr.}\
\end{center}
\end{figure}
\section{The 4-volume averaging of probabilities as a possible solution to the infrared divergence problem}
If the eternal inflation model is correct, this implies the existence of multiverse, containing infinite number of copies of every possible observer. What can we tell about probabilities of a event in such a multiverse? Suppose we are conducting an experiment to determine the value of cosmological constant with various possible outcomes $\Lambda_{n}$. The probabilities for {these outcomes} are connected by the formula
\begin{equation}
\frac{P_{\Lambda_{n}}}{P_{\Lambda_{k}}}=\frac{N(\Lambda_{n})}{N(\Lambda_{k})},
\label{Prob}
\end{equation}
where $N(\Lambda_{n, k})$ is the amount of $\Lambda_{n}$ and $\Lambda_{k}$ results in all the multiverse.
The direct calculation of probabilities by Eq. (\ref{Prob}) becomes impossible because $N(\Lambda_{k,n})\rightarrow\infty$.
The task of infinities elimination, known as "measure problem", is important for modern cosmology. Some other possibilities for the measure definition have been proposed so far \cite{Measure-1}-\cite{Measure-9}.
Among the various approaches to measure problem, one should notice the one described in \cite{Page-1}. The key idea, in \cite{Page-1}, is that one need to replace volume weighting of probabilities by volume averaging. {According to this assumption the relative probabilities are proportional to expectation values of the fraction of the number of locations in which the observation
occurs.} The latter value is proportional to a number of observers per unit 4-volume, i.e. the number of occurred observations per unit spatial volume per unit of time. Therefore one need to compare the densities of observers instead their numbers.
Initially, volume averaging was introduced in order to solve the Boltzmann brain (BB) problem \cite{Albrecht-1}, \cite{Albrecht-2}. {This problem can be described as follows. Let`s consider toy multiverse containing only two types of universe. The universe of first type (I) expands forever while the universe of second type (II) has a finite size and finite lifetime. In I-universes , ``ordinary observers'', like ourselves, should be vastly outnumbered by infinite number of BBs, arising from vacuum fluctuations. In II-universes only a finite number of ordinary observers exists, therefore the ''ordinary observers'' in such multiverse are highly atypical. As pointed in \cite{Page-1} replacing the volume weighting measure with volume averaging can avoid the BB catastrophe because the density of BBs is much less than the density of ordinary observers.} Volume averaging also has a deep link with quantum mechanics \cite{Page-2}, \cite{Page-3}, \cite{Page-4}.
One can show that volume averaging eliminates the "infrared divergence". The spatial volume corresponding to comoving volume in (\ref{Vc}) is
\begin{equation}
V_{3}=\frac{4\pi T^{3}}{3}\left(\sinh^{2/3}(\tau)\int^{\infty}_{\tau}\frac{d\tau}{\sinh^{2/3}(\tau)}\right)^{3}
\end{equation}
Obviously, for $\tau\rightarrow\infty$, $V_{3}$ converges to $4\pi(\sqrt{3/\Lambda})^3/3$, i.e. to the Hubble volume of bubble.
The 4-volume therefore is
$$
V_{4}(t)=\int_{0}^{t}V_{3}dt=\frac{4\pi T^{4}}{3}\int_{0}^{\tau}d\tau \left(\sinh^{2/3}(\tau)\int^{\infty}_{\tau}\frac{d\tau}{\sinh^{2/3}(\tau)}\right)^{3}
$$
and diverges at $\tau\rightarrow\infty$. The density of observers per unit 4-volume tends to zero for long-lived de Sitter vacua, but the relative density of observers is non-zero
\begin{equation}
\langle N_{rel}\rangle=\frac{N(\Lambda)/V_{4}(\Lambda)}{N(\Lambda_{0})/V_{4}(\Lambda_{0})}=\alpha^{2}N_{rel}.
\label{aver}
\end{equation}
So the probability to find oneself in a universe with given value of $\Lambda$ becomes a well-defined function of $\Lambda$. Combining (\ref{Nrel}) and (\ref{aver}) gives the following result
\begin{equation}
\langle N_{rel}(\alpha)\rangle\approx \left\{\begin{array}{ll} \label{Naver}
\alpha^{2-\gamma_{1}}\exp(-\alpha/\beta),\quad 0.1\Lambda_{0}\leq\Lambda\leq10\Lambda_{0}\\
\alpha^{2-\gamma_{2}(\alpha)},\quad 0<\Lambda\leq 0.1\Lambda_{0}\\
\end{array}\right.
\end{equation}
{The dependence of $N_{rel}$ on $\Lambda$ is depicted on Fig. 5 for $\Delta t=5$ (thin solid line), 7.5 (thin dotted line), 10 (thick solid line), and 12.5 Gyr (thick dotted line).} Hence the density of observers reaches its maximum for a value $\Lambda_{m}$ of the vacuum energy. {The value of this maximum decreases with increasing time delay.}
For a time delay 2.5 Gyr$<\Delta t<$12.5 Gyr, we have $\Lambda_{m}\approx5\div10 \Lambda_{0}$. For larger values of vacuum energy, $\Lambda>10\Lambda_{0}$ one has to take into account the star formation decline due to early dominance of vacuum energy.
Finally, it remains to show that volume averaging does not lead to singularities in the case $\Lambda=0$. Let`s consider the unit comoving volume in such universe. The total stellar mass within this unit volume can be estimated as
$$
M^{(1)}=\int_{0}^{t_{f}}\dot{n}(t)dt.
$$
The corresponding 4-volume increases with time according to the law
$$
V^{(1)}_{4}(t)=\int^{t}_{0} t^{2}dt=t^{3}/2.
$$
Hence the density of observers is equal to
$$
\langle N(\Lambda=0)\rangle\sim\lim_{t\rightarrow\infty}\frac{M^{(1)}}{t^{3}/2}
$$
and it is easy to see that
$$
\langle N_{rel}(0)\rangle\sim\lim_{t\rightarrow\infty}\frac{\int_{0}^{t/T_{0}}d\tau \left(\sinh^{2/3}(\tau)\int^{\infty}_{\tau}\frac{d\tau}{\sinh^{2/3}(\tau)}\right.)^{3}}{t^{3}}=0
$$
So the point $\Lambda=0$ is not singular.
\begin{figure}
\begin{center}
\includegraphics{N.eps}
\\
\caption{The relative density of observers for various $\Delta t$}
\end{center}
\end{figure}
\section{Conclusion}
Some questions remain to be answered. They are the following:
1. the observable vacuum energy density is one order of magnitude smaller than the "optimal" value $\Lambda_{m}$. Does this mean that our universe is atypical in the multiverse?
The traditional approach is that the observable universe is assumed a typical one. It seems to us that this methodology only complicates the understanding of the real universe. In our opinion, the knowledge of fundamental laws is enough to estimate parameters of a "typical" (from anthropic point of view) universe. Subsequent comparison of these parameters with the observed values in our universe help us to find an answer to the aforementioned question. One note also that according to our calculations (fig. 5) the probability to find oneself in the universe with $\Lambda=\Lambda_{m}$ is only 1.5 - 3 times higher than probability to find oneself in the universe with $\Lambda=\Lambda_{0}$. Therefore the observed value of vacuum energy lies in the reasonable region.
2. Negative values of $\Lambda$. One can notice that, for $\Lambda<0$, the volume averaging also eliminates "infrared divergence" at $|\Lambda|\rightarrow0$. Hence regularization scheme based on volume averaging gives correct answers for $\Lambda<0$.
But the following problem appears. If vacuum energy is negative, universe ends its existence in big crunch singularity. In this case the density of observers is larger than in the case of a de Sitter universe, because the 4-volume of anti de Sitter universe is much less. So, why we don't live in universe with negative $\Lambda$?
Firstly, this problem occurs in the \textit{prediction} stage, but it disappears at the \textit{explanatory} stage when cosmological constant has already been measured by the observer. According to \cite{Vilenkin}, only the observers with similar informational content can be assigned to the same equivalence class. The sign of the cosmological constant is already known for us. Hence we are representatives of the reference class which includes all observers for which $\Lambda>0$. When calculating probabilities one should consider only observers belonging to this class.
Secondly, there may exists a physical mechanism, unknown to date, imposing the bounds on the lifetime (and 4-volume) of de Sitter universe \cite{Doomsday-1}, \cite{Doomsday-2}. An interesting scenario was suggested in \cite{Page-5}, \cite{Page-6}, \cite{Page-7}. According to this scenario our vacuum should be rather unstable and should decay within 20 Gyr (which is possible if the gravitino is superheavy).
In conclusion it should be emphasized that the main result of the present paper is the proof that replacing volume weighting with volume averaging in the cosmological measure, can avoid the infrared divergence problem. Volume averaging leads to natural explanation of the reason why the observed value of vacuum energy is non-zero. Perhaps this result can be considered as argument in favor of using volume averaging measure in cosmology.
\acknowledgments
A. A. grateful to A. V. Yurov for useful discussion. Authors also grateful to anonymous referees for useful comments.
|
{
"timestamp": "2012-04-18T02:01:55",
"yymm": "1203",
"arxiv_id": "1203.2290",
"language": "en",
"url": "https://arxiv.org/abs/1203.2290"
}
|
\section{Introduction}
The discovery of the accelerating expansion of the universe
\cite{Riess,Perlmutter} has
raised a number of difficult problems in cosmology. The cosmic
acceleration can be explained via the introduction of so-called dark energy
(for recent review, see \cite{Dark-6,Cai:2009zp}) with quite strange properties
like
negative pressure and/or negative entropy, invisibility in the early universe,
etc. According to the
latest supernovae observations, the dark energy currently accounts for
73\% of the total mass energy of the universe (see, for instance,
Ref.~\cite{Kowalski}).
The equation of state (EoS) parameter $w_\mathrm{D}$ for dark energy is
negative:
\be
w_\mathrm{D}=p_\mathrm{D}/\rho_\mathrm{D}<0\, ,
\ee
where $\rho_\mathrm{D}$ is the dark energy density and $p_\mathrm{D}$ is the
pressure. Although astrophysical observations favor the standard
$\Lambda$CDM cosmology, the uncertainties in the determination of the EoS dark
energy parameter $w$ are
still too large to define which of the three cases $w < -1$, $w = -1$, and
$w >-1$ is realized in our universe: $w=-1.04^{+0.09}_{-0.10}$
\cite{PDP,Amman}.
If $w<-1$ (phantom dark energy) \cite{Caldwell}, we are dealing with the most
interesting and least understood theoretical case.
For a phantom field, the violation of all of the four energy conditions occurs.
Although this field is unstable from the quantum field theory viewpoint
\cite{Carrol}, it could be stable in classical cosmology. Some
observations \cite{P} may be understood as the indication of the crossing
of the phantom divide in the near past or in the near future.
A very unpleasant property of phantom dark energy is the Big Rip future
singularity \cite{Caldwell,Frampton,S,BR,Nojiri},
where the scale factor becomes infinite at a finite time in the future.
A less dangerous future singularity caused by phantom or quintessence dark
energy
is the sudden (Type II) singularity \cite{Barrow} where the scale factor is
finite
at Rip time. Nevertheless, the condition $w<-1$ is not sufficient for a
singularity to occur.
Mild phantom models where $w$ asymptotically tends to $-1$ and the energy
density increases with time
or remains constant but there is no finite-time future singularity are
discussed in recent works \cite{Frampton-2,Frampton-3,Astashenok}.
The key point is that if $w$ approaches $-1$ sufficiently rapidly,
then it is possible to have a model in which the time required for
the occurrence of the singularity is infinite, i.e., the
singularity effectively does not
occur. However, if the energy density grows, the disintegration of bound
structures
eventually occurs in a way similar to the case of the Big Rip singularity.
Such phenomena, called the Little Rip and the Pseudo-Rip, destroy the bound
structures in the universe at a finite time!
The most convenient formalism to construct the (scalar or fluid)
dark energy is to use the EoS:
\be
\label{EoS-0}
p=g(\rho)\, ,
\ee
where $g$ is a function of the energy density.
The evolution of the universe then depends on the choice of the EoS.
The aim of this article is to develop a general approach for the construction
of
dark energy models which are compatible with observational data and which
provide
various scenarios of evolution.
Moreover, we are most interested in models which are stable for a long period
of time before entering the Little rip/Pseudo-Rip induced dissolution of bound
structures or before entering the mild finite-time
future singularity (with a finite scale factor at the Rip).
This question is
analyzed from the viewpoint of the dark energy EoS and
corresponding description in terms of scalar field theory. In Sec.~\ref{SecII},
the general approach to this problem is developed.
The classification of cosmological models with different future evolutions is
presented. The confrontation with observational data is also given.
The non-singular Little Rip cosmology based on a massive scalar potential is
considered in Sec.~\ref{SecIII}.
It is demonstrated that it may be indistinguishable from the $\Lambda$CDM
cosmology, being stable for a long time before the disintegration of bound structures.
Although the similar models are considered in \cite{Frampton-2},
it is interesting to note that the simplest model with Klein-Gordon potential is
described apparently for the first time.
In Sec.~\ref{SecIV} the asymptotically de Sitter quintessence or phantom
cosmology is proposed. The properties of such realistic cosmologies are
similar to those of the previous section.
In comparison, for example, with \cite{Frampton-3}, our approach is based on
equation-of-state formalism instead of setting of the Hubble parameter as function of time.
The next two sections are devoted to the construction of quintessence
dark energy with a Type III future singularity and phantom/quintessence
dark energy with a Type II future singularity.
These results based on EoS formalism are novel in cosmology.
The estimation of time remaining before Type II or Type III singularity is
made for first time.
The comparison of the
predictions of such models with observational data demonstrates that they
may be indistinguishable the $\Lambda$CDM model up to the present. Moreover,
they may be
stable for billions of years before reaching a future singularity.
In this sense, such models represent a quite viable alternative to
$\Lambda$CDM. Sec.~\ref{SecVII} is devoted to the construction of Big Crush
scalar quintessence models and their comparison with the Little Rip or Type II
future
singularity cosmology. Specifically, some physical properties of the Big Crush
versus the Rip are discussed. Phantom models are briefly reviewed
in Sec.~\ref{SecPhantom}.
The main qualitative result of our study is that the current observations
make it essentially
impossible to determine whether or not the universe will end in a future
singularity.
We should note once more that the consistent application of equation-of-state formalism
is the main feature of our work.
Some summary and outlook are given in the Discussion section.
\section{Scalar dark energy models \label{SecII}}
In this section, we consider the construction of one-scalar dark energy
models and compare the models with observational data.
We mainly concentrate on different phantom theories whose classical
behavior is very similar to that of the $\Lambda$CDM model.
For the spatially-flat FRW universe with metric
\be
ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})\, ,
\ee
the cosmological equations, that is, the FRW equations are given by
\be
\label{Fried1}
\left(\frac{\dot a}{a}\right)^2 = \frac{\rho}{3}\, , \quad
\dot{\rho} = -3\left(\frac{\dot a}{a}\right)(\rho + p)\, ,
\ee
where $\rho$ and $p$ are the total energy density and pressure,
$a$ is the scale factor, $\dot{}=d/dt$,
and we use the natural system of units in which $8\pi G=c=1$.
It is convenient to write the dark energy equation of state (EoS)
in the following form:
\be \label{EoS}
p_\mathrm{D}=-\rho_\mathrm{D}-f(\rho_\mathrm{D})\, ,
\ee
where $f(\rho_\mathrm{D})$ is some function. The case $f(\rho_\mathrm{D})>0$
corresponds
to the EoS parameter $w<-1$ (phantom) while the case $f(\rho)<0$ corresponds to
the EoS
parameter $w>-1$. If dark energy dominates, one can neglect the contribution
of other components (matter, dark matter). Then from Eq.~(\ref{Fried1}), one
can get the following expression for time variable:
\be
\label{trho}
t = \frac{2}{\sqrt{3}}\int^{x}_{x_{0}} \frac{d x}{f(x)}\, , \quad
x\equiv\sqrt{\rho}\, .
\ee
Hereafter, it is convenient to omit the subscript $\mathrm{D}$. The
quintessence
energy density decreases with time ($x<x_{0}$), while the phantom energy
density increases ($x>x_{0}$).
Following Ref.~\cite{Nojiri}, one can find the following behavior for the
expression (\ref{trho}):
\begin{enumerate}
\item The integral (\ref{trho}) converges at $\rho\rightarrow\infty$. Therefore
a finite-time singularity occurs: the energy density becomes infinite at a
finite time $t_\mathrm{f}$. The expression for the scale factor
\be
\label{arho}
a = a_{0}\exp\left(\frac{2}{3}\int^{x}_{x_{0}} \frac{x d x}{f(x)}\right)\, ,
\ee
indicates that there are two possibilities:
\begin{enumerate}
\item\label{1a}
The scale factor diverges at a finite time (Big Rip singularity
\cite{Caldwell,Frampton,BR}).
\item\label{1b}
The scale factor remains finite; however, a singularity
($\rho\rightarrow\infty$) occurs.
This is a Type III singularity \cite{Nojiri-2}.
\end{enumerate}
The key difference between (\ref{1a})
and (\ref{1b})
is that for (\ref{1b})
the energy density grows so rapidly with time that
the scale factor does not reach an infinite value.
\item The integral (\ref{trho}) diverges at $\rho\rightarrow\infty$.
Such models are described in \cite{Frampton-2} (see also
Refs.~\cite{Astashenok,others}).
The energy density grows with time but
not rapidly enough for the emergence of the Big Rip singularity.
According to Ref.~\cite{Frampton-2}, we have a so-called ``Little Rip'':
eventually a dissolution of bound
structures occurs at a finite future time. Nevertheless, formally the future
singularity does not occur (or, rather, it is shifted to the infinite
future).
Such scenarios are possible only in the case of phantom dark energy. The next
two scenarios are possible for both phantom and quintessence dark
energy.
\item The integral (\ref{trho}) diverges at $\rho\rightarrow\rho_\mathrm{f}$.
The dark
energy density asymptotically tends to a constant value (``effective
cosmological constant''). Such asymptotically de Sitter theories represent
the natural alternative to the $\Lambda$CDM model, which also leads to
non-singular cosmology.
Nevertheless, even for a non-singular asymptotically de Sitter universe, the
possibility of a dramatic rip which may lead to disappearance of bound
structures in the universe remains possible \cite{Frampton-2,Astashenok}.
\item Another interesting case corresponds to $f(x)\rightarrow\pm \infty$ at
$x=x_\mathrm{f}$, i.e., the pressure of the dark energy becomes infinite at a
finite energy density. The second derivative of the scale factor diverges while
the first
derivative remains finite. It is interesting to investigate the
properties of dark energy with
such a (sudden or Type II) finite-time future
singularity \cite{Barrow,Nojiri-2}.
\end{enumerate}
What does this last type of singularity mean from the physical viewpoint?
As the
universe expands, the relative acceleration between two points separated
by a distance $l$ is given by $l \ddot a/a$.
If there is a particle with mass $m$ at each of the points, an observer at
one of the masses will measure an inertial force on the other mass of
\be
\label{i1}
F_\mathrm{iner}=m l \ddot a/a = m l \left( \dot H + H^2 \right)\, .
\ee
Let us assume that two particles are bounded by a constant force $F_0$. If
$F_\mathrm{iner}$ is positive and greater
than $F_0$, the two particles become unbounded. This is the rip produced by
the accelerating expansion.
Note that Eq.~(\ref{i1}) shows that the rip always occurs when either $H$
diverges or $\dot H$ diverges (assuming $\dot H > 0$). The first case
corresponds to
the Big Rip singularity, while
if $H$ is finite, but $\dot H$ diverges with $\dot H > 0$,
we have a Type II or sudden future singularity, which also leads to a rip.
Even if $H$ or $\dot H$ goes to infinity at the infinite future, the
inertial force becomes larger and larger,
and any bound object is ripped, i.e., the Little Rip cosmology emerges.
If $H$ is finite and $\dot H$ is negative and diverges, then all the structures
are crushed rather than ripped.
Eq.~(\ref{i1}) can be rewritten as
\be
\label{Fin}
F_\mathrm{iner}={m l}\left(\frac{x^{2}}{3}+\frac{f(x)}{2}\right)\, .
\ee
The crush corresponds to $f(x)\rightarrow -\infty$ at
$x\rightarrow x_\mathrm{f}<\infty$ while the case
$f(x)\rightarrow\infty$ at $x\rightarrow x_\mathrm{f}$
describes the sudden future singularity.
The equivalent description in
terms of scalar theory can be derived using the equations:
\be
\rho=\pm\dot{\phi}^{2}/2+V(\phi)\, , \quad p=\pm\dot{\phi}^{2}/2-V(\phi)\, ,
\ee
where $\phi$ is the scalar field with potential $V(\phi)$. The sign ``$-$''
before kinetic term corresponds to the phantom energy. For the scalar field and
its potential, one can derive the following expressions:
\bea
\label{phix}
&&
\phi(x)=\phi_{0}\pm\frac{2}{\sqrt{3}}\int_{x_{0}}^{x}\frac{dx}{\sqrt{|f(x)|}
}\, ,\\
\label{Vx}
&& V(x)=x^{2}+f(x)/2\, .
\eea
Combining Eqs.~(\ref{phix}) and (\ref{Vx}) gives the potential as function
of the scalar field.
For simplicity, we choose the sign ``$+$'' in Eq.~(\ref{phix}) for
phantom energy and ``$-$'' for quintessence.
For the crush and sudden future
singularity the potential of the scalar field has a pole, i.e. from the
mathematical viewpoint, these singularities are equivalent.
Note that singularities often correspond to the infinite value of
the scalar field $\phi\to \pm \infty$.
For the sudden future singularity
potential, we find $V(\phi)\rightarrow+\infty$, and for the big crush,
$V(\phi)\rightarrow -\infty$.
Confrontation of the theoretical models with observational data consists mainly
of comparison with the
distance modulus as a function of redshift from the Supernova Cosmology
Project \cite{Amanullah:2010vv}.
The distance modulus for a supernova with redshift $z=a_{0}/a-1$ is
\be
\mu(z)=\mbox{const}+5\log D(z)\, ,
\ee
where $D(z)$ is the luminosity distance. As is well-known, the
SNe data are well-fit by the $\Lambda$CDM cosmology. For such a model (which we
call the ``standard cosmology'' (SC)), we obtain
\be
\label{DLSC}
D^\mathrm{SC}_\mathrm{L}=\frac{c}{H_{0}}(1+z)\int_{0}^{z}
\left(\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}\right)^{-1/2}d z\, .
\ee
Here, $\Omega_{m}$ is the fraction of the total density contributed by matter,
and $\Omega_{\Lambda}$ is the fraction contributed by the vacuum energy density.
One can get also the deceleration parameter $q_0$ and jerk parameter
$j_0$ \cite{Sahni:2002fz}:
\bea
\label{DDD1}
&& q_0 = - \left. \frac{1}{a H^2} \frac{d^2 a}{dt^2}\right|_{t=t_0}
= - \left. \frac{1}{H^2} \left\{ \frac{1}{2} \frac{d \left( H^2 \right)}{dN}
+ H^2 \right\} \right|_{N=0} \, ,\nn
&& j_0 = \left. \left\{ \frac{1}{a H^3} \frac{d^3 a}{dt^3} \right|_{t=0}
= \left. \frac{1}{2H^2} \frac{d^2 \left( H^2 \right)}{dN^2}
+ \frac{3}{2H^2} \frac{d \left( H^2 \right)}{dN} + 1 \right\} \right|_{N=0}
\, .
\eea
Here $N$ is defined by
\be
\label{e-foldings}
N \equiv - \ln \left(1+z\right)\, .
\ee
For the current time $t=t_0$, we have $N=0$. It is useful to note that
\be
\frac{d}{dN}=-(1+z)\frac{d}{dz}\, , \quad
\frac{d^{2}}{dN^{2}}=(1+z)\frac{d}{dz}+(1+z)^{2}\frac{d^{2}}{dz^{2}}\, .
\ee
Measuring the deceleration parameter and especially the jerk parameter is a
much
more difficult task than measuring $H_{0}$. In order to measure the Hubble
constant, one needs to derive the distances to objects at $\sim 100$\, Mpc;
this corresponds to a redshift of $z\gtrsim 0.02$.
To obtain $q_{0}$, one needs to observe objects to redshift
$z\gtrsim 1$.
Therefore, current observational results for the deceleration and jerk
parameters are
not totally reliable. For example, Ref.~\cite{Rapetti} gives for a flat model
tight
constraints on $q_{0}=-0.81\pm0.14$ and $j_{0}=2.16^{+0.81}_{-0.76}$ from
type Ia supernovae and X-ray cluster gas mass fraction measurements.
These results are consistent with $\Lambda$CDM
at about the 1$\sigma$ confidence level.
For the $\Lambda$CDM model, one gets
\be
q_{0}=\frac{3}{2}\Omega_{m}-1\, ,\quad j_{0}=1\, .
\ee
Therefore, measuring $j_{0}$ is important in the search for deviations from
$\Lambda$CDM, since all
$\Lambda$CDM models, regardless of the matter and cosmological constant energy
densities,
are characterized by $j=1$.
This concludes our general discussion of scalar dark energy models.
Below we present several examples of such models.
\section{Scalar Little Rip cosmology \label{SecIII}}
It is known that the Little Rip cosmology can be realized in
the class of exponential or power-law scalar potentials
\cite{Frampton-3,Astashenok}.
It is interesting that even for the case of the simplest
Klein-Gordon potential, the Little Rip can occur.
Let us start from
\be
\label{LRT}
f(\rho)=\alpha^{2}=\mathrm{const}\, .
\ee
From the conservation law for the dark energy fluid (\ref{Fried1}), it is
easy
to
obtain
\be
\rho=\rho_{0}-3\alpha^{2}\ln(1+z).
\ee
The luminosity distance is given as
\be
D_\mathrm{L}=\frac{c}{H_{0}}(1+z)\int_{0}^{z}
\left(\Omega_{m}(1+z)^{3}+(1+3(w_{0}+1))\Omega_\mathrm{D}\right)^{-1/2}d z\, .
\ee
where the current EoS parameter $w_{0}$ is
\be
w_{0}=-1-\frac{\alpha^{2}}{\rho_{0}}\, ,
\ee
and $\Omega_{m}$ and $\Omega_\mathrm{D}$ are the matter and
dark energy contributions to the total energy budget. It is clear that if
$\alpha^{2}\ll \rho_{0}$ (and therefore $w_{0}\approx-1$) there is good
agreement with the observational data: such a cosmology is indistinguishable
from the
$\Lambda$CDM model. The deceleration and jerk parameters are given by
\be
q_{0} = \frac{3}{2}\Omega_{m}-1+\frac{3}{2}(w_{0}+1)\Omega_\mathrm{D} \, ,
\quad
j_{0}=1-\frac{9}{2}(w_{0}+1)\Omega_\mathrm{D}\, .
\ee
For $-1.15<w_{0}<-1$ and $\Omega_\mathrm{D}=0.72$, these parameters are in good
agreement with observational data. The scalar field grows with time as
\be
\phi=\phi_{0}+\alpha t\, ,
\ee
and the potential describes the massive scalar field:
\be
V(\phi)=\frac{m^{2}}{2}(\phi-\phi^{*})^{2}+\frac{\alpha^{2}}{2}\, ,\quad
m^{2}\equiv 3\alpha^{2}/2\, , \quad
\phi^{*}\equiv \phi_{0}-\frac{2\rho_{0}^{1/2}}{3^{1/2}\alpha}\, .
\ee
This type of potential is very popular in particle physics.
For example, the light scalar particles like dilaton and moduli appear in
superstring theories.
The dark energy density increases with time. Hence, the universe
accelerates. One can estimate the time required for disintegration of the
Sun-Earth system, as an example. The dimensionless inertial force
\be
\bar{F}_\mathrm{iner}=\frac{\ddot{a}}{aH_{0}^{2}}\, ,
\ee
can be expressed for $t\gg t_{0}$ as follows
\be
\bar{F}_\mathrm{iner}\approx\Omega_\mathrm{D}\frac{\rho}{\rho_{0}}.
\ee
Taking into account that the phantom energy density increases with time as
\be
\rho=\left(\rho_{0}^{1/2}-\frac{3^{1/2}}{2}(w_{0}+1)\rho_{0}t \right)^{2}\, ,
\ee
and that the Sun-Earth system disintegrates when $\bar{F}_\mathrm{iner}$ $\sim
10^{23}$ (see \cite{Frampton-2}),
one can find that the disintegration time is
\[
t_{dis}\approx\frac{10^{12}}{|w_{0}+1|H_{0}}\approx\frac{10^{13}}{|w_{0}+1|}\,
\mbox{Gyr}\, .
\]
Hence, we have presented a realistic Little Rip cosmology caused by scalar dark
energy with a standard particle physics massive potential.
Note that if such cosmology occurs, one can speculate on visible
reduction of galaxy clusters number in future cosmological surveys.
\section{Asymptotically de Sitter evolution: Pseudo-Rip \label{SecIV}}
It is evident that some phantom/quintessence models
may describe asymptotically de Sitter evolution.
However, even in this case the disintegration of bound structures may take
place. Such a scenario was dubbed the Pseudo-Rip cosmology
\cite{Frampton-2,Astashenok}.
Of course, not all asymptotically de Sitter phantom scenarios lead to
disintegration of bound structures.
Let us consider the example of a fluid which describes the phantom and/or
quintessence field asymptotically approaching the de Sitter regime. Let
\be
\label{EoSAsDS}
f(\rho)=\alpha^{2}\sin\left(\frac{\pi\rho}{\rho_\mathrm{f}}\right)\, .
\ee
If $2k\rho_\mathrm{f}<\rho<(2k+1)\rho_\mathrm{f}$, $k=0,1,...$, we have a
phantom case
while for $(2k+1)\rho_\mathrm{f}<\rho<(2k+2)\rho_\mathrm{f}$
Eq.~(\ref{EoSAsDS})
describes quintessence.
For $\rho\ll\rho_\mathrm{f}$, the parameter $w$ is nearly constant:
\bea
\label{asds}
w &\approx& -1-\frac{\alpha^{2}\pi}{\rho_\mathrm{f}}\, , \nn
\rho &=&
\rho_\mathrm{f}\left(\pm\frac{2}{\pi}\arctan\left(\tan\left(\frac{\pi\Delta}{2
(z+1)^{\delta}}\right)\right)+2k\right)\, ,\quad
\delta=3\alpha^{2}/\pi\rho_\mathrm{f}\, ,\quad
\Delta=\rho_{0}/\rho_\mathrm{f}\, ,
\eea
where the ``$\pm$'' corresponds to phantom and quintessence theories,
respectively.
Depending on the parameter $\Delta$, the dark energy density tends to a
single value from the set of ``effective cosmological constants''
\be
\Lambda^\mathrm{eff}=(2k+1)\rho_\mathrm{f}\, .
\ee
If the effective cosmological constant is sufficiently large ($\ddot{a}/a\gg
0$),
then disintegration of bound structures can occur.
It is obvious that this scenario occurs only for the phantom case (for
quintessence with asymptotic
de Sitter evolution, the acceleration of the universe can only decrease). The
dimensionless inertial force
\be
\bar{F}_\mathrm{iner}=3\frac{\ddot{a}}{\rho_{0}a}\, .
\ee
tends to
$\rho_{f}/\rho_{0}=\Delta^{-1}$ (for $k=0$) because $a\sim\exp(\sqrt{\rho_{f}/3}t)$ at $t\rightarrow\infty$.
Therefore if $\Delta<10^{-23}$ then
$\bar{F}_\mathrm{iner}>10^{23}$ at $t\rightarrow\infty$ and
the disintegration of the Sun-Earth system eventually happens.
Such a scenario is compatible with observational data.
For small $\Delta$, one can write with good accuracy the past dark energy
density
\be
\rho\approx\rho_{0}(1+z)^{-\delta}\, .
\ee
For small $\delta$ ($w\rightarrow-1$), the dark energy density is nearly
constant in the observable range $0<z<1.5$. Therefore, the observable relation
between modulus and redshift can be fulfilled in the model given
by Eq.~(\ref{EoSAsDS}).
The deceleration and jerk parameters are within the range of the standard
cosmology
(as $\delta\rightarrow 0$):
\bea
q_{0} &=& \frac{3}{2}\Omega_{m}-\frac{\delta}{2}\Omega_\mathrm{D}-1\, , \\
j_{0} &=& 1+\frac{3\delta+\delta^{2}}{2}\Omega_\mathrm{D}\, .
\eea
Thus, a realistic mild phantom scenario (Pseudo-Rip) may be easily realized.
\section{Dark energy models with a Type III future singularity \label{SecV}}
We shall consider in this section a flat universe which ends up in a Type III
future singularity \cite{Nojiri-2}. Let us start from
\be
\label{EoSBFS}
g(\rho)=-\beta^2 a_\mathrm{f}^{\epsilon}\rho^{1+\epsilon/3}\, ,
\ee
so that
\[
f(\rho)=\rho(-1+\beta^2 a_\mathrm{f}^{\epsilon}\rho^{\epsilon/3})\, ,
\]
where $\beta$, $a_\mathrm{f}$, and $\epsilon$ are positive constants.
One can find the dependence of the dark energy density on the scale factor
\be
\rho=\beta^{-6/\epsilon}\left(a_\mathrm{f}^{\epsilon}-a^{\epsilon}
\right)^{-3/\epsilon}\, .
\ee
Putting, for example, $\epsilon=1$, we have
\[
p=-\beta^2 a_\mathrm{f}\rho^{4/3}\, .
\]
In this case, one can find the scale factor in parametric form
\begin{equation}
\begin{array}{l}
a=a_\mathrm{f}\sin^{2}\eta\, ,\\
\\
\displaystyle{t=t_\mathrm{f}+\frac{1}{\kappa}\left(\ln\left|\tan\frac{\eta}{2}\right|
+\cos\eta+\frac{1}{3}\cos^{3}\eta\right)}\, ,
\end{array}
\label{solution}
\end{equation}
where
\[
\kappa=\frac{1}{2 sqrt{3} \beta^{3}}a_\mathrm{f}^{-3/2},\qquad
dt=\frac{\cos^4\eta}{\kappa\sin\eta}d\eta\, .
\]
We now set $t_\mathrm{f}=0$. Therefore $\eta=0$ corresponds to
$t=-\infty$, $\eta=\pi/2$ to $t=0$ (future singularity) and $\eta=\pi$ to
$t=+\infty$. Hence, this solution describes two universes: the
first one begins at $t=-\infty$ (Big Bang) and then
expands to a future singularity which takes place at $t=0$. The second
solution begins at $t=0$ (at a singularity) and then progressively contracts
until a big crunch singularity at $t=\infty$. The asymptotic
behavior of the scale factor near the future singularity in both cases is
\[
a=a_\mathrm{f}\left(1-(5\kappa)^{2/5}|t|^{2/5}\right),\qquad t\sim 0\, .
\]
The addition of dark matter allows us
to construct cosmological models in which the age of the universe is close to
the
conventional value of $10$-$20$\,Gyr.
The dependence of the dark energy density on redshift is given by
\begin{equation}
\label{chapl}
\rho=\rho_{0}(1+z)^{3}\left(\frac{N_{0}-1}{N_{0}(1+z)^{\epsilon}-1}
\right)^{3/\epsilon}\, ,
\end{equation}
where $N_{0}=(a_\mathrm{f}/a_{0})^{\epsilon}$. For the current value of the EoS
parameter $w_{0}$ we have
\begin{equation}
\label{w0}
w_{0}=-\beta^{2}N_{0}a_{0}^{3}\rho_{0}^{\epsilon/3}
=-\frac{N_{0}}{N_{0}-1}\, .
\end{equation}
One can use the standard relation between redshift and time
\bea
\label{age}
&& H_{0}^{-1}\int \frac{dz}{(1+z)\sqrt{h(z)}}=-\int dt\, , \nn
&&
h(z)=\Omega_{m}(1+z)^{3}+\Omega_\mathrm{D}(1+z)^{3}\left(\frac{N_{0}-1}{N_{0}
(1+ z)^{\epsilon}-1}\right)^{3/\epsilon}\, ,
\eea
for the calculation of the age of the universe and the estimation of the time
of the future singularity.
Integrating Eq.~(\ref{age}) from $z = \infty$ ($t=0$, Big Bang) to $z=0$
($t=t_{0}$) gives the age of the universe:
\begin{equation}
\label{age1}
t_{0}=H_{0}^{-1}\int^{\infty}_{0}\frac{dz}{(1+z)\sqrt{h(z)}}\, .
\end{equation}
For $N_{0}\gg 1$ (i.e., for $w_{0}\approx-1$) the function $h(z)$ can be
approximated by
\begin{equation}
\label{approx}
h(z)\approx\Omega_{m}(1+z)^{3}+\Omega_\mathrm{D}\, .
\end{equation}
Therefore, the age of the universe is eventually independent of $\epsilon$.
This parameter, however, may change the remaining time before the future
singularity $t_\mathrm{f}-t_{0}$.
Note that for the calculation of this time we can use
Eq.~(\ref{age}), simply assuming that the variable $z$ can take negative
values.
The lower limit of integration corresponds to the scale factor
$a=a_\mathrm{f}$,
i.e.,
$z_\mathrm{f}=N_{0}^{-1/\epsilon}-1$. Therefore for $t_\mathrm{f}-t_{0}$ one
gets the
following relation
\begin{equation}
\label{agef}
t_\mathrm{f}-t_{0}=H_{0}^{-1}\int^{0}_{z_\mathrm{f}}\frac{dz}{(1+z)\sqrt{h(z)}}\,
.
\end{equation}
It is obvious that our model can fit the Supernova Cosmological Project
data. For $N\gg 1$ the dark energy density is nearly constant in the
interval $0<t<t_{0}$, i.e. the model (\ref{EoSBFS}) mimics a cosmological
constant in the past but leads to a finite-time future singularity.
Moreover, the observational data do not
impose any significant restrictions on the lifetime of the universe.
The numerical calculation of the age of the universe, $t_0$, and the
difference between the future singularity time, $t_\mathrm{f}$, and $t_{0}$
for various values of $w_{0}$ and $\epsilon$ are presented
in Table~\ref{Table1}. The value of the Hubble parameter is chosen
to be $H_{0}^{-1}=13.6$\,Gyr for this calculation.
\begin{table}
\caption{A numerical calculation of the age of the universe, $t_0$, and the
difference between the future singularity time, $t_\mathrm{f}$, and $t_{0}$
for various values of $w_{0}$ and $\epsilon$. The time unit is $10^9$ years
(Gyr)
and we choose $H_{0}^{-1}=13.6$\,Gyr.
\label{Table1}}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& \multicolumn{2}{c}{$w_{0}=-1.01$}\vline &
\multicolumn{2}{c}{$w_{0}=-1.05$}\vline &
\multicolumn{2}{c}{$w_{0}=-1.1$}\vline \\
\hline
$\epsilon$ & $t_\mathrm{f}-t_{0}$ & $t_{0}$ & $t_\mathrm{f}-t_{0}$ & $t_{0}$ &
$t_\mathrm{f}-t_{0}$ & $t_{0}$ \\
\hline
1 & 52.95 & 13.66 & 30.48 & 13.73 & 22.17 & 13.81 \\
2 & 29.45 & 13.66 & 17.79 & 13.71 & 13.31 & 13.78 \\
5 & 12.64 & 13.65 & 7.93 & 13.69 & 6.08 & 13.73 \\
10 & 6.40 & 13.65 & 4.09 & 13.67 & 3.17 & 13.69 \\
50 & 1.27 & - & 0.83 & - & 0.65 & - \\
100 & 0.63 & - & 0.41 & - & 0.33 & - \\
1000 & 0.06 & - & 0.04 & - & 0.03 & - \\
\hline
\end{tabular}
\end{table}
The deceleration parameter is found to be
\be
q_{0}=\frac{3}{2}\Omega_{m}-1+\frac{3}{2}\Omega_\mathrm{D}(1+w_{0})
\ee
and for $w_{0}\approx -1$ $q_{0}\approx q_{0}^\mathrm{SC}$.
For the jerk parameter we have
\be
j_{0}=\frac{3}{2}(3+\epsilon)(w_{0}+w_{0}^{2})\Omega_\mathrm{D}+1\, .
\ee
For sufficiently small $\epsilon$ and $w\approx-1$, the jerk parameter is
nearly
equal to $1$. Hence, a viable quintessence dark energy model which is
compatible
with observational data and leads to a Type III future singularity is
constructed.
Of course, it may be presented in terms of a scalar field with the field value
$\phi$ and potential $V(\phi)$ written as functions of $\eta$. Furthermore, the
dependence of $V(\phi)$ on $\phi$ can be presented explicitly.
We have
\bea
&& \phi = \frac{\eta}{\alpha_{0}}+\phi_{0}\, ,\nn
&& V(\phi) =\alpha_{1}(\cos(\alpha_{0}(\phi-\phi_{0})))^{-6/\epsilon}(1
+\cos^{-2}(\alpha_{0}(\phi-\phi_{0})))\, ,\nn
&& \alpha_{0}=\sqrt{\frac{1}{12}}\epsilon\, ,\qquad
\alpha_{1}=\frac{1}{2}\beta^{-6/\epsilon}a_\mathrm{f}^{-3}\, .
\eea
Other models with similar properties can be constructed.
\section{Type II future singularity dark energy \label{SecVI}}
In this section we discuss realistic models of dark energy
which contain a Type II future singularity \cite{Nojiri-2} (or sudden future
singularity \cite{Barrow}). It is known that such evolution may be also
realized in $f(R)$ modified gravity \cite{Bamba}.
The simplest choice for an EoS producing a Type II future singularity is
\be
\label{TM}
f(\rho)=\frac{\alpha^{2}}{1-\rho/\rho_\mathrm{f}}\, ,
\ee
where $\alpha$ and $\rho_\mathrm{f}$ are positive constants.
For $\rho_{0}<\rho_\mathrm{f}$, such a model describes phantom energy.
Its energy density grows with time until
the pressure tends to infinity and a phantom sudden future singularity occurs.
If $\rho_{0}>\rho_\mathrm{f}$, the energy density decreases and a big crush
occurs when $\rho$ is equal to $\rho_\mathrm{f}$.
The time remaining before the future singularity is
\be
t_\mathrm{f}-t_{0}=\frac{2}{\sqrt{3}}\int_{x_{0}}^{x_\mathrm{f}}\frac{dx}{\alpha^2}
\left(1 -\left(\frac{x}{x_\mathrm{f}}\right)^{2}\right)\, .
\ee
The corresponding description in terms of scalar field theory can be derived
through Eqs.~(\ref{Vx}) and (\ref{phix}). The potential of the scalar field in
parametric form is:
(i) for phantom energy
\bea
\phi(y)&=&\frac{1}{\sqrt{3\gamma}}\left(\arcsin y+y(1-y^{2})^{1/2}\right)\, ,\\
V(y)& =& \frac{\rho_\mathrm{f}}{2}\left(2y^{2}+\frac{\gamma}{1-y^{2}}\right)\,
,\quad
0\leq y\leq 1\, ,
\eea
(ii) for quintessence
\bea
\phi(y)&=&-\frac{1}{\sqrt{3\gamma}}\left(\frac{\sqrt{1-y^{2}}}{y^{2}}-\ln(1-
\sqrt{1-y^{2}})+\ln y\right)\, ,\\
V(y)& =&
\frac{\rho_\mathrm{f}}{2}\left(2y^{-2}+\gamma\frac{y^{2}}{y^{2}-1}\right)\,
,\quad
0\leq y\leq 1\, ,
\eea
and $\gamma=\alpha^{2}/\rho_\mathrm{f}$. This potential is depicted in
Figure~\ref{1}.
\begin{figure}
\rotatebox{90}{\includegraphics[scale=0.5,angle=270]{1.eps}}\\
\caption{The scalar potential for the model (\ref{TM}). For a crush the
scalar field rolls down from $-\infty$ to $0$ and
$V(\phi)\rightarrow\-\infty$; for a phantom sudden future singularity the
scalar
field
rolls up and $V(\phi)\rightarrow\infty$ at some $\phi=\phi_{s}$.}\label{1}
\end{figure}
Such a model in principle can fit the latest
supernova data from the Supernova Cosmology Project.
The dependence of the dark energy density on the redshift $z$ can be derived
from
Eq.~(\ref{arho}). After a simple algebraic calculation, one can obtain
\be
\rho=\rho_\mathrm{f}\left(1\pm\left(\left(1-\Delta\right)^2
+6\gamma\ln(1+z)\right)^{1/2}\right)\, ,\quad
\Delta=\rho_{0}/\rho_\mathrm{f},\quad
\gamma=\alpha^{2}/x_\mathrm{f}^{2}\, .
\ee
The sign ``$+$'' corresponds to the case of quintessence ($\Delta>1$) while
sign
``$-$'' to that of phantom energy ($\Delta<1$).
The current EoS parameter $w_{0}$ is
\be
w_{0}=-1-\frac{\gamma}{\Delta(1-\Delta)}\, .
\ee
Therefore, the dependence of the luminosity distance $D_\mathrm{L}$ on the
redshift
$z$
is
\bea \label{DL}
D_\mathrm{L} &=& \frac{c}{H_{0}}(1+z)\int_{0}^{z}\left(\Omega_{m}
(1+z)^{3}+\Omega_\mathrm{D}h(z)\right)^{-1/2}d z\, ,\nn
h(z) &=&
\Delta^{-1}\left(1\pm\left(\left(1-\Delta\right)^2+6\gamma\ln(1+z)\right)^{1
/2}\right)\, .
\eea
Eq.~(\ref{DL}) coincides with (\ref{DLSC}) if $\gamma=0$ ($f(x)=0$). The
SNe data are available in the range $0<z<1.5$. Therefore, if the parameters
$\Delta$ and $\gamma$ are such that $(1-\Delta)^{2}\ll 6\gamma\ln(1+z)$ in the
observable range, the model under discussion is indistinguishable from
$\Lambda$CDM cosmology.
The time remaining before a future singularity is
\be
t_\mathrm{f}-t_{0}=\frac{1}{H_{0}}\int_{u}^{0}du(1+u)^{-1}
\left(\Omega_{m}(1+u)^{3}+\Omega_\mathrm{D}h(u)\right)^{-1/2}\, .
\ee
The variable $u=a_{0}/a-1$ varies from $0$ (present time) to
$\exp(-(1-\Delta)^{2}/6\gamma)-1$ ($p\rightarrow\infty$).
The function $h(u)$ coincides with $h(z)$ in
Eq.~(\ref{DL}) (after changing $z\rightarrow u$).
Numerical estimation of the difference $t_\mathrm{f}-t_{0}$ for various values
of $\Delta$ and $\gamma$ is given in Table~\ref{Table II} and
Table~\ref{Table III}.
For quintessence, the difference
$t_\mathrm{d}-t_{0}$ ($t_\mathrm{d}$ is the moment of time when $\ddot{a}=0$
and
deceleration begins) is also calculated.
We use the value of the Hubble parameter $H^{-1}_{0}=13.6$\,Gyr.
\begin{table}
\caption{Numerical estimation of the difference $t_\mathrm{f}-t_{0}$
in Gyr for various values
of $\Delta$ and $\gamma$ in the case of a phantom model. \label{Table II}}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\Delta$ & $w_{0}=-1.10$ & $w_{0}=-1.08$ & $w_{0}=-1.06$ & $w_{0}=-1.04$ &
$w_{0}=-1.02$ \\
\hline
0.5 & 22.6 & 28.4 & 38.2 & 57.6 & 116.1 \\
0.75 & 7.3 & 9.9 & 12.5 & 20.3 & 39.8 \\
0.95 & 1.1 & 1.5 & 2.0 & 3.1 & 6.3 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Numerical estimation of the difference $t_\mathrm{f}-t_{0}$
in Gyr for various values
of $\Delta$ and $\gamma$ in the case of quintessence. \label{Table III}}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{2}{c}{$w_{0}=-0.98$}\vline &
\multicolumn{2}{c}{$w_{0}=-0.96$}\vline & \multicolumn{2}{c}{$w_{0}=-0.94$}
\vline & \multicolumn{2}{c}{$w_{0}=-0.92$} \vline &
\multicolumn{2}{c}{$w_{0}=-0.90$} \vline \\
\hline
$\Delta$ & $\Delta t_\mathrm{d}$ & $\Delta t_\mathrm{f}$ & $\Delta
t_\mathrm{d}$ & $\Delta
t_\mathrm{f}$ & $\Delta t_\mathrm{d}$ & $\Delta t_\mathrm{f}$ & $\Delta
t_\mathrm{d}$ & $\Delta t_\mathrm{f}$ &
$\Delta t_\mathrm{d}$ & $\Delta t_\mathrm{f}$ \\
\hline
1.05 & 5.78 & 5.79 & 2.8 & 2.81 & 1.84& 1.86 & 1.36 & 1.38 & 1.08 & 1.10
\\
1.25 & 26.70 & 26.74 & 12.88& 12.96 & 8.34 & 8.45 & 6.10 & 6.24 & 4.76&
4.94 \\
1.5 & 46.32 & 46.42 & 22.52 & 22.72 & 14.57 & 14.85 & 10.60 & 10.96 &
8.23 & 8.65 \\
\hline
\end{tabular}\end{center}
\end{table}
The difference $\delta \mu=5\log (D/D^\mathrm{SC})$ ($\mu$ is the distance
modulus)
for
$0<z<1.5$ lies in the interval ($-0.35$ to $0.35$) for these parameter values.
Taking into account
that errors in the definition of the SNe modulus are $\sim 0.075\div 0.5$, we
conclude
that our model fits these data with excellent precision. Note that futher
observational support for quintessence which leads to a Type II future
singularity is given in Ref.~\cite{Dab}.
The deceleration and jerk parameters are given by
\bea
\label{DDD1B}
q_0 &=& \frac{9\gamma}{2\Delta(1-\Delta)}\Omega_\mathrm{D}
+ \frac{3}{2} \Omega_m - 1\, ,\nn
j_0 &=& -\frac{9\gamma}{2\Delta(1-\Delta)}\left(1
+\frac{\gamma}{(1-\Delta)^{2}}\right)\Omega_\mathrm{D} + 1\, .
\eea
It is convenient to present $q_{0}$ and $j_{0}$ through the parameters $w_{0}$
and $\Delta$:
\bea
q_0 &=& -\frac{9}{2}(w_{0}+1)\Omega_\mathrm{D}+\frac{3}{2} \Omega_m - 1\, ,\nn
j_0 &=& \frac{9}{2}(w_{0}+1)\left(1-\frac{\Delta}{1-\Delta}(w_{0}+1)\right)
\Omega_\mathrm{D} + 1\, .
\eea
For $w_{0}\approx -1$, the jerk parameter $j_0$ differs significantly from
the standard value only for $\Delta\rightarrow 1$. The deceleration parameter
$q_{0}$ for $-1.05<w<-0.95$ lies in the interval
$q_{0}^\mathrm{SC}-0.16<q_{0}<q_{0}^\mathrm{SC}+0.16$. Taking into account
the errors in the definition of $q_{0}$ and $j_{0}$ one can conclude that
dark energy models with a possible big crush or sudden future singularity fit
well the current observational data.
The example considered above is a good theoretical illustration of dark
energy models mimicking vacuum energy but leading to singularities of Type II.
Dark energy with such behavior can
be realized if the function $f(x)$ has a singularity at $x=x_\mathrm{f}$.
An important remark is in order. For quintessence
the disintegration of bound structures before a future singularity seems to be
impossible. From Eqs.~(\ref{EoS}) and (\ref{Fin}) it follows that
\be
F_\mathrm{iner}=-ml\frac{1}{2}\left(w+\frac{1}{3}\right)\rho\, ,
\ee
Therefore, the maximal value of the inertial force for quintessence is
\[
F_\mathrm{iner}^\mathrm{max}=\frac{ml\rho}{3}\, .
\]
The energy density of quintessence decreases when the universe expands and
the inertial force also decreases with time. If we consider an EoS for which
$f(\rho)$ changes sign at $x=x_\mathrm{ph}$ then the energy density increases
with
time and tends to $x_\mathrm{ph}^{2}$. Hence, such an expanding universe is a
de Sitter one.
\section{Big crush dark energy models \label{SecVII}}
Another way to construct cosmological models with various
types of evolution is
to define the Hubble parameter as a function of time.
Let us consider the Type II singularity, for example.
This case occurs when $H$ is finite and $\dot H$ diverges but is
negative. In this case,
even though the universe is expanding, all structures are crushed rather than
ripped. An example is given by
\be
\label{lc1}
H = H_0^{(0)} + H_1^{(0)} \left(t_c - t\right)^\alpha\, .
\ee
Here $H_0^{(0)}$ and $H_1^{(0)}$ are positive constants and $\alpha$ is a
constant with $0<\alpha<1$.
If we choose $\alpha$ to be given by the inverse of an odd number,
$\alpha=1/(2n+1)$,
with positive integer $n$, we can extend $H$ beyond $t=t_c$ by defining
\be
\label{H1}
H = \left\{
\begin{array}{ll}
H_0^{(0)} + H_1^{(0)} \left(t_c - t\right)^\alpha & \mbox{when}\ t<t_c \\
H_0^{(0)} - H_1^{(0)} \left(t - t_c\right)^\alpha & \mbox{when}\ t>t_c
\end{array} \right. \, .
\ee
In a sense, the $H$ obtained here is a smooth function of $t$ although $\dot H$
diverges at $t=t_c$ since there is no sharp point, that is, a point where the
line folds.
At the point $\dot H$ diverges but $H$ is the continuous, single valued,
and monotonically increasing or decreasing function of $t$.
One may consider the following model
\be
\label{H2}
H = \left\{
\begin{array}{ll}
H_0^{(0)} + H_1^{(0)} \left(\tanh\frac{\left(t_c - t\right)}{t_0}\right)^\alpha
& \mbox{when}\ t<t_c \\
H_0^{(0)} - H_1^{(0)} \left(\tanh\frac{\left(t - t_c\right)}{t_0}\right)^\alpha
& \mbox{when}\ t>t_c
\end{array} \right. \, .
\ee
In the model (\ref{H2}), we find $H\rightarrow\mbox{const}$ when
$t\to\pm\infty$,
that is, the space-time is asymptotically de Sitter.
Note that when $H$ is finite, a rip occurs only for the phantom case, since
$\dot H>0$
before the
singularity. However, a crush occurs for quintessence with $\dot H<0$.
In the models (\ref{H1}) and (\ref{H2}), a rip occurs when $H_1^{(0)}<0$
and a crush occurs when $H_1^{(0)}>0$.
From the FRW equations (\ref{Fried1}), the total energy density and pressure
of dark energy as functions of time are:
\be
\rho(t)+\rho_{m}(t)=3H^{2}\, ,\quad
p(t)=-3H^{2}-2\dot{H}\, .
\ee
Neglecting the matter density, one can easily obtain the EoS of dark energy
corresponding to model (\ref{H1}):
\be
p=\left\{
\begin{array}{ll}
-\rho+2\alpha 3^{n} H_{1}^{(0)}(\rho^{1/2}-\rho_\mathrm{f}^{1/2})^{-2n}\, ,
\quad \rho_\mathrm{f}=3H_{0}^{(0)2} & \mbox{when}\ t<t_c \\
-\rho-2\alpha 3^{n} H_{1}^{(0)}(\rho^{1/2}-\rho_\mathrm{f}^{1/2})^{-2n}\, ,
\quad \rho_\mathrm{f}=3H_{0}^{(0)2} & \mbox{when}\ t>t_c
\end{array} \right. \, .
\ee
When $H_{1}^{(0)}<0$ or $H_{1}^{(0)}>0$, $p\rightarrow-\infty$ (phantom Type II
singularity) or
$p\rightarrow+\infty$ (quintessence Type II singularity) at $t\rightarrow
t_{c}-0$.
The scale factor is finite at $t\rightarrow t-0$.
\be
a(t)=a_{0}\exp\left\{H_{0}^{(0)}t+H_{1}^{(0)}(t_{c}-t)^{\alpha+1}/(\alpha+1)
-H_{1}^{(0)}t_{c}^{\alpha+1}/(\alpha+1)\right\}\, .
\ee
In the following, we assume $\alpha$ is given by $\alpha=1/(2n+1)$
with positive integer $n$.
We now investigate if any object can be ripped or crushed at the singularity
$t=t_c$ although
the inertial force $F_\mathrm{iner}$ diverges. For this purpose, we consider
the work or the shift of
the kinetic energy of the particle. For purposes of
this estimation, we neglect all forces aside from the inertial force, and we
neglect the first term
in (\ref{i1}) since we assume only $\dot H$ diverges. Then by solving the
equation of motion
\be
\label{H3}
m \ddot x = F_\mathrm{iner}\sim m l \dot H
= - m l H_1^{(0)} \alpha\left(t_c - t\right)^{\alpha - 1}
\, ,
\ee
for the model (\ref{H1}), one finds
\be
\label{H4}
x = x_0 + v_0 t - \frac{l H_1^{(0)}}{\alpha+1}\left(t_c - t\right)^{\alpha+1}\,
.
\ee
Then the shift of the kinetic energy can be estimated to be
\be
\label{H5}
\Delta T = \int F \dot x dt \sim
- m v_0 H_1^{(0)} \alpha \int \left(t_c - t\right)^{\alpha-1} dt
+ m l^2 {H_1^{(0)}}^2 \alpha \int \left(t_c - t\right)^{2\alpha - 1} dt \, .
\ee
Since $\alpha>0$, the integration and therefore $\Delta T$ is finite.
Hence if the magnitude of the binding energy or the energy supporting the
bound object is larger than the absolute value of $\Delta T$, the object
is not ripped or crushed although $F_\mathrm{iner}$ is infinite at $t=t_c$.
In the case of a Type III singularity, $H$ behaves as in (\ref{lc1}) but
$\alpha$ is negative and greater than unity, $-1<\alpha<0$.
We also note that $H_1^{(0)}>0$ since $H>0$ when $t<t_c$.
Then in the inertial force (\ref{i1}), the first term behaves as
$\dot H\sim \left(t_c - t\right)^{\alpha-1}$ and the second one as
$H^2 \sim \left(t_c - t\right)^{2\alpha}$. Since $-1<\alpha<0$, the
first term dominates. Then in a way similar to (\ref{H5}),
the shift of the kinetic energy can
be estimated as
\be
\label{H5b}
\Delta T \sim - m v_0 H_1^{(0)} \left(t_c - t\right)^\alpha
+ \frac{m l^2 {H_1^{(0)}}^2}{2} \left(t_c - t\right)^{2\alpha} \, ,
\ee
when $t<t_c$ and it becomes positive and diverges when $t\to t_c$.
Therefore the Rip surely occurs even for a Type III singularity,
which is different from the Type II singularity in (\ref{H5}), where
all the objects are not always crushed or ripped.
By using the formulation in Ref.~\cite{Nojiri:2005pu},
we now consider what kind of scalar tensor model, whose action is given by
\be
\label{ma7}
S=\int d^4 x \sqrt{-g}\left\{
\frac{1}{2\kappa^2}R - \frac{1}{2}\omega(\phi)\partial_\mu \phi
\partial^\mu\phi - V(\phi) \right\}\, ,
\ee
can realize the evolution of $H$ given by Eq.~(\ref{lc1}).
Here, $\omega(\phi)$ and $V(\phi)$ are functions of the scalar field $\phi$.
If we consider the model where $\omega(\phi)$ and $V(\phi)$ are given by a
single function $f(\phi)$, as follows,
\be
\label{ma10}
\omega(\phi)=- \frac{2}{\kappa^2}f''(\phi)\, ,\quad
V(\phi)=\frac{1}{\kappa^2}\left(3f'(\phi)^2 + f''(\phi)\right)\, ,
\ee
the exact solution of the FRW equations has the following form:
\be
\label{ma11}
\phi=t\, ,\quad H=f'(t)\, .
\ee
Then for the model (\ref{lc1}) with $\alpha=1/(2n+1)$, we find
\be
\label{H6}
\omega(\phi) = \frac{2H^{(0)}_1 \alpha}{\kappa^2}\left( t_c - \phi
\right)^{- \frac{2n}{2n+1}}\, ,\quad
V(\phi) = \frac{1}{\kappa^2} \left\{ \left( H^{(0)}_0 + H^{(0)}_1
\left( t_c - \phi \right)^{- \frac{1}{2n+1}} \right)^2
- 2H^{(0)}_1 \alpha\left( t_c - \phi \right)^{- \frac{2n}{2n+1}} \right\}\, .
\ee
If we redefine the scalar field as
\be
\label{H7}
\varphi = - \frac{\sqrt{(2n+1) H^{(0)}_1}}{\kappa (n+1)}
\left( t_c - \phi \right)^{\frac{n+1}{2n+1}}\, ,
\ee
the kinetic term in the action (\ref{ma7}) becomes canonical
\be
\label{H8}
- \frac{1}{2}\omega(\phi) \partial_\mu \phi \partial^\mu\phi
= - \frac{1}{2} \partial_\mu \varphi \partial^\mu\varphi \, ,
\ee
and the potential is given by
\be
\label{H9}
V(\phi) = \frac{1}{\kappa^2} \left\{ \left( H^{(0)}_0 + H^{(0)}_1
\left( - \frac{\kappa (n+1)}{\sqrt{2H^{(0)}_1 (2n+1)}}\varphi
\right)^{\frac{1}{n+1}} \right)^2
- H^{(0)}_1 \alpha \left( - \frac{\kappa (n+1)}{\sqrt{2H^{(0)}_1 (2n+1)}}
\varphi\right)^{-\frac{2n}{n+1}} \right\}\, .
\ee
Note that $\varphi<0$ when $\phi=t<t_c$ and $\varphi\to 0$ when $\phi=t\to
t_c$.
Near the singularity $\phi=t\to t_c$ ($\varphi\to 0$), only the last term
in the potential (\ref{H9}) dominates:
\be
\label{H10}
V(\phi) \sim - \frac{H^{(0)}_1 \alpha }{\kappa^2} \left( - \frac{\kappa
(n+1)}{\sqrt{2H^{(0)}_1 (2n+1)}}
\varphi\right)^{-\frac{2n}{n+1}} \, .
\ee
In particular, when $n=1$, we find
\be
\label{H11}
V(\phi) \sim \frac{\alpha}{\kappa^3} \sqrt{\frac{3 {H^{(0)}_1}^3}{2}}
\varphi^{-1}\, .
\ee
Thus, the big crush occurs when the scalar field drops into the infinitely
deep potential proportional to the inverse power of the scalar potential.
We have constructed models which generate the big crush and have given the
explicit action in terms of the scalar field. After the big crush, the universe
may evolve
to asymptotic de Sitter space-time.
Hence, big crush phenomenon looks much less dangerous than disintegration of
bound structures.
\section{Phantom models and singularities \label{SecPhantom}}
One way to realize many of the models proposed here is through a
scalar field with a negative kinetic term (phantom models). The asymptotic
future evolution of such models was examined systematically in Ref.~\cite{KSS},
and we restate a number of those results here in order to show explicitly the
relation between various types of future singularity.
The simplest phantom models are characterized by a field $\phi$ with a negative
kinetic term. Such models evolve
according to the equation
\begin{equation}
\label{phiev}
\ddot{\phi} + 3 H \dot{\phi} - V^\prime(\phi) = 0\, ,
\end{equation}
where the prime denotes the derivative with respect
to $\phi$. A field evolving according to this equation rolls uphill in the
potential. The density and pressure for the phantom field are given by
\be
\rho_{\phi}=-\frac{1}{2}\dot \phi^2 + V(\phi)\, ,
\ee
and
\be
p_{\phi}=-\frac{1}{2}\dot \phi^2 - V(\phi)\, ,
\ee
respectively, so the equation of state parameter is
\begin{equation}
\label{w}
w_{\phi} =\frac{(1/2)\dot \phi^2 + V(\phi)}{ (1/2)\dot \phi^2 - V(\phi)}\, .
\end{equation}
As noted in Ref.~\cite{KSS}, the asymptotic behavior of the equation of state
parameter depends on the corresponding asymptotic behavior of $V^\prime/V$.
If $V^\prime/V \rightarrow 0$, then $w \rightarrow -1$. This set of models
displays the most diverse behavior, since it can correspond to either
a Big Rip, a Little Rip, or a Pseudo-Rip, depending on the exact functional
form for $V(\phi)$. A Big Rip (Type I singularity) occurs when \cite{KSS}
\be
\label{ripcondition}
\int \frac{\sqrt{V(\phi)}}{V^\prime(\phi)} d\phi \rightarrow \mbox{finite.}
\ee
If, instead, the integral in equation (\ref{ripcondition}) diverges, we have
either a Little Rip or a Pseudo-Rip. A Pseudo-Rip occurs if $V(\phi)
\rightarrow \mbox{const}$, while a Little Rip occurs if $V(\phi) \rightarrow
\infty$ (see also Section~\ref{SecII}).
The second set of models examined in Ref.~\cite{KSS} corresponds
to $V^\prime/V \rightarrow constant$. This gives a constant value
for $w$ with $w < -1$, and produces a Big Rip (Type I) singularity.
Finally, if $V^\prime/V \rightarrow \pm \infty$, we have
$w \rightarrow - \infty$, which can result in a Type III singularity
(see also Ref.~\cite{Sami:2003xv}).
\section{Conclusion \label{SecVIII}}
In summary, dark energy models with various scenarios of evolution have been
presented.
Specifically, we constructed scalar dark energy models with Type II and Type
III finite-time future singularities, Little Rip and Pseudo-Rip cosmologies with
finite-time disintegration of bound structures and Big Crush cosmologies. It was shown
that such models are consistent with observational data from the
Supernova Cosmology Project and therefore may be viable alternatives to the
$\Lambda$CDM
cosmology. Moreover, they may be stable for billions of years before
entering
a soft future singularity (with a finite scale factor at the Rip) or before
entering a finite-time dissolution of bound structures.
We have shown that the future evolution of the universe is
determined
by the selected EoS of dark energy. Unfortunately, current data for such
important parameters as $q_{0}$ and $j_{0}$ are not very reliable, so the
nature of the dark energy
cannot yet be determined, and one can therefore
only consider some typical models.
In the future, more accurate measurements of the
deceleration and jerk parameters as well as other cosmological parameters will
help to define the exact nature of dark energy.
Then we will acquire the information on the parameters
of the fluid description for the EoS of dark energy in this paper
and therefore also the information of the parameters in
a reconstructed scalar field theory.
In other words, more precise values of cosmological parameters may
significantly constrain the dark energy models under discussion.
The key point is that the current observational data do not answer, even in
principle,
the question of whether or not the universe will end in a future
singularity or Rip cosmology. One can construct (as we have here) models that
mimic standard
$\Lambda$CDM up the present, but evolve in the future into any number of
possible
future states, including Pseudo-Rip models, Little Rip models, and a variety of
different
future singularities. With a variety of $\Lambda$CDM-like cosmological models
in hand, one can already start to think about future cosmological
experiments to define the future of the universe more precisely.
\section*{Acknowledgments}
S.N. is supported by Global COE Program of Nagoya University (G07)
provided by the Ministry of Education, Culture, Sports, Science \&
Technology and by the JSPS Grant-in-Aid for Scientific Research (S) \# 22224003
and (C) \# 23540296.
The work by SDO has been supported in part by MICINN (Spain) project
FIS2010-15640,
by AGAUR 2009SGR-994 and by JSPS Visitor Program S11135 (Japan).
R.J.S. is supported in part by the Department of Energy (DE-FG05-85ER40226).
|
{
"timestamp": "2012-06-08T02:02:27",
"yymm": "1203",
"arxiv_id": "1203.1976",
"language": "en",
"url": "https://arxiv.org/abs/1203.1976"
}
|
\section{Introduction and the statement of results} \label{Sec-1}
Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0:=\mathbb{N}\cup \{0\}$, $\mathbb{Z}$ the ring of rational integers, $\mathbb{Q}$ the field of rational numbers, $\mathbb{R}$ the field of real numbers, $\mathbb{C}$ the field of complex numbers and $i=\sqrt{-1}$.
The Euler double zeta-function is defined by
\begin{equation}
\label{1-1}
\zeta_2(s_1,s_2)=\sum_{m=1}^\infty \frac{1}{m^{s_1}}\sum_{n=1}^\infty \frac{1}{(m+n)^{s_2}}=\sum_{k=2}^\infty \left(\sum_{m=1}^{k-1} \frac{1}{m^{s_1}}\right)\frac{1}{k^{s_2}}
\end{equation}
which is absolutely convergent
for $s_1,s_2 \in \mathbb{C}$ with $\Re s_2> 1$ and $\Re (s_1+s_2)>2$
(Theorem 3 in \cite{M1}), and can be continued meromorphically to $\mathbb{C}^2$.
The singularities are $s_2=1$ and $s_1+s_2=2,1,0,-2,-4,\ldots$
(Theorem 1 in \cite{AET}).
Euler himself considered the behaviour of this function when $s_1,s_2$ are positive
integers. It was Atkinson \cite{Atk} who first studied \eqref{1-1} from the
analytic viewpoint, and he proved the analytic continuation of it.
Recently the active research of \eqref{1-1} revived, because it is the simplest
example of multiple zeta-functions. As for the studies on the analytic side of
\eqref{1-1}, for example, upper-bound estimates were discussed in \cite{IM,KT,KTZ}, and functional equations were discovered in \cite{KMT-Debrecen,Mat-Camb}.
It is the purpose of the present paper to prove certain mean square formulas for
\eqref{1-1}.
Let
\begin{equation}
\label{1-2}
\zeta_2^{[2]}(s_1,s_2)=\sum_{k=2}^\infty \left|\sum_{m=1}^{k-1} \frac{1}{m^{s_1}}\right|^2\frac{1}{k^{s_2}}.
\end{equation}
Since the inner sum is $O(1)$ (if $\Re s_1>1$), $O(\log k)$ (if $\Re s_1=1$), or
$O(k^{1-\Re s_1})$ (if $\Re s_1<1$), the series \eqref{1-2} is convergent when
$\Re s_1\geq 1$ and $\Re s_2>1$, or when $\Re s_1<1$ and $2\Re s_1+\Re s_2>3$.
Note that $\zeta_2^{[2]}(1,q)$ $(q\in \mathbb{N}_{\geq 2})$ was already studied by Borwein et al. (see \cite{BBG}).
Hereafter we write $s_0$ and $s$ instead of $s_1$ and $s_2$, respectively, and
consider the mean square with respect to $s$, while $s_0$ is to be fixed.
\begin{theorem} \label{T-1-1} For $s_0={\sigma_0}+i{t_0}\in \mathbb{C}$ with ${\sigma_0}>1$ and $s=\sigma+it\in \mathbb{C}$ with $\sigma>1$, $t\geq 2$, we have
\begin{equation}
\begin{split}
& \int_{2}^{T}|\zeta_2(s_0,s)|^2 dt =\zeta_2^{[2]}(s_0,2\sigma)T+O(1)\qquad (T\to \infty).
\end{split}
\label{1-3}
\end{equation}
\end{theorem}
\begin{theorem} \label{T-1-2} For $s_0={\sigma_0}+i{t_0}\in \mathbb{C}$ with ${\sigma_0}>1$ and $s=\sigma+it\in \mathbb{C}$ with $\frac{1}{2}<\sigma\leq 1$, $t\geq 2$ and
$\sigma_0+\sigma>2$, we have
\begin{equation}
\begin{split}
& \int_{2}^{T}|\zeta_2(s_0,s)|^2 dt =\zeta_2^{[2]}(s_0,2\sigma)T+O\left( T^{2-2\sigma}\log T\right)+O\left( T^{1/2}\right).
\end{split}
\label{1-4}
\end{equation}
\end{theorem}
The most important result in the present paper is the following Theorem \ref{T-1-3},
which describes the situation under the condition $\frac{3}{2}<\sigma_0+\sigma\leq 2$.
\begin{theorem} \label{T-1-3} Let
$s_0={\sigma_0}+i{t_0} \in \mathbb{C}$ with
$\frac{1}{2}<{\sigma_0}< \frac{3}{2}$ and $s=\sigma+it\in \mathbb{C}$ with
$\frac{1}{2}<\sigma\leq 1$, $t\geq 2$ and $\frac{3}{2}<{\sigma_0}+\sigma\leq 2$.
Assume that when $t$ moves from $2$ to $T$, the point $(s_0,s)$ does not encounter
the hyperplane $s_0+s=2$ {\rm (}which is a singular locus of $\zeta_2${\rm )}. Then
\begin{equation}
\begin{split}
& \int_{2}^{T}|\zeta_2(s_0,s)|^2 dt =\zeta_2^{[2]}(s_0,2\sigma)T\\
& \quad +
\begin{cases}
O\left(T^{4-2\sigma_0-2\sigma}\log T\right)+O\left(T^{1/2}\right)
& (\frac{1}{2}<\sigma_0<1,\frac{1}{2}<\sigma<1)\\
O\left(T^{2-2\sigma_0}(\log T)^2\right)+O\left(T^{1/2}\right)
& (\frac{1}{2}<\sigma_0<1,\sigma=1)\\
O\left(T^{2-2\sigma}(\log T)^3\right)+O\left(T^{1/2}\right)
& (\sigma_0=1,\frac{1}{2}<\sigma<1)\\
O\left(T^{1/2}\right) & (\sigma_0=1,\sigma=1)\\
O\left(T^{2-2\sigma}\log T\right)+O\left(T^{1/2}\right)
& (1<\sigma_0<\frac{3}{2},\frac{1}{2}<\sigma<1).
\end{cases}
\end{split}
\label{1-5}
\end{equation}
\end{theorem}
\begin{remark}
In Theorems \ref{T-1-2} and \ref{T-1-3}, the error terms $O(T^{1/2})$ are coming
from the simple application of the Cauchy-Schwarz inequality.
It is plausible to expect that we can reduce these error terms by more
elaborate analysis.
\end{remark}
It is interesting to compare our theorems with the classical results on the mean
square of the Riemann zeta-function $\zeta(s)$. It is known that
\begin{align}\label{1-6}
\int_2^T|\zeta(\sigma+it)|^2 dt\sim\zeta(2\sigma)T\qquad\left(\sigma>\frac{1}{2}
\right)
\end{align}
and
\begin{align}\label{1-7}
\int_2^T|\zeta(\frac{1}{2}+it)|^2 dt\sim T\log T
\end{align}
(see Titchmarsh \cite[Theorems 7.2, 7.3]{Titch}). These simple results suggest
two important observations.
(a) First, it is trivial that $\zeta(\sigma+it)$ is
bounded with respect to $t$ in the region of absolute convergence $\sigma>1$,
but \eqref{1-6} and \eqref{1-7} suggest that $\zeta(\sigma+it)$ seems not so large
in the strip $1/2\leq\sigma\leq 1$, too. In fact, the well-known Lindel{\" o}f
hypothesis predicts that
\begin{align}\label{1-8}
\zeta(\sigma+it)=O\left(t^{\varepsilon}\right)\qquad\left( \frac{1}{2}\leq\sigma<1\right)
\end{align}
for any $\varepsilon>0$. (For $\sigma=1$, even a stronger estimate has already
been known.) Formulas \eqref{1-6} and \eqref{1-7} support this hypothesis.
(b) The second observation is that the coefficient $\zeta(2\sigma)$ on the right-hand
side of \eqref{1-6} tends to infinity as $\sigma\to 1/2$, hence the form of the
formula should be changed at $\sigma=1/2$, which is in fact embodied by \eqref{1-7}.
This is one of the special features of the ``critical line'' $\Re s=1/2$ in the theory
of the Riemann zeta-function.
Our theorems proved in the present paper may be regarded as double analogues of
\eqref{1-6}. Since the coefficient $\zeta_2^{[2]}(s_0,2\sigma)$ tends to
infinity as $\sigma_0+\sigma\to 3/2$, it is natural to raise,
analogously to the above (a) and (b), the following two conjectures:
(i) (a double analogue of the Lindel{\" o}f hypothesis) For any $\varepsilon>0$,
\begin{align}\label{1-9}
\zeta_2(s_0,s)=O\left(t^{\varepsilon}\right)
\end{align}
when $(s_0,s)$ (which is not in the domain of absolute convergence) satisfies
$\sigma_0>1/2$, $\sigma>1/2$, $t\geq 2$, $\sigma_0+\sigma\geq 3/2$ and
$s_0+s\neq 2$;
(ii) (the criticality of $\sigma_0+\sigma=3/2$)
When $\sigma_0+\sigma=3/2$, the form of the main term of the mean square formula
would not be $CT$ (with a constant $C$; most probably, some log-factor would appear).
\begin{remark}
It is not easy to find the ``correct'' double analogue of the Lindel{\" o}f
hypothesis. Nakamura and Pa{\'n}kowski \cite{Nak-Pan} raised the conjecture
\begin{align}\label{1-10}
\zeta_2\left(1/2+it,1/2+it\right)=O\left(t^{\varepsilon}\right)
\end{align}
(actually they stated their conjecture for more general multiple case), and gave
a certain result (their Proposition 6.3) which supports the conjecture.
However, the value $\zeta_2\left(1/2+it_1,1/2+it_2\right)$ is, if $t_1\neq t_2$,
not always small. In fact, Corollary 1 of Kiuchi, Tanigawa and Zhai \cite{KTZ}
describes the situation when $\zeta_2(s_1,s_2)$ is not small. For example,
if $t_2\ll t_1^{1/6-\varepsilon}$, then
$$
\zeta_2\left(1/2+it_1,1/2+it_2\right)=\Omega\left(t_1^{1/3+\varepsilon}\right).
$$
Our theorems imply that our conjecture \eqref{1-9} is true in mean. That is,
\eqref{1-9} is reasonable in view of our theorems.
\end{remark}
\begin{remark}
The above conjecture (ii) suggests that $\sigma_0+\sigma=3/2$ might be the
double analogue of the critical line of the Riemann zeta-function $\Re s=1/2$.
On the other hand, in view of the result of Nakamura and Pa{\'n}kowski mentioned
above, we see that another candidate of the double analogue of the critical line
is $\sigma_0+\sigma=1$. At present it is not clear which is more plausible.
\end{remark}
\begin{remark}
We cannot expect the analogue of the Riemann hypothesis on the location
of zeros. In fact, Theorem 5.1 of Nakamura and Pa{\'n}kowski \cite{Nak-Pan}
asserts (in the double zeta case) that for any $1/2<\sigma_1<\sigma_2<1$,
$\zeta_2(s,s)$ has $\asymp T$ non-trivial zeros in the rectangle
$\sigma_1<\sigma<\sigma_2$, $0<t<T$.
\end{remark}
The plan of the present paper is as follows. We first prove the simplest
Theorem \ref{T-1-1} in Section \ref{sec-2}. To prove the other theorems, we need
certain approximation formulas for $\zeta_2(s_0,s)$. Using the Euler-Maclaurin
formula, we show the first approximation formula (Theorem \ref{T-3-1})
in Section \ref{sec-3}, and using
it, we prove Theorem \ref{T-1-2} in Section \ref{sec-4}. In Section \ref{sec-4.25}
we introduce and discuss the double analogue of the Euler constant. The most
difficult part of the present paper is the proof of Theorem \ref{T-1-3}.
In Section \ref{sec-4.5} we show the second approximation formula (Theorem
\ref{T-5-3}), by employing
the method of Mellin-Barnes integral formula. Based on this second approximation
formula, we give the proof of Theorem \ref{T-1-3} in the final Section \ref{sec-5}.
A possible direction of future study is to search for a strong type of
approximate functional equation (that is, similar to
\cite[Theorem 4.16]{Titch})
for the double zeta-function, based on our previous results
on functional equations for the double zeta-function obtained in
\cite{KMT-Debrecen,Mat-Camb}. If we could
succeed in finding such an equation, we would be able to give
a more precise version of
mean value theorems for the double zeta-function.
A part of the results in this paper has been announced in \cite{MT-RIMS}.
\section{Proof of Theorem \ref{T-1-1}} \label{sec-2}
In this section, we give the proof of Theorem \ref{T-1-1}. Throughout this paper, we frequently use the following elementary estimations:
\begin{align*}
& \sum_{m=1}^{k-1} \frac{1}{m} \ll \int_{1}^{k} u^{-1}du = \log k,\\
& \sum_{m=1}^{k-1} \frac{1}{m^\sigma} \ll \int_{0}^{k} u^{-\sigma}du = \frac{k^{1-\sigma}}{1-\sigma}\qquad (0<\sigma<1),\\
& \sum_{m=k}^{\infty} \frac{1}{m^{\sigma}} \ll \int_{k}^{\infty} u^{-\sigma}du = \frac{k^{1-\sigma}}{\sigma-1}\qquad (\sigma>1).
\end{align*}
\begin{proof}[Proof of Theorem \ref{T-1-1}]
Let $s_0={\sigma_0}+i{t_0}\in \mathbb{C}$ with ${\sigma_0}>1$ and $s=\sigma+it\in \mathbb{C}$ with $\sigma>1$. We set
\begin{align*}
S:=\zeta_2(s_0,s) \overline{\zeta_2(s_0,{s})} & =\sum_{m_1\geq 1 \atop n_1\geq 1}\frac{1}{m_1^{s_0}(m_1+n_1)^{\sigma+it}}\sum_{m_2\geq 1 \atop n_2\geq 1}\frac{1}{m_2^{\overline{s_0}}(m_2+n_2)^{\sigma-it}}.
\end{align*}
Taking out the terms corresponding to $m_1+n_1=m_2+n_2$ and setting $k=m_1+n_1$, we have
\begin{align*}
S& =\sum_{k=2}^\infty \left( \sum_{m_1=1}^{k-1} \sum_{m_2=1}^{k-1}\frac{1}{m_1^{s_0} m_2^{\overline{s_0}}}\right) \frac{1}{k^{2\sigma}}\\
& \ +\sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1\not= m_2+n_2}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}(m_1+n_1)^{\sigma+it}(m_2+n_2)^{\sigma-it}}\\
& =\zeta_2^{[2]}({s_0},2\sigma) +\sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1\not= m_2+n_2}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\left(\frac{m_2+n_2}{m_1+n_1}\right)^{it}.
\end{align*}
Hence we have
\begin{align*}
&\int_{2}^{T}|\zeta_2({s_0},s)|^2 dt =\zeta_2^{[2]}({s_0},2\sigma)(T-2) \\
&\qquad + \sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1\not= m_2+n_2}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\int_{2}^{T}\left(\frac{m_2+n_2}{m_1+n_1}\right)^{it}dt.
\end{align*}
The second term on the right-hand side is
\begin{align*}
& \sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1\not= m_2+n_2}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\\
& \qquad \times \frac{e^{iT\log((m_2+n_2)/(m_1+n_1))}-e^{2i\log((m_2+n_2)/(m_1+n_1))}}{i\log((m_2+n_2)/(m_1+n_1))}\\
& \ll \sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1< m_2+n_2}\frac{1}{(m_1m_2)^{{\sigma_0}}(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\frac{1}{\log\frac{m_2+n_2}{m_1+n_1}}\\
& =\left(\sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_1+n_1< m_2+n_2\leq 2(m_1+n_1)}+\sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_2+n_2> 2(m_1+n_1)}\right)\frac{1}{(m_1m_2)^{{\sigma_0}}}\\
& \qquad \qquad \times \frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}\log\frac{m_2+n_2}{m_1+n_1}}.
\end{align*}
We denote the right-hand side by $V_1+V_2$. Then we have
\begin{align*}
V_2 & \ll \sum_{m_1,m_2,n_1,n_2\geq 1 \atop m_2+n_2> 2(m_1+n_1)}\frac{1}{(m_1m_2)^{{\sigma_0}}(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\\
& \ll \sum_{m_1,m_2,n_1,n_2\geq 1}\frac{1}{(m_1m_2)^{{\sigma_0}}(n_1n_2)^{\sigma}}=O(1).
\end{align*}
As for $V_1$, setting $r=(m_2+n_2)-(m_1+n_1)$, we have
\begin{align*}
V_1&=\sum_{m_1,m_2,n_1\geq 1}\frac{1}{(m_1m_2)^{{\sigma_0}}}\sum_{r=1}^{m_1+n_1}\frac{1}{(m_1+n_1)^{\sigma}(m_1+n_1+r)^{\sigma}}\frac{1}{\log\frac{m_1+n_1+r}{m_1+n_1}}.
\end{align*}
Since $m_1+n_1+r \asymp m_1+n_1$, we obtain
\begin{align*}
V_1&\ll \sum_{m_1,m_2,n_1\geq 1}\frac{1}{(m_1m_2)^{{\sigma_0}}}\frac{1}{(m_1+n_1)^{2\sigma}}\sum_{r=1}^{m_1+n_1}\frac{1}{\log\left(1+\frac{r}{m_1+n_1}\right)}\\
&\ll \sum_{m_1,m_2,n_1\geq 1}\frac{1}{(m_1m_2)^{{\sigma_0}}}\frac{1}{(m_1+n_1)^{2\sigma}}\sum_{r=1}^{m_1+n_1}\frac{m_1+n_1}{r}\\
&\ll \sum_{m_1,m_2,n_1\geq 1}\frac{1}{(m_1m_2)^{{\sigma_0}}}\frac{1}{(m_1+n_1)^{2\sigma-1}}\log(m_1+n_1)\\
&\ll \sum_{m_2\geq 1}\frac{1}{m_2^{\sigma_0}}\sum_{m_1,n_1\geq 1}\frac{\log(m_1+n_1)}{m_1^{{\sigma_0}}(m_1+n_1)^{2\sigma-1}}=O(1),
\end{align*}
because ${\sigma_0}>1$ and $\sigma>1$.
This completes the proof of Theorem \ref{T-1-1}.
\end{proof}
\begin{remark}
The fundamental idea of the above proof of Theorem \ref{T-1-1} is similar to that
of the proof of \cite[Theorem 7.2]{Titch}. The basic structure of the proofs of
Theorems \ref{T-1-2} and \ref{T-1-3} given below is the same, though the technical
details are more complicated.
\end{remark}
\section{The first approximation theorem} \label{sec-3}
Hardy and Littlewood proved the following well-known result (see \cite[Theorem 4.11]{Titch}). Let $\sigma_1>0$, $x\geq 1$ and $C>1$. Suppose $s=\sigma+it \in \mathbb{C}$ with $\sigma\geq \sigma_1$ and $|t|\leq 2\pi x/C$. Then
\begin{equation}
\label{3-1}
\zeta(s)=\sum_{1\leq n \leq x}\frac{1}{n^{s}}-\frac{x^{1-s}}{1-s}+O\left( x^{-\sigma}\right)\quad (x\to \infty).
\end{equation}
Here we prove the double series analogue of \eqref{3-1} as follows.
\begin{theorem} \label{T-3-1}
Let ${s_0}={\sigma_0}+i{t_0} \in \mathbb{C}$,
$s=\sigma+it \in \mathbb{C}\setminus\{1\}$, $x\geq 1$ and $C>1$. Suppose
$\sigma> \max(0,2-{\sigma_0})$ and $|t|\leq 2\pi x/C$. Then
\begin{equation}
\label{3-2}
\begin{split}
\zeta_2({s_0},s) & =\sum_{m=1}^\infty \sum_{1\leq n \leq x}\frac{1}{m^{{s_0}}(m+n)^{s}}-\frac{1}{1-s}\sum_{m=1}^\infty \frac{1}{m^{s_0} (m+x)^{s-1}}\\
& \qquad+
\begin{cases}
O(x^{-\sigma}) & (\sigma_0>1)\\
O(x^{-\sigma}\log x) & (\sigma_0=1)\qquad (x \to \infty).\\
O(x^{1-\sigma-\sigma_0}) & (\sigma_0<1)
\end{cases}
\end{split}
\end{equation}
\end{theorem}
In order to prove this theorem, we quote the following lemma.
\begin{lemma}[\cite{Titch} Lemma 4.10] \label{L-3-2}
Let $f(x)$ be a real function with a continuous and steadily decreasing derivative $f'(x)$ in $(a,b)$, and let $f'(b)=\alpha$, $f'(a)=\beta$. Let $g(x)$ be a real positive decreasing function with a continuous derivative $g'(x)$, satisfying that $|g'(x)|$ is steadily decreasing. Then
\begin{equation}
\begin{split}
\sum_{a<n\leq b}g(n)e^{2\pi i f(n)}& =\sum_{\nu \in \mathbb{Z} \atop
\alpha-\eta<\nu<\beta+\eta}\int_{a}^{b}g(x)e^{2\pi i(f(x)-\nu x)}dx \\
& +O\left( g(a)\log(\beta-\alpha+2)\right)+O\left( |g'(a)|\right)
\end{split}
\label{3-3}
\end{equation}
for an arbitrary $\eta\in (0,1)$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{T-3-1}]
By the Euler-Maclaurin formula (see \cite[Equation (2.1.2)]{Titch}), we have
\begin{equation}
\label{EulerMaclaurin}
\begin{split}
\sum_{a<l\leq b}\frac{1}{l^s} & =\frac{b^{1-s}-a^{1-s}}{1-s}-s\int_{a}^{b}\frac{y-[y]-1/2}{y^{s+1}}dy +\frac{1}{2}\left(b^{-s}-a^{-s}\right)
\end{split}
\end{equation}
for $0<a<b$.
At first assume $\sigma_0>1$, $\sigma>1$.
Setting $a=m+N$ (where $m \in \mathbb{N}$, $N \in \mathbb{N}_0$) in
\eqref{EulerMaclaurin} and $b\to \infty$, we have
\begin{equation*}
\begin{split}
\sum_{l=m+N+1}^{\infty}\frac{1}{l^s} & =-\frac{(m+N)^{1-s}}{1-s}-s\int_{m+N}^{\infty}\frac{y-[y]-1/2}{y^{s+1}}dy -\frac{1}{2}(m+N)^{-s}.
\end{split}
\end{equation*}
Therefore we have
\begin{align}
& \sum_{m=1}^\infty \frac{1}{m^{s_0}}\sum_{n=1}^\infty \frac{1}{(m+n)^s}\notag\\
&\ =\sum_{m=1}^\infty \frac{1}{m^{s_0}}\sum_{n=1}^{N} \frac{1}{(m+n)^s}
-\sum_{m=1}^\infty \frac{(m+N)^{1-s}}{m^{s_0}(1-s)}\notag\\
& \quad -s\sum_{m=1}^\infty \frac{1}{m^{s_0}}\int_{m+N}^{\infty}
\frac{y-[y]-1/2}{y^{s+1}}dy
-\frac{1}{2}\sum_{m=1}^\infty \frac{1}{m^{s_0}(m+N)^{s}}\notag\\
&\ =A_1-A_2-A_3-A_4,\label{3-4}
\end{align}
say. The terms $A_1$ and $A_4$ are absolutely convergent in the region
$\sigma_0+\sigma>1$, and in this region
\begin{align}\label{S_4}
A_4=O\left( \sum_{m=1}^\infty \frac{1}{m^{\sigma_0}(m+N)^{\sigma}}\right).
\end{align}
The integral in $A_3$ is absolutely convergent if $\sigma>0$, and is
$O(\sigma^{-1}(m+N)^{-\sigma})$. Therefore $A_3$ can be continued to
the region $\sigma>0$, $\sigma_0+\sigma>1$ and
\begin{align}\label{S_3}
A_3=O\left( \sum_{m=1}^\infty \frac{|s|/\sigma}{m^{\sigma_0}(m+N)^{\sigma}}\right)
\end{align}
there. The term $A_2$ is absolutely convergent for $\sigma_0+\sigma>2$, $s\neq 1$.
Therefore we see that the right-hand side of \eqref{3-4}
gives the meromorphic continuation to the desired region.
Hereafter in this proof we assume $N>x$.
The term $A_1$ can be rewritten as
\begin{equation}
\begin{split}
& \sum_{m=1}^\infty \sum_{n\leq x} \frac{1}{m^{s_0}(m+n)^s}+\sum_{m=1}^\infty \sum_{x<n\leq N} \frac{e^{-it\log(m+n)}}{m^{s_0}(m+n)^\sigma}.
\end{split}
\label{3-5}
\end{equation}
Fix $m\in \mathbb{N}$ and set
$$f(x)=\frac{t}{2\pi}\log(m+x),\quad g(x)=(m+x)^{-\sigma},$$
$(a,b)=(x,N)$ in Lemma \ref{L-3-2}. Then we have
$$(\alpha,\beta)=\left( \frac{t}{2\pi (m+N)},\ \frac{t}{2\pi (m+x)}\right).$$
We see that
$$|f'(x)|=\frac{|t|}{2\pi (m+x)}\leq \frac{|t|}{2\pi x}\leq \frac{1}{C}<1.$$
When $\sigma>0$, the function $g(x)$ is decreasing and so we can apply Lemma
\ref{L-3-2}. By taking a small $\eta$, we obtain from \eqref{3-3} that
\begin{align*}
&\sum_{x<n\leq N} \frac{e^{it\log(m+n)}}{(m+n)^\sigma}=\int_{x}^{N} \frac{1}{(m+u)^{\sigma-it}}du+O\left((m+x)^{-\sigma}\right).
\end{align*}
Considering complex conjugates on the both sides, we have
\begin{align}
\sum_{x<n\leq N}\frac{1}{(m+n)^s}&=
\sum_{x<n\leq N} \frac{e^{-it\log(m+n)}}{(m+n)^\sigma}=\int_{x}^{N} \frac{1}{(m+u)^s}du+O\left((m+x)^{-\sigma}\right)\notag\\
& =\frac{(m+N)^{1-s}-(m+x)^{1-s}}{1-s}+O\left((m+x)^{-\sigma}\right). \label{conjugate}
\end{align}
In other words, denoting the above error term by $E(s;x,m,N)$, we find that
this function is entire in $s$ (the point $s=1$ is a removable singularity)
and satisfies
\begin{align}\label{E}
E(s;x,m,N)=O\left((m+x)^{-\sigma}\right)
\end{align}
uniformly in $N$ in the region $\sigma>0$.
Using \eqref{conjugate}, we find that the second term of \eqref{3-5} is equal to
\begin{equation}
\begin{split}
& \frac{1}{1-s}\sum_{m=1}^\infty\frac{1}{m^{{s_0}}(m+N)^{s-1}}-
\frac{1}{1-s}\sum_{m=1}^\infty\frac{1}{m^{s_0} (m+x)^{s-1}}\\
& \qquad +\sum_{m=1}^\infty\frac{E(s;x,m,N)}{m^{s_0}}
\end{split}
\label{3-6}
\end{equation}
(where the first two sums are convergent in $\sigma_0+\sigma>2$,
while the last sum is convergent in $\sigma_0+\sigma>1$ because of \eqref{E}),
whose first term is cancelled with $A_2$.
Therefore now we have
\begin{align}
\zeta_2(s_0,s)&=\sum_{m=1}^{\infty}\sum_{n\leq x}\frac{1}{m^{s_0} (m+n)^{s}}
-\frac{1}{1-s}\sum_{m=1}^\infty\frac{1}{m^{s_0} (m+x)^{s-1}}\notag\\
& \qquad +\sum_{m=1}^\infty\frac{E(s;x,m,N)}{m^{s_0}}-A_3-A_4 \label{suzumenooyado}
\end{align}
in the region $\sigma>\max(0,2-\sigma_0)$, $s\neq 1$.
Letting $N \to \infty$, and noting \eqref{S_4}, \eqref{S_3} and \eqref{E},
we obtain the proof of Theorem \ref{T-3-1}.
\end{proof}
\section{Proof of Theorem \ref{T-1-2}} \label{sec-4}
In this section, using Theorem \ref{T-3-1}, we give the proof of Theorem \ref{T-1-2}.
\begin{proof}[Proof of Theorem \ref{T-1-2}]
Let ${s_0}={\sigma_0}+i{t_0}\in \mathbb{C}$ with ${\sigma_0}>1$ and
$s=\sigma+it\in \mathbb{C}\setminus\{1\}$ with $1/2<\sigma\leq 1$,
$\sigma_0+\sigma>2$.
Setting $C=2\pi$ and $x=t$
in \eqref{3-2}, we easily see that the second term on the right-hand side is
$O\left( t^{-\sigma}\right)$, so
we have
\begin{equation}
\label{4-1}
\begin{split}
\zeta_2({s_0},s) & =\sum_{m=1}^\infty \sum_{1\leq n \leq t}\frac{1}{m^{{s_0}}(m+n)^{s}}
+O\left( t^{-\sigma}\right)\quad (t\to \infty).
\end{split}
\end{equation}
We denote the first term on the right-hand side by $\Sigma_1({s_0},s)$. Let $M(n_1,n_2)=\max\{n_1,n_2,2\}$. Then
\begin{align}
& \int_{2}^{T}|\Sigma_1({s_0},s)|^2 dt \notag\\
& =\int_{2}^{T}\sum_{m_1\geq 1}\sum_{n_1\leq t}\frac{1}{m_1^{{s_0}}(m_1+n_1)^{\sigma+it}}\sum_{m_2\geq 1}\sum_{n_2\leq t}\frac{1}{m_2^{{\overline{s_0}}}(m_2+n_2)^{\sigma-it}} dt\notag\\
& =\sum_{m_1\geq 1}\sum_{m_2\geq 1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\sum_{n_1\leq T}\sum_{n_2\leq T}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^{\sigma}}\notag\\
& \qquad \times \int_{M(n_1,n_2)}^{T}\left(\frac{m_2+n_2}{m_1+n_1}\right)^{it}dt\notag\\
& =\sum_{m_1\geq 1}\sum_{m_2\geq 1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\sum_{n_1\leq T}\sum_{n_2\leq T \atop m_1+n_1=m_2+n_2}\frac{1}{(m_1+n_1)^{2\sigma}}(T-M(n_1,n_2))\notag\\
& \quad + \sum_{m_1\geq 1}\sum_{m_2\geq 1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\sum_{n_1\leq T}\sum_{n_2\leq T \atop m_1+n_1\not=m_2+n_2}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\notag\\
& \quad \times \frac{e^{iT\log((m_2+n_2)/(m_1+n_1))}-e^{iM(n_1,n_2)\log((m_2+n_2)/(m_1+n_1))}}{i\log((m_2+n_2)/(m_1+n_1))}.\label{mean-val}
\end{align}
We denote the first and the second term on the right-hand side by $S_1T-S_2$ and $S_3$, respectively. As for $S_1$, setting $k=m_1+n_1(=m_2+n_2)$, we have
\begin{align*}
S_1&= \sum_{k=2}^\infty \left(\sum_{m_1=1}^{k-1} \sum_{m_2=1}^{k-1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\right)\frac{1}{k^{2\sigma}} \\
& \ -\sum_{m_1\geq 1\atop {m_2\geq 1}}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\bigg\{ \sum_{n_1> T \atop {n_2\leq T \atop m_1+n_1=m_2+n_2}}+\sum_{n_1\leq T \atop {n_2> T \atop m_1+n_1=m_2+n_2}}+\sum_{n_1> T\atop {n_2> T \atop m_1+n_1=m_2+n_2}}\bigg\}\frac{1}{(m_1+n_1)^{2\sigma}}.
\end{align*}
We further denote the second term on the right-hand side by $-(U_1+U_2+U_3)$, which is equal to $-(U_1+U_3)-(\overline{U_1}+U_3)+U_3$ because $U_2=\overline{U_1}$. Since ${\sigma_0}>1$, we have
\begin{align*}
U_1+U_3& \ll \sum_{m_1\geq 1\atop {m_2\geq 1}}\frac{1}{(m_1m_2)^{\sigma_0}} \sum_{n_1> T}\frac{1}{(m_1+n_1)^{2\sigma}}\\
&\ll \sum_{m_1\geq 1\atop {m_2\geq 1}}\frac{1}{(m_1m_2)^{\sigma_0}} \int_{T}^\infty \frac{du}{(m_1+u)^{2\sigma}}\\
&\ll \sum_{m_1\geq 1\atop {m_2\geq 1}}\frac{1}{(m_1m_2)^{\sigma_0}(m_1+T)^{2\sigma-1}}\ll T^{1-2\sigma}.
\end{align*}
Similarly we obtain $\overline{U_1}+U_3, U_3 \ll T^{1-2\sigma}$.
Therefore we have
\begin{equation}
S_1T=\zeta_2^{[2]}({s_0},2\sigma)T+O\left( T^{2-2\sigma}\right). \label{4-2}
\end{equation}
As for $S_2$, since
$$M(n_1,n_2)=\max\{n_1,n_2\}\leq m_1+n_1 (=m_2+n_2),$$
we have
\begin{align*}
S_2& \ll \sum_{m_1\geq 1}\sum_{m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T\atop {n_2\leq T \atop m_1+n_1=m_2+n_2}}\frac{1}{(m_1+n_1)^{2\sigma-1}}\\
& \ll \sum_{m_1\geq 1}\sum_{m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{(m_1+n_1)^{2\sigma-1}}\\
& \ll \sum_{m_1\geq 1}\frac{1}{m_1^{\sigma_0}}\sum_{m_2\geq 1}\frac{1}{m_2^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{n_1^{2\sigma-1}}\\
& \ll
\begin{cases}
T^{2-2\sigma} & (1/2<\sigma < 1)\\
\log T & (\sigma=1),
\end{cases}
\end{align*}
because ${\sigma_0}>1$.
As for $S_3$, we have
\begin{align*}
S_3& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_1+n_1<m_2+n_2\leq 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\frac{1}{\log\frac{m_2+n_2}{m_1+n_1}}\\
& \ \ + \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_2+n_2> 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\frac{1}{\log\frac{m_2+n_2}{m_1+n_1}}.
\end{align*}
We denote the first and the second term by $W_1$ and $W_2$, respectively.
As for $W_2$, we have
\begin{align*}
W_2& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_2+n_2> 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\\
& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{n_1^{\sigma}}\sum_{n_2\leq T}\frac{1}{n_2^\sigma}\\
& \ll
\begin{cases}
T^{2-2\sigma} & (1/2<\sigma < 1)\\
(\log T)^2 & (\sigma=1).
\end{cases}
\end{align*}
As for $W_1$, setting $r=(m_2+n_2)-(m_1+n_1)$, we have
\begin{align*}
W_1& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T}\sum_{r=1}^{m_1+n_1}\frac{1}{(m_1+n_1)^\sigma(m_1+n_1+r)^\sigma}\frac{1}{\log\left(1+\frac{r}{m_1+n_1}\right)}\\
& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{(m_1+n_1)^{2\sigma}}\sum_{r=1}^{m_1+n_1}\frac{m_1+n_1}{r}\\
& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{(m_1+n_1)^{2\sigma-1}}\log(m_1+n_1)\\
& \ll
\begin{cases}
T^{2-2\sigma}\log T & (1/2<\sigma < 1)\\
(\log T)^2 & (\sigma=1).
\end{cases}
\end{align*}
Combining these results, we obtain
\begin{align*}
& \int_{2}^{T}|\Sigma_1({s_0},s)|^2 dt =\zeta_2^{[2]}({s_0},2\sigma)T+
\begin{cases}
O\left(T^{2-2\sigma}\log T\right) & (1/2<\sigma < 1)\\
O\left((\log T)^2\right) & (\sigma=1).
\end{cases}
\end{align*}
Therefore we have
\begin{align}
& \int_{2}^{T}|\zeta_2({s_0},s)|^2 dt \notag\\
& =\int_{2}^{T}|\Sigma_1({s_0},s)+O\left(t^{-\sigma}\right)|^2 dt \notag\\
& =\int_{2}^{T}|\Sigma_1({s_0},s)|^2 dt+O\left(\int_{2}^{T}|\Sigma_1({s_0},s)|\,t^{-\sigma}dt\right)+O\left(\int_{2}^{T}t^{-2\sigma}dt \right).\label{4-2-2}
\end{align}
We see that the third term on the right-hand side is equal to $O(1)$ because $\frac{1}{2}<\sigma\leq 1$. As for the second term, by the Cauchy-Schwarz inequality, we see that
\begin{align*}
& \int_{2}^{T}|\Sigma_1({s_0},s)|\,t^{-\sigma}dt \\
& \ll \left(\int_{2}^{T}|\Sigma_1({s_0},s)|^2 dt\right)^{1/2}\cdot \left(\int_{2}^{T}\,t^{-2\sigma}dt\right)^{1/2}\\
& =\left(
\begin{cases}
O(T)+O\left(T^{2-2\sigma}\log T\right) & (1/2<\sigma < 1)\\
O(T)+O\left((\log T)^2\right) & (\sigma=1)
\end{cases}
\right)^{1/2}\cdot O(1)^{1/2}\\
& \ll T^{1/2}.
\end{align*}
This completes the proof of Theorem \ref{T-1-2}.
\end{proof}
\section{The double analogue of the Euler constant} \label{sec-4.25}
Let $\gamma$ be the Euler constant defined by
$$\gamma=\lim_{N\to \infty}\left( \sum_{n=1}^{N}\frac{1}{n}-\log N\right),$$
which satisfies that
\begin{equation}\label{gamma}
\lim_{s\to 1}\left\{\zeta(s)-\frac{1}{s-1}\right\}=\gamma.
\end{equation}
Here we define analogues of the Euler constant corresponding to double zeta-functions as follows. For ${s_0}\in \mathbb{C}$ with $\Re {s_0}>1$, we let
\begin{equation}
\gamma_2({s_0})=\lim_{N \to \infty}\sum_{m\geq 1}\frac{1}{m^{s_0}}\left\{\sum_{1\leq n\leq N} \frac{1}{(m+n)}-\log (m+N)\right\}. \label{Eu-1}
\end{equation}
Then we obtain the following.
\begin{prop} \label{P-Euler}
For ${s_0}\in \mathbb{C}$ with $\Re {s_0}>1$,
\begin{equation}
\lim_{s\to 1}\left\{ \zeta_2(s_0,s)-\frac{\zeta(s_0)}{s-1}\right\}=\gamma_2({s_0}).\label{Eu-2}
\end{equation}
In particular,
\begin{equation}
\gamma_2({s_0})=\zeta({s_0})\gamma-\zeta_2(1,{s_0})-\zeta({s_0}+1).\label{Eu-3}
\end{equation}
\end{prop}
\begin{proof}
Applying \eqref{3-4} with $N=0$, we have
\begin{align*}
& \lim_{s\to 1}\left\{ \zeta_2(s_0,s)-\frac{\zeta(s_0)}{s-1}\right\}\\
& =\lim_{s\to 1}\bigg\{ \frac{\sum_{m\geq 1}m^{-{s_0}-s+1}-\zeta({s_0})}{s-1} -s\sum_{m\geq 1}\frac{1}{m^{s_0}}\int_{m}^\infty \frac{u-[u]-1/2}{u^{s+1}}du \\
& \qquad -\frac{1}{2}\sum_{m\geq 1}\frac{1}{m^{{s_0}+s}}\bigg\}\\
& =\zeta'({s_0})-\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\int_{m}^\infty \frac{u-[u]}{u^{2}}du \\
& \quad +\frac{1}{2}\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\int_{m}^\infty \frac{1}{u^2}du -\frac{1}{2}\zeta({s_0}+1),
\end{align*}
where the third and the fourth terms are cancelled. Hence, from
$$\zeta'({s_0})=-\sum_{m\geq 1}\frac{\log m}{m^{{s_0}}},$$
the right-hand side of the above equation can be rewritten as
\begin{align*}
& \zeta'({s_0})-\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\lim_{K\to \infty}\sum_{k=m}^{K+m-1}\int_{k}^{k+1}\frac{u-[u]}{u^{2}}du\\
& =\zeta'({s_0})-\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\lim_{K\to \infty}\sum_{k=m}^{K+m-1}\int_{k}^{k+1}\left(\frac{1}{u}-\frac{k}{u^{2}}\right)du\notag\\
& =\zeta'({s_0})-\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\lim_{K\to \infty}\left(\log(m+K)-\log m-\sum_{k=m}^{m+K-1}\frac{1}{k+1}\right)\notag\\
& =\lim_{K\to \infty}\left\{\sum_{m\geq 1}\frac{1}{m^{{s_0}}}\left(\sum_{n=1}^{K}\frac{1}{m+n}-\log(m+K) \right)\right\}=\gamma_2({s_0}),\notag
\end{align*}
which implies \eqref{Eu-2}. Note that Arakawa and Kaneko \cite[Proposition 4]{AK}
already showed that $\zeta_2({s_0},s)$, as a function in $s$, has a simple pole at
$s=1$ with its residue $\zeta({s_0})$, where ${s_0}\in \mathbb{C}$ with $\Re {s_0}>1$.
Suppose ${s_0}\in \mathbb{C}$ with $\Re {s_0}>1$ and $s>1$. Then it is well-known that
$$\zeta({s_0})\zeta(s)=\zeta_2({s_0},s)+\zeta_2(s,{s_0})+\zeta({s_0}+s).$$
By \eqref{gamma} and \eqref{Eu-2}, we have
\begin{align*}
& \zeta({s_0})\left(\frac{1}{s-1}+\gamma+o(s-1)\right)\\
& \qquad =\left(\frac{\zeta({s_0})}{s-1}+\gamma_2(s_0)+o(s-1)\right)+\zeta_2(s,{s_0})+\zeta({s_0}+s).
\end{align*}
Letting $s\to 1$, we obtain \eqref{Eu-3}.
This completes the proof.
\end{proof}
\section{The second approximation theorem} \label{sec-4.5}
In the previous section, we gave the proof of Theorem \ref{T-1-2} by use of \eqref{4-1}
which comes from Theorem \ref{T-3-1}. However Theorem \ref{T-3-1} holds under the
conditions ${\sigma}>0$ and $\sigma_0+\sigma>2$. Hence we cannot use it for
$3/2<\sigma_0+\sigma\leq 2$.
In order to prove a mean value result in the latter case, we have to prepare another
approximate formula for $\zeta_2(s_0,s)$.
We begin with \eqref{suzumenooyado}.
As was discussed in the proof of Theorem \ref{T-3-1}, all but the second term on the
right-hand side of \eqref{suzumenooyado} are convergent in
$\sigma>0, \sigma_0+\sigma>1$, so the remaining task is to study the second term.
First we assume ${\sigma_0}+\sigma>2$, $s\neq 1$. Then by the Euler-Maclaurin
formula we have
\begin{align}
& \frac{1}{1-s}\sum_{m=1}^\infty \frac{1}{m^{s_0} (m+x)^{s-1}} \notag\\
& =\frac{1}{1-s}\int_{1}^\infty \frac{dy}{y^{s_0} x^{s-1}\left(1+\frac{y}{x}\right)^{s-1}} \notag \\
& \ \ +\frac{1}{1-s}\int_{1}^\infty \left(y-[y]-\frac{1}{2}\right)\left(-\frac{{s_0}}{y^{{s_0}+1}(y+x)^{s-1}}+\frac{1-s}{y^{s_0}(y+x)^s}\right)dy \notag\\
& \ \ +\frac{1}{2(1-s)}(1+x)^{1-s}\notag\\
&=g(s_0,s;x)+Y_2+Y_3,\label{5-3}
\end{align}
say.
Obviously $Y_3$ is defined for any $s\in\mathbb{C}\setminus\{1\}$ and satisfies
$Y_3=O\left(t^{-1} x^{1-\sigma}\right)$.
Next consider $Y_2$. We have
\begin{align*}
&\frac{1}{1-s}\int_{1}^\infty \left(y-[y]-\frac{1}{2}\right)
\frac{{s_0}}{y^{{s_0}+1}(y+x)^{s-1}}dy \ll
\frac{1}{t}\int_{1}^\infty \frac{dy}{y^{{\sigma_0}+1}(y+x)^{\sigma-1}}\\
&\qquad \ll t^{-1}x^{1-\sigma}\int_{1}^\infty \frac{dy}{y^{{\sigma_0}+1}}
\ll t^{-1}x^{1-\sigma}
\end{align*}
for $\sigma_0>0$, and
\begin{align*}
\frac{1}{1-s} & \int_{1}^\infty \left(y-[y]-\frac{1}{2}\right)\frac{1-s}{y^{s_0}(y+x)^s}dy \ll \int_{1}^\infty \frac{dy}{y^{{\sigma_0}}(y+x)^{\sigma}}\\
& \ll \left(\int_{1}^{x} +\int_{x}^\infty\right)\frac{dy}{y^{{\sigma_0}}(y+x)^{\sigma}}\\
& \ll \int_{1}^{x}\frac{dy}{y^{{\sigma_0}}x^{\sigma}} +\int_{x}^\infty\frac{dy}{y^{{\sigma_0}+\sigma}}\\
& =
\begin{cases}
O\left(x^{1-{\sigma_0}-\sigma}\right)
& (0< {\sigma_0}<1;\,{\sigma_0}+\sigma>1)\\
O\left(x^{-\sigma}\log x\right)
& ({\sigma_0}=1;\,{\sigma_0}+\sigma>1)\\
O\left(x^{-\sigma}\right)
& ({\sigma_0}>1;\,{\sigma_0}+\sigma>1).
\end{cases}
\end{align*}
Therefore now we find that $Y_2+Y_3$ can be continued to the region $\sigma_0>0$,
${\sigma_0}+\sigma>1$ and $s\neq 1$, and in this region satisfies
\begin{align}\label{Y_2+Y_3}
& Y_2+Y_3 = O(t^{-1}x^{1-\sigma})+
\begin{cases}
O\left(x^{1-{\sigma_0}-\sigma}\right)
& (0< {\sigma_0}<1;\,{\sigma_0}+\sigma>1)\\
O\left(x^{-\sigma}\log x\right)
& ({\sigma_0}=1;\,{\sigma_0}+\sigma>1)\\
O\left(x^{-\sigma}\right)
& ({\sigma_0}>1;\,{\sigma_0}+\sigma>1).
\end{cases}
\end{align}
Next we consider $g(s_0,s;x)$.
Here we invoke the classical Mellin-Barnes
integral formula, that is
\begin{align}
\label{5-4}
(1+\lambda)^{-s}=\frac{1}{2\pi i}\int_{(c)}\frac{\Gamma(s+z)\Gamma(-z)}
{\Gamma(s)}\lambda^z dz,
\end{align}
where $s$, $\lambda$ are complex numbers with $\sigma=\Re s>0$, $|\arg\lambda|<\pi$,
$\lambda\neq 0$, $c$ is real with $-\sigma<c<0$, and the path
$(c)$ of integration is the vertical line $\Re z=c$.
(Formula \eqref{5-4} has already been successfully used in the theory of multiple
zeta-functions; see \cite{M1,Mat-JNT,Mat-NMJ}).
\begin{lemma} \label{L-5-2}
The function $g({s_0},s;x)$ can be continued meromorphically to the region
$\sigma_0<3/2$ and $\sigma>1/2$,
and satisfies
\begin{align*}
g({s_0},s;x)=
\begin{cases}
O\left(t^{-1}x^{1-\sigma}+t^{\sigma_0-2}x^{2-\sigma-\sigma_0}
+t^{-1/2}x^{1/2-\sigma}\right) & (s_0\neq 1)\\
O\left(t^{-1}x^{1-\sigma}(\log t+\log x)+t^{-1/2}x^{1/2-\sigma}\right) & (s_0=1)
\end{cases}
\end{align*}
in this region, except for the points on the singularities
\begin{align}\label{sing}
s=1,\quad s_0+s=2,1,0,-1,-2,-3,-4,\ldots.
\end{align}
\end{lemma}
\begin{proof}
First we assume that ${\sigma_0}>1$ and $\sigma>1$. Then, applying \eqref{5-4} with $\lambda=y/x$ and replacing $s$ by $s-1$ (because $\sigma-1>0$), we have
\begin{equation}
g({s_0},s;x) =\frac{1}{(2\pi i)(1-s)}\int_{1}^\infty \frac{1}{y^{s_0} x^{s-1}}\int_{(c)}\frac{\Gamma(s-1+z)\Gamma(-z)}{\Gamma(s-1)}\left( \frac{y}{x}\right)^z dz\,dy,
\label{5-7}
\end{equation}
where $1-\sigma<c<0$. Here we see that it is possible to change the order of the integral as follows. Since $1-\sigma<c<0<{\sigma_0}-1$, we have $-{\sigma_0}+c<-1$. This implies that \eqref{5-7} is absolutely convergent with respect to $y$. Moreover, by the Stirling formula, we can easily check that \eqref{5-7} is absolutely convergent with respect to $z$. Therefore, changing the order of the integral on the right-hand side of \eqref{5-7}, we obtain
\begin{align*}
g({s_0},s;x) &=\frac{x^{1-s}}{(2\pi i)(1-s)\Gamma(s-1)}\int_{(c)}{\Gamma(s-1+z)\Gamma(-z)}x^{-z}\int_{1}^\infty y^{z-{s_0}} dy\,dz\\
&=\frac{x^{1-s}}{(2\pi i)(1-s)\Gamma(s-1)}\int_{(c)}\frac{\Gamma(s-1+z)\Gamma(-z)}{x^{z}({s_0}-1-z)} dz.
\end{align*}
Now we temporarily assume that $1<\sigma_0<3/2$. Then the pole $z=s_0-1$
of the integrand is located in the strip $c<\Re z<1/2$.
We shift the path $(c)$ to $\Re z=1/2$.
Relevant poles are at $z=0$ and $z={s_0}-1$.
Counting the residues of those poles, we obtain
\begin{align}
g({s_0},s;x)& =\frac{x^{1-s}}{(1-s)\Gamma(s-1)}\bigg\{ \frac{\Gamma(s-1)}{{s_0}-1}+\frac{\Gamma(s+{s_0}-2)\Gamma(1-{s_0})}{x^{{s_0}-1}}\notag\\
& \qquad+\frac{1}{(2\pi i)}\int_{(1/2)}\frac{\Gamma(s-1+z)\Gamma(-z)}{x^{z}({s_0}-1-z)} dz\bigg\}\notag\\
&=\frac{x^{1-s}}{(1-s)({s_0}-1)}+\frac{x^{1-s}}{(1-s)\Gamma(s-1)}\frac{\Gamma(s+{s_0}-2)\Gamma(1-{s_0})}{x^{{s_0}-1}}\notag\\
& +\frac{x^{1-s}}{(2\pi i)(1-s)\Gamma(s-1)}\int_{(1/2)}\frac{\Gamma(s-1+z)\Gamma(-z)}{x^{z}({s_0}-1-z)} dz\notag\\
&=R_1+R_2+R_3,\label{R-tachi}
\end{align}
say. The last integral can be holomorphically continued to the region
$\sigma_0<3/2$ and $\sigma>1/2$ (because in this region the path does not meet the
poles of the integrand).
Therefore \eqref{R-tachi} gives the meromorphic
continuation of $g({s_0},s;x)$ to this region.
The possible singularities of $R_1$ and $R_2$ are $s_0=1$ and those listed as
\eqref{sing}. But $s_0=1$ is actually not a singularity. Putting $s_0=1+\delta$
and calculating the limit $\delta\to 0$, we find that
\begin{align}\label{at_1}
\biggl.R_1+R_2\biggr|_{s_0=1}&=\frac{x^{1-s}}{1-s}\left(\log x-\gamma-
\frac{\Gamma'}{\Gamma}(s-1)\right).
\end{align}
We can easily check that $R_1=O\left(t^{-1}x^{1-\sigma}\right)$ and
$R_2=O\left(t^{\sigma_0-2}x^{2-\sigma-\sigma_0}\right)$
by the Stirling formula, if $s_0\neq 1$ and
$(s_0,s)$ is not on the singularities \eqref{sing}.
If $s_0=1$, then from \eqref{at_1} we see that
$$
R_1+R_2=O\left(t^{-1}x^{1-\sigma}(\log t+\log x)\right).
$$
As for $R_3$, setting $z=1/2+iy$, we have
\begin{align*}
R_3&\ll \frac{x^{1-\sigma}e^{\pi t/2}}{t\cdot t^{\sigma-3/2}}\int_{-\infty}^\infty
\left|\frac{\Gamma(\sigma-1+it+1/2+iy)\Gamma(-1/2-iy)}{x^{1/2+iy}({\sigma_0}+it_0-1-1/2-iy)}\right|dy \\
&\ll (tx)^{1/2-\sigma}e^{\pi t/2}\int_{-\infty}^\infty (|t+y|+1)^{\sigma-1}
e^{-\pi|t+y|/2}(|y|+1)^{-2}e^{-\pi |y|/2}dy.
\end{align*}
By Lemma 4 of \cite{Mat-JNT}, we find that the above integral is
$O(t^{\sigma-1}e^{-\pi t/2})$, and hence $R_3=O(t^{-1/2}x^{1/2-\sigma})$.
This completes the proof of Lemma \ref{L-5-2}.
\end{proof}
\begin{remark}
By shifting the path more to the right, it is possible to prove that $g({s_0},s;x)$
can be continued meromorphically to the whole space $\mathbb{C}^2$.
\end{remark}
From \eqref{Y_2+Y_3} and Lemma \ref{L-5-2} we find that the right-hand side of
\eqref{5-3} can be continued to the region $\sigma_0<3/2$, $\sigma>1/2$,
$\sigma_0+\sigma>1$, and satisfies the estimates proved above.
On the other hand, the last three terms on the right-hand side of
\eqref{suzumenooyado} are estimated by \eqref{S_4}, \eqref{S_3}, and
\eqref{E}, respectively.
Now set $x=t$. Then, using \eqref{E} we have
\begin{align*}
\sum_{m=1}^{\infty}\frac{E(s;x,m,N)}{m^{s_0}}
\ll\sum_{m=1}^{\infty}\frac{1}{m^{\sigma_0}(m+t)^{\sigma}}
&\ll\sum_{m\leq t} \frac{1}{m^{\sigma_0}t^{\sigma}}
+\sum_{m> t}\frac{1}{m^{\sigma_0+\sigma}}\\
&\ll\left\{
\begin{array}{ll}
t^{1-\sigma_0-\sigma} & (0<\sigma_0<1)\\
t^{-\sigma}\log t & (\sigma_0=1)\\
t^{-\sigma} & (\sigma_0>1),
\end{array}\right.
\end{align*}
while \eqref{S_4} and \eqref{S_3} imply that the contributions of $A_3$ and $A_4$
vanish when $N\to\infty$.
Collecting all the information, we obtain the following.
\begin{theorem} \label{T-5-3}
Let ${s_0}={\sigma_0}+i{t_0}\in \mathbb{C}$ with $0< {\sigma_0}< 3/2$ and
$s=\sigma+it \in \mathbb{C}$ with $\sigma>1/2$, ${\sigma_0}+\sigma>1$,
$s\neq 1$, and $s_0+s\neq 2$. Then
\begin{equation}
\begin{split}
& \zeta_2({s_0},s)= \sum_{m=1}^\infty \sum_{n\leq t}\frac{1}{m^{s_0} (m+n)^s}+
\begin{cases}
O\left(t^{1-{\sigma_0}-\sigma}\right)& ({\sigma_0}<1 )\\
O\left(t^{-\sigma}\log t\right)& ({\sigma_0}=1)\\
O\left(t^{-\sigma}\right) & ({\sigma_0}>1).
\end{cases}
\end{split}
\label{5-8}
\end{equation}
\end{theorem}
\section{Proof of Theorem \ref{T-1-3}} \label{sec-5}
Based on these results, we finally give the proof of Theorem \ref{T-1-3}.
\begin{proof}[Proof of Theorem \ref{T-1-3}]
We let ${s_0}\in \mathbb{C}$ with $1/2<{\sigma_0}<3/2$ and
$s\in \mathbb{C}$ with $1/2<\sigma\leq 1$ with $3/2<{\sigma_0}+\sigma\leq 2$.
We further assume that $s_0+s\neq 2$.
Similarly to Section \ref{sec-4}, let
$$\Sigma_1({s_0},s)=\sum_{m=1}^\infty \sum_{1\leq n \leq t}\frac{1}{m^{{s_0}}(m+n)^{s}}.$$
Then we can again obtain \eqref{mean-val} and denote it by $S_1T-S_2+S_3$.
As for $S_1$, we similarly set $k=m_1+n_1(=m_2+n_2)$. Then we can write
\begin{align*}
S_1&= \sum_{k=2}^\infty \left(\sum_{m_1=1}^{k-1} \sum_{m_2=1}^{k-1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\right)\frac{1}{k^{2\sigma}} -(U_1+U_2+U_3),
\end{align*}
where $U_1=U_2$. We have
\begin{align*}
U_1+U_3& \ll \sum_{m_1\geq 1\atop {m_2\geq 1}}\frac{1}{(m_1m_2)^{\sigma_0}} \sum_{n_1> T}\frac{1}{(m_1+n_1)^{2\sigma}}\\
&\ll \sum_{k=2}^\infty \frac{1}{k^{2\sigma}}\sum_{m_1=1 \atop k-m_1>T}^{k-1}\sum_{m_2=1}^{k-1}\frac{1}{(m_1m_2)^{\sigma_0}},
\end{align*}
where we set $k=m_1+n_1=m_2+n_2$. Note that from the condition $k-m_1>T$, we have $k>T$. Hence we obtain
\begin{align*}
U_1+U_3&\ll \sum_{k>T}\frac{1}{k^{2\sigma}}\int_{1}^{k}u^{-{\sigma_0}}du \int_{1}^{k}v^{-{\sigma_0}}dv\\
&\ll
\begin{cases}
\sum_{k>T}k^{2-2\sigma-2{\sigma_0}}=O\left(T^{3-2\sigma-2{\sigma_0}}\right) & (\frac{1}{2}<{\sigma_0}<1)\\
\sum_{k>T}k^{-2\sigma}(\log k)^2=O\left(T^{1-2\sigma}(\log T)^2\right) & ({\sigma_0}=1)\\
\sum_{k>T}k^{-2\sigma}=O\left(T^{1-2\sigma}\right) & (1<\sigma_0<\frac{3}{2}),
\end{cases}
\end{align*}
because $2-2\sigma-2{\sigma_0}<-1$.
As for the second estimate we used the integration by parts for
$$\sum_{k>T}k^{-2\sigma}(\log k)^2\ll\int_{T}^\infty u^{-2\sigma}(\log u)^2 du.$$
Therefore
\begin{equation}
S_1=\zeta_2^{[2]}({s_0},2\sigma)+
\begin{cases}
O\left(T^{3-2\sigma-2{\sigma_0}}\right) & (\frac{1}{2}<{\sigma_0}<1)\\
O\left(T^{1-2\sigma}(\log T)^2\right) & ({\sigma_0}=1)\\
O\left(T^{1-2\sigma}\right) & (1<\sigma_0<\frac{3}{2}).
\end{cases}
\label{5-9}
\end{equation}
Next we consider $S_2$. Using $M(n_1,n_2)=\max\{n_1,n_2\}$, we have
\begin{align*}
S_2&= \sum_{m_1\geq 1 \atop m_2\geq 1}\frac{1}{m_1^{s_0}m_2^{\overline{s_0}}}\sum_{n_1\leq T \atop {n_2\leq T \atop m_1+n_1=m_2+n_2}}\frac{M(n_1,n_2)}{(m_1+n_1)^{2\sigma}}\\
&= \sum_{k=2}^\infty \frac{1}{k^{2\sigma}}\sum_{m_1\geq 1 \atop m_2\geq 1}\sum_{n_1\leq T \atop {n_2\leq T \atop {m_1+n_1=k \atop m_2+n_2=k}}}\frac{M(n_1,n_2)}{(m_1m_2)^{{s_0}}}\\
& \ll \sum_{k\leq T} \frac{k}{k^{2\sigma}}\sum_{m_1=1}^{k-1} \sum_{m_2=1}^{k-1}\frac{1}{(m_1m_2)^{\sigma_0}}+\sum_{k> T} \frac{T}{k^{2\sigma}}\sum_{m_1=1}^{k-1} \sum_{m_2=1}^{k-1}\frac{1}{(m_1m_2)^{\sigma_0}}\\
& \ll
\begin{cases}
\sum_{k\leq T} {k^{1-2\sigma}}\left(k^{1-{\sigma_0}}\right)^2+T\sum_{k> T} {k^{-2\sigma}}\left(k^{1-{\sigma_0}}\right)^2 & (\frac{1}{2}<{\sigma_0}<1)\\
\sum_{k\leq T} {k^{1-2\sigma}}\left(\log k\right)^2+T\sum_{k> T} {k^{-2\sigma}}\left(\log k\right)^2 & ({\sigma_0}=1)\\
\sum_{k\leq T} {k^{1-2\sigma}}+T\sum_{k> T} {k^{-2\sigma}} & (1<\sigma_0<\frac{3}{2}).
\end{cases}
\end{align*}
Therefore we obtain
\begin{equation}
S_2=
\begin{cases}
O\left(T^{4-2{\sigma_0}-2\sigma}\right)&
(\frac{1}{2}<{\sigma_0}<1,\frac{1}{2}<\sigma\leq 1)\\
O\left(T^{2-2\sigma}(\log T)^2\right)& ({\sigma_0}=1,\frac{1}{2}<\sigma<1)\\
O\left((\log T)^3\right)& ({\sigma_0}=1,\sigma=1)\\
O\left(T^{2-2\sigma}\right)& (1<\sigma_0<\frac{3}{2},\frac{1}{2}<\sigma<1),
\end{cases}
\label{5-10}
\end{equation}
where we have to note that $3/2<\sigma_0+\sigma<2$ in the first case, and
$\sigma\neq 1$ (because if $\sigma=1$ then $\sigma_0+\sigma>2$) in the fourth case.
Finally we consider $S_3$.
Similarly to the argument in Section \ref{sec-4}, we have
\begin{align*}
S_3& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_1+n_1<m_2+n_2\leq 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\frac{1}{\log\frac{m_2+n_2}{m_1+n_1}}\\
& \ \ + \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_2+n_2> 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\frac{1}{\log\frac{m_2+n_2}{m_1+n_1}},
\end{align*}
which we denote by $W_1+W_2$.
First estimate $W_2$. We have
\begin{align*}
W_2& \ll \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1, n_2\leq T \atop m_2+n_2> 2(m_1+n_1)}\frac{1}{(m_1+n_1)^{\sigma}(m_2+n_2)^\sigma}\\
&=\sum_{m_1\geq 1 \atop n_1\leq T}\frac{1}{m_1^{\sigma_0}(m_1+n_1)^{\sigma}}
\sum_{m_2\geq 1, n_2\leq T \atop m_2+n_2> 2(m_1+n_1)}\frac{1}
{m_2^{\sigma_0}(m_2+n_2)^{\sigma}}\\
&=\sum_{m_1\geq 1 \atop n_1\leq T}\frac{1}{m_1^{\sigma_0}(m_1+n_1)^{\sigma}}
\sum_{k>2(m_1+n_1)}\frac{1}{k^{\sigma}}\sum_{m_2\geq 1, n_2\leq T \atop
m_2+n_2=k}\frac{1}{m_2^{\sigma_0}}\\
&= \sum_{m_1\leq T \atop n_1\leq T}+\sum_{m_1> T \atop n_1\leq T}=W_{21}+W_{22},
\end{align*}
say.
Consider $W_{22}$. Since $m_1> T$, we have $k>2T$, so
$m_2=k-n_2\geq k-T>k/2$. Therefore the innermost sum of $W_{22}$ is
$$
\sum_{k-T\leq m_2\leq k-1}\frac{1}{m_2^{\sigma_0}}\ll T k^{-\sigma_0},
$$
and hence
\begin{align*}
W_{22}&\ll T\sum_{m_1> T \atop n_1\leq T}\frac{1}{m_1^{\sigma_0}(m_1+n_1)^{\sigma}}
\sum_{k>2(m_1+n_1)}k^{-\sigma_0-\sigma}\\
&\ll T\sum_{m_1> T}m_1^{-\sigma_0}\sum_{n_1\leq T}(m_1+n_1)^{1-\sigma_0-2\sigma}\\
&\leq T\sum_{m_1> T}m_1^{-\sigma_0}\sum_{n_1\leq T}m_1^{1-\sigma_0-2\sigma}\\
&\ll T^2 \sum_{m_1> T}m_1^{1-2\sigma_0-2\sigma}.
\end{align*}
Since $1-2\sigma_0-2\sigma<-1$, we have
\begin{align}\label{W_{22}}
W_{22}\ll T^2 T^{2-2\sigma_0-2\sigma}&=T^{4-2\sigma_0-2\sigma}.
\end{align}
As for $W_{21}$, we further divide the inner double sum of $W_{21}$ into two parts
$D_1$ and $D_2$ according to $2(m_1+n_1)<k\leq 2T$ and $k>2T$, respectively.
We handle the innermost sum of $D_2$ similarly to the case of $W_{22}$. We have
\begin{align*}
D_2\ll T\sum_{k>2T}\frac{1}{k^{\sigma}}\frac{1}{k^{\sigma_0}}
\ll T^{2-\sigma_0-\sigma}.
\end{align*}
The innermost sum of $D_1$ is
\begin{align*}
\ll\sum_{m_2\leq k-1}\frac{1}{m_2^{\sigma_0}}
\ll
\begin{cases}
k^{1-\sigma_0} & (\frac{1}{2}<\sigma_0<1)\\
\log k & (\sigma_0=1)\\
1 & (1<\sigma_0<\frac{3}{2}),
\end{cases}
\end{align*}
which gives
\begin{align*}
D_1\ll
\begin{cases}
T^{2-\sigma_0-\sigma} & (\frac{1}{2}<\sigma_0<1, \frac{1}{2}<\sigma\leq 1)\\
T^{1-\sigma}\log T & (\sigma_0=1,\frac{1}{2}<\sigma<1)\\
(\log T)^2 & (\sigma_0=1,\sigma=1)\\
T^{1-\sigma} & (1<\sigma_0<\frac{3}{2}, \frac{1}{2}<\sigma<1).
\end{cases}
\end{align*}
Substituting the estimates of $D_1$ and $D_2$ into $W_{21}$, and estimating the
remaining sum
$$
\sum_{m_1\leq T \atop n_1\leq T}\frac{1}{m_1^{\sigma_0}(m_1+n_1)^{\sigma}}
\leq \sum_{m_1\leq T}\frac{1}{m_1^{\sigma_0}}\sum_{n_1\leq T}\frac{1}{n_1^{\sigma}}
$$
in the obvious way, we obtain
\begin{align}\label{W_{21}}
W_{21}&\ll
\begin{cases}
T^{4-2\sigma_0-2\sigma} & (\frac{1}{2}<\sigma_0<1,\frac{1}{2}<\sigma<1)\\
T^{2-2\sigma_0}\log T & (\frac{1}{2}<\sigma_0<1,\sigma=1)\\
T^{2-2\sigma}(\log T)^2 & (\sigma_0=1,\frac{1}{2}<\sigma<1)\\
(\log T)^4 & (\sigma_0=1,\sigma=1)\\
T^{2-2\sigma} & (1<\sigma_0<\frac{3}{2},\frac{1}{2}<\sigma<1).
\end{cases}
\end{align}
Next consider $W_1$. We have
\begin{align*}
W_1&= \sum_{m_1,m_2\geq 1}\frac{1}{(m_1m_2)^{\sigma_0}}\sum_{n_1,\,n_2\leq T\atop {m_1+n_1<m_2+n_2 \atop \leq 2(m_1+n_1)}}\frac{1}{(m_1+n_1)^\sigma(m_2+n_2)^\sigma}\frac{1}{\log\left(1+\frac{m_2+n_2-m_1-n_1}{m_1+n_1}\right)}\\
& \ll \sum_{m_1\geq 1}\sum_{n_1\leq T}\frac{1}{m_1^{\sigma_0}(m_1+n_1)^{2\sigma}}\sum_{n_2\leq T}\sum_{m_2\geq 1 \atop {m_1+n_1<m_2+n_2 \atop \leq 2(m_1+n_1)}}\frac{m_1+n_1}{m_2^{\sigma_0}(m_2+n_2-m_1-n_1)}\\
& =\sum_{m_1\leq 2T}\sum_{n_1\leq T}+\sum_{m_1> 2T}\sum_{n_1\leq T}=W_{11}+W_{12},
\end{align*}
say. Consider $W_{12}$. Since $m_1>2T$, we have $n_2\leq T<m_1/2$, so $m_2>m_1+n_1-n_2>m_1/2$. Therefore, setting $r=m_2+n_2-m_1-n_1$, we have
\begin{align}
W_{12} & \ll \sum_{m_1\geq 2T}\sum_{n_1\leq T}\frac{1}{m_1^{2\sigma_0}(m_1+n_1)^{2\sigma-1}}\sum_{n_2\leq T}\sum_{m_2\geq 1 \atop {m_1+n_1<m_2+n_2 \atop \leq 2(m_1+n_1)}}\frac{1}{(m_2+n_2-m_1-n_1)}\notag\\
& \ll \sum_{m_1\geq 2T}\sum_{n_1\leq T}\frac{1}{m_1^{2\sigma_0}(m_1+n_1)^{2\sigma-1}}\sum_{n_2\leq T}\sum_{r=1}^{m_1+n_1}\frac{1}{r}\notag\\
& \ll T\sum_{m_1\geq 2T}\sum_{n_1\leq T}\frac{1}{m_1^{2\sigma_0}(m_1+n_1)^{2\sigma-1}}\log(m_1+n_1)\notag\\
&\ll T\sum_{m_1>2T}m_1^{-2\sigma_0}\times
\begin{cases}
(m_1+T)^{2-2\sigma}\log(m_1+T) & (\frac{1}{2}<\sigma<1)\\
(\log(m_1+T))^2 & (\sigma=1)
\end{cases}
\notag\\
&\ll
\begin{cases}
T^{4-2\sigma_0-2\sigma}\log T & (\frac{1}{2}<\sigma<1)\\
T^{2-2\sigma_0}(\log T)^2 & (\sigma=1).
\end{cases} \label{W_{12}}
\end{align}
Next,
since $m_2<2(m_1+n_1)$, the innermost sum of $W_{11}$ is
\begin{align*}
&\ll \sum_{m_2<2(m_1+n_1)}m_2^{-\sigma_0}\log(m_1+n_1)\\
&\ll
\begin{cases}
(m_1+n_1)^{1-\sigma_0}\log(m_1+n_1) & (\frac{1}{2}<\sigma_0<1)\\
(\log(m_1+n_1))^2 & (\sigma_0=1)\\
\log(m_1+n_1) & (1<\sigma_0<\frac{3}{2}).
\end{cases}
\end{align*}
Therefore, when $\frac{1}{2}<\sigma_0<1$, we have
\begin{align*}
W_{11}&\ll \sum_{m_1\leq 2T \atop n_1\leq T}m_1^{-\sigma_0}
(m_1+n_1)^{2-\sigma_0-2\sigma}\log(m_1+n_1)\\
&\ll \sum_{m_1\leq 2T} m_1^{-\sigma_0}(m_1+T)^{3-\sigma_0-2\sigma}\log(m_1+T)\\
&\ll T^{4-2\sigma_0-2\sigma}\log T,
\end{align*}
because $2<\sigma_0+2\sigma<3$. Similarly, we have
\begin{align*}
W_{11}&\ll \sum_{m_1\leq 2T \atop n_1\leq T}m_1^{-1}(m_1+n_1)^{1-2\sigma}
(\log(m_1+n_1))^2\\
&\ll
\begin{cases}
T^{2-2\sigma}(\log T)^3 & (\frac{1}{2}<\sigma<1)\\
(\log T)^4 & (\sigma=1)
\end{cases}
\end{align*}
when $\sigma_0=1$, and
\begin{align*}
W_{11}\ll\sum_{m_1\leq 2T \atop n_1\leq T}m_1^{-\sigma_0}(m_1+n_1)^{1-2\sigma}
\log(m_1+n_1)
\ll T^{2-2\sigma}\log T
\end{align*}
when $1<\sigma_0<\frac{3}{2}$.
By \eqref{W_{22}}, \eqref{W_{21}}, \eqref{W_{12}} and the above estimates, we
now obtain
\begin{align}
S_3&=W_1+W_2\notag\\
&\ll
\begin{cases}
T^{4-2\sigma_0-2\sigma}\log T & (\frac{1}{2}<\sigma_0<1,\frac{1}{2}<\sigma<1)\\
T^{2-2\sigma_0}(\log T)^2 & (\frac{1}{2}<\sigma_0<1,\sigma=1)\\
T^{2-2\sigma}(\log T)^3 & (\sigma_0=1,\frac{1}{2}<\sigma<1)\\
(\log T)^4 & (\sigma_0=1,\sigma=1)\\
T^{2-2\sigma}\log T & (1<\sigma_0<\frac{3}{2},\frac{1}{2}<\sigma<1).
\end{cases} \label{SSS}
\end{align}
Denote the right-hand side of the above by $\mathcal{E}(T)$.
Combining \eqref{5-9}, \eqref{5-10} and \eqref{SSS}, we obtain
\begin{align}\label{5-12}
\int_2^{T}|\Sigma_1({s_0},s)|^2dt=S_1T-S_2+S_3 &
=\zeta_2^{[2]}({s_0},2\sigma)T+O(\mathcal{E}(T)).
\end{align}
Now, using the Cauchy-Schwarz inequality, we estimate
the second term on the right-hand side of \eqref{4-2-2} with replacing $t^{-\sigma}$ by the error term
on the right-hand side of \eqref{5-8}.
Denoting by $E(t)$ the error term
on the right-hand side of \eqref{5-8},
we have
\begin{align*}
& \int_{2}^{T}|\Sigma_1({s_0},s)|\,E(t)dt \ll \left(\int_{2}^{T}|\Sigma_1({s_0},s)|^2 dt\right)^{1/2}\cdot \left(\int_{2}^{T}\,E(t)^2 dt\right)^{1/2}\\
& \ =
\left\{ O(T)+O(\mathcal{E}(T))\right\}^{1/2}\cdot O(1)^{1/2}\ll T^{1/2}.
\end{align*}
Thus we obtain the proof of Theorem \ref{T-1-3}.
\end{proof}
\
{\sc Acknowledgements.} \
The authors express their gratitude to
Mr. Soichi Ikeda, Mr. Kaneaki Matsuoka,
Mr. Akihiko Nawashiro and Mr. Tomokazu Onozuka for pointing out
some inaccuracies included in the original version of the manuscript.
\
\bibliographystyle{amsplain}
|
{
"timestamp": "2013-10-14T02:06:33",
"yymm": "1203",
"arxiv_id": "1203.2242",
"language": "en",
"url": "https://arxiv.org/abs/1203.2242"
}
|
\section{introduction}
Quantum dynamics of quantum systems coupled to
fermionic or bosonic environments has recently attracted
wide-spread interest in quantum open systems, quantum dissipative systems, quantum transport,
quantum computing, and nanoscience \cite{xx,yy,nnn,mmm}. For example, the size reduction
of quantum devices in microelectronics requires controllable systems consisting of
only a few electrons, where quantum coherence and quantum interference become dominant.
In addition, quantum dots coupled to electrons of a metal is an interesting
setup in quantum information processing where the quantum coherence of
qubits is of essential importance \cite{cc}. The non-Markovian open systems
arise in many important situations such as the strongly coupled
system-environment, structured environment, time-delayed external control etc. \cite{Hu0}.
Intuitively speaking, while the Markov evolution is an irreversible process, in the case of non-Markovian dynamics,
the system energy (or phase information) dissipated into the environment may come back to the system in a finite time \cite{Breuer2009}.
For open systems immersed in a bosonic environment, apart from the master equation and path
integral approaches \cite{Feynman-Vernon,Leggett,Hu1}, a versatile stochastic formalism for
quantum open system dynamics was developed to provide a powerful tool in studying
quantum systems in a non-Markovian regime
\cite{Diosi1,Diosi2,Strunz,YDGS99,Jing,Strunz-Yu2004,Yu2004}.
Such a stochastic pure state approach has several advantages in numerical simulations, perturbation
and the derivation of the corresponding master equations.
For a Markov environment (bosons or fermions), both the quantum state diffusion equations
\cite{Gisin,Plenio} and Lindblad master equations \cite{Quantum Noise} can be used to
describe quantum dynamics of the system of interest. Several important theoretical tools
in dealing with nonequilibrium fermionic systems have been developed including nonequilibrium
Green's function (NEGF) theory, fermionic path integral etc \cite{xx,Meir,Meir2,Jauho,Wingreen,Fransson,Zwanzig}. However, for a generic
non-Markovian fermionic environment where the system-environment coupling
is not weak or the environment cannot be treated as a broadband reservoir \cite{mmm,Meir,Meir2,Jauho,Wingreen,Fransson,Zwanzig,Ting,ZhangDQD,Wiseman},
establishing a stochastic theory analogous to the non-Markovian quantum state diffusion equation \cite{Strunz}
is a long standing problem.
The purpose of this paper is to develop a general
non-Markovian stochastic theory of electronic systems coupled to a fermionic environment.
The theory developed here is versatile enough to deal with a wide spectrum of open system problems ranging from a
``small" environment (one fermion or a few fermions) to a large environment consisting of an infinite number of fermions
irrespective of the details of spectral distribution of the fermionic environment.
As an illustration of the power of the stochastic approach developed here, we derive several exact
master equations governing the reduced density operators of the electronic
systems coupled to vacuum and finite-temperature reservoirs.
The paper is organized as follows. In Sec.~\ref{Model}, we establish the fermionic stochastic Schr\"{o}dinger equation (SSE) for a class of open system models
consisting of an electronic system coupled to a fermionic reservoir. In particular, we show how to derive the time-local SSE. We introduce a new type of Novikov
theorem emerged from the Grassmann noise. We demonstrate the derivation of the corresponding exact master equation from the SSE.
In Sec.~\ref{example1}, a many-fermion model is considered. We show explicitly that the exact fermionic SSE and the corresponding master equation can be
established. In Sec.~\ref{example2}, we establish the finite temperature fermionic SSE through a Bogoliubov transformation, and we provide an
explicit construction of the exact time-local master equation as well as the so-called $\hat{Q}$ operator for this model. Then, in Sec.~\ref{example3}, the finite temperature model is generalized into a more realistic case consisting of double quantum dots coupled to two fermionic reservoirs (source and drain).
Both the time-local fermionic SSE and the exact master equation are derived. The numerical
simulations of the double quantum dots based on the exact master equation are provided. Finally,
we conclude the paper in Sec.\ref{conclusion}. Some details about the Grassmann noise, the fermionic SSE, the $\hat{Q}$ operator, a proof of the Novikov theorems and the Heisenberg operator approach are left to Appendix.
\section{Fermionic stochastic Schr\"{o}dinger equation and non-Markovian master equation}
\label{Model}
\subsection{Model}
To begin with, we consider a simplified model involving an electronic system in contact
with a single fermionic reservoir, where the system anti-commutes with the bath
\cite{XinyuFB}. The generalization to a more physically interesting model with
two reservoirs (\emph{e.g.}, source and drain) can be established in a similar way.
With necessary modifications, the formalism is versatile enough to deal with stochastic gate
potentials and nonlinear couplings. The total Hamiltonian for the system plus environment may
be written as \cite{xx},
\begin{equation}
\hat{H}_{\mathrm{tot}}=\hat{H}_{\mathrm{S}}+\hat{H}_{\mathrm{R}}+\hat
{H}_{\mathrm{I}}, \label{Htot}%
\end{equation}
where $\hat{H}_{\mathrm{S}}$ is the Hamiltonian of the electronic system in
the absence of the environment, $\hat{H}_{\mathrm{R}}$ is the Hamiltonian for a
fermionic reservoir; $\hat{H}_{\mathrm{R}}=\sum_{\mathbf{k\alpha}}\hbar
\omega_{\mathbf{k}}\hat{b}_{\mathbf{k\alpha}}^{\dag}\hat{b}_{\mathbf{k\alpha}%
}$ where $\hat{b}_{\mathbf{k\alpha}}^{\dag},\hat{b}_{\mathbf{k\alpha}}$ are
the fermionic creation and annihilation operators $\{\hat{b}_{\mathbf{k\alpha
}},\hat{b}_{\mathbf{k^{\prime}\alpha^{\prime}}}^{\dag}\}=\delta
_{\mathbf{k\alpha,k^{\prime}\alpha^{\prime}}}$, and the interaction
Hamiltonian $\hat{H}_{\mathrm{I}}$ is given by
\begin{equation}
\hat{H}_{\mathrm{I}}=\hbar\sum_{\mathbf{k\alpha}}(t_{\mathbf{k\alpha}}\hat{L}^\dagger \hat{b}%
_{\mathbf{k\alpha}} + t_{\mathbf{k\alpha}}\hat{b}^\dagger_{\mathbf{k\alpha}}%
\hat{L}), \label{Hint}%
\end{equation}
where $\hat{L}$ is the system coupling operator and $t_{\mathbf{k\alpha}}$ are
the coupling constants. Note that $\hat{H}_{\mathrm{S}}$ is an arbitrary
Hamiltonian operator that may contain interaction terms for the system
particles (\emph{e.g.}, Coulomb interactions between two electrons). The
coupling operator $\hat{L}$ may, in general, be represented by a set of
fermionic operators which are coupled to all the participating external agents
such as the source and drain reservoirs.
The purpose of this paper is to develop a systematic stochastic theory for
the models described by Eq. (\ref{Htot}) and (\ref{Hint}), those are relevant to quantum open systems,
non-equilibrium statistical mechanics, path-integral theory and quantum
devices based on quantum dots and mesoscopic electronics
\cite{Feynman-Vernon,Leggett,Hu1,Hu0,Wiseman}.
In the framework of the stochastic Schr\"{o}dinger equation (SSE) for a bosonic bath, the state of the open
quantum system is described by a stochastic pure state, which is generated by
a complex Gaussian stochastic process. For the fermionic environment considered in this paper,
similar to the fermionic
path integral, the fermionic stochastic theory will involve a Grassmann
Gaussian stochastic process. Remarkably, we show that the reduced density matrix of the system of
interest can be reconstructed from the pure states by taking the statistical
mean over the Grassmann noise \cite{XinyuFB}. As such, in principle, the exact master
equation governing the reduced density operator for the open system can be recovered
from the SSE, as illustrated by several physically interesting models below.
\subsection{Fermionic stochastic Schr\"{o}dinger equation}
In this subsection, we will establish the fermionic stochastic Schr\"{o}dinger equation (SSE) for an open electronic
system coupled to a fermionic reservoir. Consider the model described by the total Hamiltonian in
Eq.~(\ref{Htot}), in the interaction picture with respect to the fermionic
reservoir, it becomes (setting $\hbar=1$),
\begin {eqnarray}
\hat{H}_{\mathrm{tot}}^{I}(t)=\hat{H}_{\mathrm{S}}+(\sum_{j}t_{j}\hat{L}^{\dagger}\hat{b}_{j}e^{-{\rm i}\omega_{j}t}+\mathrm{h.c.}),
\end {eqnarray}
here the subscript $\mathbf{k\alpha}$
is suppressed as $j$. In order to trace out the environmental variables, we
introduce the fermionic coherent states $|\bm{\xi}\rangle$ which is defined as
\begin {eqnarray}
|\bm{ \xi}\rangle \equiv \prod_k (1-\xi_{k} \hat{b}_k^{\dagger})|{\rm vac}\rangle_{\rm R}.
\end {eqnarray}
And this state satisfies $\hat{b}_{j}|\bm {\xi}\rangle = \xi_{j}|\bm {\xi}\rangle$. Here, $\xi_{j}$ is a
Grassmann variable, satisfying $\{\xi_i,\xi_j\}=\{\xi_i^*,\xi_j\}=0$, and $\{\xi_i,\hat
{b}_j\}=\{\xi_i,\hat{b}_j^{\dag}\}=0$ \cite{Berezin,Glauber1999}. As shown below,
the derivations and results for fermionic SSE are more complex than the
bosonic case. Using the fermionic coherent state, we define
\begin {eqnarray}
|\psi_{t}(\bm{\xi}^{\ast})\rangle \equiv \langle\bm{\xi}|{\rm e}^{{\rm i}\hat{H}_{\mathrm{R}}t} {\rm e}^{-{\rm i}\hat{H}_{\mathrm{tot}}t}|\psi_{\mathrm{tot}}(0)\rangle, \label{psit}
\end {eqnarray}
where $|\psi_{\rm tot}(0)\rangle$ is the total initial state of both
system and bath, and we assume the bath is initially prepared in vacuum
state, \emph{i.e.} $|\psi_{\rm tot}(0)\rangle=|\psi_{\rm S}(0)\rangle\wedge|\mathrm{vac}\rangle_{\rm R}$ (``$\wedge$'' stands for an antisymmetrized wave function). The thermal state case will be discussed
later. Taking the time derivative on the both sides of Eq.~(\ref{psit}), one obtains%
\begin{align}
& \partial_t|\psi_{t}(\bm{\xi}^*)\rangle \nonumber \\
=& -{\rm i}\langle{\bm\xi}|\hat{H}_{\rm tot}^{I}(t)|\psi_{\rm tot}^I(t)\rangle \nonumber \\
=& -{\rm i} [\hat{H}_{\rm S} + \sum_k (t_k\hat{L}\+\overrightarrow{\partial}_{\xi_{k}^*}{\rm e}^{-{\rm i}\omega_{k}t} + t_k\xi_k^*\hat{L}{\rm e}^{{\rm i}\omega_{k}t})]|\psi_{t}(\bm{\xi}^*)\rangle \nonumber \\
=& (-{\rm i}\hat{H}_{\rm S}-\hat{L}\+\int_0^t {\rm d} s\,\sum_k\frac{\partial\xi_t}{\partial\xi_k}\frac{\partial\xi_s^*}{\partial\xi_k^*}\overrightarrow{\delta}_{\xi_s^*}-\hat{L}\xi_t^*)|\psi_t(\bm{\xi}^*)\rangle \nonumber \\
=& [-{\rm i}\hat{H}_{\rm S} -\hat{L}\xi_t^* -\hat{L}\+ \int_0^t {\rm d} s\, K(t,s) \overrightarrow{\delta}_{\xi_{s}^*}]|\psi_t(\bm{\xi}^*)\rangle,\label{SSE}%
\end{align}
where $\xi_t^{\ast} \equiv-{\rm i}\sum_{k}t_{k} {\rm e}^{{\rm i}\omega_{k}t}\xi_k^{\ast}$
is the Grassmann Gaussian noise, $\overrightarrow{\delta}_{\xi_s^{\ast}}$ is the left functional derivative
with respect to $\xi_s^*$, and the explicit form of function $K(t,s)$ is
\begin {eqnarray}
K(t,s) \equiv \sum_{k}\frac{\partial \xi_t}{\partial \xi_k}\frac{\partial
\xi_s^{\ast}}{\partial \xi_k^{\ast}}=\sum_{k}|t_{k}|^{2} {\rm e}^{-{\rm i}\omega_{k}(t-s)}.
\end {eqnarray}
The Grassmann Gaussian process is defined by
\begin {eqnarray}
& & \mathcal{M} [\xi_k ]\equiv \int \prod_k {\rm d}\xi_{k}^{\ast}\cdot {\rm d}\xi_{k}\, {\rm e}^{-\xi_{k}^{\ast}\cdot \xi_{k}} \, \xi_k =0, \nonumber \\
& & \mathcal{M} [\xi_k \xi_k^*]\equiv \int \prod_k {\rm d}\xi_{k}^{\ast} \cdot {\rm d}\xi_{k}\, {\rm e}^{-\xi_{k}^{\ast}\cdot \xi_{k}}\, \xi_k \xi_k^*=1,
\end {eqnarray}
where ``$\mathcal{M}$'' stands for the statistical mean over the random Grassmann
variables ``$\xi_k$''. It is easy to check that the mean and the correlation function are given by: $\mathcal{M} [\xi_t ]= \mathcal{M} [\xi_t^*]=0$
and $ \mathcal{M} [\xi_t \xi_s^*]=K(t,s)$, respectively.
Note that our fermionic SSE (\ref{SSE}) is applicable to an arbitrary correlation function including both Markov and
non-Markovian environments. Unlike the complex Gaussian noise used in bosonic case, the Grassmann Gaussian
noise is a non-commutative noise at different times reflecting a fundamentally distinctive feature arising from the
fermionic environment. Our fermionic stochastic Schr\"{o}dinger equation is expected to have a close connection
with the fermionic path integral as shown in the case of bosonic case \cite{Strunz1996,Wufu}.
The fermionic SSE (\ref{SSE}) can be written in a more compact form,
\begin{equation}
\partial_{t}|\psi_{t}(\bm{\xi}^*)\rangle=-{\rm i}\hat{H}_{\mathrm{eff}}%
|\psi_{t}(\bm{\xi}^{\ast})\rangle, \label{SSE1}%
\end{equation}
where the effective Hamiltonian is given by,%
\begin{equation}
\hat{H}_{\mathrm{\mathrm{eff}}}=\hat{H}_{\mathrm{S}}-{\rm i}\hat{L}\xi_t%
^{\ast}-{\rm i}\hat{L}^{\dagger}\int_{0}^{t}{\rm d} s\, K(t,s)\overrightarrow{\delta
}_{\xi_s^{\ast}}, \label{SSE2}%
\end{equation}
Eq.~(\ref{SSE}) or (\ref{SSE1}) may serve as a fundamental equation for open fermionic
systems coupled to a fermionic environment. Our stochastic method will provide
a new insight into the individual physical process described by the SSE.
Although the stochastic method is fundamentally equivalent to density matrix or NEGF formalism,
it can be advantageous over the density operator and NEGF in several interesting cases such as fast tracking of information
for quantum coherence and entanglement \cite{Corn}. Moreover, it is known that perturbative master equations typically
lead to unphysical effect such as violation of positivity, however, the stochastic equation can yield a
systematic perturbative method that can be implemented numerically \cite{YDGS99}.
As an illustration of the power of the stochastic approach developed here, we derive several exact
master equations governing the reduced density operators of the electronic
systems coupled to vacuum and finite-temperature reservoirs.
Crucial to the practical applications of Eq.~(\ref{SSE1}) is to express the Grassmann functional
derivative under the memory integral in Eq. (\ref{SSE2}) in terms of system
operators \cite{Diosi2,Strunz,YDGS99,Jing,Strunz-Yu2004,Yu2004}. In order to
calculate the functional derivative in the stochastic Schr\"{o}dinger
equation, we introduce an operator called the fermionic $\hat{Q}$ operator
(similar to the $\hat{O}$ operator in bosonic case) as
\begin{equation}
\hat{Q}(t,s,\bm{\xi}^{\ast})|\psi_{t}(\bm{\xi}^{\ast})\rangle \equiv \overrightarrow
{\delta}_{\xi_{s}^{\ast}}|\psi_{t}(\bm{\xi}^{\ast})\rangle.
\end{equation}
With this definition, the effective
Hamiltonian in Eq.~(\ref{SSE2}) can be written as,%
\begin{equation}
\hat{H}_{\mathrm{eff}}=\hat{H}_{\mathrm{S}}-{\rm i}\hat{L} \xi_%
t^{\ast}-{\rm i}\hat{L}^{\dagger}\bar{Q}, \label{SSE3}%
\end{equation}
where
\begin{equation}
\bar{Q}(t,\bm{\xi}^{\ast}) \equiv \int_{0}^{t}{\rm d} s\, K(t,s)\hat{Q}%
(t,s,\bm{\xi}^{\ast}).
\end{equation}
The fermionic stochastic Schr\"{o}dinger equation (\ref{SSE1}) is derived directly from the
microscopic Hamiltonian without any approximation. It should be emphasized
that the system Hamiltonian $\hat{H}_{\mathrm{S}}$ and the coupling operator
$\hat{L}$ are entirely general. The evolution of the electronic system is
governed by the anti-commutative stochastic differential equation (\ref{SSE1}). Although the
mathematical form of the equation (\ref{SSE1}) is similar to the non-Markovian quantum
state diffusion equation in the bosonic case, the behavior
of the fermionic SSE can be very different due to the fermionic features of the environment
\cite{XinyuFB}. Moreover, the Grassmann stochastic process has brought about
several new features in dealing with the fermionic SSE such as a new type of
Novikov theorem (see Appendix D).
From the consistency condition for the fermionic SSE
\begin {eqnarray}
\overrightarrow{\delta}_{\xi_{s}^{\ast}}\partial_{t}|\psi_{t}(\bm{\xi}^{\ast})\rangle
=\partial_{t}\overrightarrow{\delta}_{\xi_{s}^{\ast}}|\psi_{t}(\bm{\xi}^{\ast})\rangle,
\end {eqnarray}
the fermionic $\hat{Q}$ operator can be shown to satisfy the
following equation (see Appendix B),
\begin{equation}
\partial_{t}\hat{Q}=-{\rm i}[\hat{H}_{\mathrm{\mathrm{eff}}},\hat{Q}%
]-{\rm i}\overrightarrow{\delta}_{\xi_{s}^{\ast}}(\hat{H}_{\mathrm{eff}}%
-\hat{H}_{\mathrm{S}}). \label{EQO}%
\end{equation}
Once the fermionic $\hat{Q}$ operator is determined, the SSE can be cast into
a time-local stochastic equation with the Grassmann type noise.
\subsection{Non-Markovian master equation}
Note that the
reduced density operator for the open fermionic system can be obtained by
taking the statistical average over all the Grassmann quantum trajectories
which are the solutions to the SSE (\ref{SSE1}),
\begin{align}
\hat{\rho}_{r} & =\int\prod_k {\rm d}\xi_{k}^{\ast}\cdot {\rm d}\xi_{k}%
{\rm e}^{-\xi_{k}^{\ast}\cdot\xi_{k}} \hat{P},\\
\hat{P} & =|\psi_{t}(\bm{\xi}^{\ast})\rangle\langle\psi
_{t}(-\bm{\xi})|, \label{P}%
\end{align}
and in the rest of the paper, we will use the shorthand notations $\mathcal{D}_{g}[\bm{\xi}] \equiv \prod_k {\rm d}\xi_{k}^{\ast}\cdot {\rm d}\xi_{k}%
{\rm e}^{-\xi_{k}^{\ast}\cdot\xi_{k}}$ and $|\psi_{t}\rangle\equiv|\psi_{t}(\bm{\xi}^{\ast})\rangle$, $|\psi_{t}^-\rangle\equiv|\psi_{t}(\bm{-\xi}^{\ast})\rangle$
to represent the Grassmann Gaussian measure and the quantum trajectories, respectively (for more details, see Appendix C). Then taking the time
derivative on
\begin {eqnarray}
\hat{\rho}_{r} & =\int\mathcal{D}_{g}[\bm{\xi}]|\psi_{t}\rangle \langle\psi_{t}^-|,
\end {eqnarray}
and substituting the fermionic SSE (\ref{SSE1}) into it, we can get the equation of motion for the reduced density operator,
\begin {eqnarray}
\partial_t \hat{\rho}_r &=& -{\rm i}[\hat{H}_{\rm S} , \hat{\rho}_r] + \int \mathcal D_g [\bm\xi] \{ (-\hat{L}\+ \bar{Q} - \hat{L}\xi^*_t )\hat{P} \nonumber \\
& & +\hat{P}(-\bar{Q}_- \+ \hat{L} + \xi_t\hat{L}\+ ) \}, \label{rho3}
\end {eqnarray}
where $\bar{Q}_-$ is a short hand notation of $\bar{Q}(-{\bm \xi})$.
In order to calculate the terms $\int\mathcal D_g [\bm\xi]\,\xi_t^*\hat{P}$, we need to prove an extension of Novikov theorem
for Grassmann Gaussian noise (for the bosonic case, see Ref.~\cite{YDGS99}). In the fermionic case, we have
two kinds of Novikov-type theorems corresponding to the left and right functional derivatives.
\\
Left type:
\begin{equation}
\int\mathcal D_g [\bm\xi]\, \xi^*_t \hat{P} = -\int^t_0 {\rm d} s\, \sum_k \frac{\partial\xi^*_t}{\partial\xi^*_k}\frac{\partial\xi_s}{\partial\xi_k} \int\mathcal D_g [\bm\xi]\, \hat{P} \overleftarrow{\delta}_{\xi_s};
\end{equation}
\\
Right type:
\begin{equation}
\int\mathcal D_g [\bm\xi]\, \hat{P}\xi_t = -\int_0^t {\rm d} s\, \sum_k \frac{\partial\xi_t}{\partial\xi_k}\frac{\partial\xi_s^*}{\partial\xi_k^*} \int\mathcal D_g [\bm\xi]\,\overrightarrow{\delta}_{\xi_s^*}\hat{P};
\end{equation}
where $\overleftarrow{\delta}_{\xi_s^*}$ ($\overrightarrow{\delta
}_{\xi_s^*}$) is the right (left) functional derivative with respect to $\xi_s^*$.
Applying the Novikov theorem for the Grassmann noise to Eq.~(\ref{rho3}), the formal exact master equation
can be simplified into a compact form
\begin{equation}
\partial_t\hat{\rho}_r = -{\rm i}[\hat{H}_{\rm S},\hat{\rho}_r] + \{\int\mathcal{D}_g [\bm{\xi}][\bar{Q}\hat{P},\hat{L}\+] + \rm{h.c.}\}, \label{MEQ}
\end{equation}
Similar to the derivation for the SSE, the above derivation for non-Markovian master equation is only valid for the vacuum reservoir,
in which we assume the system and environment are initially in
the state $|\psi_{\mathrm{tot}}(0)\rangle=|\psi(0)\rangle_{\rm S}\wedge
|\mathrm{vac}\rangle_{\rm R}$. However, the finite temperature case can be easily incorporated
in our approach, as shown in the examples below.
In a special case where the fermionic $\hat{Q}$ operator is independent of
noise, the master equation Eq.~(\ref{MEQ}) takes a very simple form,%
\begin{equation}
\partial_{t}\hat{\rho}_{r}=-{\rm i}[\hat{H}_{\mathrm{S}},\hat{\rho}%
_{r}] + \{ [\bar{Q}\hat{\rho_r},\hat{L}^{\dagger}]+\mathrm{h.c.} \}. \label{MEQOfree}%
\end{equation}
If we take the Markovian correlation function $K(t,s)=\Gamma\delta(t,s)$, then
$\bar{Q}$ reduces to $\bar{Q}=\Gamma\hat{L}/2$, and the master equation
will reduce to the standard Lindblad Markov master equation,
\begin {eqnarray}
\partial_{t}\hat{\rho}_{r}=-{\rm i}[\hat{H}_{\mathrm{S}},\hat{\rho}_{r}] + \{ \Gamma/2[\hat{L}\hat{\rho}_{r},\hat{L}^{\dagger}]+\mathrm{h.c.} \}.
\end {eqnarray}
\section{Many-fermion system coupled to a vacuum fermionic reservoir}
\label{example1}
The first example considers a many-fermion open system coupled to a fermionic
bath initially in the vacuum state. The total Hamiltonian is
\begin{eqnarray}
\hat{H}_{\mathrm{tot}} &=& \sum_{j=1}^{N}\Omega_{j}\hat{d}_{j}^{\dagger}\hat{d}_{j}+\sum_{k}\omega
_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k} \nonumber\\
&&+\sum_{j,k}t_{j,k}\hat{d}_{j}^{\dagger}\hat{b}_{k}+t_{j,k}\hat{b}_{k}^{\dagger}\hat{d}_{j},
\end{eqnarray}
where $\hat{d}_{j}$
and $\hat{d}_{j}^{\dagger}$ ($j=1$ to $N$) are the annihilation and
creation operators of the fermions in the system, and $\hat{b}_{k}$ and $\hat
{b}_{k}^{\dagger}$ are the fermionic annihilation and creation operators for
the bath. Here $\hat{H}_{\mathrm{S}}=\sum_{j}\Omega_{j}\hat{d}_{j}^{\dagger
}\hat{d}_{j}$, and the coupling operator is $\hat{L}=\sum_{j}\hat{d}_{j}$.
Then, the fermionic SSE is given by%
\begin{equation}
{\rm i}\partial_{t}|\psi_{t}\rangle=(\sum_{j}\Omega_{j}\hat{d}_{j}^{\dagger}\hat
{d}_{j}-{\rm i}\sum_{j}\hat{d}_{j} \xi_t^{\ast}-{\rm i}\sum_{j}\hat{d}%
_{j}^{\dagger}\bar{Q})|\psi_{t}\rangle,
\end{equation}
where the $\hat{Q}$ operator is given by $\hat{Q}=\sum_{j}f_{j}(t,s)\hat
{d}_{j}.$ Substituting the $\hat{Q}$ operator into the Eq.~(\ref{EQO}), we
obtain the differential equations for the time-dependent coefficients
$f_{j}(t,s)$ as
\begin {eqnarray}
\frac{\partial}{\partial t}f_{j}(t,s)= {\rm i}\Omega_{j}f_{j}(t,s)+\sum_{k=1}^{N}F_{k}(t)f_{j}(t,s)
\end {eqnarray}
with the final condition
$f_{j}(t,s=t)=1$. $F_{j}(t)$ is defined as $F_{j}(t)=\int_{0}^{t} {\rm d} s\,%
K(t,s)f_{j}(t,s)$. Thus, the exact $\hat{Q}$ operator is fully determined.
Then one immediately obtains the exact master equation,
\begin{equation}
\partial_{t}\hat{\rho}_r =-{\rm i}[\hat{H}_{\mathrm{S}},\hat{\rho}_r]+\{[(\sum_{j=1}%
^{N}F_{j}(t)\hat{d}_{j})\hat{\rho}_r,\sum_{j=1}^{N}\hat{d}_{j}^{\dagger
}]+\mathrm{h.c.}\}.
\end{equation}
As a special case of interest, we consider both the system and the reservoir just containing one mode with equal frequencies (resonant condition).
Then the exact master equation reduces to
\begin{equation}
\partial_{t}\hat{\rho}_r =-{\rm i}\Omega [\hat{d}^\dagger \hat{d},\hat{\rho}_r]+\{[g \tan(g t)\hat{d}\hat{\rho}_r,\hat{d}^{\dagger
}]+\mathrm{h.c.}\},
\end{equation}
where $g$ is the coupling constant. This case is an extreme case of the non-Markovian evolution with the non-Markovianity being infinity \cite{Breuer2009}.
In general, the negative values of the coefficients dictate the features of the
non-Markovian evolution of an open quantum system. A more complete studies on non-Markovian fermonic systems will be conducted in the future
publications \cite{New}.
\section{Single quantum dot (QD) coupled to a finite-temperature fermionic bath}
\label{example2}
To illustrate how to establish a fermionic SSE for the case of finite temperature reservoirs, for simplicity, we use the example of
a single QD coupled to a finite-temperature fermionic bath. As stated before, a realistic generalization to two finite temperature reservoirs is
straightforward. In the standard Hamiltonian in Eq.~(\ref{Htot}) and Eq.~(\ref{Hint}), we choose $\hat
{H}_{\mathrm{S}}=\omega_{0}\hat{d}^{\dagger}\hat{d}$ and $\hat{L}=\hat{d}$, then the total Hamiltonian is now given by,
\begin{equation}
\hat{H}_{\rm tot} =\omega_{0}\hat{d}\+\hat{d}+\sum_{k}t_{k}(\hat{d}\+\hat{b}_{k}
+\hat{b}_{k}\+\hat{d})+\sum_{k}\omega_{k}\hat{b}_{k}\+\hat{b}_{k}.
\end{equation}
It is known that the finite temperature model can be transformed into the
vacuum case by introducing a fictitious reservoir \textquotedblleft$c$" (a ``hole" system with negative energies), which is decoupled from
the system and reservoir ``$b$'', so the quantum dynamics will not be affected \cite{Strunz,Yu2004}.
With the fictitious reservoir $\hat{H}_{\mathrm{R}}^{fic}=\sum_{k}(-\omega_{k})\hat{c}_{k}^{\dagger}\hat{c}_{k}$, the total
Hamiltonian may be written as,
\begin {eqnarray}
\hat{H}_{\rm tot}' & =\omega_{0}\hat{d}\+\hat{d}+\sum_k t_k(\hat{d}\+\hat{b}_k + \hat{b}_k\+\hat{d}) \nonumber \\
& + \sum_k\omega_k(\hat{b}_k\+\hat{b}_k-\hat{c}_k\+\hat{c}_k). \label{ficH}
\end {eqnarray}
By properly choosing the parameters of bath
\textquotedblleft$c$", the combined bath \textquotedblleft$b+c$" can be
initially prepared in a pure state which is equivalent to the vacuum corresponding to bath \textquotedblleft$b'+c'$".
The relation between the original bath and the transformed baths are given by the following Bogoliubov transformations,
{\small
\begin{align}
\hat{b}_{k}'=\sqrt{1-\bar{n}_{k}}\,\hat{b}_{k}-\sqrt{\bar{n}_{k}}\,\hat{c}_{k}\+, & \quad \hat{b}_{k}'{}\+=\sqrt{1-\bar{n}_{k}}\,\hat{b}_{k}\+-\sqrt{\bar{n}_{k}}\,\hat{c}_{k},\nonumber\\
\hat{c}_{k}'=\sqrt{1-\bar{n}_{k}}\,\hat{c}_{k}+\sqrt{\bar{n}_{k}}\,\hat{b}_{k}\+, & \quad \hat{c}_{k}'{}\+=\sqrt{1-\bar{n}_{k}}\,\hat{c}_{k}\++\sqrt{\bar{n}_{k}}\,\hat{b}_{k};\nonumber\\
\hat{b}_{k}=\sqrt{1-\bar{n}_{k}}\,\hat{b}_{k}'+\sqrt{\bar{n}_{k}}\,\hat{c}_{k}'{}\+, & \quad \hat{b}_{k}\+=\sqrt{1-\bar{n}_{k}}\,\hat{b}_{k}'{}\++\sqrt{\bar{n}_{k}}\,\hat{c}_{k}^{\prime},\nonumber\\
\hat{c}_{k}=\sqrt{1-\bar{n}_{k}}\,\hat{c}_{k}'-\sqrt{\bar{n}_{k}}\,\hat{b}_{k}'{}\+, & \quad \hat{c}_{k}\+=\sqrt{1-\bar{n}_{k}}\,\hat{c}_{k}'{}\+-\sqrt{\bar{n}_{k}}\,\hat{b}_{k}';
\end{align}
}
where $\bar{n}_{k}=\frac{1}{1+{\rm e}^{\beta(\hbar\omega_{k}-\mu)}}$, and $\mu$ is
the chemical potential.
After tracing out the fictitious bath \textquotedblleft$c$" on the effective vacuum,
the real bath ``$b$" is then prepared in the initial thermal state, \emph{i.e.},
\begin {eqnarray}
\hat{\rho}_{b}(0)=\mathrm{Tr}_{c}[\hat{\rho}_{bc}(0)]={\exp[{-\frac{\hat{H}_{b}-\mu\hat{N}_b}{k_{B}T}}]}/{Z}
\end {eqnarray}
where $Z=\mathrm{Tr}\exp[{-\frac{\hat{H}_{b}-\mu\hat{N}_b}{k_{B}T}]}$ is the
partition function. In such a way, the finite temperature model can be transformed
into an effective vacuum case whose SSE has already been established in the previous sections.
Now we define the new coupling strength,
\begin{equation*}
g_{k}\equiv\sqrt{1-\bar{n}_{k}}t_{k},\quad f_{k}\equiv\sqrt{\bar{n}_{k}}t_{k},
\end{equation*}
then the total Hamiltonian takes the following form,
\begin{align}
\hat{H}_{\rm tot}' & =\omega_0\hat{d}\+\hat{d} + \sum_k (g_k\hat{d}\+\hat{b}'_k + f_k\hat{d}\+\hat{c}'_k{}\+ + g_k\hat{b}'_k{}\+\hat{d} \nonumber\\
& + f_k\hat{c}'_k\hat{d}) + \sum_k\omega_k (\hat{b}'_k{}\+\hat{b}'_k - \hat{c}'_k{}\+\hat{c}_k').
\end{align}
The coherent states for the two baths can be defined as $|\bm{\xi}\rangle\equiv\prod_{k,l}(1-\xi_{b',k}\hat{b}%
_{k}'{}\+)(1-\xi_{c',l}\hat{c}_{l}^{\prime\dagger
})|\mathrm{vac}\rangle_{\rm R}$. Thus, two independent Grassmann noises are defined,
\begin {eqnarray}
& & \xi_{b',t}^* \equiv -{\rm i}\sum_k g_k {\rm e}^{{\rm i}\omega_ k t}\xi_{b',k}^*, \nonumber \\
& & \xi_{c',t}^* \equiv -{\rm i}\sum_k f_k {\rm e}^{-{\rm i}\omega_k t}\xi_{c',k}^*.
\end {eqnarray}
Then, the corresponding $\hat{Q}$ ($\bar{Q}$) operators and correlation
functions are defined as%
\begin{eqnarray}
& & \hat{Q}_{b'}(t,s,\bm{\xi}^*)|\psi_{t}\rangle \equiv \overrightarrow{\delta}_{\xi_{b',s}^*}|\psi_{t}\rangle, \nonumber \\
& & \hat{Q}_{c'}(t,s,\bm{\xi}^*)|\psi_{t}\rangle \equiv \overrightarrow{\delta}_{\xi_{c',s}^*}|\psi_{t}\rangle, \nonumber \\
& & \bar{Q}_{b'} \equiv \int_0^t {\rm d} s\, K_{b'}(t,s)\hat{Q}_{b'}(t,s,\bm{\xi}^*), \nonumber \\
& & \bar{Q}_{c'} \equiv \int_0^t {\rm d} s\, K_{c'}(t,s)\hat{Q}_{c'}(t,s,\bm{\xi}^*), \nonumber \\
& & K_{b'}(t,s) \equiv \sum_k g_k^2\, {\rm e}^{-{\rm i}\omega_k (t-s)}, \nonumber \\
& & K_{c'}(t,s) \equiv \sum_k f_k^2\, {\rm e}^{{\rm i}\omega_k (t-s)}.
\end{eqnarray}
With the above definitions, the SSE governing $|\psi_{t}\rangle$ can be written as
\begin{equation}
{\rm i}\partial_{t}|\psi_{t}\rangle=[\hat{H}_{\rm S} +{\rm i}\hat{d}\+(\xi_{c',t}^* - \bar{Q}_{b'}) +{\rm i}\hat{d}(\bar{Q}_{c'} - \xi_{b',t}^*)]|\psi_t\rangle.
\end{equation}
Following the procedure of deriving master equation from the general SSE for the vacuum fermionic bath,
we obtain the formal exact master equation of the case of finite temperature,
\begin{align}
\partial_t\hat{\rho}_r & = -{\rm i}[\hat{H}_{\rm S},\hat{\rho}_r] + \int\mathcal{D}_g [\bm{\xi}]([\bar{Q}_{b'}\hat{P},\hat{d}\+] \nonumber \\
& +[\hat{d},\bar{Q}_{c'}\hat{P}] + \rm{h.c.}).
\end{align}
For the finite temperature case, the $\hat{Q}$ operators are not noise-free, hence we need to use the Heisenberg approach (see Appendix E)
to derive the corresponding convolutionless master equation (an example of using Heisenberg approach in
the case of bosonic bath can be found in Ref.~\cite{Strunz-Yu2004,Yu2004}).
The convolutionless master equation takes the following form,
\begin{equation}
\label{MED1d}
\partial_{t}\hat{\rho}_{r}=-{\rm i}[\hat{H}_{\mathrm{S}},\hat{\rho}_{r}%
]\newline+\{F_{1}(t)[\hat{d}\hat{\rho}_{r},\hat{d}^{\dagger}]+F_{2}%
(t)[\hat{\rho}_{r}\hat{d},\hat{d}^{\dagger}]+\text{\textrm{h.c}.}\}
\end{equation}
where the time-dependent coefficients $F_{i}(t)$ ($i=1,2$) are
\begin{align}
& F_i(t)=\int_0^t {\rm d} s\,[K_{b'}(t,s)u^{b'}_i(t,s) - K_{c'}^*(t,s)u^{c'}_i(t,s)],
\label{F1F2}%
\end{align}
and $u^\mu_i$ ($i=1,2$, $\mu=b'$ or $c'$) satisfy the
following equations
\begin {widetext}
\begin {align}
\partial_s u^{b'}_i(t,s) & = -{\rm i}\omega_0u^{b'}_i(t,s)+ [\int_s^t {\rm d} s'\, K_{c'}(s',s) -\int_0^s {\rm d} s'\, K_{b'}(s,s') ] u^{b'}_i(t,s') + \int_0^t {\rm d} s'\, K_{c'}(s',s)u^{c'}_i(t,s'), \nonumber \\
\partial_s u^{c'}_i(t,s) & = -{\rm i}\omega_0u^{c'}_i(t,s)+ [\int_s^t {\rm d} s'\, K_{b'}^*(s',s) -\int_0^s {\rm d} s'\, K_{c'}^*(s,s') ] u^{c'}_i(t,s') + \int_0^t {\rm d} s'\, K_{b'}^*(s',s)u^{b'}_i(t,s'), \label{ubc}%
\end {align}
\end {widetext}
with the final conditions: $u^{b'}_1(t,s=t) = u^{c'}_2(t,s=t) = 1$, and $u^{b'}_2(t,s=t) = u^{c'}_1(t,s=t) = 0$.
\begin{figure}[ptb]
\begin{center}
\includegraphics
[
trim=0.000000in 0.000000in 0.000000in -0.145355in,
height=2.8106in,
width=3.2621in
]
{SD_Coeff.eps}
\end{center}
\caption{Time evolution of the coefficients for the single quantum dot in a finite
temperature bath with different bandwidths. The real (imaginary) part of the
coefficients $F_{1}$ ($F_{2}$) are plotted separately. The parameters are
$T=100{\rm mK}$, $\mu=2\times10^{-5}{\rm eV}$, $\omega_{0}=3\times10^{-5}{\rm eV}$.}%
\label{sd_coeff}%
\end{figure}
\begin{figure}[ptb]
\begin{center}
\includegraphics[
trim=0.000000in 0.000000in -0.055938in 0.000000in,
height=1.6in,
width=3.2621in
]{sd.eps}
\end{center}
\caption{Time evolution of $\rho_{11}$ for the single quantum dot with different parameters of the bath.
(a) is plotted with the different temperatures $T$, and (b) is plotted with the different
bandwidths of the spectral density. The other parameters are $\mu
=2\times10^{-5}{\rm eV}$, $\omega_{0}=3\times10^{-5}{\rm eV}$.}%
\label{sd}%
\end{figure}
Generally speaking, the time-dependent coefficients of the exact master equation may evolve
in a complicated way (for example, taking negative values
as a typical manifestation of non-Markovian behaviors, see \cite{Breuer2009}) and can
be sensitively affected by the parameters of the environment. To show the temporal behavior of the exact master equation,
we plot the Fig.~\ref{sd_coeff} for the time-evolution of the coefficients $F_{1}(t)$ and $F_{2}(t)$; for simplicity, we
choose a noise-free $\hat{Q}$ operator in our numerical simulations. The
spectral density is chosen as the Lorentzian form
$$t_{k}^{2}(\omega_{k})\Delta\omega=\frac{\Gamma b^{2}}{(1-\frac{\omega_{k}}{\omega_{0}})^{2}+b^{2}}.$$
When the bandwidth $b$ is wide, which corresponds to a white noise situation,
the coefficients $F_{1}(t)$ and $F_{2}(t)$ must converge to constants rapidly, approaching the Markov limit
\cite{Wiseman}. On the contrary, if the bandwidth $b$ is narrow, the
distribution of the spectral density should represent a colored noise case, then we
could expect that the non-Markovian properties (\emph{e.g.}, time-dependent coefficients)
becomes dominant. As a direct result of using different $F_{1}(t)$ and $F_{2}(t)$, we could see that in Fig.~\ref{sd} (b), the density
matrix element $\rho_{11}$ performs differently when it converges to the steady state. The wider the bandwidth $b$ expands the faster the steady state could be approached,
and significant fluctuations would come forth when $b$ is small.
Another parameter that will affect the non-Markovian properties is the
temperature of the bath. As shown in Fig.~\ref{sd}
(a), the non-Markovian behaviors become more dominant in the low temperature
regimes.
\begin{figure}[ptb]
\begin{center}
\includegraphics[
trim=0.000000in 0.000000in 0.000000in -1.148132in,
height=2.5097in,
width=3.2621in
]{dd2.eps}
\end{center}
\caption{Dynamic evolution for the double quantum dot from the initial state $d_{1}^{\dag}|\mathrm{vac}\rangle_{\rm S}$ with
different coupling strength $g$. The other parameters are $T=100{\rm mK}$, $\mu
_{1}=2\times10^{-5}{\rm eV}$, $\mu_{2}=4\times10^{-5}{\rm eV}$, $\omega_{1}=\omega
=2.5\times10^{-5}{\rm eV}$, $\omega_{2}=3.5\times10^{-5}{\rm eV}$.}%
\label{dd}%
\end{figure}
\section{Double QDs coupled to two finite-temperature fermionic baths
\label{example3}
The model considered in this section is more involved, but physically more relevant. Here, we consider an
electronic system coupled to two fermionic baths described by the following total Hamiltonian,
\begin{align}
\hat{H}_{\mathrm{tot}} & =\omega_{1}\hat{d}_{1}^{\dagger}\hat{d}_{1}+\omega
_{2}\hat{d}_{2}^{\dagger}\hat{d}_{2}+g\hat{d}_{1}^{\dagger}\hat{d}_{2}%
+g^{\ast}\hat{d}_{2}^{\dagger}\hat{d}_{1}\nonumber\\
& +\sum_{k}\omega_{k}(\hat{b}_{1,k}^{\dagger}\hat{b}_{1,k}+\hat{b}%
_{2,k}^{\dagger}\hat{b}_{2,k})\nonumber\\
& +\{\sum_{k}t_{2,k}\hat{d}_{2}^{\dagger}\hat{b}_{2,k}+\sum_{k}t_{1,k}\hat
{d}_{1}^{\dagger}\hat{b}_{1,k}+\mathrm{h.c.}\},
\end{align}
where $\hat{d}_{i}$ and $\hat{d}_{i}^{\dagger}$ ($i=1,2$) are the fermionic
annihilation and creation operators of the two quantum dots in the system,
and $\hat{b}_{i,k}$, $\hat{b}_{i,k}^{\dagger}$ are the annihilation and creation operators
for the fermionic baths. This Hamiltonian describes a physical model that double quantum
dots coupled to two fermionic baths with different chemical potentials,
the \textquotedblleft source" and the \textquotedblleft drain". This model has been
widely studied by using the fermionic path integral and the input-output approaches
\cite{,ZhangDQD,Search2002}.
Similar to the case of the single QD model discussed before, the exact SSE for
the double QDs model can be established as,
\begin{align}
{\rm i}\partial_{t}|\psi_{t}\rangle & =(\hat{H}_{\mathrm{S}}-{\rm i}\hat{d}_{1}^{\dagger
}\bar{Q}_{1,b^{\prime}}-{\rm i}\hat{d}_{1} \xi_{1,b^{\prime},t}^{\ast}+{\rm i}\hat
{d}_{1}^{\dagger} \xi_{1,c^{\prime},t}^{\ast}+{\rm i}\hat{d}_{1}\bar
{Q}_{1,c^{\prime}}\nonumber\\
& -{\rm i}\hat{d}_{2}^{\dagger}\bar{Q}_{2,b^{\prime}}-{\rm i}\hat{d}_{2}%
\xi_{2,b^{\prime},t}^{\ast}+{\rm i}\hat{d}_{2}^{\dagger} \xi_{2,c^{\prime
},t}^{\ast}+{\rm i}\hat{d}_{2}\bar{Q}_{2,c^{\prime}})|\psi_{t}\rangle,
\end{align}
where $\bar{Q}_{\mu,\nu}|\psi_t\rangle \equiv \int_{0}^{t} {\rm d} s\, K_{\mu,\nu}%
(t,s)\overrightarrow{\delta}_{\xi_{\mu,\nu,s}^{\ast}}|\psi_t\rangle$ (the
indices $\mu=1,2$ represent the left and right baths, and $\nu=b^{\prime
}, c^{\prime}$ represent the fictitious baths $b^{\prime}$ and $c^{\prime}$ in the finite-temperature
transformation). Therefore, the exact master equation for the double quantum dots system can be derived from the
SSE,
\begin{align}
\partial_t \hat{\rho}_r & = -{\rm i}[\hat{H}_{\rm S} , \hat{\rho}_r] + \{ \sum_{j=1}^2 [ (F_1^j(t)\hat{d}_1 + F_2^j(t)\hat{d}_2) \hat{\rho}_r \nonumber \\
& + \hat{\rho}_r (F_3^j(t)\hat{d}_1 + F_4^j(t) \hat{d}_2) , \hat{d}_j\+ ] + \rm{h.c.} \}
\end{align}
where
\begin{align}
F^j_i(t) & \equiv \int_0^t {\rm d} s\, [K_{jb'}(t,s)u^{jb'}_i(t,s) - K_{jc'}^*(t,s)u^{jc'}_i(t,s)],\nonumber\\
\end{align}
where $K_{j\mu}$ ($j=1,2$, $\mu=b'$ or $c'$) are
the correlation functions, and the equations for the coefficients
$u^\mu_i$ ($i=1,2,3,4$, $\mu=1b',2b', 1c' $ or $2c'$) are
\begin {widetext}{\small
\begin {align}
\partial_s u^{1b'}_j(t,s) & = -{\rm i}\omega_1 u^{1b'}_j(t,s) -{\rm i} g\, u^{2b'}_j(t,s) + [ \int_s^t {\rm d} s'\, K_{1c'}(s',s) - \int_0^s {\rm d} s'\, K_{1b'}(s,s') ] u^{1b'}_j(t,s')
+ \int_0^t {\rm d} s'\, K_{1c'}(s',s) u^{1c'}_j(t,s') \nonumber\\
\partial_s u^{2b'}_j(t,s) & = -{\rm i}\omega_2 u^{2b'}_j(t,s) -{\rm i} g^* u^{1b'}_j(t,s) + [ \int_s^t {\rm d} s'\, K_{2c'}(s',s) - \int_0^s {\rm d} s'\, K_{2b'}(s,s') ] u^{2b'}_j(t,s')
+ \int_0^t {\rm d} s'\, K_{2c'}(s',s) u^{2c'}_j(t,s') \nonumber\\
\partial_s u^{1c'}_j(t,s) & = -{\rm i}\omega_1 u^{1c'}_j(t,s) -{\rm i} g\, u^{2c'}_j(t,s) + [ \int_s^t {\rm d} s'\, K_{1b'}^*(s',s) - \int_0^s {\rm d} s'\, K_{1c'}^*(s,s') ] u^{1c'}_j(t,s')
+ \int_0^t {\rm d} s'\, K_{1b'}^*(s',s) u^{1b'}_j(t,s') \nonumber\\
\partial_s u^{2c'}_j(t,s) & = -{\rm i}\omega_2 u^{2c'}_j(t,s) -{\rm i} g^* u^{1c'}_j(t,s) + [ \int_s^t {\rm d} s'\, K_{2b'}^*(s',s) - \int_0^s {\rm d} s'\, K_{2c'}^*(s,s') ] u^{2c'}_j(t,s')
+ \int_0^t {\rm d} s'\, K_{2b'}^*(s',s) u^{2b'}_j(t,s'), \nonumber\\
\end {align}}
\end {widetext}
with the final conditions: $u^{1b'}_1(t,s=t)=u^{2b'}_2(t,s=t)=u^{1c'}_3(t,s=t)=u^{2c'}_4(t,s=t)=1$, and the others are zero.
We omit the mathematical details of solving this model,
since the procedure is complicated however the main idea is still same to the single QD case.
Here, we only show some properties of this model by plotting the time evolution of the population; and the
detailed study of this model will be discussed elsewhere \cite{New}. In
Fig.~\ref{dd}, we plot the dynamic evolution of the probabilities of all the four
states with different coupling strength between the two QDs. In a long-time limit,
the system trends to converge to a steady state.
When $t$ is small, the electron tunneling from one dot to the other can be
significantly enhanced by the direct couplings
between the two QDs.
\section{conclusion}
\label{conclusion}
In this paper, we have developed an exact fermionic stochastic
Schr\"{o}dinger equation approach for solving the quantum open system coupled
to a fermionic environment. The fundamental dynamic equation is derived
directly from the microscopic quantum model without any approximations. By
using the Grassmann noise, the stochastic Schr\"{o}dinger equation approach is
expanded from bosonic to fermionic environments. Three examples are presented
to show the power of this approach. It is worth noting that the stochastic
Schr\"{o}dinger equation is versatile enough to deal with a generic fermionic
environment incorporating cases from strong system-reservoir interaction to structured reservoirs.
The exact stochastic approach can be applied to more realistic models when the
approximation methods are used \cite{New}.
{\it Note Added:} After completion of this work, we became aware of an independent work
by M. Chen and J. Q. You \cite{Chen-You2012}, who also derived a stochastic diffusive equation by
using Grassmann coherent state approach.
\section*{acknowledgements}
We acknowledge the grant support from the NSF PHY-0925174 and
the AFOSR No. FA9550-12-1-0001.
|
{
"timestamp": "2013-05-09T02:00:27",
"yymm": "1203",
"arxiv_id": "1203.2219",
"language": "en",
"url": "https://arxiv.org/abs/1203.2219"
}
|
\section{Conclusion and Future Work}
\label{sec:conc}
In this work, we established optimal algorithms for the worst-case
behavior of join algorithms. We also demonstrated that the join
algorithms employed in RDBMSes do not achieve these optimal bounds --
and we demonstrated families of instances where they were
asymptotically worse by factors close to the size of the largest
relation. It is interesting to ask similar questions for average case
complexity. Our work offers a fundamentally different way to approach
join optimization rather than the traditional
binary-join/dynamic-programming-based approach. Thus, our immediate
future work is to implement these ideas to see how they compare in
real RDBMS settings to the algorithms in a modern RDBMS.
Another interesting direction is to extend these results to a larger
classes of queries and to database schemata that have constraints. We
include in the appendix some preliminary results on full conjunctive
queries and simple functional dependencies (FDs). Not surprisingly,
using dependency information one can obtain tighter bounds
compared to the (FD-unaware) fractional cover technique.
We will also investigate whether our algorithm for computing
relaxed joins can be useful in related context such as those considered
in Koudas et al~\cite{Koudas06relaxingjoin}.
There are potentially interesting connections between our work and
several inter-related topics, which are all great subjects to further
explore. We algorithmically proved AGM's bound
which is equivalent to BT inequality, which in turn is essentially
equivalent to Shearer's entropy inequality. There are known combinatorial
interpretations of entropy inequalities which Shearer's is a special
case of; for example, Alon et al. \cite{DBLP:journals/ejc/AlonNSTV07}
derived some such connections
using a notion of ``sections" similar to what we used in this paper.
An analogous partitioning procedure was used in
\cite{DBLP:conf/stoc/Marx10} to compute joins by relating the number
of solutions to submodular functions.
Our lead example (the LW inequality with $n=3$) is equivalent to the problem of
enumerating all triangles in a tri-partite graph. It was known that this can be
done in time $O(N^{3/2})$ \cite{DBLP:journals/algorithmica/AlonYZ97}.
\paragraph*{Acknowledgments}
We thank Georg Gottlob for sending us a full version of his
work~\cite{GLVV09}. We thank XuanLong Nguyen for introducing us to the
Loomis-Whitney inequality. We thank the anonymous referees for many helpful
comments which have greatly improved the presentation clarity.
CR's work on this project is generously
supported the NSF CAREER Award under IIS-1054009, the Office of Naval
Research under award N000141210041, and gifts or research awards from
Google, Greenplum, Johnson Controls, LogicBlox, and Oracle.
\subsection{Relaxed Joins}
\label{sec:error}
\newcommand{\mathcal{C}}{\mathcal{C}}
\newcommand{\mathsf{LPOpt}}{\mathsf{LPOpt}}
\newcommand{\mathsf{BFS}}{\mathsf{BFS}}
We observe that our algorithm can actually evaluate a relaxed notion of join
queries. Say we are given a query $q$ represented by a hypergraph
$H=(V, E)$ where $V=[n]$ and $|E|=m$.
The $m$ input relations are $R_e$, $e\in E$.
We are also given a ``relaxation'' number $0\leq r \leq m$.
Our goal is to output all tuples that agree
with at least $m-r$ input relations.
In other words, we want to compute
$\cup_{S\subseteq E, |S|\ge m-r} \Join_{e\in S} R_e$.
However, we need to modify the problem to avoid the case that the set of
attributes of relations indexed by $S$ does not cover all the attributes
in the universe $V$.
Towards this end, define the set
\[ \mathcal{C}(q,r)= \left\{S\subseteq E \ | \ |S|\ge m-r\text{ and }
\bigcup_{e\in S} e = V \right\}.
\]
With the notations established above, we are now ready to define the
relaxed join problem.
\begin{defn}[Relaxed join problem]
Given a query $q$ represented by the hypergraph $H=(V=[n],E)$,
and an integer $0\le r\le m$, evaluate
\[ q_r\stackrel{def}{=}\bigcup_{S\in \mathcal{C}(q,r)} \left(\Join_{e\in S} R_e\right).
\]
\end{defn}
Before we proceed, we first make the following simple observation:
given any two sets $S, T\in \mathcal{C}(q,r)$ such that $S\subseteq T$,
we have $\Join_{e\in T} R_e \subseteq \Join_{e\in S} R_e$.
This means in the relaxed join problem we only need to consider
subsets of relations that are not contained in any other subset.
In particular, define
$\hat{\mathcal{C}}(q,r)\subseteq \mathcal{C}(q,r)$ to be the largest subset of $\mathcal{C}(q,r)$
such that for any $S\neq T\in \hat{\mathcal{C}}(q,r)$ neither
$S\subset T$ nor $T\subset S$.
We only need to evaluate
$q_r = \bigcup_{S\in \hat{\mathcal{C}}(q,r)} \left(\Join_{e\in S} R_e\right).$
Given an $S\in\hat{\mathcal{C}}(q,r)$, let $\mathsf{LPOpt}(S)$ denote the optimal size bound
given by the AGM's fractional cover inequality \eqref{eqn:agm08-bound}
on the join query represented by the hypergraph $(V, S)$.
In particular, $\mathsf{LPOpt}(S) = \prod_{e\in S}|R_e|^{x^*_e}$
where $\mv x^*_S = (x^*_e)_{e\in S}$ is an optimal solution to the
following linear program called LP$(S)$:
\begin{eqnarray}
\min & \sum_{e\in S} (\log |R_e|)\cdot x_e&\nonumber \\
\text{subject to}& \sum_{e\in S: i \in e} x_e \geq 1& \text{for any $i\in V$}
\label{eqn:LP(S)}\\
& x_e \geq 0 & \text{for any } e \in S.\nonumber
\end{eqnarray}
\paragraph*{Upper bounds} We start with a straightforward upper bound.
\begin{prop}
\label{prop:error-bound-loose}
Let $q$ be a join query on $m$ relations and let $0\le r\le m$ be an integer.
Then given sizes of the input relations, the number of
output tuples for query $q_r$ is upper bounded by
\[\sum_{S\in \hat{\mathcal{C}}(q,r)} \mathsf{LPOpt}(S).\]
\end{prop}
Further, Algorithm~\ref{algo:the-algo} evaluates $q_r$ with data
complexity linear in the bound above.
The next natural question is to determine how good the upper bound is.
Before we answer the question, we prove a stronger upper bound.
Given a subset of hyperedges $S\subseteq E$ which ``covers'' $V$,
i.e. $\cup_{e\in S} e = V$, let $\mathsf{BFS}(S)\subseteq S$ be the subset of
hyperedges in $S$ that gets a {\em positive} $x^*_e$ value in an {\em optimal}
basic feasible solution to the linear program LP$(S)$ defined
in \eqref{eqn:LP(S)}.
(If there are multiple such solutions, pick any one in a consistent manner.)
Call two subsets $S,T\subseteq E$ \textit{bfs-equivalent} if
$\mathsf{BFS}(S)=\mathsf{BFS}(T)$.
Finally, define $\mathcal{C}^*(q,r)\subseteq \hat{C}(q,r)$ as the collection of
sets from $\hat{\mathcal{C}}(q,r)$ which contains exactly one arbitrary
representative from each bfs-equivalence class.
\begin{thm}
\label{thm:error-bound-tight}
Let $q$ be a join query represented by $H=(V,E)$, and let $0\le r\le m$ be
an integer.
The number of output tuples of $q_r$ is upper bounded by
$\sum_{S\in \mathcal{C}^*(q,r)} \mathsf{LPOpt}(S).$
Further, the query $q_r$ can be evaluated in time
\[ O\left(\sum_{S\in \mathcal{C}^*(q,r)} \left(mn\cdot \mathsf{LPOpt}(S)+\mathrm{poly}(n,m)
\right)\right) \]
plus the time needed to compute $\mathcal{C}^*(q,r)$ from $q$.
\end{thm}
Note that since $\mathcal{C}^*(q,r)\subseteq \hat{\mathcal{C}}(q,r)$, the bound in
Theorem~\ref{thm:error-bound-tight} is no worse than that in
Proposition~\ref{prop:error-bound-loose}. We will show later that the bound
in Theorem~\ref{thm:error-bound-tight} is indeed tight.
\begin{proof}[Proof of Theorem~\ref{thm:error-bound-tight}]
We will prove the result by presenting the algorithm to compute $q_r$.
A simple yet key idea is the following.
Let $S \neq S' \in \hat{C}(q,r)$ be two different sets of hyperedges with the following property.
Define $T\stackrel{def}{=}\mathsf{BFS}(S)=\mathsf{BFS}(S')$ and let
$\mv x^*_{T}=(x^*_i)_{i\in T}$ be the projection of the corresponding
optimal basic feasible solution to the $(V,S)$ and the $(V,S')$ problems
projected down to $T$. (The two projections result in the same
vector $\mv x^*_T$.)
The outputs of the joins on $S$ and on $S'$ are both subsets of the
output of the join on $T$. We can simply run
Algorithm~\ref{algo:the-algo} on inputs $(V,T)$ and $\mv x^*_T$,
then prune the output against relations $R_e$ with $e\in S\setminus T$ or $S'\setminus T$.
In particular, we only need to
compute $\Join_{e\in T} R_e$ once for both $S$ and $S'$.
\floatname{algorithm}{Algorithm}
\begin{algorithm}
\caption{Computing Relaxed Join $q_r$}
\label{algo:relaxed-join}
\begin{algorithmic}[1]
\STATE Compute $\mathcal{C}^*(q,r)$.
\STATE $Q\leftarrow \emptyset$.
\FOR{ every $S\in \mathcal{C}^*(q,r)$}
\STATE Let $\mv x^*_S$ be an optimal BFS for LP$(S)$
\STATE Let $T=\{e\in S \ | \ x^*_e>0\}$. (Note that $T=\mathsf{BFS}(S)$.)
\STATE Run Algorithm~\ref{algo:the-algo} on $\{x^*_e\}_{e\in T}$ to compute
$\phi_{T}=\Join_{e\in T} R_e$.
\FOR{ every tuple $\mv t\in \phi_{T}$}
\IF{for at least $m-r$ hyperedges $e\in E$, $\mv t_e \in R_e$}
\STATE $Q \leftarrow Q \cup \{\mv t\}$
\ENDIF
\ENDFOR
\ENDFOR
\RETURN $Q$
\end{algorithmic}
\end{algorithm}
Other than the time to compute $\mathcal{C}^*(q,r)$ in the line 1, line 4 needs
$\mathrm{poly}(n,m)$ time to solve the LP, line 5 needs $O(m)$ time, while by
Theorem~\ref{thm:main}, line 6 will take
$O(mn\cdot\mathsf{LPOpt}(S)+m^2n)$ time. Finally,
Theorem~\ref{thm:main} shows that $|\phi_{T}|\le \mathsf{LPOpt}(S)$,\footnote{This also proves the claimed bound on the size of $q_r$.} which shows that the
loop in line 7 is repeated $\mathsf{LPOpt}(S)$ times and lines 8-9 can be
implemented in $O(m)$ time and thus, lines 7-9 will take time
$O(m\cdot \mathsf{LPOpt}(S))$.
Finally, we argue the correctness of the algorithm. We first note that by
line 8, every tuple $\mv t$ that is output is indeed a correct one.
Thus, we have to argue that we do not miss any tuple $\mv t$ that needs to be
output. For the sake of contradiction assume that there exists such a
tuple $\mv t$. Note that by definition of $\hat{\mathcal{C}}(q,r)$, this implies that
there exists a set $S'\in \hat{\mathcal{C}}(q,r)$ such that for every
$e\in S'$, $\mv t_e\in R_e$. However, note that by definition of
$\mathcal{C}^*(q,r)$, for some execution of the loop in line 3, we will consider
$T$ such that $T=\mathsf{BFS}(S')$. Further, by the correctness of
Algorithm~\ref{algo:the-algo}, we have that $\mv t\in \phi_T$.
This implies (along with the definition of $\hat{\mathcal{C}}(q,r)$) that
$\mv t$ will be retained in line 8, which is a contradiction.
\end{proof}
It is easy to check that one can compute $\mathcal{C}^*$ in time $m^{O(r)}$
(by going through all subsets of $E$ of size at least $m-r$ and
performing all the required checks). We leave open the question of
whether this time bound can be improved.
\paragraph*{Lower bound} We now show that the bound in
Theorem~\ref{thm:error-bound-tight} is tight for some query
and some database instance $I$.
We first define the query $q$. The hypergraph is $H=(V=[n],E)$
where $m=|E|=n+1$. The hyperedges are $E=\{e_1,\dots,e_{n+1}\}$
where $e_i=\{i\}$ for $i\in [n]$ and
$e_{n+1} = [n]$.
The database instance $I$ consists of
relations $R_e$, $e\in E$, all of which are of size $N$.
For each $i\in [n]$, $R_{e_i} = [N]$.
And, $R_{e_{n+1}} = \bigcup_{i=1}^N \{N+i\}^n$.
It is easy to check that for any $r>0$, $q_r(I)$ is the set
$R_{e_{n+1}} \cup [N]^n$, i.e.
$|q_r(I)| = N+N^n.$
Next, we claim that for this query instance
$\mathcal{C}^*(q,r)=\{\{n+1\}, [n]\}$. Note that $\mathsf{BFS}(\{n+1\})=\{n+1\}$ and
$\mathsf{BFS}([n])=[n]$, which implies that $\mathsf{LPOpt}(\{n+1\})=N$ and
$\mathsf{LPOpt}([n])=N^n$. This along with Theorem~\ref{thm:error-bound-tight} implies
that $|q_r(I)|\le N+N^n$, which proves the tightness of the size bound in
Theorem~\ref{thm:error-bound-tight}, as desired.
Finally, we argue that $\mathcal{C}^*(q,r)=\{ \{n+1\}, [n]\}$. Towards this end,
consider any $T\in \hat{\mathcal{C}}(q,r)$. Note that if $(n+1)\not\in T$, we have
$T=[n]$ and since $\mathsf{BFS}(T)=T$ (and we will see soon that for any other
$T\in\hat{\mathcal{C}}(q,r)$, we have $\mathsf{BFS}(T)\neq [n]$), which implies that
$[n]\in\mathcal{C}^*(q,r)$. Now consider the case when $(n+1)\in T$. Note that in this
case $T=\{n+1\} \cup T'$ for some $T'\subset [n]$ such that $|T'|\ge n-r$.
Now note that all the relations in $T$ cannot cover the $n$ attributes but
$R_{n+1}$ by itself does include all the $n$ attributes. This implies that
$\mathsf{BFS}(T)=\{n+1\}$ in this case. This proves that $\{n+1\}$ is the other
element in $\mathcal{C}^*(q,r)$, as desired.
Finally, if one wants a more general example where $m=n+k$ for $k>1$, then
one can repeat the above instance $k$ times, where each repetition has $n/k$
fresh attributes. In this case, $\mathcal{C}^*$ will consists of all subsets of
relation where in each repetition, each such subset has exactly one of
$\{n/k+1\}$ or $[n/k]$. In particular, the query output size will be
$\sum_{i=0}^r \binom{k}{i}\cdot N^{k-i}\cdot N^{n\cdot i/k}$.
\section{Extensions}
\label{sec:extensions}
In Section~\ref{sec:cc}, we describe some results on the combined
complexity of our approach. Finally, in Section~\ref{sec:error}, we
observe that our algorithm can be used to compute a relaxed notion of
join.
\subsection{Combined Complexity}
\label{sec:cc}\label{SEC:CC}
\newcommand{\mathsf{3SAT}}{\mathsf{3SAT}}
\newcommand{\mathsf{3UniqueSAT}}{\mathsf{3UniqueSAT}}
Given that our algorithms are data-optimal for
worst-case inputs it is tempting to wonder if one can obtain an join
algorithm whose run time is both query and data optimal in the worst-case.
We show that in the
special case when each input relation has arity at most $2$ we can
attain a data-optimal algorithm that is simpler than Algorithm \ref{algo:the-algo}
with an asymptotically better query complexity.
Further, given promising results in the worst case, it is natural wonder if one can obtain a join algorithm whose run time is polynomial in both the size of the query \textit{as well} as the size
of the output. More precisely, given a join query $q$ and an instance $I$, can
one compute the result of query $q$ on instance $I$ in time
$\mathrm{poly}(|q|,|q(I)|,|I|)$. Unfortunately, this is not possible unless $\textsf{NP}=\textsf{RP}$.
We briefly present a proof of this fact below.
\paragraph*{Each relation has at most $2$ attributes}
As was mentioned in the introduction, our algorithm in
Theorem~\ref{thm:main} not only has better data complexity than AGM's
algorithm (in fact we showed our algorithm has optimal worst-case data
complexity), it has a better query complexity.
In this section, we show that for the special case when the join query
$q$ is on relations with at most two attributes (i.e. the corresponding
hypergraph $H$ is a graph), we can obtain an even better query complexity as
in Theorem~\ref{thm:main} (with the same optimal data complexity).
Without loss of generality, we can assume that each relation contains
exactly $2$ attributes because a $1$-attribute relation $R_e$ needs to
have $x_e=1$ in the corresponding LP and thus,
contributes a separate factor $N_e$ to the final product.
Thus, $R_e$ can be joined with the rest of the query with any join algorithm
(including the naive Cartesian product based algorithm).
In this case, the hypergraph $H$ is a graph which can be assumed to be
simple.
We first prove an auxiliary lemma for the case when $H$ is a cycle.
We assume that all relations are indexed in advanced,
which takes $O(\sum_e N_e)$ time.
In what follows we will not include this preprocessing time in the analysis.
The following lemma essentially reduces the case when $H$ is a cycle to
the case when $H$ is a triangle, a Loomis-Whitney instance with $n=3$.
\begin{lmm}[Cycle Lemma]
If $H$ is a cycle, then $\Join_{e\in E} R_e$
can be computed in time $O(m\sqrt{\prod_{e\in H}N_e})$.
\label{lmm:cycle}
\end{lmm}
\begin{proof}
First suppose $H$ is an even cycle, consisting of consecutive
edges $e_1=(1,2)$, $e_2=(2,3)$,$\cdots$,$e_{2k'}=({2k'},1)$.
Without loss of generality, assume
\[ N_{e_1}N_{e_3} \cdots N_{e_{2k'-1}} \leq N_{e_2}N_{e_4} \cdots N_{e_{2k'}}. \]
In this case, we compute the (cross-product) join
\[ R = R_{e_1} \Join R_{e_3} \Join \cdots \Join R_{e_{2k'-1}}. \]
Note that $R$ contains all the attributes.
Then, sequentially join $R$ with each of $R_{e_2}$ to $R_{e_{2k'}}$.
The total running time is
\[ O\left(k'N_{e_1}N_{e_3} \cdots N_{e_{2k'-1}}\right)
= O\left(m\prod_{e\in H}N_e\right).
\]
Second, suppose $H$ is an odd cycle consisting of consecutive edges
$e_1=(1,2)$, $e_2=(2,3)$, $\dots$, $e_{2k'+1}=({2k'+1},1)$.
If $k'=1$ then by the Loomis-Whitney algorithm for the
$n=3$ case (Algorithm \ref{algo:LW}),
we can compute $R_{e_1} \Join R_{e_2} \Join R_{e_3}$
in time $O(\sqrt{N_{e_1}N_{e_2}N_{e_3}})$.
Suppose $k'>1$. Without loss of generality, assume
\[ N_{e_1}N_{e_3} \cdots N_{e_{2k'-1}} \leq
N_{e_2}N_{e_4} \cdots N_{e_{2k'}}.
\]
In particular,
$N_{e_1}N_{e_3} \cdots N_{e_{2k'-1}} \leq \sqrt{ \prod_{e\in H} N_e}$, which
means the following join can be computed in time
$O(m\sqrt{ \prod_{e\in H} N_e})$:
\[ X = R_{e_1} \Join R_{e_3} \Join \cdots \Join R_{e_{2k'-1}}. \]
Note that $X$ spans the attributes in the set $[2k']$.
Let $S=\{2,3,\dots,2k'-1\}$, and $X_S$ denote the projection
of $X$ down to coordinates in $S$; and define
\[ W = (\dots(X_S \Join R_{e_2}) \Join R_{e_4}) \cdots \Join R_{e_{2k'-2}}). \]
Since $R_{e_2} \Join R_{e_4} \cdots \Join R_{e_{2k'-2}}$ spans
precisely the attributes in $S$, the relation $W$ can be computed
in time $O(m|X_S|) = O(m|X|) = O(m\sqrt{ \prod_{e\in H} N_e})$.
Note that
\[ |W| \leq \min\{N_{e_1}N_{e_3}\cdots N_{e_{2k'-1}},
N_{e_2}N_{e_4}\cdots N_{e_{2k'-2}}\}.
\]
We claim that one of the following inequalities must hold:
\begin{eqnarray*}
|W| \cdot N_{e_{2k'}} & \leq & \sqrt{ \prod_{e\in H} N_e}, \text{ or}\\
|W| \cdot N_{e_{2k'+1}} & \leq & \sqrt{ \prod_{e\in H} N_e}.
\end{eqnarray*}
Suppose both of them do not hold, then
\begin{eqnarray*}
\prod_{e\in H} N_e
&=&
(N_{e_1}N_{e_3}\cdots N_{e_{2k'-1}}) \cdot (N_{e_2}N_{e_4}\cdots N_{e_{2k'-2}})
\cdot N_{e_{2k'}} \cdot N_{e_{2k'+1}}\\
&\geq&|W|^2 N_{e_{2k'}} N_{e_{2k'+1}}\\
&=&(|W| \cdot N_{e_{2k'}}) \cdot (|W| \cdot N_{e_{2k'+1}})\\
&>& \prod_{e\in H} N_e,
\end{eqnarray*}
which is a contradiction.
Hence, without loss of generality we can assume
$|W| \cdot N_{2k'} \leq \sqrt{ \prod_{e\in H} N_e}$.
Now, compute the relation
\[ Y = W \Join R_{e_{2k'}}, \]
which spans the attributes $S\cup \{2k',2k'+1\}$.
Finally, by thinking of all attributes in the set $S\cup \{2k'\}$ as a
``bundled attribute",
we can use the Loomis-Whitney algorithm for $n=3$ to compute the join
\[ X \Join Y \Join R_{e_{2k'+1}} \]
in time linear in
\begin{eqnarray*}
\sqrt{|X| \cdot |Y| \cdot N_{e_{2k'+1}}}
&\leq& \sqrt{(N_{e_1}N_{e_3}\cdots N_{e_{2k'-1}}) \cdot
(|W|\cdot N_{e_{2k'}}) \cdot N_{e_{2k'+1}}}\\
&\leq& \sqrt{(N_{e_1}N_{e_3}\cdots N_{e_{2k'-1}}) \cdot
(N_{e_2}N_{e_4}\cdots N_{e_{2k'-2}}\cdot N_{e_{2k'}}) \cdot N_{e_{2k'+1}}}\\
&=& \sqrt{ \prod_{e\in H} N_e}.
\end{eqnarray*}
\end{proof}
With the help of Lemma \ref{lmm:cycle}, we can now derive a solution
for the case when $H$ is an arbitrary graph.
Consider any {\em basic feasible solution} $\mathbf x = (x_e)_{e\in E}$
of the fractional cover polyhedron
\begin{eqnarray*}
\sum_{v\in e} x_e & \geq & 1, \ v \in V\\
x_e & \geq & 0, \ e \in E.
\end{eqnarray*}
It is known that $\mathbf x$ is {\em half-integral}, i.e.
$x_e \in \{0,1/2,1\}$ for all $e\in E$
(see Schrijver's book \cite{MR1956924}, Theorem 30.10).
However, we will also need a graph structure associated with
the half-integral solution; hence, we adapt a known proof
\cite{MR1956924} of the half-integrality property
with a slightly more specific analysis.
It should be noted, however, that the following is already implicit in
the existing proof.
\begin{lmm}
For any basic feasible solution $\mathbf x = (x_e)_{e\in E}$
of the fractional cover polyhedron above, $x_e \in \{0,1/2,1\}$ for all
$e \in E$.
Furthermore, the collection of edges $e$ for
which $x_e=1$ is a union $\mathcal S$
of stars. And, the collection of edges $e$ for which
$x_e=1/2$ form a set $\mathcal C$ of vertex-disjoint
odd-length cycles that are also
vertex disjoint from the union $S$ of stars.
\label{lmm:bfs-d=2}
\end{lmm}
\begin{proof}
First, if some $x_e= 0$, then we remove $e$ from the graph and recurse on
$G-e$.
The new $\mathbf x$ is still an extreme point of the new polyhedron.
So we can assume that $x_e > 0$ for all $e\in E$.
Second, we can also assume that $H$ is connected.
Otherwise, we consider each connected component separately.
Let $k=|V|$ and $m=|E|$.
The polyhedron is defined on $m$ variables and $k+m$ inequality constraints.
The extreme point must be the intersection of exactly $m$ (linearly independent)
tight constraints.
But the constraints $\mathbf x \geq \mathbf 0$ are not tight as
we have assumed $x_e>0, \forall e$.
Hence, there must be $m$ vertices $v$ for which
the constraints $\sum_{v\in e} x_e \geq 1$ are tight.
In particular, $m\leq k$. Since $H$ is connected, it is either a tree, or
has exactly one cycle.
Suppose $H$ is a tree, then it has at least $2$ leaves and at most one
non-tight constraint (as there must be $m=k-1$ tight constraints).
Consider the leaf $u$ whose constraint is tight.
Let $v$ be $u$'s neighbor. Then $x_{uv} = 1$ because $u$ is tight.
If $v$ is tight then we are done, the graph $H$ is just an edge $uv$.
(If there was another edge $e$ incident to $v$ then $x_e=0$.)
If $v$ is not tight then $v$ is not a leaf. We start from another tight
leaf $w \neq u$ of the tree and reason in the same way.
Then, $w$ has to be connected to $v$. Overall, the graph is a star.
Next, consider the case when $H$ is not a tree.
All $k=m$ vertices has to be tight in this case.
Thus, there cannot be a degree-$1$ vertex for the same reasoning as above.
Thus, $H$ is a cycle. If $H$ is an odd cycle then it is easy to show that the
only solution for which all vertices are tight is the all-$1/2$ solution.
If $H$ is an even cycle then $\mathbf x$ cannot be an extreme point
because it can be written as $\mathbf x = (\mathbf y+\mathbf z)/2$ for
feasible solutions $\mathbf y$ and $\mathbf z$
(just add and subtract $\epsilon$ from alternate edges to form $\mathbf y$ and
$\mathbf z$).
\end{proof}
Now, let $\mathbf x^*$ be an {\em optimal} basic feasible solution to the
following linear program.
\begin{eqnarray*}
\min & \sum_e (\log N_e) \cdot x_e\\
s.t. & \sum_{v\in e} x_e & \geq 1, \ v \in V\\
& x_e & \geq 0, \ e \in E.
\end{eqnarray*}
Then $\prod_{e\in E}N_e^{x^*_e} \leq \prod_{e\in E}N_e^{x_e}$ for any
feasible fractional cover $\mathbf x$.
Let $S$ be the set of edges on the stars
and $\mathcal C$ be the collection of disjoint cycles as shown in the
above lemma, applied to $\mathbf x^*$.
Then,
\[ \prod_{e\in E}N_e^{x^*_e}
= \left(\prod_{e\in S}N_e\right)
\prod_{C\in \mathcal C}\sqrt{\prod_{e\in C}N_e}.
\]
Consequently, we can apply Lemma \ref{lmm:cycle} to each cycle $C\in\mathcal C$
and take a cross product of all the resulting relations with the relations
$R_e$ for $e\in S$.
We just proved the following theorem.
\bth
When each relation has at most two attributes, we can compute the join
$\Join_{e\in E}R_e$ in time $O(m\prod_{e\in E}N_e^{x_e})$.
\end{thm}
\paragraph*{Impossibility of Instance Optimality}
The proof is fairly standard: we use the standard reduction of $\mathsf{3SAT}$ to
conjunctive queries but with two simple specializations: (i) We reduce from
the $\mathsf{3UniqueSAT}$, where the input formula is either unsatisfiable or has
\textit{exactly} one satisfying assignment and (ii) $q$ is a full join
query instead of a general conjunctive query. It is known that $\mathsf{3UniqueSAT}$ cannot
be solved in deterministic polynomial time unless $\textsf{NP}=\textsf{RP}$~\cite{unique-sat}.
For the sake of completeness, we sketch the reduction here.
Let $\phi=C_1\wedge C_2\wedge \dots C_m$ be a $\mathsf{3UniqueSAT}$ CNF formula on $n$
variables $a_1,\dots,a_n$. (W.l.o.g. assume that a clause does not contain
both a variable and its negation.) For each clause $C_j$ for
$j\in [m]$, create a relation $R_j$ on the variables that occur in
$C_j$. The query $q$ is \[\Join_{j\in [m]} R_j.\]
Now define the database $I$ as follows: for each $j\in [m]$, $R_j^{I}$
contains the seven assignments to the variables in $C_j$ that makes it true.
Note that $q(I)$ contains all the satisfying assignments for $\phi$: in
other words, $q(I)$ has one element if $\phi$ is satisfiable otherwise
$q(I)=\emptyset$. In other words, we have $|q(I)|\le 1$, $|q|=O(m+n)$ and
$|I|=O(m)$. Thus an instance optimal algorithm with time complexity
$\mathrm{poly}(|q|,|q(I)|,|I|)$ for $q$ would be able to determine if $\phi$ is
satisfiable or not in time $\mathrm{poly}(n,m)$, which would imply $\textsf{NP}=\textsf{RP}$.
\input{error}
\eat{
\subsection{Full queries and simple functional dependencies}
\label{sec:fqp}
\paragraph*{Full query processing}
Our goal in this section is to handle a more general class of queries
that may contain selections and joins to the same table, which we
describe now.
Our notation in this section follows Gottlob et
al's~\cite{DBLP:conf/pods/GottlobLV09} notation, and we reproduce it
here for the sake of completeness. A {\em database instance} consists
$I = (\mathcal{U}, R_1,\dots,R_m)$ consists of a finite universe of
constants $\mathcal{U}$ and relations $R_1,\dots,R_m$ each over
$\mathcal{U}$. A {\em conjunctive query} has the form $q = R(x_0)
\leftarrow R_{i_1}(u_1) \wedge \dots \wedge R_{i_m}(u_m)$, where each
$u_j$ is a list of (not necessarily distinct) variables of length
$|u_j|$. We call each $R_{i_j}$ a subgoal. Each variable that occurs
in the head $R(u_0)$ must also appear in the body. We call a
conjunctive query {\em full} if each variable that appears in the body
also appears in the head. The set of all variables in $Q$ is denoted
$\mathrm{var}(Q)$. A single relation may occur several times in the
body, and so we may have $i_{j} = i_{k}$ for some $j \neq k$. The
answer of a query $q$ over a database instance $I$ is a set of tuples
of arity $|u_0|$, which is denoted $q(I)$, and is defined to contain
exactly those tuples $\theta(x_0)$ where $\theta : \mathrm{var}(Q) \to
\mathcal{U}$ is any substitution such that for each $j=1,\dots,m$,
$\theta(u_i) \in R_{i_j}$.
We call a full conjunctive query {\em reduced} if no variable is
repeated in the same subgoal. We can assume without loss of generality
that a full conjunctive query is reduced since we can create an
equivalent reduced query within the time bound. In time $O(|R_{i_j}|)$
for each $j=1,\dots,m$, we create a new relation $R_{i_j}'$ with arity
equal to the number of distinct variables. In one scan over $R_{i_j}$
we can produce $R_{i_j}'$ by keeping only those tuples that satisfy
constants (selections) in the query and any repeated variables. We
then construct $q'$ a query over the $R_{i_j}$ in the obvious
way. Clearly $q(I) = q'(I)$ and we can construct both in a single scan
over the input. Finally, we make the observation that our method can
tolerate multisets as hypergraphs, and so our results extend our
method to full conjunctive queries. Summarizing our discussion, we
have a worst-case optimal instance for full conjunctive queries as
well.
\paragraph*{Simple Functional Dependencies}
Given a join query $(V,E)$, a (simple) functional dependency (FD) is a
triple $(e,u,v)$ where $u,v \in V$ and $e \in E$ and is written as $e.u
\to e.v$. It is a constraint in that the FD $(e,u,v)$ implies that for
any pair of tuples $\mv t,\mv t' \in R_e$, if $t_u = t'_u$ then $t_v
= t'_v$. Fix a set of functional dependencies $\Gamma$, construct a
directed (multi-)graph $G(\Gamma)$ where the nodes are the attributes $V$ and
there is an edge
$(u,v)$ for each functional dependency. The set of all nodes reachable
from a node $u$ is a set $U$ of nodes; this relationship is denoted $u
\to^{*} U$.
Given a set of functional dependencies, we propose an algorithm to
process a join query. The first step is to compute for each relation
$R_e$ for $e \in E$, a new relation $R'_{e'}$ whose attributes are the
union of the closure of each element of $v \in E$, i.e., $e' = \set{ u
\ | \ v \to u \text{ for } v \in e}$. Using the closure this can be
computed in time $|E| |V|$. Then, we compute the contents of
$R_{e}'$. Walking the graph induced by the FDs in a breadth first
manner, we can expand $R_{e}$ to contain all the attributes $R_{e'}$
in time linear in the input size. Finally, we solve the LP from
previous section and use our algorithm. It is clear that this
algorithm is a strict improvement over our previous algorithm that is
FD-unaware. It is an open question to understand its data
optimality. We are, however, able to give an example that
suggests this algorithm can be substantially better than algorithms
that are not FD aware.
Consider the following family of instances on $k+2$ attributes
$A,B_1,\dots,B_k,C$ parameterized by $N$:
\[ q = \left(\Join_{i=1}^{k} R_{i}(A,B_i) \right) \Join
\left(\Join_{i=1}^{k} S_{i}(B_i, C) \right) \]
Now we construct a family of instances such that $|R_i| = |S_i| = N$
for $i=1,\dots,k$. Suppose there are functional dependencies $A
\to B_i$.
Our algorithm will first produce a relation $R'(A,B_1,\dots,B_k)$
which can then be joined in time $N$ with each relation $S_{i}$ for
$i=1,\dots,k$. When we solve the LP, we get a bound of of $|q(I)| \leq
N^2$ -- and our algorithm runs within this time.
Now consider the original instance without functional
dependencies. Then, the AGM bound is $|q(I)| \leq N^{k}$. More
interestingly, one can construct a simple instance where half of the
join has a huge size, that is $|\Join_{i=1}^{k} S_{i}(B_i, C)| =
N^{k}$. Thus, if we choose the wrong join ordering our algorithms
running time will blow up.
\textbf{CR: Can we argue that any non-FD aware algorithm will join at
least $3$ of those nasty relations? Perhaps by blowing up some
relations? (I think I see how to do it for our algorithm\dots)}
}
\subsection{Dealing with full queries and simple functional dependencies}
\label{app:sec:fqp}
\paragraph*{Full query processing}
Our goal in this section is to handle a more general class of queries
that may contain selections and joins to the same table, which we
describe now.
Our notation in this section follows Gottlob et
al's~\cite{GLVV09} notation, and we reproduce it
here for the sake of completeness. A {\em database instance} consists
$I = (\mathcal{U}, R_1,\dots,R_m)$ consists of a finite universe of
constants $\mathcal{U}$ and relations $R_1,\dots,R_m$ each over
$\mathcal{U}$. A {\em conjunctive query} has the form $q = R(x_0)
\leftarrow R_{i_1}(u_1) \wedge \dots \wedge R_{i_m}(u_m)$, where each
$u_j$ is a list of (not necessarily distinct) variables of length
$|u_j|$. We call each $R_{i_j}$ a subgoal. Each variable that occurs
in the head $R(u_0)$ must also appear in the body. We call a
conjunctive query {\em full} if each variable that appears in the body
also appears in the head. The set of all variables in $Q$ is denoted
$\mathrm{var}(Q)$. A single relation may occur several times in the
body, and so we may have $i_{j} = i_{k}$ for some $j \neq k$. The
answer of a query $q$ over a database instance $I$ is a set of tuples
of arity $|u_0|$, which is denoted $q(I)$, and is defined to contain
exactly those tuples $\theta(x_0)$ where $\theta : \mathrm{var}(Q) \to
\mathcal{U}$ is any substitution such that for each $j=1,\dots,m$,
$\theta(u_i) \in R_{i_j}$.
We call a full conjunctive query {\em reduced} if no variable is
repeated in the same subgoal. We can assume without loss of generality
that a full conjunctive query is reduced since we can create an
equivalent reduced query within the time bound. In time $O(|R_{i_j}|)$
for each $j=1,\dots,m$, we create a new relation $R_{i_j}'$ with arity
equal to the number of distinct variables. In one scan over $R_{i_j}$
we can produce $R_{i_j}'$ by keeping only those tuples that satisfy
constants (selections) in the query and any repeated variables. We
then construct $q'$ a query over the $R_{i_j}$ in the obvious
way. Clearly $q(I) = q'(I)$ and we can construct both in a single scan
over the input. Finally, we make the observation that our method can
tolerate multisets as hypergraphs, and so our results extend our
method to full conjunctive queries. Summarizing our discussion, we
have a worst-case optimal instance for full conjunctive queries as
well.
\paragraph*{Simple Functional Dependencies}
Given a join query $(V,E)$, a (simple) functional dependency (FD) is a
triple $(e,u,v)$ where $u,v \in V$ and $e \in E$ and is written as $e.u
\to e.v$. It is a constraint in that the FD $(e,u,v)$ implies that for
any pair of tuples $\mv t,\mv t' \in R_e$, if $t_u = t'_u$ then $t_v
= t'_v$. Fix a set of functional dependencies $\Gamma$, construct a
directed (multi-)graph $G(\Gamma)$ where the nodes are the attributes $V$ and
there is an edge
$(u,v)$ for each functional dependency. The set of all nodes reachable
from a node $u$ is a set $U$ of nodes; this relationship is denoted $u
\to^{*} U$.
Given a set of functional dependencies, we propose an algorithm to
process a join query. The first step is to compute for each relation
$R_e$ for $e \in E$, a new relation $R'_{e'}$ whose attributes are the
union of the closure of each element of $v \in E$, i.e., $e' = \set{ u
\ | \ v \to u \text{ for } v \in e}$. Using the closure this can be
computed in time $|E| |V|$. Then, we compute the contents of
$R_{e}'$. Walking the graph induced by the FDs in a breadth first
manner, we can expand $R_{e}$ to contain all the attributes $R_{e'}$
in time linear in the input size. Finally, we solve the LP from
previous section and use our algorithm. It is clear that this
algorithm is a strict improvement over our previous algorithm that is
FD-unaware. It is an open question to understand its data
optimality. We are, however, able to give an example that
suggests this algorithm can be substantially better than algorithms
that are not FD aware.
Consider the following family of instances on $k+2$ attributes
$A,B_1,\dots,B_k,C$ parameterized by $N$:
\[ q = \left(\Join_{i=1}^{k} R_{i}(A,B_i) \right) \Join
\left(\Join_{i=1}^{k} S_{i}(B_i, C) \right) \]
Now we construct a family of instances such that $|R_i| = |S_i| = N$
for $i=1,\dots,k$. Suppose there are functional dependencies $A
\to B_i$.
Our algorithm will first produce a relation $R'(A,B_1,\dots,B_k)$
which can then be joined in time $N$ with each relation $S_{i}$ for
$i=1,\dots,k$. When we solve the LP, we get a bound of of $|q(I)| \leq
N^2$ -- and our algorithm runs within this time.
Now consider the original instance without functional
dependencies. Then, the AGM bound is $|q(I)| \leq N^{k}$. More
interestingly, one can construct a simple instance where half of the
join has a huge size, that is $|\Join_{i=1}^{k} S_{i}(B_i, C)| =
N^{k}$. Thus, if we choose the wrong join ordering our algorithms
running time will blow up.
\section{Introduction}
Recently, Grohe and Marx~\cite{GM06} and Atserias, Grohe, and
Marx~\cite{AGM08} (AGM's results henceforth) derived tight bounds on
the number of output tuples of a {\em full conjunctive
query}\footnote{A full conjunctive query is a conjunctive query
where every variable in the body appears in the head.} in terms of
the sizes of the relations mentioned in the query's body. As query
output size estimation is fundamentally important for efficient query
processing, these results have generated a great deal of excitement.
To understand the spirit of AGM's results, consider the following
example where we have a schema with three attributes, $A$, $B$, and
$C$, and three relations, $R(A,B)$, $S(B,C)$ and $T(A,C)$, defined
over those attributes. Consider the following natural join
query:
\begin{equation}
q = R \Join S \Join T
\label{eq:q}
\end{equation}
Let $q(I)$ denote the set of tuples that is output from applying $q$
to a database instance $I$, that is the set of triples of constants
$(a,b,c)$ such that $R(ab)$, $S(bc)$, and $T(ac)$ are in $I$.
Our goal is to bound the
number of tuples returned by $q$ on $I$, denoted by $|q(I)|$, in terms of
$|R|$, $|S|$, and $|T|$. For simplicity, let us consider
the case when $|R|=|S|=|T|=N$.
A straightforward bound is $|q(I)| \leq N^3$.
One can obtain a better bound by noticing that any pair-wise join
(say $R\Join S$) will contain $q(I)$ in it as $R$ and $S$ together contain all
attributes (or they ``cover'' all the attributes).
This leads to the bound $|q(I)|\le N^2$.
AGM showed that one can get a better upper bound of
$|q(I)|\le N^{3/2}$ by generalizing the notion of cover to a so-called
``fractional cover'' (see Section~\ref{sec:notations}).
Moreover, this estimate is tight in the sense that
for infinitely many values of $N$, one can find a database instance
$I$ that for which $|q(I)| = N^{3/2}$. These non-trivial
estimates are exciting to database researchers as they offer
previously unknown, nontrivial methods to estimate the cardinality of
a query result -- a fundamental problem to support efficient query
processing.
More generally, given an arbitrary natural-join query $q$ and given
the sizes of input relations, the AGM method can generate an upper
bound $U$ such that $|q(I)| \leq U$, where $U$ depends on the ``best''
fractional cover of the attributes. This ``best'' fractional cover
can be computed by a linear program (see Section~\ref{sec:notations} for
more details). Henceforth, we refer to this inequality as the {\em
AGM's fractional cover inequality}, and the bound $U$ as the {\em
AGM's fractional cover bound}. They also show that the bound is
essentially optimal in the sense that for infinitely many sizes of
input relations, there exists an instance $I$ such that each relation
in $I$ is of the prescribed size and $|q(I)| = U$.
AGM's results leave open whether one can compute the actual set $q(I)$
in time $O(U)$. In fact, AGM observe this issue and presented an
algorithm that computes $q(I)$ with a running time of $O(|q|^2 \cdot
U\cdot N)$ where $N$ is the cardinality of the largest input relation
and $|q|$ denotes the size of the query $q$. AGM establish that their
join-project plan can in some cases
be super-polynomially better than any join-only
plan. However, AGM's join algorithm is not optimal. Even on the
above example of \eqref{eq:q}, we can construct a family of database
instances $I_1, I_2, \dots, I_N, \dots, $ such that in the $N$th
instance $I_N$ we have $|R| = |S| = |T| = N$ and both AGM's algorithm
and any join-only plan take $\Omega(N^2)$-time even though from AGM's
bound we know that $|q(I)| \leq U = N^{3/2}$, which is the best
worst-case run-time one can hope for.
The $\sqrt N$-gap on a small example motivates our central question.
In what follows, {\em natural join queries} are defined as the
join of a set of relations $R_1,\dots,R_m$.
\noindent
\begin{quote}
\textbf{Optimal Worst-case Join Evaluation Problem} (Optimal Join
Problem). {\em Given a fixed database schema $\bar R =
\left\{R_i(\bar A_i)\right\}_{i=1}^{m}$ and an $m$-tuple of
integers $\bar N = (N_1,\dots,N_m)$. Let $q$ be the natural join
query joining the relations in $\bar R$ and let $I(\bar N)$ be the
set of all instances such that $|R_i^{I}| = N_i$ for
$i=1,\dots,m$. Define $U = \sup_{I \in I(\bar N)} |q(I)|$. Then, the
optimal worst-case join evaluation problem is to evaluate $q$ in
time $O(U + \sum_{i=1}^{m} N_i)$}.
\end{quote}
\noindent
Since any algorithm to produce $q(I)$ requires time at least $|q(I)|$,
an algorithm that solves the above problem would have an optimal
worst-case data-complexity.\footnote{In an RDBMS, one computes information,
e.g., indexes, offline that may obviate the need to read the entire
input relations to produce the output. In a similar spirit, we can
extend our results to evaluate any query $q$ in time $O(U)$,
removing the term $\sum_i N_i$ by precomputing some indices.}
(Note that we are mainly concerned with data complexity and thus the
$O(U)$ bound above ignores the dependence on $|q|$. Our results
have a small $O(|q|)$ factor.)
Implicitly, this problem has been studied for over three decades: a
modern RDBMS use decades of highly tuned algorithms to efficiently produce
query results. Nevertheless, as we described above, such systems are
asymptotically suboptimal -- even in the above simple example of
\eqref{eq:q}. Our main result is an algorithm that achieves
asymptotically optimal worst-case running times for all
conjunctive join queries.
We begin by describing connections between AGM's inequality and a family of
inequalities in geometry. In particular, we show that the
AGM's inequality is {\em equivalent} to the discrete version of a geometric
inequality proved by Boll\'obas and Thomason
(\cite{MR1338683}, Theorem 2).
This equivalence is shown in Section~\ref{sec:bt:equiv}.
Our ideas for an algorithm solving the optimal join problem begin by
examining a special case of the Boll\'obas-Thomason (BT) inequality:
the classic Loomis-Whitney (LW) inequality~\cite{MR0031538}. The LW
inequality bounds the measure of an $n$-dimensional set in terms of
the measures of its $(n-1)$-dimensional projections onto the
coordinate hyperplanes. The query \eqref{eq:q} and its bound
$|q(I)|\leq \sqrt{|R||S||T|}$ is {\em exactly} the LW inequality with
$n=3$ applied to the discrete measure. Our algorithmic development
begins with a slight generalization of the query $q$ in
\eqref{eq:q}. We describe an algorithm for join queries which have the
same format as in the LW inequality setup with $n\geq 3$. In
particular, we consider ``LW instances'' of the optimal join problem,
where the query is to join $n$ relations whose attribute sets are all
the distinct $(n-1)$-subsets of a universe of $n$ attributes. Since
the LW inequality is tight, and our join algorithm has running time
that is asymptotically data-optimal for this class of queries (e.g.,
$O(N^{3/2})$ in our motivating example), our algorithm is
data-complexity optimal in the worst case for LW instances.
Our algorithm for LW instances exhibits a key twist compared to a
conventional join algorithm. The twist is that the join algorithm
partitions the values of the join key on each side of the join into
two sets: those values that are {\em heavy} and those values that are
{\em light}. Intuitively, a value of a join key is heavy if its fanout
is high enough so that joining all such join keys could violate the
size bound (e.g., $N^{3/2}$ above). The art is selecting the precise
fanout threshold for when a join key is heavy. This per-tuple choice
of join strategy is not typically done in standard RDBMS join
processing.
Building on our algorithm for LW instances, we next describe our main
result: an algorithm to solve the optimal join problem for all join
queries. In particular, we design an algorithm for evaluating join
queries which not only {\em proves} AGM's fractional cover inequality
{\em without} using the information-theoretic Shearer's inequality,
but also has a running time that is linear in the bound (modulo
pre-processing time). As AGM's inequality implies the BT and LW
inequalities, our result is the first algorithmic proof of these
geometric inequalities as well. To do this, we must carefully select
which projections of relations to join and in which order our
algorithm joins relations on a ``per tuple'' basis as in the
LW-instance case. Our algorithm computes these orderings, and then at
each stage it performs an algorithm that is similar to the algorithm
we used for LW instances.
Our example also shows that standard join algorithms are suboptimal,
the question is, {\it when do classical RDBMS algorithms have higher
worst-case run-time than our proposed approach?} AGM's analysis of their
join-project algorithm leads to a worst case run-time complexity that is a
factor of the largest relation worse than the AGM's bound.
To investigate whether AGM's analysis is tight or not, we ask a
sharper variant of this question: {\it Given a query $q$ does there
exist a family of instances $I$ such that our algorithm runs
asymptotically faster than a standard binary-join-based plan or
AGM's join-project plan?} We give a partial answer to this question
by describing a sufficient syntactic condition for the query $q$ such
that for each $k \geq 2$, we can construct a family of instances where each
relation is of size $N$ such that any binary-join plan as well as AGM's
algorithm will need time
$\Omega(N^{2}/k^2)$, while the fractional cover bound is
$O(N^{1 + 1/(k-1)})$ -- an asymptotic gap. We then show through a more
detailed analysis that our algorithm on these instances takes
$O(k^2N)$-time.
We consider several extensions and improvements of our main result. In
terms of the dependence on query size, our algorithms are also
efficient (at most linear in $|q|$, which is better than the quadratic
dependence in AGM) for full queries, but they are not necessarily
optimal. In particular, if each relation in the schema has arity $2$,
we are able to give an algorithm with better query complexity than our
general algorithm. This shows that in general our algorithm's
dependence on the factors of the query is not the best possible. We
also consider computing a relaxed notion of joins and give worst-case
optimal algorithms for this problem as well.
\paragraph*{Outline} The remainder of the paper is organized as
follows: in the rest of this section, we describe related work.
In Section~\ref{sec:notations} we describe our notation and formulate the
main problem.
Section~\ref{sec:bt:equiv} proves the connection
between AGM's inequality and BT inequality.
In Section \ref{sec:LW-algo} we present a data-optimal join algorithm for
LW instances, and then
extend this to arbitrary join queries in Section~\ref{sec:all:j}.
We discuss the limits of performance of prior approaches and our
approach in more detail in Section~\ref{sec:limits}. In
Section~\ref{sec:extensions}, we describe several extensions. We conclude in
Section~\ref{sec:conc}.
\subsection*{Related Work}
Grohe and Marx \cite{GM06} made the first (implicit) connection between
fractional edge cover and the output size of a conjunctive query.
(Their results were stated for constraint satisfaction problems.)
Atserias, Grohe, and Marx \cite{AGM08} extended Grohe and Marx's results
in the database setting.
The first relevant AGM's result is the following inequality.
Consider a join query over relations $R_e$, $e\in E$, where $E$ is a
collection of subsets of an attribute ``universe'' $V$, and
relation $R_e$ is on attribute set $e$.
Then, the number of output tuples is bounded above by
$\prod_{e\in E}|R_e|^{x_e}$, where $\mv x = (x_e)_{e\in E}$ is an
{\em arbitrary} fractional cover of the hypergraph $H=(V,E)$.
They also showed that this bound
is tight. In particular, for infinitely many positive integers $N$ there is
a database instance with $|R_e|=N$, $\forall e\in E$, and the
upper bound gives the actual number of output tuples.
When the sizes $|R_e|$ were given as inputs to the (output size estimation)
problem, obviously the best upper bound is obtained by picking the fractional
cover $\mv x$ which minimizes the linear objective function
$\sum_{e\in E}(\log|R_e|)\cdot x_e$.
In this ``size constrained'' case, however, their lower bound is
off from the upper bound by
a factor of $2^n$, where $n$ is the total number of attributes.
AGM also presented an inapproximability result which justifies this gap.
Note, however, that the gap is only dependent on the query size
and the bound is still asymptotically optimal in the
data-complexity sense.
The second relevant result from AGM is a join-project plan with
running time $O\left(|q|^2 N_{\text{max}}^{1+ \sum x_e}\right)$,
where $N_{\text{max}}$ is the maximum size of input relations
and $|q| = |V|\cdot|E|$ is the query size.
The AGM's inequality contains as a special case the discrete versions
of two well-known inequalities in
geometry: the \textit{Loomis-Whitney} (LW)
inequality~\cite{MR0031538} and its generalization the
\textit{Bollob\'as-Thomason} (BT) inequality~\cite{MR1338683}.
There are two typical proofs of the discrete LW and BT inequalities.
The first proof is by induction using H\"older's inequality~\cite{MR1338683}.
The second proof (see Lyons and Peres~\cite{Lyons-Peres})
essentially uses ``equivalent'' entropy inequalities
by Han \cite{MR0464499} and its generalization by Shearer \cite{MR859293},
which was also the route Grohe and Marx \cite{GM06} took to prove AGM's bound.
All of these proofs are non-constructive.
There are many applications of the discrete LW and BT inequalities.
The $n=3$ case of the LW inequality was used to prove communication
lower bounds for matrix multiplication on distributed memory parallel
computers \cite{DBLP:journals/jpdc/IronyTT04}. The inequality was used
to prove submultiplicativity inequalities regarding sums of sets of
integers~\cite{MR2676833}. In~\cite{Lehman-Lehman}, a special case of
BT inequality was used to prove a network-coding bound. Recently, some
of the authors of this paper have used our \textit{algorithmic}
version of the LW inequality to design a new sub-linear time decodable
compressed sensing matrices~\cite{GNPRS} and efficient pattern
matching algorithms~\cite{NPR-pattern}.
Inspired by AGM's results, Gottlob, Lee, and Valiant~\cite{GLVV09}
provided bounds for conjunctive queries with functional
dependencies. For these bounds, they defined a new notion of
``coloring number'' which comes from the dual linear program of the
fractional cover linear program. This allowed them to generalize
previous results to all conjunctive queries, and to study several
problems related to tree-width.
Join processing algorithms are one of the most studied algorithms in
database research. A staggering number of variants have been
considered, we list a few: Block-Nested loop join, Hash-Join, Grace,
Sort-merge (see Grafe~\cite{graefe93} for a survey). Conceptually, it
is interesting that none of the classical algorithms consider
performing a per-tuple cardinality estimation as our algorithm
does. It is interesting future work to implement our algorithm to
better understand its performance.
Related to the problem of estimating the size of an output is
cardinality estimation. A large number of structures have been
proposed for cardinality
estimation~\cite{Ioannidis03,DBLP:conf/sigmod/PoosalaIHS96,wavelet,DBLP:conf/pods/AlonGMS99,DBLP:conf/vldb/KonigW99,DBLP:conf/vldb/JagadishKMPSS98},
they have all focused on various sub-classes of queries and deriving
estimates for arbitrary query expressions has involved making
statistical assumptions such as the independence and containment
assumptions which result in large estimation
errors~\cite{ioannidischristodoulakis91}. Follow-up work has
considered sophisticated probability models,
Entropy-based models~\cite{DBLP:conf/vldb/MarklMKTHS05,isomere:2006}
and graphical models~\cite{DBLP:journals/pvldb/TzoumasDJ11}. In contrast,
in this work we examine the {\em worst case behavior} of algorithms in
terms of its cardinality estimates.
In the special case when the join graph is acyclic, there are several
known results which achieve (near) optimal run time with respect to the
output size \cite{DBLP:conf/pods/PaghP06, DBLP:journals/jcss/Willard96}.
On a technical level, the work {\em adaptive query processing} is
related, e.g., Eddies~\cite{DBLP:conf/sigmod/HellersteinA00} and
RIO~\cite{DBLP:conf/sigmod/BabuBD05}. The main idea is that to
compensate for bad statistics, the query plan may adaptively be
changed (as it better understands the properties of the data). While
both our method and the methods proposed here are adaptive in some
sense, our focus is different: this body of work focuses on heuristic
optimization methods, while our focus is on provable worst-case
running time bounds. A related idea has been considered in practice:
heuristics that split tuples based on their fanout have been deployed
in modern parallel databases to handle
skew~\cite{DBLP:conf/sigmod/XuKZC08}. This idea was not used to
theoretically improve the running time of join algorithms. We are
excited by the fact that a key mechanism used by our algorithm has
been implemented in a modern commercial system.
\section{Main Results}
\newcommand{\lf}[1]{\mathcal{L}(#1)}
\algsetup{indent=2em}
\renewcommand{\algorithmiccomment}[1]{// {\em #1}}
\section{Algorithm for Loomis-Whitney instances}
\label{sec:LW-algo}
We first consider queries whose forms are slightly more general than
that in our motivating example (\ref{ex:triangle}).
This class of queries has the same setup as in LW inequality
of Theorem \ref{thm:LW}.
In this spirit, we define a {\em Loomis-Whitney
(LW) instance} of the OJ problem to be a hypergraph $H=(V,E)$ such that
$E$ is the collection of all subsets of $V$ of size $|V| - 1$. When the
LW inequality is applied to this setting, it guarantees
that $|\Join_{e\in E} R_e| \leq \left(\prod_{e\in
E}N_e\right)^{1/(n-1)}$, and the bound is tight in the
worst case. The main result of this section is the following:
\begin{thm}[Loomis-Whitney instance]
\label{thm:LW-n-1w}\label{THM:LW-N-1W}
Let $n\ge 2$ be an integer.
Consider a Loomis-Whitney instance $H=(V=[n], E)$ of the OJ problem
with input relations $R_e$, where $|R_e|=N_e$ for $e\in E$.
Then the join $\Join_{e\in E} R_e$ can be computed in time
\[ O\left(n^2\cdot \left(\prod_{e\in E}N_e \right)^{1/(n-1)}+
n^2\sum_{e\in E} N_e\right).
\]
\label{thm:lw:alg}
\end{thm}
Before proving this result, we give an example that illustrates the
intuition behind our algorithm and solve the motivating example from
the introduction (\ref{eq:q}).
\begin{example}\label{ex:lw:one}
Recall that our input has three relations $R(A,B)$, $S(B,C)$,
$T(A,C)$ and an instance $I$ such that $|R^I| = |S^I| = |T^I| = N$.
Let $J = R \Join S \Join T$.
Our goal is to construct $J$ in time $O(N^{3/2})$. For
exposition, define a parameter $\tau \geq 0$ that we will choose
below. We use $\tau$ to define two sets that effectively partition
the tuples in $R^I$.
\[ D = \{ t_{B} \in \pi_{B}(R) : |R^{I}[t_{B}] | > \tau \} \text{ and } G = \{ (t_A,t_B) \in R^I : t_B \not\in D \} \]
\noindent
Intuitively, $D$ contains the heavy join keys in $R$. Note that $|D|< N/\tau$. Observe that
$J \subseteq (D \times T) \cup (G \Join S)$ (also note that this union
is disjoint). Our algorithm will construct $D \times T$ (resp. $G \Join
S$) in time $O(N^{3/2})$, then it will filter out those tuples in
both $S$ and $R$ (resp. $T$) using the hash tables on $S$ and $R$
(resp. $T$); this process produces exactly $J$.
Since our running time is linear in the
above sets, the key question is how big are these two sets?
Observe that $|D \times T| \leq (N/ \tau) N = N^2/\tau$ while $|G
\Join S| = \sum_{t_B \in \pi_{B}(G) } |R[t_B]| |S[t_B]| \leq \tau
N$. Setting $\tau = \sqrt{N}$ makes both
terms at most $N^{3/2}$ establishing the running time of our
algorithm. One can check that if the relations are of different
cardinalities, then we can still use the same algorithm; moreover, by
setting $\tau = \sqrt{\frac{|R||T|}{|S|}}$, we achieve a running time
of $O(\sqrt{|R||S||T|} + |R| + |S| + |T|)$.
\end{example}
To describe the general algorithm underlying Theorem~\ref{thm:lw:alg}, we need to
introduce some data structures and notation.
\paragraph*{Data Structures and Notation}
Let $H=(V,E)$ be an LW instance. Algorithm~\ref{algo:LW} begins by
constructing a labeled, binary tree $\mathcal{T}$ whose set of leaves is
exactly $V$ and each internal node has exactly two children. Any
binary tree over this leaf set can be used. We denote the left child
of any internal node $x$ as $\text{\sc lc}(x)$ and its right child as
$\text{\sc rc}(x)$. Each node $x \in \mathcal{T}$ is labeled by a function $\text{\sc label}$, where
$\text{\sc label}(x) \subseteq V$ are defined inductively as follows: $\text{\sc label}(x) = V
\setminus \{x\}$ for a leaf node $x\in V$, and $\text{\sc label}(x) = \text{\sc label}(\text{\sc lc}(x))
\cap \text{\sc label}(\text{\sc rc}(x))$ if $x$ is an internal node of the tree. It is
immediate that for any internal node $x$ we have $\text{\sc label}(\text{\sc lc}(x)) \cup
\text{\sc label}(\text{\sc rc}(x)) = V$ and that $\text{\sc label}(x) = \emptyset$ if and only if $x$ is
the root of the tree. Let $J$ denote the output set of tuples of the
join, i.e. $J = \Join_{e \in E} R_e$.
For any node $x \in \mathcal{T}$, let $\mathcal{T}(x)$ denote the subtree of $\mathcal{T}$ rooted
at $x$, and $\lf{\mathcal{T}(x)}$ denote the set of leaves under this subtree.
For any three relations $R$, $S$, and $T$,
define $R \Join_S T = (R \Join T) \lJoin S$.
\paragraph*{Algorithm for LW instances}
Algorithm~\ref{algo:LW} works in two stages.
Let $u$ be the root of the tree $\mathcal{T}$.
First we compute a tuple set $\deter{u}$ containing the output
$J$ such that $\deter{u}$ has a relatively small size
(at most the size bound times $n$).
Second, we prune those tuples that cannot
participate in the join (which takes only linear time in the size of
$\deter{u}$).
The interesting part is how we compute $\deter{u}$.
Inductively, we compute a set $\deter{x}$ that at each stage contains
candidate tuples and an auxiliary set $\ndeter{x}$,
which is a superset of the projection
$\pi_{\text{\sc label}(x)}(J \setminus \deter{x})$.
The set $\ndeter{x}$ will intuitively allow
us to deal with those tuples that would blow up the size of an intermediate
relation. The key novelty in Algorithm~\ref{algo:LW} is the
construction of the set $G$ that contains all those tuples (join keys)
that are in some sense {\em light}, i.e., joining over them
would not exceed the size/time bound $P$ by much.
The elements that are not light are postponed to be processed later
by pushing them to the set $\ndeter{x}$. This is in full analogy to the
sets $G$ and $D$ defined in Example~\ref{ex:lw:one}.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\STATE An LW instance: $R_{e}$ for $e \in {V \choose |V| -1}$ and
$N_{e} = |R_e|$.
\STATE $P = \prod_{e \in E} N_e^{1/(n-1)}$ (the size
bound from LW inequality)
\STATE $u \leftarrow \text{root}(\mathcal{T})$; $(\deter{u}, \ndeter{u}) \leftarrow {\sf LW}(u)$
\STATE ``Prune'' $\deter{u}$ and return
\end{algorithmic}
\textsf{LW}$(x)$ : $x \in \mathcal{T}$ returns $(C,D)$
\begin{algorithmic}[1]
\IF{$x$ is a leaf}
\RETURN $(\emptyset,R_{\text{\sc label}(x)})$
\ENDIF
\STATE $(C_L, D_L) \leftarrow {\sc LW}(\text{\sc lc}(x))$ and $(C_R, D_R) \leftarrow {\sc LW}(\text{\sc rc}(x))$
\STATE $F \leftarrow \pi_{\text{\sc label}(x)}(D_L) \cap \pi_{\text{\sc label}(x)}(D_R)$
\STATE $G \leftarrow \left\{ \mv t \in F : |D_{L}[\mv t]| + 1 \leq \lceil P / |D_R| \rceil \right\}$
\COMMENT{$F =G= \emptyset$ if $|D_R|=0$}
\IF{$x$ is the root of $\mathcal{T}$}
\STATE $C \leftarrow (D_L \Join D_R) \cup C_L \cup C_R$
\STATE $D \leftarrow \emptyset$
\ELSE
\STATE $C \leftarrow (D_L \Join_G D_R) \cup C_L \cup C_R$
\STATE $D \leftarrow F \setminus G$.
\ENDIF
\RETURN $(C,D)$
\end{algorithmic}
\caption{Algorithm for Loomis-Whitney Instances}
\label{algo:LW}
\end{algorithm}
\begin{proof}[Proof of Theorem~\protect{\ref{thm:LW-n-1w}}]
We claim that the following three properties hold for every node $x \in \mathcal{T}$:
\begin{itemize}
\item[(1)] $\pi_{\text{\sc label}(x)}(J \setminus \deter{x}) \subseteq \ndeter{x}$;
\item[(2)] $|\deter{x}|\le (|\lf{\mathcal{T}(x)}|-1) \cdot P$; and
\item[(3)] $|\ndeter{x}|\le \min\left\{ \min_{l \in \lf{\mathcal{T}(x)}} \{ N_{[n]\setminus\{l \}}\}, \frac{\prod_{l \in \lf{\mathcal{T}(x)}} N_{[n]\setminus\{l \}}}{P^{|\lf{\mathcal{T}(x)}|-1}} \right\}$.
\end{itemize}
Assuming these three properties hold, let us first prove that that
Algorithm~\ref{algo:LW} correctly computes the join, $J$. Let $u$ denote the root of the tree
$\mathcal{T}$. By property (1),
\begin{eqnarray*}
\pi_{\text{\sc label}(\text{\sc lc}(u))}(J \setminus \deter{\text{\sc lc}(u)}) &\subseteq& \ndeter{\text{\sc lc}(u)}\\
\pi_{\text{\sc label}(\text{\sc rc}(u)))}(J \setminus \deter{\text{\sc rc}(u)}) &\subseteq& \ndeter{\text{\sc rc}(u)}
\end{eqnarray*}
Hence,
\[ J \setminus (\deter{\text{\sc lc}(u)} \cup \deter{\text{\sc rc}(u)}) \subseteq
\ndeter{\text{\sc lc}(u)} \times \ndeter{\text{\sc rc}(u)} =
\ndeter{\text{\sc lc}(u)} \Join \ndeter{\text{\sc rc}(u)}.
\]
This implies $J \subseteq \deter{u}$. Thus, from
$\deter{u}$ we can compute $J$ by keeping only tuples in $\deter{u}$
whose projection on any attribute set $e\in E = \binom{[n]}{n-1}$ is
contained in $R_e$ (the ``pruning'' step).
\newcommand{\rchld}[1]{\text{\sc rc}(#1)}
\newcommand{\lchld}[1]{\text{\sc lc}(#1)}
We next show that properties 1-3 hold by induction on each step of
Algorithm~\ref{algo:LW}. For the base case, consider
$\ell\in\lf{\mathcal{T}}$. Recall that in this case $\deter{\ell}=\emptyset$
and $\ndeter{\ell} = R_{[n]- \{\ell\}}$; thus, properties 1-3 hold.
Now assume that properties 1-3 hold for all children of an internal node $v$.
We first verify properties 2-3 for $v$. From the definition of $G$,
\[ |\ndeter{\rchld{v}} \Join_G \ndeter{\lchld{v}}|\le
\left(\left\lceil \frac{P}{|\ndeter{\rchld{v}}|}\right\rceil-1\right)
\cdot |\ndeter{\rchld{v}}|\le P.
\]
From the inductive upper bounds on $\deter{\lchld{v}}$ and $\deter{\rchld{v}}$,
property 2 holds at $v$.
By definition of $G$ and a straightforward counting argument, note that
\[ |\ndeter{v}|=|F\setminus G| \le |\ndeter{\lchld{v}}|\cdot
\frac{1}{\lceil P/|\ndeter{\rchld{v}}|\rceil}\le
\frac{|\ndeter{\lchld{v}}| \cdot |\ndeter{\rchld{v}}|}{P}. \]
From the induction hypotheses on $\lchld{v}$ and $\rchld{v}$, we have
\begin{eqnarray*}
|\ndeter{\lchld{v}}| &\le& \frac{\prod_{\ell\in\lf{\mathcal{T}(\lchld{v})}} N_{[n]-\{\ell\}}}{P^{|\lf{\mathcal{T}(\lchld{v})}|-1}}\\
|\ndeter{\rchld{v}}| &\le& \frac{\prod_{\ell\in\lf{\mathcal{T}(\rchld{v})}} N_{[n]-\{\ell\}}}{P^{|\lf{\mathcal{T}(\rchld{v})}|-1}},
\end{eqnarray*}
which implies that
\[ |\ndeter{{v}}|\le \frac{\prod_{\ell\in\lf{\mathcal{T}({v})}} N_{[n]-\{\ell\}}}{P^{|\lf{\mathcal{T}({v})}|-1}}.
\]
Further, it is easy to see that
\[ |\ndeter{{v}}|\le \min(|\ndeter{\lchld{v}}|, |\ndeter{\rchld{v}}|), \]
which by induction implies that
\[ |\ndeter{v}|\le \min_{\ell\in \lf{\mathcal{T}(v)}} N_{[n]-\{\ell\}}. \]
Property 3 is thus verified.
Finally, we verify property 1. By induction, we have
\begin{eqnarray*}
\pi_{\text{\sc label}(\lchld{v})}(J\setminus\deter{\lchld{v}}) &\subseteq&
\ndeter{\lchld{v}}\\
\pi_{\text{\sc label}(\rchld{v})}(J\setminus \deter{\rchld{v}}) &\subseteq&
\ndeter{\rchld{v}}
\end{eqnarray*}
This along with the fact that $\text{\sc label}(\lchld{v}) \cap \text{\sc label}(\rchld{v}) = \text{\sc label}(v)$
implies that
\[ \pi_{\text{\sc label}(v)}(J\setminus \deter{\lchld{v}} \cup \deter{\rchld{v}})
\subseteq \ndeter{\lchld{v}}_{\text{\sc label}(v)}\cap \ndeter{\rchld{v}}_{\text{\sc label}(v)} =
G\uplus D(v).
\]
Further, every tuple in
$(J\setminus\deter{\lchld{v}}\cup\deter{\rchld{v}})$ whose projection onto
$\text{\sc label}(v)$ is in $G$ also belongs to
$\ndeter{\rchld{v}} \Join_G \ndeter{\lchld{v}}$.
This implies that
$\pi_{\text{\sc label}(v)}(J\setminus \deter{v})= \ndeter{v}$,
as desired.
For the run time complexity of Algorithm~\ref{algo:LW}, we claim that for
every node $x$, we need time $O(n|\deter{x}|+n|\ndeter{x}|)$. To see
this note that for each node $x$, the lines 4, 5, 7, and 9 of
the algorithm can be computed in that much time using hashing.
Using property (3) above, we have a (loose) upper bound of
$O\left(nP+ n\min_{l \in \lf{\mathcal{T}(x)}} N_{[n]\setminus\{l \}}\right)$ on
the run time for node $x$.
Summing the run time over all the nodes in
the tree gives the claimed run time.
\end{proof}
\section{An Algorithm for All Join Queries}
\label{sec:all:j}\label{SEC:ALL:J}
This section presents our algorithm for proving the AGM's inequality
with running time matching the bound.
\begin{thm}
Let $H=(V,E)$ be a hypergraph representing a natural join query.
Let $n=|V|$ and $m=|E|$.
Let $\mv x = (x_e)_{e\in E}$ be an arbitrary point in the fractional
cover polytope
\begin{eqnarray*}
\sum_{e : v \in e} x_e &\geq& 1, \ \text{ for any $v \in V$}\\
x_e &\geq &0, \text{ for any $e \in E$}
\end{eqnarray*}
For each $e\in E$, let $R_e$ be a relation of size $N_e=|R_e|$
(number of tuples in the relation). Then,
\begin{itemize}
\item[(a)] The join $\Join_{e\in E}R_e$ has size (number of tuples) bounded by
\[ |\Join_{e\in E}R_e| \leq \prod_{e\in E}N_e^{x_e}. \]
\item[(b)] Furthermore, the join $\Join_{e\in E}R_e$ can be computed
in time
\[ O\left(mn\prod_{e\in E}N_e^{x_e} + n^2\sum_{e\in E}N_e + m^2n\right) \]
\end{itemize}
\label{thm:main}
\end{thm}
\begin{rmk} In the running time above, $m^2n$ is the query preprocessing time,
$n^2\sum_{e\in E}N_e$ is the data preprocessing time, and
$mn\prod_{e\in E}N_e^{x_e}$ is the query evaluation time.
If all relations in the database are indexed in advance
to satisfy three conditions (HTw), $w\in\{1,2,3\}$, below,
then we can remove the term $n^2\sum_{e\in E}N_e$ from the running time.
Also, the fractional cover solution $\mv x$ should probably be the
best fractional cover in terms of the linear objective
$\sum_e (\log N_e)\cdot x_e$.
The data-preprocessing time of $O(n^2\sum_eN_e)$ is for a single known query.
If we were to index all relations in advance without knowing which
queries to be evaluated, then the advance-indexing takes
$O(n\cdot n! \sum_e N_e)$-time.
This price is paid once, up-front, for an arbitrary number of
future queries.
\end{rmk}
Before turning to our algorithm and proof of this theorem, we observe
that a consequence of this theorem is the following algorithmic
version of the discrete version of BT inequality.
\begin{cor}
Let $S\subset \mathbb Z^n$ be a finite set of $n$-dimensional grid points.
Let $\mathcal F$ be a collection of subsets of $[n]$ in which every
$i \in [n]$ occurs in exactly $d$ members of $\mathcal F$.
Let $S_F$ be the set of projections $\mathbb Z^n \to \mathbb Z^F$ of
points in $S$ onto the coordinates in $F$. Then,
\begin{equation}
|S|^d \leq \prod_{F \in \mathcal F} |S_F|.
\label{eqn:BT}
\end{equation}
Furthermore, given the projections $S_F$ we can compute $S$ in time
\[ O\left( |\mathcal F|n\left(\prod_{F \in \mathcal F} |S_F|\right)^{1/d}
+ n^2\sum_{F\in \mathcal F}|S_F|+|\mathcal F|^2n \right) \]
\end{cor}
Recall that the LW inequality is a special case of the BT inequality.
Hence, our algorithm proves the LW inequality as well.
\subsection{Main ingredients of the algorithm}
There are three key ingredients in the algorithm
(Algorithm \ref{algo:the-algo}) and its analysis:
\begin{enumerate}
\item We first build a ``search tree'' for each relation $R_e$ which will be
used throughout the algorithm.
We can also build a collection of hash indices which functionally can
serve the same purpose. We use the ``search tree'' data structure here to
make the analysis clearer.
This step is responsible for the (near-) linear
term $O(n^2\sum_{e\in E}N_e)$ in the running time.
The search tree for each relation is built using
a particular ordering of attributes in the relation called the total order.
The total order is constructed from a data structure called
a query plan tree which also drives the recursion structure of the algorithm.
\item Suppose we have two relations $A$ and $B$ on the same set of attributes
and we'd like to compute $A\cap B$. If the search trees for $A$ and $B$
have already been built, the intersection can be computed in time
$O(k\min\{|A|,|B|\})$ where $k$ is the number of attributes in $A$,
because we can traverse every tuple of
the smaller relation and check into the search structure for the larger
relation. Also note that, for any two non-negative numbers
$a$ and $b$ such that $a+b\geq 1$, we have
$\min\{|A|, |B|\} \leq |A|^a|B|^b$.
\item The third ingredient is based on 'unrolling' sums using
generalized H\"older inequality \eqref{ineq:Holder}
in a correct way. We cannot explain it in a few lines and thus will
resort to an example presented in the next section.
The example should give the reader the correct intuition into the
entire algorithm and its analysis without getting lost
in heavy notations.
\end{enumerate}
We make extensive use of the following form of H\"older's inequality
which was also attributed to Jensen.
(See the classic book ``Inequalities'' by
Hardy, Littlewood, and P\'olya \cite{MR89d:26016},
Theorem 22 on page 29.)
\begin{lmm}[Hardy, Littlewood, and P\'olya~\cite{MR89d:26016}]
\label{lem:holder}
Let $m,n$ be positive integers. Let $y_1,\dots,y_n$ be non-negative
real numbers such that $y_1+\cdots+y_n\geq 1$. Let $a_{ij} \geq 0$ be
non-negative real numbers, for $i\in [m]$ and $j\in [n]$. With the
convention $0^0 = 0$, we have:
\begin{equation}
\sum_{i=1}^m\prod_{j=1}^n a_{ij}^{y_j}
\leq
\prod_{j=1}^n\left(\sum_{i=1}^m a_{ij}\right)^{y_j}.
\label{ineq:Holder}
\end{equation}
\end{lmm}
For each tuple $\mv t$ on attribute set $A$, we will
write $\mv t$ as $\mv t_A$ to emphasize the support of $\mv t$:
$\mv t_A = (t_a)_{a\in A}$.
Consider any
relation $R$ with attribute set $S$. Let $A\subset S$ and $\mv t_A$
be a fixed tuple.
Then, $\pi_A(R)$ denote the projection of $R$ down to attributes in $A$.
And, define the {\em $\mv t_A$-section} of $R$ to be
\[ R[\mv t_A] := \pi_{S-A}(R \lJoin \{\mv t_A\})
= \{ \mv t_{S-A} \ | \ (\mv t_A, \mv t_{S-A}) \in R\}.
\]
In particular, $R[\mv t_\emptyset] = R$.
\subsection{A complete worked example for our algorithm and its analysis}
\label{subsec:an-example}
Before presenting the algorithm and analyze it formally, we first work out a
small query to explain how the algorithm and the analysis works.
It should be noted that the following example does not cover all
the intricacies of the general algorithm, especially in the boundary cases.
We aim to convey the intuition first.
Also, the way we label nodes in the QP-tree in this example is slightly
different from the way nodes are labeled in the general algorithm,
in order to avoid heavy sub-scripting.
Consider the following instance to the OJ problem.
The hypergraph $H$ has 6 attributes $V = \{1,\dots,6\}$, and 5 relations
$R_a,R_b,R_c,R_d,R_e$ defined by the following vertex-edge incident matrix
$\mv M$:.
\[
\mv M =
\begin{array}{l||ccccc}
& a & b & c & d & e\\
\hline
\hline
\rowcolor{blue}
1& 1 & 1 & 1 & 0 & 0\\
\rowcolor{lightgray}
2& 1 & 0 & 1 & 1 & 0\\
\rowcolor{green}
3& 0 & 1 & 1 & 0 & 1\\
\rowcolor{lightgray}
4& 1 & 1 & 0 & 1 & 0\\
\rowcolor{green}
5& 1 & 0 & 0 & 0 & 1\\
\rowcolor{green}
6& 0 & 1 & 0 & 1 & 1
\end{array}
\]
We are given a fractional cover solution
$\mv x=(x_a,x_b,x_c,x_d,x_e)$, i.e.
$\mv M\mv x \geq \mv 1$.
{\bf Step 0.}
We first build something called a {\em query plan tree} (QP-tree).
The tree has nodes labeled by the hyperedge $a,b,c,d,e$,
except for the leaf nodes each of which can be labeled by a subset of
hyperedges.
(Note again that the labeling in this example is slightly different from the
labeling done in the general algorithm's description to avoid
cumbersome notations.)
Each node of the query plan tree also has an associated {\em universe}
which is a subset of attributes.
The reader is referred to Figure \ref{fig:example-join-tree} for
an illustration of the tree building process.
In the figure, the universe for each node is drawn next to the parent edge
to the node.
\begin{figure}[t]
\centerline{\includegraphics[width=4in]{join-tree.png}}
\label{fig:example-join-tree}
\caption{A query plan tree for the example OJ instance}
\end{figure}
The query plan tree is built recursively as follows.
We first arbitrarily order the hyperedges. In the example shown in Figure
\ref{fig:example-join-tree},
we have built a tree with the order $e,d,c,b,a$.
The root node has universe $V$. We visit these edges one by one
in that order.
If every remaining hyperedges contains
the universe $V$ then label the node with all remaining hyperedges and stop.
In this case the node is a leaf node.
Otherwise, consider the next hyperedge in the visiting order
above (it is $e$ as we are in the beginning).
Label the root with $e$, and
create two children of the root $e$.
The left child will have universe $V-e$, and the right child has
$e$ as its universe.
Now, we recursively build the left tree starting from the next
hyperedge (i.e. $d$) in the ordering,
but only restricting to the smaller universe
$\{1,2,4\}$.
Similarly, we build the right tree starting from the next hyperedge ($d$) in the
ordering, but only restricting to the smaller universe $\{3,5,6\}$.
Let us explain one more level of the tree building process to make things clear.
Consider the left tree of the root node $e$.
The universe is $\{1,2,4\}$. The root node will be the next hyperedge $d$
in the ordering. But we really work on the restriction of
$d$ in the universe $\{1,2,4\}$, which is $d'=d\cap\{1,2,4\}=\{2,4\}$.
Then, we create two children. The left child has universe
$\{1,2,4\}-d' = \{1\}$. The right child has universe $d'=\{2,4\}$.
For the left child, the universe has size $1$ and all three remaining
hyperedges $a$, $b$, and $c$ contain $1$, hence we label the left
child with $abc$.
By visiting all leaf nodes from left to right and print
the attributes in their universes,
we obtain something called {\em the total order} of all attributes in $V$.
In the figure, the total order is $1,4,2,5,3,6$.
(In the general case, the total order is slightly more complicated than
in this example. See Procedure \ref{algo:total-order}.)
Finally, based on the total order $1,4,2,5,3,6$ just obtained,
we build search trees for all relations respecting this ordering.
For relation $R_a$, the top level of the tree is indexed over attribute
$1$, the next two levels are $4$ and $2$, and the last level
is indexed over attribute $5$.
For $R_b$, the order is $1, 4, 3, 6$.
For $R_c$, the order is $1, 2, 3$.
For $R_d$, the order is $4, 2, 6$.
For $R_e$, the order is $5, 3, 6$.
It will be clear later that the attribute orders in the search trees
have a decisive effect on the overall running time.
This is also the step that is responsible for the
term $O(n^2 \sum_e N_e)$ in the overall running time.
{\bf Step 1.}
(This step corresponds to the left most node of the query plan tree.)
Compute the join
\begin{equation}
T_1 = \pi_{\{1\}}(R_a) \Join \pi_{\{1\}}(R_b) \Join \pi_{\{1\}}(R_c)
\label{eqn:T1-join}
\end{equation}
as follows.
This is the join over attributes {\em not} in $d$ and $e$.
If $|\pi_{\{1\}}(R_a)|$ is the smallest among
$|\pi_{\{1\}}(R_a)|$, $|\pi_{\{1\}}(R_b)|$, and
$|\pi_{\{1\}}(R_c)|$, then for each attribute
$t_1 \in \pi_{\{1\}}(R_a)$, we search the first levels of the search trees
for $R_b$ and $R_c$ to see if $t_1$ is in both
$\pi_{\{1\}}(R_b)$ and $\pi_{\{1\}}(R_b)$.
Similarly, if $|\pi_{\{1\}}(R_b)|$ or
$|\pi_{\{1\}}(R_c)|$ is the smallest then
for each $t_1 \in \pi_{\{1\}}(R_b)$ (or in $\pi_{\{1\}}(R_c)$) we search
for attribute $t_1$ in the other two search trees.
As attribute $1$ is in the first level of all three search trees,
the join \eqref{eqn:T1-join} can be computed in time
\[ O(|T_1|) = O\left(\min\left\{|\pi_{\{1\}}(R_a)|,
|\pi_{\{1\}}(R_b)|,
|\pi_{\{1\}}(R_c)|\right\}\right).
\]
Note that
\[ |T_1| \leq \min\left\{|\pi_{\{1\}}(R_a)|,
|\pi_{\{1\}}(R_b)|,
|\pi_{\{1\}}(R_c)|\right\}
\leq
|\pi_{\{1\}}(R_a)|^{x_a}
|\pi_{\{1\}}(R_b)|^{x_b}
|\pi_{\{1\}}(R_c)|^{x_c}
\leq
N_a^{x_a}N_b^{x_b}N_c^{x_c}
\]
because $x_a+x_b+x_c\geq 1$.
In particular, step 1 can be performed within the run-time budget.
{\bf Step 2.}
(This step corresponds to the node labeled $d$ on the left branch
of query plan tree.)
Compute the join
\[ T_{\{1,2,4\}} =
\pi_{\{1, 2,4\}}(R_a) \Join
\pi_{\{1,4\}}(R_b) \Join
\pi_{\{1,2\}}(R_c) \Join
\pi_{\{2,4\}}(R_d)
\]
This is a join over all attributes {\em not} in $e$.
Since we have already computed the join $T_1$ over attribute $1$
of $R_a$, $R_b$, and $R_c$, the relation
$T_{\{1,2,4\}}$ can be computed by computing
for every $t_1\in T_1$ the $t_1$-section of $T_{\{1,2,4\}}$
\[ T_{\{1,2,4\}}[t_1] =
\underbrace{\pi_{\{2,4\}}(R_a[t_1])}_{A[t_1]} \Join
\underbrace{\pi_{\{4\}}(R_b[t_1])}_{B[t_1]} \Join
\underbrace{\pi_{\{2\}}(R_c[t_1])}_{C[t_1]} \Join
\underbrace{\pi_{\{2,4\}}(R_d)}_{D}
\]
and then $T_{\{1,2,4\}}$ is simply the union of all the $t_1$-sections
$T_{\{1,2,4\}}[t_1]$.
The notations $A[t_1]$, $B[t_1]$, $C[t_1]$, and $D$ are defined for
the sake of brevity.
Fix $t_1 \in T_1$, we next describe how $T_{\{1,2,4\}}[t_1]$ is computed.
If $x_d\geq 1$ then we go directly to case 2b below.
When $x_d<1$, define
\begin{eqnarray*}
x'_a&=&\frac{x_a}{1-x_d}\\
x'_b&=&\frac{x_b}{1-x_d}\\
x'_c&=&\frac{x_c}{1-x_d}.
\end{eqnarray*}
Consider the hypergraph graph $H'$ which is the graph $H$ restricted
to the vertices $2,4$ and edges $a,b,c$.
In particular, $H'$ has vertex set $\{2,4\}$ and edges $\{2,4\}, \{4\}, \{2\}$.
It is clear that $x'_a$,$x'_b$, and $x'_c$ form a fractional cover solution
of $H'$ because $\mv x$ was a fractional cover solution for $H$.
Thus, $H'$, $\mv x'=(x'_a,x'_b,x'_c)$, and $A[t_1]$, $B[t_1]$,
and $C[t_1]$ form an instance of the OJ problem.
We will recursively solve this instance if a condition is satisfied.
{\bf Case 2a.}
Suppose
\[ |A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c} \leq |D| \]
then we (recursively) compute the join
$A[t_1]\Join B[t_1]\Join C[t_1]$.
By induction on the instance $H'$, this join can be computed in time
\[ O\left(|A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c} \right). \]
(This induction hypothesis corresponds to the node labeled $c$
on the left branch of the query plan tree.)
Here, we crucially use the fact that the search trees for $R_a$,
$R_b$, $R_c$ have been built so that the subtrees under the branch
$t_1$ are precisely the search trees for relations $A[t_1], B[t_1], C[t_1]$
and thus are readily available to compute this join.
Now, to get $T_{\{1,2,4\}}[t_1]$ we simply check whether every
tuple in $A[t_1]\Join B[t_1]\Join C[t_1]$ belongs to $D$.
{\bf Case 2b.}
Suppose either $x_d\geq 1$ or
\[ |D| \leq |A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c} \]
then for every tuple $(t_2,t_4)$ in $D$ we check whether
$(t_2,t_4)\in A[t_1]$, $t_4\in B[t_1]$, {\em and}
$t_2\in C[t_1]$. The overall running time is $O(|D|)$.
Thus, for a fixed value $t_1$, the relation $T_{\{1,2,4\}}[t_1]$
can be computed in time
\[ O\left(
\min\{|A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c}, |D|\}
\right).
\]
In fact, it is not hard to see that
\[ |T_{\{1,2,4\}}[t_1]| \leq
\min\{|A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c}, |D|\}. \]
This observation will eventually imply the inequality \eqref{eqn:agm08-bound}
(for this instance), and in the general case leads to the constructive
proof of the inequality \eqref{eqn:agm08-bound}.
Next, note that
\begin{eqnarray*}
\min\left\{|A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c}, |D| \right\}
&\leq&
\left(|A[t_1]|^{x'_a} |B[t_1]|^{x'_b} |C[t_1]|^{x'_c}\right)^{1-x_d} |D|^{x_d}\\
&=&
|A[t_1]|^{x_a} |B[t_1]|^{x_b} |C[t_1]|^{x_c} |D|^{x_d}.
\end{eqnarray*}
If $x_d\geq 1$ then the run-time is also in the order of
$O(|A[t_1]|^{x_a} |B[t_1]|^{x_b} |C[t_1]|^{x_c} |D|^{x_d})$.
Consequently, the total running time for step 2 is in the order of
\begin{eqnarray*}
\sum_{t_1\in T_1} |A[t_1]|^{x_a} |B[t_1]|^{x_b} |C[t_1]|^{x_c} |D|^{x_d}
&=&
|D|^{x_d} \sum_{t_1\in T_1} |A[t_1]|^{x_a} |B[t_1]|^{x_b} |C[t_1]|^{x_c}\\
&\leq&
|D|^{x_d}
\left(\sum_{t_1\in T_1} |A[t_1]|\right)^{x_a}
\left(\sum_{t_1\in T_1} |B[t_1]|\right)^{x_b}
\left(\sum_{t_1\in T_1} |C[t_1]|\right)^{x_c} \\
&\leq&
|D|^{x_d}
\cdot |\pi_{\{1,2,4\}}(R_a)|^{x_a}
\cdot |\pi_{\{1,4\}}(R_b)|^{x_b}
\cdot |\pi_{\{1,2\}}(R_c)|^{x_c}\\
&\leq& N_a^{x_a}N_b^{x_b}N_c^{x_c}N_d^{x_d}.
\end{eqnarray*}
The first inequality follows from generalized H\"older inequality because
$x_a+x_b+x_c\geq =1$ and $x_a,x_b,x_c \geq 0$.
The second inequality says that if we sum over the sizes of the $t_1$-sections,
we get at most the size of the relation.
In summary, step $2$ is still within the running time budget.
{\bf Step 3.} Compute the final join over all attributes
\[ T_{\{1,2,3,4,5,6\}} =
R_a \Join R_b \Join R_c \Join R_d \Join R_e. \]
Since we have already computed the join $T_{\{1,2,4\}}$ over attributes
$1,2,4$
of $R_a$, $R_b$, $R_c$, and $R_d$, the relation
$T_{\{1,2,3,4,5,6\}}$ can be computed by computing
for every $(t_1,t_2,t_4) \in T_{\{1,2,4\}}$ the join
\[
T_{\{1,2,3,4,5,6\}}[t_1,t_2,t_4] =
\underbrace{\pi_{\{5\}}(R_a[t_1,t_2,t_4])}_{A} \Join
\underbrace{\pi_{\{3,6\}}(R_b[t_1,t_4])}_{B} \Join
\underbrace{\pi_{\{3\}}(R_c[t_1,t_2])}_{C} \Join
\underbrace{\pi_{\{6\}}(R_d[t_2,t_4])}_{D} \Join
\underbrace{R_e}_{E},
\]
and return the union of these joins over all
tuples $(t_1,t_2,t_4) \in T_{\{1,2,4\}}$.
Again, the notations $A$, $B$, $C$, $D$, $E$ are introduced to
for the sake of brevity. Note, however, that they are different
from the $A$, $B$, $C$, $D$ from case 2.
This step illustrates the third ingredient of the algorithm's analysis.
Fix $(t_1,t_2,t_4) \in T_{\{1,2,4\}}$.
If $x_e\geq 1$ then we jump to case 3b; otherwise, define
\begin{eqnarray*}
x''_a&=& \frac{x_a}{1-x_e}\\
x''_b&=& \frac{x_b}{1-x_e}\\
x''_c&=& \frac{x_c}{1-x_e}\\
x''_d&=& \frac{x_d}{1-x_e}.
\end{eqnarray*}
Then define a hypergraph $H''$ on the attributes
$\{3,5,6\}$ and the restrictions of $a$, $b$, $c$, $d$
on these attributes. Clearly the vector $\mv x''$ is a fractional cover
for this instance.
{\bf Case 3a}.
Suppose $x_e\geq 1$ or
\[ |A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d} \leq |E|. \]
By applying the induction hypothesis on the $H''$ instance we can
compute the join $A \Join B \Join C \Join D$ in time
$O\left(|A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d}\right)$.
(The induction hypothesis corresponds to the node labeled $d$ on
{\em right} branch of the query plan tree.)
Again, because the search trees for all relations have been built
in such a way that the search trees for $A$, $B$, $C$, $D$
are already present on $t_1,t_2,t_4$-branches of the trees
for $R_a,R_b,R_c$, and $R_d$, there is no extra time spent on indexing
for computing this join.
Then, for every tuple $\mv t_{\{3,5,6\}}$ in the join we check
(the search tree for) $E$ to see if the tuple belongs to $E$.
{\bf Case 3b}.
Suppose
\[ |E| \leq |A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d}. \]
Then, for each tuple $\mv t_{\{3,5,6\}} =(t_3,t_5,t_6) \in E$ we check
to see whether $t_5 \in A, (t_3,t_6)\in B, t_3 \in C$, {\em and}
$t_6 \in D$.
Either way, for a fix tuple $(t_1,t_2,t_4) \in T_{\{1,2,4\}}$
the running time is
\[ \tilde O\left(
\min\left\{
|A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d},
|E|\right\}
\right). \]
Now, we apply the same trick as in case 2:
\begin{eqnarray*}
\min\left\{
|A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d},
|E|\right\}
&\leq&
\left(
|A|^{x''_a} |B|^{x''_b} |C|^{x''_c} |D|^{x''_d}
\right)^{1-x_e}|E|^{x_e}\\
&=&
|A|^{x_a}|B|^{x_b}|C|^{x_c}|D|^{x_d}|E|^{x_e}\\
&\leq&
|R_a[t_1,t_2,t_4]|^{x_a}
|R_b[t_1,t_4]|^{x_b}
|R_c[t_1,t_2]|^{x_c}
|R_d[t_2,t_4]|^{x_d}
|R_e|^{x_e}.
\end{eqnarray*}
Hence, the total running time for step $3$ is in the order of
\begin{eqnarray*}
&&\sum_{(t_1,t_2,t_4)\in T_{\{1,2,4\}}}
|R_a[t_1,t_2,t_4]|^{x_a}
|R_b[t_1,t_4]|^{x_b}
|R_c[t_1,t_2]|^{x_c}
|R_d[t_2,t_4]|^{x_d}
|R_e|^{x_e}\\
&=&
|R_e|^{x_e}
\sum_{t_1}
\sum_{t_2}
\sum_{t_4}
|R_a[t_1,t_2,t_4]|^{x_a}
|R_b[t_1,t_4]|^{x_b}
|R_c[t_1,t_2]|^{x_c}
|R_d[t_2,t_4]|^{x_d}
\end{eqnarray*}
where the first sum is over $t_1\in \pi_{\{1\}}(T_{\{1,2,4\}})$,
the second sum is over $t_2$ such that
$(t_1,t_2)\in \pi_{\{1,2\}}(T_{\{1,2,4\}})$,
and the third sum is over $t_4$
such that $(t_1,t_2,t_4)\in T_{\{1,2,4\}}$.
We apply H\"older inequality several times to ``unroll" the sums.
Note that we crucially use the fact that $\mv x$ is a fractional
cover solution ($\mv M\mv x \geq \mv 1$) to apply H\"older's inequality.
\begin{eqnarray*}
&&|R_e|^{x_e} \sum_{t_1} \sum_{t_2} \sum_{t_4}
|R_a[t_1,t_2,t_4]|^{x_a}
|R_b[t_1,t_4]|^{x_b}
|R_c[t_1,t_2]|^{x_c}
|R_d[t_2,t_4]|^{x_d}\\
&=&|R_e|^{x_e} \sum_{t_1} \sum_{t_2}
|R_c[t_1,t_2]|^{x_c}
\sum_{t_4}
|R_a[t_1,t_2,t_4]|^{x_a}
|R_b[t_1,t_4]|^{x_b}
|R_d[t_2,t_4]|^{x_d}\\
&\leq&|R_e|^{x_e} \sum_{t_1} \sum_{t_2}
|R_c[t_1,t_2]|^{x_c}
\left(\sum_{t_4}|R_a[t_1,t_2,t_4]|\right)^{x_a}
\left(\sum_{t_4}|R_b[t_1,t_4]|\right)^{x_b}
\left(\sum_{t_4}|R_d[t_2,t_4]|\right)^{x_d}\\
&\leq&|R_e|^{x_e} \sum_{t_1} \sum_{t_2}
|R_c[t_1,t_2]|^{x_c}
|R_a[t_1,t_2]|^{x_a}
|R_b[t_1]|^{x_b}
|R_d[t_2]|^{x_d}\\
&=&|R_e|^{x_e} \sum_{t_1}
|R_b[t_1]|^{x_b}
\sum_{t_2}
|R_c[t_1,t_2]|^{x_c}
|R_a[t_1,t_2]|^{x_a}
|R_d[t_2]|^{x_d}\\
&\leq&|R_e|^{x_e} \sum_{t_1}
|R_b[t_1]|^{x_b}
\left(\sum_{t_2} |R_c[t_1,t_2]|\right)^{x_c}
\left(\sum_{t_2} |R_a[t_1,t_2]|\right)^{x_a}
\left(\sum_{t_2} |R_d[t_2]|\right)^{x_d}\\
&\leq&|R_e|^{x_e} \sum_{t_1}
|R_b[t_1]|^{x_b}
|R_c[t_1]|^{x_c}
|R_a[t_1]|^{x_a}
|R_d|^{x_d}\\
&=&|R_e|^{x_e} |R_d|^{x_d}
\sum_{t_1}
|R_b[t_1]|^{x_b}
|R_c[t_1]|^{x_c}
|R_a[t_1]|^{x_a}\\
&\leq&|R_e|^{x_e} |R_d|^{x_d}
\left(\sum_{t_1} |R_b[t_1]|\right)^{x_b}
\left(\sum_{t_1} |R_c[t_1]|\right)^{x_c}
\left(\sum_{t_1} |R_a[t_1]|\right)^{x_a}\\
&\leq&|R_e|^{x_e} |R_d|^{x_d} |R_b|^{x_b} |R_c|^{x_c} |R_a|^{x_a}.
\end{eqnarray*}
\subsection{Rigorous description and analysis of the algorithm}
\label{app:analysis-of-the-algo}
Algorithm \ref{algo:the-algo} computes the join of $m$ given relations.
Beside the relations,
the input to the algorithm consists of the hypergraph $H=(V,E)$
with $|V|=n$, $|E|=m$, and a point $\mv x=(x_e)_{e\in E}$ in the
fractional cover polytope
\begin{eqnarray*}
\sum_{v \in e} x_e &\geq& 1, \ \text{for any $v \in V$}\\
x_e &\geq &0, \text{for any $e \in E$}.
\end{eqnarray*}
\begin{enumerate}
\item We first build a {\em query plan tree}. The query plan tree serves
two purposes: (a) it captures the structure of the recursions
in the algorithm where each node of the tree roughly corresponds to a
sub-problem,
(b) it gives a total order of all the attributes based on which we
can pre-build search trees for all the relations in the next step.
\item From the query plan tree, we construct a total order of all
attributes in $V$. Then, for each relation $R_e$ we construct
a search tree for $R_e$ based on the relative order of $R_e$'s attributes
imposed by the total order.
\item We traverse the query plan tree and solve some of the sub-problems
and combine the solutions to form the final answer.
It is important to note that {\bf not all} sub-problems corresponding to
nodes in the query plan trees will be solved. We decide whether to solve
a sub-problem based on a ``size check." Intuitively, if the sub-problem
is estimated to have a large output size we will try to not solve it.
\end{enumerate}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{algorithm}[h]
\caption{Computing the join $\Join_{e\in E}R_e$}
\begin{algorithmic}[1]
\REQUIRE Hypergraph $H=(V,E)$, $|V|=n$, $|E|=m$
\REQUIRE Fractional cover solution $\mv x = (x_e)_{e\in E}$
\REQUIRE Relations $R_e, e\in E$
\STATE Compute the query plan tree $\mathcal{T}$, let $u$ be $\mathcal{T}$'s root node
\STATE Compute a total order of attributes
\STATE Compute a collection of hash indices for all relations
\RETURN {\sc Recursive-Join}$(u, \mv x, \textnormal{{\sc nil}})$
\end{algorithmic}
\label{algo:the-algo}
\end{algorithm}
We repeat some of the terminologies already defined so that
this section is relatively self-contained.
For each tuple $\mv t$ on attribute set $A$, we will write
$\mv t$ as $\mv t_A$ to signify the fact that the tuple is on the attribute
set $A$: $\mv t_A = (t_a)_{a\in A}$.
Consider any relation $R$ with attribute set $S$.
Let $A\subset S$ and $\mv t_A$ be a fixed tuple.
Then $R[\mv t_A]$ denotes the ``$\mv t_A$-section" of $R$,
which is a relation on $S-A$
consisting of {\em all} tuples $\mv t_{S-A}$ such that
$(\mv t_A, \mv t_{S-A}) \in R$.
In particular, $R[\mv t_\emptyset] = R$.
Let $\pi_A(R)$ denote the projection
of $R$ down to attributes in $A$.
\subsubsection{Step (1): Building the query plan tree}
\label{subsubsec:build-QP-tree}
\begin{algorithm}
\begin{algorithmic}[1]
\STATE Fix an arbitrary order $e_1, e_2, \dots, e_m$ of all the
hyperedges in $E$.
\STATE $\mathcal{T} \leftarrow $ {\sc build-tree}$(V, m)$
\end{algorithmic}
{\sc build-tree}$(U, k)$
\begin{algorithmic}[1]
\IF {$e_i \cap U = \emptyset, \forall i\in[k]$}
\RETURN $\textnormal{{\sc nil}}$
\ENDIF
\STATE Create a node $u$ with $\text{\sc label}(u) \leftarrow k$ and $\text{\sc univ}(u) = U$
\IF {$k>1$ and $\exists i \in [k]$ such that $U\not\subseteq e_i$}
\STATE $\text{\sc lc}(u) \leftarrow $ {\sc build-tree}$(U\setminus e_k, k-1)$
\STATE $\text{\sc rc}(u) \leftarrow $ {\sc build-tree}$(U\cap e_k, k-1)$
\ENDIF
\RETURN $u$
\end{algorithmic}
\caption{Constructing the query plan tree $\mathcal{T}$}
\label{algo:QPT}
\end{algorithm}
Very roughly, each node $x$ and the sub-tree below it forms the
``skeleton'' of a sub-problem. There will be many sub-problems that
correspond to each skeleton.
The value $\text{\sc label}(x)$ points to an ``anchor'' relation for the sub-problem
and $\text{\sc univ}(x)$ is the set of attributes that the sub-problem is joining
on. The anchor relation divides the universe $\text{\sc univ}(x)$ into two
parts to further sub-divide the recursion structure.
Fix an arbitrary order $e_1, e_2, \dots, e_m$ of
{\em all} the hyperedges in $E$.
For notational convenience, for any $k\in [m]$ define
$E_k = \{e_1,\dots,e_k\}$.
The query plan tree $\mathcal{T}$ is a binary tree with the following associated
information:
\begin{itemize}
\item {\em Labels}. Each node of $\mathcal{T}$ has a ``label" $\text{\sc label}(u)$ which is an
integer $k\in [m]$.
\item {\em Universes}. Each node $u$ of $\mathcal{T}$ has a ``universe"
$\text{\sc univ}(u)$ which is a non-empty subset of attributes: $\text{\sc univ}(u)\subseteq V$.
\item Each internal node $u$ of $\mathcal{T}$ has a left child $\text{\sc lc}(u)$
or a right child $\text{\sc rc}(u)$ or both. If a child does not exist then the
child pointer points to $\textnormal{{\sc nil}}$.
\end{itemize}
Algorithm \ref{algo:QPT} builds the query plan tree $\mathcal{T}$.
Very roughly, each node $x$ and the sub-tree below it forms the
``skeleton'' of a sub-problem. There will be many sub-problems that
correspond to each skeleton.
The value $\text{\sc label}(x)$ points to an ``anchor'' relation for the sub-problem
and $\text{\sc univ}(x)$ is the set of attributes that the sub-problem is joining
on. The anchor relation divides the universe $\text{\sc univ}(x)$ into two
parts to further sub-divide the recursion structure.
Note that line 5 and 6 will not be executed if
$U\subseteq e_i, \forall i\in [k]$,
in which case $u$ is a leaf node.
When $u$ is not a leaf node, if $U\subseteq e_k$ then $u$ will not
have a left child ($\text{\sc lc}(u)=\textnormal{{\sc nil}}$).
The running time for this pre-processing step
is $O(m^2n)$.
Figure \ref{fig:sample-QPT} shows a query plan tree
produced by Algorithm \ref{algo:QPT} on an example query.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\parbox[b]{1.75in}{
\begin{align*}
q & = & R_1(A_1,A_2, A_4, A_5)\\
& \Join & R_2(A_1,A_3,A_4,A_6)\\
& \Join &R_3(A_1,A_2,A_3)\\
& \Join & R_4(A_2,A_4,A_6)\\
& \Join & R_5(A_3,A_5,A_6)
\end{align*}}&
\includegraphics[width=2.0in]{sample-QPT}
\end{tabular}
\end{center}
\caption{(a) A query $q$ and (b) a sample QP tree for $q$.}
\label{fig:sample-QPT}
\end{figure}
\subsubsection{Step (2): Computing a total order of the attributes and
building the search trees}
\label{subsubsec:subsec:total-order}
From the query plan tree $\mathcal{T}$, Procedure \ref{algo:total-order}
constructs a total order of all the attributes in $V$.
We will call this ordering {\em the total order} of $V$.
It is not hard to see that the total order satisfies the following
proposition.
\begin{prop}
The total order computed in Algorithm
\ref{algo:total-order} satisfies the following
properties
\begin{itemize}
\item[(TO1)] For every node $u$ in the query plan tree $\mathcal{T}$, all members
of $\text{\sc univ}(u)$ are consecutive in the total order
\item[(TO2)] For every internal node $u$, if $\text{\sc label}(u) = k$ and
$S$ is the set of all attributes preceding $\text{\sc univ}(u)$ in the total
order, then $S\cup\text{\sc univ}(\text{\sc lc}(u)) = S\cup(U\setminus e_k)$ is precisely
the set of all attributes preceding $\text{\sc univ}(\text{\sc rc}(u)) = e_k\cap U$ in the
total order.
\end{itemize}
\label{prop:TO-properties}
\end{prop}
\begin{algorithm}
\caption{Computing a total order of attributes in $V$}
\label{algo:total-order}
\begin{algorithmic}[1]
\STATE Let $\mathcal{T}$ be the query plan tree with root node $u$, where $\text{\sc univ}(u)=V$
\STATE {\sc print-attribs}$(u)$
\end{algorithmic}
{\sc print-attribs}$(u)$
\begin{algorithmic}[1]
\IF {$u$ is a leaf node of $\mathcal{T}$}
\PRINT all attributes in $\text{\sc univ}(u)$ in an arbitrary order
\ELSIF {$\text{\sc lc}(u) = \textnormal{{\sc nil}}$}
\STATE {\sc print-attribs}$(\text{\sc rc}(u))$
\ELSIF {$\text{\sc rc}(u) = \textnormal{{\sc nil}}$}
\STATE {\sc print-attribs}$(\text{\sc lc}(u))$
\PRINT all attributes in $\text{\sc univ}(u) \setminus \text{\sc univ}(\text{\sc lc}(u))$ in an arbitrary
order
\ELSE
\STATE {\sc print-attribs}$(\text{\sc lc}(u))$
\STATE {\sc print-attribs}$(\text{\sc rc}(u))$
\ENDIF
\end{algorithmic}
\end{algorithm}
For each relation $R_e$, $e\in E$, we order all attributes in $R_e$
such that the internal order of attributes in $R_e$ is consistent
with the total order of all attributes computed by Algorithm
\ref{algo:total-order}.
More concretely, suppose $R_e$ has $k$ attributes ordered
$a_1,\dots,a_k$, then $a_i$ must come before $a_{i+1}$ in the total order,
for all $1\leq i\leq k-1$.
Then, we build a search tree (or any indexing data structure) for
every relation $R_e$ using the internal order of $R_e$'s attributes:
$a_1$ indexes level $1$ of the tree, $a_2$ indexes the next level, $\dots$,
$a_k$ indexes the last level of the tree.
The search tree for relation $R_e$
is constructed to satisfy the following three properties.
Let $i$ and $j$ be arbitrary integers such that $1\leq i\leq j\leq k$.
Let $\mv t_{\{a_1,\dots,a_i\}} = (t_{a_1},\dots,t_{a_i})$
be an arbitrary tuple on the attributes $\{a_1,\dots,a_i\}$.
\begin{itemize}
\item[(ST1)] We can decide whether
$\mv t_{\{a_1,\dots,a_i\}} \in \pi_{\{a_1,\dots,a_i\}}(R_e)$
in $O(i)$-time (by ``stepping down" the tree along the $t_{a_1},\dots,t_{a_i}$
path).
\item[(ST2)] We can query the size
$|\pi_{\{a_{i+1},\dots,a_j\}}(R_e[\mv t_{\{a_1,\dots,a_i\}}])|$
in $O(i)$ time.
\item[(ST3)] We can list all tuples in the set
$\pi_{\{a_{i+1},\dots,a_j\}}(R_e[\mv t_{\{a_1,\dots,a_i\}}])$
in time linear in the output size if the output is not empty.
\end{itemize}
The total running time for building all the search trees is
$O(n^2\sum_e N_e)$.
\subsubsection{Step (3): Computing the join}
\label{subsubsec:subsec:computing-the-join}
\floatname{algorithm}{Procedure}
\begin{algorithm}[t]
\caption{{\sc Recursive-Join}$(u, \mv y, \mv t_S)$}
\label{proc:rec-join}
\begin{algorithmic}[1]
\STATE Let $U = \text{\sc univ}(u)$, $k = \text{\sc label}(u)$
\STATE $\text{Ret} \leftarrow \emptyset$ \COMMENT{$\text{Ret}$ is the returned tuple set}
\IF[note that $U\subseteq e_i, \forall i \leq k$]{$u$ is a leaf node of $\mathcal{T}$}
\STATE $j \leftarrow \mathop{\text{argmin}}_{i\in [k]} \left\{ |\pi_U(R_{e_i}[\mv t_{S\cap e_i}])| \right\}$
\STATE \COMMENT{By convention, $R_e[\textnormal{{\sc nil}}] = R_e$ and $R_e[\mv t_\emptyset] = R_e$}
\FOR {each tuple $\mv t_U \in \pi_U(R_{e_j}[\mv t_{S\cap e_j}])$}
\IF{$\mv t_U \in \pi_U(R_{e_i}[\mv t_{S\cap e_i}]), \text{ for all } i \in [k]\setminus\{j\}$}
\STATE $\text{Ret} \leftarrow \text{Ret} \cup \{(\mv t_S, \mv t_U)\}$
\ENDIF
\ENDFOR
\RETURN $\text{Ret}$
\ENDIF
\IF[$u$ is not a leaf node of $\mathcal{T}$]{$\text{\sc lc}(u) = \textnormal{{\sc nil}}$}
\STATE $L \leftarrow \{\mv t_S\}$
\STATE \COMMENT{note that $L \neq \emptyset$ and
$\mv t_S$ could be $\textnormal{{\sc nil}}$ (when $S=\emptyset$)}
\ELSE
\STATE $L \leftarrow $ {\sc Recursive-Join}$(\text{\sc lc}(u), (y_1,\dots,y_{k-1}), \mv t_S)$
\ENDIF
\STATE $W \leftarrow U \setminus e_k$, $W^- \leftarrow e_k \cap U$
\IF{$W^- = \emptyset$}
\RETURN $L$
\ENDIF
\FOR{each tuple $\mv t_{S\cup W} =(\mv t_S, \mv t_W) \in L$}
\IF{$y_{e_k} \geq 1$}
\STATE {\bf go to} line 27
\ENDIF
\IF {$\displaystyle{\left(\prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}} < |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|
\right)}$}
\STATE $Z \leftarrow $ {\sc Recursive-Join}$\left(\text{\sc rc}(u), \left(\frac{y_{e_i}}{1-y_{e_k}}\right)_{i=1}^{k-1}, \mv t_{S\cup W}\right)$
\FOR {each tuple $(\mv t_S, \mv t_W, \mv t_{W^-}) \in Z$}
\IF{$\mv t_{W^-} \in \pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])$}
\STATE $\text{Ret} \leftarrow \text{Ret} \cup \{(\mv t_S, \mv t_W,
\mv t_{W^-})\}$
\ENDIF
\ENDFOR
\ELSE
\FOR {each tuple $\mv t_{W^-} \in \pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])$}
\IF{$\mv t_{e_i\cap W^-} \in \pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W) \cap e_i}])$ for all $e_i$ such that $i<k$ and $e_i \cap W^- \neq \emptyset$}
\STATE $\text{Ret} \leftarrow \text{Ret} \cup \{(\mv t_S, \mv t_W,
\mv t_{W^-})\}$
\ENDIF
\ENDFOR
\ENDIF
\ENDFOR
\RETURN $\text{Ret}$
\end{algorithmic}
\end{algorithm}
At the heart of Algorithm \ref{algo:the-algo} is a recursive
procedure called {\sc Recursive-Join} (Procedure \ref{proc:rec-join})
which takes three arguments:
\begin{itemize}
\item a node $u$ from the query plan tree $\mathcal{T}$ whose label is $k$
for some $k\in [m]$.
\item a fractional cover solution $\mv y_{E_k} = (y_{e_1},\dots,y_{e_k})$
of the hypergraph $(\text{\sc univ}(u), E_k)$.
Here, we only take the restrictions of
hyperedges of $E_k$ onto the universe $\text{\sc univ}(u)$. Specifically,
\begin{eqnarray*}
\sum_{e \in E_k: i \in e} y_e &\geq& 1, \ \text{for any $i \in \text{\sc univ}(u)$}\\
y_e &\geq &0, \text{for any $e \in E_k$}
\end{eqnarray*}
\item a tuple $\mv t_{S} = (t_i)_{i\in S}$ where $S$ is the set of {\em all}
attributes in $V$ which precede $\text{\sc univ}(u)$ in the total order.
(Due to property (TO1) of Proposition \ref{prop:TO-properties},
the set $S$ is well-defined.)
If there is no attribute preceding $\text{\sc univ}(u)$ then this argument
is $\textnormal{{\sc nil}}$.
In particular, the argument is $\textnormal{{\sc nil}}$ if $u$ is a node along the left
path of QP-tree $\mathcal{T}$ from the root down to the left-most leaf.
\end{itemize}
Throughout this section, we denote the final output by $J$ which
is defined to be $J = \Join_{e \in E} R_e$.
The goal of {\sc Recursive-Join} is to compute a superset of the
relation $\{\mv t_S\} \times \pi_{\text{\sc univ}(u)}(J[\mv t_S])$,
i.e., a superset of the
output tuples that start with $\mv t_S$ on the attributes $S \cup
\text{\sc univ}(u)$. This intermediate output is analogous to the set $C$ in
Algorithm~\ref{algo:LW} for LW instances. A second similarity
Algorithm~\ref{algo:LW} is that our algorithm makes a choice per tuple
based on the output's estimated size.
Theorem \ref{thm:main} is a special case of the following lemma
where we set $u$ to be the root of the QP-tree $\mathcal{T}$,
$\mv y = \mv x$, and $S=\emptyset$ ($\mv t_S=\textnormal{{\sc nil}}$).
Finally, we observe that we need only
$O(n^2)$ number of hash indices per input relation,
which completes the proof.
\begin{lmm} \label{lmm:main}
Consider a call {\sc Recursive-Join}$(u, \mv y, \mv t_S)$ to
Procedure \ref{proc:rec-join}.
Let $k=\text{\sc label}(u)$ and $U=\text{\sc univ}(u)$. Then,
\begin{itemize}
\item[(a)] The procedure outputs a relation $\text{Ret}$ on attributes $S\cup U$
with at most the following number of tuples
\[ B(u,\mv y, \mv t_S) := \prod_{i=1}^k |\pi_{U \cap e_i}(R_{e_i}[\mv t_{S\cap e_i}])|^{y_i}. \]
(For the sake of presentation, we agree on the convention that
when $U\cap e_i=\emptyset$ we set
$|\pi_{U \cap e_i}(R_{e_i}[\mv t_{S\cap e_i}])|=1$ so that the factor
does not contribute anything to the product.)
\item[(b)] Furthermore, the procedure runs in time
$O(mn \cdot B(u,\mv y,\mv t_S))$.
\end{itemize}
\label{app:lmm:main}
\end{lmm}
\begin{proof}
We prove both $(a)$ and $(b)$ by induction on the height of the subtree
of $\mathcal{T}$ rooted at $u$. The proof will also explain in ``plain'' English
the algorithm presented in Procedure \ref{proc:rec-join}.
The procedure tries to compute the join
\[ \{\mv t_S\} \times
\left( \Join_{i=1}^k \pi_{U\cap e_i}(R_{e_i}[\mv t_{S\cap e_i}]) \right).
\]
Roughly speaking, it is computing the join of all the sections
$R_{e_i}[\mv t_{S\cap e_i}]$ inside the universe $U$.
{\bf Base case}. The height of the sub-tree rooted at $u$ is zero,
i.e. $u$ is a leaf node. In this case, lines 4-9 of Procedure
\ref{proc:rec-join} is executed.
When $u$ is a leaf node, $U \subseteq e_i, \forall i \in [k]$
and thus $U = U\cap e_i, \forall i\in [k]$.
Since $\mv y$ is a fractional cover solution to the
hypergraph instance $(U, E_k)$, we know $\sum_{i=1}^ky_i \geq 1$.
The join has size at most
\[ \min_{i\in [k]} \left\{ |\pi_U(R_{e_i}[\mv t_{S\cap e_i}])| \right\}
\leq \prod_{i=1}^k |\pi_{U\cap e_i}(R_{e_i}[\mv t_{S\cap e_i}])|^{y_i}
= B(u, \mv y, \mv t_S).
\]
To compute the join, we go over each tuple of the smallest-sized
section-projection
$\pi_U(R_{e_j}[\mv t_{S\cap e_j}])$ and check to see if the tuple
belongs to all the other section-projections.
There are at most $k$ other sections, and due to property
(ST1) each check takes time $O(n)$.
Hence, the total time spent is
$O(mn\cdot B(u,\mv y, \mv t_S))$.
{\bf Inductive step}. Now, consider the case when $u$ is not a leaf node.
If $\text{\sc lc}(u)=\textnormal{{\sc nil}}$ which means $U\subseteq e_k$ then there is no
attribute in $U\setminus e_k$ to join over (line 11).
Otherwise, we first recursively call the ``left sub-problem''
(Line 14) and store the result in $L$.
Note that the attribute set of $L$ is $S\cup W = S\cup (U\setminus e_k)$.
We need to verify that the arguments we gave to this recursive call
are legitimate. It should be obvious that $k-1=\text{\sc label}(\text{\sc lc}(u))$.
Since $\mv y=(y_1,\dots,y_k)$ is a fractional cover of
the $(U,E_k)$ hypergraph, $\mv y'=(y_1,\dots,y_{k-1})$ is a
fractional cover of the $(U\setminus e_k, E_{k-1})$ hypergraph.
And, $\text{\sc univ}(\text{\sc lc}(u)) = U\setminus e_k$.
Finally, due to property (TO2) $S$ is precisely the set of attributes
preceding $\text{\sc univ}(\text{\sc lc}(u))$ in the total order.
From the induction hypothesis, the recursive call on line 14 takes time
\[
O(mn\cdot B(\text{\sc lc}(u), \mv y', \mv t_S)) = O\left(mn \prod_{i=1}^{k-1}
|\pi_{W\cap e_i}(R_{e_i}[\mv t_{S\cap e_i}])|^{y_i} \right).
\]
Furthermore, the number of tuples in $L$ is at most
$B(\text{\sc lc}(u), \mv y', \mv t_S) =
\prod_{i=1}^{k-1} |\pi_{W\cap e_i}(R_{e_i}[\mv t_{S\cap e_i}])|^{y_i}$.
If $W^- = \emptyset$ then $L$ is returned and we are done
because in this case
$B(\text{\sc lc}(u),\mv y', \mv t_S) \leq B(u, \mv y, \mv t_S)$.
Consider the for loop from line 18 to line 29.
We execute the for loop for each tuple
$\mv t_{S\cup W} = (\mv t_S, \mv t_W) \in L$.
If $L = \emptyset$ then the output is empty and we are done.
If $L = \{\mv t_S\}$ then this for-loop is executed only once.
This is the case if the assignment in line 11 was performed,
which means $U\subseteq e_k$ and thus $W=\emptyset$.
We do not have to analyze this case separately as it is subsumed
by the general case that $L\neq \emptyset$.
Note that if $y_{e_k}\geq 1$ then we go directly to {\bf case b} below
(corresponding to line 27).
{\bf Case a.} Consider the case when $y_{e_k}<1$ and
\[ \prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{
e_i}}{1-y_{e_k}}} < |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|.
\]
In this case we first recursively solve the sub-problem
\[
Z = \text{\sc Recursive-Join}\left(\text{\sc rc}(u), \left(\frac{y_{e_i}}{1-y_{e_k}}\right)_{i=1}^{k-1}, \mv t_{S\cup W}\right).
\]
We need to make sure that the arguments are legitimate.
Note that $\text{\sc univ}(\text{\sc rc}(u)) = W^-$, and that $y_{e_k}<1$.
The sub-problem is on the hypergraph $(W^-, E_{k-1})$.
For any $v\in W^-=U\cap e_k$, because $\mv y$ is a fractional cover
of the $(U, E_k)$ hypergraph,
\[ 1 \leq \sum_{i \in [k] \ : \ v \in e_i} y_{e_i} =
y_{e_k} + \sum_{i\in [k-1] \ : \ v\in e_i} y_{e_i}.
\]
Hence,
\[ 1 \leq \sum_{i\in [k-1] \ : \ v\in e_i} \frac{y_{e_i}}{1-y_{e_k}},\]
which confirms that the solution
$\left(\frac{y_{e_i}}{1-y_{e_k}}\right)_{i=1}^{k-1}$ is a fractional cover
for the hypergraph $(W^-, E_{k-1})$.
Finally, by property (TO2) the attributes $S\cup W$ are precisely the
attributes preceding $W^-$ in the total order.
After solving the sub-problem we obtain a tuple set $Z$ over the
attributes $S\cup W\cup W^- = S\cup U$. By the induction hypothesis
the time it takes to solve the sub-problem is
\[
O\left(mn\prod_{i=1}^{k-1} |\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}}
\right)
\]
and the number of tuples in $Z$ is bounded by
\[ \prod_{i=1}^{k-1} |\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}}.
\]
Then, for each tuple in $Z$ we perform the check on line 24.
Hence, the overall running time in this case is still
$O\left(mn\prod_{i=1}^{k-1} |\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}}
\right)$
{\bf Case b.} Consider the case when either $y_{e_k}\geq 1$ or
\[ \prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{
e_i}}{1-y_{e_k}}} \geq |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|.
\]
In this case, we execute lines 27 to 29. The number of tuples output
is at most $|\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|$
and the running time is
$O(mn|\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|)$.
Overall, for both (case a) and (case b) the number of tuples output
is bounded by
\[ T =
\begin{cases}
\min\left\{\prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}}, |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])| \right\}
& y_{e_k}<1\\
|\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])| & \text{otherwise}
\end{cases}
\]
and the running time is in the order of $O(mnT)$. We bound $T$ next.
When $y_{e_k}<1$ we have
\begin{eqnarray*}
T&\leq&\min\left\{\prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}}, |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])| \right\}\\
&\leq&\left(\prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{\frac{y_{e_i}}{1-y_{e_k}}} \right)^{1-y_{e_k}} |\pi_{W^-}(R_{e_k}[\mv t_{S\cap e_k}])|^{y_{e_k}}\\
&=&|\pi_{U\cap e_k}(R_{e_k}[\mv t_{S\cap e_k}])|^{y_{e_k}} \cdot \prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{y_{e_i}}
\end{eqnarray*}
When $y_{e_k}\geq 1$, it is obvious that the same inequality holds:
\[ T \leq
|\pi_{U\cap e_k}(R_{e_k}[\mv t_{S\cap e_k}])|^{y_{e_k}} \cdot \prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{y_{e_i}}.
\]
Summing overall $(\mv t_S, \mv t_W)\in L$, the number of output tuples
is bounded by the following sum.
Without loss of generality, assume $W=\{1,\dots,d\}=[d]$.
In the following, the first sum is over
$t_1\in \pi_{\{1\}}(L)$, the second sum is over $t_2$ such that
$(t_1,t_2)\in \pi_{\{1,2\}}(L)$, and so on.
To shorten notations a little, define
\[ \bar R_i = R_{e_i}[\mv t_{S\cap e_i}]. \]
Then, the total number of output tuples is bounded by
\begin{eqnarray*}
&& \sum_{\mv t_W \in \pi_W(L)} |\pi_{U\cap e_k}(R_{e_k}[\mv t_{S\cap e_k}])|^{y_{e_k}} \cdot \prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(R_{e_i}[\mv t_{(S\cup W)\cap e_i}])|^{y_{e_i}}\\
&=& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\sum_{t_2}\cdots\sum_{t_d}
\prod_{i=1}^{k-1}
|\pi_{e_i \cap W^-}(\bar R_i[\mv t_{[d]\cap e_i}])|^{y_{e_i}}\\
&=& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\cdots \sum_{t_{d-1}}
\prod_{i<k, d\notin e_i}
|\pi_{e_i \cap W^-}(\bar R_i[\mv t_{[d]\cap e_i}])|^{y_{e_i}}
\sum_{t_d}
\prod_{i<k, d\in e_i}
|\pi_{e_i \cap W^-}(\bar R_i[\mv t_{[d]\cap e_i}])|^{y_{e_i}}\\
&\leq& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\cdots \sum_{t_{d-1}}
\prod_{i<k, d\notin e_i}
|\pi_{e_i \cap W^-}(\bar R_i[\mv t_{[d]\cap e_i}])|^{y_{e_i}}
\prod_{i<k, d\in e_i}
\left(\sum_{t_d}
|\pi_{e_i \cap W^-}(\bar R_i[\mv t_{[d]\cap e_i}])|\right)^{y_{e_i}}\\
&\leq& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\cdots \sum_{t_{d-1}}
\prod_{i<k, d\notin e_i}
|\pi_{e_i \cap (W^- \cup \{d\})}(\bar R_i[\mv t_{[d-1]\cap e_i}])|^{y_{e_i}}
\prod_{i<k, d\in e_i}
|\pi_{e_i \cap (W^- \cup \{d\})}(\bar R_i[\mv t_{[d-1]\cap e_i}])|^{y_{e_i}}\\
&=& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\sum_{t_2}\cdots\sum_{t_{d-1}}
\prod_{i=1}^{k-1}
|\pi_{e_i \cap (W^- \cup\{d\})}(\bar R_i[\mv t_{[d-1]\cap e_i}])|^{y_{e_i}}\\
&\leq&\dots\\
&\leq& |\pi_{U\cap e_k}(\bar R_k)|^{y_{e_k}}
\sum_{t_1}\sum_{t_2}\cdots\sum_{t_{d-2}}
\prod_{i=1}^{k-1}
|\pi_{e_i \cap (W^- \cup\{d-1,d\})}(\bar R_i[\mv t_{[d-2]\cap e_i}])|^{y_{e_i}}\\
&\leq&\dots\\
&=& \prod_{i=1}^{k} |\pi_{U\cap e_i}(\bar R_i)|^{y_{e_i}}
\end{eqnarray*}
\end{proof}
\section{Limits of Standard Approaches}
\label{sec:limits}
For a given join query $q$, we describe a sufficient syntactic
condition for $q$ so that when computed by any join-project plan is
asymptotically slower than the worst-case bound. Our algorithm runs
within this bound, and so for such $q$ there is an asymptotic
running-time gap.
\paragraph*{LW Instances}
Recall that an {\em LW instance} of the OJ problem is a join query $q$
represented
by the hypergraph $(V,E)$, where $V=[n]$, and $E = \binom{[n]}{n-1}$ for some
integer $n \geq 2$.
Our main result in this section is the following lemma\footnote{We thank an
anonymous PODS'12 referee for giving us the argument showing that our example
works for all join-project plans rather than just the AGM algorithm and a
join-tree algorithm.}
\begin{lmm}
Let $n\geq 2$ be an arbitrary integer.
Given any LW-query $q$ represented by a hypergraph $([n], \binom{[n]}{n-1})$,
and any positive integer $N\geq 2$, there exist relations $R_i$, $i\in[n]$,
such that $|R_i| =N,\forall i\in [n]$, the attribute set for $R_i$ is
$[n]-\{i\}$, and that {\em any} join-project plan for $q$ on these relations
runs in time $\Omega(N^2/n^2)$.
\label{LEM:BAD:INSTANCE}
\end{lmm}
Before proving the lemma, we note that both the traditional join-tree
algorithm and AGM's algorithm are join-project plans, and thus their
running times are asymptotically worse than the best AGM bound for this
instance which is
$|\Join_{i=1}^n R_i|\le \prod_{i=1}^n |R_i|^{1/(n-1)}=N^{1+1/(n-1)}.$
On the other hand, both Algorithm \ref{algo:LW} and
Algorithm \ref{algo:the-algo}
take $O(N^{1 + 1/(n-1)})$-time as we have analyzed.
In fact, for Algorithm \ref{algo:the-algo}, we are able to demonstrate
a stronger result: its run-time on this instance is $O(n^2N)$
which is better than what we can analyze for a general instance of this
type.
In particular, the run-time gap between Algorithm \ref{algo:the-algo}
and AGM's algorithm is $\Omega(N)$ for constant $n$.
\begin{proof}[Proof of Lemma \ref{LEM:BAD:INSTANCE}]
In the instances below the domain of any attribute will be
$\mathbf{D} = \{0,1,\dots,(N-1)/(n-1)\}$
For the sake of clarify, we ignore the integrality issue.
For any $i\in [n]$, let $R_i$ be the set of {\em all} tuples
in $\mathbf{D}^{[n]-\{i\}}$ each of which has at most one non-zero value.
Then, it is not hard to see that $|R_i| = (n-1)[(N-1)/(n-1)+1] - (n-2) = N$,
for all $i\in [n]$; and,
$|\Join_{i=1}^n R_i| = n[(N-1)/(n-1)+1]-(n-1) = N+(N-1)/(n-1)>N$.
A relation $R$ on attribute set $\bar A \subseteq [n]$ is called ``simple"
if $R$ is the set of {\em all} tuples in $\mathbf{D}^{\bar A}$ each of which has at
most
one non-zero value. Then, we observe the following properties.
(a) The input relations $R_i$ are simple.
(b) An arbitrary projection of a simple relation is simple.
(c) Let $S$ and $T$ be any two simple relations on attribute sets
$\bar A_S$ and $\bar A_T$, respectively.
If $\bar A_S$ is contained in $\bar A_T$ or vice versa, then
$S \Join T$ is simple.
If neither $\bar A_S$ nor $\bar A_T$ is contained in the other,
then $|S\Join T| \geq (1+(N-1)/(n-1))^2 = \Omega(N^2/n^2)$.
For an arbitrary join-project plan starting from the simple relations
$R_i$,
we eventually must join two relations whose attribute sets are not
contained
in one another, which right then requires $\Omega(N^2/n^2)$ run time.
\end{proof}
Finally, we analyze the run-time of Algorithm \ref{algo:the-algo}
directly on this instance without resorting to Lemma~\ref{lem:holder}.
H\"older's inequality lost some information about the run-time.
The following lemma shows that our algorithm and our bound can be
better than what we were able to analyze.
\begin{lmm}
On the collection of instances from the previous lemma,
Algorithm \ref{algo:the-algo} runs in time $O(n^2N)$.
\label{lmm:the-algo-on-bad-instance}
\end{lmm}
\begin{proof}
Without loss of generality, assume the hyperedge order
Algorithm~\ref{algo:the-algo} considers is
$[n]-\{1\}, \dots, [n]-{n}$.
In this case, the universe of the left-child of the root of the QP-tree
is $\{n\}$, and the universe of the right-child of the root is
$[n-1]$.
The first thing Algorithm~\ref{algo:the-algo} does is that it computes the join
$L_n = \Join_{i=1}^{n-1} \pi_{\{n\}}(R_i)$, in time $O(nN)$.
Note that $L_n = \mathbf{D}$, the domain.
Next, Algorithm~\ref{algo:the-algo} goes through each value
$a\in L_n$ and decide whether to solve a subproblem.
First, consider the case $a>0$.
Here Algorithm~\ref{algo:the-algo} estimates a bound for the join
$\Join_{j=1}^{n-1} \pi_{[n-1]}(R_j[a])$.
The estimate is $1$ because $|\pi_{[n-1]}(R_j[a])|=1$
for all $a>0$. Hence, the algorithm will recursively compute this join
which takes time $O(n^2)$ and filter the result against $R_n$.
Overall, solving the sub problems for $a>0$ takes $O(n^2N)$ time.
Second, consider the case when $a=0$. In this case
$|\pi_{[n-1]}(R_j[0])| = \frac{(n-2)N-1}{(n-1)}$.
The subproblem's estimated size bound is
\[ \prod_{i=1}^{n-1} |\pi_{[n-1]}(R_j[0])|^{\frac{1/(n-1)}{1-1/(n-1)}}
= \left[\frac{(n-2)N-1}{(n-1)}\right]^{(n-1)/(n-2)} > N
\]
if $N\geq 4$ and $n\geq 4$. Hence, in this case $R_n$ will be filtered
against
the $\pi_{[n-1]}(R_j[0])$, which takes $O(n^2N)$ time.
\end{proof}
\paragraph*{Extending beyond LW instances}
Using the above results, we give a sufficient condition
for when there exist a family of instances ${\cal I} = I_1,\dots,I_N,
\dots,$ such that on instance $I_N$ every binary join strategy takes time at
least
$\Omega(N^2)$, but our algorithm takes $o(N^2)$. Given a hypergraph
$H=(V,E)$. We first define some notation. Fix $U \subseteq V$ then
call an attribute $v \in V \setminus U$ {\em $U$-relevant} if for all $e$ such
that $v \in e$ then $e \cap U \neq \emptyset$; call $v$
{\em $U$-troublesome} if for all $e \in E$, if $v \in e$ then $U
\subseteq e$. Now we can state our result:
\begin{lmm}
Given a join query $H=(V,E)$ and some $U \subseteq V$ where $|U| \geq
2$, then if there exists $F \subseteq E$ such that $|F| = |U|$ that
satisfies the following three properties: (1) each $u \in U$ occurs in
exactly $|U|-1$ elements in $F$, (2) each $v \in V$ that is
$U$-relevant appears in at least $|U|-1$ edges in $F$, (3) there are no
$U$-troublesome attributes. Then, there is some family of
instances ${\cal I}$ such that (a) computing the join query
represented by $H$ with a join tree takes time $\Omega(N^2/|U|^2)$ while (b) the
algorithm from Section~\ref{sec:all:j} takes
time $O(N^{1 + 1/(|U|-1)})$.
\end{lmm}
Given a $(U,F)$ as in the lemma, the idea is to simply to set all
those edges in $f \in F$ to be the instances from
Lemma~\ref{LEM:BAD:INSTANCE} and extend all attributes with a single
value, say $c_0$. Since there are no $U$-troublesome attributes, to
construct the result set at least one of the relations in $F$ must be
joined. Since any pair $F$ must take time $\Omega(N^2/|U|^2)$ by the
above construction, this establishes (a). To establish (b), we need to
describe a particular feasible solution to the cover LP whose
objective value is $N^{1 + 1/(|U|-1)}$, implying that the running time
of our proposed algorithm is upper bounded by this value. To do
this, we first observe that any attribute not in $U$ takes the value
only $c_0$. Then, we observe that any node $v \in V$ that is not
$U$-relevant is covered by some edge $e$ whose size is exactly $1$
(and so we can set $x_e = 1$). Thus, we may assume that all nodes are
$U$-relevant. Then, observe that all relevant attributes can be set by
the cover $x_{e} = 1/(|U|-1)$ for $e \in F$. This is a feasible
solution to the LP and establishes our claim.
\section{Notation and Formal Problem Statement}
\label{sec:notations}
We assume the existence of a set of attribute names $\mathcal{A} =
A_1,\dots, A_n$ with associated domains $\mathbf{D}_1,\dots,\mathbf{D}_n$ and infinite
set of relational symbols $R_1,R_2, \dots$. A relational schema for
the symbol $R_i$ of arity $k$ is a tuple $\bar A_i = (A_{i_1}, \dots,
A_{i_k})$ of distinct attributes that defines the attributes of the
relation. A relational database schema is a set of relational symbols
and associated schemas denoted by $R_1(\bar A_1), \dots, R_m(\bar
A_m)$. A relational instance for $R(A_{i_1},\dots,A_{i_k})$ is a
subset of $\mathbf{D}_{i_1} \times \dots \times \mathbf{D}_{i_k}$. A relational database $I$ is
an instance for each relational symbol in schema, denoted by
$R_i^{I}$. A {\em natural join} query (or simply query) $q$ is
specified by a finite subset of relational symbols $q \subseteq
\mathbb{N}$, denoted by $\Join_{i \in q} R_i$. Let $\bar A(q)$ denote
the set of all attributes that appear in some relation in $q$, that is
$\bar A(q) = \{ A \mid A \in \bar A_i \text{ for some } i \in q\}$.
Given a tuple
$\mv t$ we will write $\mv t_{\bar A}$ to emphasize that its support
is the attribute set $\bar A$. Further, for any $\bar S\subset \bar A$
we let $\mv t_{\bar S}$ denote $\mv t$ restricted to $\bar S$.
Given a database instance $I$, the
output of the query $q$ on $I$ is denoted $q(I)$ and is defined as
\[ q(I) \stackrel{\mathrm{def}}{=} \left\{ \mv t \in \mathbf{D}^{\bar A(q)} \ | \
\mv t_{\bar A_{i}} \in R^{I}_i \text{ for each } i \in q\right\} \]
where $\mathbf{D}^{\bar A(q)}$ is a shorthand for $\times_{i : A_i \in \bar A(q)} \mathbf{D}_i$.
We also use the notion of a {\em semijoin}: Given two relations
$R(\bar A)$ and $S(\bar B)$ their semijoin $R \lJoin S$ is defined by
\[ R \lJoin S \stackrel{\mathrm{def}}{=}
\left\{ \mv t \in R : \exists \mv u \in S \text{ s.t. }
\mv t_{\bar A \cap \bar B} = \mv u_{\bar A \cap \bar B} \right\}.
\]
For any relation $R(\bar A)$, and any subset $\bar S\subseteq \bar A$
of its attributes, let $\pi_{\bar S}(R)$ denote the {\em projection} of
$R$ onto $\bar S$, i.e.
\[ \pi_{\bar S}(R) = \left\{\mv t_{\bar S} \ | \
\exists \mv t_{\bar A\setminus \bar S}, (\mv t_{\bar S}, \mv t_{\bar A\setminus \bar S})
\in R \right\}.
\]
For any tuple $\mv t_{\bar S}$,
define the {\em $\mv t_{\bar S}$-section} of $R$ as
\[ R[\mv t_{\bar S}] = \pi_{\bar A\setminus \bar S}(R \lJoin \set{\mv t_{\bar S}}).
\]
\paragraph*{From Join Queries to Hypergraphs}
A query $q$ on attributes $\bar A(q)$ can be viewed as a hypergraph
$H=(V,E)$ where $V = \bar A(q)$ and there is an edge $e_i =\bar A_i$
for each $i \in q$. Let $N_e = |R_e|$ be the number of tuples in
$R_e$. \textit{From now on we will use the hypergraph and the original notation for the query
interchangeably.}
We use this hypergraph to introduce the {\em fractional edge
cover polytope} that plays a central role in our technical
developments. The fractional edge cover polytope defined by $H$
is the set of all points $\mv x =(x_e)_{e\in E}\in \mathbb{R}^E$ such that
\begin{eqnarray*}
\sum_{v \in e} x_e &\geq& 1, \ \text{for any $v \in V$}\\
x_e &\geq &0, \text{for any $e \in E$}
\end{eqnarray*}
Note that the solution $x_e = 1$ for $e \in E$ is always feasible
for hypergraphs representing join queries.
A point $\mv x$ in the polytope is also called a {\em fractional (edge)
cover solution} of the hypergraph $H$.
Atserias, Grohe, and Marx~\cite{AGM08} establish that, for {\em any}
point $\mv x = (x_e)_{e\in E}$ in the fractional edge cover polytope
\begin{equation}
|\Join_{e\in E} R_e| \leq \prod_{e\in E}N_e^{x_e}.
\label{eqn:agm08-bound}
\end{equation}
The bound is proved nonconstructively using Shearer's entropy inequality
\cite{MR859293}.
However, AGM provide an algorithm based on join-project plans that runs
in time
$O(|q|^2\cdot N_{\max}^{1+\sum_{e} x_e})$ where $N_{\max} = \max_{e \in E} N_e$.
They observed that for a fixed hypergraph $H$ and given sizes $N_e$
the bound \eqref{eqn:agm08-bound} can be minimized by solving the linear program
which minimizes the linear objective $\sum_{e} (\log N_e)\cdot x_e$
over fractional edge cover solutions $\mv x$.
(Since in linear time we can figure out if we have an empty relation,
and hence an empty output), for the rest of the paper we are
always going to assume that $N_e\ge 1$.)
Thus, the formal
problem that we consider recast in this language is:
\begin{defn}[OJ Problem -- Optimal Join Problem]
With the notation above, design an algorithm to compute $\Join_{e\in
E} R_e$ with running time
\[ O\left( f(|V|,|E|) \cdot \left(\prod_{e\in E}N_e^{x_e} + \sum_{e\in
E}N_e\right)\right). \] Here $f(|V|,|E|)$ is ideally a
polynomial with (small) constant degree, which only depends on the
query size. The linear term $\sum_{e\in E}N_e$ is to read the input.
Hence, such an algorithm would be data-optimal in the worst case.\footnote{Following GLV~\cite{GLVV09}, we assume in this work
that given relations $R$ and $S$ one can compute $R \Join S$ in time
$O(|R| + |S| + |R \Join S|)$. This only holds in an amortized sense
(using hashing). To acheive true worst case results, one can use
sorting operations which results in a $\log$ factor increase in
running time. }
\end{defn}
We recast our motivating example from the introduction in our
notation. Recall we are given, $R(A,B), S(B,C), T(A,C)$, so $V =
\set{A,B,C}$ and three edges corresponding each to $R$, $S$, and $T$,
which are $E=\set{\set{A,B}$, $\set{B,C}$, $\set{A,C}}$
respectively. Thus, $|V| = 3$ and $|E| = 3$. If we are given that $N_e
= N$, one can check that the optimal solution to the LP is $x_e =
\frac{1}{2}$ for $e \in E$ which has the objective value $\frac{3}{2}
\log N$; in turn, this gives $\sup_{I \in I(\bar N)} |q(I)| \leq
N^{3/2}$ (recall $I(\bar N) = \set{ I : |R_e^{I}| = N_e \text{ for } e
\in E}$).
\begin{example}
\label{ex:triangle}
Given an even integer $N$, we construct an instance $I_N$
such that (1) $|R^{I_N}| = |S^{I_N}| = |T^{I_N}|=N$,
(2) $|R \Join S| = |R \Join T| = |S \Join T| = N^2/4 + N/2$, and
(3) $|R \Join S \Join T| = 0$. The following instance satisfies all three
properties:
\[ R^{I_N} = S^{I_N} = T^{I_N} =
\left\{(0,j)\right\}_{j=1}^{N/2} \cup
\left\{(j,0)\right\}_{j=1}^{N/2}.
\]
For example,
\[ R \Join S = \{(i,0,j)\}_{i,j=1}^{N/2}
\cup \{ (0,i,0) \}_{i=1,\dots,N/2} \]
and $R \Join S \Join T = \emptyset$. Thus, any standard join-based
algorithm takes time $\Omega(N^2)$.
We show later that AGM's algorithm takes $\Omega(N^2)$-time
too. Recall that the AGM bound
for this instance is $O(N^{3/2})$, and our algorithm thus takes time
$O(N^{3/2})$. In fact, as shall be shown later,
on this particular family of instances both of our
algorithms take only $O(N)$ time.
\end{example}
\section{Connections to Geometric Inequalities}
\label{sec:bt:equiv}
We describe the Bollob\'as-Thomason (BT) inequality from discrete
geometry and prove that BT inequality is equivalent to AGM's
inequality. We then look at a special case of BT inequality, the
Loomis-Whitney (LW) inequality, from which our algorithmic development
starts in the next section. We state the BT inequality:
\begin{thm}[Discrete Bollob\'as-Thomason (BT) Inequality]\label{thm:BT}
Let $S\subset \mathbb Z^n$ be a finite set of $n$-dimensional grid points.
Let $\mathcal F$ be a collection of subsets of $[n]$ in which every
$i \in [n]$ occurs in exactly $d$ members of $\mathcal F$.
Let $S_F$ be the set of projections $\mathbb Z^n \to \mathbb Z^F$ of
points in $S$ onto the coordinates in $F$. Then,
$|S|^d \leq \prod_{F \in \mathcal F} |S_F|$.
\end{thm}
To prove the equivalence between BT inequality and the AGM bound, we
first need a simple observation.
\begin{lmm}\label{lmm:whytight}\label{LMM:WHYTIGHT}
Consider an instance of the OJ problem consisting of a hypergraph $H=(V,E)$,
a fractional cover $\mv x=(x_e)_{e\in E}$ of $H$, and relations
$R_e$ for $e\in E$. Then, in linear time we can
transform the instance into another instance $H'=(V,E')$,
$\mv x'=(x'_e)_{e\in E'}$, $(R'_e)_{e\in E'}$, such that
the following properties hold:
\begin{itemize}
\item[(a)] $\mv x'$ is a ``tight'' fractional edge cover of the
hypergraph $H'$, namely $\mv x'\geq 0$ and
\[ \sum_{e\in E': v\in e} x'_e = 1, \ \ \text{ for every } v\in V. \]
\item[(b)] The two problems have the same answer:
\[ \Join_{e\in E} R_e = \ \Join_{e\in E'} R'_e. \]
\item[(c)] AGM's bound on the transformed instance is at least
as good as that of the original instance:
\[ \prod_{e\in E'} |R'_e|^{x'_e} \leq \prod_{e\in E} |R_e|^{x_e}. \]
\end{itemize}
\end{lmm}
\begin{proof}
We describe the transformation in steps. At each step properties (b) and (c)
are kept as invariants. After all steps are done, (a) holds.
While there still exists some vertex $v\in V$ such that
$\sum_{e\in E: v \in e} x_e > 1$, i.e. $v$'s constraint is not tight,
let $f$ be an arbitrary hyperedge $f\in E$ such that $v\in f$.
Partition $f$ into two parts $f = f_t \cup f_{\neg t}$, where
$f_t$ consists of all vertices $u\in f$ such that $u$'s constraint
is tight, and $f_{\neg t}$ consist of vertices $u\in f$ such that
$u$'s constraint is not tight. Note that $v\in f_{\neg t}$.
Define
$\rho = \min\left\{x_f,
\min_{u\in f_{\neg t}} \left\{ \sum_{e : u\in e} x_e - 1\right\}\right\}.
$
This is the amount which, if we were able to reduce $x_f$ by $\rho$
then we will either turn $x_f$ to $0$ or make some constraint
for $u \in f_{\neg t}$ tight.
However, reducing $x_f$ might violate some already tight
constraint $u\in f_t$.
The trick is to ``break" $f$ into two parts.
We will set $E'=E\cup\{f_t\}$,
create a ``new'' relation $R'_{f_t} = \pi_{f_t}(R_f)$, and
keep all the old relations $R'_e=R_e$ for all $e\in E$.
Set the variables $x'_e=x_e$ for all $e\in E-\{f\}$ also.
The only two variables which have not been set are
$x'_f$ and $x'_{f_t}$. We set them as follows.
\begin{itemize}
\item
When $x_f \leq \min_{u\in f_{\neg t}} \left\{ \sum_{e : u\in e} x_e -
1\right\}$,
set
$x'_f = 0$ and
$x'_{f_t} = x_f$.
\item When $x_f > \min_{u\in f_{\neg t}} \left\{ \sum_{e : u\in e} x_e -
1\right\}$,
set
$x'_f = x_f - \rho$ and $x'_{f_t} = \rho$.
\end{itemize}
Either way, it can be readily verified that the new instance is a
legitimate OJ instance satisfying properties (b) and (c).
In the first case, some positive variable in some non-tight constraint has been
reduced to $0$.
In the second case, at least one non-tight constraint has become tight.
Once we change a variable $x_f$ (essentially ``break'' it up into
$x'_{f_t}$ and $x'_f$) we won't touch it again.
Hence, after a linear number of steps in $|V|$, we will have all tight
constraints.
\end{proof}
With this
technical observation, we can now connect the two families of inequalities:
\begin{prop} BT inequality and AGM's fractional cover bound are
equivalent.
\end{prop}
\begin{proof}
To see that AGM's inequality implies BT inequality, we think of each
coordinate as
an attribute, and the projections $S_F$ as the input relations.
Set $x_F = 1/d$ for each $F\in \mathcal F$. It follows that
$\mv x = (x_F)_{F\in \mathcal F}$ is a fractional cover for
the hypergraph $H=([n], \mathcal F)$. AGM's bound then implies
that $|S| \leq \prod_{F\in\mathcal F} |S_F|^{1/d}$.
Conversely,
consider an instance of the OJ problem with hypergraph $H=(V,E)$
and a rational fractional cover $\mv x = (x_e)_{e\in E}$ of $H$.
First, by Lemma \ref{lmm:whytight},
we can assume that all cover constraints are tight, i.e.,
\[ \sum_{e: v\in e} x_e = 1, \ \ \text{ for any } v \in V. \]
By standard arguments, it can be shown that all the ``new" $x_e$ are rational values (even if the original values were not).
Second, by writing all variables $x_e$ as $d_e/d$ for a positive
common denominator $d$ we obtain
\[ \sum_{e: v\in e} d_e = d, \ \ \text{ for any } v \in V. \]
Now, create $d_e$ copies of each relation $R_e$.
Call the new relations $R'_e$. We obtain a new
hypergraph $H'=(V,E')$ where every attribute $v$ occurs in exactly $d$
hyperedges. This is precisely the Boll\'obas-Thomason's setting
of Theorem \ref{thm:BT}. Hence, the size of the join is bounded
above by
$ \prod_{e\in E'}|R'_e|^{1/d} = \prod_{e\in E}|R_e|^{d_e/d} =
\prod_{e\in E}|R_e|^{x_e}.
$
\end{proof}
\paragraph*{Loomis-Whitney} We now consider a special case of BT (or
AGM), the discrete version of a classic geometric inequality called
the {\em Loomis-Whitney inequality} \cite{MR0031538}. The setting is
that for $n \geq 2$, $V = [n]$ and $E = \binom{V}{|V|-1}$. In this
case $x_e=1/(|V|-1), \forall e\in E$ is a fractional cover solution
for $(V,E)$, and LW showed the following:
\begin{thm}[Discrete Loomis-Whitney (LW) inequality]\label{thm:LW}
Let $S\subset \mathbb Z^n$ be a finite set of $n$-dimensional grid points.
For each dimension $i \in [n]$, let $S_{[n]\setminus \{i\}}$ denote the
$(n-1)$-dimensional projection of $S$ onto the coordinates
$[n]\setminus \{i\}$. Then,
$|S|^{n-1} \leq \prod_{i=1}^n |S_{[n]\setminus \{i\}}|$.
\end{thm}
It is clear from our discussion above that LW is a special case of
BT (and so AGM), and it is with this special case that we begin
our algorithmic development in the next section.
|
{
"timestamp": "2012-03-12T01:00:18",
"yymm": "1203",
"arxiv_id": "1203.1952",
"language": "en",
"url": "https://arxiv.org/abs/1203.1952"
}
|
\section{Introduction}
In recent years, a variety of novel experimental approaches have enabled
tests of fundamental quantum physics such as superpositions, entanglement,
tunneling or quantum phase transitions in artificial devices. Prominent
examples hereof are quantum circuits, quantum dots or optical lattices~\cite{you2011atomicphysics,buluta_natural_2011}.
Apart from their fundamental relevance, quantum technologies allow
realizing quantum information processors that bring along the potential
of carrying out specific tasks at exponentially reduced computation
time~\cite{ladd2010quantum}. Furthermore, quantum simulators of
Feynman type~\cite{feynman1982simulating,houck2012onchipquantum},
employed to simulate the dynamics of one quantum system by means of
another one are a pivotal example of quantum speed-up as compared
to a classical computer. While both systems share the same dynamics,
the simulator offers far more configurability and is better accessible
for a measurement.
Generally, the possibility of observing quantum effects strongly depends
on the coherence properties of the underlying system. With regard
to superconducting circuit qubits, the advantages of versatile manufacturing,
detection and manipulation are paid for at the price of quite high
decoherence rates as compared to trapped ions or spin qubits. One
way out is given by the recently emerging field of hybrid systems~\cite{amsuss2011cavityqed,kubo2011hybridquantum,kubo2010strongcoupling,schuster2010highcooperativity,imamoglu2009cavityqed,rabl2006hybridquantum,verdu2009strongmagnetic,wesenberg2009quantum,xiang2012hybridquantum}.
The main motivation for building hybrid systems is to combine two
advantages: the addressability of artificial quantum circuits and
the long coherence times of elemental systems such as nitrogen-vacancy~(NV)
centers in diamond or in polar molecules. Usually these hybrid systems
are motivated by using the natural spins for building quantum memories.
In this work, we point out an alternative application for the exploration
of many-body physics such as quantum phase transitions. In particular,
we investigate the localized and delocalized phases that occur in
Hubbard-like models such as the Jaynes-Cummings (JC) lattice~\cite{hartmann2006strongly,greentree2006quantum,angelakis2007photonblockadeinduced,hartmann2008quantum}.
In the localized phase, excitations are localized at individual lattice
sites, whereas they are delocalized across the lattice in the delocalized
phase. We propose an intriguingly simple layout for simulating a JC-lattice:
We use the combination of an already experimentally well-proven flux
qubit array together with a single large NV-center crystal. By means
of numerical studies, we corroborate that this system exhibits localized
and delocalized phases. Finally, we demonstrate that these phases
can be identified by monitoring the signatures of the non-equilibrium
system dynamics in presence of decoherence and dissipation upon employing
experimentally accessible parameters.
\begin{figure}[!t]
\includegraphics[width=1\columnwidth]{figure1-setup} \caption{(a) Schematic of a single flux qubit coupled to a diamond crystal
with NV-centers embedded. (b) Energy diagram for the qubit with level
splitting $\omega_{q}$ and a NV-center. By applying an external magnetic
field, the level with spin projection $m=\pm1$ becomes resonant with
the upper qubit level. (c) The JC-array with tunable qubit-qubit coupling.
Here, adjacent qubits are connected via auxiliary tunable coupler
qubits. Each of the qubits couples to spatially separated regions
of the crystal. The coupler qubit does not couple to the spins because
it is far detuned from the qubits and consquently from the NV-spins
as well.\label{setup}}
\end{figure}
\section{Hybrid qubit-resonator model }
As the elementary unit for the JC-array we propose a hybrid combination
of a flux qubit and an ensemble of independent spins, given here by
the NV-centers in a diamond~\cite{marcos2010coupling,twamley2010superconducting}.
As illustrated in Fig.~\ref{setup}(a), the spin crystal is placed
in proximity of a qubit loop. The spin-qubit Zeeman interaction is
mediated by the magnetic field that stems from the qubit's persistent
currents. A weak external field splits the spin degeneracy so as to
shift one spin transition into resonance with the qubit's transition
frequency $\omega_{q}$. Reducing the spin to its two lowest levels
with mutual energy spacing $\omega_{c}$ (see Fig.~\ref{setup}~(b))
and applying the rotating wave approximation (RWA) we can express
the Hamiltonian of an ensemble of $N$ spins coupled to a single flux
qubit at its degeneracy point as~($\hbar=1$)
\begin{equation}
H_{q+s}=\omega_{q}\sigma^{+}\sigma^{-}+\omega_{c}\,\sum_{k}^{N}\tau_{k}^{+}\tau_{k}^{-}+\sum_{k}^{N}\left(g_{k}\tau_{k}^{+}\sigma^{-}+\hc\right)\,.\label{eq:H}
\end{equation}
Here, $\sigma^{\pm}=\sigma^{x}\pm i\sigma^{y}$ and $\tau_{k}^{\pm}=\tau_{k}^{x}\pm i\tau_{k}^{y}$ are raising and lowering
operators with respect to the qubit's ($\sigma$) and the spins' ($\tau_{k}$) Pauli matrices. The coupling strengths
$g_{k}$ between the qubit and the individual spins are proportional
to the magnitude of the qubit's field at the spin positions~\cite{marcos2010coupling,twamley2010superconducting}.
Assuming the case of a large spin ensemble with low polarization,
i.e., close to its ground state, we introduce a collective operator:
i.e. with $g=(\sum_{k}^{N}|g_{k}|^{2})^{1/2}$ we set $a^{\dagger}=g^{-1}\sum_{k}^{N}g_{k}\tau_{k}^{+}$
together with its hermitian conjugate $a$, yielding approximately
the bosonic commutation relation, $[a,a^{\dagger}]\cong1$ (See Appendix
A). Thus, we interpret the ensemble as an effective \emph{bosonic}
mode and arrive at the effective Jaynes-Cummings model, reading as
\begin{equation}
H_{\textrm{JC}}=\omega_{q}\sigma^{+}\sigma^{-}+\omega_{c}a^{\dagger}a+g\left(a^{\dagger}\sigma^{-}+a\sigma^{+}\right)\,.\label{eq:HJC}
\end{equation}
It describes a collective harmonic oscillator mode being coupled to
a two-level system with the interaction strength $g$. The collective
coupling $g$ is enhanced by a factor of $\sqrt{N}$ compared to the
root mean square of the individual couplings $g_{k}$, see Appendix
\ref{app:A}. Recent experiments achieved coupling strengths as strong
as $g\approx2\pi\times\unit[35]{MHz}$~\cite{zhu2011coherent}.
\section{The Jaynes-Cummings lattice with qubit-qubit coupling }
A general advantage of superconducting circuits is their scalability
and the rich variety of coupling mechanisms that can be implemented
on a chip. In particular, arrays of flux qubits with tunable coupling
strength between individual qubits have been realized using a SQUID
or ancilla flux qubit~\cite{harrabi2009engineered,hime2006solidstate,plourde2004entangling,niskanen2007quantum,vanderploeg2007controllable}.
In recent experiments manipulating coupling strengths \textit{in situ}
and the engineering of various types of circuit connectivity has become
feasible~\cite{harris2009compound,tsomokos2010usingsuperconducting,harris2010experimental,johnson2011quantum,grajcar2005directjosephson}.
In this work, we restrict ourselves to a chain of qubits with tunable
nearest neighbor interaction. This array of qubits can be readily
turned into a JC-lattice of coupled qubit-oscillator systems by putting
one NV-center crystal on top, as sketched with Fig.~\ref{setup}(c).
As argued above, the spin crystal adds an effective harmonic oscillator
degree of freedom to each site of the array. Apart from the possibility
to tune the coupling between the sites of the lattice, the most appealing
aspect of this hybrid architecture is simplicity. Furthermore, the
harmonic oscillators in the form of the spin crystal exhibit excellent
coherence properties, homogeneous transition frequencies and coupling
strengths, all implemented here within a reduced geometric dimension
as compared to coplanar waveguide resonators.
For well-separated qubits, we can neglect their mutual inductance
as well as the cross-coupling of one qubit to the spin ensemble of
another site, being even one order of magnitude smaller.{} This JC-lattice
with $M$ sites is thus described by
\begin{equation}
H_{\textrm{JCL}}=\sum_{j}^{M}H_{\textrm{JC,}j}+J\sum_{j}^{M-1}\left(\sigma_{j}^{+}\sigma_{j+1}^{-}+\hc\right)\ ,\label{eq:JC-array}
\end{equation}
with the single-site Hamiltonians $H_{\textrm{JC},j}$ given in Eq.~\eqref{eq:HJC}.
Here, $J$ denotes the uniform qubit-qubit coupling strength and the
operators $\sigma_{j}^{\pm}$ describing the creation and annihilation
of a qubit excitation at the $j$-th site.
A subtle but salient difference between our model and previously studied
JC-lattices is in the interaction mechanism between individual lattice
sites. While we propose an inter-site coupling mediated by the qubits,
previous works have dealt with the complementary approach where the
lattice sites interact via the oscillator degrees of freedom, as in
coupled cavities~\cite{hartmann2006strongly,greentree2006quantum,angelakis2007photonblockadeinduced,hartmann2008quantum,leib2010bosetextendashhubbard,makin2009timeevolution}
or in superconducting resonators~\cite{koch2009superfluidmottinsulator,schmidt2010nonequilibrium}.
In the latter case, the coupling part of the JC-lattice Hamiltonian~\eqref{eq:JC-array}
assumes the form $a_{j}^{\dagger}a_{j+1}+{\rm h.c.}$
\section{Equilibrium properties of the JC lattice }
For Bose-Hubbard--like models the occurrence of a quantum phase transition
between localized and delocalized phases has been extensively studied~\cite{sachdev2011quantum}.
Analogous transitions have been investigated with polaritons in JC-lattices~\cite{hartmann2006strongly,greentree2006quantum,angelakis2007photonblockadeinduced,hartmann2008quantum,koch2009superfluidmottinsulator,schmidt2010nonequilibrium,leib2010bosetextendashhubbard}.
Here, the term polariton refers to the eigenstates $|n,\pm\rangle$
of the single-site Hamiltonian $H_{\textrm{JC}}$ {[}Eq.~\eqref{eq:HJC}{]}.
The excitation number $n$, being the eigenvalues of the operator
$\mathcal{N}=a^{\dagger}a+\sigma^{+}\sigma^{-}$, are conserved due
to $[H_{\textrm{JC}},\mathcal{N}]=0$. Similarly, the full JC lattice
Hamiltonian~\eqref{eq:JC-array} conserves the total number of excitations
in the lattice. The ground state $|0\rangle$ has no excitations $n{=}0$,
while the states $|n,\pm\rangle$ ($n{>}0$) are each twofold degenerate
with respect to $\mathcal{N}$. If the qubit and the resonator are
in resonance, $\omega_{c}\,{=}\,\omega_{q}$, the polaritonic states
are symmetric~($+$) and antisymmetric~($-$) superpositions $|n,\pm\rangle=(|n-1\rangle|{\uparrow}\rangle\pm|n\rangle|{\downarrow}\rangle)/\sqrt{2}$
of the oscillator Fock states $|n\rangle$ and the qubit ground ($|{\downarrow}\rangle$)
and excited ($|{\uparrow}\rangle$) states, respectively.
\begin{figure}[!t]
\includegraphics[bb=0bp 0bp 247bp 369bp,width=1\columnwidth]{fig2}
\caption{Transition between the localized and delocalized phases in a qubit-coupled
JC-array. (a) Fluctuations of the number of excitations at a certain
site ${\rm var}(\mathcal{N}_{j})$ for a two-site setup (both sites
$j=1,2$ yield the same plot) as a function of the inter-site coupling
$J$ and the qubit-ensemble detuning $\Delta$. The dark shaded region
indicates that the system in the localized phase, while the brighter
areas are related to large fluctuations, i.e., the delocalized phase.
The two side panels depict a horizontal cut along $J=0.1g$ and a
vertical cut along $\Delta=2\times10^{-2}g$ (i.e., very close to qubit-oscillator
resonance), respectively. There, the solid black curves depict ${\rm var}(\mathcal{N}_{j})$
for the two-site setup as in the central panel. For comparison, we
have included the fluctuation characteristics ${\rm var}(\mathcal{N}_{j})$
for longer JC-arrays with $N{=}3$--$5$ sites in ascending order,
where $j$ denotes a central site of the array. (b) Comparison of
QQ- and CC-coupled chains. In the latter, the transition occurs at
lower detunings due to higher effective coupling of polaritons between
adjacent sites. This can be seen in the lower plot: Changing the detuning
affects both the effective repulsion $\delta$, as well as the effective
coupling $J_{\textrm{eff}}$. The transition occurs when $\delta$
and $J_{\textrm{eff}}$ cross.\label{fig2eq} }
\end{figure}
The localization-delocalization transition we consider in this work
takes place for the lowest energy state in the subspace with one average
excitation per site~\cite{hartmann2008quantum,angelakis2007photonblockadeinduced}.
As we argue with Appendix \ref{app:conservation}, for weak inter-site
coupling $J$, no inter-conversion between the $+$ and $-$ polaritons
occurs~\cite{koch2009superfluidmottinsulator,angelakis2007photonblockadeinduced}.
Therefore, in order to obtain analytical estimates, we can neglect
$|n,+\rangle$ polaritons and restrict our studies to $|n,-\rangle$
polaritons which are lower in energy. We then introduce the ``effective
repulsion'' $\delta=E_{|2,-\rangle}-2E_{|1,-\rangle}$, i.e.,
\begin{equation}
\delta=-\sqrt{2g^{2}+\frac{\Delta^{2}}{4}}+2\sqrt{g^{2}+\frac{\Delta^{2}}{4}}-\frac{\Delta}{2}\,.\label{delta}
\end{equation}
This positive-valued repulsion increases with the qubit-oscillator
coupling strength $g$ and decreases with the detuning $\Delta=\omega_{q}-\omega_{c}$.
It measures the extra energy needed to insert two polaritons into
a single site as compared to distributing them across two sites. Thus,
a large repulsion promotes an even distribution of excitations over
the lattice sites. In this case, the system eigenstate is approximately given by
a product of the local single-site eigenstates
$|1,-\rangle_{j}$.
By contrast, a large inter-site coupling quantified by $J$ favors
delocalized excitations, i.e., momentum eigenstates that are given
by a superposition of product states, each with different $n$. Thus,
modifying $J$ or the repulsion (e.g. by means of $\Delta$) one ends
up in two extreme regimes: the localized or the delocalized phase.
As stated at the beginning of this section, the Hamiltonian~(\ref{eq:JC-array})
conserves the total number of excitations. Therefore, the fluctuation
of the excitation number in a particular lattice site, ${\rm var}(\mathcal{N}_{j})=\langle\mathcal{N}_{j}^{2}\rangle-\langle\mathcal{N}_{j}\rangle^{2}$,
is used as an order parameter in JC-lattices~\cite{koch2009superfluidmottinsulator}.
For ${\rm var}(\mathcal{N}_{j})=0$, the excitations are trapped,
and the system is in a localized phase. By contrast, large fluctuations
indicate the delocalized phase.
In order to investigate the transition between the localized and the
delocalized phases numerically we calculate ${\rm var}(\mathcal{N}_{j})$
for a setup with two sites. As indicated in the main panel of Fig.~\ref{fig2eq}~(a),
a transition between the two different regimes characterized by zero
and finite ${\rm var}(\mathcal{N}_{j})$ occurs upon a change of $\Delta$
or $J$. This two-site setup already exhibits the same qualitative
features as longer arrays of finite length, as we corroborate in the
side panels of Fig.~\ref{fig2eq}(a). There, we compare the variance
for arrays with two or more sites by means of two cross sections through
the main panel for fixed values of $\Delta$ and $J$, respectively.
Thus, the elementary two-site setup -- readily feasible with present-day
experimental techniques -- already allows for a good qualitative estimate
of the transition properties of a JC-array. Experimental feasibility
will be further discussed in section \ref{sec:exp}.
To gain analytical insight, we express the coupling between
the individual sites in terms of the relevant polaritonic basis states
$|n,-\rangle_{j}$. In doing so, we can approximate $\sigma_{j}^{+}=\sum_{n}^{\infty}s_{n,--}|n+1,-\rangle_{j}\langle n,-|_{j}$,
where the coefficients $s_{n,--}$ depend on $\Delta$, $g$ and $n$
and their explicit form is detailed in Appendix~\ref{app:hopping}.
Thus, two sites initially in the state $|1,-\rangle$ are coupled
with the effective strength (see Appendix \ref{app:hopping})
\begin{equation}
J_{\textrm{eff}}=Js_{0,--}s_{1,--}\;.
\end{equation}
In the lower panel of Fig.~\ref{fig2eq}(b), we compare $J_{\textrm{eff}}$
to the effective repulsion strength $\delta$ {[}Eq.~\eqref{delta}{]},
both plotted as functions of $\Delta$ at fixed $J$. We find that
the observed crossing point of $J_{\textrm{eff}}$ and $\delta$ closely
matches the location of the localization-delocalization transition.
Furthermore, we compare our results for a JC-array with qubit-qubit
(QQ)-coupling to a similar setup with cavity-cavity (CC)-coupling,
i.e., an array in which the individual sites interact via their oscillator
degrees of freedom $J(a_{j}^{\dagger}a_{j+1}+{\rm h.c})$. For this
latter scenario, we find that the transition to the delocalized phase
already occurs at smaller $\Delta$, see Fig.~\ref{fig2eq}(b), top
panel. As in the (QQ)-coupled case we calculate $J_{\textrm{eff}}$
via the relevant polaritonic basis states and indeed find a larger
effective inter-site interaction that hence explains the observed transition
point. The interested reader can check the explicit coupling coefficients
in Appendix~\ref{app:hopping}. With increasing detuning, the $|n,-\rangle_{j}$
polaritons become more and more bosonic, i.e. only the oscillator
degree of freedom is excited, $|n,-\rangle_{j}\approx|{\downarrow}\rangle_{j}|n\rangle_{j}$.
This allows for a simple explanation of the different trends of $J_{\textrm{eff}}$
in Fig.~\ref{fig2eq}(b) when increasing the detuning. In a qubit-coupled
array, the bosonic excitations must hop via the route oscillator-qubit-qubit-oscillator
to reach the next lattice site, therefore $J_{\textrm{eff}}$ is small.
By contrast, in the cavity-coupled setup the bosonic excitations can
hop directly to the next-site oscillator and $J_{\textrm{eff}}$ is
therefore larger when the excitations are purely bosonic.
\section{Non-equilibrium dynamics }
The characterization of localized and delocalized phases at equilibrium
is helpful in exploring the physics in JC-lattices with qubit-qubit
coupling and contrasting it with JC-lattices that interact with a
cavity-cavity coupling. However, the assumption of staying in the
subspace with one mean excitation per site is not completely realistic
in practice. In particular, non-equilibrium processes such as dissipation
and decoherence are crucial in solid state devices. Next we point
out that the signatures of the localization-delocalization transition
remain preserved even in the presence of dissipation. A corresponding
measurement only requires state preparation and qubit readout.
We model dissipation for both the qubit and spin ensemble by means
of a quantum master equation, which for the JC lattice assumes the
form (at zero temperature)~\cite{scala2007cavitylosses,reuther2010tworesonator}
\begin{equation}
\dot{\varrho}(t)=-i[H_{{\rm JCL}},\varrho]+\sum_{j}\left(\gamma_{c}L_{a_{j}}[\varrho(t)]+\gamma_{q}L_{\sigma_{j}^{-}}[\varrho(t)]\right)\;.\label{qme}
\end{equation}
The Lindblad dissipators $L_{O}$ act on the density operator $\rho$
as $L_{O}[\rho]=O\rho O^{\dagger}-\frac{1}{2}(O^{\dagger}O\rho+\varrho O^{\dagger}O)$.
The operators $O=\{\sigma_{j}^{-},a_{j}\}$ describe the system-bath
coupling of the $j$-th qubit and oscillator, respectively, while
$\gamma_{q}$ and $\gamma_{c}$ are the associated, uniform decoherence
rates.
The system is initially prepared with one $|1,-\rangle$ polariton
in each site. Calculating the system dynamics numerically, we obtain
the time-dependent probability $P_{2}(t)=\textrm{Tr}\left\{ \Pi_{2}\varrho(t)\right\} $
of finding two excitations in one site, where $\Pi_{2}=|2\rangle|{\downarrow}\rangle\langle g|\langle2|+|1\rangle|{\uparrow}\rangle\langle e|\langle1|$.
If the system is in the localized phase we expect $P_{2}(t)$ to remain
close to zero. By contrast, in the the delocalized phase, $P_{2}(t)$
reaches finite values over time. This behavior is depicted in Fig.~\ref{fig3non-eq}(b).
Here, it is also visible that the system evolves eventually into its
ground state due to decoherence.
In order to quantify the phase we introduce the averaged probability
\begin{equation}
\bar{P}_{2}=\frac{1}{T}\int_{0}^{T}{\rm d}tP_{2}(t)\,.
\end{equation}
In order to take into account the dynamics before relaxation into
the ground state dominates, the integration time should fulfill $T\gg J_{{\rm eff}}^{-1}$
but $T\lesssim\min\{\gamma_{c}^{-1},\gamma_{q}^{-1}\}$. Fig.~\ref{fig3non-eq}(a)
depicts $\bar{P}_{2}$ as a function of both the hopping parameter
$J$ and the detuning $\Delta$ similar to the equilibrium analysis.
For comparison, the white dashed line marks the parameter regime where
the phase transition occurs in the equilibrium case in Fig.~\ref{fig2eq}.
While we find a good agreement for small values of $\Delta$, the
border between both phases is not resolved in the far-detuned limit
where the effective inter-site coupling strength $J_{\textrm{eff}}$
decreases below the decoherence rates $\{\gamma_{q},\gamma_{c}\}$.
\begin{figure}[!t]
\includegraphics[width=1\columnwidth]{fig3}\caption{Non-equilibrium signature of the phase transition. (a) Time-averaged
probability $\bar{P}_{2}$ to find two excitations in a single site.
The coupling is assumed as $g=2\pi\times\unit[10]{MHz}$ and the decay
rates of qubits and oscillators are $\gamma_{\mathrm{q}}=2\pi\times\unit[1]{MHz}$
and $\gamma_{\mathrm{c}}=2\pi\times\unit[0.1]{MHz}$. We choose the
integration time as $T=5\gamma_{\mathrm{q}}^{-1}$. The dotted line
marks the boundary ($\nicefrac{1}{2}\max\{{\rm var}(\mathcal{N}_{j})\}$)
where the phase transition occurs in the equilibrium case of Fig.~\ref{fig2eq}.
(b) Time evolution of $P_{2}$ in two exemplary points in the delocalized
(dashed line and dashed cross in (a)) and localized phase (solid line
and solid cross in (a)), respectively.\label{fig3non-eq} }
\end{figure}
\section{Experimental feasibility }
\label{sec:exp}
We next address the feasibility of the JC array proposed in this paper.
The case of strong coupling between an ensemble of NV-centers ($\omega_{c}\approx\unit[2\pi\times2.88]{MHz})$
and a flux qubit has previously been reported experimentally with
coupling strengths up to $g\simeq2\pi\times\unit[35]{MHz}$ \cite{zhu2011coherent}.
Thus, using our estimations, we can safely consider a coupling strength of $g\cong2\pi\times\unit[10]{MHz}$.
On the other hand, the experimental accessible tunable qubit-qubit
couplings are between $2\pi\times\unit[1]{MHz}\leq J\leq2\pi\times\unit[100]{MHz}$~\cite{vanderploeg2007controllable,hime2006solidstate,niskanen2007quantum}.
All together sets the operation range to $0.1\leq J/g\leq10$.
We next consider the decay rates of the involved subsystems. Realistic
values for the qubit decay rates are $\gamma_{q}=2\pi\times\unit[1]{MHz}$
(i.e. $\gamma_{q}/g=0.1$) and for the spin decay rates are $\gamma_{c}\leq2\pi\times\unit[0.1]{MHz}$
(i.e. $\gamma_{c}/g\leq0.01$).
These values were used to obtain the plots in Fig.~\ref{fig3non-eq}.
Therefore, current technology allows for monitoring both phases.
Finally, readout of the number of excitations at a specific site can
be performed by measuring the qubit dynamics.
\section{Conclusions}
In summary, we have introduced a novel JC-lattice based on a hybrid
combination of flux-qubits and NV-centers. In contrast to JC-lattices
based on coupled cavities or superconducting resonators, the harmonic
oscillator degree of freedom (``cavity'') is smaller in size than
the qubit (``atom''). This allows one to couple the individual JC-sites
via the qubits instead of the harmonic oscillators. We have argued
that similarly to \emph{cavity-coupled} JC-lattices a localization-delocalization
transition can be observed in these novel \emph{qubit-coupled} JC-lattices.
Even though localization-delocalization transitions in JC-lattices
have been proposed theoretically some time ago, they could not be
observed in an experiment yet. Our proposal relies on a straightforward
modification of already realized flux qubit arrays by simply mounting
a single NV-center crystal on top. This minimal modification of a
common setup opens the possibility of studying many-body phenomena
in strongly coupled hybrid architectures within state-of-the-art experimental
technology. Apart from its simplicity, further advantages are the
possibility to investigate localization-delocalization transitions
in arbitrary (even fractal) dimensions and to tune the inter-site
coupling in-situ by using common techniques for building flux qubit
networks.
\begin{acknowledgments}
We acknowledge enlightening discussions with M. Hartmann at TUM. This work
was supported by Spanish MICINN projects FIS2011-25167, CSD2007-046-
Nanolight.es, the European PROMISCE and the DFG via the Collaborative
Research Center SFB 631 and the Nanosystems Initiative Munich (NIM).
\end{acknowledgments}
|
{
"timestamp": "2012-05-11T02:03:15",
"yymm": "1203",
"arxiv_id": "1203.1857",
"language": "en",
"url": "https://arxiv.org/abs/1203.1857"
}
|
\section{Introduction}\label{sec:intro}
The Casimir force is known to depend on the electromagnetic properties of
the relevant objects (`mirrors') and on their geometric configuration, in a
rather involved way~\cite{reviews}.
To put the problem we shall deal with in context, let us consider the
Casimir force for a quite general situation, namely, we assume that the
geometry of the problem may be characterized by just two surfaces. Those
surfaces may correspond, for example, to the boundaries of two
mirrors. Alternatively, the surfaces themselves may describe zero-width
(`thin') mirrors. Yet another possibility is that those surfaces may be
the interfaces between media with different electromagnetic properties,
occupying different spatial regions.
In a situation like the ones above, one can think of the Casimir energy as
a {\em functional\/} of the functions determining the surfaces. Of course,
it is generally quite difficult to compute that functional for arbitrary
surfaces; exact results are available only for highly symmetric
configurations, the simplest of which is perhaps the case of two flat,
infinite, parallel plates.
However, when the surfaces are gently curved, almost parallel, and close to
each other, the proximity force approximation (PFA) is expected to be a
very accurate method to calculate the Casimir energy. Introduced by
Derjaguin many years ago~\cite{Derjaguin} to compute Van der Waals forces,
this approximation consists of replacing both surfaces by a set of parallel
plates. Then one calculates the energy as the sum of the Casimir energies
due to each pair of plates (each plate paired with the nearest one in the other
mirror).
The PFA has also been used successfully applied in other contexts, like nuclear
physics~\cite{nuclear} and electrostatics~\cite{electrostatics}.
In spite of the simplicity and long standing usefulness of the PFA, its validity had not
been possible to asses until quite recently, mostly because there was no
systematic way of improving the approximation. Indeed, even the next to
leading order (NTLO) correction was unknown.
In a recent work~\cite{pfa_nos}, we have shown that the PFA can
be thought of as an expansion of the Casimir energy in derivatives of the
functions that describe the shapes of the surfaces. The leading order in
this expansion, that contains no derivatives, does reproduce the PFA,
while the higher order terms contain the corrections. In Ref.~\cite{pfa_nos} we
considered the case of a flat surface in front of a gently curved one, the
latter described by a function $x_3=\psi(x_1,x_2)$. For simplicity, we computed
the vacuum energy for a massless quantum scalar field satisfying Dirichlet
boundary conditions, the result being:
\begin{equation}
E_{\rm DE} \simeq - \frac{\pi^2}{1440}\int d^2{\mathbf x_ \parallel} \;
\frac{1}{\psi^3}\left[\beta_1+\beta_2(\partial_\alpha\psi)^2\right]\, ,
\label{DE}\end{equation}
with $\beta_1=1$ and $\beta_2=2/3$.
The first term in this expression is the PFA, while the second term is the
NTLO correction. This result has been generalized by Bimonte et al to the
case of two curved, perfectly conducting surfaces, for scalar fields
satisfying Dirichlet or Neumann boundary conditions, and also to the
electromagnetic case~\cite{pfa_mit1}.
The results for the latter are, $\beta_1=2$ and $\beta_2=4/3(1-15/\pi^2)$.
As a validity check, it has been shown that, whenever analytic results are
available for particular geometries, the corresponding derivative expansion
correctly reproduces both the PFA and its NTLO
correction~\cite{pfa_nos,pfa_mit1}. Moreover, initial
discrepancies~\cite{pfa_mit1} between the improved PFA and the analytic
calculations for the particular case of a cylinder in front of a
plane~\cite{bordag_cp}, has been resolved in favor of the improved PFA
after a revision of the rather involved analytic calculation for this
particular geometry~\cite{bordagteo}.
Bimonte et al also considered the case in which the surfaces are
interfaces between different media, with frequency-dependent permittivity~\cite{pfa_mit2}.
In this case, the numerical coefficients $\beta_1$ and $\beta_2$ become
rather complicated functions of $\psi$ and of the dimensional constants that describe the
electromagnetic properties of the media.
In this paper, we will extend the improved proximity force approximation to
the case of two imperfect thin mirrors. This kind of configuration
have already been considered in several previous works; for instance, in
order to describe the interactions of plasma sheets, graphene sheets or, more
generally, arbitrary semi-transparent mirrors, both for the static and
dynamical Casimir effects~\cite{varios1,bordaggraph,thindce}.
In some derivations of the Casimir energy for perfect and imperfect
mirrors, the boundary conditions at the interfaces are represented in terms
of auxiliary scalar fields coupled to the TE and TM modes of the
electromagnetic field~\cite{golest}. We will follow here a similar
approach, but developing a new formalism, based on vector auxiliary fields
that couple to the dual of the Maxwell tensor $F_{\alpha\beta}$ evaluated
on the surfaces. In this formalism gauge invariance is more apparent at
the different stages of the calculation. Moreover, the formalism could be
useful to address problems with more complex geometries, where it could not
be possible to describe the electromagnetic field in terms of independent
TE and TM modes.
The paper is organized as follows. In the next Section we describe the
model and introduce the necessary definitions and conventions. In Section 3
we derive the formal expression of the vacuum energy for the
electromagnetic field, using the above mentioned formalism based on a
vector auxiliary field. The derivative expansion for the electromagnetic
vacuum energy is presented in Section 4. We discuss the results in Section
5, where we analyze the two limiting cases of perfectly conducting and near
transparent mirrors. In the latter, we find that the NTLO correction to the
PFA is tantamount to use the area of the curved surface in the leading
order expression. We also discuss in that section a particular class of
imperfect mirrors, in which the transmission and reflection coefficients do
not involve dimensionful constants. For these `graphene-like' mirrors,
dimensional analysis implies that the vacuum energy is of the form given in
Eq.(\ref{DE}), where $\beta_1$ and $\beta_2$ are constants and depend on the
dimensionless quantities that describe the mirrors. We present some
numerical evaluations of these coefficients that interpolate between almost
transparent and perfectly conducting mirrors.
Section 6 contains the conclusions of this work.
\section{The model: definitions and conventions}
We shall consider a model in which the role of the fluctuating vacuum field is
played by an Abelian gauge field, $A_\mu$, in $3+1$ dimensions, coupled to
two imperfect mirrors, $L$ and $R$. These are presumed to have negligible
widths, so that we shall use an idealized description whereby they are
treated as mathematical surfaces. We will, moreover, assume that one of the
surfaces ($L$) is a plane, while the other ($R$), which may be curved, can
always be described by a Monge patch. Summarizing, the two surfaces
correspond to:
\begin{equation}\label{eq:deflr}
L) \;\; x_3 \,=\, 0 \,\;\;\;\;\;\; R) \;\; x_3 \,=\, \psi(x_1,x_2) \;,
\end{equation}
where $x_i$ \mbox{($i=1,2,3$)} are the spatial coordinates. Throughout
this work we shall use Euclidean conventions, with $x^\mu \equiv x_\mu$
\mbox{($\mu=0,1,2,3$)}, $x_0$ being the imaginary time.
However, we have found it simpler to keep our treatment quite general
regarding the actual form of the surfaces, postponing the use of
(\ref{eq:deflr}) to the point when we actually need those particular expressions.
The action ${\mathcal S}$ for this model will have the following structure:
\begin{eqnarray}\label{eq:defs}
{\mathcal S} &=& {\mathcal S}(A; y_L, y_R) \nonumber\\
&=& {\mathcal S}_0(A) \,+\, {\mathcal S}_L(A; y_L)
\,+\, {\mathcal S}_R(A; y_R) \;,
\end{eqnarray}
where $A$ denotes the electromagnetic field and
${\mathcal S}_0$ its free action. ${\mathcal S}_L$ and ${\mathcal
S}_R$ are terms that couple $A$ to each mirror, with $y_L$ and $y_R$
denoting parametrizations of their respective surfaces. ${\mathcal S}_L$ and ${\mathcal S}_R$
can be different because of two
reasons: first, they correspond to different surfaces, and second, they may
also have to account for different electromagnetic (response) properties,
for example, when the mirrors are composed of different materials.
We shall then consider a rather general term, ${\mathcal S}_\Sigma$,
corresponding to the coupling to an arbitrary surface $\Sigma$,
particularizing to the $L$ and $R$ cases afterwards. Thus, we assume the
static surface $\Sigma$ to be defined in parametric form:
\begin{equation}\label{eq:defsigma}
\Sigma \big) \;\;\; (\sigma^1, \sigma^2) \; \to \; {\mathbf
y}(\sigma^1, \sigma^2) \;\in\; {\mathbb R}^3\;.
\end{equation}
Although the surface is static, to write the ($2+1$-dimensional) term
${\mathcal S}_\Sigma$ it is, however, convenient, to introduce a
parametrization for the world-volume ${\mathcal V}$ swept by the surface
$\Sigma$, since that is the spacetime region ${\mathcal V}$ where the
interaction takes place:
\begin{equation}\label{eq:defcalv}
{\mathcal V} \big) \;\;\; (\sigma^0, \sigma^1, \sigma^2) \; \to \;
y^\mu(\sigma^0, \sigma^1, \sigma^2) \, \equiv \, y^\mu(\sigma),
\;\;\;\;\mu=0,\,1,\,2,\,3,
\end{equation}
where $y^0 = \sigma^0$, and ${\mathbf y}$ as in (\ref{eq:defsigma}).
The world-volume is three-dimensional, and we adopt the convention of
using indices from the beginning of the Greek alphabet to denote components
in that space; for example, in an expression like $d\sigma^\alpha$ we
implicitly assume that $\alpha$ runs from $0$ to $2$. We do need to introduce
more objects in that space, like the induced metric, $g_{\alpha\beta}(\sigma)$,
which may be written in terms of the parametrization:
\begin{equation}\label{eq:defgab}
g_{\alpha\beta}(\sigma) = \frac{\partial y_\mu(\sigma)}{\partial \sigma^\alpha}
\frac{\partial y_\mu(\sigma)}{\partial \sigma^\beta} \;.
\end{equation}
We also need to introduce $e^\mu_\alpha$, a local basis of tangent vectors
to ${\mathcal V}$, such that \mbox{$e^\mu_\alpha = \frac{\partial
y^\mu}{\partial \sigma^\alpha}$}. They are, by construction,
normalized to satisfy the condition \mbox{$e^\mu_\alpha(\sigma) e^\mu_\beta(\sigma) =
g_{\alpha\beta}(\sigma)$}.
Before writing the explicit expression for the action, and to make contact with previous works,
let us describe a simpler model with a vacuum scalar field.
The free action is
\begin{equation}
S_0(\varphi) = \int d^4x \frac{1}{2}\partial_\mu\varphi\partial_\mu\varphi\, .
\end{equation}
Assuming that the surface action ${\mathcal S}_\Sigma$ is quadratic in $\varphi$,
its general form is
\begin{equation}
{\mathcal S}_\Sigma (\varphi; y) = \frac{1}{2} \int d^3\sigma \,
\sqrt{g(\sigma)} \, d^3\sigma' \, \sqrt{g(\sigma')} \,
\varphi(\sigma) \varphi(\sigma')
{\pi(\sigma,\sigma')} \;,
\end{equation}
where $g(\sigma)$ is the determinant of the induced metric and $\pi(\sigma,\sigma')$
describes the (nonlocal) response of the mirror. The local approximation of this action is
\begin{equation}
{\mathcal S}_\Sigma (\varphi; y) = \frac{\lambda_s}{2} \int
d^3\sigma \, \sqrt{g(\sigma)} \varphi(\sigma)^2 \;,
\label{deltapot}
\end{equation}
where $\lambda_s$ is a constant (the subindex $s$ stands for scalar).
Eq.(\ref{deltapot}) describes the so called `$\delta$-potentials', widely used
as toy models to describe imperfect mirrors. One can
check that this kind of potentials induce a discontinuity in the normal
derivative of the scalar field across the surface, i.e. ${\rm disc}
[\partial_n\varphi] = \lambda_s\varphi$. The factor $ \sqrt{g(\sigma)}$ is
crucial to produce such boundary condition \cite{sqrtg}. In the limit
$\lambda_s\to\infty$, the field must vanish on the surface in order to have
a finite discontinuity across the surface, and therefore one recovers the
usual Dirichlet boundary condition on $\Sigma$.
In the electromagnetic case, the explicit form of ${\mathcal S}_0(A)$ will be
\begin{equation}
S_0(A) \;=\; \int d^4x \big[ \frac{1}{4} F_{\mu\nu} F_{\mu\nu}
\,+\, \frac{b}{2} (\partial_\mu A_\mu)^2 \big] \;,
\end{equation}
where the term proportional to $b$ provides the gauge-fixing. In the
calculations presented in the next sections, we shall adopt the Feynman
($b=1$) gauge.
Assuming that ${\mathcal S}_\Sigma$ is quadratic in $A_\mu$, gauge invariance
implies that it will have the general form
\begin{equation}\label{eq:gralsigma}
{\mathcal S}_\Sigma (A; y) = \frac{1}{4} \, \int
d^3\sigma \, \sqrt{g(\sigma)} \, d^3\sigma' \, \sqrt{g(\sigma')}
\, F_{\alpha\beta}(\sigma) F_{\alpha'\beta'}(\sigma')
\pi^{\alpha\beta\alpha'\beta'}(\sigma,\sigma') \;,
\end{equation}
where $\pi^{\alpha\beta\alpha'\beta'}$ is a polarization tensor that
depends on the microscopic degrees of freedom on the mirror, and
\begin{equation}
F_{\alpha\beta}\,=\, \nabla_\alpha A_\beta - \nabla_\beta A_\alpha
\,=\, \partial_\alpha A_\beta - \partial_\beta A_\alpha
\;,
\end{equation}
where $\nabla_\alpha$ denotes the covariant derivative operator,
corresponding to the connection for the induced metric, acting
(in this case) on a covariant vector. We have used $A_\alpha (\sigma)$ as
a shorthand for the components of the gauge field $A_\mu(x)$ on ${\mathcal
V}$, projected along the directions defined by the local basis:
\begin{equation}\label{eq:defasigma}
A_\alpha(\sigma) \;\equiv\; A_\mu[y(\sigma)] \,
e^\mu_\alpha(\sigma) \;.
\end{equation}
As in the scalar case, we will start our discussion with a local interaction
\begin{equation}\label{eq:defssigma}
{\mathcal S}_\Sigma (A; y) \;=\; \frac{\lambda}{4} \, \int
d^3\sigma \, \sqrt{g(\sigma)}\, F_{\alpha\beta} F^{\alpha\beta} \;,
\end{equation}
where
$\lambda$ is a constant, and afterwards we shall
extend the results to include frequency-dependent couplings. Note that the constants
$\lambda$ and $\lambda_s$ have different dimensions.
As it should be evident from the
actual form of the interaction term that we are assuming for the model,
this kind of mirror involves only on the gauge field components
which are {\em parallel} to the world-volume. However, a term like this
induces discontinuities across the surface of
the component of the electric field which is normal to
$\Sigma$, and of the components of the magnetic field which are parallel to
that surface. The discontinuity is proportional to $\lambda$ and depends on
parallel components of the gauge field, producing the boundary conditions of
a perfect conductor in the limit $\lambda\to\infty$, as in the scalar case.
For instance, for the flat surface
at $x_3=0$ the boundary conditions read
\begin{equation}
\rm{disc}(F_{3\nu})=\lambda\, \partial_{\hat\mu}F_{\hat\mu\nu}\, ,
\label{bc}
\end{equation}
where the sum over $\hat\mu$ excludes $\mu=3$. These boundary conditions can be
explicitly written in terms of the field components as
\begin{eqnarray}
\rm{disc}(E_3) &=& \lambda \partial_1 E_1,\nonumber \\
\rm{disc}(B_2) &=& -\lambda \partial_0 E_1,\nonumber\\
\rm{disc}(B_1) &=& - \lambda \left( \partial_0 E_2 + \partial_1 B_3 \right ),\nonumber
\end{eqnarray}
where, for simplicity, we assumed that the fields do not depend on the coordinate $x_2$.
Note that in Eq.(\ref{eq:defssigma}) we are assuming a Lorentz-invariant interaction, which would be
produced by relativistic degrees of freedom on the mirror. One could of course consider the interaction between
the gauge fields and non-relativistic matter on the mirror, giving different boundary conditions \cite{barton2004,plb2008}.
For example, the boundary conditions obtained in the case of a fluid of nonrelativistic electrons \cite{barton2004} coincide
with ours for the normal component of the electric field, but differ for the parallel components of the magnetic field.
Accordingly, the reflection and transmission coefficients in both models will be different.
Finally, one should check that a term like
(\ref{eq:defssigma}) does preserve gauge invariance. This is indeed
the case that can be seen from the fact that under the transformation:
\mbox{$A_\mu(x) \to A_\mu(x) + \partial_\mu \omega(x)$}, which is a $U(1)$
gauge transformation in $3+1$ dimensions, one has \mbox{$A_\alpha(\sigma)
\to A_\alpha(\sigma) + \delta A_\alpha(\sigma)$}, with:
\begin{equation}
\delta A_\alpha(\sigma) = \partial_\mu\omega[y(\sigma)] \, e^\mu_\alpha(\sigma)
= \partial_\mu\omega[y(\sigma)] \partial_\alpha y^\mu(\sigma)
= \partial_\alpha \omega(\sigma)
\end{equation}
where $\omega(\sigma) \equiv \omega[y(\sigma)]$. Thus $S_{\Sigma}$ is
invariant, since $\delta F_{\alpha\beta}(\sigma) = 0$.
\section{Electromagnetic vacuum energy: auxiliary vector fields}
In the functional approach to the Casimir effect, to obtain the vacuum
energy, one usually starts from ${\mathcal Z}$, the vacuum transition
amplitude, or, equivalently, the zero temperature limit of a
finite-temperature partition function. For the vacuum field $A_\mu$, in the
presence of the two mirrors, the case at hand,
${\mathcal Z}$ may be written as follows:
\begin{equation}\label{eq:dezfda}
{\mathcal Z}\;=\; \int {\mathcal D}A_\mu \;
\exp\Big[- {\mathcal S}_0(A) - {\mathcal S}_L(A; y_L) - {\mathcal
S}_R(A; y_R) \Big]\;.
\end{equation}
To proceed, one should integrate the electromagnetic field. To that end,
it is convenient to perform first a transformation of the interaction
terms, so that $A_\mu$ only appears linearly, rather than quadratically. This
may be done at the expense of introducing auxiliary fields, a procedure
that we implement now. To simplify the procedure, we first represent
$F_{\alpha\beta}$ in terms of its dual $\widetilde{F}^\alpha$, a pseudo-vector, such that:
\begin{equation}\label{eq:dual}
F_{\alpha\beta} \,=\, \sqrt{g(\sigma)}
\,\epsilon_{\alpha\beta\gamma} \, \widetilde{F}^\gamma \;,\;\;\;
\sqrt{g(\sigma)} \widetilde{F}^\alpha \,=\,\epsilon^{\alpha\beta\gamma}
\partial_\beta A_\gamma \;,
\end{equation}
where we adopted the convention that $\epsilon_{\alpha\beta\gamma}$, as
well as $\epsilon^{\alpha\beta\gamma}$, denote the Levi-Civita permutation
symbol (i.e., without including any power of $g$ as a factor).
The generic term $S_\Sigma$ may be written as
\begin{equation}
{\mathcal S}_\Sigma (A; y) \;=\; \frac{\lambda}{2} \, \int
d^3\sigma \, \sqrt{g(\sigma)}\; \widetilde{F}_\alpha \, {\widetilde F}^\alpha \;.
\end{equation}
Then we introduce a pseudo-vector auxiliary field $\xi_\alpha(\sigma)$, so
that the exponential of the interaction term above may be
obtained as the result of a Gaussian integral:
\begin{equation}\label{eq:exponent}
\exp\left [-{\mathcal S}_\Sigma (A; y) \right] \;=\; \frac{1}{\mathcal N_\xi}\int {\mathcal D} \xi \;
\exp \big[ - {\mathcal S}_q(\xi;\lambda) + i \int d^3\sigma \sqrt{g(\sigma)} \xi_\alpha(\sigma)
\widetilde{F}^\alpha(\sigma) \big] \;,
\end{equation}
where $S_q(\xi;\lambda) = \frac{1}{2 \lambda} \int d^3\sigma \sqrt{g(\sigma)} \xi_\alpha(\sigma)
\xi^\alpha(\sigma)$ and
\begin{equation}
{\mathcal N_\xi} \;=\; \int {\mathcal D} \xi \; e^{-{\mathcal
S}_q(\xi;\lambda)} \;.
\end{equation}
Note that the representation above is not unique, in the following sense:
defining the longitudinal ($l$) and
transverse ($t$) components of $\xi$:
\begin{equation}
\xi_l^\alpha(\sigma) \;=\; \nabla^\alpha \frac{1}{\Delta}
\nabla_\beta \xi^\beta \;,\;\;\;
\xi_t^\alpha(\sigma) \;=\; \xi^\alpha(\sigma) \,-\,
\xi_l^\alpha(\sigma)\;,
\end{equation}
we see that $\xi_l$ does not couple to $A_\alpha$. Indeed, because of (\ref{eq:dual}), we see that:
\begin{equation}
\int d^3\sigma \sqrt{g(\sigma)} \xi^\alpha(\sigma)
\widetilde{F}_\alpha(\sigma) \,=\, \int d^3\sigma \sqrt{g(\sigma)} \xi_t^\alpha(\sigma)
\widetilde{F}_\alpha(\sigma)\, ,
\end{equation}
where we have used Bianchi's identity:
\mbox{$\nabla^\alpha\widetilde{F}_\alpha=\epsilon^{\alpha\beta\gamma}\partial_\alpha \partial_\beta
A_\gamma=0$}.
It is, therefore, possible to modify the auxiliary field action, for example
by adding a term depending only on $\xi_l$ to the ${\mathcal S}_q$ term,
such that:
\begin{equation}
{\mathcal S}_q \to {\mathcal S}'_q = {\mathcal S}_q +
{\mathcal S}_{\xi_\Sigma}\, ,
\end{equation}
where ${\mathcal S}_{\xi_\Sigma}$ is a function of $\xi^l$ (and not of
$\xi_t$) whose precise form will be determined in order to simplify the
calculations.
Thus, a more general (but equally valid) way to rewrite the interaction term is
\begin{equation}\label{eq:exponentg}
\exp\big[-{\mathcal S}_\Sigma (A; y) \big] = \frac{1}{\mathcal N'_\xi} \, \int {\mathcal D}
\xi \; \exp \Big[- {\mathcal S'}_q(\xi;\lambda) + i \int d^3\sigma
\sqrt{g(\sigma)} \, \xi_\alpha(\sigma) \widetilde{F}^\alpha(\sigma) \Big] \;,
\end{equation}
\begin{equation}
{\mathcal N'_\xi} \;\equiv\; \int {\mathcal D} \xi \; e^{-{\mathcal
S'}_q(\xi;\lambda)} \;.
\end{equation}
Besides, note that the term which couples linearly the gauge field to the
auxiliary field, can be reinterpreted as an interaction with a
surface-dependent `current' $J^\mu_\Sigma(x)$:
\begin{equation}
\int d^3\sigma \xi_\alpha(\sigma)
\epsilon^{\alpha\beta\gamma}\partial_\beta A_\gamma(\sigma) \,=\, \int d^4x
J^\mu_\Sigma(x) \, A_\mu(x)\;,
\end{equation}
where
\begin{equation}\label{eq:jsigma}
J^\mu_\Sigma(x) \;=\; \int d^3\sigma \, \delta^{(4)}[x-y(\sigma)] \,
e^\mu_\alpha(\sigma) \, \epsilon^{\alpha\beta\gamma}\partial_\beta
\xi_\gamma(\sigma) \;,
\end{equation}
is a `topologically conserved' current, namely, it satisfies
\mbox{$\partial_\mu J^\mu_\Sigma = 0$}, by its very form, regardless of dynamics.
The process introduced above for $\Sigma$ may be then independently applied to the two
interaction terms, ${\mathcal S}_L$ and ${\mathcal S}_R$, which are
defined as follows:
\begin{eqnarray}
{\mathcal S}_L(A;y_L) &=& {\mathcal S}_\Sigma (A; y)
\Big|_{\Sigma \to L,\, y \to y_L, \, \lambda \to \lambda_L} \nonumber\\
{\mathcal S}_R(A;y_R) &=& {\mathcal S}_\Sigma (A; y) \Big|_{\Sigma
\to R,\, y \to y_R,\,\lambda \to \lambda_R}\;.
\end{eqnarray}
Since we introduce one auxiliary field for each interaction term in the
action, the gauge field will be coupled linearly to the sum of two
currents. Indeed, the use of two auxiliary fields, $\xi_L$ and $\xi_R$, in the
partition function, yields:
$${\mathcal Z} \;=\; \frac{1}{\mathcal N'_{\xi_L} \mathcal
N'_{\xi_R}} \int {\mathcal D}\xi_L {\mathcal D}\xi_R
\; e^{- {\mathcal S'}_q(\xi_L;\lambda_L) - {\mathcal S'}_q(\xi_R;\lambda_R)}$$
\begin{equation}\label{eq:zetal}
\times \,\int {\mathcal D}A \, e^{-{\mathcal S}_0(A) + i \int d^4x \,
J^\mu(x)A^\mu(x)}
\end{equation}
where $J \equiv J_L + J_R$, with $J_L$ and $J_R$ obtained from (\ref{eq:jsigma}), replacing
$\Sigma$ by $L$ and $R$, respectively.
Integrating out $A_\mu$ in (\ref{eq:zetal}), yields:
\begin{eqnarray}\label{eq:zetal1}
{\mathcal Z} \;=\; \frac{\mathcal Z_0}{\mathcal N'_{\xi_L} \mathcal
N'_{\xi_R}} \; \int {\mathcal D}\xi_L {\mathcal D}\xi_R
\; \Big\{ e^{- {\mathcal S'}_q(\xi_L;\lambda_L) - {\mathcal S'}_q(\xi_R;\lambda_R)}
\nonumber\\
\times \exp\big[- \frac{1}{2} \int d^4x \int d^4x' J^\mu(x)
D_{\mu\mu'}(x,x') J^{\mu'}(x')\big] \Big\} \;,
\end{eqnarray}
where $D_{\mu\mu'}$ is the (free) $A$-field propagator, which in the Feynman
gauge becomes:
\begin{equation}\label{eq:propa}
D_{\mu\mu'}(x,x') \;=\; \delta_{\mu \mu'} \; D(x,x')\; .
\end{equation}
Here $D(x,x')$ is the Euclidean free scalar field propagator in $3+1$
dimensions:
\begin{equation}\label{eq:defscp}
D(x,x') \,=\, \langle x| \frac{1}{-\partial^2} |x'\rangle \,=\,
\int \frac{d^4k}{(2\pi)^4} \, \frac{e^{i k \cdot (x-x')}}{k^2} \, ,
\end{equation}
where we have used a `bra-ket' notation to denote matrix elements of
functional operators. Note that ${\mathcal Z}_0 = \int {\mathcal D}A \,
e^{-{\mathcal S}_0}$ cannot contribute to the
Casimir energy, since it is independent of the coupling to the mirrors. On
the other hand, the normalization factors ${\mathcal N'_{\xi_L}}$
${\mathcal N'_{\xi_R}}$ do not contribute either, albeit for a different
reason: each one of them depends only on the properties of one the mirrors,
being adamant to the coupling of the other. Thus, we define the vacuum
energy, $E_{\rm vac}$, in such a way that those contributions, irrelevant to
the Casimir interaction energy, are subtracted from the very beginning:
\begin{equation}\label{eq:defgamma}
E_{\rm vac} \,=\, \lim_{T \to \infty} \Big( \frac{\Gamma}{T} \Big) \;,\;\;\;\;
e^{-\Gamma} \;\equiv\; {\mathcal Z} \;
\frac{{\mathcal N'_{\xi_L}}{\mathcal N'_{\xi_R}}}{{\mathcal Z}_0}\;,
\end{equation}
where $T$ is the extent of the imaginary time interval.
We then proceed to the evaluation of $\Gamma$, defined in
(\ref{eq:defgamma}). We note that there still remain in this object
contributions that correspond to mirrors' self-interactions, depending on
only one of the mirrors. They will be neglected, since our objective is to
calculate the Casimir interaction energy between two mirrors, a physical
magnitude to which self-interaction energies cannot contribute.
We deal now with the functional integral expression for $\Gamma$,
which in view of the above has the following structure:
\begin{equation}\label{eq:gamma0}
e^{-\Gamma} \,=\, \int {\mathcal D}\xi_L {\mathcal D}\xi_R \; e^{- S_\Gamma (\xi_L, \xi_R) }
\end{equation}
where
\begin{eqnarray}
S_\Gamma (\xi_L, \xi_R) &=& {\mathcal S}_q(\xi_L;\lambda_L) + {\mathcal
S}_q(\xi_R;\lambda_R) \nonumber\\
&+& \frac{1}{2} \int d^4x \int d^4x' J_L^\mu(x) D_{\mu\mu'}(x,x')
J_L^{\mu'}(x') \nonumber\\
&+&\frac{1}{2} \int d^4x \int d^4x' J_L^\mu(x) D_{\mu\mu'}(x,x')
J_R^{\mu'}(x') \nonumber\\
&+&\frac{1}{2} \int d^4x \int d^4x' J_R^\mu(x) D_{\mu\mu'}(x,x')
J_L^{\mu'}(x') \nonumber\\
&+&\frac{1}{2} \int d^4x \int d^4x' J_R^\mu(x) D_{\mu\mu'}(x,x') J_R^{\mu'}(x') \;,
\end{eqnarray}
which is a quadratic form in the auxiliary fields. In order to perform the
integral over the auxiliary fields, we need an explicit form for the
different terms in ${\mathcal S}_\Gamma$.
Taking into account (\ref{eq:deflr}), we can find the metric tensors and
local tangent vector for each mirror; all of these are elements
that enter in the terms above. In both cases, the parameters
$\sigma^\alpha$ are chosen as $\sigma^\alpha = x_\alpha$, with
$\alpha=0,1,2$. We also refer to $(x_\alpha)$ as $x_\parallel$, reserving the
notation ${\mathbf x}_\parallel$ for $(x_1,x_2)$.
For the $L$ surface, the parametrization is then:
\begin{equation}
x_\parallel \to y_L(x_\parallel),\;\;
y_L(x_\parallel)=( x_\parallel, 0) \;,
\end{equation}
thus, for $L$ we simply have \mbox{$g_{\alpha\beta} =
\delta_{\alpha\beta}$}, $e^\mu_\alpha = \delta^\mu_\alpha$ for $\mu=0,1,2$,
while $e^3_\alpha=0$. For $R$, on the other hand:
\begin{equation}
x_\parallel \to y_R(x_\parallel),\;\;
y_R(x_\parallel)=( x_\parallel, \psi({\mathbf x}_\parallel)) \;.
\end{equation}
Therefore,
\begin{equation}
(g_{\alpha\beta}) \;=\; \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 + (\partial_1 \psi)^2 & \partial_1\psi \partial_2
\psi \\
0 & \partial_2\psi \partial_1\psi & 1 + (\partial_2\psi)^2
\end{array}
\right) \;,
\end{equation}
which implies: $\sqrt{g} = \sqrt{ 1 + ({\nabla \psi})^2}$. The tangent
vectors, on the other hand, are given by:
\begin{equation}
e^\mu_\alpha(x_\parallel) \,=\, \delta^\mu_\alpha \,+\,
\partial_\alpha\psi({\mathbf x}_\parallel) \delta^\mu_3 \;
=\; \left\{
\begin{array}{ccc}
\delta^\mu_0 & {\rm if} & \alpha = 0 \\
\delta^\mu_i + \partial_i \psi({\mathbf x}_\parallel) \delta^\mu_3 & {\rm
if} & \alpha = i = 1, 2 \;.
\end{array}
\right.
\end{equation}
Then we find that:
$$
\int_{x,x'} J_L^\mu(x) D_{\mu\mu'}(x,x') J_L^{\mu'}(x')
$$
\begin{equation}
= \int_{x_\parallel,x'_\parallel} \xi^L_\alpha(x_\parallel)
\;(\partial_\beta \partial'^\beta \delta^{\alpha\alpha'} -
\partial^{\alpha'}
{\partial'}^{\alpha})
D(x_\parallel, 0 ; x'_\parallel,0) \; \xi^L_{\alpha'}(x'_\parallel)
\end{equation}
where we have adopted the notations: $\partial^\alpha \equiv
\partial/ {\partial x_\alpha}$, ${\partial'}^{\alpha} \equiv
\partial/{\partial x'_{\alpha}}$, etc. Besides, we have written the integration
variables as a subindex of the integral.
For the analogous term that involves the $J_R$ current instead of $J_L$,
the corresponding expression is:
$$
\int_{x,x'} J_R^\mu(x) D_{\mu\mu'}(x,x') J_R^{\mu'}(x')
$$
\begin{eqnarray}
= \int_{x_\parallel,x'_\parallel} \; \xi^R_\alpha(x_\parallel)
\Big\{ \big[(\partial_\beta \partial'^\beta \delta^{\alpha\alpha'} -
\partial^{\alpha'} {\partial'}^\alpha)
D(x_\parallel, \psi({\mathbf x}_\parallel);
x'_\parallel,\psi({\mathbf x}'_\parallel)) \big]
\nonumber\\
+ \epsilon^{\alpha\beta\gamma}
\epsilon^{{\alpha'}{\beta'}{\gamma'}}
\partial_\beta\psi(x_\parallel) \, \partial'_{\beta'}
\psi({x'}_\parallel)
\big[\partial_\gamma \partial'_{\gamma'}
D(x_\parallel, \psi({\mathbf x}_\parallel) ;
x'_\parallel,\psi({\mathbf x}'_\parallel))\big] \Big\}
\xi^R_{\alpha'}(x'_\parallel) \;.
\end{eqnarray}
Finally,
$$
\int_{x,x'} J_L^\mu(x) D_{\mu\mu'}(x,x') J_R^{\mu'}(x')
$$
\begin{equation}
= \int_{x_\parallel,x'_\parallel} \xi^L_\alpha(x_\parallel)
\;(\partial_\beta \partial'^\beta \delta^{\alpha\alpha'} -
\partial^{\alpha'}
{\partial'}^{\alpha})
D(x_\parallel, 0 ; x'_\parallel,\psi({\mathbf x}'_\parallel)) \; \xi^R_{\alpha'}(x'_\parallel)
\end{equation}
and
$$
\int_{x,x'} J_R^\mu(x) D_{\mu\mu'}(x,x') J_L^{\mu'}(x')
$$
\begin{equation}
= \int_{x_\parallel,x'_\parallel} \xi^R_\alpha(x_\parallel)
\;(\partial_\beta \partial'^\beta \delta^{\alpha\alpha'} -
\partial^{\alpha'} {\partial'}^{\alpha})
D(x_\parallel, \psi({\mathbf x}_\parallel) ; x'_\parallel,0)
\; \xi^L_{\alpha'}(x'_\parallel) \;.
\end{equation}
Defining the matrix kernel ${\mathbb T}$, such that
\begin{equation}
{\mathcal S}_\Gamma \;=\; \frac{1}{2} \int_{x_\parallel,x'_\parallel}
\xi^a_\alpha(x_\parallel) \, {\mathbb T}_{\alpha \alpha'}^{ab}(x_\parallel, x'_\parallel)
\xi^b_{\alpha'}(x'_\parallel) \;,
\end{equation}
where $a,b = L, R$, the vacuum energy $E_{\rm vac}$ may be written as follows:
\begin{equation}
E_{\rm vac}\;=\; \lim_{T \to \infty} \Big[\frac{1}{2 T} {\rm Tr}\ln{\mathbb T} \Big] \;
\end{equation}
where the trace affects both continuum and discrete indices.
\section{The derivative expansion in the electromagnetic case}
As already stressed, the vacuum energy depends nontrivially on the shape of the $R$
surface, i.e. it can be thought as a nonlocal functional of $\psi$. When the R surface is gently
curved, almost parallel and close to the $L$-plane, we expect this functional to be
well approximated by a derivative expansion:
\begin{eqnarray}
E_{\rm vac} &\simeq&\, \int d^2{\mathbf x}_\parallel\left[V_{\rm eff}(\psi)+Z(\psi)(\partial_j\psi)^2+\dots\right]\, \nonumber \\
&=&\, E_{\rm vac}^{(0)} + E_{\rm vac}^{(2)} + ...
\end{eqnarray}
In order to evaluate the functions $V_{\rm eff}$ and $Z$, it is enough to consider a class of surfaces of the
form $\psi({\mathbf x}) = a + \eta({\mathbf x})$ with $\eta\ll a$. Indeed, for
these surfaces, and up to quadratic order in $\eta$, the vacuum energy
will be of the form
\begin{equation}
E_{\rm vac} \simeq\, \int d^2{\mathbf x}_\parallel\left[V_{\rm eff}(a)+V_{\rm eff}'(a)\eta+Z(a)(\partial_j\eta)^2+\dots\right]\, ,
\label{withlinear}
\end{equation}
and therefore one can obtain $V_{\rm eff}$ and $Z$ from this expression (note that the term linear in $\eta$ vanishes
if $a$ is chosen to be the mean value of the distances between surfaces).
Therefore,
it is sufficient to perform an expansion
of $\Gamma$ in powers of $\eta$, keeping terms with up to two
derivatives of $\eta$.
Denoting by $\Gamma^{(n)}$ and ${\mathbb T}^{(n)}$ the order-$n$ terms
in the respective expansions for $\Gamma$ and ${\mathbb T}$, we see that, up
to the second order, the expansion for the former is given by:
\begin{equation}
\Gamma \;=\;\Gamma^{(0)} \,+\,\Gamma^{(1)}\,+\,\Gamma^{(2)}\,+\,\ldots
\end{equation}
where the zeroth and first order terms are:
\begin{eqnarray}
\Gamma^{(0)} &=& \frac{1}{2} {\rm Tr}\ln\big[{\mathbb T}^{(0)}\big] \nonumber\\
\Gamma^{(1)} &=& \frac{1}{2} {\rm Tr}\big[ \big({\mathbb T}^{(0)}\big)^{-1}
{\mathbb T}^{(1)} \big] \;,
\end{eqnarray}
while the second order term receives two contributions $\Gamma^{(2)} =
\Gamma^{(2,1)} \,+\,\Gamma^{(2,2)}$, where:
\begin{eqnarray}
\Gamma^{(2,1)} = &\frac{1}{2}& {\rm Tr}\big[ \big({\mathbb T}^{(0)}\big)^{-1} {\mathbb T}^{(2)} \big]
\nonumber\\
\Gamma^{(2,2)} = - &\frac{1}{4}& {\rm Tr}\big[
\big({\mathbb T}^{(0)}\big)^{-1} {\mathbb T}^{(1)}
\big({\mathbb T}^{(0)}\big)^{-1} {\mathbb T}^{(1)} \big] \;.
\end{eqnarray}
Let us now write the matrices ${\mathbb T}^{(j)}$, for $j=0,1,2$:
\begin{equation}
{\mathbb T}^{(j)} \;=\;
\left(
\begin{array}{cc}
{\mathbb T}_{LL}^{(j)} & {\mathbb T}_{LR}^{(j)} \\
{\mathbb T}_{RL}^{(j)} & {\mathbb T}_{RR}^{(j)}
\end{array}
\right) \;.
\end{equation}
Those matrices are not completely defined until we adopt a specific form
for the action $S'_q$, which contains an arbitrary part that depends on
the longitudinal component of the auxiliary field $\xi_\alpha$.
In order to render the zeroth-order term as simple as possible, it is
convenient to add the following terms:
\begin{eqnarray}
S_{\xi_L} &=& \frac{1}{2}
\int_{x_\parallel,x'_\parallel}
\partial \cdot \xi_L(x_\parallel) \langle x_\parallel |
\frac{1}{2\sqrt{-\partial^2}}|x'_\parallel \rangle \partial \cdot \xi_L(x'_\parallel)
\nonumber\\
S_{\xi_R} &=& \frac{1}{2}
\int_{x_\parallel,x'_\parallel} \, \sqrt{g({\mathbf x}_\parallel)} \,
\sqrt{g({\mathbf x}'_\parallel)} \;
\nabla \cdot \xi_R(x_\parallel) \langle x_\parallel |
\frac{1}{2\sqrt{-\nabla^2}}|x'_\parallel \rangle \nabla
\cdot \xi_R(x'_\parallel) \;,
\end{eqnarray}
where we have used the `bra-ket' notation again, this time for a three dimensional space of coordinates. Besides, the derivations are understood also to act on functions defined on this space of coordinates.
The ${\mathbb T}^{(0)}$ matrix elements, which are invariant under
translations along $x_\parallel$, may be Fourier transformed:
\begin{equation}
{\mathbb T}^{(0)}(x_\parallel,x'_\parallel) = \int \frac{d^3k}{(2\pi)^3}
e^{i k \cdot (x_\parallel-x'_\parallel)} \widetilde{\mathbb T}^{(0)}(k) \;,
\end{equation}
and the explicit form of its matrix elements, for the gauge-fixing introduced above, is:
\begin{eqnarray}
\big[\widetilde{\mathbb T}_{LL}^{(0)}\big]_{\alpha\alpha'}(k) &=& \Big(
\frac{1}{\lambda_L} + \frac{|k|}{2} \Big) \delta_{\alpha \alpha'}\nonumber\\
\big[\widetilde{\mathbb T}_{RR}^{(0)}\big]_{\alpha\alpha'}(k) &=&
\Big( \frac{1}{\lambda_R} + \frac{|k|}{2} \Big) \delta_{\alpha \alpha'}\nonumber\\
\big[\widetilde{\mathbb T}_{LR}^{(0)}\big]_{\alpha\alpha'}(k) &=&
\big[\widetilde{\mathbb T}_{RL}^{(0)}\big]_{\alpha\alpha'}(k) \;=\;
\frac{|k|}{2} \big(\delta_{\alpha\alpha'} - \frac{k_\alpha
k_{\alpha'}}{k^2} \big) e^{- |k| a} \;,
\end{eqnarray}
with $|k| \equiv \sqrt{k^2}$.
Regarding the terms of order $1$ in $\eta$, it is quite straightforward to
see that:
\begin{equation}
{\mathbb T}_{LL}^{(1)} = {\mathbb T}_{RR}^{(1)} = 0 \;,
\end{equation}
so that, after evaluating the terms that mix $L$ and $R$, the result may be
put in the form:
\begin{eqnarray}
\big[{\mathbb T}^{(1)}]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel) &=& - \frac{1}{2} \big(
\partial_\beta \partial'_\beta \delta_{\alpha\alpha'} - \partial_{\alpha'}
\partial'_{\alpha} \big) \, \left( \begin{array}{cc} 0 & \eta({\mathbf x}_\parallel) \\ \eta({\mathbf x'}_\parallel)
& 0 \end{array}\right) \nonumber\\
&\times& \,
\int \frac{d^3k}{(2\pi)^3} \,
e^{i k \cdot (x_\parallel - x'_\parallel)} \, e^{- |k| a} \;.
\end{eqnarray}
Finally, we consider the second order matrix elements. We shall also
discard terms involving more than two derivatives
of $\eta$. Since the Levi-Civita connection involves at least three
derivatives of $\eta$, we replace $\nabla$ by $\partial$ in the gauge
fixing term $S_{\xi_R}$. Thus, this term will contribute to the second
order matrix element $RR$ only through the factor depending on the
determinant of the metric.
Besides, we see that ${\mathbb T}_{LL}^{(2)} \;=\; 0 $, while:
\begin{eqnarray}\label{eq:tlr2}&&
\big[{\mathbb T}_{LR}^{(2)}\big]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel)
\;=\; \big[{\mathbb T}_{RL}^{(2)}\big]_{{\alpha'} \alpha'}(x'_\parallel,x_\parallel)
\nonumber\\
&=&\frac{1}{4} \big(
\partial_\beta \partial'_\beta \delta_{\alpha\alpha'} - \partial_{\alpha'}
\partial'_\alpha \big)
[\eta({\mathbf x}_\parallel)]^2
\int \frac{d^3k}{(2\pi)^3} e^{i k \cdot (x_\parallel - x'_\parallel)} \, |k|
e^{- |k| a} \;.
\end{eqnarray}
Regarding the $RR$ matrix element, we have four different terms:
\begin{equation}
{\mathbb T}_{RR}^{(2)} \,=\, {\mathbb T}_{RR}^{(2,1)} +{\mathbb
T}_{RR}^{(2,2)}+
{\mathbb T}_{RR}^{(2,3)} + {\mathbb T}_{RR}^{(2,4)}
\end{equation}
where
\begin{eqnarray}\label{eq:t2}
\big[{\mathbb T}_{RR}^{(2,1)}\big]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel)
&=& \frac{1}{\lambda_R} \Big[\frac{1}{2}
\delta_{\alpha {\alpha'}} (\partial_j \eta({\mathbf x}_\parallel))^2 - \delta_{\alpha i}
\delta_{\alpha' j} \partial_i \eta({\mathbf x}_\parallel) \partial_j
\eta({\mathbf x}'_\parallel) \Big]
\delta^{(3)}(x_\parallel- x'_\parallel) \nonumber\\
\big[{\mathbb T}_{RR}^{(2,2)}\big]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel)
&=&\frac{1}{4} \big(\partial_\beta \partial'_\beta \delta_{\alpha\alpha'} -
\partial_{\alpha'} \partial'_\alpha \big)
[ \eta({\mathbf x}_\parallel) - \eta({\mathbf x}'_\parallel) ]^2
\int \frac{d^3k}{(2\pi)^3} e^{i k \cdot (x_\parallel - x'_\parallel)} \, |k| \nonumber\\
\big[{\mathbb T}_{RR}^{(2,3)}\big]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel)
&=& \epsilon^{i\alpha\beta} \epsilon^{j\alpha'\beta'}
\partial_i\eta({\mathbf x}_\parallel)
\partial'_j\eta({\mathbf x}'_\parallel)
\int \frac{d^3k}{(2\pi)^3} e^{i k \cdot (x_\parallel - x'_\parallel)} \,
\frac{k_\beta k_{\beta'}}{2 |k|} \nonumber\\
\big[{\mathbb T}_{RR}^{(2,4)}\big]_{\alpha {\alpha'}}(x_\parallel,x'_\parallel)
&=&\frac{1}{4} \Big(\big[\partial_j \eta({\mathbf x}_\parallel)\big]^2
+
\big[\partial_j \eta({\mathbf x'}_\parallel)\big]^2 \Big)
\int \frac{d^3k}{(2\pi)^3} e^{i k \cdot (x_\parallel - x'_\parallel)}
\frac{k_\alpha k_{\alpha'}}{|k|}\;.
\end{eqnarray}
\subsection{Evaluation of $\Gamma^{(0)}$}
We recall that $\Gamma^{(0)}=\frac{1}{2} {\rm Tr}\ln\big[{\mathbb T}^{(0)}\big]$
where the trace runs over all the indices (Lorentz and indices that label
the two mirrors).
To perform that trace it is convenient to note that
\begin{eqnarray}
\widetilde{\mathbb T}^{(0)}(k) &=& \left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & \frac{|k|}{2} e^{- |k| a} \\
\frac{|k|}{2} e^{- |k| a} & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}\right) {\mathcal P}_\perp(k) \nonumber\\
&+& \left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & 0\\
0 & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}\right) {\mathcal P}_\parallel(k) \;,
\end{eqnarray}
where we have introduced the transverse (${\mathcal P}_\perp$) and
longitudinal (${\mathcal P}_\parallel$) projectors, corresponding to the
$3$-vector $k$, namely, \mbox{$\big[{\mathcal P}_\perp\big]_{\alpha
\alpha'}(k) = \delta_{\alpha \alpha'} - \frac{k_\alpha k_{\alpha'}}{k^2}$},
and \mbox{$\big[{\mathcal P}_\parallel\big]_{\alpha \alpha'}(k) =
\frac{k_\alpha k_{\alpha'}}{k^2}$}.
Since these projectors are orthogonal,
\begin{eqnarray}
\ln\big[{\mathbb T}^{(0)}\big] &=& {\mathcal P}_\perp(k)
\ln
\left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & \frac{|k|}{2} e^{- |k| a} \\
\frac{|k|}{2} e^{- |k| a} & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}
\right)
\nonumber\\
&+& {\mathcal P}_\parallel(k) \ln
\left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & 0\\
0 & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}
\right) \;,
\end{eqnarray}
and
\begin{eqnarray}
{\rm Tr} \ln\big[{\mathbb T}^{(0)}\big] &=& 2 \times T L^2 \int \frac{d^3
k}{(2\pi)^3}
\ln \det
\left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & \frac{|k|}{2} e^{- |k| a} \\
\frac{|k|}{2} e^{- |k| a} & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}
\right)
\nonumber\\
&+& 1 \times T L^2\int \frac{d^3k}{(2\pi)^3} \ln \det
\left(
\begin{array}{cc}
\frac{1}{\lambda_L} + \frac{|k|}{2} & 0\\
0 & \frac{1}{\lambda_R} + \frac{|k|}{2}
\end{array}
\right) \;.
\end{eqnarray}
Discarding $a$-independent contributions, we see that $\Gamma^{(0)}$ may
then be written as follows:
\begin{eqnarray}\label{eq:gamma0result}
\Gamma^{(0)} \;&=&\; \frac{1}{2} \, L^2 \, T \, 2 \; \int \frac{d^3k}{(2\pi)^3} \,
\ln \Big[ 1 - \frac{ (\frac{|k|}{2})^2}{(\frac{1}{\lambda_L} +
\frac{|k|}{2}) (\frac{1}{\lambda_R} + \frac{|k|}{2})} e^{-2 |k| a} \Big]\nonumber\\
&\equiv&L^2\, T\, V_{\rm eff}(a) \;.
\end{eqnarray}
This result coincides with the vacuum energy corresponding to two imperfect, flat, and parallel mirrors
separated by a distance $a$, that we had computed previously
(\cite{plb2008}) for the particular case $\lambda_L=\lambda_R$. Note that, given the boundary conditions
produced by our relativistic model (see Eqs.(\ref{eq:defssigma}) and (\ref{bc})), up to leading order the Casimir energy
for the electromagnetic field is twice the Casimir energy for the case of a scalar field. This was already shown in Ref. \cite{plb2008},
where it was also pointed out that for nonrelativistic matter the contributions of TE and TM modes are not equal, in agreement with the
fact that the TE and TM reflection coefficients are different in this case \cite{barton2004}.
\subsection{Evaluation of $\Gamma^{(1)}$}
In the previous subsection we obtained the function $V_{\rm eff}$.
Although the evaluation of the term linear in $\eta$ is not necessary for
our next purpose of obtaining $Z$, it is useful as an internal consistency
check of the calculations.
Recalling the expression for $\Gamma^{(1)}$, we see that we need the inverse of
${\mathbb T}^{(0)}$. In Fourier space, it is given by:
\begin{eqnarray}\label{eq:t0inv}
\big[\widetilde{\mathbb T}^{(0)}(k)\big]^{-1} &=& \frac{1}{D(k)}\,
\left(
\begin{array}{cc}
\frac{1}{\lambda_R} + \frac{|k|}{2} & - \frac{|k|}{2} e^{- |k| a} \\
-\frac{|k|}{2} e^{- |k| a} & \frac{1}{\lambda_L} + \frac{|k|}{2}
\end{array}\right) {\mathcal P}_\perp(k) \nonumber\\
&+& \left(
\begin{array}{cc}
\frac{1}{\frac{1}{\lambda_L} + \frac{|k|}{2}} & 0\\
0 & \frac{1}{\frac{1}{\lambda_R} + \frac{|k|}{2}}
\end{array}\right) {\mathcal P}_\parallel(k) \;,
\end{eqnarray}
where:
\begin{equation}\label{defD}
D(k) \,=\, \big(\frac{1}{\lambda_L} + \frac{|k|}{2}\big)
\big(\frac{1}{\lambda_R} + \frac{|k|}{2}\big)
-
\big( \frac{|k|}{2} e^{- |k| a} \big)^2 \;.
\end{equation}
Then, using the notation: $\Delta_{ab} \equiv \big[{{\mathbb
T}^{(0)}}^{-1}\big]_{ab}$,
\begin{eqnarray}
\Gamma^{(1)} &=& \frac{1}{2} \, \Big[ \int_{x_\parallel, x'_\parallel}\;
\big(\Delta_{LR}\big)_{\alpha {\alpha'}} (x,x')
\;
\big({\mathbb T}^{(1)}_{RL}\big)_{{\alpha'}\alpha} (x',x) \nonumber\\
&+& \int_{x_\parallel, x'_\parallel}\; \big(\Delta_{RL}\big)_{\alpha {\alpha'}} (x,x')
\;
\big({\mathbb T}^{(1)}_{LR}\big)_{{\alpha'}\alpha} (x',x) \Big] \;.
\end{eqnarray}
This may be evaluated explicitly by introducing Fourier transforms, the
result being:
\begin{equation}
\Gamma^{(1)} \;=\; \frac{T\,L^2}{2} \, \int \frac{d^3k}{(2\pi)^3} \,
\frac{|k|^3}{D(k)} \, e^{- 2 |k| a} \; \int d^2{\mathbf x}_\parallel \, \eta({\mathbf x}_\parallel)
\;.
\label{Gamma1fin}
\end{equation}
From Eqs.~(\ref{eq:gamma0result}) and (\ref{Gamma1fin}) one can easily show that
\begin{equation}
\Gamma^{(1)} \;=\; T\, L^2 V'_{\rm eff}(a) \int d^2{\mathbf x}_\parallel \, \eta({\mathbf x}_\parallel)
\;,
\end{equation}
as expected from Eq.(\ref{withlinear}).
\subsection{Evaluation of $\Gamma^{(2)}$}\label{sec:Gamma2}
We present the evaluation of the two contributions to $\Gamma^{(2)}$
separately.
\subsubsection{ Contribution of $\Gamma^{(2,1)}$}\label{c123}
Let us consider first $\Gamma^{(2,1)}$.
It is convenient to recall the form of the inverse of ${\mathbb
T}^{(0)}$, presented in (\ref{eq:t0inv}), and of ${\mathbb T}^{(2)}$, in
equations (\ref{eq:tlr2}) and (\ref{eq:t2}).
We first note, by explicit evaluation, that the terms
$[{\mathbb T}_{LR}]^{(2)}$ and $[{\mathbb T}_{RL}]^{(2)}$ in
(\ref{eq:tlr2}) do not contribute to the force.
We also see that ${\mathbb T}_{RR}^{(2,4)}$ in (\ref{eq:t2}) can be ignored,
since its tensor structure allows it to mix only with the piece of
$\big[{\mathbb T}^{(0)}\big]^{-1}$ which is proportional to ${\mathcal P}_\parallel$.
Since neither object depends on $a$, the corresponding contribution is irrelevant to
the calculation of Casimir forces.
On the other hand, only the `transverse', i.e. proportional to ${\mathcal
P}_\perp$, term in $\big[{\mathbb T}^{(0)}\big]^{-1}$ must be retained for
the rest of the terms. Indeed, it is the only part that can produce an
$a$-dependent contribution for the terms ${\mathbb T}_{RR}^{(2,1)}$ and
${\mathbb T}_{RR}^{(2,3)}$ in (\ref{eq:t2}). On the other hand, the tensor
structure of ${\mathbb T}_{RR}^{(2,2)}$ allows it to mix only with the
transverse part of $\big[{\mathbb T}^{(0)}\big]^{-1}$.
The results due the relevant terms, after extracting the term of second order in
derivatives, shall have the following form:
\begin{equation}
\frac{1}{2}{\rm Tr}\big[\Delta_{RR} {\mathbb T}_{RR}^{(2,b)}\big]
\;=\; \frac{c_b(a)}{2} \, T \int d^2{\mathbf x}_\parallel \big[\partial_j \eta({\mathbf
x}_\parallel)\big]^2 \;,
\end{equation}
with $b = 1, 2, 3$.
Interestingly, ${T}_{RR}^{(2,1)}$ yields a local term (no need to perform a derivative
expansion):
\begin{equation}\label{c1}
c_1(a) \,=\, \frac{1}{8}\int \frac{d^3k}{(2\pi)^3}
\frac{\vert{\mathbf k}_\parallel\vert^2}{ \lambda_R (\frac{1}{\lambda_L} + \frac{|k|}{2})
(\frac{1}{\lambda_R} + \frac{|k|}{2})^2
\big[e^{2 |k| a} - \frac{(\frac{|k|}{2})^2}{(\frac{1}{\lambda_L} + \frac{|k|}{2})
(\frac{1}{\lambda_R} + \frac{|k|}{2})}\big]}
\end{equation}
(where we subtracted $a$-independent contributions). We could replace
$\vert{\mathbf k}_\parallel\vert^2$ by $2/3\vert k\vert^2$ inside the above integral.
However, in its present form, Eq.(\ref{c1}) remains valid even in the case in which
$\lambda_R$ and $\lambda_L$ are functions of $k_0$.
For ${\mathbb T}_{RR}^{(2,2)}$,
\begin{eqnarray}
c_2(a) &=& - \lim_{k \to 0}
\frac{\partial}{\partial \vert {\mathbf k}_\parallel\vert ^2} \int \frac{d^3p}{(2\pi)^3} |p+k|
|p|^2 \frac{\frac{1}{\lambda_L} +
\frac{|p|}{2}}{D(p)} \nonumber\\
&=& -\frac{1}{8}
\int \frac{d^3k}{(2\pi)^3}
\frac{ \left(1-\frac{1}{2}\frac{\vert{\mathbf k}_\parallel\vert^2}{k^2}\right)\vert k\vert^3}
{ (\frac{1}{\lambda_L} + \frac{|k|}{2})
(\frac{1}{\lambda_R} + \frac{|k|}{2})^2
\big[e^{2 |k| a} - \frac{(\frac{|k|}{2})^2}{(\frac{1}{\lambda_L} + \frac{|k|}{2})
(\frac{1}{\lambda_R} + \frac{|k|}{2})}\big]} \;,
\end{eqnarray}
where again we subtracted a term independent of $a$.
Finally, the contribution due to ${\mathbb T}_{RR}^{(2,3)}$ is also local,
and the result is $c_3(a)=-c_2(a)$.
\subsubsection{ Contribution of $\Gamma^{(2,2)}$}\label{c45}
The next point is the evaluation of $\Gamma^{(2,2)}$.
It can be shown, by using symmetries of the matrix elements appearing in
the expression, that this contribution reduces to:
\begin{equation}
\Gamma^{(2,2)} \;=\; G + U
\end{equation}
where:
\begin{equation}
G \;=\; - \frac{1}{2} {\rm Tr} \big( \Delta_{LL} T^{(1)}_{LR}
\Delta_{RR} T^{(1)}_{RL} \big) \;,\;\;
U \;=\; - \frac{1}{2} {\rm Tr} \big( \Delta_{LR} T^{(1)}_{RL}
\Delta_{LR} T^{(1)}_{RL} \big) \;.
\end{equation}
After some algebra, we see that the result for $G$ may be
written as follows:
\begin{equation}
G = \frac{1}{2} \int \frac{d^3k}{(2\pi)^3}
|\widetilde{\eta}(k)|^2 {\mathcal G}(k)
\end{equation}
\begin{equation}
{\mathcal G}(k) = - \frac{1}{4} \int \frac{d^3p}{(2\pi)^3}
e^{- 2 a |p + k|} \, \Big[\big(p \cdot (p + k) \big)^2 + p^2 (p + k)^2
\Big]
d_{LL}(p) d_{RR} (p + k)
\label{gdek}
\end{equation}
where:
\begin{equation}
d_{LL}(p) \equiv \frac{\frac{1}{\lambda_R} + \frac{|p|}{2}}{D(p)}
\;,\;\; d_{RR}(p) \equiv \frac{\frac{1}{\lambda_L} +
\frac{|p|}{2}}{D(p)} \;.
\end{equation}
Note that, as we are considering static surfaces, $\widetilde{\eta}(k)$ is proportional to
$\delta(k_0)$ and therefore ${\mathcal G}(k)={\mathcal G}(k_0=0, {\mathbf k}_\parallel)$
depends only on ${\mathbf k}_\parallel$.
Up to second order in derivatives, this term yields a contribution
with the same form we had for $\Gamma^{(2,1)}$, this time with a coefficient:
\begin{equation}
c_4(a) = \lim_{k \to 0}
\frac{\partial}{\partial \vert {\mathbf k}_\parallel\vert^2}{\mathcal G}(k)\, .
\label{c4imp}
\end{equation}
On the other hand,
\begin{equation}
U = \frac{1}{2} \int \frac{d^3k}{(2\pi)^3}
|\widetilde{\eta}(k)|^2 {\mathcal U}(k)
\end{equation}
\begin{equation}
{\mathcal U}(k) = - \frac{1}{4} \int \frac{d^3p}{(2\pi)^3}
e^{- (|p| + |p + k|) a} \, \Big[\big(p \cdot (p + k) \big)^2 + p^2 (p + k)^2
\Big]
d_{LR}(p) d_{LR} (p + k)
\label{udek}\end{equation}
where:
\begin{equation}
d_{LR}(p) \equiv - \frac{|p|}{2} \frac{e^{- |p| a}}{D(p)} \;.
\end{equation}
Up to second order in derivatives, it produces a coefficient:
\begin{equation}\label{c5imp}
c_5(a) = \lim_{k \to 0}
\frac{\partial}{\partial\vert {\mathbf k}_\parallel\vert^2}{\mathcal U}(k)\, .
\end{equation}
The explicit expressions for $c_4$ and $c_5$ can be obtained by computing
the derivatives in Eqs.~(\ref{c4imp}) and (\ref{c5imp}).
\subsection{The improved PFA in the electromagnetic case}
Using the results of the previous section, we can finally obtain our main result: the improved PFA for imperfect thin mirrors.
The zero-order contribution to the vacuum
energy is:
\begin{equation}\label{eq:e0result}
E_{\rm vac}^{(0)} \;=\; \int d^2{\mathbf x}_\parallel \, \int \frac{d^3k}{(2\pi)^3} \,
\ln \Big[ 1 - \frac{ (\frac{|k|}{2})^2}{(\frac{1}{\lambda_L} +
\frac{|k|}{2}) (\frac{1}{\lambda_R} + \frac{|k|}{2})} e^{-2 |k|
\psi({\mathbf x}_\parallel)} \Big]
\;.
\end{equation}
Putting together the results of section \ref{sec:Gamma2}, we see
that the NTLO correction to the PFA reads
\begin{equation}
E_{\rm vac}^{(2)} \,=\frac{1}{2} \int d^2{\mathbf x}_\parallel
\sum_{b=1}^{5}c_b(\psi)(\partial_j\psi)^2\, .
\end{equation}
These results can be immediately generalized to the case in which
$\lambda_{L,R}$ become frequency dependent, for which the answer is
obtained by making the replacement $\lambda_{L,R}\rightarrow
\lambda_{L,R}(k_0)$ in the final expressions for the derivative expansion of the energy. This simple
generalization is valid because we are considering static surfaces: the
time-dependence of the modes of the vacuum field is trivial and therefore
one can treat independently each frequency. By the same reason, the
generalization to the case in which the electromagnetic response of the
mirrors depends nontrivially on ${\mathbf k}_\parallel$ is not so trivial,
and would involve local momentum expansions at the different points of the
curved surface. This point deserves further investigation.
\section{Analysis of the results}
In this section we perform a general analysis of the results obtained.
Let us first consider the perfect-mirror limit. One can derive the idealized limit of perfect conductivity
by taking $\lambda_1, \lambda_2 \to \infty$. The leading term reads, in this case
\begin{equation}\label{eq:e0perf}
\Big[E_{\rm vac}^{(0)}\Big]_{\rm perf} \;=\; \int d^2{\mathbf x}_\parallel \, \int \frac{d^3k}{(2\pi)^3} \,
\ln \big[ 1 - e^{-2 |k| \psi({\mathbf x}_\parallel)} \big]
= -\frac{\pi^2}{720}\int \frac{d^2{\mathbf x}_\parallel}{\psi^3} \;,
\end{equation}
which is the well known PFA for perfect conductors.
The evaluation of the NTLO is more tedious. From the explicit expressions of
the functions $c_1, c_2$ and $c_3$ presented in Section \ref{c123} we obtain:
\begin{eqnarray}
{c_1}_{\rm perf} &=& 0\nonumber\\
{c_2}_{\rm perf}&=&-{c_3}_{\rm perf}=-\frac{\zeta(3)}{12\pi^2\psi^3}\, .
\end{eqnarray}
It is worth to remark that, in this limit, ${c_1}_{\rm perf}+{c_2}_{\rm perf}+{c_3}_{\rm perf}=0$, that is, there is no contribution from $\Gamma^{(2,1)}$ for perfect conductors.
On the other hand, the computation of the functions $c_4$ and $c_5$ defined
in Section \ref{c45} is less straightforward. We write the integrals in
Eqs.~(\ref{gdek}) and (\ref{udek})
in spherical coordinates in momentum space, perform the derivatives with respect
to $\vert {\mathbf k}_\parallel\vert^2$, and finally compute the integrals.
In this way we obtain
\begin{equation}
{c_4}_{\rm perf}={c_5}_{\rm perf}=\frac{15-\pi^2}{1080\psi^3}\, ,
\end{equation}
and therefore $Z(\psi)=\frac{15-\pi^2}{1080\psi^3}$, in agreement with the result obtained by Bimonte et al \cite{pfa_mit1}.
It is also of interest to analyze the opposite limit, in which the mirrors are almost transparent ( $\lambda_L,\lambda_R\ll \psi$).
The zeroth order vacuum energy can be easily obtained by expanding the
argument of the logarithm in Eq.~(\ref{eq:e0result}). The result is
\begin{equation}\label{eq:order0trans}
E_{\rm vac}^{(0)} \;=\; -\frac{1}{4}\int d^2{\mathbf x}_\parallel \, \int \frac{d^3k}{(2\pi)^3} \,\lambda_L\lambda_R
|k|^2 e^{-2 |k|\psi({\mathbf x}_\parallel)}
\;.
\end{equation}
The evaluation of the coefficients $c_b$ is also rather simple in this
limit. For instance, from Eq.~(\ref{c1}) we obtain
\begin{equation}\label{c1trans}
c_1(\psi) \,=\, \frac{1}{8}\int \frac{d^3k}{(2\pi)^3}
\lambda_R \lambda_L |{\mathbf k}_\parallel|^2
e^{-2 |k| \psi({\mathbf x}_\parallel)}\, .
\end{equation}
As already mentioned, $c_2+c_3=0$. For almost transparent mirrors, the function $D(p)$ defined in
Eq.~(\ref{defD}) becomes $D(p)\approx (\lambda_L\lambda_R)^{-1}$. Therefore, up to quadratic order
in $\lambda_{L,R}$ we have
\begin{equation}
c_4(\psi) \,=\, -\frac{1}{4}\int \frac{d^3k}{(2\pi)^3}
\lambda_R \lambda_L \left(\vert k\vert^2+\frac{1}{2}\vert{\mathbf k}_\parallel \vert^2\right)
e^{-2 |k| \psi({\mathbf x}_\parallel)}\, ,
\end{equation}
and $c_5=0$. Combining these results we obtain
\begin{equation}\label{eq:order2trans}
E_{\rm vac}^{(2)} \;=\; -\frac{1}{8}\int d^2{\mathbf x}_\parallel \,
\big[\partial_j \eta({\mathbf
x}_\parallel)\big]^2
\int \frac{d^3k}{(2\pi)^3} \,\lambda_L\lambda_R
|k|^2 e^{-2 |k|\psi({\mathbf x}_\parallel)}
\;,
\end{equation}
and the improved PFA is therefore
\begin{equation}\label{pfatrans}
E_{\rm vac} \;=\; -\frac{1}{4}\int d^2{\mathbf x}_\parallel \, \sqrt{g}
\int \frac{d^3k}{(2\pi)^3} \,\lambda_L\lambda_R
|k|^2 e^{-2 |k|\psi({\mathbf x}_\parallel)}
\;.
\end{equation}
Note that, in the semitransparent limit, the NTLO correction is equivalent to the insertion of the factor
$\sqrt{g}$, that is, the improved result corresponds to the use of the area of the curved surface in the usual PFA. It would be interesting to check if this result is exact, as is the case for a scalar vacuum field
\cite{scalarexact}. It is also worth to note that
the scaling of the energy with $\psi$ depends of course on the choice of $\lambda_{L,R}$: if both are constants, the integrand
in Eq.(\ref{pfatrans}) is proportional to $\psi^{-5}$.
We end up this section with a discussion on the case of graphene-like
materials, in which the electromagnetic properties of the mirrors are
described by dimensionless quantities. As is well known, the charged
degrees of freedom in graphene can be effectively described by massless
fermions confined to the surface, with a propagation velocity $v_F\approx
1/300$. When the surface is flat, the interaction between charges and the
electromagnetic field is described by a vacuum polarization tensor which,
in our notation, would correspond to a nonlocal interaction with
$\lambda(k)$ proportional to $(k_0^2+{\mathbf k}_\parallel^2)^{-1/2}$,
after a rescaling of the temporal components of all vectors and tensors in
order to take into account that the fermions have a propagation velocity
different from $c=1$. For curved surfaces, we expect an effective
interaction of the form
\begin{equation}\label{eq:graphene}
{\mathcal S}_\Sigma (A; y) \;\simeq \; \int
d^3\sigma \, \sqrt{g(\sigma)}\, F_{\alpha\beta}\, [\nabla^2]^{-1/2}F^{\alpha\beta} \;,
\end{equation}
where $\nabla^2$ is the Laplacian on the surface (the above mentioned rescaling should also be applied to
Eq.(\ref{eq:graphene})). The calculation of the Casimir energy for this particular case is of high interest, but is beyond the scope of the present paper. We will discuss here a toy model for graphene-like materials,
compatible with the assumptions we made so far. Therefore, we will consider $\lambda_{L,R}^{-1}=\xi_{L,R}\vert k_0\vert/2$,
where $\xi_{L,R}$ are dimensionless constants.
By dimensional analysis, the improved PFA will have the same functional form than the case of perfect conductivity, that is
\begin{equation}
E_{\rm vac} \simeq \int d^2{\mathbf x_ \parallel} \;
\frac{1}{\psi^3}\left[\alpha_1(\xi_L,\xi_R)+\alpha_2(\xi_L,\xi_R)(\partial_j\psi)^2\right]\, .
\label{graph like}\end{equation}
\begin{figure}
\centering
\includegraphics[width=8cm , angle=0]{alfa1.pdf}
\caption{Ratio between the coefficient of the leading term $\alpha_1(\xi)$ and the corresponding value for the perfect mirrors case, $\alpha_1(0)$,
as a function of the dimensionless parameter $\xi$. It is a monotonic decreasing function.} \label{E0}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm , angle=0]{alfa2.pdf}
\caption{Ratio between the coefficient of the NTLO correction $\alpha_2(\xi)$ and the corresponding value for the perfect case,
$\alpha_2(0)$,
as a function of the dimensionless $\xi$. The inset shows that this ratio is non-monotonous and changes sign for a particular value of $\xi$.} \label{cbs}
\end{figure}
The expressions for the dimensionless functions $\alpha_i$ can be easily
derived from our previous results. We have computed numerically these
functions for the particular case $\xi_L=\xi_R=\xi$. The results are shown
in Figs. 1 and 2. As expected, these functions approach their
perfect conductivity limits for $\xi\to 0$ and vanish in the almost transparent
limit, for $\xi\to \infty$. The leading order in the PFA has a rather simple
behavior: the absolute value of the coefficient $\alpha_1$ is a decreasing
function of $\xi$ (see Fig.1). The NTLO correction shows a qualitatively
different behavior, since it is non-monotonous and even changes sign at a
particular value of $\xi$ (Fig.2). Moreover, its absolute value falls faster
with $\xi$, that is, for this kind of materials the NTLO correction
quickly loses relevance away from the infinite conductivity limit.
\section{Conclusions}
In this paper we have computed the Casimir energy for thin and imperfect
mirrors using a derivative expansion in the shape of the surfaces. The
leading term in the expansion reproduces the usual PFA, while the term
containing two derivatives represent the NTLO correction. These results
generalize previous works that involved perfect
mirrors~\cite{pfa_nos,pfa_mit1}, and may be regarded as complementary to
those for the interaction between thick mirrors~\cite{pfa_mit2}.
The interaction between the mirrors and the vacuum field has been described
by a local effective action, which is a novel electromagnetic
generalization of the $\delta$-potentials usually considered for scalar fields.
We also discussed some nonlocal generalizations, which
could be useful to describe the interaction between curved graphene sheets.
To compute the vacuum energy we used a functional approach, and we used an
explicitly (electromagnetic) gauge invariant approach, whereby the
interaction term has been written in terms of vector auxiliary fields coupled to
the Maxwell tensor.
We have presented general expressions for the improved PFA for this model, and
checked the particular limits corresponding to perfect conductors and almost-transparent mirrors.
For the particular case of mirrors described by a single dimensionless quantity $\xi$, we computed
the leading PFA and its NTLO correction as a function of $\xi$. We have found that the NTLO
correction has a non-monotonous dependence on $\xi$, and that its absolute value drops quickly
for imperfect mirrors.
For the sake of simplicity, we considered a gently curved surface in front
of a plane. Moreover, we only considered the case in which the curved
surface can be described by a single function $x_3=\psi(x_1,x_2)$. The
results can be extended to the case of two curved surfaces described by
functions along the lines of Ref.\cite{pfa_mit1}. The generalization to
cases in which the surfaces cannot be described in this way,
for example the case of an object inside another, is far from immediate.
\section*{Acknowledgements}
This work was supported by ANPCyT, CONICET, UBA and UNCuyo.
|
{
"timestamp": "2012-06-08T02:01:12",
"yymm": "1203",
"arxiv_id": "1203.1855",
"language": "en",
"url": "https://arxiv.org/abs/1203.1855"
}
|
\section{Introduction}
The Low Mass X-ray Binary (LMXB) EXO 0748-67 was discovered in 1985 (Parmar et al. 1986) as a transient X-ray
source and its optical counterpart, UY Vol was discovered in the same year (Wade et al. 1985). EXO 0748-676 is a
well known LMXB that has been the target of detailed timing and spectroscopic observations ever since its discovery.
Unlike most X-ray transients, it did not decay back to quiescent state, but remained active and was considered part
of the persistent LMXB population. In late 2008 it started a transition to quiescence after it had been actively
accreting for more than 24 years (Hynes \& Jones 2009). During 1985 to 2008 it was a moderately bright LMXB with strong dipping
activities. It is one of the rare LMXBs that show a complete X-ray eclipse, every 3.82 h. Dips and eclipses are
due to the central X-ray source being obscured by some structure above the disk and occulted by the companion star
respectively at every orbital period. The presence of eclipses and complex dipping activity indicates that the
object is viewed at an angle of inclination in the range of 75$^\circ$-82$^\circ$ (very close to the accretion disc
plane) and the mass of the companion is constrained to be about 0.5 solar mass (Parmar et al. 1986). The compact source
is known to be a weakly magnetized neutron star because of the thermonuclear burst activities. Two burst oscillation
features at 45 Hz (Villarreal \& Strohmayer 2004) and 550 Hz (Galloway et al. 2010) were detected in this LMXB
but Jain \& Paul (2011) reported non detection of any oscillations around 45 Hz down to a very small pulse fraction.
The distance to EXO 0748-676 was derived as 7.7 kpc for a helium dominated burst photosphere and 5.9 kpc for a
hydrogen dominated burst photosphere and a strong X-ray burst from the same source is an evidence for photospheric
expansion (Wolff et al. 2005). Another unusual feature of EXO 0748-676 is presence of orbital period glitches (Wolff et al.
2009); only one other such system is known, XTE J1710-281 (Jain et al. 2011).
It is also found to show a rich variety of burst profiles and also some double and triple X-ray bursts (Boirin et al.
2007).
A consequence of the thermonuclear X-ray bursts observed in the low magnetic field accreting neutron stars
(apart from being directly received by the observer) is
reprocessing of the X-ray photons as they interact with different components of the binary, mainly the companion
star and the accretion disk. The reprocessed X-rays, often seen at optical wavelengths are delayed in time with
respect to the X-ray bursts due to light travel time and reprocessing effects. Simultaneous X-ray and optical
bursts have been detected in several LMXBs:
Ser X-1 (Hackwell et al. 1979),
4U 1536-53 (Pederson et al. 1982; Lawrence et al. 1983; Matsuoka et al. 1984 ),
GS 1826-24 (Kong et al. 2000),
MS 1603+2600 (Hakala et al. 2005) and
EXO 0748-676 (Hynes et al. 2006).
Simultaneous X-ray and UV/optical
observations offer a tremendous opportunity to probe the geometry and physics of the
X-ray binaries.
EXO 0748-676 is one of the very few eclipsing LMXBs that allows us to investigate
the reprocessing phenomena as a function of orbital phase. If the X-rays are reprocessed from the surface of the
companion star that faces the X-ray star, one expects very strong orbital phase dependence of the reprocessing
parameters that can be tested well in the case of EXO 0748-676.
The Optical Monitor on the XMM-Newton observatory allows examination of the optical and ultra-violet lightcurves
simultaneously alongside an X-ray burst. This adds an important dimension to the study of LMXBs where
one can relate the optical emission to the X-ray bursts and hence study the reprocessing mechanisms and examine
the orbital characteristics, structure of the binary and the accretion disk. Several LMXBs have been observed with
XMM with very long exposures and data from the optical monitor of XMM should be useful in investigating the optical
reprocessing of the X-ray bursts.
Compared to the literature on the thermonuclear X-ray bursts, work based on simultaneous observation of X-ray and optical bursts is quite limited owing to the difficulties in carrying out optical observations (which are mostly done from ground based observatories) simultaneously with X-ray observations (done from space).
Here we report on our examination of the archival data from simultaneous X-ray and optical monitoring of
EXO 0748-676 with the XMM-Newton Observatory.
\section{Observations, analysis and results}
The source was observed with the XMM-Newton observatory (Jansen et al. 2001) on several occasions with different instrument
configurations. XMM has 3 X-ray mirrors with differemt focal plane CCD (2 MOS CCDS and 1 PN CCD) imaging
spectrometers that cover the energy band of 0.1 - 12 keV with a collecting area of 1900 cm$^2$ at 150 eV that
decreases to 350 cm$^2$ at 10 keV for each of the the telescopes (Struder et al. 2001, Turner et al. 2001).
76 X-ray bursts were recorded during the 7 XMM-newton observations of EXO 0748-676 during September-November 2003
(Boirin et al. 2007). The
XMM Optical Monitor (XMM-OM) instrument (Mason et al. 2001) provides coverage between 180 nm and 600 nm of the
central 17 arc minute square region of the X-ray field of view, permitting multi wavelength observations of XMM
targets simultaneously in the X-ray and ultraviolet/optical bands.
The simultaneous optical observations were performed with the XMM-OM in white light.
Though
until 2008, EXO 0748-676 was considered to be a persistent LMXB, it had strong long term intenisty variations and
the source was bright during these XMM observations.
Data reduction and analysis were carried out with XMM Science Analysis Software (SAS). We processed the data using
the tasks omfchain and epchain. We extracted optical light curves
from a circular region of radius 6 arcsec (FWHM) around the optical counterpart while the X-ray light curves
were extracted from the EPIC-PN CCD from a region of radius 11 arcsec around the source position. Both the
light curves had a bin size of 1 s. All the X-ray bursts reported earlier (Boirin et al. 2007) were detected in the
energy band of 0.2-12 keV and also in two subbands of 0.2-2.5 keV and 5-12 keV. Apart from the bursts, the low
energy light curve also shows significant orbital modulation with significant variation from orbit to orbit.
The 5-12 keV light curve did not show any significant variation apart form the bursts. 63 of the X-ray bursts had
simultaneous optical data and all but one (burst no 52) of these bursts were also detected in the optical light curves. For 13 of the X-ray
bursts, there was no simultaneous optical data. The profiles of the optical bursts are shown in Figure 1 along with
the X-ray bursts. The bursts have been numbered in the same way as in Boirin et al. (2007).
A few selected bursts are shown separately in Figure 2, to clearly demonstrate some of the burst characteristics.
Some important features noticed in the light curves are as follows.
\begin{itemize}
\item Both the X-ray and the optical bursts have linear, fast rise and slow, somewhat exponential decay.
\item The start of the optical bursts have a delay compared to the X-ray bursts.
\item The optical bursts are generally of longer duration.
\item Weak correlation between the peak amplitude of the X-ray and optical bursts.
\item Some of the burst profiles show clear deviations from an exponential shape.
\end{itemize}
Echo, or reverberation mapping technique has been applied extensively on the multi-wavelength profiles of thermonuclear bursts in EXO 0748-676 (Hynes et al. 2006).
This assumes an instantaneous reprocessing of the incident X-ray flux into UV and optical light, the transfer function calculated for each brust representing the weighted geometric delays from different parts of the reprocessing region. However, for the thermonuclear bursts in EXO 0748-676, and probably for other sources as well, there is strong energy dependence of the burst profiles. The decay time scales of bursts increase from hard X-rays to soft X-rays, UV, and optical. The transfer function often extends upto 10 seconds or more, considerably larger than the light travel time accross the X-ray binary. These are indications of a complex reprocessing and temperature evolution of the reprocessed component during the bursts. Considering the temperature evolution that is seen during the thermonuclear X-ray bursts, a temperature evolution of the reprocessed emission during the bursts is certainly possible. Moreover, rather than the X-ray count rate, the reprocessed emission should be related to the X-ray flux rate.
In view of these complexities and the fact that the optical light curve in our dataset corresponds only to a single broad band, and also that the X-ray count rate is not large enough to determine the flux/temperature evolution accurately, we chose to limit the scope of this work.
In the present work therefore, instead of calculating the transfer functions, we determine the burst start times, rise times, and total burst counts in the X-ray and optical bands.
We fitted all the burst profiles with a model consisting of a constant component, a burst with a linear rise and an exponential decay. As mentioned earlier, some of the burst profiles are not fitted well with a single exponential decay. However, the model used is sufficient to investigate some important features of the X-ray bursts.
From this model fitting we were able to derive the constant emission component before the bursts, start time, peak time and the exponential decay time scale. The X-ray and optical counts of the bursts were obtained by subtracting the pre-burst component from the total photon counts during the bursts.
The orbital phase of the start time of each burst was calculated using an orbital period of 0.15933775443 day and reference mid-eclipse time of 52903.963579 MJD that is appropriate for this epoch (Wolff et al. 2009).
In the left panel of Figure 3 we have shown a plot of the optical counts of the 62 bursts detected in the XMM-OM light curves against the X-ray counts (0.2-12 keV) while the ratio of the optical counts to the X-ray counts for each burst is shown against the X-ray counts in the right panel of the same figure. Henceforth, we call the ratio of the optical counts to the X-ray counts for each burst as optical conversion factor.
A correlation is seen between the X-ray and the optical counts of the bursts (shown with a straight line) although this is accompanied by a large scatter.
In the three panels of Figure 4, we have shown the delay between the start time of the X-ray and the optical bursts, the risetime of the optical bursts and the optical conversion factor as function of the orbital phases of the bursts, with the mid-eclipse time represented as phase zero. The lines plotted through the filled circles in each panel represent the average value of the parameters in wider orbital phase intervals of 0.1. The delay between the X-ray and the optical bursts has a maximum of about 8 seconds with an average value of 3.25 seconds.
However for none of the three parameters, X-ray to optical delay, optical risetime, or optical conversion factor, is there a clear orbital phase dependence seen that can be expected for reprocessing from the surface of the companion star .
\section{Discussion}
In the LMXBs, a large fraction of the optical emission can arise from X-rays reprocessed by the material in regions
around the central compact object. Search for this reprocessed variability requires very long simultaneous X-ray and
optical observations, which is usually difficult to carry out. The thermonuclear X-ray bursts (Type-1 X-ray bursts)
in LMXBs involve a large increase in the X-ray flux, by a factor of 10 or more on timescales of a few seconds. As
well as providing insights into the conditions on the surface of the neutron star, the sudden flash lights up the
whole binary system, and they are expected to be manifested in the UV and optical via reprocessed X-ray emission. The
optical bursts almost certainly arise as a consequence of the reprocessing of a fraction of the X-ray burst energy
by matter located within a few light seconds of the neutron star. The accretion disk around the neutron star and the
surface of the companion star are plausible sites for the reprocessing. The Type-1 X-ray bursts serve as a probe to
obtain information about the geometry of the X-ray binary by illuminating the surroundings of the compact object.
The reprocessing matter absorbs part of the X-rays, get heated and re-radiates the absorbed energy with an energy
distribution related to its temperature. By comparing the features of the X-ray bursts that originate on the compact
star, with the corresponding features of optical bursts that originate from the surroundings, one can obtain
information about the surroundings (Pedersen et al. 1982). The optical burst may contain only a fraction of the flux
of the X-ray bursts depending on the solid angle that the reprocessor subtends on the compact start and the
reprocessing efficiency. From analysis of the simultaneous X-ray and optical light curves presented here it is clear
that all the X-ray bursts are accompanied by optical bursts. The resulting reprocessed variability will be delayed
in time with respect to the X-ray variability by an amount depending on the position of the reprocessing regions
which in turn depends on the geometry of the binary and its orientation with respect to the observer's line of sight. The reprocessed optical emission seen by a distant observer will be delayed in time due to a combination of light travel time differences and radiation reprocessing time. One may expect the optical radiative delays to be much smaller than the light travel time differences (Pedersen et al. 1982). But most, if not all of the optical bursts are seen to be continued several seconds after the X-ray burst ended (Figure 1 \& Figure 2). These observations indicate that there is a radiative delay and radiative smearing of the optical flux relative to the X-ray flux.
In the present work, we have used the rising parts of the X-ray and optical bursts which contain clear information
related to the light travel time differences and thus about the geometry of the reprocessing region. As different
parts of the reprocessing body give rise to different values of this geometric delay there will be a geometric
smearing also. The reprocessed radiation can be considered as a superposition of contributions from different parts
of the reprocessing body.
If the X-ray absorbing material in the disc has a significant scale height above the mid-plane so that the
companion would be effectively shielded from direct X-ray illumination, it will reduce the strength of reprocessing
from the surface of the companion star.
In agreement with the reprocessing scenario, the optical bursts were observed a few seconds after the X-ray bursts.
Average value of the delay is 3.25 seconds. The light travel time accross this compact X-ray binary system is
about 3 seconds (Hynes et al. 2006) and therefore, the average delay of the optical bursts with respect to the X-ray
bursts is compatible with the reprocessing region being the surface of the compact start or the outer accretion disk.
It has been proposed that the optical reprocessing takes place from the surface of the companion star facing the
compact star and the accretion disk. Investigations have been made for EXO 0748-676 using simultaneous X-ray and
multi band optical/UV observations (Hynes et al. 2006). In the present work, apart from detection of a large number
of simultaneous X-ray and optical bursts, an important outcome is that we did not find any convincing evidence of
orbital phase dependence of the following parameters: X-ray to optical delay, rise time of the optical bursts, and
optical conversion factor as would be expected if the optical bursts were produced by reprocessing from the surface
of the secondary star that is facing the compact star. In particular, the optical conversion factor is expected to
be very small if the orbital phase is near zero. On the other hand, if the optical bursts are produced by
reprocessing of the X-rays in the accretion disk, the onset of the bursts is not expected to have a sharp, linear
shape as is observed in a few of the bursts in EXO 0748-676.
The temperature and amplitude of the reprocessed emission is expected to evolve during the bursts and is also
expected to be different in different bursts. Thus observations carried out in a single optical/UV band is expected
to show a complex non-linear relation between the counts in the X-ray and the optical bursts. In absence of multi
band optical/UV measurement, we refrain from any discussion about the correlation between the X-ray and the optical
burst amplitudes except for examining the simple optical conversion factor as a funtion of the orbital phase of the
binary.
The average X-ray to optical flux ratio of the bursts is calculated to be about 300 with some uncertainity due to our lack of knowledge about the optical spectral characetristics of the bursts. It is evident from Figures 3 \& 4 that there is a large difference from burst to burst by a factor of as much as five. Accurate determination of this ratio from multi band optical/UV observations could be useful for determining the fraction of X-rays which are intercepted by the disk which will in turn be useful for estimation of the thickness of the disk as seen from the neutron star. The upcoming multi-wavelength observatory ASTROSAT with large area X-ray detectors (Paul 2009) and Optical-UV telescopes (Srivastava et al. 2009) will be of particular interest for such studies.
\section{Conclusions}
We have detected a large number of simultaneous X-ray and optical Type-I bursts in the eclipsing LMXB EXO 0748-676
and have investigated the delay of the optical bursts, the optical conversion factor etc. No significant orbital
phase dependence is seen for these parameters which argues against the surface of the companion star being the
reprocessing region. We emphasise the fact that simultaneous optical observations of the X-ray bursts in multiple
wavelength bands will enable further detailed investigations of the reprocessing phenomena, including any non-linear
effect of the X-ray irradiation.
\section{Acknowledgement}
This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center
(HEASARC), provided by the NASA Goddard Space Flight Center.
|
{
"timestamp": "2012-03-09T02:01:56",
"yymm": "1203",
"arxiv_id": "1203.1731",
"language": "en",
"url": "https://arxiv.org/abs/1203.1731"
}
|
\section{Introduction}
\label{s:introduction}
A Chevalley involution $C$ of a connected reductive group
over an algebraically closed field satisfies $C(h)=h\inv$ for all $h$
in some Cartan subgroup of $G$. Furthermore, $C$ takes any
semisimple\footnote{George Lusztig pointed out this is true for all
elements \cite{lusztig_remarks}.} element to a conjugate of its
inverse. Consequently, in characteristic $0$, for any algebraic representation $\pi$ of $G$,
$\pi^C$ is isomorphic to the contragredient $\pi^*$.
We are interested in the existence, and properties, of rational
Chevalley involutions.
\begin{definition}
\label{d:main}
Suppose $G$ is defined over a field $F$, and let $\overline F$ be an algebraic closure of $F$.
\begin{enumerate}[(1)]
\item
A Chevalley involution of $G(F)$ is the restriction of a Chevalley involution of $G(\overline F)$ that is defined
over $F$.
\item We say an involution of $G(F)$ is {\it dualizing}
if it takes every semisimple element of $G(F)$ to a $G(F)$-conjugate of its inverse.
\end{enumerate}
\end{definition}
We refer to a Chevalley involution of $G(\overline F)$, which is defined over $F$, as an $F$-rational
Chevalley involution, or simply a rational Chevalley involution if $F$ is understood.
If $F$ is algebraically closed
every Chevalley involution
is dualizing, and any two are conjugate by an inner automorphism.
However, if $F$ is not algebraically closed, since not all Cartan subgroups
of $G(F)$ are conjugate, neither result is true in general.
We are primarily interested in dualizing Chevalley involutions.
For certain classical groups, over any local field,
there is a dualizing Chevalley involution by \cite[Chapitre 4]{mvw}.
Our main result is the existence of dualizing Chevalley
involutions in general when $F=\mathbb R$. Not all of these
are conjugate by an inner automorphism of $G(\mathbb R)$ (see Example \ref{e:notunique}). In order to have a
uniqueness result, we impose a further restriction.
A Cartan subgroup of $G(\mathbb R)$ is said to be fundamental if
it is of minimal split rank. Such a Cartan subgroup is the ``most compact''
Cartan subgroup, and is unique up to conjugation by $G(\mathbb R)$.
\begin{theorem}
\label{t:main}
Suppose $G$ is defined over $\mathbb R$.
There is an involution $C$ of $G(\mathbb R)$ such that $C(h)=h\inv$ for all
$h$ in some fundamental Cartan subgroup of $G(\mathbb R)$. Any such
involution is the restriction of a rational Chevalley involution of
$G(\mathbb C)$, and is dualizing. Any two such involutions are conjugate by an inner
automorphism of $G(\mathbb R)$.
\end{theorem}
If $G$ is semisimple and simply connected this is due to Vogan
\cite[Chapter I, Section 7]{borelWallach}.
The proof of the Theorem is similar to the proof in \cite{borelWallach}.
See Remark \ref{r:deduce}.
If $G$ is simple all involutions (over local and finite fields) have been
classified by Helminck. In particular Theorem \ref{t:main} can be read off
from \cite{hel88}, and similar results over other fields follow from
\cite{helminck_classification}.
\begin{definition}
We refer to an involution of $G(\mathbb R)$ satisfying the conditions of the
Theorem as a {\it fundamental} Chevalley involution of $G(\mathbb R)$.
\end{definition}
Since all fundamental Chevalley involutions are conjugate by an inner
automorphism of $G(\mathbb R)$, we may safely refer to {\it the} fundamental
Chevalley involution.
\begin{corollary}
\label{c:dual}
Suppose $\pi$ is an irreducible
representation of $G(\mathbb R)$, and $C$ is the fundamental Chevalley involution.
Then $\pi^C\simeq \pi^*$.
\end{corollary}
\medskip
Over a p-adic field it is not always obvious, at least to this author,
that there is a rational Chevalley involution,
not to mention a dualizing one.
In any event, the dualizing condition is quite restrictive.
For example, if $G(F)$ is split, there are
are many $G(F)$-conjugacy classes of involutions $C$ such that
$C(h)=h\inv$ for $h$ in a split Cartan subgroup.
Most of these are not dualizing.
In fact, if $G$ is a split
exceptional group of type $G_2,F_4$ or $E_8$ over a p-adic field there is
{\it no} dualizing involution. See Example \ref{e:E8}.
\medskip
The map $\pi\rightarrow\pi^*$ defines an involution on L-packets.
The main result of \cite{contragredient} is that,
on the dual side, this involution is given by the Chevalley involution
of $\overset{L}{\vphantom{a}}\negthinspace G$. See Section \ref{s:every}.
It follows that there is an elementary condition for every L-packet to
be self-dual.
\begin{proposition}
\label{p:lpacket}
Every $L$-packet for $G(\mathbb R)$ is self-dual if and only if $-1\in W(G(\mathbb C),H(\mathbb C))$.
\end{proposition}
Here $H(\mathbb C)$ is any Cartan subgroup of $G(\mathbb C)$, and
$W(G(\mathbb C),H(\mathbb C))$ is the (absolute) Weyl group $\text{Norm}_{G(\mathbb C)}(H(\mathbb C))/H(\mathbb C)$.
Now consider the finer question, whether every irreducible representation of $G(\mathbb R)$ is self-dual.
Let $H_f(\mathbb R)$ be a fundamental Cartan subgroup of $G(\mathbb R)$,
and let $W(G(\mathbb R),H_f(\mathbb R))=\text{Norm}_{G(\mathbb R)}(H_f(\mathbb R))/H_f(\mathbb R)$.
\begin{theorem}
\label{t:every}
Every irreducible representation of $G(\mathbb R)$ is self-dual if and only
if $-1\in W(G(\mathbb R),H_f(\mathbb R))$.
\end{theorem}
The condition is equivalent to: every semisimple element of $G(\mathbb R)$
is $G(\mathbb R)$-conjugate to its inverse.
We give some information about when this condition holds
in Section \ref{s:every}.
For example, suppose $G(\mathbb R)$ is connected,
$H_f(\mathbb R)$ is compact, and let $K(\mathbb C)$ be the complexification of a maximal compact subgroup of $G(\mathbb R)$.
Then $W(G(\mathbb R),H_f(\mathbb R))$ is the Weyl group of the root system of the connected, reductive group
$K(\mathbb C)$. One can then look up this root system in a table, for example
\cite{ov}, and check if it contains $-1$.
\begin{corollary}
\label{c:every}
Every irreducible representation of $G(\mathbb R)$ is self-dual
if and only if both of these conditions hold:
\begin{enumerate}
\item[(a)] every irreducible representation of $K(\mathbb R)$ is self
dual,
\item[(b)] $-1$ is in the absolute Weyl group $W(G(\mathbb C),H(\mathbb C))$.
\end{enumerate}
If $G(\mathbb R)$ contains a compact Cartan subgroup then (a) implies (b).
\end{corollary}
It is perhaps surprising how common this is. For example:
\begin{theorem}
\label{t:everyadjoint}
If $-1\in W(G(\mathbb C),H(\mathbb C))$, and $G$ is of adjoint type,
then every irreducible representation of $G(\mathbb R)$ is self dual.
\end{theorem}
For a more precise version, and some examples, see Section
\ref{s:every}, especially Corollary \ref{c:practical}.
\medskip
We next give an application to Frobenius Schur indicators. If $\pi$ is an irreducible,
self-dual representation of $G(\mathbb R)$, the
Frobenius Schur indicator $\epsilon(\pi)$ of $\pi$ is
$\pm1$, depending on whether $\pi$ admits an invariant symmetric, or
skew-symmetric, bilinear form.
Write $\chi_\pi$ for the central character of $\pi$.
Let $\Ch\rho$ be one-half the sum of any set of positive co-roots.
Then $z(\Ch\rho)=\exp(2\pi i\Ch\rho)$ is in the center of $G(\mathbb R)$.
The Frobenius Schur indicator of a finite dimensional representation $\pi$
of $G(\mathbb C)$ is $\chi_\pi(z(\Ch\rho))$. Under an assumption the same
holds for all irreducible (possibly infinite dimensional)
representations of $G(\mathbb R)$.
\begin{theorem}
Suppose every irreducible representation of $G(\mathbb R)$ is self-dual.
Then, for any irreducible representation $\pi$,
$\epsilon(\pi)=\chi_\pi(z(\Ch\rho))$.
In particular the assumption holds if $-1\in W(G(\mathbb C),H(\mathbb C))$ and $G$ is of adjoint type (Theorem \ref{t:everyadjoint}),
in which case every irreducible representation is orthogonal.
\end{theorem}
This paper is a complement to \cite{contragredient}, which considers
the action of the Chevalley involution on the dual group, and
its relation to the contragredient. See \cite[Remark 7.5]{contragredient}.
The author would like to thank Dipendra Prasad, Dinakar Ramakrishnan
and George Lusztig for very useful discussions. He also thanks the
referees for a number of helpful suggestions which have improved the
paper.
\section{Split Groups}
\label{s:split}
Throughout this paper
$G$ denotes a connected, reductive algebraic group, defined over a field $F$.
We may identify $G$ with its points $G(\overline F)$ over an algebraic closure of $F$.
In this section $F$ is arbitary; starting in the next section $F=\mathbb R$.
For background on algebraic groups see \cite{springer_book}, \cite{borel_algebraic_groups}
or \cite{humphreys_algebraic_groups}.
We start by defining Chevalley involutions. This is well known,
although it isn't easy to find it stated in the terms we need. See
\cite[Section 2]{contragredient}.
By a {\it Chevalley involution} of $G=G(\overline F)$ we mean an involution $C$
of $G$
satisfying $C(h)=h\inv$ for all $h$ in some Cartan subgroup $H$.
Any two such involutions are conjugate by an inner automorphism.
Fix a pinning $\mathcal{P}=(H,B,\{X_\alpha\})$:
$H\subset B$ are Cartan and Borel subgroups of $G$, respectively,
and (for $\alpha$ a simple root) $X_\alpha$ is in the $\alpha$-weight
space of $\mathrm{Lie}(H)$ acting on $\mathrm{Lie}(G)$.
Pinnings always exist, and are unique up to an inner automorphism;
an inner automorphism fixes a pinning only if it is trivial.
For $\alpha$ a simple
root let $X_{-\alpha}$ be the unique $-\alpha$-weight
vector satisfying $[X_\alpha,X_{-\alpha}]=\Ch\alpha\in\mathrm{Lie}(H)$.
The choice of $\mathcal{P}$ determines a unique Chevalley involution $C$,
satisfying $C(h)=h\inv$ ($h\in H$) and $C(X_\alpha)=X_{-\alpha}$ ($\alpha$ simple).
\medskip
Now suppose $G$ is semisimple and simply connected, and $G(F)$ is split.
Generators and relations for $G(F)$ are given by
\cite[Th\'eor\`em 3.2]{steinberg} (see \cite{steinbergCollected}).
The generators are $x_\alpha(u)$ for $\alpha$ a root, and $u\in F$, and these
satisfy certain relations.
It is easy to check that the map $C(x_\alpha(u))=x_{-\alpha}(u)$ preserves
the defining relations of $G(F)$, and the resulting automorphism
satisfies $C(h)=h\inv$ for $h$ in a split Cartan subgroup.
\begin{lemma}
\label{l:split}
Suppose $G$ is semisimple and simply connected, and $G(F)$ is split.
Let $H(F)$ be a split Cartan subgroup. Then
there is a rational Chevalley involution
satisfying $C(h)=h\inv$ for all $h\in H(F)$.
\end{lemma}
\begin{remark}
The same result holds {\it a fortiori} for the (possibly) nonlinear covering group $\Delta$ of $G(F)$ of
\cite[Th\'eor\`em 3.1]{steinberg}, which is obtained by dropping some relations from
those for $G(F)$.
\end{remark}
This is a somewhat weak result. Not every rational Chevalley
involution is dualizing, and not all dualizing Chevalley involutions are
conjugate by an inner automorphism of $G(F)$.
Both these facts are illustrated by a simple example.
For $g\in G$, let $int(g)$ be conjugation by $g$.
\begin{example}
\label{e:notunique}
Let $G(F)=SL(2,F)$.
Let $H_s(F)$ be the diagonal (split) Cartan subgroup.
Let $\sigma=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$,
and let $C=\int(\sigma)$.
Then $C(g)=\hskip-6pt\phantom{a}^tg\inv$ for all $g$, and in particular
$C(g)=g\inv$ for all $g\in H_s(F)$.
Suppose $g\ne\pm I$ is contained in an anisotropic Cartan subgroup
$H_a(F)$.
Then if $-1\not\in F^{*2}$, $C(g)$ is
not conjugate to $g\inv$ (in other words, $-1$ is not in the
Weyl group of $H_a(F)$).
Therefore $C$ is not dualizing.
On the other hand let
$C'=\int(\text{diag}(i,-i)\sigma)$.
Then $C'$ is rational and dualizing. Note that
$C'$ is an outer automorphism of $G(F)$ unless $-1\in F^{*2}$.
Now replace $SL(2,F)$ with $G(F)=PGL(2,F)$.
Both $C,C'$ factor to inner automorphisms of $G(F)$.
Since every semisimple element of $G(F)$ is $G(F)$-conjugate to its
inverse, $C,C'$ are both dualizing. However it is easy to see that $C$ is
not conjugate to $C'$ by an inner automorphism of $G(F)$.
\end{example}
Surprisingly, even for split groups, which have rational Chevalley
involutions, there may be no
dualizing involution.
This is illustrated by the
following example, which was pointed out by D. Prasad \cite{prasadPreprint}.
\begin{example}
\label{e:E8}
Suppose $F$ is p-adic and $G(F)$ is the split form of $G_2,F_4$
or $E_8$. By Lemma \ref{l:split} there is a Chevalley involution
$C$ of $G(F)$.
However, $G(F)$ has no dualizing involution.
To see this, assume $\tau$ is a dualizing involution. Then
$\pi^\tau\simeq\pi^*$ for all
irreducible representations $\pi$.
Every automorphism of $G(F)$ is inner (since $\text{Out}(G)=1$ and $G$ is both simply connected and adjoint),
so $\pi^\tau\simeq\pi$,
and therefore every irreducible representation is self-dual.
However, there are irreducible representations of $G(F)$ which
are not self-dual, coming from non-self dual cuspidal unipotent representations of the group over
the residue field.
\end{example}
\section{Real Chevalley Involutions}
\label{s:construction}
From now on we take $F=\mathbb R$, and
we identify $G$ with its complex points $G(\mathbb C)$. We
recall some standard theory about real forms of $G$,
in a form convenient for our applications.
For details see
\cite[Section 5.1.4]{ov},
\cite{helgason_book}, \cite{knapp_beyond}
or \cite[Section 3]{algorithms}.
A real form $G(\mathbb R)$ of $G(\mathbb C)$ is the fixed points of an
anti-holomorphic involution.
Each complex group has two distinguished real forms: the compact one,
and the split one.
For the compact real form, fix a pinning $\mathcal{P}=(H,B,\{X_\alpha\})$
and define $\{X_{-\alpha}\}$ as at
the beginning of Section \ref{s:split}.
Let $\sigma_c$ be the unique antiholomorphic automorphism of $G$
satisfying
$\sigma(X_\alpha)=-X_{-\alpha}$. Then $G(\mathbb R)=G^{\sigma_c}$ is
compact, and $H(\mathbb R)\simeq S^{1}\times\dots \times S^1$ is a compact torus.
It is clear from the definitions that the Chevalley automorphism
$C=C_{\mathcal{P}}$ commutes with $\sigma_c$. Therefore $\sigma_s=C\sigma_c$
is an antiholomorphic involution of $G$. Furthermore $G(\mathbb R)=G^{\sigma_s}$ is
split: $H(\mathbb R)\simeq\mathbb R^{*}\times\dots\times\mathbb R^*$ is a split torus.
General real forms of $G$ may be classified either by antiholomorphic or
holomorphic involutions of $G$. The latter is provided by the theory
of the Cartan involution.
In particular, there is a bijection
\begin{equation}
\label{e:bij}
\{\text{antiholomorphic involutions }\sigma\}/G
\leftrightarrow
\{\text{holomorphic involutions }\theta\}/G
\end{equation}
(the quotients are by conjugation by $\{\int(g)\mid g\in G\}$).
If $\sigma$ is an antiholomorphic involution, after conjugating by $G$
we may assume it commutes with $\sigma_c$, and set
$\theta=\sigma\sigma_c$. The other direction is similar.
For example, by the preceding discussion,
$C$ is the Cartan involution of the split real
form of $G$ (and the Cartan involution of the compact real form is the identity).
Suppose $\sigma\leftrightarrow\theta$, and $\sigma,\theta$ commute.
Let $G(\mathbb R)=G^\sigma$, $K=G^\theta$, and $K(\mathbb R)=K\cap G(\mathbb R)=G(\mathbb R)^\theta=K^{\sigma}$.
Then $K(\mathbb R)$ is a maximal compact subgroup of $G(\mathbb R)$, with
complexification $K$.
The relationship between these groups is illustrated by a diagram.
$$
\xymatrix{
&G\ar[rd]^\sigma\ar[ld]_\theta\\
G^\theta=K\ar[rd]_\sigma&&G(\mathbb R)=G^\sigma\ar[ld]^\theta\\
&K(\mathbb R)
}
$$
\medskip
Write $\text{Aut}(G),\text{Int}(G)$ for the (holomorphic) automorphisms of $G$,
and the inner automorphisms, respectively. Let $\text{Out}(G)=\text{Aut}(G)/\text{Int}(G)$
be the group of outer automorphisms.
We say an automorphism of $G$ is {\it distinguished} if it preserves
$\mathcal{P}$.
The pinning $\mathcal{P}$ defines an injective map
$\text{Out}(G)\overset s\hookrightarrow \text{Aut}(G)$:
$s(\phi)$ is the unique distinguished automorphism mapping to $\phi$.
If $G$ is semisimple the distinguished automorphisms embed into
the automorphism group of the Dynkin diagram, and this is a bijection if
$G$ is simply connected or adjoint.
Now fix a holomorphic involution $\theta$ of $G$.
Let $\delta$ be the image of $\theta$ under the map
$\text{Aut}(G)\rightarrow\text{Out}(G)\overset s\hookrightarrow \text{Aut}(G)$, so
$\delta$ is distinguished.
\begin{lemma}
\label{l:theta}
After conjugating by $G$ we may assume
\begin{equation}
\label{e:theta}
\theta=\int(h)\delta\quad\text{for some }h\in H^\delta
\end{equation}
(where the superscript denotes the $\delta$-fixed points).
\end{lemma}
For example suppose $\delta=1$, or equivalently $\theta\in\text{Int}(G)$.
The assertion is that $\int(x)\circ\theta\circ\int(x\inv)=\int(h)$ for
some $h\in H$. Since $\theta$ is inner write $\theta=\int(g)$ for some (semisimple) element $g\in G$.
The assertion is then: $\int(xgx\inv)=\int(h)$ for some $h\in H$.
In other words this is the standard fact that any
semisimple element is conjugate to an element of $H$.
\smallskip
\begin{proof}
By the definition of $\delta$, $\theta=\int(g)\delta$ for some
semisimple element $g\in G$.
We claim $g$ is contained in a
$\delta$-stable Cartan subgroup $H_1$.
Let $L$ be the identity component of $\text{Cent}_G(g)$.
Since $\theta$ is an involution, $\delta(g)=g\inv z$ for some $z\in Z$,
and it is easy to see this implies $\delta(L)=L$.
Take $H_1$ to be a $\delta$-stable Cartan subgroup of $M$.
This contains $g$ and is clearly a Cartan subgroup of $G$.
Write $H_1=T_1A_1$ where $T_1$ (resp. $A_1$)
is the identity component of $H_1^\theta$ (resp. $H_1^{-\theta}$).
Since, for $h\in H_1$, $h(g\delta)h\inv=h\delta(h\inv)g\delta$,
we may assume the $A_1$ component of $g$ is trivial, i.e. $h\in T_1$.
Let $K_\delta=G^\delta$. Use the subscript $0$ to indicate the identity component.
Then $H_0^\delta=(H^\delta)_0$ is Cartan subgroup of
$K_{\delta,0}$. Now $T_1$ is a torus in $K_{\delta,0}$, and is therefore
$K_{\delta,0}$-conjugate to a subgroup of $H_0^\delta$. Therefore after conjugating by
$K_{\delta,0}$ we may assume $\theta=\int(h)\delta$ for $h\in H_0^\delta$.
\end{proof}
With this choice of $\theta$, $H$ is defined over $\mathbb R$, and $H(\mathbb R)$ is a fundamental
Cartan subgroup of $G(\mathbb R)$ (see the introduction).
We say $H$ is a fundamental Cartan subgroup of $G$ with respect to
$\theta$.
For example
$\delta=1$ if and only if $H(\mathbb R)$ is compact.
For later use, we single out this class of groups.
We say $G(\mathbb R)$ is of equal rank if any of the following equivalent
conditions hold:
$G(\mathbb R)$ contains a compact Cartan subgroup; $H(\mathbb R)$ is compact;
$\text{rank}(K)=\text{rank}(G)$; $\delta=1$; or $\theta$ is an inner involution.
\medskip
We now give the proof of Theorem \ref{t:main}, which we break up into
steps. We first construct an involution of $G(\mathbb R)$, restricting to $-1$ on a
fundamental Cartan subgroup.
\begin{lemma}
\label{l:step1}
Let $H(\mathbb R)$ be a fundamental Cartan subgroup.
There is a rational Chevalley involution of $G$,
satisfying $C(h)=h\inv$ for all $h\in H(\mathbb R)$.
\end{lemma}
\begin{proof}
Choose $\theta$ corresponding to $\sigma$ by the bijection
\eqref{e:bij}.
By the Lemma, after conjugating $\sigma$ and $\theta$, we may assume
$\theta=\int(h)\delta$, where $\delta$ is distinguished and
$h\in H^\delta$.
Let $C=C_{\mathcal{P}}$, the Chevalley involution defined by the splitting $\mathcal{P}$, so
$C(h)=h\inv$ for $h\in H$. We claim $C$ commutes with $\sigma$.
First of all $\theta$ and $\sigma_c$ commute.
On the one hand
\begin{subequations}
\renewcommand{\theequation}{\theparentequation)(\alph{equation}}
\label{e:commute}
\begin{equation}
\begin{aligned}
(\theta\sigma_c)(X_\alpha)=\int(h)\delta(-X_{-\alpha})=-\int(h)(X_{-\delta\alpha})=-(\delta\alpha)(h\inv)X_{-\delta\alpha}
\end{aligned}
\end{equation}
and on the other
\begin{equation}
\begin{aligned}
(\sigma_c\theta)(X_\alpha)=\sigma_c(\int(h)X_{\delta\alpha})=\sigma_c((\delta\alpha)(h)X_{\delta\alpha})=-\overline{(\delta\alpha)(h)}X_{-\delta\alpha}.
\end{aligned}
\end{equation}
\end{subequations}
Since $\theta=\int(h)\delta$ is an involution,
$h\delta(h)\in Z(G)$ (here and elsewhere $Z$ denotes the center).
But $\delta(h)=h$, so $h^2\in Z(G)$. This
implies $\beta(h)=\pm1$ for all roots, so
$(\delta\alpha)(h\inv)=\overline{(\delta\alpha)(h)}$, and (a) and (b) are equal.
Therefore, by the discussion after \eqref{e:bij}, $\sigma=\theta\sigma_c$.
Since $C$ commutes with $\sigma_c$ (see the beginning of this section),
we just need to show that $C$ and $\theta$ commute.
This is similar to \eqref{e:commute}:
$(\theta C)X_\alpha=(\delta\alpha)(h\inv)X_{-\delta\alpha}$,
$(C\theta)X_\alpha=(\delta\alpha)(h)X_{-\delta\alpha}$, and these are equal since $h^2\in Z$.
\end{proof}
Now we show the Chevalley involution just constructed is dualizing
(Definition \ref{d:main}).
\begin{lemma}
Suppose $C$ satisfies the conditions of Lemma \ref{l:step1}.
Then $C$ is dualizing, i.e. it takes every semisimple element of $G(F)$ to a
$G(F)$-conjugate of its inverse.
\end{lemma}
\begin{proof}
This is true for $g$ in the fundamental Cartan subgroup $H(\mathbb R)$. We
obtain the result on other Cartan subgroups using Cayley transforms.
We proceed by induction, so change notation momentarily, and assume
$H$ is any $\theta$ and $\sigma$-stable Cartan subgroup, such that
$C(h)$ is $G(\mathbb R)$-conjugate to $h\inv$ for all $h\in H(\mathbb R)$.
Taking $h$ regular, we see there is $g\in\text{Norm}_{G(\mathbb R)}(H(\mathbb R))$ such
that, if $\tau=\int(g)\circ C$, then $\tau|_{H(\mathbb R)}=-1$.
Suppose $\alpha$ is a root of $H$.
Let $G_\alpha$ be the derived group of the centralizer of the kernel
of $\alpha$, and set $H_\alpha=H\cap G_\alpha$.
Thus $G_\alpha$ is locally isomorphic to $SL(2)$,
and $H=\ker(\alpha)H_\alpha$.
Now assume $\alpha$ is a noncompact imaginary root, which amounts
to saying that $G_\alpha$ is $\theta,\sigma$ stable, $G_\alpha(\mathbb R)$ is
split, and $H_\alpha(\mathbb R)$ is a compact Cartan subgroup of
$G_\alpha(\mathbb R)$.
Replace $H_\alpha$ with a $\theta,\sigma$-stable
split Cartan subgroup $H'_\alpha$ of $G_\alpha$.
Since $\tau$ normalizes $G_\alpha$, and is defined over $\mathbb R$,
$\tau(H'(\mathbb R))$ is another split Cartan subgroup of $G_\alpha(\mathbb R)$.
Therefore we can find $x\in G_\alpha(\mathbb R)$
so that $x(\tau(h))x\inv=h\inv$ for all $h\in H'_\alpha(\mathbb R)$.
Let $H'=\ker(\alpha)H'_\alpha$.
Then $(\int(x)\circ\tau)(h)=h\inv$ for all $h\in H'(\mathbb R)$.
Every Cartan subgroup of $G(\mathbb R)$ is obtained, up to conjugacy by
$G(\mathbb R)$, by a series of Cayley transforms from the fundamental Cartan
subgroup. The result follows.
\end{proof}
Finally, the uniqueness statement of Theorem \ref{t:main} comes down to the
next Lemma.
\begin{lemma}
Suppose $\tau$ is an automorphism of $G(\mathbb R)$ such that the restriction of $\tau$ to a
fundamental Cartan subgroup $H(\mathbb R)$ is trivial. Then $\tau=\int(h)$ for some $h\in H(\mathbb R)$.
\end{lemma}
\begin{proof}
Since both $\mathbb R$ and $\mathbb C$ play a role here we write $G(\mathbb C)$ to
emphasize the complex group.
After complexifying, $\tau$ is an automorphism of $G(\mathbb C)$ which is
trivial on $H(\mathbb C)$. It is well known that $\tau=\int(h)$ for some
$h\in H(\mathbb C)$ (for example see \cite[Lemma 2.4]{contragredient}). It is enough
to show that $h\in H(\mathbb R)Z(G(\mathbb C))$.
Since $\tau$ normalizes $G(\mathbb R)$, $\sigma(h)=hz$ for
some $z\in Z(G(\mathbb C))$. Writing $p$ for the map to the adjoint group, this
says $p(h)\in H_{ad}(\mathbb R)$. It is well known that $H_{ad}(\mathbb R)$ is connected
(this is where we use that $H(\mathbb R)$ is fundamental), so
the map $p:H(\mathbb R)\rightarrow H_{ad}(\mathbb R)$ is surjective.
Therefore we can find $h'\in H(\mathbb R)$ with $p(h')=p(h)$, i.e. $h=h'z\in H(\mathbb R)Z(G(\mathbb C))$.
\end{proof}
\begin{lemma}
\label{l:uniqueness}
Any two automorphisms of $G(\mathbb R)$, restricting to $-1$ on
a fundamental Cartan subgroup, are conjugate by an inner automorphism
of $G(\mathbb R)$.
\end{lemma}
\begin{proof}
Suppose $\tau,\tau'$ satisfying the conditions, with respect to a
fundamental Cartan subgroup $H(\mathbb R)$.
By the previous Lemma $\tau'=\int(h)\circ\tau$ for some $h\in H(\mathbb R)$.
Since $H(\mathbb R)$ is connected, choose $x\in H(\mathbb R)$ with $x^2=h$.
Then $\tau'=\int(x)\circ\tau\circ\int(x\inv)$.
\end{proof}
This completes the proof of Theorem \ref{t:main}.
\begin{remark}
\label{r:deduce}
It is also possible to deduce Theorem \ref{t:main} from the special
case of \cite[Chapter I, Corollary 7.4]{borelWallach},
which is essentially about the Lie algebra.
According to this result (actually, its proof),
if $G(\mathbb C)$ is semisimple and simply connected, there is a
rational Chevalley $C$ involution of $G(\mathbb C)$, whose restriction to
$G(\mathbb R)$ is dualizing.
Since $C$ acts by inverse on the center of $G(\mathbb C)$, it preserves
any subgroup of the center, and therefore factors
to any quotient of $G(\mathbb C)$.
Similarly, any
complex reductive group is a quotient of a simply connected semisimple
group and a torus, and a similar argument holds in this case.
\end{remark}
\section{Groups for which every representation is self-dual}
\label{s:every}
We first consider the elementary question of when every L-packet is
self-dual (Proposition \ref{p:lpacket}).
Fix a real form $G(\mathbb R)$ of $G(\mathbb C)$, choose $\theta$ as usual, and let
$K(\mathbb C)=G(\mathbb C)^\theta$ (see Section \ref{s:construction}). Let $\mathfrak g=\mathrm{Lie}(G(\mathbb C))$.
By an irreducible representation of $G(\mathbb R)$ we mean
an irreducible $(\mathfrak g,K(\mathbb C))$-module, or equivalently
an irreducible admissible representation of $G(\mathbb R)$ on a complex Hilbert space.
See \cite[Section 0.3]{vogan_green}.
We now identify $G$ with $G(\mathbb C)$, $K$ with $K(\mathbb C)$, and similarly for other.
We will always write $\mathbb R$ to indicate a real group.
\medskip
\begin{proof}[Proof of Proposition \ref{p:lpacket}]
Suppose an L-packet $\Pi$
is defined by an admissible homomorphism $\phi:W_\mathbb R\rightarrow \overset{L}{\vphantom{a}}\negthinspace G$.
By \cite[Theorem 1.3]{contragredient} the contragredient L-packet
corresponds to $C\circ\phi$, where $C$ is the Chevalley automorphism
of $\overset{L}{\vphantom{a}}\negthinspace G$.
Therefore every L-packet is self-dual if and only if this action is trivial, up
to conjugation by $\Ch G$, i.e. the Chevalley automorphism is inner
for $\Ch G$. This is the case if and only if $-1\in W(\Ch G,\Ch
H)\simeq W(G,H)$.
\end{proof}
\begin{remark}
\label{r:-1}
By the classification of root systems, $-1$ is in the Weyl group of an irreducible
root system if and only if it is of type
$A_1$, $B_n$, $C_n$, $D_{2n}$, $F_4$, $G_2$, $E_7$, or $E_8$. It is worth noting that
if $G$ is simple and simply connected, $-1\in W(G,H)$ if and only if
$Z(G)$ is an elementary two-group (one direction is obvious, and the other is case-by-case).
\end{remark}
We are interested in real groups $G(\mathbb R)$ for which
every irreducible representation is self-dual.
By Proposition \ref{p:lpacket} an obvious necessary condition is $-1\in W(G,H)$.
We first prove Theorem \ref{t:every}, which gives
a necessary and sufficient condition, and then give more
detail in some special cases.
Let $H_f$ be the centralizer in $G$ of a Cartan subgroup of $K_0$ (the
subscript indicates identity component). Let $H_K=H_f\cap K$. This is
an abelian subgroup of the (possibly disconnected) group $K$, and
$H_{K,0}=H_K\cap K_0$ is a Cartan subgroup of $K_0$. Then $H_f$ is a
fundamental Cartan subgroup of $G$ with respect to $\theta$ (see
\cite[Definition 3.1]{khat}). For example, choose $\theta$ as in
Lemma \ref{l:theta}. Then $H_f$ is the fixed Cartan subgroup of the
pinning $\mathcal{P}$.
\medskip
\begin{proof}[Proof of Theorem \ref{t:every}]
Using standard facts about characters of representations, viewed as
functions on the regular semisimple elements, it is easy
to see that every irreducible representation is self-dual if and only if
\begin{equation}
\label{e:conj}
\text{ every regular semisimple element is $G(\mathbb R)$-conjugate to its inverse}.
\end{equation}
Assume $-1\in W(G(\mathbb R),H_f(\mathbb R))$,
so there is an inner automorphism $\tau$ of $G(\mathbb R)$ acting by $-1$ on
$H_f(\mathbb R)$.
By Theorem \ref{t:main}, if $g$ is semisimple, $\tau(g)$ is
$G(\mathbb R)$-conjugate to $g\inv$. Since $\tau$ is inner this gives
\eqref{e:conj}.
Conversely suppose \eqref{e:conj} holds. Let $h$ be a regular element
of $H_f(\mathbb R)$. Then $h\inv=xhx\inv$ for some $x\in G(\mathbb R)$, and by
regularity
$x$ normalizes $H_f(\mathbb R)$. Therefore $-1\in W(G(\mathbb R),H_f(\mathbb R))$.
\end{proof}
\begin{subequations}
\renewcommand{\theequation}{\theparentequation)(\alph{equation}}
\label{e:WKH}
It is helpful to state this result in terms of the complex group $K$,
rather than the real group $G(\mathbb R)$.
The groups
\begin{equation}
W(G(\mathbb R),H_f(\mathbb R))=\text{Norm}_{G(\mathbb R)}(H_f(\mathbb R))/H_f(\mathbb R).
\end{equation}
and
\begin{equation}
W(K,H_f)=\text{Norm}_K(H_f)/H_K
\end{equation}
are isomorphic. We reiterate that $K, H_f$ and $H_K$ are complex.
Also consider
\begin{equation}
W(K,H_K)=\text{Norm}_K(H_K)/H_K.
\end{equation}
\end{subequations}
This is defined solely in terms of $K$; the difference between (b) and (c)
is whether we consider an element to be an automorphism of $H_f$ or $H_K$ (see the next Remark).
This is also isomorphic to
(a) and (b), and is useful in computing these groups.
Some care is required here due to the fact that $K$, equivalently
$G(\mathbb R)$, may be disconnected. If $K$ is connected then $W(K,H_K)$ is
the Weyl group of the root system of $H_K$ in $K$, but otherwise
$W(K,H_K)$ may not be the Weyl group of a root system.
A key role is played by the condition $-1\in W(K,H_f)$.
We need to keep in mind
the following dangerous bend
concerning the meaning of $-1$.
\begin{remark}
\label{r:dangerous}
Suppose $-1\in W(K,H_K)$. By definition this means there is an element
$g\in \text{Norm}_K(H_K)$ such that $ghg\inv=h\inv$ for all $h\in H_K$.
However, although $g$ normalizes $H_f$,
it is not necessarily the case that $ghg\inv=h\inv$ for all $h\in
H_f\supset H_K$.
In other words, if $\text{rank}(K)\ne \text{rank}(G)$, $-1\in W(K,H_K)$ does not
imply $-1\in W(K,H_f)$, even though these two groups are isomorphic.
On the other hand $-1\in W(G(\mathbb R),H_f(\mathbb R))$ if and only if $-1\in W(K,H_f)$.
\end{remark}
\begin{example}
Let $G=SL(3,\mathbb C)$, $G(\mathbb R)=SL(3,\mathbb R)$.
Then $-1\not\in W(G,H_f)$, so {\it a fortiori} $-1\not\in W(K,H_f)$.
On the other hand $K=SO(3,\mathbb C)$,
$W(K,H_K)$ is the Weyl group of type $A_1$,
and $-1\in W(K,H_K)$.
We can choose $H_f=\{(z,w,\frac1{zw})\}\mid z,w\in\mathbb C^*\}$,
and $H_K=\{(z,\frac1z,1)\}\subset H_f$.
The nontrivial Weyl group element of $W(K,H_K)$ acts by exchanging the first
two coordinates.
This acts by inverse on $H_K$, but not $H_f$.
\end{example}
If $K$ is
connected, it is an elementary root system check to determine if $-1\in
W(K,H_K)$ (see Remark \ref{r:-1}). In the equal rank case this is all that is needed, although
in the unequal rank case some care is required to determine if
$-1\in W(K,H)$.
By the isomorphism of \eqref{e:WKH}(a) and (b),
Theorem \ref{t:every} can be stated in terms of $W(K,H_f)$.
\begin{corollary}
\label{c:everyalt}
Every irreducible representation of $G(\mathbb R)$ is self-dual if and only if
$-1\in W(K,H_f)$.
\end{corollary}
Next we prove Corollary \ref{c:every}, which gives another condition,
in terms of $K$,
for every representation of $G(\mathbb R)$ to be self dual.
\medskip
\begin{proof}[Proof of Corollary \ref{c:every}]
Every irreducible representation $\mu$ of $K(\mathbb R)$ is the unique lowest $K(\mathbb R)$-type of
an irreducible representation $\pi$ of $G(\mathbb R)$ \cite[Theorem 1.2]{khat}.
Since the lowest $K(\mathbb R)$-type of $\pi^*$ is $\mu^*$, $\pi\simeq \pi^*$
implies $\mu\simeq\mu^*$. This proves one direction.
Conversely, by Corollary \ref{c:everyalt} we need to show every
irreducible representation of $K(\mathbb R)$, equivalently $K=K(\mathbb C)$, is self-dual implies $-1\in W(K,H_f)$.
We first show that $-1\in W(K,H_K)$ and $-1\in
W(G,H_f)$ implies
$-1\in W(K,H_f)$.
This is obvious if $H_K=H_f$ (the equal rank case). Otherwise (here we
need the assumption that $-1\in W(G,H_f)$) choose $g\in G$ such that
$ghg\inv=h\inv$ for all $h\in H_f$. Also choose $k\in K$ satisfying
$khk\inv=h\inv$ for all $h\in H_K$. Then $gk\inv\in\text{Cent}_G(H_K)=H_f$.
This implies $khk\inv=h\inv$ for all $h\in H_f$.
So it is enough to show that if every irreducible representation of $K$ is
self-dual then $-1\in W(K,H_K)$.
If $K$ is connected this follows from Corollary
\ref{c:everyalt} applied to $K$.
For $\lambda\in X^*(H_{K,0})$ (the algebraic characters of the torus $H_{K,0}$)
let $\pi_\lambda$ be the irreducible
representation of $K_0$ with extremal weight $\lambda$.
Then $\pi_\lambda^*=\pi_{-\lambda}$.
Consider the induced representation
$I=\mathrm{Ind}_{K_0}^K(\pi_\lambda)$.
The restriction of $I$ to $K_0$ contains $\pi_\lambda$.
Since $I$ is self-dual by hypothesis, this restriction also contains
$\pi_{-\lambda}$.
It is easy to see
that every extremal weight of the
restriction of this representation to $K_0$ is $W(K,H_K)$-conjugate to
$\lambda$ (choose representatives of $K/K_0$ in $\text{Norm}_K(H_K)$, and
use the fact that $K_0$ is normal in $K$). Therefore $-\lambda$ is
$W(K,H_K)$-conjugate to $\lambda$. Taking $\lambda$ generic
this implies $-1\in W(K,H_K)$.
If every irreducible representation of $K$ is self-dual then $-1\in
W(K,H)$.
If $\text{rank}(G)=\text{rank}(K)$ this implies $-1\in W(G,H_f)$, giving the final assertion.
\end{proof}
\begin{remark}
Here is an example of an unequal rank group for which Condition (a) in \eqref{c:every} holds, but not (b).
Take $G(\mathbb R)=SL(2n+1,\mathbb R)$, $K=SO(2n+1,\mathbb C)$.
Then $-1\in W(K,H_K)$, and every irreducible representation of
$SO(2n+1,\mathbb C)$ is self-dual. However this is not
the case (for example for minimal principal series) for $SL(2n+1,\mathbb R)$,
since $-1\not\in W(G,H_f)$.
\end{remark}
Here is a practical way to determine if every irreducible
representation of $G(\mathbb R)$ is self-dual.
First assume $G(\mathbb R)$ is of equal rank (see the discussion after Lemma
\ref{l:theta}).
Then $\theta$ is inner, so write $\theta=\int(x)$ for some $x\in G$,
with $x^2\in Z(G)$.
Assume for the moment that $-1\in W(G,H_f)$ (recall $G=G$ and $H_f$ are complex); this implies $Z(G)$ is an
elementary two-group. We say the real form defined by $\theta$ is {\it pure} if $x^2=1$.
Since $Z(G)$ is a two-group, this condition is independent of the choice of $x$ such that
$\theta=\int(x)$.
(In other words, although purity is typically only well-defined
as a property of {\it strong} real forms \cite[Definition
5.5]{algorithms},
it is a well-defined property of real forms provided $-1\in W(G,H_f)$.)
Every real form is pure if $G$ is adjoint.
\begin{corollary}
\label{c:practical}
Assume $G(\mathbb R)$ is simple.
Every irreducible representation of $G(\mathbb R)$ is self-dual if and only
if both of these conditions hold:
\begin{enumerate}
\item[(a)] $-1\in W(G,H_f)$
\item[(b)] if $G(\mathbb R)$ is of equal rank, it is a pure real form.
\end{enumerate}
\end{corollary}
\begin{proof}
First assume we are in the equal rank case.
By Theorem \ref{t:main} we have to show
\begin{equation}
\label{e:practical}
-1\in W(G,H_f), x^2=1\Leftrightarrow -1\in W(K,H_f).
\end{equation}
After conjugating by $G$ we may assume $x\in H_f$.
Suppose $g\in G$ satisfies $ghg\inv=h\inv$ for all $h\in H_f$.
Then $\theta_x(g)=xgx\inv=x(gx\inv g\inv)g=x^2g$.
Therefore $g\in K$ if and only if $x^2=1$.
Now suppose $G(\mathbb R)$ is not of equal rank. We have to show
\begin{equation}
-1\in W(G,H_f)\Leftrightarrow -1\in W(K,H_f).
\end{equation}
The implication $\Leftarrow$ is obvious.
First assume
$G(\mathbb R)=G_1(\mathbb C)$, i.e. a complex group, viewed as a real group by
restriction of scalars.
Then, if $H_1$ is a Cartan subgroup of $G_1$,
$G=G_1\times G_1, H_f=H_1\times H_1, K=G_1^\Delta$ (embedded diagonally).
It follows immediately that
$-1\in W(K,H_f)$ if and only if $-1\in W(G,H_f)$.
Finally assume $G(\mathbb R)$ is unequal rank, but not complex.
Then
$G$ is of type $A_n$ ($n\ge 2$),
$D_n$ or $E_6$. But then $-1\in W(G,H_f)$ only in type $D_{2n}$.
This leaves only the groups locally isomorphic to $SO(p,q)$ with $p,q$
odd and $p+q=0\pmod 4$.
Let $G(\mathbb R)=Spin(p,q)$ with $p+q=4n$.
It is enough to show $-1\in W(K,H_f)$,
since $W(K,H_f)$ is, if anything, larger if $G$ is not simply
connected.
Note that $K$ is connected, of type $B_r\times B_s$, and $-1\in
W(K,H_K)$.
The only remaining issue is to check that $-1\in W(K,H_f)$; here
$\text{rank}(H_f)=\text{rank}(H_K)+1$. This is a straightforward check.
It essentially comes down to the case of $Spin(3,1)$, for which it is
easy to see, since $Spin(3,1)\simeq SL(2,\mathbb C)$.
\end{proof}
Theorem \ref{t:everyadjoint} is a special case of Corollary \ref{c:practical}.
\begin{proof}[Proof of Theorem \ref{t:everyadjoint}]
By assumption $-1\in W(G(\mathbb C),H(\mathbb C))$ so (a) of Corollary \ref{c:practical} holds.
On the other hand (b) holds since every real form of an adjoint group is pure.
By the Corollary every irreducible representation of $G(\mathbb R)$ is self-dual.
\end{proof}
With a little effort we can deduce the following list from Corollary \ref{c:everyalt} and Corollary
\ref{c:practical}.
First assume $G$ is simple, and $G(\mathbb R)$ is equal rank.
If $G$ is adjoint it is only a question of whether $-1\in W(G,H_f)$.
If $G$ is simply connected we need to check
if $-1$ is in the Weyl group of the root system of $K$, which is easy,
for example by the tables \cite[pp. 312-317]{ov}.
This leaves only the intermediate
groups of type $D_n$, which require some case-by-case checking.
In the unequal rank case, we only need to consider
complex groups, and (up to isogeny) $SO(p,q)$ with
$p,q$ odd.
\medskip
Suppose $G(\mathbb R)$ is simple. Then every irreducible representation of
$G(\mathbb R)$ is self-dual if and only if $G(\mathbb R)$ is on the following list
(see below for terminology in type $D_{2n}$).
\begin{enumerate}[(1)]
\item $A_n$: $SO(2,1)$, $SU(2)$ and $SO(3)$.
\item $B_n$: Every real form of the adjoint group, $Spin(2p,2q+1)$
($p$ even).
\item $C_n$: Every real form of the adjoint group, all $Sp(p,q)$.
\item $D_{2n+1}$: none.
\item $D_{2n}$, equal rank:
$Spin(2p,2q)$ ($p,q$ even);
all $SO(2p,2q)$ ($p+q=2n$)
$\overline{SO}(2p,2q)$ ($p,q$ even);
$\overline{SO}^*(4n)$ when disconnected;
all adjoint groups: $PSO(2p,2q)$ ($p+q=2n$) and $PSO^*(4n)$.
\item $D_{2n}$, unequal rank: all real forms, i.e. all groups locally isomorphic to
$SO(2p+1,2q+1)$ ($p+q$ odd).
\item $E_6$: none.
\item $E_7$: Every real form of the adjoint group, the simply connected compact group.
\item $G_2,F_4,E_8$: every real form.
\item complex groups of type $A_1,B_n,C_n,D_{2n},G_2,F_4,E_7,E_8$ (see
Remark \ref{r:-1}).
\end{enumerate}
In type $D_{2n}$ let
$\overline{SO}(4n,\mathbb C)$ denote the group $Spin(4n,\mathbb C)/A$
where $A\simeq\mathbb Z/2\mathbb Z$ is not fixed by the outer automorphism of $Spin(4n,\mathbb C)$.
For each $p+q=4n$ this group has a real form denoted
$\overline{SO}(p,q)$ (locally isomorphic to $SO(p,q)$).
Also it has two
subgroups locally isomorphic to $SO^*(4n)$, which we denote $\overline{SO}^*(4n)$.
These are not isomorphic: one of them is connected, and the other is not.
\section{Frobenius Schur Indicators}
Suppose $\pi$ is an irreducible self-dual representation of a group $G$.
Choosing an isomorphism $T:\pi\rightarrow\pi^*$, $\langle v,w\rangle:=T(v)(w)$
is a non-degenerate, invariant, bilinear form, unique up to scalar.
It is either symmetric or skew-symmetric. The
Frobenius Schur indicator $\epsilon(\pi)$ of $\pi$ is defined to be $1$ or $-1$, accordingly.
It is of some interest to compute this invariant. For example see \cite{prasadRamakrishnan}.
Now suppose $G$ is a connected, reductive complex group.
It is well known that if $\pi$ is a self-dual, finite dimensional representation of $G$
$\epsilon(\pi)$
is given by a particular value of its central character \cite[Ch. IX,
\S7.2, Proposition 1]{bourbakiLieGroupsLieAlgebras7-9}.
Here is an elementary proof. This is a refinement of
one of the proofs of
\cite[Section 1, Lemma 2]{prasadSelfDual};
we use the Tits group to identify the central element in question.
Let
$\Ch\rho$ be one-half the sum of any set of positive co-roots, and set
\begin{equation}
\label{e:z}
z(\Ch\rho)=\exp(2\pi i\Ch\rho).
\end{equation}
Not only is $z(\Ch\rho)$ central in $G$, it is
fixed by every automorphism of $G$.
In particular $z\in Z(G(\mathbb R))$ for any real form of $G$.
If it is necessary to specify the group in question we will write
$z(\Ch\rho_G)$.
\begin{lemma}
\label{l:w0}
Let $w_0$ be the long element of $W(G,H)$ (with respect to any set of positive roots).
There is a representative $g\in\text{Norm}_G(H)$ of $w_0$ satisfying
$g^2=z(\Ch\rho)$. Furthermore, if $w_0=-1$, this holds for any
representative of $w_0$.
\end{lemma}
\begin{proof}
We use the Tits group.
Fix a pinning $\mathcal{P}=(H,B,\{X_\alpha\})$ for $G$
(see Section \ref{s:split}). This defines the Tits group $\mathcal T$, a subgroup
of $\text{Norm}_G(H)$ mapping surjectively to $W(G,H)$. Every element $w$ of the
Weyl group has a canonical inverse image $\sigma(w)\in\mathcal T$.
See \cite[Section 5]{contragredient}.
Let $g=\sigma(w_0)$. By \cite[Lemma 5.4]{contragredient}, $g^2=z(\Ch\rho)$.
Any other representative is of the form $hg$ for some $h\in H$.
If $w_0=-1$ then $(hg)^2=h(ghg\inv)g^2=(hh\inv)g^2=g^2$.
\end{proof}
\begin{lemma}
\label{l:schurfinite}
Assume $G$ is a connected, reductive complex group.
Suppose $\pi$ is an irreducible, finite dimensional, self-dual representation of $G$.
Let $\chi_\pi$ denote the central character of $\pi$. Then
\begin{equation}
\epsilon(\pi)=\chi_\pi(z(\Ch\rho)).
\end{equation}
\end{lemma}
\begin{proof}
For any vectors $u,w$ in the space $V$ of $\pi$ we have
\begin{subequations}
\renewcommand{\theequation}{\theparentequation)(\alph{equation}}
\label{e:schur}
\begin{equation}
\langle u,w\rangle=\epsilon(\pi)\langle w,u\rangle.
\end{equation}
Suppose $g\in G,g^2\in Z(G)$, and $v\in V$. Set $u=\pi(g^2)v,w=\pi(g)v$:
\begin{equation}
\begin{aligned}
\chi_\pi(g^2)\langle v,\pi(g)v\rangle&=
\langle \pi(g^2)v,\pi(g)v)\quad\text{(since $g^2$ is central)}\\
&=\langle \pi(g)v,v\rangle\quad\text{(by invariance)}\\
&=\epsilon(\pi)\langle v,\pi(g)v\rangle\quad\text{(by (a)).}
\end{aligned}
\end{equation}
We conclude
\begin{equation}
g^2\in Z(G),\, \langle v,\pi(g)v\rangle\ne 0\Rightarrow \epsilon(\pi)=\chi_\pi(g^2).
\end{equation}
\end{subequations}
Fix a Cartan subgroup $H$, and for $\lambda\in X^*(H)$ write
$V_\lambda$ for the corresponding weight space.
It is easy to see $\langle V_\lambda,V_{-\lambda}\rangle\ne 0$.
Let $\lambda$ be the highest weight, so $V_\lambda$ is
one-dimensional. Let $w_0$ be the long element of the Weyl group.
Then $\pi^*$ has highest weight $-w_0\lambda$; since $\pi$ is
self-dual this implies $-\lambda=w_0\lambda$.
Choose $g\in \text{Norm}_G(H)$ as in Lemma \ref{l:w0}, so $g^2=z(\Ch\rho)$, and $0\ne v\in V_\lambda$.
Then $\pi(g)(v)\in V_{-\lambda}$. Since $V_{\pm\lambda}$ are
one-dimensional $\langle v,\pi(g)v\rangle\ne 0$, so apply \eqref{e:schur}(c).
\end{proof}
We now consider the Frobenius Schur indicator for infinite dimensional
representations. The basic technique is the following elementary observation,
which appears in \cite{prasadRamakrishnan}.
Suppose $H\subset G$ are groups, $\pi$ is a self-dual representation
of $G$, $\pi_H$ is a self-dual representation of $H$, and
$\pi_H$ occurs with multiplicity one in $\pi|_H$.
Then $\epsilon(\pi)=\epsilon(\pi_H)$.
We first apply this to $G$ and $K$, and later to $K$ and its identity component.
The next Lemma is a special case of the main result of this section
(Theorem \ref{t:schur}), but it is worth stating separately since
it clearly illustrates the main idea.
We continue to assume $G$ is a connected reductive complex group.
Fix a real form $G(\mathbb R)$, a corresponding Cartan
involution $\theta$, and let $K=G^\theta$.
\begin{lemma}
\label{l:schurconnected}
Suppose every irreducible representation of $G(\mathbb R)$ is
self-dual.
Also assume $G(\mathbb R)$ is
connected.
If $\pi$ is an irreducible representation then
$\epsilon(\pi)=\chi_\pi(z(\Ch\rho))$.
\end{lemma}
\begin{proof}
By Corollary \ref{c:everyalt},
the self-duality assumption implies $-1\in
W(K,H_f)$. So $-1\in W(K,H_K)$ and this implies every $K$-type is self-dual (since
$G(\mathbb R)$, and therefore $K$, is
connected).
Let $\mu$ be a lowest $K$-type of $\pi$.
Then $\mu$ has multiplicity one, and is self-dual,
so by the comment above $\epsilon(\pi)=\epsilon(\mu)$.
By Lemma \ref{l:schurfinite}, $\epsilon(\mu)=\chi_\mu(z(\Ch\rho_K))$,
where $z(\Ch\rho_K)$ is defined by \eqref{e:z} applied to $K$.
Write $\Ch\rho_G$ in place of $\Ch\rho$.
Let $g\in\text{Norm}_G(H_f)$ be a representative of $-1\in W(G,H_f)$, so by Lemma \ref{l:w0}
$g^2=z(\Ch\rho_G)$.
Now view $g$ as a representative of $-1\in W(K,H_f)$, in which case
(by Lemma \ref{l:w0} applied to to $K$) we see $g^2=z(\Ch\rho_K)$.
Therefore
$z(\Ch\rho_G)=z(\Ch\rho_K)$, and since $z(\Ch\rho_G)\in Z(G)$, $\chi_\mu(z(\Ch\rho_G))=\chi_\pi(z(\Ch\rho_G))$, independent
of $\mu$. Thus
$$
\epsilon(\pi)=\epsilon(\mu)=\chi_\mu(z(\Ch\rho_K))=\chi_\mu(z(\Ch\rho_G))=\chi_\pi(z(\Ch\rho_G)).
$$
\end{proof}
A crucial aspect of the proof is that, for $K$ connected,
$-1\in W(K,H_f)$ implies $z(\Ch\rho_G)=z(\Ch\rho_K)$.
We need the surprising fact that
this is true without the first assumption.
\begin{lemma}
\label{l:zrhoK}
Suppose $G$ is a connected, reductive complex group, $\theta$ is a Cartan involution,
$K=G^\theta$ and $H_f$ is a fundamental Cartan subgroup.
Assume $-1\in W(K,H_f)$. Then $z(\Ch\rho)=z(\Ch\rho_K)$.
\end{lemma}
This is a bit subtle, as
a simple example shows.
\begin{example}
Let $G(\mathbb R)=SL(2,\mathbb R)$, so $K=SO(2,\mathbb C)$.
Then $-1\not\in W(K,H_f)$, and $-I=z(\Ch\rho)\ne z(\Ch\rho_K)=I$.
On the other hand suppose $G(\mathbb R)=PSL(2,\mathbb R)=SO(2,1)$.
Then $K=O(2,\mathbb C)$, so $-1\in W(K,H_f)$, and now
$I=z(\Ch\rho)=z(\Ch\rho_K)$.
\end{example}
\begin{proof}
We may assume $G(\mathbb R)$ is simple.
First assume $G(\mathbb R)$ is equal rank.
Recall (see the discussion in Section \ref{s:split}) $K=\text{Cent}_G(x)$
for some $x\in H_f$. We will show $x$ is of a particular form.
We need a short digression on the
Kac classification of real forms. For details see
\cite{ov},\cite{helgason_book}.
Let $\widetilde D$ be the
extended Dynkin diagram for $G$, with nodes ${0,\dots,
m}$; roots $\alpha_0,\dots, \alpha_m$ ($-\alpha_0$ is the highest
root); and labels $n_0=1,n_1,\dots, n_m$ (the multiplicity of the
root in the highest root).
The Dynkin diagram of $K$ is
obtained from $\widetilde D$ by deleting
node $j$ with label $2$, or nodes ${j,k}$ with label $1$.
In the second case, without loss of generality, we may assume $k=0$,
so both cases may be combined, as specifying
a single node $j$ with label $n_j=1$ or
$2$.
Let $\Ch\lambda_j$ be the $j^{th}$ fundamental weight for $G$.
Then we can take
$x=\exp(\pi i\Ch\lambda_j)$.
\begin{subequations}
\renewcommand{\theequation}{\theparentequation)(\alph{equation}}
Now set $N=\sum_{i=0}^m n_i$ and let
\begin{equation}
c=
\begin{cases}
\frac N2&n_j=2\\
N-1&n_j=1.
\end{cases}
\end{equation}
Except in type $A_{2n}$, which is ruled out since $-1\in W(G,H_f)$, $N$ is even, so $c\in \mathbb Z$.
It is an exercise in root systems to see that
\begin{equation}
\Ch\rho_G-\Ch\rho_K=c\Ch\lambda_j.
\end{equation}
(For $i\ne 0,j$, both sides are $0$ when paired with
$\alpha_i$, so this amounts to computing the pairing with
$\alpha_0$ and $\alpha_j$.)
Therefore
\begin{equation}
x=\exp(\frac{\pi i}c(\Ch\rho_G-\Ch\rho_K)).
\end{equation}
\end{subequations}
Then $x^{2c}=z(\Ch\rho_G)/z(\Ch\rho_K)$.
By \eqref{e:practical} we have:
\begin{equation}
-1\in W(K,H_f)\Leftrightarrow x^2=1\Rightarrow x^{2c}=1
\Rightarrow z(\Ch\rho_G)=z(\Ch\rho_K).
\end{equation}
A similar, but more elaborate, argument holds in the unequal rank
case.
Instead, we proceed in a more case-by-case fashion.
If $G(\mathbb R)$ is complex, then $K$ is connected, and we have already
treated this case (see the proof of Lemma \ref{l:schurconnected}).
Since $-1\in W(K,H_f)$ every representation of $G(\mathbb R)$ is self-dual.
Consulting the list at the end of the previous section, this leaves
only type $D_{2n}$.
If $G$ is simply connected then by a case-by-case check (assuming unequal rank),
$-1\in W(K,H_f)$, and $K$ is connected, so again we have
$z(\Ch\rho_G)=z(\Ch\rho_K)$.
The result is then true {\it a fortiori} if $G$ is not simply
connected. This completes the proof.
\end{proof}
We also need a generalization of Lemma \ref{l:schurfinite}.
\begin{lemma}
\label{l:schurfinite2}
Assume $G$ is a connected, reductive complex group.
Let $G^\dagger=G\rtimes\langle\delta\rangle$ where $\delta^2\in Z(G)$ and $\delta$
acts on $G$ by a Chevalley involution.
Every irreducible finite dimensional representation $\pi^\dagger$ of
$G^\dagger$ is self-dual, and
if $\pi$ is an irreducible constituent of $\pi^\dagger|_G$, then
\begin{equation}
\epsilon(\pi^\dagger)=
\begin{cases}
\epsilon(\pi)&\pi\simeq\pi^*\\
\chi_\pi(\delta^2)&\pi\not\simeq\pi^*.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
The restriction of $\pi^\dagger$ is irreducible if and only if
$\pi\simeq\pi^\delta$.
Since $\delta$ acts by the Chevalley involution, this is equivalent to
$\pi\simeq\pi^*$.
If $\pi\simeq\pi^*$ the result is clear. Otherwise, let $\lambda$ be
the highest weight of $\pi$. Then $\pi^\delta$ has extremal weight
$-\lambda$, i.e. highest weight $-w_0\lambda$ where $w_0$ is the long
element of the Weyl group. Since $\pi\not\simeq\pi^*$,
$-w_0\lambda\ne\lambda$, so the $\lambda$-weight space of
$\pi^\dagger$ is one-dimensional. The proof of Lemma
\ref{l:schurfinite} now carries through using $\delta$, which interchanges
the $\lambda$ and $-\lambda$ weight spaces of $\pi^\dagger$.
\end{proof}
We need to consider finite dimensional representations of the possibly
disconnected group $K=G^\theta$. These groups are not badly
disconnected, for example the component group is an elementary abelian two-group
(this follows from \cite[Proposition 4.42(a)]{knappvogan},
and the fact that it is true for real tori),
and we
need the following property of their representations.
\begin{lemma}
\label{l:multfree}
Let $\mu$ be an irreducible, finite-dimensional, representation of $K$. Then the restriction of $\mu$ to
$K_0$ is multiplicity free.
\end{lemma}
\begin{proof}
Suppose $\mu_0$ is an irreducible summand of $\mu|_{K_0}$,
and let $K_1=\text{Stab}_{K}(\mu_0)$.
It is enough to show that $\mu_0$ extends to an irreducible
representation $\mu_1$ of $K_1$. For then, by Mackey theory,
$\mathrm{Ind}_{K_1}^K(\mu_1)$
is irreducible, so isomorphic to $\mu$, and restricts to
the sum of the distinct irreducible representations
$\{\pi_0^x\mid x\in S\}$, where $S$ is a set of representatives of $K/K_1$.
Choose Cartan and Borel subgroups $T\subset B_{K_0}$ of $K_0$.
(We can arrange that $B_{K_0}=B\cap K_0$ and $T=H\cap K_0$).
\begin{lemma}
We can choose elements $x_1,\dots, x_n\in K$ such that:
\begin{enumerate}[(1)]
\item $K=\langle K_0,x_1,\dots, x_n\rangle$;
\item $x_i$ normalizes $B_{K_0}$ and $T$;
\item The $x_i$ commute with each other.
\end{enumerate}
\end{lemma}
\begin{remark}
By a standard argument it is easy to arrange (1) and (2), the main
point is (3).
Alternatively, it is well known that we could instead choose the $x_i$ to satisfy (1),
(3) and that
each $x_i$ has order $2$.
It would be interesting to prove that one can satisfy all four
conditions simultaneously, and perhaps even that conjugation by $x_i$
is a distinguished involution of $K_0$.
\end{remark}
\begin{proof}
Choose $x\in K\backslash K_0$. Then conjugation by $x$ takes
$B_{K_0}$ to another Borel subgroup, which we may conjugate back to
$B_{K_0}$. So after replacing $x$ with another element in the same
coset of $K_0$ we may assume $x$ normalizes $B_{K_0}$.
Conjugating again by an element of $B_{K_0}$ we may assume $x$
normalizes $T$. By induction
this gives (1) and (2).
For (3), it is straightforward to reduce to the case when $G(\mathbb R)$ is
simple. Then a case-by-case check shows that $|K/K_0|\le 2$ except in
type $D_n$. Furthermore the only exception is the adjoint group
$PSO(2n,2n)$, in which case the result can be easily checked.
This is essentially \cite[Proposition 9.7]{ic4}.
\end{proof}
Let $\lambda\in X^*(T)$
be the highest weight of $\mu_0$ with respect to $B_{K_0}$.
Then $\mu_0^{x_i}$ has highest weight $x_i\lambda$.
So, after renumbering, we may write $K_1=\langle K_0,x_1,\dots,
x_r\rangle$ where $x_i\lambda=\lambda$ for $1\le i\le r$.
Let $V_\lambda$ be the (one-dimensional) highest weight space of
$\mu_0$. The group $T_1=\langle T,x_1,\dots, x_r\rangle$ acts on
$V_\lambda$.
In the terminology of
\cite[Definition 1.14(e)]{voganOrange}, $T_1$ is a large Cartan
subgroup of $K_1$, and
\cite[Theorem 1.17]{voganOrange} implies that there is an
irreducible representation $\mu_1$ of $K_1$,
containing the one-dimensional representation $V_\lambda$ of $T_1$.
Then $\mu_1|_{K_0}=\mu_0$.
\end{proof}
\begin{theorem}
\label{t:schur}
Suppose every irreducible representation of $G(\mathbb R)$ is self-dual
(see Corollary \ref{c:everyalt}).
If $\pi$ is an irreducible representation then
\begin{equation}
\epsilon(\pi)=\chi_\pi(z(\Ch\rho)).
\end{equation}
Every irreducible representation is orthogonal if
and only if
$z(\Ch\rho)=1$. This holds if $G$ is adjoint.
\end{theorem}
\begin{proof}
By Corollary \ref{c:every} every $K$-type is self-dual,
and $-1\in W(K,H_f)$.
Choose a minimal $K$-type $\mu$. Since $\mu$ is self-dual and has
multiplicity one, $\epsilon(\pi)=\epsilon(\mu)$.
Let $\mu_0$ be an irreducible summand of $\mu|_{K_0}$.
By Lemma \ref{l:multfree} $\mu_0$ has multiplicity one.
If $\mu_0$ is self-dual then $\epsilon(\mu)=\epsilon(\mu_0)$, and
by Lemma \ref{l:schurfinite}
$\epsilon(\mu_0)=\chi_{\mu_0}(z(\Ch\rho_K))$.
By Lemma \ref{l:zrhoK} this equals $\chi_{\mu_0}(z(\Ch\rho))$.
Suppose $\mu_0$ is not self-dual.
Since $-1\in W(K,H_f)$, choose a representative $g\in \text{Norm}_K(H_f)$ of
$-1\in W(K,H_f)$, and let
$K^\dagger=\langle K,g\rangle$. By Lemma \ref{l:schurfinite2}
$\mu^\dagger=\mathrm{Ind}_{K_0}^{K^\dagger}(\mu_0)$ is irreducible, self-dual,
and of multiplicity one in $\mu$, so
$\epsilon(\mu)=\epsilon(\mu^\dagger)$.
Since $\mu_0\not\simeq\mu_0^*$, by Lemma \ref{l:schurfinite},
$\epsilon(\mu^\dagger)=\chi_{\mu_0}(g^2)$.
We can also think of $g$ as a representative of $-1\in W(G,H_f)$.
Since $G$ (unlike $K$) is (necessarily) connected,
by Lemma \ref{l:w0}, $g^2=z(\Ch\rho_G)$, so again
$\epsilon(\mu)=\chi_{\mu_0}(z(\Ch\rho_G))$.
As in the proof of Lemma \ref{l:schurconnected}, since $z(\Ch\rho_G)\in Z(G(\mathbb R))$,
$\chi_{\mu_0}(z(\Ch\rho_G))=\chi_\pi(z(\Ch\rho_G))$. This completes the proof.
\end{proof}
\bibliographystyle{plain}
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|
{
"timestamp": "2014-02-14T02:06:37",
"yymm": "1203",
"arxiv_id": "1203.1901",
"language": "en",
"url": "https://arxiv.org/abs/1203.1901"
}
|
\section{Introduction}
Bolometers are detectors in which the energy from particle interactions
is converted to thermal energy and measured via their rise in temperature. They
provide excellent energy resolution, though their response is slow
compared to conventional detectors. These features make them a suitable
choice for experiments searching for rare processes, such as neutrinoless
double beta decay (0$\nu$DBD) and dark matter (DM) interactions.
The CUORE\ experiment will search for 0$\nu$DBD\ of $^{130}\mathrm{Te}$~\cite{Ardito:2005ar,
ACryo} using an array of 988 $\mathrm{TeO}_2$\ bolometers of 750\un{g} each. Operated
at a temperature of about 10\un{mK}, these detectors feature an
energy resolution of a few keV over their energy range, extending
from a few keV up to several MeV. At low energies, below 100 keV,
the resolution is of the order of 1 keV FWHM, while at 2528\un{keV}, the
0$\nu$DBD\ energy, is about 5\un{keV\,FWHM}; this,
together with the low background and the high mass of the experiment,
determines the sensitivity to the 0$\nu$DBD. CUORE\ could also search for
DM interactions, provided that the energy threshold is of
few keV~\cite{Didomizio:2010ph}. An experiment made of 62 bolometers,
CUORICINO~\cite{Andreotti:2010vj}, was operated at Laboratori Nazionali
del Gran Sasso (LNGS) in Italy between 2003 and 2008, and proved the
feasibility of the $\mathrm{TeO}_2$\ bolometric technique. CUORE\ is currently under
construction, and will start taking data in three years.
Vibrations of the cryogenic apparatus (keeping the system at
low temperatures) induce a visible noise, which limits the
energy resolution at low energies~\cite{Bellini:2010iw} and the energy
threshold~\cite{Didomizio:2010ph}. Since all bolometers are held by the same
structure, part of the vibrational noise is expected to be correlated.
In this paper we present a method to estimate the correlated
noise among different bolometers, and a method to remove it.
The application to data from CUORICINO\ shows that the
correlated noise is visible and that it can be efficiently removed.
The CUORE\ detector structure will be different from the CUORICINO\ one,
and thus a different vibrational noise is expected. If it will be unfortunately large,
the algorithms developed in this work could be used to improve the performances of CUORE\ in both 0$\nu$DBD\ and DM searches.
\section{Experimental setup}\label{sec:experimental setup}
CUORICINO\ and CUORE\ bolometers are composed of two main parts,
a $\mathrm{TeO}_2$\ crystal and a neutron transmutation doped Germanium (NTD-Ge)
thermistor ~\cite{wang,Itoh}. The crystal is cube-shaped (5x5x5~cm$^{3}$)
and held by Teflon supports in copper frames. The frames are connected
to the mixing chamber of a dilution refrigerator, which keeps the
system at a temperature of about $10\un{mK}$. The thermistor is glued
to the crystal and acts as thermometer. When energy is released in
the crystal, its temperature increases and changes the resistance of the thermistor.
To read out the signal,
the thermistor is biased with constant current, which is provided by a
voltage generator and a load resistor in series with the thermistor.
The resistance of the thermistor varies in time with the temperature,
and the voltage across it is the bolometer signal. The value of the
load resistor is chosen to be much higher than the thermistor, so that
the voltage across the thermistor is proportional to its resistance.
The wires that connect the thermistor to the electronics introduce a
non-negligible parasitic capacitance.
The typical response of CUORICINO\ and CUORE\ bolometers to particles
impinging on the crystal is of order $100\un{\mu V/MeV}$. The signal
frequency bandwidth is $0-10\un{Hz}$, while the noise components
extend to higher frequencies. The signal is amplified, filtered with
a 6-pole active Bessel filter with a cut-off frequency of 12\un{Hz}
and then acquired with an 18-bit ADC with a sampling frequency of
125\un{Hz}. The gain of the amplifiers ranges from 500 to 10000\un{V/V},
and is tuned for each bolometer to fit the signals in the ADC range,
which is $\pm10.5\un{V}$.
A typical signal recorded by the ADC, produced
by a 1461\un{keV} $\gamma$ particle fully absorbed in a CUORICINO\
bolometer, is shown in the left panel of Fig.~\ref{fig:pulse_noise}.
\begin{figure}[htbp]
\centering
\begin{minipage}{0.48\textwidth}
\includegraphics[clip=true,width=1\textwidth]{pulse_40K_ch13.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[clip=true,width=1\textwidth]{noise_ch13.pdf}
\end{minipage}
\caption{Response of a CUORICINO\ bolometer (channel 13) . Left panel: signal generated by a 1461\un{keV} $\gamma$ particle recorded by the
ADC. Right panel: noise
power spectrum expressed in $\un{mV^2/Hz}$, and energy spectrum of the signal scaled to fit in the noise range.} \label{fig:pulse_noise}
\end{figure}
The 62 bolometers of CUORICINO\ were arranged in a tower of 13 floors:
11 floors were composed by 4 5x5x5~cm$^{3}$ bolometers each, 2 floors
were composed by 9 small bolometers (3x3x6~cm$^3$) each. The small
bolometers were recycled from previous arrays, and will not be used
in CUORE, which will only use 5x5x5~cm$^{3}$ crystals arranged in
19 towers of 13 floors each (Fig.~\ref{fig:cuoricino_and_cuore}).
The front-end electronics, which provide the bias, the load resistors
and the amplifier, are placed outside the cryostat, at room temperature~\cite{AProgFE}.
In CUORICINO, 1/3 of the electronics channels used load resistors and pre-amplifiers
located inside the cryostat, at 110\un{K}. This was done to reduce the
noise from load resistors, but since no improvement was achieved, the cold
electronics will not be used in CUORE.
Three channels (1 cold, 2 warm) were lost after the initial CUORICINO\ cool-down, reducing the number of active channels
to 59.
\begin{figure}[tbp]
\centering
\begin{minipage}{0.22\textwidth}
\includegraphics[clip=true,width=1\textwidth]{cuoricino.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.6\textwidth}
\includegraphics[clip=true,width=1\textwidth]{cuore.pdf}
\end{minipage}
\caption{Arrays of $\mathrm{TeO}_2$\ bolometers. The CUORICINO\ tower (left), composed
by 62 bolometers, represented inside a schema of the cryostat; a single CUORE\ tower (middle)
and the entire CUORE\ array (right).} \label{fig:cuoricino_and_cuore}
\end{figure}
Vibrations of the detector structure generate two types of noise: thermal
and microphonic. The thermal noise is due to vibrations of the crystals
that induce temperature fluctuations of the crystals themselves. The
microphonic noise is due to vibrations of the wires that connect the thermistor
to the cryostat socket. A typical noise power spectrum is
shown in the right panel of Fig.~\ref{fig:pulse_noise}; the peaks in the
figure are attributed to microphonism, while the continuum is attributed
to crystal vibrations.
The data acquision system connected to the ADC boards implements two kind of software
triggers. The first trigger fires when a
signal is detected on a bolometer, the second fires randomly in time
to acquire noise waveforms. To estimate the noise correlation, the
acquisition was programmed to acquire simultaneously all the bolometers of
CUORICINO\ when the signal or the noise triggers fired on a bolometer.
The length of the acquisition window was set to $M=512$ samples on all channels,
corresponding to 4.096\un{s}.
The data used in this paper amount to 2 days taken from the last month of CUORICINO\ operation.
\section{Noise correlation}
The noise power spectrum is estimated from a large number of waveforms
acquired with the random trigger, removing those
that, by chance, contain signals. The power spectrum $N_i(\omega)$
of each bolometer channel $i$ is computed as:
\begin{equation}
N_i(\omega) = <n_i(\omega) \cdot n_i^*(\omega)>\,,
\label{eq:nps}
\end{equation}
where $<>$ denotes the average taken
over a large number of waveforms and $n(\omega)$ is the discrete Fourier transform
(DFT) of a noise waveform $n_i(t)$. To avoid DFT artifacts, each waveform
is weighted before the DFT using a Welch windowing function:
\begin{equation}
n_i(t) \rightarrow n_i(t) \cdot w(t)\;,\quad
w(t) = \frac{1}{G}\left\{ \frac{M}{2} - \left[\left(t-\frac{M}{2}\right)\right]^2\right\}\;.
\end{equation}
In the above expression the parameter $G$ is a normalization coefficient
chosen to satisfy the condition~\cite{Novotny2008236}:
\begin{equation}
\sum_{\omega} N_i(\omega) = \sigma_i^2
\end{equation}
where $\sigma_i$ is the standard deviation of the noise in the time domain.
The covariance between the noises on bolometers channels $i$ and $j$ is estimated
as:
\begin{equation}
N_{ij}(\omega) = <n_i(\omega) \cdot n_j^*(\omega)>
\label{eq:cov}
\end{equation}
where the definitions are the same of Eq.~\ref{eq:nps}. The magnitude
of $N_{ij}(\omega)$ is the covariance as usually intended for real variables,
while the phase accounts for any possible time delay of the frequency $\omega$
observed in the two bolometers. The correlation is also a complex
quantity, and is defined as:
\begin{equation}
\rho_{ij}(\omega) = \frac{{\rm N}_{ij}(\omega)}{\sqrt{{N}_i(\omega) {N}_j(\omega)}}\quad.
\end{equation}
As an example, we show the behaviour of $\rho_{13,1}(\omega)$ in Fig.~\ref{fig:correlation_131}; the microfonic peaks and the low frequencies are highly correlated,
while the intermediate frequencies are almost uncorrelated.
\begin{figure}
\centering
\includegraphics[clip=true,width=0.8\textwidth]{correlation_13_1.pdf}
\caption{Power spectra of CUORICINO\ channels 13 and 1 and their correlation $\rho(\omega)$.}
\label{fig:correlation_131}
\end{figure}
The pattern in the figure is visible for many $(i,j)$ pairs and indicates that the source of correlated noise has analogous origin in all bolometers.
The average correlation over all frequencies can be computed as:
\begin{equation}
\overline{\rho_{ij}} = \sqrt{
\frac{\sum_\omega |\rho_{ij}(\omega)|^2 N_i(\omega)}
{\sum_\omega N_i(\omega)}
}\;,
\label{eq:avgcorr}
\end{equation}
and is found to be around 60\% for the $(i,j)$ pair $(13,1)$. The average correlation matrix, including the correlation of all the CUORICINO\ channels,
is shown in Fig.~\ref{fig:avgcorr_space}. The matrix is not exactly triangular because Eq.~\ref{eq:avgcorr} is not symmetric under the exchange of channels $i$ and $j$, since the same frequency
may contribute with a different weight to $N_i(\omega)$ and $N_j(\omega)$.
The figure shows that the correlation is high, even tough there is no evident link with the spatial disposition of the bolometers.
Bolometers close to each other, in fact, are not necessarily the most correlated.
We also tried to order the channels by their cabling group in the cryostat or by cold/warm electronics channels, but also in this case we did not see any link.
Other variables linked to the noise correlation could be, for example, the path of the wires along the tower, the tightness of the crystals and the length of thermistor wires. These variables are not accessible now since CUORICINO\ has been disassembled. Moreover, while the understanding
of the noise sources is important, in this paper we focus on the analysis algorithms to reduce the observed noise.
\begin{figure}
\centering
\includegraphics[clip=true,width=1\textwidth]{matrix_0-60.pdf}
\caption{Noise correlation in CUORICINO. Channels are ordered as bolometers are ordered in the array. White lines indicate dead channels.}
\label{fig:avgcorr_space}
\end{figure}
\section{Noise decorrelation}
In $\mathrm{TeO}_2$\ bolometers the noise is purely additive to the signal, meaning
that the waveform $f(t)$ observed in presence of a
signal of amplitude $A$ can be expressed as:
\begin{equation}
f(t) = A\cdot s(t) + n(t)\;,
\label{eq:signal+noise}
\end{equation}
where $s(t)$ is the signal shape.
Since the noise on different bolometers is partially
correlated, it is possible to remove part of $n(t)$ from $f(t)$
using the waveforms from other bolometers in which only noise is present.
In the following we develop a method to remove the correlated noise.
Assuming that each frequency component of $n_i(\omega)$ is normally
distributed, the multidimensional probability distribution of the noises
on all bolometers $\vec{n}(\omega) = \{n_1(\omega),n_2(\omega),\ldots\}$ can be written as:
\begin{equation}
P[\vec{n}(\omega)] \propto \exp \left( -\frac{1}{2} \vec{n}^{\dag}(\omega) \hat{N}^{-1}(\omega) \vec{n}(\omega) \right)
\label{eq:gaussianamultidim}
\end{equation}
where $\hat{N}$ is the covariance matrix whose elements are defined
in Eq.~\ref{eq:cov}. The probability distribution of the noise on
bolometer $i$, $n_i(\omega)$, can be obtained integrating out the noise of other bolometers:
\begin{equation}
P[n_i(\omega)]\propto \exp \left[-\frac{N^{-1}_{ii}(\omega)}{2} \left(n_{i}(\omega) + \sum_{j \neq i} \frac{N_{ij}^{-1}(\omega)}{N_{ii}^{-1}(\omega)} n_{j}(\omega) \right)^{2} \right] \;.
\end{equation}
The above equation represents a Gaussian with mean
$\left[-\sum_{j \neq i} \frac{N_{ij}^{-1}(\omega)}{N_{ii}^{-1}(\omega)} n_{j}(\omega)\right]$
and variance
$1/{N_{ii}^{-1}(\omega)}$.
The decorrelated value of $n_{i}(\omega)$ can be
obtained for each waveform as:
\begin{equation}
n^d_{i}(\omega)=n_{i}(\omega)+\sum_{j \neq i} \frac{N_{ij}^{-1}(\omega)}{N_{ii}^{-1}(\omega)} n_{j}(\omega) \;,
\label{eq:noise_decorrelated}
\end{equation}
and its power spectrum is expected to be:
\begin{equation}
N^d_i(\omega) = {\frac{1}{N_{ii}^{-1}(\omega)}}\;.
\label{eq:nps_decorrelated}
\end{equation}
A generic waveform $f_i(t)$ on bolometer $i$, containing noise or noise+signal,
can be decorrelated using the noise from all other bolometers as:
\begin{equation}
f^d_{i} (\omega)= f_{i}(\omega) +\sum_{j \neq i} \frac{N_{ij}^{-1}(\omega)}{N_{ii}^{-1}(\omega)} n_{j}(\omega)
= A\, s_i(\omega) + n^d_i(\omega)\;.
\label{eq:signal+noise_decorrelated}
\end{equation}
Summarizing, once the covariance matrix in Eq.~\ref{eq:cov} is estimated from the data, the
waveforms on each bolometer are decorrelated using Eq.~\ref{eq:signal+noise_decorrelated}
and the effect on the noise power spectrum can be predicted with Eq.~\ref{eq:nps_decorrelated}.
\section{Application to data}
The acquired data are split in two sets, one set is used to estimate the correlation matrix and the expected
noise power spectra, the other is used to verify that the application of the decorrelation algorithm to waveforms
produces results consistent with the expectations.
Figure \ref{fig:potenzaattesatuttoilcucuzzaro} (left) shows the original
power spectrum of channel 13 (Eq.~\ref{eq:nps})
and the one expected decorrelating from
all the other CUORICINO\ bolometers (Eq.~\ref{eq:nps_decorrelated}). The power of low frequencies is reduced
and the microphonic peaks are completely removed.
In principle one could be satisfied and proceed to apply
Eq.~\ref{eq:signal+noise_decorrelated} to the single waveforms.
However, using all the bolometers to decorrelate every triggered waveform
is expensive from the computational point of view, since it requires
DFTs for each bolometer used. Moreover all the waveforms used to
decorrelate should not contain signals. This requirement is often
fulfilled in CUORICINO, where the overall counting rate was around 0.1\un{Hz}, but not in CUORE, where
the rate is expected to be larger. For this
reason we tested to see if the decorrelation is effective using a smaller number
of bolometers. For each bolometer we selected the most correlated
bolometers and we computed the expected decorrelated power spectrum. The results obtained are equivalent to those
obtained using all bolometers (Fig.~\ref{fig:potenzaattesatuttoilcucuzzaro} left). The number of bolometers used to
decorrelate was set to 11, a number sufficiently high to maximize the decorrelation and sufficiently low to
ensure good performances.
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip=true,width=0.48\textwidth]{noise_ch13_expdec.pdf}
\includegraphics[clip=true,width=0.48\textwidth]{noise_ch13_dec.pdf}
\caption{Left: Noise power spectra of bolometer 13: original (solid black),
expected decorrelated using 61 bolometers (dashed black), and expected
decorrelated using the 11 most correlated bolometers (solid red). Right: original noise
power spectrum and power spectrum of waveforms decorrelated using Eq.~\protect\ref{eq:signal+noise_decorrelated}.
The original power spectra in the figures differs slightly because different sets of data has been used
to estimate the covariance matrix (left) and the effect of the decorrelation (right).}
\label{fig:potenzaattesatuttoilcucuzzaro}
\end{center}
\end{figure}
The noise power spectrum obtained on waveforms decorrelated with
Eq.~\ref{eq:signal+noise_decorrelated} is shown in
Fig.~\ref{fig:potenzaattesatuttoilcucuzzaro} (right).
The decorrelation is found to be very effective on channel 13: the noise $\sigma$ is reduced from 2.0\un{mV} to 1.2\un{mV}
(corresponding to 3.6 and 2.2~keV, respectively),
an effect which is visible also in the time domain (Fig.~\ref{fig:decowave}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip=true,width=0.48\textwidth]{noisewave_ch13_deco.pdf}
\hfill
\includegraphics[clip=true,width=0.48\textwidth]{pulse_44keV_ch13_deco.pdf}
\caption{Left: original (solid black) and decorrelated (dashed red) noise waveform from channel 13. Right: signal generated by a 43 keV $\gamma$ particle.
The decorrelation algorithm reduces the noise leaving the signal unmodified.} \label{fig:decowave}
\end{center}
\end{figure}
We show in Fig.~\ref{fig:rms} the energy resolution on all
bolometers before and after the decorrelation: in a few bolometers the resolution remains the same, while in others
it is reduced up to 50\%.
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip=true,width=0.95\textwidth]{rms.pdf}
\caption{Noise resolution of all CUORICINO\ bolometers before
and after the decorrelation. Values are expressed in keV.} \label{fig:rms}
\end{center}
\end{figure}
\section{Combination with the optimum filter}
In the data analysis of $\mathrm{TeO}_2$\ bolometers the signal amplitude is estimated
using the optimum filter~\cite{Gatti:1986cw,Radeka:1966}, a filter that
significantly improves the energy resolution. This filter can be used when the shape
of the signal and the noise power spectrum of each channel are known.
The filter acts on a single channel and its transfer function is
\begin{equation}
H(\omega) = h \frac{s^*(\omega)}{N(\omega)}e^{-\jmath\, \omega i_M }\,,
\label{eq:of}
\end{equation}
where $i_M$ is a parameter to adjust the delay of the
filter. In this work, as in Ref.~\cite{Didomizio:2010ph}, we chose it equal
to the maximum position of $s_i$, so that the maximum of the filtered
signal is aligned to the maximum of the non-filtered one. $h$ is a normalization
constant that leaves unmodified the amplitude of the signal:
$h = \left[\sum_\omega \frac{|s(\omega)|^2}{N(\omega)}\right]^{-1}$~.
To combine the optimum filter with the decorrelation, it is sufficient to
replace in Eq.~\ref{eq:of} the original power spectrum $N(\omega)$ with
$N^d(\omega)$ defined in Eq.~\ref{eq:nps_decorrelated}, and then process
the decorrelated waveforms in Eq.~\ref{eq:signal+noise_decorrelated}. The
comparison of the filtered noise power spectrum with and without the
decorrelation for two bolometers with opposite behavior (channels 13 and 38)
is shown in Fig.~\ref{fig:of_filtered}. In channel 13, while the
decorrelated noise is lower than the original one, there is no big improvement
in the filtered-decorrelated noise, because the correlation is high only on frequencies
with low signal to noise ratio. In channel 38, on the other hand,
the decorrelation lowers the noise after the optimum filter.
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip=true,width=0.48\textwidth]{noise_ch13_decof.pdf}
\includegraphics[clip=true,width=0.48\textwidth]{noise_ch38_decof.pdf}
\caption{Comparison of original, decorrelated, optimum filtered and decorrelated/optimum filtered noises
on bolometers 13 (left) and 38 (right).}
\label{fig:of_filtered}
\end{center}
\end{figure}
To summarize we compare the expected resolution of the entire array
using both decorrelation and optimum filter with respect to optimum
filter only (Fig.~\ref{fig:rmsof}). The improvement is not significant
as when we decorrelated the original waveforms (Fig.
~\ref{fig:rms}), because in most cases the noise is not correlated in the signal frequency bandwidth.
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip=true,width=0.95\textwidth]{rmsof.pdf}
\caption{Noise resolution of all CUORICINO\
bolometers after the optimum filter before and after the decorrelation.}
\label{fig:rmsof}
\end{center}
\end{figure}
\section{Conclusions}
In this paper we developed a method to remove the correlated noise
between different detectors. The application to CUORICINO\ showed that the correlated noise,
generated by vibrations of the detector structure, can be efficiently removed. However, when the
decorrelation is combined with the optimum filter, the resolution does
not improve significantly since the correlated noise lies at frequencies higher than the
signal frequency bandwidth. CUORE, a $\mathrm{TeO}_2$\ bolometric array 20 times
larger than CUORICINO, will have a new detector structure that could
induce different vibrational noise. Moreover, while CUORICINO\ was refrigerated using
liquid helium, the CUORE\ cryostat will use pulse tubes, acoustic cryocoolers oscillating at
low frequencies (around 2 Hz). If the noise will be
unfortunately large in the signal band, the method we introduced will be a
valid tool to improve the CUORE\ performances.
\acknowledgments
We are very grateful to the members of the CUORICINO\ and CUORE\ collaborations, in particular to
F.~Bellini, L.~Cardani, T.~Gutierrez, K.~Han and G.~Pessina for their precious suggestions.
Special thanks go to Robert Joachim, for the preliminary studies he did during his summer student fellowship at INFN Roma.
\bibliographystyle{JHEP}
|
{
"timestamp": "2012-03-09T02:02:46",
"yymm": "1203",
"arxiv_id": "1203.1782",
"language": "en",
"url": "https://arxiv.org/abs/1203.1782"
}
|
\section{Introduction}
\par
In this paper, we restrict our attention to the edge ideals of graphs.
For a graph $G=(V(G),E(G))$, the edge ideal, denoted by $I(G)$, is defined by
\[
I(G) = (x_ix_j \; :\; \{x_i,x_j\} \in E(G))S,
\]
where $S = K[v : v \in V(G)] = K[x_1, \ldots, x_n]$ is
a polynomial ring over a field $K$.
Then $E(G)$ is a squarefree monomial ideal which is generated
by degree $2$ elements, and thus it can be regarded as a Stanley--Reisner ideal
and it is a radical ideal.
Then the following theorem is well-known.
\begin{thm-q}[See \cite{AV,CN, Wa}] \label{CowsikNori}
Let $S$ be a regular local ring $($resp., a polynomial ring over a field $K$$)$,
and let $I$ be a radical ideal $($resp., a homogeneous radical ideal$)$ of $S$.
Then $I$ is complete intersection if and only if $S/I^{\ell}$ is
Cohen--Macaulay for infinitely many $\ell \ge 1$.
\par
In particular, for any edge ideal $I(G)$ of a graph $G$,
$I(G)$ is a complete intersection ideal if and only if
$S/I(G)^{\ell}$ is
Cohen-Macaulay for infinitely many $\ell \ge 1$.
\end{thm-q}
\par
In what follows, let $G=(V(G),E(G))$ be a graph, and
$I(G) \subseteq S=K[v \,:\, v\in V(G)]$ the edge ideal of $G$.
\par
Recently, in \cite{TY}, the last two authors gave a generalization of
the theorem using a classification theorem for locally complete intersection
Stanley--Reinser ideals; see \cite[Theorem 1.15]{TY}.
Note that the following theorem is also true for Stanley--Reisner ideals.
\begin{thm-q}[\textrm{See \cite[Theorem 2.1]{TY}}] \label{BbmTY}
If $S/I(G)^{\ell}$ is Buchsbaum for infinitely many
$\ell \ge 1$, then $I(G)$ is complete intersection.
\end{thm-q}
\par
Moreover, the authors \cite{CRTY} gave a refinement of
the above theorem jointly with M. Crupi.
\begin{thm-q}[\textrm{See \cite[Theorem 2.1]{CRTY}}] \label{CRTY}
$I(G)$ is complete intersection if and only if
$S/I(G)^{\ell}$ is Cohen--Macaulay for some $\ell \ge \height I$.
\end{thm-q}
\par
The main purpose of this paper is to give another variation of the theorem
in this context.
Namely, we consider the following questions:
\begin{quests-q} \label{Q-CM}
Let $\ell \ge 1$ be an integer.
Let $I(G)^{(\ell)}$ denote the $\ell$th symbolic power ideal of $I(G)$.
Then$:$
\begin{enumerate}
\item When is $S/I(G)^{(\ell)}$ Cohen--Macaulay?
\item Is $I(G)$ complete intersection if $S/I(G)^{\ell}$ Cohen--Macaulay
for a fixed $\ell \ge 1$?
\end{enumerate}
\end{quests-q}
\par
The answers to these questions will give a generalization
of the original theorem described as above. For instance,
for each fixed $\ell \ge 1$,
the Cohen--Macaulayness of $S/I(G)^{\ell}$ implies
that of $S/I(G)^{(\ell)}$.
Note that the converse is \textit{not} true in general.
\par
We first consider the above question.
Let $G$ be a graph on the vertex set $V=[n]$ such that
$\dim S/I(G)=1$.
Such a graph $G$ is isomorphic to the complete graph $K_n$.
Then $S/I(G)^{(\ell)}$ is Cohen--Macaulay for every integer
$\ell \ge 1$ because the symbolic power ideal
has no embedded primes.
\par
The following theorem
characterizes graphs $G$
for which all symbolic powers $S/I(G)^{(\ell)}$ are Cohen--Macaulay
(or for $\ell \ge 3$).
\begin{thm-q}[See Theorem \ref{Main-CM}] \label{Main1}
The following conditions are equivalent$:$
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ is Cohen--Macaulay for every integer $\ell \ge 1$.
\item $S/I(G)^{(\ell)}$ is Cohen--Macaulay for some $\ell \ge 3$.
\item $S/I(G)^{(\ell)}$ satisfies Serre's condition $(S_2)$ for some $\ell \ge 3$.
\item $G$ is a disjoint union of finitely many complete graphs.
\end{enumerate}
\end{thm-q}
\par \vspace{2mm}
As an application of the theorem, we can obtain some result
for Cohen--Macaulayness of ordinary powers,
which gives an improvement of the main theorem
in \cite{CRTY}.
\begin{cor-q}[See Theorem \ref{Power-cor}] \label{Power-cor-q}
If $S/I(G)^{\ell}$ is Cohen--Macaulay
for some $\ell \ge 3$,
then $I(G)$ is complete intersection.
\end{cor-q}
\par \vspace{2mm}
Next, we consider the following question.
We need to assume that $I(G)$ is unmixed.
Then if $\dim S/I(G) \le 2$,
for every integer $\ell \ge 1$,
$S/I(G)^{(\ell)}$ is unmixed, and thus it has (FLC).
\begin{quest-q} \label{Q-FLC}
Let $\ell \ge 1$ be an integer.
When does $S/I(G)^{(\ell)}$ have $($FLC$)$?
\end{quest-q}
\par
Let $\Delta=\Delta_{n_1,\ldots,n_r}$ denote the simplicial complex
whose Stanley--Reisner ideal is equal to
the edge ideal of the disjoint union of
complete graphs $K_{n_1},\ldots,K_{n_r}$.
That is,
\[
I_{\Delta_{n_1,\ldots,n_r}} =
I(K_{n_1} \textstyle{\coprod} \cdots \textstyle{\coprod} K_{n_r}).
\]
\par
Then the following theorem gives an answer to the above question
for $\ell \ge 3$:
\begin{thm-q}[See Theorem \ref{Main-FLC}] \label{Main2}
Let $\Delta(G)$ be the simplicial complex on $V(G)$ which satisfies
$I_{\Delta(G)} = I(G)$.
Suppose that $\Delta(G)$ is pure and $d=\dim S/I(G) \ge 3$.
Let $p$ denote the number of connected components of $\Delta(G)$.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for every integer $\ell \ge 1$.
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for some $\ell \ge 3$.
\item There exist
$(n_{i1},\ldots,n_{id}) \in \bbN^d$ for every $i=1,\ldots,p$
such that
$\Delta$ can be written as
\[
\Delta = \Delta_{n_{11},\ldots,n_{1d}} \;\textstyle{\coprod}\;
\Delta_{n_{21},\ldots,n_{2d}} \;\textstyle{\coprod}
\; \ldots \;
\textstyle{\coprod} \;
\Delta_{n_{p1},\ldots,n_{pd}}.
\]
\end{enumerate}
\end{thm-q}
\par
For $\ell=2$, the problem is more complicated.
For instance, if $G$ is a pentagon, then $I(G)$ and $I(G)^2$ are Cohen--Macaulay
although $I(G)^{(\ell)}$ (and hence $I(G)^{\ell}$) is \textit{not} for any $\ell \ge 3$.
\par
After finishing this work the authors have known that N. C. Minh obtained similar results independently.
\bigskip
\section{Preliminaries}
In this section we recall some definitions and properties that we
will use later.
\subsection{Edge ideals}
Let $G$ be a graph, which means a simple finite graph without loops and
multiple edges.
Let $V(G)$ (resp., $E(G)$) denote the set of vertices (resp., edges) of $G$.
Put $V(G) =\{x_1,\ldots, x_n\}$.
Then the \textit{edge ideal} of $G$, denoted by $I(G)$,
is a squarefree monomial ideal
of $S=K[x_1,\ldots,x_n]$ defined by
\[
I(G) = (x_ix_j \,:\, \{x_i,x_j\} \in E(G)).
\]
\par
A disjoint union of two graphs $G_1$ and $G_2$, denoted by
$G_1 \coprod G_2$, is the graph $G$ which satisfies $V(G) = V(G_1) \cup V(G_2)$ and $E(G)=E(G_1) \cup E(G_2)$.
For a nonempty subset $W \subseteq V(G)$, $H=G\,|_{W}$ denotes the graph which satisfies
$V(H)=W$ and $E(H) = \{\{x,y\} \in E(G) \,:\, x,y \in W \}$.
\par \vspace{2mm}
\subsection{Stanley--Reisner ideals}
Let $V=\{x_1, \ldots, x_n\}$.
A nonempty subset $\Delta$ of the power set $2^V$
is called a \textit{simplicial complex} on $V$ if
$\{v\} \in \Delta$ for all $v \in V$,
and $F \in \Delta$, $H \subseteq F$ imply $H \in \Delta$.
An element $F \in \Delta$ is called a \textit{face} of $\Delta$.
The dimension of $\Delta$ is defined by
$\dim \Delta = \max\{\sharp(F)-1 \,:\, \text{$F$ is a face of $\Delta$}\}$.
A maximal face of $\Delta$ is called a \textit{facet} of $\Delta$.
$\mathcal{F}(\Delta)$ denote the set of all facets of $\Delta$.
The \textit{Stanley--Reisner ideal} of $\Delta$, denoted by $I_{\Delta}$,
is the squarefree monomial ideal generated by
\[
\{x_{i_1} x_{i_2} \cdots x_{i_p} \,:\, 1 \le i_1 < \cdots < i_p \le n,\;
\{x_{i_1},\ldots,x_{i_p}\} \notin \Delta \},
\]
and $K[\Delta]= K[x_1,\ldots,x_n]/I_{\Delta}$ is called
the \textit{Stanley--Reisner ring} of $\Delta$.
\par
For an arbitrary graph $G$, the simplicial complex $\Delta(G)$
with $I(G) = I_{\Delta(G)}$ is called
the \textit{complementary simplicial complex} of $G$.
\par
Put $d=\dim \Delta+1$.
A simplicial complex $\Delta$ is called \textit{pure}
if all the facets of $\Delta$ have the same cardinality $d$.
A pure simplicial complex $\Delta$ is \textit{connected in codimension $1$}
(or \textit{strongly connected}) if for every two facets $F$ and $H$ of
$\Delta$, there is a sequence of facets
$F = F_0, F_1, \ldots, F_m =H$
such that $\sharp (F_i \cap F_{i+1}) = d-1$ for each $i =0,\ldots, m-1$.
For every face $F \in \Delta$, the $\textit{star}$ and the
\textit{link} of $F$ are defined by:
\begin{eqnarray*}
\star_{\Delta} F &=& \{H \in \Delta \;:\,H \cup F \in \Delta\}, \\
\link_{\Delta} F &=& \{H \in \Delta\;:\,H \cup F \in \Delta,\, H \cap F = \emptyset\}.
\end{eqnarray*}
Note that these are also simplicial complexes.
\par \vspace{2mm}
\subsection{Serre's condition}
Let $S=K[x_1,\ldots,x_n]$ and $\frm = (x_1,\ldots,x_n)S$.
Let $I$ be a homogeneous ideal of $S$.
For a positive integer $k$,
$S/I$ satisfies \textit{Serre's condition} $(S_k)$
if $\depth (S/I)_P \ge \min\{\dim (S/I)_P,\, k \}$ for every $P \in \Spec S/I$.
\par
The ring $S/I$ is called \textit{Cohen--Macaulay}
if $\depth S/I = \dim S/I$.
This is an equivalent condition that $S/I$ satisfies Serre's condition $(S_d)$,
where $d= \dim S/I$.
Moreover, the ring $S/I$ is called (\textit{FLC}) if
$H_{\frm}^i(S/I)$ has finite length for every $i \ne \dim S/I$.
The ring $S/I$ is called \textit{Buchsbaum} if
the natural map $\Ext_S^i(S/\frm,S/I) \to H_{\frm}^i(S/I)$ is surjective
for every $i \ne \dim S/I$.
Note that any Cohen-Macaulay ring is Buchsbaum, and any Buchsbaum ring has (FLC).
\par
A simplicial complex $\Delta$ is called
\textit{Cohen--Macaulay} (resp., \textit{Buchsbaum}, \textit{FLC})
if so is $K[\Delta]$.
Note that $\Delta$ is Buchsbaum if and only if it satisfies (FLC).
Moreover,
if $\Delta$ is (FLC), then $\Delta$ is pure and
$\link_{\Delta}(F)$ is Cohen-Macaulay for every nonempty face $F \in \Delta$.
\par
We notice that $\Delta$ is pure and
connected in codimension $1$
if $K[\Delta]$ satisfies $(S_2)$ and $\dim \Delta \ge 1$.
\par \vspace{2mm}
\subsection{Takayama's formula}
Let $I$ be an arbitrary monomial ideal in $S=K[x_1,\ldots,x_n]$.
Then the $i$th local cohomology module $H_{\frm}^i(S/I)$ can be regarded as a $\bbZ^n$-module over $S/I$.
For every ${\bf a} =(a_1,\ldots,a_n) \in \bbZ^n$, we set
$G_a = \{i \;:\; a_i < 0 \}$ and define
\[
\Delta_{\bf a}(I) =
\{F \subseteq [n] \;:\; \text{$F$ satisfies $(C1)$ and $(C2)$}\},
\]
where
\begin{enumerate}
\item[(C1)] $F \cap G_{\bf a} = \emptyset$.
\item[(C2)] for every minimal generator $u=x_1^{c_1}\cdots x_n^{c_n}$ of $I$
there exists an index $i \notin F \cup G_{\bf a}$ with $c_i > a_i$.
\end{enumerate}
Moreover, we define
\[
\Delta(I) = \{F \subseteq [n] \;:\;
\textstyle{\prod_{i \in F}} x_i \notin \sqrt{I}\}.
\]
Then $\Delta(I)$ is a simplicial complex and
$\Delta_{\bf a}(I)$ is a subcomplex of $\Delta(I)$ with
$\dim \Delta_{\bf a}(I) = \dim \Delta(I)-\sharp(G_a)$
provided that $\Delta(I)$ is pure and $\Delta_{\bf a}(I) \ne \emptyset$
similarly as in \cite[Lemma 1.3]{MiT}.
\par \vspace{2mm}
Now let us recall Takayama's formula, which is a generalization of
well-known Hochster's formula.
\begin{lemma}[\textbf{Takayama's formula};
\textrm{see e.g. \cite[Theorem 1.1]{MiT}}] \label{Takayama}
Let $I$ be an arbitrary monomial ideal in $S=K[x_1,\ldots,x_n]$.
For every ${\bf a} \in \bbZ^n$, we have
\[
\dim_K H_{\frm}^i(S/I)_{\bf a} =
\left\{
\begin{array}{ll}
\dim_K \widetilde{H}_{i-\sharp(G_{\bf a})-1} (\Delta_{\bf a}(I)),
& \text{if $G_{\bf a} \in \Delta(I)$}, \\
0, & \text{else}.
\end{array}
\right.
\]
\end{lemma}
\par
Using this lemma, we obtain the following criterion for Cohen-Macaulayness
of $S/I$; see also \cite{MiT} in the case where
$I=I_{\Delta}^{(\ell)}$ and $\dim S/I_{\Delta} = 1$.
\begin{prop} \label{criterion}
The following conditions are equivalent$:$
\begin{enumerate}
\item $S/I$ is Cohen-Macaulay.
\item $S/I$ has $($FLC$)$, and for any ${\bf a} \in \bbN^n$,
we have that $\widetilde{H}_i(\Delta_{\bf a}(I))=0$ for all
$i < \dim \Delta_{\bf a}(I)$.
\end{enumerate}
\end{prop}
\begin{proof}
$(1)\Longrightarrow (2):$
Since $S/I$ is Cohen-Macaulay, it has (FLC).
For any ${\bf a} \in \bbN^n$, we have
\[
\widetilde{H}_i(\Delta_{\bf a}(I)) \cong H_{\frm}^{i+1}(S/I)_{\bf a} =0
\]
for all $i < \dim \Delta_{\bf a}(I)= \dim \Delta(I) = \dim S/I-1$
by Lemma \ref{Takayama} since $S/I$ is Cohen-Macaulay.
\par \vspace{2mm} \par \noindent
$(2)\Longrightarrow (1):$
Since $S/I$ has (FLC), $S/\sqrt{I}$ has also (FLC) and $\Delta(I)$ is pure;
see \cite{HTT}.
\par
Suppose that $S/I$ is \textit{not} Cohen-Macaulay.
For any ${\bf a} \in \bbN^n$ we have
\[
H_{\frm}^i(S/I)_{\bf a} \cong
\widetilde{H}_{i-1}(\Delta_{\bf a}(I))=0
\]
for all $i \le \dim \Delta_{\bf a}(I) = \dim \Delta(I)$.
So there exist a vector ${\bf a} \in \bbZ^n \setminus \bbN^n$ and
an index $i \le \dim \Delta(I)$ such that
\[
\widetilde{H}_{i-\sharp(G_{\bf a})-1}(\Delta_{\bf a}(I))
\cong H_{\frm}^i(S/I)_{\bf a} \ne 0.
\]
Set ${\bf a} = (a_1,\ldots,a_n)$ and $a_j < 0$.
Take any integer $k > 0$ and set ${\bf b} = {\bf a} - k {\bf e}_j$,
where ${\bf e}_j$ is the $j$th unit vector.
Then we have $\Delta_{\bf a}(I)=\Delta_{\bf b}(I)$
because $G_{\bf a} = G_{\bf b}$.
In particular, $H_{\frm}^i(S/I)_{\bf b} \ne 0$.
But this contradicts the assumption that $S/I$ has (FLC).
\end{proof}
\par \vspace{2mm}
\subsection{Symbolic power ideals}
Let $I$ be a radical ideal of $S$.
Let $\Min_S(S/I) = \{P_1,\ldots, P_r\}$ be the set of the
minimal prime ideals of $I$, and
put $W = S \setminus \bigcup_{i=1}^r P_i$.
Given an integer $\ell \ge 1$,
the \textit{$\ell$th symbolic power} of $I$
is defined to be the ideal
\[
I^{(\ell)}= I^{\ell}S_W \cap S = \bigcap_{i=1}^r P_i^{\ell}S_{P_i} \cap S.
\]
In particular, if $I$ is a squarefree monomial ideal of $S$,
then one has
\[
I^{(\ell)} = P_1^{\ell} \cap \cdots \cap P_r^{\ell}.
\]
\par
Let $\Delta$ be an arbitrary simplicial complex on $V=[n]$, and let
$I_{\Delta} \subseteq S=K[x_1,\ldots,x_n]$ denote
the Stanley-Reisner ideal of $\Delta$.
For any integer $\ell \ge 1$ and ${\bf a} \in \bbN^{n}$, we set
\[
\Delta^{(\ell)}_{\bf a} = \langle F \in \mathcal{F}(\Delta) \;:\;
\sum_{t \in V \setminus F} a_t \le \ell-1 \rangle.
\]
\par
We use the following remark and Proposition \ref{criterion} in
the proof of Theorem \ref{SPcomplete}.
\begin{remark} \label{MIT=TAK}
Under the notation above, for any ${\bf a} \in \bbN^n$, we have
\begin{enumerate}
\item $\Delta_{\bf a}^{(\ell)} = \Delta_{\bf a}(I_{\Delta}^{(\ell)})$;
see \cite[Section 1]{MiT}.
\item If $\Delta$ is pure and $\Delta_{\bf a}^{(\ell)} \ne \emptyset$ then
$\dim \Delta_{\bf a}^{(\ell)} = \dim \Delta$.
\end{enumerate}
\end{remark}
\par \vspace{2mm}
\subsection{Polarizations}
Now let $u = x_1^{a_1}x_2^{a_2} \cdots x_n^{a_n}$ be a monomial in $S$.
Then we can associate to
it a squarefree monomial $u^{\mathrm{pol}}$ as follows:
In the polarization process, each power of a variable $x_i^{a_i}$ is
replaced by a product of $a_i$ new variables $x_{i}^{(j)}$,
$i \in \{1,\ldots, n\}$, $j \in \{0, 1, \ldots, a_i-1\}$:
\[
u^{\mathrm{pol}} = x_{1}^{(0)}x_{1}^{(1)}\cdots x_{1}^{(a_1-1)}
x_{2}^{(0)} x_{2}^{(1)} \cdots x_{2}^{(a_2-1)}
\cdots
x_{n}^{(0)} x_{n}^{(1)} \cdots x_{n}^{(a_n-1)},
\]
where all $x_i^{(j)}$ are distinct variables and
$x_i^{(0)}=x_i$ for each $i$.
We call $u^{\mathrm{pol}}$ the \textit{polarization} of
$u$ (see \cite{SV}).
Let $I =(u_1,\ldots,u_s)$ be a monomial ideal of $S$,
where $\{u_1, \ldots,u_s\}$ is the minimal set
of monomial generators of $I$.
If $S^{\mathrm{pol}}$ is a polynomial ring over $K$
containing all monomials
$u_1^{\mathrm{pol}},\ldots, u_s^{\mathrm{pol}}$,
then we can consider the ideal
$I^{\mathrm{pol}} =
(u_1^{\mathrm{pol}}, \ldots, u_s^{\mathrm{pol}})$
of $S^{\mathrm{pol}}$.
It is known that, for monomial ideals $I$ and $J$, one has (\cite{SV})
\begin{equation}\label{inter}
(I \cap J)^{\mathrm{pol}} = I^{\mathrm{pol}} \cap J^{\mathrm{pol}}.
\end{equation}
\par
It is well-known that if $S/I$ is Cohen--Macaulay then
so is $S^{\mathrm{pol}}/I^{\mathrm{pol}}$.
In the proof of the first main theorem, we need a stronger result:
For a given positive integer $k$,
if $S/I$ satisfies Serre's condition $(S_k)$,
then so does $S^{\mathrm{pol}}/I^{\mathrm{pol}}$; see \cite{MuT}.
Note that a similar statement for (FLC) does not hold in general.
\par \vspace{2mm}
\subsection{Simplicial join}
Let $\Gamma$ (resp. $\Lambda$) be a non-empty
simplicial complex on $V_1$ (resp. $V_2$)
such that $V_1 \cap V_2 = \emptyset$.
Then the \textit{simplicial join} of $\Gamma$ and $\Lambda$, denoted by
$\Gamma*\Lambda$, is defined as follows:
\[
\Gamma * \Lambda = \big\{ F_1 \cup F_2 \;:\; F_1 \in \Gamma, \; F_2 \in \Lambda \big\}.
\]
Then $\Gamma * \Lambda$ is a simplicial complex on $V_1 \cup V_2$ and
$\mathcal{F}(\Gamma * \Lambda)=\big\{ F_1 \cup F_2 \;:\; F_1 \in
\mathcal{F}(\Gamma), \; F_2 \in \mathcal{F}(\Lambda) \big\}$.
In particular, $\dim \Gamma * \Lambda = \dim \Gamma + \dim \Lambda+1$.
Moreover, the $i$-th reduced homology group $\widetilde{H}_i(\Gamma * \Lambda)$
over a field $K$
of $\Gamma * \Lambda$ is given by
the so-called \textit{K\"unneth formula}:
\begin{equation} \label{Kunneth}
\widetilde{H}_i(\Gamma * \Lambda) \cong
\bigoplus_{p+q=i-1} \widetilde{H}_p(\Gamma) \otimes \widetilde{H}_q(\Lambda).
\end{equation}
Notice that $K[\Gamma * \Lambda] \cong K[\Gamma] \otimes_K K[\Lambda]$ as $K$-algebras;
see \cite[Lemma 1]{Fr}.
\par
For any disjoint union of two graphs $G_1$, $G_2$,
we have $\Delta(G_1 \coprod G_2) = \Delta(G_1) * \Delta(G_2)$.
\vspace{3mm}
\section{Symbolic powers of edge ideals of disjoint union of complete graphs}
Let $r$, $n_1,\ldots,n_r$ be
positive integers, and let
\[
S = K[x_{ij} \;:\; 1 \le i \le r,\,1 \le j \le n_i]
\]
be a polynomial ring over a field $K$.
For each integer $i$ with $1 \le i \le r$, if we put
\[
P_{ij} = (x_{i1},\ldots,\widehat{x_{ij}},\ldots, x_{in_i})S,
\quad \text{and} \quad
I_i = P_{i1}\cap P_{i2} \cap \cdots \cap P_{in_i},
\]
then $I_i$ is equal to $I(K_{n_i})S$, where
\[
I(K_{n_i})=(x_{ij}x_{ik} \,:\, 1 \le j < k \le n_i)
K[x_{i1}\ldots, x_{in_i}],
\]
denotes the edge ideal of the complete $n_i$-graph $K_{n_i}$
on the vertex set $V_i = \{x_{i1},\ldots,x_{in_i}\}$
for each $i=1,\ldots,r$.
\par \vspace{2mm}
Let $G$ be the disjoint union of complete $n_i$-graphs
for $i=1,2,\ldots,r$:
\[
G = K_{n_1} \;\textstyle{\coprod} \;K_{n_2}\;
\textstyle{\coprod}\; \cdots \;\textstyle{\coprod} \,K_{n_r}.
\]
Then the edge ideal $I(G)$ of $G$ is equal to $I_1 + I_2 + \cdots + I_r$.
Moreover, an irredundant primary decomposition of $I(G)$ is given by
\begin{equation} \label{eq-PrimeDecompo}
I(G) = \bigcap_{j_1,\ldots,j_r} (P_{1j_1} + \cdots + P_{rj_r}),
\end{equation}
where $j_1,\ldots,j_r$ move through the whole range
$1 \le j_1 \le n_1,\ldots, 1 \le j_r \le n_r$.
In particular,
\begin{equation} \label{eq-EdgePower}
I(G)^{(\ell)}
= \bigcap_{j_1,\ldots,j_r} (P_{1j_1} + \cdots + P_{rj_r})^{\ell}
\end{equation}
for every integer $\ell \ge 1$.
If we put
\[
x_i = x_{i1} + x_{i2} + \cdots + x_{in_i}
\]
for $i=1,2,\ldots,r$,
then a sequence $x_1,\ldots,x_r$ forms a system of parameters of $S/I(G)$
(and hence $S/I(G)^{(\ell)}$ for every $\ell \ge 1$).
Thus
\[
\dim S/I(G)^{(\ell)} = \dim S/I(G) = r.
\]
\par \vspace{2mm}
The main goal of this section is to prove the following theorem:
\begin{thm} \label{SPcomplete}
Let $S=K[x_{ij} \;:\; 1 \le i \le r,\,1 \le j \le n_i]$ be a polynomial ring
over a field $K$.
Let $G$ be a disjoint union of complete $n_i$-graphs$:$
$G = K_{n_1} \textstyle{\coprod} K_{n_2}
\textstyle{\coprod} \cdots \textstyle{\coprod} K_{n_r}$.
Then $S/I(G)^{(\ell)}$ is Cohen--Macaulay for every $\ell \ge 1$.
\end{thm}
\begin{remark} \label{rem-PowerRestrict}
In the above theorem, we do not need to assume that $\max\{n_1,\ldots,n_d\} \ge 2$.
\end{remark}
\par \vspace{1mm}
In order to prove Theorem \ref{SPcomplete}, we need the following key lemma.
\begin{lemma} \label{Key}
Let $\Gamma$ $($resp. $\Lambda$$)$ be a simplicial complex
on $V_1$ $($resp. $V_2$$)$
such that $V_1 \cap V_2 = \emptyset$.
Put $\Delta = \Gamma * \Lambda$ and $V = V_1 \cup V_2$.
Set $S_1 = K[V_1]$, $S_2 = K[V_2]$ and $S=S_1 \otimes_K S_2$.
If $S_1/I_{\Gamma}^{(i)}$ and $S_2/I_{\Lambda}^{(i)}$ are Cohen-Macaulay
for every $i \le \ell$, then $S/I_{\Delta}^{(\ell)}$ is Cohen-Macaulay.
\end{lemma}
\begin{proof}
We may assume that $V_1=\{1,2,\ldots,m\}$,
$V_2 = \{m+1,\ldots,n\}$ and $V=[n]$.
Note that $\Gamma$, $\Lambda$, and $\Delta$ are pure.
By an inductive argument on $n = \sharp(V)$, we may assume that
$S/I_{\Delta}^{(\ell)}$ has (FLC).
Then we must show that $\widetilde{H}_i(\Delta^{(\ell)}_{\bf a})=0$ for all
${\bf a} \in \mathbb{N}^n$ and $i < \dim \Delta_{\bf a}^{(\ell)} = \dim \Delta$
with $\Delta_{\bf a}^{(\ell)} \ne \emptyset$.
We first prove the following claim.
\begin{description}
\item[Claim 1] For each ${\bf a} \in \mathbb{N}^n$,
$\Delta_{\bf a}^{(\ell)} = \bigcup_{k=1}^{\ell}
\Gamma_{{\bf a}_1}^{(\ell+1-k)} * \Lambda_{{\bf a}_2}^{(k)}$ holds,
where ${\bf a}_1 = {\bf a} \,|_{V_1}$ and ${\bf a}_2 = {\bf a}\,|_{V_2}$.
\end{description}
\par \vspace{2mm}
Since $\mathcal{F}(\Delta)
= \{F_1 \cup F_2 \,:\, F_1 \in \mathcal{F}(\Gamma),
\,F_2 \in \mathcal{F}(\Lambda)\}$, we have
\begin{eqnarray*}
\Delta_{\bf a}^{(\ell)}
&=& \big\langle F_1 \cup F_2 \,:\; F_1 \in \mathcal{F}(\Gamma),\;
F_2 \in \mathcal{F}(\Lambda),\;
0 \le \sum_{t \in V \setminus F} a_t \le \ell-1 \big\rangle \\
&=& \bigcup_{k=1}^{\ell}
\big\langle F_1 \cup F_2 \,:\; F_1 \in \mathcal{F}(\Gamma),\;
F_2 \in \mathcal{F}(\Lambda), \; \sum_{t \in V_1 \setminus F_1} a_t \le \ell-k,\,
\sum_{t' \in V_2 \setminus F_2} a_{t'} \le k-1 \big\rangle \\
& =& \bigcup_{k=1}^{\ell} \Gamma_{{\bf a}_1}^{(\ell-k+1)} * \Lambda_{{\bf a}_2}^{(k)},
\end{eqnarray*}
as required.
We have proved the claim 1.
\par \vspace{3mm}
Put $d=\dim \Delta+1 = \dim \Gamma + \dim \Lambda+2$.
For $j = 1,\ldots,\ell$, we set
\[
\Pi_j = \bigcup_{k=1}^{j}
\Gamma_{{\bf a}_1}^{(\ell-k+1)} * \Lambda_{{\bf a}_2}^{(k)}.
\]
We next prove the following claim.
\begin{description}
\item[Claim 2] $\widetilde{H}_i(\Pi_j) =0$ holds
for every $i < d-1$ and $j=1,\ldots,\ell$.
\end{description}
We use an induction on $j$.
First consider the case where $j=1$.
Then $\Pi_1 = \Gamma_{{\bf a}_1}^{(\ell)} * \Lambda_{{\bf a}_2}$.
As $I_{\Gamma}^{(\ell)}$ and $I_{\Lambda}$ are Cohen-Macaulay by assumption,
we get
\[
p < \dim \Gamma = \dim \Gamma_{{\bf a}_1}^{(\ell)} \Longrightarrow
\widetilde{H}_p(\Gamma_{{\bf a}_1}^{(\ell)}) =0
\]
and
\[
q < \dim \Lambda = \dim \Lambda_{{\bf a}_2} \Longrightarrow
\widetilde{H}_q (\Lambda_{{\bf a}_2}) =0.
\]
Now suppose that $i < \dim \Pi_1 = \dim \Gamma + \dim \Lambda+1 = d-1$.
Then for any pair $(p,q)$ with $p+q=i-1$, either $p < \dim \Gamma$ or
$q < \dim \Lambda$ holds.
Hence the K\"unneth formula (see subsection 1.6) yields that
\[
\widetilde{H}_i (\Pi_1) \cong \bigoplus_{p+q=i-1} \widetilde{H}_p (\Gamma_{{\bf a}_1}^{(\ell)})
\otimes_K \widetilde{H}_q (\Lambda_{{\bf a}_2})=0.
\]
So we have proved the case where $j=1$.
\par
Now assume that $(\ell \ge) j \ge 2$ and $\widetilde{H}_i(\Pi_{j-1})=0$ for all $i < d-1$.
Then we must show that $\widetilde{H}_i(\Pi_{j})=0$ for all $i < d-1$.
In order to do that, we put $\Sigma = \Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j)}$.
Then $\Pi_{j-1} \cup \Sigma = \Pi_j$ and
\begin{eqnarray*}
\Pi_{j-1} \cap \Sigma
& = & \bigcup_{k=1}^{j-1} \big(\Gamma_{{\bf a}_1}^{(\ell-k+1)} * \Lambda_{{\bf a}_2}^{(k)} \big)
\cap \big(\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j)} \big) \\
& = & \bigcup_{k=1}^{j-1}
\bigg\{ \big(\Gamma_{{\bf a}_1}^{(\ell-k+1)} * \Lambda_{{\bf a}_2}^{(k)} \big) \cap
\big(\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j)} \big)
\bigg\} \\
&=& \bigcup_{k=1}^{j-1} \big(\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(k)} \big) \\
& =& \Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j-1)}.
\end{eqnarray*}
Thus the Mayer-Vietoris sequence yields the following exact sequence for each $i$:
\[
\cdots \to \widetilde{H}_i (\Pi_{j-1}) \oplus
\widetilde{H}_i (\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j)})
\to \widetilde{H}_i (\Pi_j) \to
\widetilde{H}_{i-1} (\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j-1)}) \to \cdots .
\]
By a similar argument as above, we have
\[
\widetilde{H}_i (\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j)}) =
\widetilde{H}_{i-1} (\Gamma_{{\bf a}_1}^{(\ell-j+1)} * \Lambda_{{\bf a}_2}^{(j-1)}) =0
\]
for all $i < d-1$.
Moreover, the induction hypothesis implies $\widetilde{H}_i(\Pi_{j-1})=0$ for all $i < d-1$.
Hence $\widetilde{H}_i(\Pi_{j})=0$ for all $i < d-1$, as required.
Therefore we obtain that $\widetilde{H}_i(\Delta_{\bf a}^{(\ell)}) =0$ for all $i < d-1$.
This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{SPcomplete}$]
We prove the assertion by an induction on $r$.
If $r=1$, then the assertion is clear because $\dim S/I(K_{n_1})^{(\ell)}=\dim S/I(K_{n_1}) =1$.
\par
Let $r \ge 2$.
Put
\[
S'=K[x_{ij} \,:\, 1 \le i \le r-1,\; 1 \le j \le n_i] \quad
\text{and} \quad
G'=K_{n_1} \coprod \cdots \coprod K_{n_{r-1}}.
\]
By the induction hypothesis, we may assume that
$S'/I(G')^{(i)}$ is Cohen-Macaulay for all $i \ge 1$.
As $\Delta(G) = \Delta(G' \coprod K_{n_r}) = \Delta(G')*\Delta(K_{n_r})$,
by virtue of Lemma \ref{Key}, we can conclude that
$I(G)^{(\ell)}$ is Cohen-Macaulay for all $\ell \ge 1$.
\end{proof}
\par \vspace{2mm}
In order to discuss (FLC) properties of symbolic or ordinary powers,
we generalize Theorem \ref{SPcomplete} to the following corollary.
\begin{cor} \label{Main1-extend}
Let $G = K_{n_1}\coprod \ldots \coprod K_{n_r}$ be a disjoint union of finitely many complete graphs,
and let $y_1,\ldots,y_s$ be variables which are not vertices of $G$.
Put $S=K[v \,:\, v \in G]$, $T=S[y_1,\ldots,y_s]$ and $I=I(G)+(y_1,\ldots,y_s)$.
Then $T/I^{(\ell)}$ is Cohen--Macaulay for all $\ell \ge 1$.
\end{cor}
\begin{proof}
We may assume that $s=1$, and put $y=y_1$ for simplicity.
Let $I(G)=\cap_j P_j$ be an irredundant primary decomposition of $I(G)$.
Since $I=\cap_j (P_j,y)$ gives an irredundant primary decomposition of $I$,
we have
\[
I^{(\ell)} = \bigcap_j (P_j,y)^{\ell}
= \bigcap_j \sum_{k=0}^{\ell} P_j^{k} y^{\ell-k}
= \sum_{k=0}^{\ell} \left(\cap_i P_j^{k}\right) y^{\ell-k}
= \sum_{k=0}^{\ell} I(G)^{(k)} y^{\ell-k}.
\]
Hence it follows that
\[
T/I^{(\ell)} \cong S/I(G)^{(\ell)} \oplus S/I(G)^{(\ell-1)} \oplus \cdots \oplus S/I(G)
\]
as $S$-modules.
Since all $S$-modules of the right-hand side are Cohen--Macaulay, so is $T/I^{(\ell)}$, as required.
\end{proof}
\begin{exam} \label{ex-ci}
If $G$ consists of $r$ isolated edges and $s$ isolated vertices, then
\[
S=K[x_{11},x_{12},\ldots,x_{r1},x_{r2},y_1,\ldots,y_s],\quad
I(G)=(x_{11}x_{12},\ldots,x_{r1}x_{r2}).
\]
\par
In particular, $S/I(G)^{(\ell)}=S/I(G)^{\ell}$ is Cohen--Macaulay for every $\ell \ge 1$.
\par \vspace{4mm}
\begin{center}
\begin{picture}(200,40)
\put(-40,20){$G=$}
\put(0,5){\circle*{4}}
\put(25,5){\circle*{4}}
\put(125,5){\circle*{4}}
\put(175,20){\circle*{4}}
\put(200,20){\circle*{4}}
\put(220,20){$\cdots$}
\put(250,20){\circle*{4}}
\put(0,35){\circle*{4}}
\put(25,35){\circle*{4}}
\put(125,35){\circle*{4}}
\put(0,5){\line(0,1){30}}
\put(25,5){\line(0,1){30}}
\put(125,5){\line(0,1){30}}
\put(-17,-5){$x_{11}$}
\put(22,-5){$x_{21}$}
\put(130,0){$x_{r1}$}
\put(-17,39){$x_{12}$}
\put(22,39){$x_{22}$}
\put(130,35){$x_{r2}$}
\put(70,20){$\cdots$}
\put(175,29){$y_1$}
\put(200,29){$y_2$}
\put(250,29){$y_s$}
\end{picture}
\end{center}
\par
This complete intersection complex is the boundary complex of a simplex or
an iterated cone of a cross polytope.
Namely, $I(G)=I_{\Delta(\mathcal{P})}$ holds, where
$\mathcal{P}$ is the $s$-iterated cone of the cross $r$-polytope.
\end{exam}
\par \vspace{2mm}
The next example shows that our theorem cannot be generalized
for mixed symbolic powers.
\begin{exam} \label{ex-mixedpower}
Let $G$ be a complete $n$-graph.
Then $I(G) = P_1 \cap \cdots \cap P_n$,
where $P_i=(x_1,\ldots,\widehat{x_i},\ldots,x_n)$
for each $i=1,\ldots,n$.
Since $\dim S/I(G) =1$ and $P_i^{a}$ has no embedded primes for any integer $a \ge 1$,
$S/P_1^{a_1} \cap \cdots \cap P_n^{a_n}$
is a Cohen--Macaulay ring of dimension $1$ for every positive integers $a_1,\ldots,a_n$.
\par
A similar assertion does \textit{not} hold in general for two disjoint union of
complete graphs.
For example, let $I(G) = (x_1x_2,x_1x_3,x_2x_3,y_1y_2)$ be the edge ideal of
$K_3 \;\textstyle{\coprod}\; K_2$ in $S=\QQ[x_1,x_2,x_3,y_1,y_2]$.
Then
\[
I(G) = (x_1,x_2,y_1)\cap (x_1,x_2,y_2)\cap (x_1,x_3,y_1)
\cap (x_1,x_3,y_2)\cap (x_2,x_3,y_1)\cap (x_2,x_3,y_2).
\]
Our theorem says that
\[
I(G)^{(2)} = (x_1,x_2,y_1)^2 \cap (x_1,x_2,y_2)^2 \cap (x_1,x_3,y_1)^2
\cap (x_1,x_3,y_2)^2 \cap (x_2,x_3,y_1)^2 \cap (x_2,x_3,y_2)^2
\]
is a Cohen--Macaulay ring of dimension $2$, that is, $\pd_S S/I(G)^{(2)}=3$.
Indeed, by Macaulay 2, the minimal free resolution of
$S/I(G)^{(2)}$ over $S$ is given by
\[
0 \to S^5 \to S^{12} \to S^8 \to S \to S/I(G)^{(2)} \to 0.
\]
However, this is no longer true for mixed symbolic powers.
For instance, put
\[
J_a= (x_1,x_2,y_1)^2 \cap (x_1,x_2,y_2)^2 \cap (x_1,x_3,y_1)^2
\cap (x_1,x_3,y_2)^2 \cap (x_2,x_3,y_1)^2 \cap (x_2,x_3,y_2)^a
\]
for every positive integer $a \ge 2$.
When $a \le 3$, $S/J_a$ is Cohen--Macaulay. But $S/J_4$ is \textit{not}.
\end{exam}
\par
The following question seems to be interesting.
\begin{quest} \label{Q-MixedPower}
We use the same notation as in $(\ref{eq-PrimeDecompo})$.
Let $\ell_{j_1,\ldots,j_r}$ be given integers.
When is the following mixed symbolic power ideal
\[
\bigcap_{j_1,\ldots,j_r} (P_{1,j_1}+\cdots + P_{r,j_r})^{\ell_{j_1,\ldots,j_r}}
\]
Cohen--Macaulay?
\end{quest}
\medskip
\section{Non-Cohen--Macaulayness of symbolic powers}
\subsection{Cohen--Macaulay properties of symbolic powers}
In the previous section, we proved that
all symbolic powers of the edge ideal of a disjoint union of
finitely many complete graphs are Cohen--Macaulay.
In this section, we prove the converse.
That is, the main purpose of this section is to prove Theorem \ref{Serre2}.
Using these results, we prove the first main theorem.
Moreover, as an application, we also prove an improvement of the main theorem
\cite{CRTY} with respect to Cohen--Macaulayness of ordinary powers.
\begin{thm} \label{Serre2}
Let $G$ be a graph which is not a disjoint union of finitely many complete graphs.
Then for any $\ell \ge 3$,
$S/I(G)^{(\ell)}$ does not satisfy Serre's condition $(S_2)$.
\end{thm}
\begin{remark}
The assumption that $\ell \ge 3$ is essential.
For example, let $G$ be a pentagon, and set $I(G) = (x_1x_2,x_2x_3,x_3x_4,x_4x_5,x_5x_1)$
in $S=K[x_1,x_2,x_3,x_4,x_5]$.
Then $I(G)$ is not complete intersection, but $S/I(G)^{(2)} = S/I(G)^2$ is Cohen--Macaulay.
\end{remark}
\par \vspace{2mm}
In order to study Cohen--Macaulayness of
higher symbolic powers of edge ideals, we use the notion of polarization.
Let $I$ be a monomial ideal of $S$,
and let $I^{\mathrm{pol}} \subseteq S^{\mathrm{pol}}$ denote the polarization
of $I$.
\par
Let $G$ be a graph with vertex set $V(G)=\{x_1,x_2,\ldots,x_n\}$.
Let $\Delta=\Delta(G)$ be the complementary
simplicial complex of $G$.
For a positive integer $\ell$, let $\Delta^{(\ell)}$ be the simplicial complex
such that $I_{\Delta^{(\ell)}}=(I(G)^{(\ell)})^{\mathrm{pol}}$.
\par
For a positive integer $\ell$ and for any fixed $i$,
we put $(x_i^{\ell})^{\mathrm{pol}}
=x_i^{(0)}x_i^{(1)}\cdots x_i^{(\ell-1)}$,
where $x_i^{(0)} = x_i$.
Furthermore, we put
$(x_1^{\ell_1}\cdots x_n^{\ell_n})^{\mathrm{pol}}
=(x_1^{\ell_1})^{\mathrm{pol}}\cdots (x_n^{\ell_n})^{\mathrm{pol}}$.
See Section 2 for more details.
In order to study facets of $\Delta^{(\ell)}$, we need the following lemma.
\begin{lemma} \label{pol-lemma}
Under the above notation, we have
\[
((x_1,\ldots,x_h)^{\ell})^{\mathrm{pol}}
=
\bigcap_
i_1+\cdots + i_h \le \ell-1}
(x_1^{(i_1)},\ldots,x_{h}^{(i_h)}).
\]
\end{lemma}
\begin{proof}
By the definition of polarization, we have
\begin{eqnarray*}
((x_1,\ldots,x_h)^{\ell})^{\mathrm{pol}}
& = & \big(
x_1^{j_1}\cdots x_h^{j_h} \; :\; j_1,,\ldots,j_h \ge 0,\;
j_1+\cdots +j_h = \ell \big)^{\mathrm{pol}} \\
& = & \big(\prod_{k=1}^h x_k^{(0)} x_k^{(1)} \cdots x_k^{(j_k-1)}
\;:\; j_1,\ldots,j_h \ge 0,\; j_1 + \cdots + j_h = \ell \big).
\end{eqnarray*}
So, in order to obtain the required primary decomposition, it suffices to show that
\[
((x_1,\ldots,x_h)^{\ell})^{\mathrm{pol}} \subseteq (x_1^{(i_1)},\ldots,x_{h}^{(i_h)})
\Longleftrightarrow
i_1 + \cdots + i_h \le \ell-1.
\]
Suppose $i_1 + \cdots + i_h \ge \ell$.
Take a monomial $M = \prod_{k=1}^h x_k^{(0)}x_k^{(1)}\cdots x_k^{(i_k-1)}$.
Then it is clear that $M \notin (x_1 ^{(i_1)},\ldots,x_{h}^{(i_h)})$.
On the other hand, $M$ is contained in $((x_1,\ldots,x_h)^{\ell})^{\mathrm{pol}}$
because there exists a sequence $(j_1,\ldots,j_h)$ such that
$0 \le j_k \le i_k$ for each $k$ and $j_1 + \cdots + j_h = \ell$.
\par
Next suppose that $i_1 + \cdots + i_h \le \ell-1$.
If $((x_1,\ldots,x_h)^{\ell})^{\mathrm{pol}} \not \subseteq (x_1^{(i_1)},\ldots,x_{h}^{(i_h)})$,
then there exists a monomial
$M = \prod_{k=0}^h x_k^{(0)}\cdots x_k^{(j_k-1)}$ with $j_1 + \cdots + j_h=\ell$
such that $M$ is not contained in
$(x_1^{(i_1)},\ldots,x_{h}^{(i_h)})$.
Hence $j_k \le i_k$ for each $k$.
But $\ell=j_1 + \cdots + j_h \le i_1+\cdots+ i_h \le \ell-1$. This is a contradiction.
\end{proof}
\par
By the above lemma, we get the following corollary.
\begin{cor} \label{facets-ell}
Under the above notation, we set $V^{(i)} = \{x_1^{(i)},\ldots,x_n^{(i)}\}$ for
each $i=1,2,\ldots,\ell-1$.
Then $\mathcal{F}(\Delta^{(\ell)})$ consists of
the following subsets of $V \cup V^{(1)} \cup \cdots \cup V^{(\ell-1)}:$
\begin{eqnarray*}
&& \big(F \cup \big\{x_{i_{1,1}},\ldots,x_{i_{1,j_1}},x_{i_{2,1}},\ldots,x_{i_{2,j_2}},\ldots,
x_{i_{\ell-1,1}},\ldots,x_{i_{\ell-1, j_{\ell-1}}}\big\}\big) \\
&& \cup \big(V^{(1)} \setminus \big\{x_{i_{1,1}}^{(1)},\ldots,x_{i_{1,j_1}}^{(1)} \big\}\big)
\cup \cdots \cup
\big(V^{(\ell-1)} \setminus
\big\{x_{i_{\ell-1,1}}^{(\ell-1)},\ldots,x_{i_{\ell-1,j_{\ell-1}}}^{(\ell-1)} \big\}\big),
\end{eqnarray*}
where $F$ and $x_{i}$'s run through
\[
\begin{array}{cl}
\bullet & F \in \mathcal{F}(\Delta); \\
\bullet & 0 \le j_1,j_2,\ldots,j_{\ell-1} \le n, \quad j_1 + 2 j_2 + \cdots + (\ell-1)j_{\ell-1} \le \ell-1; \\
\bullet & \big\{x_{i_{1,1}},\ldots,x_{i_{\ell-1,j_{\ell-1}}}\big\} \cap F = \emptyset,\quad
\sharp \big\{x_{i_{1,1}},\ldots,x_{i_{\ell-1,j_{\ell-1}}}\big\}
= j_1 + j_2 + \cdots + j_{\ell-1}.
\end{array}
\]
\par
In particular, if $\Delta$ is pure, then so is $\Delta^{(\ell)}$.
\end{cor}
\begin{proof}
By definition, we have
\[
I_{\Delta^{(\ell)}}
= \big((I(G))^{(\ell)}\big)^{\mathrm{pol}}
= \big(\bigcap_{F \in \mathcal{F}(\Delta)}\!\! P_F^{\ell}\big)^{\mathrm{pol}}
= \bigcap_{F \in \mathcal{F}(\Delta)}\!\! (P_F^{\ell})^{\mathrm{pol}}.
\]
If $P_F=(y_1,\ldots,y_h)$, then
\[
(P_F^{\ell})^{\mathrm{pol}} = \bigcap_{i_1+\cdots+i_h \le \ell-1} \!\!
(y_{1}^{(i_1)},\ldots,y_h^{(i_h)})
\]
by the above lemma.
\par
Let $G \in \mathcal{F}(\Delta^{(\ell)})$.
Then there exist
a facet $F \in \mathcal{F}(\Delta)$ and
integers $0 \le i_1 \le \ldots \le i_h$
with $i_1+\cdots + i_h \le \ell-1$
such that
\begin{eqnarray*}
V \cup V^{(1)} \cup \cdots \cup V^{(\ell-1)} \setminus G
&=& \{y_1^{(i_1)},\ldots,y_h^{(i_h)}\}, \\
V \setminus F &=& \{y_1,\ldots,y_h\}.
\end{eqnarray*}
Putting
\[
\big\{y_1^{(i_1)},\ldots,y_h^{(i_h)}\big\} =
\{x_{i_{0,1}}^{(0)},\ldots,x_{i_{0,j_0}}^{(0)},\ldots,
x_{i_{\ell-1,1}}^{(\ell-1)},\ldots, x_{i_{\ell-1,j_{\ell-1}}}^{(\ell-1)}\},
\]
we get a required form of $G$.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{Serre2}$]
Assume that $S/I(G)^{(\ell)}$ satisfies $(S_2)$.
As $I(G)=\sqrt{I(G)^{(\ell)}}$, $S/I(G)$ also satisfies $(S_2)$ by \cite{HTT}.
In particular, $I(G)$ is pure.
Since some connected component of $G$ is not a complete graph by assumption,
there exist $x_1,x_2,x_3 \in V(G)$ such that
\[
\{x_1, x_2\} ,\,\{x_1, x_3\} \in E(G), \quad \text{and} \; \{x_2,x_3\} \notin E(G).
\]
We may assume that
$V(G) = \{x_1,x_2,x_3,\ldots,x_n\}$, the vertex set of $G$
by renumbering if necessary.
Let $\Delta=\Delta(G)$ be the complementary
simplicial complex of $G$, and let $\Delta^{(\ell)}$ be the simplicial complex
defined as above.
Set $\widetilde{V} = V \cup V^{(1)} \cup \cdots \cup V^{(\ell-1)}$.
Note that $\Delta^{(\ell)}$ is a pure simplicial complex on $\widetilde{V}$.
\par \vspace{2mm}
Now consider the following subset of $\widetilde{V}:$
\[
F_0 = \left\{
\begin{array}{ccccccc}
x_1, & x_1^{(1)}, & x_1^{(2)}, & \ldots & x_1^{(\ell-3)}, & x_1^{(\ell-2)}, & \square \\[2mm]
x_2, & x_2^{(1)}, & x_2^{(2)}, & \ldots & x_2^{(\ell-3)}, & \square & x_2^{(\ell-1)}, \\[2mm]
x_3, & \square & x_3^{(2)}, & \ldots & x_3^{(\ell-3)}, & x_3^{(\ell-2)}, & x_3^{(\ell-1)}, \\[2mm]
\square & x_4^{(1)}, & x_4^{(2)}, & \ldots & x_4^{(\ell-3)}, & x_4^{(\ell-2)}, & x_4^{(\ell-1)},\\[2mm]
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\[2mm]
\square & x_n^{(1)}, & x_n^{(2)}, & \ldots & x_n^{(\ell-3)}, & x_n^{(\ell-2)}, & x_n^{(\ell-1)}
\end{array}
\right\}.
\]
Then $F_0$ is a face of $\Delta^{(\ell)}$.
Indeed, we can take a facet $F \in \mathcal{F}(\Delta)$ such that $\{x_2,x_3\} \subseteq F$.
Since $x_1 \notin F$,
\[
F' = (F \cup \{x_1\}) \cup V^{(1)} \cup V^{(2)} \cup \cdots \cup V^{(\ell-2)} \cup
(V^{(\ell-1)} \setminus \{x_1^{(\ell-1)}\})
\]
is a facet of $\mathcal{F}(\Delta^{(\ell)})$ by Corollary \ref{facets-ell}.
This implies that $F_0 \in \Delta^{(\ell)}$ because $F_0 \subseteq F'$.
\par
We first prove the following claim:
\begin{description}
\item[Claim] Any facet of $\link_{\Delta^{(\ell)}}(F_0)$ is given by
\[
\begin{array}{cl}
& (F \setminus \{x_2,x_3\}) \cup \{x_3^{(1)}\} \cup \{x_2^{(\ell-2)}\},
\text{where}
\, F \in \FF(\Delta) \; \text{and} \; \{x_2,x_3\} \subseteq F; \\
\text{or} & \\
& (F \setminus \{x_1\}) \cup \{x_1^{(\ell-1)}\},
\text{where}\;
F \in \FF(\Delta) \; \text{and} \; x_1 \in F.
\end{array}
\]
\end{description}
In order to prove the claim, it suffices to determine $\FF(\star_{\Delta^{(\ell)}}(F_0))$
because any facet $G$ of $\link_{\Delta^{(\ell)}}(F_0)$ can be written as
$G = \widetilde{F} \setminus F_0$
for some $\widetilde{F} \in \FF(\star_{\Delta^{(\ell)}}(F_0))$.
\par
Let $\widetilde{F} \in \FF(\star_{\Delta^{(\ell)}}(F_0))$.
Then $\widetilde{F} \in \FF(\Delta^{(\ell)})$ and
$\widetilde{F} \supseteq F_0$.
In particular, $x_i^{(1)},\ldots,x_i^{(\ell-1)} \in \widetilde{F}$
for each $i=4,\ldots,n$ and
\[
W_1 = V^{(1)} \setminus \{x_3^{(1)}\},
\quad
W_{\ell-2}=V^{(\ell-2)} \setminus \{x_2^{(\ell-2)}\},
\quad
W_{\ell-1}= V^{(\ell-1)} \setminus \{x_1^{(\ell-1)}\} \subseteq \widetilde{F}.
\]
Hence $\widetilde{F}$ is given by one of the following complexes:
{\small
\[
\begin{array}{cclllllllll}
\widetilde{F}_1 &=& (F \cup \{x_1\}) & \cup & V^{(1)} & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & V^{(\ell-2)} & \cup & W_{\ell-1}, \\[1mm]
\widetilde{F}_2 &=& (F \cup \{x_2\}) & \cup & V^{(1)} & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & W_{\ell-2} & \cup & V^{(\ell-1)}, \\[1mm]
\widetilde{F}_3 &=& (F \cup \{x_3\}) & \cup & W_1 & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & V^{(\ell-2)} & \cup & V^{(\ell-1)}, \\[1mm]
\widetilde{F}_{12} &=& (F \cup \{x_1,x_2\}) & \cup & V^{(1)} & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & W_{\ell-2} & \cup & W_{\ell-1}, \\[1mm]
\widetilde{F}_{13} &=& (F \cup \{x_1,x_3\}) & \cup & W_{1} & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & V^{(\ell-2)} & \cup & W_{\ell-1}, \\[1mm]
\widetilde{F}_{23} &=& (F \cup \{x_2,x_3\}) & \cup & W_{1} & \cup & V^{(2)} \cup \cdots \cup V^{(\ell-3)} &
\cup & W_{\ell-2} & \cup & V^{(\ell-1)}.
\end{array}
\]
}
Now suppose that $\widetilde{F} = \widetilde{F}_2$.
Then we have $x_1,x_3 \in F$.
This implies that $x_3 \notin P_F$.
Hence, $x_1x_3 \in I(G)$ yields $x_1 \in P_F$.
This contradicts $x_1 \in F$.
Therefore it does not occur that $\widetilde{F} = \widetilde{F}_2$.
Similarly, we have $\widetilde{F} \ne \widetilde{F}_3$.
\par
Next suppose that $\widetilde{F} = \widetilde{F}_{12}$.
Then $(\ell-2) j_{\ell-2} + (\ell-1) j_{\ell-1} \ge 2\ell-3 \ge \ell$ because $\ell \ge 3$.
This is impossible.
Hence $\widetilde{F} \ne \widetilde{F}_{12}$.
Similarly, we have $\widetilde{F} \ne \widetilde{F}_{13}$.
Consequently, either
\[
\widetilde{F} = \widetilde{F}_1 \quad \text{and}\quad x_2,x_3 \in F,\quad x_1 \notin F
\]
or
\[
\widetilde{F} = \widetilde{F}_{23} \quad \text{and}\quad x_1 \in F,
\quad x_2,x_3 \notin F
\]
holds.
In other words, any $G \in \FF(\link_{\Delta^{(\ell)}}(F_0))$ can be written as
\[
G' = (F \setminus \{x_2,x_3\}) \cup \{x_3^{(1)}\} \cup \{x_2^{(\ell-2)}\}
\]
for some $F \in \FF(\Delta)$ such that $x_1 \notin F$ and $x_2,\,x_3 \in F$;
or
\[
G'' = (F \setminus \{x_1\}) \cup \{x_1^{(\ell-1)}\}
\]
for some $F \in \FF(\Delta)$ such that $x_1 \in F$ and $x_2,x_3 \notin F$.
So, we proved the claim.
\par \vspace{2mm}
Choose $G'$ and $G''$ of the above type, respectively.
Note that there exist those facets as $(x_1x_2,x_1x_3) \subseteq I_{\Delta}$.
Then one can find no chain of facets in $\link_{\Delta^{(\ell)}}(F_0)$ such that
\[
G'=G_0,\,G_1, \ldots, G_r = G''
\]
with $\sharp(G_i \cap G_{i-1}) = d-1$, where
$d = \dim K[\link_{\Delta^{(\ell)}}(F_0)]$ since
both $x_3^{(1)}$ and $x_2^{(\ell-2)}$ are
contained in $G'$ but not in $G''$.
Thus $\link_{\Delta^{(\ell)}}(F_0)$ is \textit{not} connected in codimension $1$,
and hence it does \textit{not} satisfy $(S_2)$.
By the lemma below, we can conclude that $S/I(G)^{(\ell)}$ does \textit{not}
satisfy $(S_2)$, as required.
\end{proof}
\par
The following lemma was used in the proof of \cite[Theorem 4.1]{MuT}.
Moreover, it is clear that $S/I$ is Cohen--Macaulay if and only if
so is $S^{\mathrm{pol}}/I^{\mathrm{pol}}$ because
$S/I$ is isomorphic to a quotient of $S^{\mathrm{pol}}/I^{\mathrm{pol}}$
by a regular sequence.
\begin{lemma}[\textrm{See the proof of \cite[Theorem 4.1]{MuT}}]
Let $k \ge 1$ be any integer.
Let $I \subseteq S$ be a monomial ideal,
and let $I^{\mathrm{pol}} \subseteq S^{\mathrm{pol}}$
denote the polarization of $I$.
If $S/I$ satisfies $(S_k)$, then so does $S^{\mathrm{pol}}/I^{\mathrm{pol}}$.
\end{lemma}
\par
We are now ready to prove the first main theorem in this paper.
\begin{thm} \label{Main-CM}
Let $I(G)\subseteq S$ be the edge ideal of a graph $G$.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ is Cohen--Macaulay for every integer $\ell \ge 1$.
\item $S/I(G)^{(\ell)}$ is Cohen--Macaulay for some $\ell \ge 3$.
\item $S/I(G)^{(\ell)}$ satisfies Serre's condition $(S_2)$ for some $\ell \ge 3$.
\item $G$ is a disjoint union of finitely many complete graphs.
\end{enumerate}
\end{thm}
\begin{proof}
Let $I(G) \subseteq S$ be the edge ideal with $\dim S/I(G) \ge 2$.
\par \vspace{2mm}
$(1) \Longrightarrow (2):$
This is clear.
\par \vspace{2mm}
$(2) \Longrightarrow (3):$
Since any Cohen--Macaulay ring satisfies
Serre's condition $(S_2)$, it is clear.
\par \vspace{2mm}
$(3) \Longrightarrow (4):$
Now suppose that $G$ cannot be written as a disjoint union
of finitely many complete graphs.
Then, for any $\ell \ge 3$, $S/I(G)^{(\ell)}$ does not satisfy $(S_2)$
by Theorem \ref{Serre2}. This contradicts the assumption.
\par \vspace{2mm}
$(4) \Longrightarrow (1):$
By Theorem \ref{SPcomplete}, if $G$ is a disjoint union
of finitely many complete graphs, then $S/I(G)^{(\ell)}$ is Cohen--Macaulay
for every $\ell \ge 1$.
\end{proof}
\begin{remark}
By a similar argument as in Corollary \ref{Main1-extend}, we can generalize
the above theorem to the case where $I$ contains variables.
Moreover, in this case, we can replace $S$ with $S[t]$, where $t$ is an indeterminate.
\end{remark}
\par \vspace{2mm}
\subsection{Cohen--Macaulay properties of ordinary powers}
\par
Using Theorem \ref{Main-CM}, we can give an improvement
of the main theorem in \cite{CRTY}.
\begin{thm}[\textrm{cf. \cite[Theorem 2.1]{CRTY}}] \label{Power-cor}
Let $I(G)$ be the edge ideal of a graph $G$.
If $S/I(G)^{\ell}$ is Cohen--Macaulay for some $\ell \ge 3$,
then $I(G)$ is complete intersection.
\end{thm}
\begin{remark}
In \cite{CRTY}, the authors proved an analogous theorem:
$I(G)$ is complete intersection whenever
$S/I(G)^{\ell}$ is Cohen--Macaulay for some $\ell \ge \height I(G)$.
Note that it is not difficult to derive this from Theorem \ref{Power-cor}.
\end{remark}
\par
In order to prove the theorem, we need the following lemma.
\begin{lemma}[\textrm{See also \cite[Lemma 5.8, Theorem 5.9]{SVV}}] \label{SVV}
Let $I(G)$ be the edge ideal of a graph $G$.
Let $t \ge 2$ be an integer.
Then the following conditions are equivalent.
\begin{enumerate}
\item $G$ contains no odd cycles of length $2s-1$ for any $2 \le s \le t$.
\item $I(G)^{(t)}=I(G)^t$ holds.
\end{enumerate}
\end{lemma}
\begin{proof}
Put $I=I(G)$ for simplicity.
\par
$(1) \Longrightarrow (2):$
It follows from a similar argument as in the proof of
\cite[Lemma 5.8, Theorem 5.9]{SVV}.
But for the convenience of the readers, we give a sketch of the proof.
It is enough to show that
$\frm \notin \Ass_S(S/I^t)$ if $\dim S/I \ge 1$.
Now suppose \textit{not}.
Then we can take a monomial $M \notin I^t$ such that $I^t \colon M = \frm$.
Since $\depth S/I \ge 1$, we get $M \in I$.
So we can write $M = x_1x_2 L$ for some $x_1x_2 \in G(I)$ and a monomial $L$.
By definition, we have $x_2M = x_1x_2^2L \in I^t$.
It follows that $x_2^2L \in I^{t-1}$ because $I$ is generated by squarefree monomials.
This yields $M \in x_1 I^{t-1} \cap (I^t :x_1)$.
\par
On the other hand, by a similar argument as in the proof of \cite[Lemma 5.8]{SVV},
we can show that $x I^m \cap (I^{m+1} \colon x) \subseteq I^{m+1}$
for any vertex $x$ and for all $0 \le m \le t-1$ using $(1)$
(Notice that there exists a small gap in the final step of the
proof of \cite[Lemma 5.8]{SVV}.
That is, we obtain an odd cycle if only if $i$ is even.).
In particular, $M \in x_1 I^{t-1} \cap (I^t :x_1)\subseteq I^t$, which
contradicts the choice of $M$.
\par
$(2) \Longrightarrow (1):$
Suppose that $G$ contains an odd cycle of length $2s-1$ with $2 \le s \le t$;
say, $x_1x_2$, $x_2x_3, \ldots, x_{2s-2}x_{2s-1}$, $x_{2s-1}x_1$.
Put $M = x_1x_2\cdots x_{2s-1}$.
Then we show $M(x_1x_2)^{t-s} \in I^{(t)} \setminus I^t$.
Let $P$ be any associated prime ideal of $I$.
Then since $P$ is prime and $x_1x_2,x_2x_3,\ldots,x_{2s-2}x_{2s-1},x_{2s-1}x_1 \in P$,
we get $\sharp(P \cap \{x_1,x_2,\ldots,x_{2s-1}\}) \ge s$.
Hence $M \in P^{s}$ and thus $M(x_1x_2)^{t-s} \in I^{(t)}$.
On the other hand, $M (x_1x_2)^{t-s} \notin I^t$
because $\deg M(x_1x_2)^{t-s} = 2t-1 < 2t=\indeg I^t$,
where $\indeg I^t = \min\{m \in \mathbb{Z} \,:\, [I^t]_m \ne 0 \}$.
\end{proof}
\begin{cor} \label{power-not}
Let $G$ be a disjoint union of complete graphs
$K_{n_1},\ldots,K_{n_r}$.
\par
If $\max\{n_1,\ldots,n_r\} \ge 3$,
then $I(G)^{(\ell)} \ne I(G)^{\ell}$ for every $\ell \ge 2$.
In particular, $I(G)^{\ell}$ is not a Cohen--Macaulay ideal.
\end{cor}
\begin{proof}
Under the assumption, $G$ always contains a triangle ($3$-cycle).
\end{proof}
\begin{proof}[Proof of Theorem $\ref{Power-cor}$]
Now suppose that $S/I(G)^{\ell}$ is Cohen--Macaulay
for some integer $\ell \ge 3$,
and that $I(G)$ is \textit{not} complete intersection.
\par
By Theorem \ref{Main-CM}, $G$ can be written as a disjoint union
of finitely many complete graphs.
However, this contradicts the above corollary.
\end{proof}
\par
The next example shows that the Cohen--Macaulayness of symbolic power ideals
is different from that of ordinary power ideals.
\begin{exam} \label{union-triangles}
Let $G$ be a disjoint union of $d$ complete $3$-graphs.
Set
\[
I=I(G)=(x_{11}x_{12},x_{11}x_{13},x_{12}x_{13},\ldots, x_{d1}x_{d2},x_{d1}x_{d3},x_{d2}x_{d3})
\]
in a polynomial ring $S=K[x_{11},x_{12},x_{13},\ldots,x_{d1},x_{d2},x_{d3}]$.
Then
\begin{enumerate}
\item $S/I^{(\ell)}$ is Cohen--Macaulay of dimension $d$ for every $\ell \ge 1$.
\item $S/I^{\ell}$ is \textit{not} Cohen--Macaulay for any $\ell \ge 2$.
\item $I$ is \textit{not} complete intersection.
\end{enumerate}
\end{exam}
\begin{proof}
(1) follows from Theorem \ref{Main-CM}.
\par
(2) If $\ell \ge 3$, then the assertion follows from Theorem \ref{Power-cor}.
When $\ell=2$, it follows from the fact $x_{11}x_{12}x_{13} \in I^{(2)} \setminus I^2$.
\end{proof}
\par \vspace{3mm}
\subsection{Some related results}
In the final of this section, we comment a relationship between our results
and the theorem by Minh--Trung \cite{MiT}.
Minh and Trung studied Cohen--Macaulay properties of the symbolic power ideals
for $1$-dimensional simplicial complexes.
\begin{thm-q}[Minh--Trung; see \cite{MiT}]
Let $\ell \ge 3$ be an integer.
Let $I=I_{\Delta}$ be the Stanley--Reisner ideal
of a simplicial complexes of dimension $1$.
Then $S/I^{(\ell)}$ is Cohen--Macaulay if and only if
every pair of disjoint edges of $\Delta$ is contained in a cycle of length $4$.
\end{thm-q}
\par
If $I_{\Delta}$ is generated by degree $2$ monomials,
the ideal $I_{\Delta}$ can be regarded as the edge ideal of a graph $G$.
Then the required condition in the above theorem says that
$G$ is a disjoint union of two complete graphs.
So, their theorem does not conflict our theorem.
\medskip
\section{Finite local cohomology and symbolic power}
\par
In \cite{GT}, Goto and Takayama introduced the notion of generalized
complete intersection complex.
On the other hand, in \cite{TY}, the last two authors defined the notion of
locally complete intersection complex and gave a structure theorem
for those complexes.
Note that $\Delta$ is a generalized complete intersection complex
if and only if $\Delta$ is a pure, locally complete intersection complex.
\begin{defn}[\textrm{cf. \cite{TY}}]
Let $\Delta$ be a simplicial complex on the vertex set $V$.
The complex $\Delta$ is called a \textit{locally complete intersection
complex} if $K[\link_{\Delta} \{v\}]$ is complete intersection for every
vertex $v \in V$.
\end{defn}
\par
The following result gives a structure theorem for locally
complete intersection complexes.
\begin{thm}[\textrm{cf. \cite{TY}}] \label{Structure}
Let $\Delta$ be a simplicial complex on $V$ such that $V \ne \emptyset$.
Then $\Delta$ is a locally complete intersection complex if and only if
it is a finitely many disjoint union of the following connected
complexes$:$
\begin{enumerate}
\item[(a)] a complete intersection complex $\Gamma$ with $\dim \Gamma \ge 2;$
\item[(b)] $m$-gon $(m \ge 3);$
\item[(c)] $m'$-pointed path $(m' \ge 2);$
\item[(d)] a point.
\end{enumerate}
When this is the case, $K[\Delta]$ is Cohen--Macaulay
$($resp., Buchsbaum $)$
if and only if $\dim \Delta =0$ or $\Delta$ is connected $($resp., pure$)$.
\end{thm}
\par
Moreover, for any pure simplicial complex $\Delta$,
it is a locally complete intersection complex if and only if
$S/I_{\Delta}^{\ell}$ has (FLC) for all $\ell \ge 1$
(or, more generally, for infinitely many $\ell \ge 1$).
But, for a fixed $\ell \ge 1$, it is open when $S/I^{\ell}$ has (FLC).
\subsection{FLC properties of symbolic powers}
In this section, we consider the following question, which is closely related to the
above question in the case of edge ideals.
\begin{quest} \label{FLC-quest}
Let $I(G)$ be denote the edge ideal of a graph $G$.
Let $\ell \ge 1$ be an integer.
When does $S/I(G)^{(\ell)}$ have $(FLC)$?
\end{quest}
\par
As one of answers to this question, we prove the second main theorem (Theorem \ref{Main-FLC}).
We first prove the following proposition.
\begin{prop} \label{Main2-flc}
Let $\Delta_{n_1,\ldots,n_r}$ denote the simplicial complex
whose Stanley--Reisner ideal is equal to
the edge ideal of a disjoint union of
complete graphs $K_{n_1},\ldots,K_{n_r}$.
That is,
\[
I_{\Delta_{n_1,\ldots,n_r}} =
I(K_{n_1} \textstyle{\coprod} \cdots \textstyle{\coprod} K_{n_r}).
\]
Let $\Delta$ be a simplicial complex defined by
\[
\Delta = \Delta_{n_{11},\ldots,n_{1d}} \;\textstyle{\coprod}\;
\Delta_{n_{21},\ldots,n_{2d}} \;\textstyle{\coprod}
\; \ldots \;
\textstyle{\coprod} \;
\Delta_{n_{p1},\ldots,n_{pd}},
\]
where one can take all $n_{ij}=1$ when $p \ge 2$.
Put
\[
S=K \left[x_{ij}^{(k)} \;:\;
1 \le i \le d; \;
\;1 \le k \le p; \;
1 \le j \le n_{ki}
\right],
\]
a polynomial ring over $K$, and
\begin{eqnarray*}
I_{\Delta} &=& \left(x_{ij}^{(k)} x_{ij'}^{(k)} \;:\; 1 \le i \le d,\;
1 \le j < j' \le n_{ki}; \;
1 \le k \le p \right)S \\
& + & (x_{ij}^{(k)}x_{i'j'}^{(m)} \;:\; 1 \le i,\,i' \le d,\;
1 \le j \le n_{ki}, 1 \le j' \le n_{mi'},\; 1 \le k < m \le p)S.
\end{eqnarray*}
Then $S/I_{\Delta}^{(\ell)}$ has $($FLC$)$ for every $\ell \ge 1$.
\end{prop}
\begin{proof}
Put $I=I_{\Delta}$.
Since $\dim \Delta_{n_1,\ldots,n_d} =d-1$, $\Delta$ is a
pure simplicial complex of dimension $d-1$.
Hence $S/I^{(\ell)}$ is an equidimensional ring of dimension $d$.
So, it is enough to show that $(S/I^{(\ell)})_x$ is Cohen--Macaulay for
any vertex $x$.
Without loss of generality, we may assume that $x=x_{11}^{(1)}$.
Then
\begin{eqnarray*}
I_{x}&=&(x_{1j}^{(1)}\,:\, 2 \le j \le n_{11})S_x
+(x_{ij}^{(k)} \,:\, 1 \le i \le d,\;
1 \le j \le n_{ki},\; 2 \le k \le p)S_x \\
& & + (x_{ij}^{(1)}x_{ij'}^{(1)} \,:\, 2 \le i \le d,\;
1 \le j < j' \le n_{1i})S_x.
\end{eqnarray*}
By Theorem \ref{SPcomplete} and Corollary \ref{Main1-extend}, $(S/I^{(\ell)})_x$ is Cohen--Macaulay
for all $\ell \ge 1$.
\end{proof}
\par \vspace{2mm}
\begin{exam} \label{special-exam}
Let $G=K_p$ be the complete $p$-graph.
Then $\Delta_p$ is the complementary
simplicial complex of $K_p$.
Moreover, $\Delta_p$ has $p$ connected component$:$
$\Delta_p = \{x_1\} \coprod \ldots \coprod \{x_p\}$.
Then $K[\Delta_p] = K[x_1,\ldots,x_p]/(x_ix_j\,:\, 1 \le i < j \le p)$.
\par
On the other hand,
$K[\Delta_{\bf 1}] = K[x_1,\ldots,x_d]$, where ${\bf 1} = \underbrace{1,\ldots,1}_{d}$.
\end{exam}
\par \vspace{1mm}
Now suppose that $S/I(G)^{(\ell)}$ has $($FLC$)$ for some $\ell \ge 3$.
As $I(G) = \sqrt{I(G)^{(\ell)}}$, $S/I(G)$ also has $($FLC$)$ (see e.g. \cite{HTT}).
Let $\Delta=\Delta(G)$ be the complementary simplicial complex of $G$: $I_{\Delta} = I(G)$.
Then $\Delta$ is pure and
$S_x/(I_{\Delta}^{(\ell)})_x$ is Cohen--Macaulay for every vertex $x \in V$.
Put $\Gamma = \link_{\Delta} \{x\}$.
This implies that $K[V \setminus \{x\}]/I_{\Gamma}^{(\ell)}$ is Cohen--Macaulay.
Therefore, by Theorem \ref{Main-CM}, $I_x$ can be written as
\[
I_x = (y_1,\ldots,y_m)+ I(H_1)S_x + \cdots + I(H_{d-1})S_x,
\]
where $H_1,\ldots,H_{d-1}$ are disjoint complete subgraphs of $G$ and
$y_1,\ldots,y_m \in V$ such that $\{x,y_j\} \in E(G)$
and no elements of $\{y_1,\ldots,y_m\}$ are contained in $H_1 \cup \cdots \cup H_{d-1}$.
\par \vspace{2mm}
In order to prove the second main theorem (Theorem \ref{Main-FLC}), we need the following lemma.
\begin{lemma} \label{Graph}
Let $G$ be a graph, and let $\Delta$ be the complementary simplicial complex of $G$:
$I_{\Delta} = I(G)$.
Suppose $d = \dim S/I(G) \ge 3$ and $\Delta$ is pure.
Moreover, assume that for any vertex $u$, there exist vertices
$y_1,\ldots,y_m$ and complete subgraphs $H_1,\ldots, H_{d-1}$ such that
$I(G)_{u}$ can be written as
\[
I(G)_{u} = (y_1,\ldots,y_m)S_u + I(H_1)S_u+\cdots + I(H_{d-1})S_u,
\]
where $V(G) = \{u\} \coprod \{y_1,\ldots,y_m\} \coprod V(H_1) \coprod \cdots \coprod V(H_{d-1})$.
\par \vspace{2mm}
Then for any vertex $x \in V(G)$, there exist subgraphs $G_0$, $G_1$,\ldots,$G_d$
which satisfies the following conditions:
\begin{enumerate}
\item $V(G) = V(G_0) \coprod V(G_1) \coprod \cdots \coprod V(G_{d-1}) \coprod V(G_d)$ and $x \in G_d$.
\item $G\,|_{V(G_i)} = G_i$ for each $i=0,1,\ldots,d-1,d$.
\item $G_1 \coprod \ldots \coprod G_{d-1} \coprod G_d$ is a disjoint union of complete graphs.
\item For every $y \in G_0$ and for every $z_i \in G_i$ $(i=1,\ldots,d)$,
we have $\{y, z_i\} \in E(G)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Fix $x \in V(G)$.
Applying the assumption to the case of $u=x$, we can find disjoint
complete subgraphs $G_1,\ldots,G_{d-1}$ of $G$
and vertices
$y_1,\ldots,y_m$ such that
\[
I(G)_{x} = (y_1,\ldots,y_m)S_x+ I(G_1)S_x+\cdots + I(G_{d-1})S_x
\]
and $\{y_1,\ldots,y_m\}$ are contained in
$V(G) \setminus V(G_1 \cup \cdots \cup G_{d-1})$.
Then we prove the following claim.
\begin{description}
\item[Claim 1] For any $y \in \{y_1,\ldots,y_m\}$,
if $\{y,z_1\} \in E(G)$ for some $z_1 \in V(G_1)$, then
$\{y, z_i\} \in E(G)$ holds for all $i=1,2,\ldots,d-1$ and
for all $z_i \in V(G_i)$.
\end{description}
\par
Now suppose that $\{y,z_1\} \in E(G)$ for some
$z_1 \in V(G_1)$. Then $yz_1 \in I(G)$.
\par
For any $z_i \in V(G_i)$ $(i=2,\ldots,d-1)$,
if $\{y, z_i\} \notin E(G)$,
then $yz_i \notin I(G)$.
As $z_i \notin I(G)_x$, we have $xz_i \notin I(G)$.
By the choice of $G_i$, $z_1z_i \notin I(G)$.
Hence none of $y,x,z_1$ appears in $I(G)_{z_i}$.
However, since $xy,yz_1 \in I(G)_{z_i}$,
we have $xz_1 \in I(G)_{z_i}$ by assumption, and so $xz_1 \in I(G)$.
This implies that $z_1 \in I(G)_x$.
This contradicts the assumption.
Thus we have $\{y,z_i\} \in E(G)$ for all $z_i \in V(G_i)$ $(i=2,\ldots,d-1)$.
\par
As $d \ge 3$, applying $\{y,z_2\} \in E(G)$ to the above argument,
we obtain that $\{y,z'\} \in E(G)$ for all $z' \in V(G_1)$.
Hence we proved the claim.
\par \vspace{2mm}
By the above claim, by renumbering if necessary,
we may assume that there exists an integer $k$ with $1 \le k \le m$
such that
\begin{enumerate}
\item[(i)] When $1 \le j \le k$,
$\{y_j,z_i\} \in E(G)$ holds for every $1 \le i \le d-1$ and $z_i \in V(G_i)$.
\item[(ii)] When $k+1 \le j \le m$,
$\{y_j,z\} \notin E(G)$ holds for every $z \in V(G_1 \cup \cdots \cup G_{d-1})$.
\end{enumerate}
\par \vspace{2mm} \par \noindent
Then we put $V_0=\{y_1,\ldots,y_k\}$ and $V_d = \{x,y_{k+1},\ldots,y_m\}$ and
$G_0=G\,|_{V_0}$ and $G_d=G\,|_{V_d}$.
In the following, we show that these $G_j$ $(j=0,\ldots,d)$ satisfy all conditions of the lemma.
To show the condition (3), it is enough to show the following claim.
\begin{description}
\item[Claim 2] $G_d$ is a complete graph, and $G_i$ and $G_d$ are
disjoint for each $i=1,\ldots,d-1$.
\end{description}
To see that $G_d$ is a complete graph,
it is enough to show that $\{u,u'\} \in E(G)$ whenever $u,u' \in V(G_d) \setminus \{x\}$.
Suppose $\{u,u'\} \notin E(G)$. Take $z_1 \in V(G_1)$.
Then since $\{x,z_1\}, \{u,z_1\}, \{u',z_1\} \notin E(G)$ and
$xu,xu' \in I(G)_{z_1}$, we have $uu' \in I(G)_{z_1}$, and
thus $\{u,u'\}\in E(G)$.
The latter assertion immediately follows from the definition of $G_d$.
To show the condition (4), it is enough to show the following claim.
\begin{description}
\item[Claim 3] For every $y \in G_0$, $\{y,u\} \in E(G)$ for every $u \in G_d$.
\end{description}
Suppose that $\{y,u\} \notin E(G)$.
Take $z_1 \in V(G_1)$ and $z_2 \in V(G_2)$.
Then, since $d \ge 3$, $z_1,z_2,u$ are distinct vertices and $\{z_1,u\}$, $\{z_2,u\} \notin E(G)$ by Claim 2.
By definition, $\{y,z_1\}$, $\{y,z_2\} \in E(G)$.
By considering $yz_1$, $yz_2 \in I(G)_u$, we get $z_1z_2 \in I(G)_u$.
Hence we have $\{z_1,z_2\} \in E(G)$.
This is a contradiction.
Therefore we conclude that $\{y,u\} \in E(G)$.
\par \vspace{2mm}
We have finished the proof of the lemma.
\end{proof}
\par
We are now ready to prove the second main theorem
in this paper.
\begin{thm} \label{Main-FLC}
Let $G$ be a graph on $V=[n]$,
and let $I(G)\subseteq S=K[v : v \in V]$
denote the edge ideal of $G$.
Let $\Delta=\Delta(G)$ be the complementary simplicial complex of $G$, that is,
$I_{\Delta} = I(G)$.
Let $p$ denote the number of connected components of $\Delta$.
Suppose that $\Delta$ is pure and $d=\dim S/I(G) \ge 3$.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for every $\ell \ge 1$.
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for some $\ell \ge 3$.
\item There exist
$(n_{i1},\ldots,n_{id}) \in \bbN^d$ for every $i=1,\ldots,p$
such that
$\Delta$ can be written as
\[
\Delta = \Delta_{n_{11},\ldots,n_{1d}} \;\textstyle{\coprod}\;
\Delta_{n_{21},\ldots,n_{2d}} \;\textstyle{\coprod}
\; \ldots \;
\textstyle{\coprod} \;
\Delta_{n_{p1},\ldots,n_{pd}}.
\]
\end{enumerate}
\end{thm}
\begin{proof}
$(3) \Longrightarrow (1)$: It follows from Proposition \ref{Main2-flc}.
\par \vspace{1mm} \par \noindent
$(1) \Longrightarrow (2)$ is clear.
\par \vspace{1mm} \par \noindent
$(2) \Longrightarrow (3)$:
We may assume that $p \ge 2$ by Theorem \ref{Main-CM}.
Then we note that $\Delta$ satisfies the assumption of Lemma \ref{Graph}.
Fix $x \in V$.
Let $G_0,\ldots,G_d$ be subgraphs of $G$ determined by Lemma \ref{Graph}.
Then it suffices to show that the connected component containing $x$ (say, $\Delta'$)
is the following form: $\Delta' = \Delta(G')$,
where $G' = G_1 \cup \cdots \cup G_d$, which is a disjoint union of complete graphs.
\par
First we see that $V(\Delta')=V(G_1 \cup \cdots \cup G_d)$.
Let $z \in V(G_1 \cup \cdots \cup G_d)$.
If $z \in V(G_1 \cup \cdots \cup G_{d-1})$, then as $\{x,z\} \notin E(G)$, $\{x,z\} \in \Delta'$,
that is, $z \in V(\Delta')$.
Otherwise, $z \in V(G_d)$. Then there exists a vertex $z' \in V(G_1 \cup \cdots \cup G_{d-1})$
such that $\{z,z'\} \notin E(G)$.
Moreover, as $\{x,z'\} \in \Delta'$, we have $z \in V(\Delta')$.
Hence $V(G_1 \cup \cdots \cup G_d) \subseteq V(\Delta')$.
The converse follows from the condition (4) in Lemma \ref{Graph}.
\par
Next we see that $I_{\Delta'} = I(G_1 \cup \cdots \cup G_d)$.
Since $\Delta'$ is a connected component of $\Delta$, we get
\begin{eqnarray*}
I_{\Delta'} &=& (I_{\Delta} \cap K[V(G_1 \cup \cdots \cup G_d)])S \\
&=& (I(G) \cap K[V(G_1 \cup \cdots \cup G_d)])S \\
&=& I(G_1 \cup \cdots \cup G_d).
\end{eqnarray*}
This yields that $\Delta'= \Delta(G_1 \cup \cdots \cup G_d)$, as required.
\end{proof}
\begin{remark}
Let $t$ be an indeterminate over $R$.
If $R$ has $($FLC$)$ but not Cohen--Macaulay, then $R[t]$ does not have $($FLC$)$.
Hence, in the above theorem, we cannot replace $S$ with $S[t]$,
where $t$ is an indeterminate over $S$.
\end{remark}
\par
Comparing Theorem \ref{Main-CM} and Theorem \ref{Main-FLC}, we obtain the following corollary.
\begin{cor}
Suppose that $d=\dim S/I(G) \ge 3$.
Let $\Delta(G)$ denote the complementary simplicial complex of $G$.
Let $\ell \ge 3$ be an integer.
Then the following conditions are equivalent.
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ and $\Delta(G)$ is connected.
\item $S/I(G)^{(\ell)}$ is Cohen--Macaulay.
\end{enumerate}
Then $G$ is a disjoint union of finitely many complete graphs and
$S/I(G)^{(k)}$ is Cohen--Macaulay for all $k \ge 1$.
\end{cor}
\begin{remark}
In case of $\dim S/I(G)=2$, the following conditions are equivalent:
\begin{enumerate}
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for every $\ell \ge 1$.
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ for some $\ell \ge 1$.
\item $\Delta(G)$ is pure.
\end{enumerate}
In particular, we cannot remove the condition $d =\dim S/I(G) \ge 3$
from the assumption in Theorem \ref{Main-FLC}.
For example, the pentagon cannot be expressed in the form as in Theorem \ref{Main-FLC}(3).
\end{remark}
\subsection{FLC properties of ordinary powers}
\par
In the rest of this section, we consider (FLC) properties of ordinary powers.
Fix a positive integer $\ell$. Let $I=I_{\Delta}$ be a Stanley--Reisner ideal.
If $S/I^{\ell}$ has (FLC), then
$(S/I^{\ell})_x$ is Cohen--Macaulay for all vertex $x$.
Then $I^{(\ell)}/I^{\ell}$ has finite length, it is equal to
$H_{\frm}^0(S/I^{\ell})$.
Then $S/I^{(\ell)}$ also has (FLC).
Hence we have the following theorem,
which gives an improvement of Goto--Takayama theorem in \cite{GT}
in the case of edge ideals.
\begin{thm} \label{ordinary-flc}
Put $d = \dim S/I(G) \ge 1$.
Let $\Delta(G)$ denote the complementary simplicial complex of $G$.
Then the following conditions are equivalent$:$
\begin{enumerate}
\item $S/I(G)^{\ell}$ has $($FLC$)$ for every $\ell \ge 1$.
\item $S/I(G)^{\ell}$ has $($FLC$)$ for some $\ell \ge 3$.
\item $S/I(G)^{(\ell)}$ has $($FLC$)$ and $I(G)^{(\ell)}/I(G)^{\ell}$ has finite length
for some $\ell \ge 3$.
\item $\Delta(G)$ is a pure, locally complete intersection complex.
\end{enumerate}
\end{thm}
\begin{proof}
$(1) \Rightarrow (2) \Leftrightarrow (3)$ is clear.
The equivalence of (1) and (4) follows from \cite{GT}.
On the other hand, $(2) \Rightarrow (4)$ follows from
Theorem \ref{Power-cor} by a similar argument as in \cite{GT}.
\end{proof}
\begin{remark}
By Theorem \ref{Structure}, $(4)$ can be rephrased as follows:
\begin{enumerate}
\item[(4)']
When $d=2$, $\Delta(G)$ is a disjoint union of finitely many paths and $n$-gons with $n \ge 4$.
When $d \ge 3$, $\Delta(G)$ is a disjoint union of finitely many complete intersection complexes of
dimension $d-1$.
\end{enumerate}
\end{remark}
\par
The next example shows that there exists a graph $G$ for which
$S/I(G)^{\ell}$ has (FLC) but \textit{not} Cohen--Macaulay.
\begin{exam} \label{bipartite}
Under the notation as in Theorem $\ref{Main-FLC}$,
$\Delta$ is locally complete intersection if and only if
$\min\{n_{i1},\ldots,n_{id}\} \le 2$.
\par
For instance, for any positive integer $d$,
the edge ideal of the
complete bipartite graph $K_{d,d}$
\[
I=(x_iy_j \,:\, 1 \le i,j \le d) \subseteq S=K[x_1,\ldots,x_d,y_1,\ldots,y_d]
\]
satisfies the following statements$:$
\begin{enumerate}
\item $S/I^{\ell}$ has (FLC) of dimension $d$ for every $\ell \ge 1$.
\item When $d \ge 2$, $S/I^{\ell}$ is \textit{not} Cohen--Macaulay for all $\ell \ge 1$.
\end{enumerate}
\end{exam}
\begin{proof}
By \cite{SVV}, we know that $I^{(\ell)} = I^{\ell}$ for every $\ell \ge 1$; see also Lemma \ref{SVV}.
Hence our theorem says that
$S/I^{\ell}$ has $(FLC)$ for all $\ell \ge 1$.
On the other hand, as $S/I$ is not Cohen--Macaulay,
$S/I^{\ell}$ is \textit{not} Cohen--Macaulay if $d \ge 2$ and $\ell \ge 1$.
\end{proof}
\par
Even if $S/I(G)$ is Cohen--Macaulay, one can find an example of $G$ such that
$S/I(G)^{\ell}$ has (FLC) but not Cohen--Macaulay.
\begin{exam} \label{4-pointed}
Let $\Delta$ be a $4$-pointed path, and $I_{\Delta}=(x_1x_2,x_2x_3,x_3x_4)$.
Then $I_{\Delta}$ is also the edge ideal of the $4$-pointed path $G$.
Then $S/I_{\Delta}$ is Cohen--Macaulay, and $S/I_{\Delta}^{2}$ is
Buchsbaum (thus (FLC)) but \textit{not}
Cohen--Macaulay.
\par
Similarly, for the pentagon $G$, $S/I(G)$ is Cohen--Macaulay and $S/I(G)^{3}$
has (FLC) but not Cohen--Macaulay.
\end{exam}
\par \vspace{2mm}
In general, even if $S/I(G)^{(\ell)}$ has (FLC), it is not necessarily $S/I(G)^{\ell}$ has (FLC)
as the next example shows.
Note that we can construct similar examples of graphs $G$ with $\dim S/I(G)=d$
for every $d \ge 3$.
\begin{exam} \label{doubledelta}
Let $S=K[\{x_i\}_{1 \le i \le 9},\{y_j\}_{1 \le j \le 9}]$, and
let $G$ be a graph such that $\Delta(G) = \Delta_{3,3,3} \coprod \Delta_{3,3,3}$.
Set
\begin{eqnarray*}
I(G) &=& (x_1x_2,x_1x_3,x_2x_3,x_4x_5,x_4x_6,x_5x_6,x_7x_8,x_7x_9,x_8x_9) \\
&& + (y_1y_2,y_1y_3,y_2y_3,y_4y_5,y_4y_6,y_5y_6,y_7y_8,y_7y_9,y_8y_9) \\
&& + (x_iy_j \,:\, 1 \le i,j \le 9).
\end{eqnarray*}
Then
\begin{enumerate}
\item $\dim S/I(G)=3$.
\item $S/I(G)^{(\ell)}$ has (FLC) for every $\ell \ge 1$.
\item $\Delta(G)$ is not a locally complete intersection complex.
\item $S/I(G)^{\ell}$ does not have (FLC) for every $\ell \ge 3$.
\end{enumerate}
\end{exam}
\begin{flushleft}
\begin{tabular}{ccccc}
CI & & Ex.\ref{bipartite} ($K_{d,d}$) & & $4$-pointed path \\[1mm]
\framebox{\bf \large $S/I^{\ell}$ : CM} & $\Longrightarrow$ &
\framebox{\bf \large $S/I^{\ell}$ : (FLC)} & $\Longrightarrow$ &
\framebox{\bf \large $I$ : pure, LCI} \\[3mm]
$\Downarrow$ & & $\Downarrow$ & & $\Downarrow$ \\[2mm]
\framebox{\bf \large $S/I^{(\ell)}$ : CM} & $\Longrightarrow$ &
\framebox{\bf \large $S/I^{(\ell)}$ : (FLC)} & $\Longrightarrow$
& \framebox{\bf \large $S/I$ : Buchsbaum} \\[2mm]
Ex.\ref{union-triangles} ($\Delta_{3,3}$) & & Ex.\ref{doubledelta} ($\Delta_{3,3,3} \coprod \Delta_{3,3,3}$) & &
\end{tabular}
\end{flushleft}
\par \vspace{2mm}
\begin{acknowledgement}
The second author was supported by JSPS 20540047.
The third author was supported by JSPS 19340005.
The authors would like to express their gratitude to the referee for
his careful reading.
\end{acknowledgement}
|
{
"timestamp": "2012-03-12T01:00:28",
"yymm": "1203",
"arxiv_id": "1203.1967",
"language": "en",
"url": "https://arxiv.org/abs/1203.1967"
}
|
\section{Introduction}
\noindent {\bf 1.}{ \bf Introduction}
\setcounter {section}{1}
\setcounter{equation}{0}
\hspace{5.1mm}
The parabolic equation
\beq \label{11}
{\cal L}_ \varepsilon u \equiv \varepsilon \partial_{xxt} u
+ c^2 \partial_{xx}u - \partial_{tt}u = -f
\eeq
\noindent
describes a great deal of models of applied sciences and represents a typical example of hyperbolic equations perturbed by
viscous terms.
According to the meaning of $f $, examples of dissipative phenomena
related to (\ref{11}) are: motions of viscoelastic fluids
or solids \cite {jrs,M,r2}, heat conduction at low
temperature\cite{mps,fr},
sound propagation in viscous gases \cite {la}, propagation of
plane waves in perfect incompressible and electrically conducting
fluids \cite{na}. Moreover, when $f = a u_t+ sin u -\gamma$, the equation (\ref{11})
is the {\it
perturbed sine-Gordon equation} which models the
Josephson tunnel effect in Superconductivity \cite {bp}. Further applications of
(\ref{11}) arise in the study of viscoelastic plates with memory, when
the relaxation function is given by an exponential function
\cite{rf}.
At last, one remarks that even the
Navier- Stokes equations for a compressible gas with small viscosity,
in Lagrangian coordinates, can be reduced to (\ref{11}) with
$f=f(u_t,u_x,u_{xt}, u_{xx})$ \cite{mm}.
Also the meaningful analytical results
concerning the qualitative
analysis of (\ref{11}) are very
numerous and one can refer to an extensive bibliography
(e.g. \cite{mm}-\cite{s}
).
In particular, the behaviour of solutions of (\ref{11}) when $\varepsilon \rightarrow
0$ has been analized in various applications of artificial viscosity
method to
non linear second order wave equation
\cite
{kl},\cite{n}.
But, from a physical point of view, it would be interesting to
estimate the
time - intervals where the
hyperbolic or parabolic behaviour prevails, evaluating so the
influence of dissipative causes on the wave propagation.
These aspects are analyzed in this paper referring to the strip
problem
${ \cal P} _\varepsilon$ for equation (\ref{11}) with a linear $f$.
The Green function $G$ related to ${ \cal P} _\varepsilon$
has already been
determined in \cite{mda} by means of
a rapidly decreasing Fourier series, and
its asymptotic behaviour for $t \rightarrow
\infty$ has been obtained, too.
Now, in the hypotesis of $\varepsilon$ vanishing, appropriate estimates
of $G$
by the {\em slow time} $\varepsilon t$ and the {\em fast time}
$t/\varepsilon$ will be established.
As consequence, the main result is a
rigorous approximation for the solution of the problem ${ \cal P} _\varepsilon$
which holds for all
$ t < \varepsilon ^{-\eta} \
\ (\eta >0).$
\vspace{8mm}
\noindent
{\bf 2. Statement of the problem}
\setcounter{section}{2}
\setcounter{equation}{0}
\hspace{5.1mm}
If $T$ is a positive constant and
\vspace{4mm}
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D =\{(x,t) : 0 \leq x \leq
l, \ \ 0 < t \leq T \}$,
\vspace{3mm}
\noindent
let $u(x,t) $ the regular solution of the boundary initial value
problem:
\beq \label{21}
\left \{
\begin{array}{ll}
& \partial_{xx}
(\varepsilon u_t +
c^2 u) - \partial_{tt} u = \ f(x,t),\ \ \
(x,t)\in D,\vspace{2mm}\\
& u (x,0)=f_0(x), \ \ u_t (x,0)=f_1(x),
\ \ \ \ x\in [0,l],\vspace{2mm} \\
& u (0,t)=0, \ \ u (l,t)=0, \ \ \ \ 0< t \leq T,
\end{array}
\right.
\eeq
\noindent
where $f(x,t)$ is an arbitrary specified function.
Now, denote with $w(x,t)$ the solution of the reduced problem obtained
by (\ref{21}) with $\varepsilon=0$.
To establish a rigorous asymptotic approximation for $u(x,t) $ when $\varepsilon
\rightarrow 0$, we put:
\beq \label{22}
u(x,t,\varepsilon)= e^{-\varepsilon t} w(x,t) + r(x,t,\varepsilon)
\eeq
\noindent
where the {\it error} $r(x,t,\varepsilon )$ must be estimated.
By means of standard computations one verifies that $r(x,t,\varepsilon)$
is the solution of the problem:
\beq \label{23}
\left \{
\begin{array}{ll}
& \partial_{xx}
(\varepsilon r_t +
c^2 r) - \partial_{tt} r = \ F(x,t,\varepsilon),\ \ \
(x,t)\in D,\vspace{2mm}\\
& r (x,0)=0, \ \ r_t (x,0)=0,
\ \ \ \ x\in [0,l],\vspace{2mm} \\
& r (0,t)=0, \ \ r (l,t)=0, \ \ \ \ 0< t \leq T,
\end{array}
\right.
\eeq
\noindent
where the source term $F(x,t,\varepsilon)$ is:
\beq \label{24}
F(x,t,\varepsilon)=f(x,t)(1-e^{-\varepsilon t})+ \ e^{-\varepsilon t}[-\varepsilon
\lambda_t+\varepsilon^2 (w+w_{xx})],
\eeq
\noindent
with $\lambda= 2w+w_{xx}$.
The problem (\ref{23}) has already been solved in \cite{mda} and
the solution is given by:
\beq \label{25}
r(x,t,\varepsilon)= -\int_{0}^{l} d\xi \\ \int_{0}^{t}
F(\xi,\tau,\varepsilon) \\\\\\ G(x,\xi,t-\tau) \ d\tau,
\eeq
\noindent
where $G(x,\xi,t)$ is:
\beq \label{26}
G(x,\xi,t)=\frac{2}{l} \ \ \\ \ \sum_{n=1}^{\infty} \ \ H_n(t) \ \
\sin \gamma_n x \ \sin \gamma_n \xi,
\eeq
\noindent
with
\vspace{3mm}
\beq \label{27}
H_n(t)= \ \ \frac{e^{-bn^{2}t}}{bn^2 \sqrt{1-(k/n)^2}} \
\sinh \{ bn^2t \sqrt{1-(k/n)^2}\},
\eeq
\noindent
and
\beq \label{28}
b= \frac{\pi^2}{2l^2}\varepsilon = q \varepsilon, \\\\ \ \ \ \ k=\frac{2cl}{\pi\varepsilon} \
\ \ \ \\\ \ \ \gamma_n =\frac{\pi}{l} n.
\eeq
\vspace{8mm}
\noindent
{\bf 3. Estimates of the Green function by fast and slow times}
\setcounter{section}{3}
\setcounter{equation}{0}
\hspace{5.1mm}
When $\varepsilon \rightarrow 0 $, two characteristic times affect the
behaviour of $G$, i.e. $ \tau =\varepsilon t$ ({\em slow time}), and $ \theta
= t/ \varepsilon$ ({\em fast time}).
To point out the different contributions an
appropriate form of G we will considered.
For this, if
$N= [\frac {2cl}{\pi \varepsilon}]$, the G-function can be given the
form:
\beq \label{31}
G = \frac{2}{l} \ \ \\ \{ \sum_{n=1}^{N} \
+ \sum_{N +1}^{\infty} \\ \} H_n(t)
\sin(\gamma_nx) \sin(\gamma_n\xi) = G_1 + G_2.
\eeq
\noindent
where, for $n<N,$ the functions $H_n$ are:
\beq \label{32}
H_n(t)= \ \ \frac{e^{-bn^{2}t}}{bn^2 \sqrt{(k/n)^2-1}} \
\sin \{ bn^2t \sqrt{(k/n)^2-1} \}.
\eeq
\noindent
If $\alpha $ is an arbitrary constant such that:
\beq \label{33}
1/2 < \alpha <1 , \ \ \ \ \ N_\alpha
=[ \frac{2cl}{\pi \varepsilon ^\alpha}],
\eeq
\noindent
the term $G_1 $ can be written:
\beq \label{34}
G_1(x,\xi,t)= \frac{2}{l} \ \ \\ \ \{ \sum_{n=1}^{ N_\alpha} \ H_n(t) \
+ \sum_{ N \alpha+1}^{N} \\ H_n(t) \}
\sin (\gamma_n x) \ \sin (\gamma_n \xi) \ .
\eeq
\noindent
It is easy to prove that when $1\leq n \leq N_\alpha$ it results:
\vspace{2mm}
\beq \label{35}
\sqrt{(k/n)^2-1} \geq
\frac{\sqrt{1-\varepsilon^{2(1-\alpha)}}}{\varepsilon^{1-\alpha}}; \ \ \
e^{-bn^{2}t} \leq e^{-qt \varepsilon}.
\eeq
\vspace{2mm}
\noindent
Otherwise, if $ N_\alpha+1 \leq n \leq N $, one has $N =
k-\beta$ with $0<\beta<1$, and it is:
\vspace{2mm}
\beq \label{36}
\sqrt{(k/n)^2-1} \geq
\frac{ \sqrt{\pi\varepsilon \beta } \sqrt{4cl-\beta \pi \varepsilon}}{
(2cl-\pi \varepsilon\beta)}; \ \ \
e^{-bn^{2}t} \leq e^{-2 c^2 t / \varepsilon^{2\alpha-1}}.
\eeq
\vspace{2mm}
\noindent
In particular, when $k$ is an integer one has
$\beta=1$ and the term $t
e^{-2c^2 t/\varepsilon}$ must be considered too.
For all $ \varepsilon \in \ ]0,\varepsilon_0] \ (\varepsilon_0 <1)$, the
formulae
(\ref{35}) - (\ref{36}) allow to obtain the following estimate for
$G_1$:
\beq \label{37}
|G_1(x,\xi,t)| \leq A_0 \ \varepsilon^{-\alpha}
e^{-qt\varepsilon} + A_1 \ \varepsilon ^{-3/2} \
e^{-c^2t/\varepsilon^{2\alpha-1}},
\eeq
\noindent
where the constants $A_0 \ \ A_1 $ don't depend on $\varepsilon $ .
As $\varepsilon < \varepsilon^{1-2\alpha}$, the prevailing term in
(\ref{37}) is related to the slow time $\tau=\varepsilon t$. So, the
circular component $G_1$ is controlled by the slow time. On the
contrary, the hyperbolic component $G_2 $ is characterized only by the
fast time $\theta$.
In fact, let:
\beq \label{38}
C=\frac{\pi (1-\beta)4cl }{2 cl+\pi
(1-\beta)}, \ \ \ C_1 =\frac{2 \zeta(2)}{qlC}
\eeq
\noindent
with $\beta \equiv 0$ if $k$ is an integer. Observing that
$\forall n \geq N+1$, one has (see, f.i. \cite{mda}):
\vspace{2mm}
\beq \label{39}
bn^2t (1\pm \sqrt{1-(k/
n)^{2}}) \geq c^2 t / \varepsilon,\ \ \sqrt{1-(k/n)^{2}} \geq
\varepsilon \ C,
\eeq
\vspace{2mm}
\noindent
and consequently it results:
\vspace{2mm}
\beq \label{311}
|G_2(x,\xi,t)| \leq C_1 \ \varepsilon^{-2} \
e^{-c^2t/\varepsilon}.
\eeq
\noindent
So, if $M_0 = max \{A_1
, C_1 \}$ and $ 1/2 < \alpha < 1 $, the following
theorem holds:
\vspace{3mm}
{\bf Theorem
3.1}
{\em For all} $\varepsilon \in (0, \varepsilon _0] (\varepsilon
_0 < 1 ) $ and $(x,t) \in D $, {\em the Green function} $G(x,\xi,t) $
{\em verifies the following estimate}:
\vspace{2mm}
\beq \label{313}
|G(x,\xi,t)| \leq A_0 \ \varepsilon^{-\alpha}
e^{-qt\varepsilon} + M_0 \ \varepsilon
^{-3/2}
e^{-c^2t/\varepsilon^{2\alpha-1}}.
\eeq
\noindent
{\em where the constants} $A_0, \ M_0$ {\em do not depend on} $\varepsilon$.
\hbox{} \hfill \rule {1.85mm}{2.82mm}
\vspace{8mm}
\noindent
{\bf 4. On the behaviour of the solution}
\setcounter{section}{4}
\setcounter{equation}{0}
\hspace{5.1mm}
Now, the remainder term $r(x,t,\varepsilon)$ of (\ref{22}) can be
estimated. Referring to the function $f$ defined in
(\ref{24}), let
\vspace{3mm}
\beq \label{41}
\ \ ||F||= max \{\sup_{D} \ |f(x,t)|,
\sup_{D} \ [ |\lambda_t| +\varepsilon |\lambda-u|] \ \}.
\eeq
\noindent
Then, one has the following theorem.
\vspace{5mm}
{\bf Theorem 4.1 -} {\em Let} $F(x,t,\varepsilon)$
$ \in C^1(D)$
{\em and let} $ F, F_x,F_t $ {\em bounded for all t.}
{\em Then, the error term} $r(x,t,\varepsilon)$ {\em verifies the estimate: }
\beq \label{42}
|r(x,t,\varepsilon)| \ < \ k \ ||F|| \ (\varepsilon ^{ \eta} \
t)^2,
\eeq
\noindent
{\em where the constants} k {\em and} $\eta$ {\em do not depend on}
$\varepsilon $ {\em and} $
\eta \in (0,1/2) $.
\vspace{3mm}
{\bf Proof}- First, by (\ref{24})-(\ref{25}) one deduces:
\beq \label{43}
| r(x,t,\varepsilon )| \leq l \varepsilon \int _0^t e^{-\varepsilon \tau}
\{ |\lambda_t(x,\tau)|+ \varepsilon |\lambda -u|\}
|G(x,\xi,t-\tau)| d\tau
\eeq
\hspace* {2cm} \[+ l \int_0^t |f(x,\tau)|
|1-e^{-\varepsilon \tau} |G(x,\xi,t-\tau)| d\tau.\]
\noindent
Further, by means of the well- known
inequality \cite{m}:
\beq \label{44}
e^{-x} \leq [\gamma/(ex)]^\gamma \ \ \ \forall \gamma>0, \forall x>0
\eeq
\noindent
by (\ref{311}) and (\ref{43}) one can deduce:
\beq \label{45}
|r| \leq \ ||F|| \ l \ t^2
\ \{ 3/2 \ A_0 \ \varepsilon^{1-\alpha} +
\frac{2 M_0 }{(e c^2)^\gamma (1-\gamma)}
\frac{\varepsilon^{(2\alpha-1)\gamma}}{\sqrt{\varepsilon}} \}
\eeq
\noindent
So, if $\alpha$ and $\gamma$ are such that $3/4<\alpha<1$ and
$[2(2\alpha-1)]^{-1}< \gamma <1$, it suffices to put
\beq \label{48}
2\eta=min \{ (2\alpha-1)\gamma-\frac{1}{2}, \ 1-\alpha \} ;
k= max \{ \frac{3}{2}A_0, \frac{2 M_0 }{(e c^2)^\gamma (1-\gamma)} \},
\eeq
\noindent
to deduce (\ref{42}).
\hbox{} \hfill \rule {1.85mm}{2.82mm}
As consequence of Theorem 4.1, finally we can observe that
:
\vspace {3mm}
{\em When} $\varepsilon \rightarrow 0,$ {\em the solution
of the problem} (\ref{21}) {\em can be approximatated by means of the
following formula:
}
\vspace{3mm}
\beq \label{49}
u(x,t,\varepsilon)= e^{-\varepsilon t} w(x,t) + r(x,t,\varepsilon)
\eeq
\vspace{3mm}
\noindent
{\em where the error }$r(x,t,\varepsilon)$ {\em is bounded
for all } $t < (1/ \varepsilon) ^{\eta}. $
\hspace{5.1mm}
\begin {center} {\bf References}
\end {center}
\begin {thebibliography}{99}
{\small
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\vspace*{-3mm}
\bibitem {M} J. A. Morrison,{\it Wave propagations in rods of Voigt
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\bibitem {r2} P. Renno, {\it On some viscoelastic models}, Atti Acc.
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\bibitem{mps} A. Morro, L. E. Payne. B. Straughan, {\it Decay, growth,
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{fr} N. Flavin, S. Rionero {\it Qualitative Estimates for Partial
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\bibitem {la} H. Lamb, {\it Hydrodynamics}, Dover Publ. Inc., 708 (1932)
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\bibitem {bp} A.Barone, G. Paterno', {\it Physics and Application of
the Josephson Effect} Wiles and Sons N. Y. 530 (1982)
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propagations of Singularities for Viscoelastic Plates}, J.Math. Anal
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equations perturbed by viscous terms} Walter deGruyher Berlin N. Y.
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mixed problem for a class of equation of third order}, Siberian Math.
J. 22 (6) 867-872 (1981)
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\bibitem {ks} S. Kawashima and Y. Shibata, {\it Global Existence and
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}, Comm. in Math. Phys., 189-208 (1992)
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\bibitem {bb} G. I. Barenblatt, M. Bertsch, R. Del Passo, and M. Ughi, {\it
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24,(6) 1414-1439 (1993).
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\bibitem {ddr} B. D'Acunto, M. De Angelis, P. Renno, {\it Fundamental
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\bibitem {ddr} B. D'Acunto, M. De Angelis, P. Renno, {\it Estimates
for the perturbed sine-Gordon equation
}. Suppl. Rend. Cir. Palermo 57
199-204 (1998)
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\bibitem {cfl} A. T. Cousin and C. L. Frota and N. A. Lar'kin {\it
Regular Solution and Energy Decay for the Equation of Viscoelasticity
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224, 273-296 (1998).
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\bibitem {cl} A. T. Cousin and N. A. Lar'kin {\it On the nonlinear
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Nonlinear Analysis, Theory and Appl. 31 (1/2) 229-242 (1998)
\vspace*{-3mm}
\bibitem {s} Y.Shibata {\it On the Rate of Decay of Solutions to
linear viscoelastic Equation}, Math.Meth.Appl.Sci.,23 203-226 (2000)
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\bibitem{kl} A.I. Kozhanov N. A. Lar`kin, {\it Wave equation with
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\vspace*{-3mm}
\bibitem {mda} M. De Angelis {\it Asymptotic analysis for the strip
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\bibitem {r1} P. Renno {\it On a Wave Theory for the Operator
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}
\end{thebibliography}
\end{document}
|
{
"timestamp": "2012-03-13T01:01:10",
"yymm": "1203",
"arxiv_id": "1203.2253",
"language": "en",
"url": "https://arxiv.org/abs/1203.2253"
}
|
\section{Introduction}
In the experimental setup described in the previous paragraph, bouncing droplets on a fluid bed may couple with the underlying wave field to create a combined particle-wave system. Upon each bounce with the fluid bed, the droplet generates a wave field in the bath. As the droplet continues to bounce on the bed, the previously generated waves act to guide the trajectory of the particle. Following \cite{CouderFort2006, ProtiereBoudaoudCouder2006} we refer to the combination of the moving droplet dressed with a local wave as a \emph{walker}. In many cases, the underlying wave field may also act as a medium to guide droplets in the vicinity of domain boundaries or even other droplets
(see also \cite{Bush2010} for a review). For instance, in the first of a series of experiments \cite{CouderProtiereFortBoudaoud2005, ProtiereBoudaoudCouder2006}, the authors show that two droplets may orbit or scatter without direct contact, but rather through the mediation of the underlying fluid bath. In addition, \cite{ProtiereBoudaoudCouder2006} goes further to investigate the phase diagram for bouncing and walking droplets with different sizes and accelerations of the fluid bed, as well as providing the first steps towards a phenomenological model for the droplet wave system. In their model, the authors average over one droplet interaction to obtain an ODE describing droplet motion. A more detailed understanding of the droplet interactions and motion can also be found in \cite{MolacekBush1, MolacekBush2}.
Several recent experiments have also examined the trajectories of bouncing droplets in bounded domains. For instance, when a droplet is sent through a single slit scattering
experiment \cite{CouderFort2006}, the droplet may propagate with an apparently random scattering angle. Upon repeating the experiment for many droplets, the data shows that the droplets propagate with a probability distribution statistically analogous to a scattered wave amplitude. Meanwhile, in \cite{EddiFortMoisyCouder2009} Eddi et al. examine the motion of droplets in a confined billiard setting. In analogy with quantum tunneling, they show that under sufficient conditions, instead of reflecting off the boundary walls, the droplets may occasionally cross a \emph{dead zone} region which does not support stable walking trajectories. The paper also experimentally shows that one may obtain stable quasiperiodic trajectories, or the appearance of ergodic trajectories for droplets in a billiard domain. Further experiments\cite{HarrisMoukhtarFortCouderBush} examine the statistical nature of the droplet position within a confined domain. Lastly, \cite{FortEddiBoudaoudMoukhtarCouder2010} examine the trajectory of bouncing droplets in a rotating fluid. In a qualitative analogy with Landau orbits, they show that at sufficient forcing of the fluid bed, the droplets move in circular orbits with quantized radii. The papers \cite{CouderFort2006, FortEddiBoudaoudMoukhtarCouder2010} also provide a simplified phenomenological model for the droplet motion and numerically reproduce the qualitative behavior of the droplet trajectories. One drawback of the model is that the fluid wave behavior is imposed as an ansatz without a model for the evolution of the fluid bath. Subsequent theoretical improvements were included in \cite{EddiSultanMoukhtarFortRossiCouder2011} where the authors provide a more detailed description of the waves generated near the Faraday threshold.
The experiments have not only been limited to the behavior of single droplets, but also include many droplet systems. For instance, \cite{EddiTerwagneFortCouder2008} demonstrate that droplet pairs may form localized, orbiting bound states, and even small crystalline structures. Further work \cite{EddiDecelleFortCouder2009} shows that the crystalline structures may include many different Archimedean lattices, while in \cite{EddiBoudaoudCouder2011}, Eddi et al. examine the instabilities in periodic hexagonal and square arrays.
In the first section of our paper, we introduce an iterative map model for the droplet trajectories. In section (II) we outline a gravity-capillary model, similar to the one introduced in \cite{EddiSultanMoukhtarFortRossiCouder2011}, for the underlying fluid bath. Using the wave model, we show that the droplet motion is primarily a result of the most recent bounce, and then analytically examine the bouncing to walking bifurcation. We show that the sum of past droplet impacts create an outgoing standing wave, and that the model also predicts a mechanism for the transition to chaotic droplet motion. Lastly, we examine droplet trajectories in bounded, square domains. In the first case, we examine domains much larger than the fluid wavelength and show the existence of either dense trajectories or quasiperiodic orbits. In the second case, we examine small domains and show that trajectories may bounce between different regions of space.
\section{The Iterative map}
In this paper we are interested in understanding the two dimensional dynamics of the fluid droplets, as they propagate around the oscillating bath. As a result, we work in two spatial dimensions and record only the two dimensional position of the droplet, thereby ignoring the vertical (bouncing) motion of the particle.
Let $\Omega \subset \mathbbm{R}^2$ be the two dimensional domain of the fluid bath, and denote the continuous time position and velocity of the fluid droplet by $\mathbf{y}(t) \in \Omega$ and $\mathbf{v}(t)$ respectively. In addition, we let $h(\mathbf{x}, t; \mathbf{y})$ denote the surface height of the fluid bath at $\mathbf{x}$. Since the wave field depends on the past history of the particle, we include $\mathbf{y}(\tau)$ (for $\tau < t$) as a functional parameter in the wave field.
To derive a set of dynamic equations, we adopt a simplified phenomenological model for the contact interaction of the fluid bed and droplet. In doing so we do not resolve the microscopic interaction between the fluid bed and droplet, but rather assume a spontaneous interaction. Specifically, we make the following simplifying assumptions pertaining to the interaction:
\begin{enumerate}\label{Assump_1}
\item [A1.] The nondissipative forcing from the fluid bath on the particle is proportional to the wave field slope at the time and location of contact. \label{Assump_2}
\item [A2.] At each bounce, the particle provides a point source forcing to the fluid wave field.
\end{enumerate}
We note that the general assumptions (A1)--(A2) are also made in the phenomenological model taken in \cite{CouderFort2006, FortEddiBoudaoudMoukhtarCouder2010}. In general, the droplet size (droplets $\leq 1mm$ in diameter) can result in large variations in the dynamics, however (A1)--(A2) simplifies them to be point particles.
Following assumption (A1), the equations of motion for the fluid droplet take the form
\begin{eqnarray} \label{Motion1}
\dot{\mathbf{y}} &=& \mathbf{v} \\ \label{Motion2}
\dot{\mathbf{v}} &=& -\big[F \nabla h(\mathbf{x}, t; \mathbf{y}) \big|_{\mathbf{x} = \mathbf{y}} + \gamma \mathbf{v}^- \big] \delta_p(t),
\end{eqnarray}
where $F$ is the amplitude of the forcing on the particle, and $\delta_p(t) = \delta_p(t + T)$ is the periodic Dirac delta function. To capture dissipation in the droplet-bath interaction, we have added the additional term, $\gamma \mathbf{v}^- \delta_p(t)$, where $\mathbf{v}^-$ is understood to be the velocity prior to impact. Mathematically, we take $\mathbf{v}^-(t) \delta_p(t) = \mathbf{v}(t - \epsilon) \delta_p(t)$ with $\epsilon \rightarrow 0$, to correctly define the product of a distribution $\delta(t)$ with a discontinuous function $\mathbf{v}$. One should note that in the general case, the shape of the wave field $h(\mathbf{x}, t; \mathbf{y})$, depends on the previous history of the particles position $\mathbf{y}(t)$.
Since the forcing is periodic, we may integrate the equations of motion (\ref{Motion1})--(\ref{Motion2}) over one period to obtain a discrete map. To compactify the notation, we first choose the period of bouncing (T) as the natural time scale, and let $\mathbf{y}_n = \mathbf{y}(n + \epsilon)$ and $\mathbf{v}_n = \mathbf{v}(n + \epsilon)$ with ($\epsilon \rightarrow 0$), denote the droplets position immediately following the $nth$ bounce. Upon integrating equations (\ref{Motion1})--(\ref{Motion2}), we obtain the iterative map
\begin{eqnarray} \label{Map_y}
\mathbf{y}_{n+1} &=& \mathbf{y}_n + \mathbf{v}_n \\ \label{Map_v}
\mathbf{v}_{n+1} &=& (1-\gamma) \mathbf{v}_n - F \nabla h(\mathbf{x}, n+1; \mathbf{y}_n, \mathbf{y}_{n-1}, \ldots)\big|_{\mathbf{x} = \mathbf{y}_{n+1}}.
\end{eqnarray}
In the special case when $\gamma = 0$, and $h = h(\mathbf{x})$ is independent of the droplets history, then (\ref{Map_y})--(\ref{Map_v}), reduce to the \emph{standard map} \cite{Chirikov1979}. The standard map is the discrete analogue of a particle moving in a strobed Hamiltonian system, and consequently the map preserves phase-space volume. We note that the presence of a path memory, with or without dissipation, breaks the discrete analogue of Liouville's theorem, thereby allowing stable attractors in phase space.
Thus far, we have not explicitly stated a model for the evolution of the fluid bath. Hence at this point one could take a variety of models for $h(\mathbf{x}, t; \mathbf{y})$ that accurately capture various properties of the underlying fluid field.
\section{A model for the wave field}
In the following section, we introduce gravity-capillary waves \cite{LandauLifshitzFluids} as a model for the fluid wave evolution. We show that the most recent droplet impact dominates the contribution to walking, while including many previous impacts creates an outgoing standing wave centered about the droplet. We then use the approximation to analytically examine the resulting bifurcation. Lastly, we examine the effects of dissipation on the walking velocity.
In the linear theory of gravity-capillary waves, one assumes an irrotational velocity field and models the fluid with a velocity potential $\phi(\mathbf{x}, z, t)$ with $\mathbf{x} \in \Omega$. Here, $z$ is the vertical direction and $z = 0$ is aligned with the unperturbed surface height $h = 0$. It then follows that $h(\mathbf{x}, t)$ and $\phi(\mathbf{x}, z, t)$ satisfy the linearized gravity-capillary equations. Namely, $\phi(\mathbf{x}, z, t)$ is a harmonic function which vanishes at $z \rightarrow -\infty$
\begin{equation}
\begin{array}{rcll}
\Delta \phi & = & 0 \quad & \textrm{for}\;\; z < 0
\\ \rule{0ex}{2.5ex}
\phi & = & 0
\quad & \textrm{for}\;\; z \rightarrow -\infty.
\rule{0ex}{2.5ex}
\end{array}
\end{equation}
In addition, $\phi(\mathbf{x}, z, t)$ is coupled to $h(\mathbf{x}, t)$ via the kinetic and dynamic boundary conditions at $z = 0$
\begin{eqnarray} \label{Kinetic_BC}
h_t & = & \phi_z \\ \label{Dynamic_BC}
\phi_t + g h - \frac{\sigma}{\rho} \Delta_2 h + 2 \frac{\nu}{\rho} \phi_{zz} & = & -\sum_{j = 0}^{n-1} \frac{f_0}{\rho} \delta(\mathbf{x} - \mathbf{y}_{n-j}) \delta(t - t_{n-j}).
\end{eqnarray}
Here $\Delta_2$ is the 2D Laplacian in $\mathbf{x}$, while $\Delta$ is the 3D Laplacian in $(\mathbf{x}, z)$. Meanwhile, the point source forcing at time $t_n = n$ enters as a delta function, via assumption (A2), to model the instantaneous interaction with the bed. We remark that the dissipative term in equation (\ref{Dynamic_BC}) is only an effective dissipation as the assumption of an irrotational fluid field breaks down in a small viscous boundary layer near the wave field surface \cite{EddiSultanMoukhtarFortRossiCouder2011, KumarTuckerman1994, LandauLifshitzFluids}.
To nondimensionalize the equations, we again use the period of bouncing ($T$) as the time scale, and take a length scale \cite{LengthScale} set by the pure gravity waves as $L = g T^2$. Letting, $\mathbf{x} \rightarrow L \mathbf{x}$, $t \rightarrow T t$, we also let $\phi \rightarrow (\frac{L^2}{T}\phi )(\frac{f_0 T}{\rho L^4})$, $h \rightarrow (L h)(\frac{f_0 T}{\rho L^4})$ and $F \rightarrow (\frac{L}{T}F)(\frac{\rho L^4}{f_0 T})$, where the dimensionless factor $(\frac{f_0 T}{\rho L^4})$ is included to simplify the equations to
\begin{eqnarray} \label{DimensionlessEq1}
h_t &=& \phi_z \\ \label{DimensionlessEq2}
\phi_t + h - B^{-1} \Delta_2 h + 2\mu \phi_{zz} &=& - \sum_{j = 0}^{n-1} \delta(\mathbf{x} - \mathbf{y}_{n-j})\delta(t - t_{n-j}).
\end{eqnarray}
Here $\mu = \nu/ (\rho g^2 T^3)$ is the dimensionless viscosity, while $B = g\rho L^2/\sigma = g^3 \rho T^4/\sigma$ is the ratio of buoyancy to surface tension restoring forces in the wave field. Typical experimental values \cite{ProtiereBoudaoudCouder2006}, are $\sigma = .0209 Nm^{-1}$, $\rho = .965 \times 10^3 kg$ $m^{-3}$, $T = 25^{-1}s$, $\nu = 5 \times 10^{-3} Pa s$ to $0.1 Pa s$ yielding $B \sim 120$, $L \sim 16mm$ and $\mu \sim .001$ to $.016$. For calculations we will typically take $\mu = 0.008$, which is the midpoint of the viscosity range.
Along with rest conditions at $t_0$
\begin{eqnarray} \label{InitialData1}
\phi(\mathbf{x}, z, t_0) &=& 0 \\ \label{InitialData2}
h(\mathbf{x}, t_0) &=& 0
\end{eqnarray}
we take equations (\ref{DimensionlessEq1})--(\ref{DimensionlessEq2}) to describe the fluid evolution between bounce $n$ and $n+1$.
The map depends on four parameters, the dissipation of the wave ($\mu$), the dissipation of the droplet bounce ($\gamma$), the shape of the dispersion relation ($B$) and the acceleration or forcing of the particle ($F$). In the subsequent sections, we study (\ref{DimensionlessEq1})--(\ref{DimensionlessEq2}) to understand the resulting particle motion for various forcing and domains.
\subsection{Solution in free space} \label{FreeSpace}
In this section we construct the iterative map (\ref{Map_y})-(\ref{Map_v}) with (\ref{DimensionlessEq1})--(\ref{DimensionlessEq2}) in free space ($\Omega = \mathbbm{R}^2$). We do this by solving for the wave field $h(\mathbf{x}, t; \mathbf{y}_n)$ from one point source interaction at time $t_n$:
\begin{eqnarray} \label{IRDimensionlessEq1}
h_t &=& \phi_z \\ \label{IRDimensionlessEq2}
\phi_t + h - B^{-1} \Delta_2 h + 2\mu \phi_{zz} &=& -\delta(\mathbf{x} - \mathbf{y}_{n})\delta(t - t_{n}).
\end{eqnarray}
Linear superposition then allows one to add the contributions from many past bounces. With the exception of adding dissipation, the solution follows very closely that of the standard Cauchy-Poisson problem for forced gravity-capillary waves. Specifically, we seek an expansion for $\phi$ of the form ($k = |\mathbf{k}|$)
\begin{eqnarray} \label{Ansatz}
\phi(\mathbf{x}, z, t) = \int A(\mathbf{k}, t) e^{k z} e^{\imath \mathbf{k}\mathbf{x}} \du \mathbf{k}.
\end{eqnarray}
Upon substituting the ansatz (\ref{Ansatz}) into equations (\ref{DimensionlessEq1})--(\ref{DimensionlessEq2}), one obtains an initial value problem for each $A(\mathbf{k}, t)$. The solution for $h(\mathbf{x}, t; \mathbf{y}_n)$ over $t > t_n$ then becomes
\begin{eqnarray} \label{GCPointSource}
h(\mathbf{x}, t; \mathbf{y}_n) &=& - \int \frac{k}{2\pi \omega_D} \sin(\omega_D (t-t_n) ) e^{\imath \mathbf{k}(\mathbf{x} - \mathbf{y}_n)} e^{-\mu k^2 (t-t_n)} \du \mathbf{k},
\end{eqnarray}
where $\omega_D^2 = (k + B^{-1} k^3) - \mu^2 k^4$ is the dispersion relation. Note that in the case when $\omega_D^2 < 0$, the function $\frac{\sin(\omega_D)}{\omega_D} e^{-\mu k^2}$ becomes a strict exponential decay, corresponding to overdamping of the large modes.
To compute the field, we first integrate over the angular component $\theta$ in (\ref{GCPointSource}) to obtain the impulse response
\begin{eqnarray} \label{Simplified_h}
h(\mathbf{x}, t; \mathbf{y}_n) &=& h_{0}(|\mathbf{x} - \mathbf{y}_n|, t - t_n) \\
h_{0}(r, \tau) &=& -\int_{0}^{\infty} \frac{k^2}{\omega_D} \sin(\omega_D \tau) J_0(k r) e^{-\mu k^2 \tau} \du k
\end{eqnarray}
Here $J_0(x)$ is the zeroth order Bessel function. As a one-dimensional integral, we may numerically evaluate (\ref{Simplified_h}) and compute the iterative map
\begin{eqnarray} \label{Update_y}
\mathbf{y}_{n+1} &=& \mathbf{y}_n + \mathbf{v}_n \\ \label{Update_v}
\mathbf{v}_{n+1} &=& (1-\gamma) \mathbf{v}_n + F \mathbf{g}(\mathbf{y}_{n+1}, \ldots \mathbf{y}_1) \\ \label{Update_Map}
\mathbf{g}(\mathbf{y}_{n+1}, \dots, \mathbf{y}_1) &=& - \sum_{j = 0}^{n-1} \nabla h_0(r_{n-j}, j+1)\big|_{\mathbf{x} = \mathbf{y}_{n+1}}.
\end{eqnarray}
where for brevity we have let $r_n = |\mathbf{x} - \mathbf{y}_n|$.
To extract a simplified expression for the map, we now focus on computing $h_0(r, \tau)$ for the physically relevant parameters $B = 120$ and $\mu = 0.008$. Here figure (\ref{WaveFieldFreeSpace}) shows the radial wave field $h_0(r, \tau)$ for different $\tau$. In particular the point source wave both disperses and radiates outward. Within $\tau \geq 3$ periods, the wave has traveled several wavelengths and has minimal support near $r = 0$. At $\tau = 2$ and values of $r < 0.4$, the contribution of the wave $\partial_r h_0(r, 2)$ to the iterative map $(\ref{Update_Map})$ is small compared to $\partial_r h_0(r, 1)$. We therefore neglect the second impact and make the following assumption
\begin{enumerate}\label{Assump_3}
\item [A3.] The droplet walking dynamics depend only on the most recent droplet bounce. Explicitly, the assumption yields
\begin{eqnarray} \label{Map_A3}
\mathbf{g}(\mathbf{y}_{n+1}, \mathbf{y}_n) = - \nabla h_0(r_n, 1)\big|_{\mathbf{x} = \mathbf{y}_{n+1}}.
\end{eqnarray}
\end{enumerate}
The assumption (A3) is most valid provided the forcing $F$ is far below the Faraday threshold. Near the Faraday threshold, the equations (\ref{Kinetic_BC})--(\ref{Dynamic_BC}) no longer accurately describe the wave field. Instead, one must incorporate the periodic forcing of the bed into a model for $h(\mathbf{x}, t)$. Such an inclusion results in two effects: i) a different shape of the wave field radiating from point sources, namely one with a fixed wavelength, ii) a strong memory of past bounce locations.
Although the Faraday threshold does not enter into the current model, using known experimental data, we may estimate the valid region of forcing $F$. Experiments \cite{EddiSultanMoukhtarFortRossiCouder2011} show that Faraday wave memory effects become important when the bed acceleration $\tilde{\gamma}_b$ is close to $\tilde{\gamma}_f$. Specifically, data collected for a bouncing period of $T = 40$ suggests the crossover occurs when $(\tilde{\gamma}_f - \tilde{\gamma}_b )/\tilde{\gamma}_f$ is somewhere between $(0.07, 0.17)$. For this approximation, we take the crossover to be at $10^{-1}$. Experiments \cite{EddiSultanMoukhtarFortRossiCouder2011, ProtiereBoudaoudCouder2006} have also measured the walking and Faraday threshold accelerations of the bed at $\tilde{\gamma}_w \sim 3.75 g$ and $\tilde{\gamma}_f \sim 4.5g$ for $T = 25^{-1}$ and $\tilde{\gamma}_w \sim 3.2 g$ and $\tilde{\gamma}_f \sim 4.1g$ for $T = 40^{-1}$. The walking accelerations may vary depending on droplet size, however typical values for fixed $T = 25^{-1}$ are between $3.1g$ and $3.8g$. Finally, we note that $F \propto \tilde{\gamma}_b$, so that $F/F_{crit} = \tilde{\gamma}/\tilde{\gamma}_w$. Estimating the maximum value of $F/F_{crit}$ then yields
\begin{eqnarray}
1 - \frac{\tilde{\gamma}}{\tilde{\gamma}_f} &\geq& 10^{-1}\\
\frac{\tilde{\gamma}_w}{\tilde{\gamma}_f}\frac{F}{F_{crit}} &\leq& \frac{9}{10}.
\end{eqnarray}
We therefore expect that parametric effects are important when $F/F_{crit}$ is larger than $1.1$ - $1.2$.
With the simplifying assumption (A3), we may reduce the map (\ref{Update_y})-(\ref{Update_Map}) to one-dimension, and analytically examine the bifurcation from bouncing to walking. To do so, align the droplet position and velocity with the x-axis $\mathbf{y}_n = r_n \hat{\mathbf{x}}$, $\mathbf{v}_n = v_n \hat{\mathbf{x}}$ to obtain
\begin{eqnarray} \label{VelocityMap_FreeSpace}
v_{n+1} &=& (1-\gamma)v_n + F g(v_n) \\ \label{MapIntegral}
g(v) &=& \int_{0}^{\infty} \frac{k^3}{\omega_D} \sin(\omega_D) J_0'(kv) e^{-\mu k^2 } \du k\\
\label{Polynomial}
g(v) &\approx& 1150 v (1 - 4.32 v^2 + 38.86 v^4)
\end{eqnarray}
The last line (\ref{Polynomial}) is an approximate polynomial fit for $g(v)$, while $J_0'(z) = \frac{d}{dz} J_0(z)$. We note that (even regardless of $B$ and $\mu$) $v_n = 0$ is a fixed point of the map (\ref{VelocityMap_FreeSpace}), and therefore bouncing droplets with a fixed location $r_{n+1} = r_n$ are always solutions of (\ref{VelocityMap_FreeSpace}). To illustrate the nature of the bifurcation from stable bouncing to walking, we fix a value of $\gamma$ and continually vary $F$ as the bifurcation parameter.
Here figures (\ref{WaveFieldFreeSpace}) and (\ref{PoincareMapFreeSpace}) show the wave field and Poincar\'{e} map, with the associated polynomial fit (\ref{Polynomial}).
\begin{figure}
\centering
\subfloat[Gravity-capillary wave field at times $T = 1, 2,3$.]{\label{WaveFieldFreeSpace}\includegraphics[width=0.5\textwidth]{WaveField}}
\subfloat[Poincar\'{e} map of (\ref{VelocityMap_FreeSpace})-(\ref{Polynomial}) and $F > F_{crit}$.]{ \label{PoincareMapFreeSpace}\includegraphics[width=0.5\textwidth]{PoincareMapFreeSpace}}
\caption{Shows the radial wave field impulse response $h_0(r, T)$ at different times, and the Poincar\'{e} map for gravity-capillary model ($B = 120$, $\mu = 0.008$). The dashed line shows the polynomial fit (\ref{Polynomial}). }
\end{figure}
As one increases $F$, the fixed point solution $v_n = 0$ becomes unstable at which point the system undergoes a pitchfork bifurcation. For values of $F$ above the critical forcing ($F > F_{crit} = 1150^{-1}$), the points $v_n$ converge to a new fixed point solution $v^*$, indicating a bifurcation from stable bouncing to walking. For instance figure (\ref{Polynomial_Bifurcation}) shows the bifurcation diagram in the case when $\gamma = 1$. As one further increases $F$, the stable walking solution bifurcates a second time into a two-period orbit, followed by a transition to chaos (figure (\ref{Polynomial_VelocityOrbits})). We remark that the pitchfork bifurcation and transition to chaos occur for a large range of $\mu$ and $B$, however the supercritical bifurcation seen in (\ref{Polynomial_Bifurcation}) is not generic. For instance, the supercritical bifurcation to walking is due to the sign of the derivative $h^{(4)}(0) = -g^{(3)}(0) > 0$. In general, by varying $B$ (ie. the shape of the wave) one may realize both positive and negative values of $h^{(4)}(0)$, where $h^{(4)}(0) > 0$ implies a supercritical bifurcation while $h^{(4)}(0) < 0$ implies a subcritical one. Experiments have observed both sub and supercritical bifurcations, however the difference may arise from other effects not considered in the current model, such as instabilities in the vertical droplet dynamics.
The iterative map also provides predictions for the walker velocity and wave velocity which we now compare to experimental data. First, in the gravity-capillary wave model (\ref{IRDimensionlessEq1})-(\ref{IRDimensionlessEq2}), the point source forcing excites all wave lengths of $h(\mathbf{x}, t)$. Hence, the wave has a minimum group velocity $v_{g}^{min} = 0.33 = 132mm$ $s^{-1}$, obtained by the gravity-capillary dispersion relation, which approximately limits the speed of the disturbance. Meanwhile, the fixed point walking velocities are $v^* \sim 0.25 = 100mm$ $s^{-1}$, while the characteristic standing wavelength is $\lambda = 0.6 = 9.6mm$. Experiments (figure 6a. in \cite{EddiSultanMoukhtarFortRossiCouder2011}) show a wave propagating roughly $20mm$ in a time of $\sim 0.2 s$ for a minimum group velocity of $\tilde{v}_g^{min} \sim 100mm$ $s^{-1}$, while the standing wavelength $\tilde{\lambda} = 4.75mm$ . Although the gravity-capillary wave is in good agreement with the fluid experiments, the maximum \cite{ProtiereBoudaoudCouder2006} experimental droplet velocity is roughly a factor of 5 smaller: $\tilde{v}^* = 20mm$ $s^{-1}$. Despite yielding a qualitative agreement, the iterative map model over estimates the droplet velocity by locking the droplet velocity to the wave. Here the discrepancy is a result of the simplified assumptions (A1)-(A2). In particular, a detailed model for the droplet bouncing dynamics and surface interactions may account for the velocity mismatch.
\begin{figure}[htb!]
\centering
\subfloat[Bifurcation diagram.]{\label{Polynomial_Bifurcation}\includegraphics[width=0.5\textwidth]{Polynomial_Bifurcation}}
\subfloat[Droplet velocity versus forcing.]{ \label{Polynomial_VelocityOrbits}\includegraphics[width=0.5\textwidth]{Polynomial_VelocityOrbits}}
\caption{Shows the bifurcation from stable bouncing to walking for the gravity-capillary model (\ref{VelocityMap_FreeSpace})-(\ref{Polynomial}). In (a), the transition is a supercritical pitchfork where $F_{crit} = 1150^{-1}$, $B = 120$, $\mu = 0.008$. In (b), after the initial bifurcation, the droplet undergoes a transition to chaos.}
\end{figure}
Lastly, we examine the effects of including multiple bounces in the wave field. Here figure (\ref{Polynomial_VelocityOrbitsFull}) shows the velocity dynamics of including 10 past bounces, while (\ref{FullWaveField}) shows the fully developed wave field for a walking droplet. Upon the onset of walking, the superposition of many past impacts creates an apparent standing wave pattern (\ref{FullWaveField}), which qualitatively agrees with experiments\cite{EddiSultanMoukhtarFortRossiCouder2011, ProtiereBoudaoudCouder2006}. Explicitly, the wave field may be written as $\sum_{n = 1}^{10} h_0(r_n, n)$ where $r_n = r + (n-1) v^*$ and $v^*$ is the fixed point velocity.
Figure (\ref{Polynomial_VelocityOrbitsFull}) also shows that including multiple bounces can increase the threshold for walking and lower the relative forcing required for chaos. Including multiple bounces, however, only mildly changes the nature of the transition to chaos. Experimentally observing such a transition requires measuring variations in the droplet velocity. For instance, the droplet transitions from a steady velocity $v^*$, to one that jumps with alternating step size $v^* \pm \epsilon$. Experimentally, one would observe a time averaged mean velocity $v^*$ (indicating no change), while precise measurements would detect small periodic variations. We note that the transition to chaos described here does not account for possible instabilities in the angular droplet dynamics.
\begin{figure}[htb!]
\centering
\subfloat[Droplet velocity versus forcing.]{ \label{Polynomial_VelocityOrbitsFull}\includegraphics[width=0.5\textwidth]{Polynomial_VelocityOrbitsFull}}
\subfloat[Fully developed wave.]{\label{FullWaveField}\includegraphics[width=0.5\textwidth]{FullWaveField}}
\caption{ (\ref{Polynomial_VelocityOrbitsFull}) Shows the droplet velocity including 10 previous bounces, where $F_{crit} = 1150^{-1}$ is the single bounce critical forcing. Multiple bounces can increase the threshold for walking and also lower the critical forcing for chaos. (\ref{FullWaveField}) Shows the fully developed wave field for a walking droplet as a superposition of shifted sources $\sum_{n = 1}^{10} h_0(r_n, n)$ where $r_n = r + (n-1) v_n$ and $v_n = 0.05$ (thick line), $v_n = 0.25$ (thin line). The dashed line is an approximate Bessel function wave field used in section (V). The primary contribution to walking comes from the most recent bounce.}
\end{figure}
\subsection{Varying dissipation}
Although we have been using the forcing $F$ as a bifurcation parameter, the viscosity of the fluid $\mu$ can also vary depending on the vibration of the bed. In this section we examine the effect of varying dissipation in model (\ref{VelocityMap_FreeSpace})--(\ref{MapIntegral}). We first remark that one may asymptotically approximate $h(\mathbf{x}, t_{n+1}; \mathbf{y}_n)$ for $\mu \gg 1$. The result is an over-damped wave, which does not support the steady walking of droplets. For large $\mu$, the terms inside the integral may be approximated as follows: when $k > O(\mu^{-2/3})$, the value $\mu^2 k^4 > \omega_0^2$ at which point
\begin{eqnarray}
\frac{\sin(\omega_D)}{\omega_D} e^{-\mu^2 k^4} &=& \frac{\sinh( (\mu^2 k^4 -k - B^{-1} k^3)^{1/2} )}{(\mu^2 k^4 -k - B^{-1} k^3)^{1/2} } e^{-\mu^2 k^4} \\
&\approx& \frac{1}{\mu k} e^{-\frac{1}{2\mu} (B^{-1}k + k^{-1} )}
\end{eqnarray}
Here the last line is obtained via Taylor series. In addition, the factor $e^{-\frac{1}{2\mu k}}$ approaches $1$ as $k \rightarrow \infty$ and has a minor effect on the integral. We therefore approximate the wave as
\begin{eqnarray} \label{LargeMu_h}
h(\mathbf{x}, t_{n+1}; \mathbf{y}_n) &\approx& -\int \frac{1}{4\pi\mu k} e^{-\frac{1}{2\mu B}k}e^{\imath \mathbf{k}(\mathbf{x} - \mathbf{y}_n)} \du \mathbf{k} + O(\mu^{-3/2}) \\
&=& -\frac{1}{4\pi\mu} \int_{0}^{2\pi} \int_{0}^{\infty} e^{ (-\frac{1}{2\mu B} +\imath r \cos \theta) k} \du k \du \theta + O(\mu^{-3/2}) \\
&=& -\frac{1}{2\pi} \int_{0}^{2\pi} \frac{\du \theta}{(B^{-1} - \imath 2\mu r \cos \theta)} + O(\mu^{-3/2}).
\end{eqnarray}
Here, we have aligned the droplet position and velocity with the x-axis ($\mathbf{y}_n = r_n \hat{\mathbf{x}}$, $\mathbf{v}_n = v_n \hat{\mathbf{x}}$) and introduced $r = |\mathbf{x} - \mathbf{y}_n|$. To compute the last integral, we make the change of variables $z = e^{\imath \theta}$, and proceed by evaluating the residues enclosed by the unit circle $|z| = 1$. For large $\mu$ we have the over damped wave:
\begin{eqnarray} \label{LargeMu_h2}
h(r, t_{n+1}) &=& -\frac{1}{\sqrt{B^{-2} + 4 (\mu r)^2}} + O(\mu^{-3/2}).
\end{eqnarray}
Taking $\partial_r h(r)$, we have
\begin{eqnarray} \label{LargeMu_Map}
v_{n+1} = (1-\gamma) v_n - \frac{4\mu^2 F v_n}{(B^{-2} + 4\mu^2 v_n^2)^{3/2}}.
\end{eqnarray}
The iterative map (\ref{LargeMu_Map}) has only one fixed point $v_{n} = 0$, regardless of $F$ and $B$. Hence, the strongly over-damped waves do not produce a bifurcation to walking motion as one increases $F$. Physically, the effect of damping smooths out the wave curvature, thereby inhibiting a transition to stable walking.
To illustrate the effect of $\mu$ over typical experimental values ($.001$ to $.016$), figure (\ref{VaryMu}) shows the droplet velocity for a fixed $F$ and different values of $\mu$. Over most experimental values, the velocity has a non-zero fixed point, or walking solution. As $\mu$ increases, the fixed point $v = 0$ first becomes a stable attractor (around $\mu \sim 0.016$) indicating a reverse transition from walking to bouncing. At larger $\mu$, $v = 0$ destabilizes into a periodic orbit at which point the droplet bounces back and forth about one fixed location in space. Over experimental values, $\mu$ does not change the qualitative walking behavior, and only has a minor effect on the velocity of the droplet.
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.6\textwidth]{VaryMu} \\
\caption{Plot shows the droplet velocity for varying $\mu$ at a fixed $F/F_{crit} = 1.1$. The parameters are $B = 120$ and $\gamma = 1$ while $F_{crit}$ is the critical forcing for $\mu = 0.008$. At small $\mu < .006$ the fixed forcing $F$ is not large enough to induce walking. At large $\mu$ the droplet does not walk but rather oscillates about a fixed point.}
\label{VaryMu}
\end{figure}
\section{General requirements for stable walking}
In this section we examine a set of general requirements for the map (\ref{Update_y})--(\ref{Update_v}) and assumption (A3) to yield stable walking solutions in free space ($\Omega = \mathbbm{R}^2$).
Firstly, a droplet impact at position $\mathbf{y}_n$ will generate a radially symmetric wave field about $\mathbf{y}_n$. Secondly, we note that the translational symmetry in free space implies that a wave field generated by an impact at $\mathbf{y}_n$ will only depend on the difference $(\mathbf{x} - \mathbf{y}_n)$. Hence, $h(\mathbf{x}, t; \mathbf{y}_n)$ has the general form
\begin{eqnarray}
h(\mathbf{x}, t; \mathbf{y}_n) = h( |\mathbf{x} - \mathbf{y}_n|, t).
\end{eqnarray}
Letting $r = |\mathbf{x} - \mathbf{y}_n|$, we can introduce the function $g(r) = -\partial_r h(r, 1)$ where $h(r, 1)$ is the wave field generated by an impact at time $t = 0$ and evaluated at one strobe period later (ie. $t = 1$). Physically, $g(r)$ describes the radial forcing on the droplet in the iterative map (\ref{Update_y})--(\ref{Update_Map}). Using the fact that $\mathbf{y}_{n+1} - \mathbf{y}_n = \mathbf{v}_n$, we then obtain a one dimensional, iterative map for the droplet velocity in the radial direction
\begin{eqnarray} \label{General_Map}
v_{n+1} = (1-\gamma) v_n + F g(v_n).
\end{eqnarray}
Without loss of generality, we align the system with the x-axis (ie. $\mathbf{v}_n = v_n \mathbf{\hat{x}}$) and further assume the parameters $0 < \gamma \leq 1$ and $F \geq 0$. Since $h(r, t)$ is radially symmetric, it follows that $h(r, t)$ is an even function of $r$. Therefore, (provided $h(r, t)$ is regular at $r = 0$), $g(v)$ is an odd function of $v$ and $g(0) = 0$. Hence, (\ref{General_Map}) always admits $v = 0$ as a fixed point.
For stable walking solutions, we require the existence of a nonzero stable fixed point. The following criteria guarantee such a point. Let $v^* > 0$ and satisfy the following propositions
\begin{itemize}
\item[P1.] Existence of a nonzero fixed point
\begin{eqnarray}
g(v^*) > 0. \label{ExistenceFP}
\end{eqnarray}
\item[P2.] Stability of the fixed point
\begin{eqnarray} \label{StabilityFP}
0 < 1 - \frac{v^* g'(v^*)}{g(v^*)} < \frac{2}{\gamma}.
\end{eqnarray}
\end{itemize}
Here item (\ref{ExistenceFP}) implies taking a forcing $F = \gamma v^*/ g(v^*) > 0$ yields the fixed point velocity $v^*$. Physically, condition (\ref{ExistenceFP}) guarantees that the wave propels the droplet forward at each interaction. Meanwhile, condition (\ref{StabilityFP}) guarantees the stability of the linearized map at $v = v^*$. Practically, one may simply plot the function $f(v) = 1 - \frac{v g'(v)}{g(v)}$ to determine whether the corresponding wave field supports a stable walking droplet. Lastly, conditions (\ref{ExistenceFP})--(\ref{StabilityFP}) only guarantee a stable walking solution.
An additional, yet independent criteria for a bifurcation from stable bouncing to walking motion is an instability at $v = 0$
\begin{itemize}
\item[P3.] Instability of the $v = 0$ fixed point
\begin{eqnarray} \label{InstabilityFP}
g'(0) > 0
\end{eqnarray}
\end{itemize}
Here the condition (\ref{InstabilityFP}) guarantees that taking $F > \gamma/g'(0)$ yields an unstable fixed point at $v = 0$\cite{FixedPoint}. Physically, the condition (\ref{InstabilityFP}) corresponds to a concave down wave field $h''(0) < 0$ and assures that the instability will propagate the droplet in one direction.
One should note that the criteria (\ref{ExistenceFP})--(\ref{StabilityFP}) yields a stable walking solution provided $F = \gamma v^*/ g(v^*) > 0$, while (\ref{InstabilityFP}) is a separate condition which guarantees that the fixed point $v = 0$ becomes unstable for $F > \gamma/g'(0)$. In general, the simultaneous stability or instability of the $v = 0$ and $v = v^*$ fixed points depends on other details of the wave field. For example, one may have both subcritical and supercritical pitchfork bifurcations depending on the sign $h^{(4)}(0)$.
We also remark that another standard wave field that fails condition (\ref{ExistenceFP}) is the linear Green's function solution to the Helmholtz equation. Such a model is proposed in \cite{CouderFort2006}, however, they obtain walking solutions through the summation of many past bounces.
\section{Movement in a square} \label{BoundedDomainA}
In this section we examine the movement of droplets in a square domain using the model (\ref{Update_y})--(\ref{Update_v}). To capture the reflection of the fluid waves against the wall, we impose a Neumann boundary condition on the velocity potential $\frac{d \phi}{d\mathbf{n}} = 0$ corresponding to a no fluid flux boundary condition on the bath. Equivalently, such a condition corresponds to a Neumann boundary condition on $h(\mathbf{x}, t; \mathbf{y}_n)$:
\begin{eqnarray} \label{PhiBC}
\frac{dh}{d\mathbf{n}} = 0, \hspace{4mm} \mathbf{x} \in \partial \Omega
\end{eqnarray}
where $\mathbf{n}$ is the unit normal along the domain boundary. For instance, differentiating equation (\ref{DimensionlessEq1}) and projecting onto the boundary yields $\partial_t \frac{dh}{d\mathbf{n}} = \partial_z\frac{d\phi}{d\mathbf{n}} = 0$. Hence the boundary condition $\frac{d \phi}{d\mathbf{n}} = 0$ implies $\frac{dh}{dz} = C$, a constant in time. Since the constant $C = 0$ at $t = 0$, we take $\frac{dh}{d\mathbf{n}} = 0$ for all time.
To evaluate the motion of the droplet in a square domain, we must compute the wave field $h(\mathbf{x}, t)$ at each iteration of the map. To aid in the computation of the field, we may exploit the method of images \cite{TikhonovSamarskii1963} and the geometry of a square. For instance, since the field $h(\mathbf{x}, t)$ satisfies Neumann boundary conditions, and is generated by an impulse at each step, the solution may be generated by an infinite array of image points of the free space wave field $h_{0}(r, t)$ at properly chosen locations $\mathbf{x}_j^{im}$. The location of the image points depend on $\mathbf{y}_n$, the closest ones being at points reflected across the domain wall boundaries. One may then compute the wave field from the knowledge of the free space wave
\begin{eqnarray} \label{WaveSuperposition}
h(\mathbf{x}) &=& \sum_{m = 0}^{n-1}\Big( h_0(|\mathbf{x}-\mathbf{y}_{n-m}|, m+1) + \sum_j h_{0}(|\mathbf{x} - \mathbf{x}_{j}^{im}(\mathbf{y}_m)|, m+1) \Big).
\end{eqnarray}
Here the $\mathbf{x}_{j}^{im}(\mathbf{y}_m)$ are the image points $\mathbf{x}_j^{im}$ which depend on the source term $\mathbf{y}_m$. Figure (\ref{MethodImages}) illustrates the wave field with the most important image points. The addition of the image points yield correct boundary conditions for $h(\mathbf{x})$ on the bottom and right side of the square.
To simplify the expression (\ref{WaveSuperposition}), we may separate out the free space bounces at locations $\mathbf{y}_{n-j}$ as
\begin{eqnarray}
h_{FS}(\mathbf{x}) &=& \sum_{m = 0}^{n-1} h_0(|\mathbf{x}-\mathbf{y}_{n-m}|, m+1)
\end{eqnarray}
For simplicity, we may then approximate $h_{FS}(\mathbf{x})$ as a radially symmetric ansatz centered around the most recent position
\begin{eqnarray} \label{WaveAnsatz}
h_{FS}(r) &\approx& 20 J_0(11.5 r) e^{-1.15 r} \\
r &=& |\mathbf{x} - \mathbf{y}_n|.
\end{eqnarray}
Figure (\ref{FullWaveField}) compares the approximation (\ref{WaveAnsatz}) to the fully developed gravity-capillary wave in the radial direction. The approximation here is also similar to the phenomenological ansatz provided in \cite{CouderFort2006, FortEddiBoudaoudMoukhtarCouder2010}.
In making the approximation (\ref{WaveAnsatz}), one is effectively concentrating all previous bounces onto $\mathbf{y}_n$. As a result, the approximation captures contributions from previous impacts, however suppresses all memory effects. Again, such an approximation is valid at low forcing far from the Faraday threshold. The ansatz also neglects the Doppler effect present from a moving source. Experimentally, however, the Doppler effect is negligible at low forcing since the droplet velocity is small compared to the group velocity of the wave (ie. the ratio is $\sim 0.06$). In concentrating all impacts onto the previous location, (\ref{WaveAnsatz}) greatly simplifies the iterative map, and may aid in future work on developing evolution equations for the droplet probability distribution.
Finally, since $h_{FS}(r)$ decays quickly, one may truncate the sum (\ref{WaveSuperposition}) for the efficient computation of the wave field. In our case we keep the first order contributions as illustrated in figure (\ref{MethodImages}). The field $h(\mathbf{x})$ then becomes
\begin{eqnarray} \label{WaveSuperposition2}
h(\mathbf{x}) &=& h_{FS}(|\mathbf{x}-\mathbf{y}_{n}|) + \sum_j h_{FS}(|\mathbf{x} - \mathbf{x}_{j}^{im}(\mathbf{y}_n)|) \Big)
\end{eqnarray}
\subsection{Large Domain} \label{LargeDomain}
In this section we examine solutions to the map (\ref{Map_y})--(\ref{Map_v}) where the wave field is given by (\ref{WaveAnsatz})--(\ref{WaveSuperposition2}).
In our numerical evaluation of the map, we fix $\gamma = 1$ and the size of the box $D = 12$ (approximately $20 cm$) to be much larger than one wavelength. We then examine trajectories for different forcing. Since we model the droplet interaction with the boundary of the domain entirely by reflected waves, at large forcing there is a possibility that the droplet may physically collide or jump over the boundary. We therefore limit our attention to parameters which yield bounded trajectories ($F/F_{crit} < 0.869$ where $F_{crit} = 1150^{-1}$ is a normalized forcing from the gravity-capillary model), namely those which reflect off the walls.
Depending on the parameters of the underlying wave field, the long time behavior of particle trajectories may be classified into two categories depending on the nature of the limiting set: those which approach a circular quasiperiodic orbit ($0.610 < F/F_{crit} < 0.733$) , and those which continually traverse the domain ($0.733 < F/F_{crit} < 0.869$). In the second category, the trajectories appear to form a dense set throughout the spatial domain. To illustrate the different scenarios, figure (\ref{CircularOrbit}) shows the path of a droplet approaching a quasiperiodic orbit while figure (\ref{DenseTrajectory}) shows part of a dense trajectory. The emergent pattern, however, is not related to cavity modes of the square, but rather results because droplet trajectories tend to travel along paths near angles of $\pi/4$ with respect to the x-axis. For instance, although not shown, the statistics of the droplet velocity angles are centered around angles of $\pm \pi/4$. For box sizes much larger than the natural wavelength of $h(\mathbf{x})$, the droplet behaves vary much like an isolated particle. When the droplet approaches a wall, the droplet reflects off the wall through the mediation of the reflected wave field. The reflection is somewhat analogous to a billiard ball on a table since the incident and reflected angles are approximately equal.
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.6\textwidth]{MethodImages} \\
\caption{Plot shows the closest image points used to compute the wave field in a square domain. Each image point acts as a source with wave field $h_{FS}(r)$.}
\label{MethodImages}
\end{figure}
\begin{figure}
\centering
\subfloat[$\gamma = 1$, $F/F_{0} = .65$]{\label{CircularOrbit}\includegraphics[width=0.4\textwidth]{CircularOrbit}}
\subfloat[$\gamma = 1$, $F/F_{0} = .75$.]{\label{DenseTrajectory}\includegraphics[width=0.4\textwidth]{DenseTrajectory}}
\caption{The long time spatial trajectories for $D = 12$ ($\sim 20 cm$) collapse into (a) quasiperiodic orbit at lower forcing, or (b) travel throughout the domain at large forcing. Here $F_{crit} = 1150^{-1}$ is a normalization force from the free space walking threshold.}
\end{figure}
\subsection{Small domain}
In this section, we examine the long time behavior of droplet trajectories for a domain size comparable to the fluid wavelength (ie. $D \sim 1-2 cm$). Although such domains are experimentally small compared to current setups, they correspond to the classical analogy of having a quantum system with the de Broglie wavelength comparable to the domain size. Unlike the previous section, in small domains the wave field has time to respond to the geometry of the box.
Again we work well below the Faraday threshold and neglect strong memory effects where such interactions can lead to additional droplet dynamics, even in large cavities $D \gg \lambda_f$.
To determine the long time behavior of the map, we fix a set of parameter values and examine the trajectories for many different initial conditions. The data for each initial condition is chosen to survey the phase space within a bounded set by prescribing, $|\mathbf{v}_0| < 0.25$, and taking a maximum distance between $\mathbf{y}_0$ and the nearest wall to be less than $0.1$. Here the exact bounds of $0.25$ and $0.1$ are chosen somewhat arbitrarily to include a large, physically relevant, region of phase space.
\begin{figure} [htb!]
\centering
\subfloat[F = 0.22]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_0pt50}}
\subfloat[F = 0.43]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_1pt00}} \\
\subfloat[F = 0.48]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_1pt10}}
\subfloat[F = 0.52]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_1pt20}}\\
\subfloat[F = 0.57]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_1pt30}}
\subfloat[F = 0.61]{\includegraphics[width=0.4\textwidth]{D8_mu_1_10_F_1pt40}}
\caption{Attracting sets for the model response (\ref{WaveAnsatz}) and various forcing where $D = 0.7$.}
\label{fig:Box8}
\end{figure}
For each initial condition, we remove any transient effects by first evolving the trajectory for several thousand iterations. After discarding the transients, we then evolve the droplet for several thousand more iterations, and project the trajectory from the four dimensional phase space $(\mathbf{x}, \mathbf{v})$ onto the two dimensional physical domain $(\mathbf{x})$. In many cases the trajectory approaches an attracting set in the form of a periodic or quasiperiodic orbit. We now describe in detail the long time trajectories as one varies $F$ for fixed $D$. Specifically, we consider in detail the case of $D = 0.7$ ($\sim 1.1 cm$), which corresponds to a box size of roughly one wavelength, and $D = 1.05$ ($\sim 1.7 cm$) which is just under two wavelengths of the wave field.
Initially, with small values of the forcing $F$, the long time trajectories approach one of several quasiperiodic orbits. These orbits form attracting sets for different regions of phase space. For instance, the exact orbit a trajectory approaches depends only on the trajectories initial conditions. Together the collection of all attracting orbits form a symmetric array on which the precise pattern depends on the parameters $F$ and the box size $D$. Qualitatively the number of orbits depends most strongly on the box size $D$. The reason is that the droplets tend to localize near the troughs from the waves reflected off the domain boundaries. The larger box sizes allow for more wavelengths from the reflected waves. For instance, over a wide range of forcing $F$, a box size of $D = 0.7$ supports 4 quasiperiodic attracting regions, while $D = 1.05$ contains 8. Here the shape of the array, and number of orbits appear linked to the geometry of the domain.
As the forcing increasing, the spatial radii of the quasiperiodic orbits grow. For instance figures (\ref{fig:Box8}) and (\ref{fig:Box12}) show the attracting sets for box sizes $0.7$ and $1.05$ with different forcing. At sufficient forcing, the nature of the attracting sets change from thin circular orbits to thick, sets. At large forcing, the localized attracting sets break down, and the droplet wonders throughout the domain. Figures (\ref{fig:Box8}f) and (\ref{fig:Box12}f) show a shaded probability distribution for the droplets position at the large forcing. Despite the fact that there are no longer circular quasiperiodic orbits, the droplet still spends a significant time near the former quasiperiodic orbit regions. For instance, there are similarities in the dark outlines of (\ref{fig:Box8}e) and (\ref{fig:Box8}f), as well as (\ref{fig:Box12}d) and (\ref{fig:Box12}f). Although the distribution shows dark, highly traversed regions, and light, vacated regions in a regular array that appears related to the underlying wave field, the exact dependence is not completely understood. Here we defer further investigation to future work. For instance, in future work we seek to examine the relation between the dynamical systems invariant measure, and the underlying wave field.
\begin{figure}[htb!]
\centering
\subfloat[F = 0.11]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_0pt25}}
\subfloat[F = 0.22]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_0pt50}} \\
\subfloat[F = 0.43]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_1pt00}}
\subfloat[F = 0.54]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_1pt25}} \\
\subfloat[F = 0.65]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_1pt50}}
\subfloat[F = 0.66]{\includegraphics[width=0.4\textwidth]{D12_mu_1_10_F_1pt52}}
\caption{Attracting sets for the model response (\ref{WaveAnsatz}) and various forcing where $D = 1.05$.}
\label{fig:Box12}
\end{figure}
\subsection{Conclusions}
Through the introduction of an iterative map, we model the dynamics and trajectories of bouncing droplets on an oscillating fluid bed. As a first step, we examine the droplets bifurcation from a stable bouncing state to a stable walking one. In addition, we list several requirements for the underlying wave field to undergo such a bifurcation. Using the map, we then investigate the droplet trajectories for wave responses in a square (billiard ball) domain.
In the case of a large domain, we recover limit cycle and dense trajectories which appear similar to those reported in \cite{CouderFort2006}.
Lastly, in small domains we show that for low forcing, trajectories tend to approach circular attracting sets. As one increases the forcing, the attracting sets break down and the droplet tends to travel through space, jumping between the former attracting regions.
In future work we plan to further examine the statistical nature of the droplet trajectories, including their transport properties and invariant measures.
\subsection{Acknowledgments}
The author would like to thank John Bush for originally posing the problem of understanding the bouncing droplet trajectories. The author has also vastly benefited from many conversations with Renato Calleja, Tristan Gilet, Anand Oza, Jean-Christophe Nave and Ruben Rosales. Lastly, the author gratefully acknowledges the many helpful comments of an anonymous reviewer. The work was partially supported by NSERC and NSF grant $DMS�0813648$.
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
\newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
|
{
"timestamp": "2013-10-21T02:01:24",
"yymm": "1203",
"arxiv_id": "1203.2204",
"language": "en",
"url": "https://arxiv.org/abs/1203.2204"
}
|
\section{Introduction}\label{sec1}
Queueing models with many-servers are prevalent in modeling call
centers and other large-scale service systems. They are used for
optimizing staffing and making dynamic control decisions. The
complexity of the underlying queueing model renders such optimization
problems intractable for exact analysis, and one needs to resort to
approximations. A prominent mode of approximate analysis is to study
such systems in the so-called Halfin--Whitt (HW) heavy-traffic regime;
cf. \cite{HaW81}. Roughly speaking, the analysis of a queueing system
in the HW regime proceeds by scaling up the number of servers and the
arrival rate of customers in such a way that the system load approaches
one asymptotically. To be more specific, instead of considering a
single system, one considers a sequence of (closely related) queueing
systems indexed by a parameter $n$ along which the arrival rates and
the number of servers scale up so that the system traffic intensity~$\rho^n$
satisfies
\begin{equation}\label{eqHTcond}
\sqrt{n} (1-\rho^n)\rightarrow\beta\qquad\mbox{as } n\tinf.
\end{equation}
In the context of dynamic control, passing to a formal limit of the
(properly scaled) system dynamics equations as $n\tinf$ gives rise to
a \textit{limit} diffusion control problem, which is often more tractable
than the original dynamic control problem it approximates.
The approximating diffusion control problem typically provides useful
structural insights and guides the design of good policies for the
original system. Once a candidate policy is proposed for the original
problem of interest, its asymptotic performance can be studied in the
HW regime. The ultimate goal is to establish that the proposed policy
performs well. To this end, a useful criterion is the notion of
asymptotic optimality, which provides assurance that the optimality gap
associated with the proposed policy vanishes asymptotically \textit{under
diffusion scaling} as $n\tinf$. Hence, asymptotic optimality in this
context is equivalent to showing that the optimality gap is $o(\sqrt{n})$.
A central reference for our purposes is the recent paper by Atar,
Mandelbaum and Reiman \cite{AMR02}, where the authors apply all steps
of the above scheme to the important class of problems of dynamically
scheduling a multiclass queue with many identical servers in the HW
regime. Specifically, \cite{AMR02} considers a sequence of systems
indexed by the number of servers $n$, where the number of servers and
the arrival rates of the various customer classes increase with $n$
such that the heavy-traffic condition holds; cf. equation (\ref
{eqHTcond}). Following the scheme described above, the authors derive
an approximate diffusion control problem through a formal limiting
argument. They then show that the diffusion control problem admits an
optimal Markov policy, and that the corresponding HJB equation (a
semilinear elliptic PDE) has a unique classical solution. Using the
Markov control policy and the HJB equation, the authors propose
scheduling control policies for the original (sequence of) queueing
systems of interest. Finally, they prove that the proposed sequence of
policies is asymptotically optimal under diffusion scaling. Namely, the
optimality gap of the proposed policy for the $n$th system is
$o(\sqrt{n})$. A similar approach is applied to more general networks
in \cite{atar2005scheduling}. In this paper, we study a similar
queueing system (see Section \ref{secmodel}). Our goal, however, is
to provide an improved optimality gap which, in turn, requires a
substantially different scheme than the one alluded to above.
Approximations in the HW regime for performance analysis have been used
extensively for the study of fixed policies. Given a particular policy,
it may often be difficult to calculate various performance measures in
the original queueing system. Fortunately, the corresponding
approximations in the HW regime are often more tractable. The machinery
of strong approximations (cf. Cs\"{o}rgo and Horv\'{a}th \cite{csorgo})
often plays a central role in such analysis. In the context
of many-server heavy-traffic analysis, with strong approximations, the
arrival and service processes (under suitable assumptions on the
inter-arrival and service times) can be approximated by a diffusion
process so that the approximation error on finite intervals is $O(\log
n)$ (where $n$ is the number of servers as before). Therefore, it is
natural to expect that, under a given policy, the error in the
diffusion approximations of the various performance metrics is $O(\log
n)$, which is indeed verified for various settings in the literature
(see, e.g., \cite{MMR98}).
A natural question is then whether one can go beyond the analysis of
fixed policies and achieve an optimality gap that is logarithmic in $n$
also under dynamic control, improving upon the usual optimality gap of
$o(\sqrt{n})$. More specifically, can one propose a sequence of
policies (one for each system in the sequence) where the optimality gap
for the policy (associated with the $n$th system) is logarithmic
in~$n$? While one hopes to get logarithmic optimality gaps as suggested by
strong approximations, it is not a priori clear if this can be achieved
under dynamic control. The purpose of this paper is to provide a
resolution to this question. Namely, we study whether one can establish
such a strong notion of asymptotic optimality and if so, then how
should one go about constructing policies which are asymptotically
optimal in this stronger sense.
Our results show that such strengthened bounds on optimality gaps can
be attained. Specifically, we construct a sequence of asymptotically
optimal policies, where the optimality gap is logarithmic in $n$. Our
analysis reveals that identifying (a sequence of) candidate policies
requires a new approach. To be specific, we advance a sequence of
diffusion control problems (as opposed to just one) where the diffusion
coefficient in each system depends on the state and the control. This
is contrary to the existing work on the asymptotic analysis of queueing
systems in the HW regime. In that stream of literature, the diffusion
coefficient is typically a (deterministic) constant. Indeed, Borkar
\cite{borkar2005controlled} views the constant diffusion coefficient
as a characterizing feature of the problems stemming from the
heavy-traffic approximations in the HW regime. Interestingly, it is
essential in our work to have the diffusion coefficient depend on the
state and the control for achieving the logarithmic optimality gap. In
essence, incorporating the impact of control on the diffusion
coefficient allows us to track the policy performance in a more refined manner.
While the novelty of having the diffusion coefficient depend on the
control facilitates better system performance, it also leads to a more
complex diffusion control problem. In particular, the associated HJB
equation is fully nonlinear; it is also nonsmooth under a linear
holding cost structure. In what follows, we show that each of the HJB
equations in the sequence has a unique smooth solution on bounded
domains and that each of the diffusion control problems (when
considered up to a stopping time) admits an optimal Markov control
policy. Interpreting this solution appropriately in the context of the
original problem gives rise to a policy under which the optimality gap
is logarithmic in $n$. As in the performance analysis of fixed
policies,\vadjust{\goodbreak} strong approximations will be used in the last step, where we
propose a sequence of controls for the original queueing systems, and
show that we achieve the desired performance. However, it is important
to note that strong approximation results alone are not sufficient for
our results. Rather, for the improved optimality gaps we need the
refined properties of the solutions to the HJB equations. Specifically,
gradient estimates for the sequence of solutions to the HJB equations
(cf. Theorem \ref{thmHJB1sol}) play a central role in our proofs.
Our analysis restricts attention to a linear holding cost structure.
However, we expect the analysis to go through for some other cost
structures including convex holding costs. Indeed, the analysis of the
convex holding cost case will probably be simpler as one tends to get
``interior'' solutions in that case as opposed to the corner solutions
in the linear cost case, which causes nonsmoothness. One could also
enrich the model by allowing abandonment. We expect the analysis to go
through with no major changes in these cases as well; see the
discussion of possible extensions in Section \ref{secconclusions}.
For purposes of clarity, however, we chose not to incorporate these
additional/alternative features because we feel that the current set-up
enables us to focus on and clearly communicate the main idea: the use
of a novel Brownian model with state/control dependent diffusion
coefficient to obtain improved optimality gaps.
\subsection*{Organization of the paper} Section \ref{secmodel}
formulates the model and states the main result. Section \ref
{secresult} introduces a (sequence of) Brownian control problem(s),
which are then analyzed in Section \ref{secADCP}. A performance
analysis of our proposed policy appears in Section \ref{sectracking}.
The major building blocks of the proof are combined to establish the
main result in Section \ref{seccombining} and some concluding remarks
appear in Section~\ref{secconclusions}.
\section{Problem formulation}\label{secmodel}
We consider a queueing system with a single server-pool consisting of
$n$ identical servers (indexed from 1 to $n$) and a set $\I=\{1,\ldots
,I\}$ of job classes as depicted in Figure \ref{figv}. Jobs of
\begin{figure}
\includegraphics{777f01.eps}
\caption{A multiclass queue with many servers.}
\label{figv}
\end{figure}
class-$i$ arrive according to a Poisson process with rate $\lambda_i$
and wait in their designated queue until their service begins. Once
admitted to service, the service time of a class-$i$ job is distributed
as an exponential random variable with rate $\mu_i>0$. All service and
interarrival times are mutually independent.
\subsection*{Heavy-traffic scaling}
We consider a sequence of systems indexed by the number of servers $n$.
The superscript $n$ will be attached to various processes and
parameters to make the dependence on $n$ explicit. (It will be omitted
from parameters and other quantities that do not change with $n$.) We
assume\vspace*{1pt} that $\lambda_i^n=a_i\lambda^n$ for all~$n$, where $\lambda
^n$ is the total arrival rate and $a_i>0$ for $i\in\I$ with $\sum
_{i}a_i=1$. This assumption is made for simplicity of notation and
presentation. Nothing changes\vspace*{1pt} in our results if one assumes, instead,
that $\lambda_i^n/n\rightarrow\lambda_i$ and $\sqrt{n}(\lambda
_i^n/n-\lambda_i)\rightarrow\hat{\lambda}_i$ as $n\tinf$ where
$\lambda_i/\sum_{k\in\I}\lambda_k=a_i>0$.\vadjust{\goodbreak}
The nominal load in the $n$th system is then given by
\[
R^n=\sum_{i}\frac{\lambda_i^n}{\mu_i}=\lambda^n\sum_{i}\frac
{a_i}{\mu_i},
\]
so that defining $\bar{\mu}=[\sum_{i}a_i/\mu_i]^{-1}$ we have that
$R^n=\lambda^n/\bar{\mu}$, which corresponds to the nominal number
of servers required to handle all the incoming jobs. The heavy-traffic
regime is then imposed by requiring that the number of servers deviates
from the nominal load by a term that is a square root of the nominal
load. Formally, we impose this by assuming that $\lambda^n$ is such
that
\begin{equation} \label{eqHTstaff}
n= R^n+\beta\sqrt{R^n}
\end{equation}
for some
$\beta\in(-\infty,\infty)$ that does not scale with $n$. Also, we
define the relative load imposed on the system by class-$i$ jobs,
denoted by $\nu_{i}$, as follows:
\begin{equation}\label{eqrhodefin}
\nu_i =\frac{a_i/\mu_i}{\sum
_{k\in\I}a_k/\mu_k}.
\end{equation}
Note that $\sum_{i\in\I}\nu_i=1$, and $\nu_i n$ can be interpreted
as a first-order (fluid) estimate for the number of servers busy
serving class-$i$ customers.
\subsection{System dynamics}
Let $Q_i\lam(t)$ and $X_i\lam(t)$ denote the number of class-$i$ jobs
in the queue and in the system, respectively, at time $t$ in the $n$th
system. Similarly, let $Z_i\lam(t)$ denote the number of servers
working on class-$i$ jobs at time $t$. Clearly, for all $i, n,
t$, the following holds:
\[
X_i\lam(t)=Z_i\lam(t)+Q_i\lam(t).\vadjust{\goodbreak}
\]
In our setting, a control corresponds to determining how many of the
\mbox{class-$i$} jobs currently in the system are placed in queue and in
service for $i\in\I$. We take the process $Z^n$ as our control in the
$n$th system. Note that one can equivalently take the queue length
process $Q^n$ as the control. (The knowledge of either process is
sufficient to pin down the evolution of the system given the arrival,
service processes and the initial conditions.) Clearly, the control
process must satisfy certain requirements for admissibility, including
the usual nonanticipativity requirement. We defer a precise
mathematical definition of admissible controls for now (see Definition
\ref{definadmissiblecontrols}). However, it should be clear that,
given the process $Z^n$, one can construct the other processes of interest.
To be specific, consider a complete probability space $(\Omega
,\mathcal{F},\mathbb{P})$ and $2I$ mutually independent
\textit{unit-rate} Poisson processes $(\mathcal{N}_i^a(\cdot), \mathcal
{N}_i^d(\cdot), i\in\I)$ on that space. Given the \textit{primitives}
$(\mathcal{N}_i^d(\cdot),\mathcal{N}_i^a(\cdot),X_i\lam(0),Z_i\lam
(0);i\in\I)$ and the control process~$Z^n$, we construct the
processes $X^n, Q^n$ as follows: for $t \geq0$ and $i \in\mathcal{I}$
\begin{eqnarray} \label{eqdynamics1}
X_i\lam(t)&=&X_i\lam(0)+\mN_i^a(\lambda_i^n t)-\mN_i^d\biggl( \mu
_i\int_0^t Z_i\lam(s)\,ds \biggr),\\
Q_i\lam(t)&=&X_i\lam(t)-Z_i\lam(t).
\end{eqnarray}
The processes $Z^n, Q^n, X^n$ must jointly satisfy the constraints
\begin{equation}\label{eqnon-negativity}
(Q\lam(t), X\lam(t),Z\lam(t))\in\mathbb{Z}_+^{3I},\qquad
e\cdot Z\lam(t)\leq n,
\end{equation}
where $e$ is the $I$-dimensional vector of ones.
Controls can be preemptive or nonpreemptive. Under a nonpreemptive
control, a job that is assigned to a server keeps the server busy until
its service is completed. In particular, given a nonpreemptive control
$Z^n$, the process $Z_i^n$ can decrease only through service
completions of class-$i$ jobs. In contrast, the class of preemptive
controls is broader. While it includes nonpreemptive policies, it also
includes controls that (occasionally) may preempt a job's service. The
preempted job is put back in the queue and its service is resumed at a
later time (possibly by a different server). Hence, the class of
preemptive controls subsumes the class of nonpreemptive ones (which is
also immediate from Definition 1 in \cite{AMR02}) and the cost of an
optimal policy among preemptive ones gives a lower bound for that among
the nonpreemptive ones.
In what follows, we will largely focus on preemptive controls, which
are easier to work with, and derive a specific policy which is near
optimal in that class. The specific policy we derive is, however,
nonpreemptive, and therefore, is near optimal among the nonpreemptive
policies as well. More specifically, the policy we propose belongs to a
class which we refer to as \textit{tracking policies.}
To facilitate the definition of tracking policies, define $\mathcal{U}
\subset\mathbb{R}_+^I$ as
\begin{equation}\label{eqmUdefin}
\mathcal{U}=\biggl\{u\in\bbR_+^I\dvtx
\sum_{i}u_i=1\biggr\}.
\end{equation}
Also, for all $i$ and $t
\geq0$, let
\begin{equation} \label{eqtildeXdefin}
\check{X}_i\lam(t)=X_i\lam(t)-\nu_in.
\end{equation}
Hence, the process $\check{X}_i^n$ captures the
oscillations of the process $X_i\lam$ around its ``fluid''
approximation $\nu_i n$. Throughout our analysis, for $x\in\bbR$ we
let $(x)^+=\max\{0,x\}$ and $(x)^-=\max\{0,-x\}$.
\begin{defin}\label{defintracking}
Given a function $h\dvtx\bbR^I\to\mathcal{U}$, an $h$-tracking policy
makes resource allocation decisions in the $n$th system as follows:
\begin{longlist}
\item It is nonpreemptive. That is, once a server starts
working on a job, it continues without interruption until that job's
service is completed.
\item It is work conserving. That is, the number of busy
servers satisfies $e\cdot Z^n(t)=(e\cdot X^n(t))\wedge n$ for all $t>
0$. In particular, no server is idle as long as there are $n$ or more
jobs in the system.
\item When a class-$i$ job arrives to the system it joins the
queue of class $i$ if all servers are busy processing other jobs.
Otherwise, the lowest-indexed idle server starts working on that job.
\item A server that finishes processing a job at a time $t$,
idles if all queues are empty. Otherwise, she starts working on a job
of class $i\in\mathcal{K}(t-)$ with probability $\lambda_i^n/\sum
_{k\in\mathcal{K}(t-)}\lambda_k^n$, where, for $t>0$, the set
$\mathcal{K}(t-)$ is defined by
\begin{equation}\label{eqmathKdefin}
\mathcal{K}(t-)=\bigl\{k\in\I\dvtx Q_k(t)-h_k(\check
{X}^n(t-))\bigl(e\cdot\check{X}^n(t-)\bigr)^+>0\bigr\}.
\end{equation}
Finally, if $(e\cdot\check{X}^n(t-))^+>0$ and $\mathcal
{K}(t-)=\varnothing$, she picks for service a customer from the lowest
index nonempty queue.
\end{longlist}
\end{defin}
\begin{rem}\label{remrandomization}
For our optimality-gap bounds and, in particular, for the proof of
Theorem \ref{thmSSC} it is important that the policy be such that
each of the job classes in the set $\mathcal{K}(t)$ gets a sufficient
share of the capacity. This prevents excessive oscillation of the
queues that may compromise the optimality gaps. Such oscillations could
arise if, for example, the policy chooses for service a~job of class
\[
i=\min\argmax_{k\in\I}\bigl\{Q_k(t-)-h_k(\check{X}^n(t-))\bigl(e\cdot
\check{X}^n(t-)\bigr)^+\dvtx Q_k^n(t-)>0\bigr\}.
\]
Randomization is just one way
to overcome such oscillations and, as the proofs (specifically that
of Theorem \ref{thmSSC}) reveal, any choice rule that guarantees a
sufficient share of the capacity to a class in $\mathcal{K}(t-)$ will
suffice.
\end{rem}
Our main result shows that a (nonpreemptive) tracking policy can
achieve a near optimal performance among preemptive policies. Note that
in our setting under\vadjust{\goodbreak} preemption, one can restrict attention to
work-conserving policies, that is, policies under which the servers
never idle as long as there are jobs to work on.\footnote{By a coupling
argument, this can be shown to hold with general queueing costs
provided that there are no abandonments and that the service times are
exponential; see, for example, the coupling argument on page 1126 of
\cite{AMR02}.} More precisely, a control is work conserving if the
following holds for all $t>0$:
\begin{equation}\label{eqworkconservation}
e\cdot Q\lam(t)=\bigl(e\cdot\check{X}\lam(t)\bigr)^+.
\end{equation}
Hereafter, we focus on work-conserving controls. Each such control can
be mapped into a ratio control, which specifies what fraction of the
total number of jobs in queue belongs to each class. To that end, let
\begin{equation}\label{eqUQmap}
U_i\lam(t)=\frac{Q_i\lam(t)}{(e\cdot Q\lam(t))\vee1} .
\end{equation}
Note that the original control $Z^n$ can be recovered from the ratio
control~$U^n$ as follows:
\[
Z_{i}^{n}(t) = X_i^n(t) - U_{i}^{n}(t) \bigl(e\cdot\check{X}^n(t)\bigr)^+ .
\]
Equations (\ref{eqdynamics1})--(\ref{eqnon-negativity}) can then be
replaced by
\begin{eqnarray}
\label{eqdynamics2}
X_i\lam(t)&=&X_i\lam(0)+\mN_i^a(\lambda_i^n t) \nonumber\\[-8pt]\\[-8pt]
&&{} -\mN_i^d\biggl(\mu_i \int_0^t \bigl(
X_i\lam(s)-U_i\lam(s) \bigl(e\cdot\check{X}^n(t)\bigr)^+ \bigr) \,ds\biggr), \nonumber\\
Q_i\lam(t)&=&U_i\lam(t)\bigl(e\cdot\check{X}^n(t)\bigr)^+, \\
Z_i\lam(t)&=&X_i\lam(t)-Q_i\lam(t),\\
\check{X}_i\lam(t)&=&X_i\lam(t)-\nu_i n,\\
\label{eqnon-negativity2}
U\lam(t)&\in&\mathcal{U},\qquad Q\lam(t)\in\mathbb{Z}_+^I,\qquad X\lam(t)\in
\mathbb{Z}_+^I.
\end{eqnarray}
Define the filtration
\[
\bar{\mathcal{F}}_t=\sigma\{\mN_i^a(s),\mN_i^d(s);i\in\I, s\leq
t\}
\]
and the $\sigma$-field
\begin{equation}\label{eqcheckFdefin}
\bar{\mathcal{F}}_{\infty}=\bigvee
_{t\geq0} \bar{\mathcal{F}}_t.
\end{equation}
Informally,
$\bar{\mathcal{F}}_{\infty}$ contains the information about the
entire evolution of the processes $(\mN_i^a,\mN_i^d,i\in\I)$. A
natural notion of admissibility requires that the control is
nonanticipative so that it only uses historical information about the
process $X^n$ and about the arrivals and service completions up to the
decision epoch. To accommodate randomized policies (as the $h$-tracking
policy) we allow the control to use other information too as long as
this information is independent of $\bar{\mathcal{F}}_{\infty}$.
\begin{defin}\label{definadmissiblecontrols}
A process $U=(U_i(t), t\geq0, i \in\mathcal{I})$
is a ratio control for the $n$th system if there exists a process
$\mathbb{X}\lam=(X\lam,Q\lam,Z\lam,\check{X}\lam)$ such that,
together with the primitives, $(\mathbb{X}\lam,U)$ satisfies
(\ref{eqdynamics2})--(\ref{eqnon-negativity2}). The process
$U$ is an admissible ratio control if, in addition, it is adapted to
the filtration $\mathcal{G}\vee\mathcal{F}_t\lam$ where
\begin{eqnarray*}
\mathcal{F}_t\lam&=&\sigma\biggl\{\mN_i^a(\lambda
_i^n s),X_i\lam(s),\mu_i\int_0^s Z_i\lam(u)\,du,\\
&&\hphantom{\sigma\biggl\{}
\mN_i^d\biggl(\mu_i\int_0^s Z_i\lam(u)\,du\biggr); i\in
\I,0\leq s\leq t\biggr\},
\end{eqnarray*}
and $\mathcal{G}$ is a $\sigma$-field that is independent of $\bar
{\mathcal{F}}_{\infty}$. The process $\mathbb{X}^n$ is then said to
be the queueing process associated with the ratio control $U$. We let
$\Pi\lam$ be the set of admissible ratio controls for the $n$th
system.
\end{defin}
Ratio controls are work conserving by definition, but they need not be
nonpreemptive in general. However, note that given a function $h\dvtx \bbR
^I\to\mathcal{U}$, the (nonpreemptive) $h$-tracking policy
corresponds to a ratio control $U_h$, which is nonpreemptive. To be
specific, given the primitives and the $h$-tracking policy, one can
construct the corresponding queueing process $\mathbb
{X}^n=(X^n,Q^n,Z^n,\check{X}^n)$ (see the construction after Lemma
\ref{lemstrongappbounds}). Then the ratio control $U_h$ is
constructed using the relation (\ref{eqUQmap}) so that $\mathbb
{X}^n$ and $U_h$ jointly satisfy (\ref{eqdynamics2})--(\ref
{eqnon-negativity2}). Hence, one can speak of the ratio control and
the queueing process associated with an $h$-tracking policy. Note that
since the tracking policy makes resource allocation decisions using
only information on the state of the system at the decision epoch
(together with a randomization that is independent of the history), the
resulting ratio control is admissible in the sense of Definition \ref
{definadmissiblecontrols}. The terms ratio control and $h$-tracking
policy appear in several places in the paper. It will be clear from the
context whether we refer to an arbitrary ratio control or to one
associated with an $h$-tracking policy.
We close this section by stating the main result of the paper. To that
end, let
\begin{equation}\label{eqmathXdefin}
\mathcal{X}^n=\{(x,q)\in\mathbb{Z}_+^{2I}\dvtx
q=u(e\cdot
x-n)^+\mbox{ for some } u\in\mathcal{U}\}.
\end{equation}
That is, $\mathcal{X}^n$ is the set on which $(X^n, Q^n)$ can
obtain values under work conservation. In this set $e\cdot q=(e\cdot
x-n)^+$ so that positive queue and idleness do not co-exist. We let
$\Ex_{x,q}^{U}[\cdot]$ denote the expectation with respect to the
initial condition $(X^n(0),Q^n(0))=(x,q)$ and an admissible ratio
control $U$. Given a ratio control $U$ and initial conditions $(x,q)$,
the expected infinite horizon discounted cost in the $n$th system
is given by
\begin{equation}\label{eqcost1}
C\lam(x,q,U)=\Ex_{x,q}^{U}\biggl[\int_0^{\infty
}e^{-\gamma s}
c\cdot Q\lam(s)\,ds\biggr],
\end{equation}
where $c=(c_1,\ldots,c_I)'$ is the strictly positive vector of holding
cost rates and $\gamma> 0$ is the discount rate. For $(x,q) \in
\mathcal{X}^n$, the value function is given by
\[
V\lam(x,q)=\inf_{U\in\Pi\lam}\Ex_{x,q}^{U}\biggl[\int_0^{\infty
} e^{-\gamma s}c\cdot Q\lam(s)\,ds\biggr].
\]
We next state our main result.
\begin{theorem}\label{thmmain} Fix a sequence $\{(x^n,q^n),n\in\bbZ_+\}
$ such that
\mbox{$(x^n,q^n)\in\mathcal{X}^n$} and $|x^n-\nu n| \leq M \sqrt{n}$ for
all $n$ and some $M>0$. Then, there exists a sequence of tracking
functions $\{h^n,n\in\bbZ_+\}$ together with constants $C,k>0$ (that
do not depend on $n$) such that
\[
C\lam(x^n,q^n,U_h^n)\leq V\lam(x^n,q^n)+C\log^{k} n \qquad\mbox{for all
} n,
\]
where $U_h^n$ is the ratio control associated with the $h^{n}$-tracking
policy.
\end{theorem}
The constant $k$ in our bound may depend on all system and cost
parameters but not on $n$. In particular, it may depend on $(\mu
_i,c_i,a_i;i\in\I)$ and $\beta$. Its value is explicitly defined
after the statement of Theorem \ref{thmHJB1sol}.
Theorem \ref{thmmain} implies, in particular, that the optimal
performance for nonpreemptive policies is close to that among the
larger family of preemptive policies. Indeed, we identify a
nonpreemptive policy (a tracking policy) in the queueing model whose
cost performance is close to the optimal value of the preemptive
control problem.
The rest of the paper is devoted to the proof of Theorem \ref{thmmain},
which proceeds by studying a sequence of auxiliary Brownian
control problems. The next subsection offers a heuristic derivation and
a justification for the relevance of the sequence of Brownian control
problems to be considered in later sections.
\subsection{Toward a Brownian control problem}
We proceed by deriving a sequence of approximating Brownian control
problems heuristically, which will be instrumental in deriving a
near-optimal policy for our original control problem. It is important
to note that we derive an approximating Brownian control problem for
each $n$ as opposed to deriving a single approximating problem (for the
entire sequence of problems). This distinction is crucial for achieving
an improved optimality gap for $n$ large because it allows us to tailor
the approximation to each element of the sequence of systems.
To this end, let
\[
l_i^n=\lambda_i^n-\mu_i\nu_i n \qquad\mbox{for } i\in\I.
\]
Fixing an admissible control $U^n$ for the $n$th system [and
centering as in (\ref{eqtildeXdefin})], we can then write (\ref
{eqdynamics2}) as
\begin{equation}\label{eqcheckXdynamics}
\check{X}_i\lam(t)=\check{X}_i\lam(0)+l_i\lam t-\mu
_i\int_0^t
\bigl( \check{X}_i\lam(s)-U^n_i(s)\bigl( e\cdot\check{X}\lam(s)\bigr)^+
\bigr) \,ds+\check{W}_i\lam(t),\hspace*{-30pt}
\end{equation}
where
\begin{eqnarray}\label{eqWtildedefin}
\check{W}_i\lam(t)&=&\mN_i^a(\lambda_i^n t)-\lambda_i^n
t+\mu_i\int_0^t \bigl( \check{X}_i\lam(s)-U^n_i(s)\bigl( e\cdot\check
{X}\lam(s)\bigr)^+ \bigr) \,ds\nonumber\\[-8pt]\\[-8pt]
&&{}-
\mN_i^d\biggl(\mu_i\int_0^t \bigl( \check{X}_i\lam(s)+\nu_i
n-U^n_i(s)\bigl( e\cdot\check{X}\lam(s)\bigr)^+ \bigr) \,ds \biggr).\nonumber
\end{eqnarray}
In words, $\check{W}_i\lam(t)$ captures the deviations of the Poisson
processes from their means. It is\vspace*{1pt} natural to expect that an
approximation result of the following form will hold: $(\check
{X}_i^n,\check{W}_i^n;i\in\I)$ can be approximated by $(\hX
_i^n,\hW_i^n;i\in\I)$ where
\begin{eqnarray*}
\hat{X}_i\lam(t)&=&\hat{X}_i\lam(0)+l_i^n t -\mu_i\int_0^t \bigl(
\hat{X}_i\lam(s)-U^n_i(s)\bigl(e\cdot\hat{X}\lam(s)\bigr)^+ \bigr) \,ds+\hat
{W}_i\lam(t),
\\
\hat{W}_i(t)&=&\tilde{B}_i^a(\lambda_i^n t)+\tilde{B}_i^S\biggl(\mu
_i\int_0^t \bigl( \hat{X}_i\lam(s)+\nu_i n-U^n_i(s)\bigl( e\cdot\hat
{X}\lam(s)\bigr)^+ \bigr) \,ds\biggr)
\end{eqnarray*}
and $\tilde{B}^a,\tilde{B}^s$ are $I$-dimensional independent
standard Brownian motions. Moreover, by a time-change argument we can
write (see, e.g., Theorem 4.6 in \cite{KaS91})
\begin{eqnarray}\label{eqhatX}
\hat{X}_i\lam(t) &=&\hat{X}_i\lam
(0)+l_i^n t -\mu_i\int_0^t \hat{X}_i\lam(s)-U^n_i(s)\bigl(e\cdot\hat
{X}\lam(s)\bigr)^+ \,ds\nonumber\\[-8pt]\\[-8pt]
&&{}+\int_0^t \sqrt{\lambda_i^n+\mu_i\bigl( \hat
{X}_i\lam(s)+\nu_i n-U^n_i(s)\bigl( e\cdot\hat{X}\lam(s)\bigr)^+
\bigr)}\,dB_i(s),\nonumber\hspace*{-30pt}
\end{eqnarray}
where $B$ is an $I$-dimensional standard Brownian motion constructed by setting
\begin{eqnarray*}B_i(t)&=& \int_0^t\frac{d\tilde{B}_i^{S}(\mu
_i\int_0^s ( \hat{X}_i\lam(u)+\nu_i n-U^n_i(u)( e\cdot\hat
{X}\lam(u))^+ ) \,du)}{\sqrt{\mu_i ( \hat
{X}_i\lam(s)+\nu_i n-U^n_i(s)( e\cdot\hat{X}\lam(s))^+
)}}\\
&&{} + \frac{\tilde{B}_i^a(\lambda_i^nt)}{\lambda_i^n t}.
\end{eqnarray*}
Taking a leap of faith and arguing heuristically, we next consider a
Brownian control problem with the system dynamics
\begin{equation}\label{eqnbmdefn1}
\hat{X}\lam(t)=x+\int_0^t b\lam(\hat{X}\lam
(s),\hat
{U}^n(s))\,ds+\int_0^t \sigma\lam(\hat{X}\lam(s),\hat
{U}^n(s))\,dB(t),\hspace*{-30pt}
\end{equation}
where $\hat{U}^n$ will be
an admissible control for the Brownian system and
\begin{equation}
\label{eqnbmdefn30}
b_i\lam(x,u)=l_i\lam-\mu_i\bigl(x_i-u_i(e\cdot x)^+\bigr)
\end{equation}
and
\begin{equation}
\label{eqnbmdefn3}
\sigma_i\lam(x,u)=\sqrt{\lambda_i^n+\mu_i \nu_in +\mu
_i\bigl(x_i-u_i(e\cdot x)^+\bigr)}.
\end{equation}
Note that the Brownian control problem will only be used to propose a~candidate policy, whose near optimality will be verified from first
principles without relying on the heuristic derivations of this section.
To repeat, the preceding definition is purely formal and provided only
as a means of motivating our approach. In what follows, we will
directly state and analyze an auxiliary Brownian control problem
motivated by the above heuristic derivation. The analysis of the
auxiliary Brownian control problem lends itself to constructing near
optimal policies for our original control problem. To be more specific,
the system dynamics equation (\ref{eqnbmdefn1}), and in particular,
the fact that its variance is state and control dependent, is crucial
to our results. Indeed, it is this feature of the auxiliary Brownian
control problems that yields an improved optimality gap.
Needless to say, one needs to take care in interpreting (\ref
{eqnbmdefn1})--(\ref{eqnbmdefn3}), which are meaningful only up
to a suitably defined hitting time. In particular, to have $\sigma^n$
well defined, we restrict attention to the process while it is within
some bounded domain. Actually, it suffices for our purposes to fix
$\kappa>0$ and $m\geq3$ and consider the Brownian control problem
only up to the hitting time of a ball of the form
\begin{equation}\label{eqBkappadefin}
\mB_{\kappa}^n=\bigl\{x\in\mathbb{R}^I\dvtx |x|< \kappa
\sqrt{n}\log^m
n\bigr\},
\end{equation}
where \mbox{$|\cdot|$} denotes the Euclidian
norm. We will fix the constant $m$ throughout and suppress the
dependence on $m$ from the notation. Setting
\begin{equation}\label{eqnkappa}
n(\kappa)=\inf\{n\in
\bbZ_+\dvtx \sigma^n(x,u)\geq1 \mbox{ for all } x\in\mB_{\kappa}^n,
u\in\mathcal{U}\},
\end{equation}
the diffusion coefficient is strictly positive for all $n \geq n(\kappa
)$ and $x\in\mB_{\kappa}^n$. Note that, for all $i\in\I$, $x\in
\mB_{\kappa}^n$ and $u\in\mathcal{U}$,
\[
(\sigma_i^n(x,u))^2\geq\lambda_i^n+\mu_i\nu_in-2\mu_i\kappa\sqrt
{n}\log^mn,
\]
so that $(\sigma_i^n(x,u))^2\geq\mu_i\nu_in/2\geq1$ for all
sufficiently large $n$ and, consequently, $n(\kappa)<\infty$.
\begin{rem} In what follows, and, in particular, through the proof of
Theorem \ref{thmmain}, the reader should note that while choosing the
size of the ball to be $\epsilon n$ (with $\epsilon$ small enough)
would suffice for the nondegeneracy of the diffusion coefficient, that
choice would be too large for our optimality gap proofs.
\end{rem}
\section{An approximating diffusion control problem (ADCP)}\label{secresult}
Motivated by the discussion in the preceding section, we define
admissible systems as follows.
\begin{defin}[(Admissible systems)]\label{definadmissiblesystemBrownian}
Fix $\kappa>0$, $n\in\bbZ
_+$ and $x\in\mathbb{R}^I$. We refer to
$\theta=(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P},\hat
{U},B)$ as an admissible $(\kappa,n)$-system if:
\begin{longlist}[(a)]
\item[(a)] $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})$ is
a complete filtered probability space.\vadjust{\goodbreak}
\item[(b)] $B(\cdot)$ is an $I$-dimensional standard Brownian motion
adapted to $(\mF_t)$.
\item[(c)] $\hat{U}$ is $\mathcal{U}$-valued, $\mF$-measurable
and $(\mF_t)$ progressively measurable.
\end{longlist}
The process $\hat{U}$ is said to be the control associated with
$\theta$. We also say that $\hat{X}$ is a controlled process
associated with the initial data $x$ and an admissible system $\theta$
if $\hat{X}$ is a continuous $(\mF_t)$-adapted process on $(\Omega
,\mathcal{F},\mathbb{P})$ such that, almost surely, for $t\leq\hat
{\tau}_{\kappa}^n$,
\[
\hat{X}(t)=x+\int_0^t b\lam(\hat{X}(s),\hat{U}(s))\,ds+\int_0^t
\sigma\lam(\hat{X}(s),\hat{U}(s))\,d\tilde{B}(t),
\]
where $b^n(\cdot,\cdot)$ and $\sigma^n(\cdot,\cdot)$ are as
defined in (\ref{eqnbmdefn30}) and (\ref{eqnbmdefn3}),
respectively, and
$\hat{\tau}_{\kappa}^n=\inf\{t\geq0\dvtx\hat{X}(t)\notin\mB_{\kappa
}^n\}$. Given $\kappa>0$ and $n\in\bbZ_+$, we let $\Theta(\kappa
,n)$ be the set of admissible $(\kappa,n)$-systems.
\end{defin}
The Brownian control problem then corresponds to optimally choosing an
admissible $(\kappa,n)$-system with associated control $(\hat
{U}(t),t\geq0)$ that achieves the minimal cost in the optimization problem
\begin{equation}\label{eqoptbrownian}
\hat{V}\lam(x,\kappa)=\inf_{\theta\in\Theta
(\kappa,n)}\Ex
_{x}^{\theta}\biggl[\int_0^{\hat{\tau}_{\kappa}^n} e^{-\gamma s}
\sum_{i\in\I}c_i \hat{U}_i(s)\bigl(e\cdot\hat{X}(s)\bigr)^+\,ds
\biggr],
\end{equation}
where $\Ex_x^{\theta}[\cdot]$ is the
expectation operator when the initial state is $x\in\mathbb{R}^I$ and
the admissible system $\theta$. Hereafter, we refer to (\ref
{eqoptbrownian}) as the \textit{ADCP on} $\mB_{\kappa}^n$.
The following lemma shows that the Definition \ref
{definadmissiblesystemBrownian} is not vacuous. The proof appears in
the \hyperref[app]{Appendix}.
\begin{lem} \label{lemexistenceofcontrolled}
Fix the initial state $x\in\mathbb{R}^I$, $\kappa>0$,
$n\geq n(\kappa)$ and an admissible $(\kappa,n)$-system $\theta$.
Then, there exists a unique controlled process $\hat{X}$ associated
with $x$ and $\theta$.
\end{lem}
To facilitate future analysis, note from the definition of $\hat{\tau
}_k^n$ and (\ref{eqoptbrownian}) that
\begin{equation}\label{eqvaluebound}
\hat{V}^n(x,\kappa)\leq
\frac{1}{\gamma}(e\cdot c)\kappa\sqrt{n}\log^m n.
\end{equation}
\begin{defin}[(Markov controls)]
\label{definmarkoviancontrols} We say that an admissible
$(\kappa,n)$-system $\theta=(\Omega,\mathcal{F},(\mathcal
{F}_t),\mathbb{P},\hat{U},B)$ with the associated controlled process
$\hat{X}^n$ induces a Markov control if there exists a function
$g^n(\cdot)\dvtx \mathcal{B}_{\kappa}^n \to\mathcal{U}$ such that
$\hat{U}(t)=g^n(\hat{X}^n(t))$ for $t \leq\hat{\tau}_{\kappa}^n$.
We extend the function $g^n$ to $\mathbb{R}^I$ as follows:
\begin{equation}
h^n(x)= \cases{
g^n(x), &\quad$x \in\mathcal{B}_{\kappa}^{n}$, \cr
e_1, & \quad otherwise,}
\end{equation}
where $e_1$ is the $I$-dimensional vector whose first component is $1$
while the others are $0$. We refer to $h^n(\cdot)$ as the tracking
function associated with the admissible system $\theta$.
\end{defin}
In what follows, a policy $\hat{U}$ will be called optimal for the
approximating diffusion control problem (ADCP) on $\mB_{\kappa}^n$ if
there exists an admissible $(\kappa,n)$-system $\theta=(\Omega
,\mathcal{F},(\mathcal{F}_t),\mathbb{P},\hat{U},B)$ such that
\[
\hat{V}\lam(x,\kappa)=\Ex_{x}^{\theta}\biggl[\int_0^{\hat{\tau
}_{\kappa}^n} e^{-\gamma s} \sum_{i\in\I}c_i \hat{U}_i(s)\bigl(e\cdot
\hat{X}(s)\bigr)^+\,ds\biggr].
\]
Recall that $X$ and $U$ are used to denote performance relevant
stochastic processes in both the Brownian model and the original
queueing model, and that we add a hat, that is, we use $\hat{X}$ and
$\hat{U}$ in the context of the Brownian model. To avoid confusion,
the reader should keep in mind that hat-processes correspond to the
ADCP while the ones with no hats correspond to the original queueing model.
\subsection*{Roadmap for the remainder of the paper} The main result in
Theorem~\ref{thmmain} builds on the following steps:
\begin{enumerate}
\item In Section \ref{secADCP} we show that for each $n$, the HJB
equation associated with the ADCP has a unique\vadjust{\goodbreak} and sufficiently smooth
solution. Using that solution we advance an optimal Markov control for
the ADCP together with the corresponding tracking function. We also
identify useful gradient bounds on the solutions to the sequence of HJB
equations; cf. Theorem~\ref{thmHJB1sol}.
\item In Section \ref{sectracking} we conduct a performance analysis
of $h$-tracking policies in the queueing system; cf. Theorem \ref{thmSSC}.
\item The result of Theorem \ref{thmSSC} together with the gradient
estimates in Theorem \ref{thmHJB1sol} are combined in a
Taylor expansion-type argument in Section \ref{seccombining} to
complete the proof of Theorem \ref{thmmain}.
\end{enumerate}
As a convention, throughout the paper we use the capital letter $C$ to
denote a constant that does not depend on $n$. The value of $C$ may
change from line to line within the proofs but it will be clear from
the context.
\section{Solution to the ADCP} \label{secADCP} This section provides
a solution for the ADCP on $\mB_{\kappa}^n$ for each $n\in\bbZ$ and
$\kappa>0$. The HJB equation is a fully nonlinear and nonsmooth PDE.
As such, it requires extra care when compared with the usual semilinear
PDEs that arise in the analysis of \textit{asymptotically} optimal
controls in the Halfin--Whitt regime. We will build on existing results
in the theory of PDEs and proceed through the following steps: (a)
establish the existence and uniqueness of classical solutions; (b)
relate this unique solution to the value function of the ADCP and (c)
establish useful gradient estimates on the solution for the HJB
equation. The last step is not necessary for existence and uniqueness
but is important for the analysis of optimality gaps.
In what follows, we fix $\kappa>0$ and $n\geq n(\kappa)$ and suppress
the dependence of the solution to the HJB equation on $n$ and $\kappa
$. The following notation is needed to introduce the HJB equation.
Given a twice continuously differentiable function $\phi$, define
\[
\phi_i=\frac{\partial\phi}{\partial x_i}\quad\mbox{and}\quad \phi_{ii}=
\frac{\partial^2 \phi}{\partial x_i^2} .
\]
Also, define the operator $A^n_u$ for $u\in\mathcal{U}$ as follows:
\begin{equation}\label{eqgendefin}
A_{u}\lam\phi= \sum_{i\in\I} b_i\lam(\cdot
,u)\phi_i+\frac
{1}{2}\sum_{i\in\I} (\sigma_i\lam(\cdot,u))^2 \phi_{ii}.
\end{equation}
Defining
\[
L(x,u)=\sum_{i\in\I}c_iu_i (e\cdot x)^+
\]
for $x\in\bbR_+^I$ and $u\in\mathcal{U}$, the HJB equation is given by
\begin{equation} \label{eqHJB0}
0=\inf_{u \in\mathcal{U}}\{
L(x,u)+A_{u}\lam\phi
(x)-\gamma\phi(x)\}.
\end{equation}
Substituting $b\lam(\cdot,\cdot)$ and $\sigma\lam(\cdot
,\cdot)$ into (\ref{eqHJB0}) gives
\begin{eqnarray} \label{eqHJB1simp}
0&=& -\gamma\phi(x) + (e\cdot x)^+\cdot\min_{i\in\I}\biggl\{
c_i+\mu_i\phi_i(x)-\frac{1}{2}\mu_i\phi_{ii}(x)\biggr\}
\nonumber\\[-8pt]\\[-8pt]
&&{}+\sum_{i\in\I} (l_i\lam-\mu_ix_i)\phi_i(x)
+\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+x_i)\bigr)\phi_{ii}(x).\nonumber
\end{eqnarray}
Our analysis of the HJB equation (\ref{eqHJB1simp}) draws on existing
results on fully nonlinear PDEs, and, in particular, the results on
Bellman--Pucci type equations; cf. Chapter 17 of \cite{TandG}.
In what follows, fixing a set $\mB\subseteq\mathbb{R}_+^I$, $\mC
^2(\mB)$ denotes the space of twice continuously differentiable
functions from $\mB$ to $\mathbb{R}$. For $u\in\mC^{2}(\mB)$, we
let~$Du$ and $D^2u$ denote the gradient and the Hessian of $u$,
respectively. The space $\mC^{2,\alpha}(\mB)$ is then the subspace
of $\mC^{2}(\mB)$ members of which also have second derivatives that
are H\"{o}lder continuous of order $\alpha$. That is, a twice
continuously differentiable function $u\dvtx \mathbb{R}^I\to\mathbb{R}$
is in $\mC^{2,\alpha}(\mB)$ if
\[
\sup_{x,y\in\mB, x\neq y} \frac{|D^2u(x)-D^2u(y)|}{|x-y|^{\alpha
}}<\infty,
\]
where \mbox{$|\cdot|$} denotes the Euclidian norm. We define
$d_x\,{=}\,\operatorname{dist}(x,\partial\mB)\,{=}\,\inf\{|x\,{-}\,y|,\allowbreak
y\,{\in}\,\partial\mB\}$ where
$\partial\mB$ stands for the boundary of $\mB$ and we let
$d_{x,z}\,{=}\,\min\{d_x,d_z\}$. Also, we define
\begin{equation}\label{equstardefin}
|u|^*_{2,\alpha,\mB}=\sum_{j=0}^2 [u]_{j,\mB}^*+\sup_{x,y\in\mB,x\neq
y}d_{x,y}^{2+\alpha}\frac{|D^2u(x)-D^2 u(y)|}{|x-y|^{\alpha}},
\end{equation}
where $[u]_{j,\mB}^*=\sup_{x\in\mB}d_x^j
|D^j u(x)|$ for $j=0,1,2$. Note that $d_x^j$ denote the $j$th power
of $d_x$ and, similarly, $d_{x,y}^{2+\alpha}$ is the $(2+\alpha)$th
power of $d_{x,y}$. Finally, we let $|u|^*_{0,\mB}=[u]_{0,\mB
}^*=\sup_{x\in\mB}|u(x)|$.
In the statement of the following theorem, $e_j$ is the $I$-dimensional
vector with $1$ in the $j$th place and zeros elsewhere. Also, $\mB
_{\kappa}^n$, $m$ and $n(\kappa)$ are as defined in (\ref
{eqBkappadefin}) and (\ref{eqnkappa}), respectively.
\begin{theorem}\label{thmHJB1sol}
Fix $\kappa>0$ and $n\geq n(\kappa)$. Then, there
exists $0<\alpha\leq1$ (that does not depend on $n$) and a unique
classical solution $\phi_{\kappa}\lam\in\mC^{0,1}(\bar{\mB}
_{\kappa}^n)\cap\mC^{2,\alpha}(\mB_{\kappa}^n)$ to the HJB
equation (\ref{eqHJB1simp}) on $\mB_{\kappa}^n$ with the
boundary condition $\phi_{\kappa}\lam=0$ on $\partial\mB_{\kappa
}^n$. Furthermore, there exists a constant $C>0$ (that does not depend
on $n$) such that
\begin{equation}\label{eqgradients0}
|\phi_{\kappa}\lam|^*_{2,\alpha,\mB_{\kappa
}^n}\leq C\sqrt
{n}\log^{k_0} n,
\end{equation}
where $k_0=4m(1+1/\alpha)$. In turn, for any $\vartheta<1$,
\begin{equation}\label{eqgradients1}
\sup_{x\in\mB_{\vartheta\kappa}^n}|D\phi_{\kappa
}^n(x)|\leq
\frac{C}{1-\vartheta}\log^{k_1} n \quad\mbox{and}\quad\sup_{x\in\mB
_{\vartheta\kappa}^n}|D^2\phi_{\kappa}^n(x)|\leq\frac
{C}{1-\vartheta}\frac{\log^{k_2}n} {\sqrt{n}}\hspace*{-26pt}
\end{equation}
with $k_1=k_0-m$ and $k_2=k_0-2m$. Also,
\begin{eqnarray}\label{eqgenbound}
&&\sup_{u\in\mathcal{U}}\biggl|\sum_{i\in\I}\bigl((\phi_{\kappa
}^n)_{ii}(y)-(\phi_{\kappa}^n)_{ii}(x)\bigr)(\sigma_i^n(x,u))^2
\biggr|\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq\frac{C}{1-\vartheta}\log^{k_1}n\nonumber
\end{eqnarray}
for all $x,y\in\mB_{\vartheta\kappa}^n$ with $|x-y|\leq1$.
\end{theorem}
Note that (\ref{eqgradients1}) follows immediately from (\ref
{eqgradients0}) through the definition of the operation \mbox{$|\cdot
|^*_{2,\alpha,\mB_{\kappa}^n}$} in (\ref{equstardefin}).
Henceforth, we will use $k_i,i=0,1,2$ for the values given in the
statement of Theorem \ref{thmHJB1sol}. Moreover, the constant $k$
appearing in the statement of Theorem \ref{thmmain} is equal to $k_0+3$.
Theorem \ref{thmHJB1sol} facilitates a verification result, which we
state next followed by the proof of Theorem \ref{thmHJB1sol}. Below,
$\hat{V}^n(x,\kappa)$ is the value function of the ADCP; cf. equation
(\ref{eqoptbrownian}).
\begin{theorem}\label{thmBrownianverification}
Fix $\kappa>0$ and $n\geq n(\kappa)$. Let $\phi
_{\kappa}^n$ be the unique solution to the HJB equation (\ref
{eqHJB1simp}) on $\mB_{\kappa}^n$ with the boundary condition $\phi
_{\kappa}\lam=0$ on $\partial\mB_{\kappa}^n$. Then,
$\phi_{\kappa}^n(x)=\hat{V}^n(x,\kappa)$ for all $x\in\mB_{\kappa
}^n$. Moreover, there exists a Markov control which is optimal for the
ADCP on $\mB_{\kappa}^n$. The tracking function $h_{\kappa}^{*,n}$
associated with this optimal Markov control is defined by $h_{\kappa
}^{*,n}(x)=e_{i^n(x)}$, where
\begin{equation}\label{eqixdefin}
i^n(x)=\min\mathop{\argmin}_{i\in\I} \biggl\{ \biggl(c_i+\mu
_i(\phi
_{\kappa}\lam)_i(x)-\frac{1}{2}\mu_i(\phi_{\kappa}\lam
)_{ii}(x)\biggr)(e\cdot x)^+\biggr\}.\hspace*{-25pt}
\end{equation}
\end{theorem}
The HJB equation (\ref{eqHJB1simp}) has two sources of
nondifferentiability. The first source is the minimum operation and the
second is the nondifferentiability of the term $(e\cdot x)^+$. The
first source of nondifferentiability is covered almost entirely by the
results in \cite{TandG}. To deal with the nondifferentiability of the
function $(e\cdot x)^+$, we use a construction by approximations. The
proof of existence and uniqueness in Theorem \ref{thmHJB1sol} follows
an approximation scheme where one replaces the nonsmooth function
$(e\cdot x)^+$ by a smooth (parameterized by $a$) function $f_a(e\cdot
x)$. We show that the resulting ``perturbed'' PDE has a unique
classical solution and that as $a\tinf$ the corresponding sequence of
solutions converges, in an appropriate sense, to a solution to~(\ref
{eqHJB1simp}) which will be shown to be unique. Note that this argument
is repeated for each fixed $n$ and $\kappa$.
To that end, given $a>0$, define
\begin{equation}\label{eqfdefin}
f_a(y)=\cases{
y, &\quad$\displaystyle y\geq\frac{1}{4a}$,\vspace*{2pt}\cr
\displaystyle ay^2+\frac{1}{2}y+\frac{1}{16a}, &\quad$\displaystyle -\frac{1}{4a}\leq y\leq\frac
{1}{4a}$,\vspace*{2pt}\cr
0, & \quad otherwise.}
\end{equation}
Replacing $(e\cdot x)^+$ with $f_a(e\cdot x)$ in (\ref{eqHJB1simp})
gives the following equation:
\begin{eqnarray} \label{eqHJB2}
0&=& - \gamma\phi(x)+f_a(e\cdot x)\cdot\min_{i\in
\I} \biggl\{ c_i+\mu_i\phi_i(x)-\frac{1}{2}\mu_i\phi_{ii}(x)
\biggr\} \nonumber\\[-8pt]\\[-8pt]
&&{}+\sum_{i\in\I} (l_i\lam-\mu_ix_i)\phi_i(x)
+\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+x_i)\bigr)\phi_{ii}(x).\nonumber
\end{eqnarray}
To simplify this further, let $\Gamma= \mB_{\kappa}^n\times\mathbb
{R}_+\times\mathbb{R}^I\times\mathbb{R}^{I\times I}$ and for all
$y\in\Gamma$, define the function
\begin{equation}\label{eqFForm}
F_a^k[y]=\min\{F_a^1[y],\ldots, F_a^I[y]\},
\end{equation}
where for $k\in\I$ and $y=(x,z,p,r)\in\Gamma$,
\begin{eqnarray}\label{eqFdefin}
F^k_a[y]&=&f_a(e\cdot x)
\biggl[c_k+\mu_kp_k-\frac{1}{2}\mu_kr_{kk}\biggr]+\sum_{i\in\I
}(l_i\lam-\mu_ix_i)p_i
\nonumber\\[-8pt]\\[-8pt]
&&{}+\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+x_i)\bigr)r_{ii}-\gamma z.\nonumber
\end{eqnarray}
Then, (\ref{eqHJB2}) can be rewritten as
\begin{equation}\label{eqFFormPDe}
F_a[x,u(x),Du(x),D^2u(x)]=0.
\end{equation}
In the following statement we use the gradient notation introduced at
the beginning of this section.
\begin{prop}\label{propsolPHJB} Fix $\kappa>0$, $n\geq n(\kappa)$ and $a>0$. A unique
classical solution $\phi_{\kappa,a}^n\in\mC^{0,1}(\bar{\mB}
_{\kappa}^n)\cap\mC^{2,\alpha}(\mB_{\kappa}^n)$ exists for
the PDE (\ref{eqHJB2}) on $\mB_{\kappa}^n$ with the boundary
condition $\phi_{\kappa,a}^n=0$ on $\partial\mB_{\kappa}^n$. Moreover,
\begin{equation}\label{eqgradients}
|\phi^n_{\kappa,a}|^*_{2,\alpha,\mB_{\kappa
}^n}\leq C |\phi
^n_{\kappa,a}|^*_{0,\mB_{\kappa}^n}\log^{k_0}n \leq\tilde{C}
\end{equation}
for $k_0=4m(1+1/\alpha)$ where
$0<\alpha\leq1$ and $C>0$ do not depend on $a$ and $n$ and $\tilde
{C}$ does not depend on $a$. Also, $\phi^n_{\kappa,a}$ is Lipschitz
continuous on the closure $\bar{\mB}_{\kappa}^n$ with a
Lipschitz constant that does not depend on $a$ (but can depend on
$\kappa$ and $n$).
\end{prop}
We postpone the proof of Proposition \ref{propsolPHJB} to the \hyperref[app]{Appendix}
and use it to complete the proof of Theorem \ref{thmHJB1sol},
followed by the proof of Theorem \ref{thmBrownianverification}.
\subsection*{\texorpdfstring{Proof of Theorem \protect\ref{thmHJB1sol}}{Proof of Theorem 4.1}}
Since we fix $n$ and $\kappa$, they will be suppressed below. We
proceed to show the existence by an approximation argument. To that
end, fix a sequence $\{a^k;k\in\bbZ\}$ with $a^k\tinf$ as $k\tinf$
and let $\phi_{a^k}$ be the unique solution to (\ref{eqHJB2}) as
given by Proposition \ref{propsolPHJB}. The next step is to show
that $\phi_{a^k}$ has a subsequence that converges in an appropriate
sense to a function $\phi$, which is, in fact, a solution to the HJB
equation (\ref{eqHJB1simp}). To that end, let
\begin{equation}\label{eqCstar}
\mC_{*}^{2,\alpha
}(\mB)=\{u\in\mC^{2,\alpha}(\mB)\dvtx |u|_{2,\alpha,\mB}^*<\infty\}.
\end{equation}
Then, $\mC_{*}^{2,\alpha}(\mB)$ is a Banach
space (see, e.g., Exercise 5.2 in \cite{TandG}). Since the bound in
(\ref{eqgradients}) is independent of $a$, we have that $\{\phi
_{a^k}\}$ is a bounded sequence in $C_{*}^{2,\alpha}(\mB)$ and hence,
contains a convergent subsequence. Let $u$ be a limit point of the
sequence $\{\phi_{a^k}\}$. Since the gradient estimates in Proposition
\ref{propsolPHJB} are independent of $a$, they hold also for the
limit function $u$, that is,
\begin{equation}\label{equ-bound}
|u|_{2,\alpha,\mB}^*\leq C|u|_{0,\mB
}^*\log^{k_0} n\leq\tilde{C}
\end{equation}
for constants
$\alpha$ and $C$ that are independent of $n$. Proposition \ref
{propsolPHJB} also guarantees that the global Lipschitz constant is
independent of $a$ so that we may conclude that $u\in\mC
^{0,1}(\bar{\mB})$ and that $u=0$ on $\partial\mB$.
We will now show that $u$ solves (\ref{eqHJB1simp}) uniquely. To show
that $u$ solves (\ref{eqHJB1simp}), we need to show that $F[u]=0$
(where $F[\cdot]$ is defined similar to $F_a[\cdot]$ with $(e\cdot
x)^+$ replacing $f_a(e\cdot x)$). To that end, let $\{a^k,k\in\bbZ\}
$ be the corresponding convergent subsequence [i.e., such that $\phi
_{a^k}\rightarrow u$ in $\mC_*^{2,\alpha}(\mB)$]. Henceforth, to
simplify notation, we write
\[
F_{\akl}[\phi_{\akl}(x)]=F_{\akl}[x,\phi_{\akl}(x),D\phi_{\akl
}(x),D^2\phi_{\akl}(x)]
\]
(and similarly for $F[\cdot]$). Fix $\delta\,{>}\,0$ and let $\mB(\delta
)\,{=}\,\{x\,{\in}\,\bbR^I\dvtx |x|\,{<}\,\kappa\sqrt{n}\log^m n\,{-}\,\delta\}$. Note that
since $\phi_{\akl}\rightarrow u$ in $\mC_*^{2,\alpha}(\mB)$ we
have, in particular, the convergence of $(\phi_{\akl}(x),D\phi_{\akl
}(x),D^2\phi_{\akl}(x))\rightarrow(u(x),Du(x),D^2u(x))$ uniformly in
$x\in\mB(\delta)$. The equicontinuity of the function
$F^a[\cdot]$\vadjust{\goodbreak}
on $\Gamma$ guarantees then that
\begin{equation}\label{eqinterim11}
|F_{\akl}[\phi_{\akl}(x)]-F_{\akl}[u(x)]|\leq
\epsilon
\end{equation}
for all $l$ large enough and $x\in\mB(\delta)$.
Note that $\sup_{x\in\bbR^I}|f_{a}(e\cdot x)-(e\cdot x)^+|\leq
\epsilon$ for all $a$ large enough so that,
\begin{equation}\label{eqinterim12}
\sup_{x\in\mB}|F_{\akl}[u(x)]-F[u(x)]|\leq\epsilon
\end{equation}
for all $l$ large enough. Combining (\ref{eqinterim11}) and (\ref
{eqinterim12}), we then have
\[
\sup_{x\in\mB}|F_{\akl}[\phi_{\akl}(x)]-F[u(x)]|\leq2\epsilon
\]
for all $l$ large enough and $x\in\mB(\delta)$. By definition
$F^{a^k}[\phi_{\akl}(x)]=0$ for all $x\in\mB$ and since $\epsilon$
was arbitrary we have that $F[u(x)]=0$ for all $x\in\mB(\delta)$.
Finally, since $\delta$ was arbitrary we have that $F[u(x)]=0$ for all
$x\in\mB$. We already argued that $u=0$ on $\partial\mB$, so that
$u$ solves (\ref{eqHJB1simp}) on~$\mB$ with $u=0$ on~$\partial\mB
$. This concludes the proof of existence of a solution to (\ref
{eqHJB1simp}) that satisfies the gradient estimates (\ref{eqgradients0}).
Finally, the uniqueness of the solution to (\ref{eqHJB1simp}) follows
from Corollary 17.2 in \cite{TandG} noting that the function
$F[x,z,p,r]$ is indeed continuously differentiable in the $(z,p,r)$
arguments and it is decreasing in $z$ for all $(x,p,r)$.
Using Theorem \ref{thmBrownianverification} [which only uses the
existence and uniqueness of the solution $\phi_{\kappa}^n(x)$ that we
already established] together with (\ref{eqvaluebound}) we have that
\[
|\phi_{\kappa}^n|_{0,\mB_{\kappa}^n}=\sup_{x\in\mB_{\kappa
}^n}\hat{V}^n(x,\kappa)\leq\frac{1}{\gamma}\kappa\sqrt{n}\log^m n.
\]
The bounds (\ref{eqgradients0}) and (\ref{eqgradients1}) now follow
from (\ref{equ-bound}) and we turn to prove (\ref{eqgenbound}).
To that end, since $\phi_{\kappa}^n$ solves
(\ref{eqHJB1simp}), fixing $x,y\in\mB_{\kappa}^n$ we have
\begin{eqnarray}\label{eqinternational}\quad
&&\biggl|\frac{1}{2}\sum_{i\in\I}
\bigl(\lambda_i^n+\mu_i(\nu_in+x_i)\bigr)(\phi_{\kappa}^n)_{ii}(x)-
\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+y_i)\bigr)(\phi_{\kappa}^n)_{ii}(y)\biggr|\nonumber\\
&&\qquad \leq\gamma|\phi_{\kappa}^n(x)-\phi_{\kappa
}^n(y)|\nonumber\\[-8pt]\\[-8pt]
&&\qquad\quad{} +\biggl|(e\cdot x)^+\cdot\min
_{i\in\I}\biggl\{ c_i+\mu_i(\phi_{\kappa}^n)_i(x)-\frac{1}{2}\mu
_i(\phi_{\kappa}^n)_{ii}(x)\biggr\}\nonumber\\
&&\qquad\quad\hphantom{{}+\biggl|}{} -(e\cdot y)^+\cdot\min_{i\in\I}\biggl\{
c_i+\mu_i(\phi_{\kappa}^n)_i(y)-\frac{1}{2}\mu_i(\phi_{\kappa
}^n)_{ii}(y)\biggr\}\biggr|.\nonumber
\end{eqnarray}
We will now bound each of the elements on the right-hand side. To that
end, let $i(x)$ be as defined in (\ref{eqixdefin}) and for each
$x,z\in\mB_{\vartheta\kappa}^n$ define
\[
M_{i(x)}^n(z)=c_{i(x)}+\mu_{i(x)}(\phi_{\kappa}^n)_{i(x)}(z)-\tfrac
{1}{2}\mu_{i(x)}(\phi_{\kappa}^n)_{i(x)i(x)}(z).
\]
Using (\ref{eqgradients1}), we have by the mean value theorem that
\begin{equation}\label{eqinterinter}
|\phi_{\kappa}^n(x)-\phi_{\kappa}^n(y)|\leq
{|x-y|\max_{i\in\I
}\sup_{z\in\mB_{\vartheta\kappa}^n}}|(\phi_{\kappa}^n)_i(z)|\leq
C\log^{k_1} n\vadjust{\goodbreak}
\end{equation}
for all $x,y\in\mB
_{\vartheta\kappa}^n$ with $|x-y|\leq1$, and we turn to bound the
second element on the right-hand side of (\ref{eqinternational}).
Here, there are two cases to consider. Suppose first that
$i(x)=i(y)=i$. Then, using (\ref{eqgradients1}) and the mean value
theorem we have
\[
|(\phi_{\kappa}^n)_{i}(x)-(\phi_{\kappa}^n)_{i}(y)|\leq{|x-y|\max
_{i\in\I}\sup_{z\in\mB_{\vartheta\kappa}^n}}|(\phi_{\kappa
}^n)_{ii}(z)|\leq C\frac{
\log^{k_2} n}{\sqrt{n}}
\]
and, in turn, that
\begin{equation} \label{eqMbound}
|M_{i}^n(x)-M_{i}^n(y)|\leq C\frac{\log^{k_2}
n}{\sqrt{n}}
\end{equation}
for all $x,y\in\mB_{\vartheta\kappa}^n$ with $|x-y|\leq1$. Now,
$|x|\vee|y|\leq\kappa\sqrt{n}\log^m n$ for all $x,y\in\mB
_{\vartheta\kappa}^n$ and, by (\ref{eqgradients1}), $\sup_{z\in
\mB_{\vartheta\kappa}^n}|(\phi_{\kappa}^n)_{ii}(z)|\vee|(\phi
_{\kappa}^n)_{i}(z)|\leq C\log^{k_1}n$ so that
\begin{eqnarray}\label{eqinterim222222}
&& |(e\cdot x)^+ M_i^n(x)-(e\cdot y)^+
M_i^n(y)|\nonumber\\
&&\qquad
\leq
\kappa\sqrt{n}\log^m n | M_i^n(x)-M_i^n(y)|+\sup_{z\in
\mB_{\vartheta\kappa}^n}
|M_i^n(z)|\\
&&\qquad\leq C\log^{k_1}n.\nonumber
\end{eqnarray}
If, on the other hand, $i(x)\neq i(y)$ then by the definition
of $i(\cdot)$,
\begin{eqnarray*}
&&c_{i(x)}+\mu_{i(x)}(\phi_{\kappa
}^n)_{i(x)}(x)-\tfrac{1}{2}\mu_{i(x)}(\phi_{\kappa
}^n)_{i(x)i(x)}(x)\\
&&\qquad \leq
c_{i(y)}+\mu_{i(y)}(\phi_{\kappa}^n)_{i(y)}(x)-\tfrac{1}{2}\mu
_{i(y)}(\phi_{\kappa}^n)_{i(y)i(y)}(x)
\end{eqnarray*}
and
\begin{eqnarray*}
&& c_{i(y)}+\mu_{i(y)}(\phi_{\kappa
}^n)_{i(y)}(y)-\tfrac{1}{2}\mu_{i(y)}(\phi_{\kappa
}^n)_{i(y)i(y)}(y)\\
&&\qquad \leq
c_{i(x)}+\mu_{i(x)}(\phi_{\kappa}^n)_{i(x)}(y)-\tfrac{1}{2}\mu
_{i(x)}(\phi_{\kappa}^n)_{i(x)i(x)}(y).
\end{eqnarray*}
That is,
\begin{equation} \label{eqyetonemore1}
M_{i(x)}^n(x)\leq M_{i(y)}^n(x) \quad\mbox{and}\quad
M_{i(y)}^n(y)\leq M_{i(x)}^n(y).
\end{equation}
Using
(\ref{eqgradients1}) as before we have for $x,y\in\mB_{\vartheta
\kappa}^n$ with $|x-y|\leq1$ and $i(x)\neq i(y)$ that
\[
\bigl| M_{i(x)}^n(x)-M_{i(x)}^n(y)\bigr|+\bigl|
M_{i(y)}^n(x)-M_{i(y)}^n(y)\bigr| \leq C\frac{\log^{k_2} n}{\sqrt{n}}.
\]
By (\ref{eqyetonemore1}) we then have that
\begin{eqnarray*}
\bigl| M_{i(x)}^n(x)-M_{i(y)}^n(y)\bigr|&\leq&\bigl|
M_{i(x)}^n(x)-M_{i(x)}^n(y)\bigr|\\
&&{}+ \bigl|
M_{i(y)}^n(x)-M_{i(y)}^n(y)\bigr| \\
&\leq& C\frac{\log^{k_2}n}{\sqrt{n}}
\end{eqnarray*}
for all such $x$ and $y$. In turn, since $|x|\vee|y|\leq\kappa\sqrt
{n}\log^m n$,
\begin{equation}\label{eqinterim222224}
\bigl| (e\cdot x)^+M_{i(x)}^n(x)- (e\cdot
y)^+M_{i(y)}^n(y)\bigr|\leq C\log^{k_1} n
\end{equation}
for $x,y\in\mB_{\vartheta\kappa}^n$ with $|x-y|\leq1$ and
$i(x)\neq i(y)$. Plugging (\ref{eqinterinter}), (\ref{eqinterim222222})
and (\ref{eqinterim222224}) into the right-hand
side of (\ref{eqinternational}) we get
\begin{eqnarray}\label{eqtheinternational}\qquad
&&\biggl|\frac{1}{2}\sum_{i\in\I} \bigl(\lambda
_i^n+\mu_i(\nu_in+x_i)\bigr)(\phi_{\kappa}^n)_{ii}(x)-
\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+y_i)\bigr)(\phi_{\kappa}^n)_{ii}(y)\biggr|\nonumber\\[-8pt]\\[-8pt]
&&\qquad
\leq C \log^{k_1} n\nonumber
\end{eqnarray}
for all $x,y\in\mB_{\vartheta\kappa}^n$ with $|x-y|\leq1$.
Finally, recall that
\[
\sigma_i\lam(x,u)=\sqrt{\lambda_i^n+\mu_i \nu_in +\mu
_i\bigl(x_i-u_i(e\cdot x)^+\bigr)}
\]
so that for all $u\in\mathcal{U}$,
\begin{eqnarray*}
\hspace*{-4pt}&&\biggl|\sum_{i\in\I}\bigl((\phi_{\kappa
}^n)_{ii}(y)-(\phi_{\kappa}^n)_{ii}(x)\bigr)(\sigma_i^n(x,u))^2\biggr|
\\
\hspace*{-4pt}&&\qquad =\biggl|\sum_{i\in\I}(\phi_{\kappa}^n)_{ii}(y)\bigl(\lambda
_i^n+\mu_i \nu_in +\mu_i\bigl(x_i-u_i(e\cdot x)^+\bigr)\bigr)\\
\hspace*{-4pt}&&\qquad\quad\hspace*{3.5pt}{}
-(\phi_{\kappa}^n)_{ii}(x)\bigl(\lambda_i^n+\mu_i \nu_in +\mu
_i\bigl(x_i-u_i(e\cdot x)^+\bigr)\bigr)\biggr|
\\
\hspace*{-4pt}&&\qquad\leq\biggl|\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu
_i(\nu_in+x_i)\bigr)(\phi_{\kappa}^n)_{ii}(x)\\
\hspace*{-4pt}&&\qquad\quad\hspace*{2pt}{}-
\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+y_i)\bigr)(\phi_{\kappa}^n)_{ii}(y)\biggr|\\
\hspace*{-4pt}&&\qquad\quad{} +\biggl|\frac{1}{2}\sum_{i\in\I} (\phi_{\kappa
}^n)_{ii}(x)\mu_iu_i(e\cdot x)^+-(\phi_{\kappa}^n)_{ii}(y)\mu
_iu_i(e\cdot y)^+\biggr|\\
\hspace*{-4pt}&&\qquad\quad{} +\biggl|\frac{1}{2}\sum_{i\in
\I} (\phi_{\kappa}^n)_{ii}(y)\mu_i(x_i-y_i)\biggr|.
\end{eqnarray*}
The last two terms above are bounded by $C\log^{k_1}n$ by (\ref
{eqgradients1}) and using $|x|\vee|y|\leq\kappa\sqrt{n}\log^mn$.
Together with (\ref{eqtheinternational}) this establishes (\ref
{eqgenbound}) and concludes the proof of the theorem.
\subsection*{\texorpdfstring{Proof of Theorem \protect\ref{thmBrownianverification}}{Proof of Theorem 4.2}}
Fix an initial condition $x\in\mB_{\kappa}^n$ and an admissible
$(\kappa,n)$-system $\theta=(\Omega,\mathcal{F},(\mathcal
{F}_t),\mathbb{P},\hat{U},B)$ and let $\hat{X}^n$ be the associated
controlled process.\vadjust{\goodbreak} Using It\^{o}'s lemma for the function $\varphi
(t,x)=e^{-\gamma t} \phi_{\kappa}\lam(x)$ in conjunction with the inequality
\[
L(x,u)+A_{u}\phi_{\kappa}\lam(x)-\gamma\phi_{\kappa}\lam(x)\geq
0 \qquad\mbox{for all } x\in\mB_{\kappa}^n, u\in\mathcal{U}
\]
[recall that $\phi_{\kappa}^n$ solves (\ref{eqHJB1simp})] we have that
\begin{eqnarray}\label{eqinterim3}\qquad
\phi_{\kappa}\lam(x)&\leq& \Ex_x^{\theta}\int_0^{t\wedge\hat
{\tau}_{\kappa}^n}e^{-\gamma s} L(\hat{X}^n(s),\hat{U}(s))\,ds+\Ex
_{x}^{\theta}e^{-\gamma(t\wedge\hat{\tau}_{\kappa}^n)}\phi
_{\kappa}\lam\bigl(\hat{X}^n(t\wedge\hat{\tau}_{\kappa}^n)\bigr)
\nonumber\\[-8pt]\\[-8pt]
&&{}-\Ex_{x}^{\theta}\sum_{i\in\I}\int_0^{t\wedge\hat{\tau
}_{\kappa}^n}
e^{-\gamma s} (\phi_{\kappa}\lam)_i(\hat{X}^n(s))\sigma_i^n(\hat
{X}^n(s),\hat{U}(s)) \,dB(s).\nonumber
\end{eqnarray}
Here, $\hat{\tau}_{\kappa}^n$ is as defined in Definition \ref
{definadmissiblesystemBrownian} and it is a stopping time with respect
to $(\mathcal{F}_t)$ because of the continuity of $\hat{X}^n$. We now
claim that
\[
\Ex_{x}^{\theta}\bigl[e^{-\gamma t\wedge\hat{\tau}_{\kappa
}^n}\phi_{\kappa}\lam\bigl(\hat{X}^n(t\wedge\hat{\tau}_{\kappa
}^n)\bigr)\bigr]\rightarrow0 \qquad\mbox{as } t\tinf.
\]
Indeed, as $\phi_{\kappa} \lam$ is bounded on $\mB_{\kappa}^n$, on
the event $\{\hat{\tau}_{\kappa}^n=\infty\}$ we have that
\[
e^{-\gamma(t\wedge\hat{\tau}_{\kappa}^n)}\phi_{\kappa}\lam
\bigl(X(t\wedge\hat{\tau}_{\kappa}^n)\bigr)\rightarrow0 \qquad\mbox{as }t\tinf.
\]
On the event $\{\hat{\tau}_{\kappa}^n<\infty\}$ we have $\hat
{X}^n(\hat{\tau}_{\kappa}^n)\in\partial\mB$ and, by the
definition of $\hat{\tau}_{\kappa}^n$, that $\phi_{\kappa}\lam
(\hX^n(\hat{\tau}_{\kappa}^n))=0$. The convergence in expectation
then follows from the bounded convergence theorem (using again the
boundedness of $\phi_{\kappa}\lam$ on $\mB_{\kappa}^n$). The last
term in (\ref{eqinterim3}) equals zero by the optional stopping
theorem.\vadjust{\goodbreak}
Letting $t\tinf$ in (\ref{eqinterim3}) and applying the monotone
convergence theorem, we then have
\[
\phi_{\kappa}\lam(x)\leq\Ex_x^{\theta} \biggl[\int_0^{\hat{\tau
}_{\kappa}^n}e^{-\gamma s} L(\hat{X}^n(s),\hat{U}(s))\,ds\biggr].
\]
Since the admissible system $\theta$ was arbitrary, we have that $\phi
_{\kappa}\lam(x)\leq\hat{V}\lam(x,\kappa)$. To show that this
inequality is actually an equality, let
\begin{equation}\label{eqratiofunchouce}
h_{\kappa
}^n(x)=e_{i^n(x)},
\end{equation}
where $e_{i^n(x)}$ is
as defined in the statement of the theorem.
The continuity of $\phi_{\kappa}^n$ guarantees that the function
$i^n(x)$ is Lebesgue measurable, and so is, in turn, $h_{\kappa
}^n(\cdot)$. Consider now the autonomous SDE:
\begin{equation}\label{eqautonomous}
\hX^n(t)=x+\int
_0^t \hat{b}^n(\hX^n(s))\,ds+\int_0^t \hat{\sigma}^n(\hX
^n(s))\,dB(s),
\end{equation}
where $\hat
{b}^n(y)=b^n(y,h_{\kappa}^n(y))$ and $\hat{\sigma}^n(y)=\sigma
^n(y,h_{\kappa}^n(y))$ on $\mB_{\kappa}^n$. Then, $\hat{b}^n$ and
$\hat{\sigma}^n$ are bounded and measurable on the bounded domain
$\mB_{\kappa}^n$. Also, as the matrix $\hat{\sigma}^n$ is diagonal
and the elements on the diagonal are strictly positive on $\mB_{\kappa
}^n$, it is positive definite there. Hence, a weak solution exists for
the autonomous SDE (see, e.g.,\vspace*{2pt} Theorem 6.1 of \cite
{krylov2008controlled}). In particular, there exists a probability\vadjust{\goodbreak}
space $(\tilde{\Omega},\mathcal{G},\tilde{\Pd})$, a filtration
$(\mathcal{G}_t)$ that satisfies the usual conditions, a Brownian
motion $B(t)$ and a continuous process $\hX^n$---both adapted to
$(\mathcal{G}_t)$, so that $\hX^n$ satisfies the autonomous SDE (\ref
{eqautonomous}). Finally, since~$\hX^n$ has continuous sample paths
and it is adapted, it is also progressively measurable (see,
e.g.,\vspace*{1pt}
Proposition 1.13 in \cite{KaS91}) and, by measurability of $h_{\kappa
}^n(\cdot)$, so is the process $\hat{U}(t)=h_{\kappa}^n(\hX^n(t))$.
Consequently, $\theta=(\tilde{\Omega},\mathcal{G},\mathcal
{G}_t,\tilde{\Pd},\hat{U},B)$ is an admissible system in the sense
of Definition \ref{definadmissiblesystemBrownian} and $\hX^n$ is the
corresponding controlled process.
To see that $\theta$ is optimal for the ADCP on $\mB_{\kappa}^n$,
note that for $s<\hat{\tau}_{\kappa}^n$, we have by the HJB equation
(\ref{eqHJB0}) that
\[
L(\hX^n(s),\hat{U}(s))+A_{\hat{U}(s)}\phi_{\kappa}\lam(\hX
^n(s))-\gamma\phi_{\kappa}\lam(\hX^n(s))=0.
\]
Applying It\^{o}'s rule as before, together with the bounded and
dominated convergence theorems, we then have that
\[
\phi_{\kappa}\lam(x)=\Ex_x^{\theta}\biggl[\int_0^{\hat{\tau
}_{\kappa}^n}e^{-\gamma s} L(\hX^n(s),\hat{U}(s))\,ds\biggr]
\]
and the proof is complete.
\section{The performance analysis of tracking policies}\label{sectracking}
This section shows that given an optimal Markov control policy for the
ADCP together with its associated tracking function $h_{\kappa
}^{*,n}$, the nonpreemptive tracking policy imitates, in a particular
sense, the performance of the Brownian system.
\begin{theorem} \label{thmSSC}
Fix $\kappa$ and $\kappa'<\kappa$ as well as a
sequence $\{(x^n,q^n),n\in\bbZ_+\}$ such that $(x^n,q^n)\in\mathcal
{X}^n$, and $|x^n-\nu n| \leq M \sqrt{n}$ for all $n$ and some
$M>0$. Let $\phi_{\kappa}^n$ and $h_{\kappa}^{*,n}$ be as in Theorem
\ref{thmBrownianverification} and define
\[
\psi^n(x,u)=L(x,u)+A^n_u \phi_{\kappa}^n(x)-\gamma\phi_{\kappa
}^n(x) \qquad\mbox{for }x\in\mB_{\kappa}^n,u\in\mathcal{U}.
\]
Let $U_h^n$ be the ratio control associated with the $h_{\kappa
}^{*,n}$-tracking policy and let $\mathbb{X}^n=(X^n,Q^n,Z^n,\check
{X}^n)$ be the associated queueing process with the initial conditions
$Q^n(0)=q^n$ and $\check{X}^n(0)=x^n-\nu n$ and define
\[
\tau_{\kappa',T}^n=\inf\{t\geq0\dvtx\check{X}^n(t)\notin\mB_{\kappa
'}^n\}\wedge T\log n.
\]
Then,
\[
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} |\psi
^n(\check{X}^n(s),U_h^n(s))-\psi^n(\check{X}^n(s),h_{\kappa
}^{*,n}(\check{X}^n(s)))|\,ds\biggr]\leq C\log^{k_0+3}n
\]
for a constant $C$ that does not depend on $n$.
\end{theorem}
Theorem \ref{thmSSC} is proved in the \hyperref[app]{Appendix}. The proof builds on the
gradient estimates in Theorem \ref{thmHJB1sol} and on a state-space
collapse-type result for certain sub-intervals of $[0,\tau_{\kappa',T}^n]$.
\begin{rem} \label{remSSC} Typically one establishes a stronger
state-space collapse result showing that the actual queue and the
desired queue values are close in supremum norm. The difficulty with
the former approach is that the tracking functions here are nonsmooth.
While it is plausible that one can smooth these functions appropriately
(as is done, e.g., in \cite{AMR02}), such smoothing might
compromise the optimality gap. Fortunately, the weaker integral
criterion implied by Theorem
\ref{thmSSC} suffices for our purposes.
\end{rem}
\section{Proof of the main result}\label{seccombining}
Fix $\kappa>0$ and let $\phi_{\kappa}^n$ be the solution to (\ref
{eqHJB1simp}) on $\mB_{\kappa}^n$ (see Theorem \ref{thmHJB1sol}).
We start with the following lemma where $b_i^n(\cdot,\cdot)$ and
$\sigma_i^n(\cdot,\cdot)$ are as in (\ref{eqnbmdefn30}) and
(\ref{eqnbmdefn3}), respectively.
\begin{lem} \label{lemito} Let $U^n$ be an admissible ratio control
and let $\mathbb{X}^n=(X^n,Q^n$, $Z^n,\check{X}^n)$ be the queueing
process associated with $U^n$. Fix $\kappa'<\kappa$ and $T>0$ and let
\[
\tau_{\kappa',T}^n=\inf\{t\geq0\dvtx\check{X}^n(t)\notin\mB_{\kappa
'}^n\}\wedge T\log n.
\]
Then, there exists a constant $C$ that does not depend on $n$ (but may
depend on $T$, $\kappa$ and $\kappa'$) such that
\begin{eqnarray*}
\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(\tau_{\kappa',T}^n))]&\leq&\phi_{\kappa}^n(\check{X}^n(0))+
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
A_{U^n(s)}^n\phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\\
&&{}-\gamma\Ex
\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} \phi_{\kappa
}^n(\check{X}^n(s))\,ds\biggr] +C\log^{k_1+1} n\\
& \leq&
\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(\tau_{\kappa',T}^n))]+2C\log^{k_1+1}n.
\end{eqnarray*}
\end{lem}
We will also use the following lemma where $c=(c_1,\ldots, c_I)$ are
the cost coefficients (see Section \ref{secmodel}).
\begin{lem}\label{lemafterstop}
Let $(x^n,q^n)$ be as in the conditions of Theorem \ref
{thmmain}. Then, there exists a constant $C$ that does not depend on
$n$ such that
\begin{equation}\label{eqafterstop2}
\Ex_{x^n,q^n}^{U}\biggl[\int_{\tau_{\kappa
',T}^n}^{\infty} e^{-\gamma s} (e\cdot c)\bigl(e\cdot\check
{X}^n(s)\bigr)^+\,ds\biggr]\leq C\log^2n
\end{equation}
and
\begin{equation}\label{eqafterstop1}
\Ex_{x^n,q^n}^{U}[e^{-\gamma\tau_{\kappa
',T}^n}\phi
_{\kappa}^n(\check{X}^n(\tau_{\kappa',T}^n))]\leq C\log^2
n
\end{equation}
for all $n$ and any admissible ratio control
$U$.
\end{lem}
We postpone the proof of Lemma \ref{lemito} to the end of the section
and that of Lemma \ref{lemafterstop} to the \hyperref[app]{Appendix} and proceed now to
prove the main result of the paper.
\subsection*{\texorpdfstring{Proof of Theorem \protect\ref{thmmain}}{Proof of Theorem 2.1}} Let $h_{\kappa
}^{*,n}$ be the ratio function associated with the optimal Markov
control for the ADCP (as in Theorem \ref{thmHJB1sol}). Since $\kappa
$ is fixed we omit the subscript $\kappa$ and use $h^n=h_{\kappa
}^{*,n}$. Let $U_h^n$ be the ratio associated with the $h^n$-tracking policy.
The proof will proceed in three main steps. First, building on Theorem~\ref{thmSSC} we will show that
\begin{equation} \label{eqinterim2}
\Ex\biggl[\int_0^{\tau_{\kappa
',T}^n} e^{-\gamma s} L(\check{X}^n(s),U_h^n(s))\,ds\biggr] \leq\phi
_{\kappa}^n(\check{X}^n(0))+C\log^{k_0+3} n.
\end{equation}
Using Lemma \ref{lemafterstop}, this implies
\begin{eqnarray}\label{eqinterim13}
C\lam(x^n,q^n,U_h^n)&=&\Ex\biggl[\int_0^{\infty} e^{-\gamma s} L(\check
{X}^n(s),U_h^n(s))\,ds\biggr]\nonumber\\[-8pt]\\[-8pt]
&\leq&\phi_{\kappa}^n(\check
{X}^n(0))+C\log^{k_0+3} n.\nonumber
\end{eqnarray}
Finally, we will
show that for any ratio control $U^n$,
\begin{equation}\label{eqinterim303}
\phi_{\kappa}^n(\check
{X}^n(0))\leq\Ex\biggl[\int_0^{\infty} e^{-\gamma s} L(\check
{X}^n(s),U^n(s))\,ds\biggr]+C\log^{k_1+1}n,
\end{equation}
where we recall that $k_1=k_0-m$. In turn,
\[
V^n(x^n,q^n)\geq\phi_{\kappa}^n(x^n-\nu n)-C\log^{k_1+1} n \geq
C\lam(x^n,q^n,U_h^n)-2C\log^{k_1+1}n,
\]
which establishes the statement of the theorem.\vadjust{\goodbreak}
We now turn to prove each of (\ref{eqinterim2}) and (\ref{eqinterim303}).
\subsection*{\texorpdfstring{Proof of (\protect\ref{eqinterim2})}{Proof of (61)}} To simplify
notation we fix $\kappa>0$ throughout and let $h^n(\cdot
)=h_{\kappa}^{*,n}$. Using Lemma \ref{lemito} we have
\begin{eqnarray}\label{eqinterim1}
&&\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(\tau_{\kappa',T}^n))]\nonumber
\\
&&\qquad\leq \phi_{\kappa}^n(\check{X}^n(0))+
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
A_{U_h^n(s)}^n\phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\\
&&\qquad\quad{}-\gamma\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} \phi
_{\kappa}^n(\check{X}^n(s))\,ds\biggr]+ C\log^{k_1+1} n .\nonumber
\end{eqnarray}
From the definition of $h^n$ as a minimizer in the HJB equation we have that
\begin{eqnarray*}
0&=& \Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
A_{h^n(\check{X}^n(s))}^n\phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\\
&&{}-\gamma\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} \phi
_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\\
&&{}+\Ex\biggl[\int
_0^{\tau_{\kappa',T}^n} e^{-\gamma s} L(\check{X}^n(s),h^n(\check
{X}^n(s)))\,ds\biggr] .
\end{eqnarray*}
By Theorem \ref{thmSSC} we then have that
\begin{eqnarray}\label{eqinterim101}
C\log^{k_0+3}n&\geq& \Ex\biggl[\int_0^{\tau_{\kappa',T}^n}
e^{-\gamma s} A_{U_h^n(s)}^n\phi_{\kappa}^n(\check{X}^n(s))\,ds
\biggr]\nonumber\\[-2pt]
&&{} -\gamma\Ex\biggl[\int_0^{\tau_{\kappa',T}^n}
e^{-\gamma s}
\phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\nonumber\\[-9pt]\\[-9pt]
&&{} +\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} L(\check
{X}^n(s),U_h^n(s))\,ds\biggr]\nonumber\\[-2pt]
&\geq&0.\nonumber
\end{eqnarray}
Since $\phi_{\kappa}^n$ is nonnegative, combining (\ref{eqinterim1})
and (\ref{eqinterim101}) we have that
\[
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} L(\check
{X}^n(s),U_h^n(s))\,ds\biggr]\leq\phi_{\kappa}^n(\check
{X}^n(0))+C\log^{k_0+3} n,
\]
which concludes the proof of (\ref{eqinterim2}).
\subsection*{\texorpdfstring{Proof of (\protect\ref{eqinterim303})}{Proof of (63)}} We now show that
$V^n(x,q)\geq\phi_{\kappa}^n(\check{X}^n(0))-C\log^{k_1+1} n$. To
that end, fix an arbitrary ratio control $U^n$ and recall that by the
HJB equation,
\[
A_u^n\phi_{\kappa}^n(x)-\gamma\phi_{\kappa
}^n(x)+L(x,u)\geq0
\]
for all $u\in\mathcal{U}$ and $x\in\mB
_{\kappa}^n$. In turn, using the second inequality in Lemma \ref
{lemito} we have that
\begin{eqnarray*}
&&\Ex[e^{-\gamma\tau_{\kappa
',T}^n}\phi_{\kappa}^n(\check{X}^n(\tau_{\kappa',T}^n))
]\\[-2pt]
&&\qquad\geq \phi_{\kappa}^n(\check{X}^n(0))
-\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} L(\check
{X}^n(s),U^n(s))\,ds\biggr]\\[-2pt]
&&\qquad\quad{} -2C\log^{k_1+1} n.
\end{eqnarray*}
Using Lemma \ref{lemafterstop}, we have, however, that
\[
\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(\tau_{\kappa',T}^n))]\leq C\log^{2} n
\]
for a redefined constant $C$ so that
\begin{eqnarray*}C\log^{2} n &\geq& \phi
_{\kappa}^n(\check{X}^n(0))-
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} L(\check
{X}^n(s),U^n(s))\,ds\biggr]\\[-2pt]
&&{}-2C\log^{k_1+1} n\\&\geq& \phi_{\kappa
}^n(\check{X}^n(0))-
\Ex\biggl[\int_0^{\infty} e^{-\gamma s} L(\check
{X}^n(s),U^n(s))\,ds\biggr]\\[-2pt]
&&{}-2C\log^{k_1+1} n
\end{eqnarray*}
and, finally,
\[
\phi_{\kappa}^n(\check{X}^n(0))\leq\Ex\biggl[\int_0^{\infty}
e^{-\gamma s}
L(\check{X}^n(s),U^n(s))\,ds\biggr]+C\log^{k_1+1}n\vadjust{\goodbreak}
\]
for a redefined constant $C>0$. This concludes the proof of (\ref
{eqinterim303}) and of the theorem.
We end this section with the proof of Lemma \ref{lemito} in which the
following auxiliary lemma will be of use.
\begin{lem}\label{lemmartingales}
Fix $\kappa>0$ and an admissible ratio control $U^n$ and
let $\mathbb{X}\lam=(X\lam,Q\lam,Z\lam,\check{X}\lam)$ be the
corresponding queueing process. Let
\[
\tau_{\kappa,T}^n=\inf\{t\geq0\dvtx\check{X}^n(t)\notin\mB_{\kappa
}^n\}\wedge T\log n,
\]
and $(\check{W}_i^n,i\in\I)$ be as defined in (\ref{eqWtildedefin}).
Then, for each $i\in\I$, the process $\check
{W}_i^n(\cdot\wedge\tau_{\kappa,n}^n)$ is a square integrable
martingale w.r.t to the filtration $(\mathcal{F}_{t\wedge\tau
_{\kappa,T}^n}^n)$ as are the processes
\[
\mathcal{M}_i^n(\cdot)=\bigl(\check{W}_i^n(\cdot\wedge\tau_{\kappa
,T}^n)\bigr)^2-\int_0^{\cdot\wedge\tau_{\kappa,T}^n} (\sigma
_i^n(\check{X}^n(s),U^n(s)))^2\,ds
\]
and
\[
\mathcal{V}_i^n(\cdot)=\bigl(\check{W}_i^n(\cdot\wedge\tau_{\kappa
,T}^n)\bigr)^2-\sum_{s\leq\cdot\wedge\tau_{\kappa,T}^n} (\Delta\check
{W}_i^n(s))^2.
\]
\end{lem}
Lemma \ref{lemmartingales} follows from basic results on martingales
associated with time-changes of Poisson processes. The detailed proof
appears in the \hyperref[app]{Appendix}.
\subsection*{\texorpdfstring{Proof of Lemma \protect\ref{lemito}}{Proof of Lemma 6.1}}
Note that, as in
(\ref{eqcheckXdynamics}), $\check{X}^n$ satisfies
\[
\check{X}_i\lam(t)=\check{X}_i\lam(0)+\int_0^t b_i^n(\check
{X}^n(s),U^n(s))\,ds+\check{W}_i\lam(t),
\]
and is a semi martingale. Applying It\^{o}'s formula for
semimartingales (see, e.g., Theorem 5.92 in \cite{vandervaart}) we
have for all $t\leq\tau_{\kappa',T}^n$, that
\begin{eqnarray*}
e^{-\gamma t}\phi_{\kappa}^n(\check{X}^n(t))&=&\phi_{\kappa
}^n(\check{X}^n(0))\\
&&{}+
\sum_{s\leq t\dvtx |\Delta\check{X}^n(s)|> 0} e^{-\gamma s}[\phi
_{\kappa}^n(\check{X}^n(s))-\phi_{\kappa}^n(\check{X}^n(s-))]\\
&&{}-
\sum_{i\in\I}\sum_{s\leq t\dvtx |\Delta\check{X}^n(s)|> 0} e^{-\gamma
s}(\phi_{\kappa})_i^n(\check{X}^n(s))\Delta\check{X}_i^n(s)\\
&&{}+
\sum_{i\in\I}\int_0^t e^{-\gamma s} (\phi_{\kappa}^n)_i(\check
{X}^n(s-))b_i^n(\check{X}^n(s),U^n(s))\,ds\\
&&{}-\gamma\int_0^t
e^{-\gamma s} \phi_{\kappa}^n(\check{X}^n(s))\,ds
\end{eqnarray*}
and, after rearranging terms, that
\begin{eqnarray*}
&&e^{-\gamma t}\phi_{\kappa}^n(\check{X}^n(t))\\
&&\qquad=
\phi_{\kappa}^n(\check{X}^n(0))
+ \frac{1}{2}\sum_{i\in\I
}\sum_{s\leq t\dvtx |\Delta\check{X}^n(s)|>0}e^{-\gamma s} (\phi
_{\kappa}^n)_{ii}(\check{X}^n(s-))(\Delta\check{X}_i^n(s))^2\\
&&\qquad\quad{}+
\sum_{i\in\I}\int_0^t e^{-\gamma s} (\phi_{\kappa}^n)_i(\check
{X}^n(s-))b_i(\check{X}^n(s),U^n(s))\,ds\\
&&\qquad\quad{}+ C^n(t)-\gamma\int_0^t
e^{-\gamma s} \phi_{\kappa}^n(\check{X}^n(s))\,ds,
\end{eqnarray*}
where
\begin{eqnarray*} C^n(t)&=& \sum_{s\leq t\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} \biggl[\phi_{\kappa}^n(\check
{X}^n(s))-\phi_{\kappa}^n(\check{X}^n(s-))\\
&&\hphantom{\sum_{s\leq t\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} \biggl[}{}-\sum_{i\in\I}(\phi
_{\kappa}^n)_i(\check{X}^n(s-)) \Delta\check{X}_i^n(s) \\
&&\hphantom{\sum_{s\leq t\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} \biggl[}{}-\frac
{1}{2}\sum_{i\in\I} (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\Delta\check{X}_i^n(s))^2\biggr].
\end{eqnarray*}
Setting $t=\tau_{\kappa',T}^n$ as defined in the statement of the
lemma and taking expectations on both sides we have
\begin{eqnarray}\label{eqinterim404}
&&\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(t))]\nonumber\\
&&\qquad=\phi_{\kappa}^n(\check{X}^n(0))
+
\sum_{i\in\I}\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
(\phi_{\kappa}^n)_i(\check{X}^n(s-)) b_i^n(\check
{X}^n(s),U^n(s))\,ds\biggr]\nonumber\\[-8pt]\\[-8pt]
&&\qquad\quad{}+\frac{1}{2}\sum
_{i\in\I}\Ex\biggl[\sum_{s\leq t\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\Delta\check{X}_i^n(s))^2\biggr]\nonumber\\
&&\qquad\quad{}+ \Ex
[C^n(\tau_{\kappa',T}^n)]-\gamma\Ex\biggl[\int_0^{\tau_{\kappa
',T}^n} e^{-\gamma s} \phi_{\kappa}^n(\check{X}^n(s))\,ds
\biggr].\nonumber
\end{eqnarray}
We will now examine each of the elements on the right-hand side of
(\ref{eqinterim404}). First, note that $\Delta\check
{X}_i^n(s)=\Delta\check{W}_i^n(s)$ and, in particular,
\begin{eqnarray*}
&&\Ex
\biggl[\sum_{s\leq\tau_{\kappa',T}^n\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\Delta\check{X}_i^n(s))^2\biggr]\\
&&\qquad=
\Ex
\biggl[\sum_{s\leq\tau_{\kappa',T}^n\dvtx |\Delta\check
{X}^n(s)|>0}e^{-\gamma s} (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\Delta\check{W}_i^n(s))^2\biggr].
\end{eqnarray*}
Using the fact that $\mathcal{V}_i^n$, as defined in Lemma \ref
{lemmartingales}, is a martingale as well as the fact that $\phi
_{\kappa}^n(\check{X}^n(s))$ and its derivative processes are bounded
up to~$\tau_{\kappa'}^n$, we have that the processes
\begin{equation}\label{eqbarV}
\bar{\mathcal{V}}_i^n(\cdot):=\int_0^{\cdot\wedge
\tau
_{\kappa',T}^n} e^{-\gamma s}(\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))\,d\mathcal{V}_i^n(s)
\end{equation}
and
\begin{equation}\label{eqbarM}
\bar{\mathcal{M}}_i^n(\cdot):=\int_0^{\cdot\wedge
\tau
_{\kappa',T}^n} e^{-\gamma s}(\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))\,d\mathcal{M}_i^n(s)
\end{equation}
are themselves
martingales with $\bar{\mathcal{V}}_i^n(0)=\bar{\mathcal
{M}}_i^n(0)=0$ and in turn, by optional stopping, that
$\Ex[\bar{\mathcal{V}}_i^n(\tau_{\kappa',T}^n)]=\Ex[\bar
{\mathcal{M}}_i^n(\tau_{\kappa',T}^n)]$ (see,\vspace*{1pt} e.g., Lemma 5.45 in
\cite{vandervaart}). In turn, by the definition of $\mathcal
{M}_i^n(\cdot)$ and $\mathcal{V}_i^n(\cdot)$ we have
\begin{eqnarray*}
&&\Ex\biggl[\sum_{s\leq\tau_{\kappa',T}^n\dvtx
|\Delta\check{X}^n(s)|>0}e^{-\gamma s} (\phi_{\kappa
}^n)_{ii}(\check{X}^n(s-))(\Delta\check{W}_i^n(s))^2\biggr]\\
&&\qquad=\Ex\biggl[\int_0^t (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))\,d(\check{W}_i^n(s))^2\biggr]\\
&&\qquad=\Ex\biggl[\int
_0^{\tau_{\kappa',T}^n} e^{-\gamma s} (\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\sigma_i^n(\check{X}^n(s),U^n(s)))^2\,ds\biggr].
\end{eqnarray*}
Plugging this back into (\ref{eqinterim404}) we have that
\begin{eqnarray*}
&&\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(t))]\\
&&\qquad=\phi_{\kappa}^n(\check{X}^n(0))
+\sum_{i\in\I}\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
(\phi_{\kappa}^n)_{i}(\check{X}^n(s-)) b_i^n(\check
{X}^n(s),U^n(s))\,ds\biggr]\\
&&\qquad\quad{}+
\frac{1}{2}\sum_{i\in\I}
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} (\phi_{\kappa
}^n)_{ii}(\check{X}^n(s-))(\sigma_i^n(\check{X}^n(s),U^n(s)) )^2
\,ds\biggr]\\
&&\qquad\quad{}-\gamma\Ex\biggl[\int_0^{\tau_{\kappa',T}^n}
e^{-\gamma s} \phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]+ \Ex
[C^n(\tau_{\kappa',T}^n)],
\end{eqnarray*}
which, using the definition of $A_u^n$ in (\ref{eqgendefin}), yields
\begin{eqnarray*}
&&\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi_{\kappa}^n(\check
{X}^n(t))]\\
&&\qquad=\phi_{\kappa}^n(\check{X}^n(0))
+\Ex\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s}
A_{U^n(s)}^n\phi_{\kappa}^n(\check{X}^n(s))\,ds\biggr]\\
&&\qquad\quad{}-\gamma\Ex
\biggl[\int_0^{\tau_{\kappa',T}^n} e^{-\gamma s} \phi_{\kappa
}^n(\check{X}^n(s))\,ds\biggr]\\
&&\qquad\quad{}+ \Ex[C^n(\tau_{\kappa',T}^n)].
\end{eqnarray*}
To complete the proof it then remains only to show that there
exists a~constant $C$ such that
\[
|\Ex[C^n(\tau_{\kappa',T}^{n})]|\leq C\log
^{k_1+1} n.
\]
To that end, note that by Taylor's
expansion,
\begin{eqnarray*}\phi_{\kappa}^n(\check{X}^n(s))&=&\phi_{\kappa
}^n(\check{X}^n(s-))
+\sum_{i\in\I}(\phi_{\kappa}^n)_i(\check{X}^n(s-))\Delta
\check{X}_i^n(s)\\
&&{}
+\frac{1}{2}\sum_{i\in\I}(\phi_{\kappa}^n)_{ii}\bigl(\check
{X}^n(s-)+\eta_{\check{X}^n(s-)}\bigr)\Delta\check{X}_i^n(s),
\end{eqnarray*}
where
$\eta_{\check{X}^n(s-)}$ is such that $\check{X}^n(s-)+\eta_{\check
{X}^n(s-)}$ is between $\check{X}^n(s-)$ and $\check{X}^n(s-)+\Delta
\check{X}^n(s)$. In turn, adding and subtracting a term, we have that
\begin{eqnarray}\label{eqCninterim}
&&\phi_{\kappa}^n(\check{X}^n(s))-\phi_{\kappa
}^n(\check{X}^n(s-))-\sum_{i\in\I}(\phi_{\kappa}^n)_i(\check
{X}^n(s-)) \Delta\check{X}_i^n(s) \nonumber\\
&&\quad{}-\frac{1}{2}\sum_{i\in\I}(\phi_{\kappa}^n)_{ii}(\check
{X}^n(s-))(\Delta\check{X}_i^n(s))^2\\
&&\qquad = \sum_{i\in\I}\frac
{1}{2}\bigl((\phi_{\kappa}^n)_{ii}\bigl(\check{X}^n(s-)+\eta_{\check
{X}^n(s-)}\bigr) -(\phi_{\kappa}^n)_{ii}(\check{X}^n(s-))\bigr)(\Delta
\check{X}_i^n(s))^2.\nonumber
\end{eqnarray}
Since the jumps are of size $1$ and, with probability 1, there are no
simultaneous jumps, we have that $|\eta_{\check{X}^n(s-)}|\leq1$.
Adding the discounting, summing and taking expectations we have
\begin{eqnarray}\label{eqinterim2222}\quad
&&\Ex[C^n(t)]\nonumber\\
&&\qquad\leq\Ex\biggl[\sum_{s\leq t\dvtx
|\Delta\check{X}^n(s)|>0} e^{-\gamma s} \sum_{i\in\I}\frac
{1}{2}\max_{y\dvtx|y|\leq1}\bigl((\phi_{\kappa}^n)_{ii}\bigl(\check
{X}^n(s-)+y\bigr)\\
&&\qquad\quad\hspace*{163.6pt}{} -(\phi_{\kappa
}^n)_{ii}(\check{X}^n(s-))\bigr)(\Delta\check{X}_i^n(s))^2
\biggr],\nonumber
\end{eqnarray}
and a lower bound can be created by minimizing over $y$ instead of
maximizing. Using again the fact that $\Delta\check{X}_i^n(t)=\Delta
\check{W}_i^n(t)$ and that $\bar{\mathcal{M}}_i^n$ and $\bar
{\mathcal{V}}_i^n$ as defined in (\ref{eqbarM}) and (\ref{eqbarV})
are martingales, we have that
\begin{eqnarray}\label{eqinterim505}
\Ex[C^n(t)]&\leq&\Ex\biggl[\int_0^t \sum_{i\in\I
}\frac{1}{2}\max_{y\dvtx|y|\leq1}\bigl((\phi_{\kappa}^n)_{ii}\bigl(\check
{X}^n(s-)+y\bigr) \nonumber\\
&&\hspace*{92.3pt}{} -(\phi_{\kappa}^n)_{ii}(\check{X}^n(s-))\bigr)\\
&&\hspace*{60.6pt}{}\times(\sigma
_i^n(\check{X}^n(s),U^n(s)))^2\,ds\biggr].\nonumber
\end{eqnarray}
From (\ref{eqgenbound}) we have that
\begin{equation}\label{eqinterim1111}
\frac{1}{2}\biggl|\sum_{i\in\I}\bigl((\phi_{\kappa
}^n)_{ii}(y)-(\phi_{\kappa}^n)_{ii}(x)\bigr)(\sigma_i^n(x,u))^2
\biggr|\leq C\log^{k_1} n
\end{equation}
for all $u\in\mathcal{U}$ and $x,y\in\mB_{\kappa'}^n$ with
$|x-y|\leq1$. The proof is then concluded by plugging (\ref
{eqinterim1111}) into (\ref{eqinterim505}), setting $t=\tau_{\kappa
',T}^n$ and recalling that we can repeat all the above steps to obtain
a lower bound in (\ref{eqinterim505}) by replacing $\max_{y\dvtx|y|\leq
1}$ with $\min_{y\dvtx|y|\leq1}$ in~(\ref{eqinterim2222}).
\section{Concluding remarks}\label{secconclusions}
This paper proposes a novel approach for solving problems of dynamic
control of queueing systems in the Halfin--Whitt many-server
heavy-traffic regime. Its main contribution is the use of Brownian
approximations to construct controls that achieve optimality gaps that
are logarithmic in the system size. This should be contrasted with the
optimality gaps of size $o(\sqrt{n})$ that are common in the
literature on asymptotic optimality. A distinguishing feature of our
approach is the use of a \textit{sequence} of Brownian control problems
rather than a single (limit) problem. Having an entire sequence of
approximating problems allows us to perform a more refined analysis,
resulting in the improved optimality gap.
In further contrast with the earlier literature, in each of these
Brownian problems the diffusion coefficient depends on both the system
state and the control. Incorporating the impact of control on diffusion
coefficients allows us to track the performance of the policy better
but, at the same time, it leads to a more complex diffusion control
problem in which the associated HJB equation is fully nonlinear and
nonsmooth. For \textit{each} Brownian problem, we show that the HJB
equation has a sufficiently smooth solution that coincides with the
value function and that admits an optimal Markov policy. Most
importantly, we derive useful gradient estimates that apply to the
whole sequence and bound the growth rate of the gradients with the
system size. These bounds are crucial for controlling the approximation
errors when analyzing the original queueing system under the proposed
tracking control.
The motivating intuition behind our approximation scheme is that the
value functions of each queueing system and its corresponding Brownian
control problem ought to be close. In particular, the optimal control
for the Brownian problem should perform well for the queueing system.
Moreover, the optimal Markov control of the Brownian problem can be
approximated by a ratio (or tracking) control for the queueing system.
While these observations are ``correct'' at a high level, they need to
be qualified further. Our analysis underscores two sources of
approximation errors that need to be addressed in order to obtain the
refined optimality gaps.
First, the value function of the Brownian control problem may be
substantially different than that of the (preemptive) optimal control
problem for the queueing system. This difference must be quantified
relative to the system size, which we do indirectly through the
gradient estimates for the value function of the Brownian control
problem; this is manifested, for example, in the proof of Lemma \ref{lemito}.
The second source of error is in trying to imitate the optimal ratio
control of the approximating Brownian control by a tracking control in
the corresponding queueing system. The error arises because we insist
on having a~nonpreemptive control for the queueing system. Whereas
under a preemptive control, one may be able to rearrange the queues
instantaneously to match the tracking function of the Brownian system,
this is not possible with nonpreemptive controls. Instead, we carefully
construct and analyze the performance of the proposed nonpreemptive
tracking policy. In doing so, we prove that the tracking control
imitates closely the Brownian system with respect to a specific
integrated functional of the queueing dynamics (see Theorem \ref
{thmSSC} and Remark \ref{remSSC}). Here too, the gradient estimates
for the value function of the Brownian system play a key role.
While the focus of this paper has been a relatively simple model
to illustrate the key ideas behind our approach and the important steps
in the analysis, we expect that similar results can be established in
the cases of impatient customers, more general cost structures as well
as more general network structures.
As suggested by the preceding analysis, the viability of these
extensions and others will depend on whether it is possible to (a)
solve the
sequence of Brownian control problems and establish the necessary
gradient estimates and (b) establish the corresponding approximation
result for the nonpreemptive tracking control.
While we expect that the results of \cite{TandG} on fully nonlinear
elliptic PDEs can be invoked for the more general settings, extending
our analysis which builds on those results may not be always straightforward.
In particular, it is not immediately obvious how to generalize the
proof of the tracking result in Theorem \ref{thmSSC} to more general settings.
Nevertheless, we can make some observations about the extensions
mentioned above:
\begin{itemize}
\item\textit{General convex costs.} As discussed in the
\hyperref[sec1]{Introduction}, the analysis of the convex holding cost case will
probably be simpler as one tends to get ``interior'' solutions in that
case as opposed to the corner solutions in the linear cost case, which
causes nonsmoothness.
We expect that the enhanced smoothness (relative to the linear holding
cost case) will simplify the analysis of the HJB equations as well as
that of the tracking performance.
\item\textit{Abandonment.} Our starting point in the analysis is that,
among preemptive policies, work conserving policies are optimal. This
is not, in general, true when customers are impatient and may abandon
while waiting (see the discussion in Section 5.1 of \cite{AMR02}). As is
the case in \cite{AMR02}, our analysis will go through also for the
case of impatient customers provided that the cost structure is such
that work conservation is optimal among preemptive policies.
\item\textit{General networks.} Inspired by the generalization of
\cite{AMR02}, by Atar \cite{atar2005scheduling}, to tree-like networks,
we expect, for example, that such a generalization is viable in our
setting as well. Indeed, we expect that the analysis of the (sequence
of) HJB equations and the sequence of ADCPs be fairly similar for the
tree-like network setting. We expect that, in that more general
setting, it would be more complicated to bound the performance of the
tracking policies as in Theorem~\ref{thmSSC}.
\end{itemize}
\begin{appendix}\label{app}
\section*{Appendix}
\subsection*{\texorpdfstring{Proof of Lemma \protect\ref{lemexistenceofcontrolled}}{Proof of Lemma 3.1}}
Up to $\tau_{\kappa}^n$, both functions $b^n(\cdot,u)$ and $\sigma
^n(\cdot,u)$ are bounded and Lipschitz continuous (uniformly in $u$).
With these conditions satisfied, strong existence and uniqueness follow
as in Appendix D of \cite{FlemingSoner}. Specifically, strong
existence follows by successive approximations as in the proof of
Theorem~2.9 of \cite{KaS91} and uniqueness follows as in Theorem 2.5
there.
\subsection*{\texorpdfstring{Proof of Proposition \protect\ref{propsolPHJB}}{Proof of Proposition 4.1}} Fix
$\kappa>0$, $n\in\bbZ_+$ and $a>0$. Recall that (\ref{eqHJB2})
corresponds to finding $\phi_{\kappa,a}^n\in\mC^2(\mB)$ such that
\begin{equation}\label{eqHJB1min}
0=F_a[x,\phi_{\kappa,a}^n(x),D\phi_{\kappa
,a}^n(x),D^2\phi
_{\kappa,a}^n(x)],\qquad x\in\mB,
\end{equation}
and so that
$\phi_{\kappa,a}^n=0$ on $\partial\mB$ where \mbox{$F_a[\cdot]$} is as
defined in (\ref{eqFForm}). Then, Proposition~\ref{propsolPHJB}
will follow from Theorem 17.18 in \cite{TandG} upon verifying certain
conditions. The gradient estimates will also follow from \cite{TandG}
by carefully tracing some constants to identify their dependence on
$\kappa,n$ and $a$.
To that end, note that the function $F_a^i(x,z,p,r)$ [as defined in
(\ref{eqFdefin})] is linear in the $(z,p,r)$ arguments for all $k\in
\I$ and $x\in\mB$. In turn, this function is concave in these
arguments. Hence, to apply Theorem 17.18 of \cite{TandG} it remains to
establish that condition (17.53) of \cite{TandG} is satisfied for each
of these functions. In the following we suppress the constant $a>0$
from the notation. It suffices to show that there exist constants
$\underbar{\Lambda}\leq\bar{\Lambda}$ and $\eta$ such that
uniformly in $k\in\I$, $y=(x,z,p,r)\in\Gamma$, and $\xi\in\mathbb{R}^I$
\begin{eqnarray}\label{eqFcond1}
&\displaystyle
0< \underbar{\Lambda}|\xi^2|\leq\sum_{i,j}F^k_{i,j}[y] \xi_i\xi
_j\leq\bar{\Lambda} |\xi|^2,&\\
\label{eqFcond2}
&\displaystyle \max\{
|F^k_p[y]|,|F^k_z[y]|,|F^k_{rx}[y]|,|F^k_{px}[y]|,|F^k_{zx}[y]|\}
\leq\eta\underbar{\Lambda},&
\\
\label{eqFcond3}
&\displaystyle
\max\{ |F^k_x[y]|,|F^k_{xx}[y]|\}\leq\eta\underbar
{\Lambda}(1+|p|+|r|),&
\end{eqnarray}
where
\[
F^k_{i,j}(x,z,p,r)=\frac{\partial}{\partial r_{ij}}F^k(x,z,p,r),\qquad
F^k_{x_l}(x,z,p,r)=\frac{\partial}{\partial x_{l}}F^k(x,z,p,r)\vadjust{\goodbreak}
\]
and
\[
(F^k_{rx}(x,z,p,r))_{ilj}=\frac{\partial^2}{\partial r_{il}\,\partial
x_{j}}F^k(x,z,p,r).\vspace*{-2pt}
\]
The other cross-derivatives are defined similarly. We will show that we
can choose $\underbar{\Lambda}=\varepsilon_0 n$, $\bar{\Lambda}=
\varepsilon_1 n$, $\eta=\varepsilon_2$ for constants $\varepsilon
_0,\varepsilon_1$ and $\varepsilon_2$ that do not depend on $n$ and
$a$---this will be important in establishing the aforementioned
gradient estimates. To establish (\ref{eqFcond1}) note that, given
$\xi\in\mathbb{R}^I$,
\begin{equation}\label{eqFijDeriv}\quad
F^{k}_{ij}\xi_i\xi_j=\cases{
\frac{1}{2}\bigl(\lambda_i^n+\mu_i(\nu_in+x_i)\bigr)\xi
_i^2, &\quad for $i=j, i\neq k$,\vspace*{1pt}\cr
\frac{1}{2}\bigl(\lambda_i^n+\mu_i(\nu_in+x_i)\bigr)\xi_i^2 -\frac
{1}{2}f(e\cdot x), &\quad for $i=j=k$,\vspace*{1pt}\cr
0, &\quad otherwise.}\vspace*{-2pt}
\end{equation}
Hence,
\[
\sum_{i,j}F_{ij}^k \xi_i\xi_j=\frac{1}{2}\sum_{i\in\I}\bigl(\lambda
_i^n+\mu_i(\nu_in+x_i)\bigr)\xi_i^2-\frac{1}{2}f(e\cdot x)\xi_k^2.\vspace*{-2pt}
\]
Consequently, for $(x,z,r,p)\in\Gamma$ we have that
\[
\sum_{i,j}F_{ij}^k \xi_i\xi_j\leq I \bigl( \lambda+\mu_{\max
}n+\mu_{\max}\kappa\sqrt{n}\log^mn\bigr)\sum_{i\in\I}\xi_i^2+
\frac{1}{2}\kappa\sqrt{n}\log^m n\xi_k^2,\vspace*{-2pt}
\]
where $\mu_{\max}=\max_{k}\mu_k$. In particular, we can choose
$\varepsilon_1>0$ so that for all $n\in\bbZ$,
\[
\sum_{i,j}F_{ij}^k \xi_i\xi_j\leq\varepsilon_1 n.\vspace*{-2pt}
\]
To obtain the lower bound note that, for $y\in\Gamma$,
\[
\sum_{i,j}F_{ij}^k \xi_i\xi_j\geq\frac{1}{2}\Bigl(\min_{i\in\I
}\lambda_i^n+\min_{i\in\I}\mu_i\kappa\sqrt{n}\log^m n
\Bigr)\sum_{i\in\I}\xi_i^2
-\frac{1}{2}\xi_k^2\kappa\sqrt{n}\log^mn.\vspace*{-2pt}
\]
Hence, we can find $\varepsilon_0>0$ such that for all n,
\[
\sum_{i,j}F_{ij}^k \xi_i\xi_j\geq\varepsilon_0 n.\vspace*{-2pt}
\]
Note that above $\varepsilon_0$ and $\varepsilon_1$ can depend on
$\kappa$ but they do not depend on $n$ and $a$. Hence, we have
established (\ref{eqFcond1}) and we turn to (\ref{eqFcond2}). To
that end, note that
\begin{eqnarray}\label{eqFp}
F_{p_k}^k(x,z,p,r)&=&f(e\cdot x)+l_k\lam-\mu
_kx_k \quad\mbox{and }\nonumber\\[-9pt]\\[-9pt]
F_{p_i}^k(x,z,p,r)&=&l_i\lam-\mu_ix_i \qquad\mbox{for }
i\neq k.\nonumber\vspace*{-2pt}
\end{eqnarray}
Therefore,
\begin{eqnarray*} |F_{p}^k|&\leq& (e\cdot x)^+ +1+\sum_{i}|l_i\lam
+\mu_i x_i|\\[-2pt]
&\leq&
I\kappa\sqrt{n}\log^mn+1+I\max_{i}\bigl(|l_i\lam|+\mu_i\kappa\sqrt
{n}\log^m n\bigr),\vspace*{-2pt}\vadjust{\goodbreak}
\end{eqnarray*}
where we used the simple observation that $f(e\cdot x)\leq(e\cdot
x)^++1$. Clearly, we can choose $\varepsilon_2$ so that $|F_p^k|\leq
\varepsilon_2\varepsilon_0\sqrt{n}\log^m n$. Also $F_z^k=-\gamma$
and \mbox{$F_{zx}=0$} so that by re-choosing $\varepsilon_2$ large enough we
have $\max\{|F^k_z[y]|,|F^k_{zx}[y]|\}\leq\varepsilon_2\varepsilon
_0\sqrt{n}\log n$. Finally, by (\ref{eqFijDeriv}) we have that
\begin{eqnarray*}
F^k_{r_{ij}x_l}&=&0 \qquad\mbox{for } i\neq j,\\
F^k_{r_{ii}x_j}&=&0 \qquad\mbox{for } i\neq k, i\neq
j,\\
F^k_{r_{ii}x_i}&=&\frac{1}{2}\mu_i \qquad\mbox{for } i\neq k,\\
F^k_{r_{ii}x_i}&=&\frac{1}{2}\mu_i \qquad\mbox{for } i\neq
k,\\
F^k_{r_{kk}x_k}&=&\frac{1}{2}\mu_k-\frac{1}{2}\,\frac{\partial
}{\partial x_k}f(e\cdot x),\\
F^k_{r_{kk}x_j}&=&\frac{1}{2}\,\frac{\partial}{\partial x_k}f(e\cdot
x) \qquad\mbox{for } j\neq k.
\end{eqnarray*}
Thus,
\[
|F_{rx}^k|^2\leq\sum_{l\in\I}\frac{1}{2}\biggl|\frac{\partial
}{\partial x_l}f(e\cdot x)\biggr|^2+\frac{1}{2}\mu_{\max}\leq\frac
{1}{2}(1+\mu_{\max}),
\]
where we used the fact that $f(\cdot)$ is continuously differentiable
with Lipschitz constant $1$ (independently of $a$). Finally,
\[
F_{x_i}^k=\frac{\partial}{\partial x_i}f(e\cdot x)\biggl(c_k+\mu
_kp_k-\frac{1}{2}\mu_kr_{kk}\biggr)-\mu_ip_i+\frac{1}{2}\mu_ir_{ii},
\]
so that
\begin{equation}\label{eqFx}
|F_{x_i}^k|\leq|c_k|+\mu_k|p|+\tfrac{1}{2}\mu
_k|r|+\mu
_i|p|+\tfrac{1}{2}\mu_i |r|.
\end{equation}
Also, note that
\[
F_{x_ix_j}^k=\frac{\partial}{\partial x_i\,\partial x_j}f(e\cdot
x)\biggl(c_k+p_k-\frac{1}{2}r_{kk}\biggr),
\]
so that
\[
F_{x_ix_j}^k=\cases{
2\bigl[c_k+\mu_kp_k-\frac{1}{2}\mu_kr_{kk}\bigr], &\quad
if $|e\cdot x|\leq\frac{1}{4}$,\vspace*{2pt}\cr
0, &\quad otherwise.}
\]
Combining the above gives
\[
|F_{xx}^k|\leq\varepsilon_2\varepsilon_0(1+|p|+|r|)
\]
for suitably\vspace*{1pt} redefined $\varepsilon_2$ which concludes the proof that
the conditions (\ref{eqFcond1})--(\ref{eqFcond3}) hold with $\bar
{\Lambda}=\varepsilon_1n$, $\underbar{\Lambda}=\varepsilon_0n$
and $\eta=\varepsilon_2$. Having verified these conditions, the
existence and uniqueness of the solution $\phi_{\kappa,a}^n$ to
(\ref{eqHJB2}) now follows from Theorem~17.18 in \cite{TandG}.
To obtain the gradient estimates in (\ref{eqgradients})
we first outline how the solution~$\phi
_{k,a}^n$ is obtained in \cite{TandG} as a limit of solutions to
smoothed equations (we refer the reader to~\cite{TandG}, page 466, for
the more elaborate description). To that end, let
$F_a^i$ be as defined in (\ref{eqFdefin}) and for $y\in\Gamma$ define
\begin{equation}\label{eqGhPDE}
F^h[y]=G_{h}(F^1_a[y],\ldots,F^I_a[y]),
\end{equation}
where
\[
G_h(y)=h^{-I}\int_{\bar{y}\in\bbR^I}\rho\biggl(\frac{y-\bar
{y}}{h}\biggr)G_0(\bar{y})\,d\bar{y}
\]
and $G_0(x)=\min_{i\in\I}x_i$ and $\rho(\cdot)$ is a mollifier on
$\bbR^I$ (see \cite{TandG}, page 466). $F^h$ satisfies all the bounds
in (\ref{eqFcond1})--(\ref{eqFcond3}) uniformly in $h$; cf. \cite
{TandG}, page 466. Then, there exists a unique solution $u^h$ for the equations
\begin{equation} \label{eqGhPDE2}
F^h[u^h]=0
\end{equation}
on $\mB_{\kappa}^n$ with
$u^h=0$ on $\partial\mB_{\kappa}^n$.
The solution $\phi_{\kappa,a}^n$ is now obtained as a limit of $\{
u^h\}$ in the space $C_*^{2,\alpha}(\mB)$ as defined in (\ref
{eqCstar}). Moreover, since the gradient bounds are shown in \cite
{TandG} to be independent of $h$, it suffices for our purposes to fix
$h$ and focus on the construction of the gradient bounds.
Our starting point is the bound at the bottom of page 461 of \cite
{TandG} by which
\begin{equation}\label{eqonemoreinterim}
|u^h|^*_{2,\alpha,\mB_{\kappa}^n}\leq\check
{C}(a,n)(1+|u^h|^*_{2,\mB_{\kappa}^n}),
\end{equation}
where\vspace*{-1pt} $|u^h|^*_{2,\mB_{\kappa}^n}=\sum_{j=0}^2 [u^h]_{j,\mB}^*$ and
$[\cdot]_{j,\mB}^*, j=0,1,2$, are as defined in Section~\ref{secADCP}.
The constant $\alpha(a,n)$ depends only on the number of
classes $I$ and on $\bar{\Lambda}/\underbar{\Lambda}$ (see~\cite
{TandG}, top of page 461) and this fraction equals, in our context, to
$\varepsilon_1/\varepsilon_0$ and is thus constant and independent of
$n$ and $a$.
We will\vspace*{1pt} address the constant $\check{C}(a,n)$ shortly. We first argue
how one proceeds from (\ref{eqonemoreinterim}). Fix $0<\delta< 1$,
let $\epsilon=\delta/\check{C}(a,n)$ and $C(\epsilon)=2/(\epsilon
/8)^{1/\alpha}$ (see \cite{TandG}, top of page 132). Then, applying
an interpolation inequality (see~\cite{TandG}, bottom of page 461 and
Lemma 6.32 on page 130), it is obtained that
\[
|u^h|^*_{2,0,\mB_{\kappa}^n}\leq C(\epsilon)|u^h|^*_{0,\Omega
}+\epsilon|u^h|^*_{2,\alpha,\mB_{\kappa}^n}.
\]
Plugging this back into (\ref{eqonemoreinterim}) one then has
\[
|u^h|^*_{2,\alpha,\mB_{\kappa}^n}\leq\check{C}(a,n)\biggl(1+\bar
{C}\check{C}(a,n)^{1/\alpha}|u^h|^*_{0,\mB_{\kappa}^n}
+\frac{\delta}{\check{C}(a,n)}|u^h|^*_{2,\alpha,\mB_{\kappa
}^n}\biggr)
\]
for a constant $\bar{C}$ that depends only on $\delta$ and $\alpha$.
In turn,
\[
|u^h|^*_{2,\alpha,\mB_{\kappa}^n}\leq\bar{C}(\check
{C}(a,n))^{1+1/\alpha}|u^h|^*_{0,\mB_{\kappa}^n}
\]
for a constant $\bar{C}$ that\vadjust{\goodbreak} does not depend on $a$ or $n$.
Hence, to obtain the required bound in (\ref{eqgradients})
it remains only to\break bound~$\check{C}(a,n)$. Following \cite{TandG},
building on equation (17.51) of \cite{TandG}, $\check{C}(a,n)$ is the
(minimal) constant
that satisfies
\begin{equation}\label{eqCandefin}
C(1+M_2)(1+\tilde{\mu}R_0+\bar{\mu}R_0^2)\leq
\check{C}(a,n)(1+|u^h|^*_{2,\mB}),
\end{equation}
where (as stated in \cite{TandG}, bottom of page 460) the (redefined)
constant $C$ depends only on the number of class $I$ and on $\bar
{\Lambda}/\underbar{\Lambda}=\varepsilon_1/\varepsilon_0$.
The constants~$\tilde{\mu}$ and~$\bar{\mu}$ are defined in \cite{TandG}
and we will explicitly define them shortly. Here one should not
confuse $\bar{\mu}$ with the average service rate in our system. In
what follows $\bar{\mu}$ will only be used as the constant in \cite
{TandG}. We now bound constants~$\tilde{\mu}$ and~$\bar{\mu}$.
These are defined by
\begin{eqnarray*}
\tilde{\mu}&=&\frac{D_0}{\underbar{\Lambda
}(1+M_2)},\qquad
\bar{\mu} =\frac{C(I)}{\underbar{\Lambda}}
\biggl(\frac{A_0^2}{\underbar{\Lambda}\epsilon}+\frac{B_0}{1+M_2}\biggr),
\\
D_0&=&\sup_{x,y\in\mB}\{|F^h_x(y,u^h(y),Du^h(y),D^2u^h(x))|
\\
&&\hphantom{\sup_{x,y\in\mB}\{}{}+|F^h_z(y,u^h(y),Du^h(y),D^2u^h(x))||Du^h(y)|\\
&&\hphantom{\sup_{x,y\in\mB}\{}{}+|F^h_p(y,u^h(y),Du^h(y),D^2u^h(x))||D^2u^h(y)|\},
\\
A_0&=&\sup_{\mB}\{|F^h_{rx}|+|F^h_p|\}, \\
B_0&=& \sup_{\mB} \{|F_{px}||D^2 u^h|+ |F_z||D^2
u^h|+|F_{zx}||Du^h|+|F_{xx}|\},
\end{eqnarray*}
where $C(I)$ is a constant that depends only on the number of classes
$I$, $\epsilon\in(0,1)$ is arbitrary and fixed (independent of $n$
and $a$) and $M_2=\sup_{\mB}|D^2u^h|$. The constants $\bar{\mu}$,
$\tilde{\mu}$ and $M_2$ are defined in \cite{TandG}, pages 456--460,
and $A_0$ and~$B_0$ are as on page 461 there.
We note that $F^h_z$ is a constant, $F^h_p$ is bounded by $\bar
{C}\sqrt{n}\log^m n$ for some constant $\bar{C}$ [see (\ref
{eqgradients})] that depends only on $\kappa$ and, by (\ref{eqFx}),
$|F_x^h|\leq\varepsilon_2\varepsilon_0(1+|p|+|r|)$. In turn,
$D_0\leq4\varepsilon_2\varepsilon_0\sqrt{n}\log^m n\sup_{\mB
}(1+|Du^h|+|D^2u^h|)$. Arguing similarly for $A_0$ and $B_0$ we find
that there exists a constant $\bar{C}$ (that does not depend on $n$
and $a$) such that
\[
A_0\leq\bar{C}\sqrt{n}\log^mn\quad\mbox{and}\quad B^0\leq\bar{C} \sup
_{\mB}(1+|Du^h|+|D^2u^h|),
\]
which in turn implies the existence of a redefined constant $\bar{C}$
such that
\[
\tilde{\mu}\leq\frac{\bar{C}\log^{m} n}{\sqrt{n}(1+M_2)}\sup
_{\mB}(1+|Du^h|+|D^2u^h|)
\]
and
\[
\bar{\mu}\leq\frac{\bar{C}\log^{2 m} n}{n}+\frac{\bar
{C}}{n(1+M_2)}\sup_{\mB}(1+|Du^h|+|D^2u^h|).
\]
The proof of the bound is concluded by plugging these back into (\ref
{eqCandefin}) and setting $R_0=\kappa\sqrt{n}\log^m n$ there to get
that
\[
\check{C}(a,n)\leq C\log^{4m(1+{1}/{\alpha})}n
\]
for some $C$ that does not depend on $a$ and $n$.
The constant $\tilde{C}$ on the right-hand side of (\ref{eqgradients})
(which can depend on $n$ but does not depend on $a$) is
argued as in the proof of Theorem 17.17 in~\cite{TandG} and we
conclude the proof by noting that the global Lipschitz constant (that
we allow to depend on~$n$) follows from Theorem 7.2 in
\cite{trudinger1983fully}.
We next turn to proof of Theorem \ref{thmSSC}. First, we will
explicitly construct the queueing process under the $h$-tracking policy
and state a lemma that will be of use in the proof of the theorem.
Define $A_i^n(t)=\mN_i^a(\lambda_i^n t)$ so that~$A_i^n$ is the
arrival process of class-$i$ customers. Given a ratio control $U^n$ and
the associated queueing process $\mathbb{X}^n=(X^n,Q^n,Z^n,\check{X}^n)$,
$\check{W}^n$ is as defined in~(\ref{eqWtildedefin}). Also, we define
\[
D^n(t)=\sum_{i\in\I}\mN_i^d\biggl( \mu_i\int_0^t Z_i\lam(s)\,ds
\biggr).
\]
That is, $D^n(t)$ is the total number of service completions by time
$t$ in the $n$th system.
For the construction of the queueing process under the tracking policy
we define a family of processes
$\{\mathcal{A}_{i,\mH}^n, i\in\I,\mathcal{H}\subset\I\}$ as follows:
let $\{\xi_{\mK}^l; l\in\bbZ_+,\mK\subset\I\}$ be a family of i.i.d
uniform $[0,1]$ random variables independent of
$\bar{\mathcal{F}}_{\infty}$ as defined in~(\ref{eqcheckFdefin}). For
each $\mathcal{K}\subset\I$, define the processes
$(\mathcal{A}_{i,\mH}^n, i\in\I)$ by
\begin{equation}\label{eqmAdefin}
\mathcal{A}_{i,\mH}^n(t)=\sum_{l=1}^{D^n(t)}1\biggl\{\frac{\sum_{k<
i,k\in\mH}\lambda_k}{1\vee\sum_{k\in\mH}\lambda_k} < \xi_{\mK
}^l\leq
\frac{\sum_{k\leq i,k\in\mH}\lambda_k}{1\vee
\sum_{k\in\mH}\lambda_k}\biggr\}.
\end{equation}
We note that for any strict subset $\mH\subset\I$ and $i\in\mK$,
the probability that a jump of $D^n(t)$ results in a jump of $\mathcal
{A}_{i,\mH}^n$ is equal to $\lambda_i^n/\sum_{k\in\mH}\lambda
_k^n=a_i/\sum_{k\in\mH}a_k$ and is strictly greater than $\lambda
_i^n/\sum_{k\in\I}\lambda_k^n=a_i$. We define
\begin{equation}\label{eqepsilondefin}
\epsilon_i=\min
_{\mathcal{H}\subset\I}a_i-\frac{a_i}{\sum_{k\in\mH}a_k},
\end{equation}
and note that $\epsilon_i>0$ by our assumption
that $a_i>0$ for all $i\in\I$ (see Section~\ref{secmodel}). Let
$\bar{\epsilon}=\min_{i}\epsilon_i/4$.
Note that at time intervals in which $i\in\mK(\cdot)=\mH$ (see
Definition \ref{defintracking}) for some $\varnothing\neq\mH\subset
\I$, the process $\mathcal{A}_{i,\mH}^n$ jumps with probability
$\lambda_i^n/\sum_{k\in\mH}\lambda_k$ whenever a server becomes
available (i.e., upon a jump of $D^n$). In turn, we will use the
processes $\{\mathcal{A}_{i,\mH}^n, i\in\I,\mathcal{H}\subset\I\}
$ to generate (randomized) admissions to service of class-$i$ customers
under the $h$-tracking policy.
More specifically, under the $h$-tracking policy (see Definition \ref
{defintracking}) a customer from the class-$i$ queue enters service in
the following events:
\begin{longlist}
\item A class-$i$ customer that arrives at time $t$ enters
service immediately if there are idle servers, that is, if $(e\cdot
\check{X}\lam(t-))^{-}>0$.
\item If a server becomes available at time $t$ (corresponding
to a jump of~$D^n$) and $t$ is such that $i\in\mathcal{K}(t-)=\mH
\subset\I$, then a customer from the class-$i$ queue is admitted to
service at time $t$ with probability $\lambda_i^n/\sum_{k\in\mH
}\lambda_k$. This admission to service corresponds to a jump of the
process $\mathcal{A}_{i,\mH}^n$ as defined in (\ref{eqmAdefin}).
\item If a server becomes available at time $t$ (corresponding
to a jump of~$D^n$) and $t$ is such that $\mathcal{K}(t-)=\varnothing$
and $i=\min\{k\in\I\dvtx Q_i^n(t)>0\}$, then a~class-$i$ customer is
admitted to service.
\end{longlist}
Formally, the queueing process $\mathbb{X}\lam=(X\lam,Q\lam,Z\lam
,\check{X}\lam)$ satisfies
\begin{eqnarray*} Z_i\lam(t)&=&Z_i\lam(0)+\int_0^t 1\bigl\{\bigl(e\cdot
\check{X}^n(s)\bigr)^->0\bigr\}\,dA_i^n(s)\\
&&{}+\sum_{\mH\subset\I}\int
_0^t1\{i\in\mathcal{K}(s-),\mathcal{K}(s-)=\mathcal{H}\}\,
d\mathcal{A}_{i,\mH}\lam(s)\\
&&{}+ \int_0^t 1\bigl\{\mK
(s-)=\varnothing,i=\min\{k\in\I\dvtx Q_k^n(s-)>0\}\bigr\}\,dD^n(s)\\
&&{}-\mN
_i^d\biggl( \mu_i\int_0^t Z_i\lam(s)\,ds \biggr),\qquad i\in\I,\\
X_i\lam(t)&=&X_i\lam(0)+A_i^n(t)-\mN_i^d\biggl(\mu_i \int_0^t
Z_i\lam(s) \,ds \biggr),\qquad i\in\I, \\ Q_i\lam(t)&=&X_i\lam(t)-Z_i\lam
(t),\qquad i\in\I.
\end{eqnarray*}
The second, third and fourth terms on the right-hand side of the
equation for $Z_i^n$ correspond, respectively, to the events described
by items (i)--(iii) above. Finally, $\check{X}\lam$ is defined from
$X\lam$ as in (\ref{eqtildeXdefin}). The fact that the above system
of equations has a unique solution is proved by induction on arrival
and service completions times (see, e.g., the proof of Theorem 9.2 of
\cite{MMR98}). Clearly, $\mathbb{X}^n$ satisfies (\ref
{eqdynamics2})--(\ref{eqnon-negativity2}) with $U_i\lam$ there
constructed from $Q\lam$ using~(\ref{eqUQmap}).
We note that, with this construction, the tracking policy is admissible
in the sense of Definition \ref{definadmissiblecontrols}. Also, it
will be useful for the proof of Theorem~\ref{thmSSC} to note that
with this construction, if $[s,t]$ is an interval such that \mbox{$i\in
\mathcal{K}(u)\subset\I$} for all $u\in[s,t]$ then
\begin{equation}\label{eqQtrack}\qquad
Q_i^n(t)-Q_i^n(s)=A_i^n(t)-A_i^n(s)-\sum_{\mH\subset
\I}\int
_s^t1\{\mathcal{K}(u-)=\mathcal{H}\}\,d\mathcal{A}_{i,\mH
}\lam(u).
\end{equation}
Before proceeding to the proof of Theorem \ref{thmSSC} the following
lemma provides preliminary bounds for arbitrary ratio controls.
\begin{lem}\label{lemstrongappbounds}
Fix $\kappa,T>0$ and a ratio control $U^n$, let $\mathbb
{X}^n=(X^n,Q^n,\allowbreak Z^n,\check{X}^n)$ be the associated queueing process
and define
\[
\tau_{\kappa,T}^n=\inf\{t\geq0\dvtx\check{X}^n(t)\notin\mB_{\kappa
}^n\}\wedge T\log n.
\]
Then, there exist constants $C_1,C_2,K_0>0$ (that depend on $T$ and
$\kappa$ but that do not depend on $n$ or on the ratio control $U^n$)
such that for all $K>K_0$ and all $n$ large enough,
\begin{eqnarray}\label{eqstrapp1}
&&\Pd\Bigl\{\sup_{0\leq t\leq2T\log n}|\check
{W}^n(t)|> K\sqrt
{n}\log n\Bigr\}\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq C_1e^{-C_2K\log n},\nonumber\\
\label{eqstrapp2}
&&\Pd\bigl\{|\check{X}^n(t)-\check{X}^n(s)|> \bigl((t-s)+(t-s)^2
\bigr)K\sqrt{n}\log n+K\log n,
\nonumber\\
&&\hspace*{163pt}
\mbox{ for some } s<t\leq2T\log n\bigr\}\\
&&\qquad\leq C_1 e^{-C_2K\log n},\nonumber
\end{eqnarray}
\begin{eqnarray}\label{eqstrapp3}
&&\Pd\{|A_i^n(t)-A_i^n(s)-\lambda_i^n(t-s)|
> \bar{\epsilon}n(t-s)+K\log n\nonumber\\
&&\hspace*{139.3pt} \mbox{ for some }
s<t\leq\tau_{\kappa,T}^n\}\\
&&\qquad\leq C_1e^{-C_2K\log n},\qquad i\in\I,\nonumber\\
\label{eqstrapp4}
&&\Pd\biggl\{
\biggl|D^n(t)-D^n(s)-\sum_{i}\mu_i\nu_i n(t-s)\biggr|
> \bar{\epsilon}n(t-s)+K\log n\nonumber\\
&&\hspace*{174pt}\mbox{ for some } s<t\leq\tau_{\kappa,T}^n\biggr\}
\\
&&\qquad\leq C_1e^{-C_2K\log n},\nonumber\\
\label{eqstrapp5}
&&\Pd\biggl\{\mathcal{A}_{i,\mH}^n(t)-\mathcal{A}_{i,\mH
}^n(s)-\lambda_i^n(t-s)\leq\frac{\epsilon_i}{2} n(t-s)-K\log
n \nonumber\\
&&\hspace*{156.6pt}\mbox{ for some } s<t\leq\tau
_{\kappa,T}^n\biggr\}\\
&&\qquad\leq C_1e^{-C_2K\log n},\qquad i\in\I,\mK\subset\I
.\nonumber
\end{eqnarray}
\end{lem}
\begin{pf}
Equation (\ref{eqstrapp1}) follows from strong approximations (see,
e.g., Lem\-ma~2.2. in \cite{csorgo}) and known bounds on the supremum of
Brownian motion (see, e.g., equation 2.1.53 in \cite{csorgo}).
Equation (\ref{eqstrapp2}) then follows using this bound together
with (52) in \cite{AMR02} but with $X^n(t)-X^n(s)$ instead of
$X^n(t)$ (in the notation of \cite{AMR02} $\check{W}^n$ is $\hat
{W}^n$). Equations (\ref{eqstrapp3})--(\ref{eqstrapp5}) follow by
carefully constructing and bounding the increments. We outline the
proof of (\ref{eqstrapp3}) and the others follow similarly. To
that\vadjust{\goodbreak}
end, note that given $K$ and for all $n$ large enough
\begin{eqnarray*}
&& \{|A_i^n(t)-A_i^n(s)-\lambda_i^n(t-s)|\leq
\bar
{\epsilon}n(t-s)+K\log n, \\
&&\hspace*{109pt}
\mbox{ for all }
0\leq s\leq t\leq2T\log n\} \\
&&\qquad \supseteq
\biggl\{\max_{l\leq N_i^n}\max_{j\geq0\dvtx j\/\log^l n\leq3T\log
n}\frac{|A_i^{j,l,n}-\lambda_i^n/\log^l n|}{\sqrt{\lambda_i^n/\log
^l n}}\leq K\sqrt{\log n}\biggr\},
\end{eqnarray*}
where
$A_i^{j,l,n}=A_i^n((j+1)/\log^ln)-A_i^n(j/\log^ln)$ and $N=\max\{l\dvtx
\log^ln \leq\lambda_i^n/\log n\}$. Indeed, given an interval $[s,t)$
we can construct it from smaller intervals. Starting with $l=0$, we fit
as many intervals of size $1$ into $[s,t)$, we then continue to fit as
many intervals of size $1/\log n$ to the uncovered part of the interval
and continue sequentially in $l$. We omit the simple and detailed
construction. Note that with such construction, given an interval
$[s,t)$, its covering uses at most $\log n$ intervals of size $\log^l
n$ for each $l\geq0$. Also, note that $N_i^n\leq C\log n$ for all $n$
and some constant $C$. From here, using strong approximations (or
bounds for Poisson random variables as in \cite{glynn1987upper}) we
have, for each $j$ and $l$, that
\[
\Pd\biggl\{\frac{|A_i^{j,l,n}-\lambda_i^n/\log^l n|}{\sqrt{\lambda
_i^n/\log^l n}}> K\sqrt{\log n}\biggr\}\leq C_1e^{-C_2 K\log n}.
\]
Since the number of intervals considered is of the order of $n\log n$,
the bound follows with redefined constants $C_1$ and $C_2$.
\end{pf}
\subsection*{\texorpdfstring{Proof of Theorem \protect\ref{thmSSC}}{Proof of Theorem 5.1}} Since $\kappa$
is fixed throughout we use $h^n(\cdot)=h_{\kappa}^{*,n}(\cdot)$. As
in the statement of the theorem, let
\[
\psi^n(x,u)=L(x,u)+A^n_u\phi_{\kappa}^n(x)-\gamma\phi_{\kappa
}^n(x)\qquad \mbox{for } x\in\mB_{\kappa}^n, u\in\mathcal{U}
,\vadjust{\goodbreak}
\]
so that by the definition of $A^n_u(x)$ we have
\begin{eqnarray}\qquad
\psi^n(x,u)&=&-\gamma\phi_{\kappa}^n(x)\nonumber\\
&&{} + (e\cdot
x)^+\cdot\sum_{i\in\I}u_i\biggl\{ c_i+\mu_i(\phi_{\kappa
}^n)_i(x)-\mu_i\frac{1}{2}(\phi_{\kappa}^n)_{ii}(x)\biggr\}
\\
&&{}+\sum_{i\in\I} (l_i\lam-\mu_ix_i)(\phi_{\kappa}^n)_i(x)
+\frac{1}{2}\sum_{i\in\I} \bigl(\lambda_i^n+\mu_i(\nu
_in+x_i)\bigr)(\phi_{\kappa}^n){ii}(x).\nonumber
\end{eqnarray}
Defining, as before,
\[
M_{i}^n(z)=c_{i}+\mu_i(\phi_{\kappa}^n)_{i}(z)-\tfrac{1}{2}\mu
_i(\phi_{\kappa}^n)_{ii}(z),
\]
we have that
\[
\psi^n(x,u)-\psi^n(x,v)=(e\cdot x)^+\biggl(\sum
_{i\in\I}v_iM_i^n(x)-\sum_{i\in\I}u_iM_i^n(x)\biggr).
\]
Let $U^n$ be the ratio control associated with the $h^n$-tracking
policy, let $\mathbb{X}^n=(X^n,Q^n,Z^n,\check{X}^n)$ be the
associated queueing process and define
\begin{eqnarray}\label{eqpsicheckdefin}
\check{\psi}^n(s)&=&\psi^n(\check
{X}^n(s),U^n(s))-\psi^n(\check{X}^n(s),h^n(\check{X}^n(s)))\nonumber
\\
&=&
\bigl(e\cdot\check{X}^n(s)\bigr)^+\sum_{i\in\I}h_i^n(\check
{X}^n(s))M_i^n(\check{X}^n(s)) \\
&&{}-\bigl(e\cdot
\check{X}^n(s)\bigr)^+ \sum_{i\in\I}U_i^n(s)M_i^n(\check
{X}^n(s)).\nonumber
\end{eqnarray}
Recall that, by construction, $Q_i^n(s)=(e\cdot\check
{X}^n(s))^+U_i^n(s)$ so that (\ref{eqpsicheckdefin}) can be
re-written as
\begin{eqnarray*} \check{\psi}^n(s)&=&\psi^n(\check
{X}^n(s),U^n(s))-\psi^n(\check{X}^n(s),h^n(\check{X}^n(s)))
\\&=&
\bigl(e\cdot\check{X}^n(s)\bigr)^+\sum_{i\in\I}h_i^n(\check
{X}^n(s))M_i^n(\check{X}^n(s)) \\
&&{}-\sum_{i\in\I
}Q_i^n(s)M_i^n(\check{X}^n(s)).
\end{eqnarray*}
The theorem will be proved if we show that
\begin{equation}\label{eqwhatneed}
\Ex\biggl[\int_0^{\tau_{\kappa',T}^n}e^{-\gamma s}
|\check
{\psi}^n(s)|\,ds\biggr]\leq C\log^{k_0+3}n.
\end{equation}
To
that end, define a sequence of times $\{\tau_{l}^n\}$ as follows:
\[
\tau_{l+1}^n=\inf\{t> \tau_l^n\dvtx h^n(\check{X}^n(t))\neq h^n(\check
{X}^n(\tau_l^n))\}\wedge\tau_{\kappa',T}^n \qquad\mbox{for } l\geq0,
\]
where $\tau_0^n=\eta^n\wedge\tau_{\kappa',T}^n$ and
\begin{equation}\label{eqetandefin}
\eta
^n=t_0\frac{\log^mn}{\sqrt{n}}
\end{equation}
for\vspace*{1pt}
$t_0=4\kappa/\epsilon_i$ with $
\epsilon_i=\min_{\mathcal{H}\subset\I}a_i-\frac{a_i}{\sum_{k\in
\mH}a_k}$ as in (\ref{eqepsilondefin}). Finally, we define $r^n=\sup
\{l\in\bbZ_+\dvtx \tau_{l}^n\leq\tau_{\kappa',T}^n\}$ and set $\tau
_{r^n+1}^n=\tau_{\kappa',T}^n$. We then have
\begin{eqnarray*}
&&\int_0^{\tau
_{\kappa',T}^n}e^{-\gamma s} |\check{\psi}^n(s)|\,ds\\
&&\qquad=\sum
_{l=1}^{r^n+1} \int_{\tau_{l-1}^n}^{\tau_l^n}e^{-\gamma s} |\check
{\psi}^n(s)|\,ds\\
&&\qquad=\sum_{l=1}^{r^n+1}\biggl(\int_{\tau
_{l-1}^n}^{(\tau_{l-1}^n+\eta^n)\wedge\tau_{l}^n}e^{-\gamma s}
|\check{\psi}^n(s)|\,ds+\int_{\tau_{l-1}^n+\eta^n}^{\tau_l^n\vee
(\tau_{l-1}^n+\eta^n) }e^{-\gamma s} |\check{\psi}^n(s)|\,ds
\biggr).
\end{eqnarray*}
The proof is now divided into three parts. We will show that,
under the conditions of the theorem,
\begin{eqnarray}\label{eqSSC1}
\Ex\Bigl[\sup_{1\leq l\leq
r^n+1}\sup_{\tau_{l-1}^n\leq s< (\tau_{l-1}^n+\eta^n)\wedge\tau
_l^n}|\check{\psi}^n(s)|\Bigr]&\leq& C\log^{k_0+2}n,
\\
\label{eqSSC2}
\Ex\Bigl[\sup_{1\leq l\leq r^n+1}\sup_{(\tau
_{l-1}^n+\eta
^n)\leq s< \tau_{l}\vee(\tau_{l-1}^n+\eta^n)}
|\check{\psi}^n(s)|\Bigr]&\leq& C\log^{k_0+2} n,
\end{eqnarray}
where we define $\sup_{(\tau_{l-1}^n+\eta^n)\leq s< \tau_{l}^n\vee
(\tau_{l-1}^n+\eta^n)}
|\check{\psi}^n(s)|=0$ if $\tau_l^n\leq\tau_{l-1}^n+\eta^n$.
Finally, we will show that
\begin{equation}\label{eqSSC3}
\Ex\biggl[\int_0^{\eta^n\wedge\tau_{\kappa
',T}^n}|\check
{\psi}^n(s)\,ds|\biggr]\leq C\log^{k_0}n.
\end{equation}
The proof of (\ref{eqSSC1}) hinges on the fact that, sufficiently
close to a change point~$\tau_l^n$, all the customer classes, $i$, for
which $h_i^n(\check{X}^n(s))=1$ for some $s$ in a~neighborhood of
$\tau_l^n$, will have similar values of $M_i^n(\check{X}^n(s))$. This
will follow from our gradient estimates for $\phi_{\kappa}^n$. The
proof of (\ref{eqSSC2}) hinges on the fact that,~$\eta^n$ time units
after a change point $\tau_l^n$ the queues of all the classes for
which $h^n(\check{X}^n(\tau_l^n))=0$ are small because, under the
tracking policy, these classes receive a significant share of the capacity.
Toward formalizing this intuition, define the following event on the
underlying probability space:
\begin{eqnarray*} \tilde{\Omega}(K)&=&\bigl\{|\check{X}^n(t)-\check
{X}^n(s)|\leq K \sqrt{n}\log^2 n(t-s)+K\log n,\\[-0.8pt]
&&\hspace*{135.3pt}
\mbox{ for all } s<t\leq\tau_{\kappa,T}^n\bigr\}\\[-0.8pt]
&&{}\cap_{\mH\subset\I} \biggl\{\mathcal{A}_{i,\mH
}^n(t)-\mathcal{A}_{i,\mH}^n(s)-\lambda_i^n(t-s)\geq\frac{\epsilon
_i }{2} n(t-s)-K\log n \\[-0.8pt]
&&\hspace*{183.5pt}\hspace*{12.1pt} \mbox{for all } s<t\leq\tau
_{\kappa,T}^n\biggr\}\\[-0.8pt]
&&{}\cap_{i\in\I} \{
|A_{i}^n(t)-A_{i}^n(s)-\lambda_i^n(t-s)|\leq\bar{\epsilon
}n(t-s)+K\log n \\[-0.8pt]
&&\hspace*{171.7pt}\mbox{for all } s<t\leq\tau
_{\kappa,T}^n\}.
\end{eqnarray*}
For each $0\leq t\leq\tau_{\kappa',T}^n$ and $i\in\I$ let
\begin{eqnarray}
\label{eqchecktaudefin}
\check{\varsigma}_i^n(t)&=&\sup\{s\leq t\dvtx h_i^n(\check{X}^n(s))=1\},
\\
\label{eqtildetaudefin}
\tilde{\varsigma}_i^n(t)&=&\inf\{
s\geq t\dvtx h_i^n(\check{X}^n(s))=1\}\wedge\tau_{\kappa',T}^n
\end{eqnarray}
and
\begin{equation}\label{eqydefin}
\hat{\varsigma}_i^n(t)=
\cases{\check{\varsigma}_i^n(t)+\eta^n,&\quad if
$Q_i^n(t)>4K\log n$,\cr
t, &\quad otherwise.}
\end{equation}
Then, we claim that on $\tilde{\Omega
}(K)$ and for all $t$ with $\tilde{\varsigma}_i^n(t)> \hat{\varsigma
}_i^n(t)$,
\begin{eqnarray}\label{eqSSCinterim}
&&\sup_{\hat{\varsigma}_i^n(t)\leq s< \tilde
{\varsigma}_i^n(t)}\bigl|\bigl(e\cdot\check{X}^n(s)\bigr)^+U_i^n(s)-\bigl(e\cdot\check
{X}^n(s)\bigr)^+h_i^n(\check{X}^n(s))\bigr|\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq12K\log n.\nonumber
\end{eqnarray}
Note that since $h_i^n(\cdot)\in\{0,1\}$, the
above is equivalently written as
\begin{equation}\label{eqSSCinterim2}
\sup_{\hat{\varsigma}_i^n(t)\leq s< \tilde
{\varsigma}_i^n(t)}Q_i^n(s)\leq12K\log n.
\end{equation}
In words,
when the process $\check{X}^n(t)$ enters a region in which
$h^n_i(\check{X}^n(\cdot))=0$ the queue of class $i$ will be drained
up to $12 K\log n$ within at most $\eta^n$ time units and it will
remain there up to $\tilde{\varsigma}_i^n(t)$. We postpone the proof
of (\ref{eqSSCinterim}) and use it in proceeding with the proof of
the theorem.
To that end, fix $l\geq0$ and let
\[
j^*_{l}=\min\mathop{\argmin}_{i\in\I}M_i^n(\check{X}^n(\tau_l^n)).
\]
Then, by the definition of the function $h^n$ in (\ref{eqixdefin}) we
have that $h_{j^*_l}^n(\check{X}^n(\tau_l^n))=1$ and $h_i(\check
{X}^n(\tau_l^n))=0$ for all $i\neq j^*_l$. In particular,
\begin{eqnarray*}
\check{\psi}^n(s)&=&\bigl(e\cdot\check
{X}^n(s)\bigr)^+h_{j^*_l}^n(\check{X}^n(s))M_{j^*_l}^n(\check{X}^n(s))
\\
&&{}-\sum_{i\in\I}Q_i^n(s)M_i^n(\check{X}^n(s))
\end{eqnarray*}
for all $s\in[\tau_{l}^n,(\tau_{l}^n+\eta^n)\wedge\tau_{l+1}^n)$.
Let
\[
\mathcal{J}(\tau_l^n)=\{i\in\I\dvtx Q_i^n(\tau_l^n-)>4K\log n\}.
\]
Then, simple manipulations yield
\begin{eqnarray}\label{eqcheckpsiinterim}
|\check{\psi}^n(s)|&\leq&\sum_{i\notin\mathcal{J}(\tau
_l^n)\cup\{j^*_l\}}Q_i^n(s)|M_i^n(\check{X}^n(s))|\nonumber\\
&&{}+
|M_{j_l^*}^n(\check{X}^n(s)|\biggl|\bigl(e\cdot\check
{X}^n(s)\bigr)^+-\sum_{i\in\mathcal{J}(\tau_l^n)\cup\{j^*_l\}
}Q_i^n(s)\biggr|\\
&&{}+\sum_{i\in\mathcal
{J}(\tau_l^n)}Q_i^n(s)|M_i^n(\check{X}^n(s))-M_{j_l^*}^n(\check
{X}^n(s))|.\nonumber
\end{eqnarray}
We turn to bound each of the elements on the right-hand side of (\ref
{eqcheckpsiinterim}). First, note that for all $i\notin\mathcal
{J}(\tau_l^n)\cup\{j^*_l\}$ it follows from (\ref{eqSSCinterim2}) that
\[
\sup_{\tau_{l}^n\leq s<(\tau_{l}^n+\eta^n)\wedge\tau
_{l+1}^n}Q_i^n(s)\leq12K\log n.
\]
Also, by (\ref{eqgradients1}) we have for all $i\in\I$ that
\begin{equation}\label{eqMboun4}
\sup_{0\leq s\leq\tau_{\kappa',T}^n}|M_i^n(\check{X}^n(s))|\leq
C\log^{k_1}n,\vadjust{\goodbreak}
\end{equation}
so that
\begin{equation}\label{eqitai1}
\sum_{i\notin\mathcal{J}(\tau_l^n)\cup\{j^*_l\}}Q_i^n(s)|M_i^n(\check
{X}^n(s))|\leq12IKC\log^{k_1+1}n
\end{equation}
for all $s\in
[\tau_{l}^n,(\tau_{l}^n+\eta^n)\wedge\tau_{l+1}^n)$ and a constant
$C$ that does not depend on $n$. From (\ref{eqSSCinterim2}) and from
the fact that $\sum_{i\in\I}Q_i^n(s)=(e\cdot\check{X}^n(s))^+$ we
similarly have that
\begin{equation}\label{eqitai2}
|M_{j_l^*}^n(\check{X}^n(s)|
\biggl|\bigl(e\cdot\check
{X}^n(s)\bigr)^+-\sum_{i\in\mathcal{J}(\tau_l^n)\cup\{j^*_l\}
}Q_i^n(s)\biggr|\leq12IC K\log^{k_1+1}n.\hspace*{-30pt}
\end{equation}
To
bound the last element on the right-hand side of (\ref
{eqcheckpsiinterim}) note that for each $i\in\mathcal{J}(\tau_l^n)$
there exists $\tau_l^n-\eta^n\leq t\leq\tau_{l}^n$ such that
$h_j^n(\check{X}^n(t))=1$. Otherwise, we would have a contradiction to
(\ref{eqSSCinterim}). We now claim that for each \mbox{$i\in\mathcal
{J}(\tau_l^n)$},
\begin{equation}\label{eqinterim747}
|M_i^n(\check{X}^n(s))-M_{j^*_l}^n(\check
{X}^n(s))|\leq\frac{C\log^{k_1+2} n}{\sqrt{n}}
\end{equation}
for all $s$ in $[\tau_l^n-\eta^n,\tau_l^n+\eta
^n]$. Indeed, by the definition of $\tilde{\Omega}(K)$, we have that
$|\check{X}^n(t)-\check{X}^n(s)|\leq C\log^{m+2}n$ for all $s,t$ in
$[\tau_l^n-\eta^n,\tau_l^n+\eta^n]$. As in the proof of (\ref
{eqgenbound}) [see, e.g., (\ref{eqMbound})] we have that
\begin{equation}\label{eqMbound2}
|M_i^n(x)-M_i^n(y)|\leq\frac{C\log^{k_2+m+2}n}{\sqrt{n}},\qquad i\in\I,
\end{equation}
for $x,y\in\mB_{\kappa'}^n$ with $|x-y|\leq
C\log^{m+2}n$. In turn,
\begin{equation}\label{eqMbound3}
|M_i^n(\check{X}^n(t))-M_i^n(\check
{X}^n(s))|\leq\frac{C\log^{k_2+m+2} n}{\sqrt{n}}=\frac{C\log
^{k_1+2} n}{\sqrt{n}}
\end{equation}
for all $i\in\I$ and all
$s,t\in[\tau_l^n-\eta^n,\tau_l^n+\eta^n]$ where we used the fact
that \mbox{$k_1=k_2+m$}. Since, for each $j\in\mathcal{J}(\tau_l^n)$, there
exists $\tau_l^n-\eta^n\leq t\leq\tau_{l}$ such that $h_j^n(\check
{X}^n(t))=1$ we have, by the definition of $h^n$ that $j\in\argmin
_{i\in\I} M_i^n(\check{X}^n(t))$ for such $t$ so that (\ref
{eqinterim747}) now follows from (\ref{eqMbound3}). Finally, recall
that\break $\sum_{i\in\I}Q_i^n(t)=(e\cdot\check{X}^n(s))^+\leq\kappa
\sqrt{n}\log^mn$ for all $s\leq\tau_{\kappa',T}^n$ and that
$k_0=k_1+m$ so that by~(\ref{eqinterim747})
\[
\sum_{i\in\mathcal{J}(\tau_l^n)}Q_i^n(s)|M_i^n(\check
{X}^n(s))-M_{j_l^*}^n(\check{X}^n(s))|\leq C\log^{k_0+2}n.
\]
Plugging this into (\ref{eqcheckpsiinterim}) together with (\ref
{eqitai1}) and (\ref{eqitai2}) we then have that, on $\tilde{\Omega}(K)$,
\[
\sup_{\tau_{l-1}^n\leq s<(\tau_{l-1}^n+\eta
^n)\wedge\tau_l^n}|\check{\psi}^n(s)| \leq C\log
^{k_1+m+2}n=CK\log^{k_0+2}n.
\]
This argument\vspace*{1pt} is repeated for each $l$. To complete the proof of (\ref
{eqSSC1}) note that, using (\ref{eqMboun4}) together\vspace*{-1pt} with ${\sup
_{0\leq s\leq\tau_{\kappa',T}^n}}|e\cdot\check{X}^n(s)|\leq\kappa
\sqrt{n}\log^mn$, we have that ${\sup_{0\leq s\leq\tau_{\kappa
',T}^n}}|\check{\psi}^n(s)|\leq C\sqrt{n}\log^{k_1+m}n$. Applying H\"
{o}lder's inequality we have that
\begin{eqnarray*}
&&\Ex\Bigl[\sup_{1\leq l\leq
r^n+1}\sup_{\tau_{l-1}^n\leq s< (\tau_{l-1}^n+\eta^n)}|\check{\psi
}^n(s)|\Bigr]\\
&&\qquad\leq
\Ex\Bigl[\sup_{1\leq l\leq r^n+1}\sup_{\tau_{l-1}^n\leq s< (\tau
_{l-1}^n+\eta^n)}|\check{\psi}^n(s)|1\{\tilde{\Omega}(K)\}
\Bigr]\\
&&\qquad\quad{}+
\Ex\Bigl[\max_{1\leq l\leq r^n+1}\sup_{\tau_{l-1}^n\leq s< (\tau
_{l-1}^n+\eta^n)}|\check{\psi}^n(s)|1\{\tilde{\Omega}(K)^c\}
\Bigr]\\
&&\qquad\leq
C\log^{k_0+2} n + C\sqrt{n}\log^{k_1+m}n C_1e^{-(C_2
{K}/{2})\log n}
\end{eqnarray*}
for redefined constants $C_1,C_2$ and (\ref{eqSSC1}) now follows by
choosing $K$ large enough.
We turn to prove (\ref{eqSSC2}). Rearranging terms in (\ref
{eqpsicheckdefin}) we write
\[
\check{\psi}^n(s) = \sum_{i\in\I}M_i^n(\check
{X}^n(s))\bigl(\bigl(e\cdot\check{X}^n(s)\bigr)^+h_i^n(\check{X}^n(s))-\bigl(e\cdot
\check{X}^n(s)\bigr)^+ U_i^n(s)\bigr),
\]
so that equation (\ref{eqSSC2}) now follows directly from (\ref
{eqSSCinterim}) and (\ref{eqMboun4}) through an application of H\"
{o}lder's inequality
Finally, to establish (\ref{eqSSC3}), note that from the definition
of $\tau_{\kappa',T}^n$,
\begin{eqnarray*}
\sup_{0\leq t\leq\eta^n\wedge\tau_{\kappa',T}^n
} |\check{\psi}^n(s)|&\leq& I\sup_{0\leq t\leq\eta^n\wedge\tau
_{\kappa',T}^n}|\check{X}^n(t)|\sum_{i\in\I}M_i^n(\check{X}^n(t))
\\&\leq&
I\sup_{0\leq t\leq\eta^n\wedge\tau_{\kappa',T}^n}C\log
^{k_1}n|\check{X}^n(t)|\leq C \kappa\sqrt{n}\log^{k_1+m}n.
\end{eqnarray*}
In turn,
\begin{equation}\label{eqhowmanymoreinterims}
\Ex\biggl[\int_0^{\tau_0^n}e^{-\gamma t} |\check
{\psi
}^n(t)|\,dt\biggr]\leq C\log^{k_1+m}n=C\log^{k_0}n.
\end{equation}
We have thus proved (\ref{eqSSC1})--(\ref{eqSSC3}) and to conclude
the proof of the theorem it remains only to establish (\ref
{eqSSCinterim}). To that end, let $\check{\varsigma}_i^n(t)$,
$\tilde
{\varsigma}_i^n(t)$ and $\hat{\varsigma}_i^n(t)$ be as in~(\ref
{eqchecktaudefin})--(\ref{eqydefin}). Fix an interval $[l,s)\in
(\hat
{\varsigma}_i^n(t),\tilde{\varsigma}_i^n(t))$ such that
$Q_i^n(u)>2K\log n $ for all $u\in[l,s)$. By the\vadjust{\goodbreak} definition of the
tracking policy, (\ref{eqQtrack}) holds on this interval so that, on
$\omega\in\tilde{\Omega}(K)$,
\begin{eqnarray}\label{equntilwhen3}
Q_i^n(l)-Q_i^n(s)&\leq& \bar{\epsilon}n(t-s)
n-\frac{\epsilon_i}{2}n(t-s)+2K\log n\nonumber\\[-8pt]\\[-8pt]
&\leq& -\frac{\epsilon_i}{4}n(t-s)+2K\log n.\nonumber
\end{eqnarray}
Equation (\ref{eqSSCinterim}) now follows directly from (\ref
{equntilwhen3}). Indeed, note for all $t\leq\tau_{\kappa',T}$,
$Q_i^n(t)\leq(e\cdot\check{X}^n(t))^+\leq|\check{X}^n(t)|\leq
\kappa\sqrt{n}\log^mn$. Hence, $Q_i^n(\check{\varsigma}_i(t))\leq
\kappa\sqrt{n}\log^mn$. In turn, using (\ref{equntilwhen3}) and
assuming that $\tilde{\varsigma}_i^n(t)\geq\check{\varsigma
}_i^n(t)+\eta^n$ we have that $Q_i^n(\varsigma_{0,i}^n(t))\leq4K\log
n$ for some time $\varsigma_{0,i}^n(t)\leq\check{\varsigma
}_i^n(t)+\eta^n$ with $\eta^n$ as defined in (\ref{eqetandefin}).
Also, let
\[
\varsigma_{2,i}^n(t)=\inf\{t\geq\varsigma_{0,i}^n(t)\dvtx Q_i^n(t)\geq
12K\log n \}
\]
and
\[
\varsigma_{1,i}^n(t)=\sup\{t\leq\varsigma_{2,i}^n(t)\dvtx Q_i^n(t)\leq
8K\log n \}.
\]
Note that (\ref{equntilwhen3}) applies to any subinterval $[l,s)$ of
$[\varsigma_{1,i}^n(t),\varsigma_{2,i}^n(t))$. In turn, $\varsigma
_{2,i}^n(t)\leq\tilde{\varsigma}_i^n(t)$ would constitute\vspace*{1pt} a
contradiction to (\ref{equntilwhen3}) so that we must have that
$Q_i^n(s)\leq12K\log n$ for all $s\in[\varsigma_{0,i}^n(t),\tilde
{\varsigma}^n(t))$ with $\varsigma_{0,i}^n(t)\leq\tilde{\varsigma
}^n(t)+\eta^n$. Finally, note that $\varsigma_{0,i}^n(t)$ can be
taken to be $t$ if $Q_i^n(t)\leq4K\log n$.
This concludes the proof of (\ref{eqSSCinterim}) and, in turn, the
proof of the theorem.
\subsection*{\texorpdfstring{Proof of Lemma \protect\ref{lemafterstop}}{Proof of Lemma 6.2}} Let $T$,
$\tau_{\kappa',T}^n$ and $(x^n,q^n)$ be as in the statement of the
lemma. We first prove (\ref{eqafterstop2}). To that end, we claim
that, for all $T$ large enough,
\begin{equation}\label{eqthisisjustonemore}
\Ex_{x^n,q^n}\biggl[\int_{T\log
n}^{\infty} e^{-\gamma s} (e\cdot c)\bigl(e\cdot\check{X}^n(s)\bigr)^+\,ds
\biggr]\leq C\log^2n
\end{equation}
for some $C>0$ and all
$n\in\bbZ$. This is a direct consequence of Lemma 3 in~\cite{AMR02}
that, in our notation, guarantees that
\[
\Ex_{x^n,q^n}[|\check{X}^n(t)|]\leq C\bigl(1+|x^n|+ \sqrt{n}(t+t^2)\bigr)
\]
for all $t\geq0$ and some constant $C>0$. We use (\ref
{eqthisisjustonemore}) to prove Lemma \ref{lemafterstop}. The
assertion of the lemma will be established by showing that
\[
\Ex_{x^n,q^n}\biggl[\int_{\tau_{\kappa',T}^n}^{2T\log n} e^{-\gamma
s} (e\cdot c)\bigl(e\cdot\check{X}^n(s)\bigr)^+\,ds\biggr]\leq C\log^2n.
\]
To that end, applying H\"{o}lder's
inequality, we have
\begin{eqnarray}\label{eqinterim8}
&&\Ex_{x^n,q^n}\biggl[\int_{\tau_{\kappa
',T}^n}^{2T\log n} e^{-\gamma s} (e\cdot c)\bigl(e\cdot\check
{X}^n(s)\bigr)^+\,ds\biggr]\nonumber\\
&&\qquad\leq\Ex_{x^n,q^n}\Bigl[(2T\log
n-\tau_{\kappa',T}^n)^+\sup_{0\leq t\leq2T\log n}(e\cdot c)\bigl(e\cdot
\check{X}^n(t)\bigr)^+\,ds\Bigr]\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq
\sqrt{\Ex_{x^n,q^n}\bigl[\bigl((2T\log n-\tau_{\kappa
',T}^n)^+\bigr)^2\bigr]}\nonumber\\
&&\qquad\quad{}\times\sqrt{
\Ex_{x^n,q^n}\Bigl[\Bigl(\sup_{0\leq t\leq2T\log n}(e\cdot
c)\bigl(e\cdot\check{X}^n(t)\bigr)^+\,ds\Bigr)^2\Bigr]}.\nonumber
\end{eqnarray}
Using Lemma \ref{lemstrongappbounds} we have that
\begin{equation}\label{eqinterim9}
\Ex
_{x^n,q^n}\Bigl[\Bigl(\sup_{0\leq t\leq2T\log n}(e\cdot c)\bigl(e\cdot
\check{X}^n(t)\bigr)^+\,ds\Bigr)^2\Bigr]\leq Cn\log^6n
\end{equation}
for some $C>0$ (that can depend on $T$). Also, since
$m\geq3$,
\[
\Pd\{\tau_{\kappa',T}^n<2T\log n\}\leq\Pd
\Bigl\{\sup_{0\leq t\leq2T\log n}|\check{X}^n(t)|> \kappa'\sqrt{n}\log
^3 n -M\sqrt{n}\Bigr\}.
\]
Choosing $\kappa'$ (and in turn
$\kappa$ large enough) we then have, using Lemma \ref
{lemstrongappbounds}, that
\begin{equation}\label{eqtauprobbound}
\Pd\{\tau_{\kappa',T}^n<2T\log n\}
\leq\frac{C}{n^2}
\end{equation}
and hence, that
\begin{equation}\label{eqinterim10}
\Ex
_{x^n,q^n}\bigl[\bigl((2T\log n-\tau_{\kappa',T}^n)^+
\bigr)^2\bigr]\leq C.
\end{equation}
Plugging (\ref{eqinterim9}) and (\ref{eqinterim10}) into (\ref
{eqinterim8}) we
then have that
\begin{equation}\label{eqafterstopbound1}
\Ex_{x^n,q^n}\biggl[\int_{\tau_{\kappa
',T}^n}^{2T\log n}
e^{-\gamma s} (e\cdot c)\bigl(e\cdot\check{X}^n(s)\bigr)^+\,ds\biggr]\leq C\log
^2 n.
\end{equation}
To conclude the proof we will show that (\ref{eqafterstop1}) follows
from our analysis thus far. Indeed,
\begin{eqnarray*}
&&\Ex[e^{-\gamma\tau_{\kappa',T}^n}\phi
_{\kappa}^n(\check{X}^n(\tau_{\kappa',T}^n))]\\
&&\qquad\leq
\Ex_{x^n,q^n}^{U}\biggl[\int_{\tau_{\kappa',T}^n}^{2T\log
n}e^{-\gamma s} \sup_{0\leq s\leq2T\log n}(e\cdot c)\bigl(e\cdot\check
{X}^n(s)\bigr)^+ \,ds\biggr].
\end{eqnarray*}
The right-hand side here is bounded by $C\log^2n$ by the same argument
that leads to (\ref{eqafterstopbound1}).
\subsection*{\texorpdfstring{Proof of Lemma \protect\ref{lemmartingales}}{Proof of Lemma 6.3}} Recall that
$\check{W}^n$ is defined by $\check
{W}_i^n(t)=M_{i,1}^n(t)-M_{i,2}^n(t)$, where
\begin{eqnarray*}
M_{i,1}^n(t)&=&\mN_i^a(\lambda_i^n t)-\lambda_i^n t,
\\
M_{i,2}^n(t)&=&
\mN_i^d\biggl(\mu_i\int_0^t \bigl( \check{X}_i\lam(s)+\nu_i
n-U^n_i(s)\bigl( e\cdot\check{X}\lam(s)\bigr)^+ \bigr) \,ds \biggr)\\
&&{}-\mu
_i\int_0^t \bigl( \check{X}_i\lam(s)-U^n_i(s)\bigl( e\cdot\check
{X}\lam(s)\bigr)^+ \bigr) \,ds.
\end{eqnarray*}
The fact that each of the processes
$M_{i,1}^n(t)$ and $M_{i,2}^n(t)$ are square integrable martingales
with respect to the filtration $(\mathcal{F}_t^n)$ follows as in
Section 3 of \cite{pang2007martingale} and specifically as in Lemma
3.2 there.
Since, with probability 1, there are no simultaneous jumps of $\mathcal
{N}_i^a$ and $\mathcal{N}_i^d$, the quadratic variation process satisfies
\begin{eqnarray*}
[\check{W}_i^n]_t&=&[M_{i,1}^n]_t+[M_{i,2}^n]_t\\
&=&\sum_{s\leq
t}(\Delta M_{i,1}^n(s))^2+\sum_{s\leq t}(\Delta M_{i,2}^n(s))^2,
\end{eqnarray*}
where the last equality follows again from Lemma 3.1 in \cite
{pang2007martingale} (see also Example~5.65 in \cite{vandervaart}).
Finally, the predictable quadratic variation process satisfies
\begin{eqnarray*}
\langle\check{W}_i^n\rangle_t&=&\langle
M_{i,1}^n\rangle
_t+\langle M_{i,2}^n\rangle_t\\&=&\lambda_i^nt + \mu_i\int_0^t
\bigl( \check{X}_i\lam(s)+\nu_i n-U^n_i(s)\bigl( e\cdot\check{X}\lam
(s)\bigr)^+ \bigr) \,ds\\&=&
\int_0^t (\sigma_i^n(\check{X}^n(s),U^n(s)))^2\,ds,
\end{eqnarray*}
where the second equality follow again follows from Lemma 3.1 in \cite
{pang2007martingale} and the last equality from the definition of
$\sigma_i^n(\cdot,\cdot)$ [see (\ref{eqnbmdefn3})]. By Theorem
3.2 in~\cite{pang2007martingale} $((\check{W}_i^n(t))^2-[\check
{W}_i^n]_t,t\geq0])$ and $((\check{W}_i^n(t))^2-[\check
{W}_i^n]_t,t\geq0)$ are both martingales with respect to
$(\mathcal{F}_t^n)$. In turn, by the optional stopping theorem so are
the processes $\mathcal{M}_i^n(\cdot)$ and $\mathcal{V}_i^n(\cdot)$
as defined in the statement of the lemma. Finally, it is easy to verify
that these are square integrable martingales using the fact the time
changes are bounded for all finite $t$.
\end{appendix}
|
{
"timestamp": "2012-03-09T02:01:47",
"yymm": "1203",
"arxiv_id": "1203.1723",
"language": "en",
"url": "https://arxiv.org/abs/1203.1723"
}
|
\section*{Introduction}\label{sec-intro}
Let $\bk$ be an algebraically closed field of characteristic zero. Let $f\in \bk[[\bdx]][z]$ be a polynomial
with coefficients in the formal power series ring $\bk[[\bdx]]:=\bk[[x_1,\cdots, x_d]]$.
If $d=1$, its zero locus $(\{f=0\},0)\subset (\bk^2,0)$ defines a germ of plane curve singularity. One of the
initial ideas to deal with germs of plane curve singularities is the use of the Newton algorithm.
Inside the algorithm, the first step is governed by the Newton polygon of $f$
and, for each compact face of the Newton polygon, one proceeds
by doing the Newton process,
and so on. This procedure is codified in a tree called \emph{Newton tree}. Many properties can be codified in the tree, for instance,
using bi-colored Newton trees one can compute the intersection multiplicity of two plane curves.
This allows to compute from the Newton tree the Milnor number of a germ since it can be expressed
in terms of intersection multiplicities. More recently for a finite codimension ideal $I$ in $\bk[[x_1,x_2]]$
its multiplicity, its {\L}ojasiewicz exponent,
and its Hilbert-Samuel multiplicity can be computed from its Newton tree, see~\cite{cnv:12}.
For $d>1$ the well-known class of hypersurface singularities which generalizes the case of curves is the class of
\emph{quasi-ordinary hypersurface singularities}.
In that class, one of the key results is a factorization theorem given by Jung-Abhyankar, see \cite{ab:55}.
One feature of quasi-ordinary singularities is that in some coordinates their Newton polyhedron is a polygonal path, i.e.
all its compact faces have dimension one. Then we can apply the Newton process as in the case of curves.
Moreover, after Newton maps, the condition to be quasi-ordinary is preserved. In \cite{aclm:09},
we have studied a generalization of quasi-ordinary singularities, called $\nu$-quasi-ordinary
by H.~Hironaka \cite[Definition 6.1]{hr:74}.
Roughly speaking for a $\nu$-quasi-ordinary hypersurface defined by a polynomial $f\in \bk[[\bdx]][z]$ we only require the upper part of the Newton polyhedron of $f$
to be a polygonal path. We apply the Newton process associated with these faces of dimension~$1$ and we iterate the process
whenever the upper part of the corresponding Newton polyhedron is a polygonal path.
As in the case of curves we encode the
Newton process in a tree, but the tree bears leaves (arrows) and fruits (black boxes). Using this tree, we describe the
condition for two
$\nu$-quasi-ordinary series $f$ and $g\in \bk[[\bdx]][z]$ to have $z$-resultant which is a monomial times a unit.
In particular, a quasi-ordinary singularity has a Newton tree with only arrows.
The first main result of this article is that a $\nu$-quasi-ordinary polynomial $f$ is a quasi-ordinary
if and only if and only if there exists a suitable system of coordinates for $f$ such that
its Newton tree has only arrow-heads decorated with $(0)$ and $(1)$ and has no black boxes, see Theorem~\ref{main2}.
This result can also be stated as follows:
\begin{quote}\em
Let $f$ be a $\nu$-quasi-ordinary polynomial. Then, $f$ is a quasi-ordinary
polynomial if and only if $f$ is reduced, $\nu$-quasi-ordinary
and the succesive transforms by Newton maps are also $\nu$-quasi-ordinary.
\end{quote}
This gives a characterization of
quasi-ordinary polynomial which does not depend on the definition of Newton trees, that is, quasi-ordinary
is equivalent to reduced and stably (by Newton maps) $\nu$-quasi-ordinary.
Theorem~\ref{main2} is proved studying how the derivative ${\partial f}/{\partial z}$ separates from~$f$ on the Newton
tree of the product.
In the case of curves this is an algebraic elementary way to recover L\^e-Michel-Weber theorem~\cite{lmw:89}.
We also give the formula to compute the discriminant of a quasi-ordinary Weierstrass polynomial $f$ in terms
of the decorations of the Newton tree of $f$, see Proposition \ref{discrimant-tree}. This formula has two
important consequences. The first is that this formula allows to compute the discriminant avoiding the use
of determinants. The second one, if one starts with a non Weierstrass polynomial, the computation
of the Weierstrass decomposition is not needed.
This is important for applications, for instance algorithmic resolutions, see \cite{b:09,brj:03,bs:00}.
In \cite{b:09} the author gives an effective algorithm for desingularization of surfaces using Jung's method.
His Algorithms~2 and~3 use as input
a monic polynomial. If the polynomial is not monic, it is a difficult task to make effective the Weierstrass preparation theorem.
Using our method and proceeding as in \cite{alr:88}
one can avoid this problem and adapt Algorithms 2 and 3 without the monic condition.
The second main result is contained in \S\ref{sec-trans-sect}
where we compare the Newton tree of a quasi-ordinary singularity and those of its curve
transversal sections. Examples~\ref{ex-duple} and~\ref{ex-duple1} show that, in general, it is not possible to recover
the Newton tree of the quasi-ordinary singularity from the Newton trees of the curve transversal sections.
It does if we know that the Newton tree of the quasi-ordinary singularity has only one arrow, see Theorem~\ref{thm-recover}.
However the global decorations which appear
on the Newton tree of the quasi-ordinary singularities are those which appear on the Newton tree
of the transversal sections. This fact plays a crucial role in the proof of
the monodromy conjecture for quasi-ordinary singularities \cite{aclm:05}.
Moreover in \cite{GPGV:11} a description
of the motivic Milnor fibre of an irreducible quasi-ordinary polynomial
is given proving that it is a topological invariant. In~\cite{gv:12}
the notion of linear Newton tree for an irreducible quasi-ordinary polynomial is introduced and some
of its properties are discussed, as normalization or semigroup.
In \S\ref{sec-nuqonewton} we recall the definitions and some properties of
$\nu$-quasi-ordinary polynomials and Newton maps. The construction of the Newton trees introduced
in \cite{aclm:09} is recalled and explained in \S\ref{sec-trees}; the notion of comparable polynomials
and the computation of resultants are also recalled in this section. In order to discuss
the unicity of Newton trees we work with the notion of P-good coordinates in \S\ref{P-good-coor-sec}, introduced
by P.~Gonz{\'a}lez-P{\'e}rez in~\cite{go:01}. The last two sections are devoted to state and prove the main
results of the paper.
\section{On $\nu$-quasi-ordinary polynomials and Newton maps}\label{sec-nuqonewton}
\medskip
\subsection{Basic facts on Newton polyhedra}
\mbox{}
We shall follow the terminology of \cite{aclm:05,aclm:09}.
The $d$-tuples will be denoted in bold letters, e.g. $\bdx:=(x_1,\dots,x_d)$, and we will use the following
notations:
\begin{itemize}
\item $\bdx^{\balpha}:=x_1^{\alpha_1}\cdot\ldots\cdot x_d^{\alpha_d}$;
\item $\mathbf{p\cdot q}:=p_1 q_1+\dots+p_d q_d$;
\item $\mathbf{p*q}:=(p_1 q_1,\dots,p_d q_d)$;
\item $\mathbf{\dfrac{p}{q}}:=\left(\dfrac{p_1}{q_1},\dots,\dfrac{p_d}{q_d}\right)$.
\end{itemize}
For Newton theory, we shall also follow the terminology of \cite{agv,kus,W:04}; note that for the terms
\emph{polyhedron} and \emph{diagram} we follow the convention in \cite{aclm:05,aclm:09,W:04};
the terms are exchanged in \cite{agv}.
Let $\mathbb{N}\subset \mathbb{R} _+$ be the sets of non-negative integers and
non-negative real numbers respectively. Let $E\subset \mathbb{N}^{d+1}$ be a set of points and $d\geq 1$.
\begin{itemize}
\smallbreak\item The \emph{Newton diagram} $\mathcal{N}_+(E) $ is defined by the convex hull in $(\mathbb{R} _+)^{d+1}$ of the set
$E+(\mathbb{R} _+)^{d+1}$.
\smallbreak\item The \emph{Newton polyhedron} $\mathcal{N}(E)$
of $E$ is defined by the union of all
compact faces of the Newton diagram of $E$.
\smallbreak\item The smallest set $E_0$ such that
$\mathcal{N}_+(E_0) =\mathcal{N}_+(E) $ is called the \emph{set of vertices} of $E$.
\smallbreak\item A diagram is called \emph{polygonal} if the maximal dimension of its compact faces is one.
\end{itemize}
\smallbreak
Let $\bk$ be an algebraically closed field of characteristic zero. We write
$f(\bdy)=f(y_1,\ldots,y_{d+1})$ for a formal power series of several variables $\bk[[\bdy]]$.
If the Taylor expansion of $f(\bdy)$
is $\sum_{\balpha\in\bn^{d+1}} c_{\balpha} \bdy^{\balpha }$,
then the \emph{support} $\supp (f)$ is defined to be $
\{\balpha\in \mathbb{N}^{d+1} \vert c_{\balpha}\neq 0\}$.
\begin{itemize}
\smallbreak\item The \emph{Newton polyhedron} $\mathcal{N}(f)$ of $f$
(resp. the \emph{Newton diagram} $\mathcal{N}_+(f) $) of $f$ is defined
by the Newton polyhedron (resp. the Newton diagram)
of the set $\supp (f)$.
\smallbreak\item If $\gamma $ is a compact face of $\mathcal{N}_+(f)$ then the
(weighted-homogeneous polynomial) $f_{\gamma }(\bdy):=
\sum _{(\balpha )\in \gamma }
c_{\balpha }\bdy^{\alpha } \in \bk[\bdy]$ is called
the \emph{polynomial associated with $\gamma $}.
\smallbreak\item If the Newton diagram $\mathcal{N}_+(f) $ is polygonal we also say that
the polyhedron $\mathcal{N}(f)$
is a \emph{monotone polygonal path}.
\end{itemize}
If $f=f_1\cdot\ldots\cdot f_s$ in $\bk[[\bdy]]$ then
$$
\mathcal{N}_+(f)=\mathcal{N}_+(f_1)+\ldots+\mathcal{N}_+(f_s),
$$
where the sum is the Minkowski sum of Newton diagrams. This implies the following result which is well-known for
experts, e.g. see \cite[Lemma~12]{gg:05}.
\begin{lemma}\label{polygonal}
If $f\in \bk[[\bdy]]$ has a Newton polyhedron which is a monotone polygonal path, then any irreducible factor of $f$
which is not associated with $y_i$,
$i=1\ldots, {d+1}$, has a Newton polyhedron which is a monotone polygonal path.
\end{lemma}
Let $\gamma $ be a 1-dimensional compact face of $\mathcal{N}(f)$. There exist two integral points $A, A_1$ in
$\mathbb{N}^{d+1}$ such that $\gamma$ is the edge $[A,A_1]$. Let $c$ denote the greatest common divisor
of all coordinates of the vector $A_1-A$ and let
$\mathbf{u}:=\frac{1}{c}(A_1-A)\in \mathbb{Z}^{d+1}$.
The number of points with integer coordinates on~$\gamma$ is $c+1$
and any of them is of the form $A_j=A+j\mathbf{u}$,
with $0\leq j\leq c$. Then
\begin{equation}\label{factor-cara}
f_{\gamma }(\bdy)=\sum_{a\in \gamma} c_a \bdy^{a }=\bdy^A(\sum_{j=0}^{c} c_{A_j}\bdy^{j u})=
\bdy^A p(f,\gamma)(\bdy^u)
\end{equation}
where $p(f,\gamma)(t)$ is the polynomial $p(f,\gamma)(t):=\sum_{j=0}^c c_{A_j}t^{j}$
of degree $c$; since $A$ and $A_1$ are vertices of $\gamma$ we have $p(f,\gamma)(0)\ne 0$.
Let us factor $p(f,\gamma)(t)$ as
\begin{equation}\label{factor-cara-1}
p(f,\gamma)(t)=a_\gamma\prod (t-\mu_i)^{m_i},\
\sum m_i=c,\
\mu_i\in \bk^*,\
\mu_i\ne \mu_j, \text{ if }i\neq j,
\text{ and }
a_\gamma\in \bk^*.
\end{equation}
Let us summarize these facts.
\begin{proposition}\label{factor-cara-2}
Let $\gamma =[A,A_1]$ be a 1-dimensional compact face of $\mathcal{N}(f)$. With the notations
of \eqref{factor-cara} and \eqref{factor-cara-1} the
polynomial $f_{\gamma }(\bdy)$ decomposes as $a_\gamma \bdy^A \prod(\bdy^u-\mu_i)^{m_i}$ in $\bk((\bdy))$.
\end{proposition}
\begin{theorem}\cite[Theorem~3]{gg:05} If $f$ in $\bk[[\bdy]]$ is irreducible and has a Newton polyhedron
$\mathcal{N}(f)$ which is a monotone polygonal path,
then the diagram $\mathcal{N}_+(f)$ has only one compact edge $\gamma$ and the polynomial $p(f, \gamma)$ has only one
root $\mu_f$ in $\bk^*$, i.e. $p(f,\gamma)=a_\gamma(t-\mu_f)^{c},$ and $a_\gamma\in \bk^*$.
\end{theorem}
\subsection{On $\nu$-quasi-ordinary polynomials }
\mbox{}
\begin{notation}\label{inicial}
Let
$f(\bdx,z):=\sum c_{\balpha ,\beta} \bdx^{\balpha }z^{\beta } \in \bk[[\bdx]][z]$,
$\balpha\in\bn^d$, $\beta\in\bn$, be a
$z$-polynomial with coefficients in the formal
power series ring $\bk[[\bdx]]$,
$\bdx:=(x_1,\cdots, x_d)$. Since the ring $\bk[[\bdx]][z]$ is factorial,
we may assume that $f(\bdx,z)=x_1^{n_1}\cdot\ldots\cdot x_d^{n_d} g(\bdx,z)$ where $g(\bdx,z)$ is regular
of order say $n\geq 0$, that is
$g(\mathbf{0},z)=a_0 z^n+a_1 z^{n+1}+\dots $, $a_0\in \bk^*$.
Applying Weierstrass preparation theorem to $g(\bdx,z)$ there exists a unit $u(\bdx,z) \in \bk[[\bdx,z]]$
and a Weierstrass polynomial $h(\bdx,z)=z^n+a_1(\bdx)z^{n-1}+\ldots+ a_{n-1}(\bdx)z+a_0(\bdx)$
with $a_i(\textbf{0})=0$ such that $g(\bdx,z)=h(\bdx,z)u(\bdx,z)$ and $u(\textbf{0},0)=a_0\in \bk^*$.
e.g. see \cite[Chapter I, p. 11]{GLS}. Note that $\mathcal{N}(g)=\mathcal{N}(h)$ and we do not need to
know explicitely~$h$ for the constructions in this paper.
\end{notation}
\begin{remark}\label{1-vertex}
Let $f(\bdx,z):=\sum c_{\balpha ,\beta} \bdx^{\balpha }z^{\beta }=
x_1^{n_1}\cdot\ldots\cdot x_d^{n_d} g(\bdx,z) \in \bk[[\bdx]][z]$
be a polynomial and assume we are as in Notation \ref{inicial}.
Its Newton polyhedron $\mathcal{N} (f)$ consists only of one vertex
$(\bdn,n):=(n_1,\ldots,n_d,n)$ if and only if $\mathcal{N}_+ (f)=(\bdn,n)+(\mathbb{R} _+)^{d+1}$.
\end{remark}
\begin{definition}\label{eliminated}
A compact face ${\gamma }$ of the Newton polyhedron $\mathcal{N}(f)$ \emph{can be eliminated} if the polynomial
$f_{\gamma }$ associated with $\gamma $ can be written as
$f_{\gamma }=\bdx^{\mathbf{m}} (z-h(\bdx))^n \in \bk[\bdx,z]$, with $h\in \bk[\bdx], n\geq 1$.
In such a case and by applying the change of variables map
$\sigma:\bk[[\bdx]][z]\to \bk[[\bdx]][z_1]$
defined
by $z\mapsto z_1=z+h(\bdx)$ to $f$ then the face $\gamma$ is eliminated in
the new Newton diagram $\mathcal{N}_+ (f\circ \sigma)$.
\end{definition}
\begin{definition}\label{inicial2}
Let $f(\bdx,z)=\sum c_{\balpha ,\beta} \bdx^{\balpha }z^{\beta }=
x_1^{n_1}\cdot\ldots\cdot x_d^{n_d} g(\bdx,z) \in \bk[[\bdx]][z]$
be a polynomial and assume we are as in Notation \ref{inicial} and denote
$A:=(\bdn,n)\in \mathbb{Q}^{d+1}$.
Let
$$
\pi_A:\mathcal{N}_+(f)\setminus A\to \mathbb{Q}^d\equiv\mathbb{Q}^{d+1}\cap\{z=0\}
$$
be the projection into $\mathbb{Q}^d
$
with centre at $A$. We define
$\mathcal{N}_{<n}(f)$ to be the set of points
in $\mathcal{N}_+(f)$
whose $z$-coordinate is smaller than $n$. A compact face ${\Gamma}$ of $\mathcal{N}_+(f)$ will be called \emph{$\nu$-proper}
if $A\in{\Gamma}$, $A\ne{\Gamma}$ and ${\Gamma}\setminus\{A\}\subset\mathcal{N}_{<n}(f)$. We define
$\mathcal{N}^0(f)$ to be the set of all compact faces of $\pi_A(\mathcal{N}_{<n}(f))$.
\end{definition}
\begin{remark}
For a $\nu$-proper face $\Gamma$ of $\mathcal{N}(f)$, $\pi_A(\Gamma)$ is a compact
face of $\mathcal{N}^0(f)$ of dimension one less than the dimension of $\Gamma$.
In particular, $\mathcal{N}^0(f)$ consists of exactly one vertex if and only if
$\mathcal{N}(f)$ has only one $\nu$-proper compact face and this face is $1$-dimensional.
\end{remark}
\begin{definition}\label{suitable}
A regular system of parameters $x_1,\ldots,x_d,z$
of the local regular ring $\bk[[\bdx,z]]$ is called a
\emph{suitable system of coordinates} for a polynomial $f \in \bk[[\bdx]][z]$
if either
\begin{itemize}
\smallbreak\item $\mathcal{N}^0(f)$ is void, or
\smallbreak\item $\mathcal{N}^0(f)$ has more than one vertex, or
\smallbreak\item The set $\mathcal{N}^0(f)$ consists of exactly one vertex and the corresponding $\nu$-proper $1$-dimensional
compact face $\Gamma$ cannot be eliminated (Definition~\ref{eliminated}).
\end{itemize}
\end{definition}
\begin{remark}\label{suitable-change}
The condition to be a suitable system of coordinates for a polynomial $f\in \bk[[\bdx]][z]$
is weaker than the conditions of \emph{good coordinates} in \cite{aclm:05} because
it involves exactly one
$\nu$-proper $1$-dimensional
compact face $\Gamma$ of the Newton diagram $\mathcal{N}_+(f)$, see also~\S\ref{P-good-coor-sec}.
\end{remark}
\begin{definition}\label{nu-quo}
Let $f(\bdx,z): \in \bk[[\bdx]][z]$
be a polynomial and assume we are as in Notation \ref{inicial} and Definition~\ref{inicial2}.
We say that $f$ is a \emph{$\nu$-quasi-ordinary} polynomial if
\begin{itemize}
\smallbreak\item $\mathcal{N}(f)$ has only one $\nu$-proper face $\Gamma_1$,
\smallbreak\item $\Gamma_1$ is a $1$-dimensional face and
\smallbreak\item $\Gamma_1$ cannot be eliminated.
\end{itemize}
The polynomial $f_{\Gamma_1 }$ is called the \emph{initial form} of $f$.
\end{definition}
The class of \emph{$\nu$-quasi-ordinary} singularities was introduced by
H. Hironaka in \cite{hr:74} where he proved that
quasi-ordinary polynomials are indeed $\nu$-quasi-ordinary polynomials.
In comparison with the condition to be a quasi-ordinary polynomial,
$\nu$-quasi-ordinary is a very mild condition.
However, $\nu$-quasi-ordinary have interesting properties and some of them will be discussed here.
We consider $f$ to be a $\nu$-quasi-ordinary polynomial with its $\nu$-proper face $\Gamma_1$
and keep the above notations.
The initial form $f_{\Gamma_1 }$ of $f$ can be written as follows:
\begin{equation}
\label{eq-fgamma}
f_{\Gamma_1 }=
a_{\Gamma_1 }\bdx^{\mathbf{n}}z^{n_{d+1}}\prod _{j=1}^{k}(z^p-\mu _j
\bdx^{\mathbf{q}})^{m_j},
\end{equation}
where factors are irreducible in $\bk[[\bdx]][z]$,
i.e. $\gcd (\mathbf{q},p):=
\gcd (q_1,\dots , q_d,p)=1$, $\mu_j\in \bk^*$ with $\mu_j\ne \mu_i$ if $i\neq j$, and $a_{\Gamma_1 }\in \bk^*$
(see Proposition~\ref{factor-cara-2}).
\begin{definition}
A polynomial $f(\bdx,z)\in \bk[[\bdx]][z]$ is called \emph{elementary}
if its Newton polyhedron $\mathcal{N}(f)$ consists of
only one compact face which is a line segment
$[(\mathbf{0},n),(\mathbf{r},0)]\subset \mathbb{R}^{d+1}$.
More generally, if
$\Gamma $ is a compact one-dimensional face of the Newton diagram $\mathcal{N}_+(f)$ of a
polynomial $f(\bdx,z)\in \bk[[\bdx]][z]$,
we say that the polynomial $g$ is \emph{$\Gamma $-elementary} if its Newton polyhedron
$\mathcal{N}(g)$ consists of one compact face $\widetilde \Gamma$
which is a translation of the face $\Gamma $.
We denote the initial form of $g$ by $\In (g):=g_{\widetilde \Gamma }\in \bk[\bdx,z]$.
\end{definition}
In \cite[Theorem 1.5]{aclm:09} the following factorization theorem was proved.
\begin{fthm}\label{teorema1}
Let $f\in \bk[[\bdx]][z]$ be a $\nu$-quasi-ordinary polynomial with its $\nu$-proper face $\Gamma_1 $
such that $f_{\Gamma_1}$ is as in{\rm~\eqref{eq-fgamma}}.
Then there exist $k$ different
$\Gamma_1$-elementary polynomials $g_{\Gamma_1,j}\in \bk[[\bdx]][z]$, for $1\leq j \leq k$, which divide~$f$ and such that
$\In (g_{\Gamma_1,j})=
(z^p-\mu _j \bdx^{\mathbf{q}})^{m_j}$, that is
\begin{equation}\label{teorema1-factor}
f=h G^{\Gamma_1}, \,\, \textit{with }\,\, G^{\Gamma_1}:=\prod_{j=1}^k g_{\Gamma_1,j},\,\, \textit{ for some }\,\, h\in \bk[[\bdx]][z].
\end{equation}
\end{fthm}
Factorization Theorem~\ref{teorema1} has the following consequences.
\begin{cor}
Let $f$ be an irreducible polynomial which is $\nu$-quasi-ordinary.
Then it is elementary and $\In (f)=f_0^k$,
where $f_0 \in \bk[\bdx,z]$ is irreducible.
\end{cor}
\begin{cor}
Let $f$ be an elementary polynomial such that
$\In (f)=\psi_1\cdot\ldots\cdot \psi_r$ is a factorization of $\In (f)$ where the factors
are pairwise coprime. Then we can decompose $f=f_1\cdot\ldots\cdot f_r$ where, for all $i=1,\dots, r$, the series
$f_i$ is an elementary polynomial
such that $\In(f_i)=\psi _i$.
\end{cor}
\begin{cor}\label{cor-factor-gamma}
Let $f$ be a $\nu$-quasi-ordinary polynomial.
Then
$f=\bdx^{a}z^b f^{\Gamma_1 } g$
where $f^{\Gamma_1}$ is $\Gamma_1$-elementary, and $\In (f^{\Gamma_1})=f_{\Gamma_1}$.
\end{cor}
\subsection{On Newton maps and the Newton proccess }\label{sec-newton}
\mbox{}
We need to introduce more notations in order to define \emph{Newton maps}. For every $1\leq i\leq d$, let
\begin{equation}
\label{eq-gcd}
c_i:=\gcd (p,q_i),\quad
p_i:=\frac{p}{c_i},\text{ and } q'_i:=\frac{q_i}{c_i}.
\end{equation}
We will consider the following three maps.
The first map depends both on the face $\Gamma_1$ and on the root $\mu_j$, see \eqref{eq-fgamma}.
Let
$(\mathbf{u},u) \in \bn^{d+1}$ be integers such that
$1+\mathbf{u}\cdot\mathbf{q} =u p$. Let $\delta _{\Gamma_1, j}$ be the map
$$
\begin{matrix}
& \delta _{\Gamma_1, j}:&\bk[[\bdx]][z]& \longrightarrow & \bk[[\bdx]][z]\hfill\\
&&h(\bdx,z)&\mapsto &h(\mu_j^{u_1} x_1,\dots ,\mu_j^{u_d} x_d,
z).
\end{matrix}
$$
Then
$$
\delta _{\Gamma_1, j}(z^p-\mu _j
\bdx^{\mathbf{q}})=z^p-\mu_j^{up}
\bdx^{\mathbf{q}}.
$$
The second map $\epsilon _{\Gamma_1}$ depends only on the face $\Gamma_1$:
$$
\begin{matrix}
&\epsilon _{\Gamma_1 }:&\bk[[\bdx]][z]& \longrightarrow & \bk[[\bdy]][z_1]\hfill\\
&&h(\bdx,z)&\mapsto &h(y_1^{p_1},\dots ,y_d^{p_d},
\bdy^{\mathbf{q'}}z_1).
\end{matrix}
$$
It is easily seen that
$$
(\epsilon _{\Gamma_1}\circ\delta _{\Gamma_1, j})(z^p-\mu _j
\bdx^{\mathbf{q}})
=
\bdy^{p \mathbf{q'}} (z_1^p-\mu _j^{u p}).
$$
Now the third map $\tau_j$ will depend on the root $\mu _j$:
$$
\begin{matrix}
&\tau _j:&\bk[[\bdy]][z_1]& \longrightarrow & \bk[[\bdy]][z_2]\hfill\\
&&h(\bdy,z_1)&\mapsto &h(\bdy,z_2+\mu_j^u).
\end{matrix}
$$
Note that:
$$
(\tau _j\circ\epsilon _{\Gamma_1}\circ\delta _{\Gamma_1, j})(z^p-\mu _j \bdx^{\mathbf{q}})
=
\bdy^{\mathbf{p*q}}((z_2+\mu_j^u)^p-\mu _j^{u p})
$$
and the last factor is of order one in $z_2$.
Note that $p_i q_i= p q'_i$ for $1\leq i\leq d$, i.e, $\mathbf{p*q}=p\mathbf{q'}$.
\begin{definition}\label{newton-map}
The \emph{Newton map} $\sigma _{\Gamma_1 ,j}$ associated with $\Gamma_1$ and the root $\mu_j$
is the composition map $
\sigma _{\Gamma_1 ,j}:= \tau _j\circ\epsilon _{\Gamma_1}\circ\delta _{\Gamma_1, j}:
\bk[[\bdx]][z] \to \bk[[\bdy]][z_2]$ .
\end{definition}
Also the following statement was proved in \cite[Lemma 2.2]{aclm:09}.
\begin{lemma}\label{total-transf}
After the Newton map $\sigma _{\Gamma_1, j}$, the total transform $f_{\Gamma_1, j}:=\sigma _{\Gamma_1, j}(f)\in \bk[[\bdy]][z_2]$
of the polynomial $f$ can be written as
$$
f_{\Gamma_1, j}(\bdy,z_2)=
\bdy^{m\mathbf{p*q}+\mathbf{n*p}} f_{1,\Gamma_1, j}(\bdy,z_2 ),
$$
where $f_{1,\Gamma_1, j}(\mathbf{0},z)$ is regular
of order $m_j$, and $m:=\sum_{j=1}^k m_j$.
Moreover by chain rule $$\frac{\partial(f\circ \sigma _{\Gamma_1, j})}{\partial z_2}(\bdy,z_2 ) =
\bdy^\mathbf{q'} \left(\frac{\partial f}{\partial z}\circ
\sigma _{\Gamma_1, j}\right)(\bdy,z_2 ).$$
\end{lemma}
\subsubsection{Newton's process associated with the $\nu$-proper face $\Gamma_1 $}\label{new-pro-primera}
\mbox{}
\vspace{.5cm}
We can perform a change of variables of the type $z\mapsto z+h(\bdx)$ in order to have
a suitable system of coordinates for $f_{1,\Gamma_1, j}$; if $f_{1,\Gamma_1, j}$ is again $\nu$-quasi-ordinary
one can iterate the process until one gets either a monomial times a unit or a non $\nu$-quasi-ordinary polynomial.
This process is called Newton's process; note that $m_j<n$ because the face $\Gamma_1$ cannot be eliminated.
\subsubsection{Factorization and the Newton process associated with other compact 1-dimensional faces of the
Newton polyhedron}\label{new-pro-otras}
\mbox{}
\vspace{.5cm}
Since $f$ is a $\nu$-quasi-ordinary polynomial with $\nu$-proper face $\Gamma_1 $
then $\Gamma_1$ is a
$1$-dimensional face of its Newton diagram $\mathcal{N}_+(f)$. We
assume that $\Gamma_1$ is the segment $[A,A_1]$.
If the $z$-coordinate of $A_1$ is $n^1>0$, we denote
by $\mathcal{N}_{<n^1}(f)$ the set of points
in $\mathcal{N}_+(f)$
whose $z$-coordinate is smaller than $n^1$.
Let $\pi _{A_1}:\mathcal{N}_{<n^1}(f)\setminus A_1 \to \mathbb{Q}^d $ be the projection into with center $A_1$
and let
$\mathcal{N}^{0,1}(f)$ be the convex hull of the image by $\pi_{A_1}$ of $\mathcal{N}_{<n^1}(f)$.
If $\mathcal{N} ^{0,1}(f)$ has only one vertex then there is another face $\Gamma_2$
of the Newton diagram $\mathcal{N_+}(f)$ which is of dimension $1$.
We go further on this construction with $\Gamma_1=[A,A_1],\Gamma_2=[A_1,A_2],\cdots, \Gamma_s=[A_s,A_{s+1}]$ until
one of the following cases arises:
\begin{enumerate}
\enet{\rm(NW\arabic{enumi})}
\item\label{nw1} The $z$-coordinate $n^{s+1}$ of $A_{s+1}$ is zero.
\item\label{nw2} $\mathcal{N}^{0,{s+1}}(f)=\emptyset$.
\item\label{nw3} $\mathcal{N}^{0,{s+1}}(f)$ has more than one vertex.
\end{enumerate}
Moreover $\Gamma_1 \cup \Gamma_2 \cup \cdots \cup \Gamma_{s}$ is a
\emph{monotone polygonal path} in $\mathcal{N}(f)$.
\begin{lemma}
The Newton polyhedron of $f$ is not a
\emph{monotone polygonal path} if and only if $\mathcal{N}^{0,{s+1}}(f)$ has more than one vertex, i.e., \ref{nw3} arises.
\end{lemma}
For every edge $\Gamma_\ell$ of $ \Gamma_2, \cdots, \Gamma_{s}$,
the initial form $f_{\Gamma_\ell }$ of $f$ can be written as in (\ref{eq-fgamma})
as follows:
\begin{equation}
\label{eq-fgamma-ell}
f_{\Gamma_\ell }=
a_{\Gamma_\ell }\bdx^{\mathbf{n^\ell}}z^{n^\ell_{d+1}}\prod _{j=1}^{k^\ell}(z^{p_\ell}-\mu _j^\ell
\bdx^{\mathbf{q^\ell}})^{m_j^\ell},
\end{equation}
the factors being irreducible in $\bk[[\bdx]][z]$,
i.e. $\gcd (
\mathbf{q^\ell},p_\ell)=1$, $\mu_j^\ell\in \bk^*$ with $\mu_j^\ell\ne \mu_i^\ell$ and $a_{\Gamma_\ell }\in \bk^*$.
For each root $\mu _j^\ell$
of its face polynomial $f_{\Gamma_\ell }$ one applies the corresponding Newton map $\sigma _{\Gamma_\ell, \mu _j^\ell}$.
At each step, we encode the information given by the corresponding Newton diagram.
The process stops because the $z$-degree decreases since we are in a suitable system of coordinates.
In next section Newton trees of a polynomial $f(\bdx,z)\in \bk[[\bdx]][z]$ are constructed by recursion
on the number of steps of subsections~\S\ref{new-pro-primera} and~\S\ref{new-pro-otras}.
One can also apply recursively Factorization Theorem \ref{teorema1}
to $h$, see \eqref{teorema1-factor}, to get $k^\ell$ different
$\Gamma_\ell$-elementary polynomials $g_{\Gamma_\ell,j}\in \bk[[\bdx]][z]$, for $1\leq j \leq k^\ell$,
such that $G^{\Gamma_\ell}:=\prod_{j=1}^{k^\ell} g_{\Gamma_\ell,j}$
divides~$f$, that is
\begin{equation}\label{f-factor}
f=H(\bdx,z)\prod_{t=1}^s G^{\Gamma_t}, \,\,\, \text{ for some }\,\, H\in \bk[[\bdx]][z].
\end{equation}
\section{The Newton tree of a polynomial}\label{sec-trees}
\subsection{Construction of the Newton tree}\label{constr-trees}
\mbox{}
Let $f(\bdx,z)\in \bk[[\bdx]][z]$ be a polynomial
and we assume that we are as in Notation \ref{inicial}. In this section we associate with
$f(\bdx,z)$ a tree $\mathcal{T}(f)$ called
\emph{Newton tree of $f$} because its first steps are built using both its Newton diagram $\mathcal{N_+} (f)$ and
the set $\mathcal{N}^0(f)$ of compact faces of $\pi(\mathcal{N}_{<n}(f))$. Further steps will be
based on the Newton
process associated with $f$ (see subsections \ref{new-pro-primera} and \ref{new-pro-otras}).
For a given~$f$, we are going to associate a tree $\mathcal{T}_\mathcal{N}(f)$,
called \emph{vertical tree}.
First, if $f$ is not in a suitable system of coordinates (see Definition~\ref{suitable}),
we perform a change of variables such that it is the case; in order to simplify
the notations we denote again by~$f$ the resulting polynomial. We keep the notations
of Definition~\ref{inicial2}. The tree $\mathcal{T}_\mathcal{N}(f)$ is
built using its Newton diagram $\mathcal{N}_+ (f)$.
We distinguish three cases.
\begin{caso0}\label{caso1a}
The Newton polyhedron $\mathcal{N} (f)$ consists only in one vertex (see Remark \ref{1-vertex}).
Then the Newton tree $\mathcal{T}_{\mathcal{N}}(f)$ of $f$ is given in Figure~\ref{Q01}.
\end{caso0}
\begin{figure}[ht]
\centering
\subfigure[]{
\includegraphics[scale=1]{Q01}
\label{Q01}
}
\hfil
\subfigure[]{
\includegraphics[scale=1]{Q02}
\label{Q02}
}
\caption{}
\label{fig0}
\end{figure}
\begin{caso0}\label{caso2a}
The set $\mathcal{N}^0(f)$ of all compact faces of $\pi(\mathcal{N}_{<n}(f))$ has more than one vertex.
Then the Newton tree $\mathcal{T}_{\mathcal{N}}(f)$ of $f$ is given in Figure~\ref{Q02}.
\end{caso0}
\begin{caso0}\label{caso3a}
The set $\mathcal{N}^0(f)$ has exactly one
vertex.
Since we are in a suitable system of coordinates for $f$,
the face $\Gamma_1$ cannot be eliminated,
(i.e. $f$ is a $\nu$-quasi-ordinary polynomial).
In such a case we are as in subsection~\ref{new-pro-otras}, and there exist
$s$ compact edges $\Gamma_1:=[A,A_1],\Gamma_2:=[A_1,A_2],\cdots, \Gamma_s:=[A_s,A_{s+1}]$
of the Newton polyhedron $\mathcal{N}(f)$
until one of the cases~\ref{nw1},~\ref{nw2} or~\ref{nw3} happen.
Furthermore $\Gamma_1 \cup \Gamma_2 \cup \cdots \cup \Gamma_{s}$ is a
\emph{monotone polygonal path} in $\mathcal{N}(f)$.
With this monotone polygonal path we associate a decorated \textbf{vertical} graph $\mathcal{T}_{\mathcal{N}}(f)$
(which depends on $\mathcal{N}(f)$ in a suitable system of
coordinates for $f$). With each compact 1-dimensional face $\Gamma_\ell$ of the polygonal path,
we associate a vertex $v_\ell$. If two compact faces intersect at one point we draw a vertical edge from one vertex to the other.
Thus these vertices are drawn on a vertical line by the increasing order of the slopes, i.e.
$v_1, v_2,\ldots, v_s$
from above to below in order. Decorations of this \textbf{vertical line} $\mathcal{T}_{\mathcal{N}}(f)$ are as follows:
\begin{itemize}
\item On the top of the vertical line we add an arrow-head decorated with $\bdn:=(n_1,\cdots,n_d)$.
\item On the bottom of the vertical line, we add:
\begin{itemize}
\item an arrow-head decorated with $(n^{s+1})$ in cases~\ref{nw1} ($n^{s+1}=0$) and~\ref{nw2};
\item a black box decorated with $(n^{s+1})$ in case~\ref{nw3}.
\end{itemize}
\item for an edge $\Gamma_\ell$ which is defined in
coordinates $(\balpha,\beta):=(\alpha_1,\ldots,\alpha_d,\beta)$ of $\mathbb{R}^{d+1}$ by the intersection of $d$ hyperplanes of equations
\begin{equation}\label{eq-edge}
p_\ell \alpha _k+q_k^\ell \beta =N_k^\ell,\,\qquad 1\leq k\leq d,\,\qquad \gcd (\mathbf{q}^\ell,p_\ell)=1,
\end{equation}
the corresponding vertex $v_\ell$ support the following decorations:
\begin{itemize}
\item The vertex itself is decorated with $((N_1^\ell,\cdots, N_d^\ell))$.
\item The lower edge is decorated near $v_\ell$ with $p_\ell$.
\item The upper edge is decorated near $v_\ell$ with $(q_1^\ell,\cdots,q_d^\ell)$.
\end{itemize}
\end{itemize}
\end{caso0}
We describe now the construction of $\mathcal{T}(f)$. Recall that we assume
that $f$ is in suitable coordinates.
\begin{paso}
If $f$ is in either Case~\ref{caso1a} or~\ref{caso2a} then $\mathcal{T}(f):=\mathcal{T}_{\mathcal{N}}(f)$.
If $f$ is in Case~\ref{caso3a} we continue the process.
\end{paso}
\begin{paso}
For every 1-dimensional face $f_{\Gamma_\ell }$ and for each root $\mu _j^\ell$
of its face polynomial $f_{\Gamma_\ell }$ one applies
the corresponding Newton map $\sigma _{\Gamma_\ell, \mu _j^\ell}$
to $f$ and get a polynomial $f_{\Gamma_\ell, j}$;
we perform a change of coordinates to be in suitable coordinates.
\end{paso}
\begin{paso}\label{paso3}
The tree $\mathcal{T}(f)$ will be obtained by gluing in a suitable way the
tree $\mathcal{T}_{\mathcal{N}}(f)$ and $\mathcal{T}(f_{\Gamma_\ell, j})$
which may be assumed constructed recursively.
The tree $\mathcal{T}(f_{\Gamma_\ell, j})$ will be attached to the vertex $v_\ell$
of $\mathcal{T}_{\mathcal{N}}(f)$ by a \textbf{horizontal edge} which links $v_\ell$ to the top vertex of
$\mathcal{T}_{\mathcal{N}}(f_{\Gamma_\ell, j})$ where the top arrow has been deleted.
\end{paso}
Now to decorate the tree $\mathcal{T}(f)$ we need
more definitions. Let $v$ be a vertex on $\mathcal{T}(f)$. If $v$ is on the first left vertical line,
we say that $v$ has no preceding vertex.
If $v$ is not on the first left vertical line, the vertical line on which $v$ lies,
is linked by an horizontal edge ending on a vertex $v_1$, to a vertical line.
Then $v_1$ is said to be the preceding vertex of $v$. Note that the path between $v_1$ and $v$ can have many vertical edges, but has exactly one horizontal edge. Now denote by $\mathcal{S}(v)=\{v_i,v_{i-1},\cdots, v_1,v_0=v\}$ where $v_j$ is the preceding vertex of $v_{j-1}$ for $j=1,\cdots,i$ and $v_i$ has no preceding vertex. We say that $\mathcal{S}(v) \setminus \{v\}$ is the set of preceding vertices of $v$. {Now in the contruction of the Newton tree $\mathcal{T}(f)$, we glue on
$\mathcal{T}_{\mathcal{N}}(f)$ at $v_i$ a tree $\mathcal{T}(f_{\Gamma_i, l})$
where $\mathcal{S}(v)=\{v_{i-1},\cdots, v_1,v_0=v\}$.
Assume the decorations of the edges attached to $v_i$ on $\mathcal{T}_{\mathcal{N}}(f)$ are
$((q_1^i,\cdots, q_d^i),p_i)$.
We are going to decorate the edges attached to $v_0$ in $\mathcal{T}(f)$ with $((Q_1^0,\cdots,Q_d^0),p_0)$
following these rules:
\begin{itemize}
\item The decoration $p_0$ coincides with the corresponding decoration in the vertical tree containing~$v_0$.
\item Let us assume that the decorations of $v_j$ on $\mathcal{T}(f_{\Gamma_i, l})$
are $((Q_1^{j,i},\cdots,Q_d^{j,i}),p_j)$ for $j=0,\cdots,i-1$.
Then
\begin{equation}\label{eq-newton-tree-Q}
Q_k^0:=\frac{p_i q_k^{i} p_{i-1}^2\cdot\ldots\cdot p_1^2 p_0}
{\gcd(p_i,q_k^i)\cdot\gcd(p_{i-1},Q_k^{i-1,i})\cdot\ldots\cdot\gcd(p_1,Q_k^{1,i})}+Q_k^{0,i},\,\,\, k\in \{1,\cdots,d\}.
\end{equation}
Note that, in particular, $Q_k^i=q_k^i$.
\end{itemize}
It is useful to have another way for computing these decorations, see~\cite[(5.1)]{aclm:09}. Let us assume with the above notations that
the decorations $((Q_1^j,\cdots,Q_d^j),p_j)$ for $\mathcal{T}(f)$ have been defined for $v_j$, $j=0,1,\dots,i-1$.
Recall that $Q_k^i=q_k^i$. Let us denote $((q_1^0,\cdots, q_d^0),p_0)$ the decorations of $v_0$ in its vertical tree.
\begin{lemma}\label{lema-newton-tree-Q}
With the above notations
$$
Q_k^{0}=\frac{p_1 Q_k^1 p_0}{\gcd(p_1,Q_k^1)}+q_k^{0}.
$$
\end{lemma}
\begin{proof}
Let us assume first that $i=1$. In this case $Q_k^{0,1}=q_k^0$ since
$\mathcal{T}(f_{\Gamma_1, l})=\mathcal{T}_{\mathcal{N}}(f_{\Gamma_1, l})$ and $Q_k^1=q_k^1$
since $i=1$. The result follows from a direct substitution in~\eqref{eq-newton-tree-Q}.
If $i>1$ then, by induction hypothesis, we have
$
Q_k^{0,i}=\frac{p_1 Q_k^{1,i} p_0}{\gcd(p_1,Q_k^{1,i})}+q_k^{0}
$.
Hence,
\begin{equation*}
\begin{split}
Q_k^0=&\frac{p_i q_k^{i} p_{i-1}^2\cdot\ldots\cdot p_1^2 p_0}
{\gcd(p_i,q_k^i)\cdot\gcd(p_{i-1},Q_k^{i-1,i})\cdot\ldots\cdot\gcd(p_1,Q_k^{1,i})}+Q_k^{0,i}\\
=& \frac{p_i q_k^{i} p_{i-1}^2\cdot\ldots\cdot p_1^2 p_0}
{\gcd(p_i,q_k^i)\cdot\gcd(p_{i-1},Q_k^{i-1,i})\cdot\ldots\cdot\gcd(p_1,Q_k^{1,i})}
+\frac{p_1 Q_k^{1,i} p_0}{\gcd(p_1,Q_k^{1,i})}+q_k^{0}\\
=&\left(\frac{p_i q_k^{i} p_{i-1}^2\cdot\ldots\cdot p_2^2 p_1}
{\gcd(p_i,q_k^i)\cdot\gcd(p_{i-1},Q_k^{i-1,i})\cdot\ldots\cdot\gcd(p_1,Q_k^{1,i})}
+Q_k^{1,i}\right)\frac{p_1 p_0}{\gcd(p_1,Q_k^{1,i})}+q_k^{0}.
\end{split}
\end{equation*}
Using \eqref{eq-newton-tree-Q}
$$
Q_k^1=\frac{p_i q_k^{i} p_{i-1}^2\cdot\ldots\cdot p_2^2 p_1}
{\gcd(p_i,q_k^i)\cdot\gcd(p_{i-1},Q_k^{i-1,i})\cdot\ldots\cdot\gcd(p_1,Q_k^{1,i})}
+Q_k^{1,i}
$$
and the result follows.
\end{proof}
\begin{remark}\label{vert-hori}
By construction the Newton tree $\mathcal{T}(f)$ has \textbf{vertical parts} and \textbf{horizontal parts}:
vertical parts correspond to Newton diagrams of total transforms by Newton maps and horizontal edges
are edges used for connecting vertical parts.
\end{remark}
\begin{definition}\label{ends}
An \emph{end} of the Newton tree $\mathcal{T}(f)$ is either an arrow-head or a black box. The arrow-heads or black boxes
decorated with~$(0)$ will be called \emph{dead ends}.
\end{definition}
\begin{remark}\label{end-arrows}
By construction black boxes in the Newton tree $\mathcal{T}(f)$ appear if and only if
we reach at some step Case~\ref{caso2a}.
This means that the Newton tree $\mathcal{T}(f)$ ends with arrow-heads if and only if
we never reach Case~\ref{caso2a}.
\end{remark}
\begin{definition}\label{gl-data}
For every vertex $v_j$ of the Newton tree $\mathcal{T}(f)$, the
\emph{global numerical data} of the vertex $v_j$ is
$(\mathbf{N}_{v_j},p_j)$ where $\mathbf{N}_{v_j}:=(N^j_{1},\cdots ,N^j_{d})$.
\end{definition}
\begin{remark}
Global numerical data
of the first vertex satisfy the following property. If
$\sigma$ is a Newton map associated with this vertex, then
$$
\sigma (f)=\bdy^{\mathbf{\frac{N}{c}}}
f_1(\bdy, z_1)
$$
where no $y_i$ divides $f_1$ and $c_i=\gcd(p,q_i)$ as in \eqref{eq-gcd}.
\end{remark}
\begin{definition}\label{lc-data}
The numerical data
$((Q^j_{1},\cdots,Q^j_{d}),p_j)$
associated with each vertex $v_j$ of the Newton tree $\mathcal{T}(f)$
are called the \emph{local numerical data} of the vertex $v_j$.
The \emph{gcd of the vertex}~$v_j$ is $\mathbf{c}:=(\gcd(Q^j_{1},p_j),\dots,\gcd(Q^j_{d},p_j))$.
\end{definition}
\begin{remark}\label{rem-growth}
It is easy to show that,
if $((Q_1,\cdots,Q_{d}),p)$ are the local numerical data
of a vertex~$v$ and $((Q'_{1},\cdots,Q'_{d}),p_j)$ are the local numerical data
of a vertex $v'$ such that $v$ is the preceding vertex of $v'$.
Then, applying Lemma~\ref{lema-newton-tree-Q}, the following strict inequalities hold:
$$
Q'_i>\frac{p Q_i }{\gcd(p,Q_i)} p_j, \,\,\text{for}\,\,
i=1,\cdots, d, \,\,\text{and}\,\, j=1,\cdots, r.
$$
This condition is called \emph{the growth condition on the local numerical data}.
\end{remark}
\begin{remark}
In the case for $d=1$, and if we forget about vertical and horizontal edges the Newton trees
decorated with local numerical data are the Eisenbud-Neumann diagrams of the corresponding germ defined in \cite{en:85}.
\end{remark}
\begin{definition}
The \emph{valency} of a vertex in a Newton tree is the number of edges attached to the vertex.
\end{definition}
\subsection{Comparable polynomials and coloured Newton trees}
\mbox{}
Let $f(\bdx,z), g(\bdx,z)\in \bk[[\bdx]][z]$ be polynomials.
We assume that we are as in Notation~\ref{inicial} and Definition~\ref{inicial2}.
\begin{definition}\label{def-comp}
Two polynomials $f$ and $g$ in $\bk[[\bdx]][z]$ are called \emph{comparable} if
their resultant $\res_z(f,g)(\bdx)$ of
$f$ and $g$ with respect to $z$ is equal to a monomial times a unit, that is,
$$
\res_z(f,g)(\bdx)=\bdx^\bdn u(\bdx), \,\,\textrm{ with }\,\, \,\,\bdn\in{\mathbb{N}}^d,\,\,u(\bdx)\in \bk[[\bdx]], \textrm{ and }
\,\, u(\textbf{0} )\ne 0.
$$
\end{definition}
\subsubsection{Coloured Newton trees}
\mbox{}
\vspace{.5cm}
\emph{Coloured Newton trees} are associated with the product of two
polynomials $f(\bdx,z), g(\bdx,z)\in \bk[[\bdx]][z]$. We assume $fg$ is in suitable coordinates. Take the Newton tree $\mathcal{T}(fg)$
of the product $f g=f(\bdx,z)\cdot g(\bdx,z)$
and add two colours to it, say red and blue.
Blue colour is associated with $f$ and red colour with $g$.
The coloured Newton tree $\mathcal{T}(fg)$ can have blue parts, red parts and blue-red parts.
\begin{definition}\label{coloured}
We consider two polynomials
$f(\bdx,z):=\bdx^\bdn f_1(\bdx,z),g(\bdx,z):=\bdx^\bdm g_1(\bdx,z)\in \bk[[\bdx]][z]$ such that
$f_1(\bdx,z)$ and $g_1(\bdx,z)$ are regular
of order say $m,n\geq 0$. In such a case the product
$$
f_1 g_1(\textbf{0},z)=a_0z^{m+n}+\cdots
$$
satisfies $a_0 \neq 0$. In this case $A:=(\mathbf{0},m+n)\in \mathcal {N}_+ (f _1g_1)$ and
let $\pi$ be the projection into
$\bq ^d$ with centre $A$ as before and we consider $\mathcal {N}_0 (f_1 g_1)$ as in Definition~\ref{inicial2}.
We consider the Newton diagram
$\mathcal {N}_+(f g)=\mathcal {N}_+(\bdx^{\bdn+\bdm})+\mathcal {N}_+(f_1 g_1)$
which is used to construct the first vertical part $\mathcal{T}_\mathcal{N}(fg)$ of the Newton tree $\mathcal{T}(fg)$
of $f g$. Three possible cases may arise either
\begin{enumerate}
\item $\mathcal{T}_\mathcal{N}(fg)$ is
blue coloured if $\deg g_1(\textbf{0},z)=0$ or,
\item $\mathcal{T}_\mathcal{N}(fg)$ is
red coloured if $\deg f_1(\textbf{0},z)=0$ or,
\item $\mathcal{T}_\mathcal{N}(fg)$ is bi-coloured blue-red otherwise.
\end{enumerate}
We apply the same rule for every steps of the Newton process.
In particular every vertical line in the Newton tree $\mathcal{T}(fg)$ of $f g$ has the same (bi)color.
Bicoloured vertices of the bicoloured Newton tree $\mathcal{T}(fg)$ will be called \emph{common vertices} of $f$ and $g$.
\end{definition}
\begin{example}
The bi-coloured Newton tree $\mathcal{T}(fg)$ of $f g$ where $f=z^2-x^3$ and $g=z^3-x^2$ is as in Figure~\ref{QOnov1} where all vertical lines are bi-coloured, the above horizontal line is red and the below one is blue.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1]{QOnov1}
\caption{}
\label{QOnov1}
\end{center}
\end{figure}
\end{example}
\begin{definition}\label{separated}
Two polynomials $f$ and $g$ in $\bk[[\bdx]][z]$
are called \emph{separated} if there exists a suitable system of coordinates for $fg$ such that all ends which are not dead-ends of the bi-coloured Newton tree
$\mathcal{T}(fg)$ are either red coloured or blue coloured, see Definition~\ref{ends}.
\end{definition}
\begin{remark}
If $d=1$, $f$ and $g$ are separated if they do not share a common component.
\end{remark}
\begin{definition}\label{separated-vertex}
Let $f,g\in\bk[[\bdx]][z]$ such that the system of coordinates is suitable for $fg$.
Let $v$ a vertex of the Newton tree
$\mathcal{T}(fg)$ corresponding to an edge $\Gamma$. Let $\hat {f}_\Gamma$ obtained from $f_\Gamma$ by taking away the powers of $x_i$ and $z$.
We say that $f$ and $g$ \emph{separate at $v$} if $\gcd(\hat{f}_\Gamma,\hat{g}_\Gamma)$ is not equal to
either $\hat{f}_\Gamma$ or $\hat{g}_\Gamma$.
The \emph{order of separation} of $f$ at $v$ is the
$z$-degree of $\frac{\hat{f}_\Gamma}{\gcd(\hat{f}_\Gamma,\hat{g}_\Gamma)}$.
\end{definition}
\begin{example}
We illustrate the two kinds of separation of $f$ and $g$. In Figure~\ref{QO44nna}
after two Newton maps the total transform of $fg$, see Lemma~\ref{total-transf}, has
a Newton polyhedron with two different compact 1-dimensional faces
and each of them corresponds either to (the total transform of) $f$ and the other to (the total transform of)
$g$. In Figure~\ref{QO45nna} they have different polynomials on the same
face, see \eqref{eq-fgamma}.
\begin{figure}[ht]
\centering
\subfigure[]{
\includegraphics[scale=1]{QO44nna}
\label{QO44nna}
}
\hfil
\subfigure[]{
\includegraphics[scale=1]{QO45nna}
\label{QO45nna}
}
\caption{}
\label{fig4}
\end{figure}
\end{example}
\begin{remark}\label{vertical-separated}
If two polynomials $f,g\in\bk[[\bdx]][z]$
are \emph{separated} then there is at least a vertical line in the bi-coloured Newton tree $\mathcal{T}(fg)$
which corresponds
to a Newton polygon of a step of its Newton process, after which its Newton tree $\mathcal{T}(fg)$ is not bi-coloured,
e.g see Figures~\ref{QOnov1},~\ref{QO44nna} or~\ref{QO45nna}.
\end{remark}
\begin{remark}
Let $f,g\in\bk[[\bdx]][z]$ be two polynomials. Assume there exists a suitable system of coordinates of each of them. For a common vertex of the bicoloured Newton tree $\mathcal{T}(fg)$
its local numerical data is the same as its local numerical data as vertex of $\mathcal{T}(f)$ and the same as
its local numerical data as vertex of $\mathcal{T}(g)$.
\end{remark}
Next theorem is one of the main results in~\cite{aclm:09}.
\begin{theorem}[{\cite[Theorem 1.6]{aclm:09}}]\label{thm-sepcom}
If two polynomials $f$ and $g$ in $\bk[[\bdx]][z]$ are separated
then they are comparable, i.e. their resultant $\res_z(f,g)=\bdx^{\balpha} \eta(\bdx)$
with $\eta(\bdx)\in \bk[[\bdx]]$,
$\balpha\in \bn^d$ and $\eta(\textbf{0})\ne 0$.
\end{theorem}
\subsection{Computation of resultants}
\mbox{}
Let $f$ and $g$ be two Weierstrass polynomials which are comparable polynomials and
such that their Newton trees have only one end which is not a dead-end. The main result in this section is to show
that the resultant $\res_z(f,g)$
can be read from the coloured Newton tree of $f g$ decorated with its local numerical data.
This result can be seen as a generalization of \cite[Corollaire 5]{pgp:00}
where P.~Gonz{\'a}lez-P{\'e}rez gave information about the Newton diagram of the
resultant of two quasi-ordinary hypersurfaces satisfying an
appropriate non-degeneracy condition.
Let $f$ and $g$ be two Weierstrass polynomials in $\bk[[\bdx]][z]$
and asume we are as in Notation~\ref{inicial} and the system of coordinates is suitable for both
$f$ and $g$.
We consider the coloured Newton tree $\mathcal{T}(fg)$ of $f g$ decorated with its local numerical data.
We recall the following result
\cite{aclm:09} where it was shown that to compute their resultant $\res_z(f,g)$
one can use the factorization of total transforms of $f$ and $g$
after Newton process
even if they are not factorizations of $f$ and $g$.
\begin{proposition}[{\cite[Proposition 5.11]{aclm:09}}]\label{prop-sep}
Let $f,g\in \bk[[\bdx]][z]$ be two Weierstrass polynomials in suitable coordinates which are comparable and
such that their Newton trees $\mathcal{T}(f)$ and $\mathcal{T}(g)$
have only one end which is not a dead-end. Then, $\res_z(f,g)$
can be read from the coloured Newton tree $\mathcal{T}(fg)$
of $f g$ decorated with its local numerical data.
More precisely, for every $1\leq i\leq d$ the multiplicity of $x_i$ in $\res_z(f,g)$ is computed as follows:
\begin{enumerate}
\item Consider the path from the blue-end representing
$f$ to the red-end representing $g$.
\item Take the product of all the decorations that are
adjacent to the path (the $i^{\rm\text{th}}$ coordinate in the case of $d$-uples), including
decorations of these ends.
\item Multiply it by the the $i^{\text{th}}$ coordinate of the $\gcd$ of the vertices before the vertex where they
separate, see Definition{\rm~\ref{lc-data}}.
\end{enumerate}
\end{proposition}
\section{P-Good coordinates}\label{P-good-coor-sec}
Let $f(\bdx,z)\in \bk[[\bdx]][z]$ be a polynomial and we assume that we are as in Notation \ref{inicial}
and in a suitable system of coordinates for $f$.
There is a system of good coordinates, introduced by P.~Gonz\'alez-P\'erez in \cite{go:01}
and \cite[Lemma 3.2]{GMN:03}, that is called P-good coordinates.
\begin{definition}
A polynomial $f(\bdx,z)\in \bk[[\bdx]][z]$ is in \emph{P-good coordinates}
if
\begin{enumerate}
\item its Newton polyhedron $\mathcal{N}(f)$ is a monotone polygonal path,
\item if
there exists a face $\Gamma$ of $\mathcal{N}(f)$ whose line supporting $\Gamma$ is given by
equations $\{\alpha _i +q_i \beta =N_i$, $i=1,\cdots d\}$, then $\Gamma$ is unique and
hits the hyperplane $\beta =0$,
\item if
there exists a face $\Gamma$ of $\mathcal{N}(f)$ whose line supporting $\Gamma$ is given by
equations $\{\alpha _i +q_i \beta =N_i$, $i=1,\cdots d\}$, then the corresponding polynomial
$f_\Gamma$ is not of the form $a_{\Gamma}\bdx^{\mathbf{m}} (z-h(\bdx))^n $
with $h(\bdx) \in \bk[\bdx]$ and $a_{\Gamma}\in \bk^*$, see~(\ref{eq-fgamma}).
\end{enumerate}
\end{definition}
Let us assume that there is a suitable system of coordinates such that the Newton polyhedron of $f$ is a monotone polygonal path.
Let $\mathcal{T}(f)$ be the Newton tree of $f$ in this suitable system of coordinates.
\begin{proposition}\label{prop-p-coord}
Starting from a suitable system of coordinates such that the Newton polyhedron of $f$ is a monotone polygonal path
we find a P-good system of coordinates for $f$ by a change of coordinates of the type $z\mapsto z+a\bdx ^{\mathbf{q}}$.
The Newton tree in this system of coordinates can be deduce from $\mathcal{T}(f)$ in a unique way.
\end{proposition}
\begin{proof}
Let $\mathcal{T}_{\mathcal{N}}(f)$ be the first vertical line of $\mathcal{T}(f)$. The vertices are denoted by $v_1, \cdots, v_n$ from top to bottom decorated with $(\mathbf{q},p)$.
\begin{caso1}
None of the decorations $p$ of the vertices of $\mathcal{T}_{\mathcal{N}}(f)$ is equal to 1
\end{caso1}
In this case, $f$ is already in P-good coordinates.
\begin{caso1} There is a decoration $p$ of some vertex which is equal to one.
\end{caso1}
Let $i$ be the smallest index such that $p_i=1$. We have two cases to consider.
\begin{subcaso}\label{subcaso1} $i\neq n$.
\end{subcaso}
In this case, $f$ is not in P-good coordinates. Assume the decorations of the vertex $v_i$
in $\mathcal{T}_{\mathcal{N}}(f)$ are
$(q_1^i,\cdots,q_d^i,p_i=1)$ and the face polynomial is
$$
\bdx^{k'}z^k\prod_j (z-\mu_j\bdx^{\mathbf{q}^i})^{m_j}.
$$
Consider the change of coordinates
$$
z=z'+a\bdx ^{\mathbf{q}^i},
$$
where $a\neq \mu_j$ for all $j$.
Under this change of variables neither the faces $\Gamma_1,\cdots, \Gamma_{i-1}$
nor their face polynomials do change. The face $\Gamma_i$
has the same equation but now the face polynomial is
$$
\bdx^{k'}(z'+a\bdx^{\mathbf{q}^i})\prod_j (z-(\mu_j-a)\bdx^{\mathbf{q}^i})^{m_j}.
$$
The new face $\Gamma_i'$ hits the $\bdx$-hyperplane.
Then $f$ is in P-good coordinates.
Now we compare the tree $\mathcal{T}(f)$ with the tree $\mathcal{T}'(f)$ in these new coordinates.
The Newton maps corresponding to the vertices $v_1, \cdots,v_{i-1}$ are the same.
At $v_i$, we have to consider the Newton maps
$$
\bdx=\bdy, \quad z=\bdy^{\mathbf{q}^i}(z_1-\mu_j).
$$
The Newton maps to be considered at $v_i'$ are given by the equations
$$
\bdx=\bdy, \quad z'=\bdy^{\mathbf{q}^i}(z'_1-(\mu_j-a)).
$$
Then $z_1=z_1'$ and along the corresponding edge nothing is changed.
Now at $v_i'$ we have also to consider the Newton map
$$
\bdx=\bdy, \quad z'=\bdy^{\mathbf{q}^i}(z'_1-a)
\Longrightarrow
\bdx=\bdy, \quad z=\bdy^{\mathbf{q}^i}z_1'.
$$
In the change of variables, the monomial $\bdx ^{\balpha}z^{\beta}$ becomes $\bdy ^{\balpha+\beta\mathbf{q}^i}z_1'^{\beta}$.
We consider the transformation in the affine space
$$
(\balpha,\beta)\mapsto (\balpha+\beta\mathbf{q}^i,\beta).
$$
A hyperplane with equations $\{P\alpha_j+Q_j\beta=0\}$ maps to $\{P\alpha_j'+(Q_j-P q_j^i)\beta=0\}$. Then the faces
$\Gamma_{i+1},\cdots,\Gamma_n$, map to faces of the Newton diagram in the coordinates $(\bdy,z_1')$
and since $\Gamma_{i+1},\cdots,\Gamma_n$
was a monotone polygonal path, it transforms to a monotone polygonal path.
Now consider a face polynomial $\bdx^{k'}z^k \prod_l (z^P-\mu_l\bdx ^{\mathbf{Q}})^{m_l}$. In the change of coordinates it becomes
$$
\bdy^{k'+\mathbf{q}^ik"}z_1'^{k}\prod_l (z^P-\mu_l\bdy^{\mathbf{Q}-\mathbf{q}^iP})^{m_l}.
$$
We consider the following Newton maps.
Let $(\mathbf{u},u)$ be integers such that $1+\mathbf{u}\cdot\mathbf{Q}=uP$.
We have $1+\mathbf{u}\cdot(\mathbf{Q}-P \mathbf{q}^i)=(u-\mathbf{u}\cdot\mathbf{q}^i)P$.
The Newton maps are
$$
\bdx=\mu_l^{\mathbf{u}}\bdx'^P,z=\bdx'^{\mathbf{Q}}(z_2+\mu_j^{u})\quad\text{and}
\quad
\bdy=\mu_l^{\mathbf{u}}\bdy'^P,z'_1=\bdy'^{(\mathbf{Q}-P \mathbf{q}^i)}(z'_2+\mu_j^{(u-\mathbf{u}\cdot\mathbf{q}^i)}).
$$
Since $z=\bdx^{\mathbf{q}^i}z_1'$, we have
$\bdx'=\bdy', z_2=\mu_l^{\mathbf{u}.\mathbf{q}^i}z_2'$. Then we have essentially the same coordinates.
In conclusion, to get the tree of $f$ in P-good coordinates from
$\mathcal{T}(f)$, we have to cut the edge~$e_i$ under $v_i$.
We get two trees, $\mathcal{T}_a(f)$, the part which contains $v_i$ and $\mathcal{T}_u(f)$ containing~$e_i$.
Then we stick again $\mathcal{T}_u(f)$ to $\mathcal{T}_a(f)$, sticking the edge $e_i$ on $v_i$ as an horizontal edge.
We add a new vertical edge under $v_i$ decorated with $1$ and ending by an arrowhead decorated with~$(0)$.
Since $v_i$ has a valency greater or equal to $3$ on $\mathcal{T}(f)$, it has a valency greater or equal to~$4$ on $\mathcal{T}'(f)$.
To put $f$ in P-good coordinates we made the choice of $a$.
It is easy to verify that actually the tree doesn't depend on the choice of $a$.
Then in this case we have unicity of $\mathcal{T}'(f)$ from $\mathcal{T}(f)$.
\begin{subcaso}\label{subcaso2}
$i=n$.
\end{subcaso}
The edge $e_n$ under the vertex $v_n$ ends with an arrow decorated with $(k)$. Three cases may arise:
\begin{itemize}
\item If $k\neq 0$ then we are in P-good coordinates.
\item If $k=0$ and the valency of $v_n$ is strictly greater than~$3$ then we are in P-good coordinates.
\item If $k=0$ and the valency of $v_n$ is equal $3$ then we are not in P-good coordinates.
\end{itemize}
We study this last case.
The face polynomial at $v_n$ is
$$\bdx ^{k'}(z-a\bdx ^{\mathbf{q}^n})^m.$$
To eliminate the face $\Gamma_n$, we have to perform the (unique) change of variables
$$z=z_1'+a\bdx^{\mathbf{q}^n}.$$
We considered the Newton map:
$$\bdx=\bdy, \quad z=\bdy^{\mathbf{q}^n}(z_1+a)$$
and we have $z_1'=\bdx^{\mathbf{q}^n}z_1$. The computation are the same than in Sub-Case~\ref{subcaso1}.
To get $\mathcal{T}'(f)$ from $\mathcal{T}(f)$, we do the inverse operation than in Sub-Case~\ref{subcaso1}.
We delete the edge $e_n$.
Denote by $e$ the horizontal edge attached to $v_n$. We cut $\mathcal{T}(f)$ in two pieces, separating $v_n$ and $e$
and we stick it back so that $e$ becomes the vertical edge under $v_n$.
The vertex $v_n$ has now valency $2$ and has to be eliminated.
Note that we are not necessarily in P-good coordinates yet. We illustrate this fact in several examples:
\begin{itemize}
\item In Figure~\ref{fig:QO082a} we are in suitable coordinates and in Sub-Case~\ref{subcaso2}.
\item In Figure~\ref{fig:QO082c} we have eliminated the face $\Gamma_n$ but we are not yet in P-good
coordinates. We are in Sub-Case~\ref{subcaso1}.
\item In Figure~\ref{fig:QO082b} we are in P-good coordinates.
\end{itemize}
We have proven that if $f$ is in suitable coordinates such that its Newton polyhedron is a monotone polygonal path,
then there is essentially a unique way to find P-good coordinates.
\end{proof}
\begin{figure}[ht]
\centering
\subfigure[]{
\includegraphics[scale=1]{QO082a}
\label{fig:QO082a}
}
\hfil
\subfigure[]{
\includegraphics[scale=1]{QO082c}
\label{fig:QO082c}
}
\caption{Newton tree in suitable coordinates and not P-good}
\label{fig1}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1]{QO082b}
\caption{Newton tree in P-good coordinates}
\label{fig:QO082b}
\end{center}
\end{figure}
\subsection{On $\nu$-Quasi-ordinary polynomials whose Newton tree
ends only with arrow-heads.}
\mbox{}
Now we assume that $f$ is such that there exists a system of suitable coordinates such that $\mathcal{T} (f)$ has only arrowheads (no black boxes).
In that case, the Newton polyhedron of $f$ is a monotone polygonal path.
We can find a system of P-good coordinates for $f$.
Let $\mathcal{T}_1 (f)$ denote the Newton tree in this system.
It has also only arrowheads. Consider the Newton transforms of $f$ in this system of P-good coordinates;
their Newton polyhedra are monotone polygonal paths.
Then we can find a system of P-good coordinates for them, and so on, see Proposition~\ref{prop-p-coord}.
We can summarize it as follows.
\begin{definition}
A Newton tree is said to be a \emph{P-good tree} if
the vertices with decoration
$p=1$ are at the bottom of the tree, connected to a vertical edge ending with an arrowhead decorated with $(0)$,
and with valency strictly greater than $3$.
\end{definition}
\begin{proposition}
Starting from a Newton tree $\mathcal{T} (f)$ (for suitable system of coordinates),
we can associate to $f$ a unique P-good tree.
\end{proposition}
\begin{qst}\label{qst-1}
Assume there are two suitable systems of coordinates such that the Newton trees
$\mathcal{T} (f)$ and $\mathcal{T}' (f)$ have only arrowheads. Do we get the same P-good tree?
\end{qst}
In order to answer this question, we distinguish three cases:
\begin{enumerate}
\item The change of variables doesn't change the Newton diagram of $f$.
It doesn't change either the face polynomials, then $\mathcal{T} (f)=\mathcal{T}' (f)$.
Then there is one P-good tree.
\item The change of variables modifies the Newton diagram as follows:
The edges $\Gamma_1,\cdots,\Gamma_i$ remain unchanged and
$\Gamma_{i+1},\cdots, \Gamma_n$ are replaced by a newface $\Gamma$.
In any case $\mathcal{T}'(f)$ is not in P-good coordinates
and we are in Sub-Case~\ref{subcaso2} in the proof of Proposition~\ref{prop-p-coord}.
In order to get P-good coordinates we first have to come back to $\mathcal{T} (f)$. Then we have the same P-good tree.
\item The change of coordinates changes the Newton diagram as follows: It keeps unchanged $\Gamma_1,\cdots,\Gamma_i$, and replace $\Gamma_{i+1}$ by a face with same equations which hits the hyperplane $\{z=0\}$. Either one of
$\mathcal{T} (f)$ or $\mathcal{T}' (f)$ is P-good and is the P-good tree of the other, or the two trees only differ by exchanging one vertical edge and one horizontal edge. They have the same P-good tree because in the P-good tree they are all horizontal.
\end{enumerate}
Hence, if the Newton tree $\mathcal{T} (f)$ for $f$, in suitable coordinates, has no black box
then we associate to it a unique P-good tree denoted by
$\mathcal{T}_P (f)$ and the answer to Question~\ref{qst-1} is positive.
\begin{remark}
A P-good tree is not a minimal tree in the sense of Eisenbud and Neumann~\cite{en:85}.
A minimal tree is unique if we forget about the direction of the edges, but not in the strong sense. A P-good tree is unique in the strong sense.
\end{remark}
This allows us to define the depth of $f$ in suitable coordinates such that $\mathcal{T} (f)$ has no black box.
\begin{definition}
The \emph{depth} of $f$ is the maximal length of horizontal paths in $\mathcal{T}_P (f)$,
denoted by~$\depth(f)$.
\end{definition}
\section {Quasi-ordinary polynomials}
\begin{notation}\label{inicial3}
Let
$g(\bdx,z) \in \bk[[\bdx]][z]$ be a
polynomial of degree $m$ and regular of order $n\leq m$
with coefficients in the formal
power series ring $\bk[[\bdx]]$, that is
$$
g(\bdx,z):=z^m+a_1(\bdx)z^{m-1}+\ldots+a_{m-n}(\bdx)z^n+a_{m-n+1}(\bdx)z^{n-1}+\ldots+ a_{m-1}(\bdx)z+a_m(\bdx),
$$
with $a_i(\textbf{0})=0$ for $m-n+1\leq i\leq m$ and $a_{m-n}(\textbf{0})\in \bk^*$ and
let $f(\bdx,z):=a_0 x_1^{n_1}\cdot\ldots\cdot x_d^{n_d} g(\bdx,z)$ where $ a_0\in \bk^*$.
Applying Weierstrass Preparation Theorem to $g(\bdx,z)$ there exists a unit $u(\bdx,z) \in \bk[[\bdx]][z]$
and a Weierstrass polynomial
$$
h(\bdx,z):=z^n+B_1(\bdx)z^{n-1}+\ldots+ B_{n-1}(\bdx)z+B_n(\bdx)
$$
such that $g(\bdx,z)=h(\bdx,z)u(\bdx,z)$. In fact
$\deg_z u(\bdx,z)=m-n$,
e.g. see \cite[Chapter I,~p.~11]{GLS}.
\end{notation}
\begin{definition}
A polynomial $f\in \bk[[\bdx]][z]$ as in Notation \ref{inicial3} is called \emph{quasi-ordinary} if its
discriminant $\Delta_z(f):=\res_z(f,\frac{\partial f}{\partial z})(\bdx)$ is
$$\res_z(f,\frac{\partial f}{\partial z})(\bdx)=\bdx^{\balpha} \eta(\bdx)$$ with $\balpha\in \bn^d$ and $\eta(\textbf{0})\ne 0$.
\end{definition}
\begin{remark}
Since $\res_z(f,f_1f_2)=\res_z(f,f_1)\res_z(f,f_2)$,
a polynomial $f\in \bk[[\bdx]][z]$ as in Notation \ref{inicial3} is {quasi-ordinary} if and only if
$g$ is quasi-ordinary.
If $f=f_1f_2\in \bk[[\bdx]][z]$ then $f_i$ is also quasi-ordinary because $\bk[[\bdx]]$ is a factorial ring and
the well-known property of discriminants $\Delta_z(f_1f_2)=\Delta_z(f_1)\Delta_z(f_2)(\res_z(f_1,f_2))^2$.
\end{remark}
In this section we prove the following theorem.
\begin{theorem}\label{main2}
Let $f\in \bk[[\bdx]][z]$ be a polynomial as in Notation {\rm~\ref{inicial3}}.
Then
$f$ is quasi-ordinary if and only if there exists a suitable system of coordinates of $f$ such that
its Newton tree $\mathcal{T}(f)$ has only arrow-heads decorated with $(0)$ and $(1)$ and has no black boxes.
\end{theorem}
Quasi-ordinary power series were introduced by Zariski
using the discriminant. In fact quasi-ordinary power
series are the natural generalization, in the sense of the
factorization theorem given by Jung and Abhyankar~\cite{ab:55}, of algebraic curves.
The fact that at each stage of the Newton process, using eventually automorphisms,
its Newton polyhedron is a monotone polygonal path is very useful.
It is one of the main ingredients in the proof in~\cite{aclm:05} of the monodromy
conjecture for hypersurfaces defined by quasi-ordinary power series in arbitrary dimension.
First we prove that if $f\in \bk[[\bdx]][z]$ is quasi-ordinary,
the Newton process ends with monomials multiplied by a unit.
\begin{proposition}[\cite{lu:83}]\label{prop-lu}
Let $f\in \bk[[\bdx]][z]$ be a quasi-ordinary polynomial, then there exists a power series $b(\bdx)\in \bk[[\bdx]]$ such that
$f(\bdx,z-b(\bdx))\in \bk[[\bdx]][z]$ is a $\nu$-quasi-ordinary polynomial.
\end{proposition}
\begin{lemma}[{\cite[Lemma 3.16]{GMN:03}}]
If $f$ is a quasi-ordinary polynomial in $\bk[[\bdx]][z]$,
then there exists a system of coordinates
such that its Newton polyhedron $\mathcal{N}(f)$
is a monotone polygonal path.
\end{lemma}
\begin{proof}
Since $f(\bdx,z)=\bdx^\bdn h(\bdx,z)u(\bdx,z)$ the Newton diagram $\mathcal{N}_+(f)$ is the Minkowski sum ot each
of the Newton diagram of its factors it is enough to prove for $h$ which is regular of order $n$ in $z$.
We will work by induction on the order of $h$ in $z$. It is true for $n=1$.
We assume that it is true for all regular polynomial of order strictly less than $n$.
Since $h$ is a regular quasi-ordinary polynomial, by Proposition~\ref{prop-lu}, there exists a change of coordinates
such that $h$ is $\nu$-quasi-ordinary with a $\nu$-proper face $\Gamma_1$.
By Factorization Theorem~\ref{teorema1} and Corollary \ref{cor-factor-gamma}
there exists an elementary polynomial $f_1$ in $\bk[[\bdx]][z]$ with Newton diagram parallel to $\Gamma_1$ and
a polynomial $q(\bdx,z)$ in $\bk[[\bdx]][z]$ such that $h=f_1q$.
Since $h$ is quasi-ordinary and $q$ is one of its factors then $q$ is a quasi-ordinary polynomial of order strictly less than $n$.
From the hypothesis, there exists a system of coordinates such that $\mathcal{N} (q)$ is a monotone
polygonal path, and this change of coordinates does not change the Newton diagram of $f_1$. Using
the fact that the Newton diagram of a product is the Minkowski sum of the Newton diagrams of each factors,
we deduce that the Newton polyhedron of $h$ is a monotone polygonal path.
\end{proof}
In particular there exist P-good coordinates for quasi-ordinary polynomials $f$.
\begin{lemma}\cite[Chapter~3, Lemma 3.21]{aclm:05}
If $f$ is quasi-ordinary, after a Newton map $\sigma _{\Gamma, j}$,
the total transform $f_{\Gamma_1, j}:=\sigma _{\Gamma_1, j}(f)\in \bk[[\bdy]][z_2]$
is a quasi-ordinary polynomial in $\bk[[\bdy]][z_2]$.
\end{lemma}
Furthermore the \textbf{depth} of a quasi-ordinary polynomial $f$ in P-good coordinates
was defined similarly as in \cite[Chapter~3, Definition~24]{aclm:05}, and it decreases after Newton maps.
\begin{lemma}\label{lema-ends}
Let $f\in \bk[[\bdx]][z]$ be a polynomial as in Notation{\rm~\ref{inicial3}}.
If
$f$ is quasi-ordinary then there exists a system of suitable coordinates such that
its Newton tree $\mathcal{T}(f)$ has only arrow-heads decorated with $(0)$ and $(1)$ and has no black boxes.
\end{lemma}
\begin{proof}
In fact in \cite[Chapter~3, Section~3.26]{aclm:05} it was defined Newton tree $\mathcal{T}(f)$ for
of a quasi-ordinary polynomials $f$ in P-good coordinates and in this tree all ends
are arrow-heads and if an arrow-head is not a dead-end then it is decorated with~$(1)$.
\end{proof}
\mbox{}
To prove the converse
it is enough to prove that
$f\in \bk[[\bdx]][z]$ and its \emph{polar} $f_z:=\frac{\partial f}{\partial z}\in \bk[[\bdx]][z]$
are separated polynomials because after Theorem \ref{thm-sepcom} they are comparable series.
We assume that there is a system of suitable coordinates such that
$\mathcal{T} (f)$ has no black box and arrowheads decorated with $(0)$ and $(1)$. We consider coordinates such that the Newton tree is
$\mathcal{T}_P (f)$. Since P-good coordinates are suitable coordinates, these coordinates are suitable for $ff_z$.
Given a linear form $D$ given by $\sum_{j=1}^d a_j \alpha_j+ b\beta$, $a_j,b\in\bn$,
we define $f_D(\bdx,z)$ as the sum of the monomials $c_{\balpha,\beta}\bdx^{\alpha}z^{\beta}$,
$c_{\balpha,\beta}\neq 0$, for which $D(\balpha,\beta)$ is minimal. Note that $f_D$ is not a monomial
if and only there is a face of the Newton polyhedron contained in an affine hyperplane defined by~$D$.
\begin{lemma}
Assume that $D$ is a linear form such that
$$
f_D(\bdx,z)=\bdx^{\mathbf{A}}z^B,
$$
with $B\neq 0$, then there is no face of $\mathcal{N}(f_z)$ contained in an affine hyperplane defined by~$D$.
\end{lemma}
\begin{proof}
Denote by $D_+$ the space limited by the first quadrant and the affine hyperplane defined by~$D$
which hits the Newton polyhedron of $f$ and doesn't contain the origin.
We have
$$
f(\bdx,z)=f_D(\bdx,z)+\sum_{(\balpha,\beta)\in D_+,\beta>0}
c_{\balpha,\beta}\bdx^{\balpha}z^{\beta}+\sum_{(\balpha,\beta)\in D_+,\beta=0}
c_{\balpha,\beta}\bdx^{\balpha}z^{\beta},
$$
$$f_z(\bdx,z)=B\bdx^{\mathbf{A}}z^{B-1}+\sum_{(\balpha,\beta)\in D_+,\beta>0}
\beta c_{\balpha,\beta}\bdx^{\balpha}z^{\beta-1}.
$$
Then, there is no face of the Newton diagram of $f_z$ in an affine hyperplane defined by~$D$.
\end{proof}
\begin{lemma}
Assume $\Gamma_\ell$ is a face of the Newton diagram of $f$
which doesn't hit the hyperplane $\{z=0\}$ with face polynomial,
see{\rm~\eqref{eq-fgamma-ell}},
$$
f_{\Gamma_\ell }=
a_{\Gamma_\ell }\bdx^{\mathbf{n^\ell}}z^{n^\ell_{d+1}}\prod _{j=1}^{k^\ell}(z^{p_\ell}-\mu _j^\ell
\bdx^{\mathbf{q^\ell}})^{m_j^\ell},
$$
the factors being irreducible in $\bk[[\bdx]][z]$,
i.e. $\gcd
\mathbf{q^\ell},p_\ell)=1$, $\mu_j^\ell\in \bk^*$ with $\mu_j^\ell\ne \mu_i^\ell$ and $a_{\Gamma_\ell }\in \bk^*$.
Then the Newton diagram of $f_z$ has a face $\Gamma'_\ell$ parallel to $\Gamma_\ell$ and
the corresponding initial form $(f_{z})_{\Gamma'_\ell }$ is
$$
a_{\Gamma_\ell }\bdx^{\mathbf{n^\ell}}\!\! z^{n^\ell_{d+1}-1}
\!\!\!\left(\prod _{j=1}^{k^\ell}(z^{p_\ell}-
\mu _j^\ell
\bdx^{\mathbf{q^\ell}})^{m_j^\ell-1}\right)\!\!\!
\left(\sum _{i=1}^{k^\ell}\left((n^\ell_{d+1}+p_\ell m_i^\ell)z^{p_\ell}
-n^\ell_{d+1}\mu_i^\ell \bdx^{\mathbf{q^\ell}}\right)\prod _{j\neq i}
(z^{p_\ell}-\mu _j^\ell \bdx^{\mathbf{q^\ell}})\right)\!\!.
$$
\end{lemma}
\begin{proof}
Same computation as before.
\end{proof}
\begin{lemma}
Assume $\Gamma_\ell$ is the face of the Newton diagram of $f$ which hits the hyperplane $\{z=0\}$ with face polynomial
$$
f_{\Gamma_\ell}=a_{\Gamma_\ell}\bdx ^{\bdn_\ell} \prod_{j=1}^{k^\ell} (z^{p_\ell}-\mu_j^\ell\bdx ^{\mathbf{q}_\ell})^{m_j^\ell}.
$$
Then if $k^l>1$ or if $k^l=1$ and $m^l_1>1$ the Newton diagram of $f_z$ has a face parallel to $\Gamma_\ell$,
denoted by $\Gamma'_\ell$ and
\begin{equation}\label{eq-newton-fz}
(f_z)_{\Gamma'_\ell}= p_\ell a_{\Gamma_\ell}\bdx^{\bdn_\ell}z^{p_\ell-1}
\prod _{j=1}^{k^\ell} (z^{p_\ell}-\mu_j^\ell\bdx ^{\mathbf{q}_\ell})^{m_j^\ell-1}
\left(\sum_{i=1}^{k^\ell}m_i^\ell\prod_{j\neq i}(z^{p_\ell}-\mu_j^\ell\bdx^{\mathbf{q}}_\ell)\right).
\end{equation}
If $k^\ell=1$ and $m_1^\ell=1$ then the Newton diagram of $f_z$ has no face parallel to $\Gamma_\ell$.
\end{lemma}
\begin{remark}
If $k^\ell=1$ and $m_1^\ell=1$ then $p_\ell>1$ since we are in P-good coordinates for $f$.
Then either $f_z$ is divisible by $z^{p_\ell-1}$ or $\mathcal{N}(f_z)$ has faces which are not faces of $f$.
The proof of the above Lemma is straightforward.
\end{remark}
\begin{cor}
Consider the bicoloured Newton tree $\mathcal{T}(ff_z)$ coloured blue for $f$ and red for~$f_z$.
Denote by $v_1,\cdots,v_n$ the vertices on the first vertical line.
There exists $n_0$, $1\leq n_0\leq n$ such that for $1\leq j\leq n_0$
there is a blue or blue-red horizontal edge attached to $v_j$ and for $n_0<j\leq n$
there is no blue, neither blue-red horizontal edge attached to $v_j$.
Attached to $v_j$, $1\leq j\leq n_0$, there are eventually blue horizontal edges ending with an arrow and/or blue-red horizontal edges ending with a vertex and always red horizontal edges.
\end{cor}
The horizontal edges corresponding to
the roots coming from the factor
$$\sum _{i=1}^{k^\ell}\left((n^\ell_{d+1}+p_\ell m_i^\ell)z^{p_\ell}
-n^\ell_{d+1}\mu_i^\ell \bdx^{\mathbf{q^\ell}}\right)\prod _{j\neq i}
(z^{p_\ell}-\mu _j^\ell \bdx^{\mathbf{q^\ell}}),
$$ of $(f_{z})_{ \Gamma'_\ell }$
are red-coloured. The degree in $z$ is $k^\ell p_{\ell}$.
\bigskip
\begin{proof}[Proof of $\Leftarrow)$ of Theorem{\rm~\ref{main2}}]
We use induction on $\delta:=\depth(f)$.
Let us start with the case $\delta=0$.
We have $f=\bdx^{\bdn}z$ up to a unit and $f_z=\bdx^{\bdn}$. Then $f$ and $f_z$ are separated.
For $\delta=1$,
consider a vertex $v_j$ of the tree of $ff_z$.
If $1<j\leq n_0$, there are exactly $k_j$ blue horizontal edges attached to $v_j$ ending by an arrow, and some red edges, but no blue-red edges. For $n_0<j\leq n$, there are only red edges. Then $f$ and $f_z$ are separated.
Assume that for any $g$ in P-good coordinates with $\mathcal{T}_P(g)$
with no black box and arrowheads decorated with $(0)$ and $(1)$ of depth $\delta'<\delta$, $g$ and $g_z$ are separated.
Consider $f$ in P-good coordinates of depth $\delta$ and assume that $\mathcal{T}(f)$
has no black box and has arrowheads decorated with $(0)$ and $(1)$.
If $\delta>1$ there is a vertex $v_j$, $1\leq j\leq n_0$ such that there is a blue-red horizontal edge attached to $v_j$.
We consider the corresponding Newton map $\sigma$.
Consider the polynomials $(f_z)_{\sigma}(\bdy,z_1)$ and $f_{\sigma}(\bdy,z_1)$, after the
change of variables to have $f_{\sigma}(\bdy,z_1)$ in P-good coordinates.
Using see Lemma \ref{total-transf} we have
$(f_{\sigma})_{z_1}=\bdx^{\mathbf{q}}(f_z)_{\sigma}$.
Therefore, $\mathcal{T}(f_{\sigma}\cdot(f_z)_{\sigma})$ and
$\mathcal{T}(f_{\sigma}\cdot(f_{\sigma})_{z_1})$ are the same except for the decoration of the top arrow.
Since $\depth(f_{\sigma})<\delta$, then $f_{\sigma}$ and $(f_{\sigma})_{z_1}$ are separated on
$\mathcal{T}(f_{\sigma}\cdot(f_{\sigma})_{z_1})$. The tree $\mathcal{T}(f\cdot f_z)$ is obtained by
gluing $\mathcal{T}(f_{\sigma}\cdot(f_z)_{\sigma})$ on $v_j$.
We can apply that to every vertex where we have a blue-red edge attached. Then $f$ and $f_z$ are separated.
\end{proof}
\bigskip
This description allows us to study how $f$ and its derivative $f_z$
separate on the Newton tree of $f$.
\begin{definition}
The edges at the bottom
of the polyhedron ending with an arrow-head of multiplicity~$(0)$
(dead ends) are called {\it leaves}, and the vertices at the other end of the arrow-head \emph{leaf vertices}.
\end{definition}
\begin{theorem}\label{thm-ends}
Let $f\in \bk[[\bdx]][z]$ be a polynomial as in Notation{\rm~\ref{inicial3}} in a P-good system of coordinates and
we consider the Newton tree
$\mathcal{T}(f)$ whose first vertical line
has $v_1,\ldots,v_s$ as vertices.
\begin{enumerate}
\enet{\rm(D\arabic{enumi})}
\item\label{D1} The polar $f_z$ doesn't separate on the edges of the Newton tree of
$f$ which are not leaves of $f$.
\item\label{D2} On each vertex $v_\ell$ of the Newton tree of $f$, different from a leaf vertex, the polar
$f_z$ separates
with order equal to $k^\ell p_{\ell}$, see~\eqref{eq-newton-fz}; recall $k^\ell +2$
is the valency of the vertex and $p_\ell$ the decoration under the vertex.
\item\label{D3} The polar $f_z$
separates on a leaf vertex or on a leaf with total order
$\frac{k^s-1}{p_s}+p_s-1$
\end{enumerate}
\end{theorem}
\begin{proof}
It follows from the previous discussion.
\end{proof}
\begin{remark}
We don't know
anything on $f_z$ after its separation with $f$. In particular in general the derivative is
not quasi-ordinary. In the case $d=1$ this gives L\^e-Michel-Weber Theorem~\cite{lmw:89}.
\end{remark}
We can compute the discriminant of $f$ from its Newton tree. The following
result follows from a recursive application of \cite[Lemmas~5.7,~5.10]{aclm:09}, Proposition~\ref{prop-sep} and Theorem~\ref{thm-ends}.
\begin{proposition}\label{discrimant-tree}
Let $f$ be a quasi-ordinary Weierstrass polynomial of degree~$n$.
Its discriminant can be computed from the polyhedron of $f$ by
$$
\Delta_z(f):=\res_z\left(f,f_z\right)(\bdx)=\bdx^{\mathbf{D}} u(\bdx), \textit{ with } u(0)\ne 0,
$$
where
$$
\mathbf{D}:=(D_1,\ldots,D_d)=\sum _v (\delta _v -2)\mathbf{N}'_v -\sum _{v\text{\rm\ leaf vertex}}\frac{\mathbf{N}'_{v}}{p_{v}},
$$
where
$\delta_v$ is the valency of the vertex~$v$,
and $\mathbf{N}'_v=\mathbf{\rho}*\mathbf{N}_v$ where $\mathbf{\rho}:=\prod_{w\text{\rm\ before }v}\mathbf{c}_w$
(recall that $\mathbf{c}_w$ is the $\gcd$'s of the vertex $w$).
\end{proposition}
This can be seen as a generalisation of Kouchnirenko theorem~\cite{kus,cn:96} and
as a generalization of \cite[Corollaire 5]{pgp:00} where P.~Gonz{\'a}lez-P{\'e}rez
gave information about the Newton polyhedron of the
resultant of two quasi-ordinary hypersurfaces satisfying an
appropriate non-degeneracy condition.
\section{Transversal sections}\label{sec-trans-sect}
Consider a quasi-ordinary polynomial $f\in \bk[[\bdx]][z]$. Fix $i\in \{1,\cdots , d\}$. We follow this convention: if $\mathbf{p}$ is
a $d$-tuple, then $\mathbf{p}^i$ is the $(d-1)$-tuple $(p_1,\dots,p_{i-1},p_{i+1},\dots,p_d)$.
Let $K_i$ be an algebraic closure of $K((x_i))$.
The $i$-\emph{transversal section} of $h$ is the quasi-ordinary polynomial
$h^i\in K_i[[\bdx^i]] [z]$ obtained from $h$ considering $x_i$ as a generic constant.
For $I\subset \{1,\cdots,d\}$ we denote $I':=\{1,\cdots,d\}\setminus I$;
we can define recursively the $I$-\emph{transversal section} $h^I$ and if
$I'$ has one element, say $j$, we call it \emph{$j$-curve transversal section} $h^{I_j}$.
Consider $f$ a quasi-ordinary polynomial in suitable coordinates such that its Newton tree has no black box. We first study the Newton trees of the transversal sections in the same system of coordinates.
Then we assume the Newton tree of $f$ has only one arrow-head with positive multiplicity.
We show that we can retrieve its
decorated Newton tree from the Newton tree of its curve
transversal sections in the same system of coordinates.
We give examples where this is no
more true when the Newton tree of $f$ has more than one arrow-head with positive multiplicity. Nevertheless we can prove
that we can retrieve the global decorations of the vertices of~$f$ from the decorations
of the curve transversal sections. This result was a crucial ingredient
in the proof of the monodromy conjecture for quasi-ordinary singularities in arbitrary dimension~\cite{aclm:05}.
Let $f$ be a quasi-ordinary polynomial in suitable coordinates such that the Newton tree $\mathcal{T}(f)$
has no black box. We are going to describe how we can find the tree of the $i$-transversal section $f^i$
in the same coordinates.
\subsection{Construction of $\mathcal{T}_{\mathcal{N}}(f^i)$.}
\mbox{}
The general procedure is that we can copy $\mathcal{T}_{\mathcal{N}}(f)$
and erase the $i^{\text{th}}$-component of the local numerical data except in $3$ cases
which we develop below.
\begin{caso}\label{caso1}
Along $\mathcal{T}_{\mathcal{N}}(f)$, there is a sequence of consecutive vertices where
the local numerical data coincide after erasing the $i^{\text{th}}$-coordinate (and dividing
by the gcd).
This means that the two faces corresponding to these vertices project on the same face.
Then the chain of vertices \emph{project}
on $\mathcal{T}_{\mathcal{N}}(f^i)$ on the same vertex decorated with the common normalized numerical data.
\end{caso}
\begin{caso}\label{caso2} The local numerical data at $v$ on $\mathcal{T}_{\mathcal{N}}(f)$ satisfy
$\gcd (\mathbf{q}^i,p)=:d^i$,
with $d^i>1$, and let $q'_j:=\frac{q_j}{d^i}, j \in I'$, and $p':=\frac{p}{d^i}$.
Assume that the face polynomial at $v$ is
$$
\prod _j (z^p -\mu _j\mathbf{x}^{\mathbf{q}})^{m_j}.
$$
On $\mathcal{T}_{\mathcal{N}}(f^i)$, there is a vertex $v^i$ decorated with $(\mathbf{q'},p')$ and
with face polynomial
$$
\prod _j \prod _{\{\zeta \vert \zeta^{d^i}=1\}}\left(z^{p'} -\zeta \mu _jx_i^{\frac{q_i}{d_i}}(\mathbf{x}^i)^{\mathbf{q'}}\right)^{m_j}.
$$
Then if $k$ horizontal edges arise from $v$, $kd^i$ arise from $v^i$.
\end{caso}
\begin{remark}
Case~\ref{caso1} and~\ref{caso2} are not exclusive.
The number of horizontal edges at $v^i$ is obtained as the sum, along
the vertices $v$ projecting on $v^i$,
of $d^i(v)$ times the number of horizontal edges arising from $v$.
\end{remark}
\begin{example}\label{ex-sec-trans-1}
We have in Figure~\ref{fig:desQO8nn1} the tree of
$$
f(x_1,x_2,z):=(z^2-x_1^3x_2)(z^2-x_1^3x_2^4)(z^2-x_1^5x_2^6)\in K[[x_1,x_2]][z];
$$
Figures~\ref{fig:desQO8nn2} and~\ref{fig:desQO8nn3} show the trees
for $f(x_1,x_2,z)\in K_1[[x_2]][z]$ and $f(x_1,x_2,z)\in K_2[[x_1]][z]$, respectively.
This example illustrates Cases~\ref{caso1} and~\ref{caso2}.
More explicitely, the tree for $K_2[[x_1]][z]$ illustrates Case~\ref{caso1}
and the tree for $K_1[[x_2]][z]$ illustrates Case~\ref{caso2}.
\begin{figure}[ht]
\centering
\subfigure[{$K[[x_1,x_2]][z]$}]{
\includegraphics[scale=1]{desQO8nn1}
\label{fig:desQO8nn1}
}
\hfil
\subfigure[{$K_1[[x_2]][z]$}]{
\includegraphics[scale=1]{desQO8nn2}
\label{fig:desQO8nn2}
}
\hfil
\subfigure[{$K_2[[x_1]][z]$}]{
\includegraphics[scale=1]{desQO8nn3}
\label{fig:desQO8nn3}
}
\caption{Newton trees of Example~\ref{ex-sec-trans-1}}
\label{desQO8nn}
\end{figure}
\end{example}
\begin{caso}\label{caso3}
We have $\mathbf{q}_\ell^i=\mathbf{0}^i$ for the first $h$ vertices of
$\mathcal{T}_{\mathcal{N}}(f)$.
That means that the projection of the face $\Gamma_\ell$ of $\mathcal{N}(f)$ has equation $p x_j=0$,
$j \in I'$;
the polynomial $f_{\Gamma_\ell} (\bdx,z )$ doesn't depend on $x_j$, $j\in I'$, that is
$$
f_{\Gamma_\ell} (\bdx^i,z )=z^{k_\ell}\bdx^{\mathbf{N}}\prod_{j=1}^{r_\ell}(z^{p_\ell}-\mu_j^\ell x_i^{q_\ell})^{m_j^\ell},
$$
i.e., $f^i$ decomposes in $K_i[[\bdx^i]][z]$ in $m:=\sum_{\ell=1}^h p_\ell\sum_{j=1}^{r_\ell} m_j^\ell$
factors based at the $p_\ell$-roots $\mu_j^\ell x_i^{q_\ell}$ of $z$-degree $m_j^\ell$, and one
extra factor based at $z=0$ of degree $k_h$ (if $k_h>0$).
We proceed as follows. We keep $m$
void trees and if $k_h>0$
we consider also a tree for $z=0$, where we erase
the vertices $v_1,\dots,v_h$ and we keep the upper arrow.
\end{caso}
\subsection{Construction of $\mathcal{T}(f^i)$ without decorations}
\mbox{}
Once we have obtained the vertical Newton trees we explain how to obtain
the trees $\mathcal{T}^i(f)$, when we see $f\in K_i[[\bdx^i]][z]$.
We proceed by induction of $\depth(f)$. If the depth is one we have
$\mathcal{T}^i(f)=\mathcal{T}_{\mathcal{N}}^i(f)$ with the following exception.
If we are in Case~\ref{caso3}, we replace the empty trees by the ones
in~Figure~\ref{Q01}.
If the depth is greater than one, we start with $\mathcal{T}_{\mathcal{N}}(f^i)$.
Each arrow of this tree is related to one Newton transformation
$\sigma:=\sigma_{\Gamma,\mu}$ and by the induction hypothesis we may
assume that $\mathcal{T}(f_\sigma^i)$ are constructed $\forall\Gamma,\mu$.
In order to recover $\mathcal{T}(f^i)$ we proceed
as in Step~\ref{paso3} of~\S\ref{constr-trees} with the following caveats.
If we are not in Case~\ref{caso3} and $\mathcal{T}(f_\sigma^i)$ is a disjoint union
of $k_{\Gamma,\mu}$ trees, then we perform Step~\ref{paso3} for each connected component.
Hence we will obtain $\prod_{\Gamma,\mu} k_{\Gamma,\mu}$ disconnected trees.
If we are in Case~\ref{caso3}, we proceed in the same way for the tree corresponding
to~$z=0$ (if $k_h>0$). Each void tree is associated to a pair $(\Gamma_\ell,\mu_j^\ell)$,
$\ell\leq h$, and we simply take $p_\ell$ copies of the tree $\mathcal{T}(f_\sigma^i)$,
$\sigma=\sigma_{\Gamma_\ell,\mu_j^\ell}$.
\begin{remark}
In order to construct Newton maps, we followed some conventions in \S\ref{sec-newton} which may not pass
to transversal sections. It is easily seen that the results do not depend
on the particular choice of conventions. In particular, everything works if
one does not choose a tuple $\mathbf{u}$ in \S\ref{sec-newton} but a $p^{\text{th}}$-root
of $\mu_j$ instead.
\end{remark}
\subsection{Decorations of $\mathcal{T}(f^i)$.}
\mbox{}
In order to finish the construction of the Newton trees $\mathcal{T}^i(f)$ we need
to compute the decorations. In order to do it in a simpler way we define
new decorations to the Newton tree.
\begin{definition}
The \emph{prime local decorations} $\langle R_1^v,\dots,R_d^v;p^v\rangle$
of a vertex~$v$ are defined as follows. They coincide with the local decoration
if $v$ is in the first vertical tree. If not, they are computed with the following
recursive formula (as in Lemma~\ref{lema-newton-tree-Q}) where $w$ is the preceding
vertex:
$$
R_j^v:=q_j^v+\frac{p^v R_j^w p^w}{\gcd(R_j^w,p^w)^2}.
$$
\end{definition}
Note that it is possible to compute local decorations from prime local decorations
(and viceversa). Moreover, they coincide if the decorations are \emph{coprime enough},
e.g., in the curve case. The reason to define these decorations is that they behave
better when passing to transversal section since they really depend on the
quotients $\frac{q_j}{p}$ and not on the pairs $(q_j,p)$.
In order to decorate the Newton trees of the transversal sections the strategy
is as follows:
\begin{enumerate}
\enet{\rm(TSNT\arabic{enumi})}
\item In $\mathcal{T}(f)$ compute the prime local decorations from the local decorations.
\item Pick a vertex $v^i$ in $\mathcal{T}^i(f)$, and consider the prime local decorations
of the vertex $v$ in $\mathcal{T}(f)$ which originates~$v^i$. Forget the $i^{\text{th}}$
coordinate and make the decoration coprime as in Case~\ref{caso2}.
\item Obtain the local decoration of $v^i$ from the prime local decorations.
\end{enumerate}
\subsection{Examples of transversal sections.}
\mbox{}
We illustrate the above theory with some examples.
\begin{example}\label{ex-sec-trans-2}
The tree of Figure~\ref{fig:desQO9nn1} corresponds to
$$
f(x_1,x_2,x_3,z)=(z^7-x_1^2)^2(z^3-x_1^5x_2x_3)+x_1^{10}x_2x_3\in K[[x_1,x_2,x_3]][z]
$$
and illustrates Case~\ref{caso3}.
\begin{figure}[ht]
\centering
\subfigure[{$K[[x_1,x_2,x_3]][z]$}]{
\includegraphics[scale=1]{desQO9nn1}
\label{fig:desQO9nn1}
}
\hfil
\subfigure[{$f_0$}]{
\includegraphics[scale=1]{desQO9nn2}
\label{fig:desQO9nn2}
}
\hfil
\subfigure[{$f_j$}]{
\includegraphics[scale=1]{desQO9nn3}
\label{fig:desQO9nn3}
}
\caption{Newton trees of Example~\ref{ex-sec-trans-2}.}
\label{desQO9nn}
\end{figure}
If we consider $f\in K_1[[x_2,x_3]][z]$, we have $f(x_1,0,0,z)=(z^7-x_1^2)^2 z^3$, i.e,
$f=f_0\prod_{j=1}^7 f_j$ where $f_0(x_1,0,0,0)=0$ and $\{f_j(x_1,0,0,0)\}_{j=1}^7=\{b\in K_1\mid b^7=x_1^2\}$.
Hence we obtain $8$ disjoint Newton trees, one for $f_0$ and $7$ equal trees for $f_j$, $j=1,\dots,7$.
\end{example}
These following examples illustrate Case~\ref{caso3} at the second step of the algorithm.
\begin{example}\label{ex-sec-trans-3}
Figure~\ref{fig:desQO10nn1} shows the tree of
$$
f(x_1,x_2,z)=((z^7-x_1^2x_2^3)^2-x_1^5x_2^6)^2+x_1^{11}x_2^{13}\in K[[x_1,x_2]][z],
$$
while Figure~\ref{fig:desQO10nn2} shows the one for $f\in K_1[[x_2]][z]$.
\begin{figure}[ht]
\centering
\subfigure[{$K[[x_1,x_2]][z]$}]{
\includegraphics[scale=1]{desQO10nn1}
\label{fig:desQO10nn1}
}
\hfil
\subfigure[{$K_1[[x_2]][z]$}]{
\includegraphics[scale=1]{desQO10nn2}
\label{fig:desQO10nn2}
}
\caption{Newton trees of Example~\ref{ex-sec-trans-3}.}
\label{desQO10nn}
\end{figure}
\end{example}
\begin{example}\label{ex-sec-trans-4}
Figure~\ref{fig:desQO11nn1} shows the tree of
$$
f(x_1,x_2,z)=(z^2-x_1^2 x_2^3)^6+ (z^2-x_1^2 x_2^3)^3 x_1^7 x_2^9+x_1^{15} x_2^{19}\in K[[x_1,x_2]][z].
$$
while Figure~\ref{fig:desQO11nn2} shows the one for $f\in K_1[[x_2]][z]$.
\begin{figure}[ht]
\centering
\subfigure[{$K[[x_1,x_2]][z]$}]{
\includegraphics[scale=1]{desQO11nn1}
\label{fig:desQO11nn1}
}
\hfil
\subfigure[{$K_1[[x_2]][z]$}]{
\includegraphics[scale=1]{desQO11nn2}
\label{fig:desQO11nn2}
}
\caption{Newton trees of Example~\ref{ex-sec-trans-4}.}
\label{desQO11nn}
\end{figure}
\end{example}
\subsection{Curve transversal sections}
\mbox{}
We summarize the previous process showing the computations for curve transversal sections.
Now we are describing the Newton trees of the curve transversal sections in the case where, in suitable coordinates,
the Newton tree of $f$ has only one arrow-head with positive multiplicity and no black box.
Consider the first vertex and fix $i\in\{1,\dots,d\}$. There are three cases.
\begin{vertex1}
$q_i\neq 0 ,\ \gcd(q_i,p)=1$.
\end{vertex1}
In this case, we have a first vertex decorated with $(q_i,p)$
and with a unique horizontal edge starting from this first vertex.
\begin{vertex1}
$q_i\neq 0,\ \gcd(q_i,p)=:c_i> 1$.
\end{vertex1}
We have a first vertex decorated with
$(q'_i,p_i)$, $q'_i:=\frac{q_i}{c_i}$, $p_i:=\frac{p_i}{c_i}$, and $c_i$ horizontal edges starting from the vertex.
\begin{vertex1}\label{fv3}
$q_i=0$.
\end{vertex1}
We are not studying the transversal section at the origin for $z$, but at some other points.
There are $p$ of them. In this case we have $p$ Newton trees which begin eventually with the next vertex
(as far as this situation is not repeated again).
Now we apply a Newton map $\sigma_{\Gamma,\mu}$ to go to the following vertex
whose decorations in $\mathcal{T}_{\mathcal{N}}(f_{\Gamma,\mu})$ are
$((r_1,\dots,r_n),p^1)$.
The local numerical data $(Q_1,\cdots,Q_d,p^1)$
and the prime local numerical data
$\langle R_1,\dots,R_d;p^1\rangle$ of the second vertex
satisfy
$$
Q_i=p^1 p \frac{q_i}{c_i}+r_i\quad
R_i=p^1 p \frac{q_i}{c_i^2}+r_i.
$$
Again we have three cases.
\begin{vertex2}
$\gcd(Q_i,p^1)=1$, $\forall i$.
\end{vertex2}
For all the transversal sections with a first vertex decorated with $(q_i^t,p^t)$
we add a new vertex to each of the edges
decorated with $(r_i+q_i^tp^tp^1,p^1)$ and we add a unique edge to this vertex. If we do not have a first vertex
(the case of First Vertex~\ref{fv3})
we begin the tree
with a first vertex decorated with $(r_i,p^1)$
and one horizontal edge starting from the vertex.
\begin{vertex2}
For some $i$, $\gcd(Q_i,p^1)=:c_i^1>1$ and $r_i\neq 0$
\end{vertex2}
For the
corresponding transversal section, if we have a first vertex decorated with
$(q_i^t,p^t)$ we add a new vertex decorated $(r_i+p^t p^1 \frac{q_i^t}{c_i},\frac{p^1}{c_i^1})$ to all the edges and
$c_i^1$ horizontal edges starting from the vertex.
If we do not have a first vertex we begin the tree
with a first vertex decorated with $(\frac{r_i}{c_i^1},\frac{p^1}{c_i^1})$ and we add $c_i^1$ horizontal edges.
\begin{vertex2}
For some $i$ we have $r_i=0$.
\end{vertex2}
If we have a first vertex decorated with
$(q_i^t,p^t)$, we stay on the same vertex. Then if we had already $p$
edges we should have $pp^1$ edges starting from the vertex.
If we don't have already any vertex then we will have $pp^1$ trees starting eventually
with the next vertex.
The global process continue with all their corresponding Newton maps. Note that in the curve
case prime local decorations and local decorations coincide.
\subsection{Obtaining $\mathcal{T}(f)$ from $\mathcal{T}(f^{I_i})$.}
\mbox{}
Now we will see how one can retrieve the quasi-ordinary
singularity from the transversal sections.
This is not always possible as shown in the following examples.
\begin{example}\label{ex-duple}
The quasi-ordinary polynomial
\begin{align*}
f_1=&((z^3-x_1^2)^2+x_1^{25} x_2^{11})((z^3-x_1^4)^2+x_1^{25}x_2^5)\\
f_2=&((z^3-x_1^2)^2+x_1^{25}x_2^5)((z^3-x_1^4)^2+x_1^{25} x_2^{11})
\end{align*}
have the same transversal sections but they do not have the same decorated Newton tree.
which are displayed in Figure~\ref{dessinduple}.
\begin{figure}[ht]
\centering
\subfigure[{$f_1$}]{
\includegraphics[scale=1]{dessinduple-1}
\label{fig:dessinduple-1}
}
\hfil
\subfigure[{$f_2$}]{
\includegraphics[scale=1]{dessinduple-2}
\label{fig:dessinduple-2}
}
\caption{}
\label{dessinduple}
\end{figure}
\end{example}
\begin{example}
Let $f_n=z^n-x_1 x_2$. If $n_1\neq n_2$ then the decorated Newton tree of $f_{n_1}$ is not the same than the decorated
Newton tree of $f_{n_2}$. If we do not keep the system of coordinates,
i.e. exchanging the roles of $z$ and $x_i$ in each transversal section, we get
the same trees, see Figure~\ref{dessinduple2}, for any~$n$.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1]{dessinduple2}
\caption{}
\label{dessinduple2}
\end{center}
\end{figure}
\end{example}
\begin{example}\label{ex-duple1}
The transversal sections of the quasi-ordinary polynomial
\begin{align*}
f_1=&((z^2-x_1^3 x_2)^2+x_1^5 x_2^3 z)((z^2-x_1^3 x_2^4)^2+x_1^6x_2^9 z)\\
f_2=&((z^2-x_1^3 x_2^4)^2+x_1^5 x_2^9z)((z^2-x_1^3 x_2)^2+ x_1^6 x_2^3 z)
\end{align*}
have the same decorated Newton trees but the two germs
have different decorated Newton trees, see Figure~\ref{dessibduple1}.
\begin{figure}
\begin{center}
\includegraphics[scale=1]{dessibduple1}
\caption{}
\label{dessibduple1}
\end{center}
\end{figure}
\end{example}
\begin{definition}
The \emph{multiplicity} of a tree is the sum over all vertices~$v$ of the number
of horizontal edges arising from~$v$.
\end{definition}
\begin{theorem}\label{thm-recover}
Assume that $f$ is a quasi-ordinary polynomial in $\bk[[\bdx]][z]$ whose Newton tree in
suitable coordinates has only one arrow-head with positive multiplicity and no black box.
Then, there is a unique way to recover
the decorated Newton tree of $f$ from the
decorated Newton trees, in the same system of coordinates, of all its curve transversal sections.
\end{theorem}
\begin{proof}
The proof can be done by induction on the sum of the multiplicities of the curve transversal sections.
If this sum is equal to one, it means that we may assume
$f_\Gamma(\bdx,z)=\bdx^{\mathbf{N}}(z^p-x_1^q)$ and the result follows immediately.
Let us assume that the result is true when the sum of the multiplicities is less than $n$, and assume
this sum for $f$ equals $n>1$. In this case
$$
f_\Gamma(\bdx,z)=\bdx^{\mathbf{N}}(z^p-\bdx^{\mathbf{q}})^{m},
$$
for $q_1\geq\dots q_k>q_{k+1}=\dots= q_d=0$; recall that $f$ hasn only
one arrow-head with positive multiplicity and no black box. Then we must have either $m\geq 2$ or $k\geq 2$.
If $k=1$ the results follows easily by induction. Let us consider the case $k\geq 2$.
s we have seen before,
after the Newton map we get in general
bunches of disconnected trees.
Moreover, all the trees in the bunch corresponding to the $i^{\text{th}}$-coordinate
are equal, so we need only to retain one of them and the number of trees in the bunch.
We start the construction of the tree of $f$ from the trees of the curve transversal sections.
We consider all curve transversal sections with a complete tree. There is at least one. We consider the first vertices of these trees. We consider the decorations
$(q_i,p_i)$ of these vertices.
Let $p=\max \{p_1,\dots,p_d\}$. We know that $p_i$ divides $p$.
Let $p=p_i c_i$. Then the first vertex of the tree of $f$ is decorated by $((q_1 c_1,\cdots,q_k c_k,0\cdots),p)$.
We consider the Newton tree
of the transforms of the curve transversal sections
of~$f$. The sum of the multiplicities is strictly less than~$n$ and hence
we can apply induction hypothesis and all the results in this sections.
Then we can reconstruct our decorated tree from the transversal sections.
There are many branches in the transversal sections, but because there are all the same
there is a unique way to reconstruct the tree of $f$.
\end{proof}
Now we prove that we have all the information of the global numerical data can be found
in the trees of the transversal sections.
\begin{proposition}
For all vertex $v$ of the tree of a quasi ordinary polynomial, and all $i=1,\cdots ,d$,
the terms $\frac{N_{i,v}}{c_i}$ can be retrieved from the decorations
of the trees of the transversal sections, except
the order $N$ in $z$ of $f$ when the transversal section is not divisible by the variables $x_j$.
\end{proposition}
\begin{proof}
The proof relies on the fact that the global decorations are given in terms of the equations
of the faces of the Newton diagrams.
If the vertex we are considering corresponds to a face of the Newton diagram of $f$,
we have the following cases:
\enumerate
\item
By projection on $\{x_i=0\}$ the face is parallel to the $z$-axis:
In this case, the vertex is the first vertex on
the top of the tree and its decoration is $((0,\cdots, N_i, \cdots ,0))$. The vertex will
disappear, but the decorations are $0$.
\item
By projection the face $\Gamma$ gives a face of the Newton diagram of the transversal section.
Either this face is different from the projections of faces intersecting $\Gamma$, and it will give
a vertex with the decoration $((N_1,\cdots ,N_d))$,
or it is the same face as one of the projections, then the vertex will disappear, but the vertex
representing the face have the desired decorations.
Now, if we are on a Newton diagram which appeared in the Newton process.
We have again different cases:
\begin{enumerate}
\item The Newton process is a linear change of variable for $z$.
In this case there are two consecutive vertices
which appear with the same $N_j$, $j\neq i$.
Only one appears in the transversal section, but its bears the decoration we are
interested in.
\item The Newton process is a Newton map for the transversal section,
with the same roots. In this case, either we find the vertex in the transversal sections,
or it disappears because there are two faces which project on the same. In any case
the remaining vertex gives the decoration we need.
\item The Newton process is a Newton map for the transversal sections
with more roots. This case is the same as the case above but we have many copies
of the Newton diagram. Then the decoration appears many times.\qedhere
\end{enumerate}
\end{proof}
We can notice that not only we recover the decorations of the tree of $f$, but
no new decorations appear in the transversal sections.
\begin{remark}
A particular useful case of the preceeding proposition is that all the $N_i$'s can be retrieved
using the germs of curves obtained setting all variables $x_j$ except one as constant. This
is a fundamental fact in the proof of the monodromy
conjecture for quasi-ordinary polynomial \cite{aclm:05}.
\end{remark}
\bibliographystyle{amsplain}
|
{
"timestamp": "2012-03-09T02:01:23",
"yymm": "1203",
"arxiv_id": "1203.1704",
"language": "en",
"url": "https://arxiv.org/abs/1203.1704"
}
|
\section{Introduction}\mylabel{sec:intro}
Graphs in this paper are allowed to have loops and multiple edges.
A graph is a {\em minor} of another if the first can be obtained from
a subgraph of the second by contracting edges. An {\em $H$ minor}
is a minor isomorphic to $H$.
A graph $G$ is {\em apex} if it has a vertex $v$ such that
$G\backslash v$ is planar.
(We use $\backslash$ for deletion.) J\o rgensen~\cite{Jor} made the following beautiful conjecture.
\begin{conjecture}
\mylabel{con:jorgensen}
Every $6$-connected graph with no $K_6$ minor is apex.
\end{conjecture}
In a companion paper~\cite{KawNorThoWollarge}
we prove that Conjecture~\ref{con:jorgensen}
holds for all sufficiently big $6$-connected graphs. Here we
establish the first step toward that goal, the following.
\begin{theorem}
\mylabel{main}
For every integer $w\ge1$ there exists an integer $N$ such that
every $6$-connected graph on at least $N$ vertices and tree-width
at most $w$ with no $K_6$ minor is apex.
\end{theorem}
We define tree-width later in this section, but let us discuss
J\o rgensen's conjecture first.
It is related to Hadwiger's conjecture~\cite{Had}, the following.
\begin{conjecture}
\mylabel{con:hadwiger}
For every integer $t\ge1$, if a loopless graph has no $K_t$ minor, then it
is $(t-1)$-colorable.
\end{conjecture}
Hadwiger's conjecture is known for $t\le6$.
It is trivial for $t\le3$, and is still fairly easy for $t=4$,
as shown by Dirac~\cite{Dirproperty}.
However, for $t\ge5$ Hadwiger's conjecture implies the Four-Color Theorem.
Wagner~\cite{Wag37} gave a structural characterization of graphs
with no $K_5$ minor, which implies that Hadwiger's conjecture for $t=5$
is actually equivalent to the Four-Color Theorem.
The same conclusion has been obtained for $t=6$ in~\cite{RobSeyThoHad} by showing
that a minimal counterexample to Hadwiger's conjecture for $t=6$
is apex.
The proof uses an earlier result of Mader~\cite{MadHomkrit}
that every minimal counterexample to Conjecture~\ref{con:hadwiger}
is $6$-connected.
Thus Conjecture~\ref{con:jorgensen}, if true, would give more
structural information.
Furthermore, the structure of all graphs with no $K_6$ minor is not known,
and appears complicated and difficult.
Thus obtaining a structural characterization of graphs with no
$K_6$ minor, an analogue of Wagner's theorem mentioned above,
appears beyond reach at the moment.
On the other hand, Conjecture~\ref{con:jorgensen} provides a nice
necessary and sufficient condition for $6$-connected graphs.
Unfortunately, it, too, appears to be a difficult problem.
Let us turn to tree-width and our proof method.
Tree-width of a graph was first defined by Halin~\cite{HalSfunctions},
and was later rediscovered in~\cite{RobSeyGM3}, and, independently,
in~\cite{ArnProLinear}. The definition is as follows.
A {\em tree-decomposition} of a graph $G$ is a pair $(T,Y)$, where $T$
is a tree and $Y$ is a family $\{Y_t \mid t \in V(T)\}$ of vertex sets
$Y_t\subseteq V(G)$, such that the following two properties hold:
\begin{enumerate}
\item[(W1)] $\bigcup_{t \in V(T)} Y_t = V(G)$, and every edge of $G$ has
both ends in some $Y_t$.
\item[(W2)] If $t,t',t''\in V(T)$ and $t'$ lies on the path in $T$
between $t$ and $t''$, then $Y_t \cap Y_{t''} \subseteq Y_{t'}$.
\end{enumerate}
The {\em width} of a tree-decomposition $(T,Y)$ is $\max_{t\in V(T)} (|Y_t|-1)$,
and the {\em tree-width} of a graph $G$ is the minimum width of a
tree-decomposition of $G$.
Our proof of Theorem~\ref{main} proceeds as follows. We choose a tree-decomposition $(T,W)$ of
$G$ of width $w$ with no ``redundancies".
It follows easily that if $T$ has a vertex of large degree, then
$G$ has a $K_6$ minor, and so we may assume that $T$ has a long path.
For the rest of the proof we concentrate our effort on this long path.
Since other branches of $T$ are inconsequential, we convert $(T,W)$
to a ``linear decomposition", which is really just a tree-decomposition,
where the underlying tree is a path, but we find it more convenient
to number the sets of vertices $W_0,W_1,\ldots,W_l$, rather than
index them by the vertices of a path. At this point we no longer
require that the width be bounded; all we need is that the intersections
$W_{i-1}\cap W_i$ are bounded and that $l$ is sufficiently large.
Thus we may assume (by trimming our linear decomposition) that
all the sets $W_{i-1}\cap W_i$ have the same size, say $q$. Furthermore,
it can be arranged (by invoking the result from~\cite{RThoMenger}
or by a direct argument) that there exist $q$ disjoint paths
$P_1,P_2,\ldots,P_q$
from $W_{0}\cap W_1$ to $W_{l-1}\cap W_l$.
We apply the pigeon hole principle many times, each time trimming
the linear decomposition, but still keeping it sufficiently long,
to make sure that if the subgraph $G[W_i]$ has some useful property for some
$i\in\{1,2,\ldots,l-1\}$, then all the graphs $G[W_i]$ have that property
for all $i\in\{1,2,\ldots,l-1\}$.
A prime example of a useful property is the existence of
two disjoint paths $Q_1,Q_2$ in $G[W_i]$,
internally disjoint from $P_1,P_2,\ldots,P_q$,
with ends $u_1,v_1$ and $u_2,v_2$, respectively, such that
$u_1,v_2\in V(P_1)$, $u_2,v_1\in V(P_2)$ and they appear on those
paths in the order listed as $P_1$ and $P_2$ are traversed from
$W_{0}\cap W_1$ to $W_{l-1}\cap W_l$.
In those circumstances we say that $P_1$ and $P_2$ {\em twist} in $W_i$.
Thus, in particular, we can arrange that if two paths $P_j$ and $P_{j'}$ twist in
$W_i$ for some $i\in\{1,2,\ldots,l-1\}$, then they twist in
$W_i$ for all $i\in\{1,2,\ldots,l-1\}$.
On the other hand, if two paths $P_j$ and $P_{j'}$ twist in
$W_i$ for all $i\in\{1,2,\ldots,l-1\}$ and $l$ is not too small, then
$G$ has a $K_6$ minor.
This is the sort of argument we will be using, but the details are too
numerous to be described in their entirety here.
In~\cite{KawNorThoWollarge} we use Theorem~\ref{main} to prove
J\o rgensen's conjecture for sufficiently big graphs, formally the
following:
\begin{theorem}
\mylabel{future}
There exists an integer $N$ such that
every $6$-connected graph on at least $N$ vertices with no $K_6$ minor is apex.
\end{theorem}
How does Theorem~\ref{main} help in the proof of Theorem~\ref{future}?
By the excluded grid theorem of Robertson and Seymour~\cite{RobSeyGM5}
(see also \cite{DieGorJenTho,ReedBCC,RobSeyThoQuickPlanar})
\nocite{DieGorJenTho}
\nocite{ReedBCC}
\nocite{RobSeyThoQuickPlanar}
it suffices to prove Theorem~\ref{future} for graphs that have a sufficiently large
grid minor. We then analyze how the remainder of the graph attaches to
the grid. We refer to~\cite{KawNorThoWollarge} for details.
The paper is organized as follows.
In Section~\ref{sec:reroute} we state a few lemmas, mostly from other papers.
In Section~\ref{sec:linear} we convert a tree-decomposition into a
linear decomposition,
as described above, and we prove that the linear decomposition can be
chosen with some additional useful properties.
In Section~\ref{sec:auxil} we introduce the auxiliary graph---its vertices
are the paths $P_1,P_2,\ldots,P_q$, and two of them are adjacent if
they are joined by a path avoiding all the other paths $P_1,P_2,\ldots,P_q$.
By joined we mean in some or every $W_i$; by now the two are equivalent.
We use the auxiliary graph to further refine the linear decomposition.
A {\em core} is a component of the subgraph of the auxiliary graph induced
by those of the paths $P_1,P_2,\ldots,P_q$ that have at least one edge.
We show, among other things, that every core is a path or a cycle.
In Section~\ref{sec:pin} we use the theory of ``non-planar extensions"
of planar graphs from~\cite{RobSeyThoExt} to get under control adjacencies
in the auxiliary graph of those paths $P_i$ that are trivial.
In Section~\ref{sec:taming} we further refine our linear decomposition
to arrange that the part of $G$ that corresponds to a core can be drawn
either in a disk or in a cylinder, depending on whether the core is
a path or a cycle.
In Section~\ref{sec:control} we digress and prove a slight extension
of a result of DeVos and Seymour~\cite{DevSeyExt3col}.
Finally, in Section~\ref{sec:cyl} we essentially complete the proof of Theorem~\ref{main} in the case when some
core of the auxiliary graph is a cycle, and in Section~\ref{sec:planar}
we do the same when some core is a path.
\section{Rerouting and rural societies}\mylabel{sec:reroute}
Let $S$ be a subgraph of a graph $G$. An {\em $S$-bridge} in $G$ is a
connected subgraph $B$ of $G$ such that $E(B)\cap E(S)=\emptyset$
and either $E(B)$ consists of a unique edge with both ends in $S$, or for
some component $C$ of $G\backslash V(S)$ the set $E(B)$ consists of all
edges of $G$ with at least one end in $V(C)$. The vertices in
$V(B)\cap V(S)$ are called the {\em attachments} of $B$. We say that an $S$-bridge $B$ \emph{attaches to} a subgraph $H$ of $S$ if $V(H) \cap V(B) \neq \emptyset$.
Now let $S$ be such that no block of $S$ is a cycle.
By a {\em segment of $S$} we mean a maximal subpath $P$ of $S$ such that
every internal vertex of $P$ has degree two in $S$.
It follows that the segments
of $S$ are uniquely determined. Now if $B$ is an $S$-bridge of $G$, then we
say that $B$ is {\em unstable} if some segment of $S$ includes all the
attachments of $B$, and otherwise we say that $B$ is
{\em stable}. Our next lemma says that it is possible to make all
bridges stable by making the following ``local" changes. Let $G$ and $S$ be
as before, let $P$ be a segment of $S$ of length at least two,
and let $Q$ be a path
in $G$ with ends $x,y\in V(P)$ and otherwise disjoint from $S$.
Let $S'$ be obtained from $S$ by replacing the path $xPy$
(the subpath of $P$ with ends $x$ and $y$) by $Q$;
then we say that
$S'$ was obtained from $S$ by {\em rerouting} $P$ along $Q$, or
simply that $S'$ was obtained from $S$ by {\em rerouting}. Please
note that $P$ is required to have length at least two,
and hence this relation is not symmetric. We say that the
rerouting is {\em proper} if
all the attachments
of the $S$-bridge that contains $Q$ belong to $P$.
The following lemma is essentially due to Tutte.
\begin{lemma}
\mylabel{prestable}
Let $G$ be a simple graph,
and let $S$ be a subgraph of $G$
such that no block of $S$ is a cycle.
Then there exists a subgraph $S'$ of $G$
obtained from $S$ by a sequence of proper reroutings
such that if an $S'$-bridge $B$ of $G$ is unstable, say all its attachments
belong to a segment $P$ of $S'$, then there exist
vertices
$x,y\in V(P)$ such that
some component of $G\backslash\{x,y\}$ includes a vertex of $B$
and is disjoint from $S'\backslash V(P)$.
\end{lemma}
{\noindent\bf Proof. } We may choose a subgraph $S'$ of $G$ obtained from $S$ by a
sequence of proper reroutings such that the number of vertices that
belong to stable $S'$-bridges is maximum, and, subject to that,
$|V(S')|$ is minimum. We will show that $S'$ is as
desired. To that end we may assume that
$B$ is an $S'$-bridge of $G$ with all its attachments in a
segment $P$ of $S'$.
Let $v_0, v_1,\dots, v_k$ be distinct vertices of $P$, listed in order of
occurrence on $P$ such that $v_0$ and $v_k$ are the ends of $P$ and
$\{v_1,\dots, v_{k-1}\}$ is the set of all internal vertices of $P$
that are attachments of a stable $S'$-bridge.
We claim that if
$u,v$ are two attachments of $B$, then no $v_i$ belongs to
the interior of $uPv$.
To prove this suppose to the contrary that $v_i$ is an internal
vertex of $uPv$.
But then replacing $uPv$ by an induced subpath of $B$ with ends $u,v$
and otherwise disjoint from $S'$ is a proper rerouting that produces
a graph $S''$ with strictly more vertices belonging to stable
$S''$-bridges, contrary to the choice of $S'$.
This proves our claim that no $v_i$ belongs to the interior of $uPv$.
But then for some $i=1,2,\ldots,k$ the path $v_{i-1}Pv_i$
includes all attachments of $B$.
Since $G$ has no parallel edges, the same argument (using the minimality
of $|V(S')|$) now shows that $V(B)-\{v_{i-1},v_i\} \neq \emptyset$.
Consequently some component $J$ of $G\backslash \{v_{i-1},v_i\}$
includes a vertex of $B$. It follows that $B\backslash \{v_{i-1},v_i\}$
is a subgraph of $J$.
Now $B$ has all its attachments in
$v_{i-1}Pv_i$, the interior of $v_{i-1}Pv_i$ includes no
attachment of a stable $S'$-bridge, and
(by what we have shown about $B$) every unstable $S'$-bridge with
an attachment in the interior of $v_{i-1}Pv_i$ has all its attachments
in $v_{i-1}Pv_i$.
It follows that $J$ is disjoint
from $S'\backslash V(P)$, as desired.~$\square$\bigskip
We deduce the following corollary.
\begin{theorem}
\mylabel{stable}
Let $G$ be a $3$-connected graph,
and let $S$ be a subgraph of $G$
with at least two segments such that no block of $S$ is a cycle.
Then there exists a subgraph $S'$ of $G$
obtained from $S$ by a sequence of proper reroutings
such that every $S'$-bridge is stable.
\end{theorem}
We will need the following lemma, a special case of
\cite[Lemma 3.2]{KawNorThoWollarge}.
A {\em linkage} in a graph is a set $\cal P$ of disjoint paths.
If $A,B$ are sets such that each $P\in\cal P$ has one end in $A$ and the
other in $B$, then we say that $\cal P$ is a {\em linkage from $A$ to $B$}.
The {\em graph of the linkage $\cal P$} is the union of all $P\in\cal P$.
Occasionally we will use $\cal P$ in reference to the graph of $\cal P$;
thus we will use $V({\cal P})$ to denote the vertex-set of the graph of $\cal P$
and we will also speak of $\cal P$-bridges.
A path is {\em trivial} if it has exactly one vertex and {\em non-trivial}
otherwise.
By a $\cal P$-path we mean a non-trivial path with both ends in
$V({\cal P})$ and otherwise disjoint from the graph of $\cal P$.
\begin{lemma}
\mylabel{planarstrip}
Let $k\ge1$ be an integer,
let ${\cal P}=\{P_1,P_2,\ldots,P_k\}$ be a linkage in a graph $G$,
where $P_i$ has distinct ends $u_i$ and $v_i$, and let every $\cal P$-bridge
of $G$ be stable.
Assume that $G$ cannot be drawn in a disk with
$u_1,u_2,\ldots,u_k,v_k,v_{k-1},\ldots,v_1$ drawn on the boundary
of the disk in order and the paths $P_1$ and $P_k$ also drawn
on the boundary, and assume also that
there is no set $X\subseteq V(G)$ of size at most three
such that some component of $G\backslash X$ is disjoint from
$\{u_1,u_2,\ldots,u_k,v_1,v_2,\ldots,v_k\}$.
Then either
\par\hangindent0pt\mytextindent{(i)} there exist integers $i,j\in\{1,2,\ldots,k\}$ with
$|i-j|>1$ and a $\cal P$-path $Q$
in $G$ with one end in $V(P_{i})$ and
the other end in $V(P_{j})$, or
\par\hangindent0pt\mytextindent{(ii)} there exist an integer $i\in\{1,2,\ldots,k-1\}$ and two
disjoint $\cal P$-paths $Q_1$, $Q_2$ in $G$ such that $Q_j$ has ends $x_j,y_j$,
the vertices
$u_{i},x_1,x_2,v_i$ occur on $P_i$ in the order listed and
$u_{i+1},y_2,y_1,v_{i+1}$ occur on $P_{i+1}$ in the order listed, or
\par\hangindent0pt\mytextindent{(iii)} there exist an integer $i\in \{2,3,\ldots,k-1\}$ and three
$\cal P$-paths $Q_0$, $Q_{1}$, $Q_2$ such that $Q_j$ has ends $x_j$ and $y_j$,
we have
$x_0,y_0\in V(P_i)$, the vertices $x_1,x_2$ are internal vertices of $x_0P_i y_0$,
$y_1\in V(P_{i-1})$, $y_2\in V(P_{i+1})$,
and the paths $Q_0$, $Q_{1}$, $Q_2$ are pairwise disjoint, except possibly
for $x_1=x_2$.
\end{lemma}
By a {\em cylinder} we mean the surface obtained from a sphere by
removing the interiors of two disjoint closed disks $\Delta_1,\Delta_2$.
By a clockwise ordering of the boundary of $\Delta_i$ we mean
the cyclic ordering that traverses around $\Delta_i$ in clockwise
direction.
We need a slight variation of the previous lemma. We omit its proof,
because it is completely analogous.
\begin{lemma}
\mylabel{cylinderstrip}
Let $k\ge3$ be an integer,
let ${\cal P}=\{P_1,P_2,\ldots,P_k\}$ be a linkage in a graph $G$,
where $P_i$ has distinct ends $u_i$ and $v_i$, and let every $\cal P$-bridge
of $G$ be stable.
Assume that $G$ cannot be drawn in a cylinder with
$u_1,u_2,\ldots,u_k$ drawn on one boundary component in the clockwise
cyclic order listed and $v_k,v_{k-1},\ldots,v_1$ drawn on the other boundary
component in the clockwise cyclic order listed, assume also that
there is no set $X\subseteq V(G)$ of size at most three
such that some component of $G\backslash X$ is disjoint from
$\{u_1,u_2,\ldots,u_k,v_1,v_2,\ldots,v_k\}$,
and finally assume that if $k=3$, then no $\cal P$-bridge has vertices
of attachment on all three members of $\cal P$.
Then either
\par\hangindent0pt\mytextindent{(i)} there exist integers $i,j\in\{1,2,\ldots,k\}$ with
$|i-j|>1$ and $\{i,j\}\ne\{1,k\}$ and a $\cal P$-path $Q$
in $G$ with one end in $V(P_{i})$ and
the other end in $V(P_{j})$, or
\par\hangindent0pt\mytextindent{(ii)} there exist an integer $i\in\{1,2,\ldots,k-1\}$ and two
disjoint $\cal P$-paths $Q_1$, $Q_2$ in $G$ such that $Q_j$ has ends $x_jy_j$,
the vertices
$u_{i},x_1,x_2,v_i$ occur on $P_i$ in the order listed and
$u_{i+1},y_2,y_1,v_{i+1}$ occur on $P_{i+1}$ in the order listed, or
\par\hangindent0pt\mytextindent{(iii)} there exist an integer $i=1,2,\ldots,k$ and three
$\cal P$-paths $Q_0$, $Q_{1}$, $Q_2$ such that $Q_j$ has ends $x_j$ and $y_j$,
we have
$x_0,y_0\in V(P_i)$, the vertices $x_1,x_2$ are internal vertices of $x_0P_i y_0$,
$y_1\in V(P_{i-1})$, $y_2\in V(P_{i+1})$,
and the paths $Q_0$, $Q_{1}$, $Q_2$ are pairwise disjoint, except possibly
for $x_1=x_2$, where $P_0$ means $P_k$ and $P_{k+1}$ means $P_1$.
\end{lemma}
We finish the section by introducing several notions and a theorem from~\cite{RobSeyGM9}. We
will make use of them in the last two sections. Let $\Omega$ be a cyclic permutation of the elements of some set;
we denote this set by $V(\Omega)$. A {\em society} is a pair $(G,\Omega)$, where $G$ is a graph, and $\Omega$
is a cyclic permutation with $V(\Omega)\subseteq V(G)$. A society $(G,\Omega)$ is {\em rural} if $G$ can be drawn in a disk with $V(\Omega)$ drawn on the boundary of the disk in the order given by $\Omega$. A {\em cross} in $(G, \Omega)$ is a pair of disjoint non-trivial paths $P_1$ and $P_2$ with ends $u_1$, $v_1$ and $u_2$,$v_2$ respectively, so that $u_1, u_2, v_1, v_2 \in V(\Omega)$ appear in $\Omega$ in this or reverse order, and $P_1$ and $P_2$ are otherwise disjoint from $V(\Omega)$.
A {\em separation} of a graph $G$ is a pair $(A,B)$ such that $A\cup B=V(G)$
and there is no edge with one end in $A-B$ and the other end in $B-A$.
The order of $(A,B)$ is $|A\cap B|$.
We say that a society $(G,\Omega)$ is {\em $4$-connected}
if there is no separation $(A,B)$ of $G$ of order at most three
with $V(\Omega)\subseteq A$ and $B-A\ne\emptyset$.
The next theorem follows from Theorems (2.3) and (2.4) in~\cite{RobSeyGM9}.
\begin{theorem}
\mylabel{thm:RSsociety}
Let $(G,\Omega)$ be a $4$-connected society with no cross. Then $(G,\Omega)$ is rural.
\end{theorem}
\section{Linear decompositions}\mylabel{sec:linear}
In this section we show that it suffices to prove Theorem~\ref{main}
for graphs that have a ``linear decomposition" of bounded ``adhesion".
Similar techniques have been developed and used
in~\cite{BohMahMoh,BohKawMahMoh,OpoOxlTho}.
A linear decomposition is really a tree-decomposition, where the
underlying tree is a path, but it is more convenient to number the
sets by integers rather than vertices of a path. Thus a linear
decomposition of a graph $G$ is a family of sets
${\cal W}=(W_0,W_1,\ldots,W_l)$ such that
\begin{enumerate}
\item[(L1)] $\bigcup_{i=0}^l W_i = V(G)$, and every edge of $G$ has
both ends in some $W_i$, and
\item[(L2)] if $0\le i\le j\le k\le l$, then
$W_i \cap W_{k} \subseteq W_{j}$.
\end{enumerate}
We say that the \emph{length} of $\cal W$ is $l$.
In the proof of Theorem~\ref{main} we will need linear decompositions
satisfying the following additional properties:
\begin{enumerate}
\item[(L3)] there is an integer $q$ such that
$|W_{i-1} \cap W_{i}| = q$ for all $i = 1,2, \dots, l$,
\item[(L4)] for every $i = 1,2, \dots, l$,
$W_{i-1}\ne W_{i-1} \cap W_{i} \neq W_{i}$,
\item[(L5)] there exists a linkage
from $W_0 \cap W_1$ to $W_{l-1} \cap W_{l}$ of cardinality $q$.
\end{enumerate}
\noindent
If a linear decomposition satisfies (L3), then we say that it has
{\em adhesion} $q$.
A linkage as in (L5) will be called a {\em foundational linkage}
and its members will be called {\em foundational paths}.
We will need more properties,
but first we show that we can assume that our graph has a linear
decomposition satisfying (L1)--(L5).
In the proof we will need the following additional properties of
tree-decompositions, stated using the same notation as (W1)--(W2):
\begin{enumerate}
\item[(W3)] for every two vertices $t, t'$ of $T$ and every positive
integer $k$, either there are $k$ disjoint paths in $G$ between $Y_t$
and $Y_{t'}$, or there is a vertex $t''$
of $T$ on the path between $t$ and $t'$ such that $|Y_{t''}| < k$,
\item[(W4)] if $t,t'$ are distinct vertices of $T$, then $Y_t \not= Y_{t'}$,
and
\item[(W5)] if $t_0 \in V(T)$ and $W$ is a component of $T - t_0$, then
$ \bigcup_{t\in V(W)} Y_t \setminus Y_{t_0} \not = \emptyset$.
\end{enumerate}
\begin{lemma}
\mylabel{lem:l5}
For all integers $k,l,p,w\ge0$ there exists an integer $N$ with
the following property.
If $G$ is a $p$-connected graph of tree-width at most $w$ with
at least $N$ vertices, then either $G$ has a minor isomorphic to
$K_{p,k}$, or $G$ has a linear decomposition of
length at least $l$ and adhesion at most $w$ satisfying {\rm (L1)--(L5)}.
\end{lemma}
{\noindent\bf Proof. }
Let $k,l,w\ge0$ be given integers. We will use the proof technique
of~\cite[Theorem~3.1]{OpoOxlTho} with the constants
$n_1,n_6,n_7,n_8$ and $n_9$ redefined as follows:
Let $n_1:=w$, $n_6:=l$, $n_7:=n_6^{n_1+1}$,
$n_8:={n_1\choose p}(k-1)$, and
$$n_9 := 2+n_8+n_8(n_8-1)+\cdots+n_8(n_8-1)^{\lceil n_7/2\rceil-2}.$$
We will show that $N:=n_1n_9$ satisfies the lemma.
To this end let $G$ be as stated.
The argument in Claims (1)--(4) of~\cite[Theorem~3.1]{OpoOxlTho}
shows that $G$ either has a minor isomorphic to
$K_{p,k}$, or a tree-decomposition $(T,Y)$
satisfying (W1)--(W5) such that $T$ has a path $R$ that includes
distinct vertices $r_1,r_2,\ldots,r_l$, appearing on $R$ in the order listed,
such that for some integer $q$ with $p\le q\le w$ we have that
$|Y_{r_i}|=q$ for all $i=1,2,\ldots,l$ and $|Y_r|\ge q$ for every
$r\in V(R)$ between $r_1$ and $r_l$.
It is easy to see that there exist subtrees
$T_0,T_1,\ldots,T_l$ of $T$ such that
\par\hangindent0pt\mytextindent{(i)} $T_0\cup T_1\cup\cdots\cup T_l=T$,
\par\hangindent0pt\mytextindent{(ii)} $T_i$ and $T_j$ are disjoint whenever $|i-j|\ge2$, and
\par\hangindent0pt\mytextindent{(iii)} $V(T_{i-1})\cap V(T_i)=\{r_i\}$ for all $i=1,2,\ldots, l$.
\noindent For $i=0,1,\ldots, l$ let $W_i$ be the union of $Y_t$
over all $t\in V(T_i)$. We claim that $(W_0,W_1,\ldots,W_l)$ is a linear decomposition of $G$
satisfying (L1)--(L5).
Property (L1) is satisfied by (W1) and (i). If $0\le i < j < k\le l$, then for every $t \in V(T_i)$ and $t' \in V(T_k)$
the path from $t$ to $t'$ in $T$ contains the path from $r_{i+1}$ to $r_k$. Therefore,
by (W2) and (iii), we have $Y_{t} \cap Y_{t'} \subseteq Y_{r_{j}}$ and, consequently, $W_{i} \cap W_{k} \subseteq Y_{r_j} \subseteq W_j$. Thus (L2) is satisfied. Similarly, we have $W_{i-1} \cap W_i = Y_{r_i}$, and, therefore, we have $|W_{i-1} \cap W_i|=q$, implying (L3). For $1 < i \leq l$ we have $|Y_{r_{i-1}}|=|Y_{r_i}|=q$, and $Y_{r_i} \neq Y_{r_{i-1}}$ by (W4). Therefore $W_{i-1} - W_{i} \supseteq Y_{r_{i-1}} - Y_{r_i} \neq \emptyset$. By construction,
$T_0\setminus r_1$ is the union of components of $T \setminus r_1$ disjoint from $R$. It follows from (W5) that $W_{0} - W_1 = W_0 - Y_{r_1} \neq \emptyset$. By symmetry, $W_i - W_{i-1} \neq \emptyset$ for every $1 \leq i \leq l$, and (L4) holds. Finally, by (W3) and the choice of $r_1,r_2,\ldots,r_l$, there exists a linkage from $W_0 \cap W_1= Y_{r_1}$ to $W_{l-1} \cap W_l = Y_{r_l}$, implying (L5).
~$\square$\bigskip
Let $\cal P$ be a foundational linkage for a linear decomposition
${\cal W}=(W_0,W_1,\ldots,W_l)$ of a graph $G$, and let $i\in \{1,2,\ldots,l-1\}$.
We say that distinct foundational paths $P,P'\in\cal P$
are {\em bridge adjacent} in $W_i$ if there exists a $\cal P$-bridge in
$G[W_i]$ with an attachment in both $P$ and $P'$.
Given a fixed integer $p$ we
wish to consider the following properties of $\cal W$ and $\cal P$.
In our applications we will always have $p=6$.
\begin{enumerate}
\item[(L6)] for all $i\in \{1,2,\ldots,l-1\}$ and all non-trivial
paths $P\in\cal P$, if some $\cal P$-bridge of $G[W_i]$ has
at least one attachment in $P$ and no attachment in a non-trivial
foundational path other than $P$,
then $P$ is bridge adjacent in $W_i$
to at least $p-2$ trivial members of $\cal P$,
\item[(L7)] for every $P\in \cal P$,
if there exists an index $i\in\{1,2,\ldots,l-1\}$ such that
$P[W_i]$ is a trivial path,
then $P[W_k]$ is a trivial path for all $k = 1,2, \dots, l-1$,
\item[(L8)] for every two distinct paths $P,P'\in \cal P$,
if there exists an integer $k \in \{1, \dots, l-1\}$ such that
$P$ and $P'$ are bridge adjacent in $W_k$, then they are
bridge adjacent in $W_{k'}$ for all $k' \in \{1, \dots, l-1\}$.
\end{enumerate}
With respect to condition (L8) it may be helpful to point out that
for all $i = 1,2, \dots, l$ we have $W_{i-1}\cap W_i\subseteq V(\cal P)$,
and hence each $\cal P$-bridge $H$ of $G$ satisfies $V(H)\subseteq W_k$
for some $k\in\{0,1,\ldots,l\}$, even though this index $k$ need not be unique.
\begin{lemma}
\mylabel{lem:l6}
Let $p\ge 0$ be an integer, and
let $\cal W$ be a linear decomposition of a $p$-connected graph
satisfying {\rm (L1)--(L5)}.
Then $\cal W$ has a foundational linkage $\cal P$ satisfying {\rm (L6)}.
\end{lemma}
{\noindent\bf Proof. }
Let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be as stated.
By (L5) there exists a linkage $\cal P$ from $W_0\cap W_1$ to
$W_{l-1}\cap W_{l}$ of cardinality $q$.
Let $S$ be the union of all non-trivial paths in $\cal P$,
and let $H$ be obtained from $G[W_1\cup W_2\cup\cdots\cup W_{l-1}]$
by deleting all trivial paths in $\cal P$.
By Lemma~\ref{prestable} applied to $H$ and $S$ we may assume
(by changing $\cal P$) that $S$
satisfies the conclusion of that lemma.
We claim that the linkage $\cal P$ then satisfies (L6).
To prove this claim suppose that
$i\in\{1,2,\ldots,l-1\}$ and some $S$-bridge $B$ of $H[W_i]$
has all its attachments in $V(P)$ for some non-trivial $P\in\cal P$;
then there are vertices $x,y\in V(P)$
such that some component $J$ of $H\backslash\{x,y\}$ has at least three vertices, includes
a vertex of $B$ and is disjoint from $V(S)-V(P)$.
Since $G$ is $p$-connected the set $V(J)$ has at least $p-2$ neighbors
among the trivial paths in $\cal P$.
Hence $P$ is bridge adjacent in $W_i$ to those trivial paths, as required.
This proves that $\cal P$ satisfies (L6).~$\square$\bigskip
We will make use of the following easy lemma, whose proof we omit.
\begin{lemma}
\mylabel{lem:merge}
Let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be a linear decomposition
of a graph $G$ of length $l\ge2$, and let $i\in\{1,2,\ldots,l\}$.
Then
${\cal W'}:=(W_0,W_1,\ldots,W_{i-2},W_{i-1}\cup W_i,W_{i+1},W_{i+2},\ldots,W_l)$
is also a linear decomposition of $G$. Furthermore, if $\cal W$ satisfies
any of the properties {\rm (L3)--(L8)}, then so does $\cal W'$.
\end{lemma}
If $\cal W$ and $\cal W'$ are as in Lemma~\ref{lem:merge}, then we
say that $\cal W'$ was obtained from $\cal W$ by an {\em elementary
contraction}. Let $\cal P$ be a foundational linkage for $\cal W$. If $i \not \in \{1,l\}$, then let ${\cal P}':=\cal P$. If $i=1$, then let ${\cal P}'$ be the linkage obtained from $\cal P$ by restricting each $P \in \cal P$ to $W_2 \cup W_3 \cup \ldots \cup W_l$, and if $i=l$, then let ${\cal P}'$ be obtained by restricting $\cal P$ to $W_1 \cup W_2 \cup \ldots \cup W_{l-1}$. Then ${\cal P}'$ is a foundational linkage for ${\cal W}'$. It will be referred to as the \emph{corresponding restriction} of $\cal P$.
If a linear decomposition $\cal W''$ is obtained from $\cal W$ by
a sequence of elementary contractions, then we say that $\cal W''$
is obtained from $\cal W$ by a {\em contraction}.
We will also need the following lemma about sequences of sets.
\begin{lemma}\mylabel{lem:3}
Let $l,n,\lambda\ge0 $ be integers such that $\lambda\ge l^{n+1} n!$.
For all sequences $S_1, S_2,\dots, S_\lambda$ of subsets of $\{1, \dots, n\}$
there exist
integers $1 \le i_0 < i_1 < \dots < i_l \le \lambda+1$ such that
$$S_{i_0} \cup S_{i_0 + 1} \cup \dots \cup S_{i_1 - 1} =
S_{i_1} \cup S_{i_1 + 1} \cup \dots \cup S_{i_2 - 1} = \dots =
S_{i_{l-1}} \cup \dots \cup S_{i_l - 1}.$$
\end{lemma}
{\noindent\bf Proof. }
We proceed by induction on $n$.
The lemma clearly holds when $n=0$, and so we assume that $n>0$ and
that the lemma holds for all smaller values of $n$.
If $l$ consecutive sets $S_i$ are empty, say $S_i, S_{i+1},\ldots,S_{i+l-1}$,
then the lemma holds with $i_j=i+j$ for $j=0,1,\ldots,l$.
Thus we may assume that this is not the case, and hence there
is an integer $x\in\{1,2,\ldots,n\}$ such that at least
$\lambda':=\lambda/(ln)\ge l^n (n-1)!$ of the sets $S_i$ include the element $x$.
Thus $\{1, \dots, \lambda\}$ can be partitioned into consecutive intervals
$I_1, I_2,\ldots,I_{\lambda'}$ such that each interval includes an
index $i$ with $x\in S_i$.
For $i=1,2,\ldots,\lambda'$ let $S'_i$ be the union of $S_j-\{x\}$ over
all $j\in I_i$.
By the induction hypothesis applied to the sets $S'_i$ there exist
required indices $1\le i'_0<i'_1< \cdots <i'_l\le\lambda'+1$
for the sets $S_i'$.
For $j=0,1,\ldots,l$ let $i_j:=\min I_{i'_j}$.
It follows from the construction
that these indices satisfy the conclusion of the lemma.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:l8}
For every triple of integers $l,p,q\ge 0$ there exists an integer
$\lambda$ such that
the following holds.
If a
graph $G$ has a linear decomposition $\cal W$ of length $\lambda+1$ and
adhesion $q$ and a foundational linkage $\cal P$
satisfying {\rm (L1)--(L6)}, then it has
a linear decomposition $\cal W'$ of length $l$ and adhesion $q$
obtained from $\cal W$ by a contraction such that $\cal W'$ and the corresponding restriction of $\cal P$
satisfy {\rm (L1)--(L8)}.
\end{lemma}
{\noindent\bf Proof. }
Let $l,q\ge 0$ be given, let $s:={q\choose 2}$, and let
$\mu:= l^{s+1} s!$. We will show that $\lambda:=\mu^{q+1} q!$ satisfies the
conclusion of the lemma.
Let ${\cal W}=(W_0,W_1,\ldots,W_{\lambda+1})$ be as stated.
We wish to apply Lemma~\ref{lem:3} with $q$ playing the role of $n$
and $\mu$ playing the role of $l$. For $i=1,2,\ldots,\lambda$
let $S_i$ be the set of all $P\in\cal P$
such that $P[W_i]$ is a non-trivial path.
By Lemma~\ref{lem:3} there exist indices
$1 \le i_0 < i_1 < \dots < i_\mu \le \lambda+1$ as stated in that lemma.
Let $i_{-1}:=0$ and $i_{\mu+1}:=\lambda+1$ and
for $t=-1,0,\ldots,\mu$ define
$$W'_{t+1}:= W_{i_t}\cup W_{i_t+1}\cup\dots\cup W_{i_{t+1}-1}.$$
By Lemma~\ref{lem:merge} ${\cal W}':=(W'_0,W'_1,\ldots,W'_{\mu+1})$
is a linear decomposition of $G$ satisfying (L1)--(L6).
It follows from the construction that it also satisfies (L7).
To construct a linear decomposition satisfying (L1)--(L8) we apply
the same argument again, as follows. For a $2$-element subset
$X\subseteq\cal P$ let $S_X$ be the set of integers
$j\in\{1,2,\ldots,q\}$
such that some $\cal P$-bridge $H$ of $G$ has attachments in $P$ for
both elements $P\in X$ and satisfies $V(H)\subseteq W_j$.
By applying Lemma~\ref{lem:3} with $n:={q\choose 2}$ and $\lambda$
replaced by $\mu$
to the linear decomposition $\cal W'$
and using the same construction
we arrive at the desired linear decomposition of $G$.~$\square$\bigskip
Let $\cal W$ be a linear decomposition of a graph $G$
of length $l\ge2$ with foundational
linkage $\cal P$ satisfying (L1)--(L8).
We define the {\em auxiliary graph}
of the pair $({\cal W},{\cal P})$ to be the graph with vertex-set
$\cal P$ in which two paths $P,P'\in \cal P$ are adjacent if
they are bridge adjacent in $W_i$ for some (and hence every)
$i\in\{1,2,\ldots,l-1\}$.
We will need one more property of a linear decomposition $\cal W$ and its
foundational linkage ${\cal P}$.
The parameter $p$ is the same as in (L6).
\begin{enumerate}
\item[(L9)]
Let ${\cal P}_1\subseteq{\cal P}_2 \subseteq {\cal P}$ such
that $|{\cal P}_1|+|{\cal P}_2|\le p$ and each member of ${\cal P}_1$
is non-trivial.
Then there exists a linkage $\cal Q$ in $G$ of cardinality $|{\cal P}_1|$
from $W_0\cap W_1\cap V({\cal P}_1)$ to $W_{l-1}\cap W_l\cap V({\cal P}_1)$
such that its graph is a subgraph of
$H:=G[W_0\cup W_l]\cup \bigcup_{P\in{\cal P}-{\cal P}_2} P$.
\end{enumerate}
Our objective is to show that if a graph has a linear decomposition
satisfying (L1)--(L8), then it also has one satisfying (L9).
For the proof we need a definition and a lemma.
Let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be a linear decomposition
of a graph $G$ with foundational linkage ${\cal P}$ satisfying (L1)--(L8).
We say that a set ${\cal P}'$ of components of ${\cal P}$ is {\em well-connected} if for every
two paths $P,P'\in\cal P'$ there exists a path $\cal Q$ in the auxiliary
graph of $({\cal W},{\cal P})$ such that every internal vertex of $\cal Q$
is a non-trivial foundational path belonging to ${\cal P}'$.
The lemma we need is the following.
\begin{lemma}
\mylabel{lem:prel9}
Let $l,s,q\ge0$ be integers, and let $G$ be a graph
with a linear decomposition ${\cal W}=(W_0,W_1,\ldots,W_l)$
of length $l$, adhesion $q$
and foundational linkage ${\cal P}$ satisfying {\rm (L1)--(L8)}.
Let $\cal Q$ be a well-connected set of foundational paths,
and let $X_{ij}:= (W_{i-1}\cap W_{i}\cap V({\cal Q}))\cup
(W_j\cap W_{j+1}\cap V({\cal Q}))$.
Then for every two integers $i,j$ with $1\le i\le i+2q\le j< l$
and every two sets $A,B\subseteq X_{ij}$ of size $s$ there exist $s$
disjoint paths, each with one end in $A$,
the other end in $B$ and no internal vertex in any $W_k$
for $k\in\{0,1,\ldots,l\}-\{i,i+1,\ldots,j\}$.
\end{lemma}
{\noindent\bf Proof. }
Let $H$ be the subgraph of $G$ obtained by deleting $W_j-A-B$
for all $j\in\{0,1,\ldots,l\}-\{i,i+1,\ldots,j\}$.
If the paths do not exist, then by Menger's theorem there exists
a set $Y\subseteq V(H)$ of size at most $s-1$ such that $H\backslash Y$
has no path from $A$ to $B$.
We may assume that $A\cap B=\emptyset$, for otherwise we may proceed
by induction by deleting $A\cap B$.
Since $|W_{i-1}\cap W_{i}|=|W_{j}\cap W_{j+1}|=q$ we deduce that $s\le q$.
Let $Z$ be the union of the vertex-sets of the trivial paths in $\cal P$. By (L7) and the fact that $W_i \cap W_{i+1} \subseteq V({\cal P})$ for all $i=1,2,\ldots,l-1$, the sets $W_{i+1}-Z,W_{i+3}-Z,\ldots,W_{i+2q-1}-Z$ are pairwise disjoint,
and so one of them, say $W_{m}-Z$, is disjoint from $Y$.
For $x\in X_{ij}$ let $P_x$ be the member of $\cal Q$ that includes $x$.
If $x\in W_{i-1}\cap W_i$, then let $P'_x$ denote the restriction
of $P_x$ to $W_i\cup W_{i+1}\cup\cdots\cup W_{m-1}$, and
if $x\in W_{l}\cap W_{l+1}$, then let $P'_x$ denote the restriction
of $P_x$ to $W_{m+1}\cup W_{m+2}\cup\cdots\cup W_l$.
Since the paths $P_x'$ are pairwise vertex-disjoint,
there exist $a\in A$ and $b\in B$ such that $P'_a$ and $P_b'$
are disjoint from $Y$. Since $\cal Q$ is well-connected
it follows that $P'_a\cup G[W_m]\cup P'_b$ includes a path in $H$ from
$a$ to $b$ with no internal vertex in $Z$.
That path is disjoint from $Y$, a contradiction.~$\square$\bigskip
\begin{lemma}\mylabel{lem:l9}
Let $p,q\ge0$ and $l\ge3$ be integers, and let $G$ be a $p$-connected graph
with a linear decomposition
${\cal W}=(W_0,W_1,\ldots,W_{l+4q+2})$ of length $l+4q+2$, adhesion $q$
and foundational linkage $\cal P$ satisfying {\rm (L1)--(L8)}.
Let
${\cal W}':=(W'_0,W'_1,\ldots,W'_l)$,
where $W'_0:=W_0\cup W_1\cup \cdots\cup W_{2q+1}$,
$W'_i:= W_{i+2q+1}$ for $i=1,2,\ldots,l-1$ and
$W'_l:=W_{l+2q+1}\cup W_{l+2q+2}\cup \cdots\cup W_{l+4q+2}$,
and let $\cal P'$ be the corresponding restriction of $\cal P$.
Then $\cal W'$ is a linear decomposition of $G$ of length $l$ and adhesion $q$,
and $\cal P'$ is a foundational linkage for $\cal W'$
such that conditions {\rm (L1)--(L9)} hold.
\end{lemma}
{\noindent\bf Proof. }
The linear decomposition $\cal W'$ satisfies (L1)--(L8) by
Lemma~\ref{lem:merge}, and so it remains to show that it satisfies (L9).
Since $l\ge3$ we may choose an index $s$ with $2q+2<s<l+2q+1$.
Let ${\cal P}_1\subseteq{\cal P}_2$ be two sets of foundational paths
such that every member of ${\cal P}_1$ is non-trivial and
$|{\cal P}_1|+|{\cal P}_2|\le p$.
Let $H:=G[W_0'\cup W'_l]\cup \bigcup_{P\in {\cal P}-{\cal P}_2} P$.
We must show that there exist $|{\cal P}_1|$ disjoint paths in $H$
from $X_0:=W_0'\cap W_1'\cap V({\cal P}_1)$ to
$X_l:=W_{l-1}'\cap W_l'\cap V({\cal P}_1)$.
Since $G$ is $p$-connected and
$|W_j\cap W_{j+1}\cap V({\cal P}_2)|=|{\cal P}_2|$ we deduce
that there exists a linkage of size $|{\cal P}_1|$
from $X_0$ to $X_l$ in $G\backslash (W_s\cap W_{s+1}\cap V({\cal P}_2))$.
Let us choose such linkage, say $\cal Q$,
such that it uses the least number of edges not in $H$.
We will prove that $\cal Q$ is as desired. To do so
we may assume for a contradiction
that $\cal Q$ uses an edge
$e\in E(G)-E(H)$. By considering the linear decomposition
$(W'_l,W'_{l-1},\ldots,W'_0)$ we may assume that $e$ has both ends in
$W_i$ for some $i\in \{2q+2,2q+3,\ldots,s\}$.
By an {\em annex} we mean a maximal well-connected set
of foundational paths that includes at least one non-trivial foundational
path.
Let $\cal R$ be an annex.
We define $H_1({\cal R})$ to be the subgraph
of $J:=G[W_{1}\cup W_{2}\cup\cdots\cup W_{s}]$ consisting of the graph
of $\cal R$ restricted to $J$ and all $\cal R$-bridges that are the subgraphs of $J$ and have all vertices of attachment in $V({\cal R})$.
We define $H_0({\cal R})$ analogously as a subgraph of
$G[W_{1}\cup W_{2}\cup\cdots\cup W_{2q+1}]$.
It follows that $e$ is an edge of $H_1({\cal R})$ for some
maximal well-connected set $\cal R$ of foundational paths.
Let us assume that $e$ belongs to $H_1({\cal R})$ for some annex $\cal R$.
Thus we fix $\cal R$ and denote $H_0({\cal R})$ and
$H_1({\cal R})$ by $H_0$ and $H_1$, respectively. We will modify the linkage $\cal Q$ within $H_1$, and will obtain
a contradiction to its choice that way.
Let $\cal Q'$ be the subset of $\cal Q$ consisting of those paths
that use at least one vertex of $H_1$. For $Q\in{\cal Q}'$ let
$a(Q)$ be its end in $X_0$, let $d(Q)$ be its end in $X_l$,
and let $b(Q)$ and $c(Q)$ be two vertices of $Q\cap H_1$ such that
the subpath of $Q$ from $b(Q)$ to $c(Q)$ is maximum and
$a(Q),b(Q),c(Q),d(Q)$ occur on $Q$ in the order listed.
It follows that $b(Q),c(Q)$ belong to $(W_0\cap W_1)\cup
(W'_0\cap W'_1)\cup (W_s\cap W_{s+1})$,
but if one of them belongs to $W'_0\cap W'_1$, then it is equal to $a(Q)$.
If $b(Q)\in W_0\cap W_1$ or $b(Q)\in W_0'\cap W_1'$ we define $b'(Q):=b(Q)$
and let $B(Q)$ be the null graph; otherwise
$b(Q)$ belongs to a foundational path $P\not \in{\cal P}_2$,
and we define $b'(Q)$ to be the unique member of
$W_{2q+1}\cap W_{2q+2}\cap V(P)$, and we let
$B(Q):=P[W_{2q+2}\cup W_{2q+3}\cup\cdots\cup W_s]$.
We define $c'(Q)$ and $C(Q)$ analogously.
By Lemma~\ref{lem:prel9} applied to $\cal W$ and $\cal P$ with
$i=0$ and $j=2q+1$
there exists a linkage $\cal S$ in $H_0$ of size $|{\cal Q}'|$ from
$\{b'(Q):Q\in{\cal Q}'\}$ to $\{c'(Q):Q\in{\cal Q}'\}$.
The fact that $\cal R$ was chosen to be a maximal well-connected set
implies that members of this linkage are disjoint
from the members of ${\cal Q}-{\cal Q'}$.
For each $Q\in{\cal Q}'$ we delete the interior of the subpath of
$Q$ between $b(Q)$ and $c(Q)$, and add
the linkage $\cal S$ and the paths
$B(Q)$ and $C(Q)$ for all $Q\in{\cal Q}'$.
Thus we obtain a new linkage with the same properties
as $\cal Q$, but with fewer edges not in $H$, contrary to the choice
of $\cal Q$.
This completes the case when $e$ belongs to $H_1({\cal R})$
for some annex $\cal R$, and so from now on we may assume the opposite.
Let $K$ denote the union of the trivial paths in ${\cal P}$.
Since $e$ belongs to $H_1({\cal R})$ for no annex $\cal R$
it follows that the $K$-bridge $B$ of $H$ containing $e$ includes
no non-trivial foundational path.
Let $Q\in \cal Q$ be the path containing $e$, and let $b,c\in V(Q)$
be such that $bQc$ is a maximal subpath of $B$ containing $e$.
Since $Q$ is disjoint from $W_{s}\cap W_{s+1}\cap V({\cal P}_2)$,
and hence from the the trivial paths in ${\cal P}_2$,
we deduce that $b,c\not\in V({\cal P}_2)$.
It follows more generally
(from the fact that $e$ belongs to $H_1({\cal R})$ for no annex $\cal R$)
that every $K$-bridge $B'$ of $H$
that has $b$ and $c$ as attachments includes no non-trivial foundational
path.
Consequently, if $B'$ includes a non-trivial subpath of some member
of $\cal Q$, then this subpath uses two vertices of $V(K)$.
On the other hand the foundational paths with vertex-sets $\{b\}$
and $\{c\}$ are adjacent in the auxiliary graph, and hence
for each $i=1,2,\ldots,q$ there exists a $K$-bridge of $G[W_i]$ whose
attachments include $b$ and $c$.
By the conclusion of the sentence before the previous one we deduce
that there is $i\in\{1,2,\ldots,q\}$ such that $W_i$ includes no
non-trivial subpath of a member of $\cal Q$.
Thus we can replace $bQc$ by a subpath of $W_i$, contrary to the choice
of $\cal Q$.
This completes the proof that $\cal W'$ and $\cal P'$ satisfy (L9).~$\square$\bigskip
We are now ready to state the main result of this section.
\begin{theorem}
\mylabel{thm:lindec}
For all integers $k,l,p,w\ge0$ there exists an integer $N$ with
the following property.
If $G$ is a $p$-connected graph of tree-width at most $w$ with
at least $N$ vertices, then either $G$ has a minor isomorphic to
$K_{p,k}$, or $G$ has a linear decomposition of
length at least $l$ and adhesion at most $w$ satisfying {\rm (L1)--(L9)}.
\end{theorem}
{\noindent\bf Proof. } Let $k,l,p,w\ge0$ be integers,
and let $l_1:=l+4w+2$.
Let $l_2$ be the minimum value of $\lambda$ such that Lemma~\ref{lem:l8}
holds for $l=l_1$, $p$ and all $q\le w$.
Finally, let $N$ be such that Lemma~\ref{lem:l5} holds for
$l=l_2$, $k,p$, and $w$. We claim that $N$ satisfies the theorem.
To prove the claim let $G$ be a $p$-connected graph of tree-width at most $w$ with
at least $N$ vertices. By Lemma~\ref{lem:l5} it has either a
minor isomorphic to $K_{p,k}$, or a linear decomposition ${\cal W}_2$
of length at least $l_2$ and adhesion $q\le w$ satisfying (L1)--(L5), and so we may assume the latter.
By Lemma~\ref{lem:l6} there is a foundational linkage ${\cal P}_1$
satisfying (L6).
By Lemma~\ref{lem:l8} the graph $G$ has a linear decomposition
${\cal W}_1$ of length $l_1$ and adhesion $q$ such that
${\cal W}_1$ and ${\cal P}_1$ satisfy (L1)--(L8).
Finally, by Lemma~\ref{lem:l9} there exist a linear decomposition
$\cal W$ of length $l$ and adhesion $q$ and a foundational linkage
satisfying (L1)--(L9).~$\square$\bigskip
We will need the following special case.
\begin{corollary}
\mylabel{cor:normdec}
For all integers $l,w\ge0$ there exists an integer $N$ with
the following property.
If $G$ is a $6$-connected graph of tree-width at most $w$ with
at least $N$ vertices, then either $G$ has a minor isomorphic to
$K_6$, or $G$ has a linear decomposition of
length at least $l$ and adhesion at most $w$ satisfying {\rm (L1)--(L9)} for
$p=6$.
\end{corollary}
\section{Analyzing the auxiliary graph}\mylabel{sec:auxil}
Let $G$ be a $6$-connected graph with no $K_6$ minor, and let $\cal W$
and $\cal P$ be as before and satisfy (L1)--(L9).
In this section we establish several properties of the auxiliary graph
of the pair $({\cal W}, {\cal P})$.
The first main result is Lemma~\ref{maxdeg2} stating
that if $\cal W$ is sufficiently long, then every component of
the subgraph of the auxiliary
graph induced by the non-trivial foundational paths is either a path or
a cycle.
The second main result of this section, Lemma~\ref{lem:l10},
allows us to modify the pair $({\cal W}, {\cal P})$ such that in the
new pair every non-trivial $\cal P$-bridge attaches to exactly two
non-trivial foundational paths.
Let $k,l\ge3$ be integers.
For $i\in \{1,2,\ldots,k\}$ let $P_i$ be a path with vertices
$v^i_1, \dots, v^i_{l}$ in order.
We define the \emph{linked k-cylinder of length $l$} to be the graph
with vertex-set $\bigcup_{i=1}^k V(P_i)$ and edge-set
$\bigcup_{i=1}^k E(P_i) \cup \left\{ v^i_j v^{i+1}_j: 1 \le i
\le k, 1 \le j \le l \right \} \cup \{q_1, q_2\}$,
where the index notation is taken modulo $k$ and
the edges $q_1$ and $q_2$ have no common end
and each have one end in $\{ v^1_1, v^2_1, \dots, v^k_1\}$ and the other end
in $\{ v^1_{l}, v^2_{l}, \dots, v^k_l\}$.
Figure~\ref{fig:3cyl} shows a linked $3$-cylinder of length six.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page=8]{jorg_figs.pdf}
\end{center}
\caption{Finding a $K_6$ minor in a linked $3$-cylinder of length six.}
\label{fig:3cyl}
\end{figure}
\begin{lemma}\mylabel{lem:HK6}
For all integers $k \ge 3$, a linked $k$-cylinder of length twelve
has a $K_6$ minor.
\end{lemma}
{\noindent\bf Proof. }
By finding two suitable paths with vertex-sets in
$\{ v^i_j: 1 \le i \le k, 1 \le j \le 3 \}$, and
two paths with vertex-sets in
$\{\{ v^i_j : 1 \le i
\le k, 10 \le j \le 12 \}$, we see that a linked $k$-cylinder of length twelve
has a minor isomorphic to a linked $3$-cylinder of length six with the
additional property that the ends of the edge $q_i$
are $v_1^i$ and $v_6^i$ for $i = 1, 2$.
This graph has a $K_6$ minor as indicated in Figure~\ref{fig:3cyl}.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:3att}
Let $l \ge 2$ and $q \ge 3$ be integers, and let ${\cal W}=(W_0,W_1,\ldots,W_l)$
be a linear decomposition of length $l$
and adhesion $q$ of a graph $G$, and let $\cal P$ be a foundational
linkage for $\cal W$ such that {\rm (L1)--(L5)} and {\rm (L9)} hold.
If for at least $48\binom{q}{3}$ indices $i\in\{1,2,\ldots,l-1\}$ there exists
a $\cal P$-bridge in $G[W_i]$ with attachments on at least three
non-trivial paths in $\cal P$, then $G$ has a $K_6$ minor.
\end{lemma}
{\noindent\bf Proof. }
Let $l, q$ be integers and $\mathcal{W} = (W_0, \dots, W_l)$ and $\mathcal{P}$
be given. If there exist $48\binom{q}{3}$ distinct indices $i$ with $1 \le i \le l-1$
such that $G[W_i]$
contains a $\mathcal{P}$-bridge attaching to at least three non-trivial foundational
paths, then there exist $48$ distinct indices $i$ and three distinct non-trivial
foundational paths $P_1, P_2, P_3 \in \mathcal{P}$ such that
$G[W_i]$ contains a $\mathcal{P}$-bridge
attaching to $P_j$ for $j = 1, 2, 3$. Then there exists a subset of
indices $I \subseteq \{1, \dots, l-1\}$ with $|I| = 24$ such that $|i - j| >2$
for all distinct $i, j \in I$, and furthermore, $G[W_i]$ contains
a bridge $B_i$ attaching to $P_j$ for all $i \in I$ and $j = 1, 2, 3$. By property
(L9), there exist two disjoint paths $Q_1$ and $Q_2$ each with one
end in $V(P_1\cup P_2\cup P_3) \cap W_1 \cap W_2$
and one end in $V(P_1\cup P_2\cup P_3) \cap W_{l-1} \cap W_l$.
Moreover, the paths $Q_1$ and $Q_2$ do not have an internal vertex
in either $B_i \setminus V({\cal P})$ or $P_j$ for all $i \in I$ and $1 \le j \le 3$.
It follows that $G$ has a minor isomorphic to a linked $3$-cylinder of length
twelve since
each pair of successive bridges $B_i$ can be contracted to a single
cycle of length three.
By Lemma~\ref{lem:HK6} the graph $G$ has a $K_6$ minor, as desired.~$\square$\bigskip
The following will be a hypothesis common to several forthcoming
lemmas. In order to avoid unnecessary repetition we give it a name.
\begin{hypothesis}
\mylabel{hypot}
Let $p=6$, $l \ge 2$ and $q \ge 6$ be integers,
let $G$ be a $6$-connected graph with no $K_6$ minor,
and let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be a linear decomposition
of $G$ of length $l$ and adhesion $q$ with a foundational
linkage $\cal P$ such that conditions (L1)--(L9) hold.
\end{hypothesis}
\begin{lemma}\mylabel{lem:nontriv2}
Assume Hypothesis~\ref{hypot}. Then there do not exist
$6 \binom{q}{6}$ distinct indices $i$ with $1 \le i \le l-1$ such
that $G[W_i]$ contains a non-trivial $\mathcal{P}$-bridge attaching only
to trivial foundational paths.
\end{lemma}
{\noindent\bf Proof. }
Let $G$, $\cal{W}$, $\cal{P}$, $q$, and $l$ be as stated. If
the conclusion of the lemma does not hold, then there exist
six distinct indices $i$ such that $G[W_i]$ contains a non-trivial
$\mathcal{P}$-bridge $B_i$ attaching to the same subset of
six trivial foundational paths. By contracting the internal vertices of
each $B_i$
to a single vertex, we see $G$ would have a $K_6$ minor, a contradiction.~$\square$\bigskip
\begin{lemma}
\mylabel{nontrivpath}
Assume Hypothesis~\ref{hypot}. If $l> 6 \binom{q}{6}$, then $\cal P$ includes
at least one non-trivial path.
\end{lemma}
{\noindent\bf Proof. }
Let $G$, $\cal{W}$, $\cal{P}$, $q$, and $l$ be as stated, and suppose
for a contradiction that every path in $\mathcal{P}$ is trivial.
For every $i$, $1 \le i \le l-1$,
$G[W_i]$ contains a non-trivial bridge $B_i$, as $W_i \nsubseteq W_{i+1}$,
$W_i \nsubseteq W_{i-1}$ by (L4), in contradiction with Lemma~\ref{lem:nontriv2}.
$\square$\bigskip
Let $\cal W$ be a linear decomposition of a graph $G$ and let
$\cal P$ be a foundational linkage such that $\cal W$ and $\cal P$
satisfy (L1)--(L8).
By a {\em core} of the pair $({\cal W},{\cal P})$
we mean a component of the graph obtained from
the auxiliary graph of $({\cal W},{\cal P})$ by deleting all
trivial foundational paths.
The next lemma is the first main result of this section.
\begin{lemma}
\mylabel{maxdeg2}
Assume Hypothesis~\ref{hypot}. If $l \ge 48$, then
every core of the pair $({\cal W},{\cal P})$ is a path or a cycle.
\end{lemma}
{\noindent\bf Proof. } Let $G$, $\cal{W}$, $\cal{P}$, $q$, and $l$ be as stated.
Suppose for a contradiction that there exists a non-trivial
foundational path $P_1 \in \cal{P}$
adjacent in the auxiliary graph to three
non-trivial paths $P_2, P_3, P_4 \in \cal{P}$.
By property (L9), there exist two disjoint paths $Q_1$ and $Q_2$ each
with one end in $V(P_2\cup P_3\cup P_4) \cap W_0 \cap W_1$
and one end in $V(P_2\cup P_3\cup P_4) \cap W_{l-1} \cap W_l$.
Furthermore, $Q_1$ and $Q_2$ avoid any internal vertex of $P_i$ for $1 \le i \le 4$
as well as any internal vertex of a $\mathcal{P}$-bridge in $G[W_j]$ for $1 \le j \le l-1$.
For all $i\in\{1,2,\ldots, 24\}$, we contract to a single vertex $b_i$
the set of vertices consisting
of $P_1[W_{2i-1}]$ and the
internal vertices of every non-trivial
bridge attaching to $P_1$ in $G[W_{2i-1}]$. Note that no vertex of $Q_i$ for
$i = 1, 2$ is contained in the contracted set of $b_{2j-1}$ for any $1 \le j \le 24$.
Each vertex $b_i$ has a neighbor in each of $P_2$, $P_3$, and $P_4$.
Also, the neighbors of $b_i$ and $b_j$ are distinct for $i \neq j$. It follows
that $G$ has a minor isomorphic to a linked
3-cylinder of length twelve, contrary to Lemma \ref{lem:HK6}.~$\square$\bigskip
\begin{lemma}\mylabel{lem:4trivatt}
Assume Hypothesis~\ref{hypot}. If $l \ge 12$, then every non-trivial path in $\cal P$
is adjacent in the auxiliary graph to at most three trivial paths
in $\cal P$.
\end{lemma}
{\noindent\bf Proof. } Let $G$, $\cal{W}$, $\cal{P}$, $q$, and $l$ be as stated. Assume, to
reach a contradiction, that $P_1 \in \cal{P}$ is a non-trivial path and is
adjacent to four trivial foundational paths in the auxiliary graph.
Let the vertices comprising
the four trivial foundational paths be $v_1,v_2,v_3,v_4$.
For each $i\in\{1,2,\ldots,6\}$
we contract to a single vertex $b_i$ the vertex set containing $P_1[W_{2i-1}]$ and
the internal vertices of all non-trivial bridges of $G[W_{2i-1}]$ attaching to $P_1$.
It follows that $G$ has as a minor isomorphic to
the graph with vertex set $\{v_i: 1 \le i \le 4\}
\cup \{ b_i : 1 \le i \le 6\}$ and edges $\{v_ib_j: 1 \le i \le 4, 1 \le j \le 6\} \cup
\{b_i b_{i+1} : 1 \le i \le 5\}$.
This graph has a $K_6$ minor, and hence so does $G$, a contradiction.~$\square$\bigskip
\begin{corollary}\mylabel{cor:induced} Assume Hypothesis~\ref{hypot}. If $l \geq 12$, then every member of $\cal P$ is an induced path.
\end{corollary}
{\noindent\bf Proof. } If some non-trivial $P \in \cal P$ is not induced, then by (L6) the path $P$ is adjacent to at least $4$ trivial foundational paths in the auxilliary graph, contrary to Lemma~\ref{lem:4trivatt}.
~$\square$\bigskip
\begin{lemma}\mylabel{lem:3trivatt}
Assume Hypothesis \ref{hypot}. If $l \ge 12$, then no non-trivial foundational path
is adjacent in the auxiliary graph to three or more trivial foundational paths.
\end{lemma}
{\noindent\bf Proof. } Let $G$, $\cal{W}$, $\cal{P}$, $q$, and $l$ be as stated. As above,
assume to reach a contradiction, that
$P_1 \in \cal{P}$ is a non-trivial path and is adjacent to
three trivial foundational paths in the auxiliary graph. By the $6$-connectivity
of $G$, $P_1$ must be adjacent to another foundational path in the
auxiliary graph. By Lemma \ref{lem:4trivatt},
such a path, call it $P_2$, must be non-trivial.
For each $i$, $1 \le i \le 6$,
we contract to a single vertex the vertex set containing $P_1[W_{2i-1}]$ and
the internal vertices of any non-trivial bridge of $G[W_{2i-1}]$ attaching to $P_1$.
It follows that $G$ has a minor isomorphic to the graph in
Figure~\ref{fig:3trivatt}, which has a
$K_6$ minor as indicated in that figure, a contradiction.~$\square$\bigskip.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page = 12]{jorg_figs.pdf}
\end{center}
\caption{Finding a $K_6$ minor when a non-trivial foundational path is bridge
adjacent to three trivial foundational paths.}
\label{fig:3trivatt}
\end{figure}
In the next lemma, the second main result of this section, we show that
we can assume that our linear decomposition ${\cal W}=(W_0,W_1,\ldots,W_l)$
and foundational linkage $\cal P$ satisfy the following property.
\begin{itemize}
\item[(L10)] For all $i\in\{1,2,\ldots,l-1\}$,
every non-trivial $\mathcal{P}$-bridge
of $G[W_i]$ attaches to exactly two non-trivial foundational paths.
\end{itemize}
\begin{lemma}\mylabel{lem:l10}
Assume Hypothesis~\ref{hypot}. If
$l \ge \left ( 6\binom{q}{6} + 48 \binom{q}{3}\right )l'$,
then there exist a contraction $\mathcal{W}'$
of $\cal W$ of length $l'$ and adhesion $q$
and a foundational linkage ${\cal P}'$ for ${\cal W}'$
satisfying {\rm (L1)--(L10)}.
\end{lemma}
{\noindent\bf Proof. }
By Lemma \ref{lem:nontriv2} and Lemma \ref{lem:3att} and our choice of $l$,
there exists an index $\alpha$ such that for all $i\in\{1,2,\ldots,l'-1\}$,
$G[W_{\alpha + i}]$ contains neither a non-trivial $\cal P$-bridge
attaching only to trivial foundational paths nor a $\cal P$-bridge
attaching to three or more
non-trivial foundational paths. Moreover, Lemma \ref{lem:4trivatt} and property
(L6) imply that no non-trivial bridge attaches to exactly one non-trivial
foundational path.
The lemma follows from considering the contraction
$\mathcal{W}' = \left ( \bigcup_{i = 0}^{\alpha} W_i, W_{\alpha +1},
W_{\alpha+2}, \dots, W_{\alpha + l'-1}, \bigcup_{i = \alpha + l' }^l W_i\right)$
of $\cal{W}$ and the corresponding restriction of $\cal P$.~$\square$\bigskip
\section{Finding and eliminating a pinwheel}\mylabel{sec:pin}
Let us assume Hypothesis~\ref{hypot}.
In the previous section we have shown that $\cal W$ and $\cal P$ can
be chosen so that for every $i\in\{1,2,\ldots,l-1\}$, every non-trivial
$\cal P$-bridge $B$ of $G[W_i]$ attaches to exactly two non-trivial
foundational paths.
The main result of this section will be used in Section~\ref{sec:taming} to show that if $G$ is not an apex graph then
$\cal W$ and $\cal P$ can be chosen so that every such bridge attaches to no trivial foundational path.
The proof technique is different, and relies on a theory
of ``non-planar extensions" of planar graphs, developed
in~\cite{RobSeyThoExt}.
A \emph{pinwheel with $t$ vanes} is the graph defined as follows. Let $C^1$ and $C^2$
be two disjoint cycles of length $2t$, where the vertices of $C^i$ are $v_1^i,v_2^i,\ldots,v_{2t}^i$ in order. Let $w_1, w_2, \dots, w_t, x$
be $t+1$ distinct vertices. The pinwheel with $t$ vanes has
vertex-set $V(C^1) \cup V(C^2) \cup \{ w_1, w_2, \dots, w_t, x\}$ and edge-set
\begin{align*}
E(C^1) &\cup E(C^2) \cup \{v^1_{2j}v^2_{2j} : 1 \le j \le t\} \\
& \cup \{ w_jv_{2j-1}^i: 1 \le j \le t, i=1,2\} \cup \{x w_j : 1 \le j \le t\}
\end{align*}
The cycles $C^1$ and $C^2$ form the \emph{rings} of the pinwheel. A pinwheel with four vanes is pictured in Figure~\ref{fig:pin}.
A \emph{M\"obius pinwheel with $t$ vanes} is obtained from a pinwheel with $t$ vanes by deleting the edges $v_{2t}^1v_1^1$ and
$v_{2t}^2v_1^2$ and adding the edges $v_{2t}^1v_1^2$ and $v_{2t}^2
v_1^1$. The cycle formed by $V(C^1) \cup V(C^2)$ in a M\"obius pinwheel
is the \emph{ring} of the M\"obius pinwheel. A M\"obius pinwheel with $4$ vanes contains $K_6$ as a minor as shown on Figure~\ref{fig:pin}.
\begin{figure}[h!]
\begin{center}
\includegraphics[viewport= 0 0 400 200]{pinwheel2.pdf}
\end{center}
\caption{(a) A pinwheel with four vanes, (b) A M\"obius pinwheel with $4$ vanes and a $K_6$ minor in it.}
\label{fig:pin}
\end{figure}
\begin{lemma}\mylabel{lem:pin1}
Let $q$, $l$, and $p=6$, $t\ge 4$ be positive integers. Let $\mathcal{W} = (W_0, W_1, \dots, W_l)$
be a linear decomposition of a 6-connected graph $G$ of length $l$ and adhesion $q$ with foundational linkage ${\cal P}$
satisfying {\rm (L1)--(L9)}. Let $P_1, P_2, P_3, Q \in {\cal P}$ be distinct, let $Q$ be trivial, and let $P_i$ be non-trivial for $i = 1, 2, 3$.
Furthermore, let
$P_2$ be adjacent to $P_1$, $P_3$, and $Q$ in the auxiliary graph. If
$l \ge 4t + 1$, then $G$ has a subgraph
isomorphic to a subdivision of a pinwheel or a M\"obius pinwheel with $t$ vanes.
\end{lemma}
{\noindent\bf Proof. } Let $V(Q) = \{x\}$, let $P_i \cap W_0 \cap W_1 = \{s_i\}$ for $i = 1, 3$, and let $P_i \cap W_{l-1} \cap W_l = \{t_i\}$
for $i = 1, 3$. Let $\bar {\cal P} = {\cal P} - \{P_1,P_2,P_3,Q\}$.
By property (L9),
there exist two disjoint paths $R_1$ and $R_2$ in $G[W_0 \cup W_l] \cup \bigcup_{P \in \bar{\cal P}}P$ each with one end in
$\{s_1, s_3\}$ and one end in $\{t_1, t_3\}$. The rings of our pinwheel will be formed by
$R_1 \cup R_2 \cup P_1 \cup P_3$. If the paths $R_1$ and $R_2$ cross, i.e. the ends of
$R_1$ are $s_1$ and $t_3$ and the ends of $R_2$ are $s_3$ and $t_1$, we construct
a M\"obius pinwheel. Otherwise, we simply construct a pinwheel on $t$ vanes.
Note that for every $j=1,\ldots, l-1$ there exists a path $S_j$ with one end in $W_j \cap V(P_1)$ and the other end in $W_j \cap V(P_3)$, such that
$V(S_j) \subseteq W_j$, and $S_j$ is internally disjoint from
$\bigcup_{P \in {\cal P} - P_2} P$. Also, for every $j=1,\ldots, l-1$ there exists a vertex $v_j \in W_j$ and three paths $T^1_j, T^2_j$ and $T^3_j$, internally disjoint from each other and from $\bigcup_{P \in {\cal P} - P_2} P$, satisfying the following. Each of $T^1_j, T^2_j$ and $T^3_j$ has one end $v_j$, the second end of $T^1_j$ is in $V(P_1)$, the second end of $T^3_j$ is in $V(P_3)$ and the second end of $T^2_j$ is $x$. The paths $S_j, T^1_j, T^2_j$ and $T^3_j$ are internally disjoint from the rings of our pinwheel by construction, and the paths, corresponding to the sets $W_i$ with non-consecutive indices, are also disjoint. Therefore we can use the paths corresponding to the sets $W_i$ with odd indices to construct a subgraph of $G$ isomorphic to a subdivision of a pinwheel or a M\"obius pinwheel, with rings of the pinwheel as prescribed above.
$\square$\bigskip
As we have seen above a M\"obius pinwheel with sufficiently many vanes contains a $K_6$ minor. A pinwheel is, however, an apex graph.
In order to prove that graphs containing a subdivision of a pinwheel with many vanes satisfy
Theorem \ref{main}, we will need the following lemma concerning subdivisions of apex graphs
contained in larger non-apex graphs.
The lemma is proved in~\cite[Theorem (9.2)]{RobSeyThoExt}.
\begin{lemma}\mylabel{lem:apexsub}
Let $J$ be an internally $4$-connected triangle-free
planar graph not isomorphic to the cube, and let $F\subseteq E(J)$ be
a nonempty set of edges such that no two edges of $F$
are incident with the same face of $J$.
Let $J'$ be obtained from $J$ by subdividing each edge in $F$
exactly once, and let
$H$ be the graph obtained from $J'$ by adding a new vertex $v\not\in V(J')$
and joining it by an edge to all the new vertices of $J'$.
Let a subdivision of $H$ be isomorphic to a subgraph of $G$, and let
$u\in V(G)$ correspond to the vertex $v$.
If $G\backslash u$ is internally $4$-connected and non-planar,
then there exists an edge $e\in E(H)$ incident with $v$ such that
either
\begin{itemize}
\item[(i)] there exist vertices $x,y\in V(J')$ not belonging
to the same face of $J'$ such that $(H\backslash e)+xy$ is
isomorphic to a minor of $G$, or
\item[(ii)] there exist vertices $x_1,x_2,x_3,x_4\in V(J')$ appearing
on some face of $J'$ in order such that $(H\backslash e)+x_1x_3+x_2x_4$ is
isomorphic to a minor of $G$.
\end{itemize}
\end{lemma}
\begin{lemma}\mylabel{lem:pin2}
If a $5$-connected graph $G$ with no $K_6$
minor contains a subdivision of a pinwheel with $20$ vanes as a subgraph, then $G$ is apex.
\end{lemma}
{\noindent\bf Proof. }
We will show that for every positive integer $t$ every $5$-connected non-apex graph $G$ containing a subdivision of a pinwheel with $4t$ vanes
contains a M\"obius pinwheel with $t-1$ vanes as a minor. A M\"obius pinwheel with $4$ vanes contains a $K_6$ minor, as observed above, and so the lemma will follow.
We apply Lemma~\ref{lem:apexsub}, where the graphs $H$ and $J$, the vertex $v \in V(H)$ and the set of edges $F \subseteq E(J)$ are defined as follows.
Let $H$ be the pinwheel with $4t$ vanes, and let $v$ be the ``hub"
of the pinwheel (denoted by $x$ in the definition of a pinwheel). Let the graph $J$ consist of two disjoint cycles $C^1$ and $C^2$ of length $8t$ with the vertices of $C^i = \{v_j^i: 1 \le j \le 8t\}$
for $i = 1, 2$ and $v_j^i$ adjacent to $v_{j+1}^i$ and $v_{j}^{i+1}$ for all $1 \le j \le 8t$ and $i = 1, 2$ with the
subscript addition taken modulo $8t$ and the superscript addition taken modulo $2$. Finally, let $F=\{v_{2j-1}^1v_{2j-1}^2: 1 \le j \le 4t\}$.
Suppose that outcome (ii) of Lemma~\ref{lem:apexsub} holds (the case when outcome (i) holds is analogous). If the boundary of the face of $J$ containing the vertices
$x_1,x_2,x_3$ and $x_4$ is not one of the cycles $C_1$ and $C_2$, then without loss of generality we have $x_1=v_1^1, x_2 = v_1^2, x_3=v_2^2$ and $x_4=v_2^1$. Clearly, for every edge $e \in E(H)$ incident to $v$ the graph $(H\backslash e)+x_1x_3+x_2x_4$ contains a M\"obius pinwheel with $4t-1$ vanes as a subgraph.
Therefore, by symmetry, we assume that the vertices $x_1,x_2,x_3$ and $x_4$ are contained in $C_1$, i.e. $x_i = v_{k_i}^1$ for $i =1,2,3,4$, where, without loss of generality, $t \leq k_1, k_2,k_3, k_4 \leq 4t$. Then the subgraph $J_0$ of $J+x_1x_3+x_2x_4$ induced on $\{v_i^j : t \le i \le 4t, j=1,2\}$ contains two disjoint paths, one with ends $v_t^1$ and $v_{4t}^2$, and another with ends $v_t^2$ and $v_{4t}^1$. Now consider the graph $(H\backslash e)+x_1x_3+x_2x_4$, where $e \in E(H)$ is an edge incident to $v$, and delete all the edges of subdivision of $J_0$ from this graph, except for those that belong to the paths constructed above.
If is easy to see that the resulting graph contains a subdivision of a M\"obius pinwheel with $t-1$ vanes, as claimed.
$\square$\bigskip
The next corollary follows immediately from Lemmas~\ref{lem:pin1}
and~\ref{lem:pin2}.
\begin{corollary}
\mylabel{cor:mainpin}
Assume Hypothesis~\ref{hypot}.
If $l\ge81$ and some non-trivial foundational path is adjacent in
the auxiliary graph to two non-trivial and at least one trivial
foundational path, then $G$ is apex.
\end{corollary}
\section{Taming the bridges}\mylabel{sec:taming}
In Lemma~\ref{lem:l10} we have modified $\cal W$ and $\cal P$ so that
for every $i\in\{1,2,\ldots,l-1\}$
every non-trivial $\cal P$-bridge $B$ of $G[W_i]$
attaches to exactly two non-trivial
foundational paths.
Let us recall that a core is a component of the subgraph of the
auxiliary graph restricted to non-trivial foundational paths.
In this section we show that the graph consisting of all paths of a
core of $({\cal W},{\cal P})$ and all bridges that attach to two paths
of the core can be drawn in either a disk or a cylinder, depending
on whether the core is a path or a cycle.
The following lemma follows easily from the definition of properties
(L1)--(L5) and (L9).
\begin{lemma}\mylabel{lem:reroute}
Let $l \ge 2$, $q \ge 0$, and $p \ge 0$ be integers, and let ${\cal W}=(W_0,W_1,\ldots,W_l)$
be a linear decomposition of length $l$
and adhesion $q$ of a graph $G$, and let $\cal P$ be a foundational
linkage for $\cal W$ such that {\rm (L1)--(L5)} and {\rm (L9)} hold.
Let $i$ be fixed
with $1\le i \le l-1$ and let $Q$ be a path in $G[W_i]$ with ends $x$ and $y$
such that $x,y\in V(P)$ for some $P\in\cal P$ and
$Q$ is otherwise disjoint from $V(\cal{P})$.
Let $P'$ be obtained from $P$ by replacing $xPy$ by $Q$.
Then the linkage $\mathcal{P}' =( \mathcal{P} -\{ P\}) \cup \{P'\}$ satisfies
{\rm (L1)--(L5)} and {\rm (L9)}.
\end{lemma}
Let $G$ be a graph and $\mathcal{W} = (W_0, \dots, W_l)$ be a
linear decomposition of length $l$ and adhesion $q$ of $G$, and let
$\cal P$ be a foundational linkage such that (L1)--(L5) hold.
Let $i\in\{1,2,\ldots,l-1\}$, let $P,P'\in\cal P$ be two non-trivial
foundational paths, let
$W_{i-1} \cap W_i \cap V(P)=\{x\}$, $W_{i-1} \cap W_i \cap V(P')=\{x'\}$,
$W_{i} \cap W_{i+1} \cap V(P)=\{y\}$, and $W_{i} \cap W_{i+1} \cap V(P')=\{y'\}$. Let
$Q_1, Q_2$ be two disjoint paths where $Q_i$ has ends $u_i$ and $v_i$ for $i=1,2$.
If the paths $Q_1$ and $Q_2$ are internally disjoint from $V(\cal{P})$,
the vertices $x$, $u_1$, $u_2$, $y$ occur
on $P$ in that order, and $x'$, $v_2$, $v_1$, $y'$ occur on $P'$ in that order,
then we say that the foundational paths $P$ and $P'$ {\em twist}.
Let $P_1$, $P_2$ and $P_3$ be three non-trivial foundational paths and let $Q_1$, $Q_2$,
and $Q_3$ be three internally disjoint paths such that $Q_j$ is also
internally disjoint from each member of $\cal{P}$ for each $j\in\{1,2,3\}$.
Let the ends of
$Q_j$ be $x_j$, $y_j$ for $1 \le j \le 3$. The paths $Q_1$, $Q_2$, and $Q_3$
form a \emph{$P_1$-tunnel} if $x_1,y_1\in V(P_1)$,
the vertices $x_2, x_3 \in V(x_1P_1y_1) -\{x_1, y_1\}$
and $y_j \in V(P_j)$ for $j = 2, 3$. The path $Q_1$ is called the \emph{arch}
of the tunnel.
\begin{lemma}\mylabel{lem:elimtunnels}
Let $l \ge 2$, $q \ge 3$, and $p=6$ be integers, and let ${\cal W}=(W_0,W_1,\ldots,W_l)$
be a linear decomposition of length $l$
and adhesion $q$ of a graph $G$, and let $\cal P$ be a foundational
linkage for $\cal W$ such that {\rm (L1)--(L5)} and {\rm (L9)} hold.
If there exist $48\binom{q}{3}$ distinct indices $i\in\{1,2,\ldots,l-1\}$
such that $G[W_i]$
contains a $P$-tunnel for some non-trivial foundational path $P \in \cal{P}$,
then $G$ has a $K_6$ minor.
\end{lemma}
{\noindent\bf Proof. }
Let $l$, $q$, $p$, $\mathcal{W}$ and $\mathcal{P}$ be given.
Assume, to reach a contradiction, that there exist $48\binom{q}{3}$ indices
$i\in\{1,2,\ldots,l-1\}$ such that $G[W_i]$ has
a $P_i$-tunnel for some non-trivial foundational path $P_i\in\cal P$.
Reroute the paths $P_i$ along
the arches of the $P_i$-tunnels to get a linkage $\mathcal{P}'$.
By Lemma \ref{lem:reroute} $\cal W$ and ${\cal P}'$ satisfy (L1)--(L5) and (L9).
Moreover, for each of the above $48\binom{q}{3}$ distinct indices $i$
there exists a non-trivial ${\cal P}'$-bridge in $G[W_i]$
that attaches to at least three non-trivial foundational paths.
It follows from Lemma \ref{lem:3att}
that $G$ has a $K_6$ minor, as desired.~$\square$\bigskip
\begin{lemma}\mylabel{lem:elimtwists}
Let $l \ge 2$, $q \ge 3$, and $p=6$ be integers, and let $\mathcal{ W}=(W_0,W_1,\ldots,W_l)$
be a linear decomposition of length $l$
and adhesion $q$ of a graph $G$, and let $\cal P$ be a foundational
linkage for $\cal W$ such that {\rm (L1)--(L5)} and {\rm (L9)} hold.
If there exist $12\binom{q}{2}$ distinct indices $i \in \{1,2,\ldots,l-1\}$ such that $G[W_i]$ contains
a pair of twisting non-trivial foundational paths, then $G$ has a $K_6$ minor.
\end{lemma}
{\noindent\bf Proof. }
Let $l$, $q$, $p$, $\mathcal{W}$ and $\mathcal{P}$ be given. Assume
there exist $12\binom{q}{2}$ distinct indices $i \in \{1,2,\ldots,l-1\}$ such that $G[W_i]$ contains a pair of twisting
non-trivial foundational paths. It follows that there exists a subset $\mathcal{I} \subseteq
\{1, 2, \dots, l-1\}$ of cardinality $12$
and non-trivial paths $P_1, P_2 \in \mathcal{P}$ such that $P_1$ and $P_2$
twist in $G[W_i]$ for all $i \in \mathcal{I}$. We use the twisting
paths to contract three disjoint $K_4$ subgraphs onto $P_1$ and $P_2$ to find
a minor isomorphic to the graph in Figure \ref{fig:3K4}.
The edges $r_1$ and $r_2$ in the figure exist by applying property (L9)
to the ends of $P_1$ and $P_2$.
The numbering in Figure~\ref{fig:3K4} shows a $K_6$ minor,
implying that $G$ also has a $K_6$ minor, as desired.~$\square$\bigskip
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page = 22]{jorg_figs.pdf}
\end{center}
\caption{Finding a $K_6$ minor when there exist a pair of non-trivial foundational
paths that twist in twelve distinct $W_i$. The edges $r_1$ and $r_2$ are depicted
as not crossing, however, if they cross the graph still contains $K_6$ as a minor.}
\label{fig:3K4}
\end{figure}
\begin{lemma}\mylabel{lem:elimattach} Let $G$ be a $6$-connected graph with
no $K_6$ minor. Let $l \ge 2$, $q \ge 3$, and $p=6$ be integers, let $\mathcal{ W}=(W_0,W_1,\ldots,W_l)$
be a linear decomposition of length $l$
and adhesion $q$ of $G$, and let $\cal P$ be a foundational
linkage for $\cal W$ such that {\rm (L1)--(L9)} hold.
If there exist $40\binom{q}{3}$ distinct indices $i \in \{1,2,\ldots,l-1\}$ such that $G[W_i]$ contains
a non-trivial $\cal P$-bridge attaching to a trivial foundational path, then
$G$ is apex.
\end{lemma}
{\noindent\bf Proof. } Let $l$, $q$, $p$, $\mathcal{W}$ and $\mathcal{P}$ be given. Assume that there exist $40\binom{q}{3}$ distinct indices $i \in \{1,2,\ldots,l-1\}$ such that $G[W_i]$ contains a non-trivial $\cal P$-bridge attaching to a trivial foundational path. By (L10) each such bridge attaches to two non-trivial foundational paths. Therefore, there exist distinct non-trivial paths $P, P' \in \cal P$ and a trivial path $Q \in \cal P$ such that $G[W_i]$ contains a $\cal P$-bridge attaching to $P,P'$ and $Q$ for at least $40$ distinct indices $i \in \{1,2,\ldots,l-1\}$. The argument used in the proof of Lemma~\ref{lem:pin1} implies that $G$ contains a
subgraph isomorphic to a subdivision of a pinwheel with $20$ vanes or a M\"{o}bius pinwheel with $20$
vanes. Note that the M\"{o}bius pinwheel with $20$
vanes contains a $K_6$ minor, and, thus, $G$ is apex by Lemma~\ref{lem:pin2}, as desired.~$\square$\bigskip
Let us assume Hypothesis~\ref{hypot}, and let $\cal C$ be a core
of $({\cal W},{\cal P})$.
We define \emph{the $i^{\hbox{th}}$ section of} $\cal C$, denoted by
$G({\cal C},i)$, to be the subgraph of $G[W_i]$, obtained from the union of
the paths in $\cal C$ and all $\cal P$-bridges of $G[W_i]$ that
attach to a member of $\cal C$ by deleting the trivial foundational paths.
By Lemma~\ref{maxdeg2} the graph $\cal C$ is a path or a cycle.
Let $P_1,P_2,\ldots,P_t$ be the vertices of $\cal C$, listed in order,
let $W_{i-1}\cap W_i\cap V(P_j)=\{u_j\}$ and
let $W_{i}\cap W_{i+1}\cap V(P_j)=\{v_j\}$.
If $\cal C$ is a path, then we say that $\cal C$ is {\em flat in $W_i$}
if $G({\cal C},i)$ can be drawn in a disk with the vertices
$u_1,u_2,\ldots,u_t,v_t,v_{t-1},\ldots,v_1$ drawn on the boundary of the
disk in order, and the paths $P_1$ and $P_t$ also drawn on the
boundary of the disk.
If $\cal C$ is a cycle, then we say that $\cal C$ is {\em flat in $W_i$}
if $G({\cal C},i)$ can be drawn in a cylinder with the vertices
$u_1,u_2,\ldots,u_t$ drawn on one of the boundary components of the
cylinder in the clockwise order listed, and
$v_t,v_{t-1},\ldots,v_1$ drawn on the other boundary component
in the clockwise order listed.
Our next objective is to find a linear decomposition
${\cal W}=(W_0,W_1,\ldots,W_l)$ and a foundational linkage $\cal P$
such that
\begin{itemize}\item[(L11)]
Every core of $({\cal W},{\cal P})$ is flat in $W_i$
for every $i\in\{1,2,\ldots,l-1\}$.
\item[(L12)] For every $i\in\{1,2,\ldots,l-1\}$, no non-trivial $\cal P$-bridge of $G[W_i]$ attaches to a trivial foundational path.
\end{itemize}
\begin{lemma}\mylabel{lem:planarbridges}
Let $G$ be a 6-connected non-apex graph not containing $K_6$ as a minor. Let $p=6$, $l \ge 2$, $q \ge 6$
be integers, and let $\mathcal{W} = (W_1, W_2, \dots, W_l)$ be a linear decomposition of $G$
of adhesion $q$ and length $l$
satisfying {\rm (L1)--(L10)}. If $l > \left ( 88 \binom{q}{3} + 12\binom{q}{2}\right) l'$, then there exists a
contraction $\mathcal{W}'$ of $\mathcal{W}$ of length $l'$ such that ${\cal W}'$ and the corresponding restriction of $\cal P$ satisfy
{\rm (L1)--(L12)}.
\end{lemma}
{\noindent\bf Proof. }
Let $G$, $p$, $q$, $l$, $\mathcal{W}$, and $\cal P$ be given.
By our choice of $l$ and Lemmas ~\ref{lem:elimtwists},~\ref{lem:elimtunnels} and~\ref{lem:elimattach},
there exists an index $\alpha$ such that
for all $i\in\{0,1,\ldots,l'\}$ the graph $G[W_{\alpha+i}]$
does not contain a $P$-tunnel for any $P$ in $\mathcal{P}$,
nor does it contain a pair of non-trivial twisting foundational paths, nor does it contain
a non-trivial bridge attaching to a trivial foundational path. We claim that the contraction
$\left ( \bigcup_{i = 0}^{\alpha-1} W_i, W_{\alpha}, W_{\alpha +1}, \dots,
W_{\alpha+l'}, \bigcup_{i = \alpha + l' +1}^l W_i\right)$ of $\mathcal{W}$ is as desired. Condition (L12) follows from the construction, and hence it suffices to prove (L11).
Fix an index $i\in\{0,1\ldots, l'\}$ and a core $\cal C$ of the auxiliary graph.
We wish to apply Lemma~\ref{planarstrip} or~\ref{cylinderstrip},
depending on whether $\cal C$ is a path or cycle, to the graph $H:=G({\cal C},\alpha + i)$
and linkage $\cal C$. Let $P_j,u_j,v_j$ for $j \in \{1,2,\ldots,t\}$ be as in the definition of flat.
By Corollary~\ref{cor:induced} and (L10) every $\cal C$-bridge of $H$ is stable, and by (L10) no $\cal C$-bridge of $H$ attaches to three or more members of $\cal C$. If there exists a set $X \subseteq V(H)$ of size at most three such that some component $J$ of $G \setminus X$ is disjoint from $\{u_1,u_2,\ldots,u_t,v_1,v_2, \ldots, v_t\}$, then by $6$-connectivity of $G$ the vertices of $J$ include a neighbor of at least three distinct trivial paths of $\cal P$. We conclude that some member of $\cal C$ is adjacent in the auxiliary graph to at least three trivial foundational paths, contrary to Lemma~\ref{lem:3trivatt}. Thus no such set $X$ exists. Next we show that none of the outcomes (i)--(iii) of Lemmas~\ref{planarstrip} and~\ref{cylinderstrip} hold. Outcome (i) does not hold by the definition of $\cal C$, and outcomes (ii) and (iii) do not hold by the choice of $\alpha$ and $i$. Thus it follows from Lemma~\ref{planarstrip} if $\cal C$ is a path or Lemma~\ref{cylinderstrip} if $\cal C$ is a cycle that $H$ can be drawn in a disk or a cylinder as described in that lemma, which is precisely the definition of $\cal C$ being flat in $W_{\alpha+i}$. Thus $\cal W'$ satisfies (L11) as well.~$\square$\bigskip
\section{Controlling the boundary of a planar graph}\mylabel{sec:control}
Let $G$ be a simple plane graph with the infinite region bounded by
a cycle $C$, and such that the degree of every
vertex in $V(G) - V(C)$ is at least six.
DeVos and Seymour~\cite{DevSeyExt3col} proved that
$|V(G)|\le |V(C)|^2/12 + O(|V(C)|)$.
In this section we digress to prove
a similar result under the weaker hypothesis that $G$ has
deficiency at most five, where the {\em deficiency} of a plane graph
$G$ with the infinite region bounded by a cycle $C$
is defined
as $\sum_{v \in V(G) - V(C)} \max\{ 6- \deg(v),0\}$.
We denote the deficiency of $G$ by $\hbox{def}(G)$.
The proof is an adaptation of the argument
from~\cite{DevSeyExt3col}, but we include it,
because the details are different.
We begin with a couple of definitions and a lemma.
A {\em quilt} is a simple plane graph $G$ with the infinite region bounded
by a cycle $C$, such that $G$ has deficiency at most five and every finite
region of $G$ is bounded by a triangle.
If exactly one vertex of $C$ has degree three, and all other vertices
have degree exactly four, then we say that $C$ is a {\em convenient graph}.
Otherwise, a {\rm convenient graph} is a subpath of $C$
with at least one edge, with both ends
of degree exactly three, and all internal vertices of degree exactly four.
\begin{lemma}\mylabel{lem:ds1}
Every quilt with no vertices of degree two has a convenient graph.
\end{lemma}
{\noindent\bf Proof. }
Let $G$ be a quilt with no vertices of degree two, and
let the deficiency of $G$ be $d$. Consider the planar graph $G'$ obtained
by adding a vertex $v$ to $G$ adjacent to every vertex of $C$.
Let $|V(G)| = n$ and $|V(C)| = m$. Then
\begin{align*} 6(n+1) - 12 & = \sum_{v \in V(G')} \deg_{G'}(v) \\
& = \sum_{v \in V(C)} (\deg_G (v) + 1) + m + \sum_{v \in V(G) - V(C)} \deg_G(v)\\
& \ge \sum_{v \in V(C)} \deg_G(v) + 6(n-m) - d + 2m.
\end{align*}
It follows that $\sum_{v \in V(C)} \deg_G(v) \le 4m - 6 + d$.
Since $d \le 5$ we deduce that there are strictly more vertices in $C$ of degree
three than of degree at least five. Thus, a convenient graph exists.~$\square$\bigskip
The main theorem of this section follows easily from the next lemma.
If $G$ is a quilt, we define $\mu(G)$ to be $1$ if $G$ has a vertex of
degree two, and otherwise we define $\mu(G)$ to be the minimum number of
edges in a convenient graph.
Thus $\mu(G)$ is at least one, and at most the length of the cycle
bounding the infinite region of $G$.
\begin{lemma}
\mylabel{lem:quilt}
Let $G$ be a quilt on at least four vertices with the infinite region bounded
by a cycle of length $k$.
Then $|V(G)|\le k^2/2+k/2+\mu(G)+\hbox{\rm def}(G)-6$.
\end{lemma}
{\noindent\bf Proof. } Let $G$ and $k$ be as stated. We proceed by induction on $|V(G)|$.
If $G$ has exactly four vertices, then it is isomorphic to $K_4$, or $K_4$ minus an edge.
We have $k=3$, $\mu(G)=1$, $\hbox{def}(G)=3$, or $k=4$, $\mu(G)=1$, $\hbox{def}(G)=0$, and the lemma holds.
Thus we may assume that $G$ has at least five vertices, and that
the lemma holds for all quilts on fewer than $|V(G)|$ vertices.
Let $C$ be the cycle bounding the infinite region of $G$.
If $C$ has a chord, then the chord divides $G$ into two
quilts $G_1$ and $G_2$ in the obvious way.
Let the infinite region of $G_i$ have length $k_i$.
Assume first that $G_2$ has exactly three vertices.
Then by induction
\begin{align*}
|V(G)| &= |V(G_1)|+1\le k_1^2/2+k_1/2+\mu(G_1)+\hbox{def}(G_1)-6+1\\
& = k^2/2+k/2 +\mu(G_1)-k+1+\hbox{def}(G_1)-6 \\
& \le k^2/2+k/2 +\mu(G)+\hbox{def}(G)-6,
\end{align*}
as desired.
Thus we may assume that both $G_1$ and $G_2$ have at least four vertices.
Since $k_1,k_2\ge 3$ we have $3(k_1+k_2)\le k_1k_2+9$,
and hence by induction
\begin{align*}
|V(G)| &= |V(G_1)|+|V(G_2)|- 2\\
& \le k_1^2/2+k_1/2+k_1+\hbox{def}(G_1)-6+k_2^2/2+k_2/2+k_2+\hbox{def}(G_2)-6-2\\
& = (k_1+k_2-2)^2/2 + (k_1+k_2-2)/2 +\hbox{def}(G_1)+\hbox{def}(G_2)
-k_1k_2+3k_1+3k_2-15 \\
& \le k^2 + k/2 + \mu(G)+\hbox{def}(G)-6,
\end{align*}
as desired.
Thus we may assume that $C$ has no chord.
In particular, $G$ has no vertex of degree two.
By Lemma~\ref{lem:ds1} the quilt $G$ has a convenient graph.
Let $P$ be a convenient graph with the smallest number of edges.
Let us assume first that $P$ has exactly one edge.
Then $P$ is a path with ends $u$ and $v$, say.
Since $C$ does not have any chords and $G$ has at least five vertices,
the graph $G':=G\backslash \{u,v\}$ is a quilt.
If $G'$ has exactly three vertices, then $G$ is the wheel on five
vertices, $k=4$, $\mu(G)=1$, $\hbox{def}(G)=2$, and the lemma holds.
Thus we may assume that $G'$ has at least four vertices, and hence by
induction
\begin{align*}
|V(G)| &= |V(G')|+2\le (k-1)^2/2+(k-1)/2+\mu(G')+\hbox{def}(G')-6+2\\
& = k^2/2+k/2 +\mu(G')-k+2+\hbox{def}(G')-6 \\
& \le k^2/2+k/2 +\mu(G)+\hbox{def}(G)-6,
\end{align*}
as desired.
Thus we may assume that $P$ has at least two edges.
If $P=C$, then let $u$ be the unique vertex of $C$ of degree three;
otherwise $P$ is a path, and we let $u$ be an end of $P$.
Let $u'$ be the unique neighbor of $u$ that does not belong to $C$.
Then $G':=G\backslash u$ is a quilt on at least four vertices
with the infinite region bounded
by a cycle $C'$, where $C'$ has length $k$.
Since $C$ has no chords and $G$ has at least five vertices we deduce
that $\deg_{G'}(u')\ge 3$.
If equality holds, then $u$ has degree four in $G$, and hence
$\hbox{def}(G')=\hbox{def}(G)-2$.
Otherwise $\mu(G')\le\mu(G)-1$. In either case we have by induction
\begin{align*}
|V(G)| &= |V(G')|+1\le k^2/2+k/2+\mu(G')+\hbox{def}(G')-6+1\\
& \le k^2/2+k/2 +\mu(G)+\hbox{def}(G)-6,
\end{align*}
as desired.~$\square$\bigskip
\begin{theorem}\mylabel{thm:bdeddisc}
Let $G$ be a simple graph drawn in a disk, let $X$ be the set of vertices of $G$
drawn on the boundary of the disk, and assume that
$\sum_{v\in V(G)-X} \max\{6-\deg(v),0\}\le 5$. If $|X| \geq 3$,
then $|V(G)|\le |X|^2/2+3|X|/2-1$.
\end{theorem}
{\noindent\bf Proof. } Let $G$ and $X$ be as stated. We may assume, by adding edges to $G$, that $G$ is a quilt with the infinite region bounded by a cycle with vertex set $X$. By Lemma~\ref{lem:quilt} we have $|V(G)| \leq |X|^2/2 + |X|/2 + \mu(G) +\hbox{def}(G) -6 \leq |X|^2/2+3|X|/2-1$, as desired.~$\square$\bigskip
\section{Cylindrical tube}\mylabel{sec:cyl}
Lemma~\ref{nontrivpath} guarantees the existence of a non-empty core in a sufficiently long linear decomposition of any $K_6$-minor-free $6$-connected
graph $G$ of bounded tree-width, assuming that such a decomposition satisfies conditions (L1)--(L9).
Lemma~\ref{maxdeg2} implies that, under the same conditions, each core is a path or a cycle. In this section we handle the case when some core of a linear decomposition of the graph $G$ is a cycle.
Before introducing the main result of this section, we need to present one more definition and a related lemma.
Let $k$, $l$ be positive integers, $k, l \ge 3$. A \emph{double crossed
$k$-cylinder of length $l$} is the graph defined as follows. Let
$P_1, \dots, P_k$ be $k$ vertex disjoint paths with the vertex set of $P_i =
\{v^i_j: 1 \le j \le l\}$ for all $1 \le i \le k$ with $v^i_j$ adjacent to $v^i_{j+1}$
for all $1 \le j \le l-1$. The double crossed
$k$-cylinder of length $l$ has vertex set $\{v^i_j: 1 \le j \le l, 1 \le i \le k\}$ and edge set $$\left (
\bigcup_{i = 1}^k E(P_i) \right) \cup \{ v^i_j v^{i+1}_j : 1 \le i \le k, 1 \le j \le l\} \cup
\{ q_1, q_2, r_1, r_2\},$$ where the superscript addition is taken
modulo $k$. Furthermore, the
ends of $q_i$ are $u_i, v_i\in \{v^j_1: 1 \le j \le k\}$ for $i = 1, 2$ and
the vertices $u_1, u_2, v_1, v_2$ occur in that order in the cyclic order
$(v^1_1, v^2_1, \dots, v^1_k)$. Similarly, the edges $r_1$ and $r_2$
cross in the cyclic order $(v^1_l, v^2_l, \dots, v^k_l)$. Explicitly, the ends of
$r_i$ are $x_i, y_i \in \{v^j_l: 1 \le j \le k\}$ for $i = 1, 2$ and occur
in the order $x_1, x_2, y_1, y_2$ in the cyclic order
$(v^1_l, v^2_l, \dots, v^k_l)$.
\begin{lemma}\mylabel{lem:dblextube}
Let $t$ and $l$ be integers, $t \ge 5$, $l \ge 16$. A double crossed $t$-cylinder
of length $l$ contains $K_6$ as a minor.
\end{lemma}
{\noindent\bf Proof. }
Let $G$ be a doubled crossed $t$-cylinder
of length $l$ with vertex set $\{v^i_j: 1 \le j \le l, 1 \le i \le t\}$.
By possibly routing the crossing edges $q_1$ and $q_2$ in the first five cycles
on vertices $\{v^i_j: 1 \le j \le 5, 1 \le i \le t\}$ and routing the edges $r_1$ and $r_2$ on the final
five cycles with vertex set $\{v^i_j: l-5 \le j \le l, 1 \le i \le t\}$, we see that $G$ contains as a minor
a doubled crossed $5$-cylinder $G'$
of length $6$ and moreover, with the additional property that
the ends of $q_1$ are $v^1_1$ and $v^3_1$ and the ends
of $q_2$ are $v^2_1$ and $v^4_1$. Similarly, the edges $r_1$ and $r_2$ of $G'$
have ends $v^1_6$, $v^3_6$ and $v^2_6$, $v^4_6$, respectively. The graph $G$
then contains $K_6$ as a minor, as indicated in Figure \ref{fig:cyl2x}.~$\square$\bigskip
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page = 19]{jorg_figs.pdf}
\end{center}
\caption{A double crossed $5$-cylinder of length $6$ contains $K_6$ as a minor}
\label{fig:cyl2x}
\end{figure}
We now give the main result of this section.
\begin{lemma}\mylabel{lem:cylcase}
Let $p=6$, $l \ge 2$, and $q \ge 6$ be integers.
Let $G$ be a $6$-connected graph with no $K_6$ minor,
and let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be a linear decomposition
of $G$ of length $l$ and adhesion $q$ with a foundational
linkage $\cal P$ satisfying {\rm(L1)--(L12)}. Further, assume
that some core of $({\cal W},{\cal P})$ is a cycle. If
$l \ge 2q + 32$, then $G$ is apex.
\end{lemma}
{\noindent\bf Proof. }
Let $p$, $l$, $q$, and $\cal W$ be given, let $\cal C$ be a core of $({\cal W},{\cal P})$ that is a cycle, and assume for a contradiction that $G$ is not apex.
Let $P_1,P_2,\ldots, P_t$ be the vertices of $\cal C$ listed in order. For $i=1,2,\ldots, l-1$ let $H_i$ denote the graph $G({\cal C}, i)$, and for $j =1,2,\ldots,t$ let $u_j$ be the unique element of $V(P_j) \cap W_q \cap W_{q+1}$ and $v_j$ the unique element of $V(P_j) \cap W_{q+32} \cap W_{q+33}$. Let $A=\{u_1,u_2,\ldots,u_t\}$, $B=\{v_1,v_2,\ldots,v_t\}$, let $K$ denote the graph $H_{q+1} \cup H_{q+2} \cup \ldots \cup H_{q+32}$, and let $L$ denote the graph $G \setminus (V(K)-A-B)$. Since $G$ is not apex and $\cal C$ is a cycle, by Corollary~\ref{cor:mainpin} the core $\cal C$ forms a component of the auxiliary graph. Therefore, we have $K \cup L = G$ and $V(K \cap L)=A \cup B$.
We claim that $L$ does not include two disjoint paths from $A$ to $B$. Indeed, otherwise by contracting $P_i[W_{q + 2j}]$ to a single vertex for $1 \le i \le t$ and $0 \le j \le 11$,
we see that $G$ contains a linked $t$-cylinder of length twelve.
Lemma \ref{lem:HK6} then contradicts our choice of $G$. Thus there exist subgraphs $L_1,L_2$ of $L$ such that $L_1 \cup L_2=L$, $A\subseteq V(L_1)$, $B \subseteq V(L_2)$ and $|V(L_1 \cap L_2)| \leq 1$. Now property (L9) applied to $\cal C$ and a subset of $\cal C$ of size two implies that $t \geq 5$.
Let $\Omega_1$ be the cyclic permutation $(u_1,u_2,\ldots, u_t)$, and let $\Omega_2$ be the cyclic permutation $(v_1,v_2,\ldots, v_t)$. Thus $(L_1, \Omega_1)$ and
$(L_2, \Omega_2)$ are societies. Let $X= V(L_1 \cap L_2)$. By (L11) the graph $K$ can be drawn in a cylinder with $u_1,u_2,\ldots, u_t$ drawn in one boundary component in the clockwise order listed, and $v_1,v_2,\ldots, v_t$ drawn in the other boundary component in the clockwise order listed. Thus if both societies $(L_1 \setminus X, \Omega_1 \setminus X)$ and $(L_2 \setminus X, \Omega_2 \setminus X)$ are rural, then $G$ is apex, so we may assume that $(L_1 \setminus X, \Omega_1 \setminus X)$ is not rural and hence by Theorem~\ref{thm:RSsociety} it has a cross. The society $(L_2 , \Omega_2)$ is not rural by Theorem~\ref{thm:bdeddisc}, because each vertex of $V(L_2) - B - X$ has degree at least $6$ and $|V(L_2)| \geq qt \geq t^2= |B|^2$, because $V(L_2)$ includes each of the pairwise disjoint sets $W_i \cap W_{i+1}\cap V({\cal C})$ for $i=q+32,q+33,\ldots, 2q+31$. Likewise, $(L_2, \Omega_2)$ has a cross by Theorem~\ref{thm:bdeddisc}.
We have shown that there exist four pairwise disjoint paths, two of them forming a cross in
$(L_1, \Omega_1)$ and two forming a cross in
$(L_2, \Omega_2)$. Let $j \in \{0,1,\ldots, 15 \}$. By the definition of core the graph $G({\cal C}, q+2j+1)$ has internally disjoint paths $Q_1,Q_2,\ldots, Q_t$ such that $Q_i$ has one end in $P_i$, the other end in $P_{i+1}$ (where $P_{t+1}$ means $P_1$), and is otherwise disjoint from $\cal C$. Since for $j\neq j'$ the graphs $G({\cal C}, q+2j+1)$ and $G({\cal C}, q+2j'+1)$ are vertex disjoint,
we conclude that $G$ contains as a minor a double crossed $t$-cylinder
of length at least 16. This observation contradicts Lemma \ref{lem:dblextube} and completes the
proof of the lemma.
$\square$\bigskip
\section{Planar strip}\mylabel{sec:planar}
We now examine the case when some core of the auxiliary graph
is a path.
\begin{lemma}\mylabel{lem:finalplan}
Let $p=6$, $l \ge 2$ and $q \ge 6$ be integers.
Let $G$ be a $6$-connected graph with no $K_6$ minor,
and let ${\cal W}=(W_0,W_1,\ldots,W_l)$ be a linear decomposition
of $G$ of length $l$ and adhesion $q$ with a foundational
linkage $\cal P$ satisfying (L1)--(L12). Further, assume
that some core of $({\cal W},{\cal P})$ is a path. If $l \ge \max\{4q+11, 48\}$,
then $G$ is an apex graph.
\end{lemma}
{\noindent\bf Proof. } Let $p$, $l$, $q$, and $\cal W$ be given, let $\cal C$ be a core of $({\cal W},{\cal P})$ that is a path, and assume for a contradiction that $G$ is not apex.
Let $P_1,P_2,\ldots, P_t$ be the vertices of $\cal C$ listed in order. As in the proof of Lemma~\ref{lem:cylcase}, for $i=1,2,\ldots, l-1$ let $H_i$ denote the graph $G({\cal C}, i)$, and for $j =1,2,\ldots,t$ let $u_j$ be the unique element of $V(P_j) \cap W_0 \cap W_{1}$ and $v_j$ the unique element of $V(P_j) \cap W_{l-1} \cap W_{l}$. Let $A=\{u_1,u_2,\ldots,u_t\}$, $B=\{v_1,v_2,\ldots,v_t\}$, and let $\cal Q$ denote the set of trivial foundational paths adjacent in the auxiliary graph to paths in $\cal C$. Let $K$ denote the subgraph of $G$ induced on $V(H_{1} \cup H_{2} \cup \ldots \cup H_{l-1}) \cup V({\cal Q})$, and let $L$ denote the graph $G \setminus (V(K)-A-B-V({\cal Q}))$. Note that $K \cup L = G$ and $V(K) \cap V(L) = A \cup B \cup V(\cal Q)$.
We claim that either $P_1$ or $P_t$ is adjacent in the auxiliary graph to at least two paths in $\cal Q$. Suppose for a contradiction that both $P_1$ and $P_t$ are adjacent to at most one such path. We assume that $P_i$
is adjacent to exactly one trivial foundational path $S_i \in \cal Q$ for $i = 1,i=t$. The argument is similar in the case when one or both of $P_1$ and $P_t$ are not adjacent to any paths in $\cal Q$. Note that by (L12) and Corollary~\ref{cor:mainpin} all the neighbors of $V(S_1)$ and $V(S_2)$ lie on $P_1 \cup P_2$.
If $S_1 \neq S_t$, we let $\{s_i\} = V(S_i)$
for $i = 1$, $i = t$ and $K'=K$. If $S_1 = S_t$ with $V(S_1) = V(S_t) = \{s\}$, let $K'$ be obtained from $K$ by deleting $s$, and adding new vertices $s_1$ and $s_2$, where $s_1$ is adjacent to every neighbor of $s$ on $P_1$, and
$s_t$ is adjacent to every neighbor of $s$ on $P_t$.
By property (L11), the graph
$K'$ is planar and embeds in a disk with exactly the
vertices $\{s_1, s_t\} \cup
A \cup B$ on the boundary.
Moreover, every vertex not on the boundary of the disk has degree at least six. This is a contradiction to Theorem \ref{thm:bdeddisc},
as $|V(K')| \geq lt > (2t+2)^2$, because $l \geq 4q+11$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page=13]{jorg_figs.pdf}
\end{center}
\caption{Finding a $K_6$ minor when there exist four distinct trivial foundational
paths with neighbors in $\cal C$.}
\label{fig:plan2by2}
\end{figure}
Using the above claim and Lemma~\ref{lem:3att} we assume without loss of generality that $P_1$ is adjacent in the auxiliary graph to exactly two paths in $\cal Q$, say $Q_1$ and $Q_2$. Let $V(Q_1)=\{q_1\}$ and $V(Q_2)=\{q_2\}$. We claim that the graph $G'=G \setminus \{q_1, q_2\}$ is planar and that $P_1$ is a subset of a facial boundary of $G'$. Suppose that $P_t$ is adjacent to at least two paths in ${\cal Q} - \{Q_1,Q_2\}$. Then $G$ contains as a minor the graph in Figure \ref{fig:plan2by2}. The horizontal paths in the figure correspond to contractions of $P_1$ and $P_t$ and the vertical edges correspond to paths in $H_{2i+1}$ for $i=1,2,\ldots, 6$ with ends on $P_1$ and $P_t$, which exist by the definition of $\cal C$. The graph in Figure \ref{fig:plan2by2} contains a $K_6$ minor, as indicated, a contradiction.
Therefore $P_t$ is adjacent to at most one path in ${\cal Q} - \{Q_1,Q_2\}$. By (L11), (L12) and Corollary~\ref{cor:mainpin}, the graph
$K$ is planar and embeds in the disk with $P_1$ forming part of its boundary. Let $\Omega$ be a cyclic permutation of the set $V(\Omega)= A \cup B \cup (V({\cal Q})-\{q_1,q_2\})$ ordered $u_t, u_{t-1},\ldots,u_1,v_1,\ldots,v_t$ followed by the element of $V({\cal Q})-\{q_1,q_2\}$ if $V({\cal Q})-\{q_1,q_2\} \neq \emptyset$. If the society $(L, \Omega)$ contains a cross, then $G$ contains as a minor one
of the configurations pictured in Figure \ref{fig:strip3}. As each of this configurations contains a $K_6$ minor as indicated in Figure \ref{fig:strip3}, we conclude by Theorem~\ref{thm:RSsociety} that $(L, \Omega)$ is rural. Combined with the planarity of $K$ this implies our claim that $G'$ is planar and $P_1$ is a subset of a facial boundary.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page = 16]{jorg_figs.pdf}
\hfill
\includegraphics[scale = .65, page = 15]{jorg_figs.pdf}
\vskip 12pt
\vfill
\includegraphics[scale = .65, page = 14]{jorg_figs.pdf}
\end{center}
\caption{Finding $K_6$ minor when the society $(L, \Omega)$ is not rural.}
\label{fig:strip3}
\end{figure}
Let ${\cal P}_2=\{Q_1,Q_2,P_1,P_2\}$. By property (L9), there exist two disjoint paths $R_1$ and $R_2$ in $G[W_0\cup W_l]\cup \bigcup_{P\in{\cal P}-{\cal P}_2} P$ linking the set $\{u_1, u_2\}$ to the set $\{v_1, v_2\}$. By the claim in the previous paragraph we assume without loss of generality that $R_i$ has ends $u_i$ and $v_i$ for $i=1,2$, and that $R_1 \cup P_1$ forms a facial cycle of $G'$. As $G$ is not apex, both $q_1$ and $q_2$ must have some neighbor not contained in $R_1 \cup P_1$. Let $q'_i$
be such a neighbor of $q_i$ for $i = 1, 2$. The cycle $R_1 \cup P_1$ is a facial cycle in the $4$-connected planar graph $G'$, and hence there is a unique $(R_1 \cup P_1)$-bridge in $G-\{q_1,q_2\}$. It follows that for each $q'_i$ there exists a path from $q'_i$ to $R_2 \cup P_2$ avoiding $R_1 \cup P_1$. Let $R'_i$ for $i = 1, 2$ be such paths
from $q'_i$ to $R_2 \cup P_2$. Since $l \geq 48$ there exists an index $\alpha$ such that $W_{\alpha+i}$
is disjoint from $R'_1$ and $R'_2$ for $0 \le i \le 14$. By considering
$P_1$ and $P_2$ and the bridges attaching to $P_1$ and $P_2$ in $H_\alpha, H_{\alpha + 1}, \ldots, H_{\alpha + 14}$, we see that
$G$ contains as a minor the graph in Figure \ref{fig:plan2}, and consequently, a $K_6$ minor, as indicated in Figure \ref{fig:plan2}. This contradiction completes the proof of the lemma.~$\square$\bigskip
\begin{figure}[h!]
\begin{center}
\includegraphics[scale = .65, page = 18]{jorg_figs.pdf}
\end{center}
\caption{Configurations giving $K_6$ minors when the trivial foundational paths
$Q_1$ and $Q_2$ have a neighbor not contained in the boundary of the face defined by $R_1 \cup P_1$}
\label{fig:plan2}
\end{figure}
Lemma~\ref{lem:finalplan} represents the final step in our analysis of the structure of the auxiliary graph. We are now
ready to prove Theorem~\ref{main}.
\vskip 5pt
{\noindent\bf Proof of Theorem~\ref{main}.} Let $w\geq 1$ be an integer. Let $l_1= \max\{4w+11, 2w+32, 58\}$, let $l_2= \left (88 \binom{w}{3} + 12\binom{w}{2}\right)l_1$, and let $l_3 = \left( 6\binom{w}{6} + 48 \binom{w}{3}\right )l_2.$ By Corollary~\ref{cor:normdec} there exists an integer $N$ such that every $6$-connected graph $G$ of tree-width at most $w$ with no $K_6$ minor has a linear decomposition of length at least $l_3$ and adhesion at most $w$ satisfying properties {\rm (L1)--(L9)} for $p=6$. We claim that such an integer $N$ satisfies Theorem~\ref{main}.
Let $G$ be a $6$-connected graph of tree-width at most $w$ with at least $N$ vertices and no $K_6$ minor.
By Lemma~\ref{lem:l10} the graph $G$ has a linear decomposition of length at least $l_2$ and adhesion at most $w$ satisfying properties (L1)--(L10), and thus by Lemma~\ref{lem:planarbridges} the graph $G$ has a linear decomposition $\cal W$ of length at least $l_1$ and adhesion at most $w$ and a foundational linkage $\cal P$ satisfying properties (L1)--(L12). By Lemma~\ref{nontrivpath} $\cal P$ includes a non-trivial foundational path. By Lemma~\ref{lem:3trivatt} every non-trivial foundational path of $\cal P$ attaches to at most $2$ trivial foundational paths in the auxiliary graph. Therefore, by the $6$-connectivity of $G$, every core of $({\cal W}, {\cal P})$ has at least two vertices, and by Lemma~\ref{maxdeg2} every core is a path or a cycle. If some core of $({\cal W}, {\cal P})$ is a cycle, then $G$ is apex by Lemma~\ref{lem:cylcase}. Otherwise, $G$ is apex by Lemma~\ref{lem:finalplan}.~$\square$\bigskip
\section*{Acknowledgment}
We would like to acknowledge the contributions of Matthew DeVos and
Rajneesh Hegde, who worked
with us in March 2005 and contributed to this paper,
but did not want to be included as a coauthors.
|
{
"timestamp": "2013-04-09T02:00:21",
"yymm": "1203",
"arxiv_id": "1203.2171",
"language": "en",
"url": "https://arxiv.org/abs/1203.2171"
}
|
\section{Introduction}
Calorimetry is a powerful tool that allows complete thermodynamic characterization of materials. In particular, nanocaloric measurements are suitable to study phase transitions and specific heat dependencies of new superconductors, often available in just $\upmu$g quantities. In the past two decades much attention was devoted to studies of high-$T_\mathrm{c}$ superconductors \cite{Fisher} and, more recently, the iron-based superconductors \cite{Paglione}. The electronic specific heat is one of the crucial parameters for understanding high-temperature superconductivity, but it is very hard to measure with good absolute accuracy. Furthermore, it represents just a fraction of the total heat capacity and therefore requires high resolution techniques. The AC steady state method \cite{SullivanSeidel} is a very sensitive technique which usually gives only relative heat capacity values \cite{Fisher} because of the difficulties related to the choice of the working frequency.
\begin{figure}[ht]
\includegraphics[width=0.43\linewidth]{Tagliati_Figure1.png}\hspace{0.04\linewidth}%
\begin{minipage}[b]{0.53\linewidth}\caption{\label{Fig1_Sketch}%
Layout of one of the $\mathrm{Si}_3\mathrm{N}_4$ membranes which host the calorimetric cell. A Ti heater and $\mathrm{Ge}_\mathrm{1-x}\mathrm{Au}_\mathrm{x}$ thermometer are fabricated in a pile in the central $110\times80\,\upmu\mathrm{m}^2$ area onto which the sample is placed. Two layers of $\mathrm{AlO}_\mathrm{x}$ ensure electrical insulation and a third $\mathrm{SiO}_2$ layer protects the surface of the $\mathrm{Ge}_\mathrm{1-x}\mathrm{Au}_\mathrm{x}$ sensor. The pile and the membrane underneath compose the platform with heat capacity $C_\mathrm{0}$. The membrane outside the platform thermally connects the sample area to the thermal bath (thermal conductance $K_\mathrm{e}$) and contributes with a heat capacity $C_{\mathrm{m,eff}}$.}
\end{minipage}
\end{figure}
In this method, a certain power $P_0\sin\omega t$ modulates the temperature of sample and calorimetric cell that oscillate with amplitude $T_{\mathrm{ac},0}$ and a phase lag $\phi$. The heat capacity is given by \cite{Gmelin}:
\begin{equation}{}\label{EqCK}
C =\frac{P_\mathrm{0}}{\omega T_\mathrm{ac,0}}\sin\phi.
\end{equation
The optimal frequency range is usually quite narrow and depends on several factors, such as temperature, sample heat capacity and device thermal link. If $\omega$ is too high the sample becomes thermally disconnected and the signal probes just the heat capacity of the cell. On the other hand, the resolution degrades at too low frequencies \cite{Rydh_Manuscript}.
We have implemented a measurement method which avoids these problems and furnishes absolute values \cite{Tagliati}. It requires continuous tuning of the frequency based on the reading of the phase.
The measured heat capacity can be written as:
\begin{equation}
C = C_0+C_\mathrm{m,eff} + (1-g)C_\mathrm{s},
\end{equation}\label{Eq_CKtotal
where $C_0+C_\mathrm{m,eff}$ is the background heat capacity (see Fig.~\ref{Fig1_Sketch}) and $g$ is a frequency dependent function.
Inaccuracies in the evaluation of the sample heat capacity $C_\mathrm{s}$ are due to non-zero values of $g$ which, in turn, are caused by a finite thermal conductance $K_\mathrm{i}$ between sample and cell. The tangent of the phase $\tan\phi$ is an excellent indicator of the working conditions. It has a local maximum at a certain frequency $\omega_\mathrm{max}$, after which the absolute accuracy decreases quickly and the working conditions become unstable. The key parameter which controls this maximum is $\beta=K_\mathrm{i}/K_\mathrm{e}$: $\tan\phi_{\max}\approx\sqrt{\beta}/2$. In general, the absolute error at the maximum is $g|_{\omega_{\max}} \approx 1/\beta$. By tuning $\omega$ to keep a constant phase the error can be minimized while maintaining a good resolution. To demonstrate the effectiveness of the technique we here characterize a microgram Pb sample.
\section{Experimental}
The calorimetric cell used for measuring the Pb sample is shown in Fig.~\ref{Fig1_Sketch}.
The crystal is selected to match the dimensions of the calorimeter central area. To reach a uniform temperature along the sample thickness at a given frequency, the crystal should be thinner than the length of the thermal wave. It was chosen to be less than $50\,\upmu\mathrm{m}$ thick.
Apiezon grease, with thermal conductance $K_\mathrm{i}$, is used to ensure thermal contact between the sample and the pile of heater and thermometer underneath. Because of the design with all layers stacked on top of each other, the relaxation time within the calorimetric cell is negligible.
In our case, $\beta \approx 700$ at room temperature. During each temperature scan the frequency is automatically varied to maintain the phase constant well below $\phi_\mathrm{max}$. Between $300\,\mathrm{K}$ and $0.5\,\mathrm{K}$ the drive frequency $f=\omega/4\pi$ spans the range $1\,\mathrm{Hz}<f<60\,\mathrm{Hz}$.
The temperature oscillation amplitude $T_{\mathrm{ac},0}$ is chosen to be a certain fraction of the absolute temperature $T$, and $P_0$ is changed accordingly. For a certain phase, the resolution improves with higher temperature oscillation amplitudes. For example, a typical resolution $\delta C/C=3\cdot10^{-4}$ is achieved at $T_{\mathrm{ac},0}/T=5\cdot10^{-3}$ for a $1\,\mathrm{s}$ integration time constant. The drawback of a high $T_{\mathrm{ac},0}/T$ ratio is a smearing effect of the absolute temperature scale. By decreasing $T_{\mathrm{ac},0}$ it is possible to contain this effect still maintaining a reasonable resolution. To measure the superconducting transition we used $T_{\mathrm{ac},0}=4.8\,\mathrm{mK}$ and $5\,\mathrm{s}$ time constant which gave $\delta C/C=6.8\cdot10^{-4}$.
The empty cell, and cell with grease were pre-characterized to be able to subtract each contribution from the total signal.
To explain the residual $\gamma$ in the superconducting state and obtain absolute agreement with the specific heat at $T_\mathrm{c}$, $c=1.09\,\mathrm{J}/{\mathrm{mol}\,\mathrm{K}}$ reported by Shiffman \textit{et al.} \cite{Schiffman}, $5\%$ of the normal state heat capacity had to be subtracted in addition to the device background. Whether this unexplained sample contribution is in reality due an absolute accuracy issue is left as an open question.
\section{Results}
The Pb sample was characterized down to $0.5\,\mathrm{K}$. To obtain a pure superconducting state curve a small field was applied to compensate the remanent field of the magnet. The normal state was measured in $120\,\mathrm{mT}$. The curves $C(T)/T$ of the Pb sample are shown in Fig.~\ref{Fig2_HeatCapacity}a. The low temperature data (inset of Fig.~\ref{Fig2_HeatCapacity}a) were fitted by a linear function to extrapolate the normal state electronic heat capacity at zero temperature $\gamma=39.0\,\mathrm{pJ}/\mathrm{K}^2$.
The zero field superconducting transition displayed in Fig.~\ref{Fig2_HeatCapacity}b, has a $20\,\mathrm{mK}$ width and no upturn in $C$.
\begin{figure}
\begin{center}
\includegraphics[height=12.2pc]{Tagliati_Figure2a}\hspace{2pc}%
\includegraphics[height=11.5pc]{Tagliati_Figure2b}%
\end{center}
\caption{\label{Fig2_HeatCapacity}\textbf{(a)} Temperature dependence of $C/T$ of the Pb sample in the superconducting (circles) and normal (squares) states. The inset shows $C/T$ vs $T^2$ at low $T$ for the normal state. The line represent the best fit to $C/T=\gamma+\beta T^2$. \textbf{(b)} Heat capacity in zero field and in $120\,\mathrm{mT}$ around $T_\mathrm{c}$.}
\end{figure}
From the step height $\Delta C=0.754\,\mathrm{nJ}/\mathrm{K}$ at $T_\mathrm{c}$ and the Sommerfeld term $\gamma$, we find $\Delta C/\gamma T_\mathrm{c}=2.68$ in agreement with the values obtained from magnetic \cite{Decker} and other calorimetric \cite{Schiffman, Neighbor} measurements. The normalized slope of the specific heat discontinuity at $T_\mathrm{c}$ is an indicator of the coupling strength of the superconductor. From the data in Fig.~\ref{Fig2_HeatCapacity} we obtain $(T_\mathrm{c}/\Delta C)(d\Delta C/dT)_{T_\mathrm{c}}\approx4.5$, close to the value $4.6$ obtained from strong-coupling theory \cite{Carbotte}.
\noindent Using the experimentally determined $\Delta C(T)$, one can calculate $\Delta S(T)$ and $\Delta U(T)$ by integrating $\Delta C(T)/T$ and $\Delta C(T)$ respectively, from $0\,\mathrm{K}$ to $T_\mathrm{c}$. By imposing the conservation law $\Delta F=F_\mathrm{s}-F_\mathrm{n}=0$ at $T_\mathrm{c}$ on the free energy difference $\Delta F=\Delta U-T\Delta S$, the condensation energy $\upmu_0H_\mathrm{c}^2 V/2$, is obtained.
To obtain the most precise measurement of the sample volume, $H_\mathrm{c}(0)$ was taken to be equal to the literature value $80.3\,\mathrm{mT}$ \cite{Carbotte}. The volume is equal to $2\Delta F (0)/\upmu_0 H^2_\mathrm{c}(0)$, giving $N=12.75\,\mathrm{nmol}$.
The temperature dependence of the deviation of $H_\mathrm{c}$ from the two fluid model, $D=H_\mathrm{c}(T)/H_\mathrm{c}(0)-[1-(T/T_\mathrm{c})^2]$ is plotted in Fig.~\ref{Fig3_Hc&L}a. Pb shows a positive deviation curve related to the strong electron-phonon coupling, in contrast with the negative $D$ values of the weak-coupling superconductors. The position of the maximum deviation is at $T/T_\mathrm{c}=0.65$, close to the experimentally determined functional form provided by Decker \textit{et al.} \cite{Decker}, the experimental value by Chanin \textit{et al.} \cite{Chanin}, and the strong-coupling theory calculation of Swihart \textit{et al.} \cite{Swihart}. The maximum deviation for our data, $2.3\%$, is slightly lower than the the values of Decker, $2.4\%$, and Swihart, $2.7\%$, but higher than the $2.1\%$ found by Chanin.
The latent heat $L(T)=T\Delta S(T)$ is shown in Fig.~\ref{Fig3_Hc&L}b. Our data match the curve derived using the power series expansion of $H_\mathrm{c}^2(T)$ by Decker \textit{et al.} \cite{Decker}. The maximum falls at $T/T_\mathrm{c}=0.53$ for both curves.
\begin{figure}
\begin{center}
\includegraphics[height=11.5pc]{Tagliati_Figure3a}\hspace{2pc}%
\includegraphics[height=11.3pc]{Tagliati_Figure3b}%
\end{center}
\caption{\label{Fig3_Hc&L}\textbf{(a)} Deviation of the reduced critical field from a parabolic curve as a function of the reduced temperature. \textbf{(b)} Transitional latent heat as a function of reduced temperature. In both figures the squares are values obtained in this experiment and the line is obtained from the analytical expression for $H_\mathrm{c}(T)$ (a) and $H_\mathrm{c}^2(T)$ (b) proposed by Decker \textit{et al.} \cite{Decker}.}
\end{figure}
\section{Summary and conclusions}
We recently proposed an experimental procedure to obtain both good resolution and absolute accuracy in AC calorimetry \cite{Tagliati}. The results reported in this paper on a $\sim2.6\,\upmu\mathrm{g}$ Pb sample demonstrate the feasibility and effectiveness of this method. From the presented low temperature superconducting and normal state heat capacity curves, thermodynamic properties such as Sommerfeld term $\gamma=3.06\,\mathrm{mJ/mol K}$, reduced jump anomaly $\Delta C/\gamma T_\mathrm{c}=2.68$ and normalized slope $(T_\mathrm{c}/\Delta C)(d\Delta C/dT)_{T_\mathrm{c}}\approx4.5$, are found in good agreement with literature values. The accuracy achieved is further confirmed by the obtained temperature dependence of $H_\mathrm{c}(T)$ and latent heat.
\ack
Support from the the Swedish Research Council and the Knut and Alice Wallenberg Foundation is acknowledged. We would like to thank V.~M.~Krasnov for useful discussions and for providing the Pb sample.
\section*{References}
|
{
"timestamp": "2012-03-12T01:01:20",
"yymm": "1203",
"arxiv_id": "1203.2052",
"language": "en",
"url": "https://arxiv.org/abs/1203.2052"
}
|
\section{Data to Monte Carlo comparison of transverse helicity}
\label{sect:datamccomp}
\begin{figure*}
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\centering
\subfigure[$\mu^+$]{
\includegraphics[width=0.38\textwidth]{fig_03a.pdf}
}\hspace*{-0.5cm}
\subfigure[$\mu^-$]{
\includegraphics[width=0.38\textwidth]{fig_03b.pdf}
}
\subfigure[e$^+$]{
\includegraphics[width=0.38\textwidth]{fig_03c.pdf}
}\hspace*{-0.5cm}
\subfigure[e$^-$]{
\includegraphics[width=0.38\textwidth]{fig_03d.pdf}
}
\end{changemargin}
\caption{The \ctdd\ distributions for 35 $<$ \ptw\ $<$ 50 \GeV. The data (dots) are compared to the distributions from \pow\ (dashed line),
\mcnlo\ (solid line), and for unpolarised \Wbosons\ (dotted line) in the muon (top) and electron (bottom) channel,
split by charge. The bottom parts of each plot represent the ratio of data, \pow\ and unpolarised distributions to \mcnlo.
\label{fig:NoTemplFitEl35}}
\end{figure*}
As shown in Ref.\ \cite{Aad2011dm}, \mcnlo\ and \pow\ give a rather good description of inclusive \Wboson\ production. However both generators
were shown\ \cite{Belloni1361975} to underestimate the fraction of events at large \ptw\ (see also \Tab{Statistics}). While this affects the
relative fraction of data versus Monte Carlo events retained in the two \ptw\ bins of the analysis, it should not significantly impact the angular
distributions used to measure the \Wboson\ polarisation. This is discussed in more detail in Section\ \ref{sect:ptwrew}.
\begin{figure*}
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\centering
\subfigure[$\mu^+$]{
\includegraphics[width=0.38\textwidth]{fig_04a.pdf}
}\hspace*{-0.5cm}
\subfigure[$\mu^-$]{
\includegraphics[width=0.38\textwidth]{fig_04b.pdf}
}
\subfigure[e$^+$]{
\includegraphics[width=0.38\textwidth]{fig_04c.pdf}
}\hspace*{-0.5cm}
\subfigure[e$^-$]{
\includegraphics[width=0.38\textwidth]{fig_04d.pdf}
}
\end{changemargin}
\caption{The \ctdd\ distributions for \ptw\ $>$ 50 \GeV. The data (dots) are compared to the distributions from \pow\ (dashed line),
\mcnlo\ (solid line), and for unpolarised \Wbosons\ (dotted line) in the muon (top) and electron (bottom) channel, split
by charge. The bottom parts of each plot represent the ratio of data, \pow\ and unpolarised distributions to \mcnlo. \label{fig:NoTemplFitEl}}
\end{figure*}
\begin{table*}
\centering
\caption{The $\chi^2$ values from the comparison of the data with the \mcnlo, \pow\ and unpolarised predictions for the \ctdd\ distributions
(see \Figs[NoTemplFitEl35]{NoTemplFitEl}). The number of degrees of freedom in the fits is 19. Only statistical uncertainties are considered.}
\begin{tabular}{|c|c|c|c|c||c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{\minitab[c]{$\chi^2$ between\\ data and}}}& \multicolumn{4}{c||}{35 $<$ \ptw\ $<$ 50 \GeV \bigstrut} & \multicolumn{4}{c|}{\ptw\ $>$ 50 \GeV} \\
\cline{2-9}
& $\mu^+$ & $\mu^-$ & $e^+$ & $e^-$ & $\mu^+$ & $\mu^-$ & $e^+$ & $e^-$ \\
\hline
\mcnlo\ Monte Carlo & 20.0 & 25.0 & 17.0 & 32.1 & 36.2 & 31.5 & 28.6 & 17.3 \\
\hline
\pow\ Monte Carlo & 12.8 & 22.9 & 10.7 & 25.5 & 40.3 & 32.7 & 30.3 & 16.3 \\
\hline
Unpolarised & 23.6 & 33.5 & 28.0 & 79.5 & 62.4 & 44.2 & 129.2 & 42.9 \\
\hline
\end{tabular}
\label{tab:chi2values}
\end{table*}
\begin{figure*}
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\captionsetup{type=figure}
\centering
\subfigure[\fl\ for $\ell^+$]{
\tikz{ \node (f0p) { \includegraphics[width=0.38\textwidth]{fig_05a.pdf}};}
}
\subfigure[\fl\ for $\ell^-$]{
\tikz{ \node (f0m) { \includegraphics[width=0.38\textwidth]{fig_05b.pdf}};}
}
\subfigure[\fLmR\ for $\ell^+$]{
\tikz{ \node (fLmRp) { \includegraphics[width=0.38\textwidth]{fig_05c.pdf}};}
}
\subfigure[\fLmR\ for $\ell^-$]{
\tikz{ \node (fLmRm) { \includegraphics[width=0.38\textwidth]{fig_05d.pdf}};}
}
\end{changemargin}
\vspace*{0.3cm}
\captionsetup{type=figure}
\caption{Computed values of \fl\ (top) and \fLmR\ (bottom) using fits with \Eqn{fracdef} to \mcnlo\ samples in (\absyw, \ptw) bins,
split by charge. These values are used to calculate the weights needed to create helicity templates.\label{fig:ComputedFractions}}
\centering
\end{figure*}
\FFigs[NoTemplFitEl35]{NoTemplFitEl} show the \ctdd\ distributions for electrons and muons and both charges, compared to the predictions from
\mcnlo\ and \pow\ and to the expected behaviour of unpolarised \Wbosons\ (the unpolarised distributions are obtained by averaging the longitudinal,
left- and right-handed \mcnlo\ templates with equal weights, see Section\ \ref{templates}). The good agreement of both the \mcnlo\ and \pow\
distributions with data is demonstrated also by the $\chi^2$ values reported in \Tab{chi2values}. It is also clear from \Tab{chi2values} and
\Figs[NoTemplFitEl35]{NoTemplFitEl} that the production of unpolarised \Wbosons\ does not match the data.\par
For the electron channel, the jet background clusters around \ctdd=1, which supports the assumption that these were two-jet events, where one of
the jets was mis-identified as an electron. On the other hand, in the muon channel, the jet background clusters around \mbox{\ctdd=$-$1}, in
agreement with the assumption that the background originates mainly from semi-leptonic decay of heavy-flavour in jets. \par
\section{Detector, data and simulation}
\label{sect:atlas}
\subsection{The ATLAS detector}
The ATLAS detector\ \cite{atlas} at the LHC covers nearly the entire solid angle around the collision region. It consists of an inner
tracking system surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer
incorporating three large superconducting toroid magnets.\par
The inner detector (ID) is immersed in a 2~T axial magnetic field and allows charged particle tracking in the range $|\eta| <
2.5$. The high-granularity silicon pixel detector covers the vertex region and typically provides three measurements per track. It is
followed by the silicon microstrip tracker which usually provides four two-dimensional measurement points per track. These silicon
detectors are complemented by the transition radiation tracker, which enables radially extended track reconstruction up to $|\eta| = 2.0$.
The transition radiation tracker also provides electron identification information based on the fraction of hits (typically 30 per track)
above an energy threshold corresponding to transition radiation.\par
The calorimeter system covers the pseudorapidity range $|\eta|< 4.9$. Within the region $|\eta|< 3.2$, electromagnetic calorimetry is based on
barrel and end-cap high-granularity lead liquid-argon (LAr) electromagnetic calorimeters, with an additional thin LAr presampler covering
$|\eta| < 1.8$ to correct for energy loss in material upstream of the calorimeters. Hadronic calorimetry is provided by a steel/scintillating-tile
detector, segmented into three structures within $|\eta| < 1.7$, and two copper/LAr hadronic endcap calorimeters. The solid angle
coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimised for electromagnetic and hadronic measurements
respectively.\par
The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic
field generated by superconducting air-core toroids. The precision chamber system covers the region $|\eta| < 2.7$, with three layers of
monitored drift tubes complemented by cathode strip chambers in the region beyond $|\eta| = 2.0$ where the background is highest. The muon
trigger system covers the range $|\eta| < 2.4$ with resistive plate chambers in the barrel, and thin gap chambers in the endcap regions.\par
A three-level trigger system is used to select interesting events\ \cite{Aad2011xs}. The Level-1 trigger is implemented in hardware and uses
a subset of detector information to reduce the event rate to a design value of at most 75~kHz. This is followed by two software-based trigger
levels which together reduce the event rate to about 200~Hz.
\subsection{Data sample}
\label{sect:datasample}
The data used in this analysis were collected from August to October 2010. Requirements on beam, detector and trigger conditions, as well as
on data quality, were used in the event selection, resulting in integrated luminosities of 37.3 \ipb\ for the electron channel and 31.4 \ipb\
for the muon channel (data where the muon trigger conditions varied too rapidly were not included). \par
\begin{table*}
\centering
\caption{Numbers of events in data and signal Monte Carlo samples, after standard and analysis cuts (see text), classified according to lepton flavour
and charge. The remaining numbers of events after standard plus analysis cuts are also represented as a percentage of the numbers of events
passing the standard selection.}
\label{tab:Statistics}
\footnotesize
\begin{tabular}{|c|c|c|c||c|c|}
\hline
\multicolumn{2}{|c|}{ }& \mupl & \mumi & \epl & \emi \\
\hline \hline
\multirow{3}{*}{Data} & Standard cuts & 79713 & 52186 & 67130 & 45690 \\
&Analysis cuts (35 $<$ $\ptw$ $<$ 50 \GeV) & 4459 (5.6\%) & 3018 (5.8\%) & 3778 (5.6\%) & 2656 (5.8\%) \\
&Analysis cuts ($\ptw$ $\ge$ 50 \GeV) & 3921 (4.9\%) & 2640 (5.1\%) & 3573 (5.3\%) & 2572 (5.6\%) \\
\hline
\multirow{3}{*}{\mcnlo} & Standard cuts & 1484062 & 1041818 & 1054705 & 774952 \\
&Analysis cuts (35 $<$ $\ptw$ $<$ 50 \GeV) & 76807 (5.2\%) & 52781 (5.1\%) & 54044 (5.1\%) & 39528 (5.1\%) \\
&Analysis cuts ($\ptw$ $\ge$ 50 \GeV) & 57699 (3.9\%) & 39114 (3.8\%) & 43509 (4.1\%) & 31283 (4.0\%) \\
\hline
\multirow{3}{*}{\pow} & Standard cuts & 1498352 & 1056697 & 1056561 & 775894 \\
&Analysis cuts (35 $<$ $\ptw$ $<$ 50 \GeV) & 82174 (5.5\%) & 59788 (5.7\%) & 58423 (5.5\%) & 44276 (5.7\%) \\
&Analysis cuts ($\ptw$ $\ge$ 50 \GeV) & 66674 (4.5\%) & 47115 (4.6\%) & 50705 (4.8\%) & 37792 (4.9\%) \\
\hline
\end{tabular}
\end{table*}
The integrated luminosity measurement has an uncertainty of 3.4\%\ \cite{Lumi,Aad2011dr}.\par
\subsection{Simulation}
\label{MonteCarlo}
Signal and background samples were processed through a \textsc{geant}4\ \cite{geant4} simulation of the ATLAS detector\ \cite{simulation} and
reconstructed using the same analysis chain as the data.\par
The signal samples were generated using \mcnlo~3.4.2\ with \textsc{herwig}\ \cite{herwig} parton showering, and with \pow~1.0\ and \pyth\ parton
showering. Both used the CTEQ~6.6\ \cite{Pumplin2002vw} PDF set. All background samples were generated with \pyth~6.4.21\ \cite{pythia}
except \ttbar\ for which \mcnlo\ was used. In order to study the sensitivity of the angular distributions to different NLO PDF sets,
the \mcnlo\ sample was reweighted\ \cite{Tricoli2005nx} according to MSTW~2008\ \cite{MSTW2008} and HERAPDF~1.0\ \cite{2009wt} PDF sets. \par
The radiation of photons from charged leptons was simulated using PHOTOS\ \cite{Photos}, and TAUOLA\ \cite{Tauola} was used for $\tau$
decays. The underlying event\ \cite{jimmy} was simulated according to the ATLAS tune\ \cite{mbtuneJan}. The Monte Carlo samples were generated with,
in average, two soft inelastic collisions overlaid on the hard-scattering event. Events were subsequently reweighted so that the distribution
of the number of reconstructed vertices matched that in data, which was 2.2 on average.\par
\subsection{Event selection}
\label{sect:selection}
Events in this analysis are first selected using either a single-muon trigger with a requirement on the transverse momentum \ptl\ of at least
13 \GeV, or a single-electron trigger, with a \ptl\ requirement of at least 15 \GeV\ \cite{Aad2011xs}. Subsequent selection criteria closely
follow those used for the \Wboson\ boson inclusive cross-section measurement reported in Ref.\ \cite{Aad2011dm}.\par
Events from $pp$ collisions are selected by requiring a reconstructed vertex compatible with the beam-spot position and with at least three
associated tracks each with transverse momentum greater than 0.5 \GeV.\par
Electron candidates are required to satisfy \ptl\ $>$ 20 \GeV, $|\eta|$ $<$ 2.47 (but removing the region where barrel and endcap calorimeters
overlap, i.e. 1.37 $<$ $|\eta|$ $<$ 1.52) and to pass the ``tight'' identification criteria described in Ref.\ \cite{Aad2011mk}. This selection rejects
charged hadrons and secondary electrons from conversions by fully exploiting the electron identification potential of the detector. It makes
requirements on shower shapes in the electromagnetic calorimeter, on the angular matching between the calorimeter energy cluster and the ID
track, on the ratio of cluster energy to track momentum, and on the number of hits in the pixels (in particular a hit in the innermost layer
is required), in the silicon microstrip tracker and in the transition radiation tracker. \par
Muon candidates are required to be reconstructed in both the ID and the MS, with transverse momenta satisfying the conditions $|(p_{\rm T}^{\rm MS}-
p_{\rm T}^{\rm ID})/p_{\rm T}^{\rm ID}|<0.5$ and $p_{\rm T}^{\rm MS}>10\ \GeV$. The two measurements are then combined, weighted by their respective
uncertainties, to form a \emph{combined muon}. The \Wboson\ candidate events are required to have at least one combined muon track with \ptl\ $>$
20 \GeV, within the range $|\eta|$ $<$ 2.4. This muon candidate must also satisfy the isolation condition $(\Sigma p_{\rm T}^{\rm ID})/\ptl$ $<$ 0.2,
where the sum is over all charged particle tracks around the muon direction within a cone of size $\Delta R$ = $\sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$
= 0.4. Finally, to reduce the contribution of cosmic-ray events, and beam-halo induced by proton losses from the beam, the analysis requires the
reconstructed vertex position along the beam axis to be within 20 cm of the nominal interaction point.\par
The \MTE\ (\met) is reconstructed as the negative vector sum of calibrated ``objects'' (jets, electrons or photons, muons) to which the energies
of calorimeter cells not associated to any of the objects are added. \met\ is required to be larger than 25 \GeV. A cut \trm\ $>$ 40 \GeV\ is
finally applied.\par
In addition to these cuts, called in the following \emph{standard cuts}, additional selections are used for this analysis. A low \trm\ cut at 50\
\GeV\ is applied to minimise backgrounds, and a high \trm\ cut at 110 \GeV\ is applied to remove tails of badly reconstructed events. Finally a
\ptw\ selection in two bins (35 $<$ $\ptw$ $<$ 50 \GeV, and \ptw $>$ 50 \GeV) is made. The numbers of events passing these cuts are shown in
\Tab{Statistics}.\par
\begin{table*}
\caption{Background fractions (with respect to the expected signal) obtained from Monte Carlo simulations (electroweak and \ttbar) normalised
to state-of-the-art signal cross-section predictions (see text) and from data (jet background) by fitting \met\ distributions with templates.
\label{tab:QCDfractions}}
\centering
\footnotesize
\begin{tabular}{|cc|c|c||c|c|}
\cline{2-6}
\multicolumn{1}{c|}{}&Fractions (\%) & $\mu^+$ & $\mu^-$ & e$^+$ & e$^-$ \\
\hline
\multirow{5}*{Standard cuts} & jet & 2.1 $\pm$ 0.1 & 3.1 $\pm$ 0.2 & 2.4 $\pm$ 0.1 & 3.6 $\pm$ 0.1 \\
& \ttbar & 0.2 & 0.4 & 0.3 & 0.5\\
& \Wtaun & 2.6 & 2.8 & 2.3 & 2.5\\
& \Ztautau & 0.1 & 0.2 & 0.1 & 0.1\\
& \Zll & 2.9 & 3.9 & 0.1 & 0.2\\
\hline
\multirow{5}*{\minitab[c]{Analysis cuts\\ (35 $<$ \ptw\ $<$ 50 \GeV)}} & jet & 2 $\pm$ 2 & 2 $\pm$ 2 & 2.4 $\pm$ 0.4 & 2.5 $\pm$ 0.5 \\
& \ttbar & 0.5 & 0.7 & 0.6 & 0.9 \\
& \Wtaun & 2.1 & 2.4 & 1.8 & 1.9 \\
& \Ztautau & 0.1 & 0.1 & 0.1 & 0.1\\
& \Zll & 2.9 & 3.9 & 0.3 & 0.4\\
\hline
\multirow{5}*{\minitab[c]{Analysis cuts \\(\ptw\ $>$ 50 \GeV)}} & jet & 2 $\pm$ 2 & 2 $\pm$ 2 & 2.0 $\pm$ 0.3 & 2.5 $\pm$ 0.4 \\
& \ttbar & 2.8 & 4.1 & 3.5 & 5.0\\
& \Wtaun & 2.1 & 2.0 & 1.9 & 2.0 \\
& \Ztautau & 0.1 & 0.1 & 0.1 & 0.1 \\
& \Zll & 2.6 & 3.5 & 0.3 & 0.4 \\
\hline
\end{tabular}
\end{table*}
The data are compared to expectations based on Monte Carlo simulations. In addition to the signal (\Wboson\ production followed by leptonic
decay to an electron or a muon), the following electroweak backgrounds are considered: \Wtaun, \Zee, \Zmm\ and \Ztautau, as well as \ttbar\
events with at least one semi-leptonic decay. Jet production via QCD was also simulated, but the final estimate of this background is obtained
from data, as explained in Section\ \ref{sect:backgrounds}.\par
\section{Signal normalisation and background estimate}
\label{sect:normalisatiom}
\subsection{Signal normalisation}
The \Wln\ production cross-sections and the decay branching ratios used in this study are normalised to the NNLO predictions of the FEWZ program\
\cite{fewz} with the MSTW~2008 PDF set:
\begin{linenomath*}
\begin{gather*}
\hspace*{2.8cm} \sigma^{NNLO}_{\ensuremath{W^+} \rightarrow \ell \nu} = 6.16\ \mathrm{nb}\,, \hfill \\
\hspace*{2.8cm} \sigma^{NNLO}_{\ensuremath{W^-} \rightarrow \ell \nu} = 4.30\ \mathrm{nb}\,. \hfill
\end{gather*}
\end{linenomath*}
The estimated uncertainties on each cross-section coming from the factorisation and renormalisation scales as well as from the parton distribution
functions are expected to be approximately 5\%\ \cite{Aad2011dm}.
\subsection{Background estimates}
\label{sect:backgrounds}
\Wboson\ events decaying into $\tau$-leptons with subsequent leptonic $\tau$ decays contribute as background to both electron and muon channels.
Contributions from \Zmm\ decays are significant in the muon channel, where the limited $\eta$ coverage of the tracking and muon systems can result
in fake \met\ when one of the muons is missed. On the contrary, the \Zee\ background is almost negligible in the electron channel due to the nearly
hermetic calorimeter coverage over $|\eta|< 4.9$. For both the electron and the muon channels, contributions from \Ztautau\ decays and from
\ttbar\ events involving at least one leptonic \Wboson\ decay are also taken into account. The latter is particularly relevant for the large
transverse momentum \Wbosons\ studied here.\par
The normalisation of electroweak and \ttbar\ backgrounds is based on their total theoretical cross-sections. These cross-sections are calculated
at NLO (plus next-to-next-to-leading-log corrections) for \ttbar\ \cite{topxs,Langenfeld2009tc}, and at NNLO for the others. The contributions of
these backgrounds to the final data sample have been estimated using simulation to model acceptance effects.\par
One of the major background contributions, especially in the electron channel, is from dijet production via QCD processes. The selected leptons
from these processes have components from semi-leptonic decays of heavy quarks, hadrons mis-identified as leptons, and, in the case of the
electron channel, electrons from conversions. The \MTE\ is due mainly to jet mis-measurement. For both the electron and muon channels, these
sources of background are obtained from the data. Monte Carlo simulated samples are also used for cross-checks.\par
The jet background is obtained by fitting the \met\ data distributions to the sum of the \Wln\ signal and the electroweak and \ttbar\ backgrounds,
normalised as described above and called hereafter the ``electroweak template'', plus a ``jet event template'' derived from control samples in the
data.\par
In the electron case, the jet event template is obtained by selecting electron candidates passing the ``loose'' selection\ \cite{Aad2011mk}, but
failing one or more of the additional criteria required to flag an electron as ``medium'' as well as an isolation cut (which removes signal
events).\par
In the muon case, the jet event template is obtained by inverting the track isolation requirement.\par
In both cases, the relative normalisation of the jet event and electroweak templates is determined by fitting the two templates to the
\met\ distribution in the data down to 10 \GeV. The jet event fraction is then obtained from the (normalised) jet event template
by counting events above \met=\ 25 \GeV.\par
The background fractions determined with the methods described above, for the standard cuts and for the standard plus analysis cuts,
are shown in \Tab{QCDfractions}. These results were obtained with \mcnlo\ for the signal simulation, and are in agreement with those obtained with
\pow. For the muon channel, as jet event fractions are small and measured with larger uncertainties than for electrons, a value of 2\% with an
uncertainty of $\pm$ 2\% is used for both \Wp\ and \Wm. \Tab{QCDfractions} shows the statistical uncertainties from the jet template method.
Uncertainties on the measurement due to background modelling are described in Section\ \ref{sect:backnorm}.\par
\section{Introduction}
\label{sect:intro}
This paper describes a measurement with the ATLAS detector of the polarisation of \Wbosons\ with transverse momenta greater
than 35 \GeV, using the electron and muon decay modes, in data recorded at 7 \TeV\ centre-of-mass energy, with a total
integrated luminosity of about 35 \ipb. The results are compared with theoretical predictions from \mcnlo\ \cite{Frixione2002ik}
and \pow\ \cite{Nason2004rx,Frixione2007vw,Alioli2010xd,Alioli2008gx}.
The paper is organised as follows. Section\ \ref{sect:theor} describes the theoretical framework of this analysis. Section\
\ref{sect:atlas} reviews the relevant components of the ATLAS detector, the data, the corresponding Monte Carlo simulated data sets, and
the event selection. The estimation of backgrounds after this selection is explained in Section\ \ref{sect:normalisatiom}, and the comparison
of data and Monte Carlo simulations for the most relevant variable (\ctdd) is given in Section\ \ref{sect:datamccomp}. The construction of
helicity templates and its validation using Monte Carlo samples is described in Section\ \ref{sect:templatesandclosure}, while the uncorrected
results are given in Section\ \ref{sect:fitresults}. The systematic uncertainties associated with the fitting procedure are discussed in
Section\ \ref{sect:systematic} and the final results, corrected for reconstruction effects, are given in Section\ \ref{sect:final}. Section\
\ref{sect:conclusions} is devoted to the conclusions.
\section{Theoretical framework and analysis procedure}
\label{sect:theor}
Measuring the polarisation of particles is crucial for understanding their production mechanisms.\par
At hadron colliders, \Wbosons\ with small transverse momentum are mainly produced through the leading order electroweak processes
\begin{linenomath*} $$u\bar{d}\ra \Wp \qquad \mbox{ and } \qquad d\bar{u}\ra \Wm$$ \end{linenomath*} At
the LHC the quarks generally carry a larger fraction of the momentum of the initial-state protons than the antiquarks. This causes
the \Wbosons\ to be boosted in the direction of the initial quark. In the massless quark approximation, the quark must be left-handed
and the antiquark right-handed. As a result the \Wbosons\ with large rapidity (\yw) are purely left-handed.\par
For more centrally produced \Wbosons, there is an increasing probability that the antiquark carries a larger momentum fraction than
the quark, so the helicity state of the \Wbosons\ becomes a mixture of left- and right-handed states whose proportions are respectively
described with fractions \fL\ and \fR.\par
For \Wboson\ bosons with large transverse momentum, three main processes contribute (taking the \Wp\ as example): \begin{linenomath*}
$$ug\ra \Wp d \;\; \mbox{, }\;\; u\bar{d}\ra \Wp g \;\; \mbox{ and } \;\; g\bar{d}\ra \Wp \bar{u}$$\end{linenomath*} Given the vector
nature of the gluon, present in all three reactions, the simple argument used at low \ptw\ no longer applies. Predictions require
detailed helicity state calculations. Leading-order (LO) and next-to-leading-order (NLO) QCD predictions have been available for
$p\bar{p}$ interactions for some time\ \cite{Mirkes1992hu} and more recently for proton-proton interactions\ \cite{Bern2011ie}. At high
transverse momenta more complex production mechanisms contribute, and polarisation in longitudinal states is also possible (the proportion
of longitudinal \Wbosons\ is hereafter described by \fl). This state is particularly interesting as it is directly connected to the massive
character of the gauge bosons.
\subsection{Theoretical framework}
\label{sect:theorinput}
The general form for inclusive \Wboson\ production followed by its leptonic decay can be written
as\ \cite{Mirkes1992hu}:
\begin{changemargin}{-0.025\textwidth}{0cm}
\begin{eqnarray}
\label{eqn:xsectot}
\frac{d \sigma}{d(\ptw)^2 d\yw d\cos \theta d \phi} &=& \frac{3}{16 \pi}\frac{d \sigma^{u}}{d(\ptw)^2d\yw} \times \big[(1+\cos^{2} \theta) \nonumber\\
& + &\frac{1}{2}A_{0}(1-3\cos^{2}\theta) + A_{1}\sin 2\theta \cos\phi \nonumber \\
&+& \frac{1}{2}A_{2} \sin^2 \theta \cos2\phi + A_3 \sin\theta \cos\phi \nonumber \\
&+& A_4 \cos\theta + A_5 \sin^2\theta \sin2\phi \nonumber \\
&+& A_6 \sin 2\theta \sin\phi + A_7 \sin\theta \sin\phi\big]
\end{eqnarray}
\end{changemargin}
where $\sigma^u$ is the unpolarised cross-section and $\phi$ and $\theta$ are the azimuthal and polar angles of the charged lepton
in a given \Wboson\ rest frame. The $A_i$ coefficients are functions of \ptw\ and \yw\ and depend on the parton distribution functions
(PDFs). For $\ptw\rightarrow0$ all reference frames used in Refs.\ \cite{Mirkes1992hu,Korner1990im,Collins1977iv,Lam1978pu,Mirkes1994eb,
Berger2007jw,Bern2011ie} become identical, with the $z$-axis directed along the beam axis. In these conditions the dependence on $\phi$
disappears and only the term with $(1+\cos^{2} \theta)$ and the terms proportional to $A_{0}$ and $A_{4}$ remain.\par
The $A_0$ to $A_4$ coefficients in \Eqn{xsectot} receive contributions from QCD at leading and higher orders, while $A_5$ to $A_7$ appear
only at next-to-leading order. Their expression as a function of \ptw\ and \yw\ depends on the reference frame used for the
calculation.\par
Several papers have been published to discuss and predict these coefficients, first for $p\bar{p}$ colliders\ \cite{Mirkes1992hu,Korner1990im,
Collins1977iv,Lam1978pu,Mirkes1994eb,Berger2007jw} and more recently for the LHC\ \cite{Bern2011ie}. While at $p\bar{p}$ colliders,
because of CP invariance, the $A_i$ coefficients are either equal ($A_0$, $A_2$, $A_3$, $A_5$, $A_7$) or opposite ($A_1$, $A_4$, $A_6$)
for \Wp\ and \Wm\ production, there is no such simple relationship at $pp$ colliders. However it has been observed\ \cite{Bern2011ie} that
$A_3$ and $A_4$ change sign between \Wp\ and \Wm, while the other coefficients ($A_0$, $A_1$, $A_2$, $A_5$, $A_6$, $A_7$) do not and are
similar in magnitude between \Wp\ and \Wm. In all cases, the pure NLO coefficients ($A_5$ to $A_7$) are small. They are neglected in this analysis.\par
Experimental measurements have been reported from the Tevatron by CDF\ \cite{CDF2005angle}, from HERA by H1\ \cite{HERA} and recently from
the LHC by CMS\ \cite{Chatrchyan2011ig}.
\subsection{Helicity fractions}
\label{sect:polafrac}
Helicity is normally measured by analysing the distribution of the cosine of the helicity angle ($\theta_{\rm 3D}$ in the following), defined
as the angle between the direction of the \Wboson\ in the laboratory frame and the direction of the decay charged lepton in the \Wboson\
rest frame. The distribution of this angle as generated by \mcnlo\ is shown in \Fig{Trcosine} without phase space restriction, as well as
with the acceptance (\ptl, \etal\ and \ptn)\footnote{ATLAS uses a right-handed coordinate system with its origin at the nominal interaction
point (IP) in the centre of the detector and the $z$-axis along the beam pipe. The $x$-axis points from the IP to the centre of the LHC ring,
and the $y$-axis points upward. Cylindrical coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle around
the beam pipe. The pseudorapidity is defined in terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$.} and \Wboson\ transverse mass
\trm\ cuts (where \trm=$\sqrt{ 2( \ptl \ptn - \vptl \cdot \vptn )}$), described in Section\ \ref{sect:selection}.\par
\begin{figure}[h!]
\begin{changemargin}{-0.025\textwidth}{0cm}
\begin{center}
\includegraphics[width=0.52\textwidth]{fig_01.pdf}
\end{center}
\end{changemargin}
\caption{Cosine of the helicity angle of the lepton from \Wboson\ decay at generator-level for positive charge (left) and negative charge
(right). Solid lines are without selection, dashed lines are after all acceptance plus \trm\ cuts except the \etal\ cuts and dotted lines
are after all acceptance plus \trm\ cuts. ``All events'' distributions are normalised to unity.\label{fig:Trcosine}}
\end{figure}
The differential cross-section in the helicity frame\footnote{The helicity frame is the \Wboson\ rest frame with the $z$-axis along the \Wboson\
laboratory direction of flight and the $x$-axis in the event plane, in the hemisphere opposite to the recoil system.} is expressed by using
$\theta_{\rm 3D}$ and $\phi_{\rm 3D}$ in \Eqn{xsectot}. Integrated over \yw\ and $\phi_{\rm 3D}$, \Eqn{xsectot} then takes the form:
\begin{eqnarray}
\label{eqn:Aioverphihel}
\frac{1}{\sigma}\frac{d\sigma}{d\ctd} = \frac{3}{8} [(1+\cct_{\rm 3D})& + &A_0\frac{1}{2}(1-3\cct_{\rm 3D}) \nonumber\\
& + &A_4\ctd].
\end{eqnarray}
Comparing \Eqn{Aioverphihel} to the standard form\ \cite{Ellis1991qj} using helicity fractions:
\begin{eqnarray}
\label{eqn:fracdef}
\frac{1}{\sigma}\frac{d\sigma}{d\ctd} = \frac{3}{8}\fL (1\mp\ctd)^2 & + & \frac{3}{8}\fR (1\pm\ctd)^2\nonumber\\
& + & \frac{3}{4}\fl \sct_{\rm 3D}
\end{eqnarray}
yields the relations between the $A_i$ coefficients and the helicity fractions:
\begin{linenomath*}\begin{alignat}{2}
\label{eqn:fracAi}
\fL(\yw,\ptw) & = &&\, \frac{1}{4} (2-A_0(\yw,\ptw)\mp A_4(\yw,\ptw)) \nonumber \\
\fR(\yw,\ptw) & = &&\, \frac{1}{4} (2-A_0(\yw,\ptw)\pm A_4(\yw,\ptw)) \nonumber \\
\fl(\yw,\ptw) & = &&\, \frac{1}{2} A_0(\yw,\ptw)
\end{alignat}\end{linenomath*}
where the upper (lower) sign corresponds to \Wp\ (\Wm) boson production respectively. It is interesting to notice that the difference
between the left- and right-handed fraction is proportional to $A_4$ only, as:
\begin{linenomath*}\begin{equation}
\label{eqn:a4rel}
\hfill \fLmR = \mp \frac{A_4}{2}. \hfill
\end{equation}\end{linenomath*}
From general considerations, the longitudinal helicity fraction \fl\ is expected to vanish for $\ptw\rightarrow 0$ as well as for $\ptw
\rightarrow\infty$, with a maximum expected around 45 \GeV\ \cite{Bern2011ie}.\par
\subsection{Analysis principle and variable definitions}
\label{sect:analyprinciple}
When analysing data, a major difficulty arises from the incomplete knowledge of the neutrino momentum. The large angular coverage of the
ATLAS detector enables measurement of the \MTE, which can be identified with the transverse momentum of the neutrino. The longitudinal
momentum can be obtained thro\-ugh the \Wboson\ mass constraint. However, solving the corresponding equation leads to two solutions, between
which it is not possible to choose in an efficient way. The approach taken in this analysis is to work in the transverse plane only, using the
``transverse helicity'' angle $\theta_{\rm 2D}$ defined by:
\begin{linenomath*}\begin{equation}
\hfill \ctdd=\frac{\vptls \cdot \vptw}{|\vptls| \; |\vptw|}\,,\hfill
\end{equation} \end{linenomath*}
where \vptls\ is the transverse momentum of the lepton in the transverse \Wboson\ rest frame and \vptw\ is the transverse momentum of the
\Wboson\ boson in the laboratory frame. The angle $\theta_{\rm 2D}$ is a two dimensional projection of the helicity angle $\theta_{\rm 3D}$.
Its determination uses only fully measurable quantities, defined in the transverse plane. Its use is limited to sizeable values of \ptw,
which corresponds to the physics addressed in this work.\par
The correlations between \ctdd\ and \ctd\ for events where \ptw\ $>$ 50 \GeV\ are represented in \Figs[ctdres1]{ctdres2} for positive and
negative leptons respectively. This information is obtained using a sample of events simulated with \mcnlo\ after applying acceptance and
\trm\ cuts.\par
The enhancement near $-$1 for positive leptons reflects that the maximum of the left-handed part of the decay distribution
(first term in \Eqn{fracdef}) falls within detector acceptance, as opposed to the case of negative leptons where the maximum (near +1) falls
largely beyond the \etal\ acceptance, resulting in a more ``symmetric'' distribution between forward and backward hemispheres. This effect is also
seen in \Fig{Trcosine} when comparing \ctd\ distributions at generator-level, before and after the lepton pseudorapidity cut.\par
\begin{figure}[h!]
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\centering
\subfigure[$\ell^+$]{
\includegraphics[width=0.27\textwidth]{fig_02a.pdf}
\label{fig:ctdres1}
}\hspace*{-0.4cm}
\subfigure[$\ell^-$]{
\includegraphics[width=0.27\textwidth]{fig_02b.pdf}
\label{fig:ctdres2}
}
\end{changemargin}
\caption{Representation of \ctdd\ as a function of \ctd\ in events where the \Wboson\ transverse momentum is greater
than 50 \GeV, for (a) positive and (b) negative leptons. Events are simulated with \mcnlo\ after applying the acceptance
and \trm\ cuts, as defined in Section\ \ref{sect:selection}.\label{fig:ctdresidu}}
\end{figure}
The measurement of helicity fractions is made by fitting \ctdd\ distributions with a weighted sum of templates obtained from Monte Carlo
simulations, which correspond to longitudinal, left- and right-handed states. This is described in detail in Section\ \ref{sect:templatesandclosure}.
\section{Fit results}
\label{sect:fitresults}
The raw helicity fractions for each of the four analysed channels were obtained by fitting the experimental \ctdd\ distributions, after
background subtraction, with a sum of templates (see \Eqn{fracdef}) corresponding to longitudinal, left- and right-handed states.\par
In order to correct for systematic effects associated with the choice of the variable used in the fit (\ctdd), and for resolution effects,
the raw results have been corrected in a second step by the differences observed in Monte Carlo events between the fits at the generator level
with the \ctd\ distribution after acceptance plus \trm\ cuts and the fit on on \ctdd\ distributions after full simulation. The two sets of
templates obtained from \mcnlo\ or from \pow\ were used, and their bias corrected for accordingly. Differences between the results obtained with
the two Monte Carlo generators were used to estimate a systematic uncertainty associated with the choice of templates (see Section\
\ref{sect:systemp}).\par
The minimisation \cite{Barlow1993dm} gives the uncertainties and correlations between the parameters. The $\chi^2$ values, in
\Tab{chi2valuesfit}, obtained using \mcnlo\ and \pow\ templates, are similar. They are significantly lower, in most cases, than in
\Tab{chi2values}, especially for muons, even taking into account that the number of degrees of freedom is reduced from 19 to 17.
\begin{table*}
\centering
\caption{Values of the $\chi^2$ from the fit of data with \mcnlo\ and \pow\ helicity templates (see \Figs[electronMedPT]{electronHPT} for
\mcnlo). The number of degrees of freedom in the fits is 17.}
\begin{tabular}{|c|c|c|c|c||c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{\minitab[c]{$\chi^2$ between\\ data and}}}& \multicolumn{4}{c||}{35 $<$ \ptw\ $<$50 \GeV \bigstrut} & \multicolumn{4}{c|}{\ptw\ $>$50 \GeV} \\
\cline{2-9}
& $\mu^+$ & $\mu^-$ & $e^+$ & $e^-$ & $\mu^+$ & $\mu^-$ & $e^+$ & $e^-$ \\
\hline
\mcnlo\ templates & 13.5 & 23.1 & 7.6 & 25.3 & 29.3 & 21.1 & 24.8 & 16.9 \\
\hline
\pow\ templates & 11.1 & 20.7 & 8.2 & 20.8 & 30.1 & 26.6 & 20.9 & 13.1 \\
\hline
\end{tabular}
\label{tab:chi2valuesfit}
\end{table*}
The values of the fitted parameters, using \mcnlo\ and \pow\ templates, are reported in \Tab{rawdatatot35}. The contributions
of the individual fitted helicity states, and their sum, are also shown, for the \mcnlo\ case, in \Fig{electronMedPT} for 35 $<$
\ptw\ $<$ 50 \GeV, and in \Fig{electronHPT} for \ptw\ $>$ 50 \GeV. These histograms show the contributions of each polarisation
state (separately and summed together), with a normalisation which, in addition to the value of \fl, \fL\ and \fR, also takes into
account the relative average acceptance for each of the three polarisation states. The data show a dominance of the left-handed over
the right-handed fraction in about the same proportion as in the Monte Carlo simulations.\par
\begin{table*}
\begin{changemargin}{-0.3cm}{-0.3cm}
\centering
\footnotesize
\caption{Summary of raw data results for helicity fractions (as percentages) for 35 $<$ \ptw\ $<$ 50 \GeV\ and \ptw\ $>$ 50 \GeV\
obtained with \mcnlo\ or with \pow\ template fits (see \Figs[electronMedPT]{electronHPT} for \mcnlo). The errors represent the
statistical uncertainties only.\label{tab:rawdatatot35}}
\begin{tabular}{|c|c|c|c||c|c|}
\cline{3-6}
\multicolumn{2}{c|}{ }& \mupl & \mumi & \epl & \emi \\
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{35 $<$ \ptw\ $<$ 50 \GeV} \bigstrut \\
\hline
\multirow{2}{*}{\minitab[c]{Data \\ with \mcnlo}} & \fl\ $(\%)$ & 26.6 $\pm$ 5.1 & 10.9 $\pm$ 5.6 & 23.2 $\pm$ 5.7 & 9.9 $\pm$ 10.2 \\
& \fLmR\ $(\%)$& 20.6 $\pm$ 3.9 & 27.1 $\pm$ 4.3 & 17.9 $\pm$ 4.2 & 33.0 $\pm$ 4.0 \\
\hline
\multirow{2}{*}{\minitab[c]{Data \\ with \pow\ }} & \fl\ $(\%)$ & 42.8 $\pm$ 5.1 & 35.1 $\pm$ 5.7 & 36.9 $\pm$ 9.1 & 26.5 $\pm$ 6.1 \\
& \fLmR\ $(\%)$& 25.6 $\pm$ 3.9 & 21.8 $\pm$ 4.3 & 21.3 $\pm$ 5.3 & 25.1 $\pm$ 4.3 \\
\hline
\multicolumn{6}{c}{ } \\
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{\ptw\ $>$ 50 \GeV} \bigstrut \\
\hline
\multirow{2}{*}{\minitab[c]{Data \\ with \mcnlo}} & \fl\ $(\%)$& 8.3 $\pm$ 5.0 & -0.0 $\pm$ 7.3 & 9.7 $\pm$ 5.7 & 20.0 $\pm$ 5.6 \\
& \fLmR\ $(\%)$& 27.5 $\pm$ 3.3 & 29.9 $\pm$ 3.4 & 29.3 $\pm$ 3.5 & 19.7 $\pm$ 3.9 \\
\hline
\multirow{2}{*}{\minitab[c]{Data s\\ with \pow\ }} & \fl\ $(\%)$& 15.3 $\pm$ 4.4 & 13.0 $\pm$ 5.0 & 19.6 $\pm$ 5.7 & 26.6 $\pm$ 6.9 \\
& \fLmR\ $(\%)$& 27.7 $\pm$ 3.2 & 19.9 $\pm$ 3.6 & 29.5 $\pm$ 3.6 & 13.3 $\pm$ 4.2 \\
\hline
\end{tabular}
\end{changemargin}
\end{table*}
The \fl\ values obtained with the \pow\ templates are in general larger (see \Tab{rawdatatot35}). For the negative charges, the increase
of \fl\ is correlated with a decrease of \fLmR, while for positive charges the reverse is observed, though with a smaller increase, especially
in the higher \ptw\ bin.\par
\begin{figure*}
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\centering
\subfigure[\mupl]{
\includegraphics[width=0.38\textwidth]{fig_07a.pdf}
}\hspace*{-0.5cm}
\subfigure[\mumi]{
\includegraphics[width=0.38\textwidth]{fig_07b.pdf}
}
\subfigure[\epl]{
\includegraphics[width=0.38\textwidth]{fig_07c.pdf}
}\hspace*{-0.5cm}
\subfigure[\emi]{
\includegraphics[width=0.38\textwidth]{fig_07d.pdf}
}
\end{changemargin}
\caption{Results of the fits to \ctdd\ distributions using helicity templates (built from \mcnlo), for \Wmn\ (top) and \Wen\ (bottom) events in
data with 35 $<$ \ptw\ $<$ 50 \GeV, after background subtraction. Each template distribution is represented: left-handed contribution (dashed
line), longitudinal contribution (dotted-dashed line) and right-handed contribution (dotted line).\label{fig:electronMedPT}}
\end{figure*}
\begin{figure*}
\begin{changemargin}{-0.05\textwidth}{-0.05\textwidth}
\centering
\subfigure[\mupl]{
\includegraphics[width=0.38\textwidth]{fig_08a.pdf}
}\hspace*{-0.5cm}
\subfigure[\mumi]{
\includegraphics[width=0.38\textwidth]{fig_08b.pdf}
}
\subfigure[\epl]{
\includegraphics[width=0.38\textwidth]{fig_08c.pdf}
}\hspace*{-0.5cm}
\subfigure[\emi]{
\includegraphics[width=0.38\textwidth]{fig_08d.pdf}
}
\end{changemargin}
\caption{Results of the fits to \ctdd\ distributions using helicity templates (built from \mcnlo), for \Wmn\ (top) and \Wen\ (bottom) events in
data with \ptw\ $>$ 50 \GeV, after background subtraction. Each template distribution is represented: left-handed contribution (dashed
line), longitudinal contribution (dotted-dashed line) and right-handed contribution (dotted line).\label{fig:electronHPT} }
\end{figure*}
\section{Systematic effects}
\label{sect:systematic}
In addition to the choice of templates, which is treated separately, the measurement suffers from systematic effects due to limited knowledge
of backgrounds, charge mis-identification, choice of PDF sets, uncertainties on the lepton energy scale and resolution, and uncertainties on
the recoil system energy scale and resolution. The uncertainties on helicity fractions have been estimated using \mcnlo\ and are reported in
\Tab{systtot35}, in absolute terms.\par
The effect of reweighting simulated events to restore a \ptw\ distribution closer to that observed \cite{Belloni1361975} was also assessed.
\subsection{Backgrounds}
\label{sect:backnorm}
The electroweak and \ttbar\ backgrounds have been studied previously and found to be well modelled by Monte Carlo simulations\ \cite{Aad2011dm,
Aad2011fu,Aad2011kt,Aad2012qf}. As these backgrounds are subtracted from data for the final fit, an associated systematic uncertainty has been
estimated by changing the global normalisation of the subtracted distributions by $\pm$ 6.8\% ($\pm$ 3.4\% to take into account the uncertainty
on the integrated luminosity, $\pm$ 5\% for the uncertainty on background cross-sections relative to signal, and $\pm$ 3\% for the influence of
PDFs on the acceptance\ \cite{Aad2010yt}).\par
Furthermore, the amount of jet background was varied inside the uncertainty estimated by the dedicated fit (see \Tab{QCDfractions}).\par
\subsection{Charge mis-identification}
Since charge mis-identification is well reproduced by simulations\ \cite{Aad2011mk}, the possible associated effect on the results presented
here has been measured by comparing helicity fractions extracted from fully simulated events where the charge assignment was taken either from
ge\-ne\-ra\-tor-level information or after full reconstruction. The effect on \fl\ and \fLmR\ is estimated to be about 0.4\% in the electron case, and
is negligible for muons.\par
\subsection{Reweighting of \ptw\ distribution}
\label{sect:ptwrew}
\mcnlo\ and, to a lesser extent \pow, underestimate the fraction of \Wboson\ events at high \ptw. In order to investigate the
possible consequences of such a bias on this measurement, the \mcnlo\ Monte Carlo signal sample, weighted event-by-event so as to restore a \ptw\
spectrum compatible with data, was fitted using unchanged helicity templates (both \pow\ and \mcnlo\ templates were used for this test). The effect
of the reweighting was found to have a small impact on the fitted values of \fl\ (less than 2\%). For \fLmR\ sizeable effects were observed (up
to 5\% in the low \ptw\ bin). However, they are of opposite sign for the positive and negative lepton charges, and almost perfectly cancel when
analysing charge-averaged values (see \Tab{systtot35}).
\subsection{PDF sets}
Using the PDF reweighting method, the uncertainty associated with PDFs was estimated by keeping the templates unchanged and using MSTW~2008 and
HERAPDF~1.0 instead of the CTEQ~6.6 PDFs for the simulation of the signal distributions. The impact on \fl\ and \fLmR\ is in the range of 1\% to 2\%.
\begin{table*}
\begin{changemargin}{-0.3cm}{-0.3cm}
\centering
\footnotesize
\caption{Summary of systematic uncertainties on helicity fractions for 35 $<$ \ptw\ $<$ 50 \GeV\ and \ptw\ $>$
50 \GeV. The effect of lepton and recoil energy scales, and of \ptw\ reweighting, on \fLmR\ is also estimated on the mean
between the two charges. The larger errors appear with the $\pm$ ($\mp$) sign if they vary in the same (opposite) direction
as the parameter studied, in order to highlight the correlations used in calculating the errors on the means.\label{tab:systtot35}}
\begin{tabular}{|c|c|c|c||c|c|c|c|c||c|c|}
\cline{3-6} \cline{8-11}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{35 $<$ \ptw\ $<$ 50 \GeV} & \multicolumn{1}{c|}{} & \multicolumn{4}{c|}{\ptw\ $>$ 50 \GeV} \bigstrut \\
\cline{3-6} \cline{8-11}
\multicolumn{2}{c|}{ }& \mupl & \mumi & \epl & \emi & \multicolumn{1}{c|}{} & \mupl & \mumi & \epl & \emi \\
\cline{1-6} \cline{8-11}
\multirow{2}{*}{EW background} & $\delta \fl$ $(\%)$ & 0.5 & 0.6 & 0.3 & 0.4 & \multicolumn{1}{c|}{} & 0.6 & 0.6 & 0.3 & 0.5 \\
& $\delta(\fLmR)$ $(\%)$ & 0.2 & 0.3 & 0.2 & 0.2 & \multicolumn{1}{c|}{} & 0.2 & 0.3 & 0.2 & 0.2 \\
\cline{1-6} \cline{8-11}
\multirow{2}{*}{jet background} & $\delta \fl$ $(\%)$ & 1.5 & 1.5 & 1.5 & 1.5 & \multicolumn{1}{c|}{} & 2.3 & 1.3 & 2 & 2 \\
& $\delta(\fLmR)$ $(\%)$ & 0.3 & 0.7 & 1.5 & 1.5 & \multicolumn{1}{c|}{} & 1.2 & 1.3 & 1.5 & 1.5 \\
\cline{1-6} \cline{8-11}
\multirow{3}{*}{\ptl\ scale} & $\delta \fl$ $(\%)$ & $\mp$ 4.5 & $\mp$ 5.0 & $\mp$ 4.5 & $\mp$ 4.5 & \multicolumn{1}{c|}{} & $\mp$ 3.5 & $\mp$ 3.5 & $\mp$ 3.5 & $\mp$ 4.5 \\
& $\delta(\fLmR)$ $(\%)$ & $\mp$ 2.5 & $\pm$ 2.0 & $\mp$ 2.5 & $\pm$ 2.0 & \multicolumn{1}{c|}{} & $\mp$ 1.5 & $\pm$ 1.5 & $\mp$ 2.0 & $\pm$ 1.5 \\
\cline{3-6} \cline{8-11}
& $\delta(\fLmR)_{\rm mean}$ $(\%)$ & \multicolumn{2}{c||}{1.1} & \multicolumn{2}{c|}{0.4} & \multicolumn{1}{c|}{} & \multicolumn{2}{c||}{0.1} & \multicolumn{2}{c|}{0.4} \\
\cline{1-6} \cline{8-11}
\multirow{3}{*}{Recoil scale} & $\delta \fl$ $(\%)$ & $\pm$ 12.5 & $\pm$ 16.8 & $\pm$ 12.5 & $\pm$ 13.3 & \multicolumn{1}{c|}{} & $\pm$ 8.1 & $\pm$ 10.2 & $\pm$ 9.4 & $\pm$ 11.1 \\
& $\delta(\fLmR)$ $(\%)$ & $\pm$ 9.9 & $\mp$ 10.4 & $\pm$ 10.9 & $\mp$ 9.5 & \multicolumn{1}{c|}{} & $\pm$ 7.7 & $\mp$ 7.7 & $\pm$ 8.2 & $\mp$ 8.2 \\
\cline{3-6} \cline{8-11}
& $\delta(\fLmR)_{\rm mean}$ $(\%)$ & \multicolumn{2}{|c||}{3.0} & \multicolumn{2}{c|}{2.9} & \multicolumn{1}{c|}{} & \multicolumn{2}{c||}{1.2} & \multicolumn{2}{c|}{0.7} \\
\cline{1-6} \cline{8-11}
\multirow{2}{*}{PDF set} & $\delta \fl$ $(\%)$ & 2.0 & 2.0 & 0.4 & 0.8 & \multicolumn{1}{c|}{} & 2.0 & 2.0 & 0.2 & 0.8 \\
& $\delta(\fLmR)$ $(\%)$ & 1.5 & 1.5 & 0.5 & 1.5 & \multicolumn{1}{c|}{} & 1.5 & 1.5 & 0.4 & 1.1 \\
\cline{1-6} \cline{8-11}
\multirow{2}{*}{Charge mis-ID} & $\delta \fl$ $(\%)$ & \multirow{2}{*}{ $-$ } & \multirow{2}{*}{ $-$ } & 0.2 & 0.4 & \multicolumn{1}{c|}{} & \multirow{2}{*}{ $-$ } & \multirow{2}{*}{ $-$ } & 0.2 & 0.2 \\
& $\delta(\fLmR)$ $(\%)$ & & & 0.3 & 0.4 & \multicolumn{1}{c|}{} & & & 0.2 & 0.3 \\
\cline{1-6} \cline{8-11}
\multirow{2}{*}{\ptl\ resolution} & $\delta \fl$ $(\%)$ & 0.1 & 0.1 & 0.5 & 0.5 & \multicolumn{1}{c|}{} & 0.1 & 0.2 & 0.2 & 1.2 \\
& $\delta(\fLmR)$ $(\%)$ & 0.1 & 0.1 & 0.3 & 0.3 & \multicolumn{1}{c|}{} & 0.1 & 0.2 & 0.2 & 0.2 \\
\cline{1-6} \cline{8-11}
\multirow{3}{*}{\ptw\ reweighting} & $\delta \fl$ $(\%)$ & 2.5 & 1.1 & 0.6 & 0.9 & \multicolumn{1}{c|}{} & 1.9 & 1.6 & 0.5 & 1.2 \\
& $\delta(\fLmR)$ $(\%)$ & $\mp$ 4.9 & $\pm$ 5.2 & $\mp$ 4.2 & $\pm$ 4.0 & \multicolumn{1}{c|}{} & $\mp$ 2.7 & $\pm$ 2.9 & $\mp$ 2.6 & $\pm$ 2.3 \\
\cline{3-6} \cline{8-11}
& $\delta(\fLmR)_{\rm mean}$ $(\%)$ & \multicolumn{2}{|c||}{0.2} & \multicolumn{2}{c|}{0.1} & \multicolumn{1}{c|}{} & \multicolumn{2}{c||}{0.1} & \multicolumn{2}{c|}{0.2} \\
\cline{1-6} \cline{8-11}
\end{tabular}
\end{changemargin}
\end{table*}
\subsection{Energy scales}
\label{sect:escale}
While a coherent change of the lepton and recoil energy scales would leave the angles in the transverse plane unchanged, both in the laboratory
and in the transverse \Wboson\ rest frame, an effect on \ctdd\ arises when only one of the two measured objects (lepton, recoil) changes, or if
they change by different amounts.\par
Using simulated events, it has been observed that an increase of the lepton transverse momentum alone gives a positive slope to the \ctdd\
distribution, which in turn induces an increase of the left-handed fraction in the negative lepton sample, and a decrease of the left-handed
fraction in the positive lepton sample. As expected, the reverse happens for an increase of the recoil transverse energy.\par
The value of \fLmR\ when averaged over the two charges is largely independent of the lepton and recoil energy scales, as can be seen in
\Tab{systtot35}.\par
The same compensation mechanism is however not present for \fl, for which an increase in the recoil energy scale induces an increase of \fl\ for
both charges.\par
The lepton energy scale is precisely determined from \Zll\ decays: using the precisely-known value of the \Zboson\ boson mass, scale factors
have been extracted by \etal\ regions, which in the muon case depend also on the muon charge\ \cite{Aad2011mk,muonscale}. The reconstructed
\Zboson\ boson mass spectrum has also been used to derive smearing corrections to be applied to Monte Carlo electrons and muons in order to
reproduce the observed \Zboson\ mass peak resolution. The resulting uncertainties are about 3\% to 5\% on \fl\ and around 2\% on \fLmR.\par
For the rather large \pt\ of the \Wbosons\ studied here, the recoil system in general contains one or several jets with \pt\ $>$ 20 \GeV, and may
also include additional ``soft jets'' (7 $<$ \pt\ $<$ 20 \GeV), and clusters of calorimeter cells not included in the above objects. The uncertainty
on the energy scale of these objects (typically 3\% for jets, 10.5\% for soft jets and 13.5\% for isolated clusters) was propagated as described in\
\cite{Aad2011re}. This is the largest systematic uncertainty on the helicity fractions measured in this study. In the worst case (muons in the low
\ptw\ bin), the resulting uncertainty on \fl\ is 16\%. This uncertainty is largely correlated between the muon and electron channels.\par
Given the anti-correlation observed between the impacts on positive and negative leptons, the uncertainties from energy scale variations enter with
$\pm$\ or $\mp$\ in \Tab{systtot35}, depending on whether the effect goes in the same direction as an energy increase or in the opposite direction. As
already pointed out, in the case of \fLmR\ the effects largely cancel when considering the average between negative and positive charges.\par
\subsection{Choice of the Monte Carlo generator}
\label{sect:systemp}
The results of the template fits to real and fully simulated data are affected by the imperfect correlation between \ctdd\ and \ctd\ and by
resolution effects.\par
In order to compare results directly to theoretical models, the raw results from Section\ \ref{sect:fitresults} are corrected by adding the
difference, found using simulations, between the ``true'' values which would be given by fits to \ctd\ distributions obtained at the generator
level within acceptance and \trm\ cuts as used here, and the results obtained using fully-simulated \ctdd\ distributions. In order to be able to
average results from muons and electrons, the electron results are corrected to the same \etal\ acceptance as for muons (i.e. without
the barrel-endcap calorimeters overlap region around 1.5, and with a maximum $|\etal|$ value of 2.4).\par
The corrections for results obtained using \mcnlo\ templates were determined from the difference between:
\begin{itemize}
\item results of a fit of \mcnlo\ (3D) templates to \ctd\ distributions of the \pow\ Monte Carlo samples at the generator-level with acceptance
and \trm\ cuts.
\item results of a fit of \mcnlo\ (2D) templates to \ctdd\ distributions of the same \pow\ Monte Carlo samples, after full simulation and with standard
plus analysis cuts.
\end{itemize}
The corrections for results obtained using \pow\ templates were derived in the same way as above, interchanging the roles of \mcnlo\ and \pow.
\begin{table}
\centering
\caption{Percentage values of \fLmR\ and \fl\ averaged over charges, separately for electrons and muons, obtained by averaging results with
templates from \mcnlo\ (see \Figs[electronMedPT]{electronHPT}) and from \pow. The first uncertainty is statistical, the second covers the
systematic uncertainties from instrumental and analysis effects, and the last one the differences between templates constructed with the two
generators.\label{tab:correctedfLmR}}
\begin{changemargin}{-0.3cm}{0cm}
\begin{tabular}{|c|c||c|}
\cline{2-3}
\multicolumn{1}{c|}{}& 35$<$\ptw$<$50 \GeV & \ptw$>$50 \GeV \bigstrut\\
\cline{2-3}
\multicolumn{1}{c|}{}& \multicolumn{2}{c|}{ \fLmR\ $(\%)$} \bigstrut\\
\hline
muon average & 21.7$\pm$3.0$\pm$3.6$\pm$2.0 & 25.0$\pm$2.5$\pm$2.3$\pm$2.5 \\
\hline
electron average & 26.0$\pm$2.8$\pm$3.4$\pm$2.0 & 25.5$\pm$2.6$\pm$2.0$\pm$2.0 \\
\hline
\multicolumn{3}{c}{} \\
\cline{2-3}
\multicolumn{1}{c|}{}& \multicolumn{2}{c|}{ \fl\ $(\%)$ } \bigstrut\\
\hline
muon average & 23.6$\pm$3.8$\pm$12.0$\pm$7.2 & 7.6$\pm$4.8$\pm$9.0$\pm$5.2 \\
\hline
electron average & 20.1$\pm$6.9$\pm$12.0$\pm$5.0 & 17.7$\pm$4.3$\pm$9.0$\pm$6.0 \\
\hline
\end{tabular}
\end{changemargin}
\end{table}
In a further step, after averaging over the charges for each lepton flavour:
\begin{itemize}
\item the corrected data result, for \fLmR\ and \fl, was obtained by averaging the numbers obtained with \mcnlo\ and with \pow\ templates.
\item the systematic uncertainty associated with the choice of templates was taken as half the difference between the two numbers, with
a minimum value of 2\%.
\end{itemize}
The corrected results and the associated systematic uncertainties are shown in \Tab{correctedfLmR} for \fLmR\ and \fl.\par
The systematic uncertainty associated with the differences between the two sets of templates is large for \fl, for
which other systematic effects are also large.\par
Another correction procedure was tried, using the same Monte Carlo generator for producing the templates and calculating the corrections.
The resulting central values of the helicity fractions are very close to those shown in \Tab{correctedfLmR} (within less than 2\%), but the systematic
uncertainties of the corrections are slightly larger (by about 10\% in relative terms).\par
Finally, a full simulation based on \sher~1.2.2\ \cite{sherpa}, made only for the electron channel, was also used to obtain, similarly as
above, first raw results, and then correction terms found by applying \sher\ templates to simulated data produced with both \mcnlo\ and \pow. The
corrected measurement obtained in this way are shown in \Tab{sherresults}, together with the ``electron average'' results from \Tab{correctedfLmR}. In
the case of \sher, only the uncertainty associated with the choice of template is reported. A very good agreement is observed.
\begin{table}[h!]
\footnotesize
\caption{Corrected values of \fLmR\ and \fl\ (as percentages) obtained using \sher\ templates, compared to the standard result (\Tab{correctedfLmR}),
for the electron channels averaged over charges. In the \sher\ case the only uncertainty quoted is associated with the two ways of calculating the
correction term: applying \sher\ templates either to \mcnlo\ or to \pow\ simulated data.\label{tab:sherresults}}
\begin{changemargin}{-0.6cm}{-0.6cm}
\centering
\begin{tabular}{|c|c||c|}
\cline{2-3}
\multicolumn{1}{c|}{}& \multicolumn{1}{c|}{35 $<$ \ptw\ $<$ 50 \GeV} & \ptw\ $>$ 50\GeV\bigstrut \\
\cline{2-3}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\fLmR\ $(\%)$} \\
\hline
Data (\sher) & 25.5$\pm$2.2 & 26.6$\pm$2 \\
\hline
Data (standard) & 26.0$\pm$2.8$\pm$3.4$\pm$2.0 & 25.5$\pm$2.6$\pm$2.0$\pm$2.0 \\
\hline
\multicolumn{3}{c}{}\\
\cline{2-3}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\fl\ $(\%)$} \\
\hline
Data (\sher) & 21.0 $\pm$ 9.1 & 15.6 $\pm$ 6.1 \\
\hline
Data (standard) & 20.1$\pm$6.9$\pm$12.0$\pm$5.0 & 17.7$\pm$4.3$\pm$9.0$\pm$6.0 \\
\hline
\end{tabular}
\end{changemargin}
\end{table}
\vspace*{-0.3cm}
\section{Results}
\label{sect:final}
The corrected final measurements of \fLmR, already shown in \Tab{correctedfLmR}, are compared in \Tab{fLmRaverage35} to the values obtained from the
\mcnlo\ and \pow\ samples, at the generator-level with the acceptance and \trm\ cuts, using a template fit to the \ctd\ distributions.\par
In the low \pt\ bin the data lie in between the \mcnlo\ and \pow\ predictions, slightly closer to the former. For \ptw\ $>$ 50
\GeV, the data are close to the \mcnlo\ values, while \pow\ predicts a somewhat smaller difference between left- and right-handed states than
observed in the data.\par
The same good agreement between data and \mcnlo\ remains after averaging results over lepton flavours (\Tab{fLRaverage}). While the complete NNLO
cross-section calculation of Ref. \cite{Bern2011ie} has not been implemented in a Monte Carlo generator, it can be seen in \Fig{ComputedFractions}
and its equivalent (not shown) for BlackHat, that at the particle level, without any cuts, the \fLmR\ values from\ \cite{Bern2011ie} are on average
about 5\% lower (in absolute terms) than the \mcnlo\ predictions. They are thus quite close to \pow\ and somewhat lower than the data.\par
The measurements shown in \Tab{fLRaverage}, where all systematic uncertainties have been combined, are the main result of this study concerning \fLmR,
and the directly related coefficient $A_4$ (\Eqn{a4rel}).\\
\begin{table*}
\centering
\caption{Corrected values, of \fLmR\ (as percentages), averaged over charge, separately for electrons and muons, for the data, \mcnlo\ and \pow,
and for 35 $<$ \ptw\ $<$ 50 \GeV\ and \ptw\ $>$ 50 \GeV. For data the first uncertainty is statistical, the second covers
the systematic uncertainties from instrumental and analysis effects, and the last one the differences between templates constructed with the two
generators. For \mcnlo\ and \pow\ the uncertainties are only statistical.\label{tab:fLmRaverage35}}
\begin{tabular}{|c|c||c|c|c||c|}
\cline{2-3} \cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{35 $<$ \ptw\ $<$ 50 \GeV} \bigstrut & \multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\ptw\ $>$ 50 \GeV} \\
\cline{2-3} \cline{5-6}
\multicolumn{1}{c|}{} & muon average & electron average & \multicolumn{1}{c|}{} & muon average & electron average \\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{Data } & 21.7 $\pm$ 3.0 $\pm$ 3.6 $\pm$ 2.0 &26.0 $\pm$ 2.8 $\pm$ 3.4 $\pm$ 2.0& \multicolumn{1}{c|}{} &25.0 $\pm$ 2.5 $\pm$ 2.3 $\pm$ 2.5& 25.5 $\pm$ 2.6 $\pm$ 2.0 $\pm$ 2.0\\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{\mcnlo\ } & 27.2 $\pm$ 0.8& 27.1 $\pm$ 1.0 & \multicolumn{1}{c|}{} & 26.4 $\pm$ 0.8& 26.1 $\pm$ 0.9 \\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{\pow\ } & 19.9$\pm$ 0.8& 19.9 $\pm$ 1.0 & \multicolumn{1}{c|}{} & 21.2 $\pm$ 0.8& 21.2 $\pm$ 0.9 \\
\cline{1-3} \cline{5-6}
\multicolumn{6}{c}{}\\
\end{tabular}
\caption{Corrected values of \fLmR\ and \fl\ (as percentages), averaged over charges and lepton flavours, for the data, \mcnlo\ and \pow, and for
35 $<$ \ptw\ $<$ 50 \GeV\ and \ptw\ $>$ 50 \GeV (\Fig{Final}). For data the first uncertainty is statistical, the second covers all systematic
uncertainties. For \mcnlo\ and \pow\ the uncertainties are only statistical.\label{tab:fLRaverage}}
\begin{tabular}{|c|c||c|c|c||c|}
\cline{2-3} \cline{5-6}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\fLmR\ $(\%)$} & \multicolumn{1}{c|}{} & \multicolumn{2}{c|}{\fl\ $(\%)$} \\
\cline{2-3} \cline{5-6}
\multicolumn{1}{c|}{}& 35 $<$ \ptw\ $<$ 50 \GeV & \ptw\ $>$ 50 \GeV & \multicolumn{1}{c|}{} & 35 $<$ \ptw\ $<$ 50 \GeV & \ptw\ $>$ 50 \GeV \bigstrut \\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{Data} & 23.8 $\pm$ 2.0 $\pm$ 3.4 & 25.2 $\pm$ 1.7 $\pm$ 3.0 & \multicolumn{1}{c|}{} & 21.9 $\pm$ 3.3 $\pm$ 13.4 &12.7 $\pm$ 3.0 $\pm$ 10.8 \\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{\mcnlo\ } & 27.1 $\pm$ 0.7& 26.2 $\pm$ 0.5 & \multicolumn{1}{c|}{} & 17.9 $\pm$ 1.2& 21.0 $\pm$ 1.0 \\
\cline{1-3} \cline{5-6}
\multirow{1}{*}{\pow\ } & 19.9 $\pm$ 1.0& 21.2 $\pm$ 0.8 & \multicolumn{1}{c|}{} & 22.9 $\pm$ 1.0& 19.4 $\pm$ 0.8 \\
\cline{1-3} \cline{5-6}
\multicolumn{6}{c}{}
\end{tabular}
\captionsetup{type=figure}
\centering
\includegraphics[width=0.48\textwidth]{fig_09a.pdf}
\includegraphics[width=0.48\textwidth]{fig_09b.pdf}
\caption{Measured values of \fl\ and \fLmR after corrections (\Tab{fLRaverage}), within acceptance cuts, for 35 $<$ \ptw\ $<$ 50 \GeV\ (left)
and \ptw\ $>$ 50 \GeV\ (right), compared with the predictions of \mcnlo\ and \pow. The ellipses around the data points correspond to one
standard deviation.\label{fig:Final}}
\end{table*}
For \fl, and the directly related coefficient $A_0$ (\Eqn{fracAi}), the systematic uncertainties associated with the recoil and lepton energy scales
do not cancel between negative and positive charges. In order to reduce the statistical uncertainties, which are also large, and the uncorrelated
instrumental and analysis systematic uncertainties, the measurements in each \ptw\ bin were averaged over charges and lepton flavours. The uncertainties
from the recoil energy scale were taken to be fully correlated among all four measurements. The uncertainty associated with the template model
(\Tab{correctedfLmR}) was combined quadratically with the other systematic uncertainties.\par
A comparison between the corrected experimental results and the predicted values, within the acceptance and \trm\ cuts (\Tab{fLRaverage}),
indicates that:
\begin{itemize}
\item
in the low \ptw\ bin the data are compatible with both \mcnlo\ and \pow\ predictions, which are mutually consistent.
\item
in the high \ptw\ bin, the data favour \fl\ values smaller than the predictions of \mcnlo\ and \pow, which are close to each other.
\end{itemize}
Due to the large uncertainties on the measurements, however, no stringent constraints nor clear inconsistencies can be deduced.
The measured values of \fl\ and \fLmR\ are plotted in \Fig{Final} within the triangular region allowed by the constraint \fL+\fl+\fR=1,
together with the predictions from \mcnlo\ and \pow.\par
\section{Summary and conclusions}
\label{sect:conclusions}
The results presented in this paper show that \mcnlo\ and \pow\ reproduce well the shape of the angular distributions in the transverse
plane of charged leptons from high-\pt\ \Wboson\ boson decays (\ptw\ $>$ 35 \GeV), a regime where the leading-quark effect in quark-antiquark
annihilation is subordinate to the dynamics of quark-gluon interactions producing \Wbosons.\par
The variable used for the analysis in terms of helicity fractions (respectively \fl, \fL\ and \fR) is the cosine of the ``transverse helicity''
angle \ctdd. Given that the three helicity fractions are constrained to sum to unity, the independent variables chosen in this study are \fl\
and \fLmR. Their values have been derived by fitting \ctdd\ distributions with templates representing longitudinal, left- and right-handed
\Wbosons. Two sets of templates were used, obtained from \mcnlo\ and \pow.\par
The experimental results have been corrected for the difference between the distribution of the measured quantity, the ``transverse helicity''
angle \ctdd, and the distribution of the true helicity angle, \ctd. The correction includes resolution effects, as well as systematic differences
between the two sets of templates. Corrected results correspond to the following acceptance region: $|\etal|$ $<$ 2.4, \ptn\ $>$ 25 \GeV, \ptl\ $>$
20 \GeV\ and 50 $<$ \trm\ $<$ 110 \GeV.\par
The longitudinal fraction is the most difficult to extract and has rather large systematic uncertainties, especially in the low \ptw\ bin, mostly
associated with the recoil energy scale and with the choice of Monte Carlo generator. In the low \ptw\ bin the data are compatible with both
\mcnlo\ and \pow\ predictions while in the high \ptw\ bin, they favour lower values than predicted by either of the simulations, which agree well
with each other.\par
When averaging over charges, \fLmR\ is measured with a small statistical uncertainty and a relatively small systematic uncertainty. The agreement
between data and \mcnlo, separately for the four measurements (two lepton flavours and two\ \ptw\ bins) is good. Predictions by \pow\ are somewhat
smaller than data, especially in the high \ptw\ bin.\par
\begin{acknowledgement}
~\\
We thank L.Dixon and D.Kosower for stimulating discussions, and S.Hoeche for providing data from BlackHat.\par
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be
operated efficiently.
We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP,
Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic;
DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET and ERC, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG
and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM
and NWO, Netherlands; RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation;
JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF
and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF,
United States of America.
The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at
TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain),
ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.
\end{acknowledgement}
\section{Helicity templates and Monte Carlo closure test}
\label{sect:templatesandclosure}
\subsection{Construction of helicity templates}
\label{templates}
In order to measure the helicity fractions, it is necessary to construct \ctdd\ distributions corresponding to samples of longitudinal, left-
and right-handed \Wbosons\ that decay into a lepton and a neutrino. As a check at the generator-level, and for the correction procedure (see
Section\ \ref{sect:systemp}), \ctd\ distributions corresponding to the three polarisation states were also made. All these distributions are
called helicity templates in the following. The templates were built independently from \mcnlo\ and from \pow\ using the following reweighting
technique.\par
It was first verified that, at the generator-level, and in bins of limited size in \ptw\ and \yw, \Wboson\ decays generated with the Monte Carlo
simulations are well described by \Eqn{fracdef}. The generator-level \ctd\ distributions were then fitted with the distribution corresponding to
this equation, which gave the values of \fL, \fl\ and \fR\ in \yw\ and \ptw\ bins. The results, in terms of \fl\ and \fLmR, are shown in
\Fig{ComputedFractions} for \mcnlo. The size of the bins results from a compromise between the rate of variation of the coefficients and the size
of the available samples.\par
Several conclusions may be drawn from \Fig{ComputedFractions}. The longitudinal fraction, which is very small for low \ptw, grows with \ptw\
(especially at low \absyw), before flattening out and then starting to decrease. The difference between the fractions of left- and right-handed
\Wboson\ bosons is small for low \absyw\ and grows quickly with \absyw, reaching up to 70\% for \absyw\ = 3. As already explained in Section\
\ref{sect:intro}, a smaller left-right difference is expected for negative than for positive \Wbosons; however in the \ptw\
range analysed here, these differences differ by at most a few percent. The analysis of systematic uncertainties described in Section\
\ref{sect:escale}, shows that it is experimentally advantageous to average the measured values of \fLmR\ between the two charges. As an
anticipation of this observation, it can be seen in \Fig{ComputedFractions} that this averaging is physically meaningful.\par
An equivalent analysis for \pow\ shows a similar trend for \fLmR\ as observed for \mcnlo. For \fl, in the \ptw\ range analysed here, \pow\
exhibits a much flatter dependence on \yw\ than \mcnlo, the average values being, however, very close to each other. Analytical calculations
at NNLO reported in Ref.\ \cite{Bern2011ie} by the BlackHat collaboration are very close to \pow. This is illustrated in \Fig{CompSherpa}.\par
\begin{figure}[h!]
\begin{changemargin}{-0.05\textwidth}{0cm}
\begin{center}
\includegraphics[width=0.49\textwidth]{fig_06.pdf}
\end{center}
\end{changemargin}
\caption{Evolution of the longitudinal polarisation fraction as a function of \absyw, in \mcnlo, \pow\ and a calculation based on BlackHat, for
\Wp\ (top) and \Wm\ (bottom) for two \ptw\ bins.\label{fig:CompSherpa}}
\end{figure}
Samples representing longitudinal, left- and right-handed states are obtained by reweighting the \mcnlo\ or \pow\ simulated events according to :
\begin{changemargin}{-0.015\textwidth}{-0.015\textwidth}
\begin{equation}
\label{eqn:weicompu}
\frac{ \left. \displaystyle\frac{1}{\sigma^\pm}\frac{d\sigma^\pm}{d\ctd} \right|_{L/0/R}}{\frac{3}{8}\fL (1\mp\ctd)^2 + \frac{3}{8}\fR (1\pm\ctd)^2 + \frac{3}{4}\fl \sct_{\rm 3D}}
\end{equation}
\end{changemargin}
where
\begin{equation}
\hfill \left. \frac{1}{\sigma^\pm}\frac{d\sigma^\pm}{d\ctd} \right|
\begin{array}{l}
{\scriptstyle L}\\
{\scriptstyle 0}\\
{\scriptstyle R}
\end{array}
= \frac{3}{8} \left \lbrace
\begin{array}{l}
(1\mp\ctd)^2 \\
2 \std^2 \\
(1\pm\ctd)^2
\end{array}
\right . \hfill
\end{equation}
and where the denominator corresponds to the general form of the differential cross-section in which the coefficients are taken from
\Fig{ComputedFractions} (or its equivalent from \pow), for the corresponding value of \ptw\ and \absyw. In these equations, the upper (lower)
sign corresponds to \Wp\ (\Wm) boson.
\subsection{Fit procedure applied to Monte Carlo samples}
The fitting procedure with templates was first applied to the simulated samples, at three different levels:
\begin{itemize}
\item all events using generator information for \ctd\ distributions;
\item events remaining after applying acceptance and \trm\ cuts using generator information for \ctd\ distributions;
\item events after the complete event selection (standard plus analysis cuts), using fully simulated information followed by reconstruction
for \ctdd\ distributions.
\end{itemize}
The fits of \ctd\ and \ctdd\ distributions were performed using a binned maximum-likelihood fit\ \cite{Barlow1993dm,ROOT}. Since the parameters
of the fit, \fl, \fL\ and \fR, must sum to 1, only two independent parameters, chosen to be \fl\ and \fLmR, are reported. The parameters were not
individually constrained to be between 0 and 1.\par
For the second and third steps, numerical results for \fl\ and \fLmR\ fits are summarised in \Tab{MCfitresults} for 35 $<$ \ptw\ $<$ 50
\GeV\ and \ptw\ $>$ 50 \GeV. In \Tab{MCfitresults} and in the following, the coefficients \fl\ and \fLmR\ represent helicity fractions, averaged
over \yw, within a given \ptw\ bin.\par
\begin{table*}
\centering
\caption{Results (as percentages) of fitting \ctd\ and \ctdd\ distributions from \mcnlo\ simulated samples using helicity templates. The fits are
performed at generator-level, after applying acceptance and \trm\ cuts, and on fully simulated events, after applying standard plus analysis
selections using \ctdd. \label{tab:MCfitresults}}
\begin{tabular}{|c|c|c|c||c|c|}
\cline{3-6}
\multicolumn{2}{c|}{} & $\mu^+$ & $\mu^-$ & $e^+$ & $e^-$ \\
\cline{3-6}
\multicolumn{2}{c|}{}&\multicolumn{4}{|c|}{35$<$ \ptw\ $<$ 50 \GeV \bigstrut} \\
\hline
\multirow{2}{*}{\ctd\ generator-level} & \fl\ $(\%)$& 14.6 $\pm$ 0.8 & 20.9 $\pm$ 0.8 & 15.3 $\pm$ 0.8 & 20.4 $\pm$ 0.9 \\
& \fLmR\ $(\%)$& 27.9 $\pm$ 0.7 & 26.5 $\pm$ 0.8 & 28.2 $\pm$ 0.7 & 26.4 $\pm$ 0.8 \\
\hline
\multirow{2}{*}{\ctdd\ fully simulated} & \fl\ $(\%)$& 30.1 $\pm$ 2.4 & 19.5 $\pm$ 2.2 & 26.9 $\pm$ 2.2 & 21.6 $\pm$ 2.3 \\
& \fLmR\ $(\%)$& 31.8 $\pm$ 1.4 & 26.5 $\pm$ 1.2 & 27.3 $\pm$ 1.4 & 22.5 $\pm$ 1.4 \\
\hline
\multicolumn{6}{c}{} \\
\cline{3-6}
\multicolumn{2}{c|}{}&\multicolumn{4}{|c|}{\ptw\ $\ge$ 50 \GeV \bigstrut} \\
\hline
\multirow{2}{*}{\ctd\ generator-level} & \fl\ $(\%)$& 18.3 $\pm$ 1.0 & 22.7 $\pm$ 1.0 & 19.0 $\pm$ 0.9 & 22.1 $\pm$ 1.0 \\
& \fLmR\ $(\%)$& 26.9 $\pm$ 0.8 & 25.8 $\pm$ 0.9 & 27.6 $\pm$ 0.8 & 25.9 $\pm$ 0.9 \\
\hline
\multirow{2}{*}{\ctdd\ fully simulated} & \fl\ $(\%)$& 25.1 $\pm$ 1.9 & 20.7 $\pm$ 2.2 & 24.9 $\pm$ 1.8 & 22.5 $\pm$ 2.0 \\
& \fLmR\ $(\%)$& 29.7 $\pm$ 1.1 & 26.2 $\pm$ 1.2 & 25.6 $\pm$ 1.2 & 22.6 $\pm$ 1.3 \\
\hline
\end{tabular}
\end{table*}
Template fit results using the \ctd\ distributions at the generator-level, without any cut, reproduce the average value of the numbers
quoted in the relevant \ptw\ bin of \Fig{ComputedFractions}. With respect to these fit results, the numbers shown in the first lines of
\Tab{MCfitresults} for the two \ptw\ bins reflect the effect of the acceptance and \trm\ cuts, which is small on \fl\ but is sizeable on
\fLmR, typically reducing it by 25\% (relative). Indeed, the detector has a small acceptance for the events produced at high \absyw, for
which \fLmR\ is largest.\par
Comparisons of the first row of each part of \Tab{MCfitresults} (\ctd\ at generator-level, within acceptance) to the second row (\ctdd\
after full simulation) indicates that the values of \fl\ are rather stable for \Wm\, while for \Wp\ there is in several cases a
significant increase. Similar effects are observed with \pow. Corrections applied at the analysis level (see Section\ \ref{sect:systemp})
are intended to remove these effects to obtain the final, corrected results.\par
|
{
"timestamp": "2012-05-24T02:03:42",
"yymm": "1203",
"arxiv_id": "1203.2165",
"language": "en",
"url": "https://arxiv.org/abs/1203.2165"
}
|
\section{Introduction}
Extending the Morse potential by the addition of some nonsingular function, in such a way that the resulting potential remains exactly solvable (ES), has a long history and has been done along several paths.\par
In the context of unbroken supersymmetric quantum mechanics (SUSYQM) \cite{cooper}, an extended potential has been built in terms of the confluent hypergeometric function and its derivative and shown to be ES under certain conditions on the potential parameters \cite{junker}. For such a reason, the new potential has been termed `conditionally exactly solvable' \cite{dutra, dutt}.\par
In the framework of backward Darboux transformations \cite{darboux} (equivalent to the previous approach of unbroken SUSYQM) of (translationally) shape invariant (SI) potentials \cite{gendenshtein}, algebraic deformations of the Morse potential have been considered \cite{gomez04}. These deformations are characterized by the fact that the superpotential is a rational function or a composition of a rational function with an exponential. Special attention has been devoted to the polynomials appearing in the partner wavefunctions and to their properties.\par
Another method has employed ground- or excited-state wavefunctions of the Morse potential to construct nonsingular isospectral potentials \cite{berger, dutta} by resorting to the well-known nonuniqueness of factorization \cite{mielnik}. The latter indeed allows one to avoid the singularities arising from the use of excited-state wavefunctions in standard SUSYQM \cite{cooper}.\par
After the introduction of the first families of exceptional orthogonal polynomials (EOP) in the context of Sturm-Liouville theory \cite{gomez10a, gomez09}, the realization of their usefulness in constructing new SI extensions of ES potentials in quantum mechanics \cite{cq08a, bagchi09a, cq09}, and the rapid developments that followed in this area \cite{odake09, odake10a, odake10b, ho11a, gomez10b, gomez11a, sasaki, grandati11a, ho11b, gomez12, odake11, cq11a, cq11b, grandati11b, cq11c}, it soon appeared that only some of the well-known SI potentials led to rational extensions connected with EOP. In this category, one finds the radial oscillator \cite{cq08a, cq09, odake09, odake10a, odake10b, sasaki, grandati11a, ho11b}, the Scarf I (also called trigonometric P\"oschl-Teller or P\"oschl-Teller I) \cite{cq08a, cq09, odake09, odake10a, sasaki, ho11b}, and the generalized P\"oschl-Teller (also termed hyperbolic P\"oschl-Teller or P\"oschl-Teller II) \cite{bagchi09a, odake09, odake10a}.\par
In such a context, the Morse potential has been recently re-examined by making use of two different (but equivalent) approaches, the Darboux-B\"acklund transformation \cite{grandati11c} and the prepotential method \cite{ho11c}. Both studies have recovered the previously found algebraic deformations \cite{gomez04}.\par
The purpose of the present work is twofold. First, we will review the construction of rationally-extended Morse potentials by using a standard SUSYQM approach, similar to that previously employed for some other potentials \cite{bagchi09a, cq09, cq11b, cq11c}. This will enable us to point out the existence of a whole family of strictly isospectral extensions for some range of parameter values. Second, we will generalize to rationally-extended potentials the point canonical transformation (PCT) known to connect the radial oscillator to the Morse potential \cite{cooper, haymaker, de, cq08b}. This will lead us to quasi-exactly solvable (QES) \cite{turbiner, shifman, ushveridze} rational extensions.\par
\section{Rationally-Extended Morse Potentials in First-Order SUSYQM}
\subsection{General results}
As well known, the Morse potential
\begin{equation}
V_{A,B}(x) = B^2 e^{-2x} - B (2A+1) e^{-x}, \qquad - \infty < x < \infty, \label{eq:Morse-pot}
\end{equation}
has a minimum $V_{A,B}(x_{\rm min}) = - \frac{1}{4} (2A+1)^2$, for $x_{\rm min}$ such that $e^{-x_{\rm min}} = (2A+1)/(2B)$, provided this quantity is positive. It then has a finite number of bound states with energy \cite{cooper}
\begin{equation}
\epsilon^{(A)}_{\nu} = - (A-\nu)^2, \qquad \nu = 0, 1, \ldots, \nu_{\rm max}, \qquad A-1 \le \nu_{\rm max} <
A, \label{eq:Morse-spectrum}
\end{equation}
provided $A > 0$ (hence $B > 0$ too).\footnote{In this paper, we take units wherein $\hbar = 2m = 1$.} The corresponding bound-state wavefunctions can be expressed in terms of (generalized) Laguerre polynomials as
\begin{equation}
\varphi^{(A)}_{\nu}(x) \propto \exp[- (A-\nu) x - B e^{-x}] L^{(2A-2\nu)}_{\nu}(2B e^{-x}) \propto \xi_A(z)
L^{(2A-2\nu)}_{\nu}(z), \label{eq:Morse-wf}
\end{equation}
with
\begin{equation}
z = 2B e^{-x}, \qquad \xi_A(z) = z^{A-\nu} e^{- \frac{1}{2} z}, \qquad 0 < z < \infty. \label{eq:z}
\end{equation}
\par
In first-order SUSYQM \cite{cooper}, one considers a pair of SUSY partners
\begin{equation}
\begin{split}
H^{(+)} & = \hat{A}^{\dagger} \hat{A} = - \frac{d^2}{dx^2} + V^{(+)}(x) - \epsilon, \\
H^{(-)} & = \hat{A} \hat{A}^{\dagger} = - \frac{d^2}{dx^2} + V^{(-)}(x) - \epsilon, \\
\hat{A}^{\dagger} & = - \frac{d}{dx} + W(x), \\
\hat{A} & = \frac{d}{dx} + W(x), \\
V^{(\pm)}(x) & = W^2(x) \mp W'(x) + \epsilon ,
\end{split} \label{eq:SUSY}
\end{equation}
which intertwine with the first-order differential operators $\hat{A}$ and $\hat{A}^{\dagger}$ as $\hat{A} H^{(+)} = H^{(-)} \hat{A}$ and $\hat{A}^{\dagger} H^{(-)} = H^{(+)} \hat{A}^{\dagger}$. Here $W(x)$ is the superpotential, which can be expressed as $W(x) = - \bigl(\log \phi(x)\bigr)'$ in terms of a (nodeless) seed solution $\phi(x)$ of the initial Schr\"odinger equation
\begin{equation}
\left(- \frac{d^2}{dx^2} + V^{(+)}(x)\right) \phi(x) = \epsilon \phi(x), \label{eq:SE}
\end{equation}
$\epsilon$ is the factorization energy, assumed smaller than or equal to the ground-state energy $\epsilon^{(+)}_0$ of $V^{(+)}(x)$, and a prime denotes a derivative with respect to $x$. For $\epsilon = \epsilon^{(+)}_0$ and $\phi(x) = \varphi^{(+)}_0(x)$ corresponding to the ground state of $V^{(+)}(x)$, the partner potential $V^{(-)}(x)$ has the same bound-state spectrum as $V^{(+)}(x)$, except for the ground-state energy which is removed (case {\it i}). For $\epsilon < \epsilon^{(+)}_0$, in which case $\phi(x)$ is a nonnormalizable function, $V^{(-)}(x)$ has the same spectrum as $V^{(+)}(x)$ if $\phi^{-1}(x)$ is also nonnormalizable (case {\it ii} or isospectral case) or it has an extra bound-state energy $\epsilon$ below $\epsilon^{(+)}_0$, corresponding to the wavefunction $\phi^{-1}(x)$, if the latter is normalizable (case {\it iii}).\par
{}For $V^{(+)}(x) = V_{A,B}(x)$, it is well known \cite{cooper} that $\epsilon = \epsilon^{(+)}_0 = \epsilon^{(A)}_0$ and $\phi(x) = \varphi^{(+)}_0(x) = \varphi^{(A)}_0(x)$ lead to $V^{(-)}(x) = V_{A-1,B}(x)$, showing that the Morse potential is SI \cite{gendenshtein}.\par
To construct rational extensions of the Morse potential, we have to determine all well-behaved (i.e., nodeless) solutions $\phi(x)$ of (\ref{eq:SE}) with $\epsilon < \epsilon^{(A)}_0 = - A^2$ that are of polynomial type. For such a purpose, let us use the ansatz
\begin{equation}
\phi\bigl(x(z)\bigr) = z^{\lambda} e^{- \frac{1}{2} z} f(z)
\end{equation}
in the Schr\"odinger equation (\ref{eq:SE}), rewritten in the variable $z$ defined in (\ref{eq:z}),
\begin{equation}
\left[- z^2 \frac{d^2}{dz^2} - z \frac{d}{dz} + \frac{1}{4} z^2 - \left(A + \frac{1}{2}\right) z - \epsilon\right]
\phi\bigl(x(z)\bigr) = 0.
\end{equation}
Here $\lambda$ and $f(z)$ denote some constant and some function, respectively. Provided
\begin{equation}
\epsilon = - \lambda^2, \qquad a = \lambda - A, \qquad b = 2\lambda + 1, \label{eq:conditions}
\end{equation}
the resulting equation for $f(z)$ reduces to the confluent hypergeometric equation
\begin{equation}
\left[z \frac{d^2}{dz^2} + (b-z) \frac{d}{dz} - a \right] f(z) = 0, \label{eq:confluent}
\end{equation}
whose regular solution is ${}_1F_1(a; b; z)$. Equation (\ref{eq:confluent}) admits four polynomial-type solutions, expressed in terms of Laguerre polynomials, if and only if either $a$ or $b-a$ is an integer \cite{erdelyi} (see also Ref.\ \cite{gomez04}),
\begin{equation}
\begin{split}
f_1(z) & = {}_1F_1(a; b; z) \propto L^{(b-1)}_m(z) \qquad \text{for\ } a = - m, \\
f_2(z) & = z^{1-b} {}_1F_1(a-b+1; 2-b; z) \propto z^{1-b} L^{(1-b)}_m(z) \qquad \text{for\ } b-a = m+1, \\
f_3(z) & = e^z {}_1F_1(b-a; b; -z) \propto e^z L^{(b-1)}_m(-z) \qquad \text{for\ } b-a = - m, \\
f_4(z) & = z^{1-b} e^z {}_1F_1(1-a; 2-b; -z) \propto z^{1-b} e^z L^{(1-b)}_m(-z) \qquad \text{for\ } a = m+1.
\end{split} \label{eq:f}
\end{equation}
\par
It remains to combine Eq.\ (\ref{eq:conditions}) with the condition found for $a$ or $b-a$ in (\ref{eq:f}) and to look for those cases where $\epsilon < - A^2$ and the Laguerre polynomial has no zero for $z \in (0, \infty)$. To check the last condition, we use Kienast-Lawton-Hahn's theorem on the zeros of Laguerre polynomials \cite{erdelyi} (see also Ref.\ \cite{grandati11a}). From $f_1(z)$ and $f_3(z)$ (or, equivalently, from $f_2(z)$ and $f_4(z)$), we arrive at the following two acceptable factorization functions
\begin{eqnarray}
\phi^{\rm II}_{A,m}(x) & = & \chi^{\rm II}_{A,m}(z) L^{(2A-2m)}_m(z) \nonumber \\
& \propto & \exp[- (A-m)x - Be^{-x}] L^{(2A-2m)}_m(2Be^{-x}) \nonumber \\
&& \text{if\ } m = 1, 2, 3, \ldots \text{\ and\ } A < \frac{m}{2}, \label{eq:phi-II}
\end{eqnarray}
\begin{eqnarray}
\phi^{\rm III}_{A,m}(x) & = & \chi^{\rm III}_{A,m}(z) L^{(-2A-2m-2)}_m(-z) \nonumber \\
& \propto & \exp[(A+m+1)x + Be^{-x}] L^{(-2A-2m-2)}_m(-2Be^{-x}) \nonumber \\
&& \text{if\ } m = 2, 4, 6, \ldots, \label{eq:phi-III}
\end{eqnarray}
with
\begin{equation}
\chi^{\rm II}_{A,m}(z) = z^{A-m} e^{- \frac{1}{2} z}, \qquad \chi^{\rm III}_{A,m}(z) = z^{-A-m-1}
e^{\frac{1}{2} z}, \label{eq:chi}
\end{equation}
and corresponding energies $\epsilon^{\rm II}_{A,m} = - (A-m)^2$, $\epsilon^{\rm III}_{A,m} = - (A+m+1)^2$, respectively. The inverse of $\phi^{\rm III}_{A,m}(x)$ is normalizable, in contrast with that of $\phi^{\rm II}_{A,m}(x)$.\par
A superscript II or III has been introduced in Eqs.\ (\ref{eq:phi-II}), (\ref{eq:phi-III}), and (\ref{eq:chi}), by analogy with what is often done in the case of extended radial oscillator potentials pertaining to the L2 and L3 series, respectively \cite{grandati11a, cq11c}. The former series is associated with Laguerre polynomials with negative argument and positive variable, while for the latter both the argument and the variable are negative. For the extended Morse potentials, no counterpart of the L1 series, corresponding to positive argument and negative variable, is obtained. It is worth stressing that $\phi^{\rm III}_{A,m}(x)$ has been first derived in Ref.\ \cite{gomez04}, then reconsidered in Refs.\ \cite{grandati11c} and \cite{ho11c}, but that to the best of the author's knowledge, $\phi^{\rm II}_{A,m}(x)$ has not been mentioned so far.\par
To obtain some rationally-extended Morse potentials $V_{A,B,{\rm ext}}(x)$ with given $A$ and $B$, we have to start from a conventional Morse potential $V_{A',B}(x)$ with some different $A'$, but the same $B$ (hence $z$ remains unchanged). From Eqs.\ (\ref{eq:SUSY}), (\ref{eq:phi-II}), (\ref{eq:phi-III}), and (\ref{eq:chi}), it is straightforward to get
\begin{equation}
\begin{split}
& V^{(+)}(x) = V_{A',B}(x), \qquad V^{(-)}(x) = V_{A,B,{\rm ext}}(x) = V_{A,B}(x) + V_{A,B,{\rm rat}}(x), \\
& V_{A,B,{\rm rat}}(x) = - 2z \biggl\{\frac{\dot{g}^{(A)}_m}{g^{(A)}_m} + z
\biggl[\frac{\ddot{g}^{(A)}_m}{g^{(A)}_m} - \biggl(\frac{\dot{g}^{(A)}_m}
{g^{(A)}_m}\biggr)^2\biggr]\biggr\},
\end{split} \label{eq:partners}
\end{equation}
where a dot denotes a derivative with respect to $z$. According to the choice made for the factorization function $\phi(x)$, we may distinguish the two cases
\begin{eqnarray}
& ({\rm II}) \; & A' = A+1, \quad \phi = \phi^{\rm II}_{A+1,m}, \quad g^{(A)}_m(z) = L^{(2A+2-2m)}_m(z),
\nonumber \\
&& m = 1, 2, 3, \ldots, \quad -1 < A < \frac{m-2}{2}; \label{eq:type-II} \\
& ({\rm III}) \; & A' = A-1, \quad \phi = \phi^{\rm III}_{A-1,m}, \quad g^{(A)}_m(z) = L^{(-2A-2m)}_m(-z),
\nonumber \\
&& m = 2, 4, 6, \ldots, \quad A>1. \label{eq:type-III}
\end{eqnarray}
Note that $B>0$ everywhere.\par
\subsection{Type II rationally-extended Morse potentials}
In type II case, $V^{(+)}(x)$ and $V^{(-)}(x)$ are isospectral (case {\it ii} of SUSYQM) and their common bound-state spectrum is given by
\begin{equation}
\epsilon^{(+)}_{\nu} = \epsilon^{(-)}_{\nu} = - (A + 1 - \nu)^2, \qquad \nu = 0, 1, \ldots, \nu_{\rm max},
\qquad A \le \nu_{\rm max} < A+1.
\end{equation}
When $A$ varies in the range $-1 < A < \frac{m-2}{2}$, the number of bound states $\nu_{\rm max} + 1$ goes from one to $\left[\frac{m+1}{2}\right]$.\par
{}From the bound-state wavefunctions $\varphi^{(+)}_{\nu}(x) \propto \xi_{A+1}(z) L^{(2A+2-2\nu)}_{\nu}(z)$, $\nu = 0$, 1, \ldots, $\nu_{\rm max}$, of $V^{(+)}(x)$, those of $V^{(-)}(x)$ are obtained as
\begin{equation}
\varphi^{(-)}_{\nu}(x) \propto \hat{A} \varphi^{(+)}_{\nu}(x) \propto \frac{\xi_{A+1}(z)}{g^{(A)}_m(z)}
y^{(A)}_n(z), \qquad \nu = 0, 1, \ldots, \nu_{\rm max}, \label{eq:partner-wf}
\end{equation}
with
\begin{equation}
\hat{A} = - z \left(\frac{d}{dz} + \frac{1}{2} - \frac{A+1-m}{z} - \frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\right).
\end{equation}
In (\ref{eq:partner-wf}), $y^{(A)}_n(z)$ denotes some $n$th-degree polynomial in $z$, defined by
\begin{equation}
y^{(A)}_n(z) = \left[g^{(A)}_m \left(- z \frac{d}{dz} + \nu - m\right) + z \dot{g}^{(A)}_m\right]
L^{(2A+2-2\nu)}_{\nu}(z). \label{eq:def-y}
\end{equation}
This definition seems to imply that $n = m + \nu$. Nevertheless, from the relation
\begin{equation}
y^{(A)}_n(z) = (2A + 2 - \nu) g^{(A)}_m(z) L^{(2A+2-2\nu)}_{\nu-1}(z) - (2A + 2 - m) g^{(A-1)}_{m-1}(z)
L^{(2A+2-2\nu)}_{\nu}(z),
\end{equation}
directly obtainable from the right-hand side of (\ref{eq:def-y}) and some elementary properties of Laguerre polynomials \cite{gradshteyn}, we actually deduce that
\begin{equation}
n = m + \nu - 1.
\end{equation}
Note that with the normalization assumed in (\ref{eq:def-y}), the highest-degree term of $y^{(A)}_{m+\nu-1}(z)$ is given by $(m-\nu) (m+\nu-2A-2) (-z)^{m+\nu-1}/(m!\, \nu!)$. As a special case, the ground-state wavefunction of $V^{(-)}(x)$ can be written as
\begin{equation}
\varphi^{(-)}_0(x) \propto \frac{\xi_{A+1}(z)}{g^{(A)}_m(z)} g^{(A-1)}_{m-1}(z). \label{eq:gs-wf}
\end{equation}
\par
On the other hand, by directly inserting Eq.\ (\ref{eq:partner-wf}) in the Schr\"odinger equation for $V^{(-)}(x)$, we arrive at the following second-order differential equation for $y^{(A)}_{m+\nu-1}(z)$,
\begin{eqnarray}
&& \biggl\{z \frac{d^2}{dz^2} + \biggl[2A+3-2\nu - z\biggl(1 + 2\frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\biggr)
\biggr] \frac{d}{dz} - 2 (2A+2-m-\nu-z) \frac{\dot{g}^{(A)}_m}{g^{(A)}_m} \nonumber \\
&& \quad - (m-\nu+1)\biggr\} y^{(A)}_{m+\nu-1}(z) = 0, \qquad \nu = 0, 1, \ldots, \nu_{\rm max}.
\label{eq:diff-eq}
\end{eqnarray}
\par
Let us illustrate the results obtained here by considering the $m=1$, 2, and 3 special cases. The rational part of the extended potentials can be written as
\begin{equation}
V_{A,B,{\rm rat}}(x) = \frac{N_1(x)}{D(x)} + \frac{N_2(x)}{D^2(x)}, \label{eq:V-rat}
\end{equation}
where
\begin{equation}
\begin{split}
N_1(x) & = 2 (2A+1), \\
N_2(x) & = 2 (2A+1)^2, \\
D(x) & = 2Be^{-x} - 2A - 1,
\end{split}
\end{equation}
with $-1 < A < -\frac{1}{2}$ for $m=1$,
\begin{equation}
\begin{split}
N_1(x) & = 8A \bigl(Be^{-x} + 1\bigr), \\
N_2(x) & = 8A^2 \bigl(4Be^{-x} - 2A + 1\bigr), \\
D(x) & = 2B^2 e^{-2x} - 4ABe^{-x} + A (2A-1),
\end{split} \label{eq:example-II}
\end{equation}
with $-1 < A < 0$ for $m=2$, and
\begin{equation}
\begin{split}
N_1(x) & = 3 (2A-1) \bigl[4B^2 e^{-2x} - 2 (2A-5) Be^{-x} + 3 (2A+1)\bigr], \\
N_2(x) & = 9 (2A-1)^2 \bigl[2 (2A+7) B^2 e^{-2x} - 4 (A-1)(2A+3) Be^{-x} \\
& \quad + (A-1)(2A-3)(2A+1)\bigr], \\
D(x) & = 4B^3 e^{-3x} - 6(2A-1) B^2 e^{-2x} + 6(A-1)(2A-1) Be^{-x} \\
& \quad - (A-1)(2A-1)(2A-3),
\end{split}
\end{equation}
with $-1 < A < \frac{1}{2}$ for $m=3$. In the first two cases, there is a single bound state with energy $\epsilon^{(-)}_0 = - (A+1)^2$ and wavefunction
\begin{equation}
\varphi^{(-)}_0(x) \propto \frac{\exp\bigl[- (A+1)x - Be^{-x}\bigr]}{D(x)}
\end{equation}
or
\begin{equation}
\varphi^{(-)}_0(x) \propto \frac{\exp\bigl[- (A+1)x - Be^{-x}\bigr]}{D(x)} \bigl(2Be^{-x} - 2A + 1\bigr),
\end{equation}
respectively. In contrast, in the third case, there may be up to two bound states
\begin{equation}
\varphi^{(-)}_0(x) \propto \frac{\exp\bigl[- (A+1)x - Be^{-x}\bigr]}{D(x)} \bigl[2B^2 e^{-2x} - 4(A-1) Be^{-x}
+ (A-1)(2A-3)\bigr]
\end{equation}
and
\begin{equation}
\begin{split}
\varphi^{(-)}_1(x) & \propto \frac{\exp\bigl[- (A+1)x - Be^{-x}\bigr]}{D(x)} \bigl[8B^3 e^{-3x} - 12(2A-1) B^2
e^{-2x} \\
& \quad + 6(2A-1)^2 Be^{-x} - (2A+1)(2A-1)(2A-3)\bigr],
\end{split}
\end{equation}
corresponding to $\epsilon^{(-)}_0 = - (A+1)^2$ and $\epsilon^{(-)}_1 = - A^2$, provided $0 < A < \frac{1}{2}$. Only the first one of them, however, exists for $-1 < A \le 0$.\par
It is worth observing that the addition of $V_{A,B,{\rm rat}}(x)$ to $V_{A,B}(x)$ has the effect of increasing the number of bound states by one, since for the allowed values of parameter $A$, the core part of the extended potential has no bound state for $m=1$ nor for $m=2$, and may have zero or one bound state for $m=3$ according to whether $-1 < A \le 0$ or $0 < A < \frac{1}{2}$. This observation remains true for higher $m$ values.\par
\subsection{Type III rationally-extended Morse potentials}
In type III case, $V^{(+)}(x)$ and $V^{(-)}(x)$ are not isospectral anymore (case {\it iii} of SUSYQM). Their bound-state spectra are given instead by
\begin{equation}
\epsilon^{(+)}_{\nu} = - (A-1-\nu)^2, \quad \nu = 0, 1, \ldots, \nu_{\rm max}, \quad A-2 \le \nu_{\rm max}
< A-1,
\end{equation}
and
\begin{equation}
\epsilon^{(-)}_{\nu} = - (A-1-\nu)^2, \quad \nu = -m-1, 0, 1, \ldots, \nu_{\rm max}, \quad A-2 \le
\nu_{\rm max} < A-1,
\end{equation}
the ground state of $V^{(-)}(x)$ corresponding to $\epsilon^{(-)}_{-m-1} = \epsilon^{\rm III}_{A-1,m} = - (A+m)^2$.\par
The bound-state wavefunctions of $V^{(-)}(x)$ can be written as
\begin{equation}
\varphi^{(-)}_{\nu}(x) \propto \frac{\xi_{A-1}(z)}{g^{(A)}_m(z)} y^{(A)}_n(z), \qquad n = m+\nu+1, \qquad
\nu = -m-1, 0, 1, \ldots, \nu_{\rm max},
\end{equation}
where $y^{(A)}_n(z)$ is an $n$-th-degree polynomial in $z$. For the ground state,
\begin{equation}
\varphi^{(-)}_{-m-1}(x) \propto \left(\phi^{\rm III}_{A-1,m}(x)\right)^{-1}, \qquad y^{(A)}_0(z) = 1,
\end{equation}
while for the excited states
\begin{equation}
\varphi^{(-)}_{\nu}(x) \propto \hat{A} \varphi^{(+)}_{\nu}(x), \qquad \nu = 0, 1, \ldots, \nu_{\rm max},
\end{equation}
with $\varphi^{(+)}_{\nu}(x) \propto \xi_{A-1}(z) L^{(2A-2-2\nu)}_{\nu}(z)$ and
\begin{equation}
\hat{A} = - z \left(\frac{d}{dz} - \frac{1}{2} + \frac{A+m}{z} - \frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\right).
\end{equation}
For $n = m+1$, $m+2$, \ldots, $m+1+\nu_{\rm max}$, we may therefore define $y^{(A)}_n(z)$ as
\begin{equation}
y^{(A)}_n(z) = \left[g^{(A)}_m \left(-z \frac{d}{dz} - 2A + 1 - m + \nu + z\right) + z \dot{g}^{(A)}_m\right]
L^{(2A-2-2\nu)}_{\nu}(z). \label{eq:def-y-bis}
\end{equation}
Standard properties of Laguerre polynomials \cite{gradshteyn} may be used to rewrite Eq.\ (\ref{eq:def-y-bis}) in either of the two equivalent forms
\begin{equation}
y^{(A)}_n(z) = (m+1) g^{(A-1)}_{m+1}(z) L^{(2A-2-2\nu)}_{\nu}(z) + (2A-2-\nu) g^{(A)}_m(z)
L^{(2A-2-2\nu)}_{\nu-1}(z)
\end{equation}
or
\begin{equation}
y^{(A)}_n(z) = (2A+m) g^{(A+1)}_{m-1}(z) L^{(2A-2-2\nu)}_{\nu}(z) - (\nu+1) g^{(A)}_m(z)
L^{(2A-2-2\nu)}_{\nu+1}(z).
\end{equation}
The latter expression was already given in previous studies \cite{grandati11c, ho11c}, but the former is a new one, which has the advantage of directly providing us with the polynomial appearing in the first-excited state wavefunction,
\begin{equation}
y^{(A)}_{m+1}(z) = (m+1) g^{(A-1)}_{m+1}(z) = (m+1) L^{(-2A-2m)}_{m+1}(-z).
\end{equation}
We may also observe that with the normalization chosen in (\ref{eq:def-y-bis}), the highest-degree term of $y^{(A)}_{m+\nu+1}(z)$ is given by $(-1)^{\nu} z^{m+\nu+1}/(m!\, \nu!)$ for $\nu = 0$, 1, \ldots, $\nu_{\rm max}$.\par
{}Finally, for type III polynomials, the counterpart of the second-order differential equation (\ref{eq:diff-eq}) reads
\begin{eqnarray}
&& \biggl\{z \frac{d^2}{dz^2} + \biggl[2A-1-2\nu - z\biggl(1 + 2\frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\biggr)
\biggr] \frac{d}{dz} + (m+\nu+1) \biggl(1 + 2\frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\biggr)\biggr\}
\nonumber \\
&& \quad \times y^{(A)}_{m+\nu+1}(z) = 0, \qquad \nu = -m-1, 0, 1, \ldots, \nu_{\rm max}.
\end{eqnarray}
\par
The results obtained here may be illustrated by considering the lowest allowed $m$ value, namely $m=2$. In such a case, the rational part of the extended potential takes the form (\ref{eq:V-rat}) with
\begin{equation}
\begin{split}
N_1(x) & = 8 (A+1) \bigl(Be^{-x} - 1\bigr), \\
N_2(x) & = - 8 (A+1)^2 \bigl(4Be^{-x} - 2A - 3\bigr), \\
D(x) & = 2B^2 e^{-2x} - 4 (A+1) Be^{-x} + (A+1) (2A+3),
\end{split} \label{eq:example-III}
\end{equation}
where $A>1$. Equation (\ref{eq:example-III}) may be compared with Eq.\ (\ref{eq:example-II}), corresponding to the other quadratic-type extended potential.\par
\subsection{Partner of rationally-extended Morse potentials in unbroken SUSYQM}
If we take any of the rationally-extended Morse potentials obtained so far as the starting potential $\bar{V}^{(+)}(x) = V^{(-)}(x)$ in first-order SUSYQM, it is interesting to determine its partner $\bar{V}^{(-)}(x)$ when its ground state is deleted (case {\sl i} of SUSYQM). In type III case, this is of course the inverse transformation of that carried out in Secs.\ 2.1 and 2.3, so that $\bar{V}^{(-)}(x) = V^{(+)}(x)$. Hence it only remains to consider type II extended potentials.\par
In such a case, the new factorization function corresponds to the ground-state wavefunction (\ref{eq:gs-wf}) of $V^{(-)}(x)$, which leads to the new superpotential
\begin{equation}
\bar{W}(x) = A+1 - \frac{1}{2}z + z\left(\frac{\dot{g}^{(A-1)}_{m-1}}{g^{(A-1)}_{m-1}} -
\frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\right),
\end{equation}
written in terms of the variable $z$. The searched for partner is then given by
\begin{equation}
\bar{V}^{(-)}(x) = \bar{V}^{(+)}(x) + 2 \bar{W}'(x)
\end{equation}
with $\bar{V}^{(+)}(x) = V^{(-)}(x)$ expressed in terms of $g^{(A)}_m(z)$ and its derivatives as in (\ref{eq:partners}). A straightforward calculation yields
\begin{equation}
\bar{V}^{(-)}(x) = V_{A-1,B}(x) - 2z \biggl\{\frac{\dot{g}^{(A-1)}_{m-1}}{g^{(A-1)}_{m-1}} + z
\biggl[\frac{\ddot{g}^{(A-1)}_{m-1}}{g^{(A-1)}_{m-1}} - \biggl(\frac{\dot{g}^{(A-1)}_{m-1}}
{g^{(A-1)}_{m-1}}\biggr)^2\biggr]\biggr\}.
\end{equation}
\par
It is remarkable that although $V_{A,B,{\rm ext}}(x)$ is not translationally SI, its partner belongs to an enlarged family of extended Morse potentials, wherein both the potential parameter $A$ and the polynomial degree $m$ are translated: $A \to A-1$, $m \to m-1$. In other words, what we have done here is to go from $\bar{V}^{(+)}(x) = V^{(m)}_{A,B,{\rm ext}}(x)$ to $\bar{V}^{(-)}(x) = V^{(m-1)}_{A-1,B,{\rm ext}}(x)$, where we have appended a superscript to specify the polynomial degree. The final potential $\bar{V}^{(-)}(x)$ having one bound state less than the initial one $\bar{V}^{(+)}(x)$, its spectrum is given by $- (A-\nu)^2$, $\nu=0$, 1, \ldots, $\nu_{\rm max}-1$ ($A \le \nu_{\rm max} < A+1$). As a consequence, for $m=1$ and $m=2$, we arrive at a conventional Morse potential and an extended Morse one with no bound state, respectively. It is only from $m=3$ upwards that extended Morse potentials with at least one bound state may be obtained.\par
Putting together the first step from $V^{(+)}(x)$ to $V^{(-)}(x)$ and the second one from $\bar{V}^{(+)}(x) = V^{(-)}(x)$ to $\bar{V}^{(-)}(x)$ (see Ref.\ \cite{bagchi09a} and references quoted therein), we arrive at a reducible second-order SUSYQM transformation from a conventional Morse potential $V_{A+1,B}(x)$ to an extended one $V^{(m-1)}_{A-1,B,{\rm ext}}(x)$. Since going from $V_{A+1,B}(x)$ to $V^{(m-1)}_{A-1,B,{\rm ext}}(x)$ can be achieved along another path by combining the usual unbroken SUSYQM transformation relating the two conventional Morse potentials $V_{A+1,B}(x)$ and $V_{A,B}(x)$ \cite{cooper} with the broken one connecting $V_{A,B}(x)$ to $V^{(m-1)}_{A-1,B,{\rm ext}}(x)$ (obtained by substituting $A-1$ and $m-1$ for $A$ and $m$ in Secs.\ 2.1 and 2.2), we finally get the following commutative diagram:
\begin{equation}
\begin{CD}
V_{A+1,B}(x) @>\text{unbroken}>> V_{A,B}(x)\\
@V\text{broken}VV @VV\text{broken}V\\
V_{A,B,{\rm ext}}^{(m)}(x) @>>\text{unbroken}> V_{A-1,B,{\rm ext}}^{(m-1)}(x)
\end{CD}
\end{equation}
This is another example of the possible existence of different intermediate Hamiltonians in higher-order SUSYQM \cite{cq11a} or in type A $\cal N$-fold supersymmetry \cite{bagchi09b}.\par
\section{Point Canonical Transformation Relating Rationally-Extended Radial Oscillator and Morse Potentials}
\subsection{Going from the radial oscillator to the Morse potential}
To start with, let us briefly review the case of conventional potentials and consider the Schr\"odinger equation for a radial oscillator potential
\begin{equation}
\left(- \frac{d}{dr^2} + V_l(r)\right) \psi^{(l)}_{\nu}(r) = E^{(l)}_{\nu} \psi^{(l)}_{\nu}(r), \qquad 0 < r < \infty,
\label{eq:SE-RO}
\end{equation}
where
\begin{equation}
\begin{split}
V_l(r) & = \frac{1}{4} \omega^2 r^2 + \frac{l(l+1)}{r^2}, \\
E^{(l)}_{\nu} & = \omega \left(2\nu + l + \frac{3}{2}\right), \qquad \nu=0, 1, 2, \ldots, \\
\psi^{(l)}_{\nu}(r) & \propto r^{l+1} e^{-\frac{1}{4}\omega r^2} L^{(l+\frac{1}{2})}_{\nu}\left(\frac{1}{2}
\omega r^2\right) \propto \eta_l(z) L^{(\alpha)}_{\nu}(z), \qquad \nu=0, 1, 2, \ldots,
\end{split} \label{eq:RO-pot}
\end{equation}
with
\begin{equation}
z = \frac{1}{2} \omega r^2, \qquad \alpha = l + \frac{1}{2}, \qquad \eta_l(z) = z^{\frac{1}{4}(2\alpha+1)}
e^{- \frac{1}{2}z}.
\end{equation}
\par
The changes of variable and of function \cite{cooper, haymaker, de, cq08b}
\begin{equation}
r = e^{- \frac{1}{2}x}, \qquad \psi^{(l)}_{\nu}(r) = e^{- \frac{1}{4}x} \varphi_{\nu, A_0}(x) \label{eq:PCT}
\end{equation}
transform Eq.\ (\ref{eq:SE-RO}) into the Schr\"odinger equation for a Morse potential
\begin{equation}
\left(- \frac{d^2}{dx^2} + V_{A_{\nu},B}(x)\right) \varphi_{\nu,A_0}(x) = \epsilon \varphi_{\nu,A_0}(x),
\label{eq:SE-Morse-map}
\end{equation}
for some fixed energy $\epsilon$ defined by
\begin{equation}
\epsilon = - A_0^2 = - \frac{1}{4}\left(l + \frac{1}{2}\right)^2. \label{eq:epsilon-map}
\end{equation}
Here $V_{A_{\nu},B}(x)$ is given by (\ref{eq:Morse-pot}) with $A$ replaced by the $\nu$-dependent parameter
\begin{equation}
A_{\nu} = A_0 + \nu, \qquad A_0 = \frac{1}{2}\left(l + \frac{1}{2}\right) = \sqrt{|\epsilon|},
\end{equation}
while
\begin{equation}
B = \frac{1}{4} \omega
\end{equation}
remains constant. The wavefunction $\varphi_{\nu,A_0}(x)$, corresponding to the energy $\epsilon$, can be obtained by applying (\ref{eq:PCT}) to the radial oscillator wavefunction $\psi^{(l)}_{\nu}(r)$ and is given by\footnote{It should be noted that transformation (\ref{eq:PCT}) results in functions $\varphi_{\nu,A_0}(x)$ that are normalized with respect to an unconventional scalar product \cite{cq08b}.}
\begin{equation}
\varphi_{\nu,A_0}(x) \propto \exp\bigl(- A_0 x - Be^{-x}\bigr) L^{(2A_0)}_{\nu} (z), \qquad z = 2Be^{-x}.
\end{equation}
\par
It is important to stress that the Hamiltonian for a single radial oscillator $V_l(r)$, with a given frequency $\omega$ and a given angular momentum quantum number $l$, is transformed into a hierarchy of Hamiltonians of the Morse family, corresponding to $V_{A_{\nu},B}(x)$ with $A_{\nu} = A_0 + \nu$, $\nu=0$, 1, 2,~\ldots, and constant $A_0$, $B$.\par
Although, for a given $V_{A_{\nu},B}(x)$, we get only a single eigenvalue $\epsilon$ and the corresponding eigenfunction $\varphi_{\nu,A_0}(x)$, it is possible to retrieve the whole Morse spectrum in the following way: on forgetting the map for a moment and focusing on a single Morse potential with given values of $A_{\nu} = \bar{A}$ and $B$, it is obvious that such a potential appears in a finite number of equations of type (\ref{eq:SE-Morse-map}) since, for $A_0$ in (\ref{eq:epsilon-map}), we may choose any of the values $\bar{A} - \bar{\nu}$ with $\bar{\nu} = 0$, 1, \ldots, $\bar{\nu}_{\rm max}$ ($\bar{A}-1 \le \bar{\nu}_{\rm max} < \bar{A}$). Hence, the resulting energy spectrum $- (\bar{A} - \bar{\nu})^2$, $\bar{\nu} = 0$, 1, \ldots, $\bar{\nu}_{\rm max}$, coincides with the standard one $\epsilon^{(\bar{A})}_{\bar{\nu}}$, as given in (\ref{eq:Morse-spectrum}).\par
\subsection{Going from extended radial oscillators to extended Morse potentials}
In Eq.\ (\ref{eq:SE-RO}), let us now replace the conventional radial oscillator potential by some rationally-extended one. There exist three different types of such extended potentials,
\begin{equation}
\begin{split}
V_{l,{\rm ext}}(r) & = V_l(r) + V_{l,{\rm rat}}(r), \\
V_{l,{\rm rat}}(r) & = - 2\omega \biggl\{\frac{\dot{g}^{(\alpha)}_m}{g^{(\alpha)}_m} + 2z
\biggl[\frac{\ddot{g}^{(\alpha)}_m}{g^{(\alpha)}_m}
- \biggl(\frac{\dot{g}^{(\alpha)}_m}{g^{(\alpha)}_m}\biggr)^2\biggr]\biggr\},
\end{split}
\end{equation}
where
\begin{equation}
g^{(\alpha)}_m(z) =
\begin{cases}
L^{(\alpha-1)}_m(-z), \quad m=1, 2, 3, \ldots, & \text{for type I}, \\
L^{(-\alpha-1)}_m(z), \quad m=1, 2, 3, \ldots, \quad \alpha > m-1, & \text{for type II}, \\
L^{(-\alpha-1)}_m(-z), \quad m=2, 4, 6, \ldots, \quad \alpha > m-1, & \text{for type III},
\end{cases}
\end{equation}
and
\begin{equation}
z = \frac{1}{2} \omega r^2, \qquad \alpha = l + \frac{1}{2}.
\end{equation}
Type I and type II extended potentials are related to EOP families \cite{cq08a, cq09, odake09, odake10a, odake10b, sasaki, grandati11a, ho11b}, while type III ones are not \cite{cq09, grandati11a}. For the former, the spectrum remains the same as for the conventional potential, in contrast with what happens for the latter. So Eq.\ (\ref{eq:SE-RO}) is changed into
\begin{equation}
\left(- \frac{d}{dr^2} + V_{l,{\rm ext}}(r)\right) \psi_{l,\nu}(r) = E_{l,\nu} \psi_{l,\nu}(r), \qquad 0 < r < \infty,
\label{eq:SE-RO-ext}
\end{equation}
where
\begin{equation}
E_{l,\nu} =
\begin{cases}
\omega \left(2\nu + l + \frac{3}{2}\right), \quad \nu = 0, 1, 2, \ldots, & \text{for type I or II}, \\
\omega \left(2\nu + l + \frac{7}{2}\right), \quad \nu = -m-1, 0, 1, 2, \ldots, & \text{for type III},
\end{cases} \label{eq:RO-ext-spectrum}
\end{equation}
and
\begin{equation}
\psi_{l,\nu}(r) \propto \frac{\eta_l(z)}{g^{(\alpha)}_m(z)} y^{(\alpha)}_n(z). \label{eq:RO-ext-wf}
\end{equation}
In (\ref{eq:RO-ext-wf}), $y^{(\alpha)}_n(z)$ denotes a $n$th-degree polynomial in $z$, where $n=m+\nu$, $\nu=0$, 1, 2,~\dots, for type I or II and $n=m+\nu+1$, $\nu=-m-1$, 0, 1, 2,~\dots, for type III.\par
On performing the PCT
\begin{equation}
r = e^{- \frac{1}{2}x}, \qquad \psi_{l,\nu}(r) = e^{- \frac{1}{4}x} \varphi_{\nu, A_0}(x)
\end{equation}
on Eq.\ (\ref{eq:SE-RO-ext}), we get the Schr\"odinger equation for some extended Morse potential,
\begin{equation}
\left(- \frac{d^2}{dx^2} + V_{A_{\nu},B,{\rm ext}}(x)\right) \varphi_{\nu,A_0}(x) = \epsilon
\varphi_{\nu,A_0}(x), \label{eq:SE-Morse-ext-map}
\end{equation}
where
\begin{equation}
\begin{split}
V_{A_{\nu},B,{\rm ext}}(x) & = V_{A_{\nu},B}(x) + V_{A_0,B,{\rm rat}}(x), \\
V_{A_0,B,{\rm rat}}(x) & = - z \biggl\{\frac{\dot{g}^{(\alpha)}_m}{g^{(\alpha)}_m} + 2z
\biggl[\frac{\ddot{g}^{(\alpha)}_m}{g^{(\alpha)}_m}
- \biggl(\frac{\dot{g}^{(\alpha)}_m}{g^{(\alpha)}_m}\biggr)^2\biggr]\biggr\},
\end{split} \label{eq:V-Morse-ext-map}
\end{equation}
and
\begin{equation}
\begin{split}
& A_{\nu} = A_0 + \nu, \qquad A_0 = \frac{\alpha}{2} \text{\ (for type I or II)}, \qquad A_0 = \frac{\alpha}{2}
+ 1 \text{\ (for type III)}, \\
& B = \frac{1}{4} \omega, \qquad z = 2Be^{-x}, \qquad \epsilon = - \frac{\alpha^2}{4}.
\end{split}
\end{equation}
The eigenfunction $\varphi_{\nu,A_0}(x)$ in (\ref{eq:SE-Morse-ext-map}) can be written as
\begin{equation}
\varphi_{\nu,A_0}(x) \propto
\begin{cases}
\frac{\exp\left(- A_0 x - Be^{-x}\right)}{g^{(2A_0)}_m(z)} y^{(2A_0)}_{m+\nu}(z) & \text{for type I or II},
\\[0.2cm]
\frac{\exp\left[- (A_0-1) x - Be^{-x}\right]}{g^{(2A_0-2)}_m(z)} y^{(2A_0-2)}_{m+\nu+1}(z) &
\text{for type III},
\end{cases} \label{eq:Morse-ext-wf-map}
\end{equation}
and corresponds to $\epsilon = - A_0^2$ or $\epsilon = - (A_0-1)^2$, respectively.\par
As for the conventional potentials, a single rationally-extended radial oscillator potential $V_{l,{\rm ext}}(r)$ is mapped onto a hierarchy of rationally-extended Morse potentials $V_{A_{\nu},B,{\rm ext}}(x)$ with $A_{\nu} = A_0 + \nu$ and $\nu$ running over the range given in (\ref{eq:RO-ext-spectrum}), while $A_0$ and $B$ remain fixed. However, since when $\nu$ is varied, only the core part $V_{A_{\nu},B}(x)$ of the potential is changed and not the whole potential, we cannot deduce other eigenvalues from $\epsilon$ by a reasoning similar to that carried out in Sec.\ 3.1. As a consequence, the potentials (\ref{eq:V-Morse-ext-map}) are QES ones with a single known eigenvalue $\epsilon$ and corresponding eigenfunction $\varphi_{\nu,A_0}(x)$ \cite{turbiner, shifman, ushveridze}. From the known zeros of the polynomials appearing in (\ref{eq:Morse-ext-wf-map}), it is clear that $\varphi_{\nu,A_0}(x)$ is a ground-state wavefunction if $\nu=0$ for type I or II and if $\nu=-m-1$ for type III. Furthermore, if $\nu > 0$ or $\nu \ge 0$, it is a $\nu$th- or $(\nu+1)$th-excited state, respectively.\par
The results obtained here can be illustrated by considering some special cases. On writing $V_{A_0,B,{\rm rat}}(x) - m$ in a form similar to (\ref{eq:V-rat}), we get
\begin{equation}
\begin{split}
N_1(x) & = - 3A_0, \\
N_2(x) & = 2A_0^2, \\
D(x) & = Be^{-x} + A_0,
\end{split}
\end{equation}
with $A_0 > 0$ for $m=1$ and type I or II,
\begin{equation}
\begin{split}
N_1(x) & = -2 (2A_0+1) \bigl(3Be^{-x} + A_0 - 2\bigr), \\
N_2(x) & = -4 (2A_0+1)^2 \bigl(2Be^{-x} + A_0\bigr), \\
D(x) & = 2B^2 e^{-2x} + 2 (2A_0+1) Be^{-x} + A_0 (2A_0+1),
\end{split}
\end{equation}
with $A_0 > 0$ for $m=2$ and type I,
\begin{equation}
\begin{split}
N_1(x) & = -2 (2A_0-1) \bigl(3Be^{-x} + A_0 + 2\bigr), \\
N_2(x) & = 4 (2A_0-1)^2 \bigl(2Be^{-x} + A_0\bigr), \\
D(x) & = 2B^2 e^{-2x} + 2 (2A_0-1) Be^{-x} + A_0 (2A_0-1),
\end{split}
\end{equation}
with $A_0 > \frac{1}{2}$ for $m=2$ and type II, and
\begin{equation}
\begin{split}
N_1(x) & = 2 (2A_0-3) \bigl(3Be^{-x} - A_0 - 1\bigr), \\
N_2(x) & = -4 (2A_0-3)^2 \bigl(2Be^{-x} - A_0 + 1\bigr), \\
D(x) & = 2B^2 e^{-2x} - 2 (2A_0-3) Be^{-x} + (A_0-1)(2A_0-3),
\end{split}
\end{equation}
with $A_0 > \frac{3}{2}$ for $m=2$ and type III.\par
{}For completeness' sake, in Appendix A we review the inverse transformation from the Morse potential to the radial oscillator and apply it to the extended Morse potentials, defined in (\ref{eq:partners}), to generate some QES extended radial oscillators.\par
\section{Conclusion}
In the present work, we have reconsidered the construction of ES rationally-extended Morse potentials, previously carried out in the framework of the Darboux-B\"acklund transformation \cite{grandati11c} or the prepotential method \cite{ho11c}. On using a first-order SUSYQM approach \cite{cooper} and building on the concept of algebraic deformations of SI potentials \cite{gomez04}, we have obtained the already known family of extended Morse potentials, corresponding to case {\it iii} of SUSYQM. We have called it type III family because it is similar to the L3 series of rationally-extended radial oscillators, for which the polynomial arising in the potential denominator is an $m$th-degree Laguerre polynomial with $m=2$, 4, 6,~\ldots\ and with both negative argument and negative variable.\par
More importantly, we have pointed out the existence of another family of extensions, corresponding to case {\it ii} of SUSYQM. The members $V^{(m)}_{A,B,{\rm ext}}(x)$ of this family, isospectral to the conventional Morse potential $V_{A+1,B}(x)$, are constructed in terms of $m$th-degree Laguerre polynomials with $m=1$, 2, 3,~\ldots\ and with negative argument, but positive variable, as the L2 series of rationally-extended radial oscillators. For this reason, we have called it type II family. As in the case of extended radial oscillators, the range of parameter values has to be restricted to get singularity-free potentials.\par
In contrast with what happens for the corresponding extended radial oscillators, however, type II extended Morse potentials are not SI. Nevertheless, they exhibit a kind of `enlarged' SI property in the sense that their partner $V^{(m-1)}_{A-1,B,{\rm ext}}(x)$, in case {\it i} of SUSYQM, is obtained by translating both the parameter $A$ and the polynomial degree $m$, and therefore belongs to the same family of extended potentials.\par
{}Finally, we have applied the PCT relating the radial oscillator and the Morse potential \cite{cooper, haymaker, de, cq08b} to the rationally-extended radial oscillators belonging to the three known families \cite{cq08a, cq09, odake09, odake10a, odake10b, sasaki, grandati11a, ho11b}. We have shown that to each of the latter potentials, we can associate an infinite hierarchy of extended Morse potentials $V_{A_{\nu},B,{\rm ext}}(x)$, $A_{\nu} = A_0 + \nu$, $\nu=0$, 1, 2,~\ldots, for each one of which one bound state is determined (as in the conventional case). The potentials of this hierarchy contain the same rational extension $V_{A_0,B,{\rm rat}}(x)$, but a different core potential $V_{A_{\nu},B}(x)$, $\nu=0$, 1, 2,~\ldots. As a consequence, the whole spectrum of $V_{A_{\nu},B,{\rm ext}}(x)$ cannot be determined in contrast with what happens in the conventional case. The constructed potentials are therefore QES with a single known bound state \cite{turbiner, shifman, ushveridze}.\par
Apart from the Morse potential, considered in this paper, and the three potentials mentioned in Sec.\ 1, the Coulomb potential has been rationally extended in two different ways by using the Darboux-B\"acklund transformation \cite{grandati11c}. Our discovery of type II Morse extensions draws a parallel between the Morse and Coulomb potentials, which previously seemed compromised.\par
In a future work, we hope to be able to carry out a similar study for those SI potentials associated with Jacobi polynomials and whose solvable rational extensions have not been constructed so far.\par
\section*{Appendix A. Going from Extended Morse Potentials to Extended Radial Oscillators}
\renewcommand{\theequation}{A.\arabic{equation}}
Let us start from the Schr\"odinger equation for the conventional Morse potential (\ref{eq:Morse-pot}) with eigenvalues $\epsilon^{(A)}_{\nu}$ and wavefunctions $\varphi^{(A)}_{\nu}(x)$, given in (\ref{eq:Morse-spectrum}) and (\ref{eq:Morse-wf}), respectively. Then the map
\begin{equation}
x = - 2 \log r, \qquad \varphi^{(A)}_{\nu}(x) = r^{-1/2} \psi_{\nu,l_0}(r) \label{eq:PCT-bis}
\end{equation}
gives rise to the Schr\"odinger equation for a conventional radial oscillator potential
\begin{equation}
\left(- \frac{d^2}{dr^2} + V_{l_{\nu}}(r)\right) \psi_{\nu,l_0}(r) = E \psi_{\nu,l_0}(r), \label{eq:SE-RO-map}
\end{equation}
for some fixed energy $E$ defined by
\begin{equation}
E = \omega \left(l_0 + \frac{3}{2}\right) = 4B (2A+1).
\end{equation}
The potential $V_{l_{\nu}}(r)$ is the same as in (\ref{eq:RO-pot}), but with $l$ replaced by the $\nu$-dependent parameter
\begin{equation}
l_{\nu} = l_0 - 2\nu, \qquad l_0 = 2A - \frac{1}{2} = \frac{E}{\omega} - \frac{3}{2},
\end{equation}
while the frequency
\begin{equation}
\omega = 4B
\end{equation}
remains fixed. The wavefunction $\psi_{\nu,l_0}(r)$, associated with the energy $E$, is obtained from (\ref{eq:RO-pot}) and (\ref{eq:PCT-bis}) in the form
\begin{equation}
\psi_{\nu,l_0}(r) \propto r^{l_{\nu}+1} e^{- \frac{1}{4} \omega r^2}
L^{(l_{\nu} + \frac{1}{2})}_{\nu}(z), \qquad z = \frac{1}{2} \omega r^2.
\end{equation}
\par
The Hamiltonian for a single Morse potential $V_{A,B}(x)$, with given parameters $A$, $B$, is mapped onto a hierarchy of radial oscillator Hamiltonians, corresponding to $V_{l_{\nu}}(r)$ with $l_{\nu} = l_0 - 2\nu$, $\nu=0$, 1, \ldots, $\nu_{\rm max}$, $\nu_{\rm max} = \bigl[\frac{l_0}{2}\bigr]$, and constant $\omega$, $l_0$.\footnote{Note that only some discrete $A$ values lead to $l_0 \in \mathbb{N}$.} Although, for a given $V_{l_{\nu}}(r)$, we get a single eigenvalue $E$ and eigenfunction $\psi_{\nu,l_0}(r)$, the whole spectrum of the radial oscillator potential can be obtained by observing that $V_{l_{\nu}}(r)$ with a given $l_{\nu} = \bar{l}$ may occur in an infinite number of equations of type (\ref{eq:SE-RO-map}), corresponding to $l_0 = \bar{l} + 2\bar{\nu}$, $\bar{\nu}=0$, 1, 2, \ldots, thus leading for $E$ to the well-known expression $\omega \bigl(2\bar{\nu} + \bar{l} + \frac{3}{2}\bigr)$, $\bar{\nu}=0$, 1, 2, \ldots.\par
Let us now replace the conventional Morse potential by some rationally-extended one, as defined in Eqs.\ (\ref{eq:partners}), (\ref{eq:type-II}), and (\ref{eq:type-III}). We therefore start from the Schr\"odinger equation
\begin{equation}
\left(- \frac{d^2}{dx^2} + V_{A,B,{\rm ext}}(x)\right) \varphi_{A,\nu}(x) = \epsilon_{A,\nu} \varphi_{A,\nu}(x),
\qquad - \infty < x < \infty, \label{eq:SE-Morse}
\end{equation}
where
\begin{equation}
\epsilon_{A,\nu} =
\begin{cases}
- (A+1-\nu)^2, \quad \nu=0, 1, \ldots, \nu_{\rm max}, & \\
\quad A \le \nu_{\rm max} < A+1, & \text{for type II}, \\
- (A-1-\nu)^2, \quad \nu=-m-1, 0, 1, \ldots, \nu_{\rm max}, & \\
\quad A-2 \le \nu_{\rm max} < A-1, & \text{for type III},
\end{cases} \label{eq:Morse-ext-spectrum}
\end{equation}
and
\begin{equation}
\varphi_{A,\nu}(x) \propto
\begin{cases}
\frac{\xi_{A+1}(z)}{g^{(A)}_m(z)} y^{(A)}_n(z), \quad n = m+\nu-1, & \\
\quad \nu=0, 1, \ldots, \nu_{\rm max}, & \text{for type II}, \\
\frac{\xi_{A-1}(z)}{g^{(A)}_m(z)} y^{(A)}_n(z), \quad n = m+\nu+1, & \\
\quad \nu=-m-1, 0, 1, \ldots, \nu_{\rm max}, & \text{for type III},
\end{cases}
\end{equation}
with $z$ and $\xi_A(z)$ as given in (\ref{eq:z}), while $y^{(A)}_n(z)$ is defined in (\ref{eq:def-y}) or (\ref{eq:def-y-bis}), respectively.\par
On applying to (\ref{eq:SE-Morse}) a transformation similar to (\ref{eq:PCT-bis}), we get the Schr\"odinger equation for some extended radial oscillator potential,
\begin{equation}
\left(- \frac{d^2}{dr^2} + V_{l_{\nu},{\rm ext}}(r)\right) \psi_{\nu,l_0}(r) = E \psi_{\nu,l_0}(r),
\label{eq:SE-RO-ext-map}
\end{equation}
where
\begin{equation}
\begin{split}
V_{l_{\nu},{\rm ext}}(r) & = V_{l_{\nu}}(r) + V_{l_0,{\rm rat}}(r), \\
V_{l_0,{\rm rat}}(r) & = - 4\omega\biggl\{\frac{\dot{g}^{(A)}_m}{g^{(A)}_m} + z \biggl[
\frac{\ddot{g}^{(A)}_m}{g^{(A)}_m} - \biggl(\frac{\dot{g}^{(A)}_m}{g^{(A)}_m}\biggr)^2\biggr]\biggr\},
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
l_{\nu} & = l_0 - 2\nu, \qquad l_0 = 2A + \frac{3}{2} \text{\ (for type II}), \qquad l_0 = 2A - \frac{5}{2}
\text{\ (for type III}), \\
\omega & = 4B, \qquad z = \frac{1}{2} \omega r^2, \qquad E = 2\omega \left(A + \frac{1}{2}\right).
\end{split}
\end{equation}
The eigenfunction $\psi_{\nu,l_0}(r)$ in (\ref{eq:SE-RO-ext-map}) can be written as
\begin{equation}
\begin{split}
\psi_{\nu,l_0}(r) \propto
\begin{cases}
\frac{r^{l_{\nu}+1} \exp(- \frac{1}{4} \omega r^2)}{g^{(\frac{1}{2}(l_0 - \frac{3}{2}))}_m(z)}
y^{(\frac{1}{2}(l_0 - \frac{3}{2}))}_{m+\nu-1}(z) & \text{for type II}, \\[0.4cm]
\frac{r^{l_{\nu}+1} \exp(- \frac{1}{4} \omega r^2)}{g^{(\frac{1}{2}(l_0 + \frac{5}{2}))}_m(z)}
y^{(\frac{1}{2}(l_0 + \frac{5}{2}))}_{m+\nu+1}(z) & \text{for type III},
\end{cases}
\end{split}
\end{equation}
and corresponds to $E = \omega \bigl(l_0 - \frac{1}{2}\bigr)$ or $E = \omega \bigl(l_0 + \frac{7}{2}\bigr)$, respectively.\par
Hence, a single rationally-extended Morse potential $V_{A,B,{\rm ext}}(x)$ is mapped onto a hierarchy of rationally-extended radial oscillators $V_{l_{\nu},{\rm ext}}(r)$ with $l_{\nu} = l_0 - 2\nu$ and $\nu$ running over the range given in (\ref{eq:Morse-ext-spectrum}), while $l_0$ and $\omega$ remain fixed. When $\nu$ is varied, only the core part $V_{l_{\nu}}(r)$ of $V_{l_{\nu},{\rm ext}}(r)$ is changed, so that we get a QES potential with a single known eigenvalue $E$ again.\par
\section*{Acknowledgments}
The author would like to thank Y.\ Grandati for several useful discussions.\par
\newpage
|
{
"timestamp": "2012-06-01T02:04:42",
"yymm": "1203",
"arxiv_id": "1203.1812",
"language": "en",
"url": "https://arxiv.org/abs/1203.1812"
}
|
\section{Stripe 82 and Statistical Variability}
A small section of sky observed by the Sloan Digital Sky Survey (SDSS) known as Stripe 82 (S82) was specifically targeted for multiple photometric observations. The stripe covers the area 22h 24m$ < $ RA$ < $ 04h 08m and $|$Dec$| <$ 1.27 degrees, approximating 290 square degrees on the sky. S82 has been observed over 50 times with revisit time ranging from 3 hours to 8 years. There are 9275 spectroscopically confirmed quasars in the Stripe 82 SDSS Data Release 7 \nocite{MIK10}({MacLeod} et~al. 2010). We sorted these into BAL and non-BAL quasars based on the classifications done in \nocite{GJB09}{Gibson} et~al. (2009) and \nocite{AHM11}{Allen} et~al. (2011) (DR5 and DR6, respectively).
Our working hypothesis is that BAL quasars will exhibit higher variability in their colour than non-BAL quasars as a result of the changes in absorption troughs. We use the time-series nature of the S82 photometry to extract a signal of higher variability among the BAL quasars. Nearly all BAL quasars exhibit absorption due to {C\,{\sc iv}} 1550, and so we build our preliminary study around this transition.
Using the possible range in blueshift of a BAL trough (-25 000 km s$^{-1}$ to -3000 km s$^{-1}$, as defined in \nocite{WM91}{Weymann} et~al. 1991) and the wavelength range of the SDSS filters, we sort the quasars of S82 into redshift bins based on where the {C\,{\sc iv}} trough will occur. For example, in the redshift range $1.73 < z < 2.51$, any {C\,{\sc iv}} broad absorption trough will land entirely within the SDSS $g$ filter. We can therefore use the colour $g-r$ to compare the colour variability of BAL and non-BAL quasars in that redshift range.
\section{Preliminary Results}
To quantify the overall variability of a given colour, we use the $\chi$ distribution $\chi_i = (c_i - c_0)/(\sigma_i)$, where $c_i$ is the $i$th colour in a series, $\sigma_i$ is the observational uncertainty on the colour, and $c_0$ is the mean colour. In S82, we observe a higher number of large positive and negative $\chi$ values in the BAL colours compared to non-BALs (significance: $10^{-15}$), as well as larger values of the reduced $\chi^2$ statistic. This indicates that, overall, BAL quasars exhibit larger deviation from their mean colours than non-BAL quasars.
As an example of a variable BAL quasar (selected for large reduced $\chi^2$), in Fig. \ref{example}a, we have plotted the time-series colours $g-r$ and $r-i$ (as a comparison), along with estimated colour measurements from spectra taken at the given MJD, of SDSSJ 213138.07-002537.8, a BAL quasar at $z=1.837\pm0.002$. It is clear that large variations have occured in $g-r$, where the {C\,{\sc iv}} trough is, but not in $r-i$. In Fig. \ref{example}b, we have plotted the three spectra indicated by the square symbols. There are clear changes in the {C\,{\sc iv}} trough which contributes to the colour changes observed. The continuum has also changed by a significant amount.
The S82 photometry provides an excellent testbed to compare colour variability in different types of quasars. We have found BAL colour to be more variable than non-BAL colour, a feature attributed to the variable nature of absorption troughs. Such a result hints towards a photometric monitoring program that would trigger spectroscopic followup and coverage when a colour begins to vary. These data also provide a basis from which to track the colours of non-BAL quasars that may turn into BALs.
\begin{figure}
\epsscale{1.0}
\plottwo{rogerson_fig1a.eps}{rogerson_fig1b.eps}
\caption{SDSSJ 213138.07-002537.8 photometry (left) and spectra (right).}
\label{example}
\end{figure}
\acknowledgements JAR and PBH are supported by NSERC. We acknowledge support by NSF grant AST-0807500 to the University of Washington, and NSF grant AST-0551161 to LSST for design and development activity.
|
{
"timestamp": "2012-03-12T01:02:37",
"yymm": "1203",
"arxiv_id": "1203.2162",
"language": "en",
"url": "https://arxiv.org/abs/1203.2162"
}
|
\section{Introduction}\label{sec1}
Continuous-time diffusion processes defined by stochastic
differential equations [\citet{r25}, \citet{r29}, \citet{r31}] are the
basic stochastic modeling tools in the
modern financial theory and applications. Diffusion models are
commonly employed to describe the price dynamics of a financial
asset or a portfolio of assets. An eminent application is in
deriving the price of a derivative contract on an asset or a group
of assets.
The celebrated Black--Scholes--Merton option pricing formula [\citet{r13},
\citet{r26}]
was obtained by assuming that the
underlying asset followed a geometric Brownian motion such that the
log price process of the underlying asset followed an
Ornstein--Uhlenbeck diffusion process.
The widely used \citet{r37} and \citet{r15} pricing
formulas for the zero coupon bond
were developed based on two specific mean-reverting diffusion
processes with a~constant or the square root [\citet{r21}]
diffusion functions, respectively.
Other pricing formulas have also
been developed for assets defined by other processes; see \citet{r9} and \citet{r17}. In the
implementations of the aforementioned pricing formula,
the parameters of the diffusion processes which describe
the underlying assets dynamics have to be estimated based on empirical
observations. \citet{r35} gave a comprehensive survey on the financial
applications of continuous-time stochastic models which
were largely the diffusion processes. \citet{r18} provided an
overview on nonparametric estimation for diffusion processes. Other
related works include \citet{r11}, \citet{r38}, \citet{r20}, \citet{r19}, \citet{r27} and
\citet{r8}.
There are several challenges to be faced when estimating parameters of
diffusion processes. One challenge is that despite being
continuous-time models, the
processes are only observed at discrete time points rather than
observed continuously over time. The discrete observations prevent
the use of the relatively straightforward likelihood expressions
[\citet{r30}] available for continuously observed diffusion
processes. Another challenge is that despite the fact that the
diffusion processes
are Markovian, their transition densities from one time point to the
next do not have finite analytic expressions, except for only a few
specific processes. This means that the efficient maximum likelihood
estimation (MLE) cannot be readily implemented for most of these
processes.
In ground-breaking works, A{\"{\i}}t-Sahalia (\citeyear{r1}, \citeyear{r2}) established
series expansions to approximate the
transition densities of univariate diffusion processes. Similar
expansions have been proposed for multivariate processes in
\citet{r3}.
These density approximations, as advocated by A{\"{\i}}t-Sahalia, are then
employed to form approximate likelihood functions, which are maximized
to obtain the approximate maximum likelihood estimators (AMLEs).
A{\"{\i}}t-Sahalia (\citeyear{r2}, \citeyear{r3})
demonstrated that the approximate likelihood converges to the true
likelihood as the number of terms in the series expansions goes to
infinity. He also provided some results on the consistency of the AMLEs.
Numerical evaluations of the transition density approximations as
conducted in \citet{r1}, Stramer and Yan (\citeyear{r33}, \citeyear{r34}) and
others, have shown good performance in the numerical approximation of
the underlying transition densities.
The approach has opened a very accessible route for obtaining parameter
estimators for diffusion processes, and for estimating other quantities
which are functions of the transition density, as commonly encountered
in finance.
Indeed, A{\"{\i}}t-Sahalia and Kimmel (\citeyear{r5}, \citeyear{r6})
demonstrated two such applications in stochastic volatility models and
the affine term structure models, respectively.
\citet{r36} provided some results on the AMLE based on the
one-term expansion for the mean-reverting processes. They revealed
that there was an extra leading order bias term in the AMLE due to
the density approximation.
Although the above-mentioned results on the transition density
approximation and the AMLE had been provided, there are some key
questions that remain to be addressed. One is on the consistency of the
AMLE. While A{\"{\i}}t-Sahalia (\citeyear{r2}, \citeyear{r3})
contained some results on consistency,
there is {more} to be explored. There are two key ingredients in
A\"{\i}t-Sahalia's density approximation. One is $J$, the number of
terms used
in the
approximation, and the other is $\delta$, the length
of the sampling interval between successive observations. In this
paper, we study explicitly the roles played by $J$ and $\delta$ on
the consistency of the AMLE, and quantify their roles on the
convergence rate. Another question is under what conditions on $J$
and~$\delta$, does the AMLE have the same asymptotic distribution as the
full MLE. Here, we consider two regimes: (i) $\delta$ is fixed, and
$J \to\infty$; (ii) $J$ is fixed, but $\delta\to0$, representing
two views of asymptotics.
In the case of $\delta\to0$, it is found that $J \geq2$ is
necessary to ensure the AMLE having the same asymptotic normality as
the MLE. Like the transition density, the Fisher information
matrix, the quantity that defines the efficiency of the full MLE, is
unknown analytically; even the underlying transition density is
known. We show in this paper that an approximation to the Fisher
information matrix can be obtained based on the one-term density
approximation.
The paper is organized as follows. In Section \ref{sec2}, we outline the
transition density approximations of A{\"{\i}}t-Sahalia (\citeyear{r1}, \citeyear{r2}).
Some preliminary analysis is needed for studying the AMLE is presented
in Section \ref{sec3}.
Section \ref{sec4} establishes the consistency and convergence rates of the
AMLE. Asymptotic normality of the AMLE and its equivalence to the
full MLE are addressed in Section \ref{sec5}. Section \ref{sec6} discusses the
approximation for the Fisher information matrix.
Simulation results are reported in Section \ref{sec7}. Technical conditions
and details of proofs are relegated to the \hyperref[app]{Appendix}.
\section{Transition density approximation}\label{sec2}
Consider a univariate diffusion process $(X_t)_{t \ge0}$ defined by
a stochastic differential equation
\begin{equation} \label{eq11}
dX_t=\mu(X_t;\theta)\,dt+\sigma(X_t;\theta)\,dB_t,
\end{equation}
where $\mu$ and $\sigma$ are, respectively, the drift and diffusion
functions and $B_t$ is the standard Brownian motion. Both the drift and
diffusion functions are known except for an unknown parameter
vector $\theta$ taking values in a set
$\Theta\subseteq\mathbb{R}^d$.
Given a sampling interval $\delta> 0$, let $f_X(x | x_0, \delta;
\theta)$ be the transition density of $X_{t +\delta}$ given
$X_t = x_0$ for $(x_0,x) \in{\mathcal{X}\times\mathcal{X}}$, where
${\mathcal{X}}$ is the domain of $X_t$. Despite the parametric forms of
the drift and the diffusion functions that are available
in~(\ref{eq11}), a closed-form expression for $f_X(x | x_0, \delta;
\theta)$ is not generally available for most of the processes. In
most cases, the density is only known to satisfy the Kolmogorov
backward and forward partial differential equations. In
ground-breaking works, A{\"{\i}}t-Sahalia (\citeyear{r1}, \citeyear{r2}) proposed asymptotic
expansions to approximate the transition density.
The approach of A{\"{\i}}t-Sahalia is the following. He first
transformed $X_t$ to a diffusion process with unit diffusion
function by
\begin{equation}\label{eqYt}
Y_t=\gamma(X_t;\theta):=
\int^{X_t}\frac{du}{\sigma(u;\theta)},
\end{equation}
which satisfies
$
dY_t=\mu_Y(Y_t;\theta)\,dt+dB_t$, where
\[
\mu_Y(y;\theta)=\frac{\mu(\gamma^{-1}(y;\theta);\theta)}{\sigma(\gamma
^{-1}(y;\theta);\theta)}-\frac{1}{2}\,\frac{\partial\sigma}{\partial
x}(\gamma^{-1}(y;\theta);\theta).
\]
Let $f_Y(y|y_0,\delta;\theta)$ be the transition density of
$Y_{t + \delta}$ given $Y_t = y_0$. The two density functions are
related according to
\begin{equation}\label{eq2}
f_X(x_t|x_{t-1},\delta;\theta)=\sigma^{-1}(x_t;\theta)\cdot
f_Y(\gamma(x_t;\theta) |\gamma(x_{t-1};\theta),
\delta;\theta).
\end{equation}
To ensure convergence of the expansions, A\"{\i}t-Sahalia standardized
$Y_{t+\delta}$ by $Z_{t+\delta} = \delta^{-1/2} (Y_{t+\delta} -
y_0)$. Let $f_Z(z| y_0, \delta; \theta)$ denote the conditional
density of $Z_{t+\delta}$ given $Z_{t} = 0$, which is related to
$f_Y$ by
\[
f_Z(z| y_0, \delta; \theta) = \delta^{1/2} f_Y(\delta^{1/2} z + y_0 |
y_0,\delta; \theta).
\]
Let $\{ H_j(z)\}_{j=1}^{\infty}$ be the Hermite polynomials
\[
H_j(z) = \phi^{-1}(z) \,{d^j \phi(z) \over d z^j},
\]
which are orthogonal with respect to the standard normal density
$\phi$, namely $\int H_j(z) H_k(z) \phi(x) \,dx = 0$ if $j \ne k$. A
formal Hermite orthogonal series expansion to the density $f_Z(z|
y_0, \delta; \theta)$ is
\begin{equation}\label{eqHermite}
f_Z^{H}(z | y_0, \delta; \theta) =
\phi(z) \sum_{j=0}^{\infty} \eta_j(y_0, \delta; \theta) H_j(z),
\end{equation}
where the coefficients
\begin{eqnarray*}
\eta_j(y_0, \delta; \theta) &=& (j !)^{-1} \int H_j(z) f_Z(z| y_0,
\delta; \theta) \,dz \\
&=&(j !)^{-1} \mathbb{E}\bigl[ H_j\bigl(\delta^{-1/2} (Y_{t+\delta}-y_0)\bigr) |
Y_t=y_0;\theta\bigr].
\end{eqnarray*}
The last conditional
expectation has no analytic expression in general, although it may
be simulated using the method proposed in \citet{r10}.
A\"{\i}t-Sahalia proposed
Taylor expansions for this conditional expectation with respect to
the sampling interval $\delta$ based on the infinitesimal generator
of $Y_t$. For twice continuously differentiable function $g$, the
infinitesimal generator of $Y_t$ is
\begin{equation}\label{eqgenerator}
\mathcal{A}_{\theta} g(y) = \mu_Y(y; \theta) \,{\partial g \over
\partial y} + \frac{1}{2} \,{\partial^2 g \over\partial
y^2}.
\end{equation}
A $K$-term Taylor series expansion to $\mathbb{E} [
H_j(\delta^{-1/2} (Y_{t+\delta}-y_0)) | Y_t=y_0;\theta]$ is
\begin{eqnarray} \label{eqTaylor}
&&\mathbb{E}\bigl[ H_j\bigl(\delta^{-1/2} (Y_{t+\delta}-y_0)\bigr) | Y_t=y_0;\theta
\bigr]\nonumber\\
&&\qquad= \sum_{k=0}^K \mathcal{A}^k_{\theta} H_j\bigl(\delta^{-1/2}
(y-y_0)\bigr)
\big|_{y=y_0} \frac{\delta^k}{k !}\\
&&\qquad\quad{}+ \mathbb{E}\bigl[\mathcal{A}_{\theta}^{k+1}
H_j\bigl(\delta^{-1/2} (Y_{t+\delta^*}-y_0)\bigr) \big|
Y_t=y_0;\theta\bigr] \frac{\delta^{k+1}}{(k+1) !}. \nonumber
\end{eqnarray}
Substituting (\ref{eqTaylor}) to the orthogonal
expansion (\ref{eqHermite}) followed by gathering terms according
to the powers of $\delta$, a $J$-term expansion to the
transition density $f_Y(y, \delta| y_0; \theta)$ is
\[
f^{(J)}_Y(y |y_0, \delta;\theta) = \delta^{-1/2} \phi\biggl({y-y_0
\over\delta^{1/2}}\biggr) \exp\biggl(\int_{y_0}^y \mu_Y(u; \theta)
\,du \biggr) \sum_{j=0}^J c_{j}(y|y_0; \theta)\frac{ \delta^j}{j !} ,
\]
where $c_0(y|y_0 ; \theta) \equiv1$ and for $j \geq1$,
\begin{eqnarray*}
c_j(y|y_0;\theta)&=&j(y-y_0)^{-j}\\
&&{}\times\int^y_{y_0}(w-y_0)^{j-1}\\
&&\hspace*{26pt}{}\times\biggl\{\lambda_Y(w;\theta)c_{j-1}(w|y_0;\theta)+\frac{1}{2}\,\frac
{\partial^2c_{j-1}(w|y_0;\theta)}{\partial
w^2}\biggr\}\,dw.
\end{eqnarray*}
Here
$\lambda_Y(y;\theta)=-\{\mu^2_Y(y;\theta)+\partial\mu_Y(y;\theta
)/\partial
y\}/2$.
Transforming back from $y$ to $x$ via (\ref{eqYt}) and
(\ref{eq2}), the $J$-term expansion to $f_X(x|x_0, \delta; \theta)$
is
\begin{eqnarray*}
&&
f^{(J)}_X(x|x_0,\delta; \theta) \\
&&\qquad= \sigma^{-1}(x; \theta) \delta^{-1/2}
\phi\biggl({\gamma(x; \theta)- \gamma(x_0; \theta)\over\delta^{1/2}}\biggr) \\
&&\qquad\quad{} \times \exp\biggl\{\int_{x_0}^x
\frac{\mu_Y(\gamma(u;\theta);\theta)} {\sigma(u;\theta)} \,du \biggr\}
\sum_{j=0}^J c_{j}(\gamma(x; \theta) |\gamma(x_0; \theta); \theta)
\frac{\delta^j}{j!} .
\end{eqnarray*}
Although it employs the Hermite polynomials and has the Gaussian
density as the leading term as an Edgeworth expansion does, the
transition density expansion is not an Edgeworth expansion. This
is because the latter is for density functions of statistics
admitting the central limit theorem, which differs from the current
context of expanding the transition density.
\citet{r2} demonstrated that
as $J \to\infty$,
\begin{equation} \label{eqconv1}
f_X^{(J)}(x | x_0, \delta; \theta) \to f_X(x
| x_0, \delta; \theta)
\end{equation}
uniformly with respect
to $\theta\in\Theta$ and $x_0$ over compact subsets of
${\mathcal{X}}$. The convergence is also uniform with respect to $x$ over
subsets of ${\mathcal{X}}$ depending on the property of $\sigma(x;
\theta)$.
Define
\begin{eqnarray*}
A_1(x|x_0, \delta;\theta)&=&-\log
\{\sigma(x;\theta)\}-\frac{1}{2\delta}
\{\gamma(x;\theta)-\gamma(x_0;\theta)\}^2,\\
A_2(x|x_0, \delta;\theta) &=&
\int_{x_0}^{x}\frac{\mu_Y(\gamma(u;\theta);\theta)}
{\sigma(u;\theta)}\,du
\end{eqnarray*}
and
\[
A_3(x|x_0, \delta;\theta) =\log
\Biggl\{\sum_{j=0}^{J}c_j(\gamma(x; \theta)| \gamma(x_0;
\theta);\theta)\delta^j/j!\Biggr\}.
\]
If ${\sum_{j=0}^\infty}
|c_j(y|y_0,\delta;\theta)|\delta^j/j!<\infty$ on $\mathcal
{Y}\times\mathcal{Y}$ with probability one, where $\mathcal{Y}$ is
the domain of $Y_t$, we can define $\tilde{A}_3(x|x_0,
\delta;\theta)=\log
\{\sum_{j=0}^{\infty}c_j(y|y_0;\theta)\delta^j/j!\}$. Then the
result in (\ref{eqconv1}) implies that
\begin{eqnarray}\label{eqlog-exp}
&&\log f_X(x|x_0, \delta;\theta) \nonumber\\
&&\qquad=-\log\sqrt{2\pi\delta}+A_1(x|x_0,
\delta;\theta)+A_2(x|x_0, \delta;\theta)\\
&&\qquad\quad{}+\tilde{A}_3(x|x_0,
\delta;\theta).\nonumber
\end{eqnarray}
Expression (\ref{eqlog-exp}) is the starting point for our analysis.
Given a set of discrete observations $\{X_{t \delta}\}_{t=1}^n$ with
equal sampling length~$\delta$ of the diffusion process $(X_t)_{t
\ge0}$, to simplify notations, we write $X_t$ for~$X_{t \delta}$
and hide $\delta$ in the expressions for the transition density
$f_X$ and its approximations. At the same time, we use $f$ and
$f^{(J)}$ to express $f_X$ and $f_X^{(J)}$, respectively. Based on
the $J$-term expansion to the true transition density, the $J$-term
approximate log-likelihood function given in \citet{r2} is
\begin{eqnarray*}
{\ell_{n,\delta}^{(J)}}(\theta
&=&- n \log\sqrt{2\pi\delta}+ \sum_{t=1}^n
A_1(X_t|X_{t-1},\delta;\theta)\\
&&{} + \sum_{t=1}^n A_2(X_t|X_{t-1},\delta;\theta)+ \sum_{t=1}^n
A_3(X_t|X_{t-1},\delta;\theta).
\end{eqnarray*}
Let ${\hat{\theta}_{n,\delta}^{(J)}} = \arg\max_{\theta\in\Theta}
{\ell_{n,\delta}^{(J)}}(\theta)$ be the approximate MLE (AMLE) and
${\hat{\theta}_{n,\delta}}$ be the true MLE that maximizes the full
likelihood
\[
{\ell_{n,\delta}}(\theta)=\sum_{t=1}^n \log f(X_{t} |
X_{t-1},\delta; \theta).
\]
To keep the notation simple, we write
$\hat{\theta}^{(J)}_n = \hat{\theta}_{n,\delta}^{(J)}$ and
$\hat{\theta}_n = \hat{\theta}_{n,\delta}$ by suppressing $\delta$
in subscripts.
\section{Preliminaries}\label{sec3}
Under regular circumstances as assumed by condi-\break tion~(A.2)(ii)~in~the
\hyperref[app]{Appendix}, the full MLE $\hat{\theta}_{n}$ and the $J$-term
approximate MLE $\hat{\theta}^{(J)}_{n}$ satisfy their respective
likelihood score equations so that
\begin{equation}\label{eq03}
\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;{\hat{\theta}_{n}})=\sum_{t=1}^n\nabla_\theta\log
f^{(J)}\bigl(X_t|X_{t-1},\delta;{\hat{\theta}^
{(J)}_{n}}\bigr)=0.\hspace*{-22pt}
\end{equation}
Subtracting $\sum_{t=1}^n \nabla_\theta\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0)$ from both sides of
(\ref{eq03}),
\begin{eqnarray}\label{eq04}
&&\sum_{t=1}^n\nabla_\theta\log
f^{(J)}\bigl(X_t|X_{t-1},\delta;{\hat{\theta}^
{(J)}_{n}}\bigr)-\sum_{t=1}^n\nabla_\theta\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0)\nonumber\\
&&\qquad=\sum_{t=1}^n\nabla_\theta[\tilde{A}_3(X_t|X_{t-1},\delta;\theta
_0)-A_3(X_t|X_{t-1},\delta;\theta_0)]
\\
&&\qquad\quad{}+\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;{\hat{\theta}_{n}})-\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1};\theta_0).
\nonumber
\end{eqnarray}
Carrying out Taylor expansions on both sides of (\ref{eq04}), we
can get
\begin{eqnarray}\label{eqtaylor-eq}\quad
&&\frac{1}{n} \sum_{t=1}^n\nabla^2_{\theta\theta}\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0)\cdot\bigl({\hat{\theta
}^{(J)}_{n}}-\theta_0\bigr) \nonumber\\
&&\quad{}+\frac{1}{2} \bigl[E_d\otimes\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)'\bigr]\cdot
\frac{1}{n}\sum_{t=1}^n\nabla^3_{\theta\theta\theta}\log
f^{(J)}(X_t|X_{t-1},\delta;\tilde{\theta})\cdot\bigl({\hat{\theta
}_{n}^{(J)}}-\theta_0\bigr) \nonumber\\
&&\qquad=\frac{1}{n} \sum_{t=1}^{n}\nabla_\theta[\tilde
{A}_3(X_t|X_{t-1},\delta;\theta_0)-A_3(X_t|X_{t-1},\delta;\theta_0)]\\
&&\qquad\quad{}+\frac{1}{n}\sum_{t=1}^{n}\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0) \cdot({\hat{\theta}_{n}}-\theta_0) \nonumber\\
&&\qquad\quad{}+\frac{1}{2} [E_d\otimes
({\hat{\theta}_{n}}-\theta_0)']\cdot
\frac{1}{n}\sum_{t=1}^n\nabla^3_{\theta\theta\theta}\log
f(X_t|X_{t-1},\delta;\bar{\theta})\cdot({\hat{\theta}_{n}}-\theta_0),
\nonumber
\end{eqnarray}
where $E_d$ is the $d\times d$ identity matrix, $\tilde{\theta}$ is
on the joint line between ${\hat{\theta}_{n}^{(J)}}$ and~$\theta_0$
and $\bar{\theta}$ is on the joint line between $\hat{\theta}_{n}$
and $\theta_0$. Here we define
\[
\nabla^3_{\theta\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta):=\pmatrix{
\partial^3\log
f(X_t|X_{t-1},\delta;\theta)/\partial\theta\,\partial\theta'\,\partial\theta
_1 \cr
\vdots\cr
\partial^3\log
f(X_t|X_{t-1},\delta;\theta)/\partial\theta\,\partial\theta'\,\partial\theta
_d },
\]
which is a $d^2\times d$ matrix, and
$\nabla^3_{\theta\theta\theta}\log
f^{(J)}(X_t|X_{t-1},\delta;\theta)$ is similarly defined.
Furthermore, let
\begin{eqnarray*}
F_n(\theta_0,J,\delta) &=& n^{-1}
\sum_{t=1}^{n}\nabla^2_{\theta\theta}
[\tilde{A}_3(X_t|X_{t-1},\delta;\theta_0)-A_3(X_t|X_{t-1},\delta
;\theta_0)], \\
U_n(\theta_0,J,\delta) &=& n^{-1}
\sum_{t=1}^{n}\nabla_{\theta}
[\tilde{A}_3(X_t|X_{t-1},\delta;\theta_0)-A_3(X_t|X_{t-1},\delta
;\theta_0)]
\end{eqnarray*}
and
\[
N_n(\theta_0,J,\delta)= n^{-1}
\sum_{t=1}^{n}\nabla^2_{\theta\theta}\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0).
\]
Then (\ref{eqtaylor-eq}) can be written as
\begin{eqnarray}\label{eqkeyeq}
&&N_n(\theta_0,J,\delta)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)+ \Delta_{n
1}\bigl({\hat{\theta}^{(J)}_{n}}, \theta_0\bigr) \nonumber\\
&&\qquad=U_n(\theta_0,J,\delta) + [N_n(\theta_0,J,\delta) +F_n(\theta
_0,J,\delta) ]({\hat{\theta}_{n}}-\theta_0)\\
&&\qquad\quad{}+ \Delta_{n 2}({\hat{\theta
}_{n}}, \theta_0),
\nonumber
\end{eqnarray}
where
$\Delta_{n 1}({\hat{\theta}^{(J)}_{n}}, \theta_0)$
and $\Delta_{n 2}({\hat{\theta}_{n}}, \theta_0)$ denote the
{remainder terms whose explicit expressions can be obtained by
matching (\ref{eqtaylor-eq}) with (\ref{eqkeyeq}).
Expansion (\ref{eqkeyeq}) is the starting point in our studies
for the consistency and asymptotic distribution of the AMLE.
Indeed, the asymptotic properties of the AMLE will be evaluated under
two regimes regarding $J$ and $\delta$.
The first one is that
\begin{equation}\label{eqregime1}
\mbox{$\delta$ is fixed} \qquad\mbox{but $J \to\infty$,}
\end{equation}
which is the situation considered in \citet{r2}.
The second regime allows that
\begin{equation} \label{eqregime2}
\mbox{$J$ is fixed},\qquad \mbox{$\delta\rightarrow0$}\qquad\mbox{but $n\delta\rightarrow\infty
$,}
\end{equation}
which is more tuned with an implementation of the density
approximation with a fixed number of terms.
We will first present some results which are valid for any fixed $J$
and $\delta$.
Let $\|A\|_2=\{\rho(A'A)\}^{1/2}$ be the spectral norm of a matrix
$A$, where $\rho(A'A)$ denotes the largest eigen-value of $A' A$.
The following proposition describes properties for the quantities that
appear in (\ref{eqkeyeq}).
\begin{pn}\label{prop1}
Under conditions \textup{(A.1), (A.3), (A.4), (A.6), (A.7)} given in the \hyperref[app]{Appendix},
there exists a positive constant $\Delta$ such that for any positive
integer $J$ and $\delta\in(0,\Delta)$:\vspace*{8pt}
\textup{(a)} $\mathbb{E}\{F_n(\theta_0,J,\delta)\}$,
$\mathbb{E}\{U_n(\theta_0,J,\delta)\}$ and
$\mathbb{E}\{N_n(\theta_0,J,\delta)\}$ exist;\vspace*{1pt}
\textup{(b)} $ \Delta_{n 1}({\hat{\theta}^{(J)}_{n}},
\theta_0)=O_p\{\|{\hat{\theta}_{n}^{(J)}}-\theta_0\|_2^2\}$ and
$\Delta_{n 2}({\hat{\theta}_{n}},
\theta_0)=O_p\{\|{\hat{\theta}_{n}}-\theta_0\|_2^2\}$ as $n \to
\infty$.
\end{pn}
Let $I(\delta) = -\mathbb{E}\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0)$ be the Fisher information matrix,
which we assume is invertible in condition (A.5). It is expected
that the expected value of $N_n(\theta_0, J, \delta)$, denoted by
$N(\theta_0, J, \delta)$, will converge to $-I(\delta)$, as $J \to
\infty$ for each fixed $\delta$ or $J$ being fixed but
$\delta\rightarrow0$. The following proposition bounds the
difference between $N(\theta_0, J, \delta)$ and $-I(\delta)$ for
each fixed $J$ and $\delta$.
\begin{pn}\label{prop2}
Under conditions \textup{(A.1), (A.4), (A.6), (A.7)} given in the
\hyperref[app]{Appendix}, there exist two positive constants $\bar{\Delta}$ and $C$,
that are not dependent on $J$ and $\delta$,
such that for any positive integer $J$ and
$\delta\in(0,\bar{\Delta})$,
\[
\|N(\theta_0,J,\delta)+I(\delta)\|_2\leq C\delta^{J+1}.
\]
\end{pn}
As $I(\delta)$ is invertible for each fixed
$\delta
>0$, $N_n(\theta_0, J, \delta)$ will be invertible with probability
approaching one as $J \to\infty$ for a fixed $\delta$. However,
{if} $\delta\to0$, the limit of the Fisher information
$I(0):=\lim_{\delta\to0}I(\delta)$, as well as~$N(\theta_0, J,
0)$, may be singular. This is the case for some Ornstein--Uhlenbeck
processes as shown in Section~\ref{sec6}. The following proposition provides
another account on $N(\theta_0,J,\delta)$ and its deviation from
$-I(\delta)$, as well as the convergence of
$N^{-1}(\theta_0,J,\delta)U(\theta_0,J,\delta)$, where
$U(\theta_0,J,\delta)$ denotes the expected value of~$U_n(\theta_0,J,\delta)$ for each pair of fixed $J$ and $\delta$.
\begin{pn}\label{prop3}
Under conditions \textup{(A.1), (A.3)--(A.7)} given in the \hyperref[app]{Ap-}
\hyperref[app]{pendix}, there exist
two constants $C_1, C_2$, that are not dependent on $J$ and
$\delta$, and a constant $\underline{\Delta}>0$ such that for any
positive integer $J$ and $\delta\in(0,\underline{\Delta})$,
\[
\|N^{-1}(\theta_0,J,\delta)I(\delta)+E_d\|_2\leq
C_1\delta^J \quad\mbox{and}\quad\|N^{-1}(\theta_0,J,\delta)U(\theta_0,J,\delta
)\|_2\leq
C_2\delta^J.
\]
\end{pn}
The next proposition describes the convergence rate for the
difference between the first derivatives of the full log-likelihood
and the approximate log-likelihood.
\begin{pn}\label{prop4}
Under\vspace*{1pt} conditions \textup{(A.1), (A.4), (A.6), (A.7)} given in the \hyperref[app]{Appendix}, there
exist two finite\vspace*{1pt} positive constants $\tilde{\Delta}$ and $C$,
not dependent on $J$ and~$\delta$, such that for any $J$,
$\delta\in(0,\tilde{\Delta}]$ and $n$,
\[
\mathbb{E}\Bigl\{\sup_{\theta\in\Theta}\bigl\|n^{-1}\cdot\nabla_\theta\bigl[\ell
_{n,\delta}(\theta)-\ell_{n,\delta}^{(J)}(\theta)\bigr]\bigr\|_2\Bigr\}\leq
C\delta^{J+1}.
\]
\end{pn}
The following proposition together with Proposition \ref{prop4}
is needed to establish the consistency of the AMLE.
\begin{pn}\label{prop5}
Under conditions \textup{(A.1), (A.3), (A.4), (A.6), (A.7)} given in the \hyperref[app]{Appendix},
there exists a constant $\dot{\Delta}>0$ such that
\[
\sup_{\theta\in\Theta}\Biggl\|\frac{1}{n}\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;\theta)-\mathbb{E}\nabla_\theta\log
f(X_t|X_{t-1},\delta;\theta)\Biggr\|_2\stackrel{p}{\rightarrow}0
\]
for \textup{(i)} $\delta\in(0,\dot{\Delta}]$ being fixed,
$n\rightarrow\infty$, or \textup{(ii)} $n\rightarrow\infty$,
$\delta\rightarrow0$ but $n\delta\rightarrow\infty$.
\end{pn}
As the full MLE $\hat{\theta}_{n}$ is a key bridge for the AMLE, we
report in the following proposition
the asymptotic normality of the
MLE which covers both cases of fixed $\delta$ and diminishing
$\delta$ case.
\begin{pn}\label{prop6}
Under conditions \textup{(A.1)--(A.7)}
given in the \hyperref[app]{Appendix},
\[
\sqrt{n}I^{1/2}(\delta)(\hat{\theta}_n-\theta_0)\stackrel{d}{\rightarrow
}N(0,E_d)\qquad\mbox{as }n\delta^3\rightarrow\infty,
\]
where $E_d$ is $d\times d$ identity matrix.
\end{pn}
The requirement of $n\delta^3 \to\infty$ in the above proposition
is to cover the case where $I(0) = \lim_{\delta\to0} I(\delta)$ is
singular, as spelled out in the proof given in the \hyperref[app]{Appendix}. If such
case is ruled out, for instance, via the so-call Jacobsen condition
[\citet{r24}, \citet{r32}], the more standard $n \delta
\to\infty$ is sufficient; see also \citet{r23} for related
results.
\section{Consistency}\label{sec4}
We consider in this section the consistency of
the\break AMLE~$\hat{\theta}_{n}^{(J)}$ and establish its convergence rate under
the two asymptotic regimes given in (\ref{eqregime1}) and
(\ref{eqregime2}), respectively. The two asymptotic regimes were
also considered in A\"{\i}t-Sahalia (\citeyear{r2}, \citeyear{r3}). For a fixed
sampling interval~$\delta$, \citet{r2} proved that
there existed a sequence $J_n \to\infty$ such that
${\hat{\theta}_{n}^{(J_n)} - \hat{\theta}_{n}} \stackrel{p} \to0$
under $P_{\theta_0}$ as $n \to\infty$, where $P_{\theta_0}$ is the
underlying probability measure.
Based on the
consistency of $\hat{\theta}_{n}$, we know that the consistency of
$\hat{\theta}_{n}^{(J_n)}$ is hold.
For a fixed $J$, \citet{r3} proved that there existed a
sequence $\{\delta_n\}$ vanishing to zero such that
$\sqrt{n}I^{1/2}(\delta_n)({\hat{\theta}_{n,\delta_n}^{(J)}}-\theta_0)=O_p(1)$.
In this paper, we will give more explicit guidelines on how to
select the afore-mentioned sequences $J_n$ and $\delta_n$ so that
the AMLE is consistent. Our study here begins with (\ref{eq03}),
which together with Propositions \ref{prop4} and \ref{prop5} lead to the following {result}
on the consistency of the AMLE under the two asymptotic
regimes, respectively.
\begin{tm}\label{theo1}
Under conditions \textup{(A.1)--(A.4), (A.6), (A.7)} given in the \hyperref[app]{Appendix},
$\hat{\theta}_{n}^{(J)}-\theta_0\stackrel{p}{\rightarrow}0$ under
either: \textup{(i)}
$\delta\in(0,\tilde{\Delta}\wedge\dot{\Delta}]$ being fixed,
$J\rightarrow\infty$ and $n\rightarrow\infty$, or \textup{(ii)} $J$ being
fixed, $n\rightarrow\infty$, $\delta\rightarrow0$ but
$n\delta\rightarrow\infty$.\vadjust{\goodbreak}
\end{tm}
By Proposition \ref{prop2} and condition (A.5),
multiply $N^{-1}(\theta_0,J,\delta)$ on both sides of~(\ref{eqkeyeq}),
we have
\begin{eqnarray}\label{eq41}
&&{\hat{\theta}_{n}^{(J)}}-\theta_0\nonumber\\
&&\qquad=N^{-1}U_n+N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta
_0)-N^{-1}(N_n-N)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\\
&&\qquad\quad{}-N^{-1}\Delta_{n1}\bigl({\hat{\theta}_{n}^{(J)}},\theta_0\bigr)+N^{-1}\Delta
_{n2}({\hat{\theta}_{n}},\theta_0).\nonumber
\end{eqnarray}
From this together with Proposition \ref{prop4} and Theorem \ref{theo1}, we can
establish the convergence rate of the AMLE.
\begin{tm}\label{theo2}
Under conditions \textup{(A.1)--(A.7)} given
in the \hyperref[app]{Appendix},
\[
{\hat{\theta}_{n}^{(J)}}-\theta_0=\cases{
O_p\{\delta^{J+1}+(n\delta)^{-1/2}\}, &\quad if $\delta\in
(0,\tilde{\Delta}\wedge\dot{\Delta}]$ is fixed and $J \to\infty$;
\vspace*{2pt}\cr
O_p\{\delta^{J}+(n\delta)^{-1/2}\}, &\quad if $J$ is fixed, $\delta
\rightarrow0$ but $n\delta^3\rightarrow\infty$.}
\]
\end{tm}
The above theorem reveals the impacts of the sampling interval
$\delta$ and the number of terms $J$ used in the density
approximation on the convergence rate. In particular, the rate of
AMLE has an extra $\delta^{J+1}$ or $\delta^J$ term in addition to
the standard rate $(n \delta)^{-1/2}$ of the full MLE. This extra
term is the result of the density approximation, and its particular
form suggests\vspace*{1pt} that the sampling interval $\delta$ has to be less
than 1 in order to make the AMLE~$\hat{\theta}_{n}^{(J)}$ converge
to~$\theta_0$. It is apparent\vspace*{1pt} that the higher the $J$ is, the less
impact the extra term has on the AMLE $\hat{\theta}_{n}^{(J)}$.
\section{Asymptotic distribution}\label{sec5}
In this section, we consider the asymptotic distribution of the AMLE
$\hat{\theta}_{n}^{(J)}$. Our investigations are organized according
to two asymptotic regimes: (i) $\delta$ fixed, $J\rightarrow\infty$
and (ii) $J$ fixed, $\delta\rightarrow0$ but
$n\delta\rightarrow\infty$.
\subsection{\texorpdfstring{Fixed $\delta$, $J \to\infty$}{Fixed delta, J to infinity}}\label{sec51}
This is a simple case to treat. Under this setting, we note from
Proposition \ref{prop2} and condition (A.5) that
$N^{-1}(\theta_0,J,\delta)=O(1)$ uniformly for any $J$. Utilizing
the result in Theorem \ref{theo2}, expansion (\ref{eq41}) becomes
\[
{\hat{\theta}_{n}^{(J)}}-\theta_0
=N^{-1}U_n+({\hat{\theta}_{n}}-\theta_0)+O_p(n^{-1/2}\delta
^{J-1/2}+n^{-1}\delta^{-1} + \delta^{2J+2}).
\]
Hence, note that $U_n=O_p(\delta^{J+1})$,
\begin{eqnarray*}
&&\sqrt{n}I^{1/2}(\delta)
\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\\
&&\qquad=\sqrt{n}I^{1/2}(\delta) ({\hat{\theta}_{n}}-\theta_0)+O_p(\delta
^{J-1/2}+n^{-1/2}\delta^{-1} + n^{1/2}\delta^{J+1}).
\end{eqnarray*}
If $n\delta^{2J+2}\rightarrow0$, then
\[
\sqrt{n}I^{1/2}(\delta)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\stackrel
{d}{\rightarrow}N(0,E_d).
\]
Therefore, the AMLE has the same asymptotic distribution as the full
MLE~$\hat{\theta}_{n}$.
This is attained by requesting $n\delta^{2J+2}\rightarrow0$ in addition
to $J \to\infty$.
If $n \delta^{2J+2} \to c > 0$, the AMLE is still
asymptotically normal but would have an inflated variance due to the
contribution from the first term involving $U_n$. Apart from this,
the asymptotic mean will no longer be zero. Hence, it is much desirable
to have $ n \delta^{2J+2}
\to0$. The latter condition prescribes a rule on the selection
of the $J=J_n(\delta)$. By choosing an $\varepsilon> 0$ so that
$\delta^{2J+2}=n^{-1-\varepsilon}$ for each pair of $n$ and $\delta$,
then
\[
J=J_n(\delta)= \frac{-1-\varepsilon}{2\log\delta}\log
n-1>\frac{-1}{2\log\delta}\log n-1.
\]
The integer truncation of the above lower bound plus one can be used
as a~reference value for the number of terms used in the density
approximation for each given pair of $(n,\delta)$.
\begin{table}
\tablewidth=240pt
\caption{The least approximation term selection to guarantee the
AMLE has the same asymptotic distribution as the full MLE for
special sampling interval $\delta$ and sample size $n$}
\label{table1}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lcccc@{}}
\hline
$\bolds{\delta}$ & $\bolds{n=500}$ & $\bolds{n=1\mbox{\textbf{,}}000}$
& $\bolds{n=2\mbox{\textbf{,}}000}$ & $\bolds{n=4\mbox{\textbf{,}}000}$\\
\hline
$1/252$ & \hphantom{0}1 & \hphantom{0}1 & \hphantom{0}1 &\hphantom{0}1\\
$1/52$ & \hphantom{0}1 & \hphantom{0}1 & \hphantom{0}1&\hphantom{0}1\\
$1/12$ & \hphantom{0}1 & \hphantom{0}1 & \hphantom{0}1&\hphantom{0}1\\
$1/4$ & \hphantom{0}2 & \hphantom{0}2 & \hphantom{0}2 &\hphantom{0}2\\
$1/2$ & \hphantom{0}4 & \hphantom{0}4 & \hphantom{0}5 &\hphantom{0}5\\
$3/4$ & 10 & 12 & 13 &14\\
\hline
\end{tabular*}
\end{table}
Table \ref{table1} reports such reference values of $J$ assigned by the above
formula for a set of $(n,\delta)$ combinations commonly encountered
in empirical studies. It shows that for monthly frequency or less
($\delta\leq1/12$), one term approximation is adequate, and for
$\delta=1/4$, $J=2$ is needed. However, there is a dramatic increase
in $J$ as the sampling length is larger than $1/4$: demanding at
least four terms for $\delta=1/2$ (half yearly) or at least ten
terms for $\delta= 3/4$. The number of terms also increases for
these higher $\delta$ values as $n$ increases, although the rate of
this increase is much slower than that as $\delta$ is increased. The
latter may be understood that for a given~$\delta$, as $n$
increases, the chance of having extreme values in the tails of the
transition distribution increases. As the density approximation
is less accurate in the tails than in the main body of the
distribution, there is a need for having more terms in the density
approximation.
\subsection{\texorpdfstring{$J$ fixed, $\delta\rightarrow0$ but $n\delta\rightarrow\infty$}
{J fixed, delta rightarrow 0 but n delta rightarrow infinity}}
\label{sec52}
Our starting point is the expansion~(\ref{eq41}). As
$N_n-N=O_p\{(n\delta)^{-1/2}\}$, $N^{-1} (N_n - N) = o_p(1)$ if $n
\delta^3 \to\infty$, which is also required in the asymptotic
normality of the full MLE
as outlined in Proposition \ref{prop6}.
{We will show in the following that $n \delta^3 \to\infty$ is also necessary
to ensure AMLE sharing the same asymptotic distribution
as the full MLE. It is understood that in order for
$\hat{\theta}_{n}^{(J)}$ having the same asymptotic distribution as
$\hat{\theta}_{n}$,
it is required that
\[
N^{-1} U_n,
N^{-1}\Delta_{n1}\bigl({\hat{\theta}_{n}^{(J)}},\theta_0\bigr)\mbox{ and }
N^{-1}\Delta_{n2}({\hat{\theta}_{n}},\theta_0)\qquad\mbox{are all }
o_p\bigl\{\bigl\|{\hat{\theta}_{n}^{(J)}}-\theta_0\bigr\|_2\bigr\}.
\]
We will demonstrate in the following that the above requirements can
be attained by $n \delta^3 \to\infty$ and $J \geq2$. Hence,
under these circumstances, $\hat{\theta}_{n}^{(J)}$ has the same
asymptotic distribution as $\hat{\theta}_{n}$. Later we will
demonstrate that this equivalence in the asymptotic distribution is
quite unlikely for $J=1$.
Our analysis needs to expand (\ref{eq04}) to the quadratic terms.
To this end, let us define
\[
M_n(\theta_0,J,\delta)=n^{-1}\sum_{t=1}^n\nabla^3_{\theta\theta\theta
}\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0)
\]
\mbox{and}
\[
T_n(\theta_0,J,\delta)=n^{-1}\sum_{t=1}^n\nabla^3_{\theta\theta\theta
}\log
f(X_t|X_{t-1},\delta;\theta_0).
\]
By further expanding to quadratic terms, (\ref{eq41}) can be
written as
\begin{eqnarray}\label{eqtaylor2}\quad
&&{\hat{\theta}_{n}^{(J)}}-\theta_0\nonumber\\
&&\qquad=N^{-1}U_n+N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta
_0)-N^{-1}(N_n-N)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\nonumber\\
&&\qquad\quad{}-\tfrac{1}{2} N^{-1}\bigl[E_d\otimes\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)'\bigr]
M_n\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\\
&&\qquad\quad{}+\tfrac{1}{2} N^{-1}[E_d\otimes({\hat{\theta}_{n}}-\theta_0)'] T_n({\hat
{\theta}_{n}}-\theta_0)\nonumber\\
&&\qquad\quad{}-N^{-1}\tilde{\Delta}_{n1}\bigl({\hat{\theta}^{(J)}_{n}},
\theta_0\bigr)+N^{-1}\tilde{\Delta}_{n2}({\hat{\theta}_{n}},
\theta_0),
\nonumber
\end{eqnarray}
where $\tilde{\Delta}_{n 1}({\hat{\theta}^{(J)}_{n}}, \theta_0)$ and
$\tilde{\Delta}_{n 2}({\hat{\theta}_{n}}, \theta_0)$ are remainder
terms. Using the same method in the proof of Proposition \ref{prop1}, it can
be shown that $\tilde{\Delta}_{n 1}({\hat{\theta}^{(J)}_{n}},
\theta_0)=O_p\{\|{\hat{\theta}^{(J)}_{n}}- \theta_0\|_2^3\}$ and
$\tilde{\Delta}_{n 2}({\hat{\theta}_{n}},
\theta_0)=O_p\{\|{\hat{\theta}_{n}}-\theta_0\|_2^3\}$.
In order to make $\hat{\theta}_{n}^{(J)}$ have the same asymptotic
distribution as $\hat{\theta}_{n}$, the two quadratic terms on the
right-hand side of (\ref{eqtaylor2}) have to be smaller order of
$\hat{\theta}_{n}^{(J)}-\theta_0$ and $\hat{\theta}_{n} - \theta_0$,
respectively, namely
\[
N^{-1}\bigl[E_d\otimes\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)'\bigr]
M_n\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)=o_p\bigl\{\bigl\|\hat{\theta
}_{n}^{(J)}-\theta_0\bigr\|_2\bigr\}
\]
or equivalently
\begin{equation}\label{eqcond1}
N^{-1}\bigl[E_d\otimes
\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)'\bigr]=o_p(1)
\end{equation}
and
\[
N^{-1}[E_d\otimes({\hat{\theta}_{n}}-\theta_0)']
T_n({\hat{\theta}_{n}}-\theta_0)=o_p\{\|\hat{\theta}_{n}-\theta_0\|_2\}\vadjust{\goodbreak}
\]
or equivalently
\begin{equation}\label{eqcond2}
n\delta^3\rightarrow\infty
\end{equation}
since ${\hat{\theta}_{n}} - \theta_0 = O_p\{(n\delta)^{-1/2}\}$ and
$N^{-1} = O(\delta^{-1})$.
As ${\hat{\theta}_{n}^{(J)}}-\theta_0 = O_p\{\delta^J +
(n\delta)^{-1/2}\}$, (\ref{eqcond1}) requires that $\delta^{J-1} +
n^{-1/2} \delta^{-3/2} \to0$. Hence, in order to make
$\hat{\theta}_{n}^{(J)}$ have the same asymptotic distribution as
$\hat{\theta}_{n}$, it is necessary to have
\begin{equation}\label{eqcond3}
J \geq2\quad\mbox{and}\quad n\delta^3 \to\infty.
\end{equation}
Now we consider the case of $J=1$. To ensure the remainder terms
$N^{-1}\times\Delta_{n1}({\hat{\theta}_{n}^{(J)}},\theta_0)$ and
$N^{-1}\Delta_{n2}({\hat{\theta}_{n},\theta_0})$ are negligible, by
a similar argument applied above for the case of $J \geq2$, it
is also necessary to assume $n \delta^3 \to\infty$.
From Theorem \ref{theo2}, ${\hat{\theta}_{n}^{(1)}}-\theta_0=O_p\{\delta+
(n\delta)^{-1/2}\}$. To gain insight on the situation, we need to
find out the order of magnitude of the quadratic term in~(\ref{eqtaylor2}), namely the order of magnitude of
\[
S_n=N^{-1}\bigl[E_d\otimes\bigl({\hat{\theta}_{n}^{(1)}}-\theta_0\bigr)'\bigr]M_n\bigl({\hat
{\theta}_{n}^{(1)}}-\theta_0\bigr)
-N^{-1}[E_d\otimes({\hat{\theta}_{n}}-\theta_0)']T_n({\hat{\theta
}_{n}}-\theta_0).
\]
With this notation, (\ref{eqtaylor2}) can be written as
\begin{eqnarray}\label{eqtaylor3}
{\hat{\theta}_{n}^{(J)}}-\theta_0&=&N^{-1}U_n+N^{-1}(N_n+F_n)({\hat
{\theta}_{n}}-\theta_0) -\tfrac{1}{2} S_n \nonumber\\[-8pt]\\[-8pt]
&&{}+ o_p\{(n\delta)^{-1/2}\} + O_p(\delta^2).
\nonumber
\end{eqnarray}
Define an operator between two vectors $A$ and $B$,
\[
A*B=[E_d\otimes A']M_nB+[E_d\otimes B'] M_n A.
\]
By repeated substitutions, it can be shown that
\begin{eqnarray*}
S_n&=
\tfrac{1}{2} N^{-1}[(N^{-1}U_n)*(N^{-1}U_n)]+\tfrac{1}{2} N^{-1}\bigl[\bigl(\tfrac{1}{2}
S_n\bigr)*\bigl(\tfrac{1}{2} S_n\bigr)\bigr]\\
&&{}-N^{-1}\bigl[(N^{-1}U_n)*\bigl(\tfrac{1}{2} S_n\bigr)\bigr]+o_p(\delta).
\end{eqnarray*}
As $U_n= O_p(\delta^2)$ for $J=1$ and $N^{-1} = O(\delta^{-1})$,
it can be deduced from the above equation that $S_n=O_p(\delta)$.
Hence, for $J=1$ if we require $n\delta^3\rightarrow\infty$,
the quadratic term $S_n$ will contribute to the leading order of
$\hat{\theta}_{n}^{(1)}-\theta_0$. If we do not require
$n\delta^3\rightarrow\infty$, then the sum of remainder terms,
$N^{-1}\tilde{\Delta}_{n1}({\hat{\theta}^{(J)}_{n}},
\theta_0)+N^{-1}\tilde{\Delta}_{n2}({\hat{\theta}_{n}}, \theta_0)$
will not be controlled. Hence, if $J=1$, it is very likely that the
asymptotic distribution of $\hat{\theta}_{n}^{(J)}$ will differ from
that of $\hat{\theta}_{n}$ unless $U_n =0$ with probability one. In
the rare case of $U_n = 0$, it is possible for
$\hat{\theta}_{n}^{(1)}$ and~$\hat{\theta}_{n}$ to share the same
limiting distribution.
Therefore, in order to guarantee that $\hat{\theta}_{n}^{(J)}$ has
the same asymptotic distribution as $\hat{\theta}_{n}$ under
$\delta\rightarrow0$, we need to use the AMLE based on at least
two-term expansions, while satisfying
$n\delta^3\rightarrow\infty$, which we will assume in the rest of
this section.\vadjust{\goodbreak
Note that
${\hat{\theta}_{n}^{(J)}}-\theta_0=O_p\{\delta^{J}+(n\delta)^{-1/2}\}$.
Then,
\begin{eqnarray*}
{\hat{\theta}_{n}^{(J)}}-\theta_0
&=&N^{-1}U_n+({\hat{\theta}_{n}}-\theta_0)\\
&&{}+O_p(n^{-1/2}\delta^{J-3/2})+N^{-1}\cdot O_p(\delta^{2J}+n^{-1}\delta
^{-1}).
\end{eqnarray*}
Furthermore,
\begin{eqnarray*}
&&\sqrt{n}I^{1/2}(\delta)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)\\
&&\qquad=\sqrt{n}I^{-1/2}(\delta)I(\delta) N^{-1}U_n+\sqrt{n}I^{1/2}(\delta)
({\hat{\theta}_{n}}-\theta_0)+O_p(\delta^{J-3/2})\\
&&\qquad\quad{}+\sqrt{n}I^{-1/2}(\delta)I(\delta)N^{-1}\cdot O_p(\delta
^{2J}+n^{-1}\delta^{-1})\\
&&\qquad=\sqrt{n}I^{1/2}(\delta)({\hat{\theta}_{n}}-\theta_0)+O_p(\delta
^{J-3/2}+n^{-1/2}\delta^{-3/2}+n^{1/2}\delta^{J+1/2}).
\end{eqnarray*}
Hence, for any $J\geq2$ such that $n\delta^3\rightarrow\infty$
and $n\delta^{2J+1}\rightarrow0$,
\[
\sqrt{n}I^{1/2}(\delta)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)
\stackrel{d}{\rightarrow}N(0,E_d).
\]
This result shows that, when $\delta$ vanishes to zero, in order to
guarantee the AMLE has the same asymptotic distribution as full MLE,
we need
to pick the approximation order $J\geq2$, while maintaining
$n\delta^3\rightarrow\infty$ and $n\delta^{2J+1}\rightarrow0$.
The following theorem summarizes the asymptotic normality under both
asymptotic regimes.
\begin{tm}\label{theo3}
Under conditions \textup{(A.1)--(A.7)} given in the \hyperref[app]{Appendix},
\[
\sqrt{n}I^{1/2}(\delta)\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)
\stackrel{d}{\rightarrow}N(0,E_d)
\]
for: \textup{(i)} $\delta\in(0,\tilde{\Delta}\wedge\dot{\Delta}]$ being
fixed, $n\rightarrow\infty$, $J\rightarrow\infty$ but
$n\delta^{2J+2}\rightarrow0$ or \textup{(ii)}~$J\geq2$ being fixed,
$n\rightarrow\infty$, $\delta\rightarrow0$ but
$n\delta^3\rightarrow\infty$ and $n\delta^{2J+1}\rightarrow0$.
\end{tm}
\subsection{Asymptotic bias and variance}\label{sec53}
The remainder of this section is devoted to the consideration of the
asymptotic bias and variance of the AMLE under the two asymptotic
regimes. Given our analysis in the early part of this section, our
consideration will be focused on the situations where the asymptotic
normality of the AMLE can be assumed, namely under: (i) $\delta$
being fixed, $J \to\infty$, $n\to\infty$ but $n\delta^{2J+2}\to0$
or (ii) $J \geq2 $ being fixed, $\delta\to0$, $n \delta^3 \to
\infty$ but $n\delta^{2J+1}\to0$.
In the case of $\delta$ being fixed and $J\rightarrow\infty$,
from (\ref{eqtaylor2}) and provided $n\delta^{2J+2}\rightarrow0$,
we have
\begin{eqnarray*}
{\hat{\theta}_{n}^{(J)}}-\theta_0
&=&N^{-1}U_n+N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta
_0)-N^{-1}(N_n-N)N^{-1}U_n\\
&&{}-N^{-1}(N_n-N)N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta_0)\\
&&{}-\tfrac{1}{2} N^{-1}\{E_d\otimes[N^{-1}U_n+N^{-1}(N_n+F_n)({\hat{\theta
}_{n}}-\theta_0)]'\}\\
&&\hspace*{11.2pt}{}\times M_n[N^{-1}U_n+N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta_0)]\\
&&{}+\tfrac{1}{2} N^{-1}[E_d\otimes({\hat{\theta}_{n}}-\theta_0)']T_n({\hat{\theta
}_{n}}-\theta_0)+O_p(n^{-3/2})\\
&=&N^{-1}U_n+[E_d-N^{-1}(N_n-N)]N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta
_0)\\
&&{}+O_p(n^{-1/2}\delta^{J+1})+O_p(n^{-3/2}).
\end{eqnarray*}
Then, the leading order bias of $\hat{\theta}_{n}^{(J)}$ is
\begin{equation}\label{eqasybias}
B(\theta_0,J,\delta)\,{=}\,N^{-1}U\,{+}\,\mathbb{E}\{[E_d\,{-}\,N^{-1}(N_n\,{-}\,N)]N^{-1}(N_n\,{+}\,F_n)({\hat{\theta}_{n}}\,{-}\,\theta_0)\},\hspace*{-35pt}
\end{equation}
and the leading order variance is
\begin{equation}\label{eqasyvar}
V(\theta_0,J,\delta)=N^{-1}I(\delta)
\operatorname{Var}({\hat{\theta}_{n}})I(\delta) N^{-1}.
\end{equation}
In the case of {$J \geq2$ being fixed, $\delta\rightarrow0$
and $n \delta^3 \to\infty$ but $n \delta^{2J+1} \to0$, it can be
shown by a similar argument to that for the fixed $\delta$ case
above, the asymptotic bias and variance have the same forms as
(\ref{eqasybias}) and (\ref{eqasyvar}), respectively. Both
(\ref{eqasybias}) and (\ref{eqasyvar}) will be used to calibrate
with the simulated bias and variance in the simulation study in
Section \ref{sec7}. For $J=1$ and $\delta\to0$, there are difficulties in
obtaining an expression for the bias {of} the AMLE in general
due to the same dilemma in controlling the reminder terms and the
quadratic term $S_n$ as outlined in Section~\ref{sec52}.
\section{Approximating Fisher information matrix}\label{sec6}
We demonstrate in this section that the approximation of the
transition density provides a way to approximate the Fisher
information matrix. Fisher information matrix~$I(\delta)$ is a key
quantity associated with inference based on the full MLE. It defines
the asymptotic efficiency and {convergence rate}.
From Proposition~\ref{prop2}, a~natural candidate to approximate
$I(\delta)$ is $-N( \theta_0, J,\delta)$ based on the $J$-term
expansion.
To simplify our expedition, our consideration here is
focused under the following diffusion process:
\begin{equation}\label{eq51}
dX_t=\mu(X_t;\eta)\,dt+\sigma(X_t;\xi)\,dB_t,
\end{equation}
where $\eta= (\eta_1, \ldots, \eta_{d_1})'$ and $\xi=(\xi_1, \ldots,
\xi_{d_2})'$ are distinct drift and diffusion parameters,
respectively. The whole parameter $\theta=(\eta',\xi')'$.
Here, we provide an explicit expression $N(\theta_0, 1, \delta)$ based
on the one-term density expansion. Expressions
for higher $J$ values may be made via more extensive derivations.
Let $\mu_i$, $\mu_{ij}$ and so on denote partial derivatives with
respect to $\eta_i$, $\eta_{i}$ and~$\eta_j$, respectively; and
$\sigma_i$ and $\sigma_{x, j}$ and so on denote partial derivatives
with respect to~$\xi_i$, and $x$ and $\xi_j$, respectively. By the
one-term ($J=1$) transition density approximation, derivations given
in \citet{r14} show that
\[
\mathbb{E}\biggl(\frac{\partial^2\log
f^{(1)}}{\partial\eta_i\,\partial\eta_j}\biggr)=:\delta\cdot
N_{11}^{(1)} +O(\delta^2),\qquad \mathbb{E}\biggl(\frac{\partial^2\log
f^{(1)}}{\partial\eta_i\,\partial\xi_j}\biggr)=:\delta\cdot
N_{12}^{(1)} +O(\delta^2)
\]
and
\[
\mathbb{E}\biggl(\frac{\partial^2\log
f^{(1)}}{\partial\xi_i\,\partial\xi_j}\biggr)
=:-2\mathbb{E}(\sigma^{-2}\sigma_i\sigma_j) + \delta\cdot
N^{(1)}_{22} + O(\delta^2),
\]
where
\begin{eqnarray*}
N_{11}^{(1)} &=& \mathbb{E}\bigl\{-\sigma^{-2}\mu_i\mu_j-\mu\sigma^{-2}\mu
_{ij}+\sigma^{-1}\mu_{ij}\sigma_x-\tfrac{1}{2}\mu_{xij}\bigr\},\\
N_{12}^{(1)} &=& \mathbb{E}\{2\mu\sigma^{-3}\mu_i\sigma_j-\sigma^{-2}\mu
_i\sigma_x\sigma_j+\sigma^{-1}\mu_i\sigma_{xj}\},\\
N^{(1)}_{22} &=& \mathbb{E}\bigl\{-6\mu^2\sigma^{-4}\sigma_i\sigma_j+16\mu
\sigma^{-3}\sigma_x\sigma_i\sigma_j+2\mu^2\sigma^{-3}\sigma_{ij}-3\sigma
^{-2}\mu_x\sigma_i\sigma_j\\
&&\hphantom{\mathbb{E}\bigl\{}{}
-\tfrac{19}{2}\sigma^{-2}\sigma_x^2\sigma_i\sigma_j-\tfrac{9}{2}\mu
\sigma^{-2}\sigma_x\sigma_{ij}-5 \mu\sigma^{-2}\sigma_{xi}\sigma_j -
5\mu\sigma^{-2}\sigma_{xj}\sigma_i\\
&&\hphantom{\mathbb{E}\bigl\{}{}
+ \sigma^{-1}\mu_x\sigma_{ij}+4\sigma^{-1}\sigma_{xx}\sigma_i\sigma
_j+\tfrac{11}{2}\sigma^{-1}\sigma_x\sigma_{xi}\sigma_j+\tfrac{11}{
2}\sigma^{-1}\sigma_x\sigma_{xj}\sigma_i\\
&&\hphantom{\mathbb{E}\bigl\{}{}
+ \tfrac{3}{2}\sigma^{-1}\sigma_x^2\sigma_{ij}+\tfrac{5}{2}\mu\sigma
^{-1}\sigma_{xij}-\tfrac{3}{4}\sigma_{xx}\sigma_{ij}-\tfrac{5}{
2}\sigma_{xi}\sigma_{xj}-\tfrac{3}{2}\sigma_x\sigma_{xij}\\
&&\hphantom{\mathbb{E}\bigl\{}\hspace*{153pt}{}
-\sigma_{xxi}\sigma_j-\sigma_{xxj}\sigma_i+\tfrac{3}{4}\sigma\sigma
_{xxij}\bigr\}.
\end{eqnarray*}
Thus
\begin{equation}\label{eqfisherapprox}
N(\theta_0,1,\delta) = \pmatrix{
\delta\cdot N_{11}^{(1)} & \delta\cdot N_{12}^{(1)}
\vspace*{2pt}\cr
\delta\cdot{N_{12}^{(1)}}' & -2\cdot\mathbb{E}(\sigma^{-2}\sigma
_i\sigma_j) + \delta\cdot N^{(1)}_{22}}
+ O(\delta^2).
\end{equation}
We learn from Proposition \ref{prop2} that $-N(\theta_0,1,\delta)$ provides a
leading order approximation to $I(\delta)$ with a reminder term at
the order of $\delta^2$. Equation~(\ref{eqfisherapprox}) confirms
that as $\delta\to0$, given the asymptotic normality of the full
MLE $\hat{\theta}_{n}$ as conveyed by Proposition \ref{prop6}, that the
convergence rate of the full MLE for the drift parameters $\eta$ is
$(n\delta)^{-1/2}$ whereas that for the diffusion parameters~$\xi$
is $n^{-1/2}$, faster than the drift parameter estimator. Our study
confirms the results of \citet{r23}, \citet{r32} and \citet{r36}.
In the rest of the section, we will derive the Fisher information
matrix approximation for two specific diffusion processes. Both are
widely employed in {modeling} of the interest rate dynamics.
\subsection{Vasicek model}\label{sec61}
Consider the Vasicek (\citeyear{r37}) model,
\begin{equation}
dX_t=\kappa(\alpha-X_t)\,dt+\sigma \,dB_t,
\end{equation}
which is also the Ornstein--Uhlenbeck process. The conditional
distribution of $X_t$ given $X_{t-1}$ is
\[
X_t|X_{t-1}\sim
N\bigl\{X_{t-1}e^{-\kappa\delta}+\alpha(1-e^{-\kappa\delta}),\tfrac{1}{2}
\sigma^2\kappa^{-1}(1-e^{-2\kappa\delta})\bigr\}
\]
and the stationary distribution of $\{X_t\}$ is $
N(\alpha,\frac{\sigma^2}{2\kappa}).\label{eqvasicekstationarydistribution}
$
It yields that the information matrix of
$\theta=(\kappa,\alpha,\sigma)'$ is $I(\delta)=(
I_{ij})_{3 \times3}$
where
\begin{eqnarray*}
I_{11}&=&\frac{1}{2\kappa^2}+\frac{\delta[\kappa\delta+\kappa\delta
e^{2\kappa\delta}-2e^{2\kappa\delta}+2]}{\kappa(e^{2\kappa\delta
}-1)^2}=\frac{\delta}{2\kappa}+O(\delta^2),\qquad I_{12}=I_{21}=0,
\\
I_{13}&=&I_{31}=\frac{(1+2\kappa\delta)-e^{2\kappa\delta}}{\sigma\kappa
(e^{2\kappa\delta}-1)}=-\frac{\delta}{\sigma}+O(\delta^2),
\\
I_{22}&=&\frac{2\kappa(e^{\kappa\delta}-1)^2}{\sigma^2(e^{2\kappa\delta
}-1)}=\frac{\kappa^2\delta}{\sigma^2}+O(\delta^2),\qquad
I_{23}=I_{32}=0\quad\mbox{and}\quad I_{33}=\frac{2}{\sigma^2}.
\end{eqnarray*}
These mean that
\begin{equation} \label{eqfishervas}
I(\delta) = \pmatrix{
\delta\cdot(2\kappa)^{-1} & 0 & -\delta\cdot\sigma^{-1} \cr
0 & \delta\cdot\kappa^2\sigma^{-2} & 0 \cr
-\delta\cdot\sigma^{-1} & 0 & 2\sigma^{-2}}
+O(\delta^2).
\end{equation}
Hence $I(0) =
\lim_{\delta\to0} I(\delta)$ is singular, an issue we have raised
earlier, which makes us assume that $\delta I^{-1}(\delta)$'s largest
eigenvalue is bounded in condition~(A.5).
Using the approximation formula in (\ref{eqfisherapprox}), we
have
\[
N(\theta, 1, \delta) = \pmatrix{
-\delta\cdot(2\kappa)^{-1} & 0 & \delta\cdot\sigma^{-1} \cr
0 & -\delta\cdot\kappa^2\sigma^{-2} & 0 \cr
\delta\cdot\sigma^{-1} & 0 & -2\sigma^{-2} }+O(\delta^2).
\]
This means the leading order term of $-N(\theta, 1, \delta)$ is
identical with that of the true Fisher information matrix in
(\ref{eqfishervas}).
\subsection{Cox--Ingersoll--Ross model}\label{sec62}
Consider the Cox--Ingersoll--Ross (CIR) model
[Cox, Ingersoll and Ross (\citeyear{r15})],
\begin{equation}
dX_t=\kappa(\alpha-X_t)\,dt+\sigma\sqrt{X_t}\,dB_t,
\end{equation}
which is also Feller's (\citeyear{r21}) square root {process}.
Let $\theta=(\kappa,\alpha,\sigma)'$ and
$c=4\kappa\sigma^{-2}(1-e^{-\kappa\delta})^{-1}$. The conditional
distribution of $cX_t$ given $X_{t-1}$ is
\[
cX_t|X_{t-1}\sim\chi^2_\nu(\lambda),
\]
where the distribution is a noncentral $\chi^2$ distribution with
degree of freedom $\nu=4\kappa\alpha\sigma^{-2}$ and noncentral
parameter $\lambda=cX_{t-1}e^{-\kappa\delta}$. The transition
density of $X_{t}$ given $X_{t-1}$ is
\[
f(X_t|X_{t-1},\delta;\theta)=\frac{c}{2}e^{-u-v}\biggl(\frac
{v}{u}\biggr)^{q/2}I_q\bigl(2\sqrt{uv}\bigr),
\]
where $u=cX_{t-1}e^{-\kappa\delta}/2$, $v=cX_t/2$,
$q=2\kappa\alpha/\sigma^2-1\geq0$, and $I_q$ is the modified
Bessel function of the first kind of order $q$. If
$2\kappa\alpha>\sigma^2$, then the stationary distribution of
$\{X_t\}$ is
$\Gamma(\frac{2\kappa\alpha}{\sigma^2},\frac{\sigma^2}{2\kappa}).\label
{eqstationary}
$
Although the second partial derivations of the log transition
density function can be derived after some labor that is involved with
differentiating the modified Bessel function of {the} first kind,
acquiring an expression for the Fisher information matrix is a
rather hard task, largely due to the difficulty in deriving the
expectations.
In contrast, using the approximation formula~(\ref{eqfisherapprox}), we can obtain the approximation for the
opposite Fisher information matrix,
\[
N(\theta_0, 1, \delta) = \pmatrix{
N_{11} & N_{12} & N_{13} \cr
N_{21} & N_{22} & N_{23} \cr
N_{31} & N_{32} & N_{33} }
+O(\delta^2),
\]
where
\begin{eqnarray*}
N_{11}&=&\delta\cdot\frac{\alpha^2\sigma^2-2\kappa\alpha^2+\alpha\sigma
^2}{\sigma^4-2\kappa\alpha\sigma^2},\\
N_{12}&=&N_{21}=\delta\cdot\frac
{4\kappa\alpha\sigma^2-\sigma^4-8\kappa^2\alpha+4\kappa\sigma^2}{2\sigma
^4-4\kappa\alpha\sigma^2},\\
N_{13}&=&N_{31}=-\delta\cdot\frac{2\kappa\alpha^2\sigma^2-4\kappa^2\alpha
^2+2\kappa\alpha\sigma^2}{\sigma^5-2\kappa\alpha\sigma^3},\\
N_{22}&=&\delta\cdot\frac{\kappa^2}{\sigma^2-2\kappa\alpha},\\
N_{23}&=&-\delta\cdot\frac{2\kappa^2\alpha\sigma^2-4\kappa^3\alpha+2\kappa
^2\sigma^2}{\sigma^5-2\kappa\alpha\sigma^3}
\end{eqnarray*}
and
\begin{eqnarray*}
N_{33}&=&\frac{-2}{\sigma^2}
+\delta
\cdot\fracb{24\kappa^2\alpha^2\sigma^2-48\kappa^3\alpha^2+48\kappa
^2\alpha\sigma^2\\
&&\hspace*{44pt}{}-24\kappa\alpha\sigma^4+36\kappa\sigma^4+4\sigma
^5+9\sigma^6}{4\sigma^6-8\kappa\alpha\sigma^4}.
\end{eqnarray*}
Using $-N(\theta_0, 1, \delta)$, we can get the approximation of the
Fisher information matrix. This approximation may be used in
carrying out statistical inference on the CIR processes.
\subsection{Observed Fisher information}\label{sec63}
The major application for the asymptotic normality of both the full
and approximate MLEs is for statistical inference of $\theta$, which
include confidence regions and testing hypotheses for $\theta$. For
such purposes, the Fisher information $I(\delta)$ needs to be
estimated. A natural candidate would be $-N_n(\hat{\theta}_n^{(J)},
J, \delta)$. Although\vspace*{1pt} it converges to $I(\delta)$ at the rate of
$O_p\{ (n\delta)^{-1/2} + \delta^J\}$ or $O_p\{ (n\delta)^{-1/2} +
\delta^{J+1}\}$, depending on whether $\delta$ is fixed or diminishing,
$-N_n(\hat{\theta}_n^{(J)}, J, \delta)$ may not be nonnegative
definite, which can hinder the acquisition of
$\{-N_n(\hat{\theta}_n^{(J)}, J, \delta)\}^{1/2}$. To get around
this issue, by noticing that $I(\delta)$ is the variance of the
likelihood score, we consider
\[
\tilde{I}_n(\theta,J,\delta)=\frac{1}{n}\sum_{t=1}^n\bigl[\nabla_\theta
\log
f^{(J)}(X_t|X_{t-1},\delta;\theta)\bigr]\bigl[\nabla_\theta\log
f^{(J)}(X_t|X_{t-1},\delta;\theta)\bigr]'
\]
as an estimator of $I(\delta)$. The following theorem shows this by
replacing~$I(\delta)$ with
$\tilde{I}_n(\hat{\theta}_n^{(J)},J,\delta)$ in Theorem \ref{theo3}.
\begin{tm}\label{theo4}
Under conditions \textup{(A.1)--(A.7)} given in the \hyperref[app]{Appendix},
\[
\sqrt{n}\tilde{I}_n^{1/2}\bigl(\hat{\theta}_n^{(J)},J,\delta\bigr)\bigl(\hat{\theta
}_n^{(J)}-\theta_0\bigr)
\stackrel{d}{\rightarrow}N(0,E_d)
\]
for: \textup{(i)} $\delta\in(0,\tilde{\Delta}\wedge\dot{\Delta}]$ being
fixed, $n\rightarrow\infty$, $J\rightarrow\infty$ but
$n\delta^{2J+2}\rightarrow0$ or \textup{(ii)}~\mbox{$J\geq2$} being fixed,
$n\rightarrow\infty$, $\delta\rightarrow0$ but
$n\delta^3\rightarrow\infty$ and $n\delta^{2J+1}\rightarrow0$.
\end{tm}
Confidence regions and testing hypotheses can be readily carried out by
utilizing the above results.
\section{Simulation}\label{sec7}
We report results from simulation studies which are designed to
confirm the theoretical findings on the AMLE as reported in the
earlier sections. To allow verification with the full MLE, we
considered the Vasicek and CIR diffusion models reported in the
previous section as both models permit the full MLE. The two
asymptotic regimes were experimented: the fixed $\delta$ and the
diminishing $\delta$
with $n\delta^3\rightarrow\infty$.
The first part of the simulation is about the case in which $\delta$ is
fixed. The parameters used in the simulated Vasicek and CIR models
were $\theta=(\kappa,\alpha,\sigma)'=(0.858,0.0891,0.0468)'$ and
$\theta=(\kappa,\alpha,\sigma)'=(0.892,0.09,0.1817)'$, respectively.
The sampling interval $\delta$ was $1/12$ and $1/4$, and the order
of the density approximation $J$ was 1 and 2, respectively. For each
$\delta$ and $J$, the sample size $n$ was set at $500$, 1,000 and
2,000, respectively. In addition to bias and standard deviation, we
consider
\[
\mathrm{RMSD}(n,J,\delta)=\sqrt{\mathbb{E}\bigl\|{\hat{\theta}_{n}^{(J)}-\hat
{\theta}_{n}}\bigr\|^2_2}
\]
the\vspace*{1pt} square root of the expected square of modulated deviations
between $\hat{\theta}_{n}^{(J)}$ and~$\hat{\theta}_{n}$, as an overall
performance measure.
\begin{table
\caption{Simulated average bias (Bias) and
standard deviations (SD) of the full MLE and two AMLEs
with $J=1$ and $2$ for Vasicek
model ($\kappa=0.858, \alpha=0.0891, \sigma=0.0468$); A.Bias and
A.SD are asymptotic bias and SD based on formulas (\protect\ref{eqasybias})
and (\protect\ref{eqasyvar}); RMSD is the root of mean square deviation
between $\hat{\theta}_{n}$ and $\hat{\theta}_{n}^{(J)}$}
\label{table2}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{2.5}d{2.5}d{2.5}d{1.5}d{2.5}d{1.5}@{}}
\hline
& &&\multicolumn{3}{c}{$\bolds{\delta=1/12}$}
& \multicolumn{3}{c@{}}{$\bolds{\delta=1/4}$}\\[-4pt]
& &&\multicolumn{3}{c}{\hrulefill}&\multicolumn{3}{c@{}}{\hrulefill}\\
$\bolds{n}$
& \multicolumn{2}{c}{\textbf{Statistics}} &
\multicolumn{1}{c}{\textbf{MLE}} & \multicolumn{1}{c}{$\bolds{J=1}$}
& \multicolumn{1}{c}{$\bolds{J=2}$} & \multicolumn{1}{c}{\textbf{MLE}}
& \multicolumn{1}{c}{$\bolds{J=1}$} & \multicolumn{1}{c@{}}{$\bolds{J=2}$}\\
\hline
500& Bias &$\kappa$&0.0992 & 0.0896 &0.0992 &0.0380 &0.0127 & 0.0396 \\
&&$\alpha$&0.0002 &0.0002 &0.0002 &4.09\mbox{e--}5 &5.63\mbox{e--}5 &4.17\mbox{e--}5 \\
& &$\sigma$&4.39\mbox{e--}5 &4.14\mbox{e--}5 &4.39\mbox{e--}5 &9.12\mbox{e--}5 &7.13\mbox{e--}5 &9.43\mbox{e--}5 \\
[3pt]
&A.Bias &$\kappa$& &0.0908 &0.1016 & & 0.0174 &0.0376 \\
&&$\alpha$& &0.0003 &0.0002 & &0.0002 &0.0001 \\
& &$\sigma$& & 4.55\mbox{e--}5 &4.55\mbox{e--}5 & &0.0001 &0.0001 \\
[3pt]
&SD &$\kappa$&0.2307 &0.2255 &0.2309 &0.1366 &0.1290 &0.1386 \\
& &$\alpha$&0.0085 &0.0085 &0.0085 &0.0050 &0.0050 &0.0050 \\
& &$\sigma$&0.0016 &0.0016 &0.0016 &0.0016 &0.0016 &0.0016 \\
[3pt]
&A.SD &$\kappa$& &0.2251 &0.2366 & &0.1215 &0.1403 \\
&&$\alpha$& &0.0084 &0.0085 & &0.0047 &0.0050 \\
& &$\sigma$& &0.0016 &0.0016 & &0.0016 &0.0016 \\
[3pt]
&RMSD &$\kappa$& &0.0173 &0.0062 & &0.0332 &0.0316 \\
&&$\alpha$& &0.0002 &1.28\mbox{e--}5 & &0.0005 &0.0002 \\
& &$\sigma$& &1.36\mbox{e--}5 &1.05\mbox{e--}5 & &0.0001 &0.0001 \\
[6pt]
1,000& Bias&$\kappa$&0.0518 & 0.0419 &0.0520 &0.0170 &-0.0095 &0.0186 \\
&&$\alpha$&-0.0002 &-0.0002 &-0.0002 &1.83\mbox{e--}5 &2.81\mbox{e--}5 &1.58\mbox{e--}5 \\
& &$\sigma$&7.05\mbox{e--}5 &6.68\mbox{e--}5 &7.06\mbox{e--}5 &3.66\mbox{e--}5 &6.83\mbox{e--}6 &3.96\mbox{e--}5 \\
[3pt]
& A.Bias&$\kappa$& & 0.0446 &0.0529 & &-0.0097 &0.0161 \\
&&$\alpha$& &-0.0001 &-0.0002 & &1.69\mbox{e--}5 &1.45\mbox{e--}5 \\
& &$\sigma$& &0.0001 &0.0001 & &3.29\mbox{e--}5 &4.55\mbox{e--}5 \\
[3pt]
&SD &$\kappa$&0.1624 &0.1586 &0.1625 &0.0957 &0.0905 &0.0966 \\
& &$\alpha$&0.0058 &0.0058 &0.0058 &0.0034 &0.0034 &0.0034 \\
& &$\sigma$&0.0011 &0.0011 &0.0011 &0.0012 &0.0012 &0.0012 \\
[3pt]
&A.SD &$\kappa$& &0.1585 &0.1666 & &0.0849 &0.0982 \\
&&$\alpha$& &0.0057 &0.0058 & &0.0032 &0.0034 \\
& &$\sigma$& &0.0011 &0.0011 & &0.0012 &0.0012 \\
[3pt]
&RMSD &$\kappa$& &0.0100 &0.0008 & &0.0316 &0.0063 \\
&&$\alpha$& &0.0001 &9.14\mbox{e--}6 & &0.0004 &0.0001 \\
& &$\sigma$& &7.39\mbox{e--}6 &7.80\mbox{e--}7 & &0.0001 &1.59\mbox{e--}5 \\
\hline
\end{tabular*}
\end{table}
\setcounter{table}{1}
\begin{table}
\caption{(Continued)}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{2.5}
d{2.5}d{2.5}d{2.5}d{2.5}d{2.5}@{}}
\hline
& &&\multicolumn{3}{c}{$\bolds{\delta=1/12}$}
& \multicolumn{3}{c@{}}{$\bolds{\delta=1/4}$}\\[-4pt]
& &&\multicolumn{3}{c}{\hrulefill}&\multicolumn{3}{c@{}}{\hrulefill}\\
$\bolds{n}$
& \multicolumn{2}{c}{\textbf{Statistics}} &
\multicolumn{1}{c}{\textbf{MLE}} & \multicolumn{1}{c}{$\bolds{J=1}$}
& \multicolumn{1}{c}{$\bolds{J=2}$} & \multicolumn{1}{c}{\textbf{MLE}}
& \multicolumn{1}{c}{$\bolds{J=1}$} & \multicolumn{1}{c@{}}{$\bolds{J=2}$}\\
\hline
2,000&Bias &$\kappa$&0.0245 &0.0149 &0.0246 &0.0084 &-0.0191 &0.0100 \\
&&$\alpha$&-3.97\mbox{e--}5 &-3.34\mbox{e--}5 &-4.01\mbox{e--}5 &-5.72\mbox{e--}5 &-4.90\mbox{e--}5
&-5.80\mbox{e--}5 \\
& &$\sigma$&2.69\mbox{e--}5 &2.30\mbox{e--}5 &2.70\mbox{e--}5 &4.00\mbox{e--}5 &9.21\mbox{e--}6 &4.34\mbox{e--}5 \\
[3pt]
&A.Bias &$\kappa$& &0.0179 &0.0249 & &-0.0085 & 0.0071 \\
&&$\alpha$& &-2.63\mbox{e--}5 &-2.98\mbox{e--}5 & &0.0001 &-0.0001 \\
& &$\sigma$& &4.55\mbox{e--}5 &4.55\mbox{e--}5 & &4.55\mbox{e--}5 & 4.55\mbox{e--}5 \\
[3pt]
&SD &$\kappa$&0.1114 &0.1091 &0.1115 &0.0647 &0.0611 &0.0652 \\
& &$\alpha$&0.0042 &0.0041 &0.0042 &0.0024 &0.0024 &0.0024 \\
& &$\sigma$&0.0008 &0.0008 &0.0008 &0.0008 &0.0008 &0.0008 \\
[3pt]
&A.SD &$\kappa$& &0.1088 &0.1143 & &0.0576 &0.0665 \\
&&$\alpha$& &0.0041 &0.0042 & &0.0023 &0.0024 \\
& &$\sigma$& &0.0008 &0.0008 & &0.0008 &0.0008 \\
[3pt]
&RMSD &$\kappa$& &0.0100 &0.0006 & &0.0300 &0.0042 \\
&&$\alpha$& &0.0001 &7.37\mbox{e--}6 & &0.0003 &4.68\mbox{e--}5 \\
& &$\sigma$& &6.27\mbox{e--}6 &7.80\mbox{e--}7 & &0.0001 &1.02\mbox{e--}5 \\
\hline
\end{tabular*}
\end{table}
Tables \ref{table2} and \ref{table3} summarize the simulation for the fixed $\delta$ case.
They report the average bias and standard deviation (SD) for the
full MLE and AMLEs with $J=1$ and $J=2$, as well as the RMSD between
the AMLEs and the full MLE, for both the Vasicek and the CIR models.
To give the simulation results more perspective and to confirm the
derived {approximate} bias and variance formulas in Section \ref{sec5}, we
also computed the asymptotic bias and standard deviation based on
formulas (\ref{eqasybias}) and (\ref{eqasyvar}). We observe
from Tables \ref{table2} and \ref{table3} that at each $\delta$ ($1/12$ and $1/4$)
experimented,
the bias and the standard deviation of all {the} estimators for {the}
three parameters became smaller as $n$ increased. These confirmed the
consistency of the estimators. The tables also showed that there was a
good agreement among the three estimators in terms of the performance
measures. It appeared that the bias and the variance of the AMLE
with $J=1$ and $J=2$ were quite comparable to each other. However, by
comparing RMSD, it was clear that in most of the cases (except for
$n=500$ of CIR model), the RMSD for $J=2$ was smaller than $J=1$,
signaling the AMLE with $J=2$ was closer to the full MLE than that of
the AMLE with $J=1$. This indicates that the AMLEs with $J=2$ were
indeed closer to those with $J=1$, as confirmed by our early analysis.
The asymptotic bias and standard deviation predicted for the AMLE with
$J=1$ and $2$ offer more insights, and show good agreement between the
simulated results and the predicted values by the theory, which is very
assuring.
We also {observe} that for $\delta=1/4$, the AMLE with $J=2$
performs better than AMLE with $J=1$, which somehow reflects Table \ref{table1}
which shows that $J=2$ is preferred to $J=1$ at this frequency.
When $\delta$ was fixed at $1/12$, we see the performance between
$J=1$ and $J=2$ was largely similar.
\begin{table}
\caption{Simulated average bias (Bias) and standard deviations (SD)
of the full MLE and two AMLEs with $J=1$ and $2$ for CIR model
($\kappa=0.892, \alpha=0.09, \sigma=0.1817$); A.Bias and A.SD are
asymptotic bias and SD based on formulas (\protect\ref{eqasybias}) and
(\protect\ref{eqasyvar}); RMSD is the root of mean square deviation between
$\hat{\theta}_{n}$ and $\hat{\theta}_{n}^{(J)}$}
\label{table3}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{2.5}d{1.5}d{2.5}d{2.5}cd{2.4}@{}}
\hline
& &&\multicolumn{3}{c}{$\bolds{\delta=1/12}$}&\multicolumn{3}{c@{}}{$\bolds{\delta
=1/4}$}\\[-4pt]
& &&\multicolumn{3}{c}{\hrulefill}&\multicolumn{3}{c@{}}{\hrulefill}\\
$ \bolds{n} $& \multicolumn
{2}{c}{\textbf{Statistics}}&\multicolumn{1}{c}{\textbf{MLE}}
&\multicolumn{1}{c}{$\bolds{J=1}$}&\multicolumn{1}{c}{$\bolds{J=2}$}
&\multicolumn{1}{c}{\textbf{MLE}}&\multicolumn{1}{c}{$\bolds{J=1}$}
&\multicolumn{1}{c@{}}{$\bolds{J=2}$}\\
\hline
500& Bias&$\kappa$&0.0980 &0.0910 &0.0978 &0.0371 &0.0234 & 0.0388 \\
&&$\alpha$&0.0001 &0.0004 &0.0001 &-6.38\mbox{e--5} &0.0008 &-0.0001 \\
& &$\sigma$&0.0003 &0.0003 &0.0003 &0.0004 &0.0005 &0.0003 \\
[3pt]
&A.Bias &$\kappa$& &0.0818 &0.0984 & &0.0207 & 0.0513 \\
&&$\alpha$& &0.0005 &0.0001 & &0.0008 &-0.0001 \\
& &$\sigma$& &0.0003 &0.0003 & &0.0004 &0.0002 \\
[3pt]
&SD &$\kappa$&0.2389 &0.2340 &0.2405 &0.1437 &0.1338 &0.2256 \\
& &$\alpha$&0.0093 &0.0093 &0.0093 &0.0055 &0.0054 &0.0055 \\
& &$\sigma$&0.0060 &0.0060 &0.0060 &0.0065 &0.0065 &0.0069 \\
[3pt]
&A.SD &$\kappa$& &0.2169 &0.2389 & &0.1159 &0.1938 \\
&&$\alpha$& &0.0091 &0.0093 & &0.0064 &0.0055 \\
& &$\sigma$& &0.0060 &0.0060 & &0.0067 &0.0065 \\
[3pt]
&RMSD &$\kappa$& &0.0200 &0.0224 & &0.0447 &0.1622 \\
&&$\alpha$& &0.0009 &0.0004 & &0.0018 &0.0004 \\
& &$\sigma$& &0.0004 &0.0004 & &0.0017 &0.0021 \\
[6pt]
1,000& Bias&$\kappa$&0.0521 &0.0435 &0.0521 &0.0218 &0.0070 &0.0186 \\
&&$\alpha$&-1.54\mbox{e--}5 &0.0002 &-2.22\mbox{e--}5 &-0.0002 &0.0007 &-0.0003 \\
& &$\sigma$&3.86\mbox{e--}5 &4.35\mbox{e--}5 &3.81\mbox{e--}5 &0.0003 &0.0006 &0.0003 \\
[3pt]
&A.Bias &$\kappa$& &0.0411 &0.0525 & &0.0095 &0.0262 \\
&&$\alpha$& &0.0004 &-3.43\mbox{e--}5 & &0.0007 &-0.0003 \\
& &$\sigma$& &3.17\mbox{e--}5 &2.69\mbox{e--}5 & &0.0003 &0.0001 \\
[3pt]
&SD &$\kappa$&0.1596 &0.1558 &0.1603 &0.0968 &0.0861 &0.0980 \\
& &$\alpha$&0.0067 &0.0067 &0.0067 &0.0039 &0.0037 &0.0039 \\
& &$\sigma$&0.0043 &0.0043 &0.0043 &0.0045 &0.0045 &0.0045 \\
[3pt]
&A.SD &$\kappa$& &0.1452 &0.1596 & &0.0823 &0.0969 \\
&&$\alpha$& &0.0066 &0.0067 & &0.0044 &0.0039 \\
& &$\sigma$& &0.0040 &0.0043 & &0.0047 &0.0045 \\
[3pt]
&RMSD &$\kappa$& &0.0173 &0.0141 & &0.0447 &0.0200 \\
&&$\alpha$& &0.0003 &2.66\mbox{e--}5 & &0.0020 &0.0001 \\
& &$\sigma$& &0.0002 &3.91\mbox{e--}5 & &0.0021 &0.0002 \\
\hline
\end{tabular*}
\vspace*{-5pt}
\end{table}
\setcounter{table}{2}
\begin{table}
\caption{(Continued)}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{2.4}cd{2.4}d{2.5}d{2.4}d{2.5}@{}}
\hline
& &&\multicolumn{3}{c}{$\bolds{\delta=1/12}$}&\multicolumn{3}{c@{}}{$\bolds{\delta
=1/4}$}\\[-4pt]
& &&\multicolumn{3}{c}{\hrulefill}&\multicolumn{3}{c@{}}{\hrulefill}\\
$ \bolds{n} $& \multicolumn
{2}{c}{\textbf{Statistics}}&\multicolumn{1}{c}{\textbf{MLE}}
&\multicolumn{1}{c}{$\bolds{J=1}$}&\multicolumn{1}{c}{$\bolds{J=2}$}
&\multicolumn{1}{c}{\textbf{MLE}}&\multicolumn{1}{c}{$\bolds{J=1}$}
&\multicolumn{1}{c@{}}{$\bolds{J=2}$}\\
\hline
2,000&Bias &$\kappa$&0.0295 &0.0199 &0.0294 &0.0103 &-0.0057 &0.0069 \\
&&$\alpha$&-0.0002 &0.0001 &-0.0002 &-3.06\mbox{e--}5 &0.0010 &-9.87\mbox{e--}5 \\
& &$\sigma$&0.0002 &0.0002 &0.0002 &3.05\mbox{e--}5 &0.0006 &1.33\mbox{e--}5 \\
[3pt]
&A.Bias &$\kappa$& &0.0213 &0.0299 & &-0.0011 &0.0147 \\
&&$\alpha$& &0.0002 &-0.0002 & &0.0006 &-0.0001 \\
& &$\sigma$& &0.0002 &0.0002 & &0.0005 &1.06\mbox{e--}5 \\
[3pt]
&SD &$\kappa$&0.1082 &0.1053 &0.1088 &0.0696 &0.0607 &0.0698 \\
& &$\alpha$&0.0048 &0.0048 &0.0048 &0.0028 &0.0027 &0.0028 \\
& &$\sigma$&0.0030 &0.0031 &0.0030 &0.0033 &0.0037 &0.0033 \\
[3pt]
&A.SD &$\kappa$& &0.1181 &0.1105 & &0.0592 &0.0697 \\
&&$\alpha$& &0.0047 &0.0048 & &0.0027 &0.0028 \\
& &$\sigma$& &0.0030 &0.0030 & &0.0034 &0.0033 \\
[3pt]
&RMSD &$\kappa$& &0.0173 &0.0068 & &0.0424 &0.0100 \\
&&$\alpha$& &0.0004 &0.0001 & &0.0020 &0.0001 \\
& &$\sigma$& &0.0005 &0.0003 & &0.0027 &0.0001 \\
\hline
\end{tabular*}
\end{table}
The second part of the simulation was devoted to the diminishing
$\delta$ case. Here we wanted to confirm the differential behavior
of the AMLEs in the limiting distribution between $J=1$ and $J
\geq2$, as revealed in Section \ref{sec5}. The Vasicek model with
$\theta=(\kappa,\alpha,\sigma)'=(0.892,0.09,0.1817)'$ was
considered. We tried to create two scenarios: (i)
$n\delta^3\rightarrow\infty$ and (ii) $n\delta^3\rightarrow0$,
while $\delta\rightarrow0$. They were created by choosing
$\delta=n^{-1/6}$ and $\delta=n^{-1/2}$, respectively, whiling
selecting $n=500, 1\mbox{,}000, 2\mbox{,}000, 4\mbox{,}000$ and 8,000, respectively, to
create two streams of asymptotic sequences. For each $n$ and
$\delta$, we generated repeatedly the Vasicek sample paths 1,000
times. For each simulated sample path, we obtained the AMLEs
$\hat{\theta}_{n}^{(J)}$ for $J=1$ and $2$, respectively, and computed
the Wald statistics
\[
W_n(J)=n\bigl({\hat{\theta}_{n}^{(J)}}-\theta_0\bigr)'I(\delta)\bigl({\hat{\theta
}_{n}^{(J)}}-\theta_0\bigr).
\]
If $\sqrt{n}I^{1/2}(\delta)({\hat{\theta}_{n}^{(J)}}-\theta_0)$ is
asymptotically standard normally distributed in~$\mathbb{R}^d$,
then the Wald statistic $W_n(J) \stackrel{d}{\rightarrow} \chi_3^2$.
Based on the 1,000 Wald statistics from the simulations, we then
performed the Kolmogorov--Smirnov \mbox{(K--S)} test to test $H_0\dvtx W_n(J)
\sim\chi_3^2$, or not,
for each of the designed sequences of~$(n, \delta)$\vadjust{\goodbreak} generated under
the two scenarios.
Table \ref{table4} reports the $p$-values of the test, which show that for $J=1$,
under both scenarios, the $p$-values of the K--S test became smaller, and
hence the above null hypothesis was rejected as $n$ increased. For
$J=2$, the $p$-values of the K--S test were sharply {different} between
the two scenarios.
\begin{table}[b]
\caption{$p$-values of Kolmogorov--Smirnov test for
$W_n(J)\sim\chi^2_3$}
\label{table4}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccd{1.5}d{1.5}@{}}
\hline
\textbf{Situation} &$\bolds{n}$ & $\bolds{\delta}$
& \multicolumn{1}{c}{$\bolds{J=1}$}
& \multicolumn{1}{c@{}}{$\bolds{J=2}$} \\
\hline
$\delta=n^{-1/6}$& \hphantom{0,}500 & 0.3550 & 0.3524 & 0.0587 \\
&1,000 & 0.3162 & 0.4595 & 0.5830 \\
&2,000 & 0.2817 & 0.1149 & 0.2710 \\
&4,000 & 0.2510 & 0.0019 & 0.8309\\
&8,000 & 0.2236 & 5.74\mbox{e--}8 & 0.6002 \\
[4pt]
$\delta=n^{-1/2}$&\hphantom{0,}500 & 0.0447 & 5.04\mbox{e--}7 & 2.45\mbox{e--}8 \\
&1,000 & 0.0316 & 0.0003 & 9.72\mbox{e--}5\\
&2,000 & 0.0224 & 0.0006 & 0.0003\\
&4,000 & 0.0158 & 0.1109 & 0.0851\\
&8,000 & 0.0112 & 0.0470 & 0.0367\\
\hline
\end{tabular*}
\end{table}
In particular, the $p$-values were mostly quite large under the
scenario of $n \delta^3 \to\infty$, and they were\vspace*{1pt} largely significant
(small) when $\delta$ was diminishing at the faster rate of $n^{-1/2}$
such that $n\delta^3\rightarrow0$. These were consistent with our
theoretical findings in Section \ref{sec5}.
\begin{appendix}\label{app}
\section*{Appendix}
We need the following technical assumptions in our analysis.
(A.1) (i) $\Theta$ is a compact set in $\mathbb{R}^d$, and the true
parameter $\theta_0$ is an interior point of $\Theta$; (ii) for all
values of the parameters $\theta$, Assumption 1--3 in
\citet{r2} hold; (iii) the drift function $\mu(x;
\theta)$ is a bona fide function of~$\theta$ for each $x$.
(A.2) (i) For every $\delta> 0$, $ \mathbb{E}\nabla_\theta\log
f(X_t|X_{t-1},\delta;\theta_0)=0
$ and $\theta_0$ is the only root of $\mathbb{E}\nabla_\theta\log
f(X_t|X_{t-1},\delta;\theta)=0$. (ii) the MLE $\hat{\theta}_{n}$
and the $J$-term approximate MLE $ \hat{\theta}^{(J)}_{n}$ satisfy,
respectively,
\[
\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;{\hat{\theta}_{n}})=0
\quad\mbox{and}\quad\sum_{t=1}^n\nabla_\theta\log
f^{(J)}\bigl(X_t|X_{t-1},\delta;{\hat{\theta}^ {(J)}_{n}}\bigr)=0.
\]
(A.3) There exist finite positive constants $\Delta$ and $K_1$
such that, for $l=1,2,3$, any $\delta\in(0,\Delta]$, $i_1, i_2, i_3
\in\{1,\ldots,d\}$ and $j=1$ and $2$,
\[
\mathbb{E}\sup_{\theta\in\Theta}\biggl\{\biggl|\frac{\partial^l
A_j(X_t|X_{t-1},\delta;\theta)}{\partial\theta_{i_1} \cdots
\partial\theta_{i_l} }\biggr|^3\biggr\}\leq
K_1.
\]
(A.4) There exist finite positive constants $\nu_l$ for $l=0, 1,2$
and $3$, $\Delta>0$ and $K_2$ such that $\nu_0 > 3$,
$\nu_2>\nu_1>3$, $\nu_3 > 1$ and
for any $i_1, \ldots, i_3 \in\{1,\ldots,d\}$ and $\delta\in(0,\Delta]$,
\[
\mathbb{E}\Biggl\{\sup_{\theta\in\Theta}\Biggl[\sum_{l=0}^\infty\biggl|\frac{\partial^q
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta);\theta)}{\partial\theta_{i_1}
\cdots\partial\theta_{i_q}
}\biggr|\frac{\Delta^l}{l!}\Biggr]^{\nu_l}\Biggr\}\leq K_2.
\]
(A.5) For any $\delta>0$, the Fisher information matrix
\[
I(\delta):=-\mathbb{E}\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0)
\]
is invertible and as $\delta\rightarrow0$ the largest eigenvalues
of $\delta I^{-1}(\delta)$ is bounded away from infinity.
(A.6) For each positive integer $K$, which may be infinite, and
any $\delta\in(0,\Delta]$,
\[
\mathbb{P}\Biggl\{\inf_{\theta\in\Theta}\Biggl|\sum_{l=0}^K
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta);\theta)\frac{\delta
^l}{l!}\Biggr|=0\Biggr\}=0.
\]
(A.7) For any $\beta>1$ and $\eta>0$, there exists
$\Delta(\beta,\eta)>0$, then for any
$\delta\in(0,\Delta(\beta,\eta)]$ and $K$, where $K$ may be
infinite,
\[
\mathbb{P}\Biggl\{\inf_{\theta\in\Theta}\Biggl|\sum_{l=0}^Kc_l(\gamma(X_t;\theta
)|\gamma(X_{t-1};\theta);\theta)\frac{\delta^l}{l!}\Biggr|<\eta^{1/\beta}\Biggr\}
<\eta.
\]
Assumptions (A.1) and (A.2) are standard requirements for maximum
likelihood estimators. In particular, (A.1) (ii) contains conditions on
the smoothness of the drift and the diffusion which ensures the
existence of a~unique solution to (\ref{eq11}) as well as the infinite
differentiability of the transition density $f(x|x_0, \delta; \theta)$
with respect to $x$, $x_0$ and $\delta$, and three {times}
differentiable with respect to $\theta$ [\citet{r22}]. The second part of
(A.2) is the simplified approach of \citet{r16} assuming the MLEs
are the solutions of the likelihood score equations.
Assumption (A.3) is needed to guarantee the third derivative of $\log
f(X_t|X_{t-1},\delta;\theta)$ with respect to $\theta$ can be
controlled by an integrable function, while condition (A.4)
ensures the absolute convergence of the infinite series
${\sum_{l=0}^\infty}|c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta)|\delta^l/l!
= \exp\{\tilde{A}_3(x|x_0, \delta;\theta)\}$ as
\citet{r2} has provided conditions on the nondegeneracy
of the
diffusion function and the boundary condition, which together with
the third part of condition (A.1) leads to the convergence of the
above infinite series $\exp\{\tilde{A}_3(x|x_0,
\delta;\theta)\}$. Condition (A.4) is also needed to allow exchange of
differentiation and summation for the infinite series.
The first part of the (A.5) is of standard in likelihood inference.
Its second part reflects the fact that for some processes
$\lim_{\delta\to0}I(\delta)$ may be singular, as conveyed in our
discussion in Section \ref{sec6} for the Vasicek process.
Condition~(A.6) is needed to guarantee the derivatives of log
transition density and log approximate transition density
exist with probability one. Condition~(A.7) is needed to manage the
denominators in the derivatives of the log approximate transition density,
ensuring that the probability of their taking small values can be
controlled uniformly.
We shall give the proofs for the propositions and theorems mentioned
in Sections \ref{sec3}--\ref{sec6}. We first present some lemmas about the true
transition density and its approximations, which we will use in
later proofs. The proofs for the lemmas can be found in \citet{r14}.
\begin{la}\label{lem1}
Under \textup{(A.1)} and \textup{(A.4)}, for any $\delta\in(0,\Delta)$, the infinite
series
\[
\sum_{l=0}^\infty
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta))\frac{\delta^l}{l!}
\]
absolutely converges with probability 1, and for $k=1,2$ and $3$,
and ${i_1}, i_2, i_3 \in\{1,\ldots,d\}$,
\begin{eqnarray*}
&& \frac{\partial^k}{\partial\theta_{i_1} \cdots\partial
\theta_{i_k}}\sum_{l=0}^\infty
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta))\frac{\delta^l}{l!} \\
&&\qquad= \sum_{l=0}^\infty
\frac{\partial^k}{\partial\theta_{i_1} \cdots\partial\theta_{i_k}}
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta))\frac{\delta^l}{l!}.
\end{eqnarray*}
\end{la}
\begin{la}\label{lem2}
Under \textup{(A.6)} and \textup{(A.7)}, for any positive $\beta>1$, there exist two
constants $m(\beta)<\infty$ and $\Delta_1(\beta)>0$ such that for
any $\delta\in(0,\Delta_1(\beta)]$ and~$J$, where~$J$ can be
infinity, then
\[
\mathbb{E}\Biggl\{\sup_{\theta\in\Theta}\Biggl|\sum_{l=0}^J
c_l(\gamma(X_t;\theta)|\gamma(X_{t-1};\theta))
\frac{\delta^l}{l!}\Biggr|^{-\beta}\Biggr\}<m(\beta).
\]
\end{la}
\begin{la}\label{lem3}
Under \textup{(A.1), (A.3), (A.4), (A.6), (A.7)}, there exist two constants
$M_1<\infty$ and $\Delta_2>0$ such that, for any $J$, where $J$ can
be infinity, $\delta\in(0,\Delta_2)$ and $i,j,k\in\{1,\ldots,d\}$,
\[
\mathbb{E}\biggl\{\sup_{\theta\in\Theta}\biggl|\frac{\partial^3\log
f^{(J)}(X_t|X_{t-1},\delta;\theta)}{\partial\theta_i\,\partial\theta
_j\,\partial\theta_k}\biggr|\biggr\}<M_1.
\]
\end{la}
\begin{pf*}{Proof of Proposition \ref{prop1}}
Using the same method in the proof
of Lemma~\ref{lem3}, we know (a) holds. On the other hand, Lemma \ref{lem3}
implies
(b).
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{prop2}}
See the proof of Proposition 2 in \citet{r14}.
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{prop3}}
Recall Proposition \ref{prop2}, then
\[
\|I^{-1}(\delta) N(\theta_0, J,
\delta)+E_d\|_2\leq\|I^{-1}(\delta)\|_2\cdot\| N(\theta_0, J,
\delta)+I(\delta)\|_2\leq C\delta^{J}.
\]
If $C\delta^{J}<1$, then
\[
\|N^{-1}(\theta_0,J,\delta)I(\delta)+E_d\|_2\leq\frac
{\|I^{-1}(\delta)
N(\theta_0, J, \delta)+E_d\|_2}{1-\|I^{-1}(\delta) N(\theta_0, J,
\delta)+E_d\|_2}.
\]
From Proposition \ref{prop2}, if $C\delta^{J+1}<1$, then
\[
\|N^{-1}(\theta_0,J,\delta)+I^{-1}(\delta)\|_2\leq\frac
{\|I^{-1}(\delta)\|_2^2\|N(\theta_0,J,\delta)+I(\delta
)\|_2}{1-\|I^{-1}(\delta)\|_2\|N(\theta_0,J,\delta)+I(\delta)\|_2}.
\]
On the other hand, using the same method in the proof of Proposition
\ref{prop2}, we have
\[
\|U(\theta_0,J,\delta)\|_2\leq C\delta^{J+1}
\]
for any positive $J$ and $\delta\in(0,\bar{\Delta})$. Hence, we can
find the constants $C_1, C_2$ and $\underline{\Delta}>0$ such that
\[
\|N^{-1}(\theta_0,J,\delta)I(\delta)+E_d\|_2\leq
C_1\delta^{J} \quad\mbox{and}\quad \|N^{-1}(\theta_0,J,\delta)U(\theta_0,J,\delta
)\|_2\leq
C_2\delta^{J}
\]
for any positive integer $J$ and $\delta\in(0,\underline{\Delta})$.
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{prop4}} Use the same method in
the proof of Proposition~\ref{prop2}.
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{prop5}} We'll use Corollary 2.1
in Newey (\citeyear{r28}) to prove this proposition. We only need to verify three
conditions under two situations mentioned in Proposition \ref{prop5}:
\begin{longlist}
\item
for any $i\in\{1,\ldots,d\}$,
\[
\mathbb{E}\biggl\{\frac{\partial}{\partial\theta_i}\log
f(X_t|X_{t-1},\delta;\theta)\biggr\}\qquad \mbox{is equicontinuous};
\]
\item for any $i\in\{1,\ldots,d\}$,
\[
\sup_{\theta\in\Theta}\Biggl\|\frac{1}{n}\sum_{t=1}^n\frac{\partial
^2}{\partial\theta_i\,\partial\theta'}\log
f(X_t|X_{t-1},\delta;\theta)\Biggr\|_2=O_p(1);
\]
\item for any $i\in\{1,\ldots,d\}$ and $\theta\in\Theta$,
\[
\frac{1}{n}\sum_{t=1}^n\frac{\partial}{\partial\theta_i}\log
f(X_t|X_{t-1},\delta;\theta)-\mathbb{E}\biggl\{\frac{\partial}{\partial\theta
_i}\log
f(X_t|X_{t-1},\delta;\theta)\biggr\}\stackrel{p}{\rightarrow}0.
\]
\end{longlist}
For any $\theta^*,\theta^{**}\in\Theta$, note that
\begin{eqnarray*}
&& \mathbb{E}\biggl\{\frac{\partial}{\partial\theta_i}\log
f(X_t|X_{t-1},\delta;\theta^*)\biggr\}-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_t|X_{t-1},\delta;\theta^{**})\biggr\}\\
&&\qquad =\mathbb{E}\biggl\{\frac{\partial^2}{\partial\theta_i\,\partial\theta'}\log
f(X_t|X_{t-1},\delta;\bar{\theta})\biggr\}\cdot(\theta^*-\theta^{**}),
\end{eqnarray*}
where $\bar{\theta}$ is on the joint line between $\theta^*$ and
$\theta^{**}$. Then
\begin{eqnarray*}
&& \biggl|\mathbb{E}\biggl\{\frac{\partial}{\partial\theta_i}\log
f(X_t|X_{t-1},\delta;\theta^*)\biggr\}-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_t|X_{t-1},\delta;\theta^{**})\biggr\}\biggr|\\
&&\qquad \leq\biggl\|\mathbb{E}\biggl\{\frac{\partial^2}{\partial\theta_i\,\partial
\theta'}\log
f(X_t|X_{t-1},\delta;\bar{\theta})\biggr\}\biggr\|_2
\cdot\|\theta^{*}-\theta^{**}\|_2.
\end{eqnarray*}
For any $j\in\{1,\ldots,d\}$, use the same method in the proof of
Lemma \ref{lem3}, we know that there exists a constant $C$, which is not
dependent on $J$ and $\delta$, and $\hat{\Delta}>0$ such that, for
any $J$ and $\delta\in(0,\hat{\Delta}]$,
\[
\mathbb{E}\biggl\{\sup_{\theta\in\Theta}\biggl|\frac{\partial^2}
{\partial\theta
_i\,\partial\theta_j}\log
f(X_t|X_{t-1},\delta;\theta)\biggr|\biggr\}<C.
\]
Hence, (i) and (ii) can be established.
To verify (iii), from (A.3) [Lemmas 3 and 4 in
\citet{r7}], we know that there exists a positive constant
$\kappa$ such that for any $t_1<t_2$,
\begin{eqnarray*}
& &\biggl|\mathbb{E}\biggl\{\biggl[\frac{\partial}{\partial\theta_i}\log
f(X_{t_1}|X_{t_1-1},\delta;\theta)-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_{t_1}|X_{t_1-1},\delta;\theta)\biggr\}\biggr]\\
&&\hspace*{6pt}\quad \times\biggl[\frac{\partial}{\partial\theta_i}\log
f(X_{t_2}|X_{t_2-1},\delta;\theta)-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_{t_2}|X_{t_2-1},\delta;\theta)\biggr\}\biggr]\biggr\}\biggr|\\
&&\hspace*{6pt}\qquad \leq C\cdot\exp\{-\kappa(t_2-t_1)\delta\},
\end{eqnarray*}
where
\[
C=\mathbb{E}\biggl\{\biggl[\frac{\partial}{\partial\theta_i}\log
f(X_{t}|X_{t-1},\delta;\theta)-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_{t}|X_{t-1},\delta;\theta)\biggr\}\biggr]^2\biggr\}.
\]
Then
\begin{eqnarray*}
&& \mathbb{E}\Biggl\{\frac{1}{n}\sum_{t=1}^n\biggl[\frac{\partial}{\partial\theta
_i}\log
f(X_{t}|X_{t-1},\delta;\theta)-\mathbb{E}\biggl\{\frac{\partial}{\partial
\theta_i}\log
f(X_{t}|X_{t-1},\delta;\theta)\biggr\}\biggr]\Biggr\}^2\\
&&\qquad \leq\frac{C}{n}+\frac{C}{n}\cdot\frac{\exp\{-\kappa\delta\}
}{1-\exp\{-\kappa\delta\}}\\
&&\qquad \leq3\biggl[2K_1+K_2\cdot
m\biggl(\frac{2\nu_1}{\nu_1-2}\biggr)\biggr]
\cdot\biggl\{\frac{1}{n}+\frac{1}{n[\exp(\kappa
\delta)-1]}\biggr\}\rightarrow0,
\end{eqnarray*}
under the two situations mentioned in the statement of Proposition
\ref{prop5}. Hence we complete the proof.
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{prop6}} From (A.2), we can get
$n^{-1}\nabla_\theta\ell_{n,\delta}(\hat{\theta}_n)=0$. Expanding it
at $\theta_0$,
\[
0=\frac{1}{n}\sum_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;\theta_0)+\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta
\theta}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})\cdot(\hat{\theta}_n-\theta_0).
\]
Then
\[
\hat{\theta}_n-\theta_0=\Biggl\{-\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta
}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})\Biggr\}^{-1}\cdot\frac{1}{n}\sum
_{t=1}^n\nabla_\theta
\log f(X_t|X_{t-1},\delta;\theta_0).
\]
Define\vspace*{1pt} $I_n(\delta)=-n^{-1}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0)$. From Lemma \ref{lem3}, an\break
$-{n}^{-1}\times\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})=I_n(\delta)\cdot\{1+o_p(1)\}$.
Using the same way as that in the verification of (iii) in the proof
of Proposition \ref{prop5}, we can get
$I_n(\delta)-I(\delta)=O_p\{(n\delta)^{-1/2}\}$. If
$n\delta^{3}\rightarrow\infty$, by (A.5),
\begin{eqnarray*}
\Biggl\{-\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})\Biggr\}^{-1}
&=&\bigl\{I(\delta)\cdot\{1+o_p(1)\}+O_p\{(n\delta)^{-1/2}\}\bigr\}
^{-1}\\
&=&I^{-1}(\delta)\cdot\{1+o_p(1)\}.
\end{eqnarray*}
Then
\[
\sqrt{n}I^{1/2}(\delta)(\hat{\theta}_n-\theta_0)=I^{-1/2}(\delta)\frac
{1}{n^{1/2}}\sum_{t=1}^n\nabla_\theta
\log f(X_t|X_{t-1},\delta;\theta_0)\cdot\{1+o_p(1)\}.
\]
We will use the martingale central limit theorem [\citet{r12},
page~476] to show that the first part on the right-hand side of the
above equation
converges to a standard normal distribution. For any\vspace*{1pt}
$\alpha\in\mathbb{R}^d$ with unit $L_2$ norm, to simplify notations,
let $ U_{n,m}=\alpha'I^{-1/2}(\delta)n^{-1/2}\nabla_\theta\log
f(X_m|X_{m-1},\delta;\theta_0)$ and
$\mathscr{F}_{n,m}=\sigma(X_1,\ldots,X_m)$. It is easy to check
$(U_{n,m},\mathscr{F}_{n,m})$ is a martinga\-le~difference array. By the
Markov property and Birkhoff's Ergodic theo\-rem,
$V_{n,n}=\sum_{m=1}^n\mathbb{E}(U_{n,m}^2|\mathscr{F}_{n,m})
\stackrel{p}{\rightarrow}\mathbb{E}U_{n,m}^2=1$.
On the other hand,\break $\sum_{m=1}^n|U_{n,m}|^3\leq
C(n\times\delta^3)^{-1/2}\rightarrow0$. This implies the asymptotic
normality of
$\sqrt{n}\alpha'I^{1/2}(\delta)(\hat{\theta}_n-\theta_0)$. Then we
complete the proof.
\end{pf*}
\begin{pf*}{Proof of Theorem \ref{theo1}}
From Propositions \ref{prop4} and \ref{prop5}, we can get
\[
\bigl\|\mathbb{E}\nabla_\theta\log
f\bigl(X_t|X_{t-1},\delta;{\hat{\theta}_{n}^{(J)}}\bigr)\bigr\|_2
\stackrel{p}{\rightarrow}0
\]
for either: (i) $\delta\in(0,\tilde{\Delta}\wedge\dot{\Delta}]$
being fixed, $J\rightarrow\infty$ and $n\rightarrow\infty$, or (ii)
$J$ being fixed, $n\rightarrow\infty$, $\delta\rightarrow0$ but
$n\delta\rightarrow\infty$. Hence, noting condition (A.2)(i), we
have the consistency of the AMLE $\hat{\theta}_{n}^{(J)}$.
\end{pf*}
\begin{pf*}{Proof of Theorem \ref{theo2}}
For fixed $\delta$, from Theorem \ref{theo1} and
(\ref{eq41}), we know that the leading order term of
$\hat{\theta}_{n}^{(J)}-\theta_0$ contains two parts: one is
$N^{-1}U_n$, and the other is
$N^{-1}(N_n+F_n)({\hat{\theta}_{n}}-\theta_0)$. Hence,
${\hat{\theta}_{n}^{(J)}}-\theta_0=O_p\{\delta^{J+1}+(n\delta)^{-1/2}\}$.
For $J$ fixed and $\delta\rightarrow0$, Proposition \ref{prop4} implies
\begin{eqnarray*}
&& \mathbb{E}\Biggl\{\Biggl\|\frac{1}{n}\sum_{t=1}^n\nabla_\theta\log
f\bigl(X_t|X_{t-1},\delta;{\hat{\theta}_{n}^{(J)}}\bigr)-\frac{1}{n}\sum
_{t=1}^n\nabla_\theta\log
f(X_t|X_{t-1},\delta;{\hat{\theta}_{n}})\Biggr\|_2\Biggr\}\\
&&\qquad \leq C\delta^{J+1}.
\end{eqnarray*}
This means that
\[
\mathbb{E}\Biggl\{\Biggl\|\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})\cdot\bigl({\hat{\theta}_{n}^{(J)}-\hat
{\theta}_{n}}\bigr)\Biggr\|_2\Biggr\}\leq
C\delta^{J+1},
\]
where $\tilde{\theta}$ is on the joining line between
$\hat{\theta}_{n}^{(J)}$ and $\hat{\theta}_{n}$. Hence
\[
\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\tilde{\theta})\cdot\bigl({\hat{\theta}_{n}^{(J)}-\hat
{\theta}_{n}}\bigr)=O_p(\delta^{J+1}).
\]
Since $\tilde{\theta}\stackrel{p}{\rightarrow}\theta_0$ and
${\hat{\theta}_{n}^{(J)}-\hat{\theta}_{n}}=o_p(1)$,
\[
\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0)\cdot\bigl({\hat{\theta}_{n}^{(J)}-\hat{\theta
}_{n}}\bigr)=O_p(\delta^{J+1}).
\]
On the other hand, from Proposition \ref{prop2}, we know
\begin{eqnarray*}
&&\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta\theta}\log
f(X_t|X_{t-1},\delta;\theta_0)-\frac{1}{n}\sum_{t=1}^n\nabla^2_{\theta
\theta}\log
f^{(J)}(X_t|X_{t-1},\delta;\theta_0)\\
&&\qquad=O_p(\delta^{J+1}).
\end{eqnarray*}
Then\vspace*{1pt}
$N_n({\hat{\theta}_{n}^{(J)}-\hat{\theta}_{n}})=O_p(\delta^{J+1})$.
Using the same way of verifying (iii) in the proof of Proposition \ref{prop5},
we know $N_n-N=O_p\{(n\delta)^{-1/2}\}$. As
$n\delta^3\rightarrow\infty$, then
$N({\hat{\theta}_{n}^{(J)}-\hat{\theta}_{n}})=O_p(\delta^{J+1})$.
Hence, $ {\hat{\theta}_{n}^{(J)}-\hat{\theta}_{n}}=O_p(\delta^J). $
At the same time, we know
${\hat{\theta}_{n}}-\theta_0=O_p\{(n\delta)^{-1/2}\}$. Then
\[
{\hat{\theta}_{n}^{(J)}}-\theta_0=O_p\{\delta^J+(n\delta)^{-1/2}\}.
\]
This completes the proof of Theorem \ref{theo2}.
\end{pf*}
\begin{pf*}{Proof of Theorem \ref{theo4}} We only need to prove following result:
\[
\sqrt{n}\tilde{I}_n^{1/2}\bigl(\hat{\theta}_n^{(J)},J,\delta\bigr)\bigl(\hat{\theta
}_n^{(J)}-\theta_0\bigr)=\sqrt{n}I^{1/2}(\delta)\bigl(\hat{\theta}_n^{(J)}-\theta
_0\bigr)+o_p(1)
\]
under\vspace*{1pt} the two situations mentioned in Theorem \ref{theo4}. Using the approach
in the proof of Lemma \ref{lem3}, we have
$\tilde{I}_n(\hat{\theta}_n^{(J)},J,\delta)-\tilde{I}_n({\theta
}_0,J,\delta)=O_p\{\|\hat{\theta}_n^{(J)}-\theta_0\|_2\}$.
Also,\vspace*{1pt} using the same way of verifying (iii)\vspace*{1pt} in the proof of
Proposition \ref{prop5},
$\tilde{I}_n({\theta}_0,J,\delta)-\mathbb{E}\tilde{I}_n({\theta
}_0,J,\delta)=O_p\{(n\delta)^{-1/2}\}$.
By the same argument in the proof of Proposition~\ref{prop2},
$\mathbb{E}\tilde{I}_n({\theta}_0,J,\delta)-I(\delta)=O(\delta^{J+1})$.
Hence, if $n\delta^3\rightarrow\infty$, under either asymptotic
regime in Theorem \ref{theo4},
\[
\tilde{I}_n^{1/2}\bigl(\hat{\theta}_n^{(J)},J,\delta\bigr)=I^{1/2}(\delta
)\cdot\{1+o_p(1)\}.
\]
Then we complete the proof
\end{pf*}
\end{appendix}
\section*{Acknowledgments}
We thank the Associate Editor for very constructive comments and
suggestions which have improved the presentation of the paper.
The first author thanks Department of Statistics at Iowa State
University for hospitality during his visits.
|
{
"timestamp": "2012-03-12T01:00:49",
"yymm": "1203",
"arxiv_id": "1203.2004",
"language": "en",
"url": "https://arxiv.org/abs/1203.2004"
}
|
\section{Introduction}
\label{intro}
We study a class of special solutions to the
nonlinear Schr\"odinger equation
\beq
\label{1.1}
i \frac{ \p \Phi}{\p t} + \Delta \Phi + F( | \Phi |^2) \Phi = 0
\qquad \mbox{ in } \R^N,
\eeq
where $ \Phi$ is a complex-valued function on $\R^N$ satisfying the "boundary condition"
$|\Phi | \lra r_0 $ as $ |x| \lra \infty $,
$r_0 >0$ and $F$ is a real-valued function on $\R_+$ such that $ F(r_0^2) = 0$.
Equation (\ref{1.1}),
with the considered non-zero conditions at infinity, arises in the modeling of a
great variety of physical phenomena such as superconductivity, superfluidity
in Helium II, phase transitions and Bose-Einstein condensate
(\cite{AHMNPTB}, \cite{barashenkov2}, \cite{barashenkov1}, \cite{berloff}, \cite{coste}, \cite{GR},
\cite{gross}, \cite{IS}, \cite{JR}, \cite{JPR}, \cite{RB}).
In nonlinear optics, it appears in the context of dark solitons
(\cite{KL}, \cite{KPS}).
Two important model cases for (1.1) have been extensively studied
both in the physical and mathematical literature:
the Gross-Pitaevskii equation (where $F(s)= 1-s$)
and the so-called "cubic-quintic" Schr\"odinger equation (where
$F(s) = - \al _1 + \al _3 s - \al _5 s ^2 $, $\; \al_1, \, \al _3, \, \al _5$
are positive and
$F$ has two positive roots).
In contrast to the case of zero boundary conditions at infinity
(when the dynamics associated to (\ref{1.1}) is essentially governed
by dispersion and scattering), the non-zero boundary conditions allow a
much richer dynamics and
give rise to a remarkable variety of special solutions, such as
traveling waves, standing waves or vortex solutions.
\medskip
Using the Madelung transformation
$\Phi(x,t) = \sqrt{\rho(x,t)} e^{i\theta (x,t )}$
(which is well-defined in any region where $ \Phi \neq 0$),
equation (\ref{1.1}) is equivalent to a system of Euler's equations
for a compressible inviscid fluid of density $ \rho $ and velocity
$ 2 \nabla \theta$.
In this context it has been shown that, if $F$ is $C^1$ near $ r_0^2$
and $ F'(r_0^2) < 0$, the sound velocity at infinity associated to (\ref{1.1}) is
$ v_s = r_0 \sqrt{ - 2 F'(r_0^2) } $ (see the introduction of \cite{M8}).
If $ F'( r_0^2) <0 $ (which means that (\ref{1.1}) is defocusing),
a simple scaling enables us to assume that $ r_0 = 1 $ and $ F'(r_0 ^2) = -1$;
we will do so throughout the rest of this paper. The sound velocity at infinity is then $ v_s = \sqrt{2}$.
Equation (\ref{1.1}) has a Hamiltonian structure.
Indeed, let $ V( s) = \ii_s^{1} F( \tau ) \, d \tau$.
It is then easy to see that, at least formally, the "energy"
\beq
\label{1.2}
E(\Phi ) = \int_{\R^N} | \nabla \Phi |^2 \, dx
+ \int_{\R^N} V( |\Phi |^2) \, dx
\eeq
is conserved.
Another quantity which is conserved by the flow of (\ref{1.1}) is the momentum,
$ \mathbf{P} ( \Phi) = ( P_1( \Phi), \dots , P_N( \Phi ))$.
A rigorous definition of the momentum will be given in the next section.
If $\Phi$ is a function sufficiently localized in space,
we have $P_k( \Phi ) = \ii _{\R^N} \langle i \Phi_{x_k}, \Phi \rangle\, dx $,
where $ \langle \cdot , \cdot \rangle $ is the usual scalar product in
$\C \simeq \R^2$.
In a series of papers (see, e.g., \cite{barashenkov2}, \cite{barashenkov1},
\cite{GR}, \cite{JR}, \cite{JPR}),
particular attention has been paid to the traveling waves of (\ref{1.1}).
These are solutions of the form $ \Phi (x, t) = \psi (x + ct \omega)$, where
$ \omega \in S^{N-1}$ is the direction of propagation and $ c \in \R^*$ is
the speed of the traveling wave.
They are supposed to play an important role in the dynamics of (\ref{1.1}).
We say that $\psi $ has finite energy
if $ \nabla \psi \in L^2(\R^N)$ and $ V(|\psi |^2) \in L^1( \R^N)$.
Since the equation (\ref{1.1}) is rotation invariant, we may assume that
$ \omega = (1, 0, \dots, 0)$. Then a traveling wave of speed $c $ satisfies
the equation
\beq
\label{1.3}
i c \frac{\p \psi}{\p x_1} + \Delta \psi + F(|\psi|^2) \psi = 0
\qquad \mbox{ in } \R^N.
\eeq
It is obvious that a function $ \psi $ satisfies (\ref{1.3}) for some velocity
$c$ if and only if $\psi( - x_1, x')$ satisfies (\ref{1.3}) with $c$ replaced
by $-c$. Hence it suffices to consider the case $ c \geq 0$.
\medskip
In view of formal computations and numerical experiments, it has been conjectured
that finite energy traveling waves of speed $c$ exist only for subsonic speeds:
$ c < v_s$. The nonexistence of traveling waves for supersonic speeds ($c > v_s$)
has been proven first in \cite{gravejat1} in the case of the Gross-Pitaevskii
equation, then in \cite{M8} for a wide class of nonlinearities.
More qualitatively, the numerical investigation of the traveling waves
of the Gross-Pitaevskii equation ($F(s) = 1-s$) has been carried out
in \cite{JR}. The method used there was a continuation argument with
respect to the speed, solving \eqref{1.3} by Newton's algorithm.
Denoting $Q(\psi ) = P_1( \psi)$ the momentum of $\psi $ with respect
to the $x_1-$direction, the representation of solutions in the
energy vs. momentum diagram gives the following curves (the straight
line is the line $E = v_s Q$).
\begin{figure}[H]
\centerline{\psfig{figure=diagJR2.pdf,width=6.6cm,height=6.2cm}
\quad \quad \quad
\psfig{figure=diagJR3b.pdf,width=6.6cm,height=6.2cm} }
\caption{$(E,P)$ diagrams for (GP): (a) dimension $N=2$; (b) dimension $N=3$.}
\label{diaggrospit}
\end{figure}
The rigorous proof of the existence of traveling waves has been a
long lasting problem and was considered in a series of papers,
see \cite{BS}, \cite{BOS}, \cite{chiron}, \cite{BGS}, \cite{M10}.
At least formally, traveling waves are critical points of the functional
$E - cQ$. Therefore, it is a natural idea to look for such solutions
as minimizers of the energy at fixed momentum, the speed $c$ being then the
Lagrange multiplier associated to the minimization problem. In the case of
the Gross-Pitaevskii equation, in view of the above diagrams, this method
is expected to give the full curve of traveling waves if $N=2$ and
only the lower part that lies under the line $E = v_s Q$ if $N=3$ (because it is clear that minimizers of $E$
at fixed $Q$ cannot lie on the upper branch). On a rigorous level,
minimizing the energy at fixed momentum was used in \cite{BOS} to
construct a sequence of traveling waves with speeds $ c_n \lra 0 $
in dimension $N\geq 3$. Minimizing the energy $E$ at fixed momentum
$Q$ has the advantage of providing orbitally stable traveling waves,
and this is intimately related to the concavity of the curve $Q \mapsto E $.
On the other hand, if $Q \mapsto E $ is convex, as it is the case
on the upper branch in figure \ref{diaggrospit} (b), one expects
orbital instability.
More recently, the curves describing the minimum of the energy at fixed
momentum in dimension $2$ and $3$ have been obtained in \cite{BGS}, where
the existence of minimizers of $E$ under the constraint $Q = constant$
is also proven for any $ q >0$ if $N=2$,
respectively for any $ q \in (q_0, \infty)$ (with $q_0 >0$) if $N=3$.
The proofs in \cite{BGS} depend on the special algebraic structure of
the Gross-Pitaevskii nonlinearity and it seems difficult to extend them
to other nonlinearities. The existence of minimizers has been shown by
considering the corresponding problem on tori $\left( \R / 2 n \pi \Z \right)^N$,
proving a priori bounds for minimizers on tori, then passing to the
limit as $ n \lra \infty$. Although this method gives the existence of minimizers
on $\R^N$, it does not imply the precompactness of all minimizing sequences,
and therefore leaves the question of the orbital stability of
minimizers completely open.
\medskip
The existence of traveling waves for (\ref{1.1}) under general conditions
on the nonlinearity, in any space dimension $N \geq 3$ and for any speed
$c \in (0, v_s)$ has been proven in \cite{M10} by minimizing the action
$E - cQ$ under a Pohozaev constraint. The method in \cite{M10} cannot be
used in space dimension two (there are no minimizers under Pohozaev constraints).
Although the traveling waves obtained in \cite{M10} minimize the action $ E - cQ $
among all traveling waves of speed $c$, the constraint used to prove their
existence is not conserved by the flow of (\ref{1.1}) and consequently it
seems very difficult to prove their orbital stability (which is expected
at least for small speeds $c$).
\medskip
In the present paper we adopt a different strategy. If the nonlinear potential
$V$ is nonnegative, we consider the problem of minimizing the energy at
fixed momentum $Q = q$ and we show that in any space dimension $ N \geq 2$
there exist minimizers for any $ q \in (q_0, \infty)$, with $ q_0 \geq 0$.
The minimizers are traveling waves and their speeds are
the Lagrange multipliers associated to the variational problem.
These speeds tend to zero as $ q \lra \infty$. If $ N = 2$ and $F$ has a
good behavior near $1$ (more precisely, if assumption (A4) below is satisfied
and the "nondegeneracy condition" $ F''(1) \not= 3 $ holds), we prove that $ q_0 = 0 $ and that the speeds of the
traveling waves that we obtain
tend to $v_s$ as $ q \lra 0 $. For general nonlinearities we obtain the properties
of the minimum of the energy vs. momentum curve and this is in agreement
with the results in \cite{JR}, \cite{JPR} and \cite{BGS}. We also prove
the precompactness of all minimizing sequences for the above mentioned problem,
which implies the orbital stability of the set of traveling waves obtained in this way.
If $V$ achieves negative values (this happens, for instance, in the case of
the cubic-quintic NLS), the infimum of the energy in the set of functions of
constant momentum is always $ - \infty $. In this case we minimize the functional
$E - Q$ in the set of functions $ \psi $ satisfying
$ \ii_ { \R^N} |\nabla \psi |^2 \, dx = k$.
In space dimension $N \geq 2$ we prove that minimizers exist for any $ k $ in
some interval $ (k_0, k_{\infty})$
and, after scaling, they give rise to traveling waves.
Moreover, if $ N = 2$ and $F$ behaves nicely near $1$ we have $ k_0 = 0 $ and
the speeds of traveling waves obtained in this way tend to $ v_s$ as $ k \lra 0 $.
Let us emphasize that the result of \cite{M10}, which holds for any $N \geq 3 $,
does not require any sign assumption on the potential $V$.
In space dimension two, even if $V$ takes negative values
it is still possible to find local minimizers of the energy
under the constraint $ Q = q= constant $ if $q$ is not too large.
If $F$ satisfies assumption (A4) below and $ F''(1) \not= 3 $ this can be done
for any $q$ in some
interval $(0, q_{\infty})$ and the speeds of traveling waves obtained
in this way tend to $ v_s$ as $ q \lra 0 $. Moreover, we get the
precompactness of all minimizing sequences, and consequently the
orbital stability of the set of local minimizers.
Our results cover as well nonlinearities of Gross-Pitaevskii type and of
cubic-quintic type. To the best of our knowledge, all previous results in
the literature about the existence of traveling waves for (\ref{1.1})
in space dimension two are concerned only with the Gross-Pitaevskii equation
and the proofs make use of the specific algebraic properties of this nonlinearity.
The main disadvantage of the present approaches is that although we get
minimizers for any momentum in some interval $(q_0, \infty) $ or $(0 , q_{\infty}) $
or for any kinetic energy in some interval $(k_0, k_{\infty})$,
the speeds of the traveling waves obtained in this way are Lagrange multipliers, so we cannot
guarantee that these speeds cover a whole interval.
However, in all cases it can be proved that we get an uncountable set of speeds.
One might ask whether there is a relationship between the families of
traveling waves obtained from different minimization problems.
In dimension $ N \geq 3$ we prove that all traveling waves found in
the present paper also minimize the action $ E - cQ$ under the Pohozaev
constraint considered in \cite{M10}. The converse is, in general, not true.
For instance, in the case of the Gross-Pitaevskii equation in dimension
$ N \geq 3 $, it was proved in \cite{BGS, dL} that there are no traveling
waves of small energy, and we generalize that result in the present paper; this
implies that there is $ c_0 < v_s$ such that there are no traveling waves
of speed $c \in (c_0, v_s)$ which minimize the energy at fixed momentum.
However, if $N \geq 3$ the existence of traveling waves as minimizers of $E - cQ$ under
a Pohozaev constraint has been proven for any $ c \in (0, v_s)$. This is in
agreement with the energy-momentum diagram of figure \ref{diaggrospit} (b),
where the traveling waves with speed $ c $ close to the speed of sound $v_s$
are expected to be on the upper branch. We also prove that all minimizers of
the energy at fixed momentum are (after scaling) minimizers of
$ E - Q$ at fixed kinetic energy. It is an open question whether
the converse is true or not. An affirmative answer to this question
would imply that the set of speeds of traveling waves which minimize
the energy at fixed momentum is an interval.
However, this last fact might not be true, at least if we do not impose further conditions on the nonlinearity $ F $.
Indeed, in the case of general nonlinearities (as those studied
in dimension one in \cite{C1d}), the two-dimensional traveling waves to
(\ref{1.1}) have been studied numerically in \cite{CS}.
The numerical algorithms in \cite{CS} allow to perform the constrained
minimization procedures used is the present paper. It appears that for $N=2$,
even if the potential $V$ is nonnegative, it is not true in general that minimizing $E$ at
fixed $Q$ or minimizing $ E - Q$ at fixed kinetic energy provides
a single interval of speeds; for instance, it may provide the union
of two disjoint intervals.
\medskip
We will consider the following set of assumptions:
\medskip
{\bf (A1) } The function $F$ is continuous on $[0, \infty)$,
$C^1$ in a neighborhood of $1$, $F(1) = 0$ and $F'(1) < 0$.
\smallskip
{\bf (A2) }
There exist $C > 0$ and $ p _0 < \frac{2}{N-2} $ (with $p_0 < \infty $ if $ N =2$)
such that $|F(s) | \leq C(1 + s^{p_0}) $ for any $ s \geq 0$.
\smallskip
{\bf (A3) } There exist $C, \, \al _0> 0$ and $r_* > 1 $ such that $F(s) \leq - C s^{\al _0} $
for any $ s \geq r_*$.
\medskip
{\bf (A4) } $F$ is $C^2$ near $ 1 $ and
$$
F(s) = - ( s - 1) + \frac 12 F''( 1) ( s - 1) ^ 2 + \Oo (( s - 1)^3)
\qquad \mbox{ for $s$ close to } 1.
$$
If (A1) and (A3) are satisfied, it is explained in the introduction of \cite{M10}
how it is possible to modify $F$ is a neighborhood of infinity in such a
way that the modified function $\tilde{F}$ satisfies also (A2) and (\ref{1.1}) has
the same traveling waves as the equation obtained from it by replacing $F$
with $\tilde{F}$. If (A1) and (A2) hold, we get traveling waves as minimizers of
some functionals under constraints. However, if (A1) and (A3) are verified
but (A2) is not, the above argument implies only the existence of
such solutions, and not the fact that they are minimizers.
\medskip
If $F$ satisfies (A1), using Taylor's formula for $s$ in a neighborhood of $ 1 $ we have
\beq
\label{1.4}
V(s) = \frac 12 V''(1) ( s - 1)^2 + ( s - 1)^2 \e (s - 1)
= \frac 12 ( s - 1)^2 + ( s - 1)^2 \e (s - 1) ,
\eeq
where $ \e (t) \lra 0 \mbox{ as } t \lra 0.$
Hence for $|\psi |$ close to $ 1$, $V(|\psi |^2) $ can be approximated by
the Ginzburg-Landau potential $\frac 12 ( |\psi |^2 - 1)^2$.
\medskip
{\bf Energy and function spaces. }
We fix an odd function $ \ph \in C^{\infty } (\R)$ such that $ \ph (s) = s $
for $ s \in [0, 2 ]$, $ 0 \leq \ph ' \leq 1 $ on $ \R$ and
$ \ph (s) = 3 $ for $ s \geq 4 $.
If assumptions (A1) and (A2) are satisfied, it is not hard to see that
there exist $C_1, \, C_2, \, C_3 > 0$ such that
\beq
\label{i1}
\begin{array}{l}
|V(s)| \leq C_1 ( s - 1)^2 \quad \mbox{ for any } s \leq 9 ;
\\
\mbox{in particular, }
| V(\ph ^2(\tau )) | \leq C_1 (\ph ^2(\tau ) - 1) ^2 \mbox{ for any } \tau,
\end{array}
\eeq
\vspace*{-11pt}
\beq
\label{i2}
|V(b) - V(a)| \leq C_2 |b-a| \max (a^{p_0} , b^{p_0} )
\qquad \mbox{ for any } a, \, b \geq 2 .
\eeq
Given $ \psi \in H_{loc}^1 ( \R^N) $ and an open set $ \Om \subset \R^N$,
the modified Ginzburg-Landau energy of $\psi $ in $ \Om $ is defined by
\beq
\label{1.5}
E_{GL }^{\Om } (\psi ) = \ds \int_{\Om } |\nabla \psi |^2 \, dx
+ \frac 12 \int_{\Om } \left( \ph ^2(|\psi |) - 1 \right)^2 \, dx .
\eeq
We simply write $ E_{GL}(\psi) $ instead of $E_{GL}^{\R^N} (\psi)$.
The modified Ginzburg-Landau energy will play a central role in our analysis.
We denote
$ \dot{H}^1(\R^N) = \{ \psi \in L_{loc}^1(\R^N) \; | \; \nabla \psi \in L^2(\R^N) \}$ and
\beq
\label{E}
\begin{array}{rcl}
\Eo & = & \{ \psi \in \dot{H}^1(\R^N) \; \big| \; \ph^2(|\psi |) - 1 \in L^2( \R^N) \}
\\
& = & \{ \psi \in \dot{H}^1(\R^N) \; \big| \;
E_{GL}(\psi) < \infty \}.
\end{array}
\eeq
Let $\DR$ be the completion of $ C_c^{\infty } (\R^N) $
for the norm $\|v \| = \|\nabla v \| _{L^2 (\R^N) }$ and let
\beq
\label{X}
\begin{array}{rcl}
\Xo & = & \{ u \in \DR \; \big| \; \ph ^2(|1+ u|) - 1 \in L^2( \R^N) \}
\\
& = & \{ u \in \dot{H}^1(\R^N) \; \big| \;
u \in L^{2^*} (\R^N), \; E_{GL}(1+ u) < \infty \} \quad \mbox{ if } N \geq 3.
\end{array}
\eeq
If $ N \geq 3 $ and $ \psi \in \Eo$,
there exists a constant $ z_0 \in \C$ such that $ \psi - z_0 \in L^{2^*}(\R^N)$, where $ 2^* = \frac{2N}{N-2}$ (see, for instance, Lemma 7 and Remark 4.2 pp. 774-775 in \cite{PG}).
It follows that $ \ph (|\psi | ) - \ph(|z_0|) \in L^{2^*}(\R^N)$.
On the other hand, the fact that $E_{GL}(\psi ) < \infty $ implies $ \ph (|\psi | ) - 1 \in L^2(\R^N)$,
thus necessarily $ \ph(|z_0|) = 1$, that is $ |z_0 | = 1 $.
Then it is easily seen that there exist $ \al _0 \in [0, 2\pi)$ and $ u \in \Xo$, uniquely determined by $ \psi$, such that $ \psi = e^{i \al _0} ( 1+ u).$
In other words, if $ N \geq 3$ we have
$\Eo = \{ e^{i \al _0 } (1+ u) \; | \; \al _0 \in [0, 2\pi), \; u \in \Xo \}.$
It is not hard to see that for $N \geq 2$ we have
\beq
\label{L2}
\Eo = \{ \psi : \R^N \lra \C\; \big| \;
\psi \mbox{ is measurable, } |\psi | - 1 \in L^2( \R^N), \nabla \psi \in L^2(\R^N) \}.
\eeq
Indeed, we have $\big| \ph ^2 ( |\psi |) - 1\big| \leq 4 \big| \, | \psi | - 1 \big|$, hence
$ \ph ^2 ( |\psi |) - 1 \in L^2( \R^N)$ if $ |\psi | - 1 \in L^2( \R^N) $.
Conversely, let $ \psi \in \Eo$.
If $ N =2$, it follows from Lemma \ref{L2.1} below that $ |\psi |^2 - 1 \in L^2 ( \R^2)$
and we have
$\big| \, | \psi | - 1 \big| = \frac{1}{ |\psi | + 1} \big| \, | \psi |^2 - 1 \big| \leq \big| \, | \psi |^2 - 1 \big|.$
If $ N \geq 3$, we know that $ \ph (|\psi | )- 1 \in L^2( \R^N)$ and
$ 0 \leq |\psi | - \ph(|\psi |) \leq |\psi | \1_{\{ |\psi | \geq 2 \} }
\leq 2 ( |\psi | - 1) \1_{\{ |\psi | \geq 2 \} }
\leq 2 \big| \, |\psi | - 1 \big| ^{ \frac{ 2^*}{2} } \1_{\{ |\psi | \geq 2 \} } $
and the last function belongs to $ L^2( \R^N)$ by the Sobolev embedding.
Moreover, one may find bounds for $\| \, |\psi | - 1 \|_{L^2( \R^N)} $ in terms of $E_{GL}(\psi)$ (see Corollary \ref{C4.2} below).
Proceeding as in \cite{PG2}, section 1, one proves that $ \Eo \subset L^2 + L^{\infty}(\R^N)$
and that $ \Eo$ endowed with the distance
\beq
\label{distance}
d_{\Eo}(\psi _1, \psi _2) = \| \psi _1 - \psi _2 \| _{L^2 + L^{\infty}(\R^N)}
+ \| \nabla \psi _1 - \nabla \psi _2 \| _{L^2 (\R^N)}
+ \| \, |\psi _1| - |\psi _2 |\, \| _{L^2(\R^N)}
\eeq
is a complete metric space. We recall that, given two Banach spaces $X$ and $Y$ of distributions
on $ \R^N$, the space $X + Y$ with norm defined by
$\| w \|_{X + Y} = \inf \{ \| x \|_X + \| y \|_Y \; \big| \; w = x + y, x \in X , y \in Y \}$ is a Banach space.
We will also consider the following semi-distance on $ \Eo$:
\beq
\label{semidistance}
d_0(\psi _1, \psi _2) =\| \nabla \psi _1 - \nabla \psi _2 \| _{L^2 (\R^N)}
+ \| \, |\psi _1| - |\psi _2 |\, \| _{L^2(\R^N)} .
\eeq
If $\psi _1, \psi _2 \in \Eo $ and $d_0(\psi _1, \psi _2) = 0$,
then we have $|\psi _1 | = |\psi _2 |$ a.e. on $\R^N$
and $ \psi _1 - \psi _2$ is a constant (of modulus not exceeding $ 2 $) a.e. on $\R^N$.
In space dimension $ N = 2, 3, 4$, the Cauchy problem for the Gross-Pitaevskii
equation has been studied by Patrick G\'erard (\cite{PG, PG2}) in the space
naturally associated to that equation, namely
$$
\E = \{ \psi \in H_{loc}^1(\R^N) \; | \; \nabla \psi \in L^2( \R^N), |\psi |^2 - 1 \in L^2( \R^N) \}
$$
endowed with the distance
$$
d_{\E} (\psi _1, \psi _2) = \| \psi _1 - \psi _2 \| _{L^2 + L^{\infty}(\R^N)}
+ \| \nabla \psi _1 - \nabla \psi _2 \| _{L^2 (\R^N)}
+ \| \, |\psi _1| ^2 - |\psi _2 | ^2 \, \| _{L^2(\R^N)} .
$$
If $ N = 2, 3 $ or $4$
it can be proven that $ \E = \Eo $ and the distances $ d_{\Eo} $
and $ d_{\E}$ are equivalent on $ \Eo$.
Global well-posedness was shown in \cite{PG, PG2} (see section \ref{sectionorbistab}) if
$ N \in \{ 2, 3 \} $ or if $N= 4$ and the initial data is small.
In the case $ N=4$, global well-posedness for any initial data in $\E$ was recently proven in \cite{KOPV}.
\medskip
{\bf Notation. }
Throughout the paper, $ \Lo ^N$ is the Lebesgue measure on $ \R^N$ and
$\mathcal{H}^{s}$ is the $s-$dimensional Hausdorff measure on $ \R^N$.
For $ x = ( x_1, \dots, x_N) \in \R^N$,
we denote $ x' = ( x_2, \dots , x_N ) \in \R^{N-1}$.
We write $\langle z_1, z_2 \rangle $ for the scalar product
of two complex numbers $z_1, z_2$.
Given a function $f$ defined on $ \R^N$ and $ \la , \, \si > 0$,
we denote
\beq
\label{scale}
f_{\la, \si } (x) = f\left( \frac{x_1}{\la }, \frac{x'}{\si} \right).
\eeq
If $ 1 \leq p <N$, we write $ p^*$ for the Sobolev exponent
associated to $p$, that is $ \frac{1}{p^*} = \frac 1p - \frac 1N$.
\medskip
{\bf Main results. } Our most important results can be summarized as follows.
\begin{Theorem}
\label{T1.1}
Assume that $N\geq 2$, (A1) and (A2) are satisfied and $V \geq 0$ on $[0, \infty)$.
For $q \geq 0$, let
$$
E_{min}(q) = \inf \{ E( \psi) \; \big| \; \psi \in \Eo, \; Q( \psi ) = q \}.
$$
Then:
\medskip
(i) The function $E_{min}$ is concave, increasing on $[0, \infty)$,
$ E_{min}(q) \leq v_s q $ for any $ q \geq 0$, the right derivative of
$E_{min} $ at $0$ is $ v_s$, and $ E_{min}(q) \lra \infty$ and
$ \frac{ E_{min}(q) }{q} \lra 0$ as $ q \lra \infty$.
\medskip
(ii) Let $ q_0 = \inf \{ q > 0 \; | \; E_{min}(q) < v_s q \}.$
For any $ q > q_0$, all sequences $(\psi_n)_{n \geq 1} \subset \Eo $
satisfying $Q( \psi _n) \lra q $ and $E(\psi _n) \lra E_{min}(q)$ are
precompact for $d_0$ (modulo translations).
The set $ \So _q = \{ \psi \in \Eo \; | \; Q( \psi ) = q, \; E( \psi ) = E_{min}(q) \}$
is not empty and is orbitally stable (for the semi-distance
$d_0$) by the flow associated to (\ref{1.1}).
\medskip
(iii) Any $ \psi_q \in \So _q $ is a traveling wave for (\ref{1.1}) of speed
$ c( \psi _q) \in [ d^+ E_{min}(q), d^- E_{min}(q) ]$, where we denote by
$d^- $ and $d^+$ the left and right derivatives. We have
$ c( \psi _q) \lra 0 $ as $ q \lra \infty$.
\medskip
(iv) If $ N \geq 3 $ we have always $ q_0 > 0 $. Moreover, if $ N =2$ and
assumption (A4) is satisfied, we have $ q_0 = 0$ if and only if $F''(1) \not = 3 $,
in which case $ c( \psi _q) \lra v_s $ as $ q \lra 0$.
\end{Theorem}
If $V$ achieves negative values, the infimum of $E$ on the set
$\{ \psi \in \Eo \; | \; Q( \psi ) = q \} $ is $- \infty$ for any $q$.
In this case we prove the existence of traveling waves by minimizing the functional
$I(\psi ) = - Q( \psi ) + \ii_{\R^N} V(|\psi |^2) \, dx $
(or, equivalently, the functional $E - Q$)
under the constraint $ \ii_{\R^N} |\nabla \psi |^2 \, dx = k.$
More precisely, we have the following results:
\begin{Theorem}
\label{T1.2}
Assume that $N\geq 2$ and (A1), (A2) are satisfied. For $ k \geq 0 $, let
$$
I_{min}(k) = \inf \Big\{ I( \psi) \; \big| \; \psi \in \Eo, \;
\ii_{\R^N} |\nabla \psi |^2 \, dx = k \Big\}.
$$
Then there is $k_{\infty} \in (0, \infty]$ such that the following holds:
\medskip
(i) For any $ k > k_\infty $, $I_{min}(k) = - \infty$. The function
$I_{min}$ is concave, decreasing on $[0, k_\infty)$,
$ I_{min}(k) \leq - k / v_s^2 $ for any $ k \geq 0$, the right derivative
of $ I_{min} $ at $0$ is $ - 1/ v_s^2 $, and $ \frac{ I_{min}(k) }{k} \lra - \infty$
as $ k \lra \infty$.
\medskip
(ii) Let $ k_0 = \inf \{ k > 0 \; | \; I_{min}(k) < - k / v_s^2 \} \in [ 0, k_\infty]$.
For any $ k \in (k_0, k_{\infty})$, all sequences $(\psi_n)_{n \geq 1} \subset \Eo $
satisfying $\ii_{\R^N} |\nabla \psi_n |^2 \, dx \lra k $ and
$ I(\psi _n) \lra I_{min}(k)$ are precompact for $d_0$ (modulo translations).
If $ \psi_k \in \Eo $ is a minimizer for $I_{min}(k)$, there exists
$ c = c( \psi _k) \in \left[ \sqrt{ - 1 / d^+ I_{min}(k)}, \sqrt{ - 1 / d^- I_{min}(k)} \right]$
such that $ \psi_k ( \frac{\cdot}{c} )$ is a non constant traveling wave
of (\ref{1.1}) of speed $c(\psi_k)$.
\medskip
(iii) We have $ k_{\infty} < \infty$ if and only if ($N= 2$ and $ \inf V < 0$).
If $ k_{\infty} = \infty$, the speeds of the traveling waves obtained
from minimizers of $I_{min}(k)$ tend to $0$ as $ k \lra \infty$.
\medskip
(iv) For $ N \geq 3$, we have $ k_0 > 0 $. If $N=2$ and assumption (A4)
is satisfied we have $ k_0 = 0$ if and only if $F''(1) \not= 3 $, in which
case the speeds of the traveling waves obtained from minimizers of $I_{min}(k)$
tend to $ v_s $ as $ k \lra 0 $.
\end{Theorem}
In space dimension two, the tools developed to prove Theorem \ref{T1.2}
enable us to find minimizers of $E$ at fixed momentum on a subset of
$ \Eo$ even if $V$ achieves negative values. We have:
\begin{Theorem}
\label{T1.3}
Assume that $N = 2$ and that (A1), (A2) are satisfied. Let
$$ E_{min}^{\sharp}(q) = \inf \Big\{ E( \psi) \; | \; \psi \in \Eo, \; Q( \psi ) = q \;
\mbox{ and } \; \ii_{\R^2} V(|\psi|^2) dx \geq 0 \Big\}.$$
Then:
\medskip
(i) The function $E_{min}^{\sharp}$ is concave, nondecreasing on $[0, \infty)$,
$ E_{min}^{\sharp}(q) \leq v_s q $, $d^+E_{min}^{\sharp}(0) = v_s$ and
$ E_{min }^{\sharp}(q) \leq k_{\infty}$ for any $ q > 0$, where
$ k_{\infty}$ is as in Theorem \ref{T1.2}.
\medskip
(ii) Let $ q_0^\sharp = \inf \{ q > 0 \; \big| \; E_{min}^\sharp(q) < v_s q \} \in [ 0, \infty]$
and $ q_\infty^\sharp = \sup \{ q > 0 \; \big| \; E_{min}^{\sharp}(q) < k_\infty \} \in ( 0, \infty]$.
Then $ q_0^\sharp \leq q_\infty^\sharp $ and for any $q \in ( q_0^\sharp , q_\infty^\sharp)$,
all sequences $(\psi_n)_{n \geq 1} \subset \Eo $ satisfying
$Q( \psi _n) \lra q $ and $E(\psi _n) \lra E_{min}^{\sharp}(q)$ are
precompact for $d_0$ (modulo translations).
The set $ \So _q^{\sharp} = \{ \psi \in \Eo \; \big| \; Q( \psi ) = q, \;
E( \psi ) = E_{min}^{\sharp}(q) \}$ is not empty and is orbitally stable
by the flow of (\ref{1.1}) for the semi-distance $d_0$.
\medskip
(iii) Any $ \psi_q \in \So _q ^{\sharp}$ verifies $ \ii_{\R^2} V(|\psi_q|^2) dx > 0 $,
hence minimizes $E$ under the constraint $Q=q$ in the open set
$ \{ w\in \Eo \; \big| \; \ii_{\R^2} V(|w|^2) dx > 0 \} $. Therefore, it is a
traveling wave for (\ref{1.1}) of speed
$ c( \psi _q) \in [ d^+ E_{min}^\sharp(q), d^- E_{min}^\sharp(q) ]$.
\medskip
(iv) If assumption (A4) is satisfied, we have $ q_0^\sharp = 0 $ if and
only if $F''(1) \not = 3$, and in this case $ c( \psi _q) \lra v_s $ as $q \lra 0 $.
\end{Theorem}
We may observe that in Theorem \ref{1.2} it may happen that
$ k_0 = k_\infty $, and then (ii) never occurs. Statements (iii)
and (iv) in Theorem \ref{1.2} provide sufficient conditions to have
$ k_0 < k_\infty $. Actually, this is always the case if $ N \geq 3 $.
In the case $ N = 2$, we have $ k_0 < k_\infty $ if $\inf V \geq 0 $,
or if ($\inf V < 0 $, $F$ verifies assumption (A4) and $F''(1) \not = 3$).
Notice that the main physical example of nonlinearity satisfying $ \inf V < 0 $
is the cubic-quintic nonlinearity, for which one has $ F''(1) \neq 3 $.
In the same way, in Theorem \ref{1.3} it may happen that
$ q_0^\sharp = q_\infty^\sharp $, in which case (ii) never holds, but
here again, under assumption (A4), this is possible only if $ F''(1) = 3 $.
\bigskip
We conclude with a result concerning the nonexistence of small energy solutions
to ({\ref{1.3}). This is a sharp version of a result proven in \cite{BGS} for the Gross-Pitaevskii
nonlinearity in dimension $N=3$, then extended to $N \geq 4$
in \cite{dL}.
The cases where $ q_0 > 0 $, $ k_0 > 0 $ or $ q_0^\sharp > 0 $ in the above
theorems follow directly from this result.
\begin{Proposition}
\label{smallE}
Assume that $N \geq 2 $ and that $F$ verifies (A1) and ((A2) or (A3)). Suppose that either
$\bullet$ $N \geq 3$, or
$\bullet$ $N = 2$, $F$ satisfies (A4) and $ F''(1) = 3 $.
\\
The following holds.
\medskip
(i) There is $ k_* > 0$, depending only on $ N $ and $F$, such that
if $ c \in [0, v_s] $ and if $ U \in \Eo $ is a solution
to (\ref{1.3}) satisfying $ \ii_{\R^N} |\nabla U |^2 \, dx < k_*$,
then $ U $ is constant.
\medskip
(ii) Assume, moreover, that $F$ satisfies (A2) with $ p_0 < \frac 2N$ or $F$ satisfies (A3).
There is $ \ell_* > 0$, depending only on $ N $ and $F$, such that any solution $ U \in \Eo $ to (\ref{1.3})
with $ c \in [0, v_s]$ and $ \ii_{\R^N} \left( |U|^2 - 1 \right)^2 \, dx < \ell_* $ is constant.
\end{Proposition}
{\bf Outline of the paper.}
In the next section we give a convenient definition of the momentum and we study
its basic properties. In section \ref{reg} we present a regularization procedure which
enables us to eliminate the small-scale topological defects of functions in $ \Eo$.
The tools introduced in sections \ref{mom} and \ref{reg} will be crucial for the variational
machinery developed later. In section \ref{minem} we consider the problem of minimizing
the energy at fixed momentum and we prove Theorem \ref{T1.1}.
We also develop some analytical tools that will be useful elsewhere.
In section \ref{another} we consider the problem of minimizing the functional $E-Q$ when
the kinetic energy is fixed and we prove Theorem \ref{T1.2}.
Section \ref{local} is devoted to the proof of Theorem \ref{T1.3}.
The orbital stability of the set of traveling waves provided by Theorems \ref{T1.1} and \ref{T1.3}
is proven in section \ref{sectionorbistab}.
In section \ref{3fam} we investigate the relationship between the traveling waves given by
Theorems \ref{T1.1}, \ref{T1.2}, \ref{T1.3} above and those found in \cite{M10}.
In section \ref{slow} we show that if the stationary variant of (\ref{1.1}) admits nontrivial solutions,
the traveling waves found in the present paper converge to the ground states of the stationary equation as $ c \lra 0$.
The small energy solutions are studied in section \ref{sectionsmallenergy}, where we prove Proposition \ref{smallE}.
\section{The momentum}
\label{mom}
The momentum (with respect to the $x_1$ direction) should be a functional
defined on $ \Eo$ whose "G\^ateaux differential"\footnote{We did not introduce a
manifold structure on $ \Eo$, although this can be done in a natural way,
see \cite{PG, PG2}. However, it will be clear (see (\ref{gateaux})) what
we mean here by "G\^ateaux differential." } is $2 i \p_{x_1}$.
In dimension $ N \geq 3$, it has been shown in \cite{M10} how to define
the momentum on $\Xo$ (and, consequently, on $\Eo$). In this section we
will extend that definition in dimension $N=2$.
It is clear that on the affine space $ 1 + H^1(\R^N) \subset \Eo$,
the momentum should be defined by
$Q(1+ u) = \ii_{\R^N} \langle i u _{x_1}, u \rangle \, dx .$
In order to define the momentum on the whole $ \Eo$,
we introduce the space $ \Yo = \{ \p _{x_1} \phi \; | \; \phi \in \dot{H}^1(\R^N) \}$.
It is easy to see that $\Yo $ endowed with the norm
$\| \p _{x_1} \phi \|_{\Yo} = \| \nabla \phi \| _{L^2(\R^N )} $ is a Hilbert space.
In dimension $ N \geq 3$, it follows from Lemmas 2.1 and 2.2 in \cite{M10}
that for any $ u \in \Xo $ we have $\langle i u_{x_1} , u \rangle \in L^1(\R^N) + \Yo.$
If $ N \geq 3 $ and $ \psi \in \Eo$, we have already seen there are
$ u \in \Xo $ and $ \al _0 \in [0, 2 \pi) $ such that $\psi = e^{i \al _0 } (1+ u).$
An easy computation gives
$\langle i \psi_{x_1} , \psi \rangle = Im (u_{x_1} ) + \langle i u_{x_1} , u \rangle $ and it is obvious that
$ Im (u_{x_1} ) \in \Yo$, thus $ \langle i \psi_{x_1} , \psi \rangle \in L^1(\R^N) + \Yo.$
The next Lemma shows that a similar result holds if $N=2$.
\begin{Lemma} \label{L2.1}
Let $ N \! = \!2. $ For any $ \psi \in \Eo $ we have $ |\psi |^2 - 1 \in L^2(\R^2)$ and~$\langle i \psi _{x_1}, \psi \rangle \in L^1(\R^2) + \Yo.$
\end{Lemma}
{\it {Proof.}}
The following facts, borrowed from \cite{brezis-lieb}, will be useful here and in the sequel:
for any $ q \in [2, \infty )$ there is $C_q >0$ such that for all $ \phi \in L_{loc}^1(\R^2) $ satisfying
$\nabla \phi \in L^2(\R^2)$ and $\Lo ^2( supp (\phi ) ) < \infty $ we have
\beq
\label{ineq1}
\| \phi \| _{L^q(\R^2)} \leq
C_q \| \nabla \phi \|_{L^1(\R^2)} ^{\frac 2q} \| \nabla \phi \|_{L^2(\R^2)} ^{1- \frac 2q}
\eeq
(see inequality (3.12) p. 108 in \cite{brezis-lieb}).
Since $ \nabla \phi = 0 $ a.e. on $ \{ \phi = 0 \}$,
(\ref{ineq1}) and the Cauchy-Schwarz inequality give
\beq
\label{ineq2}
\| \phi \| _{L^q(\R^2)} \leq
C_q \| \nabla \phi \|_{L^2(\R^2)} \left( \Lo ^2(\{ \phi (x) \neq 0 \} ) \right)^{\frac 1q}.
\eeq
Notice that (\ref{ineq2}), which is a variant of inequality (3.10) p. 107 in \cite{brezis-lieb}, holds for any $q \in [1, \infty)$.
Let $ \psi \in \Eo$. It is clear that
\beq
\label{e1}
\ds \int_{\{ | \psi | \leq 2 \}} (|\psi |^2 - 1)^2 \, dx
= \int_{\{ | \psi | \leq 2 \}} (\ph ^2(|\psi | ) - 1 )^2 \, dx < \infty .
\eeq
Obviously,
$\Lo ^2(\{ |\psi | \geq \frac 32 \}) < \infty$ (because $E_{GL}(\psi ) < \infty$) and
$ |\psi |^2 - 1 \leq C ( |\psi |- \frac 32 ) ^2 $ on $\{ |\psi | \geq 2 \}$.
Using (\ref{ineq2}) for $\phi = ( |\psi |- \frac 32 ) _+$ (which satisfies
$ |\nabla \phi | \leq | \nabla \psi | \1_{ \{ |\psi | \geq \frac 32 \} }$ a.e.)~we~get
\beq
\label{e2}
\ds \int_{\{ | \psi | > 2 \}} \left(|\psi |^2 - 1 \right)^2 dx
\leq C \int \left( |\psi | - \frac 32 \right)_+ ^4 \, dx
\leq C \| \nabla \psi \|_{L^2(\R^2)} ^4 \Lo ^2( \{ |\psi | \geq \frac 32 \}) < \infty.
\eeq
Thus $ |\psi |^2 - 1 \in L^2(\R^2)$.
It follows from Theorem 1.8 p. 134 in \cite{PG2} that there exist $w \in H^1(\R^2)$ and
a real-valued function $\phi$ on $\R^2$
such that $\phi \in L_{loc}^2(\R^2)$, $\; \p^{\al } \phi \in L^2(\R^2)$
for any $ \al \in \N^2$ with $|\al | \geq 1$ and
\beq
\label{2.0}
\psi = e^{i \phi } + w.
\eeq
A simple computation gives
\beq
\label{2.1}
\langle i \psi _{x_1}, \psi \rangle
= - \frac{\p \phi}{\p x_1}
+ \frac{\p }{\p x_1} \left( \langle iw, e^{i \phi} \rangle \right)
- 2 \langle \frac{ \p \phi}{\p x_1} e^{ i \phi } , w \rangle
+ \langle i w_{x_1}, w \rangle.
\eeq
The Cauchy-Schwarz inequality implies that
$\langle \phi _{x_1} e^{i \phi } , w \rangle $ and $ \langle i w _{x_1}, w \rangle $
belong to $L^1(\R^2)$. It is obvious that $\frac{\p \phi}{\p x_1} \in \Yo$.
We have $ \langle iw, e^{i \phi} \rangle \in L^2 (\R^2)$ and
$$
\frac{\p }{\p x_j} \left( \langle iw, e^{i \phi} \rangle \right)
= \langle i\frac{\p w}{\p x_j}, e^{i \phi} \rangle + \langle w, \frac{\p \phi}{\p x_j}
e^{i \phi} \rangle.
$$
The fact that $ w $ and $ \frac{ \p \phi}{\p x_j}$ belong to $H^1(\R^2)$ and the Sobolev
embedding give
$w, \; \frac{ \p \phi}{\p x_j} \in L^p(\R^2)$ for any $ p \in [2, \infty)$, hence
$\langle w, \frac{\p \phi}{\p x_j} e^{i \phi} \rangle \in L^p(\R^2)$ for any
$ p \in [1, \infty)$.
Since $\langle i \frac{\p w }{\p x_j}, e^{i \phi} \rangle \in L^2(\R^2)$, we get
$\frac{\p }{\p x_j} \left( \langle iw, e^{i \phi} \rangle \right) \in L^2(\R^2)$, hence
$\langle iw, e^{i \phi} \rangle \in H^1(\R^2)$ and consequently
$ \frac{\p }{\p x_1} \left( \langle iw, e^{i \phi} \rangle \right) \in \Yo.$
The proof of Lemma \ref{L2.1} is complete.
\hfill $\Box$
\medskip
For $ v \in L^1(\R^N)$ and $ w \in \Yo $, let $ L(v + w ) = \ii _{\R^N} v (x) \, dx$.
It follows from Lemma 2.3 in \cite{M10} that $L$ is well-defined and that it is a
continuous linear functional on $ L^1(\R^N) + \Yo$. Taking into account Lemma
\ref{L2.1} and the above considerations, for any $N \geq 2$ we give the following
\begin{Definition}
\label{D2.2}
Given $ \psi \in \Eo$, the momentum of $\psi$ with respect to the
$x_1-$direction is
$$
Q(\psi) = L( \langle i \psi_{x_1}, \psi \rangle ).
$$
\end{Definition}
Notice that the momentum (with respect to the
$x_1-$direction) has been defined in \cite{M10}
for functions $u \in \Xo $ by $ \tilde{Q}(u) = L( \langle i \frac{ \p u}{\p x_1} , u \rangle)$.
If $\psi = e^{i \al _0} ( 1+ u)$, it is easy to see that
$Q(\psi )= \tilde{Q}(u) $.
\medskip
If $ \psi \in \Eo $ is symmetric with respect to $ x_1$ (in particular, if $\psi $ is radial), then
$
Q(\psi) = Q( \psi (- x_1, x')) = - Q(\psi) ,
$ hence $ Q(\psi) = 0$.
\medskip
If $ \psi \in \Eo $ has a lifting $ \psi = \rho e^{ i \theta } $ with $ \rho ^2 - 1 \in L^2( \R^N)$
and $ \theta \in \dot{H}^1( \R^N)$
(note that if $ 2 \leq N \leq 4$ we have always $|\psi |^2 - 1 \in L^2 ( \R^N)$ by (\ref{L2}) and
the Sobolev embedding), then
\beq
\label{lift}
Q( \psi ) = L( - \rho ^2 \theta _{x_1}) = - \ds \int_{\R^N} ( \rho ^2 - 1) \theta _{x_1} \, dx.
\eeq
The next Lemma is an "integration by parts" formula.
\begin{Lemma}
\label{L2.3}
For any $ \psi \in \Eo$ and $ v \in H^1 ( \R^N) $ we have
$\langle i\psi _{x_1}, v \rangle \in L^1( \R^N)$,
$\langle i\psi , v_{x_1} \rangle \in L^1( \R^N) + \Yo $ and
\beq
\label{}
L(\langle i\psi_{x_1}, v \rangle + \langle i\psi , v_{x_1} \rangle) = 0.
\eeq
\end{Lemma}
{\it Proof.} If $ N \geq 3$ this follows immediately from Lemma 2.5 in \cite{M10}.
We give the proof in the case $N=2$. The Cauchy-Schwarz inequality implies
$\langle i\psi_{x_1}, v \rangle \in L^1(\R^2).$
Let $w \in H^1(\R^N)$ and $ \phi $ be as in (\ref{2.0}), so that
$ \psi = e^{i \phi } + w$. Then
\beq
\label{2.5}
\langle i\psi , v_{x_1} \rangle
= \frac{\p }{\p x_1} \left( \langle i e ^{i \phi } , v \rangle \right)
+ \langle \phi _{x_1} e ^{i \phi } , v \rangle
+ \langle i w, v _{x_1} \rangle .
\eeq
From the Cauchy-Schwarz inequality we have
$\langle \phi _{x_1} e ^{i \phi } , v \rangle \in L^1(\R^2) $ and
$ \langle i w, v _{x_1}\rangle \in L^1(\R^2) $.
As in the proof of Lemma \ref{L2.1} we obtain
$\langle i e ^{i \phi } , v \rangle \in H^1(\R^2)$, hence
$ \frac{\p }{\p x_1} \left( \langle i e ^{i \phi } , v \rangle \right) \in \Yo$.
We conclude that $\langle i\psi , v_{x_1} \rangle \in L^1( \R^N) + \Yo $.
Using (\ref{2.0}), (\ref{2.5}) and the definition of $L$ we get
$$
L(\langle i\psi_{x_1}, v \rangle + \langle i\psi , v_{x_1} \rangle)
= L(\langle i w_{x_1}, v \rangle + \langle i w , v_{x_1} \rangle)
= \int_{\R^N} \langle i w_{x_1}, v \rangle + \langle i w , v_{x_1} \rangle \ dx
$$
and the last quantity is zero by the standard integration by parts formula for
functions in $H^1(\R^2)$ (see, e.g., \cite{brezis} p. 197). \hfill $\Box$
\begin{Corollary}
\label{C2.4}
Let $ \psi_1, \; \psi_2 \in \Eo $ be such that $ \psi _1 -\psi _2 \in L^2( \R^N)$.
Then
\beq
\label{2.7}
|Q(\psi _1) - Q(\psi _2) | \leq \|\psi _1 - \psi _2\|_{L^2(\R^N)}
\left( \Big\| \frac{\p \psi _1}{\p x_1} \Big\| _{L^2(\R^N)}
+ \Big\| \frac{\p \psi _2}{\p x_1} \Big\| _{L^2(\R^N)} \right)
\eeq
\end{Corollary}
{\it Proof. } The same as the proof of Corollary 2.6 in \cite{M10}.
\hfill
$\Box$
\medskip
Let $ \psi \in \Eo$.
It is easy to see that for any function with compact support $ \phi \in H^1(\R^N)$
we have $ \psi + \phi \in \Eo$ and using Lemma \ref{L2.3} we get
\beq
\label{gateaux}
\lim_{t \ra 0 } \frac 1t (Q(\psi + t \phi ) - Q(\psi ) )
= L(\langle i \psi_{x_1}, \phi \rangle + \langle i \phi_{x_1}, \psi \rangle )
= 2 \int_{\R^N} \langle i \psi _{x_1} , \phi \rangle \, dx.
\eeq
The momentum has a nice behavior with respect to dilations:
for $ \psi \in \Eo$, $ \la, \; \si > 0$ we have
\beq
\label{2.9}
Q(\psi_{\la, \si}) = \si ^{N-1} Q(\psi).
\eeq
\section{A regularization procedure}
\label{reg}
The regularization procedure described below will be an important tool
for our analysis. It was first introduced in
\cite{AB}, then developed in \cite{M10}, where it was a key ingredient in proofs.
It enables us to
get rid of the small-scale topological defects of functions and in
the meantime to control the Ginzburg-Landau energy and the momentum of the regularized functions.
In this section $\Om $ is an open set in $ \R^N$.
We do not assume $ \Om $ bounded, nor connected.
If $ \p \Om \neq \emptyset$, we assume that $ \p \Om $ is $ C^2$.
Fix $ \psi \in \Eo $ and $ h > 0$.
We consider the functional
$$
G_{h, \Om }^{\psi} (\zeta) = \left\{
\begin{array}{ll}
\ds E_{GL}^{\Om } (\zeta) + \frac{1}{h^2} \int_{\Om} |\zeta - \psi |^2 \ dx \qquad
& \mbox{ if } N=2,
\\ \\
\ds E_{GL}^{\Om } (\zeta) +
\frac{ 1}{h^2} \ds \int_{\Om } \ph \left( |\zeta - \psi |^2 \right) \, dx
& \mbox{ if } N \geq 3.
\end{array}
\right. $$
Note that $G_{h, \Om }^{\psi} (\zeta) $ may equal $ \infty $ for some $ \zeta \in \Eo $;
however, $G_{h, \Om }^{\psi} (\zeta) $ is finite whenever $ \zeta \in \Eo $ and
$ \zeta - \psi \in L^2( \Om )$.
We denote
$ H_0^1(\Om ) = \{ u \in H^1( \R^N) \; | \; u = 0 \mbox{ on } \R^N \setminus \Om \}$
and
$$
H_{\psi} ^1 ( \Om ) = \{ \zeta \in \Eo \; | \; \zeta - \psi \in H_0^1 ( \Om ) \}.
$$
Assume that $ N \geq 3$ and $ \psi = e^{i \al _0 } ( 1+ u ) \in \Eo$,
where $ \al _ 0 \in [0, 2 \pi ) $ and $ u \in \Xo$.
Then
$$
H_{\psi} ^1 ( \Om ) = \{ e^{i \al _0 } ( 1+ v) \; | \; v \in H_u^1(\Om) \}.
$$
Let
$$
\tilde{G}_{h, \Om }^u (w) = E_{GL}^{\Om } (1+ w) +
\frac{ 1}{h^2} \ds \int_{\Om } \ph \left( |w-u|^2 \right) \, dx.
$$
It is obvious that $ \zeta = e^{i \al _0 } ( 1+ v)$ is a minimizer of
$ G_{h, \Om }^{\psi} $ in $ H_{\psi}^1(\Om)$ if and only if $v$ is a minimizer
of $\tilde{G}_{h, \Om }^u $ in $ H_u^1(\Om)$, hence the results proved
in \cite{M10} for minimizers of $\tilde{G}_{h, \Om }^u $ also hold
for minimizers of $ G_{h, \Om }^{\psi} $.
\medskip
The next three lemmas are analogous to Lemmas 3.1, 3.2 and 3.3 in \cite{M10}.
For the convenience of the reader we give the full statements in any space
dimension, but for the proofs in the case $ N \geq 3 $ we refer to \cite{M10};
we only indicate here what changes in proofs if $N=2$.
\begin{Lemma}
\label{L3.1}
(i) The functional $ G_{h, \Om }^{\psi} $ has a minimizer in $H_{\psi} ^1 ( \Om ) $.
(ii) Let $ \zeta _h$ be a minimizer of $ G_{h, \Om }^{\psi} $ in $H_{\psi} ^1 ( \Om ) $.
There exist constants $C_i >0$, depending only on $N$, such that:
\beq
\label{3.1}
E_{GL}^{\Om } ( \zeta_h )\leq E_{GL}^{\Om } ( \psi );
\eeq
\vspace{-10pt}
\beq
\label{3.2}
\| \zeta_h - \psi \|_{L^2( \Om ) } ^2 \leq
\left\{
\begin{array}{ll}
h^2 E_{GL}^{\Om } ( \psi ) & \qquad \mbox{ if } N=2,
\\
\\
h^2 E_{GL}^{\Om } ( \psi )
+ C _1 \left( E_{GL}^{\Om } ( \psi ) \right)^{1 + \frac 2N} h^{\frac 4N}
& \qquad \mbox{ if } N \geq 3.
\end{array}
\right.
\eeq
\vspace{-10pt}
\beq
\label{3.3}
\ds \int_{\Om } \Big\vert \left( \ph ^2( |\zeta_h|) - 1 \right)^2 -
\left( \ph ^2( |\psi |) - 1 \right)^2 \Big\vert \, dx
\leq C_2 h E_{GL}^{\Om } ( u ) ;
\eeq
\vspace{-10pt}
\beq
\label{3.4}
| Q(\zeta_h) - Q( \psi) | \leq
\left\{
\begin{array}{ll}
2h E_{GL}^{\Om } ( \psi ) & \qquad \mbox{ if } N=2,
\\
\\
C_3 \left( h^2 + \left( E_{GL}^{\Om } ( \psi ) \right)^{ \frac 2N}
h^{\frac 4N} \right)^{\frac 12}
E_{GL}^{\Om } ( \psi ) & \qquad \mbox{ if } N \geq 3.
\end{array}
\right.
\eeq
(iii) For $ z \in \C$, denote
$H(z) = \left( \ph ^2( |z |) - 1 \right) \ph ( |z |) \ph '( |z |)
\frac{z }{ |z |}$ if $ z \neq 0$ and $H(0) = 0$.
Then any minimizer $\zeta_h $ of $G_{h, \Om }^{\psi} $ in $H_{\psi}^1( \Om )$
satisfies in $\Do '(\Om )$ the equation
\beq
\label{3.5}
\left\{
\begin{array}{ll}
\ds - \Delta \zeta_h + H( \zeta_h )
+ \frac{ 1}{ h^2 } (\zeta _h - \psi ) = 0 & \qquad \mbox{ if } N=2,
\\
\\
\ds - \Delta \zeta_h + H( \zeta_h )
+ \frac{ 1}{ h^2 } \ph ' \left( |\zeta _h - \psi|^2 \right)
(\zeta _h - \psi ) = 0 & \qquad \mbox{ if } N\geq 3.
\end{array}
\right.
\eeq
Moreover, for any $ \omega \subset \subset \Om $ we have $ \zeta_h \in W^{2, p} ( \omega ) $
for $ p \in [1, \infty )$; thus, in particular, $ \zeta_h \in C^{1, \al } (\omega )$
for $ \al \in [0, 1)$.
(iv) For any $ h>0$, $ \de >0$ and $ R> 0$ there exists a constant
$K = K ( N, h, \de, R) > 0 $ such that for any $ \psi \in \Eo $ with
$E_{GL}^{\Om } ( \psi) \leq K$ and for any minimizer
$\zeta_h$ of $G_{h, \Om }^{\psi} $ in $H_{\psi}^1( \Om )$~we~have
\beq
\label{3.6}
1 - \de < |\zeta _h(x) | < 1 + \de
\qquad \mbox{ whenever } x \in \Om \mbox{ and } dist(x, \p \Om ) > 4R.
\eeq
\end{Lemma}
{\it Proof. } Let $N =2$.
(i) The existence of a minimizer is proven exactly as in Lemma 3.1~in~\cite{M10}.
(ii) Let $\zeta _h$ be a minimizer. We have $G_{h, \Om }^{\psi} (\zeta _h ) \leq
G_{h, \Om }^{\psi} (\psi ) = E_{GL } (\psi ) $ and this gives (\ref{3.1}) and
(\ref{3.2}). It is obvious that
$$
\big\vert \left( \ph^2(|z_1|) - 1 \right)^2 - \left( \ph^2(|z_2|) - 1 \right)^2
\big\vert
\leq 6 \big\vert \ph (|z_1 |) - \ph (|z_2 |) \big\vert \cdot
\big\vert \ph (|z_1|^2) + \ph (|z_2|^2) - 2 \big\vert
$$
and $ |\ph (|z_1 |) - \ph (|z_2 |) | \leq |z_1 - z_2|$.
Using the Cauchy-Schwarz inequality and (\ref{3.2}) we get
$$
\begin{array}{l}
\ds \int_{\Om } \Big\vert \left( \ph ^2( |\zeta_h|) - 1 \right)^2 -
\left( \ph ^2( |\psi |) - 1 \right)^2 \Big\vert \, dx
\\
\\
\leq 6 \| \zeta _h - \psi \|_{L^2(\Om )}
\left( \ds \int_{\Om }
\Big\vert \ph ^2( | \zeta _h|) + \ph ^2( | \psi |) -2 \Big\vert ^2\, dx \right)^{\frac 12}
\\
\\
\leq 6 h \left( E_{GL }^{\Om } (\psi ) \right)^{\frac 12} \cdot
\left( 2 \ds \int_{\Om } \left( \ph ^2( |\zeta_h|) - 1 \right)^2 +
\left( \ph ^2( |\psi |) - 1 \right)^2 \, dx \right)^{\frac12}
\leq 12 \sqrt{2} h E_{GL }^{\Om } (\psi )
\end{array}
$$
and (\ref{3.3}) is proven.
Finally, (\ref{3.4}) follows from Corollary \ref{C2.4}, (\ref{3.1}) and (\ref{3.2}).
\medskip
(iii) For any
$ \phi \in C_c^{\infty }(\Om )$ we have $ \zeta_h + \phi \in H_{\psi} ^1( \Om )$
and the function $ t \longmapsto G_{h , \Om }^{\psi} ( \zeta_h + t \phi )$ is
differentiable and achieves its minimum at $ t =0$.
Hence $\frac{d}{dt }_{\big\vert _{t =0}} \left(G_{h , \Om }^{\psi} ( \zeta_h + t \phi )\right) =0$
for any $ \phi \in C_c^{\infty }(\Om ) $ and this is precisely (\ref{3.5}).
For any $ z \in \C$ we have
\beq
\label{3.7}
|H(z) | \leq 3 | \ph ^2 ( |z |) - 1| \leq 24 .
\eeq
Since $ \zeta_h \in \Eo$, we have $ \ph ^2 (| \zeta _ h |) - 1 \in L^2 ( \R^2) $
and the previous inequality gives $ H( v_h ) \in L^{2}\cap L^{\infty } ( \R^2)$.
We have $\zeta_h, \psi \in H_{loc}^1(\R^2)$ and from the Sobolev embedding theorem
we get $\zeta_h, \psi \in L_{loc}^p(\R^2)$ for any $ p \in [2, \infty)$.
Using (\ref{3.5}) we infer that $ \Delta \zeta_h \in L_{loc}^p ( \Om )$
for any $ p \in [2, \infty)$. Then (iii) follows from standard elliptic
estimates (see, e.g., Theorem 9.11 p. 235 in \cite{GT}).
\medskip
iv) Using (\ref{3.7}) we get
$$
\|H(\zeta_h)\|_{L^2(\Om )} \leq 3 \| \ph ^2 ( |\zeta _h |) - 1 \|_{L^2(\Om )}
\leq 3 \sqrt{2} \left( E_{GL}^{\Om } (\zeta _h ) \right)^{\frac 12}
\leq 3 \sqrt{2} \left( E_{GL}^{\Om } (\psi ) \right)^{\frac 12}.
$$
From (\ref{3.5}), (\ref{3.2}) and the above estimate we get
\beq
\label{3.8}
\| \Delta \zeta _h \| _{L^2(\Om ) } \leq \left( 3 \sqrt{2} + \frac 1h \right)
\left( E_{GL}^{\Om } (\psi ) \right)^{\frac 12}.
\eeq
For a measurable set $ \omega \subset \R^N$ with $ \Lo ^N ( \omega ) < \infty$ and for
$ f \in L^1( \omega)$, we denote by
$m( f, \omega) = \frac{1}{\Lo ^N ( \omega ) } \ds \int_{\omega} f (x) \, dx $
the mean value of $f$ on $ \omega$. In particular, if $ f \in L^2(\omega)$ using
the Cauchy-Schwarz inequality we get
$| m(f, \omega)| \leq \left( \Lo ^N (\omega) \right)^{- \frac 12 } \| f \| _{L^2(\omega)} $
and consequently
\beq
\label{3.9}
\| m(f, \omega) \| _{L^q (\omega) }
= \left( \Lo ^N (\omega) \right)^{\frac 1q } | m(f, \omega)|
\leq \left( \Lo ^N (\omega) \right)^{\frac 1q - \frac 12} \| f \| _{L^2(\omega)} .
\eeq
Let $ x_0 $ be such that $B( x_0 , 4R) \subset \Om $.
Using the Poincar\'e inequality and (\ref{3.1}) we have
\beq
\label{3.10}
\| \zeta_h - m( \zeta_h, B( x_0, 4R)) \| _{L^2( B(x_0, 4R))}
\leq C_P R \| \nabla \zeta_h \| _{L^2( B(x_0, 4R))}
\leq C_P R \left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}.
\eeq
It is well-known (see Theorem 9.11 p. 235 in \cite{GT})
that for $ p \in (1, \infty )$ there exists $ C = C(N, r, p ) > 0$ such that
for any $ w \in W^{2, p} ( B(a, 2r)) $ we have
\beq
\label{3.11}
\| w \| _{W^{2, p}(B(a, r))} \leq C \left(
\| w\| _{L^ p(B(a, 2 r))} + \| \Delta w\| _{L^ p(B(a, 2 r))} \right).
\eeq
From (\ref{3.8}), (\ref{3.10}) and (\ref{3.11}) we get
\beq
\label{3.12}
\| \zeta_h - m( \zeta_h, B( x_0, 4R)) \| _{W^{2,2 } (B(x_0, 2R))}
\leq C( h, R) \left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}
\eeq
and in particular
\beq
\label{3.13}
\forall 1 \leq i, j \leq 2, \quad \quad \quad
\Big\| \frac{ \p ^2 \zeta _h}{\p x_i \p x_j } \Big\|_{L^2 (B(x_0, 2R))}
\leq C( h, R) \left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}.
\eeq
We will use the following variant of the Gagliardo-Nirenberg inequality:
\beq
\label{3.14}
\| w - m( w,B ( a, r) ) \|_ {L^p ( B(a, r))}
\leq C(p, q, N, r) \|w \| _ {L^q ( B(a,2 r))} ^{\frac qp}
\|\nabla w \| _ {L^N ( B(a,2 r))} ^{1- \frac qp}
\eeq
for any $ w \in W^{1, N}(B(a, 2r))$, where $ 1 \leq q \leq p < \infty $
(see, e.g., \cite{kavian} p. 78).
Using (\ref{3.14}) with $N=2$, $ p =4$, $ q = 2$, then (\ref{3.1}) and (\ref{3.13})
we find
\beq
\label{3.15}
\begin{array}{l}
\| \nabla \zeta _h - m( \nabla \zeta _h , B(x_0, R)) \|_{L^4(B(x_0, R)) }
\leq C \| \nabla \zeta _h \| _{L^2 (B(x_0, 2R ) )} ^{\frac 12 }
\| \nabla ^2 \zeta _h \| _{L^2 (B(x_0, 2R ) )} ^{\frac 12 }
\\
\\
\leq C( h , R ) \left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}.
\end{array}
\eeq
By (\ref{3.9}) and (\ref{3.1}) we have
$\| m( \nabla \zeta _h , B(x_0, R)) \|_{L^4(B(x_0, R)) } \leq (\pi R ^2)^{- \frac 14}
\left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}.$
Together with (\ref{3.15}), this gives
\beq
\label{3.16}
\| \nabla \zeta _h \|_{L^4(B(x_0, R) )}
\leq C( h , R ) \left( E_{GL}^{\Om } (\psi) \right)^{\frac 12}.
\eeq
We will use the Morrey inequality which asserts that, for any
$ w \in C^0 \cap W^{1, p}(B(x_0, r))$ with $ p >N$ we have
\beq
\label{3.17}
| w(x) - w(y) | \leq C(p, N) | x-y|^{1 - \frac Np} ||\nabla w ||_{L^p(B(x_0, r))}
\qquad \mbox{ for any } x, y \in B(x_0, r)
\eeq
(see the proof of Theorem IX.12 p. 166 in \cite{brezis}).
The Morrey inequality and (\ref{3.16}) imply that
\beq
\label{3.18}
|\zeta_h (x) - \zeta_h (y) | \leq C_*( h , R )
\left( E_{GL}^{\Om } (\psi) \right)^{\frac 12} |x -y|^{\frac 12}
\quad \mbox{ for any } x, y \in B(x_0, R).
\eeq
Fix $ \de > 0 $.
Assume that there exists $ x_0 \in \Om $ such that $ dist( x_0, \p \Om ) > 4R $
and $\big| \; | \zeta _h (x_0)| - 1 \big| \geq \de $.
Since
$\Big| \; \big| \; | \zeta _h (x)| - 1 \big| - \big| \; | \zeta _h (y)| - 1 \big| \; \Big|
\leq |\zeta_h (x) - \zeta _h (y) |$,
using (\ref{3.18}) we infer that
$$
\big| \; | \zeta _h (x)| - 1 \big| \geq \frac{\de }{2}
\quad \mbox{ for any } x \in B(x_0, r_{\de }),
$$
where $ r_{\de }
= \min \left( R, \frac{\de ^2}{4 C_* ^2 ( h , R ) E_{GL}^{\Om } (\psi)} \right).
$
Let
\beq
\label{3.19}
\eta ( s) = \inf \{ ( \ph ^2 ( \tau ) - 1 )^2 \; | \;
\tau \in ( - \infty, 1 - s ] \cup [1 + s, \infty) \}.
\eeq
It is clear that $ \eta $ is nondecreasing and positive on $(0, \infty )$.
We have:
\beq
\label{3.20}
\begin{array}{l}
E_{GL}^{\Om }(\psi ) \geq E_{GL}^{\Om }(\zeta _h)
\geq \frac 12 \ds \int_{B(x_0, r_{\de } )} \left( \ph ^2 (|\zeta _h |) - 1 \right)^2 \, dx
\\
\\
\geq \frac 12 {\ds \int_{B(x_0, r_{\de } )} } \eta (\frac{\de }{2} ) \, dx = \frac{\pi}{2} \eta (\frac{\de }{2} ) r_{\de}^2
= \frac{\pi}{2} \eta (\frac{\de }{2} )
\min \left( R, \frac{\de ^2}{4 C_* ^2 ( h , R ) E_{GL}^{\Om } (\psi)} \right)^2.
\end{array}
\eeq
It is clear that there exists a constant $ K= K( h, R, \de ) $ such
that (\ref{3.20}) cannot hold if $E_{GL}^{\Om }(\psi ) \leq K$.
We infer that $ \big| \; | \zeta _h (x_0)| - 1 \big| < \de $ whenever
$ x_0 \in \Om, $ $ dist( x_0, \p \Om ) > 4R$ and $E_{GL}^{\Om }(\psi ) \leq K$.
\hfill $\Box $
\begin{Lemma}
\label{vanishing}
Let $ (\psi_n)_{n \geq 1} \subset \Eo $ be a sequence of functions satisfying:
\smallskip
(a) $ (E_{GL}(\psi_n))_{n \geq 1} $ is bounded and
\smallskip
(b) $ \ds \lim_{n \ra \infty } \Big( \sup_{y \in \R^N} E_{GL}^{B(y, 1)} (\psi_n) \Big) =0.$
\smallskip
There exists a sequence $ h_n \lra 0 $ such that for any minimizer $ \zeta_n $ of
$G_{h_n, \R^N}^{\psi _n}$ in $H_{\psi _n} ^1 (\R^N)$ we have
$ \| \, | \zeta_n | - 1 \|_{L^{\infty }(\R^N)} \lra 0 $ as $ n \lra \infty $.
\end{Lemma}
{\it Proof. } Let $N=2$. We split the proof into several steps.
\medskip
{\it Step 1. Choice of the sequence $(h_n)_{n \geq 1}$.}
Let $M = \ds \sup_{n \geq 1} E_{GL}(\psi_n)$.
For $ n \geq 1 $ and $ x \in \R^2$ we denote
$$
m_n (x) = m(\psi _n, B(x, 1)) = \frac{1}{ \pi } \int_{B(x, 1)} \psi _n (y) \, dy.
$$
The Poincar\'e inequality implies that there exists $C_P > 0$ such that
$$
\ds \int_{B(x, 1)} | \psi _n (y) - m_n (x) |^ 2 \, dy \leq C_P \int_{B(x,1)} | \nabla \psi _n |^2 \, dy.
$$
Using assumption (b) we find
\beq
\label{3.21}
\sup_{x \in \R^2} \| \psi _n - m_n (x) \| _{L^2(B(x,1))} \lra 0 \quad
\mbox{ as } n \lra \infty.
\eeq
Proceeding exactly as in the proof of Lemma 3.2 in \cite{M10} (see the
proof of (3.35) there) we get
\beq
\label{3.22}
\lim_{ n \ra \infty} \sup_{x \in \R^2} |H(m_n(x))| = 0.
\eeq
Let
\beq
\label{3.23}
h_n = \max \left( \left( \sup_ {x \in \R^2} \| \psi _n - m_n (x) \| _{L^2(B(x,1))} \right)^{\frac 13} ,
\sup _{x \in \R^2} |H(m_n(x))| \right).
\eeq
From (\ref{3.21}) and (\ref{3.22}) it follows that
$ h_n \lra 0 $ as $ n \lra \infty $.
Hence we may assume that $ 0 < h_n < 1$ for each $n$
(if $ h_n = 0$ then $ \psi _n$ is constant a.e.
and any minimizer $ \zeta_n $ of
$G_{h_n, \R^2}^{\psi _n}$ equals $\psi _n$ a.e.).
Let $ \zeta_n $ be a minimizer of $G_{h_n, \R^2}^{\psi _n}$
(as given by Lemma \ref{L3.1} (i)).
It follows from Lemma \ref{L3.1} (iii) that $\zeta_n $ satisfies (\ref{3.5})
and $\zeta _n \in W_{loc}^{2,2}(\R^2)$.
\medskip
{\it Step 2. We prove that $\| \Delta \zeta _n \| _{L^2(B(x, \frac 12 ))}$ is
bounded independently on $n$ and on $x$. }
There is no loss of generality to assume that $ x = 0 $.
Then we observe that (\ref{3.5}) can be written as
\beq
\label{3.24}
- \Delta \zeta _n + \frac{1}{h_n^2} ( \zeta _n - m_n(0)) = f_n \qquad \mbox{ in }
\Do' (\R^2),
\eeq
where
\beq
\label{3.25}
f_n = \frac{1}{h_n^2} ( \psi _n - m_n(0)) - (H(\zeta _n) - H( m_n(0))) - H(m_n(0)).
\eeq
From (\ref{3.2}) we have
$\| \zeta _n - \psi _n \| _{L^2(\R^2)} \leq h_n E_{GL}(\psi _n)^{\frac 12}
\leq h_n M^{\frac 12}$
and from (\ref{3.23}) we obtain $ \| \psi _n - m_n(0) \|_{L^2(B(0,1))}
\leq h_n ^3 \leq h_n$,
hence
\beq
\label{3.26}
\| \zeta _n - m_n(0) \|_{L^2(B(0,1))} \leq (M^{\frac 12} +1) h_n.
\eeq
Since $H$ is Lipschitz, we get
\beq
\label{3.27}
\| H(\zeta _n ) - H( m_n(0) ) \| _{L^2(B(0,1)}
\leq C_1 \| \zeta _n - m_n(0) \|_{L^2(B(0,1))}
\leq C_2 h_n.
\eeq
Using (\ref{3.25}), (\ref{3.23}) and (\ref{3.27}) we get
\beq
\label{3.28}
\begin{array}{l}
\| f_n \| _{L^2(B(0,1))}
\\
\leq \frac{1 }{h_n^2} \| \psi _n - m_n (0) \| _{L^2(B(0,1))}
+ \| H(\zeta _n ) - H( m_n(0) ) \| _{L^2(B(0,1))}
+ \pi ^{\frac 12} | H(m_n(0) ) |
\\
\leq C_3 h_n.
\end{array}
\eeq
It is obvious that for any bounded domain $ \Om \subset \R^2$, each term in
(\ref{3.24}) belongs to $H^{-1}(\Om)$.
Let $ \chi \in C_c^{\infty}(\R^2)$ be such that $ supp( \chi ) \subset B(0,1)$,
$ 0 \leq \chi \leq 1$ and $ \chi = 1 $ on $B(0, \frac 12)$.
Taking the duality product of (\ref{3.24}) by $ \chi ( \zeta _n - m_n(0))$ we find
\beq
\label{3.29}
\int_{\R^2} \chi | \nabla \zeta _n |^2 \, dx - \frac 12 \int_{\R^2} (\Delta \chi ) |
\zeta _n - m_n(0) |^2 \, dx
+ \frac{1}{h_n ^2} \int_{\R^2} \chi | \zeta _n - m_n(0) |^2 \, dx
= \int_{\R^2} \langle f_n , \zeta_n - m_n(0) \rangle \chi \, dx.
\eeq
Using (\ref{3.29}), the Cauchy-Schwarz inequality and (\ref{3.26}), (\ref{3.28}) we infer that
\beq
\label{3.30}
\begin{array}{l}
\ds \frac{1}{h_n ^2} \int_{B(0, \frac 12)} | \zeta _n - m_n(0) |^2 \, dx
\\
\\
\ds \leq \| \Delta \chi \|_{L^{\infty} (\R^2) } \int_{B(0, 1)} | \zeta _n - m_n(0) |^2 \, dx
+ \| f_n \| _{L^2(B(0,1))} \| \zeta _n - m_n(0) \|_{L^2(B(0,1))}
\leq C_4 h_n ^2.
\end{array}
\eeq
Now (\ref{3.24}), (\ref{3.28}) and (\ref{3.30}) imply that there is $C_5 >0 $
such that $\| \Delta \zeta _n \| _{L^2(B(0, \frac 12))} \leq C_5$. Thus we
have proved that for any $n$ and $x$,
\beq
\label{3.31}
\| \Delta \zeta _n \| _{L^2(B(x, \frac 12))} \leq C_5,
\qquad \mbox{ where } C_5 \mbox{ does not depend on $x$ and $n$.}
\eeq
\medskip
{\it Step 3. A H\"older estimate on $ \zeta _n$. } It follows from (\ref{3.11}) that
\beq
\label{3.32}
\| \zeta _n - m_n\| _{W^{2,2} (B(x, \frac 14) )} \leq
C ( \| \Delta \zeta _n \|_{L^2(B(x, \frac 12 ))} +
\| \zeta _n - m_n \| _{L^2(B(x, \frac 12 ))} ) \leq C _6.
\eeq
From (\ref{3.14}) and (\ref{3.32}) we find
\beq
\label{3.33}
\| \nabla \zeta _n - m ( \nabla \zeta _n , B(x, \frac 18 )) \| _{L^4(B(x, \frac 18))}
\leq C \| \nabla \zeta _n \| _{L^2(B(x, \frac 14))} ^{\frac 12}
\| \nabla ^2 \zeta _n \| _{L^2(B(x, \frac 14))} ^{\frac 12} \leq C_7.
\eeq
It is clear that
$| m ( \nabla \zeta _n , B(x, \frac 18 )) | \leq \left( \Lo ^2( B(x, \frac 18 )) \right)^{- \frac 12}
\| \nabla \zeta _n \| _{L^2 ( B(x, \frac 18 )) } \leq C_8.$
Then (\ref{3.33}) implies that $ \| \nabla \zeta _n \| _{L^4(B(x, \frac 18))} $
is bounded independently on $n$ and $x$. Using the Morrey inequality
(\ref{3.17}) we infer that there is $C_9 >0$ such that
\beq
\label{3.34}
|\zeta _n (x) - \zeta _n (y) | \leq C_9 |x-y |^{\frac 12} \qquad
\mbox{ for any $ n \in \N^*$ and any $x, y \in \R^2$ with } |x-y| < \frac 18.
\eeq
\medskip
{\it Step 4. Conclusion. }
Let $ \de _n = \| \, |\zeta _n | - 1 \|_{L^{\infty}(\R^2)}$ if $\zeta _n $
is bounded, and $ \de _n = 1$ otherwise.
Choose $ x_0^n \in \R^2$ such that $\big|\, |\zeta _n ( x_0 ^n) | - 1 \big| \geq \frac{\de _n }{2} .$
From (\ref{3.34}) we infer that $\big|\, |\zeta _n (x) | - 1 \big| \geq \frac{\de _n }{4}$
for any $ x \in B(x_0 ^n, r_n)$,
where $ r_n = \min \left( \frac 18, \left(\frac{\de _n }{ 4 C_9 } \right)^2 \right).$
Let $ \eta $ be as in (\ref{3.19}). Then we have
\beq
\label{3.35}
\int_{B(x_0 ^n, r_n) } \left( \ph ^2(|\zeta _n |) - 1 \right)^2\, dx \geq
\int_{B(x_0 ^n, r_n) } \eta \left( \frac{\de _n}{4} \right) \, dx =
\eta \left( \frac{\de _n}{4} \right) \pi r_n ^2.
\eeq
On the other hand,
the function $ z\longmapsto \left( \ph ^2(| z|) - 1 \right)^2 $ is
Lipschitz on $\C$.
From this fact, the Cauchy-Schwarz inequality, (\ref{3.2}) and assumption
(a) we get
$$
\begin{array}{l}
\ds \int_{B(x, 1)} \Big\vert
\left( \ph ^2(|\zeta _n (y)|) - 1 \right)^2
- \left( \ph ^2(|\psi _n (y) |) - 1 \right)^2 \Big\vert \, dy
\\
\leq C \ds \int_{B(x, 1)} | \zeta_n (y) - \psi_n (y) | \, dy
\leq C \pi ^{\frac 12} \| \zeta_n - \psi_n \| _{L^2 ( B(x, 1) )}
\leq C \pi ^{\frac 12} \| \zeta_n - \psi_n \|_{L^2 ( \R^2) }
\leq C_{10} h_n .
\end{array}
$$
Then using assumption (b) we infer that
\beq
\label{3.36}
\ds \sup_{ x \in \R^2} \ds \int_{B(x, 1)}
\left( \ph ^2(|\zeta _n (y)|) - 1 \right)^2 \, dy
\lra 0 \qquad \mbox{ as } n \lra \infty.
\eeq
From (\ref{3.35}) and (\ref{3.36}) we get
$ {\ds \lim_{n \ra \infty }} \eta \left( \frac{\de_n}{4} \right) r_n ^2 = 0$
and this clearly implies $ \ds \lim_{n \ra \infty } \de _n = 0$.
This completes the proof of Lemma \ref{vanishing}.
\hfill
$\Box $
\medskip
The next result is based on Lemma \ref{L3.1} and will be very useful
in the next sections to
prove the "concentration" of minimizing sequences.
For $0 < R_1 < R_2 $ we denote
$ \Om _{R_1, R_2} = B(0, R_2) \setminus \ov{B}(0, R_1)$.
\medskip
\begin{Lemma}
\label{splitting}
Let $ A > A_3 > A_2 >1$.
There exist $ \e _0 >0$ and
$C_i >0$, depending only on $ \; N, \; A, \; A_2, \; A_3$
(and $F$ for (vi))
such that for any $ R \geq 1$,
$ \e \in (0, \e _0)$ and $ \psi \in \Eo $ verifying
$E_{GL}^{\Om _{R, AR}} (\psi) \leq \e, $
there exist two functions $ \psi_1, \, \psi_2 \in \Eo $ and a constant $ \theta _0 \in [0, 2 \pi)$
satisfying the following properties:
\smallskip
(i) $\psi _1 = \psi $ on $B(0, R)$ and $ \psi _1 = e^{i \theta _0}$ on
$\R^N \setminus B(0, A_2 R) $,
\smallskip
(ii) $ \psi_2 = \psi $ on $ \R^N \setminus B(0, AR)$ and
$ \psi _2 = e^{ i \theta _0 }= constant$
on $B(0, A_3R)$,
\medskip
(iii) $ \ds \int_{\R^N} \Big\vert \,
\Big\vert \frac{\p \psi}{\p x_j } \Big\vert ^2 -
\Big\vert \frac{\p \psi_1}{\p x_j } \Big\vert ^2 -
\Big\vert \frac{\p \psi_2}{\p x_j } \Big\vert ^2 \, \Big\vert \, dx
\leq C_1 \e $ for $ j =1, \dots, N$,
\medskip
(iv) $ \ds \int_{\R^N} \Big\vert
\left( \ph ^2(| \psi |) - 1 \right)^2
- \left( \ph ^2(|\psi_1|) - 1 \right)^2
- \left( \ph ^2(|\psi_2|) - 1 \right)^2
\Big\vert \, dx \leq C_2 \e $,
\medskip
(v) $ | Q(\psi) - Q(\psi_1) - Q( \psi_2) | \leq C_3 \e $,
\medskip
(vi) If assumptions (A1) and (A2) in the introduction hold, then
$$
\ds \int_{\R^N} \Big\vert V( |\psi |^2) - V( |\psi _1|^2) - V( |\psi _2 |^2) \Big\vert \, dx
\leq
\left\{
\begin{array}{l}
C_4 \e + C_5 \sqrt{\e } \left( E_{GL}(\psi) \right)^{\frac{2^* -1}{2}} \quad \mbox{ if } N \geq 3,
\\
\\
C_6 \e + C_7 \sqrt{\e } \left( E_{GL}(\psi ) \right) ^{ p_0 + 1} \quad \mbox{ if } N =2.
\end{array}
\right.
$$
Furthermore, the same estimate holds with $ V_+$ (respectively $V_-$) instead of $V$.
\end{Lemma}
{\it Proof. }
If $ N \geq 3$, this is Lemma 3.3 in \cite{M10}.
Let $ N =2$.
Fix $ k > 0$, $ A_1 $ and $ A_4 $ such that
$ 1 + 4k < A_1 < A_2 < A_3 < A_4 < A - 4k. $
Let $ h = 1 $ and $ \de = \frac{1}{2}$.
Let $K( N, h, \de, r )$ be as in Lemma \ref{L3.1} (iv).
We will prove that Lemma \ref{splitting} holds for
$\e _0 = \min \left( K( 2, 1, \frac{1}{2}, k ),
\frac{\pi}{8} \ln \left( \frac{A - 4k}{1 + 4k} \right) \right) .$
Fix $ \e < \e _0 $. Consider $\psi \in \Eo $ such that
$ E_{GL}^{\Om _{R, AR}}(\psi ) \leq \e$.
Let $\zeta $ be a minimizer of $G_{1, \Om _{R, AR}}^{\psi }$ in the space $H_{\psi }^1(\Om _{R, AR})$.
Such minimizers exist by Lemma \ref{L3.1} (but are perhaps not unique).
From Lemma \ref{L3.1} (iii) we have $\zeta \in W_{loc}^{2,p} (\Om _{R, AR}) $
for any $ p \in [1, \infty)$, hence $ \zeta \in C^1 (\Om _{R, AR}) $.
Moreover, Lemma \ref{L3.1} (iv) implies that
\beq
\label{3.37}
\frac{1}{2} \leq |\zeta (x) | \leq \frac{ 3 }{2}
\qquad \mbox{ for any } x \mbox{ such that } R + 4k \leq |x| \leq AR - 4k.
\eeq
Therefore, the topological degree $ deg( \frac{ \zeta }{|\zeta |} , \p B(0, r))$ is well defined for any
$ r \in [R+ 4k, AR - 4k]$ and does not depend on $r$.
It is well-known that $ \zeta $ admits a $C^1$ lifting $\theta$ (i.e.
$ \zeta = |\zeta |e^{i \theta} $) on $\Om_{R+ 4k, AR- 4k} $
if and only if
$deg( \zeta , \p B(0, r))= 0$ for $ r \in (R + 4k, AR - 4k)$.
Denoting by $ \tau = (- \sin t, \cos t) $ the unit tangent vector at
$\p B(0, r)$ at a point $ r e^{i t} = (r \cos t, r \sin t ) \in \p B(0, r)$, we get
\beq
\label{3.38}
\begin{array}{l}
| deg( \zeta , \p B(0, r)) |
= \bigg\vert \ds \frac{1}{2 i \pi} \int_0 ^{2 \pi}
\frac{\frac{\p }{\p t } ( \zeta (r e^{ i t}))}
{\zeta (r e^{i t })} \, d t
\bigg\vert
= \bigg\vert \ds \frac{r}{2 i \pi} \int_0 ^{2 \pi}
\frac{\frac{\p \zeta }{\p \tau} (r e^{ i t})}
{\zeta (r e^{i t })} \, d t
\bigg\vert
\\
\\
\leq \ds \frac{r}{2 \pi} \int_0^{2 \pi} 2 | \nabla \zeta (r e^{i t})| \, d t
\leq \frac{r}{\pi } \sqrt{ 2 \pi} \left(
\int_0^{2 \pi} | \nabla \zeta (re^{i t })| ^2 \, d t \right)^{\frac 12}.
\end{array}
\eeq
On the other hand,
$$
\ds \int_{ \Om_{R+ 4k, AR- 4k} } |\nabla \zeta (x)|^2 \, dx =
\int_{R + 4k }^{AR - 4k } r \int_0^{2 \pi} | \nabla \zeta (re^{i t})| ^2 \, d t \, dr.
$$
We have
${\ds \int_{\Om_{R+ 4k, AR- 4k} } } |\nabla \zeta (x)|^2 \, dx \leq E_{GL}^{\Om_{R, AR}} (\zeta )
\leq E_{GL}^{\Om_{R, AR}} (\psi) < \e _0 \leq \frac{\pi }{8} \ln \left( \frac{AR - 4k}{R + 4k} \right)$
and we infer that there exists
$r _* \in (R + 4k, AR - 4k) $ such that
$
\ds r _* \int_0^{2 \pi} | \nabla \zeta (R_* e^{i t})| ^2 \, d t
< \frac{ \pi }{8} \frac {1}{r _*}.
$
From (\ref{3.38}) we get
$$
| deg( \zeta , \p B(0, r _* )) | < \frac{r _*}{\pi } \sqrt{ 2 \pi}
\left( \frac{ \pi }{8} \frac {1}{r _* ^2} \right)^{\frac 12}
= \frac 12.
$$
Since the topological degree is an integer, we have necessarily
$deg( \zeta , \p B(0, r _*)) =0$.
Consequently $deg( \zeta , \p B(0, r)) =0$ for any
$ r \in ( R+ 4k, AR- 4k) $ and $ \zeta $ admits a $C^1$ lifting
$\zeta = \rho e^{i \theta }$. In fact,
$\rho, \, \theta \in W_{loc}^{2, p}(\Om_{R+ 4k, AR- 4k} ) $
because $\zeta \in W_{loc}^{2, p}(\Om_{R+ 4k, AR- 4k}) $
(see Theorem 3 p. 38 in \cite{BBM}).
Consider $ \eta _1, \eta _2 \in C ^{\infty }(\R)$ satisfying
the following properties:
$$
\begin{array}{l}
\eta_1 = 1 \mbox{ on } (-\infty, A_1], \quad \eta_1 = 0 \mbox{ on } [A_2, \infty),
\quad \eta_1 \mbox{ is nonincreasing, }
\\
\eta_2 = 0 \mbox{ on } (-\infty, A_3], \quad \eta_2 = 1 \mbox{ on } [A_4, \infty),
\quad \eta_2 \mbox{ is nondecreasing. }
\end{array}
$$
Denote $ \theta _0 = m( \theta, \Om _{A_1 R, A_4 R })$.
We define $ \psi _1$ and $ \psi _2$ as follows:
\beq
\label{3.39}
\psi _1 (x) = \left\{ \begin{array}{l}
\psi (x) \quad \mbox{ if } x \in \ov{B}(0, R),
\\
\zeta (x) \quad \mbox{ if } x \in {B}(0, A_1R) \setminus \ov{B}(0, R),
\\
\left(1 + \eta_1 (\frac{|x|}{R} ) (\rho (x) - 1 )\right)
e^{i \left(\theta _0 + \eta_1 (\frac{|x|}{R} ) (\theta (x) - \theta_0 )\right)}
\\
\qquad \qquad \qquad \qquad \qquad \quad \mbox{ if } x \in {B}(0, A_4R) \setminus B(0, A_1R),
\\
e^{i \theta _0} \quad \mbox{ if } x \in \R^2 \setminus {B}(0, A_4R ),
\end{array}
\right.
\eeq
\beq
\label{3.40}
\psi _2 (x) = \left\{ \begin{array}{l}
e^{i \theta _0} \quad \mbox{ if } x \in \ov{B}(0, A_1R ),
\\
\left(1 + \eta_2 (\frac{|x|}{R} ) (\rho (x) - 1 )\right)
e^{i \left(\theta _0 + \eta_2 (\frac{|x|}{R} ) (\theta (x) - \theta_0 )\right)}
\\
\qquad \qquad \qquad \qquad \quad \qquad \mbox{ if } x \in {B}(0, A_4R) \setminus \ov{B}(0, A_1R),
\\
\zeta (x) \quad \mbox{ if } x \in {B}(0, AR) \setminus {B}(0, A_4 R),
\\
\psi (x) \quad \mbox{ if } x \in \R^2 \setminus {B}(0, AR ).
\end{array}
\right.
\eeq
Then $ \psi _1, \; \psi _2 \in \Eo $ and satisfy (i) and (ii).
The proof of (iii), (iv) and (v) is exactly as in \cite{M10}.
Next we prove (vi).
Assume that (A1) and (A2) are satisfied and let $W(s) = V(s) - V(\ph^2(s))$.
Then $W(s) = 0 $ for $ s \in [0,4]$ and it is easy to see that $W$ satisfies
\beq
\label{i3}
|W(b^2) - W(a^2) | \leq C_3 |b-a|
\left( a^{2p _0 +1} \1_{\{ a > 2 \} } + b^{2p _0 +1} \1_{\{ b > 2 \} } \right)
\; \; \mbox{ for any } a, b \geq 0.
\eeq
Using (\ref{i1}) and (\ref{i3}), then H\"older's inequality we obtain
\beq
\label{3.41}
\begin{array}{l}
\ds \int _{\R ^2 }
\big\vert V( | \psi|^2) - V( | \zeta |^2) \big\vert \, dx
\\
\\
\leq
\ds \int _{\Om _{ R, \, A R} }
\big\vert V( \ph ^2(| \psi |) ) \! - \! V( \ph ^2(|\zeta | )) \big\vert
+ \big\vert W( |\psi |^2) \! - \! W( |\zeta |^2) \big\vert dx
\\
\\
\leq C \ds \int _{\Om _{ R, \, A R} }
\left( \ph ^2(| \psi |) - 1 \right)^2 + \left( \ph ^2(| \zeta |) - 1 \right)^2 \, dx
\\
\\
\qquad + C \ds \int _{\Om _{ R, \, A R} }
\big\vert \, | \psi | - | \zeta | \, \big\vert
\left( | \psi| ^{2 p_0 + 1} \1 _{\{ | \psi | > 2 \} }
+ | \zeta | ^{2 p_0 + 1} \1 _{\{ |\zeta | > 2 \} } \right)\, dx
\\
\\
\leq \! C' \e +
\| \psi \! - \! \zeta \|_{L^2(\Om _{ R, \, A R} ) } \! \!
\left[ \! \! {\left( \ds \int _{\Om _{ R, \, A R} } \! \! \! \! \! \!
| \psi| ^{4 p_0 + 2} \1 _{\{ | \psi | > 2 \} } dx \! \right) \! \! } ^{\frac 12}
\! \!+ \!
{\left( \ds \int _{\Om _{ R, \, A R} } \! \! \! \! \! \!
| \zeta | ^{4 p_0 + 2} \1 _{\{ | \zeta | > 2 \} } dx \! \right) \! \! }^{\frac 12} \right] \! \! .
\end{array}
\eeq
Using (\ref{ineq2}) we get
\beq
\label{3.42}
\int_{\R^2 } | \psi| ^{4 p_0 + 2} \1 _{\{ | \psi | > 2 \} } \, dx
\leq C \| \nabla \psi \| _{L^2(\R^2 )} ^{ 4 p_0 + 2}
\Lo ^2\left( \{ x \in \R^2 \; | \; |\psi (x) | \geq 2 \} \right).
\eeq
On the other hand,
\beq
\label{3.43}
9
\Lo ^2\left( \{ x \in \R^2 \; \big| \; |\psi (x) | \geq 2 \} \right)
\leq \int_{\R^2} \left( \ph ^2 ( |\psi |) - 1 \right)^2 \, dx
\leq 2 E_{GL}(\psi )
\eeq
and a similar estimate holds for $ \zeta $.
We insert (\ref{3.42}) and (\ref{3.43}) into (\ref{3.41}) to discover
\beq
\label{3.44}
\int _{\R ^2 }
\big\vert V( | \psi|^2) - V( | \zeta |^2) \big\vert \, dx \leq C '\e + C \sqrt{\e } \left( E_{GL}(\psi ) \right)^{p_0 +1 }.
\eeq
Proceeding exactly as in \cite {M10} (see the proof of (3.88) p. 144 there) we obtain
\beq
\label{3.45}
\int_{\R^2} \big\vert V(|\zeta |^2 ) - V( |\psi _1|) - V(|\psi _2|) \big\vert \, dx \leq C \e.
\eeq
Then (vi) follows from (\ref{3.44}) and (\ref{3.45}).
\hfill
$\Box $
\begin{Corollary}
\label{C3.4}
For any $ \psi \in \Eo$, there is a sequence of functions
$(\psi _n )_{n \geq 1} \subset \Eo $ satisfying:
(i) $\psi _n = \psi $ on $B(0, 2^n)$ and $ \psi _n = e^{i \theta _n } = constant $
on $\R^N \setminus B(0, 2^{n+1})$,
(ii) $\| \nabla \psi _n - \nabla \psi \|_{L^2(\R^N)} \lra 0$ and
$\| \ph ^2 (|\psi _n|) - \ph ^2(|\psi |) \| _{L^2(\R^N)} \lra 0$,
(iii) $ Q(\psi _n ) \lra Q(\psi)$, $\ds \int_{\R^N} \big\vert V(|\psi _n|^2 ) - V(|\psi |^2) \big\vert \, dx \lra 0 $
and \\
$\ds \int_{\R^N} \big\vert \left( \ph ^2(|\psi _n |) - 1 \right)^2
- \left( \ph ^2(|\psi |) - 1 \right)^2 \big\vert \, dx \lra 0 $ as $ n \lra \infty.$
\end{Corollary}
{\it Proof. } Let $ \e _n = E_{GL}^{\R^N \setminus B(0, 2^n) }(\psi)$, so that $ \e _n \lra 0 $ as $ n \lra \infty$.
Let $ A = 2$, fix $ 1 < A_2 < A_3 < 2 $ and use Lemma \ref{splitting} with $ R = 2^n$
to obtain two functions $ \psi _1^n$, $\psi _2^n$ with properties (i)-(vi) in that Lemma.
Let $ \psi _n = \psi _1^n$. It is then straightforward to prove that $(\psi _n)_{n \geq 1}$
satisfies (i)-(iii) above.
\hfill
$\Box$
\medskip
The next Lemma allows to approximate functions in $ \Eo $ by functions
with higher regularity.
\begin{Lemma}
\label{L3.5}
(i) Assume that $ \Om = \R^N$ or that $ \p \Om $ is $ C^1$.
Let $ \psi \in \Eo $. For each $ h > 0$, let $ \zeta_h $ be a minimizer of
$G_{h, \Om } ^{ \psi }$ in $H_{\psi }^1( \Om)$. Then
$
\| \zeta _h - \psi \| _{H^1( \Om )} \lra 0 $
as $ h \lra 0. $
\medskip
(ii) Let $ \psi \in \Eo $. For any $ \e > 0$ and any $ k \in \N$ there is $ \zeta \in \Eo $ such that
$ \nabla \zeta \in H^k( \R^N )$, $E_{GL}(\zeta ) \leq E_{GL}(\psi) $ and
$\| \zeta - \psi \|_{
H^1( \R^N)} < \e$.
\end{Lemma}
{\it Proof. } (i) It suffices to prove that for any sequence $ h_n \lra 0 $ and any choice of a minimizer
$ \zeta _{n}$ of $G_{h _n, \Om } ^{ \psi }$ in $H_{\psi }^1( \Om)$, there is a subsequence
$ (\zeta_{n_k})_{k \geq 1}$ such that
$
\ds \lim_{k \ra \infty} \| \zeta _{h_{n_k}} - \psi \| _{H^1( \Om )} = 0. $
Let $ h_n \lra 0 $ and let $ \zeta _{n}$ be as above.
By (\ref{3.2}) we have $ \zeta_{n} - \psi \lra 0 $ in $L^2( \Om )$ and it is clear that
$ \zeta_{n} - \psi $ is bounded in $H_0^1( \Om)$.
Then there are $v \in H_0^1( \Om )$ and a subsequence $ (\zeta_{n_k})_{k \geq 1}$
such that
$$
(\zeta_{n_k} - \psi) \rightharpoonup v \quad \mbox{ weakly in } H_0^1( \Om)
\quad \mbox{ and } \quad
(\zeta_{n_k} - \psi) \lra v \quad \mbox{ a.e. on } \Om.
$$
Since $ \zeta_{n_k} - \psi \lra 0 $ in $L^2( \Om)$ we infer that $ v = 0 $ a.e., therefore
$\nabla \zeta_{n_k} \rightharpoonup \nabla \psi $
weakly in $ L^2( \Om )$ and $ \zeta_{n_k} \lra \psi $ a.e on $ \Om$.
By weak convergence we have
$
\ii _{\Om } |\nabla \psi |^2 \, dx
\leq {\ds \liminf _{k \ra \infty }} \ii_{\Om } |\nabla \zeta_{n _k} |^2 \, dx
$
and Fatou's Lemma gives
$
\ii _{\Om } \left( \ph^2(|\psi |) - 1 \right)^2 \, dx \leq
{\ds \liminf _{k \ra \infty } } \ii _{\Om } \left( \ph^2(|\zeta _{n_k} |) - 1 \right)^2 \, dx .
$
Thus we get $ E_{GL}^{\Om} ( \psi ) \leq \ds \liminf _{k \ra \infty } E_{GL}^{\Om} (\zeta_{n_k}).$
On the other hand we have $ E_{GL}^{\Om} (\zeta_{n_k}) \leq E_{GL}^{\Om} ( \psi ) $ for all $k$.
We infer that necessarily $ \ds \lim _{k \ra \infty } E_{GL}^{\Om} (\zeta_{n_k}) = E_{GL}^{\Om} ( \psi ) $
and
$ {\ds \lim _{k \ra \infty } } \ii _{\Om } |\nabla \zeta_{n _k} |^2 \, dx = \ii _{\Om } |\nabla \psi |^2 \, dx.$
Taking into account that $\nabla \zeta_{n_k} \rightharpoonup \nabla \psi $
weakly in $ L^2( \Om )$, we deduce that $\nabla \zeta_{n_k} \lra \nabla \psi $
strongly in $ L^2( \Om )$, thus $( \zeta_{n _k } - \psi ) \lra 0 $ in $H_0^1( \Om)$, as desired.
\medskip
(ii) Let $ h > 0$ and let $ \zeta _h $ be a minimizer of $ G_{h, \R^N} ^{\psi }.$
Then $ \zeta _h$ satisfies (\ref{3.5}) in $ \Do '(\R^N)$, thus $ \Delta \zeta _h \in L^2( \R^N)$
and this implies $ \frac{ \p ^2 \zeta _h }{\p x_i \p x_j } \in L^2( \R^N)$ for any $i, j$,
hence $ \nabla \zeta _h \in H^1( \R^N)$.
Moreover, if $ \nabla \psi \in H^{\ell} ( \R^N)$ for some $ \ell \in \N$,
taking successively the derivatives of (\ref{3.5}) up to order $ \ell $ and repeating the above argument we get
$ \nabla \zeta _h \in H^{\ell +1}( \R^N)$.
Fix $ \psi \in \Eo$, $ k \in \N$ and $ \e > 0$.
Using (i), there are $ h_1 > 0$ and a minimizer $ \zeta _1 $ of $G_{h_1, \R^N}^{\psi } $ such that
$\| \zeta _1 - \psi \| _{H^1( \R^N ) } < \frac{\e}{2}$ and $ \nabla \zeta _1 \in H^1( \R^N)$.
Then there are $ h_2 > 0$ and a minimizer $ \zeta _2 $ of $G_{h_2, \R^N}^ {\zeta _1}$ such that
$\| \zeta _2 - \zeta _1 \|_{H^1( \R^N ) } < \frac{\e}{2^2}$ and $ \nabla \zeta _2 \in H^2( \R^N)$,
and so on.
After $k$ steps we find $ h_k $ and $ \zeta _k $ such that $ \zeta _k $ is a minimizer of $G_{h_k, \R^N}^{\zeta_{k-1}}$,
$\| \zeta _k - \zeta _{k-1} \|_{H^1( \R^N ) } < \frac{\e}{2^k}$, and
$ \nabla \zeta _k \in H^k( \R^N)$. Then
$ \| \zeta _k - \psi \| _{H^1( \R^N ) } < \| \zeta _k - \zeta_{k-1} \| _{H^1( \R^N ) } +
\dots + \| \zeta _2 - \zeta _1 \| _{H^1( \R^N ) } + \| \zeta _1 - \psi \| _{H^1( \R^N ) } < \e$.
Moreover,
$ E_{GL}(\zeta_{k}) \leq E_{GL}(\zeta_{k-1}) \leq \dots \leq E_{GL}(\psi).$
\hfill
$\Box$
\section{Minimizing the energy at fixed momentum}
\label{minem}
The aim of this section is to investigate the existence of minimizers of the
energy $E$ under the constraint $Q = q >0$. If such minimizers exist, they are
traveling waves to (\ref{1.1}) and their speed is precisely the Lagrange
multiplier appearing in the variational problem.
We start with some useful properties of the functionals $E$, $E_{GL}$ and $Q$.
\begin{Lemma}
\label{L4.1} If (A1) and (A2) in the Introduction hold, then
$V(|\psi|^2) \in L^1(\R^N)$ whenever $ \psi \in \Eo $.
Moreover, for any $ \de > 0$ there exist $C_1 (\de), \, C_2 ( \de ) >0$ such that
for all $ \psi \in \Eo $ we have
\beq
\label{4.1}
\begin{array}{l}
\ds \frac{1 - \de }{2} \ds \int_{\R^N}
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx
- C_1 ( \de ) \| \nabla \psi \|_{L^2(\R^N)}^{2^*}
\leq
\ds \int_{\R^N} V(| \psi |^2 ) \, dx
\\
\\
\ds \leq
\frac{1 + \de }{2} \ds \int_{\R^N}
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx
+ C_2 ( \de ) \| \nabla \psi \|_{L^2(\R^N)}^{2^*} \
\qquad \mbox{ if } N \geq 3 ,
\end{array}
\eeq
respectively
\beq
\label{4.2}
\begin{array}{l}
\ds \left( \frac{1 - \de }{2} - C_1(\de) \| \nabla \psi \|_{L^2(\R^2)} ^{2 p_0 + 2} \right)
\ds \int_{\R^2} \left( \ph^2(|\psi |) - 1 \right)^2 \, dx
\leq
\ds \int_{\R^2} V(| \psi |^2 ) \, dx
\\
\\
\ds \leq
\left( \frac{1 + \de }{2} + C_2(\de) \| \nabla \psi \|_{L^2(\R^2)} ^{2 p_0 + 2} \right)
\ds \int_{\R^2} \left( \ph^2(|\psi |) - 1 \right)^2 \, dx
\qquad \mbox{ if } N =2.
\end{array}
\eeq
These estimates still hold if we replace the condition $ F \in C^0([0, \infty)) $ in (A1) by
$F \in L_{loc}^1([0, \infty))$ and if we replace $V$ by $|V|$.
\end{Lemma}
{\it Proof. }
Inequality (\ref{4.1}) follows from Lemma 4.1 p. 144 in \cite{M10}. We only prove (\ref{4.2}).
Fix $ \de > 0$. There exists $ \beta = \beta (\de ) \in (0, 1] $ such that
\beq
\label{4.3}
\frac{ 1 - \de }{2} ( s - 1) ^2 \leq V(s) \leq \frac{ 1 + \de }{2} ( s - 1) ^2
\qquad \mbox{ for any } s \in ( ( 1 - \beta )^2, (1 + \beta )^2).
\eeq
Let $ \psi \in \Eo$. It follows from (\ref{4.3}) that $V(|\psi |^2) \1 _{ \{ 1 - \beta \leq |\psi | \leq 1 + \beta \} } \in L^1(\R^2)$ and
\beq
\label{4.4}
\begin{array}{l}
\ds \frac{1 - \de }{2} \int_{\{ 1 - \beta \leq |\psi | \leq 1 + \beta \} }
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx
\leq \int_{\{ 1 - \beta \leq |\psi | \leq 1 + \beta \} } V(|\psi |^2) \, dx
\\
\\
\ds \leq \frac{1 + \de }{2} \int_{\{ 1 - \beta \leq |\psi | \leq 1 + \beta \} }
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx .
\end{array}
\eeq
Using (A2) we infer that there exists $ C(\de ) > 0 $ such that
\beq
\label{4.5}
\Big\vert V( s^2) - \frac{ 1 \pm \de }{2} \left( \ph^2(s) - 1 \right)^2 \Big\vert
\leq C(\de ) \left( |s - 1 | - \frac 12 \beta \right)^{ 2 p_0 + 2}
\eeq
for any $ s \geq 0 \mbox{ satisfying } |s - 1 | \geq \beta.$
Let $K = \{ x \in \R^2 \; \big\vert \; \; \big| \, | \psi (x)| - 1 \big| \geq \frac{\beta }{2} \}. $
Let $ \eta $ be as in (\ref{3.19}). Then $ \left( \ph^2(|\psi |) - 1 \right)^2 \geq \eta (\frac{ \beta }{2} )$ on $K$, hence
\beq
\label{4.6}
\Lo ^2 ( K) \leq \frac{ 1}{\eta (\frac{ \beta }{2} ) } \int_{\R^2} \left( \ph^2(|\psi |) - 1 \right)^2 \, dx .
\eeq
Let $ \tilde{\psi } = \left( \big| \, | \psi | - 1 \big| - \frac{\beta }{2} \right)_+ \! .$
Then $\tilde{\psi} \in L_{loc}^1(\R^2)$,
$|\nabla \tilde{\psi}| \leq |\nabla \psi |$ a.e. on $ \R^2$ and using (\ref{ineq2}) we get
\beq
\label{4.7}
\int_{\R^2} |\tilde{\psi } |^{2 p_0 + 2} \, dx \leq C \| \nabla \tilde{\psi} \| _{L^2(\R^2)} ^{2 p_0 + 2}
\Lo ^2(K).
\eeq
Using (\ref{4.5}), (\ref{4.6}) and (\ref{4.7}) we obtain
\beq
\label{4.8}
\begin{array}{l}
\ds \int_{\R^2 \setminus \{ 1 - \beta \leq |\psi | \leq 1 + \beta \} }
\Big\vert V( |\psi |^2) - \frac{ 1 \pm \de }{2} \left( \ph^2(|\psi |) - 1 \right)^2 \Big\vert \, dx
\\
\\
\ds \leq C(\de) \int_{\R^2} |\tilde{\psi } |^{2 p_0 + 2} \, dx
\leq C'(\de) \| \nabla \tilde{\psi} \| _{L^2(\R^2)} ^{2 p_0 + 2}
\ds \int_{\R^2} \left( \ph^2(|\psi |) - 1 \right)^2 \, dx .
\end{array}
\eeq
From (\ref{4.4}) and (\ref{4.8}) we infer that $V (|\psi |^2) \in L^1(\R^2)$ and (\ref{4.2}) holds.
\hfill
$\Box$
\medskip
The following result is a direct consequence of (\ref{4.2}).
\begin{Corollary}
\label{C4.1}
Assume that $ N =2$ and (A1) and (A2) hold.
There is $ k_1 > 0 $ such that for any $ \psi \in \Eo $ satisfying
$ \ii_{\R^2} |\nabla \psi |^2 \, dx \leq k_1 $ we have
$
\ii_{\R^2} V( |\psi |^2 ) \, dx \geq 0.
$
\end{Corollary}
If $N \geq 3 $ and there exists $ s_0 \geq 0 $ satisfying $V(s_0 )<0$, Corollary \ref{C4.1} is not valid anymore.
Indeed, if $V$ achieves negative values it easy to see that there
exists $\psi \in \Eo $ such that $ \ii_{\R^N} V( |\psi |^2 ) \, dx < 0.$
Then $ \ii_{\R^N} V( |\psi _{\si, \si }|^2 ) \, dx = \si ^N \ii_{\R^N} V( |\psi |^2 ) \, dx < 0$
for any $ \si > 0 $ and
$ \ii_{\R^N} |\nabla \psi _{\si, \si } |^2 \, dx
= \si ^{N-2} \ii_{\R^N} |\nabla \psi |^2 \, dx \lra 0 $ as $ \si \lra 0 $.
\begin{Corollary}
\label{C4.2}
Let $ N \geq 2$. There is an increasing function $ m : \R_+ \lra \R_+$ such that
$ \ds \lim_{\tau \ra 0 } m ( \tau ) = 0 $ and
$$
\| \, | \psi | - 1 \|_{L^2(\R^N)} \leq m ( E_{GL}(\psi ) ) \qquad
\mbox{ for any } \psi \in \Eo.
$$
\end{Corollary}
{\it Proof.} Let $ \tilde{F}(s) = \frac{ 1}{\sqrt{s}} - 1$. It is obvious that $ \tilde{F}$
satisfies the assumptions (A1) and (A2) in the introduction (except the continuity at $0$, but this plays no role here).
Let $ \tilde{V}(s) = \ii_s ^{1} \tilde{F} (\tau ) \, d \tau $, so that
$ \tilde{V}(s) = ( \sqrt{s} - 1) ^2 $ and $\ii_{\R^N} V(|\psi |^2) \, dx
= \| \, |\psi | - 1 \|_{L^2(\R^N)}^2$.
The conclusion follows by using the second inequalities in (\ref{4.1}) and (\ref{4.2})
with $ \tilde{F}$ and $\tilde{V}$ instead of $F$ and $V$.
\hfill
$\Box $
\medskip
\begin{Lemma}
\label{L4.2}
(i) Let $ \de \in (0, 1)$ and let $ \psi \in \Eo $ be such that
$ 1 - \de \leq |\psi | \leq 1 + \de $ a.e. on $\R^N$. Then
$$
| Q(\psi) | \leq \frac{ 1}{\sqrt{2} (1 - \de )} E_{GL}(\psi).
$$
(ii) Assume that $ 0 \leq c < v_s $ and let $ \e \in (0, 1 - \frac{c}{v_s})$.
There exists a constant $ K _1 = K_1( F, N, c, \e ) > 0$
such that for any $ \psi \in \Eo $ satisfying $E_{GL}(\psi) < K_1$ we have
$$
\ds \int_{\R^N} |\nabla \psi |^2 \, dx + \int_{\R^N} V(| \psi |^2 ) \, dx
- c | Q(\psi) | \geq \e E_{GL}(\psi).
$$
\end{Lemma}
{\it Proof.} If $N\geq 3$, (i) is precisely Lemma 4.2 p. 145 and (ii) is Lemma 4.3 p. 146
in \cite{M10}. In the case $N=2$ the proof is similar and is left to the reader.
\hfill
$\Box$
\medskip
For any $ q \in \R$ we denote
$$
E_{min}(q) = \inf \left\{ \ii_{\R^N} |\nabla \psi |^2\, dx + \ii_{\R^N} \big| V(|\psi |^2) \big| \, dx \; \; \; \Big| \; \psi \in \Eo, Q(\psi ) = q \right\}.
$$
Notice that if $ V \geq 0$, the above definition of $E_{min}$ is the same as the one given in Theorem \ref{T1.1}.
For later purpose we need this more general definition. To simplify the notation, we denote
$$
\ov{E}(\psi) = \ii_{\R^N} |\nabla \psi |^2\, dx + \ii_{\R^N} \big| V(|\psi |^2) \big| \, dx \qquad \mbox{ for any } \psi \in \Eo.
$$
There are functions $ \psi \in \Eo $ such that $ Q(\psi ) \neq 0$
(see for instance Lemma 4.4 p. 147 in \cite{M10}).
For any $ \psi \in \Eo$, the function $ \tilde{\psi }(x) = \psi ( - x_1, x')$
also belongs to $ \Eo$ and satisfies $\ov{E}(\tilde{\psi}) = \ov{E}({\psi})$,
$Q(\tilde{\psi}) = - Q(\psi)$.
Taking into account (\ref{2.9}), it is clear that for any $q$ the set
$ \{ \psi \in \Eo \; | \; Q(\psi ) = q \}$ is not empty and
$E_{min}(- q) = E_{min}(q).$
Thus it suffices to study $E_{min} (q)$ for $q \in [0, \infty)$.
\medskip
If there is $ s_0 $ such that $ V(s_0^2) <0$, then
$ \inf \{ E(\psi ) \; | \; \psi \in \Eo, Q(\psi ) = q \} = - \infty $ for all $ q \in \R$.
(This is one reason why we use $\ov{E}$, not $E$, in the definition of $E_{min}$.)
Indeed, fix $ q \in \R$. From Corollary \ref{C3.4} and (\ref{2.9})
we see that there is $\psi _* \in \Eo$ such that
$Q(\psi _* ) = q $ and $ \psi _* = 1$ outside a ball $B(0, R_*)$.
It is easy to construct a radial, real-valued function $\psi _0$ such that
$E(\psi _0) < 0 $ and $ \psi _0 = 1$ outside a ball $B(0, R_0)$
(for instance, take $ R_0$ sufficiently large, let $\psi _0 = s_0$ on $B(0, R_0 -1)$,
$\psi _0 = 1$ on
$ \R^N \setminus B(0, R_0) $ and $\psi _0$ affine in $|x|$ for $R_0 - 1 \leq |x|\leq R_0$).
Then $ Q( \psi _0 ) = 0$.
Let $ e_1 = (1, 0 , \dots 0 )$. For $ n \geq 1$, we define $ \psi _n $ by
$ \psi _n = \psi _* $
on $B(0, R_*)$, and $ \psi _n (x) = \psi _0 (\frac xn - n ^2 (R_0 + R_*) e_1)$
on $ \R^N \setminus B(0, R_*)$.
Then $Q( \psi _n ) = Q(\psi _* ) + n^{N-1} Q ( \psi_0) = q$
and $ E(\psi _n) = E(\psi _*) + n^{N-2} \ii_{\R^N} |\nabla \psi _0 |^2 \, dx
+ n^N \ii _{\R^N} V(|\psi _0 |^2) \, dx \lra - \infty $ as $ n \lra \infty$.
\medskip
\medskip
The next Lemmas establish the properties of $E_{min}$.
\begin{Lemma}
\label{L4.3}
Assume that $N \geq 2$.
For any $ q > 0 $ we have $E_{min}(q) \leq v_s q. $
Moreover, there is a sequence $ (\psi _n)_{n \geq 1}$ such that
$ \psi _n - 1 \in C_c^{\infty} (\R^N)$, $ V(\psi_n) \geq 0$,
$Q(\psi _n ) = q $, $E(\psi_n ) \lra v_s q$, $E_{GL} (\psi_n ) \lra v_s q$
and $\ds \sup_{x \in \R^N} | \p ^{\al } \psi _n(x )| \lra 0 $ as $ n \lra \infty $
for any $ \al \in \N^N$, $ |\al | \geq 1$.
\end{Lemma}
{\it Proof.}
Fix
$ \chi \in C_c ^{\infty } (\R^N ),$ $ \chi \neq 0$.
We will consider three parameters $ \e, \, \la , \, \si > 0 $ such that $ \e \lra 0 $,
$ \la \lra \infty$, $ \si \lra \infty$ and $\la \ll \si$.
We put
$$
\rho_{\e, \la , \si } (x) = 1 + \frac{ \e }{\sqrt{2} \la } \frac{\p \chi }{\p x _1} \left(\frac{ x_1}{\la}, \frac{ x'}{\si } \right),
\quad
\theta_{\la, \si } (x) = \chi \left( \frac{ x_1}{\la}, \frac{ x'}{\si } \right),
\quad
\psi_{\e, \la, \si } (x) = \rho_{\e, \la , \si } (x) e^{- i \e \theta_{\la, \si } (x) }.
$$
It is clear that $V(\rho_{\e, \la , \si }^2) \geq 0$ if $\frac{\e}{\la}$ is small enough. A straightforward computation gives
$$
\int_{\R^N} \Big\vert \frac{ \p \rho_{\e, \la , \si } }{\p x_1 } \Big\vert ^2 \, dx
= \frac{ \e^2 \si^{N-1}}{2 \la^3 } \int_{\R^N} \Big\vert \frac{ \p ^2 \chi }{\p x_1 ^2} \Big\vert ^2\, dx,
$$
$$
\int_{\R^N} \Big\vert \frac{ \p \rho_{\e, \la , \si } }{\p x_j } \Big\vert ^2 \, dx
= \frac{ \e^2 \si^{N-3}}{2 \la } \int_{\R^N} \Big\vert \frac{ \p ^2 \chi }{\p x_1 \p x_j} \Big\vert ^2\, dx,
\qquad j = 2, \dots, N,
$$
$$
\int_{\R^N} \rho_{\e, \la , \si } ^2 \Big\vert \frac{ \p \theta_{ \la , \si } }{\p x_1 } \Big\vert ^2 \, dx
= \frac{\si^{N-1}}{\la } \int_{\R^N} \left( 1 +\frac{ \e}{ \sqrt{2} \la } \frac{ \p \chi}{\p {x_1} } \right) ^2
\Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx
\simeq
\frac{\si^{N-1}}{\la } \int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx,
$$
$$
\int_{\R^N} \rho_{\e, \la , \si } ^2 \Big\vert \frac{ \p \theta_{ \la , \si } }{\p x_j } \Big\vert ^2 \, dx
= \si^{N-3} {\la } \int_{\R^N} \left( 1 +\frac{ \e}{ \sqrt{2} \la } \frac{ \p \chi}{\p {x_1} } \right) ^2
\Big\vert \frac{ \p \chi}{\p {x_j} } \Big\vert^2 \, dx
\simeq
{\si^{N-3}}{\la } \int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_j} } \Big\vert^2 \, dx,
$$
$$
\int_{\R^N} V( \rho_{\e, \la , \si } ^2 ) \, dx \simeq \frac{\e ^2 \si^{N-1}}{\la }
\int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx,
$$
$$
\int_{\R^N} \left( \ph ^2( \rho_{\e, \la , \si }) - 1 \right)^2 \, dx
\simeq \frac{2 \e ^2\si^{N-1}}{ \la }
\int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx,
$$
$$
Q(\psi_{\e, \la, \si } )
= \e \int_{\R^N} ( \rho_{\e, \la , \si } ^2 - 1 ) \frac{ \p \theta_{ \la , \si } }{\p x_1} \, dx
\simeq \frac{\sqrt{2} \e ^2 \si^{N-1}}{ \la }
\int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx.
$$
Now fix $ q >0$.
Then choose sequences of positive numbers $(\e _n)_{n \geq 1}$,
$(\la _n)_{n \geq 1}$, $(\si _n)_{n \geq 1}$ such that $ \e _n \lra 0 $,
$ \la _n \lra \infty$, $ \si _n \lra \infty$, $\frac{ \la _n}{\si _n} \lra 0 $
and $Q( \psi_{\e_n, \la_n , \si _n } ) = q $ for each $n$.
Such a choice is possible in view of the last estimate above. In particular,
this gives
$ \ds \frac{ \e _n ^2 \si _n ^{N-1}}{ \la _n }
\int_{\R^N} \Big\vert \frac{ \p \chi}{\p {x_1} } \Big\vert^2 \, dx \lra \frac{q }{\sqrt{2}}.$
Let $ \psi _ n = \psi_{\e_n, \la_n , \si _n }$.
It follows from the above estimates that
$$
\ov{E}(\psi _n) = E(\psi _n) = \int_{\R^N} |\nabla \rho_{\e, \la , \si } | ^2
+ \e ^2 \rho_{\e, \la , \si } ^2 |\nabla \theta_{ \la , \si } |^2
+ V( \rho_{\e, \la , \si } ^2) \, dx
\lra \sqrt{2} q = v_s q
$$
and similarly $
E_{GL} (\psi _n) \lra v_s q $ as $ n \lra \infty$.
The other statements are obvious.
Notice that a similar construction can be found in the proof of Lemma 3.3 p. 604 in \cite{BGS}.
\hfill
$\Box $
\begin{Lemma}
\label{L4.4}
Let $N\geq 2$. For each $ \e > 0$ there is $ q_{\e}>0$ such that
$$
E_{min}(q) > ( v_s - \e) q \qquad \mbox{ for any } q \in (0, q _{\e}).
$$
\end{Lemma}
{\it Proof.}
Fix $ \e > 0$.
It follows from Lemma \ref{L4.2} (ii) that there is $ K_1( \e ) > 0 $ such
that for any $ \psi \in \Eo $ satisfying $ E_{GL}(\psi ) < K_1 ( \e ) $ we have
$$
\ov{E}(\psi ) \geq \left ( v_s - \frac{ \e }{2} \right) | Q(\psi )|.
$$
Using Lemma \ref{L4.1} we infer that there exists $ K_2 ( \e ) > 0 $ such
that for any $ \psi \in \Eo $
satisfying $ \ov{E}(\psi ) < K_2 ( \e ) $ we have $ E_{GL}(\psi ) < K_1 ( \e ) $.
Take $ q _{\e}= \frac{ K_2 (\e)} { v_s + 1 }.$
Let $ q \in (0, q_{\e})$.
There is $ \psi \in \Eo $ such that
$ Q(\psi ) = q $ and $ \ov{E}(\psi ) < E_{min}(q) + q $.
Since $E_{min}(q) \leq v_s q $ by Lemma \ref{L4.3},
for any such $ \psi $ we have $\ov{E}(\psi ) < ( v_s + 1) q_{\e} = K_2 ( \e ) $ and we infer that
$E_{GL}(\psi ) < K_1 ( \e) $, thus
$\ov{E}(\psi ) \geq \left ( v_s - \frac{ \e }{2} \right) | Q(\psi )| =
\left ( v_s - \frac{ \e }{2} \right) q .$
This clearly implies $E_{min}(q) \geq \left ( v_s - \frac{ \e }{2} \right) q.$
\hfill
$\Box $
\begin{Lemma}
\label{L4.5}
Assume that $N\geq 2$.
(i) The function $E_{min} $ is subadditive: for any $ q_1, \, q_2 \geq 0 $ we have
$E_{min}(q_1 + q_2) \leq E_{min}(q_1) + E_{min}(q_2).$
(ii) The function $E_{min} $ is nondecreasing on $[0, \infty)$, concave, Lipschitz
continuous and its best Lipschitz constant is $ v_s$.
Moreover, for $ 0 < q_1 < q_2 $ we have
$E_{min}(q_1) \leq \left( \frac{ q_1}{q_2} \right)^{\frac{N-2}{N-1}} E_{min}(q_2).$
(iii) For any $ q > 0 $ we have the following alternative:
$\qquad \bullet $ either $E_{min}(\tau ) = v_s \tau $ for all $ \tau \in [0, q], $
$\qquad \bullet $ or $E_{min}(q) < E_{min}(\tau) + E_{min}(q - \tau ) \quad $
for all $ \tau \in (0, q)$.
\end{Lemma}
{\it Proof. } (i) Fix $ \e > 0$. From Corollary \ref{C3.4} and (\ref{2.9}) it
follows that there exist
$ \psi _1, \psi _2 \in \Eo$ such that $Q(\psi _i ) = q_i $,
$\ov{E}(\psi _i) < E_{min}(q_i) + \frac{ \e}{2}$
and $ \psi _i = 1 $ outside a ball $B(0, R_i)$, $ i = 1,2$.
Let $ e \in \R^N$ be a vector of length $1$. Define
$ \psi (x) = \left\{ \begin{array}{l} \psi _1 (x) \quad \mbox{ if } |x| \leq R_1, \\
\psi _2 (x- 4(R_1 + R_2) e)\quad \mbox{ otherwise. }
\end{array}
\right. $
Then $ \psi \in \Eo$, $Q(\psi ) = Q(\psi _1) + Q(\psi _2 ) = q_1 + q_2 $ and
$E_{min}(q_1+ q_2) \leq \ov{E}(\psi ) = \ov{E}(\psi _1) + \ov{E}(\psi _2 ) <
E_{min}(q_1) + E_{min}(q_2) + \e.$
Letting $ \e \lra 0 $ we get $E_{min}(q_1 + q_2) \leq E_{min}(q_1) + E_{min}(q_2).$
\medskip
(ii) From Lemma \ref{L4.3}
we obtain
$ 0 \leq E_{min }(q) \leq v_s q $ for any $ q \geq 0$.
For $ \psi \in \Eo$ we have $ \psi _{\si, \si } = \psi \left( \frac{\cdot }{\si } \right)
\in \Eo$,
\beq
\label{4.9}
Q(\psi _{\si, \si } ) = \si ^{N-1} Q(\psi )\quad \mbox{ and } \quad
\ov{E}(\psi _{\si, \si } ) = \si ^{N-2} \int_{\R^N} |\nabla \psi |^2 \, dx
+ \si ^N \int_{\R^N} \big| V(|\psi |^2) \big| \, dx.
\eeq
Assume that $0 < q_1 < q_2$. Let $ \si _0 =\left(
\frac{ q_1}{q_2} \right) ^{\frac{1}{N-1}} < 1$.
For any $ \psi \in \Eo$ satisfying $ Q(\psi ) = q_2$ we have $Q(\psi_{\si _0, \si _0}) = q_1 $
and from (\ref{4.9}) we see that $E_{min} (q_1) \leq \ov{E}(\psi _{\si _0, \si _0}) \leq
\si _0 ^{N-2} \ov{E}(\psi )$.
Passing to the infimum over all $\psi $ verifying $ Q(\psi ) = q_2$ we find
$ E_{min}(q_1) \leq \left( \frac{ q_1}{q_2} \right)^{\frac{N-2}{N-1}} E_{min} (q_2).$
In particular, $E_{min}$ is nondecreasing.
Using (i) and Lemma \ref{L4.3} we get
$$
0 \leq E_{min} (q_2) - E_{min} (q_1) \leq E_{min} (q_2- q_1) \leq v_s( q_2 - q_1).
$$
Hence $E_{min}$ is Lipschitz continuous and $ v_s$ is a Lipschitz constant for $E_{min}$.
Lemma \ref{L4.4} implies that $ v_s$ is indeed the best Lipschitz constant of $E_{min}$.
Given a function $f$ defined on $ \R^N$ and $ t \in \R$, we denote
by $S_t^+ f$ and $S_t^-f$, respectively, the functions
\beq
\label{4.10}
S_t^+ f(x) = \left\{
\begin{array}{l}
f(x) \quad \mbox{ if } x_N \geq t,
\\
f( x_1, \dots, x_{N-1}, 2t - x_N ) \mbox{ if } x_N < t,
\end{array}
\right.
\eeq
\vspace*{-12pt}
\beq
\label{4.11}
S_t^- f(x) = \left\{
\begin{array}{l}
f( x_1, \dots, x_{N-1}, 2t - x_N ) \quad \mbox{ if } x_N \geq t,
\\
f( x ) \mbox{ if } x_N < t.
\end{array}
\right.
\eeq
It is easy to see that for all $ \psi \in \Eo $ and $ t \in \R$
we have $S_t^+ \psi , S_t^- \psi \in \Eo, \quad$
$\ov{E}(S_t^+\psi ) + \ov{E}(S_t^-\psi ) = 2 \ov{E}( \psi ) $ and
$\langle i (S_t^{\pm} \psi )_{x_1}, S_t^{\pm} \psi \rangle =
S_t^{\pm} (\langle i \psi _{x_1}, \psi \rangle ).$
Moreover, if $ \phi \in \dot{H}^1(\R^N)$ then
$S_t^{\pm} \phi \in \dot{H}^1(\R^N)$ and
$ \p _{x_1} (S_t^{\pm} \phi) = S_t^{\pm} ( \p _{x_1} \phi).$
If $ \psi \in \Eo$, there are $ \phi \in \dot{H}^1(\R^N)$ and $ g \in L^1 ( \R^N) $
such that $\langle i \psi _{x_1}, \psi \rangle = \p _{x_1}\phi + g$ (see Lemma \ref{L2.1}
and the remarks preceding it). Then
$\langle i (S_t^{\pm} \psi )_{x_1}, S_t^{\pm} \psi \rangle = S_t^{\pm} (\langle i \psi _{x_1}, \psi \rangle )
= \p_{x_1} ( S_t^{\pm} \phi ) + S_t^{\pm} g$ and Definition \ref{D2.2} gives
$Q(S_t^{\pm} \psi ) = \ii _{\R^N} S_t^{\pm} g \, dx$.
It follows that $Q(S_t^{+}\psi ) + Q(S_t^{-}\psi ) = 2 Q( \psi )$
and the mapping $ t \longmapsto Q(S_t^{+}\psi ) = \ii _{\R^N} S_t^{+} g \, dx
= 2 \ii _{\{ x_n \geq t \} } g \, dx $ is continuous on $ \R$, tends to $0 $ as $ t \lra \infty $
and to $ 2 \ii _{\R} g \, dx = 2 Q( \psi ) $ as $t \lra - \infty$.
Fix $ 0 < q_1 < q_2 $ and $ \e > 0$.
Let $ \psi \in \Eo $ be such that $ Q( \psi) = \frac{ q_1 + q_2}{2} $ and
$\ov{E} (\psi ) < E_{min}\left( \frac{q_1 + q_2}{2} \right) + \e$.
The continuity of $ t \longmapsto Q(S_t^{+}\psi ) $ implies that
there exists $ t_0 \in \R $ such that $Q( S_{t_0}^{+}\psi ) = q_1$.
Then necessarily $Q( S_t^{-}\psi ) = q_2$ and we infer that
$\ov{E}( S_t^{+}\psi ) \geq E_{min}( q_1) $, $\ov{E}( S_t^{-}\psi ) \geq E_{min}(q_2)$,
and consequently
$$
E_{min}\left( \frac{q_1 + q_2}{2} \right) + \e > \ov{E}(\psi )
= \frac 12 ( \ov{E}( S_t^{+}\psi ) + \ov{E}( S_t^{-}\psi ) )
\geq \frac 12 ( E_{min}( q_1) + E_{min}( q_2)).
$$
Passing to the limit as $ \e \lra 0 $ in the above inequality we discover
\beq
\label{4.12}
E_{min}\left( \frac{q_1 + q_2}{2} \right) \geq \frac 12 ( E_{min}( q_1) + E_{min}( q_2)).
\eeq
It is an easy exercise to prove that any continuous function satisfying (\ref{4.12})
is concave.
\medskip
(iii) Fix $ q > 0 $.
By the concavity of $E_{min}$ we have $E_{min}(\tau ) \geq \frac{ \tau}{q} E_{min} (q) $
for any
$ \tau \in (0, q) $ and equality may occur if and only if $E_{min} $ is linear on $[0, q]$.
Therefore for any $ \tau \in (0, q ) $ we have
$E_{min}(\tau ) + E_{min}(q - \tau ) \geq
\frac{ \tau}{q} E_{min} (q) + \frac{ q - \tau}{q} E_{min} (q) = E_{min} (q) $
and equality occurs if and only if $ E_{min} $ is linear on $[0, q]$, that is
$E_{min}(\tau ) = a \tau $ for $\tau \in [0, q]$ and some $ a \in \R$.
Then Lemma \ref{L4.3} gives $ a \leq v_s$ and Lemma \ref{L4.4} implies $ a \geq v_s - \e $
for any $ \e >0$, hence $ a = v_s$.
\hfill
$\Box $
\medskip
The function $ q \longmapsto \frac{E_{min}(q)}{q} $ is nonincreasing (because $E_{min}$
is concave), positive and by Lemma 4.4 in \cite{M10} there is
a sequence $ q_n \lra \infty $ such that
$ {\ds \lim_{n \ra \infty} }\frac{E_{min}(q_n)}{q_n } = 0 $, hence
$ {\ds \lim_{q \ra \infty} }\frac{E_{min}(q)}{q} = 0 $. Let
$$
q_0 = \inf \{ q > 0 \; | \; E_{min}(q) < v_s q \} ,
$$
so that $ q_0 \in [0, \infty)$, $E_{min}(q) = v_s q $ for $ q \in [0, q_0 ]$
and $E_{min}(q) < v_s q $ for any $q > q_0$.
\begin{Lemma}
\label{L4.6}
Let $ N \geq 2 $. Assume that (A1), (A2) hold
Then for any $ m , \, M > 0 $ there exist $ C_1(m), \, C_2(M) > 0 $ such that
for all $ \psi \in \Eo $ satisfying $ m \leq \ov{E}( \psi ) \leq M$ we have
$$
C_1(m) \leq E_{GL}(\psi) \leq C_2(M).
$$
\end{Lemma}
{\it Proof. }
If $ N \geq 3$, Lemma \ref{L4.6} follows directly from (\ref{4.1}) with $|V|$ instead of $V$.
If $ N=2$, the second inequality in (\ref{4.2}) implies that there is $C_1(m) > 0 $
such that $E_{GL}(\psi ) \geq C_1(m) $ if $ \ov{E}( \psi ) \geq m$. All we have
to do is to prove that
$ \ii _{\R^2} \left( \ph^2( |\psi |) - 1 \right)^2 \, dx $ remains bounded
if $ \ov{E}( \psi ) \leq M$.
This would be trivial if $ \inf \{ V(s^2) \; | \; s \geq 0, \; |s - 1 | \geq \de \} > 0 $
for any $ \de > 0$; however, our assumptions do not prevent $V$ to vanish somewhere
on $[0, \infty)$ or to tend to zero at infinity.
Since the proof is the same if $ N =2$ or if $ N \geq 2$, let us consider the general case.
Fix $ \de \in (0, 1 ] $ such that
$V(s^2) \geq \frac 14 ( s^2 - 1) ^2 $ for $ s \in [1 - \de, 1 + \de]$.
Consider $ \psi \in \Eo $ such that $ \ov{E}(\psi ) \leq M$.
Clearly, $ \ii _{\{ | \, |\psi | - 1 |
\leq \de \} } \left( \ph^2 (|\psi | ) - 1 \right)^2 \, dx
\leq 4 \ii _{\{ | \, |\psi | - 1 | \leq \de \} } V( | \psi |^2) \, dx \leq 4M$ and
we have to prove that
$ \ii _{\{ | \, |\psi | - 1 | > \de \} } \left( \ph^2 (|\psi | ) - 1 \right)^2 \, dx $
is bounded. Since $ \ph $ is bounded, it suffices to prove that
$ \Lo ^N ( \{ \big| \, |\psi | - 1 \big| > \de \} ) $ is bounded.
Let $ w = |\psi | - 1$. Then $|\nabla w | \leq |\nabla \psi | $ a.e., hence
$ \nabla w \in L^2( \R^N)$, and $ \Lo ^N ( \{ | w| > \al \} ) $ is finite for all $ \al > 0$
(because $ \psi \in \Eo $).
Let $ w_1 ( x) = \phi_1 (|x|) $ and $ w_2 ( x) = \phi_2 (|x|) $ be the symmetric
decreasing rearrangements of $ w_+$ and $w_-$, respectively.
Then $ \ph _1$ and $ \ph _2$ are finite, nonincreasing on
$ (0, \infty)$ and tend to zero at infinity.
From Lemma 7.17 p. 174 in \cite{lieb-loss} it follows that
$ \| \nabla w _1 \|_{L^2(\R^N)} \leq \| \nabla w _+ \|_{L^2(\R^N)} $ and
$ \| \nabla w _2 \|_{L^2(\R^N)} \leq \| \nabla w _- \|_{L^2(\R^N)} $.
In particular, $ w_1, \, w_2 \in H^1 ( \Om _{R_1, R_2})$ for any $0< R_1 < R_2 < \infty$,
where $ \Om _{R_1, R_2} = B(0, R_2) \setminus \ov{B(0, R_1)} $.
Using Theorem 2 p. 164 in \cite{EG} we infer that $ \phi _1 ,\, \phi _2 \in H_{loc}^1((0, \infty)) $,
hence are continuous on $(0, \infty)$.
Let $t_i = \inf \{ t \geq 0 \; | \; \phi_i(t) \leq \de \}$, $i = 1,2$, so that
$0 \leq \phi _i (t) \leq \de $ on $[t_i, \infty)$ and, if $ t_i >0$, then $ \phi_i (t_i) = \de$.
It is clear that
\beq
\label{4.13}
\begin{array}{l}
\Lo ^N ( \{ \big| \, | \psi | - 1 \big| > \de \})
= \Lo ^N ( \{ w_+ > \de \}) + \Lo ^N ( \{ w_ - > \de \})
\\
= \Lo ^N ( \{ w_1 > \de \}) + \Lo ^N ( \{ w_ 2 > \de \})
= ( t_ 1^N + t_2 ^N) \Lo ^N (B(0,1)).
\end{array}
\eeq
Define $h_1(s) = s^2 + 2 s$, $H_1(s) = \frac 13 s^3 + s^2$,
$h_2(s) = - s^2 + 2 s$, $H_2(s) = - \frac 13 s^3 + s^2$, so that $ H_1 ' = h_1$ and $ H_2' = h_2$.
If $ t_1 > 0$ we have:
\beq
\label{4.14}
\begin{array}{l}
\ov{E}(\psi ) \geq \ds \int_{\R^N} |\nabla \psi |^2 \, dx
+ \frac{1}{4} \int_{\R^N} \left( \ph^2( |\psi |) - 1 \right)^2 \1_{\{ 1 \leq |\psi |\leq 1 + \de \}}\, dx
\\
\\
\geq \ds \int_{\R^N} |\nabla w_+ |^2 \, dx
+ \ds \frac 14 \int_{\{ w_+ \leq \de \}} \left( (w_+ + 1 )^2 - 1 \right)^2 \, dx
\\
\\
\geq \ds \int_{\R^N} |\nabla w_1 |^2 \, dx
+ \ds \frac 14 \int_{\{ w_1 \leq \de \}} \left( (w_1 + 1 )^2 - 1 \right)^2 \, dx
\\
\\
\geq \ds \int_{\R^N \setminus B(0, t_1)} |\nabla w_1|^2 + \frac 14 h_1^2 ( w_1) \, dx
\\
\\
= | S^{N-1} | \ds \int_{t_1}^{\infty}
\left( |\phi _1 '(s) | ^2 + \frac 14 h_1^2 ( \phi _1 (s)) \right) s^{N-1} \, ds
\\
\\
\geq t_1^{N-1} |S^{N-1}| \ds \int_{t_1}^{\infty}
|\phi _1 '(s) | ^2 + \frac 14 h_1^2 ( \phi _1 (s)) \, ds
\\
\\
\geq t_1^{N-1} | S^{N-1} |\ds \int_{t_1}^{\infty}
- h_1( \phi_1 (s)) \phi _1'(s) \, ds
\\
\\
= t_1^{N-1} |S^{N-1} | \left[ - H_1 ( \phi _1(s)) \right]_{s = t _0 }^{\infty}
= |S^{N-1} | H_1 ( \de ) t_1^{N-1} ,
\end{array}
\eeq
where $ |S^{N-1}| $ is the surface measure of the unit sphere in $ \R^N$.
From (\ref{4.14}) we get $ t_1 ^{N-1} \leq C E(\psi)$, where $C$ depends only on $N$ and $V$.
It is clear that a similar estimate holds for $t_2$. Then using (\ref{4.13}) we obtain
$$
\Lo ^N ( \{ \big| \, | \psi | - 1 \big| > \de \}) \leq C \left( \ov{E}(\psi) \right)^{\frac{N}{N-1}},
$$
where $C$ depends only on $N$ and $V$, and the proof of Lemma \ref{L4.6} is complete.
\hfill
$\Box $
We can now state the main result of this section, showing precompactness
of minimizing sequences for $E_{min}(q)$ as soon as $q > q_0$.
\begin{Theorem}
\label{T4.7}
Assume that $ q > q_0$, that is $E_{min}(q) < v_s q$.
Let $ (\psi _n )_{n \geq 1 } $ be a sequence in $ \Eo $ satisfying
$$
Q( \psi _ n ) \lra q \quad \mbox{ and } \quad \ov{E}( \psi _n ) \lra E_{min}(q).
$$
There exist a subsequence $(\psi_{n_k})_{k \geq 1}$, a sequence
of points $(x_k)_{k \geq 1} \subset \R^N$,
and $ \psi \in \Eo $ such that $Q( \psi )\! = \!q$, $\ov{E}( \psi )\! =\! E_{min}(q) $,
$\psi_{n _k} ( \cdot + x_k ) \lra \psi $ a.e. on $ \R^N$ and
$ d_0( \psi_{n _k} ( \cdot + x_k ), \psi) \lra 0 $, that~is
$$
\| \nabla \psi_{n _k} ( \cdot + x_k ) - \nabla \psi \|_{L^2(\R^N)} \lra 0 ,
\qquad
\| \, | \psi_{n _k} |( \cdot + x_k ) - | \psi | \, \|_{L^2(\R^N)} \lra 0
\qquad \mbox{ as } k \lra \infty.
$$
\end{Theorem}
{\it Proof. }
Since $\ov{E}(\psi _n) \lra E_{min}(q) > 0$,
it follows from Lemma \ref{L4.6} that there are two positive constants $M_1, \, M_2$
such that $M_1 \leq E_{GL}(\psi _n ) \leq M_2$ for all sufficiently large $n$.
Passing to a subsequence if necessary, we may assume that $E_{GL}(\psi _n) \lra \al _0> 0$.
We will use the concentration-compactness principle \cite{lions}.
We denote by $ \Lambda_n(t)$ the concentration function associated to
$E_{GL}(\psi _n)$, that is
\beq
\label{4.15}
\Lambda_n(t ) = \sup_{y \in \R^N} \int_{B(y, t)}
|\nabla \psi_n |^2 + \frac 12 \left( \ph ^2( |\psi _n |) - 1 \right)^2 \, dx.
\eeq
Proceeding as in \cite{lions}, it is straightforward to prove that there
exists a subsequence of $((\psi_n, \Lambda_n))_{n \geq 1}$, still denoted
$((\psi_n, \Lambda_n))_{n \geq 1}$, there exists a nondecreasing
function $ \Lambda : [0, \infty ) \lra \R$ and there is $\al \in [0, \al _0] $
such that
\beq
\label{4.16}
\Lambda_n (t) \lra \Lambda(t)\; \mbox{ a.e on } [0, \infty ) \; \mbox{ as } n \lra \infty
\qquad \mbox{ and } \qquad
\Lambda(t) \lra \al \mbox{ as } t \lra \infty.
\eeq
As in the proof of Theorem 5.3 in \cite{M10}, we see that there is a nondecreasing sequence $t_n \lra \infty$
such that
\beq
\label{annulus}
\lim_{n \ra \infty }\Lambda_n (t_n) = \lim_{n \ra \infty }\Lambda_n \left(\frac{t_n}{2} \right) = \al.
\eeq
Our aim is to prove that $\al = \al _0$.
The next lemma implies that $ \al > 0$.
\begin{Lemma}
\label{L4.8}
Assume that $N \geq 2$ and assumptions (A1) and (A2) in the Introduction hold.
Let $(\psi_n) _{n \geq 1} \subset \Eo $ be a sequence satisfying:
\medskip
(a) $ E_{GL}(\psi _n) \leq M $ for some positive constant $M$.
\medskip
(b) $ \ds \liminf_{n \ra \infty} Q(\psi _n ) \geq q \in \R \cup \{ \infty \} $
as $ n \lra \infty.$
\medskip
c) $\ds \limsup_{n \ra \infty } E(\psi_n) < v_s q$.
\\
Then there exists $ k > 0$ such that
$ \ds \sup_{ y \in \R^N } E_{GL}^{B(y,1)} ( \psi _n ) \geq k
$
for all sufficiently large $n$.
\end{Lemma}
{\it Proof. } We argue by contradiction and we suppose that the conclusion is false.
Then there is a subsequence (still denoted $(\psi_n) _{n \geq 1} $) such that
\beq
\label{4.17}
\lim_ {n \ra \infty } \sup_{ y \in \R^N } \int_{B(y, 1)}
|\nabla \psi_n |^2 + \frac 12 \left( \ph ^2( |\psi_n |) - 1 \right)^2 \, dx =0.
\eeq
The first step is to prove that
\beq
\label{4.18}
\lim_ {n \ra \infty } \int_{\R^N}
\Big\vert V(|\psi _n |^2) - \frac 12 \left( \ph ^2( |\psi _n |) - 1 \right)^2 \Big\vert \, dx = 0.
\eeq
If $ N \geq 3 $ this is done exactly as in the proof of Lemma 5.4 p. 156 in \cite{M10}.
We consider here only the case $ N =2$.
Fix $ \e > 0 $. By (A1) there is $ \de (\e) > 0 $ such that
$$
\Big\vert V(s^2) - \frac 12 \left( \ph ^2( s) - 1 \right)^2 \Big\vert \leq \frac{\e }{2} \left( \ph ^2( s) - 1 \right)^2
\qquad \mbox{ for any } s \in [ 1 - \de ( \e), 1 + \de ( \e)],
$$
hence
\beq
\label{4.19}
\begin{array}{l}
\ds \int_{\{ 1 - \de ( \e) \leq |\psi | \leq 1 + \de ( \e) \} }
\Big\vert V(|\psi _n |^2) - \frac 12 \left( \ph ^2( |\psi _n |) - 1 \right)^2 \Big\vert \, dx
\\
\\
\ds \leq \frac{\e}{2} \int_{\{1 - \de ( \e) \leq |\psi | \leq 1 + \de ( \e) \} }
\left( \ph ^2( |\psi _n |) - 1 \right)^2 \, dx \leq \e M.
\end{array}
\eeq
Using (A2) we infer that there is $ C(\e) > 0 $ such that
\beq
\label{4.20}
\Big\vert V(s^2) - \frac 12 \left( \ph ^2( s) - 1 \right)^2 \Big\vert \leq C(\e) ( |s| - 1 ) ^{2 p_0 + 2 }
\quad \mbox{ for any } s
\mbox{ satisfying } | s - 1 | \geq \de (\e).
\eeq
Let $ w_n = \big| \, |\psi _n | - 1 \big|$.
Then $ w_n \in L_{loc}^1(\R^N)$ and $|\nabla w_n | \leq | \nabla \psi _n|$ a.e., hence
$ \| \nabla w_n \|_{L^2( \R^2)} \leq \| \nabla \psi _n \|_{L^2( \R^2)} \leq \sqrt{M}$.
Using (\ref{ineq2}) for $ \left( w_n - \frac{ \de (\e )}{2} \right)_+ $ we obtain
\beq
\label{4.21}
\begin{array}{l}
\ds \int_{\{ w_n > \de ( \e ) \} } w_n ^{2 p_0 + 2 } \, dx
\leq 2^{ 2 p_0 + 2} \int_{\{ w_n > \de ( \e ) \} } \left( w_n - \frac{ \de (\e )}{2} \right)_+ ^{ 2 p_0 + 2} \, dx
\\
\\
\leq C \| \nabla w_n \|_{L^2( \R^2)} ^{ 2 p_0 + 2 }
\Lo ^2 ( \{ w_n > \frac{\de ( \e )}{2} \} )
\leq C M^{ p_0 + 1} \Lo ^2 ( \{ w_n > \frac{\de ( \e )}{2} \} ) .
\end{array}
\eeq
We claim that for any $ \de > 0 $ we have
\beq
\label{4.22}
\lim_{n \ra \infty } \Lo ^2 ( \{ w_n > \de \} ) = 0.
\eeq
The proof of (\ref{4.22}) relies on Lieb's Lemma
(see Lemma 6 p. 447 in \cite{lieb} or Lemma 2.2 p. 101 in \cite{brezis-lieb})
and is the same as the proof of (5.20) p. 157 in \cite{M10}, so we omit it.
From (\ref{4.20}), (\ref{4.21}) and (\ref{4.22}) we get
\beq
\label{4.23}
\int_{ \{ |\, |\psi | - 1 | > \de ( \e ) \} }
\Big\vert V(s^2) - \frac 12 \left( \ph ^2( s) - 1 \right)^2 \Big\vert \, dx
\leq
C(\e ) \int_{\{ w_n > \de ( \e ) \} } w_n ^{2 p_0 + 2 } \, dx \lra 0
\eeq
as $ n \lra \infty.$ Then (\ref{4.18}) follows from (\ref{4.19}) and (\ref{4.23}).
From (\ref{4.17}) and Lemma \ref{vanishing} we infer that there exists a sequence $ h_n \lra 0 $
and for each $n $ there is a minimizer $ \zeta _n $ of $G_{h_n, \R^N} ^{\psi _n} $ in
$H_{\psi _n }^1( \R^N) $ such that
\beq
\label{4.24}
\de _n := \| \, |\zeta _n | - 1 \|_{L^{\infty} (\R^N)} \lra 0 \qquad \mbox{ as } n \lra \infty.
\eeq
Then Lemma \ref{L4.2} (i) implies
\beq
\label{4.25}
E_{GL } ( \zeta_n ) \geq \sqrt{2} ( 1 - \de _n ) | Q( \zeta _n) |.
\eeq
From (\ref{3.4}) we obtain $ \ds \lim_{n \ra \infty} | Q( \zeta _n ) - Q( \psi _n) | = 0$,
hence $\ds \liminf_{n \ra \infty} Q( \zeta _n ) = \liminf_{n \ra \infty} Q( \psi _n) \geq q.$
Using (\ref{4.18}), the fact that $E_{GL} ( \zeta _n) \leq E_{GL} ( \psi _n) $ and (\ref{4.25}) we get
$$
\begin{array}{l}
E(\psi _n) = E_{GL} (\psi _n) + \ds \int_{\R^N} V(|\psi_n |^2) - \frac 12 ( \ph ^2(|\psi _n|) - 1)^2\, dx
\\
\geq
E_{GL} (\zeta _n) + \ds \int_{\R^N} V(|\psi_n |^2) - \frac 12 ( \ph ^2(|\psi _n|) - 1)^2\, dx
\\
\geq
\sqrt{2} ( 1 - \de _n ) |Q(\zeta _n )| + \ds \int_{\R^N} V(|\psi_n |^2) - \frac 12 ( \ph ^2(|\psi _n|) - 1)^2\, dx .
\end{array}
$$
Passing to the limit as $ n \lra \infty $ in the above inequality we get
$$
\liminf_{n \ra \infty} E(\psi _n) \geq \sqrt{2} q = v_s q,
$$
which contradicts assumption c) in Lemma \ref{L4.8}. This ends the
proof of Lemma \ref{L4.8}.
\hfill
$\Box $
\medskip
Next we prove that $ \al \not\in (0, \al _0)$.
We argue again by contradiction and we assume that
$ 0 < \al < \al _0$.
Let $ t_n$ be as in (\ref{annulus}) and let $ R_n= \frac{t_n}{2}$.
For each $n \geq 1$, fix $ y_n \in \R^N$ such that
$E_{GL}^{B(y_n, R_n)} ( \psi_n) \geq \Lambda_n ( R_n) - \frac 1n$.
Using (\ref{annulus}), we have
\beq
\label{4.27}
\e _n : = E_{GL}^{B(y_n, 2R_n) \setminus B(y_n, R_n)} ( \psi _n)
\leq \Lambda_n ( 2R_n) - \left( \Lambda_n( R_n) - \frac 1n \right) \lra 0
\mbox{ as } n \lra \infty .
\eeq
After a translation, we may assume that $ y_n = 0$.
Using Lemma \ref{splitting} with $ A = 2$, $ R = R_n$, $ \e = \e_n$,
we infer that for all $n$ sufficiently large there exist
two functions $ \psi_{n, 1}$, $ \psi_{n, 2}$ having the properties (i)-(vi)
in Lemma \ref{splitting}.
In particular, we have
$E_{GL}(\psi _{n, 1} ) \geq E_{GL}^{B(0, R_n)} ( \psi _n ) \geq q( R_n) - \frac 1n$,
$E_{GL}(\psi _{n, 2} ) \geq E_{GL}^{\R^N \setminus B(0,2 R_n)} ( \psi _n ) \geq E_{GL}(\psi _{n} ) - q(2 R_n) $
and
$ | E_{GL}(\psi _{n} ) - E_{GL}(\psi _{n, 1} ) - E_{GL}(\psi _{n, 2} )| \leq C \e _n \lra 0 $ as $ n \lra \infty.$
Taking into account (\ref{annulus}), we conclude that necessarily
\beq
\label{4.28}
E_{GL}(\psi _{n, 1} ) \lra \al
\qquad
\mbox{ and }
\qquad
E_{GL}(\psi _{n, 2} ) \lra \al _0 - \al
\qquad
\mbox{ as } n \lra \infty.
\eeq
From Lemma \ref{splitting} (iii)-(vi) we get
\beq
\label{4.29}
| \ov{E}(\psi _n) - \ov{E}(\psi _{n, 1}) - \ov{E}(\psi _{n, 2}) | \lra 0 \qquad \mbox{ and }
\eeq
\beq
\label{4.30}
| Q(\psi _n) - Q(\psi _{n, 1}) - Q(\psi _{n, 2}) | \lra 0 \qquad \mbox{ as } n \lra \infty.
\eeq
In particular, $\ov{E}(\psi _{n, i}) $ is bounded, $ i =1,2$.
Passing to a subsequence if necessary, we may assume that $ \ov{E}(\psi _{n, i}) \lra m_i \geq 0 $
as $ n \lra \infty$.
Since $\ds \lim_{n \ra \infty } E_{GL}(\psi _{n,i} ) > 0$, it follows from Lemma \ref{L4.1} that
$ m_i > 0$, $i=1,2$.
Using (\ref{4.29}) we see that $ m_1 + m_2 = E_{min}(q)$, hence $ m_1, m_2 \in (0, E_{min}(q)).$
Assume that $\ds \liminf_{n \ra \infty } Q( \psi_{n,1}) \leq 0.$
Then (\ref{4.30}) implies $\ds \limsup_{n \ra \infty } Q( \psi_{n,2}) \geq q.$
It is obvious that
$$
\ov{E}(\psi_{n, 2} ) \geq E_{min}(Q(\psi _{n,2})).
$$
Passing to $\ds \limsup$ in the above inequality and using the continuity
and the monotonicity of $E_{min}$ we get
$ m_2 \geq E_{min}(q), $ a contradiction.
Thus necessarily $\ds \liminf_{n \ra \infty } Q( \psi_{n,1}) > 0$ and similarly
$\ds \liminf_{n \ra \infty } Q( \psi_{n,2}) > 0$.
From (\ref{4.30}) we get
$\ds \limsup_{n \ra \infty } Q( \psi_{n,i})< q$, $ i = 1,2$.
Passing again to a subsequence, we may assume that
$ Q( \psi_{n,i}) \lra q_i$ as $ n \lra \infty$, $ i = 1,2$, where $ q_1, q_2 \in (0, q)$.
Using (\ref{4.30}) we infer that $ q_1 + q_2 = q$.
Since $\ov{E}(\psi_{n,i} ) \geq E_{min}(Q(\psi _{n,i}))$,
passing to the limit we get $ m_i \geq E_{min}(q_i)$, $ i =1,2$ and consequently
$$
E_{min}(q) = m_1 + m_2 \geq E_{min}(q_1) + E_{min}(q_2).
$$
Since $ E_{min}(q) < v_s q$, the above inequality is in contradiction
with Lemma \ref{L4.5} (iii). Thus we cannot have $ \al \in (0, \al _0)$.
\medskip
So far we have proved that $ \al = \al _0$.
Then it is standard to prove that there is a sequence $(x_n)_{n \geq 1} \subset \R^N$ such that
for any $ \e > 0 $ there is $ R_{\e} > 0 $ satisfying
$E_{GL} ^{ \R^N\setminus B(x_n, R_{\e})} (\psi _n) < \e$ for all sufficiently large $ n$.
Denoting $ \tilde{\psi}{_n} = \psi _n ( \cdot + x_n)$, we see that
for any $ \e > 0 $ there exist $ R_{\e} > 0 $ and $ n_{\e } \in \N$ such that
\beq
\label{4.31}
E_{GL} ^{ \R^N\setminus B(0, R_{\e})}(\tilde{\psi}{_n}) < \e \qquad \mbox{ for all } n \geq n_{\e}.
\eeq
Obviously, $ (\nabla \tilde{\psi}_{n})_{n \geq 1} $ is bounded in $L^2(\R^N)$ and it is easy to see that
$( \tilde{\psi}{_n})_{n \geq 1} $ is bounded in $L^2 (B(0,R))$ for any $R >0$ (use (\ref{e1}) and
(\ref{e2}) if $ N=2$, respectively (\ref{e1}) and the Sobolev embedding if $ N \geq 3$).
By a standard argument, there exist a function $ \psi \in H_{loc}^1(\R^N)$
such that $ \nabla \psi \in L^2(\R^N)$ and a subsequence $ (\tilde{\psi}_{n _k})_{k \geq 1} $
satisfying
\beq
\label{4.32}
\begin{array}{l}
\nabla \tilde{\psi}_{n_k } \rightharpoonup \nabla \psi \qquad \mbox{ weakly in } L^2( \R^N),
\\
\tilde{\psi}_{n_k } \rightharpoonup \psi \qquad \mbox{ weakly in } H^1(B(0,R)) \mbox{ for all } R > 0,
\\
\tilde{\psi}_{n_k } \lra \psi \; \; \mbox{ strongly in } L^p(B(0,R)) \mbox{ for } R > 0 \mbox{ and }
p \in [1, 2^* ) \; ( p \in [1, \infty) \mbox{ if } N =2),
\\
\tilde{\psi}_{n_k } \lra \psi \; \; \mbox{ a.e. on } \R^N.
\end{array}
\eeq
By weak convergence we have
\beq
\label{4.33}
\int_{\R^N} |\nabla \psi |^2 \, dx
\leq \liminf_{k \ra \infty }
\int_{\R^N} |\nabla \tilde{\psi }_{n_k} |^2 \, dx .
\eeq
The a.e. convergence and Fatou's Lemma imply
\beq
\label{4.34}
\int_{\R^N}
\left( \ph^2 (|\psi |) - 1 \right)^2 \, dx
\leq \liminf_{k \ra \infty }
\int_{\R^N} \left( \ph^2 (|\tilde{\psi }_{n_k}|) - 1 \right)^2 \, dx
\quad \mbox{ and }
\eeq
\beq
\label{4.35}
\quad
\int_{\R^N} \big| V(|\psi |^2) \big| \, dx \leq \liminf_{k \ra \infty } \int_{\R^N} \big| V(|\tilde{\psi} _{n_k}|^2) \big| \, dx.
\eeq
From (\ref{4.33}), (\ref{4.34}) and (\ref{4.35}) we obtain
\beq
\label{4.36}
E_{GL} ( \psi ) \leq \liminf_{k \ra \infty } E_{GL} ( \tilde{\psi } _{n_k}) = \al _0
\quad
\mbox{ and }
\quad
\ov{E} ( \psi ) \leq \liminf_{k \ra \infty } \ov{E} ( \tilde{\psi } _{n_k}) = E_{min}(q).
\eeq
Similarly, for any $ \e > 0 $ we get
\beq
\label{4.37}
E_{GL}^{\R^N \setminus B(0, R_{\e})} ( \psi )
\leq \liminf_{k \ra \infty } E_{GL}^{\R^N \setminus B(0, R_{\e})} ( \tilde{\psi } _{n_k})
\leq \limsup_{k \ra \infty } E_{GL}^{\R^N \setminus B(0, R_{\e})} ( \tilde{\psi } _{n_k}) \leq \e.
\eeq
The following holds.
\begin{Lemma}
\label{L4.10}
Assume that $N \geq 2$ and assumptions (A1) and (A2) are verified.
Let $ ( \g _n ) _{n \geq 1} \subset \Eo $ be a sequence satisfying:
\smallskip
(a) $ (E_{GL}( \g _n)) _{n \geq 1}$ is bounded and for any $ \e > 0 $
there are $ R_{\e } > 0$ and $n_\e \in \N$ such that
$E_{GL}^{\R^N \setminus B(0, R_{\e })} ( \g _n ) < \e$ for $n \geq n_\e$.
\smallskip
(b) There exists $ \g \in \Eo $ such that
$ \g _n \lra \g $ strongly in $L^2( B(0, R))$ for any $ R > 0$,
and $ \g_n \lra \g $ a.e. on $ \R^N$ as $ n \lra \infty$.
\smallskip
Then $ \| \, |\g_n | - | \g | \, \| _{L^2( \R^N)} \lra 0 $ and
$ \| V( |\g _n |^2) - V( |\g |^2) \| _{L^1( \R^N)} \lra 0 $ as $ n \lra \infty$.
\end{Lemma}
{\it Proof of Lemma \ref{L4.10}.}
Fix $ \e > 0 $. Let $ R_{\e}$ and $n_\e \in \N$ be as in assumption (a).
Then
\beq
\label{4.38}
\| \ph ( |\g _n |) - 1 \|_{L^2(\R^N \setminus B(0, R_{\e }))} ^2
\leq
\int_{\R^N \setminus B(0, R_{\e })} \left( \ph ^2 ( |\g _n |) - 1 \right)^2 dx
\leq 2 \e
\eeq
for $ n \geq n_\e $. It is clear that a similar estimate holds for $ \g$. Let
$$
\begin{array}{ll}
\tilde{\g }_n = |\g _n | - \ph (|\g _n|), & \tilde{\g } = |\g | - \ph (|\g |),
\\
\\
A_n = \{ x \in \R^N \; \big| \; \; | \g _n (x) | \geq 2 \}, & A = \{ x \in \R^N \; \big| \; \; | \g (x) | \geq 2 \},
\\
\\
A_n ^{\e }= \{ x \in \R^N \setminus B(0, R_{\e}) \; \big| \; | \g _n (x) | \geq 2 \},
& A ^{\e} = \{ x \in \R^N \setminus B(0, R_{\e}) \; \big| \; | \g (x) | \geq 2 \}.
\end{array}
$$
We have
$$
9 \Lo ^N ( A_n^{\e})
\leq \int_{\R^N \setminus B(0, R_{\e })} \left( \ph ^2 ( |\g _n |) - 1 \right)^2 dx
\leq 2 E_{GL} ^{\R^N \setminus B(0, R_{\e })} ( \g _n )
\leq 2 \e
$$
and similarly $ 9 \Lo ^N ( A^{\e}) \leq 2 \e.$
In the same way $ \Lo ^N ( A_n ) \leq \frac{2}{9 } E_{GL}(\g_n)$ and
$ \Lo ^N ( A ) \leq \frac{2}{9 } E_{GL}(\g)$.
Since $ 0 \leq \ph' \leq 1 $, it is easy to see that $|\nabla \tilde{\g} _n | \leq | \nabla \g _n|$ a.e. and
$|\nabla \tilde{\g} | \leq | \nabla \g |$ a.e., hence
$(|\nabla \tilde{\g} _n | )_{n \geq 1}$ and $ \nabla \tilde{\g}$ are bounded in $L^2( \R^N)$.
If $ N \geq 3$, the Sobolev embedding implies that $(\tilde{\g} _n)_{n \geq 1}$
is bounded in $ L^{2^*}(\R^N)$.
Then using the fact that $ \tilde{\g}_n = 0 $ on $ \R^N \setminus A_n$ and H\"older's inequality we infer that
$ \tilde{\g}_n$ is bounded in $ L^p(\R^N)$ for $1 \leq p \leq 2^*$.
If $ N = 2$, by (\ref{ineq2}) we get
$$
\| \tilde{\g}_n \|_{L^p(\R^2)} ^p \leq C_p^p \| \nabla \tilde{\g }_n \|_{L^2(\R^2)} ^p \Lo ^ 2( A_n),
$$
hence $(\tilde{\g} _n)_{n \geq 1}$ is bounded in $L^p(\R^N)$ for any $ 2 \leq p < \infty . $
Let $ p = 2^* $ if $ N \geq 3$ and let $p > 2 p_0 + 2 $ if $ N = 2$.
Using H\"older's inequality ($p> 2 p_0 + 2 > 2$) we have
\beq
\label{4.39}
\| \tilde{\g }_n \|_{L^2(\R^N \setminus B(0, R_{\e }))} ^2
= \int_{A_n^{\e} } |\tilde{\g}_n |^2 \, dx
\leq
\| \tilde{\g }_n \|_{L^p (\R^N)} ^2 \Lo ^N ( A_n ^{\e } )^{1 - \frac 2p} \leq C _1\e ^{ 1 - \frac 2p},
\eeq
where $C_1 $ does not depend on $n$.
It is clear that a similar estimate holds for $ \tilde{\g}$.
In the same way, using (A2) and H\"older's inequality ($p> 2 p_0 + 2 $) we get
\beq
\label{4.39bis}
\begin{array}{l}
\ds \int_{(\R^N \setminus B(0, R_{\e })) \cap \{ |\g _n | \geq 4 \} } | V ( |{\g } _n | ^2 ) | \, dx
\leq C' \int_{(\R^N \setminus B(0, R_{\e })) \cap \{ |\g _n | \geq 4 \} } |\g _n |^{ 2 p_0 + 2} \, dx
\\
\\
\leq C''
\ds \int_{A_n^{\e} } |\tilde{\g}_n |^{2 p_0 + 2} \, dx
\leq C '' \| \tilde{\g }_n \|_{L^p (\R^N)} ^{2 p_ 0 + 2} \Lo ^N ( A_n ^{\e } )^{1 - \frac{ 2p_0 + 2}{p}}
\leq C_2 \e ^{ 1 - \frac{2p_0 + 2}{p} },
\end{array}
\eeq
and (A1) implies
\beq
\label{4.39ter}
\ds \int_{(\R^N \setminus B(0, R_{\e })) \cap \{ |\g _n | \leq 4 \} } | V ( |{\g } _n | ^2 ) | \, dx
\leq C''' \int_{\R^N \setminus B(0, R_{\e }) }
\left( \ph ^2 ( |\g _n |) - 1 \right)^2 dx
\leq C_3 \e,
\eeq
where the constants $C_2, C_3$ do not depend on $n$.
The same estimates are obviously valid for $ \g$.
From (\ref{4.38}) and (\ref{4.39}) we get
\beq
\label{4.39qua}
\begin{array}{l}
\| \, | \g _n | - |\g | \, \|_{L^2( \R^N \setminus B(0, R_{\e}\! ))}
\! \leq \!
\| \ph (|\g _n |) \! - \! 1 \| _{L^2( \R^N \setminus B(0, R_{\e}))}
\! + \! \| \ph (|\g |) \! - \! 1 \| _{L^2( \R^N \setminus B(0, R_{\e}\! ))}
\\
\quad + \| \tilde{\g} _n \| _{L^2( \R^N \setminus B(0, R_{\e}))}
+ \| \tilde{\g} \| _{L^2( \R^N \setminus B(0, \R_{\e}))}
\leq 2 \sqrt{ 2 } \e + 2 C_1 \e ^{1 - \frac 2p }.
\end{array}
\eeq
Using (\ref{4.39bis}) and (\ref{4.39ter}) we obtain
\beq
\label{4.39cinq}
\ds \int_{\R^N \setminus B(0, R_{\e})} | V(|\g _n |^2) | \, dx
\leq C_2 \e ^{ 1 - \frac{2p_0 + 2}{p} } + C_3 \e.
\eeq
It is obvious that $ \g $ also satisfies (\ref{4.39cinq}).
Since $ |\g _n | = \ph (|\g _n |) + \tilde{\g }_n $ is bounded in $L^p( B(0,R))$ for any $ p \in [2, 2^*]$ if $ N \geq 3$,
respectively $ p \in [2, \infty )$ if $ N = 2$, and
$ \g_n \lra \g$ in $L^2( B(0, R))$ by assumption (b), using interpolation we infer that
$ \g_n \lra \g$ in $L^p( B(0, R))$ for any $p \in [1, 2^*)$ (with $ 2^* = \infty$ if $N=2$).
This implies that $V(|\g _n |^2) \lra V(|\g |^2) $ in $L^1( B(0, R))$
(see, for instance, Theorem A2 p. 133 in \cite{willem}).
Thus we have $ \| \, |\g _n | - |\g | \, \|_{L^2( B(0, R_{\e}))} \leq \e $
and $ \| V( |\g _n |^2) - V(|\g |^2) \|_{L^1( B(0, R_{\e}))} \leq \e $ for all sufficiently large $n$.
Together with inequalities (\ref{4.39qua}) and (\ref{4.39cinq}), this implies
$\| \, |\g _n | - |\g | \, \|_{L^2(\R^N)} \leq 2 \sqrt{2} \e + 2 C_1 \e ^{1 - \frac 2p } + \e$
and
$ \| V( |\g _n |^2) - V(|\g |^2) \|_{L^1( \R^N)}
\leq 2 C_2 \e ^{ 1 - \frac{2p_0 + 2}{p} } + ( 2C_3 +1) \e $
for all sufficiently large $n$.
Since $ \e $ is arbitrary, Lemma \ref{L4.10} follows.
\hfill
$\Box $
\medskip
We come back to the proof of Theorem \ref{T4.7}.
From (\ref{4.31}), ({\ref{4.32}) and Lemma \ref{L4.10} we obtain
$\| \, |\tilde{\psi}_{n _k} | - |\psi | \, \| _{L^2( \R^N) } \lra 0 $ as $ k \lra \infty.$
Clearly, this implies
$\| \ph^2(| \tilde{\psi}_{n_k } |) - \ph^2(| \psi |) \|_{L^2(\R^N)} \lra 0$.
\medskip
We will use the following result:
\begin{Lemma}
\label{L4.11}
Let $N \geq 2$ and assume that $ ( \g _n ) _{n \geq 1} \subset \Eo $ is
a sequence satisfying:
\smallskip
(a) $ (E_{GL}( \g _n) ) _{n \geq 1} $ is bounded and for any $ \e > 0 $
there are $ R_{\e } > 0$ and $n_\e \in \N $ such that
$E_{GL}^{\R^N \setminus B(0, R_{\e })} ( \g _n ) < \e$ for $n \geq n_\e$.
\smallskip
(b) There exists $ \g \in \Eo $ such that
$ \nabla \g _n \rightharpoonup \nabla \g $ weakly in $ L^2(\R^N)$ and
$ \g _n \lra \g $ strongly in $L^2( B(0, R))$ for any $ R > 0$
as $ n \lra \infty$.
\smallskip
Then $ Q( \g_n ) \lra Q( \g )$ as $ n \lra \infty$.
\end{Lemma}
We postpone the proof of Lemma \ref{L4.11} and we complete the proof of Theorem \ref{T4.7}.
From (\ref{4.31}), (\ref{4.32}) and Lemma \ref{L4.11} it follows that
$Q(\psi ) = \ds \lim_{k \ra \infty} Q(\tilde{\psi}_{n_k } ) = q.$
Then necessarily
$ \ov{E}(\psi ) \geq E_{min}(q) = \ds \lim_{k\ra \infty} \ov{E}(\tilde{\psi}_{n_k } ).$
From (\ref{4.36}) we get $ \ov{E}(\psi ) = E_{min}(q)$, hence $ \psi $ is a minimizer of $\ov{E}$
under the constraint $ Q (\psi) = q$. Taking into account (\ref{4.33}), (\ref{4.35})
and the fact that $ \ov{E}(\tilde{\psi }_{n _k} ) \lra \ov{E}( \psi ) $,
we infer that
$\ii _{\R^N} |\nabla \tilde{\psi }_{n_k}| ^2 \, dx \lra \ii _{\R^N} |\nabla \psi | ^2 \, dx .$
Together with the weak convergence
$\nabla \tilde{\psi}_{n_k } \rightharpoonup \nabla \psi $ in $ L^2( \R^N)$,
this gives the strong convergence
$\| \nabla \tilde{\psi}_{n_k } - \nabla \psi \|_{L^2(\R^N)} \lra 0 $ as $ k \lra \infty$
and Theorem \ref{T4.7} is proven.
\hfill
$\Box$
\medskip
{\it Proof of Lemma \ref{L4.11}.}
It follows from Lemma \ref{L4.1} and Lemma \ref{L4.2} (ii) that
there are $ \e _ 0 > 0 $ and $C_0>0$ such that for
any $ \phi \in \Eo $ satisfying $E_{GL}( \phi ) \leq \e_0 $ we have
\beq
\label{4.40}
|Q(\phi )| \leq C _0 E_{GL}(\phi).
\eeq
Fix $ \e \in (0, \frac{ \e _0}{2})$.
Let $R_{\e}$ and $n_\e$ be as in assumption (a).
We will use the conformal transform. Let
\beq
\label{4.41}
v_k (x) = \left\{
\begin{array}{ll}
\g _k (x) & \quad \mbox{ if } |x| \geq R_{\e} ,
\\
\g _k \left( \frac{ R_{\e}^2}{ |x|^2 } x \right) & \quad \mbox{ if } |x| < R_{\e},
\end{array}
\right.
\qquad
v (x) = \left\{
\begin{array}{ll}
{\g } (x) & \quad \mbox{ if } |x| \geq R_{\e} ,
\\
{\g} \left( \frac{ R_{\e}^2}{ |x|^2 } x \right) & \quad \mbox{ if } |x| < R_{\e}.
\end{array}
\right.
\eeq
A straightforward computation gives
\beq
\label{4.43}
\int_{B(0, R_{\e})} |\nabla v_k |^2 \, dx = \!
\int_{\R^N \setminus B(0, R_{\e})} \! |\nabla \g_k (y) |^2
\left( \frac{ R_{\e} ^2}{|y |^2} \right)^{N-2} \! dy
\leq \!
\int_{\R^N \setminus B(0, R_{\e})} |\nabla \g_k (y) |^2 \, dy ,
\eeq
\beq
\label{4.44}
\begin{array}{l}
\ds \int_{B(0, R_{\e})} \left( \ph ^2 (| v_k | ) - 1 \right)^2 \, dx =
\int_{\R^N \setminus B(0, R_{\e})} \left( \ph ^2 (|\g _k (y) | ) - 1 \right)^2
\left( \frac{ R_{\e} ^2}{|y |^2} \right)^{N} dy
\\
\\
\leq
\ds \int_{\R^N \setminus B(0, R_{\e})} \left( \ph ^2 (|\g _k | ) - 1 \right)^2 \, dy,
\end{array}
\eeq
so that $ v_k \in \Eo $ and $E_{GL}(v_k) < 2\e < \e _0 $.
Similarly $ v \in \Eo $ and $E_{GL}(v) < 2\e.$
From (\ref{4.40}) we get
\beq
\label{4.45}
| Q( v_k) | \leq 2 C _0 \e \qquad \mbox{ and } \qquad | Q( v) | \leq 2 C _0 \e.
\eeq
Since $ \nabla \g _k \rightharpoonup \nabla \g $ weakly in $L^2( \R^N)$,
a simple change of variables shows that for any fixed $ \de \in (0, R_{\e}) $ we have
$ \nabla v _k \rightharpoonup \nabla v $ weakly in $L^2( B(0, R_{\e}) \setminus B(0, \de ))$.
On the other hand,
$$
\ds \int_{B(0, \de)} |\nabla v_k |^2 \, dx = \!
\int_{\R^N \setminus B(0, \frac{R_{\e}^2}{\de})} \! |\nabla \g _k (y) |^2
\left( \frac{ R_{\e} ^2}{|y |^2} \right)^{N-2} \! dy
\leq \int_{\R^N \setminus B(0, \frac{R_{\e}^2}{\de})} \! |\nabla \g _k (y) |^2 \, dy
$$
and $ \ds \sup_{k \geq 1}
\int_{\R^N \setminus B(0, \frac{R_{\e}^2}{\de})} \! |\nabla \g _k (y) |^2 \, dy
\lra 0 $
as $ \de \lra 0 $ by assumption (a). We conclude that
\beq
\label{4.46}
\nabla v_k \rightharpoonup \nabla v \qquad \mbox{ weakly in } L^2( B(0, R_{\e})).
\eeq
Since $ \g _k \lra \g $ in $L^2( B(0,R)) $ for any $R>0$,
we have for any fixed $ \de \in (0, R_{\e})$,
$$
\int_{B(0, R_{\e}) \setminus B(0, \de)} |v_k - v|^2 \, dx
= \int_{B(0, \frac{R_{\e}^2}{\de}) \setminus B(0, R_{\e})} |\g _k (y)- \g (y) |^2
\left( \frac{ R_{\e} ^2}{|y |^2} \right)^{N} dy
\lra 0 \mbox{ as } k \lra \infty.
$$
It is easy to see that there is $ p > 2$ such that
$\left( \left(|v_k | - 2 \right)_+ \right)_{k \geq 1}$
is bounded in $L^p(\R^N)$.
(If $ N \geq 3$ this follows for $ p = 2^*$ from the Sobolev embedding
because $\|\nabla v_k \|_{L^2(\R^N)}^2 \leq E_{GL} (v_k)\leq 2 \e$.
If $ N =2$, the fact that $E_{GL} (v_k)\leq 2 \e$ implies that
$\Lo ^2( \{ | v_k | \geq 2 \})$ and $\|\nabla v_k \|_{L^2(\R^2)} $
are bounded and the conclusion follows from (\ref{ineq2}).)
Using H\"older's inequality we obtain
$$
\int_{B(0, \de )}
\left(|v_k | - 2 \right)_+ ^2 \, dx \leq \| \left(|v_k | - 2 \right)_+ \| _{L^p(\R^N)} ^{2}
\left( \Lo ^N ( B(0, \de )) \right)^{1 - \frac 2p}
$$
and the last quantity tends to zero as $ \de \lra 0 $ uniformly with respect to $k$.
This implies
$$
\int_{B(0, \de )} | v_k |^2 \, dx \lra 0 \qquad \mbox{ as } \de \lra 0
\mbox{ uniformly with respect to } k
$$
and we conclude that
\beq
\label{4.47}
v_k \lra v \qquad \mbox{ in } L^2(B(0, R_{\e})).
\eeq
Let
\beq
\label{4.42}
w_k = \g _k - v_k, \qquad w = \g - v.
\eeq
It is obvious that $ w_k, \, w \in H_0^1 (B(0, R_{\e}))$, $ \g _k = v_k + w_k $ and $ \g = v + w$.
From assumption (b), (\ref{4.46}) and (\ref{4.47}) it follows that
\beq
\label{4.48}
w_k \lra w \quad \mbox{ strongly $ \quad $ and } \quad
\nabla w_k \rightharpoonup \nabla w \quad \mbox{ weakly in } L^2(B(0, R_{\e})).
\eeq
Using Definition \ref{D2.2} we have
\beq
\label{4.49}
\begin{array}{l}
| Q( \g _k ) - Q( \g ) | \leq |Q(v_k ) - Q(v) |
+ \big\vert L ( \langle i \frac{ \p v_k}{\p x_1 } , w_k \rangle - \langle i \frac{ \p v}{\p x_1 } , w \rangle ) \big\vert
\\
\\
+ \big\vert L ( \langle i \frac{ \p w_k}{\p x_1 } , v_k \rangle - \langle i \frac{ \p w}{\p x_1 } , v \rangle ) \big\vert
+ \big\vert L ( \langle i \frac{ \p w_k}{\p x_1 } , w_k \rangle - \langle i \frac{ \p w}{\p x_1 } , w \rangle ) \big\vert.
\end{array}
\eeq
From (\ref{4.45}) we get $ |Q(v_k ) - Q(v) | \leq 4 C _0 \e$.
Since $ w_k = 0 $ and $ w = 0 $ outside $\ov{B}(0, R_{\e})$, using the definition of $L$ we obtain
$$
\begin{array}{l}
L ( \langle i \frac{ \p v_k}{\p x_1 } , w_k \rangle - \langle i \frac{ \p v}{\p x_1 } , w \rangle )
= {\ds \int _{B(0, R_{\e})} }
\langle i \frac{ \p v_k}{\p x_1 } - i \frac{ \p v}{\p x_1 }, w \rangle
+ \langle i \frac{ \p v_k}{\p x_1 }, w_k - w \rangle \, dx
\lra 0 \quad \mbox{ as } k \lra \infty
\end{array}
$$
because $\frac{ \p v_k}{\p x_1} - \frac{ \p v}{\p x_1} \rightharpoonup 0 $
weakly and $ w_k - w \lra 0 $ strongly in $L^2(B(0, R_{\e}))$.
Similarly the last two terms in (\ref{4.49}) tend to zero as $ k \lra \infty$.
Finally we get
$
| Q( \g _k ) - Q( \g ) | \leq ( 4 C _0 + 1) \e
$ for all sufficiently large $ k$.
Since $ \e \in (0, \frac{\e_0}{2})$ is arbitrary, the conclusion of Lemma \ref{L4.11} follows.
\hfill
$\Box $
\begin{Corollary}
\label{C4.13}
Assume that $ N \geq 2 $ and (A1), (A2) are satisfied.
If $ (\g _n ) _{n \geq 1} \subset \Eo $, $ \g \in \Eo $ are such that
$ d_0 ( \g _n , \g ) \lra 0 $, then
$\ds \lim_{n \ra \infty} Q( \g _n ) = Q( \g)$ and
$ \ds \lim_{n \ra \infty} \| V(|\g _n |^2) - V(|\g |^2) \|_{L^1( \R^N) } = 0 .$
In particular, $Q$ and $E$ are continuous functionals on $ \Eo $ endowed with
the semi-distance $d_0$.
\end{Corollary}
{\it Proof.}
We have $ \nabla \g _n \lra \nabla \g $ and $(|\g _n| - |\g |) \lra 0 $
in $L^2( \R^N)$ as $ n \lra \infty$, hence
$
|\nabla \g _n |^2 + \frac 12 \left( \ph^2( |\g _n |) - 1 \right)^2
\lra
|\nabla \g |^2 + \frac 12 \left( \ph^2( |\g |) - 1 \right)^2
$
in $L^1( \R^N)$, and consequently $ (\g _n) _{n \geq 1}$ satisfies
assumption (a) in Lemma \ref{L4.11}.
Consider a subsequence $(\g_{n_{\ell}})_{\ell \geq 1}$ of $(\g_n)_{n \geq 1}$.
Then there exist a subsequence
$(\g_{n_{\ell _k}})_{k \geq 1} $ and $ \g _0 \in \Eo $ that satisfy (\ref{4.32}).
Since $ \nabla \g _{n _{\ell _k}} \rightharpoonup \nabla \g _0 $ weakly
in $L^2( \R^N)$ and $ \nabla \g _{n _{\ell _k}} \lra \nabla \g $ in $L^2( \R^N)$
we see that $\nabla \g _0 = \nabla \g$ a.e.
on $ \R^N$, hence there is a constant $ \beta \in \C$ such that
$ \g_0 = \g + \beta $ a.e. on $ \R^N$. The convergence
$|\g_{n _{\ell _k}}| \lra |\g _0|$ in $L_{loc}^2( \R^N) $ gives
$ |\g _0 | = |\g | $ a.e. on $ \R^N$. By the definition of $Q$ it follows that
$Q(\g _0) = Q( \g + \beta ) = Q(\g)$.
Using Lemma \ref{L4.11} we get $Q( \g_{n _{\ell_k}}) \lra Q( \g _0 ) = Q(\g)$ as $ k \lra \infty$
and Lemma \ref{L4.10} implies that
$V(| \g_{n_{\ell _k}} |^2) \lra V(|\g _0|^2) = V(|\g |^2) $ in $ L^1( \R^N)$ as $ k \lra \infty$.
Hence any subsequence $(\g_{n_{\ell}})_{\ell \geq 1}$ of $(\g_n)_{n \geq 1}$ contains a subsequence
$(\g_{n_{\ell _k}})_{k \geq 1} $ such that $Q( \g_{n _{\ell_k}}) \lra Q( \g )$
and $ \| V(| \g_{n_{\ell _k}} |^2) - V(|\g |^2) \|_{L^1( \R^N)} \lra 0 $,
and this
clearly implies the desired conclusion.
\hfill
$\Box$
\medskip
Assume that for some $ q >0$ there is $ \psi \in \Eo $ such that $ Q( \psi ) = q \, $
and $ \ov{E}(\psi) = E_{min}(q)$.
Using Corollary \ref{C4.13}, for any sequence $(\psi_n )_{n \geq 1} \subset \Eo $
such that $ d_0 ( \psi _n, \psi ) \lra 0 $ and
for any sequence of points $(x_n )_{n \geq 1} \subset \R^N$ we have
$Q(\psi _n(\cdot + x_n)) \lra q $ and $\ov{E}(\psi _n(\cdot + x_n)) \lra E_{min}(q). $
Hence the convergence result provided by Theorem \ref{T4.7} for minimizing
(sub)sequences of $\ov{E}$ under the constraint $Q= q$ is optimal.
\medskip
Next we show that if $ V \geq 0 $ on $[0, \infty)$, the minimizers of $ \ov{E} = E$ at fixed momentum are traveling waves to (\ref{1.1}).
We denote by $ d^- E_{min}(q) $ and $ d^+ E_{min}(q) $ the left and right derivatives
of $E_{min}$ at $ q>0 $ (which exist and are finite for any $ q > 0 $ because
$E_{min}$ is concave). We have:
\begin{Proposition}
\label{P4.12}
Let $ N \geq 2 $ and $ q > 0$.
Assume that $V(s) \geq 0 $ for any $ s \geq 0$ and $\psi $ is a minimizer of $E$ in the set
$ \{ \phi \in \Eo\; | \; Q( \phi ) = q \}$. Then:
\smallskip
(i) There is $ c \in [d^+ E_{min}(q), d^- E_{min}(q) ] $ such that $ \psi $ satisfies
\beq
\label{4.50}
i c \psi _{x_1} + \Delta \psi + F(|\psi |^2) \psi = 0 \qquad \mbox{ in } \Do' ( \R^N).
\eeq
\smallskip
(ii) Any solution $\psi \in \Eo $ of (\ref{4.50}) satisfies
$ \psi \in W_{loc}^{2, p } ( \R^N)$ and $\nabla \psi \in W^{1, p } ( \R^N)$
for any $ p \in [2, \infty)$, $\psi $ and $ \nabla \psi $ are bounded
and $ \psi \in C^{1, \al } (\R^N)$ for any $ \al \in [0, 1)$.
\smallskip
(iii) After a translation, $ \psi $ is axially symmetric with respect
to the $ x_1-$axis if $ N \geq 3$.
The same conclusion holds for $N=2$ if we assume in addition that $F$ is $C^1$.
(iv) For any $ q > q_0 $ there are $ \psi ^+, \psi ^- \in \Eo $ such that
$Q( \psi ^+ ) = Q( \psi ^- )=p$, $E( \psi ^+ ) = E( \psi ^- )= E_{min}(p)$ and
$ \psi ^+, \psi ^-$ satisfy (\ref{4.50}) with speeds $c^+ = d^+E_{min}(p)$ and
$c^- = d^-E_{min}(p)$, respectively.
\end{Proposition}
{\it Proof. }
(i) It is easy to see that
$ \Delta \psi + F( |\psi |^2 ) \psi \in H^{-1}(\R^N)$, $ i \psi _{ x_1} \in L^2( \R^N)$
and for any $ \phi \in C_c^{\infty }( \R^N)$ we have $ \psi + \phi \in \Eo $,
$ \ds \lim_{t \ra 0 } \frac 1t ( Q( \psi + t \phi ) - Q( \psi))
= 2 \langle i \psi _{x _1}, \phi \rangle_{L^2( \R^N)}$ and
\beq
\label{gateaux2}
\begin{array}{rcl}
\ds \lim_{t \ra 0 } \frac 1t ( E( \psi + t \phi ) - E( \psi)) & = &
2 \ds \int_{\R^N} \langle \nabla \psi , \nabla \phi \rangle
- F( |\psi |^2 ) \langle \psi, \phi \rangle \, dx
\\
& = & -2 \langle \Delta \psi + F( |\psi |^2 ) \psi , \phi \rangle_{H^{-1}(\R^N), H^1 (\R^N)}.
\end{array}
\eeq
Denote $ E' ( \psi) . \phi = -2 \langle \Delta \psi + F( |\psi |^2 ) \psi , \phi \rangle_{H^{-1}(\R^N), H^1 (\R^N)}$ and
$ Q' ( \psi) . \phi = 2 \langle i \psi _{x _1}, \phi \rangle_{L^2( \R^N)}$.
We have
$ E( \psi + t \phi ) \geq E_{min}(Q( \psi + t \phi))$, hence for all $ t > 0$
\beq
\label{4.52}
\frac 1t ( E( \psi + t \phi ) - E ( \psi ) ) \geq \frac 1t ( E_{min}( Q(\psi + t \phi ))
- E_{min} ( q ) ).
\eeq
If $ Q' ( \psi) . \phi >0$, we have $ Q( \psi + t \phi ) > Q ( \psi ) = q $
for $ t > 0$ and $t$ close to $0$,
then passing to the limit as $ t \downarrow 0 $ in (\ref{4.52}) we get
$ E' ( \psi) . \phi \geq d^+ E_{min} (q) Q' ( \psi) . \phi $.
If $ Q' ( \psi) . \phi < 0$, we have $ Q( \psi + t \phi ) < Q ( \psi ) = q $
for $t$ close to $0$ and $ t > 0$, then
passing to the limit as $ t \downarrow 0 $ in (\ref{4.52}) we get
$ E' ( \psi) . \phi \geq d^- E_{min} (q) Q' ( \psi) . \phi $.
Putting $ - \phi $ instead of $ \phi$ in the above, we discover
\beq
\label{4.53}
\begin{array}{l}
d^+ E_{min} (q) Q' ( \psi) . \phi \leq E' ( \psi) . \phi
\leq d^- E_{min} (q) Q' ( \psi) . \phi
\qquad \mbox{ if } Q' ( \psi) . \phi > 0, \mbox{ and }
\\
d^- E_{min} (q) Q' ( \psi) . \phi \leq E' ( \psi) . \phi
\leq d^+ E_{min} (q) Q' ( \psi) . \phi
\qquad \mbox{ if } Q' ( \psi) . \phi < 0.
\end{array}
\eeq
Let $ \phi _0 \in C_c^{\infty}(\R^N)$ be such that $ Q' ( \psi) . \phi _0 = 0$. We claim that $ E' ( \psi) . \phi _0 = 0$.
To see this, consider $ \phi \in C_c^{\infty}(\R^N)$ such that $ Q' ( \psi) . \phi \neq 0$.
(Such a $\phi$ exists for otherwise, we would have
$ 0 = Q' ( \psi) . \phi = 2 \langle i \psi _{x _1}, \phi \rangle_{L^2( \R^N)} $
for any $ \phi \in C_c^{\infty}(\R^N)$, yielding $ \psi _{x _1} = 0 $,
hence $ Q ( \psi) = 0 \not = q $.)
Then for any $n\in \N$ we have $ Q' ( \psi) . ( \phi + n \phi _0 ) = Q' ( \psi) . \phi $.
From (\ref{4.53}) it follows that
$E' ( \psi) . (\phi + n \phi _0 )= E' ( \psi) . \phi + n E' ( \psi) . \phi _0 $
is bounded, thus necessarily $ E' ( \psi) . \phi _0 = 0$.
Take $ \phi _1 \in C_c^{\infty}(\R^N)$ such that $ Q' ( \psi) . \phi _1 = 1$.
Let $ c = E' ( \psi) . \phi _1.$
Using (\ref{4.53}) we obtain $ c \in [d^+ E_{min}(q), d^- E_{min}(q) ] $.
For any $ \phi \in C_c^{\infty} (\R^N)$ we have
$Q' ( \psi) .( \phi - (Q' ( \psi) .\phi) \phi _1 ) = 0$, hence
$E' ( \psi) .( \phi - (Q' ( \psi) .\phi) \phi _1 ) = 0$, that is
$E' ( \psi) . \phi = c Q' ( \psi) .\phi$
and this is precisely (\ref{4.50}).
\medskip
(ii) If $ N \geq 3$ this is Lemma 5.5 in \cite{M10}.
If $ N =2$ the proof is very similar and we omit it.
\medskip
(iii) If $ N \geq 3$, the axial symmetry follows from the fact that the
minimizers are $C^1$ and from Theorem 2' p. 329 in \cite{M7}.
We use an argument due to O. Lopes \cite{lop1} to give a proof which requires
$F$ to be $C^1$, but works also for $N =2$.
Let $S_t^+$ and $S_t ^-$ be as in (\ref{4.10}) and (\ref{4.11}), respectively.
Proceeding as in the proof of Lemma \ref{L4.5} (ii),
we find $ t \in \R$ such that $ Q( S_t^+ \psi) = Q( S_t^- \psi) = q.$
This implies $E(S_t^+ \psi ) \geq E_{min}(q) $ and $E(S_t^- \psi ) \geq E_{min}(q) $.
On the other hand $E(S_t^+ \psi ) + E(S_t^- \psi ) = 2 E( \psi )= 2 E_{min}(q)$,
thus necessarily $E(S_t^+ \psi ) = E(S_t^- \psi ) = E_{min}(q) $
and $S_t^+ \psi $ and $S_t^- \psi $ are also minimizers.
Then $S_t^+ \psi $ and $S_t^- \psi $ satisfy (\ref{4.50})
(with some coefficients $c_+$ and $c_-$ instead of $c$)
and have the regularity properties given by (ii).
Since $S_t^+ \psi = \psi $ on $\{ x_N > t \}$ and $S_t^- \psi = \psi $
on $\{ x_N < t \}$, we infer that necessarily $ c_+ = c_- = c$.
Let $ \phi _ 0 (x) = e^{ \frac{ i c x_1}{2}} \psi (x)$,
$ \phi _ 1 (x) = e^{ \frac{ i c x_1}{2}}S_t^+ \psi (x)$,
$ \phi _ 2 (x) = e^{ \frac{ i c x_1}{2}} S_t^- \psi (x)$.
Then $ \phi _0, \; \phi _1 $ and $ \phi _2$ are bounded, belong
to $W_{loc}^{2, p}(\R^N)$ for any $ p\in [2,\infty)$ and solve the equation
$$
\Delta \phi + \left( \frac{ c^2}{4} + F(|\phi |^2) \right) \phi = 0 \qquad \mbox{ in } \R^N.
$$
Since $F$ is $ C^1$ and $ \phi _0$, $\phi _1$ are bounded, the function
$ w = \phi _1 - \phi _0$ satisfies an equation
$$
\Delta w + A(x) w = 0 \qquad \mbox{ in } \R^N,
$$
where $A(x)$ is a $2 \times 2$ matrix and $ A \in L^{\infty}(\R^N)$.
Since $w \in H_{loc}^2( \R^N)$ and $w = 0 $ in $ \{ x_N > t \}$, the
Unique Continuation Theorem (see, for instance, the appendix of \cite{lop1})
implies that $ w = 0 $ on $ \R^N$, that is
$S_t^+ \psi = \psi $ on $ \R^N$. We have thus proved that $ \psi $ is
symmetric with respect to the hyperplane $\{ x_N = t \}$.
Similarly we prove that for any $ e \in S^{N-1}$ orthogonal to $ e_1 = (1, 0 , \dots, 0)$
there is $ t_e \in \R$ such that $ \psi $ is symmetric with respect to the hyperplane
$ \{ x \in \R^N \; | \; x.e = t _e \}$.
Then it is easy to see that after a translation $ \psi $ is symmetric
with respect to $O x_1$.
\medskip
iv) Consider a sequence $ q_n \uparrow q $. We may assume $ q_n > q_0 $
for each $n$. By Theorem \ref{T4.7} there is $ \psi _n \in \Eo $ such that
$ Q(\psi _n ) = q_n \lra q $ and $E(\psi _n) = E_{min}(q_n ) \lra E_{min}(q)$
by continuity of $E_{min}$. Since $ q > q_0 $ we have $E_{min}(q) < v_s q$ and using Theorem \ref{T4.7} again
we infer that there are a subsequence $(\psi_{n_k})_{k \geq 1}$,
a sequence $(x_k)_{k \geq 1} \subset \R^N$ and $ \psi ^- \in \Eo$ such that
$Q( \psi ^- ) = q$, $E(\psi^-) = E_{min}(q)$ and, denoting
$\tilde{\psi}_{n_k} = \psi_{n_k} ( \cdot + x_n)$, we have
$\tilde{\psi}_{n_k} \lra \psi^- $ a.e. on $ \R^N$ and
$ d_0 (\tilde{\psi}_{n_k} , \psi ^-) \lra 0 $ as $ k \lra \infty$.
By (i) we know that each $ \tilde{\psi} _{n_k}$ satisfies (\ref{4.50}) for some
$c_{n_k} \in [ d^+E_{min}(q_{n_k}), d^-E_{min}(q_{n_k}) ]$.
Since $E_{min}$ is concave, we have $c_{n_k} \lra d^-E_{min}(q)$ as $ k \lra \infty$.
It is easily seen that $ \tilde{\psi} _{n_k} \lra \psi ^-$ and
$F(| \tilde{\psi} _{n_k} |^2) \tilde{\psi} _{n_k} \lra F(|\psi ^-|^2) \psi ^-$
in $ \Do '(\R^N)$. Writing (\ref{4.50}) for each $\tilde{\psi} _{n_k}$ and
passing to the limit as $ k\lra \infty$ we infer that
$\psi ^- $ satisfies (\ref{4.50}) in $\Do'( \R^N)$ with $ c = d^- E_{min}(q)$.
The same argument for a sequence $ q_n \downarrow q$ gives the existence
of $ \psi ^+$.
\hfill
$\Box $
\medskip
If $F$ satisfies assumption (A4) in the introduction and $ F''(1) \neq 3$,
we prove that in space dimension $N=2$ we have $ q_0 = 0$. This implies that we can minimize $E$ under
the constraint $Q= q$ for any $ q >0$. The traveling waves obtained
in this way have small energy and speed tending to $ v_s $ as $ q \lra 0 $.
For the two-dimensional Gross-Pitaevskii equation, the numerical and formal
study in \cite{JR} suggests that these traveling waves are rarefaction
pulses asymptotically described by the ground states of the Kadomtsev-Petviashvili I (KP-I) equation.
The rigorous convergence, up to rescaling and renormalization, of the
traveling waves of (\ref{1.1}) in the transonic limit
to the ground states of the (KP-I) equation
has been proven in \cite{BGS1} in the case of the two-dimensional Gross-Pitaevskii equation.
That result has been extended in \cite{CM} to a general
nonlinearity satisfying (A1), (A2) and (A4) with $ F''(1) \neq 3 $.
A result similar to Theorem \ref{T4.13} below is {\it not } true in higher dimensions:
in view of Proposition \ref{smallE} we have $ q_0 > 0 $ for any $N \geq 3$.
If $ N \geq 3$, the existence of traveling waves with speed close to $ v_s$ is guaranteed by
Theorem 1.1 and Corollary 1.2 p. 113 in \cite{M10}.
In space dimension three, the convergence of the traveling waves constructed in \cite{M10}
to the ground states of the three-dimensional (KP-I) equation as $ c \to v_s$ has been rigorously justified
under the same assumptions as in dimension two (see Theorem 6 in \cite{CM}).
It was also shown in \cite{CM} that these solutions have high energy and momentum
(of order $1/\sqrt{v_s ^2 - c^2 } $ as $ c \to v_s$) and thus lie on the upper branch in figure \ref{diaggrospit} (b).
\begin{Theorem}
\label{T4.13}
Suppose that $ N =2$, the assumption (A4) in the introduction holds
and $F''(1) \neq 3 $. Then $E_{min}(q) < v_s q $ for any $ q > 0$.
In other words, $q_0 = 0$.
\end{Theorem}
\begin{remark} \label{R4.16} \rm
If $N = 2$, $ V \geq 0$ and (A1), (A2) and (A4) hold with $ F''(1) \neq 3 $,
it follows from
Theorems \ref{T4.13} and \ref{T4.7} that for any $ q > 0 $ there is
$ \psi _ q \in \Eo $ such that $ Q( \psi _q ) = q $ and $ E( \psi _p) = E_{min}(q)$.
Proposition \ref{P4.12} (i) implies that $ \psi_q $ is a traveling wave of (\ref{1.1})
of speed $ c ( \psi _q ) \in [ d^+E_{min}(q), d^-E_{min}(q) ].$
Using Lemmas \ref{L4.3} and \ref{L4.4} we infer that $ c( \psi _q) \lra v_s $
as $ q \lra 0 $. In particular, we see that there are traveling waves of
arbitrarily small energy whose speeds are arbitrarily close to $ v_s$.
\medskip
In view of the formal asymptotics given in \cite{JR}, it is natural to try to
prove Theorem \ref{T4.13} by using test functions constructed from
an ansatz related to the (KP-I) equation.
\end{remark}
{\it Proof of Theorem \ref{T4.13}.}
Fix $ \g > 0 $ (to be chosen later).
We consider the (KP-I) equation
\beq
\label{KP-I}
u_t - \g u u _x + \frac {1}{ v_s ^2} u_{xxx} - \p_x^{-1} u _{yy} = 0 ,
\qquad t \in \R, ( x, y ) \in \R^2 ,
\eeq
where $u$ is real-valued. Let $ Y$ be the completion of
$ \{ \p_x \phi \; | \; \phi \in C_c^{\infty}(\R^2, \R) \}$ for the norm
$\| \p _ x \phi \|_Y ^2 = \| \p _ x \phi \|_{L^2(\R^2)} ^2 +
v_s ^2 \| \p _ y \phi \|_{L^2(\R^2)} ^2 + \| \p _ {xx} \phi \|_{L^2(\R^2)} ^2. $
A traveling wave for (\ref{KP-I}) moving with velocity $ \frac{1}{v_s ^2}$ is a
solution of the form
$u( t, x, y ) = v ( x - \frac{t }{ v_s ^2}, y)$, where $ v \in Y$.
The traveling wave profile $v$ solves the equation
$$
\frac{1}{v_s ^2} v_x + \g v v_x - \frac{1}{v_s ^2} v_{xxx} + \p_x^{-1} v _{yy} = 0
\qquad \mbox{ in } \R^2,
$$
or equivalently, after integrating in $x$,
\beq
\label{4.55}
\frac{1}{v_s ^2} v + \frac{\g}{2} v ^2 - \frac{1}{v_s ^2} v_{xx} + \p_x^{-2} v _{yy} = 0
\qquad \mbox{ in } \R^2.
\eeq
It is a critical point of the functional (called the {\it action})
$$
\Sr (v) = \int_{\R^2 } \frac{1}{v_s ^2} |v|^2 + \frac{1}{v_s ^2} |v _x|^2 + | \p _x ^{-1} v _y |^2 \, dx \, dy
+ \frac{ \g }{3} \int_{\R^2} v ^3 \, dx \, dy
= \frac{1}{v_s ^2} \| v \|_Y ^2 + \frac{ \g }{3} \int_{\R^2} v ^3 \, dx \, dy.
$$
Equation (\ref{4.55}) is indeed nonlinear if $\g \not = 0 $. The existence of a
nontrivial traveling wave solution $w$ for (KP-I) follows
from Theorem 3.1 p. 217 in \cite{dBSIHP}. The solution found in \cite{dBSIHP}
minimizes $ \| \cdot \|_Y$ in the set
$ \{ v \in Y \; | \; \ds \int_{\R^2} v ^3 \, dx \, dy = \int_{\R^2} w ^3 \, dx \, dy\}$.
It was also proved (see Theorem 4.1 p. 227 in \cite{dBSIHP}) that
$ w \in H^{\infty}(\R^2) := \ds \cap_{m \in \N} H^m( \R^2)$, $ \p _x ^{-1}w _y \in H^{\infty}(\R^2) $
and $w$ minimizes the action $ \Sr $ among all nontrivial solutions of (\ref{4.55})
(that is, $w$ is a {\it ground state}).
Moreover, $w$ satisfies the following integral identities:
\beq
\label{identites}
\left\{\begin{array}{ll}
\ds \int_{\R^2} \frac{1}{v_s^2} w^2 + \frac {\g}{2} w^3 + \frac{1}{v_s^2} | \p _x w |^2
+ | \p _x^{-1} w _y |^2 \, dx \, dy = 0,
\\
\\
\ds \int_{\R^2} \frac{1}{v_s^2} w^2 + \frac {\g}{3} w^3 - \frac{1}{v_s^2} | \p _x w |^2
+ 3 | \p _x^{-1} w _y |^2 \, dx \, dy = 0,
\\
\\
\ds \int_{\R^2} \frac{1}{v_s^2} w^2 + \frac {\g}{3} w^3 + \frac{1}{v_s^2} | \p _x w |^2
- | \p _x^{-1} w _y |^2 \, dx \, dy = 0.
\end{array}\right.
\eeq
The first identity is obtained by multiplying (\ref{4.55}) by $w$ and integrating,
while the two other are Pohozaev identities associated to the scalings in $x$,
respectively in $y$. They are formally obtained by multiplying (\ref{4.55}) by
$ xw$, respectively by $ y \p _x^{-1} w _y$ and integrating by parts; see the
proof of Theorem 1.1 p. 214 in \cite{dBSIHP} for a rigorous justification.
Comparing $ \Sr(w) $ to the last equality in (\ref{identites}) we get
\beq
\label{4.57}
\int_{\R^2} | \p _x^{-1} w _y |^2 \, dx \, dy = \frac 12 \Sr (w).
\eeq
In particular, $ \Sr (w) > 0$.
Then from the three identities (\ref{identites}) we obtain
\beq
\label{4.58}
\frac{1}{v_s ^2} \! \int_{\R^2} \! |w|^2 \, dx \, dy = \frac 32 \Sr (w), \quad
\frac{1}{v_s ^2} \! \int_{\R^2} \! |w_x |^2 \, dx \, dy = \Sr (w), \quad
\frac{ \g }{6} \! \int_{\R^2} \! w^3 \, dx \, dy = - \Sr (w).
\eeq
Let $ w$ be as above and let $ \phi = v_s \p _x ^{-1} w$, so that $ \p _x \phi = v_s w$.
For $ \e > 0 $ small we define
$$
\rho _{\e } (x, y ) = 1 + \e ^2 w ( \e x , \e ^2 y), \qquad
\theta_{\e } (x,y) = \e \phi ( \e x, \e ^2 y ) , \qquad
U_{\e} = \rho _{\e } e^{-i \theta _{\e}}.
$$
Then $ U_{\e} \in \Eo $ (because $w \in H^{\infty}(\R^2)$).
For $\e$ sufficiently small we have $V(|U_{\e}|^2) = V(\rho_{\e}^2) \geq 0$, hence $\ov{E}(U_{\e}) = {E}(U_{\e})$.
A straightforward computation and (\ref{4.57}), (\ref{4.58}) give
$$
\int_{\R^2} \Big\vert \frac{\p \rho_{\e}}{\p x } \Big\vert ^2 dx \, dy
= \e ^3 \int_{\R^2} \Big\vert \frac{\p w}{\p x } \Big\vert ^2 dx \, dy
= \e ^3 v_s ^2 \Sr (w) = 2 \e ^3 \Sr (w),
$$
$$
\int_{\R^2} \Big\vert \frac{\p \rho_{\e}}{\p y } \Big\vert ^2 dx \, dy
= \e ^5 \int_{\R^2} \Big\vert \frac{\p w}{\p y } \Big\vert ^2 dx \, dy ,
$$
$$
\begin{array}{l}
\ds \int_{\R^2} \rho _{\e } ^2 \Big\vert \frac{\p \theta_{\e}}{\p x } \Big\vert ^2 dx \, dy
= \e \int_{\R^2} ( 1 + \e ^2 w ) ^2 \vert \phi _x \vert ^2 dx \, dy
\\
= \ds \e v_s ^2 \int_{\R^2} ( 1 + \e ^2 w ) ^2 w^2 dx \, dy
= \frac 32 v_s ^4 \Sr (w) \e - \frac{12}{\g } v_s ^2 \Sr (w) \e ^3
+ v_s ^2 \e ^5 \int_{\R^2} w^4 dx \, dy,
\end{array}
$$
$$
\begin{array}{l}
\ds \int_{\R^2} \rho _{\e } ^2 \Big\vert \frac{\p \theta_{\e}}{\p y } \Big\vert ^2 dx \, dy
= \e ^3 \int_{\R^2} ( 1 + \e ^2 w ) ^2 \vert \phi _y \vert ^2 dx \, dy
\\
= \ds \e^3 v_s ^2 \int_{\R^2} ( 1 + \e ^2 w ) ^2 | \p _x ^{-1}w_y |^2 dx \, dy
\\
= \ds \frac 12 v_s ^2 \Sr (w) \e ^3
+ 2 \e ^5 v_s ^2 \int_{\R^2} w | \p _x ^{-1}w_y |^2 dx \, dy
+ \e^ 7 v_s ^2 \int_{\R^2} w ^2| \p _x ^{-1}w_y |^2 dx \, dy .
\end{array}
$$
Using (\ref{lift}) we get
\beq
\label{4.59}
\begin{array}{l}
Q( U_{\e}) = \ds \int_{\R^2} ( \rho _{\e }^2 - 1 ) \frac{ \p \theta _{\e}}{\p x } dx \, dy
= \e \int_{\R^2} ( 2 w + \e^2 w^2) \phi _x \, dx \, dy
\\
= \e v_s \ds \int_{\R^2} ( 2 w + \e^2 w^2) w \, dx \, dy
= 3 v_s ^3 \Sr (w) \e - \frac { 6}{\g } v_s \Sr (w) \e ^3.
\end{array}
\eeq
If (A4) holds we have the expansion
\beq
\label{4.60}
V(s) = \frac 12 ( s - 1)^2 - \frac 16 F''(1) ( s - 1)^3 + H(s),
\eeq
where $|H(s) | \leq C(s - 1 )^4 $ for $ s $ close to $ 1$.
Using (\ref{4.60}) and the fact that $ w \in L^p( \R^2)$ for any $ p \in [2, \infty]$,
for small $ \e $ we may expand $V(\rho _{\e})$ and integrate to get
$$
\begin{array}{l}
\ds \int_{\R^2} V( \rho _{\e }^2) \, dx \, dy
= 2 \e \int_{\R^2} w^2 dx \, dy
+ \e ^3 \left( 2 - \frac 43 F''( 1) \right) \int_{\R^2} w^3 dx \, dy + \Oo ( \e ^5)
\\
\\
= \frac 32 v_s ^4 \Sr (w) \e - \frac{6}{\g } \left( v_s ^2 - \frac 43 F''( 1) \right)\Sr (w) \e^3
+ \Oo ( \e ^5) .
\end{array}
$$
From the previous computations we find
\beq
\label{4.61}
E( U_{\e}) - v_s Q(U_{\e}) = v_s ^2 \Sr (w) \left( \frac 32 - \frac{12 - 4 F''(1)}{\g } \right) \e ^3 + \Oo ( \e^5).
\eeq
If $ F''(1) \neq 3$, choose $ \g \in \R$ such that $\frac 32 - \frac{12 - 4 F''(1)}{\g } < 0$
(take, for instance, $ \g = 6 - 2 F''(1)$).
Let $w$ be a ground state of (\ref{4.55}) for this choice of $ \g$.
It follows from (\ref{4.61}) that there is $ \e _0 > 0 $ such that
$ E( U_{\e }) - v_s Q(U_{\e}) < 0 $ for any $ \e \in (0, \e _0)$ (since $\Sr (w) > 0 $).
On the other hand, using (\ref{4.59}) we infer that there is $ \e _1 < \e _0 $
such that the mapping $ \e \longmapsto Q(U_{\e})$ is a homeomorphism from $(0, \e _1)$
to an interval $(0, q_1)$. Since $E_{min}(Q(U_{\e})) \leq \ov{E}(U_{\e}) = E(U_{\e}) < v_s Q (U_{\e})$,
we see that $ E_{min}(q) < v_s q $ for any $ q \in (0, q_1)$.
Then the concavity of $E_{min}$ implies $ E_{min}(q) < v_s q $ for any $ q >0$.
\hfill
$\Box$
\medskip
We pursue with some qualitative properties of $E_{min}$ for large $q$.
Theorem \ref{T4.16} (a) below implies that the speeds of traveling waves
obtained from Theorem \ref{T4.7} tend to $0$ as $ q \lra \infty$.
\begin{Theorem}
\label{T4.16}
$\; $
(a) If (A1) holds and $N\geq 2$, there is $C > 0$ such that
$ E_{min}(q) \leq C q^{ \frac{N-2}{N-1}} \ln q \; $ for large $q$.
(b) If $ N\geq 2$ and (A1) and (A2) hold we have
$ \ds \lim_{q \ra \infty} E_{min}(q) = \infty$. Moreover, if
$N \geq 3$ there is $ C> 0$ such that $ E_{min}(q) \geq C q^{ \frac{N-2}{N-1}}$.
\end{Theorem}
{\it Proof. }
(a) Using Lemma 4.4 p. 147 in \cite{M10} we see that there is
a continuous mapping $R \longmapsto v_R$ from $[2, \infty )$ to
$H^1( \R^N)$ and constants $C_i > 0$, $i=1$, $2$, $3$, such that
\beq
\label{4.62}
\int_{\R^N} |\nabla v_R| ^2 dx \leq C_1 R^{N-2} \ln R,
\quad
\int_{\R^N} \| V(|1+ v_R|^2) \| \, dx \leq C_2 R^{N-2},
\eeq
\beq
\label{4.63}
C_3 (R-2) ^{N-1} \leq Q( 1+ v_R) \leq C_3 R^{N-1}.
\eeq
Let $ q_R = Q( 1+ v_R)$.
The set $ \{ q_R \; | \; R \geq 2 \}$ is an interval of the form $[q_*, \infty)$.
By (\ref{4.63}) we have
$ C_3 ^{ - \frac{1}{N-1}} q_R^{\frac{1}{N-1}}
\leq R \leq
2 + C_3 ^{ - \frac{1}{N-1}} q_R^{\frac{1}{N-1}} .$
Then using (\ref{4.62}) we get for $R$ sufficiently large
$$
E_{min}(q_R) \leq \ov{E}( 1+ v_R) \leq C_1 R^{N-2} \ln R + C_2 R^{N-2}
\leq C q_R^{\frac{N-2}{N-1}} \ln q_R.
$$
(b) As in the proof of Lemma \ref{L4.5} (ii), using (\ref{4.9}) we get
$E_{min}(q_2) \geq \left( \frac{ q_2}{q_1} \right) ^{\frac{N-2}{N-1}} E_{min}(q_1)$
for any $ q_2 > q_1 > 0$. This is the second statement of (b), and it
implies that $ \ds \lim_{q \ra \infty} E_{min}(q) = \infty$ if $N \geq 3$.
Let $ N =2$.
We argue by contradiction and we assume that $ \ds \lim_{q \ra \infty} E_{min}(q) $
is finite. Using Theorem \ref{T4.7} for $q$ sufficiently large, we may choose
$ \psi _q \in \Eo $ such that $ Q( \psi _q ) = q $ and $ \ov{E} ( \psi _q) = E_{min}(q)$.
Consider a sequence $ q_n \lra \infty$. From Lemma \ref{L4.6} it follows that
$E_{GL} ( \psi _{q_n} )$ is bounded and stays away from $0$.
Passing to a subsequence we may assume that $E_{GL} ( \psi _{q_n} )\lra \al _0 > 0$.
Let $ \Lambda_n(t)$ be the concentration function associated to $E_{GL}(\psi _{q_n})$
(as in (\ref{4.15})). Arguing as in the proof of Theorem \ref{T4.7} and passing
to a subsequence (still denoted $(q_n)_{n \geq 1}$), we see that there exist
a nondecreasing function $ \Lambda : [0, \infty ) \lra \R$, $ \al \in [0, \al _0 ] $
and a sequence $ t_n \lra \infty $ satisfying (\ref{4.16}) and (\ref{annulus}).
Then we use Lemma \ref{L4.8} to infer that $ \al > 0$.
If $ \al \in (0, \al _0)$, proceeding as in the proof of Theorem \ref{T4.7} and using
Lemma \ref{splitting} for $ \psi _{q_n}$ we see that there exist functions
$ \psi _{n,1}, \psi _{n,2} \in \Eo $ such that (\ref{4.28})$-$(\ref{4.30}) hold.
Passing to a subsequence if necessary, we may assume that
$ \ov{E}(\psi _{n, i}) \lra m_i \geq 0 $ as $ n \lra \infty$.
Since $\ds \lim_{n \ra \infty } E_{GL}(\psi _{n,i} ) > 0$, it follows from Lemma
\ref{L4.1} that $ m_i > 0$, $i=1,2$. Using (\ref{4.29}) we see that
$ m_1 + m_2 = \ds \lim_{q \ra \infty} E_{min}(q)$, hence
$ 0 < m_i < \ds \lim_{q \ra \infty} E_{min}(q).$
Since $ Q( \psi _{q_n} ) = q_n \lra \infty$, from (\ref{4.30}) it follows that at
least one of the sequences $(Q(\psi_{n, i}))_{n \geq 1}$ contains a subsequence
$(Q(\psi_{n_k, i}))_{k \geq 1}$ that tends to $\infty$. Then
$ \ov{E}(\psi_{n_k, i}) \geq E_{min}(Q(\psi_{n_k, i})) $ and passing to the limit
as $ k \lra \infty $ we find $ m_i \geq \ds \lim_{q \ra \infty} E_{min}(q)$,
a contradiction. Thus we cannot have $ \al \in (0, \al _0)$.
We conclude that necessarily $ \al = \al _0 $.
Proceeding again as in the proof of Theorem \ref{T4.7} we infer that
there is a sequence $(x_n)_{n \geq 1} \subset \R^N$ such that
$\tilde{ \psi}_n = \psi _{q_n} ( \cdot + x_n)$ satisfies (\ref{4.31}).
Then there is a subsequence $(\tilde{ \psi}_{n _k})_{k \geq 1}$ and
$ \psi \in \Eo $ such that (\ref{4.32}) holds.
Using Lemma \ref{L4.11} we infer that $ Q( \tilde{ \psi}_{n _k}) \lra Q( \psi ) \in \R$
and this is in contradiction with $ Q( \tilde{ \psi}_{n _k})= q_{n _k} \lra \infty.$
Thus necessarily $ E_{min}(q) \lra \infty $ as $ q \lra \infty.$
\hfill
$\Box $
\medskip
An alternative proof of the fact that $ E_{min}(q) \lra \infty $ as $ q \lra \infty$
is to show that for $ \psi \in \Eo $ we may write
$ \langle i \psi _{x_1}, \psi \rangle = f + g$, where $ g \in \Yo $ and $ f $
is bounded in $ L^1( \R^N)$ if $E_{GL}(\psi )$ is bounded, then to use Lemma
\ref{L4.6} to infer that $Q( \psi )$ remains bounded if $\ov{E}(\psi )$ is bounded.
\medskip
From Theorem \ref{T4.16} and Lemma \ref{L4.6} we obtain the following:
\begin{Corollary}
\label{C4.17}
For all $ M >0$, the functional $Q$ is bounded on the set
$ \{ \psi \in \Eo \; | \; E _{GL} ( \psi ) \leq M \} .$ \\
If (A1) and (A2) hold, $Q$ is also bounded on
$\{ \psi \in \Eo \; | \; \ov{E} ( \psi ) \leq M \} .$
\end{Corollary}
\section{Minimizing the action at fixed kinetic energy}
\label{another}
Although in many important physical applications the nonlinear potential $V$
may achieve negative values (this happens, for instance, for the cubic-quintic NLS),
there are no results in the literature that imply the existence of
finite energy traveling waves for (\ref{1.1}) in space dimension two for this
kind of nonlinearity. We develop here a method that works if $N \geq 2$ and
$V$ takes negative values. The method used in \cite{M10}
(minimization of $E_c$ under a Pohozaev constraint) does not require any assumption on sign of the potential $V$,
hence can be applied for
the cubic-quintic NLS if $ N \geq 3$, but does not work in space dimension two.
Throughout this section we assume that (A1) and
(A2) are satisfied.
We begin with a refinement of Lemma \ref{L4.2}.
\begin{Lemma}
\label{L5.1}
Assume that $|c| < v_s $ and let $ \e \in (0, 1 - \frac{|c|}{v_s})$. There is $k > 0$ such that for any
$ \psi \in \Eo $ satisfying $ \ii _{\R^N} |\nabla u |^2 \, dx \leq k$ we have
$$ E(\psi) - \e E_{GL}(\psi) \geq |c Q(\psi)|. $$
\end{Lemma}
{\it Proof. }
Fix $ \e _1 > 0 $ such that $ \e + \e _1 < 1 - \frac{|c|} {v_s}$.
It follows from Lemma \ref{L4.1} that there is $ k_1 > 0 $ such that
\beq
\label{5.1}
( 1 - \e _1 ) E_{GL} (\psi ) \leq E( \psi ) \qquad
\mbox{ for any } \psi \in \Eo \mbox{ satisfying } \ds \int_{\R^N} |\nabla \psi |^2 \,dx \leq k_1.
\eeq
Let
$ \tilde{F}(s) = ( 1 - \ph ^2 ( \sqrt{s}) ) \ph ( \sqrt{ s}) \ph '( \sqrt{ s}) \frac{1}{\sqrt{s}}$.
Then $ \tilde{F}(s) = 1 - s $ for $ s \in [0, 4 ]$ and $\tilde{F}$ satisfies
(A1) and (A2).
Let $ \tilde{V} (s) = \ii _s ^{ 1} \tilde{F} (\tau ) \, d \tau
= \frac 12 \left( \ph ^2( \sqrt{s}) - 1 \right) ^2. $
Using Lemma \ref{L4.2} (ii) with $\tilde{F}$ and $\tilde{V}$ instead of $F$ and $V$
we infer that there is $ k \in (0, \frac{k_1}{2}) $ such that for any $ \psi \in \Eo $ with
$ E_{GL}(\psi ) \leq 2 k$ we have
\beq
\label{5.2}
( 1 - \e - \e _1) E_{GL}(\psi ) \geq | c Q (\psi )|.
\eeq
Let $ \psi \in \Eo $ be such that $ \ii _{\R^N} |\nabla \psi |^2 \, dx \leq k .$
If $ \frac 12 \ii _{\R^N} \left( \ph ^2 (|\psi |) ^2 - 1 \right)^2 \, dx \leq k$ we have
$E_{GL}(\psi ) \leq 2 k$ and (\ref{5.2}) holds.
Then using (\ref{5.1}) we obtain
$ E(\psi) - \e E_{GL}(\psi) \geq ( 1 - \e - \e _1 ) E_{GL}(\psi ) \geq | c Q ( \psi ) | . $
If $ \frac 12 \ii_{\R^N} \left( \ph ^2 (|\psi |) ^2 - 1 \right)^2 \, dx > k$, let
$ \si = \left( \ii _{\R^N} |\nabla \psi |^2 \, dx \right)^{\frac 12}
\left( \frac 12 \! \ii _{\R^N} \! \left( \ph ^2 (|\psi |) ^2 - 1 \right)^2 \, dx \! \right)^{ - \frac 12} \!$.
Then $ \si \in (0, 1)$ and
$$\frac 12 \ds \int_{\R^N} \left( \ph ^2 (|\psi _{\si, \si } |) ^2 - 1 \right)^2 \, dx
= \int_{\R^N} |\nabla \psi _{\si, \si } |^2 \, dx
= \frac 12 E_{GL}(\psi_{\si, \si})
= \si ^{N-2} \int_{\R^N} |\nabla \psi |^2 \, dx < k.$$
Using (\ref{5.1}) and (\ref{5.2}) we get $E(\psi ) \geq ( 1 - \e_1) E_{GL}(\psi )$ and
$ ( 1 - \e - \e _1) E_{GL}(\psi _{\si, \si }) \geq | c Q (\psi _{\si, \si })|.$
Then we have
$$
\begin{array}{l}
E(\psi) - \e E_{GL}(\psi) - |c Q(\psi)| \geq ( 1 - \e - \e _1 ) E_{GL}(\psi) - |c Q(\psi)|
\\
\\
\geq \ds ( 1 - \e - \e _1 ) \left( \frac{1}{\si ^{N-2}} \! \int_{\R^N} \! |\nabla \psi _{\si, \si } |^2 \, dx
+ \frac{1}{2 \si ^N} \! \int_{\R^N} \! \left( \ph ^2 (|\psi _{\si, \si } |) ^2 - 1 \right)^2 \, dx \right)
- \frac{1}{\si ^{N-1}} |c Q(\psi _{\si, \si})|
\\
\\
\geq \ds \frac{1 - \e - \e _1}{2} \left( \frac{1}{\si ^{N-2}}
+ \frac{1}{\si ^N} \right) E_{GL}(\psi _{\si, \si}) - \frac{1- \e - \e _1}{\si^{N-1} } E_{GL}(\psi _{\si, \si})
\geq 0.
\end{array}
$$
$\; $
\vspace*{-18pt}
\hfill
$\Box$
\medskip
Let $ I( \psi ) = - Q( \psi ) + \ii_{\R^N} V(|\psi |^2) \, dx = E( \psi) - Q( \psi ) - \ii _{\R^N} |\nabla \psi |^2 \, dx.$
\smallskip
\\
We will minimize $I(\psi)$ under the constraint $\| \nabla \psi \|_{L^2( \R^N) } \!= \! constant.$
For any $ k > 0$~we~define
$$
I_{min}(k) = \inf \Big\{ I(\psi ) \; \big| \; \; \psi \in \Eo,
\int_{\R^N} |\nabla \psi |^2 \, dx = k \Big\}.
$$
The next Lemmas establish the basic properties of the function $I_{min}$.
\begin{Lemma}
\label{L5.2}
(i) For any $ k > 0 $ we have $I_{min}(k) \leq - \frac{ 1}{v_s ^2} k $.
(ii) For any $ \de > 0 $ there is $ k( \de ) > 0 $ such that
$I_{min}(k) \geq - \frac{ 1 + \de }{v_s ^2} k $ for any $ k \in (0, k ( \de )).$
\end{Lemma}
{\it Proof.}
i) Let $ N \geq 3$. Let $ q = 2 k v_s ^{ N-3}$. In the proof of Lemma \ref{L4.3}
we have constructed a sequence $(\psi _n )_{n \geq 1} \subset \Eo $ such that
$$
Q( \psi _n ) = q, \qquad \int_{\R^N} |\nabla \psi _n |^2 \, dx \lra \frac 12 v_s q = k v_s^{N-2} ,
\qquad \int_{\R^N} V(|\psi _n |^2) \, dx \lra \frac 12 v_s q
$$
and $ \psi _n $ is constant outside a large ball.
Let $ \si _n = k^{\frac{1}{N-2}} \left( \ii _{\R^N} |\nabla \psi _n |^2 \, dx \right)^{- \frac{1}{N-2}} .$
Then $ \si _n \lra \frac{1}{v_s}$ as $ n \lra \infty$. We get
$$
\int_{\R^N} |\nabla ( (\psi _n )_{\si _n, \si _n }) |^2 \, dx = \si _n ^{N-2} \int_{\R^N} |\nabla \psi _n |^2 \, dx = k,
$$
$$
Q((\psi_n)_{\si_n, \si _n}) = \si _n ^{N-1} Q( \psi _n ) \lra
\frac{q}{v_s ^{N-1}} = \frac{ 2 k }{v_s ^2},
$$
$$
\int_{\R^N} V(|(\psi _n)_{\si _n , \si _n }|^2) \, dx
= \si _n ^N \int_{\R^N} V(|\psi _n |^2) \, dx
\lra \frac{1}{v_s ^N} \cdot \frac{v_s q }{2} = \frac{k}{v_s^2}.
$$
We have $ I_{min}(k) \leq I( (\psi _n )_{\si _n, \si _n })$ for each $n$
and passing to the limit as $ n \lra \infty $ we obtain $ I_{min}(k) \leq - \frac{1}{v_s ^2} k .$
If $ N =2$, let $ q = \frac{2k}{v_s}$, choose $ \psi _n $ as in the proof
of Lemma \ref{L4.3} such that
$$
\int_{\R^2} |\nabla \psi _n |^2 \, dx = k, \qquad
Q( \psi _n ) \lra q = \frac{ 2k}{v_s} \qquad \mbox{ and } \qquad
\int_{\R^2} V(|\psi _n |^2) \, dx \lra k .
$$
Let $ \si = \frac{1}{v_s}.$
Then $ \ii _{\R^2} |\nabla ((\psi _n)_{\si, \si}) |^2 \, dx = k$,
$ Q((\psi _n)_{\si, \si}) = \si Q( \psi _n) \lra \frac{ 2k}{v_s ^2}$ and \\
${ \ii _{\R^2} } V(|(\psi _n )_{\si, \si} |^2) \, dx
=\si ^2 { \ii _{\R^2} } V(|\psi _n |^2) \, dx \lra \frac{k}{v_s ^2}$,
hence $I( (\psi _n)_{\si, \si}) \lra - \frac{ k}{v_s ^2}$.
\medskip
(ii) Fix $ \de > 0 $ and let $ c = \frac{v_s}{\sqrt{ 1 + \de}}.$
Lemma \ref{L5.1} implies that there is $ k > 0 $ such that for any $ \psi \in \Eo $
with $ \ii _{\R^N} |\nabla \psi |^2 \, dx \leq k $ we have
\beq
\label{5.3}
\ds \int_{\R^N} |\nabla \psi |^2 \, dx - c Q( \psi) + \int_{\R^2} V(|\psi |^2) \, dx \geq 0.
\eeq
Let $ \psi \in \Eo $ be such that ${ \ii _{\R^N} } |\nabla \psi |^2 \, dx \leq \frac{k}{c^{N-2}}.$
Then
$ \ii _{\R^N} |\nabla \psi _{c,c }|^2 \, dx = c^{N-2} \ii _{\R^N} |\nabla \psi |^2 \, dx \leq k$,
hence $ \psi_{c, c}$ satisfies (\ref{5.3}), that is
$ c^{N-2} \ii _{\R^N} |\nabla \psi |^2 \, dx + c^N I(\psi ) \geq 0$ or equivalently
$$
I(\psi) \geq - \frac{1}{c^2} \ds \int_{\R^N} |\nabla \psi |^2 \, dx = - \frac{1 + \de}{v_s ^2} \ds \int_{\R^N} |\nabla \psi |^2 \, dx.
$$
Hence (ii) holds with $ k( \de) = \frac{k}{c^{N-2}}.$
\hfill
$\Box $
\medskip
We give now global properties of $I_{min}$.
\begin{Lemma}
\label{L5.3}
The function $I_{min}$ has the following properties:
\medskip
(i) $I_{min}$ is concave, decreasing on $[0, \infty)$ and
${\ds \lim_{k \ra \infty}} \frac{ I_{min}(k) }{k} = - \infty. $
\medskip
(ii) $I_{min}$ is subadditive, that is $I_{min}(k_1 + k_2) \leq I_{min} (k_1)+ I_{min} (k_2) $
for any $ k_1, k_2 \geq 0$.
\medskip
(iii) If either $N\geq 3$ or ($ N=2$ and $V\geq 0 $ on $[0, \infty)$),
we have $I_{min}(k) > - \infty$ for any $ k > 0$.
\medskip
(iv) If ($ N = 2$ and $ \inf V < 0$), then $ I_{min}(k) = - \infty$ for
all sufficiently large $k$.
\medskip
(v) Assume that $ N =2$, (A4) holds and $ F''(1) \not= 3 $.
Then $I_{min}(k) < - \frac{1}{v_s ^2} k $ for any $ k > 0$.
\end{Lemma}
{\it Proof. }
i) We prove that for any $ k > 0$,
\beq
\label{5.4}
I_{min}(k) \geq \limsup_{h \downarrow k } I_{min}(h).
\eeq
Fix $ \psi \in \Eo $ such that $\ii _{\R^N} |\nabla \psi |^2\, dx = k$.
At least one of the mappings $ t \longmapsto \ii_{\R^N} |\nabla \psi_{t,t} |^2\, dx$,
$ t \longmapsto \ii_{\R^N} |\nabla \psi_{1,t} |^2\, dx$ or
$ t \longmapsto \ii_{\R^N} |\nabla \psi_{t,1} |^2\, dx$
is (strictly) increasing on $[1, \infty)$.
Let $ \psi ^t $ be either $\psi_{t,t}$ or $\psi_{1,t}$ or $\psi_{t,1}$, in such a way that
$ t \longmapsto \ii_{\R^N} |\nabla \psi ^{t} |^2\, dx$
is continuous and increasing on $[0, \infty)$.
It is easy to see that $ I(\psi ^t) \lra I(\psi )$ as $ t \lra 1$.
Let $(k_n)_{n\geq 1}$ be a sequence satisfying $ k_n \downarrow k$.
There is a sequence $ t_n \downarrow 1 $ such that $ \ii _{\R^N} |\nabla \psi ^{t_n} |^2 \, dx = k_n$.
For each $n$ we have $ I_{min}(k_n) \leq I(\psi^{t_n}) $
and passing to the limit as $ n \lra \infty $ we find $\ds \limsup_{n \ra \infty} I_{min}(k_n) \leq I(\psi)$.
Since this is true for any sequence $ k_n \downarrow k$ and any $ \psi \in \Eo $ satisfying
$ \ii _{\R^N} |\nabla \psi |^2\, dx = k$, (\ref{5.4}) follows.
Proceeding exactly as in the proof of Lemma \ref{L4.5} (see the proof of (\ref{4.12}) there) we find
\beq
\label{5.5}
I_{min}\left(\frac{k_1 + k_2}{2} \right) \geq \frac 12 (I_{min}(k_1) + I_{min}(k_2))
\qquad \mbox{ for any } k_1, \, k_2 >0.
\eeq
Let $ 0 \leq k_1 < k_2$. Using (\ref{5.5}) and a straightforward induction we find
\beq
\label{5.6}
I_{min} ( \al k_1 + (1 - \al ) k_2) \geq \al I_{min}(k_1) + (1 - \al ) I_{min}(k_2)
\qquad \mbox{ for any } \al \in [0,1] \cap \Q.
\eeq
Let $ \al \in (0, 1). $ Consider a sequence $ (\al _n)_{n \geq 1} \subset [0,1] \cap \Q$
such that $ \al _n \uparrow \al $. Using (\ref{5.4}) and (\ref{5.6}) we get
$$
\begin{array}{l}
I_{min} ( \al k_1 + (1 - \al ) k_2)
\geq \ds \limsup_{n \ra \infty} I_{min} ( \al _n k_1 + (1 - \al _n ) k_2)
\\
\geq \ds \limsup_{n \ra \infty} \left(\al_n I_{min}(k_1) + (1 - \al _n ) I_{min}(k_2) \right)
= \al I_{min}(k_1) + (1 - \al ) I_{min}(k_2).
\end{array}
$$
Thus $ I_{min} $ is concave on $[0, \infty)$. Since $I_{min}(0) = 0$,
by Lemma \ref{L5.2} $I_{min}$ is continuous at $0$ and negative on an
interval $(0, \de)$ and we infer that $I_{min}$ is negative and decreasing
on $(0, \infty)$.
The concavity of $I_{min}$ implies that the function
$k \longmapsto \frac{I_{min}(k)}{k}$ is nonincreasing on $(0, \infty)$.
Using Lemma 4.4 in \cite{M10} we find a sequence
$(\psi_n)_{n \geq 3}\subset \Eo $ such that
$$
k_n := \ds \int_{\R^N } |\nabla \psi _n |^2 \, dx \leq C_1 n^{N-2} \ln n, \quad
\Big\vert \int_{\R^N } V(|\psi _n |^2) \, dx \Big\vert \leq C_2 n^{N-2} \; \mbox{ and } \; \;
Q( \psi _n ) \geq C_3 n^{N-1},
$$
where $C_1,\, C_2, \, C_3 >0$ do not depend on $n$. Then
${\ds \lim_{k \ra \infty}} \frac{ I_{min}(k) }{k}
\leq {\ds \lim_{n \ra \infty}} \frac{ I(\psi _n)}{k_n } = - \infty.$
\medskip
(ii) By concavity we have $I_{min}(k_i ) \geq \frac{k_i}{k_1 + k_2}I_{min}(k_1 + k_2), \; i = 1,2$,
and the subadditivity follows.
\medskip
(iii) Consider first the case $ N \geq 3$. Fix $ k >0$.
Argue by contradiction and assume that there is a sequence $(\psi_n)_{n\geq 1}\subset \Eo $ such that
$ \ii _{\R^N} |\nabla \psi _n|^2 \, dx = k $ and
\beq
\label{5.7}
I(\psi _n) = - Q( \psi _n) + \int_{\R^N} V(|\psi _n|^2) \, dx \lra - \infty \qquad \mbox{ as } n \lra \infty .
\eeq
Let $ c = \frac{v_s}{2}.$
By Lemma \ref{L5.1} there exists $ k _2 > 0 $ such that $ \frac{k_2}{k} < \left( \frac{v_s}{2} \right)^{N-2}$
and (\ref{5.3}) holds for any $ \psi \in \Eo $ with
$ \ii _{\R^N} |\nabla \psi |^2 \, dx \leq k_2 $.
Let $ \si = k_2^{\frac{1}{N-2}} k^{-\frac{1}{N-2}} < \frac{v_s}{2}$.
Then $\ii _{\R^N} |\nabla((\psi_n)_{\si, \si }|^2 \, dx = k_2$,
hence $(\psi _n)_{\si, \si} $ satisfies (\ref{5.3}), that is
\beq
\label{5.8}
\int_{\R^N} |\nabla \psi _n|^2 \, dx - \si \frac{v_s}{2} Q( \psi _n)
+ \si ^2 \int_{\R^N} V(|\psi _n|^2) \, dx \geq 0.
\eeq
From (\ref{5.7}) and (\ref{5.8}) we get
$$
- \int_{\R^N} |\nabla \psi _n|^2 \, dx
+ \left(\si \frac{v_s}{2} - \si ^2 \right)\int_{\R^N} V(|\psi _n|^2) \, dx
\lra - \infty,
$$
which implies $ \ii_{\R^N} V(|\psi _n|^2) \, dx \lra - \infty $ as $ n \lra \infty$.
Since $ \ii _{\R^N} |\nabla \psi _n|^2 \, dx = k$, this contradicts the first inequality in (\ref{4.1}).
Next assume that $ N =2$ and $V\geq 0 $ on $[0, \infty)$. Fix $ k >0$.
By Corollary \ref{C4.17} there is $q_k > 0$ such that $ |Q(\psi ) | \leq q_k $
for any $ \psi \in \Eo $ satisfying $E(\psi) \leq k+1$. Let $ \psi \in \Eo $
be such that $ \ii _{\R^2} |\nabla \psi |^2 \, dx = k$.
If $ \ii _{\R^2} V(|\psi |^2) \, dx = 0$ we infer that $ |\psi | = 1 $ a.e.
on $ \R^2$ and then (\ref{lift}) implies $Q(\psi ) = 0$, hence $I(\psi) = 0$.
If $ \ii _{\R^2} V(|\psi |^2) \, dx > 0$
let $ \si = \left( \ii _{\R^2} V(|\psi |^2) \, dx \right)^{- \frac 12}$ and
$ \tilde{\psi} = \psi_{\si, \si}$, so that
$\ii _{\R^2} V(|\tilde{\psi }|^2) \, dx = 1 $ and
$ \ii _{\R^2} |\nabla \tilde{\psi} |^2 \, dx = k$.
We infer that $ |Q(\tilde{\psi})| \leq q_k $.
Since $ \psi = \tilde{\psi}_{\frac{1}{\si}, \frac{1}{\si}}$ we have
by scaling $ I(\psi) = \si^{-2} \ii _{\R^2} V(|\tilde{\psi} |^2) \, dx
- \si^{-1} Q (\tilde{\psi}) \geq \si^{-2} - \si^{-1} q_k \geq - \frac{q_k^2}{4}. $
We conclude that $I_{min}(k ) \geq - \frac{q_k^2}{4} > - \infty.$
\medskip
iv) If $V$ achieves negative values, it is easy to see that there exists
$ \psi _1 \in \Eo $ such that $ \ii _{\R^2} V(|\psi _1 |^2) \, dx < 0$.
Let $k_1 = \ii _{\R^2} |\nabla \psi _1|^2 \, dx .$ Then, for any $t > 0 $,
$ \ii _{\R^2} |\nabla (\psi _1)_{t, t}|^2 \, dx = \ii _{\R^2} |\nabla \psi _1|^2 \, dx
= k_1 $ because $N=2$, thus
$$
I_{min}(k_1) \leq I((\psi_1)_{t, t}) =
- t Q( \psi _1) + t^2 \ds \int_{\R^2} V(|\psi _1 |^2) \, dx \lra - \infty
$$
as $t \to \infty $. By concavity we have $I_{min}(k) = - \infty $ for any $ k \geq k_1$.
\medskip
v) The proof relies on the comparison maps constructed in the proof
of Theorem \ref{T4.13} from the (KP-I) ground state. Notice first
that if $ \psi \in \Eo $ is such that $ \ii_{\R^2} |\nabla \psi |^2 \, dx = k$, $\ii_{\R^2} V(|\psi |^2) \, dx > 0$
and $ Q( \psi ) > 0 $, then the function $ t \longmapsto I(\psi_{t,t}) =
t^2 \ii_{\R^2} V(|\psi |^2) \, dx - t Q( \psi ) $ achieves its minimum at
$ t_0 = \frac 12 Q( \psi ) \left( \ii _{\R^2} V(|\psi |^2) \, dx \right)^{-1}$.
Since $ \int_{\R^2} |\nabla \psi |^2 \, dx = \int_{\R^2} |\nabla \psi_{t,t} |^2 \, dx
= k $ in dimension $N=2$, it follows that
\beq
\label{rajout}
I_{min} (k) \leq \inf_{t > 0 } I(\psi_{t,t})
= I(\psi_{t_0, t_0}) = - \frac { Q^2( \psi)}{ 4 \int_{\R^2} V(|\psi |^2) \, dx } .
\eeq
Fix $ \gamma \not = 0 $ (to be chosen later), and let $w$ be a ground state
for (\ref{4.55}). Then, for $\e$ small enough, we have seen in the proof
of Theorem \ref{T4.13} how to construct from $w$ a comparison map $U_\e \in \Eo $
satisfying
$$
\begin{array}{l}
\ds Q( U_{\e}) = 3 v_s ^3 \Sr (w) \e - \frac { 6}{\g } v_s \Sr (w) \e ^3 ,
\\ \\
\ds \ii_{\R^2} V(|U_\e|^2) \ dx = \frac32 v_s^4 \Sr (w) \e
- \frac{6}{\gamma}\left( v_s^2 - \frac43 F''(1) \right) \Sr (w) \e^3
+ \mathcal{O}(\e^5) ,
\\ \\
\ds \ii_{\R^2} | \nabla U_\e|^2 \ dx = \frac32 v_s^4 \Sr (w) \e
+ v_s^2 \Sr (w) \left( \frac32 - \frac{12}{\g} \right) \e^3
+ \mathcal{O}(\e^5) .
\end{array}
$$
Let $k_\e = \ii_{\R^2} | \nabla U_\e|^2 \ dx $. Since $ Q(U_\e) > 0 $
and $ \ii_{\R^2} V(|U_\e|^2) \ dx > 0 $ for $\e$ small, we infer from
(\ref{rajout}) that
$$
\begin{array}{rl}
\ds I_{min} (k_\e) \leq & \ds - \frac { Q^2( U_\e)}{ 4 \int_{\R^2} V(|U_\e |^2) \, dx }
= - \frac32 v_s^2 \Sr (w) \e
+ \frac{ 4 F''(1) }{\gamma} \Sr (w) \e^3
+ \mathcal{O}(\e^5)
\\ \\
= & \ds - \frac{1}{v_s^2} \left[ \frac32 v_s^4 \Sr (w) \e
- v_s^2 \frac{4F''(1) }{\gamma} \Sr (w) \e^3
+ \mathcal{O}(\e^5) \right] .
\end{array}$$
Therefore, we have
$$ I_{min} (k_\e) < - \frac{k_\e}{v_s^2} $$
for all $\e $ sufficiently small
provided that $- \frac{4}{\g} F''(1) \geq \frac 32 - \frac{12}{\g}$, that is $ \frac{4( 3 - F''(1))}{\g } > \frac 32$
(take, for instance, $\g = 3 - F''(1)$).
\hfill
$\Box$
\medskip
Let
\beq
\label{k}
k_0 = \inf \Big\{ k \geq 0 \; \big| \; I_{min} (k) < - \frac{1}{v_s ^2} k \Big\}
\qquad \mbox{ and } \qquad
k_{\infty} = \inf \{ k > 0 \; \big| \; I_{min} (k) = - \infty \}.
\eeq
By Lemmas \ref{5.2} and \ref{5.3} (i) we have $0 \leq k_0 < \infty$ and
$ 0 < k_{\infty} \leq \infty$.
It is clear that $ k_0 \leq k_{\infty}$. If either $ N \geq 3$ or $N =2$
and $ V \geq 0 $ on $[0, \infty)$ we have $ k_{\infty} = \infty$, while if
$ N=2$ and (A4) holds with $ F''(1) \not= 3 $, we have $ k_0 = 0$; obviously, in all these cases we have $k_0 < k_{\infty}$.
The next Lemma gives further
information in the case when $N = 2$ and $V$ achieves negative values.
It brings into light the relationship between $k_\infty $ and the Dirichlet energy of
the stationary solutions of (\ref{1.1}) with minimal energy, the so-called ground
states or bubbles.
\begin{Lemma}
\label{L5.3bis}
Assume that $N = 2$, (A1), (A2) are satisfied and $\inf V < 0$. Let
$$
T = \inf \left\{ \int_{\R^2}| \nabla \psi |^2 \, dx \; \Big\vert \; \psi \in \Eo, \;
|\psi | \mbox{ is not constant and } \int_{\R^2 } V(|\psi |^2) \, dx \leq 0 \right\}.$$
Then:
\medskip
(i) We have $T > 0$ and the infimum is achieved for some $ \psi _0 \in \Eo $.
Moreover, any such $ \psi _0 $ satisfies the equation $ \Delta \psi _0 + \si ^2 F(|\psi _0|^2) \psi _0 = 0 $ in $ \Do '(\R^2)$
for some $ \si > 0$, $ \int_{\R^2} V(|\psi _0|^2) \, dx = 0$,
$\psi _0 $ belongs to $C^{1, \al }(\R^2)$ for any $ \al \in (0, 1)$ and, after a translation,
$\psi _0$ is radially symmetric.
\medskip
(ii) For any $k < T $ and any $M >0$, $E_{GL}$ is bounded on the set
$$
\Eo_{k, M} := \left \{ \psi \in \Eo \; \Big\vert \; \int_{\R^2} |\nabla \psi |^2 \, dx \leq k, \;
\int_{\R^2} V(|\psi |^2) \, dx \leq M \right\}.
$$
(iii) We have $ k_{\infty } = T.$
\end{Lemma}
{\it Proof.}
(i) It follows from Corollary \ref{C4.1} that $ T>0$.
The proof of the existence and regularity of minimizers is rather classical and is similar to the proof of
Theorem 3.1 p. 106 in \cite{brezis-lieb}, so we omit it.
Notice that any minimizer of the considered problem is also a minimizer of
$\int_{\R^2} V(|\psi |^2) \, dx $
under the constraint $\int_{\R^2} |\nabla \psi |^2 \, dx = T$ and then the radial symmetry follows from Theorem
2 p. 314 in \cite{M7}.
\medskip
(ii)
Fix $ \beta \in (0, 1]$ such that
\beq
\label{5.13}
V(s^2) \geq \frac{1}{4} ( s^2 - 1)^2 \qquad \mbox{ for any } s \in ( ( 1 - \beta)^2, ( 1 + \beta )^2).
\eeq
It suffices to prove that for any sequence $(\psi _n ) _{n \geq 1 } \subset \Eo _{k, M}$,
$E_{GL}(\psi _n)$ is bounded. Let $(\psi _n ) _{n \geq 1 } \subset \Eo _{k, M}$.
Let $ K_n = \{ x \in \R^2 \; \big | \; \; \big| \, |\psi _n (x)| - 1 \big| \geq \frac{ \beta}{2} \}.$
We claim that it suffices to prove that $ \Lo ^2( K_n)$ is bounded.
Indeed, assume $ \Lo ^2( K_n)$ bounded.
Let $ \tilde{\psi} _n = \left( \big| \, |\psi _n | - 1 \big| - \frac{ \beta}{2} \right)_+.$
Then $ \tilde{\psi }_n \in L_{loc}^1 (\R^2),$ $|\nabla \tilde{\psi }_n | \leq | \nabla \psi _n|$
a.e. on $ \R^2$ and by (\ref{ineq2}) we have
$$
\int_{\R^2} |\tilde{\psi}_n|^{2 p_0 + 2} \, dx
\leq C_{2 p_0 + 2} \| \nabla \tilde{\psi } _n \|_{L^2(\R^2)}^{2 p_0 + 2} \Lo ^2 ( K_n)
\leq C_{2 p_0 + 2} \| \nabla {\psi } _n \|_{L^2(\R^2) }^{2 p_0 + 2} \Lo ^2 ( K_n) .
$$
By (A1) and (A2) there is $C_0 >0$ such that $|V(s^2) | \leq C_0 \left( | s - 1 | - \frac{\beta}{2} \right)^{2 p_0 + 2}$
for any $ s $ satisfying $|s - 1 | \geq \beta$. Hence
$$
\int_{\R^2 \setminus \{ 1 - \beta \leq |\psi _n | \leq 1 + \beta \} }
| V(|\psi _n |^2) | \, dx
\leq C_0 \int_{\R^2} |\tilde{\psi}_n|^{2 p_0 + 2} \, dx \leq C_0 C_{2p_0 + 2}
\| \nabla {\psi } _n \|_{L^2 (\R^2) }^{2 p_0 + 2} \Lo ^2 ( K_n)
$$
and the last quantity is bounded. Since $ \int_{\R^2} V(|\psi _n|^2) \, dx $ is bounded,
we infer that \\
$ \int_{\{ 1 - \beta \leq |\psi _n | \leq 1 + \beta \} } V(|\psi _n|^2) \, dx $ is bounded,
and by (\ref{5.13}),
$ \int_{\{ 1 - \beta \leq |\psi _n | \leq 1 + \beta \} } \left( \ph ^2( |\psi_n |) - 1 \right)^2 \, dx $ is bounded.
On the other hand,
$ \int_{\R^2 \setminus \{ 1 - \beta \leq |\psi _n | \leq 1 + \beta \} }
\left( \ph ^2( |\psi_n |) - 1 \right)^2 \, dx \leq \int_{ K_n}\left( \ph ^2( |\psi_n |) - 1 \right)^2 \, dx \leq
64 \Lo ^2 ( K_n)$
and the conclusion follows.
\medskip
It remains to prove the boundedness of $ \Lo ^2 ( K_n)$. Let
$$
\psi _n ^+ = \left\{
\begin{array}{ll}
|\psi _n | & \quad \mbox{ if } |\psi _n | \geq 1
\\
1 & \quad \mbox{ otherwise, }
\end{array}
\right.
\qquad \qquad \mbox{and}
\qquad \qquad
\psi _n ^- = \left\{
\begin{array}{ll}
|\psi _n | & \quad \mbox{ if } |\psi _n | \leq 1
\\
1 & \quad \mbox{ if } |\psi _n | \geq 1.
\end{array}
\right.
$$
It is clear that $ \psi_n^+, \psi _n ^- \in \Eo$,
$ \int_{\R^2} |\nabla \psi _n ^+|^2 + |\nabla \psi _n ^-| ^2 \, dx
= \int_{\R^2} |\nabla | \psi _n | \,|^2 \, dx \leq k $ and
$ \int_{\R^2} V(|\psi _n ^+|^2) + V(|\psi _n ^-|^2) \, dx = \int_{\R^2} V(|\psi _n |^2)\, dx.$
If $ \int_{\R^2} V(|\psi _n ^+|^2) \, dx <0$,
by (i) we have $\int_{\R^2} |\nabla \psi _n^+|^2 \, dx \geq T > k$,
a contradiction.
Thus necessarily $ \int_{\R^2} V(|\psi _n ^+|^2) \, dx \geq 0$
and similarly $ \int_{\R^2} V(|\psi _n ^-|^2) \, dx \geq 0$, hence
$ \int_{\R^2} V(|\psi _n ^{\pm}|^2) \, dx \in [0, M].$
Let $K_n ^+ = \{ x \in \R^2 \; \big| \; \; |\psi _n (x) | \geq 1 + \frac{ \beta}{2} \}, $
$K_n ^- = \{ x \in \R^2 \; \big| \; \; |\psi _n (x) | \leq 1 - \frac{ \beta}{2} \}. $
Let $w_n ^+ = \phi_n^+(|x|) $ and $w_n ^- = \phi_n^-(|x|) $ be the symmetric decreasing rearrangements of
$(|\psi_n| - 1 )_+ = \psi_n ^+ - 1$ and of $(|\psi_n| - 1 )_ -= 1 - \psi _n ^- $, respectively.
As in the proof of Lemma \ref{L4.6} we have $ \phi_n^{\pm} \in H_{loc}^1( (0, \infty)).$ Let
$$
\begin{array}{c}
t_n = \inf \{ t \geq 0 \; | \; \phi_n ^+(t)< \frac{ \beta}{2} \} \qquad \mbox{ and } \qquad
s_n = \inf \{ t \geq 0 \; | \; \phi_n ^- (t)< \frac{ \beta}{2} \} .
\end{array}
$$
Then $ \Lo ^2 (K_n^+) = \Lo ^2 ( \{ (|\psi_n| - 1 )_+ \geq \frac{\beta}{2} \} )
= \Lo ^2 ( \{ w_n ^+ \geq \frac{\beta}{2} \} ) = \Lo ^2 ( \ov{B} (0, t_n )) = \pi t_n ^2$
and similarly $ \Lo ^2 (K_n^-) = \pi s_n ^2$, so that $ \Lo ^2( K_n) = \pi( t_n^2 + s_n ^2)$.
Assume that there is a subsequence $t_{n_j} \lra \infty$.
Let $ \tilde{w}_j = (w_{n_j}^+)_{\frac{1}{t_{n_j}}, \frac{1}{t_{n_j}} }
= \phi_{n_j} ^+ ( t_{n_j} |\cdot|), $
so that $\tilde{w}_j \geq \frac{\beta}{2}$ on $ \ov{B} (0, 1)$ and
$ 0 \leq \tilde{w}_j < \frac{\beta}{2}$ on $ \R^2 \setminus \ov{B} (0, 1)$.
Then $ \int_{\R^2} |\nabla \tilde{w}_j |^2 \, dx
= \int_{\R^2} |\nabla {w}_{n_j} |^2 \, dx \leq k $
and using (\ref{ineq2}) we see that $(\tilde{w}_j - \frac{\beta}{2} )_+$
is uniformly bounded in $L^p( B(0,1)) $ for any $ p < \infty$, and consequently
$(\tilde{w}_j)_{j \geq 1} $ is bounded in $L^p(B(0, R))$
for any $ p < \infty $ and any $ R \in(0, \infty)$.
Then there is a subsequence of $(\tilde{w}_j )_{j \geq 1}$, still denoted $(\tilde{w}_j )_{j \geq 1}$,
and there is $ \tilde{w} \in H_{loc}^1(\R^2)$ such that $ \nabla \tilde{w} \in L^2 (\R^2)$ and
$(\tilde{w}_j )_{j \geq 1}$, $\tilde{w}$ satisfy (\ref{4.32}).
It is easy to see that $ 1 + \tilde{w} \in \Eo $ and $ \tilde{w} \geq \frac{\beta}{2}$ on $\ov{B}(0,1).$
By weak convergence we have
$$
\tilde{k} := \ds \int_{\R^2} |\nabla \tilde{w} |^2\, dx \leq \liminf_{ j \ra \infty}
\int_{\R^2} |\nabla \tilde{w} _j |^2\, dx \leq k.
$$
Using (A2), the convergence $ \tilde{w} _j \lra \tilde{w}$ in $L^{2 p_0 + 2}(B(0, 1))$
and Theorem A2 p. 133 in \cite{willem} we get
$ \int_{B(0,1)} V((1 + \tilde{w}_j )^2) \, dx \lra \int_{B(0,1)} V((1 + \tilde{w} )^2) \, dx.$
Since $ \tilde{w}_j \in [0, \frac{\beta}{2}]$ on $\R^2 \setminus {B}(0,1) $ and
$ V( s^2 ) \geq 0$ for $ s \in [1 , 1 + \frac{\beta}{2} ]$,
using Fatou's Lemma we obtain
$ \int_{\R^2 \setminus B(0,1)} V((1 + \tilde{w} )^2) \, dx
\leq {\ds \liminf_{j \ra \infty}} \int_{\R^2 \setminus B(0,1)} V((1 + \tilde{w}_j )^2) \, dx.$
Therefore
$$
\begin{array}{l}
\ds \int_{\R^2} V((1 + \tilde{w} )^2) \, dx \leq
\liminf_{j \ra \infty} \int_{\R^2 } V( (1 + \tilde{w}_j )^2) \, dx
= \liminf_{j \ra \infty} \frac{1}{t_{n_j} ^2} \int_{\R^2 } V((1 + {w}_{n_j}^+ )^2) \, dx
\\
= \ds \liminf_{j \ra \infty} \frac{1}{t_{n_j} ^2} \int_{\R^2 } V((\psi_{n_j}^+ )^2) \, dx \leq 0
\end{array}
$$
because $\int_{\R^2 } V((\psi_{n_j}^+ )^2) \, dx \leq M$ and $t_{n_j} \to +\infty$
by our assumption. Since $1 + \tilde{w} \geq 1 + \frac{ \beta}{2}$ on $B(0,1)$
we infer that $ \int_{\R^2} |\nabla \tilde{w} |^2 \, dx \geq T > k$, a contradiction.
So far we have proved that $(t_n)_{n \geq 1}$ is bounded. Similarly $(s_n)_{n \geq 1}$ is bounded,
thus $(\Lo ^2( K_n))_{n \geq 1}$ is bounded and the proof of (ii) is complete.
\medskip
(iii) Consider a radial function $ \psi _0 \in \Eo $ such that
$|\psi _0|$ is not constant, $\int_{\R^2} V(|\psi _0|^2) \, dx = 0 $ and
$\int_{\R^2} |\nabla \psi _0 |^2 \, dx = T$.
Since $F(|\psi _0 |^2 ) \psi _0$ does not vanish a.e. on $ \R^2$,
there exists a radial function $ \phi \in C_c^{\infty}(\R^2)$ such that
$\int_{\R^2} \langle F(|\psi _0 |^2 ) \psi _0 , \phi \rangle \, dx > 0$.
It follows that $\frac{d}{dt} _{| t =0} \int_{\R^2} V(|\psi _0 + t \phi |^2) \, dx
= - 2 \int_{\R^2} \langle F(|\psi _0 |^2 \psi _0 , \phi \rangle \, dx <0$,
consequently there is $ \e > 0$ such that
$ \int_{\R^2} V(|\psi _0 + t \phi |^2) \, dx < \int_{\R^2} V(|\psi _0 |^2) \, dx = 0$ for any $ t \in (0, \e)$.
Denote $ k(t) = \int_{\R^2} |\nabla ( \psi _0 + t \phi ) |^2 \, dx.$
It follows from the proof of Lemma \ref{L5.3} (iv) that
$I_{min}(k(t) ) = - \infty $ for any $ t \in (0, \e)$, thus
$k_\infty \leq k(t) $ for any $ t \in (0, \e)$. Since $ k(t) \lra T $
as $ t \lra 0$, we infer that $ k_{\infty} \leq T$.
Let $ k < T$.
Consider $ \psi \in \Eo $ such that $\int_{\R^2} |\nabla \psi |^2 \, dx = k$.
If $|\psi | = 1 $ a.e. we have $V(|\psi |^2) = 0 $ a.e. and
$ Q( \psi ) = 0 $ by (\ref{lift}), hence $I(\psi) = 0$.
If $|\psi |$ is not constant, then we have necessarily $\int_{\R^2} V(|\psi|^2) \, dx > 0$.
If $Q( \psi ) \leq 0$, it is obvious that $I(\psi ) > 0$.
If $ Q(\psi ) > 0$ we have ${\ds \inf_{t > 0}} I(\psi_{t,t})
= - \frac 14 Q^2(\psi) \left( \int_{\R^2} V(|\psi|^2) \, dx \right)^{-1} $ and the infimum is achieved for
$t_{min} = \frac 12 Q(\psi) \left( \int_{\R^2} V(|\psi|^2) \, dx \right)^{-1} $.
There exists $ t_1 > 0$ such that $\int_{\R^2} V(|\psi _{t_1, t_1}|^2) \, dx = 1$. Then
$\int_{\R^2} |\nabla \psi _{t_1, t_1 } |^2 \, dx = k$ and
$$
I(\psi) \geq \inf_{t > 0} I(\psi_{t,t})
= - \frac {Q^2(\psi) }{4 \int_{\R^2} V(|\psi|^2) \, dx }
= - \frac { Q^2(\psi_{t_1, t_1}) }{ 4\int_{\R^2} V(|\psi_{t_1, t_1}|^2) \, dx }
= - \frac 14 Q^2(\psi_{t_1, t_1}).
$$
This implies $I(\psi) \geq \inf \{ - \frac 14 Q^2( \phi ) \; | \; \phi \in \Eo _{k, 1} \}.$
By (ii) we know that $E_{GL}$ is bounded on $\Eo _{k, 1}$ and Corollary \ref{C4.17} implies that $Q$
is also bounded on $\Eo _{k, 1}$. We conclude that $ I_{min}(k) > - \infty$, hence $ k < k_{\infty }$.
Since this is true for any $ k < T$, we infer that $ k_{\infty } \geq T$.
Thus $ k_{\infty} = T$.
\hfill
$\Box$
\begin{Lemma}
\label{L5.4}
Assume that $ 0 < k < k_{\infty }$ and $(\psi_n)_{n \geq 1} \subset \Eo $ is a sequence such that
$ \int_{\R^N} |\nabla \psi _n |^2 \, dx \leq k $ for all $n$.
Suppose that $(I(\psi_n))_{n \geq 1}$ is bounded in the case $N \geq 3$,
respectively that $I(\psi _n ) < 0$ for all $n$ in the case $N = 2$.
Then $(Q(\psi_n))_{n \geq 1}$, $ \left( \int_{\R^N} V(|\psi _n |^2) \, dx \right)_{n \geq 1}$ and
$(E_{GL}(\psi_n))_{n \geq 1}$ are bounded.
\end{Lemma}
{\it Proof.}
Consider first the case $ N \geq 3$.
Let us show that $ \int_{\R^N} V(|\psi _n |^2) \, dx $ is bounded from above.
We argue by contradiction and assume that this is false.
Then there is a subsequence, still denoted $(\psi_n)_{n \geq 1}$, such that
$ s_n : = \int_{\R^N} V(|\psi _n |^2) \, dx \lra \infty $ as $ n \lra \infty$.
Let $ \si _n = s_n ^{- \frac 1N}$. Since
$ \int_{\R^N}|\nabla ((\psi _n )_{\si _n, \si _n}) |^2 \, dx
= \si_n ^{N-2} \int_{\R^N}|\nabla \psi _n |^2 \, dx\lra 0 $ as $ n \lra \infty$,
Lemma \ref{L5.1} implies that $(\psi_n)_{\si_n, \si _n}$ satisfies (\ref{5.3}) with $ c = \frac{v_s}{2}$
for all sufficiently large $n$, that is
$$
\int_{\R^N}|\nabla \psi _n |^2 \, dx - \frac{v_s}{2} s_n ^{- \frac 1N} ( s_n - I(\psi _n)) + s_n^{ 1 - \frac{2}{N}} \geq 0.
$$
Since $ \int_{\R^N}|\nabla \psi _n |^2 \, dx $ and $ I(\psi _n)$ are bounded and $ s_n \lra \infty$,
the left-hand side of the above inequality tends to $- \infty $ as $ n \lra \infty$, a contradiction.
We conclude that there is $M > 0 $ such that $ \int_{\R^N} V(|\psi _n |^2) \, dx \leq M$ for all $M$.
Then (\ref{4.1}) implies that $ \ii_{\R^N} \left( \ph ^2( |\psi _n|) - 1 \right)^2 \, dx $ is bounded.
By (\ref{4.1}), $\ii _{\R^N} V(|\psi|^2) \, dx $ is bounded from below.
Using Corollary \ref{C4.17} we infer that $(Q(\psi_n))_{n \geq 1}$ is bounded.
\medskip
Consider next the case $ N =2$.
Since $ \ii _{\R^2}|\nabla \psi _n |^2 \, dx \leq k < k_{\infty}$,
using Lemma \ref{L5.3bis} (i) and (iii)
we see that either $ \ii _{\R^N} V(|\psi _n |^2) \, dx > 0$
or $ \ii _{\R^N} V(|\psi _n |^2) \, dx = 0 $ and $ |\psi _n | = 1 $ a.e. on $\R^2$.
In the latter case (\ref{lift}) implies $ Q( \psi _n) = 0$, hence $ I( \psi _n) = 0$,
contrary to the assumption that $ I( \psi _n ) < 0$.
Thus necessarily $ 0< \ii _{\R^N} V(|\psi _n |^2) \, dx < Q( \psi _n) $ for all $n$ because
$ I( \psi _n) < 0$.
Since $\ii _{\R^2} |\nabla (\psi _n) _{\si, \si} |^2 \, dx = \ii _{\R^2} |\nabla \psi _n|^2 \, dx $
for any $ \si >0$, as in the proof of Lemma \ref{L5.3bis} (iii) we have
$$
- \frac {Q^2(\psi _n ) }{4 \int_{\R^2} V(|\psi _n |^2) \, dx }
= \inf_{\si > 0} I((\psi_n)_{\si,\si}) \geq I_{min}
\left( \int_{\R^2} |\nabla \psi _n |^2 \, dx \right) \geq I_{min}(k)
$$
and this implies
$$
Q^2 ( \psi _n) \leq -4 I_{min}(k) \int_{\R^N} V(|\psi _n |^2) \, dx .
$$
Combining this with the inequality $ 0< \int_{\R^2} V(|\psi _n |^2) \, dx < Q( \psi _n) $, we get
\beq
\label{5.12}
0< \ds \int_{\R^2} V(|\psi _n |^2) \, dx < Q( \psi _n) \leq -4 I_{min}(k).
\eeq
We have thus proved that
$(Q(\psi_n))_{n \geq 1}$ and $ \left( \int_{\R^2} V(|\psi _n |^2) \, dx
\right)_{n \geq 1}$ are bounded.
The boundedness of $ \int_{\R^2} \left( \ph ^2( |\psi_n |) - 1 \right)^2 \, dx $
follows from Lemma \ref{L4.6} if $V \geq 0 $ on $[0, \infty)$,
respectively from Lemma \ref{L5.3bis} (ii) if $V$ achieves negative values.
\hfill
$\Box$
\medskip
We now state the main result of this section, which shows precompactness
of minimizing sequences for $I_{min}(k)$ as soon as $ k_0 < k < k_\infty $.
\begin{Theorem}
\label{T5.5}
Assume that $N \geq 2$ and (A1), (A2) hold. Let $ k \in ( k_0 , k_{\infty})$ and let
$(\psi_n )_{n \geq 1} \subset \Eo $ be a sequence such that
$$
\int_{\R^N} |\nabla \psi _n |^2 \, dx \lra k \qquad \mbox{ and } \qquad I(\psi _n) \lra I_{min}(k).
$$
There exist a subsequence $(\psi_{n_k})_{k \geq 1}$, a sequence of points $(x_k)_{k \geq 1} \subset \R^N$,
and $ \psi \in \Eo $ such that $ \int_{\R^N} |\nabla \psi |^2 \, dx = k$,
$I( \psi ) = I_{min}(k) $, $\psi_{n_k} ( x_k + \cdot ) \lra \psi $ a.e. on $\R^N$ and
$$
\| \nabla \psi_{n _k} ( \cdot + x_k ) - \nabla \psi \|_{L^2(\R^N)} \lra 0 ,
\qquad
\| \, | \psi_{n _k} |( \cdot + x_k ) - | \psi | \, \|_{L^2(\R^N)} \lra 0
\qquad \mbox{ as } k \lra \infty.
$$
\end{Theorem}
{\it Proof. }
Since $I_{min}(k) < 0$, we have $I(\psi_n) < 0$ for all sufficiently large $n$.
By Lemma \ref{L5.4} the sequences
$(Q(\psi_n))_{n \geq 1}$, $ \left( \int_{\R^N} V(|\psi _n |^2) \, dx \right)_{n \geq 1}$ and
$(E_{GL}(\psi_n))_{n \geq 1}$ are bounded. Passing to a subsequence
if necessary, we may assume that $E_{GL}(\psi _n) \lra \al _0 \geq k > 0$
and $Q( \psi _n) \lra q $ as $ n \lra \infty$.
We use the Concentration-Compactness Principle (\cite{lions})
and we argue as in the proof of Theorem \ref{T4.7}.
Let $\Lambda_n (t)$ be the concentration function associated to $E_{GL}(\psi _n) $, as in (\ref{4.15}).
It is standard to prove that there exist a subsequence of
$((\psi_n, \Lambda_n))_{n \geq 1}$, still denoted $((\psi_n, \Lambda_n))_{n \geq 1}$,
a nondecreasing function $ \Lambda : [0, \infty ) \lra \R$, $ \al \in [0, \al _0 ]$,
and a nondecreasing sequence $t_n \lra \infty$ such that (\ref{4.16}) and (\ref{annulus}) hold.
The next result implies that $ \al >0$.
\begin{Lemma}
\label{L5.6}
Let $(\psi_n) _{n \geq 1} \subset \Eo $ be a sequence satisfying:
\medskip
(a) $ E_{GL}(\psi _n) \leq M $ for some positive constant $M$.
\medskip
(b) $ \int_{\R^N} |\nabla \psi _n |^2\, dx \lra k $ and $ Q(\psi _n ) \lra q $
as $ n \lra \infty.$
\medskip
(c) ${\ds \limsup_{n \ra \infty }}\, I (\psi_n) < - \frac{1}{v_s ^2} k$.
\\
Then there exists $ \ell > 0$ such that
$ \ds \sup_{ y \in \R^N } E_{GL}^{B(y,1)} ( \psi _n ) \geq \ell $
for all sufficiently large $n$.
\end{Lemma}
{\it Proof. }
It is obvious that the sequence $(\psi_n) _{n \geq 1}$ satisfies the
conclusion of Lemma \ref{L5.6} if and only if
$((\psi_n)_{v_s, v_s}) _{n \geq 1}$ satisfies the same conclusion.
By (a) we have $E_{GL}((\psi_n)_{v_s, v_s}) \leq \max( v_s^{N-2}, v_s^N) M = 2^{\frac N2} M. $
Assumption (b) implies
$$\ds \int_{\R^N} |\nabla (\psi _n)_{v_s, v_s} |^2\, dx \lra v_s^{N-2}k \qquad \mbox{ and }
\qquad Q((\psi _n) _{v_s, v_s} ) \lra v_s^{N-1} q = \tilde{q} .
$$
Using (c) we find
$$
\begin{array}{l}
\ds \limsup_{n \ra \infty} E((\psi _n)_{v_s, v_s}) - v_s \tilde{q}
= \ds \limsup_{n \ra \infty} \left( \! v_s^{N-2} \! \! \int_{\R^N} |\nabla \psi _n |^2\, dx
+ v_s^N \! \! \int_{\R^N} V(|\psi _n|^2) \, dx - v_ s ^N Q( \psi _n) \! \right)
\\
= v_s ^N \left( \frac{1}{v_s ^2} k + \ds \limsup_{n \ra \infty} I( \psi _n ) \right) < 0.
\end{array}
$$
Then the result follows directly from Lemma \ref{L4.8}.
\hfill
$\Box $
\medskip
Next we prove that $ \al \not\in (0, \al _0)$.
We argue again by contradiction and we assume that
$ 0 < \al < \al _0$.
Arguing as in the proof of Theorem \ref{T4.7} and using Lemma \ref{splitting}
for each $n $ sufficiently large we construct two functions
$\psi_{n,1}, \psi_{n,2} \in \Eo $ such that
\beq
\label{5.14}
E_{GL}(\psi _{n, 1} ) \lra \al
\qquad
\mbox{ and }
\qquad
E_{GL}(\psi _{n, 2} ) \lra \al _0 - \al ,
\eeq
\vspace*{-15pt}
\beq
\label{5.15}
\int_{\R^N} \big\vert |\nabla \psi_{n} |^2 - |\nabla \psi_{n, 1} |^2 - |\nabla \psi_{n, 2} |^2 \big\vert \, dx \lra 0,
\eeq
\vspace*{-15pt}
\beq
\label{5.16}
\int_{\R^N} \big\vert V (| \psi_{n} |^2) - V( | \psi_{n, 1} |^2 ) - V( | \psi_{n, 2} |^2 ) \big\vert \, dx \lra 0,
\eeq
\vspace*{-15pt}
\beq
\label{5.17}
| Q(\psi _n) - Q(\psi _{n, 1}) - Q(\psi _{n, 2}) | \lra 0 \qquad \mbox{ as } n \lra \infty.
\eeq
Passing to a subsequence if necessary, we may assume that
$ \ii _{\R^N} |\nabla \psi_{n, i} |^2 \, dx \lra k_i \geq 0 $
as $ n \lra \infty$, $ i=1,2$. By (\ref{5.15}) we have $ k_1 + k_2 = k$. We claim that $ k_1 > 0 $ and $ k_2 >0$.
To prove the claim assume, for instance, that $ k_1 = 0$. From (\ref{5.14}) it follows that \\
$ \frac 12 \ii _{\R^N} \left( \ph^2 (|\psi_{n,1}|) - 1 \right)^2 \, dx \lra \al.$
Using Lemma \ref{L4.1} we find $ \ii _{\R^N} V( | \psi_{n, 1} |^2 ) \, dx \lra \al $.
From Lemma \ref{L4.2} (ii) we infer that there is $ \kappa > 0$
such that
$
E(\psi) \geq \frac{ v_s}{2} | Q( \psi )|
$
for any $ \psi \in \Eo $ satisfying $ E_{GL}(\psi ) \leq \kappa.$
It is clear that there are $ n_0 \in \N$ and $ \si _0 > 0$ such that
$E_{GL}( (\psi_{n,1})_{\si , \si }) \leq \kappa $ for any $ n \geq n_0 $ and any $ \si \in (0, \si _0]$.
Then $E((\psi_{n,1})_{\si , \si } ) \geq
\frac{ v_s}{2} | Q( (\psi_{n,1})_{\si , \si }) |$, that is
$$
\frac{ v_s}{2} | Q( \psi_{n,1}) | \leq \frac{1}{\si } \int_{\R^N} |\nabla \psi_{n, 1} |^2 \, dx
+ \si \int_{\R^N} V (|\psi_{n,1}| ^2) \, dx
$$
for any $n \geq n_0 $ and $ \si \in (0, \si_0].$
Passing to the limit as $ n \lra \infty $ in the above inequality we discover
$\frac{v_s}{2} \ds \limsup_{n \ra \infty} | Q( \psi_{n,1}) | \leq \si \al $ for any $ \si \in (0, \si _0]$,
that is $\ds \lim_{n \ra \infty} | Q( \psi_{n,1}) | =0$.
As a consequence we find $\ds \lim_{n \ra \infty} I( \psi_{n,1}) = \al$.
Since $|I(\psi _n ) - I(\psi _{n, 1}) - I(\psi _{n, 2}) | \lra 0 $ by (\ref{5.16}) and (\ref{5.17}),
we infer that $I(\psi _{n, 2}) \lra I_{min}(k) - \al$ as $ n \lra \infty$.
Since $ \ii _{\R^N} |\nabla \psi_{n, 2} |^2 \, dx \lra k_2 = k$,
this contradicts the definition of $I_{min}$ and the fact that $I_{min}$ is continuous at $k$.
Thus necessarily $ k_1 >0$. Similarly we have $ k_2 >0$, that is $ k_1, \, k_2 \in (0, k)$.
\medskip
We have $I(\psi_{n,i}) \geq I_{min} ( \int_{\R^N} |\nabla \psi_{n, i} |^2 \, dx )$ and passing to the limit we get
$\ds \liminf_{n \ra \infty} I(\psi_{n,i}) \geq I_{min} (k_i )$, $ i =1,2$.
Using (\ref{5.16}), (\ref{5.17}) and the fact that $I(\psi _n) \lra I_{min} (k)$
we infer that $I_{min} (k) \geq I_{min} (k_1) + I_{min} (k_2)$.
On the other hand, the concavity of $I_{min} $ implies $I_{min}(k_i) \geq \frac{k_i}{k} I_{min} (k)$,
hence $I_{min}(k_1) + I_{min}(k_2) \geq I_{min}(k) $ and equality may occur if and only if $I_{min}$
is linear on $[0, k]$. Thus there is $ A \in \R$ such that $I_{min}(s) = As $ for any $ s \in [0, k]$.
By Lemma \ref{L5.2} we have $ A = - \frac{1}{v_s ^2}$, hence $I_{min}(k) = - \frac{k}{v_s ^2}$,
contradicting the fact that $ k > k_0$.
Thus we cannot have $ \al \in (0, \al _0)$, and then necessarily $ \al = \al _0$.
\medskip
As in the proof of Theorem \ref{T4.7},
there is a sequence $(x_n)_{n \geq 1} \subset \R^N$ such that
for any $ \e > 0 $ there is $ R_{\e} > 0 $ satisfying
$E_{GL} ^{ \R^N\setminus B(x_n, R_{\e})} (\psi _n) < \e$ for all $ n$ sufficiently large.
Let $ \tilde{\psi}{_n} = \psi _n ( \cdot + x_n)$. Then
for any $ \e > 0 $ there exist $ R_{\e} > 0 $ and $ n_{\e } \in \N$ such that
$(\tilde{\psi}_n)_{n \geq 1}$ satisfies (\ref{4.31}).
It is standard to prove that there is a function $ \psi \in H_{loc}^1(\R^N)$
such that $ \nabla \psi \in L^2(\R^N)$ and a subsequence $ (\tilde{\psi}_{n _j})_{j \geq 1} $
satisfying (\ref{4.32})-(\ref{4.34}) and (\ref{4.37}).
\medskip
Lemmas \ref{L4.10} and \ref{L4.11} imply that
$ \| \, |\tilde{\psi} _{n _j} | - |\psi | \, \|_{L^2( \R^N) } \lra 0 $,
$Q (\tilde{\psi} _{n_j}) \lra Q( \psi) $ and
$ \ii _{\R^N} V(|\tilde{\psi }_{n_j} |^2) \, dx \lra \ii _{\R^N} V(|\psi |^2) \, dx $
as $ j \lra \infty. $
Therefore $I (\tilde{\psi} _{n_j}) \lra I( \psi) $, and consequently $I(\psi ) = I_{min}(k).$
On the other hand, by (\ref{4.33}) we have $\ii _{\R^N} |\nabla \psi |^2 \, dx \leq k$.
Since $I_{min}$ is strictly decreasing, we infer that necessarily
$\ii _{\R^N} |\nabla \psi |^2 \, dx = k
= {\ds \lim_{j \ra \infty}} \ii _{\R^N} |\nabla \tilde{\psi}_{n_j} |^2 \, dx.$
Combined with the weak convergence $ \nabla \tilde{\psi}_{n_j} \rightharpoonup \nabla \psi$ in
$L^2( \R^N)$, this gives the strong convergence
$ \nabla \tilde{\psi}_{n_j} \lra \nabla \psi$ in $L^2( \R^N)$
and the proof of Theorem \ref{T5.5} is complete.
\hfill
$\Box $
\medskip
Denote by $ d^- I_{min}(k) $ and $ d^+ I_{min}(k) $ the left and right
derivatives of $I_{min}$ at $ k>0 $ (which exist and are finite for
any $ k > 0 $ because $I_{min}$ is concave). We have:
\begin{Proposition}
\label{P5.7}
(i) Let $ c > 0 $. Then the function $ \psi $ is a minimizer of $I$
in the set $ \{ \phi \in \Eo \; | \; \int_{\R^N} |\nabla \phi |^2 \, dx = k \}$
if and only if $ \psi_{c,c}$ minimizes the functional
$$ I_c (\phi) = - cQ ( \phi ) + \int_{\R^N} V(|\phi|^2) \, dx $$
in the set $ \{ \phi \in \Eo \; | \; \int_{\R^N} |\nabla \phi |^2 \, dx = c^{N-2} k \}$.
\medskip
(ii) If $ \psi \in \Eo $ satisfies $ \int_{\R^N} |\nabla \psi |^2 \, dx = k $
and $ I(\psi) = I_{min}(k)$, there is $ \vartheta \in [d^+ I_{min}(k), d^- I_{min}(k) ] $
such that
\beq
\label{5.18}
i \psi _{x_1} - \vartheta \Delta \psi + F(|\psi |^2) \psi = 0 \qquad
\mbox{ in } \Do' ( \R^N).
\eeq
Then for $ c = \frac{1}{\sqrt{- \vartheta}}$ the function $ \psi _{c,c}$
satisfies (\ref{4.50}) and minimizes $E_c = E - cQ $ in the set
$ \{ \phi \in \Eo \; | \; \int_{\R^N} |\nabla \phi |^2 \, dx = c^{N-2} k \}$.
Moreover, $ \psi \in W_{loc}^{2, p } ( \R^N)$ and
$\nabla \psi \in W^{1, p } ( \R^N)$ for any $ p \in [2, \infty)$.
\medskip
(iii) After a translation, $ \psi $ is axially symmetric with respect
to the $ x_1-$axis if $ N \geq 3$. The same conclusion is true if
$N=2$ and we assume in addition that $F$ is $C^1$.
\medskip
(iv) For any $ k \in (k_0, k_{\infty}) $ there are $ \psi ^+, \psi ^- \in \Eo $
such that
$\int_{\R^N} |\nabla \psi ^+|^2 \, dx = \int_{\R^N} |\nabla \psi ^-|^2 \, dx = k$,
$I(\psi ^+) = I(\psi ^-) = I_{min}(k)$ and $\psi ^+$, $\psi ^-$ satisfy (\ref{5.18})
with $ \vartheta^+ = d^+ I_{min}(k)$ and $ \vartheta^- = d^- I_{min}(k)$, respectively.
\end{Proposition}
{\it Proof. }
For any $ \phi \in \Eo $ we have $I_c( \phi_{c,c}) = c^N I(\phi)$,
$ \ii _{\R^N} |\nabla \phi_{c,c} |^2 \, dx = c^{N-2} \ii _{\R^N} |\nabla \phi |^2 \, dx$
and (i) follows.
The proofs of (ii), (iii) and (iv) are very similar to the proof of Proposition \ref{P4.12}
and we omit them.
\hfill
$\Box $
\medskip
We will establish later (see Proposition \ref{P5.9} below) a relationship between the traveling waves constructed in
section \ref{minem} and those given by
Theorem \ref{T5.5} and Proposition \ref{P5.7} above.
The next remark shows that, in some sense,
there is equivalence between the inequalities $E_{min}(q) < v_s q$ and $I_{min}(k) < - \frac{k}{v_s^2}$.
\begin{remark}
\label{R5.8}
(i) \rm
Let $ \psi \in \Eo $ be such that $ E(\psi ) < v_s Q(\psi)$ and let
$ k = \ii _{\R^N} |\nabla \psi |^2 \, dx .$
Then $I_{min}(\frac{k}{v_s^{N-2}}) < - \frac{k}{v_s^ N}.$
Indeed, we have
$ { \ii _{\R^N} } |\nabla \psi _{\frac{1}{v_s}, \frac{1}{v_s}} |^2 \, dx = \frac{1}{v_s ^{N-2}} k $ and
$$
I_{min}\left(\frac{k}{v_s^{N-2}} \right) + \frac{k}{v_s^ N} \leq I \left( \psi_{\frac{1}{v_s}, \frac{1}{v_s}} \right)
+ \frac{1}{v_s ^2} \int_{\R^N} |\nabla \psi _{\frac{1}{v_s}, \frac{1}{v_s}} |^2 \, dx
= \frac{1}{ v_ s ^N} ( E(\psi) - v_s Q(\psi ) ) < 0.
$$
{\it (ii)} Conversely, let $\psi \in \Eo $ be such that
$I(\psi ) < - \frac{1}{v_s ^2} \ii _{\R^N} |\nabla \psi |^2 \, dx$
and denote $ q = v_s^{N-1} Q(\psi)$. Then $ E_{min}(p) < v_s q.$
Indeed, we have $ Q(\psi_{v_s, v_s}) = v_s^{N-1} Q(\psi) = q $ and
$$
E_{min}(q) - v_s q \leq E(\psi_{v_s, v_s}) - Q (\psi_{v_s, v_s})
= v_s ^N \left( \frac{1}{v_s ^2} \int_{\R^N} |\nabla \psi |^2 \, dx + I(\psi) \right)
< 0.
$$
\end{remark}
\section{Local minimizers of the energy at fixed momentum ($N=2$)}
\label{local}
We will use the results in the previous section
to find traveling waves to (\ref{1.1}) in space dimension $N=2$ which
are {\it local} minimizers of the energy at fixed momentum
even when $V$ achieves negative values.
If $ N =2$ and $q \geq 0$, define
\beq
\label{5.40}
E_{min}^{\sharp }(q) = \inf \Big\{ E( \psi ) \; \big| \; \psi \in \Eo, \; Q( \psi) = q
\mbox{ and } \int_{\R^2} V(|\psi |^2) \, dx \geq 0 \Big\}.
\eeq
This definition agrees with the one given in section \ref{minem} in the case $ V \geq 0.$
\begin{Lemma}
\label{L5.10}
Assume that $N=2$ and (A1), (A2) are satisfied.
The function $E_{min}^{\sharp }$ has the following properties:
\medskip
(i) $E_{min}^\sharp(q) \leq v_s q $ for any $ q \geq 0. $
\medskip
(ii) For any $ \e > 0 $ there is $ q_{\e } > 0 $ such that
$E_{min}^\sharp (q) > (v_s - \e ) q $ for any $ q \in (0, q_{\e})$.
\medskip
(iii) $E_{min}^\sharp$ is subadditive on $[0, \infty)$, nondecreasing,
Lipschitz continuous and its best Lipschitz constant is $v_s$.
\medskip
(iv) If $ \inf V < 0 $, then for any $ q > 0 $ we have
$E_{min}^\sharp (q) \leq k_{\infty}$, where $k_{\infty}$ is as in (\ref{k}) or in Lemma \ref{L5.3bis} (iii).
\medskip
(v) $E_{min}^\sharp$ is concave on $[0, \infty)$.
\end{Lemma}
{\it Proof. } If $ V \geq 0$ on $[0, \infty)$, the statements of Lemma
\ref{L5.10} have already been proven in section \ref{minem}.
We only consider here the case when $V$ achieves negative values.
The estimate (i) follows from Lemma \ref{L4.3}.
For (ii) proceed as in the proof of Lemma \ref{L4.4} and use Lemma \ref{L5.1} instead of Lemma \ref{L4.2}.
The proof of (iii) is the same as that of Lemma \ref{L4.5} (i).
\medskip
(iv) Let $ q > 0$. Fix $ \e > 0$, $ \e $ small.
By (ii) there is $ \psi \in \Eo $ such that $ \ii _{\R^2} |\nabla \psi |^2 \, dx \leq \frac{\e }{4} $ and
$ Q( \psi ) \geq \frac{ \e}{ 8 v_s }$.
It is obvious that
$ \ii _{\R^2} |\nabla (\psi _{\si, \si}) |^2 \, dx = \ii _{\R^2} |\nabla \psi |^2 \, dx \leq \frac{\e }{4} $
for any $ \si > 0 $ and there is $ \si _0 > 0 $ such that $ Q( \psi _{\si _0, \si _0 }) > q$.
Using Corollary \ref{C3.4} and (\ref{2.9}), we see that there is $ \psi _1 \in \Eo $ such that
$ Q( \psi _1 ) = q$, $ \ii _{\R^2} |\nabla \psi _1 |^2 \, dx \leq \frac{\e }{2} $ and
$ \psi _1 = 1 $ outside a large ball $B(0, R_1)$.
Let $M_1 = \ii _{\R^2} V( |\psi _1 |^2 ) \, dx. $
Let $ \psi _0 $ be as in Lemma \ref{L5.3bis} (i).
Proceeding as in the proof of Lemma \ref{L5.3bis} (iii) we see that there exists a radial function
$ \phi \in C_c^{\infty} ( \R^2)$ and there is $ \e _1 > 0$ such that
$\ii _{\R^2} V( |\psi _0 + t \phi |^2) \, dx < 0 $ for any $ t \in (0, \e_1)$.
Taking $ t \in (0, \e _1) $ sufficiently small and using a radial cut-off and scaling
it is not hard to construct a radial function $ \psi _2 \in \Eo $ such that
$ \ii _{\R^2} |\nabla \psi _2 |^2 \, dx \leq k_{\infty} + \frac{\e}{4}$,
$ \ii _{\R^2} V( |\psi _2 |^2) \, dx = - M_2 < 0 $
and $ \psi _2 = 1 $ outside a large ball $B(0, R_2)$.
Since $ \psi _2$ is radial, we have $ Q( \psi _2 ) = 0$.
Let $ t = \left( \frac{ M_1 - \frac{ \e}{4}}{M_2} \right)^{\frac 12}$.
Choose $ x_0 \in \R^2$ such that $|x_0| > 2( R_1 + t R_2)$ and define
$$
\psi _* (x) = \left\{
\begin{array}{ll}
\psi _1 (x) & \quad \mbox{ if } |x| \leq R_1, \\
\psi _2 \left( \frac{ x - x_0}{t } \right) & \quad \mbox{ if } |x| > R_1.
\end{array}
\right.
$$
Then $ \psi _* \in \Eo$,
$ Q( \psi _*) = Q( \psi_1) + t Q(\psi_2) = q $,
$ \ii _{\R^2} |\nabla \psi _* |^2 \, dx = \ii _{\R^2} |\nabla \psi _1 |^2 \, dx
+ \ii _{\R^2} |\nabla \psi _2 |^2 \, dx \leq k_{\infty} + \frac{ 3 \e}{4}$,
and
$\ii _ {\R^2} V(|\psi _*|^2) \, dx = \ii _ {\R^2} V(|\psi _1|^2) \, dx + t^2 \ii _ {\R^2} V(|\psi _2|^2) \, dx
= M_1 - t^2 M_2 =\frac{ \e}{4 } > 0$.
Thus $E_{min}^\sharp(q) \leq E(\psi _*) \leq k_{\infty} + \e.$
Since $ \e $ is arbitrary, the conclusion follows.
\medskip
(v) The idea is basically the same as in the proof of Lemma \ref{L4.5} (ii) but we have to be more careful
because the functions $ \psi \in \Eo $ that satisfy $ \ii_{\R^2} V(|\psi |^2) \, dx \geq 0 $ do not necessarily satisfy
$ \ii_{\R^2} V(|S_t^{\pm}\psi |^2) \, dx \geq 0 $ for all $t$, where $S_t^{\pm}$ are as in (\ref{4.10}) - (\ref{4.11}).
Let $E^\sharp = \ds \sup_{q \geq 0 } E_{min}^\sharp(q).$
By (iv) we have $ E^\sharp \leq k_{\infty}$. Denote
\beq
\label{qsharp}
q^\sharp = \sup \{ q > 0 \; | \; E_{min}^\sharp(q) < E^\sharp \}.
\eeq
Define $E_{min}^{\sharp,-1} (k) = \sup \{ q \geq 0 \; | \; E_{min}^\sharp(q) \leq k \}.$
Then $E_{min}^{\sharp,-1}$ is finite, increasing, right continuous on $[0, E^\sharp )$ and
$E_{min}^{\sharp} (E_{min}^{\sharp,-1} (k)) = k $ for all $ k \in [0, E^{\sharp})$.
By convention, put $E_{min}^{\sharp,-1} (k) = 0$ if $ k <0$.
For any $ \phi \in \Eo $ with $\ii _{\R^2} V(|\phi |^2) \, dx \geq 0$
we have $ E_{min}^{\sharp }( Q(\phi) ) \leq E(\phi)$, thus
\beq
\label{Einverse}
Q( \phi ) \leq E_{min}^{\sharp,-1} (E(\phi)).
\eeq
We will prove that for any fixed $ q \in (0, q^{\sharp})$ there are $ q_1 < q$ and $ q_2 > q$ such that $E_{min}^{\sharp}$
is concave on $[q_1, q_2]$.
Let $ q \in (0, q^\sharp )$. Fix an arbitrary $ \e > 0 $ such that
$ E_{min}^\sharp (q) + 4 \e < E^\sharp .$
Choose $ \psi \in \Eo $ such that $ Q( \psi ) = q $,
$ \ii _{\R^2} V(|\psi |^2) \, dx > 0 $ and
$E(\psi) < E_{min}^\sharp(q) + \e$.
We may assume that $ \psi $ is symmetric with respect to $ x_2$.
Indeed, let $S_t^+$ and $S_t^-$ be as in (\ref{4.10})-(\ref{4.11}).
Arguing as in the proof of Lemma \ref{L4.5} (ii), there is $ t_0 \in \R$ such that
$\ii _{\R^2} |\nabla (S_{t_0 }^+(\psi) ) |^2 \, dx = \ii _{\R^2} |\nabla (S_{t_0 }^-(\psi) ) |^2 \, dx
= \ii _{\R^2} |\nabla \psi |^2 \, dx < k_{\infty}$.
After a translation, we may assume that $ t_0 = 0$.
Let $ \psi _1 = S_{0 }^-(\psi)$, $ \psi _2 = S_{0 }^+(\psi)$, denote $q_i = Q( \psi _i)$ and
$ v_i = \ii _{\R^2} V(|\psi _i |^2) \, dx $, $ i = 1,2$ and $ v= \ii _{\R^2} V(|\psi |^2)$,
so that $ q_1 + q_2 = 2 Q(\psi) = 2q $ and $ v_1 + v_2 = 2 v$. Since
$\ii _{\R^2} |\nabla \psi_i ) |^2 \, dx < k_\infty = T $, by Lemma
{\ref{L5.3bis} we have $ v_1 \geq 0$ and $ v_2 \geq 0$ and consequently $v_1, v_2 \in [0, 2v]$.
If $ q_ 1 \leq 0 $ we have $ q_2 \geq 2 q$ and then for
$\si _2 = \frac{ q}{q_2} \leq \frac 12$, we get
$Q((\psi_2)_{\si _2, \si_2}) = q$ and $E((\psi_2)_{\si _2, \si_2}) \leq E(\psi)
< E_{min}^\sharp ( q ) + \e $,
hence we may choose $(\psi_2)_{\si _2, \si_2} $ instead of $ \psi$, and $(\psi_2)_{\si _2, \si_2} $ is
symmetric with respect to $x_2$. A similar argument works if $ q_2 \leq 0$.
If $ q_1 > 0 $ and $ q_2 > 0$, let $\si _1 = \frac{ q}{q_1}$ and
$\si _2 = \frac{ q}{q_2}$, so that $ \frac{1}{\si _1} + \frac{1}{\si_2} = 2$.
We claim that there is $ i \in \{ 1, 2 \}$ such that $ \si _i ^2 v_i \leq v$,
and then we may choose $(\psi_i)_{\si_i, \si _i}$,
which is symmetric with respect to $x_2$, instead of $ \psi$ .
Indeed, if the claim is
false we have $ v_i > \frac{1}{\si _i ^2} v$ and taking the sum we get
$ 2 > \frac{1}{\si_1^2} + \frac{1}{\si_2^2}$,
which is impossible because $\frac{1}{\si _1} + \frac{1}{\si_2} = 2$.
Since $ \psi $ is symmetric with respect to $ x_2$, we have
$Q(S_0^{\pm } \psi ) = q $ and
$E(S_0^{\pm } \psi ) = E( \psi) < k_{\infty} - 3 \e .$ As in Lemma
\ref{L4.5} (ii), the mapping $ t \longmapsto E(S_t^- \psi) $ is continuous
and tends to $ 2 E(\psi)$ as $ t \lra \infty$.
Let
$$
t_{\infty} = \inf \{ t \geq 0 \; | \; E(S_t^- \psi) \geq k_{\infty} \}
\qquad (\mbox{with possibly } t_{\infty} = \infty \mbox{ if } E(\psi) \leq \frac 12 k_{\infty}).
$$
For any $ t \in [0, t_{\infty})$ we have $E(S_t^- \psi) < k_{\infty} .$
If there is $ t \in [0, t_{\infty})$ such that $\ii _{\R^2} V( |S_t^-\psi |^2) \, dx = 0$,
we have necessarily $ \ii _{\R^2} |\nabla (S_t^-\psi )|^2 \, dx \geq k_{\infty}$,
thus $E(S_t^- \psi) \geq k_{\infty} $, a contradiction.
We infer that the function $t \longmapsto \ii _{\R^2} V( |S_t^-\psi |^2) \, dx $
is continuous, positive at $ t =0$ and cannot vanish on $[0, t_{\infty})$,
hence $ \ii_{\R^2} V(|S_t^-\psi |^2) \, dx >0$ for all $ t \in [0, t_{\infty})$. Consequently we have
\beq
\label{5.41}
E(S_t^- \psi) \geq E_{min}^\sharp (Q(S_t^- \psi)) \qquad
\mbox{ for any } t \in [0, t_{\infty}).
\eeq
For any $ t \geq 0$ we have
$\ii _{\R^2} |\nabla (S_t^+ \psi ) |^2 \, dx = 2 \ii _{\{x_2 \geq t \} } |\nabla \psi |^2 \, dx
\leq 2 \ii _{\{ x_2 \geq 0 \}} |\nabla \psi |^2 \, dx \leq E(\psi ) < k_{\infty}$,
hence $\ii _{\R^2} V( |S_t^+\psi |^2) \, dx \geq 0$ (by Lemma \ref{L5.3bis})
and therefore
\beq
\label{5.42}
E(S_t^+ \psi) \geq E_{min}^\sharp (Q(S_t^+ \psi)) \qquad \mbox{ for any } t \geq 0.
\eeq
The mapping $ t \longmapsto Q(S_t^+ \psi)$ is continuous, tends to $0$ as
$ t \lra \infty$ and $Q(S_0^+ \psi) =q$. If $ t_{\infty}= \infty$,
for any $ q_1 \in (0, q)$ there is $ t_{q_1} > 0$ such that
$Q(S_{t_{q_1}}^+ \psi) = q_1.$ Then $Q(S_{t_{q_1}}^- \psi) = 2q -q_1$
and using (\ref{5.40}), (\ref{5.41}) we get
$$
E_{min}^\sharp(q) + \e > E(\psi) = \frac 12 \left( E(S_{t_{q_1}}^+ \psi) + E(S_{t_{q_1}}^- \psi) \right)
\geq \frac 12 \left( E_{min}^\sharp( q_1) + E_{min}^\sharp(2q - q_1) \right) .
$$
In the case $ t_{\infty } < \infty$ we have $E(S_{t_{\infty}}^ - \psi) = k_{\infty}$,
hence $E(S_{t_{\infty}}^ + \psi) = 2 E(\psi) - E(S_{t_{\infty}}^ - \psi) < 2 E_{min}^{\sharp}(q) + 2 \e - k_{\infty} < E_{min}^\sharp(q) $
and by (\ref{Einverse}) it follows that
$$
Q(S_{t_{\infty}}^ + \psi) \leq E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q)
+ 2 \e - k_{\infty}) < q.
$$
For any $ q_1 \in [Q(S_{t_{\infty}}^ + \psi) , q]$ there is
$ t_{q_1 } \in [0, t_{\infty}]$ such that $Q(S_{t_{q_1}}^+ \psi) = q_1.$
As above, we obtain
$$
E_{min}^\sharp(q) + \e > \frac 12 \left( E_{min}^\sharp( q_1)
+ E_{min}^\sharp(2q - q_1) \right)
\qquad \mbox{ for any } q_1 \in [E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q)
+ 2 \e - k_{\infty}), \; q ].
$$
Since $ \e \in (0, \; \frac 14( E^\sharp - E_{min}^\sharp(q) )) $
is arbitrary and $E_{min}^{\sharp,-1} $ is right continuous we infer
that for any $ q \in (0, q^\sharp)$ there holds
\beq
\label{5.43}
E_{min}^\sharp(q) \geq \frac 12 \left( E_{min}^\sharp( q_1) + E_{min}^\sharp(2q - q_1) \right)
\quad \mbox{ for all } q_1 \in (E_{min}^{\sharp,-1}
( 2 E_{min}^\sharp(q) - k_{\infty}), \; q ].
\eeq
The function $ q \longmapsto E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q) - k_{\infty}) $ is nondecreasing and right continuous on $(0, q^\sharp)$. Fix
$ q_* \in (0, q^\sharp)$. We have
$$
\lim_{q \downarrow q_* } E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q) - k_{\infty})
= E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q _* ) - k_{\infty}) < q_*
$$
because $ 2 E_{min}^{\sharp} ( q_*) - k_{\infty} < E_{min}^{\sharp} ( q_*)$.
It is then easy to see that there are $ q_* ' < q_* $ and $ q_* '' \in (q_* , q^\sharp)$
such that for any $ q \in [q_* ', q_* ''],$
\beq
\label{5.44}
E_{min}^{\sharp,-1} ( 2 E_{min}^\sharp(q) - k_{\infty}) < q_* ' .
\eeq
Using (\ref{5.43}) we see that for any $ q_1, q_2 \in [ q_* ', q_* ''] $
we have
$$
E_{min}^\sharp \left( \frac{ q_1 + q_2}{2} \right) \geq
\frac 12 (E_{min}^\sharp( q_1) + E_{min}^\sharp( q_2) ).
$$
Since $E_{min}^\sharp$ is continuous, we infer that $E_{min}^\sharp$
is concave on $ [q_* ', q_* ''] $. Thus any point $ q_* \in (0, q^\sharp)$
has a neighborhood where $E_{min}^\sharp$ is concave
and then it is not hard to see that $E_{min}^\sharp$ is concave on $[0, q^\sharp)$.
If $ q^\sharp < \infty$ we have $E_{min}^\sharp = E^\sharp $ on
$[q^\sharp, \infty)$, hence $E_{min}^\sharp$ is concave on $[0, \infty)$.
\hfill
$\Box$
\medskip
Let
\beq
\label{5.45}
q_0^\sharp = \inf \{ q > 0 \; | \; E_{min}^{\sharp} (q) < v_s q \}
\qquad \mbox{ and } \qquad
q_{\infty}^{\sharp } = \sup \{ q > 0 \; | \; E_{min}^\sharp(q) < k_{\infty} \}.
\eeq
It is obvious that $ q_0^\sharp \leq q_{\infty}^\sharp $ and
$ q_{\infty} > 0$ because $E_{min}^{\sharp }(q) \to 0 < k_\infty$ as $q \lra 0$.
If $F$ satisfies assumption (A4) and $ F''(1) \not= 3 $, it follows from Theorem \ref{T4.13}
that $ q_0^\sharp = 0$ (notice that
the test functions
$U_{\e}$ constructed in the proof of Theorem \ref{T4.13} satisfy $ V(|U_{\e}|^2) \geq 0$ in $ \R^2$).
\medskip
Our next result shows the
precompactness of minimizing sequences for $E_{min}^\sharp (q)$.
\begin{Theorem}
\label{T5.11}
Assume that $N=2$, (A1), (A2) are satisfied, and $ \inf V < 0$.
Let $ q \in (q_0^\sharp, q_{\infty}^\sharp)$ and assume that
$ (\psi _n )_{n \geq 1 } \subset \Eo $ is a sequence satisfying
$$
\ii_{\R^2} V(|\psi_n|^2) \, dx \geq 0, \quad Q( \psi _ n ) \lra q
\quad \mbox{ and } \quad E( \psi _n ) \lra E_{min}^\sharp(q).
$$
There exist a subsequence $(\psi_{n_k})_{k \geq 1}$, a sequence of
points $(x_k)_{k \geq 1} \subset \R^N$,
and $ \psi \in \Eo $ such that $Q( \psi ) = q$, $E( \psi ) = E_{min}^\sharp(q) $,
$\psi_{n_k} ( x_k + \cdot ) \lra \psi $ a.e. on $ \R ^2$ and
$ \ds \lim_{k \ra \infty} d_0(\psi_{n_k} ( x_k + \cdot ) , \; \psi ) = 0 .$
Furthermore, $ \ii_{\R^2} V(|\psi|^2) \, dx > 0 $, hence
$\psi \in \Eo $ is a local minimizer in the sense that
$$ E(\psi) = E_{min}^\sharp (q) = \inf \Big\{ E(w)\; \big| \;
w \in \Eo, \qquad Q(w) = q , \qquad \ii_{\R^2} V(|w|^2) \, dx > 0 \Big\} .$$
Moreover, the conclusions of Proposition \ref{P4.12} hold true with
$E_{min}$ replaced by $E_{min}^\sharp$.
\end{Theorem}
{\it Proof. } Fix $ k_1, k_2$ such that $ 0< k_1 < E_{min}^\sharp(q) < k_2 < k_{\infty}$.
We may assume that $ k_1 < E( \psi _n) < k_2$ for all $n$.
By Lemma \ref{L4.1} there is $ C_1( k_1) > 0 $ such that
$E_{GL}( \psi _n) \geq C_1( k_1).$ Since
$\ii_{\R^2} V(|\psi_n|^2) \, dx \geq 0, $
we have $ \psi_n \in \Eo_{k_2, k_2}$ and using Lemma \ref{L5.3bis} we
infer that $E_{GL}( \psi _n)$ is bounded. Passing to a subsequence
if necessary, we may assume that $E_{GL}(\psi _n) \lra \al _0 > 0$.
Then we proceed as in the proof of Theorem \ref{T4.7} and we use the Concentration-Compactness Principle
for the sequence of functions
$ f_n = |\nabla \psi _n |^2 + \frac 12 \left( \ph^2( |\psi _n|)- 1 \right)^2$.
We rule out vanishing thanks to Lemma \ref{L4.8}.
If dichotomy occurs for a subsequence (still denoted $(\psi_n)_{n \geq 1}$),
using Lemma \ref{splitting} for all $n$ sufficiently large we construct
two functions $ \psi_{n, 1}, \; \psi_{n , 2} \in \Eo $
such that
$\big\vert \ii _{\R^2} |\nabla \psi_n |^2 \,dx - \ii _{\R^2} |\nabla \psi_{n , 1} |^2 \,dx
- \ii _{\R^2} |\nabla \psi_{n , 2} |^2 \,dx \big\vert \lra 0$, and
(\ref{4.28}), (\ref{4.29}), (\ref{4.30}) hold for some $ \al \in (0, \al _0)$.
In particular, we have $\ii _{\R^2} |\nabla \psi_{n , i} |^2 \,dx < k_2 < k_{\infty }$, $ i =1,2$ for all $n$ sufficiently large and this implies
$\ii _{\R^2} V(|\psi_{n, i}|^2) \, dx \geq 0$,
so that $E(\psi_{n, i}) \geq E_{min}^\sharp (Q (\psi_{n, i}))$.
Since $ q \in (q_0^\sharp, q_{\infty}^\sharp)$, using
the concavity of $E_{min}^{\sharp }$ and Lemma \ref{L5.10} (i) and (ii)
we infer that $E_{min}^\sharp(q) < E_{min}^\sharp(q') + E_{min}^\sharp(q- q')$
for any $ q' \in (0, q)$.
Then arguing as in the proof of Theorem \ref{T4.7} we rule out dichotomy and we conclude that concentration occurs.
Hence there is a sequence $(x_n)_{n \geq 1} \subset \R^N$
such that, denoting $ \tilde{\psi}_{n} = \psi_{n} ( x_{n} + \cdot ) $,
(\ref{4.31}) holds.
Consequently there are a subsequence $(\tilde{\psi}_{n_k})_{k \geq 1}$ and
$ \psi \in \Eo $ that satisfy (\ref{4.32}) and (\ref{4.33}).
Using Lemmas \ref{L4.10} and \ref{L4.11} we get
$ {\ds \lim_{k \ra \infty} } \| \, | \tilde{\psi}_{n_k} | - | \psi | \, \| _{L^2( \R^N)} = 0, $
\beq
\label{5.46}
\lim_{k \ra \infty} \ii _{\R^2} V( | \tilde{\psi}_{n_k} |^2) \, dx = \ii _{\R^2} V( | \psi |^2) \, dx
\quad \mbox{ and } \quad \lim_{k \ra \infty} Q( \tilde{\psi}_{n_k}) = Q( \psi).
\eeq
In particular, we have $\ii _{\R^2} V( | \psi |^2) \, dx \geq 0$, $Q( \psi) = q$
and this implies $E(\psi) \geq E_{min}^\sharp(q)$.
Combining this information with (\ref{4.33}) and (\ref{5.46}) we see that necessarily
$\ii _{\R^2} |\nabla \tilde{\psi}_{n_k} |^2\, dx \lra \ii _{\R^2} |\nabla \psi |^2\, dx.$
Together with the weak convergence $\nabla \tilde{\psi}_{n_k} \rightharpoonup \nabla \psi$ in $L^2( \R^2)$,
this implies the strong convergence
$\| \nabla \tilde{\psi}_{n_k} - \nabla \psi \| _{L^2( \R^2)} \lra 0 $.
Hence $ d_0( \tilde{\psi}_{n_k} , \psi) \lra 0$ as $ k \lra \infty$.
The fact that $ \ii _{\R^2} V( | \psi |^2) \, dx > 0 $ comes from
the fact that
$|\psi |$ is not constant (because $Q(\psi ) = q >0$) and
$ \ii_{\R^2} |\nabla \psi|^2 \, dx < k_\infty $.
The last part is proved in the same way as Proposition \ref{P4.12}.
\hfill
$\Box$
\medskip
If $ q_{\infty}^\sharp < \infty$ we have $E_{min}^\sharp(q) = k_{\infty} $
for all $ q \geq q_{\infty}^\sharp$.
The conclusion of Theorem \ref{T5.11} is not valid for $ q \geq q_{\infty}^\sharp $.
Indeed, for such $q$ the argument used in the proof of Lemma \ref{L5.10} (iv)
leads to the construction of a minimizing sequence $(\psi_n)_{n \geq 1} \subset \Eo$ satisfying the assumptions of Theorem \ref{T5.11},
but $E_{GL}(\psi_n) \lra \infty$. Furthermore, if
$ \ii_{\R^2} |\nabla \psi|^2 \, dx \geq k_\infty $, Lemma {\ref{L5.3bis}
does not guarantee that the potential energy $ \ii _{\R^2} V(|\psi |^2) \, dx $ is positive.
\section{Orbital stability}
\label{sectionorbistab}
It is beyond the scope of the present paper to study the Cauchy problem associated
to (\ref{1.1}). Instead, we will content ourselves to assume in the sequel that
the nonlinearity $F$ satisfies (A1), (A2) and is such that the following holds:
\medskip
{\bf (P1)} (local well-posedness) For any $ M > 0$ there is $T(M) > 0$
such that for any $ \psi _0 \in \Eo $ with $E_{GL}(\psi _0) \leq M$ there exist
$T_{\psi_0} \geq T(M) $ and a unique solution
$ t \longmapsto \psi (t) \in C([0, T_{\psi_0}), (\Eo, d))$ such that $ \psi(0) = \psi_0$.
Moreover, $\psi(\cdot)$ depends continuously on the initial data in the following sense:
if $ d( \psi _0 ^n, \psi _0) \lra 0$ and $ t \longmapsto \psi _n(t) $ is the solution
of (\ref{1.1})
with initial data $\psi_0^n$,
then for any $T < T_{\psi_0} $ we have $T < T_{\psi_0^n}$ for all sufficiently large $n$
and $d( \psi _n(t), \psi (t)) \lra 0$ uniformly on $[0, T]$ as $ n \lra \infty$.
\medskip
{\bf (P2)} (conservation of phase at infinity)
We have $\psi(\cdot) - \psi _0 \in C([0, T_{\psi_0}), H^1( \R^N)).$
\medskip
{\bf (P3)} (conservation of energy) We have $E(\psi(t) ) = E( \psi _0) $ for any
$t \in [0, T_{\psi_0})$.
\medskip
{\bf (P4)} (regularity) If $\Delta \psi _0 \in L^2( \R^N)$, then
$ \Delta \psi(\cdot) \in C([0, T_{\psi_0}), L^2( \R^N) ) $.
\medskip
In space dimension $N = 2,3,4$, the Cauchy problem for the Gross-Pitaevskii equation
(that is (\ref{1.1}) with $F(s) = 1- s$)
has been studied in \cite{PG, PG2} and it was proved that the flow has the properties
(P1)-(P4) above. Moreover, the solutions found in \cite{PG, PG2} are global in
time if $ N=2, 3$ or if $N=4$ and the initial data has sufficiently small energy.
This comes from the conservation of energy and from the fact that the Gross-Pitaevskii
equation is subcritical if $N=2, \; 3$ and it is critical if $N=4$.
It seems that the proofs in \cite{PG, PG2} can be easily adapted to more general
subcritical nonlinearities provided that the associated nonlinear potential $V$
is nonnegative on $[0, \infty)$. Notice that any nonlinearity satisfying (A2)
is subcritical.
Recently it has been proved in \cite{KOPV} that the Gross-Pitaevskii equation is globally well-posed on the whole energy space $ \Eo$ in space dimension $N=4$
and that the cubic-quintic NLS is globally
well-posed on $ \Eo$ if $N=3$,
despite the fact that both problems are critical.
\medskip
Assume that (P1) and (P3) hold.
If $ V \geq 0$, using the conservation of energy and Lemma \ref{L4.6} it is
easy to prove that all solutions are global.
If $N=2$ and $ \inf V < 0 $, any solution $ t \longmapsto \psi (t)$
with initial data $ \psi _0 $ satisfying
$ \ii _{ \R^2} |\nabla \psi _0 | ^2 \, dx < k_{\infty}$ and
$E( \psi _0) < k_{\infty}$ is global.
Indeed, the mapping $ t \longmapsto \ii_{\R^2} V(|\psi (t) |^2) dx $ is continuous;
if it changes sign at some
$ t_0 \in (0, T_{\psi _0})$, there are two possibilities:
either $ \psi(t_0)$ is constant (and then $E(\psi (t_0)) = 0$,
hence $E(\psi (t)) = 0 $ for all $t$ and $ \psi (t)$ is constant)
or Lemma \ref{L5.3bis} (i) implies that
$\ii_{\R^2} V(|\psi (t_0) |^2) dx = 0 $ and
$ \ii _{ \R^2} |\nabla \psi _0 | ^2 \, dx \geq k_{\infty}$, thus $E(\psi(t_0) ) \geq k_{\infty}$,
contradicting the fact that, by conservation of the energy,
$E(\psi(t_0) ) = E( \psi _0 ) < k_{\infty}$.
Consequently $ 0 \leq \ii_{\R^2} V(|\psi (t) |^2) dx \leq E( \psi _0)$ and
$ 0 \leq \ii_{\R^2} |\nabla \psi (t) |^2 dx \leq E( \psi _0)$
as long as the solution exists. Then Lemma \ref{L5.3bis} (ii) implies that $E_{GL}( \psi (t))$
remains bounded and using (P1) we see that the solution is global.
\medskip
In the case of more general nonlinearities, the Cauchy problem for (\ref{1.1})
has been considered by C. Gallo in \cite{gallo}.
In space dimension $N=1,\, 2,\, 3, \, 4 $ and under suitable assumptions on $F$, he proved the following
(see Theorems 1.1 and 1.2 pp. 731-732 in \cite{gallo}):
\medskip
{\bf (P1')} For any $ \psi _0 \in \Eo$ and any $ u_0 \in H^1 ( \R^N)$, there exists a unique global
solution $ \psi _0 + u(t)$, where $ u(\cdot) \in C([0, \infty), H^1( \R^N))$
and $ u(0) = u_0$.
The solution depends continuously on the initial data $ u_0 \in H^1( \R^N)$.
\medskip
Notice that the solutions in \cite{gallo} satisfy (P2) by construction
and they also satisfy (P3) and (P4).
Moreover, it is proved (see Theorem 1.5 p. 733 in \cite{gallo}) that
any solution $ \psi \in C([0, T], \Eo )$
automatically satisfies (P2).
\begin{Lemma}
\label{L6.1} (conservation of the momentum)
Assume that $F$ is such that (A1), (A2), ((P1) or (P1')) and (P2)$-$(P4) hold.
Let $ \psi _0 \in \Eo $ and let $ \psi $ be the solution of (\ref{1.1})
with initial data $ \psi _0$, as given by (P1) or (P1')). Then
$$
Q( \psi (t)) = Q( \psi _0) \qquad \mbox{ for any } t \in [0, T_{\psi_ 0}).
$$
\end{Lemma}
{\it Proof. }
Assume that $\psi _0 \in \Eo $ is such that $\Delta \psi _0 \in L^2( \R^N)$.
Let $\psi(\cdot)$ be the solution of (\ref{1.1}) with initial data $ \psi _0$.
By (P1) and (P4) we have $ \psi_{x_j}(\cdot ) \in C([0, T_{\psi_0}), H^1( \R^N ))$, $ j = 1, \dots, N$.
Let $t, t+s \in [0, T_{\psi _0} )$.
Since $ \psi( t+s) - \psi (t) \in H^1( \R^N)$ by (P2),
the Cauchy-Schwarz inequality implies
$\langle i \psi_{x_1}( t+s) + i \psi _{x_1}(t) , \psi( t+s) - \psi (t) \rangle \in L^1( \R^N)$.
Using the definition of the momentum and Lemma \ref{L2.3} we get
$$
\begin{array}{l}
\frac 1s \left( Q (\psi( t+s)) - Q( \psi (t)) \right)
= \frac 1s L ( \langle i \psi_{x_1}( t+s) + i \psi _{x_1}(t) , \; \psi( t+s) - \psi (t) \rangle )
\\
\\
= {\ds \ii _{\R^N} } \langle i \psi_{x_1}( t+s) + i \psi _{x_1}(t) , \; \frac 1s ( \psi( t+s) - \psi (t) ) \rangle\, dx.
\end{array}
$$
Letting $ s \lra 0 $ in the above equality and using (\ref{1.1}) we get
\beq
\label{6.1}
\frac{d}{dt} ( Q( \psi (t))) = 2 \ii_{\R^N } \langle \frac{ \p \psi (t)}{\p x_1 } , \;
\Delta \psi (t) + F( |\psi |^2) \psi (t) \rangle \, dx.
\eeq
Since $ \frac{ \p \psi (t)}{\p x_j } \in H^1( \R^N)$, using the integration by parts formula for $H^1$ functions
(see, e.g., \cite{brezis} p. 197) we have
\beq
\label{6.2}
\ii_{\R^N } \langle \frac{ \p \psi (t)}{\p x_1 } , \; \Delta \psi (t) \rangle \, dx
= - \ii_{\R^N} \sum_{j = 1}^N
\langle \frac{ \p ^2 \psi (t) }{ \p x_1 \p x_j } , \; \frac{ \p \psi (t) }{ \p x _j} \rangle \, dx
= - \frac 12 \ii_{\R^N} \frac{ \p } {\p x_1} \left( |\nabla \psi(t) |^2 \right) \, dx.
\eeq
We have $|\nabla \psi(t) |^2 \in L^1( \R^N)$ and
$ \frac{ \p } {\p x_k} \left( |\nabla \psi(t) |^2 \right)= 2 \sum_{j = 1}^N
\langle \frac{ \p ^2 \psi (t)}{\p x_k \p x_j },\; \frac{ \p \psi (t)}{ \p x_j } \rangle \in L^1( \R^N)$,
hence $ |\nabla \psi(t) |^2 \in W^{1,1}( \R^N)$.
It is well-known that for any $ f \in W^{1,1}(\R^N)$ we have $ \ii _{\R^N} \frac{\p f}{\p x_j} (x) \, dx = 0$
and using (\ref{6.2}) we get
$\ii_{\R^N } \langle \frac{ \p \psi (t)}{\p x_1 } , \; \Delta \psi (t) \rangle \, dx = 0.$
On the other hand, $ 2 \langle \psi _{x_1} (t) ,\; F( |\psi |^2) \psi (t) \rangle
= - \frac{ \p }{\p x_1} \left( V(|\psi(t )|^2) \right).$
We have $ V(|\psi(t )|^2) \in L^1( \R^N)$ by Lemma \ref{L4.1}. Using the fact that
$ \psi _{x_j} ( t) \in H^1( \R^N)$, (A1), (A2) and the Sobolev embedding it is easy to see that
$ \frac{ \p }{\p x_j} \left( V(|\psi(t )|^2) \right) = -2 \langle \psi _{x_j} (t) ,\; F( |\psi |^2) \psi (t) \rangle \in L^1( \R^N)$
for all $j$, hence $ V(|\psi(t )|^2) \in W^{1,1}( \R^N)$ and therefore
$\ii _{\R^N} \frac{ \p }{\p x_1} \left( V(|\psi(t )|^2) \right) \, dx = 0.$
Then using (\ref{6.1}) we obtain $\frac{d}{dt} ( Q( \psi (t))) = 0$ for any $t$,
consequently $ Q( \psi ( \cdot))$ is constant on $[0, T_{\psi _0}).$
\medskip
Let $\psi _0 \in \Eo$ be arbitrary.
By Lemma \ref{L3.5}, there is a sequence $ (\psi _0 ^n)_{n \geq 1} \subset \Eo $
such that $ \nabla \psi _0 ^n \in H^2 ( \R^N)$
and $\| \psi _0 ^n - \psi _0 \|_{H^1( \R^N)} \lra 0 $ as $ n \lra \infty$
(thus, in particular, $d( \psi _0 ^n, \psi _0) \lra 0 $).
Fix $ T \in (0, T_{\psi _0}).$ It follows from (P1) or (P1')
that for all sufficiently large $n$, the solution $\psi _n (\cdot ) $ of (\ref{1.1})
with initial data $ \psi _0 ^n$ exists at least on $[0, T]$ and $ d ( \psi _n (t) , \psi (t) ) \lra 0 $
uniformly on $[0, T]$.
Using Corollary \ref{C4.13} we infer that for any fixed $ t \in [0, T]$ we have
$Q( \psi _n (t) ) \lra Q(\psi (t))$.
From the first part of the proof and Corollary \ref{C2.4} we get
$Q( \psi _n(t)) = Q (\psi _0^n) \lra Q( \psi _0) $ as $ n\lra \infty$.
Hence $Q( \psi (t)) = Q( \psi _0)$.
\hfill
$\Box $
\medskip
We now state our orbital stability result, which is based on the argument
in \cite{CL}.
\begin{Theorem}
\label{T6.2}
Assume that (A1), (A2), ((P1) or (P1')) and (P2)$-$(P4) hold.
$\bullet$ We assume $N \geq 2$ and $V\geq 0$ on $[0, \infty )$. Let
$ q > q_0 $, and define $ \So_q = \{ \psi \in \Eo \; | \; Q( \psi ) = q,
\mbox{ and } E( \psi ) = E_{min}(q) \}. $
\smallskip
Then, $\So _q$ is not empty
and is orbitally stable by the flow of (\ref{1.1}) for the semi-distance $d_0$
in the following sense: for any $ \e > 0$ there is $ \de _{\e } >0$
such that any solution of (\ref{1.1}) with initial data $\psi _0$
such that $d_0( \psi_0, \So _q)< \de _{\e} $ is global and
satisfies $d_0 (\psi(t), \So _q) < \e$ for any $ t > 0 $.
\smallskip
$\bullet$ Assume that $N=2$ and $ \inf V < 0 $. Let
$ q \in (q_0^\sharp, q_{\infty}^\sharp)$, where $q_0^\sharp, q_{\infty}^\sharp $ are as in (\ref{5.45}), and define
$ \So_q^\sharp = \{ \psi \in \Eo \; | \; Q( \psi ) = q, \;
\ii_{\R^2 } V(|\psi|^2) dx \geq 0 \mbox{ and } E( \psi ) = E_{min}^\sharp(q) \}. $
\smallskip
Then $\So _q^\sharp$ is orbitally stable by the flow of (\ref{1.1})
for the semi-distance $d_0$.
\end{Theorem}
{\it Proof. } We argue by contradiction and we assume that the
statement is false. Then there is some $ \e _0> 0$ such that
for any $ n \geq 1$ there is $ \psi _0^n \in \Eo$ satisfying
$ d_0( \psi_0^n, \So _q) < \frac 1n$ (resp. $ d_0( \psi_0^n, \So_q^\sharp) < \frac 1n$)
and there is $ t_n >0$ such that
$d_0 ( \psi_n( t_n) , \So _q ) \geq \e _0 $ (resp.
$d_0 ( \psi_n( t_n) , \So _q^\sharp ) \geq \e _0 $), where $\psi_n$ is
the solution of the Cauchy problem associated to (\ref{1.1}) with
initial data $ \psi _0^n$.
\medskip
We claim that $Q( \psi _0^n ) \lra q $ and $E(\psi _0^n) \lra E_{min}(q) $
(resp. $E(\psi _0^n) \lra E_{min}^\sharp(q) $).
Indeed, for each $n$ there is $ \phi _n \in \So _q$ (resp. $\in \So _q^\sharp$)
such that $ d_0 ( \psi _0^n , \phi _n ) < \frac 2n$.
If $N=2$ and $V$ achieves negative values, we have
$$
\limsup_{n \ra \infty} \ii _{\R^2} |\nabla \psi _0^n | ^2 \, dx
= \limsup_{n \ra \infty} \ii _{\R^2} |\nabla \phi ^n | ^2 \, dx
\leq \limsup_{n \ra \infty} E(\phi_n) = E_{min}^{\sharp}(q) < k_{\infty } ,
$$
hence $ \ii _{\R^2} V(|\psi _0^n |^2) \, dx \geq 0$ for all sufficiently large $n$.
Consider an arbitrary subsequence $(\psi_0^{n_{\ell}})_{\ell \geq 1} $
of $(\psi_0^n)_{n \geq 1}$. Using either Theorem \ref{T4.7} or
Theorem \ref{T5.11} we infer that there exist a subsequence
$(\phi_{n_{\ell_k}})_{ k \geq 1}$ of $ (\phi_n)_{n \geq 1}$,
a sequence $ (x_k)_{k \geq 1} \in \R^N$ and $ \phi \in \So_q$ (resp. $\in \So_q^\sharp$) such that
$ d_0( \phi_{n_{\ell_k}} ( \cdot + x_k ), \, \phi ) \lra 0$ as $ k \lra \infty$.
Then $ d_0( \psi_0^{n_{\ell_k}} ( \cdot + x_k ), \, \phi ) \leq
d_0( \phi_{n_{\ell_k}} ( \cdot + x_k ), \, \phi ) + \frac{2}{n_{\ell_k} } \lra 0$
and using Corollary \ref{C4.13} we get
$ Q( \psi_0^{n_{\ell_k}} ) = Q(\psi_0^{n_{\ell_k}} ( \cdot + x_k ) )
\lra Q( \phi) = q $ and
$ E( \psi_0^{n_{\ell_k}} ) = E(\psi_0^{n_{\ell_k}} ( \cdot + x_k ) ) \lra E( \phi) = E_{min}(q) $
(resp. $ E( \psi_0^{n_{\ell_k}} ) \lra E( \phi) = E_{min}^\sharp (q)$). Since any subsequence
of $(\psi_0^n)_{n \geq 1}$ contains a subsequence as above, the claim follows.
\medskip
By (P3) and Lemma \ref{L6.1} we have
$E(\psi_n(t_n) ) = E(\psi _0^n) \lra E_{min}(q)$ (resp. $E(\psi_n(t_n) ) \lra E_{min}^\sharp (q)$)
and $Q(\psi_n(t_n) ) = Q(\psi _0^n) \lra q.$
Moreover, if $N=2$ and $ \inf V < 0 $, we have already seen that
$ \ii _{\R^2} V(|\psi _n (t) |^2) \, dx$ cannot change sign, hence
$ \ii _{\R^2} V(|\psi _n (t_n) |^2) \, dx \geq 0$.
Using again either Theorem \ref{T4.7} or Theorem \ref{T5.11} we see that
there are a subsequence $ (n_k)_{k \geq 1}$, $ y _k \in \R^N$ and
$ \zeta \in \So _q $ (resp. $\in \So_q^\sharp$) such that
$ d_0 (\phi_{n_{k}} (t_{n_k}) , \, \zeta ( \cdot - y_k) ) \lra 0 $
as $ k \lra \infty$, and this contradicts the assumption
$d_0 ( \psi_n( t_n) , \So _q ) \geq \e _0 $ (resp.
$d_0 ( \psi_n( t_n) , \So _q^\sharp ) \geq \e _0 $) for all $n$.
The proof of Theorem \ref{T6.2} is thus complete.
\hfill
$\Box $
\section{Three families of traveling waves}
\label{3fam}
If the assumptions (A1), (A2) are satisfied and $V\geq 0$ on $[0, \infty)$, Theorem \ref{T4.7} and Proposition \ref{P4.12}
provide finite energy traveling waves to (\ref{1.1}) with any momentum $ q > q_0$; denote by $ \mathscr{M}$ the family of these traveling waves.
Theorem \ref{T5.5} and Proposition \ref{P5.7} provide traveling waves that minimize the action $E - cQ$ at constant kinetic energy;
let $ \mathscr{K}$ be the family of those solutions.
If $N=2$, we have also a family $ \mathscr{M}^{\sharp}$ of traveling waves given by Theorem \ref{T5.11}.
Finally, let $\mathscr{P} $ be the family of traveling waves found in \cite{M10};
the elements of $\mathscr{P} $ are minimizers of the action
$E - cQ$ under a Pohozaev constraint (see Theorem \ref{T4.19} below for a precise statement).
Our next goal is to establish relationships between these families of solutions.
We will prove that $ \mathscr{M} \subset \mathscr{K}$ and $ \mathscr{K} \subset \mathscr{P}$ if $ N \geq 3$, and that
$ \mathscr{M} \subset \mathscr{K}$ and $ \mathscr{M}^{\sharp} \subset \mathscr{K}$ if $ N =2$.
Besides, we find interesting characterizations of the minima of the associated functionals.
\medskip
Let
\beq
\label{functionals}
A(\psi ) \!=\! \int_{\R^N} \sum_{j = 2}^N \Big\vert \frac{ \p \psi}{\p x_j } \Big\vert ^2 dx,
\quad \;
E_ c ( \psi ) = E(\psi ) - c Q ( \psi),
\quad \;
P_c(\psi) = E_c( \psi) - \frac{2}{N-1} A(\psi).
\eeq
It follows from Proposition 4.1 p. 1091 in \cite{M7} that any finite-energy
traveling wave $\psi $ of speed $c$ of (\ref{1.1}) satisfies the Pohozaev
identity $P_c(\psi) = 0$. Denote
\beq
\label{Tc}
\Co _c= \{ \psi \in \Eo \; | \; \psi \mbox{ is not constant and } P_c(\psi ) = 0 \}
\quad \mbox{ and } \quad
T_c = \inf \{ E_c( \psi ) \; | \; \psi \in \Co _c\}.
\eeq
We summarize below the main results in \cite{M10}.
\begin{Theorem}\label{T4.19}(\cite{M10}) Assume that $ N \geq 3 $ and (A1) and (A2) hold. Then:
(i) For any $ c \in (0, v_s)$ the set $ \Co _c $ is not empty and $ T_c > 0 $.
\medskip
(ii) Let $(\psi_n)_{n \geq 1} \subset \Eo $ be a sequence such that
$$
P_c(\psi_n ) \lra 0 \qquad \mbox{ and } \qquad E_c (\psi _n ) \lra T_c \qquad \mbox{ as } n \lra \infty.
$$
If $ N=3$ we assume in addition that there is a positive constant $ d $ such that
$$
D(\psi_n) \lra d \quad \mbox{ as } n \lra \infty, \qquad \mbox{ where }
D(\phi ) = \ii_{\R^N} \Big\vert \frac{\p \phi }{\p x_1} \Big\vert ^2 + \frac 12 \left(\ph^2(|\phi|) - 1 \right)^2 \, dx .
$$
Then there exist a subsequence $(\psi_{n_k})_{k \geq 1}$, a sequence
$(x_k)_{k \geq 1} \subset \R^N$,
and $ \psi \in \Co_c $ such that $E_c ( \psi ) = T_c$, that is, $\psi $ is a minimizer of $E_c$ in $\Co_c $,
$\psi_{n _k} ( \cdot + x_k ) \lra \psi $ in $L_{loc}^p(\R^N) $ for $1 \leq p < \infty$ and a.e. on $ \R^N$ and
$$
\| \nabla \psi_{n _k} ( \cdot + x_k ) - \nabla \psi \|_{L^2(\R^N)} \lra 0 ,
\qquad
\| \, | \psi_{n _k} |( \cdot + x_k ) - | \psi | \, \|_{L^2(\R^N)} \lra 0
\qquad \mbox{ as } k \lra \infty.
$$
\medskip
(iii) Let $ \psi $ be a minimizer of $E_c$ in $\Co_c $.
Then $\psi $ satisfies (\ref{1.3}) if $N \geq 4$, respectively there exists $ \si > 0 $ such that $\psi_{1, \si }$ satisfies
(\ref{1.3}) if $N =3$. Moreover, $ \psi $ (respectively $\psi_{1, \si }$)
is a minimum action solution of (\ref{1.3}), that is it minimizes the action $E_c$ among all finite energy solutions.
Conversely, any minimum action solution to (\ref{1.3}) is a minimizer of $E_c$ in $ \Co _c$.
\end{Theorem}
Part (i) is Lemma 4.7 in \cite{M10},
part (ii) follows from Theorems 5.3 and 6.2 there and part (iii) follows from Propositions 5.6 and 6.5
in the same paper and from the fact that any solution $\psi$ satisfies the Pohozaev identity $P_c(\psi) = 0$.
\begin{remark}\label{R4.20} \rm
As already mentioned in \cite{M10} p. 119, all the conclusions of Theorem \ref{T4.19} above are valid if $ c = 0 $ provided that the set
$ \Co_0 = \{ \psi \in \Eo \; | \; \psi \mbox{ is not constant and } P_0 (\psi ) = 0 \}$
is not empty.
We will see later in section \ref{slow} that $ \Co _0 \neq \emptyset $ if and only if $V$ achieves negative values.
\end{remark}
\begin{Proposition}
\label{P4.18}
Assume that $N\geq 3$, (A1) and (A2) hold and $V\geq 0$ on $[0, \infty)$. Then:
(i) $ T_c \geq E_{min}(q) - cq $ for any $ q > 0 $ and $ c \in (0, v_s)$.
(ii) $ T_c \lra \infty $ as $ c \lra 0$.
(iii) Let $ \psi \in \Eo $ be a minimizer of $E$ under the constraint
$ Q = q_* > 0$. Assume that $ \psi $ satisfies an Euler-Lagrange equation
$ E'(\psi ) = c Q'( \psi)$ for some $c \in (0,v_s)$. Then $\psi $ is a
minimizer of $ E_c$ in $ \Co _c$.
\end{Proposition}
{\it Proof. } For $ \psi \in \Eo$ denote
\beq
\label{Bc}
B_c( \psi ) = \ii_{\R^N} \big\vert \frac{ \p \psi}{\p x_1 } \big\vert ^2 \, dx - c Q ( \psi ) + \ii_{\R^N} V( |\psi |^2 ) \, dx.
\eeq
Then $E_c( \psi) = A( \psi ) + B_c( \psi) = \frac{2}{N-1} A(\psi) + P_c( \psi)$ and $P_c( \psi) = \frac{N-3}{N-1} A( \psi) + B_c(\psi)$.
\medskip
i) Consider first the case $ N \geq 4$. Fix $ \psi \in \Co _c$.
It is clear
that $A(\psi) > 0 $, hence
$ B_c( \psi ) = P_c( \psi) - \frac{N-3}{N-1} A( \psi)
= - \frac{N-3}{N-1} A( \psi) <0$. Since $ V \geq 0 $ by hypothesis,
it follows that $ c Q( \psi ) = \ii_{\R^N} V( |\psi |^2 ) \, dx
+ \ii_{\R^N} \big\vert \frac{ \p \psi}{\p x_1 } \big\vert ^2 \, dx
- B_c(\psi) > 0$, hence $ Q( \psi ) > 0 $ because $c>0$. It is easy to see that the function
$ \si \longmapsto E_c( \psi_{1, \si}) = \si^{ N-3} A( \psi) + \si^{N-1} B_c( \psi)$
achieves its maximum at $ \si = 1$. Fix $ q > 0$. Since
$ Q( \psi_{1, \si} ) = \si ^{N-1} Q( \psi)$, there is $ \si _q > 0$ such that
$ Q(\psi_{1, \si_q }) = q$. We have obviously $ E( \psi _{1, \si _q }) \geq E_{min}(q) $
and
$$
E_{min}(q) - cq \leq E( \psi _{1, \si _q }) - c Q( \psi _{1, \si _q })
= E_c( \psi _{1, \si _q }) \leq E_c( \psi_{1,1}) = E_c(\psi).
$$
Taking the infimum as $ \psi \in \Co _c$, then the supremum as $ q >0$ in the
above inequality we get $\ds \sup_{q >0} ( E_{min}(q) - cq) \leq T_c.$
Now consider the case $N=3$. Let $ \psi \in \Co _c$. Then
$P_c(\psi) = B_c( \psi) = 0 $, $Q( \psi ) > 0$ and
$E_c ( \psi_{1, \si}) = A ( \psi) + \si ^2 B_c( \psi) = A( \psi)$ for any
$ \si >0$. Fix $ q >0$. Since $Q( \psi_{1, \si }) = \si ^2 Q( \psi)$, there
is $ \si _q >0$ such that $ Q(\psi_{1, \si_q }) = q$ and this implies
$ E( \psi _{1, \si _q }) \geq E_{min}(q) $. We have
$$
E_{min}(q) - cq \leq E( \psi _{1, \si _q }) - c Q( \psi _{1, \si _q })
= E_c( \psi _{1, \si _q }) = A( \psi) = E_c( \psi_{1,1}) = E_c(\psi).
$$
Since this is true for any $ \psi \in \Co _c$ and any $ q>0$, we conclude
again that $\ds \sup_{q >0} ( E_{min}(q) - cq) \leq T_c.$
\smallskip
(ii) Fix $ q > \frac{1}{v_s}$. We have $E_{min}(q) - cq > E_{min}(q) - 1 $ for
any $ c \in (0, \frac 1q)$. Using (i) we get
$$
T_c \geq E_{min}(q) - cq > E_{min} (q) - 1 \qquad \mbox{ for any } c \in (0, \frac 1q).
$$
Since $E_{min}(q) \lra \infty $ as $ q \lra \infty $ by Theorem \ref{T4.16} (b),
the conclusion follows.
\smallskip
(iii) We know that $\psi $ is a traveling wave of speed $c$ and by
Proposition 4.1 p. 1091 in \cite{M7} we have $P_c(\psi ) = 0$,
that is $ \psi \in \Co _c$. Using (i) we obtain
\beq
\label{4.68}
E_c(\psi) \geq T_c \geq \sup_{q >0} ( E_{min}(q) - cq) .
\eeq
On the other hand, we have
$$
E_c( \psi)= E( \psi) - c Q( \psi) = E_{min}(q_*) - c q_* .
$$
Therefore all inequalities in (\ref{4.68}) have to be equalities.
We infer that $\psi$ minimizes $ E_c$ in $ \Co _c$, $ T_c = E_{min}(q_*) - c q_*$
and the function $ q \longmapsto E_{min}(q) - cq $ achieves its maximum
at $ q_*$.
\hfill
$\Box$
\medskip
The next result shows that the minimizers of
$E_{min}$ or $E_{min}^\sharp$ are also minimizers for $I_{min}$ (after scaling).
\begin{Proposition}
\label{P5.9}
Let $ N \geq 2$. Assume that (A1), (A2) hold and either
(a) $V\geq 0$ on $[0, \infty )$ and $ q > q_0$, or
(b) $N=2$, $ \inf V < 0 $ and $ q \in (q_0^\sharp, q_{\infty}^\sharp).$
Consider $ \psi \in \Eo$ such that $Q(\psi ) = q$ and $E(\psi) = E_{min}(q) $ in case (a), respectively $E(\psi) = E_{min}^{\sharp }(q) $ in case (b),
and $ \psi $ satisfies (\ref{4.50}) for some $ c \in ( 0, v_s )$
(the existence of $\psi$ follows from
Theorem \ref{T4.7} in case (a) and from Theorem \ref{T5.11}) in case (b)).
Let $ k = \int_{\R^N} |\nabla \psi |^2 \, dx $.
\smallskip
Then $\frac{ k }{c^{N-2}} > k_0$
and $ \psi_{\frac 1c, \frac 1c}$ is a minimizer of $I$ in the set
$ \{ \phi \in \Eo \; | \; \int_{\R^2} | \nabla \phi |^2 \, dx = \frac{k}{c^{N-2}} \}$, that is
$ I( \psi_{\frac 1c, \frac 1c}) = I_{min} \left( \frac{k}{c^{N-2}} \right) . $
Equivalently, $\psi $ is a minimizer of $I_c$ (and of $E_c $) in the set
$ \{ \phi \in \Eo \; | \; \int_{\R^2} | \nabla \phi |^2 \, dx =k \}$).
Moreover, if $ N\geq 3$ the function $I_{min}$ is differentiable
at $\frac{k}{c^{N-2}}.$
\end{Proposition}
{\it Proof. } By Remark \ref{R5.8} (i) we have $I_{min}(\frac{k}{v_s^{N-2}}) < - \frac{k}{v_s^{N}}$ and
Proposition \ref{P4.12} (i) implies $ c \in (0, v_s)$, hence
$ \frac{k}{c^{N-2}} > \frac{k}{v_s^{N-2}} > k_0$.
Using Theorem \ref{T5.5} we infer that there is a minimizer
$\tilde{\psi} \in \Eo$ of the functional $I$
under the constraint $ \int_{\R^N} |\nabla \tilde{\psi} | ^2 \, dx = \frac{k}{c^{N-2}}$.
By Proposition \ref{P5.7} (ii) there is $ c_1 \in (0, v_s)$ such that $\tilde{\psi}_{c_1, c_1}$ satisfies
(\ref{4.50}) with $c_1$ instead of $c$.
Let $ \psi _1 = \tilde{\psi}_{c,c}$, so that $\int_{\R^N} |\nabla \psi _1 |^2 \, dx =
c^{N-2} \int_{\R^N} |\nabla \tilde{ \psi } |^2 \, dx = k= \int_{\R^N} |\nabla \psi |^2 \, dx$.
Denote $ q_1 = Q( \psi _1) = c^{N-1} Q( \tilde{\psi} )$.
It follows from Proposition 4.1 p. 1091-1092 in \cite{M8} that $\psi $ and $\tilde{\psi}_{c_1, c_1}$ satisfy
the following Pohozaev identities:
\beq
\label{5.19}
-(N-2) \int_{\R^N} |\nabla \psi |^2 \, dx + c (N-1) Q( \psi) = N \int_{\R^N} V(|\psi |^2 ) \, dx ,
\eeq
respectively
$
- (N-2) \ii _{\R^N} |\nabla \tilde{ \psi }_{c_1, c_1} |^2 \, dx
+ c _1 (N-1) Q( \tilde{\psi}_{c_1, c_1}) = N \ii _{\R^N} V(|\tilde{\psi }_{c_1, c_1}|^2 ) \, dx .
$
Since $ \tilde{\psi}_{c_1, c_1} = (\psi _1)_{\frac{c_1}{c}, \frac{c_1}{c}}$,
the latter equality is equivalent to
\beq
\label{5.20}
- (N-2) \frac{ c_1^{N-2}}{c^{N-2}} \int_{\R^N} |\nabla \psi _1 |^2 \, dx
+ (N-1) \frac{ c_1^N}{c^{N-1}} Q( \psi _1) = N \frac{ c_1^N}{c^N} \int_{\R^N} V(|\psi _1|^2 ) \, dx .
\eeq
Since $\ii _{\R^N} |\nabla \psi_{\frac 1c, \frac 1c} |^2 \, dx = \frac{k}{c^{N-2}}
= \ii _{\R^N} |\nabla \tilde{\psi }|^2 \, dx $
we have $I(\tilde{\psi}) \leq I(\psi_{\frac 1c, \frac 1c})$, that is
\beq
\label{5.21}
- \frac{1}{c^{N-1}} Q( \psi _1) + \frac{1}{c^N} \int_{\R^N} V(|\psi _1 |^2) \, dx
\leq - \frac{1}{c^{N-1}} Q( \psi ) + \frac{1}{c^N} \int_{\R^N} V(|\psi |^2) \, dx .
\eeq
Replacing $\ii _{\R^N} V(|\psi |^2) \, dx $ and $\ii _{\R^N} V(|\psi _1 |^2) \, dx $
from (\ref{5.19}) and (\ref{5.20}) into (\ref{5.21}) we get
\beq
\label{5.22}
cq + (N-2 ) k \leq c q_1 + (N-2) \frac{c^2}{c_1^2} k.
\eeq
Let $\si = \left( \frac{q}{q_1} \right)^{\frac{1}{N-1}}$.
Then $ Q ( (\psi_1 )_{\si, \si } ) = q$ and consequently
$E(\psi ) \leq E((\psi_1 )_{\si, \si } )$, that is
\beq
\label{5.23}
k + \int_{\R^N} V(|\psi |^2) \, dx \leq \si^{N-2} k + \si ^N \int_{\R^N} V(|\psi _1 |^2).
\eeq
We plug (\ref{5.19}) and (\ref{5.20}) into (\ref{5.23}) to obtain
\beq
\label{5.24}
c q_1 + (N-2) \frac{c^2}{c_1^2} k \leq N c q_1 - \frac{N-1}{\si ^N} cq
+ \left( \frac{N}{\si ^2} - \frac{2}{\si ^N} \right) k.
\eeq
Combining (\ref{5.24}) with (\ref{5.22}) we infer that
$cq + (N-2) k \leq N c q_1 - \frac{N-1}{\si ^N} c q
+ \left( \frac{N}{\si ^2} - \frac{2}{\si ^N} \right) k.$
Since $ q = \si^{N-1} q_1$, the last inequality can also be written as
\beq
\label{5.25}
\frac{ c q_1}{\si } ( \si ^N - N \si + N -1)
+ \frac{ k}{\si ^N} ( ( N-2) \si ^N - N \si^{N-2} + 2) \leq 0.
\eeq
If $ N=2$, (\ref{5.25}) is equivalent to $\frac{ cq_1}{\si} ( \si - 1 )^2 \leq 0$
and it implies that $ \si = 1$, thus $ q = q_1$.
If $ N \geq 3 $ we have $ \si ^N - N \si + N -1 = ( \si - 1)^2 \ds \sum_{j = 0}^{N-2} (N-1-j) \si ^j $ and
$( N-2) \si ^N - N \si^{N-2} + 2 = (\si - 1)^2 \left[ (N-2) \si^{N-2} + 2 \ds \sum_{j=0}^{N-3} (j+1) \si ^j \right].$
Inserting these identities into (\ref{5.25}) and using the fact that
$ \si, c, q_1, k$ are positive we infer that
$ \si = 1$, hence $ q = q_1$. Then using (\ref{5.22}) we obtain $ c_1 ^2 \leq c^2$.
On the other hand, from (\ref{5.24}) and the fact that $ q = q_1$, $\si = 1$
we obtain $ c^2 \leq c_1^2$.
Since $ c$ and $ c_1$ are positive, we have necessarily $ c = c_1$.
Then using (\ref{5.19}) and (\ref{5.20}) it is easy to see that $I(\psi _{\frac 1c, \frac 1c} ) = I(\tilde{\psi})$,
hence $ I(\psi _{\frac 1c, \frac 1c} ) = I_{min}(\frac{k}{c^{N-2}})$.
Moreover, we have proved that {\it any} minimizer $\tilde{\psi }$ of $I$ under
the constraint $\ii _{\R^N} |\nabla \tilde{\psi}|^2 \, dx = \frac{k}{c^{N-2}}$
satisfies (\ref{5.18}) with $ \vartheta = - \frac{1}{c^2}$.
It follows from Proposition \ref{P5.7} (iv) that
$d^+ I_{min}\left( \frac{k}{c^{N-2}} \right) = d^- I_{min}\left( \frac{k}{c^{N-2}} \right) $, hence
$I_{min} $ is differentiable at $\frac{k}{c^{N-2}}$ and
$I_{min}' (\frac{k}{c^{N-2}}) = - \frac{1}{c^2}. $
\hfill
$\Box$
\medskip
The next result establishes the relationship, if $N\geq 3$, between the traveling
waves obtained from minimizers of $I_{min}$ and the traveling wave solutions
given by Theorem \ref{T4.19}.
\begin{Proposition}
\label{P5.13} Assume that $N \geq 3$ and (A1), (A2) hold.
Let $ \Co _c$ and $ T_c$ be as in (\ref{Tc}). Then:
(i) $ T_c \geq k + c^N I_{min}\left( \frac{ k}{c^{N-2}} \right) $ for any
$ k >0 $ and any $ c \in ( 0, v_s)$.
\medskip
(ii) Let $ \psi $ be a minimizer of $I$ under the constraint
$ \ii_ {\R^N} |\nabla \psi |^2 \, dx = k$
and let $ c \in ( 0, v_s)$ be such that $\psi_{c,c}$ satisfies (\ref{4.50}).
Then $\psi _{c,c}$ minimizes $E_c = E - cQ$ in $ \Co _c$.
\end{Proposition}
{\it Proof. }
We keep the same notation as in the proof of Proposition \ref{P4.18}.
\medskip
(i) Consider the case $N \geq 4$. Fix $ \psi \in \Co _c$ and $ k >0$.
Since $A(\psi) > 0 $, the function
$ \si \longmapsto \ii _ {\R^N} |\nabla \psi _{1, \si } |^2 \, dx
= \si ^{N-3}A ( \psi ) + \si^{N-1} \ii _{\R^N} | \frac{ \p \psi}{\p x_1} |^2 \, dx $ is one-to-one from
$(0, \infty )$ to $(0, \infty )$,
so there is $ \si _k$ such that $\ii _ {\R^N} |\nabla \psi _{1, \si _k} |^2 \, dx = k, $
that is $\ii _ {\R^N} |\nabla \psi _{\frac 1c, \frac { \si _k}{c}} |^2 \, dx = \frac{ k}{c^{N-2}} $.
This implies $I\left( \psi _{\frac 1c, \frac { \si _k}{c}} \right) \geq I_{min} \left (\frac{ k}{c^{N-2}} \right)$.
We have $ 0 = P_c( \psi) = A(\psi) + B_c( \psi)$, thus $ A(\psi) > 0 > B_c(\psi ) $ and the function
$ \si \longmapsto E_c ( \psi _{1, \si} ) = \si^{N-3} A( \psi ) + \si ^{N-1} B_c( \psi ) $
achieves its maximum at $ \si = 1$. Then we have
$$
\begin{array}{l}
E_c( \psi) = E_c( \psi_{1, 1}) \geq E_c( \psi_{1, \si_k})
= \ii _ {\R^N} |\nabla \psi _{1, \si _k} |^2 \, dx + I_c( \psi_{1, \si _k})
\\
\\
= k + c^N I ( \psi _{\frac 1c, \frac { \si _k}{c}}) \geq k + c^N I_{min}\left( \frac{ k}{c^{N-2}} \right).
\end{array}
$$
The above inequality is valid for any $\psi \in \Co _c$ and $ k >0$, hence
$T_c \geq {\ds \sup_{k >0} } \left( k + c^N I_{min}\left( \frac{ k}{c^{N-2}} \right) \right).$
Next consider the case $N =3$.
Let $ \psi \in \Co _c $ and let $ k > 0$.
Then $P_c( \psi) = B_c( \psi) = 0 $ and for any $ \si > 0 $ we have $ E_c(\psi_{1, \si}) = E_c ( \psi ) = A( \psi )$ and
$\ii _ {\R^3} |\nabla \psi _{1, \si } |^2 \, dx = A( \psi ) +
\si ^2 \ii _{\R^3} | \frac{ \p \psi}{\p x_1} |^2 \, dx .$
If $ A( \psi ) \geq k$ we have, taking into account that $I_{min}$ is negative on $(0, \infty)$,
$$
E_c ( \psi) = A( \psi) \geq k > k + c^3 I_{min}\left( \frac{ k}{c} \right).
$$
If $A(\psi ) < k$, there is $ \si _k > 0 $ such that
$\ii _ {\R^3} |\nabla \psi _{1, \si _k} |^2 \, dx = k, $ which means
$\ii _ {\R^3} |\nabla \psi _{\frac 1c, \frac { \si _k}{c}} |^2 \, dx = \frac{ k}{c} $.
This implies
$I_c (\psi _{1, \si _k } ) = c^3 I\left( \psi _{\frac 1c, \frac { \si _k}{c}} \right)
\geq c^3 I_{min} \left (\frac{ k}{c} \right)$.
Thus we get
$$
E_c( \psi ) = E_c (\psi _{1, \si _k} ) =
\ii _ {\R^3} |\nabla \psi _{1, \si _k} |^2 \, dx + I_c (\psi _{1, \si _k} )
\geq k + c^3 I_{min} \left (\frac{ k}{c} \right) .
$$
Hence $ E_c ( \psi ) \geq k + c^3 I_{min} \left (\frac{ k}{c} \right) $ for any $ \psi \in \Co _c$ and
$ k>0$, and the conclusion follows.
\medskip
(ii) Since $ \psi_{c, c}$ satisfies (\ref{4.50}),
by Proposition 4.1 p. 1091 in \cite{M7} we have $ \psi_{c,c} \in \Co _c$. Then
\beq
\label{5.50}
E_c( \psi _{c,c}) \geq T_c \geq \sup_{\kappa >0}
\left( \kappa + c^N I_{min}\left( \frac{ \kappa}{c^{N-2}} \right) \right).
\eeq
On the other hand,
$$
E_c( \psi _{c,c}) = c^{N-2} \ii_{\R^N} |\nabla \psi |^2 \, dx + c^N I( \psi )
= c^{N-2} k + c^N I_{min}(k) \leq \sup_{\kappa > 0 }
\left( \kappa + c^N I_{min}\left( \frac{ \kappa}{c^{N-2}} \right) \right).
$$
Therefore all inequalities in (\ref{5.50}) are equalities,
$ \psi_{c,c}$ minimizes $ E_c $ in $ \Co _c$,
$T_c = c^{N-2} k + c^N I_{min}(k) $ and the function
$ \kappa \longmapsto \kappa + c^N I_{min}\left( \frac{ \kappa}{c^{N-2}} \right) $
achieves its maximum at $ \kappa = c^{N-2} k$.
\hfill
$\Box$
\section{Small speed traveling waves}
\label{slow}
Theorem \ref{T4.16} implies that $ \frac{E_{min}(q)}{q} \lra 0$ as $ q \lra \infty$.
Since $E_{min}$ is concave and positive, necessarily
$ d^+ E_{min}(q) \lra 0 $ and $ d^- E_{min}(q) \lra 0$ as $ q \lra \infty$ and we infer that
the traveling waves provided by Theorem \ref{T4.7} and Proposition \ref{P4.12} have speeds close to zero as $ q \ra \infty$.
Similarly, using Lemma \ref{L5.3} (i) and (iii) we find that $I_{min}$ is finite for all $ k > 0 $ and
$ d^+ I_{min}(k) \lra - \infty $, $ d^- I_{min}(k) \lra -\infty$ as $ k \lra \infty$
if either $ N \geq 3$ or ($N=2$ and $V\geq 0$).
Hence the traveling waves given by Theorem \ref{T5.5} and Proposition \ref{P5.7}
have speeds that tend to zero as $ k \lra \infty$.
This section is a first step in understanding the behavior of traveling waves in the limit $ c \lra 0 $.
As one would expect, this
is related to the existence of finite energy solutions to the stationary version of (\ref{1.1}), namely to the equation
\beq
\label{5.650}
\Delta \psi + F(|\psi |^2) \psi = 0 \qquad \mbox{ in } \R^N .
\eeq
Clearly, the solutions of (\ref{5.650}) are precisely the critical points of $E$.
We call {\it ground state } of (\ref{5.650}) a solution that minimizes the energy $E$ among all nontrivial solutions.
Assume that $ N \geq 2 $ and the assumptions (A1) and (A2) are satisfied.
Then (\ref{5.650}) admits nontrivial solutions $ \psi \in \Eo $ if and only if the nonlinear potential
$V$ achieves negative values.
The existence follows from Theorem 2.1 p. 100 and Theorem 2.2 p. 103 in \cite{brezis-lieb} if $ N \geq 3$,
respectively from Theorem 3.1 p. 106 in \cite{brezis-lieb} if $ N =2$.
Moreover, the solutions found in \cite{brezis-lieb} are ground states.
On the other hand, any solution $ \psi \in \Eo $ of (\ref{5.650}) has the regularity provided by Proposition \ref{P4.12} (ii)
and this is enough to prove that $ \psi $ satisfies the Pohozaev identity
\beq
\label{5.66}
(N-2) \ii_{\R^N} |\nabla \psi |^2 \, dx + N \ii_{\R^N} V(|\psi |^2) \, dx = 0
\eeq
(see Lemma 2.4 p. 104 in \cite{brezis-lieb}). In particular, (\ref{5.66}) implies that (\ref{5.650}) cannot have finite energy solutions if $ V \geq 0$.
We will prove in the sequel that if $N\geq 3$ and $V$ achieves negative values, the traveling waves constructed in this paper
tend to the ground states of (\ref{5.650}) as their speed goes to zero.
If $ N \geq 3$, we have shown in section \ref{3fam} that all traveling waves found here
also belong to the family of traveling waves given by Theorem \ref{T4.19}, hence it suffices to establish the result for
the solutions provided by Theorem \ref{T4.19}.
If $ N=2$ and $V $ takes negative values, we were not able to prove that $d^{\pm} I_{min}(k) \lra -\infty$ as $ k \lra k_{\infty}$.
Numerical computations in \cite{CS} indicate that this is indeed the case, at least for some model nonlinearities (including the cubic-quintic one).
If $\ds \lim_{k \uparrow k_{\infty}} d^{\pm} I_{min}(k) = -\infty $, the speeds of the traveling waves given by Theorem \ref{T5.5} and Proposition \ref{P5.7}
tend to zero as $ k \lra k_{\infty}$ and we are able to prove a result similar to Proposition \ref{P5.14} below
(although the proof is very different because minimization under Pohozaev constraints is no longer possible).
If $V\geq 0$ on $[0, \infty)$, equation (\ref{5.650}) does not have finite energy solutions.
Then the traveling waves of (\ref{1.1}) have large energy (see Proposition \ref{P4.18} (ii)) and
are expected to develop vortex structures in the limit $ c \lra 0 $.
This is the case for the traveling waves to the Gross-Pitaevskii equation: in dimension two the solutions found in
\cite{BS} have two vortices of opposite sign located at a distance of order $\frac 2c$,
and in dimension three the traveling waves found in \cite{BOS} and \cite{chiron} have vortex rings.
If $ V\geq 0$, the behavior of traveling waves in the limit $ c \lra 0 $
still needs to be investigated.
\begin{Proposition}
\label{P5.14}
Let $ N \geq 3$. Suppose that (A1) and (A2) are satisfied and there exists $ s_0 \geq 0 $ such that $V( s_0 ) < 0$.
Let $ (c_n)_{n \geq 1} $ be any sequence of numbers in $(0, v_s)$ such that $ c_n \lra 0 $.
For each $ n $, let $ \psi _n \in \Eo$ be any minimizer of $ E_{c_n} = E - c_n Q$ in $ \Co _{c_n}$ such that
$\psi _n$ is a traveling wave of (\ref{1.1}) with speed $ c_n $. Then:
(i) There are a subsequence $(c_{n_k})_{k \geq 1}$, a sequence $ (x_k ) _{k \geq 1} \subset \R^N$
and a ground state $ \psi $ of
(\ref{5.650}) such that $\psi_{n _k} ( \cdot + x_k ) \lra \psi $ in $ L_{loc}^p( \R^N) $ for $ 1 \leq p < \infty $ and a.e. on $ \R^N$ and
$$
\| \nabla \psi_{n _k} ( \cdot + x_k ) - \nabla \psi \|_{L^2(\R^N)} \lra 0 ,
\qquad
\| \, | \psi_{n _k} |( \cdot + x_k ) - | \psi | \, \|_{L^2(\R^N)} \lra 0
\qquad \mbox{ as } k \lra \infty.
$$
(ii) There is a sequence $(a_k)_{k \geq 1}$ of complex numbers of modulus $1$ such that $a_k \lra 1$ as $ k \lra \infty$ and
$$
\| a_k \psi_{n_k} ( \cdot + x_k) - \psi \|_{W^{2, p }(\R^N)} \lra 0 \qquad \mbox{ as } k \lra \infty \quad \mbox{ for any } p \in [2^*, \infty).
$$
In particular,
$\| a_k \psi_{n_k} ( \cdot + x_k) - \psi \|_{C^{1, \al }(\R^N)} \lra 0 $ as $ k \lra \infty $ for any $ \al \in [0,1).$
\end{Proposition}
If $F $ is $C^k$ it can be proved that the convergence in (ii) holds in $W^{k+2, p }(\R^N)$, $ 2^* \leq p < \infty.$
\medskip
{\it Proof. }
(i) Let $ \psi _0 $ be any ground state of (\ref{5.650}). By (\ref{5.66}) we have
$ \ii_{\R^N} V(|\psi _0 |^2) \, dx = - \frac{N-2}{N} \ii_{\R^N} |\nabla \psi |^2 \, dx <0$.
It is shown in \cite{brezis-lieb} that $ \psi _0$ is a minimizer of the functional $ J(\phi ) = \ii_{\R^N} |\nabla \phi|^2 \, dx $
subject to the constraint $ \ii_{\R^N} V(|\phi |^2) \, dx = \ii_{\R^N} V(|\psi _0 |^2) \, dx$;
conversely, any minimizer of this problem is a ground state to of (\ref{5.650}), and Proposition \ref{P4.12} (ii)
implies that any minimizer is $C^1$ on $ \R^N$.
It follows from Theorem 2 p. 314 in \cite{M7} that any ground state of (\ref{5.650}) is, after translation, radially symmetric.
In particular, the radial symmetry implies that $Q(\psi _0 ) = 0$.
Let $A$, $E_c = E - cQ$, $P_c$ be as in (\ref{functionals}) and $ \Co _c $ and $ T_c$ as in ({\ref{Tc}).
Since $ \psi _0$ is a solution of (\ref{5.650}), it satisfies the Pohozaev identity $P_0 (\psi _0) = 0 $ and then we get
$ P_c( \psi _0) = P_0 (\psi _0) - c Q( \psi _0) = 0$ for any $c$, that is $ \psi _0 \in \Co _c $ for any $c$.
Therefore
\beq
\label{5.67}
A( \psi _n) = \frac{N-1}{2} \left( E_{c_n}(\psi _n) - P_{c_n}(\psi_n) \right)
= \frac{N-1}{2} E_{c_n}(\psi _n) = \frac{N-1}{2} T_{c_n} \leq \frac{N-1}{2} E_{c_n}(\psi _0)
= A( \psi _0).
\eeq
On the other hand, by Proposition 10 (ii) in \cite{CM} the function $ c \longmapsto T_c$ is decreasing on $(0, v_s)$.
Fix $ c_* \in (0, v_s)$. For all sufficiently large $n$ we have $ c_n < c^*$, hence
\beq
\label{5.68}
A(\psi _n) = \frac{N-1}{2} T_{c_n} \geq \frac{N-1}{2} T_{c_*} > 0.
\eeq
Consider first the case $N\geq 4$. We claim that $E_{GL}(\psi_n)$ is bounded.
To see this we argue by contradiction and we assume that there is a subsequence, still denoted $(\psi_n)_{n\geq1}$, such that
$E_{GL}(\psi_n)\lra \infty$.
By (\ref{5.67}) we have
\beq
\label{5.69}
D(\psi_n) = \ii_{\R^N} \Big| \frac{\p \psi_n}{\p x_1} \Big| ^2 + \frac 12 \left( \ph ^2 (|\psi_n|)-1 \right)^2 \, dx \lra \infty \qquad \mbox{ as } n \lra \infty.
\eeq
Using Lemma \ref{L4.2} (ii) we see that there are two positive constants $ k_0, \ell _0$ such that
for any $ \psi \in \Eo$ satisfying $ E_{GL}(\psi) = k_0 $ and for any $c \in (0, c_*) $
(where $c_*$ is as in (\ref{5.68})) there holds
\beq
\label{5.70}
E_c(\psi) \geq E(\psi) - c |Q(\psi)| \geq \ell_0.
\eeq
It is easy to see that for each $n$ there is $ \si _n >0$ such that
\beq
\label{5.71}
E_{GL}((\psi_n)_{\si_n, \si_n}) =
\si_n ^{N-3} A( \psi_n) + \si_n ^{N-1} D(\psi_n)
= k_0.
\eeq
In particular, $(\psi_n)_{\si_n, \si_n} $ satisfies (\ref{5.70}).
We recall that the functional $B_c$ was defined in (\ref{Bc}). We have
$ B_{c_n}(\psi_n) = P_{c_n}( \psi_n) - \frac{N-3}{N-1}A(\psi_n)$.
Then the fact that $P_{c_n}(\psi _n) = 0 $ and (\ref{5.67}) imply that $ B_{c_n}(\psi_n)$ is bounded.
From (\ref{5.69}) and (\ref{5.71}) it follows that $ \si _n \lra 0 $ as $ n \lra \infty$,
hence
$$
E_{c_n} ( (\psi _n)_{\si_n, \si_n}) = \si_n ^{N-3} A( \psi_n) + \si_n ^{N-1} B_{c_n} (\psi_n) \lra 0 \qquad \mbox{ as } n \lra \infty.
$$
This contradicts the fact that $ E_{c_n} ( (\psi _n)_{\si_n, \si_n}) \geq \ell_0$ for all $n$ and the claim is proven.
Using Corollary \ref{C4.17} we infer that $Q(\psi_n)$ is bounded.
Since $ c_n \lra 0 $, using (\ref{5.67}) we find
\beq
\label{5.72}
P_0(\psi_n) = P_{c_n}(\psi_n) + c_n Q(\psi_n) \lra 0 \qquad \mbox{ and }
\eeq
\beq
\label{5.73}
\begin{array}{l}
E(\psi_n) = E_{c_n}(\psi_n) + c_n Q(\psi_n)= \frac{2}{N-1}A(\psi_n) + P_{c_n}(\psi_n) + c_n Q(\psi_n)
\\
\\
\leq \frac{2}{N-1}A(\psi_0) + c_n Q(\psi_n)= E( \psi _0 ) + c_n Q(\psi_n) \lra E(\psi _0)
\qquad \mbox{ as } n \lra \infty.
\end{array}
\eeq
Then the conclusion follows from Theorem \ref{T4.19} (with $c=0$) and Remark \ref{R4.20}.
\medskip
Next consider the case $ N = 3$. For all $n$ and all $ \si > 0$ we have
$$
P_{c_n} ((\psi_n)_{1, \si}) = \si^2 P_{c_n} ( \psi_n ) = 0
\; \; \mbox{ and } \; \;
E_{c_n} ((\psi_n)_{1, \si}) = A ((\psi_n)_{1, \si}) + P_{c_n} ((\psi_n)_{1, \si}) = A(\psi_n) = T_{c_n},
$$
hence $(\psi_n)_{1, \si} $ is also a minimizer of $ E_{c_n}$ in $ \Co_{c_n}$.
For each $n$ there is $ \si _n > 0 $ such that $D((\psi_n)_{1, \si_n}) = \si _n ^2 D(\psi_n) = 1$.
We denote $\tilde{\psi}_n= (\psi_n)_{1, \si_n}$.
Then $\tilde{\psi}_n $ is a minimizer of $E_{c_n}$ in $ \Co_{c_n}$,
$E_{GL} (\tilde{\psi}_n) = A(\tilde{\psi}_n) +1 = A(\psi_n) + 1$ is bounded by (\ref{5.67})
and then Corollary \ref{C4.17} implies that $Q(\tilde{\psi}_n) $ is bounded.
As in the case $N\geq 4$ we find that $(\tilde{\psi}_n)_{n\geq 1}$ satisfies (\ref{5.72}) and (\ref{5.73}).
From Theorem \ref{T4.19} and Remark \ref{R4.20} it follows that there exist a subsequence
$(\tilde{\psi}_{n_k})_{k\geq 1}$, a sequence $ (x_k)_{k \geq 1} \subset \R^3$ and a minimizer $\tilde{\psi}$ of $ E$ on $ \Co _0$ that
satisfy the conclusion of Theorem \ref{T4.19} (ii).
Moreover, there is $ \si >0$ such that $\tilde{\psi}$ satisfies the equation
\beq
\label{5.74}
\frac{\p^2 \tilde{\psi}}{\p x_1 ^2 }
+ \si ^2 \sum_{j=2}^3 \frac{\p ^2\tilde{\psi}}{\p x_j ^2 } + F(|\tilde{\psi}|^2) \tilde{\psi} = 0
\qquad \mbox{ in } \Do '(\R^3).
\eeq
Let $ \psi_k ^* = \tilde{\psi}_{n_k} ( \cdot + x_k). $
Since $ \psi _n$ solves (\ref{1.3}) with $ c_n$ instead of $c$, it is obvious that $ \psi _k^*$ satisfies
\beq
\label{5.75}
i c_{n_k} \frac{\p \psi _k^*}{\p x_1 } + \frac{\p^2 \psi _k^* }{\p x_1 ^2 }
+ \si _{n_k} ^2 \sum_{j=2}^3 \frac{\p ^2\psi _k^*}{\p x_j ^2 } + F(|\psi _k^*|^2) \psi_k^* = 0
\qquad \mbox{ in } \Do '(\R^3).
\eeq
It is easy to see that $ \psi_k^* \lra \tilde{\psi} $ and $ F(|\psi _k^*|^2) \psi_k^* \lra F(|\tilde{\psi}|^2) \tilde{\psi}$
in $ \Do'(\R^3)$.
We show that $ (\si_{n_k})_{k \geq 1}$ is bounded.
We argue by contradiction and we assume that it contains a subsequence, still denoted the same, that tends to $\infty$.
Multiplying (\ref{5.75}) by $\frac{1}{ \si _{n_k} ^2}$ and passing to the limit as $ k \lra \infty$ we get
\beq
\label{5.76}
\frac{\p ^2\tilde{\psi}}{\p x_2 ^2 } + \frac{\p ^2\tilde{\psi}}{\p x_3 ^2 } = 0 \qquad \mbox{ in } \Do '(\R^3).
\eeq
Since $ \frac{\p ^2 \tilde{\psi}}{\p x_j \p x_k} \in L_{loc}^p(\R^3)$ for any $ p \in [1, \infty)$, we infer that the above
equality holds in $ L_{loc}^p(\R^3)$ for any $ p \in [1, \infty)$.
By the Sobolev embedding (see Lemma 7 and Remark 4.2 p. 774-775 in \cite{PG}) we know that there is $ \al \in \C$ such that $ |\al | = 1$ and $ \tilde{\psi} - \al \in L^6(\R^3)$.
Let $ \chi \in C_c^{\infty} ( \R^3) $ be a cut-off function such that $ \chi = 1 $ on $B(0,1)$ and $supp(\chi) \subset B(0,2)$.
Taking the scalar product (in $\C$) of (\ref{5.76}) by $ \chi(\frac xn) (\psi - \al)$ and letting $ n\lra \infty$ we find
$\ii_{\R^3} \big| \frac{\p \tilde{\psi}}{\p x_2} \big| ^2 + \big| \frac{\p \tilde{\psi}}{\p x_3} \big| ^2 \, dx = 0.$
Since $ \psi \in C^{1, \al}(\R^3)$, we conclude that $ \frac{\p \tilde{\psi}}{\p x_2} = \frac{\p \tilde{\psi}}{\p x_3} = 0$,
hence $ \tilde{\psi }$ depends only on $x_1$. Together with the fact that $ \frac{\p \tilde{\psi}}{\p x_1} \in L^2( \R^3)$
this implies that $\tilde{\psi}$ is constant, a contradiction. Thus $ (\si_{n_k})_{k \geq 1}$ is bounded.
If there is a subsequence $( \si_{n_{k_j}})_{j \geq 1}$ such that $ \si_{n_{k_j}} \lra \si _*$ as $ j \lra \infty$,
passing to the limit in (\ref{5.75}) we discover
$$
\frac{\p^2 \tilde{\psi}}{\p x_1 ^2 }
+ \si _* ^2 \sum_{j=2}^3 \frac{\p ^2\tilde{\psi}}{\p x_j ^2 } + F(|\tilde{\psi}|^2) \tilde{\psi} = 0
\qquad \mbox{ in } \Do '(\R^3).
$$
If $ \si _* \neq \si$, comparing the above equation to (\ref{5.74}) we find
$\frac{\p ^2\tilde{\psi}}{\p x_2 ^2 } + \frac{\p ^2\tilde{\psi}}{\p x_3 ^2 } = 0 $ in $ \Do'(\R^3)$
and arguing as previously we infer that $\tilde{\psi}$ is constant, a contradiction.
We conclude that necessarily $\si_{n_k}\lra \si $ as $ k \lra \infty$.
Denoting $ \psi = \tilde{\psi}_{1, \frac{1}{\si}}$, we easily see that $ \psi $ minimizes $ E$ in $ \Co _0$ and is a ground state of
(\ref{5.650}). Then $(\psi_{n_k})_{k \geq 1}$ and $ \psi $ satisfy the conclusion of Proposition \ref{P5.14} (i).
\medskip
(ii) By the Sobolev embedding there are $ \al, \, \al _k \in \C$ of modulus $1$ and $ C_S >0 $ such that
$$
\| \psi_{n_k} - \al _k \|_{L^{2^*}(\R^N)} \leq C_S \| \nabla \psi_{n_k} \|_{L^{2}(\R^N)} \qquad \mbox{ and } \qquad
\| \psi - \al \|_{L^{2^*}(\R^N)} \leq C_S \| \nabla \psi \|_{L^{2}(\R^N)} .
$$
We may assume that $ \al =1$ for otherwise we multiply $ \psi_{n_k}$ and $ \psi $ by $ \al ^{-1}$.
(In fact we have $\psi = \al \psi_0$, where $ \psi _0$ is real-valued, but we do not need this observation.)
Let $ R > 0 $ be arbitrary, but fixed. By (i) there exists $ k(R) \in \N$ such that for all $ k \geq k(R)$ we have
$\| \psi_{n_k} ( \cdot + x_k) - \psi \|_{L^{2^*} (B(0,R))} < 1$. Then we find
$$
\| \al _k - 1\| _{L^{2^*} (B(0,R))} \leq \| \psi_{n_k} ( \cdot + x_k) - \al _k \|_{L^{2^*} (\R^N )}
+ \| \psi_{n_k} ( \cdot + x_k) - \psi \|_{L^{2^*} (B(0,R))}
+ \| \psi - 1 \| _{L^{2^*} (\R^N )} \leq C
$$
for any $ k \geq k(R)$, where $C$ does not depend on $k$. This implies that $ \al _k \lra 1$.
Let $ \psi _k ^* = \al _k^{-1} \psi_{n_k} ( \cdot + x_k) $, so that
$ \psi _k ^* - \psi \in L^{2^*} (\R^N)$. Using (i) and the Sobolev embedding we get
\beq
\label{5.77}
\| \psi_k ^* - \psi \|_{L^{2^*} (\R^N )} \leq C_S \| \nabla \psi _k ^* - \nabla \psi \|_{L^{2} (\R^N )} \lra 0 \qquad \mbox{ as } k \lra \infty.
\eeq
By (i), $ \nabla \psi _{k}^* $ is bounded in $ L^2( \R^N)$ and $ \psi _k^*$ is a traveling wave to (\ref{1.1}) of speed $ c_{n_k}$.
It follows from Step 1 in the proof of Lemma \ref{L7.1} below that there is $L>0$, independent of $k$, such that
$$
\| \nabla \psi _k ^* \| _{L^{\infty}(\R^N)} \leq L \qquad \mbox{ and } \qquad \| \nabla \psi \| _{L^{\infty}(\R^N)} \leq L.
$$
By interpolation we get
\beq
\label{5.78}
\| \nabla \psi_k ^* - \nabla \psi \|_{L^{p} (\R^N )} \lra 0 \qquad \mbox{ as } k \lra \infty \quad \mbox{ for any } p \in [2, \infty).
\eeq
Using (\ref{5.77}), (\ref{5.78}) and the Sobolev embedding we infer that
\beq
\label{5.79}
\| \psi_k ^* - \psi \|_{L^{p} (\R^N )} \lra 0 \qquad \mbox{ as } k \lra \infty \quad \mbox{ for any } p \in [2^*, \infty].
\eeq
We claim that $ \|F(|\psi_k ^*|^2) \psi _k ^* - F(|\psi |^2) \psi \|_{L^{p} (\R^N )} \lra 0 $
as $ k \lra \infty $ for any $ p \in [2^*, \infty).$
To see this fix $ \de > 0 $ such that $ F$ is $ C^1 $ on $[1 - 2 \de, 1 + 2 \de]$ (such $ \de $ exists by (A1)).
Since $ \psi - 1 \in L^{2^*}( \R^N)$ and $ \| \nabla \psi \| _{L^{\infty}(\R^N)} \leq L$
we have $ \psi \lra 1 $ as $ |x | \lra \infty$, hence there exists $ R(\de) > 0 $ verifying $ \big| \, |\psi | - 1 \big| < \de$
on $ \R^N \setminus B(0, R(\de))$.
By (\ref{5.79}) there is $ k_{\de} \in \N$ such that $\| \psi_k ^* - \psi \|_{L^{\infty} (\R^N )} < \de $ for $ k \geq k_{\de}$.
The mapping $ z \longmapsto F(|z|^2) z $ is Lipschitz on $\{ z \in \C \; | \; 1 - 2 \de \leq |z| \leq 1 + 2 \de \}$,
hence there is $C>0$ such that
\beq
\label{5.80}
\big| F(|\psi_k ^*|^2) \psi _k ^* - F( |\psi |^2) \psi \big| \leq C | \psi_k^* - \psi | \qquad
\mbox{ on } \R^N \setminus B(0, R(\de)) \mbox{ for all } k \geq k_{\de}.
\eeq
Since $ F(|\psi_k ^*|^2) \psi _k ^* - F(|\psi |^2) \psi $ is bounded and tends a.e. to zero,
using Lebesgue's dominated convergence theorem we get
\beq
\label{5.81}
\|F(|\psi_k ^*|^2) \psi _k ^* - F(|\psi |^2) \psi \|_{L^{p} (B(0, \de))} \lra 0
\qquad \mbox{ for any } p \in [1, \infty).
\eeq
Now the claim follows from (\ref{5.79}) - (\ref{5.81}).
Using the equations satisfied by $ \psi_k^*$ and $\psi$ we get
$$
\Delta ( \psi_k^* - \psi ) = - i c_{n_k} \frac{ \p \psi_k^*}{\p x_1} - \left( F(|\psi_k ^*|^2) \psi _k ^* - F(|\psi |^2) \psi \right).
$$
From the above we infer that $\| \Delta ( \psi_k^* - \psi) \|_{L^p(\R^N)} \lra 0 $ for any $ p \in [2^*, \infty)$, then using
(\ref{5.79}) and the inequality $\| f \|_{W^{2,p}( \R^N ) } \leq C_p \left( \| f \|_{L^p( \R^N ) } + \| \Delta f \|_{L^p( \R^N ) } \right)$
we get the desired conclusion.
\hfill
$\Box$
\section{Small energy traveling waves}
\label{sectionsmallenergy}
The aim of this section is to prove Proposition \ref{smallE}.
The next lemma shows that
the modulus of traveling waves
of small energy is close to $1$.
\begin{Lemma}
\label{L7.1} Let $ N \geq 2$.
Assume that (A1) and ((A2) or (A3)) hold.
(i) For any $ \e >0 $ there exists $ M(\e ) > 0$ such that for any $c \in [0, v _s]$ and for
any solution $ \psi \in \Eo$ of (\ref{1.3}) with $\| \nabla \psi \|_{L^2( \R^N) } < M(\e) $ we have
\beq
\label{7.1}
\big| \, | \psi (x) | - 1 \big| < \e \qquad \mbox{ for all } x \in \R^N.
\eeq
(ii) Let $ p > N p_0$, where $ p_0 $ is as in (A2) (respectively $ p \geq 1$ if (A3) is satisfied).
For any $ \e >0 $ there exists $ \ell _p(\e ) > 0$ such that for any
$c \in [0, v _s]$ and for
any solution $ \psi \in \Eo$ of (\ref{1.3}) with $\| \, | \psi | - 1 \|_{L^p( \R^N) } < \ell_p(\e) $,
(\ref{7.1}) holds.
\end{Lemma}
{\it Proof.}
Assume first that (A1) and (A2) are satisfied.
We will prove that there is $L>0$ such that any solution $ \psi \in \Eo $ of (\ref{1.3}) such that
$\| \nabla \psi \|_{L^2( \R^N) }$ is sufficiently small
(respectively $\| | \psi | - r_0 \|_{L^2( \R^N) } $ is sufficiently small)
satisfies
\beq
\label{7.2}
\| \nabla \psi \|_{L^{\infty}(\R^N)} \leq L.
\eeq
{\it Step 1. } We prove (\ref{7.2}) if $ N \geq 3$ and $\| \nabla \psi \|_{L^2( \R^N) } \leq M$,
where $M>0$ is fixed.
Using the Sobolev embedding, for any $ \phi \in \Eo $ such that $\| \nabla \phi \|_{L^2( \R^N)} \leq M$
we get
$$
\| ( |\phi | - 2 )_+ \| _{L^{2^*}( \R^N) } \leq C_S \| \nabla | \phi | \, \|_{L^2( \R^N)}
\leq C_S \| \nabla \phi \|_{L^2( \R^N)}.
$$
Since $ |\phi| \leq 2 + ( |\phi | - 2 )_+, $
we see that $ \phi $ is bounded in $ L^{2^*} + L^{\infty}(\R^N)$.
It follows that for any $ R > 0 $ there exists $C_{R, M } > 0 $ such that for any
$ \phi \in \Eo $ as above we have
$$
\| \phi \|_{H^1( B(x, R )) } \leq C_{R, M} \qquad \mbox{ for all } x \in \R^N.
$$
If $ c \in [0, v_s]$, $ \psi \in \Eo $ is a solution of (\ref{1.3}) and $\| \nabla \psi \|_{L^2( \R^N) } \leq M$,
using (\ref{3.11}) and a standard bootstrap argument (which works thanks to (A2))
we infer that for any $ p \in [2, \infty)$
there is $ \tilde{C}_p > 0 $ (depending only on $F$, $N$, $p$ and $M$) such that
$$
\| \psi \|_{W^{2,p}( B(x, 1 ))} \leq \tilde{C}_p \qquad \mbox{ for all } x \in \R^N.
$$
Then the Sobolev embedding implies that $ \psi \in C^{1, \al }(\R^N) $ for all $ \al \in [0, 1)$
and there is $ L>0$ such that (\ref{7.2}) holds.
\medskip
{\it Step 2.} Proof of (i) in the case $N \geq 3$.
Fix $ \e > 0$.
There is $L>0$ such that any solution $ \psi \in \Eo $ of (\ref{1.3})
with $\| \nabla \psi \| _{L^2( \R^N)} \leq 1$ satisfies (\ref{7.2}).
If $ \psi $ is such a solution and $ \big|\, | \psi (x_0) | - 1 \big| \geq \e $ for some $ x _0 \in \R^N$,
from (\ref{7.2}) we infer that
$ \big|\, | \psi (x) | - 1 \big| \geq \frac{\e}{2} $ for any $ x \in B(x_0, \frac{ \e}{2L})$.
Then using the Sobolev embedding we get
$$
C_S \| \nabla \psi \| _{L^2( \R^N)} \geq \| \, | \psi | - 1 \|_{L^{2^*}(\R^N)}
\geq \| \, | \psi | - 1 \|_{L^{2^*}(B(x_0, \frac{\e}{2L}))}
\geq \frac{ \e}{2} \left( \left(\frac{ \e}{2 L} \right) ^N \Lo ^N (B(0,1)) \right)^{\frac{1}{2^*}}.
$$
We conclude that if
$\| \nabla \psi \| _{L^2( \R^N)} < \min \left(\! 1,
\frac{ \e}{2 C_S} \left( \left(\frac{ \e}{2 L} \right) ^N \Lo ^N (B(0,1)) \right)^{\frac{1}{2^*}} \right)$,
then
$ \psi $ satisfies~(\ref{7.1}).
\medskip
{\it Step 3.} Proof of (\ref{7.2}) if $N=2$ and $\| \nabla \psi \| _{L^2( \R^2)} $ is sufficiently small.
By (\ref{4.2}) there is $ M_1 >0$ such that for any $ \phi \in \Eo $
with $ \| \nabla \phi \|_{L^2( \R^2)} \leq M_1$ we have
\beq
\label{7.3}
\frac 14 \int_{\R^2} \left( \ph^2( |\phi|) - 1 \right) ^2 \, dx
\leq \int_{\R^2} V(| \phi |^2) ) \, dx
\leq \frac 34 \int_{\R^2} \left( \ph^2( |\phi|) - 1 \right) ^2 \, dx .
\eeq
Let $ \psi \in \Eo $ be a solution of (\ref{1.3}).
By Proposition \ref{P4.12} (ii) we have $ \psi \in W_{loc}^{2, p} ( \R^2) $ and this regularity is enough to
prove that $ \psi $ satisfies the Pohozaev identity
\beq
\label{7.4}
- \int_{\R^2} \bigg\vert \frac{ \p \psi}{\p x_1} \bigg\vert ^2 \, dx
+ \int_{\R^2} \bigg\vert \frac{ \p \psi}{\p x_2} \bigg\vert ^2 \, dx
+ \int_{\R^2} V(| \psi |^2) ) \, dx = 0
\eeq
(see Proposition 4.1 p. 1091 in \cite{M8}).
In particular, if $ \| \nabla \psi \|_{L^2( \R^2)} \leq M_1$ by (\ref{7.3}) and (\ref{7.4}) we get
\beq
\label{7.5}
\int_{\R^2} \left( \ph^2( |\psi|) - 1 \right) ^2 \, dx
\leq 4 \int_{\R^2} V(| \psi |^2) ) \, dx
\leq 4 \int_{\R^2} \bigg\vert \frac{ \p \psi}{\p x_1} \bigg\vert ^2 \, dx \leq 4 M_1
\eeq
and Corollary \ref{C4.2} implies that there is some $ M_2 > 0 $ (independent on $\psi $) such that
$\| \, | \psi | - 1 \| _{L^2( \R^2)} \leq M_2$.
We infer that for any $ R > 0 $ there is $ M_3(R) > 0 $ (independent on $\psi $) such that
$ \| \psi \| _{H^1 (B(x, R) ) } \leq M_3(R)$ and hence, by the Sobolev embedding,
$ \| \psi \| _{L^p (B(x, R) ) } \leq C_p (R)$ for all $ x \in \R^2$ and
$ p \in [2, \infty)$.
Using (\ref{3.11}) and an easy bootstrap argument we get
$ \| \psi \| _{W^{2,p} (B(x, 1) ) } \leq \tilde{C}_p $ for all $ x \in \R^2$ and
$ p \in [1, \infty)$.
As in Step 1 we conclude that there is $ L>0$ such that any solution $ \psi \in \Eo $ of (\ref{1.3}) with
$ \| \nabla \psi \|_{L^2( \R^2)} \leq M_1$ satisfies (\ref{7.2}).
\medskip
{\it Step 4.} Proof of (i) if $N=2$.
Fix $ \e > 0 $. Let $ \eta $ be as in (\ref{3.19}) and $M_1$ as in step 3.
If $ \psi \in \Eo $ is a solution of (\ref{1.3}) with $\| \nabla \psi \|_{L^2( \R^2)} \leq M_1 $
and there is $ x_0 \in \R^2$ such that $ \big|\, | \psi (x_0) | - 1 \big| \geq \e $,
using (\ref{7.2}) we infer that
$ \big| \, | \psi (x) | - 1 \big| \geq \frac{\e}{2} $ for any $ x \in B(x_0, \frac{ \e}{2L})$, hence
$ \left( \ph^2(|\psi |) - 1 \right)^2 \geq \eta( \frac{\e}{2}) $ on $ B(x_0, \frac{ \e}{2L})$
and therefore
$$
\int_{\R^2}
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx
\geq \int_{B(x_0, \frac{ \e}{ 2L} ) } \left( \ph^2(|\psi |) - 1 \right)^2 \, dx
\geq \pi \left( \frac{ \e}{ 2L} \right)^2 \eta \left( \frac{\e}{2} \right) .
$$
On the other hand, by (\ref{7.5}) we have
$ \ii_{\R^2}
\left( \ph^2(|\psi |) - 1 \right)^2 \, dx \leq 4 \| \nabla \psi \|_{L^2( \R^2)} ^2$.
We conclude that necessarily $\big| \, | \psi | - 1 \big| < \e $ on $ \R^2$ if
$\| \nabla \psi \|_{L^2( \R^2)} ^2 < \frac {\pi}{4} \left( \frac{ \e}{ 2L} \right)^2 \eta( \frac{\e}{2}).$
\medskip
{\it Step 5.} Proof of (\ref{7.2}) if $\| \, |\psi | - 1 \|_{L^p(\R^N)} \leq M$ and $ p > Np_0$.
By Proposition \ref{P4.12} (ii) we know that $ \psi $ and $ \nabla \psi $ belong to $ L^{\infty}(\R^N)$.
We will prove that $\| \psi \|_{L^{\infty} (\R^N)}$ and $\| \nabla \psi \|_{L^{\infty} (\R^N)}$
are bounded uniformly with respect to $\psi$.
The constants $ C_j$ below depend only on $ M, F, p, N$, but not on $\psi$.
Let $ \phi (x) = e^{\frac{i c x_1}{2}} \psi (x)$,
so that $ |\phi | = |\psi |$ and $ \phi $ satisfies the equation
\beq
\label{7.6}
\Delta \phi + \left( \frac{ c^2}{4} + F(|\phi|^2) \right) \phi = 0 \qquad \mbox{ in } \R^N.
\eeq
For all $ x \in \R^N$ we have $ \| \phi \|_{L^p(B(x,2))} \leq C_1$, where $C_1$ depends only
on $M$. Fix $ r = (\frac{p}{ 2 p_0} )^-$ such that $ N p_0 < 2r p_0 < p$ and
$ (2 p_0 + 1) r > p$. In particular, we have $ r > \frac N2 \geq 1$. Since
$ \big\vert \left( \frac{ c^2}{4} + F(|\phi|^2) \right) \phi \big\vert \leq C_2 + C_3 |\phi |^{2 p_0 + 1}$,
using (\ref{7.6}) we find that for all $ x \in \R^N$ we have
\beq
\label{7.7}
\| \Delta \phi \| _{L^r(B(x,2))}
\leq C_4 + C_5 \| \phi \|_{L^{\infty} (B(x,2))} ^{2 p_0 + 1 - \frac pr } \| \phi \|_{L^{p} (B(x,2))} ^{ \frac pr }
\leq C_6 + C_7 \| \phi \|_{L^{\infty} (\R^N)} ^{2 p_0 + 1 - \frac pr }.
\eeq
It is obvious that $ |\phi | \leq C_8 + C_9 \| \phi \|^{ 2 p_0 + 1 - \frac pr }_{L^\infty(\R^N)}
| \phi |^{ \frac pr }$, hence $ \phi $ satisfies
$$
\| \phi \| _{L^r(B(x,2))} \leq (\mathcal{L}^N (B(0,2) )^{\frac 1r} C_8
+ C_9 C_1^{ \frac pr } \| \phi \|^{ 2 p_0 + 1 - \frac pr }_{L^\infty(\R^N)} .
$$
Then, using (\ref{3.11}) we infer that for all $ x \in \R^N$,
$$
\| \phi \|_{W^{2,r}(B(x,1))} \leq C_{10} + C_{11} \| \phi \|_{L^{\infty} (\R^N)} ^{2 p_0 + 1 - \frac pr }.
$$
Since $ r > \frac N2$, the Sobolev embedding implies
$ \| \phi \|_{L^{\infty} (B(x,1))} \leq C_s \| \phi \|_{W^{2,r}(B(x,1))}.$
Choose $ x _0 \in \R^N$ such that $ \| \phi \|_{L^{\infty} (B(x_0,1))} \geq
\frac 12 \| \phi \|_{L^{\infty} (\R^N)} .$ We have
$$
\frac{1}{2 C_s} \| \phi \|_{L^{\infty} (\R^N)}
\leq \frac{1}{ C_s} \| \phi \|_{L^{\infty} (B(x_0,1))}
\leq \| \phi \|_{W^{2,r}(B(x_0,1))}
\leq C_{10} + C_{11} \| \phi \|_{L^{\infty} (\R^N)} ^{2 p_0 + 1 - \frac pr }.
$$
Since $ 2 p_0 + 1 - \frac pr < 1 $ by the choice of $ r$,
the above inequality implies that there is $ C_{12} > 0$ such that $\| \phi \|_{L^{\infty} (\R^N)} \leq C_{12}$.
Then using (\ref{7.6}) and (\ref{3.11}) we infer that
$ \| \phi \|_{W^{2,q}(B(x,1))} \leq C(q)$ for all $ x \in ( \R^N)$ and all $ q \in (1, \infty)$,
and the Sobolev embedding implies
$\| \nabla \phi \|_{L^{\infty} (\R^N)} \leq C_{13}$ for some $ C_{13} > 0$.
Since $ \psi (x)= e^{- \frac{ i c x_1}{2}} \phi (x)$,
the conclusion follows.
\medskip
{\it Step 6.} Proof of (ii).
Let $ \psi $ be a solution of (\ref{1.3}) such that $\| \, |\psi | - 1 \|_{L^p( \R^N)} \leq 1$.
By step 5, there is $ L > 0$ (independent on $ \psi $) such that $(\ref{7.2})$ holds.
If there is $ x_0 \in \R^N$ such that $\big|\, |\psi( x_0)| - 1 \big| \geq \e$,
we have $\big|\, |\psi | - 1 \big| \geq \frac{\e }{2} $ on $ B ( x_0, \frac{\e}{2 L})$ and consequently
$$
\| \, |\psi | - 1 \|_{L^p( \R^N)}
\geq \| \, |\psi | - 1 \|_{L^p( B(x_0, \frac{\e}{2L}))}
\geq \frac{ \e}{2} \left( \left(\frac{ \e}{2L} \right)^N \Lo ^N ( B(0,1)) \right) ^{\frac 1p}.
$$
Thus necessarily $\big| \, |\psi( x)| - 1 \big| < \e$ on $\R^N$ if
$\| \, |\psi | - 1 \|_{L^p( \R^N)} < \min \left( 1 , \frac{ \e}{2} \left( \left( \frac{ \e}{2L} \right)^N \Lo ^N ( B(0,1)) \right) ^{\frac 1p} \right).$
\medskip
If (A1) and (A3) hold, it follows from the proof of Proposition 2.2 (i) p. 1078 in \cite{M8} that
there is $ L > 0$ such that (\ref{7.2}) holds for any $ c \in [0, v_s]$ and any solution $ \psi \in \Eo $ of (\ref{1.3}).
Therefore the conclusions of steps 1, 3 and 5 are automatically satisfied.
The rest of the proof is exactly as above.
\hfill
$\Box$
\medskip
By (A1) we may fix $ \beta _* > 0 $ such that
$ \frac 14 ( s - 1) ^2 \leq V( s) \leq \frac 34 ( s - 1) ^2$
if $ | \sqrt{ s} - 1 | \leq \beta_*$.
Let $U\in \Eo $ be a traveling wave to (\ref{1.1}) such that
$ 1 - \beta _* \leq |U | \leq 1 + \beta _* $. It is clear that
\beq
\label{equival}
\frac 14 ( |U |^2 - 1) ^2 \leq V( |U|^2) \leq \frac 34 ( |U|^2 - 1) ^2 \qquad \mbox{ on } \R^N.
\eeq
It is an easy consequence of
Theorem 3 p. 38 and of Lemma C1 p. 66 in \cite{BBM} that there exists a lifting
$ U = \rho e^{i \theta} $ on $ \R^N$, where
$ \rho, \theta \in W_{loc}^{2,p}(\R^N) $ for any $p \in [1, \infty)$.
Then (\ref{1.3}) can be written in the form
\beq
\label{syst}
\left\{
\begin{array}{l}
\ds \Delta \rho - \rho |\nabla \theta|^2 + \rho F(\rho^2)
= c \rho \frac{\partial \theta}{\partial x_1},
\\
\\
\ds \mbox{div} ( \rho^2 \nabla \theta)
= - \frac{c}{2} \frac{\partial }{\partial x_1}(\rho^2 - 1 ) .
\end{array}
\right.
\eeq
Multiplying the first equation in (\ref{syst}) by $\rho$ we get
\beq
\label{7.9}
\frac 12 \Delta ( \rho ^2 - 1) - |\nabla U|^2 + \rho ^2 F( \rho ^2) - c ( \rho ^2 - 1) \frac{ \p \theta }{\p x _1} = c \frac{ \p \theta }{\p x _1} .
\eeq
The second equation in (\ref{syst}) can be written as
\beq
\label{7.10}
\mbox{div} (( \rho ^2 - 1) \nabla \theta ) + \frac{c}{2} \frac{\partial }{\partial x_1}(\rho^2 - 1 )
= - \Delta \theta.
\eeq
We set $ \eta = \rho^2 - 1 $
and define $ g : [-1 , +\infty ) $ by
$ g(s) =v_s^2 s + 2 ( 1 + s) F(1 + s ) $, so that
$ g(s) = \mathcal{O}(s^2) $ for $ s \lra 0 $.
Taking the Laplacian of (\ref{7.9}) and applying the operator $ c \frac{ \p }{\p x_1}$ to (\ref{7.10}),
then summing up the resulting equalities we find
\beq
\label{7.11}
\left[ \Delta^2 \!- v_s^2 \Delta + c^2 \partial_{x_1}^2 \right] \eta =
\Delta \left( 2 |\nabla U|^2 \!- g(\eta) + 2 c \eta \partial_{x_1} \theta \right)
- 2 c \partial_{x_1} (\mbox{div} ( \eta \nabla \theta))
\quad \mbox{ in } \So '( \R^N).
\eeq
Notice that
the right-hand side of (\ref{7.11}) contains terms that are
(at least) quadratic.
We write (\ref{7.11}) using the Fourier transform as
\beq
\label{magique}
\hat{\eta} (\xi) = \mathcal{L}_c (\xi) \hat{\Upsilon}(\xi) ,
\eeq
where
\beq
\label{7.14}
\wh{\Upsilon}(\xi) = - \mathcal{F} ( 2 |\nabla U|^2 - g(\eta) )
- 2 c \frac{|\xi|^2-\xi_1^2}{|\xi|^2} \mathcal{F}(\eta \p_{x_1} \phi)
+ 2c \sum_{j=2}^N \frac{\xi_1 \xi_j}{|\xi|^2} \mathcal{F} ( \eta \p_{x_j} \phi )
\eeq
and
\beq
\label{7.15}
\mathcal{L}_c (\xi) = \frac{|\xi|^2}{ |\xi|^4 + v_s^2 |\xi|^2 - c^2 \xi_1^2} .
\eeq
On the other hand, we know that $U$ satisfies the Pohozaev identity (\ref{5.19}).
Using (\ref{lift}) and the Cauchy-Schwarz identity we have
$$
| Q( U) | = \Big\vert \ii_{\R^N} (\rho ^2 - 1) \theta _{x_1} \, dx \Big\vert
\leq \| \eta \|_{L^2( \R^N)} \| \theta_{x_1} \|_{L^2( \R^N)}
\leq \frac{1}{ 1 - \beta_* } \| \eta \|_{L^2( \R^N)} \| \nabla U \|_{L^2( \R^N)} .
$$
Inserting this estimate into
(\ref{5.19}), using (\ref{equival}) and the fact that
$|c| \leq v_s $ we get
\beq
\label{7.16}
(N-2) \| \nabla U \|_{L^2( \R^N )}^2
- \frac{(N-1)v_s}{ 1 - \beta_* } \| \eta \|_{L^2( \R^N)} \| \nabla U \|_{L^2( \R^N)}
+ \frac{N}{4} \| \eta \|_{L^2( \R^N)} ^2 \leq 0.
\eeq
{\bf The case $\boldsymbol{N \geq 3}.$ }
If $ N \geq 3$, let $ a_1 \leq a_2$ be the two roots of the equation
$(N-2) y^2 - \frac{(N-1)v_s}{ 1 - \beta_* } y + \frac{N}{4} = 0.$
It is obvious that $ a_1$ and $a_2$ are positive and from (\ref{7.16}) we infer that
\beq
\label{7.17}
a_1 \| \eta \|_{L^2( \R^N)} \leq \| \nabla U \|_{L^2( \R^N )} \leq a_2 \| \eta \|_{L^2( \R^N)}.
\eeq
\medskip
{\it Proof of Proposition \ref{smallE} for $N \geq 3$}.
We use the ideas introduced in \cite{BGS} and \cite{dL}.
In the following $C_j$ and $ K_j$ are positive constants depending only on $N$ and $F$.
Let $ \beta_*$ be as above.
By Lemma \ref{L7.1}, there are $ M_1, \, \ell_1 > 0 $ such that any solution $ U \in \Eo $ to (\ref{1.3}) with
$\ii_{\R^N} |\nabla U|^2 \, dx \leq M_1$
(respectively with $\ii_{\R^N} \left(|U|^2 - 1 \right)^2\, dx \leq \ell _1$ if (A3)
holds or if (A2) holds and $ p_0 < \frac 2N$) satisfies $ 1 - \beta_* \leq |U| \leq 1 + \beta _*$ and, in addition, (\ref{7.2}) is verified.
Then we have a lifting $U = \rho e^{i \theta } $ and (\ref{equival})-(\ref{7.17}) hold.
Since $ g ( \eta) = \Oo( \eta ^2)$, it follows from (\ref{7.17}) that
\beq
\begin{array}{rcl}
\label{7.18}
\| 2 |\nabla U|^2 - g ( \eta ) \| _{L^1( \R^N)}
& \leq & 2 \| \nabla U \|_{L^2( \R^N)} ^2 + C_1 \| \eta \|_{L^2(\R^N)} ^2
\leq C_2 \| \eta \|_{L^2(\R^N)} ^2
\\
& \leq & C_3 \| \nabla U \|_{L^2( \R^N)} ^2.
\end{array}
\eeq
On the other hand, from $ 1 - \beta_* \leq |U| \leq 1 + \beta_*$ and (\ref{7.2}) we get
$ \| 2 |\nabla U|^2 - g ( \eta ) \| _{L^{\infty}( \R^N)} \leq C_4$ and then, by interpolation,
\beq
\label{7.19}
\| 2 |\nabla U|^2 - g ( \eta ) \| _{L^p( \R^N)}
\leq C_1 (p) \| \eta \|_{L^2(\R^N)} ^{\frac 2p} ,
\eeq
respectively
\beq
\label{7.20}
\| 2 |\nabla U|^2 - g ( \eta ) \| _{L^p( \R^N)}
\leq K_1 (p) \| \nabla U \|_{L^2(\R^N)} ^{\frac 2p}
\eeq
for any $ p \in [1, \infty)$.
It is obvious that
$| \eta \p_{x_j} \theta | \leq \frac{1}{1 - \beta_* } |\eta | \cdot |\nabla U | $ and, as above, we find
\beq
\label{7.21}
\| \eta \p_{x_j} \theta \| _{L^p( \R^N)}
\leq C_2 (p) \| \eta \|_{L^2(\R^N)} ^{\frac 2p} ,
\quad \mbox{ and } \quad
\| \eta \p_{x_j} \theta \| _{L^p( \R^N)}
\leq K_2 (p) \| \nabla U \|_{L^2(\R^N)} ^{\frac 2p} .
\eeq
By the standard theory of Riesz operators (see, e.g., \cite{stein}), the functions
$ \xi \longmapsto \frac{ \xi_j \xi_k}{|\xi |^2}$ are Fourier multipliers from $L^p(\R^N)$ to $L^p(\R^N)$,
$1 < p < \infty$.
Using (\ref{7.14}) and (\ref{7.19})-(\ref{7.21}) we infer that
$\Upsilon \in L^p(\R^N)$ for $1 < p < \infty$ and
\beq
\label{7.22}
\| \Upsilon \| _{L^p( \R^N)}
\leq C_3 (p) \| \eta \|_{L^2(\R^N)} ^{\frac 2p} ,
\quad \mbox{ respectively } \quad
\| \Upsilon \| _{L^p( \R^N)}
\leq K_3 (p) \| \nabla U \|_{L^2(\R^N)} ^{\frac 2p} .
\eeq
We will use the following result, which is Lemma 3.3 p. 377 in \cite{dL} with
$ \alpha = \frac{2}{2N-1} $ and $q=2$. Notice that $ \frac{1}{1-\alpha} = \frac{2N-1}{2N-3} <2 $
if $ N \geq 3 $.
\begin{Lemma} (\cite{dL})
\label{multiplier}
Let $N \geq 3$ and let $p_N = \frac{2(2N-1)}{2N+3} \in (1 , 2) $.
There exists a constant $K_N$, depending only on $N$,
such that for any $ c \in [ 0 ,v_s ] $ and any $f \in L^{p_N}(\R^N)$ we have
$$
\| \mathcal{F}^{-1} (\mathcal{L}_c (\xi) \mathcal{F}(f) ) \|_{L^2(\R^N)}
\leq K _N\| f \|_{L^{p_N}(\R^N)} .
$$
\end{Lemma}
From (\ref{magique}), Lemma \ref{multiplier} and (\ref{7.22}) we get
\beq
\label{7.23}
\| \eta \|_{L^2(\R^N)} \leq K_N \| \Upsilon \|_{L^{p_N}(\R^N)} \leq K _NC_3(p_N) \| \eta \|_{L^2(\R^N)} ^{\frac{2}{p_N}}.
\eeq
Since $ \frac{2}{p_N} > 1$, (\ref{7.23}) implies that there is $ \ell_* > 0 $ (depending only on $N$ and $F$)
such that $ \| \eta \|_{L^2(\R^N)} \geq \ell_*$, or $ \| \eta \|_{L^2(\R^N)} = 0$.
In the latter case from (\ref{7.17}) we get $ \| \nabla U \|_{L^2(\R^N)} = 0$, hence $U$ is constant.
From (\ref{7.23}) and (\ref{7.17}) we obtain
$$
\| \nabla U \|_{L^2(\R^N)} \leq a_2 \| \eta \|_{L^2(\R^N)}
\leq
a_2 K_N C_3(p_N) \| \eta \|_{L^2(\R^N)} ^{\frac{2}{p_N}}
\leq a_1 ^{- \frac{2}{p_N}} a_2 K _NC_3(p_N) \| \nabla U \|_{L^2(\R^N)} ^{\frac{2}{p_N}}.
$$
As above we infer that there is $ k_* > 0$ such that either $ \| \nabla U \|_{L^2(\R^N)} \geq k_* $, or $U$ is constant.
\hfill
$\Box$
\medskip
{\bf The case $\boldsymbol{N = 2}.$ }
If $N=2$,
from (\ref{7.16}) we infer that
$\| \eta\|_{L^2(\R^N)} \leq \frac{2 v_s}{1 - \beta_*} \| \nabla U \|_{L^2(\R^N)} $.
However,
the Pohozaev identities alone do not imply an estimate of the form
$ \| \nabla U \|_{L^2(\R^N)} \leq C \| \eta\|_{L^2(\R^N)}$.
To prove this we need the following two identities, which are valid in any space dimension and are of independent interest.
\begin{Lemma}
\label{L7.3}
Let $ U = \rho e^{ i \theta } \in \Eo $ be a solution of (\ref{1.3}), where $ \inf \rho >0$ and $ \rho $ is bounded.
Then we have
\beq
\label{grincheux}
2 \int_{\R^N} \rho^2 |\nabla \theta |^2 \ dx =
- c \int_{\R^N} ( \rho^2 - 1 ) \p_{x_1} \theta \ dx
\qquad \mbox{ and }
\eeq
\beq
\label{dormeur}
\int_{\R^N} 2 \rho |\nabla \rho|^2
+ \rho (\rho^2 -1) |\nabla \theta|^2
- \rho (\rho^2 -1) F(\rho^2) \ dx
= - c \int_{\R^N} \rho (\rho^2 -1) \p_{x_1} \theta \ dx .
\eeq
\end{Lemma}
{\it Proof. }
Formally, $U$ is a critical point of the functional $E_c = E - cQ$.
Denoting $U(s) = \rho e^{ i s \theta}$ one would expect that
$\frac{d}{ds} _{\mid s=1} (E_c (U(s)) = 0$ and this is precisely (\ref{grincheux}).
In the case of the Gross-Pitaevskii equation, (\ref{grincheux}) was proven in \cite{BGS}
(see Lemma 2.8 p. 594 there)
by multiplying the second equation in (\ref{syst}) by $\theta$, then integrating by parts.
The integrations are justified by the particular decay at infinity of traveling waves for the Gross-Pitaevskii equation.
Since such decay properties have not been rigorously established for other nonlinearities, we proceed as follows.
For $R > 0 $, we denote
$ \bar{\theta} = \frac{1}{\mathcal{H}^{N-1} (\p B(0,R) )} \ii_{\p B_R} \theta \ d \mathcal{H}^{N-1} ,$
we multiply the second equation in (\ref{syst})
by $\theta - \bar{\theta}$ and
integrate by parts over $B(0,R)$. We get
\beq
\label{7.26}
\begin{array}{l}
\ds 2 \ii_{B(0,R)} \rho^2 |\nabla \theta |^2 \ dx
- 2 \ii_{\p B(0,R)} \rho^2 \frac{\p \theta}{\p \nu} (\theta - \bar{\theta}) \ d\mathcal{H}^{N-1} \\
\\
= \ds - c \ii_{B(0,R)} ( \rho^2 - 1 ) \p_{x_1} \theta \ dx
+ c \int_{\p B(0,R)} (\rho^2-1 ) (\theta - \bar{\theta}) \nu_1\ d\mathcal{H}^{N-1} ,
\end{array}
\eeq
where $ \nu $ is the outward unit normal to $ \p B(0, R)$.
By the Poincar\'e inequality we have for some constant $C$
independent of $R$,
$$ \| \theta - \bar{\theta} \|_{L^2(\p B(0,R))} \leq
C R \| \nabla \theta \|_{L^2(\p B(0,R))} . $$
Using the boundedness of $ \rho $ and the Cauchy-Schwarz inequality we have for $ R \geq 1$
\begin{align*}
\Big| 2 \ii_{\p B(0,R)} \rho^2 (\theta - \bar{\theta}) \frac{\p \theta}{\p \nu}
\ d\mathcal{H}^{N-1} \Big|
+ \Big| c \ii_{\p B(0,R)} (\rho^2-1) (\theta - \bar{\theta}) \nu_1\ d\mathcal{H}^{N-1} \Big|
\\
\leq C R \ii_{\p B(0,R)} ( \rho^2 - 1 )^2
+ | \nabla \theta |^2 \ d \mathcal{H}^{N-1} .
\end{align*}
Since $ \rho^2 - 1 \in L^2(\R^N) $ and $\nabla \theta \in L^2(\R^N)$, we have
$$ \ii_1^{+\infty} \left( \int_{\p B(0,R)} ( \rho^2 - 1 )^2
+ |\nabla \theta|^2 \ d\mathcal{H}^{N-1} \right) \ dR =
\ii_{ \{ |x| \geq 1 \} } ( \rho^2 - 1 )^2 + |\nabla \theta|^2 \ dx < \infty , $$
hence there exists a sequence $ R_j \lra + \infty $ such that
$$ \ii_{\p B (0, R_j)} ( \rho^2 - 1 )^2 + |\nabla \theta|^2 \ d\mathcal{H}^{N-1}
\leq \frac{1}{R_j \ln R_j} . $$
Writing (\ref{7.26}) for each $j$, then passing to the limit as $ j \lra \infty$
we obtain \eqref{grincheux}.
It is easily seen that $ \rho ^2 - 1 \in H^1( \R^N)$.
Multiplying the first equation in
\eqref{syst} by $\rho^2 -1 $ and using the standard integration by parts formula for
$H^1$ functions (cf. \cite{brezis} p. 197) we get \eqref{dormeur}.
\hfill
$\Box$
\medskip
Using \eqref{grincheux} and the Cauchy-Schwarz inequality we get
$$ 2 \ii_{\R^N} \rho^2 |\nabla \theta |^2 \ dx =
- c \ii_{\R^N} ( \rho^2 - 1 ) \p_{x_1} \theta \ dx
\leq C \left( \ii_{\R^N} ( \rho^2 - 1 )^2 \ dx \right)^{1/2}
\left( \ii_{\R^N} \rho^2 |\nabla \theta |^2 \ dx \right)^{1/2} , $$
from which it comes
$$ \ii_{\R^N} \rho^2 |\nabla \theta |^2 \ dx \leq C
\int_{\R^N} \eta^2 \ dx . $$
Using \eqref{dormeur}, the fact that
$ 0 < 1 - \beta_* \leq \rho \leq 1 + \beta_*$,
the inequality $ 2 ab \leq a^2 + b^2$ and the above estimate we find
\begin{align*}
2( 1 - \beta_*) \ii_{\R^N} |\nabla \rho|^2 \ dx
\leq & \ \ii_{\R^N} 2 \rho |\nabla \rho|^2 \ dx \\
= & \ - \int_{\R^N} \rho \eta |\nabla \theta |^2 \ dx
- c \int_{\R^N} \eta \rho \p_{x_1} \theta \ dx
+ \int_{\R^N} \rho^2 \eta F(\rho^2) \ dx \\
\leq & \ C \int_{\R^N} \rho^2 |\nabla \theta |^2 + \eta^2 \ dx \leq
C \int_{\R^N} \eta^2 \ dx .
\end{align*}
It follows from the above inequalities that in the case $N = 2$, there exist two positive constants $a_1, \, a_2$ such that any solution $ U \in \Eo $ to (\ref{1.3})
with $ 0 \leq c \leq v_s$ and $ 1 - \beta_* \leq |U| \leq 1 + \beta_*$ satisfies (\ref{7.17}).
\medskip
{\it Proof of Proposition \ref{smallE} if $N=2$, (A4) holds and $ F''(1) = 3$.}
The strategy used in the case $N \geq 3$ has to be adapted: small energy traveling waves
do exist when $N=2$ and $ F''(1) \not = 3 $ (see Theorem \ref{T4.7}, Proposition \ref{P4.12}
and Theorem \ref{T4.13}). This is related to
the fact that Lemma \ref{multiplier} does not apply if $N=2$. The proof relies on an expansion in the small parameter
$\eta$ and the observation that when the energy is small, we must have
$ \p_{x_1} \phi \simeq - c \eta / 2 $.
Since $ v_s^2 = 2= - 2 F'(1) $ and
$ F''(1) = 3 $, by (A4) the function $g$ has the expansion as $ s \lra 0 $
\begin{align*}
g(s) = & \, v_s^2 s + 2 (1 +s) F (1 +s)
= v_s^2 s + 2 (1 +s) \Big( s F'(1 ) + \frac12 s^2 F''(1 )
+ \mathcal{O} (s^3) \Big)
\\
= & \, s^2 + \mathcal{O} (s^3) .
\end{align*}
By Lemma \ref{L7.1}, there are $ M_1, \, \ell_1 > 0 $ such that any solution $ U \in \Eo $ to (\ref{1.3}) with $ c \in [0, v_s]$ and
$\ii_{\R^2} |\nabla U|^2 \, dx \leq M_1$ (respectively $\ii_{\R^2} \left(|U|^2 - 1 \right)^2\, dx \leq \ell _1$
if (A3) holds or if (A2) holds and $ p_0 < 1$) satisfies $ 1 - \beta_* \leq |U| \leq 1 + \beta _*$, the estimate (\ref{7.2}) is verified,
we have a lifting $U = \rho e^{i \theta } $ and
all the statements above are valid.
Recalling that $\Upsilon$ is defined by (\ref{7.14}),
we observe that in the expression of $ 2 |\nabla U|^2 - g(\eta) $
we have the almost cancellation of two quadratic terms:
$ 2 \rho^2 (\p_{x_1} \theta)^2 - \eta^2
\simeq 2 ( ( \p_{x_1} \theta)^2 - \frac{v_s^2}{4} \eta^2 ) $
is much smaller than quadratic if $ \p_{x_1} \theta \simeq v_s \eta / 2 $.
We now quantify this idea and split the proof into 7 steps. We denote
\beq
\label{7.27}
h = \p_{x_1} \theta + \frac c2 \eta .
\eeq
By Lemma \ref{L7.1}, $ \| \eta \|_{L^\infty(\R^2)}$ can be made arbitrarily small
by taking $M_1$ (respectively $\ell _1$) sufficiently small.
Moreover, using (\ref{7.17}) we get
\beq
\label{tookiki}
\| \eta \|_{L^4(\R^2)}^4 \leq
\| \eta \|_{L^\infty(\R^2)}^2 \| \eta \|_{L^2(\R^2)}^2 \leq C \| \eta \|_{L^2(\R^2)}^2
\leq C \| \nabla U \|_{L^2(\R^2)}^2 .
\eeq
\medskip
{\it Step 1.} There is $ C >0$, depending only on $F$,
such that if $M_1$ (respectively $ \ell_1$) is small enough,
$$ \ii_{\R^2} h^2 + (\p_{x_2} \theta)^2 + ( v_s ^2 - c^2) \eta^2 \ dx
\leq C \| \eta \|^4_{L^4(\R^2)} . $$
The starting point is the integral identity
$$
\ii_{\R^2} \rho^2 |\nabla \theta|^2 + V(\rho^2) + c (\rho^2 - 1) \p_{x_1} \theta
\ d x = 0 ,
$$ which comes from the combination of (\ref{grincheux}) and the
Pohozaev identity
$ \ii_{\R^2} 2 V(\rho^2) + c (\rho^2 - 1) \p_{x_1} \theta \ dx = 0 $
(see Proposition 4.1 in \cite{M8}).
From (A4) with $ F''(1) = 3 $ we have the Taylor expansion of the potential
$$
V (\rho^2 ) = V (1+ \eta )
= \frac{v_s^2}{4} \eta^2 - \frac16 F''(1) \eta^3 + \mathcal{O}(\eta^4)
= \frac{v_s^2}{4} \eta^2 - \frac{v_s^2}{4 } \eta^3 + \mathcal{O}(\eta^4) .
$$
Therefore, the above integral identity gives
$$
\ii_{\R^2} ( 1 + \eta ) (\p_{x_2} \theta)^2
+ (\p_{x_1} \theta)^2 + \eta (\p_{x_1} \theta)^2
+ \frac{v_s^2}{4} \eta^2 - \frac{v_s^2}{4 } \eta^3 + \mathcal{O}(\eta^4)
+ c \eta \p_{x_1} \theta
\ d x = 0 .
$$
Then the identity $ h^2 = (\p_{x_1} \theta )^2 + c \eta \p_{x_1} \theta
+ \frac{c^2}{4 } \eta^2 $ gives
$$
\ii_{\R^2} ( 1 + \eta ) (\p_{x_2} \theta)^2
+ h^2 + \frac{v_s^2-c^2}{4 } \eta^2
+ \eta (\p_{x_1} \theta)^2
- \frac{v_s^2}{4 } \eta^3
\ d x = - \ii_{\R^2} \mathcal{O}(\eta^4) \ dx ,
$$
hence, rearranging the cubic terms,
\beq
\label{eqrase}
\ii_{\R^2} ( 1 + \eta ) (\p_{x_2} \theta)^2 + h^2
+ \frac{v_s^2-c^2}{4 } \eta^2 \left( 1 - \eta \right) \ dx
= - \ii_{\R^2} \eta h \left( h - c \eta \right)
+ \mathcal{O}(\eta^4) \ dx .
\eeq
For the left-hand side, we have $ 1 + \eta \geq \frac 12 $ and
$ 1 - \eta \geq \frac12 $ if $M_1$ or $ \ell _1$ are sufficiently small (because
$ \| \eta \|_{L^\infty(\R^2)} $ is small). We now estimate the right-hand side.
Since $ \| \eta \|_{L^\infty(\R^2)}$ is small, we have
$ | \ii_{\R^2} \mathcal{O}(\eta^4) \ dx | \leq C \| \eta \|_{L^4(\R^2)}^4 $
and by Cauchy-Schwarz and the inequality $ 2 a b \leq a^2 + b^2 $,
\begin{align*}
\Big| \ii_{\R^2} \eta h \left( h - c \eta \right) \ dx \Big|
\leq & \, \| \eta \|_{L^\infty(\R^2)} \ii_{\R^2} h^2 \ dx
+ c \Big( \ii_{\R^2} h^2 \ dx \Big)^{\frac12}
\Big( \ii_{\R^2} \eta^4 \ dx \Big)^{\frac12} \\
\leq & \, \frac{1}{2} \ii_{\R^2} h^2 \ dx + C \| \eta \|^4_{L^4(\R^2)} ,
\end{align*}
provided that $M_1 $ or $ \ell _1$ are small enough, where $C$ depends only on $F$. Inserting
these estimates into (\ref{eqrase}) yields the result.
\medskip
{\it Step 2.} There exists $ C $, depending only on $F$,
such that for $M1$ (respectively $\ell _1$) small enough,
$$
\int_{\R^2} | \nabla \rho |^2 + ( v_s^2 - c^2 ) \eta^2 \ dx
\leq C \| \eta \|^2_{L^4(\R^2)} .
$$
We start from (\ref{dormeur}), that we write in the form
$$
\int_{\R^2} 2 \rho |\nabla \rho|^2 \ dx
= - \int_{\R^2} \rho \eta \left( ( \p_{x_1} \theta)^2 + ( \p_{x_2} \theta)^2 \right)
- \rho \eta F(\rho^2) + c \rho \eta \p_{x_1} \theta \ dx .
$$
Using the expansion $ F(\rho^2) = \eta F'(1) + \mathcal{O}(\eta^2)
= - \frac{v_s^2 \eta}{2 } + \mathcal{O}(\eta^2) $, this gives
\beq
\label{eqlabousse}
\int_{\R^2} 2 \rho |\nabla \rho|^2 + \frac{v_s^2 - c^2}{2 } \rho \eta^2 \ dx
= - \int_{\R^2} \rho \eta \left( ( \p_{x_1} \theta)^2 + ( \p_{x_2} \theta)^2 \right)
+ c \rho \eta h + \mathcal{O} ( |\eta|^3 )\ dx .
\eeq
Note that by the Cauchy-Schwarz inequality,
\beq
\label{eta3}
\| \eta \|_{L^3(\R^2)}^3 \leq \| \eta \|_{L^4(\R^2)}^2 \| \eta \|_{L^2(\R^2)} .
\eeq
Since either $ \|\eta \| _{L^2( \R^2)} ^2\leq \ell_1$ or
$ \|\nabla U \| _{L^2( \R^2)} ^2\leq M_1$ and then, by (\ref{7.17}),
$ \|\eta \| _{L^2( \R^2)} ^2\leq \frac{ M_1}{a_1^2}$, we get
\beq
\label{7.32}
\Big| \int_{\R^2} \mathcal{O} ( |\eta|^3 )\ dx \Big|
\leq C \| \eta \|_{L^3(\R^2)}^3
\leq C \| \eta \|_{L^2(\R^2)} \| \eta \|_{L^4(\R^2)}^2
\leq C \| \eta \|_{L^4(\R^2)}^2.
\eeq
Recall that $1 - \beta_* \leq \rho \leq 1 + \beta_*$ and using step 1 we find
$$ \Big| \int_{\R^2} \rho \eta ( \p_{x_2} \theta)^2 \ dx \Big|
\leq C \int_{\R^2} ( \p_{x_2} \theta)^2 \ dx
\leq C \| \eta \|_{L^4(\R^2)}^4 .$$
Since $ c\in [0,v_s]$, from step 1 and the Cauchy-Schwarz inequality we obtain
$$ \Big| \int_{\R^2} c \rho \eta h \ dx \Big|
\leq C \| \eta \|_{L^2(\R^2)} \| h \|_{L^2(\R^2)}
\leq C \| \eta \|_{L^4(\R^2)}^2 , $$
Using the definition of $h$, step 1 and (\ref{7.32}) we now estimate
\beq
\label{cool}
\begin{array}{l}
\ds \Big| \ii_{\R^2} \rho \eta ( \p_{x_1} \theta)^2 \ dx \Big|
\leq C \ii_{\R^2} |\eta| \left( h - \frac{c}{2} \eta \right)^2 \ dx
\\
\\
\ds \leq C \| \eta \|_{L^\infty(\R^2)} \int_{\R^2} h^2 \ dx
+ C \int_{\R^2} |\eta|^3 \ dx
\leq C \| \eta \|_{L^4(\R^2)}^2 .
\end{array}
\eeq
Summing up the above estimates and using (\ref{eqlabousse}) yields the conclusion.
\medskip
In steps 1 and 2 we have not used the fact that $ F''(1) = 3 $.
Since $ g(s) = \frac{v_s^2}{2 } s^2 + \mathcal{O}(s^3)$ when
$ F''(1) = 3 $, it is natural to write (\ref{magique}) in the form
\begin{align}
\label{magic}
\wh{\eta} (\xi) = & \,
- \mathcal{L}_c (\xi) \mathcal{F} \left(
2 (\p_{x_1} \theta)^2 - \frac{v_s^2}{2 } \eta^2 \right)
\nonumber \\
& \, - \mathcal{L}_c (\xi) \mathcal{F} \left(
2 \eta (\p_{x_1} \theta)^2 + 2 \rho^2 (\p_{x_2} \theta)^2 + 2 | \nabla \rho|^2
- \left[ g(\eta) - \frac{v_s^2}{2 } \eta^2 \right] \right)
\\
& \, - 2 c \mathcal{L}_c (\xi) \frac{\xi_2^2}{|\xi|^2} \mathcal{F}(\eta \p_{x_1} \theta)
+ 2c \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \mathcal{F} ( \eta \p_{x_2} \theta )
. \nonumber
\end{align}
where we recall that
$ \mathcal{L}_c (\xi) $ is given by (\ref{7.15}).
We expect the term in the first line of (\ref{magic}) to be much smaller than quadratic.
By the Riesz-Thorin Theorem we have
$ \| \eta \|_{L^4(\R^2)} \leq C \| \wh{\eta} \|_{L^{4/3}(\R^2)} $.
We will estimate the $L^{4/3} $ norm of all the terms in the right-hand side of (\ref{magic}) and we will show that
they are bounded by $C \| \eta \|_{L^4(\R^2)} ^2$.
\medskip
{\it Step 3.} We have, for some constant $C$ depending only
on $F$,
$$
\Big\| 2c \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \mathcal{F}
( \eta \p_{x_2} \theta ) \Big\|_{L^{4/3}(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}^2 .
$$
Indeed, by the continuity of $ \mathcal{F} : L^1(\R^2) \lra L^\infty(\R^2) $
and the Cauchy-Schwarz inequality one has
\begin{align*}
\Big\| 2c \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi)
\mathcal{F} ( \eta \p_{x_2} \theta ) \Big\|_{L^{4/3}(\R^2)}
\leq & \, C \| \mathcal{F} ( \eta \p_{x_2} \theta ) \|_{L^\infty(\R^2)}
\Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)}
\\ \leq & \, C \| \eta \p_{x_2} \theta \|_{L^1(\R^2)}
\Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)}
\\ \leq & \, C \| \eta \|_{L^2(\R^2)} \| \p_{x_2} \theta \|_{L^2(\R^2)}
\Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)}
\\ \leq & \, C \| \eta \|_{L^4(\R^2)}^2
\Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)} ,
\end{align*}
where we have used the estimate
$\| \p _{x_2} \theta \|_{L^2(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}$ (see Step 1)
and the fact that $\| \eta \|_{L^2(\R^2)}$ is bounded.
Thus it suffices to prove that $ \Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)} $ is bounded independently on $c$.
Using polar coordinates, we find for all $ q >1$,
$$
\begin{array}{l}
\ds \| \mathcal{L}_c (\xi) \|_{L^{q}(\R^2)} ^{q}
= \ii_{\R^2} \frac{|\xi|^{2q} d \xi }{ ( |\xi|^4 + v_s^2 |\xi|^2 - c^2 \xi_1^2)^{q}}
= 4 \ii_0^{\pi/2} \int_0^{+\infty} \frac{ r\, d r \, d \vartheta }{
( r^2 + v_s^2 - c^2 \cos^2\vartheta ) ^{q}}
\\
\\
= \frac{2}{q-1}
{\ds \ii _0^{\pi/2} \frac{ d \vartheta }{ ( v_s^2 - c^2 \cos^2 \vartheta ) ^{q-1}} }
\leq \frac{2}{q-1}
{\ds \ii_0^{\pi/2} \frac{ d \vartheta }{ ( v_s^2 - v_s^2 \cos^2 \vartheta ) ^{q-1}} }
= \frac{2}{(q-1)v_s^{2(q-1)}}
{\ds \ii_0^{\pi/2} \frac{ d \vartheta }{ (\sin \vartheta )^{2(q-1)}} .}
\end{array}
$$
Since the last integral is finite and does not depend on $c$ if $ 2(q-1) < 1$, we get
\beq
\label{enorme43}
\sup_{0 \leq c \leq v_s} \| \mathcal{L}_c (\xi) \|_{L^{q}(\R^2)} \leq C_q < \infty
\qquad \mbox{ for any } q \in \left(1, \frac 32\right) .
\eeq
In particular we have
$ \Big\| \frac{\xi_1 \xi_2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)}
\leq \| \mathcal{L}_c (\xi) \|_{L^{4/3}(\R^2)} \leq C_{\frac 43} $
for $ 0 \leq c \leq v_s $ and this concludes the proof of step 3.
\medskip
{\it Step 4.} There holds
$$
\Big\| 2c \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \mathcal{F} ( \eta \p_{x_1} \theta )
\Big\|_{L^{4/3}(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}^2 .
$$
From the definition of $ h $ we have
$ \eta \p_{x_1} \theta = \eta h - \frac{c \eta^2}{2 } $, thus
$$
\Big\| 2c \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi)
\mathcal{F} ( \eta \p_{x_1} \theta ) \Big\|_{L^{4/3}(\R^2)}
\leq C \Big\| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \mathcal{F} ( \eta^2 )
\Big\|_{L^{4/3}(\R^2)} + C \| \mathcal{L}_c (\xi) \mathcal{F} ( \eta h ) \|_{L^{4/3}(\R^2)} . $$
The second term is estimated as in Step 3, using (\ref{enorme43}), step 1 and the fact that $\| \eta \|_{L^2(\R^2)}$ is bounded:
$$ \| \mathcal{L}_c (\xi) \mathcal{F} ( \eta h ) \|_{L^{4/3}(\R^2)}
\leq \| \mathcal{L}_c (\xi) \|_{L^{4/3}(\R^2)}
\| \mathcal{F} ( \eta h ) \|_{L^{\infty}(\R^2)}
\leq C \| \eta \|_{L^2(\R^2)} \| h \|_{L^2(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}^2 . $$
For the first term we first observe that, since $ c^2 \leq v_s^2 $,
$$ \Big| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \Big| =
\frac{\xi_2^{2} }{ |\xi|^4 + v_s^2 |\xi|^2 - c^2 \xi_1^2}
\leq \frac{ \xi_2^2 }{ v_s^2 |\xi|^2 - v_s^2 \xi_1^2} = \frac{1}{v_s^2} . $$
Hence, using the estimate $ \| f\|_{L^4} \leq \|f\|_{L^{\infty}}^{\frac 23} \|f\|_{L^{4/3}}^{\frac 13}$,
we get for $0 \leq c \leq v_s$,
$$
\Big\| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^4(\R^2)}
\leq \frac{1}{v_s^{4/3}} \Big\| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^{4/3}(\R^2)} ^{\frac 13}
\leq \frac{1}{v_s^{4/3}} \| \mathcal{L}_c (\xi) \|_{L^{4/3}(\R^2)} ^{\frac 13}
\leq C
$$
(Warning: $ \mathcal{L}_c $ is not uniformly bounded in $ L^4(\R^2) $ as $ c \ra v_s $.)
As a consequence, using the generalized H\"older inequality with
$ \frac{1}{4/3} = \frac{1}{4} + \frac{1}{2} $ and the Plancherel formula,
$$ \Big\| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \mathcal{F} ( \eta^2 )
\Big\|_{L^{4/3}(\R^2)}
\leq \Big\| \frac{ \xi_2^2}{|\xi|^2} \mathcal{L}_c (\xi) \Big\|_{L^4(\R^2)}
\| \mathcal{F} ( \eta^2 ) \|_{L^{2}(\R^2)}
\leq
C \| \eta \|_{L^4(\R^2)}^2 . $$
Combining the above estimates gives the desired conclusion.
\medskip
{\it Step 5.} If $F''(1) = 3$ we have
$$ \Big\| \mathcal{L}_c (\xi) \mathcal{F} \left(
2 \eta (\p_{x_1} \theta)^2 + 2 \rho^2 (\p_{x_2} \theta)^2 + 2 | \nabla \rho|^2
- \left[ g(\eta) - \frac{v_s^2}{2 } \eta^2 \right] \right)
\Big\|_{L^{4/3}(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}^2 . $$
By (\ref{enorme43}) and the inequality
$$ \| \mathcal{L}_c (\xi) \mathcal{F}( H ) \|_{L^{4/3}(\R^2)} \leq
\| \mathcal{L}_c (\xi) \|_{L^{4/3}(\R^2)} \|\mathcal{F}( H ) \|_{L^{\infty}(\R^2)}
\leq C_{\frac43} \| H \|_{L^1(\R^2)}
$$
it suffices to estimate
$$
\Big\| 2 \eta (\p_{x_1} \theta)^2 + 2 \rho^2 (\p_{x_2} \theta)^2
+ 2 | \nabla \rho|^2 - \left[ g(\eta) - \frac{v_s^2}{2 } \eta^2 \right]
\Big\|_{L^{1}(\R^2)} .
$$
We estimate each term separately.
We have already seen that $ g(s) = \frac{v_s^2}{2 } s^2 + \mathcal{O}(s^3)$
as $s \ra 0$ because $F''(1) = 3$. By (\ref{eta3}) we obtain
$$ \Big\| g(\eta) - \frac{v_s^2}{2 } \eta^2 \Big\|_{L^{1}(\R^2)}
\leq C \| \eta \|_{L^3(\R^2)}^3 \leq C \| \eta \|_{L^4(\R^2)}^2 .
$$
From step 2 we have
$$ \Big\| |\nabla \rho |^2 \Big\|_{L^{1}(\R^2)}
= \int_{\R^2} | \nabla \rho |^2 \ dx \leq C \| \eta \|_{L^4(\R^2)}^2 $$
and from step 1 we get
$$ \Big\| \rho^2 (\p_{x_2} \theta)^2 \Big\|_{L^{1}(\R^2)}
\leq C \int_{\R^2} ( \p_{x_2} \theta )^2 \ dx \leq C \| \eta \|_{L^4(\R^2)}^4
\leq C \| \eta \|_{L^4(\R^2)}^2 . $$
Finally, as in (\ref{cool}) we infer that
$$ \Big\| \eta (\p_{x_1} \theta )^2 \Big\|_{L^{1}(\R^2)}
\leq C \| \eta \|_{L^4(\R^2)}^2 . $$
Gathering the above estimates we get the conclusion.
\medskip
{\it Step 6.} The following estimate holds:
$$ \Big\| \mathcal{L}_c (\xi) \mathcal{F} \left(
2 (\p_{x_1} \theta)^2 - \frac{v_s^2}{2 } \eta^2 \right)
\Big\|_{L^{4/3}(\R^2)} \leq C \| \eta \|_{L^4(\R^2)}^2 .
$$
Indeed, arguing as is step 5 and using the definition of $h$, the Cauchy-Schwarz inequality and step 1 we deduce
$$
\begin{array}{l}
\ds \Big\| \mathcal{L}_c (\xi) \mathcal{F} \left(
2 (\p_{x_1} \theta)^2 - \frac{v_s^2}{2 } \eta^2 \right)
\Big\|_{L^{4/3}(\R^2)}
\leq C_{\frac43} \Big\| 2 \left( h - \frac{ c}{2} \eta \right)^2 - \frac{v_s^2}{2 } \eta^2 \Big\|_{L^1(\R^2)}
\\
\\
\ds
\leq C \| h^2 \|_{L^1(\R^2)} + C \| \eta \|_{L^2(\R^2)} \| h \|_{L^2(\R^2)} +
\frac{v_s ^2 - c^2}{2 } \ii_{\R^2} \eta ^2 \, dx
\leq C \| \eta \|_{L^4(\R^2)}^2 .
\end{array}
$$
\medskip
{\it Step 7.} Conclusion.
Using the Riesz-Thorin
theorem, we have $ \| \eta \|_{L^{4}(\R^2)} \leq C \| \hat{\eta} \|_{L^{4/3}(\R^2)} $.
Coming back to (\ref{magic}) and gathering the estimates in steps 3-6, we deduce
$$
\| \eta \|_{L^{4}(\R^2)} \leq C \| \hat{\eta} \|_{L^{4/3}(\R^2)}
\leq C \| \eta \|_{L^{4}(\R^2)}^2,
$$
where $C$ depends only on $F$.
Consequently, either $\| \eta \|_{L^{4}(\R^2)} = 0 $, or there is a constant $\kappa > 0$ such that $\| \eta \|_{L^{4}(\R^2)} \geq \kappa$.
If $\| \eta \|_{L^{4}(\R^2)} = 0 $ we have $ \eta = 0 $ a.e. and from (\ref{7.17}) we get
$\| \nabla U \|_{L^{2}(\R^2)} = 0 $, hence $U$ is constant.
If $\| \eta \|_{L^{4}(\R^2)} \geq \kappa$, (\ref{tookiki}) implies that there are $ \ell_*> 0$
and $ k_* > 0$ such that $\| \eta \|_{L^{2}(\R^2)} \geq \ell_* $ and $\| \nabla U \|_{L^{2}(\R^2)} \geq k_*. $
The proof of Proposition \ref{smallE} is complete.
\hfill
$\Box$
\bigskip
\noindent
{\bf Acknowledgement. } We acknowledge the support of the French ANR (Agence
Nationale de la Recherche) under Grant ANR JC ArDyPitEq.
|
{
"timestamp": "2013-09-05T02:09:02",
"yymm": "1203",
"arxiv_id": "1203.1912",
"language": "en",
"url": "https://arxiv.org/abs/1203.1912"
}
|
\section{Introduction}\label{sec:intro}
The BL Lac objects constitute a rare class of Active Galactic Nuclei (AGNs).
Their observational features include: weak or absent emission lines,
high radio and optical polarization, superluminal motions,
and a typical double-humped spectral energy distribution (SED, $\nu F_\nu$).
Their continuum emission is dominated by non-thermal radiations
from radio to $\gamma$-ray frequencies, that make them the most
frequently detected class of extragalactic sources at TeV energies.
The observed broadband emission is widely interpreted as arising in a
jet of relativistic particles closely aligned to our line of sight (l.o.s.) \citep{blandford78}.
In the widely accepted framework
of leptonic Synchrotron Self-Compton (SSC) radiation,
the low-energy bump is constituted by synchrotron emission
from ultrarelativistic electrons accelerated in the jets; the high-energy
component is due to inverse-Compton
scattering of these synchrotron photons by the same electron population
\citep[e.g.,][]{marscher85,inoue96}.
The BL Lacs come in two subclasses: the ``low-frequency peaked BL Lacs"
(LBLs) in which the synchrotron peak
falls in the IR-optical range, and the ``high-frequency peaked BL Lacs" (HBLs)
where it falls in the UV-X-ray bands \citep{padovani95}.
To mark the HBLs detected at TeV energies from those undetected,
we refer to the former as TBLs and to the latter as UBLs \citep{massaro11a}.
A convenient description of the BL Lac SEDs has been suggested by Landau et al. (1986) in terms
of a {\it log-parabolic} (LP) model, i.e., a curved, parabolic shape in a double-log plot.
Subsequently, the LP model has been frequently adopted to describe the X-ray spectral continuum
in HBLs \citep{tanihata04,massaro04,massaro08} as well as the TeV
emission from the TBLs \citep{massaro06,aharonian09}; it has been also used
\citet{gonzalez2010,giommi11} to describe BL Lac SEDs in the sub-mm and infrared bands.
Such curved spectra are known
to arise both by synchrotron or inverse-Compton radiations
from electron distributions featuring in turn a log-parabolic shape
\citep{massaro04,tramacere07,paggi09}.
Recently, we carried out an extensive investigation of the X-ray synchrotron
emission of both TBL and UBL subclasses,
based on archival observations carried out by {\it BeppoSAX}, {\it XMM-Newton}~and {\it Swift}~ between
1997 and 2010 \citep{massaro11a}.
On adopting the LP model, the X-ray SED of HBLs is described
in terms of 3 parameters:
the SED peak position $E_p$, its maximum flux $S_p$ evaluated at
$E_p$ (or the corresponding peak luminosities
$L_p$ $\simeq$ $4\pi D^2_L S_p$ in terms of the luminosity distance $D_L$),
and the spectral curvature $b$ around $E_p$
\citep{tramacere07,massaro08}.
Comparing the spectral properties of TBLs and UBLs, we found that:
(i) the $E_p$ distributions of UBLs and TBLs are \textit{similarly}
symmetric around a value of few keVs for both subclasses; and
(ii) the X-ray spectral curvature $b$ is systematically lower in UBLs than in TBLs,
so that the former feature \textit{narrower} spectral shapes \citep{massaro11a}.
In this Letter we compare the X-ray synchrotron
luminosities $L_p$ of the TBLs and the UBLs,
as derived from our previous analysis \citep{massaro11a}.
Motivated by the observational results recalled above,
we interpret both the TBLs and UBLs $E_p$ distributions in terms of a
coherent electron acceleration scenario,
and those of $b$ as due to accompanying stochastic acceleration.
Finally, we provide a relation between the X-ray spectral curvatures $b$
and the IC luminosities.
We use cgs units throughout this Letter
and assume a flat cosmology with $H_0=72$ km s$^{-1}$ Mpc$^{-1}$,
$\Omega_{M}=0.26$ and $\Omega_{\Lambda}=0.74$ \citep[e.g.,][]{dunkley09}.
In the following, the parameters $E_p$, $S_p$ and $L_p$ refer to the observer reference frame,
while all unprimed quantities refer to the jet frame.
\section{Log-parabolic Synchrotron Spectra}\label{sec:spectra}
Log-parabolic electron energy distributions (PEDs;
i.e., number of particles per unit volume and Lorentz factor $\gamma$) are generally written in the form
$n\left({\gamma}\right)\,=\,n_0\, (\gamma/\gamma_p)^{-2-r\log{\left(\gamma/\gamma_p\right)}}$,
where $\gamma$ is the electron Lorentz factor, $n_0$ the normalization,
$\gamma_p = \langle{\gamma^2}\rangle^{1/2}$
the mean particle energy (i.e., the peak of $\gamma^2~n(\gamma)$) and $r$ the electron curvature parameter.
Such PEDs represent the general solution of the energy and time dependent Fokker-Planck
kinetic equation, that includes systematic (e.g., electrostatic or electrodynamic)
and stochastic (e.g., turbulent) accelerations, together with radiative and adiabatic cooling as well as
particle escape and injection terms \citep{kardashev62,paggi09,tramacere11}.
In general, both the peak energy $E_p$ and the peak luminosity $L_p$ of a synchrotron SED
emitted by a curved electron distribution
depend on $\gamma_{3p}$, the peak energy of the distribution $\gamma^3~n(\gamma)$.
But in the case of a log-parabolic spectrum $\gamma_{3p}$ is proportional to $\gamma_p$ itself to imply
\begin{equation}
\gamma_{3p} = \gamma_p\,10^{1/2r} = \gamma_p\,10^{1/10b}\, ,
\label{eq:gammap}
\end{equation}
given that the relation between the curvature parameter $r$ of the PED and the homologous $b$
of the synchrotron SED reads simply $b =r/5$ \citep[e.g.,][]{massaro06}.
Then for a typical TBL {with} $b \simeq$ 0.3 or UBL {with} $b \simeq$ 0.7 \citep{massaro11a}
the ratio of $\gamma_{3p}$ to $\gamma_p$ is always $\lesssim$ 3.
\begin{figure}[!b]
\includegraphics[height=6.6cm,width=8.5cm,angle=0]{./histograms.pdf}
\caption{The $L_p$ distribution of UBLs (red) and TBLs (black).
TBL data do not include Mrk 421 and PKS 2155-304
and the giant flares of Mrk 501 and 1H 1426+421 \citep{massaro11a}.
The maximal separation $D_{KS}$ between the two cumulative distributions,
(i.e, the KS test variable) is reported.}
\label{fig:histograms}
\end{figure}
The synchrotron peak energy $E_p$ is generally proportional to $\gamma_{3p}^2\,B\,\delta$, where
$B$ is the mean magnetic field and $\delta$ the beaming factor $\delta = \Gamma^{-1}\,(1-\beta cos(\vartheta))^{-1}$,
with $\Gamma$ the bulk Lorentz factor in the jet and $\vartheta$ its opening angle
\citep[e.g.,][]{paggi09,tramacere09}.
In the case of a log-parabolic PED, this relation simplifies to the
proportionality $E_p\propto\gamma_{p}^2\,B\,\delta\,10^{1/r}$ {on account of Equation~\ref{eq:gammap}} \citep[e.g.,][]{massaro10,massaro11b}.
In addition, the synchrotron peak luminosity $L_p$ scales proportionally to
$n(\gamma_{3p})\,\gamma_{3p}^3\,B^2\,\delta^4$ when considering
variations of the PED, spectral shape \citep{tramacere11}. But,
in the case of log-parabolic PEDs, for the product
$n(\gamma_{3p})\,\gamma_{3p}^3\,=\,N_e\,\gamma_p^2\,f(r)$ obtains, where $N_e$ is the total number of emitting electrons, and
$f(r)$ is a spectral factor depending only on $r$ and ranging from 2.5 to 1.3
for the typical values of spectral curvature of the TBLs and UBLs, respectively.
We compare the $L_p$ distributions of the two HBL subclasses by
performing a KS test, and find that they are indistinguishable
at a confidence level of 99\% (see Figure \ref{fig:histograms}).
The $L_p$ distribution for the two HBL populations may still differ within one standard deviation.
As complementary test, we simulate the two distributions of $log(L_p)$
with the same number of events (i.e., 55 for TBLs and 76 for UBLs),
adopting the statistical approach described in Massaro et al. (2011a).
We considered two gaussian simulated distributions,
having the same median ($<log(L_p)>$(UBLs) = 43.80 and $<log(L_p)>$(TBLs) 44.17),
the same variance ($\sigma_{L_p}$(UBLs) = 0.06 and $\sigma_{L_p}$(TBLs) = 0.11) of
the observed distributions and spanning the same range of $L_p$.
Then, we measured the KS variable, $D_{\rm KS,simul}$, between the two simulated distributions.
We built the distribution of $D_{\rm KS,simul}$ repeating the simulations 8000 times and
we found that the probability to obtain, randomly, the observed $D_{\rm KS}$ is 86\%.
Thus, UBLs and TBLs have the same $E_p$ and $L_p$ distributions,
and differ, significantly, only as for the spectral curvature $b$.
Accordingly, we can assume that $\gamma_p$, $B$, and $\delta$ have close
values in both TBLs and UBLs.
Then, given the proportionality relations for $L_p$ and $E_p$ written above,
these observational results allow to consider a similar
number $N_e$ of emitting electrons for both HBL subclasses.
Moreover, the impact of limited variations of $L_p$ is softened on $\gamma_p$ and even more on $B$,
since we expect $B \sim \gamma_p^{1/2}$ (see Section~\ref{sec:systematic});
furthermore, the values of $E_p$ are closely the same for the two populations,
and so $L_p$ turns out to be proportional to $\gamma_p^{3/2}$.
\section{Electron acceleration in BL Lac objects}\label{sec:acceleration}
The characteristic energy \(\gamma_p\) of the PED, simply relates
to the synchrotron SED peak energy by \(E_p\propto \gamma_p^2 \)
as said in Section~\ref{sec:spectra},
is mainly set by the systematic acceleration component.
On the other hand, the stochastic acceleration mechanism
is responsible for the curvature \(r\) of the PED, related to the spectral width $b$.
In the following, we assume the acceleration mechanisms effective in BL Lac jets
to be a combination of systematic coherent acceleration,
responsible for the energy peak position of the PED,
and of stochastic acceleration, which accounts for the broadened
PED around its peak related to the lower spectral curvature.
We consider inverse Compton radiative losses
to be subdominant compared to those by synchrotron emission, consistent with the
observational lack of inverse Compton dominance in the HBL SEDs,
(e.g., PKS 0548-322, \citealt{aharonian10}, 1ES 0806+524, \citealt{acciari08}),
for which any external component appears to be necessary to describe their spectral evolution.
\subsection{Systematic acceleration}\label{sec:systematic}
According to \citet{cavaliere02}, BL Lac jets are likely to be powered by the
Blandford \& Znajek (BZ, \citeyear{blandford77}) {or {the} \citet{blandford82} {mechanism}}
(see also \citealt{lovelace76,ghosh1997,krolik1999,livio1999}), i.e., {by} the extraction of rotational {power}
from a spinning supermassive black hole via the Poynting flux associated
with the adjacent magnetosphere.
As discussed by the above authors, the simple force-free condition
${\underline E} \cdot {\underline B} = 0 $ governing these magnetospheres
is likely to break down at the jet boundaries, due to considerable electric fields
$E \leq B$ parallel to magnetic field (e.g. \citealt{cavaliere80});
such fields are present in the BZ configuration, especially at the jet boundary.
Alternatively, they may result from magnetic field reconnections
in current layers at the jet boundary (e.g. \citealt{litvinenko96,litvinenko99});
the related systematic acceleration mechanism is primarily electrostatic
(see \citealt{massaro11b} for the case of gamma-ray bursts (GRBs).
However, these electric fields will be electrodynamically screened
out by the embedding plasma at distances that exceed
the screening length $d$ defined by
\begin{equation}\label{eq4}
d = \frac{c}{\omega_p} = \left(\frac{\gamma_p~m_e~c^2}{4\pi~e^2~n}\right)^{1/2} = 5.3 \cdot 10^6 \left(\frac{\gamma_p}{n}\right)^{1/2}\mbox{ cm}\,,
\end{equation}
\citep{cavaliere02}. Here $\omega_p$ is the plasma frequency, $\gamma_p$ is the characteristic electron Lorentz factor,
$m_e$ is the electron mass, $e$ its electric charge, $c$ the speed of light, and $n$ the total electron
(e.g., \citealt{massaro11b}).
Accordingly, the electron energy gained for each acceleration step writes
\begin{equation}\label{eq5}
\gamma_a~m_e~c^2 \simeq e~B~d.
\end{equation}
{Assimilating \(\gamma_a=\gamma_p\),} from Equations \ref{eq4} and
\ref{eq5} we obtain the expression for the typical Lorentz factor of the accelerated electrons
\begin{equation}\label{gammap}
\gamma_p \simeq \frac{1}{4~\pi~m_e~c^2} \frac{B^2}{n} = 9.8 \cdot 10^4 \frac{B^2}{n}\, .
\end{equation}
The peak energy $E_p$
of the synchrotron emission for an electron of Lorentz factor $ \gamma_p \sim$ 10$^5$ falls
in the X-ray band, {on} adopting {for} the parameters standard values for HBLs
$\delta \sim$ 25, $n\sim 1$ cm$^{-3}$ and $B\sim$ 1 G
\citep[e.g.,][]{massaro06,paggi09,celotti08,acciari08,aharonian09}.
The maximal energy radiated by an electron with Lorentz factor 10$^5$ is $E\sim$ 0.05 TeV.
Then, considering again $\delta\simeq$ 25 we expect the peak of the inverse Compton emission to lie {at} around $E\sim$ 0.5 TeV.
This is consistent with all TBL observations that have a $\gamma$-ray photon index typically 2 or higher in the TeV energy range,
implying the energy peak of their inverse Compton component to lie below a few TeVs.
To complete the above scenario, the limiting electron Lorentz factor $\gamma_{M}$ attained by an electron corresponds to the condition
where the acceleration compensates the radiative {(mainly synchrotron)} losses. This occurs for
\begin{equation}
\gamma_{M} = \left(\frac{3\,e}{2\,\pi\,\sigma_T\,B}\right)^{1/2} =1.9 \cdot 10^{7} B^{-1/2}\, ,
\end{equation}
{at values considerably higher than given by Equation \ref{gammap} for our standard values of $B$ and $n$.}
Thus, the maximal energy available to the bulk of the electron population
is of order $\gamma_{M}\,m_e\,c^2$ $\sim$ 1 TeV for $B \sim$ 1G, in agreement with the observed TeV spectral ``tail" of TBLs.
Finally, we note that lower values of the magnetic field coupled with high Doppler factors have also been found to provide good fits
of the HBL spectra during high luminous states \citep[e.g.][]{finke08}.
\subsection{Geometry and timescales}\label{sec:geometry}
The simplest source condition obtains when the acceleration and the emitting region
are cospatial.
In particular, we suggest that the emission arises from thin sheaths of thickness $\Delta\,R$
that bound the jet {with radius \(R\gg \Delta R\),} as shown in Figure~\ref{fig:geometry}. There the
particle density is low, and the
screening length is sufficiently long as to allow the electrons to attain the required high energies
to emit up to TeV energies as discussed above.
In this scenario the relativistic aberration of light concentrates radiation isotropically emitted
in the comoving frame into a cone with opening angle $\vartheta \sim \Gamma^{-1}\ll 1$,
where $\Gamma$ is the bulk Lorentz factor of the jet.
On the other hand, only photons emitted within $\vartheta$ around the l.o.s. will be
detected by the observer.
The typical delay time $\tau_{d}$ between two photons emitted simultaneously in the
comoving frame from different points on the jet surface is
\begin{equation}\label{delay}
\tau_{d} = \frac{l}{c}\,[1-\cos(\Gamma^{-1})] \simeq 2\Gamma^2 \frac{l}{c}
\end{equation}
The variability timescale $t_{v}$ consistent with the delay time $t_{d}$
of the photons in the observer's frame implies an upper limit on the physical length
of the emitting region of order
\begin{equation}\label{delay}
l \simeq 2\Gamma^2 t_{v}\,c~.
\end{equation}
This differs from the one usually adopted based on the motion of the jet bulk toward
the observer and implying $l \simeq \delta t_{v}\,c$.
Such a ``flashlight" effect is analogous to the one presented by Ryde \& Petrosian (2002) in the case of GRBs,
but adapted to a cylindrical geometry, and so more like a flash along
the jet axis over $l \sim 10^{15}$ cm, produced by a relativistic shear instability.
Observed variability timescales $t'_{v}$ of order 10$^3$s imply
$l$ $\sim$ $R$ $\sim$ 10$^{15}$cm to hold for a bulk beaming
factor $\delta\sim$ 25.
This is also consistent with the synchrotron loss length
\begin{equation}
l_{s} \simeq 1.5 \cdot 10^{19} \gamma^{-1}B^{-2} cm\, ,
\label{eq:syn}
\end{equation}
of order 10$^{14}$cm for an electron with
$\gamma_p\,\sim$10$^5$ in a {field} $B\simeq$ 1 G.
Considering these emitting regions to lie at the base of the jet close to the supermassive black hole central to the AGN,
the jet radius $R$ can match the height $l$ (i.e., $l\leq R$, see Figure~\ref{fig:geometry}).
Assuming for the thickness of the effective acceleration and emitting region $\Delta R \sim$10$^{-2}\,R$,
the total volume filled by the emitting electrons is $V \sim \pi/2\,\Delta R\,l\,R \sim$10$^{44}$ cm$^{3}$,
close to the standard values considered for leptonic radiation processes of HBLs
(e.g., \citealt{massaro06,aharonian09}).
\begin{figure}[!htp]
\includegraphics[height=8.7cm,width=6.2cm,angle=-90]{./geometry.pdf}
\caption{A schematic view of the jet emitting region.}
\label{fig:geometry}
\end{figure}
\subsection{Stochastic acceleration}\label{sec:stochastic}
The observational evidence reported in Section \ref{sec:intro} that UBLs feature systematically narrower spectra compared to
TBLs may be interpreted in terms of {a} less efficient stochastic acceleration occurring in the former's jets.
In fact, the curvature parameter of the PED is related to the stochastic acceleration term in
a Fokker-Planck kinetic equation \citep[e.g.,][]{kardashev62,tramacere07,tramacere09,paggi09,tramacere11},
and is inversely proportional to the stochastic acceleration rate;
thus the synchrotron SEDs are relatively broader when the stochastic acceleration is more efficient.
Specifically, the PED curvature $r$
is directly related to the diffusion coefficient $D$ in the Fokker-Planck kinetic equation by
$r \propto D^{-1}$. Higher values of $D$ and faster diffusion also imply
less time spent in the acceleration region \citep[see][]{ginzburg64,protheroe04}.
In a simple statistical picture, $r$ is proportional to the energy gain $\epsilon$ itself,
while it is inversely proportional to the number $n_s$ of acceleration steps, and
to the variance $\sigma^2_{\epsilon}$ of the energy gain; in sum,
$r\propto \epsilon / (\sigma^2_{\epsilon}\,n_s)$ \citep{massaro04,tramacere11}.
As TBLs and UBLs show similar $E_p$ distributions, we can assume that both subclasses
have similar values of $B$ and of {the} $\epsilon/\sigma^2_{\epsilon}$ ratio.
Thus higher values of $n_s$ correspond to lower values of $r$;
we suggest such high values of $n_s$ to be comparable with smaller acceleration regions,
since each acceleration step is shorter.
So, the observational evidence that $b$ is systematically larger
in UBLs than in TBLs is consistent with larger volumes $V$ for the former than for the latter.
Finally, we remark that while the average magnetic field is comparable in TBLs and UBLs,
the difference is due to the small scale fluctuations in the power spectrum.
On large scales B is related to the electric field (see Section~\ref{sec:systematic})
which is responsible for the systematic acceleration;
on small scales, a turbulent component gives rise to stochastic diffusion
(see e.g., Brunetti \& Lazarian 2011, for a related approach concerning radio volumes).
The latter component
yields different numbers of acceleration steps in TBLs and UBLs, but averages out on large scales.
On the other hand, considering that the similarities between the $E_p$ and $L_p$ distributions of TBLs and UBLs
have been interpreted in terms of similar
numbers of emitting electrons $N_e$ (see Sect.~\ref{sec:spectra}),
the curvature-volume relation described above suggests that the electron density $n_e\,=\,N_e/V$
is larger in TBLs
than in UBLs, making a \textit{brighter} inverse Compton peak in a SSC scenario.
We conclude that the TBLs, not only have
\textit{wider} X-ray spectra, but are also expected to be \textit{brighter} in $\gamma$-rays than UBLs.
In addition, the diffusion/acceleration timescale is inversely proportional to the diffusion coefficient; so low values of $D$
will correspond to less variable sources.
This feature is also consistent with the lower variability and the lack of giant X-ray flares found in the whole sample of UBLs
in comparison with the TBLs \citep{massaro11a}.
\section{{Conclusions and} Discussion}\label{sec:discussion}
In \citet{massaro11a}, we analyzed and compared the X-ray spectral properties of TBLs and UBLs,
finding that they have \textit{similar} $E_p$ distributions,
{both} symmetric around a value of few keVs, while
the X-ray spectral curvature $b$ is systematically lower in the former.
In this Letter, we have compared the $L_p$ distributions of the two HBL subclasses, finding them
similar at high level of confidence level.
Then UBLs and TBLs differ mainly as for the spectral curvature $b$;
these observational results likely imply similar numbers $N_e$ of emitting
electrons for both subclasses.
We have proposed to interpret the $E_p$ and $b$ distributions on
assuming that the electron energy gain is due to
both coherent and stochastic particle accelerations.
The scenario is based on re-acceleration rather than
continuous injection of fresh highly relativistic electrons;
re-acceleration occurs via both systematic and stochastic mechanisms,
with equilibrium occurring between the overall acceleration rate and the radiative losses.
Describing the coherent acceleration in terms of energy gain from an electric field, we have derived a relation
for the expected particle Lorentz factors $\gamma_p$ $\lesssim$ 10$^5$.
Thus for a typical HBL with magnetic {fields} B $\sim$1 G, plasma density $n\sim$ 1 cm$^{-3}$
and a beaming factor $\delta$ $\sim$ 25, the expected synchrotron peak energy is at $E_p\sim$ 1 keV,
as in fact observed in {the} X-ray SEDs of HBLs \citep{massaro08,massaro11a}.
On the other hand, the stochastic acceleration component is mainly responsible for
spectral broadening around $E_p$.
In fact, the curvature $b$ of the X-ray spectra is \textit{only} dependent on
the stochastic acceleration term in a Fokker-Planck equation, and thus is inversely proportional to the diffusion coefficient $D$
and to the stochastic acceleration rate $\rho_{acc}$, that is, $b \propto \rho_{acc}^{-1}$.
Thus, we interpret the narrow X-ray SEDs of UBLs in terms of less efficient stochastic acceleration compared to TBLs.
Finally, pursuing the stochastic acceleration scenario
we have linked the curvature parameter $b$ to the volume of the emitting region, through its
inverse proportionality to the number of acceleration steps $n_s$ (see Section~\ref{sec:stochastic}).
This curvature-volume relation, combined with the above consideration
of similar values of $N_e$, indicates the emitting electron density to be larger in
TBLs than in UBLs, making the inverse Compton peak \textit{brighter} in the former than in the latter.
Thus electron energies sufficiently \textit{high} to radiate in the TeV range are related to
sufficient \textit{bright} luminosities for effective detection. Conversely, {narrower}
SEDs and lower fluxes make UBLs harder to
detect in the TeV range than TBLs, in agreement
with our previous results concerning their X-ray observations.
\acknowledgements
We thank the referee for the specific suggestions that improved our manuscript.
FM thanks M. Elvis, M. Petrera, J. E. Grindlay, M. Murgia, G. Brunetti and A. Tramacere for fruitful discussions.
FM acknowledges the Fondazione Angelo Della Riccia for the grant awarded him during 2011;
The work at SAO is supported by the NASA grant NNX10AD50G and by the Foundation BLANCEFLOR Boncompagni-Ludovisi, n'ee Bildt .
~
|
{
"timestamp": "2012-03-12T01:00:06",
"yymm": "1203",
"arxiv_id": "1203.1924",
"language": "en",
"url": "https://arxiv.org/abs/1203.1924"
}
|
\section{Stirling's formula and binomial coefficients}
For a nonnegative integer $m$, Stirling's formula
\begin{align*}
m! \sim \sqrt{2\pi m}\Bigl(\frac{m}{e}\Bigr)^m,
\end{align*}
where $e$ is Euler's number, yields an approximation
of the central
binomial coefficient $\binom{m}{m/2}$ (assuming that $m$ is
even)
using $\binom{m}{k}=\frac{m!}{k!(m-k)!}$
as
\begin{align*}
\binom{m}{m/2} \sim \frac{2^{m+1}}{\sqrt{2\pi m}},
\end{align*}
where we write $a_m\sim b_m$
as a short-hand for $\lim_{m\goesto\infty} \frac{a_m}{b_m}=1$. In the
current note, we derive an asymptotic formula for the central
\emph{multinomial triangle} or,
\emph{polynomial}, \emph{coefficient} (cf. \cite{caiado}, \cite{comtet}),
where multinomial triangles
are a generalization of binomial triangles, where entries in row $k$ are
defined as coefficients of the polynomial $(1+x+x^2+\dotsc+x^l)^k$ for
$l\ge 0$. Our derivation
is not based upon asymptotics of factorials but
upon the limiting distribution of
the sum of discrete uniform random variables.\footnote{Throughout, we assume that $m$ or $l$
are even. If this is not the case, replace respective
quantities, e.g. $\frac{ml}{2}$,
with their floor, $\lfloor\frac{ml}{2}\rfloor$.}
Polynomial coefficients are important, for example, because they
denote the number of \emph{integer compositions} of the nonnegative
integer $n$ with parts in the set
$A:=\set{a,a+1,\dotsc,b}$, where $a,b$, $0\le a\le b$, are
nonnegative integers
(cf. \cite{eger}), i.e. the number of possibilities to write
$n$ as a sum of
integers $p_1,\dotsc,p_k$, with $p_i\in A$, for $i=1,\dotsc,k$.
\section{Multinomial triangles}
In generalization to binomial triangles, $(l+1)$-nomial
triangles, $l\ge 0$, are defined in the following way. Starting with a
$1$ in row zero, construct an entry in row $k$, $k\ge 1$, by adding the
overlying $(l+1)$ entries in row $(k-1)$ (some of these entries are
taken as zero if not defined); thereby, row $k$ has
$(kl+1)$ entries. For example, the monomial ($l=0$), binomial ($l=1$),
trinomial ($l=2$) and
quadrinomial triangles ($l=3$) start as follows,
\begin{table}[!h]
\begin{tabular}{r}
1\\ 1\\ 1\\ 1
\end{tabular}\hspace{0.5cm}
\begin{tabular}{rrrr}
1\\
1 & 1\\
1 & 2 & 1\\
1 & 3 & 3 & 1\\
\end{tabular}\hspace{0.5cm}
\begin{tabular}{rrrrrrr}
1\\
1 & 1 & 1\\
1 & 2 & 3 & 2 & 1\\
1 & 3 & 6 & 7 & 6 & 3 & 1\\
\end{tabular}\hspace{0.5cm}
\begin{tabular}{rrrrrrrrrr}
1\\
1 & 1 & 1 & 1\\
1 & 2 & 3 & 4 & 3 & 2 & 1\\
1 & 3 & 6 & 10 & 12 & 12 & 10 & 6 & 3 & 1\\
\end{tabular}
\end{table}
In the $(l+1)$-nomial triangle, entry $n$, $0\le n\le kl$, in row $k$, which we denote
by $\binom{k}{n}_{l+1}$, has the following interpretation. It
is the coefficient of $x^n$ in the expansion of
\begin{align}\label{eq:multinomial_coeff1}
(1+x+x^2+\dotsc+x^l)^k = \sum_{n=0}^{kl} \binom{k}{n}_{l+1}x^n.
\end{align}
In recent work, Eger \cite{eger} has shown that $\binom{k}{n}_{l+1}$ denotes
the number of {integer compositions} of the nonnegative integer $n$ with
$k$ parts $p_1,\dotsc,p_k$, each from the set $\set{0,1,\dotsc,l}$,
and allows the
following representation,
\begin{align}\label{eq:repr}
\binom{k}{n}_{l+1} = \sum_{\overset{k_0+\dotsc+k_l=k}{0\cdot k_0+1\cdot
k_1+\dotsc+l\cdot
k_l=n}} \begin{pmatrix}k\\ k_0,\dotsc,k_l\end{pmatrix},
\end{align}
where $\begin{pmatrix}k\\ k_0,\dotsc,k_l\end{pmatrix}$ is a
\emph{multinomial coefficient}, defined as $\frac{k!}{k_0!\dotsc
k_l!}$, for nonnegative integers $k_0,\dotsc,k_l$. We can easily
verify representation \eqref{eq:repr} by noting that for real
numbers $x_0,\dotsc,x_l$, it holds that (cf. \cite{weisstein})
\begin{align*}
(x_0+x_1+\dotsc+x_l)^k = \sum_{\overset{k_0,\dotsc,k_l\ge
0}{k_0+\dotsc+k_l=k}}\binom{k}{k_0,\dotsc,k_l}x_0^{k_0}\dotsc x_l^{k_l}.
\end{align*}
Then setting $x_i=x^i$ for $i=0,\dotsc,l$,
\begin{align}\label{eq:multinomial_coeff2}
(1+x+x^2+\dotsc+x^l)^k = \sum_{\overset{k_0,\dotsc,k_l\ge
0}{k_0+\dotsc+k_l=k}}\binom{k}{k_0,\dotsc,k_l}x^{0\cdot
k_0+\dotsc+l\cdot k_l},
\end{align}
and comparing coefficients of the right-hand sides of
\eqref{eq:multinomial_coeff1} and \eqref{eq:multinomial_coeff2}
leads to \eqref{eq:repr}.
\section{Generalized Stirling's formula}
Our strategy for deriving approximation formulae for
polynomial coefficients is as follows. First, we determine the
asymptotic
distribution of the sum of discrete uniform variables, which we easily
find to be a normal distribution by the Central Limit Theorem (CLT). Then we
determine the exact distribution, which turns out to yield the
normalized polynomial coefficients $\binom{m}{n}_{l+1}$. This
distribution has recently been found by Caiado and Rathie
\cite{caiado} using characteristic functions, residue theory and
Cauchy's theorem, but we give an elementary two-line proof that almost
requires no effort at all. By relating the asymptotic distribution to
the exact distribution, we obtain a multinomial analogue of Stirling's
approximation to the central binomial coefficient.
\subsection{Distribution of the sum of discrete uniform variables}
Let $m$ be a positive integer and let $l$ be a nonnegative
integer.
Let $S_j$, $j=1,\dotsc,m$, be identically and independently
distributed random draws from the discrete
uniform distribution on the set $\set{0,\dotsc,l}$, and let $S^{(m)}$ be
their sum,
\begin{align*}
S^{(m)} = \sum_{j=1}^m S_j.
\end{align*}
Obviously, the mean and variance of $S^{(m)}$ are given by
\begin{align*}
\mu_{m,l}:=\Exp[S^{(m)}] = m\Exp[S_j] = \frac{ml}{2},\quad\quad\sigma^2_{m,l}:=\Var[S^{(m)}] = m\Var[S_j] =
m\frac{(l+1)^2-1}{12},
\end{align*}
by standard moments of the uniform distribution. Moreover, by the
CLT
\begin{align*}
{\sqrt{m}}\Bigl(\frac{S^{(m)}}{m}-\frac{l}{2}\Bigr)\goesto
\mathcal{N}(0,\frac{(l+1)^2-1}{12}) \quad\text{as } m\goesto\infty,
\end{align*}
so that for large $m$, $S^{(m)}$ is approximately normally distributed
with mean $\frac{ml}{2}$ and variance $m\frac{(l+1)^2-1}{12}$.
However, we can also exactly determine the probability that $S^{(m)}$
takes on the integer value $n$, for $0\le n\le ml$.
For this purpose, interpret all $S_j$,
$j=1,\dotsc,m$,
as
independently and identically \emph{multinomially}
distributed with probabilities
$p_0=\dotsc=p_l=\frac{1}{l+1}$ of `events' $0$ to $l$ occurring,
i.e. each $S_j$ is vector-valued, $S_j=(A_0,\dotsc,A_l)$, with
$P[S_j=(a_0,\dotsc,a_l)]=\frac{1}{l+1}$ for $a_0+\dotsc+a_l=1$,
$a_k\ge 0$, where $A_k$ denotes the number of times event $k$ occurs,
for $k=0,\dotsc,l$.
Hence,
the
sum $S^{(m)}$ is multinomially distributed with probability mass
functio
\begin{align*}
P[S^{(m)}=(a_0,\dotsc,a_l)]=P[A_0=a_0,\dotsc,A_l=a_l] =
\binom{m}{a_0,\dotsc,a_l}\Bigl(\frac{1}{l+1}\Bigr)^m,
\end{align*}
where
$a_0+\dotsc+a_l=m$.
Then, if $S^{(m)}=(a_0,\dotsc,a_l)$ for nonnegative integers
$a_0,\dotsc,a_l$ with $a_0+\dotsc+a_l=m$, $S^{(m)}$ `represents' the
integer $0\cdot
a_0+\dotsc+l\cdot a_l$.
Thus, for $n$ such that $0\le n\le ml$,
we obviously have, reinterpreting $S^{(m)}$ as scalar-valued,
\begin{align*}
P[S^{(m)}=n] = \sum_{\overset{a_0+\dotsc+a_l=m}{0\cdot
a_0+\dotsc+l\cdot
a_l=n}}P[S^{(m)}=(a_0,\dotsc,a_l)]=\Bigl(\frac{1}{l+1}\Bigr)^m\sum_{\overset{a_0+\dotsc,a_l=m}{0\cdot
a_0+\dotsc+l\cdot
a_l=n}}\binom{m}{a_0,\dotsc,a_l}=\Bigl(\frac{1}{l+1}\Bigr)^m\binom{m}{n}_{l+1}.
\end{align*}
\subsection{Approximation Formulae}
From our above derivations, we know that the normalized
random variable $S^{(m)}$
converges in distribution to a normally distributed random variable,
which precisely means that the distribution function of $S^{(m)}$
converges pointwise to the distribution function of the corresponding
normally distributed random variable.
Let $0\le n\le ml$ and consider
\begin{align*}
P[S^{(m)}=n] = P[S^{(m)}\le n]-P[S^{(m)}\le n-1].
\end{align*}
By convergence of distribution functions, we hence know that, for
large $m$,
\begin{align}\label{eq:approx}
P[S^{(m)}=n] \approx P[n-1\le X\le n] \approx P[n-1/2\le X\le n+1/2],
\end{align}
where $X$ has a normal distribution with the same mean and variance as
$S^{(m)}$, and applying a continuity correction
(cf. \cite{walsh}). Rewriting the
right hand side of \eqref{eq:approx} in terms of the standard normal
variable $Z=\frac{X-\mu_{m,l}}{\sigma_{m,l}}$, we thus obtain for
$n=\mu_{m,l}$,
\begin{align*}
P[S^{(m)}=\frac{ml}{2}] \approx P[-\frac{1}{2\sigma_{m,l}}\le Z\le
+\frac{1}{2\sigma_{m,l}}] = \Phi\left(\frac{1}{2\sigma_{m,l}}\right)-\Phi\left(-\frac{1}{2\sigma_{m,l}}\right),
\end{align*}
where $\Phi$ is the standard normal cumulative distribution
function. Geometrically,
$\Phi\left(\frac{1}{2\sigma_{m,l}}\right)-\Phi\left(-\frac{1}{2\sigma_{m,l}}\right)$
denotes the area under the normal curve
$\phi(z)=1/\sqrt{2\pi}\exp(-z^2/2)$ from $-\frac{1}{2\sigma_{m,l}}$ to
$+\frac{1}{2\sigma_{m,l}}$. For large $m$, this area can be
approximated by the area of the rectangle with width
$\frac{1}{\sigma_{m,l}}$ and
height $\phi(0)$, that is,
\begin{align*}
P[S^{(m)}=\frac{ml}{2}] \approx \frac{1}{\sqrt{2\pi
m\frac{(l+1)^2-1}{12}}}.
\end{align*}
Using the expression for $P[S^{(m)}=n]$ from above,
we hence have
\begin{align*}
\binom{m}{\frac{ml}{2}}_{l+1} \sim \frac{(l+1)^m}{\sqrt{2\pi
m\frac{(l+1)^2-1}{12}}}.
\end{align*}
For $l=1$, Pascal's case, $l=2$, $l=3$, and $l=4$, we therefore have
the approximations
\begin{align*}
\binom{m}{\frac{m}{2}} \sim \frac{2^{m+1}}{\sqrt{ 2\pi m}},\quad\:\:\:
\binom{m}{m}_{3} \sim \frac{3^m}{\sqrt{\frac{4}{3}\pi m}},\quad\:\:\:
\binom{m}{\frac{3}{2}m}_{4} \sim \frac{4^m}{\sqrt{\frac{5}{2}\pi
m}},\quad\:\:\:
\binom{m}{2m}_{5} \sim \frac{5^m}{2\sqrt{\pi m}}.
\end{align*}
\begin{figure}[!h]
\centering
\includegraphics[scale=.21]{normalApproxl4_m5.png}
\includegraphics[scale=.21]{normalApproxl4_m10.png}
\includegraphics[scale=.21]{normalApproxl4_m20.png}
\label{figure:plots1}
\caption{Distributions $P[S^{(m)}=n]$ for $m=5,10,20$ for $l=4$ fixed;
and normal approximations.}
\end{figure}
In Figure \ref{figure:plots1}, we show for $l=4$ the distributions $P[S^{(m)}=n]$
for $m=5,10,20$ and their respective normal approximations.
These plots suggest to more generally consider a pointwise
approximation of $P[S^{(m)}=n]$, for $0\le n\le ml$, by the value
$f(n)$ of the density of the corresponding normally distributed random
variable $X$, leading to the following approximation formulae for
polynomial coefficients $\binom{m}{n}_{l+1}$,
\begin{align*}
\binom{m}{n}_{l+1} \approx \frac{(l+1)^m}{\sqrt{2\pi
\sigma_{m,l}^2}}\exp(-\frac{(n-\mu_{m,l})^2}{2\sigma_{m,l}^2}).
\end{align*}
How well are these `normal approximations' $\hat{a}_{m,n,l}$ of
$a_{m,n,l}:=\binom{m}{n}_{l+1}$ for various
values of $n$ for finite $m$? In Figure \ref{figure:plots}, we depict
the relative
error $\abs{\frac{\hat{a}_{m,n,l}-a_{m,n,l}}{a_{m,n,l}}}$ for
all values of $n$ between $0$ and $ml$ for $m=100,200$ and $l=3,4$. We
see that the relative error becomes very low for $n$ `around'
$\frac{ml}{2}$ and increases symmetrically
as $n$ approaches the boundary values
$0$ and $ml$. In the same figure, we show the decrease of
the relative error for the central polynomial coefficient as a
function of $m$.
\begin{figure}[!h]
\centering
\includegraphics[scale=.33]{relativeError.png}
\includegraphics[scale=.33]{centralMultinomial.png}
\label{figure:plots}
\caption{Left: Relative error of normal approximation of
$\binom{m}{n}_{l+1}$ for
$m=100,200$ and $l=3,4$ as a function of $n$. Right: Relative
error for central polynomial coefficient as a function of $m$ for
$l=4$.}
\end{figure}
|
{
"timestamp": "2012-03-12T01:02:02",
"yymm": "1203",
"arxiv_id": "1203.2122",
"language": "en",
"url": "https://arxiv.org/abs/1203.2122"
}
|
\section{Introduction}
Disorder---an inevitable feature of nature---is known to induce
striking slowing down phenomena in transport processes and
in the relaxation of
systems with many degrees of freedom \cite{havlin,bouchaud,im}.
The source of complexity in such systems is twofold.
First, the dynamics can be regarded as
a random walk in the configuration space, which is, in general,
rather complicated. This problem also arises directly in
the context of transport processes, which usually take place on
inhomogeneous structures in reality \cite{bouchaud}.
Second, the system is frequently subject to an external
source of disorder, such
as a random force-field, which can be modeled by quenched
(i.e. time-independent) random transition rates.
The theoretical description of the dynamics of such systems
is a challenging task
since, in most cases, the complexity induced by disorder makes
the application of analytical techniques of clean systems extremely hard.
Diffusion in the presence of the most general form of disorder, where
transitions between pairs of states are non-symmetric, is non-trivial
even on regular $d$-dimensional lattices.
In the simplest case of $d=1$, where a single-valued potential exists,
the position $x(t)$ of the walker varies ultra-slowly with time, obeying
\be
\overline{\langle x^2(t)\rangle}\sim (\ln t)^{2/\psi},
\label{log}
\end{equation}
when the average force acting on the walker is zero \cite{sinai}.
Here, $\langle\cdot\rangle$ denotes an average over different stochastic
histories,
whereas the overbar denotes an average over the random transition rates.
This is Sinai's diffusion law, where the value $\psi=1/2$ of the
barrier exponent is related to the Gaussian fluctuations of the potential
landscape.
In higher dimensions, no potential can be defined, and the system
is a genuinely non-equilibrium process.
For increasing $d$, the effects of disorder are expected to be less pronounced
as there are more and more paths connecting the initial and final state.
Weak disorder expansions \cite{dl} and perturbative renormalization group
analyses \cite{luck,fisher} agree in that,
for $d\ge d_c=2$, the diffusion remains
normal, i.e. $\overline{\langle {\bf x}^2(t)\rangle}\sim t$, with logarithmic
corrections at the critical dimension $d_c=2$, while for $d<2$ the disorder is
relevant and results in sub-diffusion.
Besides studying directly $x(t)$, the inverse question of how large the mean first-passage
time (MFPT) from a given initial state to a final one is frequently asked in
such systems \cite{redner}. In probing the dynamics, the finite-size scaling
of the MFPT is an alternative possibility as, in general, it obeys the same
dynamical relation between time and length scales as $x(t)$ does.
In transition networks other than regular lattices neither a general
criterion for the relevance of asymmetric disorder
nor the dynamics in case of relevance are known.
Our aim in this letter is to study these questions in general.
We will prove an exact relationship between the strength of weak asymmetry in the transition
rates and the effective resistance of the corresponding resistor network,
by which a relevance criterion can be formulated in terms of
the sign of the resistance exponent $\zeta$.
This will be then demonstrated by numerical calculations
in various random and non-random fractal lattices.
In the case of relevance ($\zeta>0$), the dynamics are found to be logarithmic
given by Eq.(\ref{log}) with a barrier exponent $\psi$ that is characteristic
of the underlying lattice.
\section{The model and its renormalization}
To formulate the above statements precisely,
let us assume that the system has a finite number of states labeled
by the integers
$i=1,2,\dots,N$ and consider a continuous-time random walk on them,
where the transition
rates ($p_{ij},p_{ji}$) are from link to link independent random variables.
The rates $p_{ij}$ and $p_{ji}$ are
allowed to be different but their distributions are required to be
identical, so that
the average local 'force' is zero,
i.e.
\be
\overline{F_{ij}}\equiv\overline{\ln(p_{ij}/p_{ji})}=0
\label{symm}
\end{equation}
on each link $(ij)$.
For the sake of simplicity, we assume, furthermore, that
every state is reachable from every other one.
The central quantity in a fixed realization of the
transition network is the MFPT, $\tau_i^{(A)}$, which is
the expected value of the time needed to first reach any state in a fixed
set $A$ of target sites when starting from state $i$.
These quantities with different starting state $i$ obey the
following backward master equation \cite{redner}:
\be
\sum_ip_{ji}\tau_i^{(A)}-\sum_ip_{ji}\tau_j^{(A)}=-K_j,
\label{backward}
\end{equation}
with $K_i=1$ for all $i$ and the boundary conditions $\tau_i^{(A)}=0$ for all
$i\in A$.
As it has been recently pointed out in ref. \cite{monthus},
this form of the equation enables one to calculate $\tau_1^{(A)}$
by recursively eliminating all states other than $1$ and $A$ one after the
other in a way closely related to strong disorder
renormalization group (SDRG) methods \cite{im}.
When eliminating state $k$,
direct transitions between states that were linked with state $k$
are generated with the rates:
\be
\tilde p_{ij}^0 = p_{ik}p_{kj}/\sum_ip_{ki}. \qquad (generation)
\label{p_ruleA}
\end{equation}
If the link $(ij)$ has already existed before the elimination of $k$ than its
rates $p_{ij}$ are added to the newly generated ones, so
that the final renormalized rates read as:
\be
\tilde p_{ij} = p_{ij} +\tilde p_{ij}^0. \qquad (addition)
\label{p_ruleB}
\end{equation}
The parameters $K_i$ at those states that were linked
with state $k$ are also renormalized
as $\tilde K_i = K_i + p_{ik}K_k/\sum_ip_{ki}$.
When all states except of $1$ and $A$ have been eliminated, the
MFPT can be calculated by
$\tau_1^{(A)}=\tilde K_1/\sum_{i\in A}\tilde p_{1i}$
\footnote{The denominator here has a direct probabilistic interpretation:
$\sum_{i\in A}\tilde p_{1i}/\sum_{i}p_{1i}$ is the probability that
the system, leaving state $1$, will reach state $A$ earlier than
state $1$.}.
In the simple case $d=1$, the generation rule in Eq. (\ref{p_ruleA})
is the only operation and its
product form results in a rapid decrease of effective rates under the
elimination (or renormalization) procedure that signals slow dynamics.
In more complex networks, also the addition rule given in Eq. (\ref{p_ruleB})
is applied since alternative
paths from state $i$ to $j$ which do not go through $k$ may exist.
The more connected the network is the more frequently the addition occurs and the slower the effective rate decreases under renormalization.
\section{Renormalization of the weakly asymmetric model}
In general, the recursions described above cannot be solved analytically.
Nevertheless, when merely seeking an answer to whether disorder is
relevant, the analysis can be greatly simplified as follows.
It is known in the case $d=1$ that any weak disorder
drives the system to the infinite-randomness fixed point of the SDRG
\cite{im} that describes the logarithmic behavior in Eq. (\ref{log}).
Moreover, the behavior of the asymmetry in the effective transition rates
alone reflects the logarithmic dynamics as
$|\ln(\tilde p_{1N}/\tilde p_{N1})|\sim N^{1/2}$,
and indicates the relevance of weak disorder.
Therefore, we shall deal only with a weak,
asymmetric perturbation of the symmetric model of
the form $p_{ij}/p_{ji}\equiv 1+\epsilon_{ij}$ with $\epsilon_{ij}$
infinitesimally small, and
keep track of the renormalization of $p_{ij}$ and $\epsilon_{ij}$.
The transformation rules then read in leading order in $\epsilon_{ij}$ as
\beqn
\tilde\epsilon_{ij}^0 = \epsilon_{ik} + \epsilon_{kj}, \qquad (generation)
\label{e_ruleA} \\
\tilde p_{ij}\tilde\epsilon_{ij} = p_{ij}\epsilon_{ij} + \tilde p_{ij}^0\tilde\epsilon_{ij}^0, \qquad (addition)
\label{e_ruleB}
\end{eqnarray}
whereas the rates $p_{ij}$ still obey
Eqs. (\ref{p_ruleA}-\ref{p_ruleB}), however, they are
now symmetric, $p_{ij}=p_{ji}$
and transform identically to the reduction rules
of a resistor network with resistances $r_{ij}\equiv 1/p_{ij}$ on links $(ij)$.
Let us consider an ensemble of networks with fixed resistances $r_{ij}$ and random variables $\epsilon_{ij}$ on each link, for which
Eq. (\ref{symm}) implies $\overline{\epsilon_{ij}}=0$, the overbar now
denoting the average over $\epsilon_{ij}$ on link $(ij)$.
In order to prove general statements we need
to allow $\epsilon_{ij}$ to be non-identically distributed on different links
and to require that
$\overline{\epsilon^2_{ij}}=\alpha r_{ij}$, where $\alpha$ is an
infinitesimally small global constant.
\section{Relationship with the two-point resistance}
First, let us consider a special class of networks which
can be reduced to two fixed states, $a$ and $b$,
by exclusively eliminating states with {\it two} links.
An example is the hierarchical diamond lattice illustrated
in Fig. \ref{fractal}.
It is easy to see that, here, the generation and addition steps
are equivalent to the well-known reduction of
resistors in series and in parallel, respectively, and follow the simple rules:
\beqn
\tilde r=r_1+r_2, \quad \tilde\epsilon = \epsilon_{1} + \epsilon_{2}
\qquad \qquad (generation)
\\
\tilde r^{-1}=r_1^{-1}+r_2^{-1},
\quad \tilde r^{-1}\tilde\epsilon = r_1^{-1}\epsilon_{1} + r_2^{-1}\epsilon_{2}
\quad (addition).
\end{eqnarray}
When any of the above operations is performed,
$\epsilon_{1}$ and $\epsilon_{2}$ are always
independent, $\overline{\epsilon_1\epsilon_2}=0$, so one easily obtains that
\be
\overline{\tilde\epsilon_{ij}^2}=\alpha\tilde r_{ij}
\label{relation}
\end{equation}
remains valid at any stage of the renormalization procedure,
all the way to the last link connecting states $a$ and $b$:
\be
\overline{\tilde\epsilon_{ab}^2}=\alpha\tilde r_{ab}.
\label{last}
\end{equation}
For arbitrary networks, the elimination of states with more than two links
cannot be avoided in general. When decimating a state with
$n>2$ links, the relation
Eq. (\ref{relation}) will be broken for the modified $n(n-1)/2$ links,
moreover, their rates will become correlated with each other.
Surprisingly, when the network is reduced to two (arbitrary) states,
$a$ and $b$, the relationship
in Eq. (\ref{last}) will still hold.
In the following, it will be convenient to introduce the scaled asymmetry
parameters $\omega_{ij}\equiv p_{ij}\epsilon_{ij}$,
which satisfy initially $\overline{\omega_{ij}^2}=\alpha p_{ij}$.
It is sufficient to prove Eq. (\ref{last}) for complete networks (in which every state
is linked with every other one) and for an arbitrary set of
rates $\{p_{ij}\}$,
then it applies to any other network by
setting $p_{ij}=\overline{\omega_{ij}^2}=0$ for appropriate links,
which amounts to deleting that link.
For $N=3$, the statement is true since this network
belongs to the special class,
while for $N>3$ we shall prove it by induction.
Assume the Eq. (\ref{last}) holds for a fixed $N$.
It follows from Eqs. (\ref{e_ruleA}-\ref{e_ruleB}) that the final
asymmetry parameter is a linear combination of the initial ones:
$\tilde \omega_{ab}^{(N)}(\{p_{ij}\})=\sum_{i<j}C_{ij}^{(N)}(\{p_{ij}\})\omega_{ij}$, where the
coefficients are functions of the set of rates and the summation goes over all
links.
Using this, Eq. (\ref{last}) can be rewritten as
$\sum_{i<j}[C_{ij}^{(N)}(\{p_{ij}\})]^2p_{ij}=\tilde p_{ab}^{(N)}(\{p_{ij}\})$.
Now let us extend the network to a one state larger one and keep the
notation of parameters on links of the $N$-state subgraph, while
denoting the rates and the scaled asymmetry parameters on the links
from the old state $i$ to the new one by $q_{i}$ and $\phi_i$,
$i=1,2,\dots,N$, respectively.
If the new state is eliminated, then effective rates
$P_{ij}=q_iq_j/\sum_{i=1}^Nq_i$ and scaled asymmetry parameters
$\Omega_{ij}=(q_j\phi_i-q_i\phi_j)/\sum_{i=1}^Nq_i$ are generated according to
Eqs. (\ref{p_ruleA}) and (\ref{e_ruleA}) at all old links, and are added to
the old parameters as given in Eqs. (\ref{p_ruleB}) and (\ref{e_ruleB}).
This yields for the parameters of the system with $N+1$ states:
\beqn
\tilde p_{ab}^{(N+1)}(\{p_{ij}\},\{q_{i}\})=
\textstyle\sum_{i<j}C_{ij}^2(p_{ij}+P_{ij}),
\label{pN1} \\
\tilde\omega_{ab}^{(N+1)}(\{p_{ij}\},\{q_i\})=\textstyle\sum_{i<j}C_{ij}(\omega_{ij}+\Omega_{ij}),
\label{omN1}
\end{eqnarray}
where we have used the shorthand notation
$C_{ij}$ for $C_{ij}^{(N)}(\{p_{ij}+P_{ij}\})$.
It is to be shown that the expected value of the
square of Eq. (\ref{omN1}) is
equal to Eq. (\ref{pN1}) (multiplied by $\alpha$) for any $\{p_{ij}\},\{q_i\}$.
Using that
$\{\omega_{ij}\}$ and $\{\Omega_{ij}\}$ are independent and
$\overline{\omega_{ij}^2}=\alpha p_{ij}$, we obtain that
this is equivalent to
\be
\alpha\textstyle\sum_{i<j}C_{ij}^2P_{ij}=
\overline{(\textstyle \sum_{i<j}C_{ij}\Omega_{ij})^2}.
\label{lemma}
\end{equation}
Expanding $\Omega_{ij}$ in terms of $\phi_i$ and using
$\overline{\phi_i^2}=\alpha q_i$,
the r.h.s. assumes the form
$\alpha\sum_{i=1}^NG_i^2q_i$ with the coefficients
$G_i=\sum_{j\neq i}q_jC_{ij}/\sum_jq_j$, where $C_{ij}$ for $i>j$ is
defined as $C_{ij}\equiv -C_{ji}$.
Then Eq. (\ref{lemma}) can be rewritten after some algebra in the form
\be
\sum_{i<j<k}q_iq_jq_k(C_{ij}+C_{jk}+C_{ki})^2=0,
\label{sumprod}
\end{equation}
where the summation goes over all triangles of links.
A sufficient condition for Eq. (\ref{sumprod}) to hold is that
\be
C^{(N)}_{ij}(\{p_{ij}\})+C^{(N)}_{jk}(\{p_{ij}\})+C^{(N)}_{ki}(\{p_{ij}\})=0
\label{closed}
\end{equation}
for all triangles $i<j<k$ (and, in fact, for all closed
paths) and for any set $\{p_{ij}\}$.
Up to $N=4$ this can be justified by direct calculations.
The key observation for the induction is that when extending the network by
adding a new state then the coefficients of the old links in the
enlarged network are modified only by a shift in their argument as
$C_{ij}^{(N)}(\{p_{ij}\})\to C_{ij}^{(N)}(\{p_{ij}+P_{ij}\})$, see
Eq. (\ref{omN1}). Therefore the identity will hold in the enlarged network
for all closed paths consisting of old links.
The same network with $N+1$ states can, however, be built from another
starting set of $N$ states by adding one extra state, as well, in which case
the identity (\ref{closed}) can be extended to another set of links.
It is easy to see that applying this
reasoning to all possible sets of $N$ states (note that $a$ and $b$ are always part of this set), the old links cover all possible
triangles of the extended network and the identity will be valid for $N+1$.
So, we have proved that Eq. (\ref{last}) holds for arbitrary networks.
This means that, although the two-point resistance depends
on the particular network,
the variance of the effective asymmetry parameter, which can be
interpreted as the strength of disorder, depends on the resistance
{\it universally} at least for weak disorder.
If the resistance is increasing(decreasing) with the system size then
the effective asymmetry is getting stronger(weaker) on larger scales.
So, the dynamical relation between time and length scale on lattices
with a decreasing resistance
is expected to be stable against weak disorder; otherwise,
it may be altered compared to that of the homogeneous system.
Measuring the distance $l$ between $a$ and $b$ in terms of
the linear size if the
network can be embedded in a $d$-dimensional lattice and in terms of the
length of shortest path otherwise,
the resistance scales in many cases as $\tilde r_{ab}(l)\sim l^{\zeta}$, where
$\zeta$ is the resistance exponent.
The strength of disorder then scales with $l$ in leading order as
\be
|\ln (\tilde p_{ab}/\tilde p_{ba})|=|\tilde\epsilon_{ab}|
\sim \sqrt{\tilde r_{ab}}\sim l^{\zeta/2}.
\end{equation}
For $\zeta>0$ this is the same type of logarithmic behavior
as known for $d=1$, with the barrier exponent $\psi^0=\zeta/2$.
But unlike for $d=1$,
this law will, in general, be modified for finite disorder
by higher order correction terms which are
expected to be non-universal. Indeed, numerical results that will be presented
in the rest of this work indicate,
in the case of $\zeta>0$, a logarithmic scaling law but with $\psi\neq\psi^0$.
The results obtained here are in accordance with previous ones on regular
lattices.
For $d>2$, the two-point resistance tends to a constant as
$\tilde r_{ab}(l)\sim l^{\zeta}+{\rm const}$, with
$\zeta=2-d<0$; at $d=2$, where the diffusion is normal with a logarithmic
correction, it still increases but just barely (formally $\zeta=0$),
while for $d=1$, $\zeta=1$.
So, in general, we expect disorder to be relevant(irrelevant)
if $\zeta>0$($\zeta<0$) while, in the marginal case $\zeta=0$,
it will induce at most corrections to the clean behavior.
\section{Numerical analysis}
We have tested this criterion on fractal lattices,
where, in the case of homogeneous rates, the law of diffusion is, in general,
of the form $\langle{\bf x}^2(t)\rangle\sim t^{2/d_w}$, characterized by
an anomalous diffusion exponent, $d_w>2$ \cite{havlin}.
This problem has been thoroughly investigated but, apart from
early Monte Carlo simulation studies \cite{pandey,mc},
not in the presence of asymmetric disorder.
The Einstein relation between
diffusion and conduction implies that $\zeta=d_w-d_f$, where $d_f$ is the
fractal dimension. From this, one
can see that $\zeta>0$ for all fractals in $d=2$ and, in fact,
$\zeta$ is positive for most fractals.
We have numerically calculated the MFPT by recursively solving
Eq. (\ref{backward}) on three different fractal lattices
with $\zeta>0$, namely, the Sierpinski triangle (ST) shown in
Fig. \ref{fractal} and bond percolation
clusters in $2$ and $3$ dimensions at the percolation threshold.
In the former case, the MFPT from one of the tips of the
largest triangle to the other two tips was considered, whereas
in the latter case, percolating clusters that connect opposite
$d-1$-dimensional surfaces of a $d$-dimensional cube have been generated (with
periodic boundaries in the other directions), and the MFPT from
a given site on one surface to the opposite surface was considered.
\begin{figure}[h]
\onefigure[scale=0.5]{fig1.eps}
\caption{
Iteration rule and the $4$th generation of the Sierpinski (a) and the
hierarchical diamond lattice (b). The starting(target) sites are denoted by
green(red) dots.
}
\label{fractal}
\end{figure}
The rates on each link were $p_{ij}=1$, $p_{ji}=\lambda<1$ with a
random (equiprobable) orientation.
The distributions of the logarithm of the MFPT determined
in $10^6$ realizations for different
linear system sizes $L$ are found to broaden with $L$ in all three cases,
see Figs. \ref{st}-\ref{2d} and to obey the scaling law
\be
\rho[\ln(\tau/\tau_0),L]=L^{-\psi}\tilde\rho[\ln(\tau/\tau_0)L^{-\psi}],
\end{equation}
where the barrier exponents $\psi$ are independent of the
strength of disorder $\lambda$, and are estimated for the
ST and $d=2$ and $d=3$ percolation clusters to be, in order,
$0.296(2)$, $0.46(2)$ and $0.63(1)$.
\begin{figure}[h]
\onefigure[scale=0.6]{fig2.ps}
\caption{Scaling plot of the distribution
of the logarithm of the MFPT on Sierpinski triangles with different linear
size $L$. The parameters are $\psi=0.296$ and $\ln\tau_0=-1.47$. The strength of disorder was $\lambda=0.1$.
Inset: Unscaled distributions.
}
\label{st}
\end{figure}
\begin{figure}[h]
\onefigure[scale=0.6]{fig3a.ps}
\caption{The same plot for $d=3$ percolation
clusters as in Fig. \ref{st} with $\psi=0.63$ and $\ln\tau_0=-5.7$.
}
\label{2d}
\end{figure}
\begin{figure}[h]
\onefigure[scale=0.6]{fig3b.ps}
\caption{
Finite-size behavior of the typical MFPT in the hierarchical diamond lattice
with $\lambda=0.1$.
}
\label{hdl}
\end{figure}
We have the most accurate estimate for the ST, due to
an efficient renormalization, essentially by reversing the
construction procedure. Here, the barrier exponent $\psi=0.296(2)$
can be seen to be significantly different from
$\psi_0=\zeta/2=\ln(5/3)/\ln 4\approx 0.368$.
Concerning marginal structures ($\zeta=0$), we have considered the
hierarchical diamond lattice (see Fig. \ref{fractal}), where $d_f=d_w=2$ and,
in fact, $\tilde r_{ab}=1$ in each generation.
The distribution of $\tau/L^{d_w}$
is found to converge to a limit
distribution for $L\to\infty$, but very slowly;
the typical MFPT, $\tau_{\rm typ}\equiv \exp(\overline{\ln\tau})$, has
a logarithmic correction of the same form as in $d=2$ regular lattices:
$L^{d_w}/\tau_{\rm typ}\simeq D(1+a/\ln\tau_{\rm typ})$, see Fig. \ref{hdl}.
\section{Discussion}
In summary, we have revealed a
close relationship between the effective strength of asymmetric disorder
in the diffusion problem and the electric resistance.
This yields a simple relevance criterion
in terms of the resistance exponent.
If $\zeta<0$, the dynamics are stable against weak asymmetric disorder,
while if $\zeta>0$, they are unstable and, as numerical results
show, the logarithmic scaling is not a peculiarity of one dimension but
it is the general rule, characterized by the exponent $\psi$
of the underlying structure that is independent of other
known dynamical exponents. In the marginal case, $\zeta=0$,
slow corrections to the clean system behavior are expected to be general.
As aforementioned, stochastic processes can be regarded as
random walks in their configuration spaces.
In disordered stochastic systems with many degrees of freedom,
logarithmically slow dynamical behavior indeed appears, such as in
many-particle transport processes with zero-range or exclusion interaction
\cite{jsi}, or in the contact process at criticality in one \cite{hiv} and
higher \cite{vfm} dimensions.
The effect of disorder is, however, not restricted to appear in one single
(critical) point of the parameter space of these systems (including the random walk in
one dimension) but the latter point is surrounded by an extended phase, called
Griffiths phase \cite{griffiths}, where the dynamical exponents are finite and vary with the
parameters of the model non-universally \cite{im}.
In the random walk representation, the average force acting on the walker
becomes non-zero in some direction in this phase, therefore this is
out of the scope of the present treatment (cf. Eq. (\ref{symm})).
Nevertheless, it is plausible to expect systems with a positive resistance
exponent to have a Griffiths phase,
when a biased force-field is applied.
The non-universal behavior with a finite dynamical exponent is, however,
suppressed if the lattice contains macroscopic 'dead-ends' (such as
in critical percolation clusters) from which the walker escapes after an
exponentially large trapping time \cite{havlin}, leading to
logarithmically slow dynamics also in the driven phase.
The analyses of these phenomena is left for future research.
\acknowledgments
This work was supported by the J\'anos Bolyai Research Scholarship of the
Hungarian Academy of Sciences and by the National Research Fund
under grant no. K75324.
|
{
"timestamp": "2012-05-11T02:02:21",
"yymm": "1203",
"arxiv_id": "1203.1735",
"language": "en",
"url": "https://arxiv.org/abs/1203.1735"
}
|
\section{Introduction}
Supersymmetric models are considered as possible extensions of the Standard Model, see {\it e.g.} \cite{Signer:2009dx}, and supersymmetry is then assumed to be a symmetry of Nature at a sufficiently high energy scale. Experiments at the Large Hadron Collider may soon show whether this is the case. This paper is devoted to a plasma system with the dynamics governed by the ${\cal N} =4$ supersymmetric Yang-Mills theory \cite{Brink:1976bc,Gliozzi:1976qd}. In the models with an extended (${\cal N} > 1$) supersymmetry, the left- and right-handed fermions interact in the same way, in conflict with the Standard Model where the left- and right-handed matter particles are coupled differently. Consequently, the ${\cal N} =4$ super Yang-Mills is not treated as a serious candidate for a theory to describe the world of elementary particles. Nevertheless, the theory attracts a lot of attention because of its unique features. The ${\cal N} =4$ super Yang-Mills appears to be finite and thus it is conformally invariant not only at the classical but at the quantum level as well.
A great interest in the ${\cal N} =4$ super Yang-Mills theory was stimulated by a discovery of the AdS/CFT duality of the five-dimensional gravity in the anti de Sitter geometry and the conformal field theories \cite{Maldacena:1997re}, for a review see \cite{Aharony:1999ti} and the lecture notes \cite{Klebanov:2000me} as an introduction. The duality offered a unique tool to study strongly coupled field theories. Since the gravitational constant and the coupling constant of dual conformal field theory are inversely proportional to each other, some problems of strongly coupled field theories can be solved via weakly coupled gravity. In this way some intriguing features of strongly coupled systems driven by the ${\cal N} =4$ super Yang-Mills dynamics were revealed, see the reviews \cite{Son:2007vk,Janik:2010we}, but relevance of the results for non-supersymmetric systems, which are of our actual interest, remains an open issue. In particular, one asks how properties of the ${\cal N} =4$ super Yang-Mills plasma (SYMP) are related to those of quark-gluon plasma (QGP) studied experimentally in relativistic heavy-ion collisions. While such a comparison is, in general, a difficult problem, some comparative analyses have been done in the domain of weak coupling where perturbative methods are applicable \cite{CaronHuot:2006te,Huot:2006ys,CaronHuot:2008uh,Blaizot:2006tk,Chesler:2006gr,Chesler:2009yg}.
We undertook a task of systematic comparison of supersymmetric perturbative plasmas to their non-supersymmetric counterparts. We started with the ${\cal N} =1$ SUSY QED, analyzing first collective excitations of ultrarelativistic plasma which, in general, is out of equilibrium \cite{Czajka:2010zh}. We computed the one-loop retarded self-energies of photons, photinos, electrons and selectrons in the Hard Loop Approximation using the Keldysh-Schwinger formalism. The self-energies, which we also analyzed in the context of effective action, enter the dispersion equations of photons, photinos, electrons and selectrons, respectively. The collective modes of ${\cal N} =1$ SUSY QED plasma appear to be essentially the same as those in ultrarelativistic electromagnetic plasma of photons, electrons and positrons. In particular, a spectrum of photino modes coincides with that of quasi-electrons. Therefore, independently whether photon modes are stable or unstable, there are no unstable photino excitations. The supersymmetry, which is obviously broken in the plasma medium, does not induce any instability in the photino sector.
In the subsequent paper \cite{Czajka:2011zn} we discussed collisional characteristics of ${\cal N} =1$ SUSY QED plasma. For this purpose we computed cross sections of all elementary processes which occur at the lowest non-trivial order of $\alpha\equiv e^2/4\pi$. We found that some processes, {\it e.g.} the Compton scattering on selectrons, are independent of momentum transfer. The processes are qualitatively different from those of usual electromagnetic interactions dominated by small momentum transfers. Further on we discussed collisional characteristics of equilibrium ${\cal N} =1$ SUSY QED plasma, observing that parameters of ultrarelativistic plasmas are strongly constrained by dimensional arguments, as the temperature is the only dimensional quantity of equilibrium system. Then, transport coefficients like viscosity are proportional to appropriate powers of temperature and the coefficients characterizing different plasmas can differ only by numerical factors. So, we derived the energy loss and momentum broadening of a particle traversing the equilibrium plasma, which depend not only on the plasma temperature but on the energy of test particle as well. We found that the two quantities have very similar structure (in the limit of high energy of the test particle) even for very different elementary cross sections. Our findings presented in \cite{Czajka:2010zh,Czajka:2011zn} show that the plasmas of ${\cal N} =1$ SUSY QED and of QED are surprisingly similar to each other. In this paper we discuss properties of the ${\cal N} =4$ super Yang-Mills plasma, analyzing both collective excitations and collisional characteristics of the system.
Our main aim is to confront the weakly coupled plasma driven by ${\cal N} =4$ super Yang-Mills with the perturbative quark-gluon plasma governed by QCD. We do not attempt to compare our results to those obtained in strong coupling regime using either the AdS/CFT duality or lattice QCD. Some plasma characteristics we discuss, {\it e.g.} the energy loss, are computed in both strongly and weakly coupled systems but it is rather unclear how to study collective excitations representing colored quasiparticles in the setting of AdS/CFT duality or lattice QCD. The paper \cite{Bak:2007fk} demonstrates that even the definition of Debye screening mass, which has a very simple meaning in perturbative plasmas, is far not straightforward in strongly interacting systems. For these reasons we escape from discussing our results in the context of strong coupling.
Our paper is organized as follows. In the next section, we discuss the Lagrangian of ${\cal N} = 4$ super Yang-Mills and the field content of the system under consideration. The vertexes of ${\cal N} = 4$ super Yang-Mills are collected in Appendix \ref{sec-SYM-vertex}. In Sec.~\ref{sec-basic} basic characteristics of SYMP such as energy density and Debye mass are discussed and compared to those of QGP. Then, we move to plasma collective excitations. The general dispersion equations of gauge bosons, fermions and scalars are written down in Sec.~\ref{sec-dis-eqs} and the self-energies, which enter the equations, are obtained in the subsequent section. We apply here the Keldysh-Schwinger approach which allows one to study equilibrium and non-equilibrium systems. The free Green's functions of Keldysh-Schwinger formalism are given in Appendix \ref{sec-Green-fun}. Since we are interested in collective modes, the self-energies are obtained in the long wavelength limit corresponding to the Hard Loop Approximation. The effective action of the Hard Loop Approach is derived in Sec.~\ref{sec-eff-action} and possible structures of self-energies are considered in this context. In Sec.~\ref{sec-modes} we present a qualitative discussion of collective modes in SYMP. Sec.~\ref{sec-collisions} is devoted to collisional characteristics of the plasma - elementary processes and transport coefficients are briefly discussed here. Finally, we conclude our study in Sec.~\ref{sec-conclusions}.
As we have intended to make our paper complete and self-contained, there is inevitably some repetition of the content of our previous publications \cite{Czajka:2010zh,Czajka:2011zn}, mostly in Secs.~\ref{sec-modes}, \ref{sec-collisions}. Throughout the paper we use the natural system of units with $c= \hbar = k_B =1$; our choice of the signature of the metric tensor is $(+ - - -)$.
\section{${\cal N}=4$ Super Yang-Mills Theory}
\label{sec-lagrangian}
We start our considerations with a discussion of the Lagrangian of ${\cal N}=4$ super Yang-Mills theory \cite{Brink:1976bc,Gliozzi:1976qd}. We follow here the presentation given in \cite{Yamada:2006rx}.
The gauge group is assumed to be ${\rm SU}(N_c)$ and every field of the ${\cal N}=4$ super Yang-Mills theory belongs to its adjoint representation. The field content of the theory, which is summarized in Table~\ref{table-field-content}, is the following. There are gauge bosons (gluons) described by the vector field $A_\mu^a$ with $a, b, c, \dots = 1, 2, \dots N_c^2 -1$. There are four Majorana fermions represented by the Weyl spinors $\lambda^\alpha$ with $\alpha = 1,2$ which can be combined in the Dirac bispinors as
\begin{equation}
\label{Majorana-bispinor}
\Psi = \left(
\begin{matrix}
\lambda^\alpha \cr
\bar{\lambda}_{\dot{\alpha}} \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\bar{\Psi} = ( \lambda_\alpha, \bar{\lambda}^{\dot{\alpha}} ) ,
\end{equation}
where $\bar{\lambda}_{\dot{\alpha}} \equiv [ \lambda_\alpha]^\dagger$ with $\dagger$ denoting Hermitian conjugation. To numerate the Majorana fermions we use the indices $i, j = 1,2,3,4$ and the corresponding bispinor is denoted as $\Psi_i$. Finally, there are six real scalar fields which are assembled in the multiplet $\Phi = (X_1, Y_1, X_2, Y_2, X_3, Y_3)$. The components of $\Phi$ are either denoted as $X_p$ for scalars, and $Y_p$ for pseudoscalars, with $p,q =1,2,3$ or as $\Phi_A$ with $A, B=1,2, \dots 6$.
\begin{table}[b]
\caption{\label{table-field-content} Field content of ${\cal N} =4$ super Yang-Mills theory.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
Field's symbol & Type of the field & Range of the field's index & Spin & Number
of degrees of freedom $(N^{\rm dof})$
\\[4mm]
$A^\mu$ & vector &$\mu, \nu = 0,1,2,3$ & 1& $2 \times (N_c^2-1)$
\\[2mm]
$\Phi_A$ & real (pseudo-)scalar & $A, B = 1,2, 3,4,5, 6$ & 0 & $6 \times (N_c^2-1)$
\\[2mm]
$\lambda_i$ & Majorana spinor & $i, j = 1,2,3,4$ & 1/2 & $8 \times (N_c^2-1)$
\end{tabular}
\end{ruledtabular}
\end{table}
The Lagrangian density of ${\cal N} =4$ super Yang-Mills theory can be written as
\begin{eqnarray}
\label{Lagrangian-1}
{\cal L}
&=&
-\frac{1}{4}F^{\mu \nu}_a F_{\mu \nu}^a
+\frac{i}{2}\bar \Psi_i^a (D\!\!\!\!/ \, \Psi_i)^a
+\frac{1}{2}(D_\mu \Phi_A)_a (D^\mu \Phi_A)_a
\\ [2mm] \nn
&&
-\frac{1}{4} g^2f^{abe} f^{cde} \Phi_A^a \Phi_B^b \Phi_A^c \Phi_B^d
-i\frac{g}{2} f^{abc} \Big( \bar \Psi_i^a \alpha_{ij}^p X_p^b \Psi_j^c
+i\bar \Psi_i^a \beta_{ij}^p\gamma_5 Y_p^b \Psi_j^c \Big),
\end{eqnarray}
where
$F^{\mu \nu}_a = \partial^\mu A^\nu_a - \partial^\nu A^\mu_a + g f^{abc} A^\mu_b A^\nu_c$ and the covariant derivatives equal $(D\!\!\!\!/ \, \Psi_i)^a = (\partial \,\!\!\!\!/ \, \delta_{ab} +g f^{abc} A_c \!\! \!\!\!/ \,) \Psi_i^b$ and $(D^\mu \Phi)_a = D^\mu_{ab} \Phi_b = (\partial^\mu \delta_{ab} + gf^{abc}A^\mu_c)\Phi_b$; $g$ is the coupling constant; $f^{abc}$ are the structure constants of ${\rm SU}(N_c)$ group; the $4 \times 4$ matrices $\alpha^p, \beta^p$ satisfy the relations
\begin{equation}
\label{alpha-beta-relations}
\{\alpha^p, \alpha^q \} = - 2 \delta^{p q},
\;\;\;\;\;\;\;
\{\beta^p, \beta^q \} = - 2 \delta^{p q},
\;\;\;\;\;\;\;
[ \alpha^p, \beta^q] = 0 ,
\end{equation}
and their explicit form can be chosen as
\begin{eqnarray}
\label{alphas}
\alpha^1 &=& \left(
\begin{matrix}
0 & \sigma_1 \cr
- \sigma_1 & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\alpha^2 = \left(
\begin{matrix}
0 & -\sigma_3 \cr
\sigma_3 & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\alpha^3 = \left(
\begin{matrix}
i \sigma_2 & 0 \cr
0 & i \sigma_2 \cr
\end{matrix}
\right) ,
\\[2mm]
\label{betas}
\beta^1 &=& \left(
\begin{matrix}
0 & i\sigma_2 \cr
i \sigma_2 & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\beta^2 = \left(
\begin{matrix}
0 & \sigma_0 \cr
-\sigma_0 & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\beta^3 = \left(
\begin{matrix}
-i \sigma_2 & 0 \cr
0 & i \sigma_2 \cr
\end{matrix}
\right) ,
\end{eqnarray}
where the $2 \times 2$ Pauli matrices read
\begin{eqnarray}
\label{Pauli}
\sigma^0 = \left(
\begin{matrix}
1 & 0 \cr
0 & 1 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\sigma^1 = \left(
\begin{matrix}
0 & 1 \cr
1 & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\sigma^2 = \left(
\begin{matrix}
0 & -i \cr
i & 0 \cr
\end{matrix}
\right) ,
\;\;\;\;\;\;\;
\sigma^3 = \left(
\begin{matrix}
1 & 0 \cr
0 & -1 \cr
\end{matrix}
\right) .
\end{eqnarray}
As seen, the matrices $\alpha^p, \beta^p$ are antiHermitian: $(\alpha^p)^\dagger = -\alpha^p$ , $(\beta^p)^\dagger = -\beta^p$. The vertexes of ${\cal N} = 4$ super Yang-Mills, which can be inferred from the Lagrangian (\ref{Lagrangian-1}), are collected in Appendix \ref{sec-SYM-vertex}. The vertexes are used in perturbative calculations presented in the subsequent sections.
The Lagrangian (\ref{Lagrangian-1}) is sometimes \cite{CaronHuot:2008uh,Chesler:2006gr,Chesler:2009yg} extended by adding a fundamental ${\cal N} = 2$ hypermultiplet to mimic a behavior of quarks in QCD plasma. The hypermultiplet is typically massive to study heavy flavors but it can be massless as well. We do not consider any extension of the Lagrangian (\ref{Lagrangian-1}) but at the end of Sec.~\ref{sec-eff-action} we briefly comment on a possible structure of self-energies of fields belonging to the fundamental ${\cal N} = 2$ hypermultiplet.
\section{Basic plasma characteristics}
\label{sec-basic}
We start our discussion of SYMP with basic characteristics of the equilibrium plasma. Specifically, we consider the energy and particle densities, Debye mass and plasma parameter of SYMP comparing the quantities to those of QGP. For the beginning, however, a few comments are in order.
In QGP there are several conserved charges: baryon number, electric and color charges, strangeness. The net baryon number and electric charge are typically non-zero in QGP produced in relativistic heavy-ion collisions while the total strangeness and color charge vanish. Actually, the color charge is usually assumed to vanish not only globally but locally as well. It certainly makes sense as the whitening of QGP appears to be the relaxation process of the shortest time scale \cite{Manuel:2004gk}. In SYMP, there are conserved charges carried by fermions and scalars associated with the global ${\rm SU}(4)$ symmetry. One of these charges can be identified with the electric charge to couple ${\cal N} = 4$ super Yang-Mills to electromagnetic field \cite{CaronHuot:2006te}. In the forthcoming the average ${\rm SU}(4)$ charges of SYMP are assumed to vanish and so are the associated chemical potentials. The constituents of SYMP carry color charges but we further assume that the plasma is globally and locally colorless.
Since there are conserved supercharges in supersymmetric theories, it seems reasonable to consider a statistical supersymmetric system with a non-zero expectation value of the supercharge. However, it is not obvious how to deal with a partition function customary defined as ${\rm Tr}e^{-\beta (H - \mu Q)}$ where $\beta \equiv T^{-1}$ is the inverse temperature, $H$ is the Hamiltonian, $Q$ is the supercharge operator and $\mu$ is the associated chemical potential. The problem is caused by a fermionic character of the supercharge $Q$. If $\mu$ is simply a number, as, say, the baryon chemical potential, the partition function even of non-interacting system does not factorize into a product of partition functions of single momentum modes because the supercharges of different modes do not commute with each other. The supercharge is not an extensive quantity \cite{Kapusta:1984cp}. There were proposed two ways to resolve the problem. Either the chemical potential remains a number but the supercharge is modified by multiplying it by an additional fermionic field $c$ \cite{Kapusta:1984cp,Mrowczynski:1986cu} or the chemical potential by itself is a fermionic field \cite{Kovtun:2003vj}. Then, $\mu c Q$ and $\mu Q$ are both bosonic and the partition function can be computed in a standard way. The two formulations, however, are not equivalent to each other. According to the former one \cite{Kapusta:1984cp,Mrowczynski:1986cu}, properties of a supercharged system vary with an expectation value of the supercharge, within the latter one \cite{Kovtun:2003vj}, the partition function appears to be effectively independent of $Q$. Because of the ambiguity, we further consider SYMP where the expectation values of all supercharges vanish both globally and locally.
In view of the above discussion, SYMP is comparable to QGP where the conserved charges are all zero and so are the associated chemical potentials. We adopt the assumption whenever the two plasma systems are compared to each other.
When the chemical potentials are absent, the temperature $(T)$ is the only dimensional parameter, which characterizes the equilibrium plasma, and all plasma parameters are expressed through the appropriate powers of $T$. Taking into account that the right numbers of bosonic and fermionic degrees of freedom in SYMP and QGP, the energy densities of equilibrium non-interacting plasmas equal
\begin{equation}
\varepsilon = \frac{\pi^2}{60} \, {30 (N_c^2 -1) \choose 4(N_c^2 -1) + 7 N_f N_c} T^4.
\end{equation}
where the upper expression is for SYMP and the lower one for QGP with $N_f$ light quark flavors. The quark is {\em light} when its mass is much smaller than the plasma temperature. For $N_c=N_f =3$, the energy density of SYMP is approximately 2.5 times bigger than that of QGP at the same temperature. The same holds for the pressure $p$ which, obviously, equals $\varepsilon/3$.
The particle densities in SYMP and QGP are found to be
\begin{equation}
n = \frac{2\zeta(3)}{\pi^2} \, {7 (N_c^2 -1) \choose 2(N_c^2 -1) + 3 N_f N_c} T^3,
\end{equation}
where $\zeta(3) \approx 1.202$ is the Riemann zeta function. For $N_c=N_f =3$ we have $n_{\rm SYMP}/n_{\rm QGP} \approx 1.3$ at the same temperature.
As we show in Sec.~\ref{subsec-polar-tensor}, the gluon polarization tensor has exactly the same structure in SYMP and QGP, and consequently the Debye mass in SYMP is defined in the same way as in QGP. The masses in both plasmas equal
\begin{equation}
m_D^2 = \frac{g^2}{6} \, {12 N_c \choose 2 N_c + N_f} T^2,
\end{equation}
where, as previously, the upper case is for SYMP and the lower one for QGP. For $N_c=N_f =3$, the ratio of Debye masses squared is 2.4 at the same value of $gT$. The Debye mass determines not only the screening length $r_D = 1/m_D$ but it also gives the plasma frequency $\omega_p = m_D/\sqrt{3}$ which is the minimal frequency of longitudinal and transverse plasma oscillations corresponding to the zero wave vector. The plasma frequency is also called the gluon thermal mass.
Another important quantity characterizing the equilibrium plasma is the so-called plasma parameter $\lambda$ which equals the inverse number of particles in the sphere of radius of the screening length. When $\lambda$ is decreasing, the behavior of plasma is more and more collective while inter-particle collisions are less and less important. For $N_c=N_f =3$, we have
\begin{equation}
\lambda \equiv \frac{1}{\frac{4}{3} \pi r_D^3 n} \approx {0.257 \choose 0.042} g^3 .
\end{equation}
As seen, the dynamics of QGP is more collective than that of SYMP.
The differences of $\varepsilon$ and $n$ for SYMP and QGP merely reflect the difference in numbers of degrees of freedom in the two plasma systems. In case of $m_D$ and $\lambda$ it also matters that (anti-)quarks in QGP and fermions in SYMP belong to different representations - fundamental and adjoint, respectively - of the ${\rm SU}(N_c)$ gauge group.
\section{Dispersion equations}
\label{sec-dis-eqs}
Dispersion equations determine dispersion relations of quasi-particle excitations. Below
we write down the dispersion equations of quasi-gluons, quasi-fermions, and quasi-scalars.
\subsection{Gluons}
Since the equation of motion of the gluon field $A^{\mu}_a(k)$ can be written in the form
\begin{equation}
\label{eq-motion-A}
\Big[ k^2 g^{\mu \nu} -k^{\mu} k^{\nu} - \Pi^{\mu \nu}(k) \Big]
A_{\nu}(k) = 0 ,
\end{equation}
where color indices are dropped, $\Pi^{\mu \nu}(k)$ is the retarded polarization tensor and $k\equiv (\omega, {\bf k})$ is the four-momentum, the general gluon dispersion equation is
\begin{equation}
\label{dis-photon-1}
{\rm det}\Big[ k^2 g^{\mu \nu} -k^{\mu} k^{\nu} - \Pi^{\mu \nu}(k) \Big]
= 0 .
\end{equation}
Strictly speaking, one should consider the equation of motion not of the gluon field but of the gluon propagator. Then, Eq.~(\ref{dis-photon-1}) determines the poles of the propagator. Due to the transversality of $\Pi^{\mu \nu}$ ($k_\mu \Pi^{\mu \nu}(k) =0$), which is required by the gauge covariance, not all components of $\Pi^{\mu \nu}$ are independent from each other, and consequently the dispersion equation (\ref{dis-photon-1}) can be much simplified by expressing the polarization tensor through the dielectric tensor $\varepsilon^{ij}(k)$ which is the $3 \times 3$ not $4 \times 4$ matrix.
\subsection{Fermions}
The fermion field $\psi_i (k)$ obeys the equation
\begin{equation}
\Big[ k\!\!\!/ \, - \Sigma (k) \Big] \psi (k) =0 ,
\end{equation}
where any indices are neglected and $\Sigma (k)$ is the retarded fermion self-energy.
The dispersion equation thus is
\begin{equation}
\label{dis-electron-1}
{\rm det}\Big[ k\!\!\!/ \, - \Sigma (k) \Big] = 0 .
\end{equation}
Further on we assume that the spinor structure of
$\Sigma(k)$ is
\begin{equation}
\label{structure-electron}
\Sigma (k) = \gamma^{\mu} \Sigma_{\mu}(k) .
\end{equation}
Then, substituting the expression (\ref{structure-electron}) into Eq.~(\ref{dis-electron-1}) and computing the determinant as explained in Appendix 1 of \cite{Mrowczynski:1992hq}, we get
\begin{equation}
\label{dis-electron-2}
\Big[\big( k^{\mu} - \Sigma^{\mu}(k)
\big) \big(k_{\mu} - \Sigma_{\mu}(k) \big)\Big]^2 = 0 .
\end{equation}
\subsection{Scalars}
The scalar field $\Phi_A (k)$ obeys the Klein-Gordon equation
\begin{equation}
\label{dis-eq-selectron}
\Big[ k^2 + P(k) \Big] \Phi(k) =0 ,
\end{equation}
where $P(k)$ is the retarded self-energy of scalar field and any indices are dropped. The dispersion equation is
\begin{equation}
\label{dis-selectron}
k^2 + P(k) = 0 .
\end{equation}
As seen, the whole dynamical information about plasma medium is contained in the self-energies which are computed perturbatively in the next section.
\section{Self-energies}
\label{sec-self-energies}
We compute here the self-energies which enter the dispersion equations (\ref{dis-photon-1}, \ref{dis-electron-1}, \ref{dis-selectron}). The vertexes of ${\cal N} = 4$ super Yang-Mills, which are used in our perturbative calculations, are listed in Appendix \ref{sec-SYM-vertex}. The plasma is assumed to be homogeneous (translationally invariant), locally colorless but the momentum distribution is, in general, different from equilibrium one. Therefore, we adopt the Keldysh-Schwinger or real-time formalism which allows one to describe both equilibrium and non-equilibrium many-body systems. The free Green's functions, which are labeled with the indices $+,-, >, <, {\rm sym}$, are collected in Appendix \ref{sec-Green-fun}. What concerns the Keldysh-Schwinger formalism we follow the conventions explained in \cite{Mrowczynski:1992hq}. The computation is performed within the Hard Loop Approach, see the reviews \cite{Thoma:1995ju,Blaizot:2001nr,Kraemmer:2003gd}, which was generalized to anisotropic systems in \cite{Mrowczynski:2000ed}.
\subsection{Polarization tensor}
\label{subsec-polar-tensor}
The gluon polarization tensor $\Pi^{\mu \nu}$ can be defined by means of the Dyson-Schwinger equation
\begin{equation}
i{\cal D}^{\mu \nu} (k) = i D^{\mu \nu} (k)
+ i D^{\mu \rho}(k) \, i\Pi_{\rho \sigma}(k) \, i{\cal D}^{\sigma \nu}(k) ,
\end{equation}
where ${\cal D}^{\mu \nu}$ and $D^{\mu \nu}$ is the interacting and free gluon propagator, respectively. The lowest order contributions to gluon polarization tensor are given by six diagrams shown in Fig.~\ref{fig-gluon}. The curly, plain, doted and dashed lines denote, respectively, gluon, fermion, ghost, and scalar fields.
\begin{figure}[t]
\centering
\includegraphics*[width=0.7\textwidth]{gluon_se.pdf}
\caption{Contributions to the gluon self-energy. }
\label{fig-gluon}
\end{figure}
Using the vertexes given in Appendix \ref{sec-SYM-vertex}, the contribution to the {\em contour} polarization tensor of Keldysh-Schwinger formalism, which comes from the fermion loop corresponding to the graph in Fig.~\ref{fig-gluon}a, is immediately written down in the coordinate space as
\begin{equation}
\label{contour-Pi}
_{(a)}\Pi^{\mu \nu}_{ab}(x,y)
= -ig^2 N_c \delta_{ab}
{\rm Tr} [\gamma^\mu S_{ij}(x,y) \gamma^\nu S_{ji}(y,x)] .
\end{equation}
where the trace is taken over spinor indices. The factor $(-1)$ due to the fermion loop is included and the relation $f_{acd }f_{bcd}=\delta_{ab} N_c$ is used here.
We are interested in the retarded polarization tensor which is expressed
through $\Pi^{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~$ as
\begin{equation}
\Pi^+ (x,y) = \Theta(x_0 - y_0)
\Big( \Pi^> (x,y) - \Pi^< (x,y) \Big).
\end{equation}
The polarization tensors $\Pi^{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~$ are found from the contour tensor (\ref{contour-Pi}) by locating the argument $x_0$ on the upper (lower) and $y_0$ on the lower (upper) branch of the contour. Then, one gets
\begin{equation}
\label{Pi->-<-x-y}
\big( {_{(a)} \Pi^{{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~} (x,y)} \big)^{\mu \nu}_{ab}
= -ig^2 N_c \delta_{ab}
{\rm Tr} [\gamma^\mu S^{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~_{ij} (x,y) \gamma^\nu S^{\mathchoice{~\raise.58ex\hbox{$>$}\mkern-12.8mu\lower.52ex\hbox{$<$}~_{ji} (y,x)] .
\end{equation}
As already mentioned, the system under study is assumed to be translationally invariant. Then, the two-point functions as $S(x,y)$ effectively depend on $x$ and $y$ only through $x-y$. Therefore, we put $y=0$ and we write $S(x,y)$ as $S(x)$ and $S(y,x)$ as $S(-x)$. Then, Eq.~(\ref{Pi->-<-x-y}) is
\begin{equation}
\label{Pi->-<-x-y-1}
\big( {_{(a)} \Pi^{{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~} (x)} \big)^{\mu \nu}_{ab}
= -\frac{i}{2} g^2 N_c \delta_{ab}
{\rm Tr} [\gamma^\mu S^{\mathchoice{~\raise.58ex\hbox{$<$}\mkern-14.8mu\lower.52ex\hbox{$>$}~_{ij} (x) \gamma^\nu S^{\mathchoice{~\raise.58ex\hbox{$>$}\mkern-12.8mu\lower.52ex\hbox{$<$}~_{ji} (-x)] .
\end{equation}
Since
\begin{equation}
S^{\pm} (x) = \pm \Theta(\pm x_0)
\Big(S^> (x) - S^< (x) \Big)
\end{equation}
the retarded polarization tensor $\Pi^+ (x)$ is found as
\begin{equation}
\label{Pi-x-1}
\big({_{(a)} \Pi^+(x)} \big)^{\mu \nu}_{ab}
= -\frac{i}{2} g^2 N_c \delta_{ab}
{\rm Tr}\big[\gamma^\mu S^+_{ij}(x) \gamma^\nu S^{\rm sym}_{ji}(-x)
+ \gamma^\mu S^{\rm sym}_{ji}(x) \gamma^\nu S^-_{ij}(-x) \big],
\end{equation}
which in the momentum space reads
\begin{equation}
\label{Pi-k-e-1}
\big({_{(a)} \Pi^+(k)} \big)^{\mu \nu}_{ab}
= -\frac{i}{2} g^2 N_c \delta_{ab}
\int \frac{d^4p}{(2\pi)^4}
{\rm Tr} \big[\gamma^\mu S^+_{ij}(p+k) \gamma ^\nu S^{\rm sym}_{ji}(p)
+ \gamma^\mu S^{\rm sym}_{ji}(p) \gamma^\nu S^-_{ij}(p-k) \big].
\end{equation}
Further on the index $+$ is dropped and $\Pi^+$ is denoted as $\Pi$, as only the {\em retarded} polarization tensor is discussed. Substituting the functions $S^{\pm}, S^{\rm sym}$ given by Eqs.~(\ref{S-pm}, \ref{S-<}, \ref{S->}) into the formula (\ref{Pi-k-e-1}), one finds
\begin{eqnarray}
\label{Pi-k-e-4}
_{(a)}\Pi^{\mu \nu}_{ab}(k)
&=& - 4 g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi)^3} \, \frac{2n_f({\bf p})-1}{E_p}
\\ \nonumber
&& \times
\bigg(
\frac{2p^\mu p^\nu + k^\mu p^\nu + p^\mu k^\nu - g^{\mu \nu} (k \cdot p)}
{(p+k)^2 + i\, {\rm sgn}\big((p+k)_0\big)0^+}
+ \frac{2p^\mu p^\nu - k^\mu p^\nu - p^\mu k^\nu + g^{\mu \nu} (k \cdot p)}
{(p-k)^2 - i\, {\rm sgn}\big((p-k)_0\big)0^+}
\bigg) ,
\end{eqnarray}
where $p^\mu \equiv (E_p, {\bf p})$ with $E_p \equiv |{\bf p}|$, the traces of gamma matrices are computed and it is taken into account that $p^2 =0$. We also note that after performing the integration over $p_0$, the momentum ${\bf p}$ was changed into $-{\bf p}$ in the negative energy contribution.
In the Hard Loop Approximation, when $p \gg k$, we have
\begin{eqnarray}
\label{HLA-plus}
\frac{1}{(p+k)^2 + i0^+}
+ \frac{1} {(p-k)^2 - i0^+}
&=& \frac{2k^2}{(k^2)^2 - 4 (k\cdot p)^2 - i {\rm sgn}(k\cdot p) 0^+}
\approx -\frac{1}{2} \frac{k^2}{(k\cdot p + i 0^+)^2} ,
\\ [2mm]
\label{HLA-minus}
\frac{1}{(p+k)^2 + i0^+}
- \frac{1} {(p-k)^2 - i0^+}
&=& \frac{4(k \cdot p)}{(k^2)^2 - 4 (k\cdot p)^2 - i {\rm sgn}(k\cdot p) 0^+}
\approx \frac{k\cdot p}{(k\cdot p + i 0^+)^2}.
\end{eqnarray}
We note that $(p+k)_0 > 0$ and $(p-k)_0 > 0$ for $p \gg k$. With the formulas (\ref{HLA-plus}, \ref{HLA-minus}), Eq.~(\ref{Pi-k-e-4}) gives
\begin{eqnarray}
\label{Pi-k-e-final}
_{(a)} \Pi^{\mu \nu}_{ab}(k)
&=& 4 g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi)^3} \, \frac{2n_f({\bf p})-1}{E_p} \,
\frac{k^2 p^\mu p^\nu - \big(k^\mu p^\nu + p^\mu k^\nu
- g^{\mu \nu} (k \cdot p) \big) (k \cdot p)}
{(k\cdot p + i 0^+)^2} ,
\end{eqnarray}
which has the well-known structure of the polarization tensor of gauge bosons in ultrarelativistic QED and QCD plasmas. As seen, $\Pi(k)$ is symmetric with respect to Lorentz indices $ {_{(a)}\Pi}^{\mu \nu}_{ab}(k) = {_{(a)}\Pi}^{\nu \mu}_{ab}(k)$ and transverse $k_\mu {_{(a)} \Pi}^{\mu \nu}_{ab}(k) = 0$, as required by the gauge invariance. In the vacuum limit, when the fermion distribution function $n_f({\bf p})$ vanishes, the polarization tensor (\ref{Pi-k-e-final}) is still nonzero (actually infinite). As we will see, the vacuum contribution to the complete polarization tensor exactly vanishes due to the supersymmetry.
In analogy to the fermion-loop expression (\ref{Pi-k-e-1}), one finds the gluon-loop contribution to the retarded polarization tensor shown in Fig.~\ref{fig-gluon}b as
\begin{eqnarray}
\nonumber
_{(b)}\Pi^{\mu \nu}_{ab}(k) &=& - i\frac{g^2}{4} N_c \delta_{ab}
\int \frac{d^4p}{(2\pi )^4} \int \frac{d^4q}{(2\pi )^4} D^{\rm sym}(p)
\Big[ (2\pi)^4 \delta^{(4)}(k+p-q)
M^{\mu \nu} (k,q,p) D^+(q)
\\ [2mm]
\label{Pi-gluon-loop-2}
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
+ (2\pi)^4 \delta^{(4)}(k-p+q)
M^{\mu \nu} (k,-q,-p) D^-(q)
\Big],
\end{eqnarray}
where the gluon Green's functions $D^\pm$ and $D^{\rm sym}$ are given by Eqs.~(\ref{D-pm}, \ref{D-sym}), the combinatorial factor $1/2$ is included and
\begin{equation}
\label{tensor-M-def}
M^{\mu \nu} (k,q,p) \equiv
\Gamma^{\mu \sigma \rho} (k,-q,p)
\Gamma^{\;\;\,\nu}_{\sigma \;\; \rho} (q,-k,-p)
\end{equation}
with
\begin{equation}
\label{3-g-vertex-2}
\Gamma^{\mu \nu \rho} (k,p,q) \equiv
g^{\mu \nu }(k-p)^\rho
+g^{\nu \rho}(p-q)^\mu +g^{\rho \mu}(q-k)^\nu .
\end{equation}
Within the Hard Loop Approximation the tensor (\ref{tensor-M-def}) is computed as
\begin{equation}
\label{tensor-M-F-HL}
M^{\mu \nu} (k,p \pm k, \pm p) \approx \pm 2 g^{\mu \nu} (k\cdot p)
+ 10 p^\mu p^\nu
\pm 5(k^\mu p^\nu + p^\mu k^\nu),
\end{equation}
where we have taken into account that $p^2=0$.
Substituting the expressions (\ref{tensor-M-F-HL}) into Eq.~(\ref{Pi-gluon-loop-2}), using the explicit form of the functions $D^\pm$ and $D^{\rm sym}$ given by Eqs.~(\ref{D-pm}, \ref{D-sym}), and applying the Hard Loop Approximation (\ref{HLA-plus}, \ref{HLA-minus}) we get
\begin{eqnarray}
\label{Pi-gluon-loop-5}
_{(b)}\Pi^{\mu \nu}_{ab}(k) =
\frac{g^2}{4} N_c \delta_{ab}
\int \frac{d^3p}{(2\pi )^3} \frac{2n_g({\bf p})+1}{E_p}
\frac{5k^2 p^\mu p^\nu - 2 g^{\mu \nu} (k\cdot p)^2
- 5(k^\mu p^\nu + p^\mu k^\nu)(k\cdot p)}{(k\cdot p + i 0^+)^2} .
\end{eqnarray}
The gluon-tadpole contribution to the retarded polarization tensor, which shown in Fig.~\ref{fig-gluon}c,
equals
\begin{equation}
\label{Pi-gluon-tadpole-1}
_{(c)}\Pi^{\mu \nu}_{ab}(k) = - \frac{g^2}{2}
\int \frac{d^4p}{(2\pi )^4}
\Gamma^{\mu \nu \rho}_{abcc \rho} D^<(p) ,
\end{equation}
where the combinatorial factor $1/2$ is included and $\Gamma^{\mu \nu \rho \sigma }_{abcd}$ equals
\begin{equation}
\label{4-g-vertex}
\Gamma^{\mu \nu \rho \sigma }_{abcd} \equiv
f_{abe}f_{ecd}(g^{\mu \sigma} g^{\nu \rho} - g^{\mu \rho} g^{\nu \sigma})
+ f_{ace}f_{edb}(g^{\mu \rho} g^{\nu \sigma} - g^{\mu \nu} g^{\rho \sigma})
+ f_{ade}f_{ebc}(g^{\mu \nu} g^{\rho \sigma} - g^{\mu \sigma} g^{\nu \rho}).
\end{equation}
With the explicit form of the function $D^<(p)$ given by Eq.~(\ref{D-<}), the formula (\ref{Pi-gluon-tadpole-1}) provides
\begin{equation}
_{(c)}\Pi^{\mu \nu}_{ab}(k) = \frac{3}{2} g^2 N_c \, \delta_{ab} g^{\mu \nu}
\int \frac{d^3p}{(2\pi )^3} \frac{2 n_g({\bf p}) +1}{E_p} .
\end{equation}
The ghost-loop contribution to the retarded polarization tensor, which is shown in Fig.~\ref{fig-gluon}d, equals
\begin{eqnarray}
\label{Pi-ghost-loop-2}
_{(d)}\Pi^{\mu \nu}_{ab}(k) &=& i\frac{g^2}{2} N_c \delta_{ab}
\int \frac{d^4p}{(2\pi )^4} \;G^{\rm sym}(p)
\Big[ (p+k)^\mu p^{\nu} G^+(p+k)
+ p^\mu (p-k)^\nu G^-(p-k) \Big].
\end{eqnarray}
where the factor $(-1)$ is included as we deal with the fermion loop. Using the explicit form of the functions $G^\pm$ and $G^{\rm sym}$ given by Eqs.~(\ref{G-pm}, \ref{G-sym}), the formula (\ref{Pi-ghost-loop-2}) is manipulated to
\begin{equation}
\label{Pi-ghost-loop-4}
_{(d)}\Pi^{\mu \nu}_{ab}(k) = -\frac{g^2}{4} N_c \delta_{ab}
\int \frac{d^3p}{(2\pi )^3} \; \frac{2n_g({\bf p})+1}{E_p}
\frac{k^2 p^\mu p^\nu - (k^\mu p^\nu + p^\mu k^\nu) (k\cdot p)}{(k\cdot p + i 0^+)^2} .
\end{equation}
which holds in the Hard Loop Approximation.
As already mentioned, the quark-loop contribution to the polarization tensor is symmetric and transverse with respect to Lorentz indices. The same holds for the sum of gluon-loop, gluon-tadpole and ghost-loop contributions which gives the gluon polarization tensor in pure gluodynamics (QCD with no quarks). The sum of the three contributions equals
\begin{equation}
\label{Pi-b-c-d}
_{(b)+(c)+(d)}\Pi^{\mu \nu}_{ab}(k)
= g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi )^3} \frac{2n_g({\bf p})+ 1}{E_p}
\frac{k^2 p^\mu p^\nu + g^{\mu \nu} (k\cdot p)^2
- (k^\mu p^\nu + p^\mu k^\nu) (k\cdot p)}{(k\cdot p + i 0^+)^2}.
\end{equation}
To our best knowledge this is the first computation of the QCD polarization tensor in Hard Loop Approximation performed in the Keldysh-Schwinger (real time) formalism which explicitly demonstrates the transversality of the tensor. In Refs.~\cite{Weldon:1982aq,Mrowczynski:2000ed}, where the equilibrium and non-equilibrium anisotropic plasmas were considered, respectively, the transversality of $\Pi^{\mu \nu}(k)$ was actually assumed. In case of imaginary time formalism, the computation of the gluon polarization tensor in Hard Loop Approximation is the textbook material \cite{lebellac,Kapusta-Gale}.
The contribution to the polarization tensor coming from the scalar loop depicted
in Fig.~\ref{fig-gluon}e is given by
\begin{equation}
\label{Pi-s-l-1}
_{(e)}\Pi^{\mu \nu}_{ab}(k) = - i \frac{g^2}{2} \delta_{ab} N_c \delta^{AA}
\int \frac{d^4p}{(2\pi)^4}
\big[(2p+k)^\mu (2p+k)^\nu \Delta^+ (p+k) \Delta^{\rm sym}(p)
+ (2p-k)^\mu (2p-k)^\nu \Delta^{\rm sym} (p) \Delta^-(p-k) \big],
\end{equation}
which changes into
\begin{equation}
\label{Pi-s-l-3}
_{(e)}\Pi^{\mu \nu}_{ab}(k) = 3g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi )^3} \,
\frac{2n_s({\bf p})+1}{E_p} \,
\frac{k^2 p^\mu p^\nu - (p^\mu k^\nu + k^\mu p^\nu)(k \cdot p)}{(k \cdot p + i0^+)^2}.
\end{equation}
when the functions $\Delta^{\pm}$ and $\Delta^{\rm sym}$ given by Eqs.~(\ref{Del-pm}, \ref{Del-sym}) are used and the Hard Loop Approximation is adopted.
The contribution to the polarization tensor coming from the scalar tadpole depicted
in Fig.~\ref{fig-gluon}f is
\begin{equation}
\label{Pi-s-t-1}
_{(f)}\Pi^{\mu \nu}_{ab}(k) = - \frac{1}{2} 2ig^2 \delta^{ab} N_c \delta^{AA} g^{\mu \nu}
\int \frac{d^4p}{(2\pi)^4} \Delta^<(p) ,
\end{equation}
where the combinatorial factor $1/2$ is included. With the function $\Delta^<$ given by
Eq.~(\ref{Del-<}) we have
\begin{equation}
\label{Pi-s-t-2}
_{(f)}\Pi^{\mu \nu}_{ab}(k) = 3 g^2 N_c \delta_{ab} g^{\mu \nu}
\int \frac{d^3p}{(2\pi )^3} \,
\frac{2n_s({\bf p})+1}{E_p} .
\end{equation}
We get the complete contribution from a scalar field to the polarization tensor by summing up the scalar loop and scalar tadpole. Thus, one finds
\begin{equation}
\label{Pi-s-total}
_{(e+f)}\Pi^{\mu \nu}_{ab}(k) = 3 g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi )^3} \,
\frac{2n_s({\bf p})+1}{E_p} \,
\frac{k^2 p^\mu p^\nu - \big(p^\mu k^\nu + k^\mu p^\nu - g^{\mu \nu} (k \cdot p)\big)(k \cdot p)}
{(k \cdot p + i0^+)^2},
\end{equation}
which has the structure corresponding to the scalar QED. Then, it is not a surprise that the polarization tensor (\ref{Pi-s-total}) is symmetric and transverse.
After summing up all contributions, we get the final expression of gluon polarization tensor
\begin{equation}
\label{Pi-k-final}
\Pi^{\mu \nu}_{ab}(k)
= g^2 N_c \delta_{ab}
\int \frac{d^3p}{(2\pi)^3}
\frac{f({\bf p})}{E_p}
\frac{k^2 p^\mu p^\nu - (k^\mu p^\nu + p^\mu k^\nu - g^{\mu \nu} (k\cdot p))
(k\cdot p)}{(k\cdot p + i 0^+)^2},
\end{equation}
where
\begin{equation}
\label{f-def}
f({\bf p}) \equiv 2n_g({\bf p}) + 8n_f({\bf p}) + 6n_s({\bf p})
\end{equation}
is the effective distribution function of plasma constituents. We observe that the coefficients in front of the distributions functions $n_g({\bf p})$, $n_f({\bf p})$, $n_s({\bf p})$ equal the numbers of degrees of freedom (except colors) of, respectively, gauge bosons, fermions and scalars, {\it cf.} Table \ref{table-field-content}. This is obviously a manifestation of supersymmetry. Another effect of the supersymmetry is vanishing of the tensor (\ref{Pi-k-final}) in the vacuum limit when $f({\bf p}) = 0$. Needles to say, the polarization tensor (\ref{Pi-k-final}) is symmetric and transverse in Lorentz indices and thus it is gauge independent.
In case of QCD plasma, one gets the polarization tensor of the form (\ref{Pi-k-final}) after the vacuum contribution is subtracted. For the QGP with the number $N_f$ of massless flavors, the effective distribution function equals
\begin{equation}
\label{f-def-QGP}
f_{\rm QGP}({\bf p}) \equiv 2n_g({\bf p}) + \frac{N_f}{N_c} \big(n_q({\bf p}) + n_{\bar q}({\bf p}) \big) ,
\end{equation}
where $n_q({\bf p})$, $n_{\bar q}({\bf p})$ are the distribution functions of quarks and antiquarks which contribute differently to the polarization tensor than fermions of the ${\cal N} = 4$ super Yang-Mills. This happens because (anti-)quarks of QCD belong to the fundamental representation of ${\rm SU}(N_c)$ while the fermions belong to the adjoint representation.
\subsection{Fermion self-energy}
The fermion self-energy $\Sigma$ can be defined by means of the Dyson-Schwinger
equation
\begin{equation}
i{\cal S} (k) = i S (k) + i S(k) \, \big(-i\Sigma (k) \big) \, i{\cal S}(k) ,
\end{equation}
where ${\cal S}$ and $S$ is the interacting and free propagator, respectively. The lowest order contributions to fermion self-energy are given by diagrams shown in Fig.~\ref{fig-fermion}. The curly, plain, and dashed lines denote, respectively, gluon, fermion, and scalar fields.
\begin{figure}[t]
\centering
\includegraphics*[width=0.7\textwidth]{fermion_se.pdf}
\caption{Contributions to the fermion self-energy. }
\label{fig-fermion}
\end{figure}
The contribution to the fermion self-energy corresponding to the graph depicted in Fig.~\ref{fig-fermion}a is given by
\begin{equation}
\label{Si-k-a-1}
_{(a)}\Sigma^{ij}_{ab}(k) = \frac{i}{2} g^2
N_c \delta_{ab} \delta^{ij}
\int \frac{d^4p}{(2\pi )^4}
\big[\gamma_\mu S^+(p+k) \gamma^\mu D^{\rm sym}(p)
+ \gamma_\mu S^{\rm sym}(p) \gamma^\mu D^-(p-k)
\big].
\end{equation}
With the functions $D^{\pm}$, $D^{\rm sym}$ and $S^{\pm}$, $S^{\rm sym}$ given by Eqs.~(\ref{D-pm}, \ref{D-sym}, \ref{S-pm}, \ref{S-sym}), one obtains
\begin{eqnarray}
\label{Si-k-a-3}
_{(a)}\Sigma^{ij}_{ab}(k) &=& g^2 N_c \delta_{ab} \delta^{ij}
\int \frac{d^3p}{(2\pi )^3} \, \frac{ n_g ({\bf p}) + n_f ({\bf p})}{E_p} \,
\frac{p\!\!\!/ \,}{k\cdot p + i 0^+} ,
\end{eqnarray}
where the traces over gamma matrices are computed and the Hard Loop Approximation is applied.
Eq.~(\ref{Si-k-a-3}) has the well-known form of electron self-energy in QED.
Since there are scalar and pseudoscalar fields $X_p$ and $Y_p$, there are two contributions to the fermion self-energy corresponding to the graphs depicted in Figs.~\ref{fig-fermion}b, \ref{fig-fermion}c. The first one corresponding to the $X_p$ field equals
\begin{equation}
\label{Si-k-b-1}
{_{(b)}\Sigma}^{ij}_{ab}(k) = i \frac{g^2}{2} N_c \delta^{ab} \alpha^p_{ik}\alpha^p_{kj}
\int \frac{d^4p}{(2\pi )^4}
\big[ S^+(p+k) \Delta^{\rm sym}(p) + S^{\rm sym}(p) \Delta^-(p-k) \big].
\end{equation}
Due to the relations (\ref{alpha-beta-relations}), one finds that $\alpha^p_{ik}\alpha^p_{kj}=-3\delta_{ij}$.
Using the result and substituting the functions $S^{\pm}$, $S^{\rm sym}$ and $\Delta^{\pm}$, $\Delta^{\rm sym}$ given by Eqs.~(\ref{S-pm}, \ref{S-sym}, \ref{Del-pm}, \ref{Del-sym}) into Eq.~(\ref{Si-k-b-1}), one obtains the following result
\begin{eqnarray}
\label{Si-k-b-2}
{_{(b)}\Sigma}^{ij}_{ab} (k) &=& \frac{3}{2} g^2 N_c \delta_{ab} \delta^{ij}
\int \frac{d^3p}{(2\pi)^3}
\frac{ n_f ({\bf p}) + n_s ({\bf p})}{E_p} \,
\frac{p\!\!\!/ \,}{k\cdot p + i 0^+} ,
\end{eqnarray}
which holds in the Hard Loop Approximation.
The contribution due to the pseudoscalar field $Y_p$ is
\begin{equation}
\label{Si-k-c-1}
{_{(c)}\Sigma}^{ij}_{ab}(k) = i\frac{g^2}{2} N_c \delta^{ab} \beta^p_{ik}\beta^p_{kj}
\int \frac{d^4p}{(2\pi )^4} \big[ \gamma_5 S^+(p+k) \gamma_5 \Delta^{\rm sym}(p)
+ \gamma_5 S^{\rm sym}(p) \gamma_5 \Delta^-(p-k) \big].
\end{equation}
Because $\beta^p_{ik}\beta^p_{kj}=-3\delta_{ij}$, $\gamma_\mu \gamma_5=-\gamma_5 \gamma_\mu$, and $\gamma_5^2 = 1$, we again obtain the result (\ref{Si-k-b-2}).
Summing up all the contributions, we get the final expression for the fermion self-energy
\begin{eqnarray}
\label{Si-k-final}
\Sigma^{ij}_{ab}(k) &=& \frac{g^2}{2} \, N_c \delta_{ab}\delta^{ij}
\int \frac{d^3p}{(2\pi )^3}
\frac{f({\bf p})}{E_p} \, \frac{p\!\!\!/ \,}{k\cdot p + i 0^+}.
\end{eqnarray}
which, as the polarization tensor (\ref{Pi-k-final}), depends on the effective distribution function (\ref{f-def}).
\subsection{Scalar self-energy}
The scalar self-energy $P(k)$ can be defined by means of the Dyson-Schwinger equation
\begin{equation}
i \tilde \Delta (k) = i \Delta(k)
+ i \Delta (k) \, i P(k) \, i \tilde\Delta(k) ,
\end{equation}
where $\tilde\Delta$ and $\Delta$ is the scalar interacting and free propagator,
respectively. The lowest order contributions to the scalar self-energy are given
by the diagrams shown in Fig.~\ref{fig-scalar}. The curly, plain, and dashed lines
denote, respectively, gluon, fermion, and scalar fields.
\begin{figure}[t]
\centering
\includegraphics*[width=0.7\textwidth]{scalar_se.pdf}
\caption{Contributions to the scalar self-energy. }
\label{fig-scalar}
\end{figure}
Since there are scalar ($X_p$) and pseudoscalar ($Y_p$) fields, we have to consider separately the self-energies of $X_p$ and $Y_p$. However, one observes that only the coupling of scalars to fermions differs for $X_p$ and $Y_p$. The self-interaction and the coupling to the gauge field are the same. Therefore, only the fermion-loop contribution to the scalar self-energy, which is shown in the diagram Fig.~\ref{fig-scalar}a, needs to be computed separately for the $X_p$ and $Y_p$ fields.
In case of the scalar $X_p$ field, the diagram Fig.~\ref{fig-scalar}a provides
\begin{equation}
\label{P-k-a-1}
_{(a)} P^{pq}_{ab}(k) = i \frac{g^2}{4} \, N_c \delta^{ab} \alpha^p_{ij} \alpha^q_{ji}
\int \frac{d^4p}{(2\pi)^4}
{\rm Tr}\big[ S^+ (p+k) S^{\rm sym} (p)
+ S^{\rm sym} (p) S^- (p-k) \big].
\end{equation}
where the symmetry factor $1/2$ and the extra minus sign due to the fermionic character of the loop are included. With the explicit form of the functions $S^{\pm}, \; S^{\rm sym}$ given by Eqs.~(\ref{S-pm}, \ref{S-sym}) and the identity
$\alpha^p_{ij}\alpha^q_{ji}=-4\delta^{pq}$ which follows from the relations (\ref{alpha-beta-relations}), one finds
\begin{equation}
\label{P-k-a-2}
_{(a)}P^{pq}_{ab}(k) = -4 g^2 N_c \delta_{ab} \delta^{pq}
\int \frac{d^3p}{(2\pi )^3} \, \frac{ 2n_f({\bf p})-1}{E_p}.
\end{equation}
The result holds in the Hard Loop Approximation. For the pseudoscalar $Y_p$ we obtain the same expression because
$\beta^p_{ij}\beta^q_{ji}=-4\delta^{pq}$, $\gamma_5 \gamma_\mu = -\gamma_\mu \gamma_5$ and $\gamma_5^2 = 1$.
Therefore, we replace the indices $p,q$ by $A,B$ and we write down the result (\ref{P-k-a-2}) as
\begin{equation}
\label{P-k-a-5}
_{(a)}P^{AB}_{ab}(k) = -4 g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^3p}{(2\pi )^3} \, \frac{ 2n_f({\bf p})-1}{E_p}.
\end{equation}
The contribution represented by the graph depicted in Fig.~\ref{fig-scalar}b equals
\begin{eqnarray}
\label{P-k-b-1}
{_{(b)}}P^{AB}_{ab}(k) = - i \frac{1}{2} g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^4p}{(2\pi)^4}
\big[(p+2k)^2 \Delta^+(p+k) \, D^{\rm sym}(p)
+ (p+k)^2 \Delta^{\rm sym}(p) \, D^-(p-k) \big] ,
\end{eqnarray}
which after the substitution of the functions $D^{\pm}, \, D^{\rm sym}$
and $\Delta^{\pm}, \Delta^{\rm sym}$ in the form (\ref{D-pm}, \ref{D-sym}, \ref{Del-pm},
\ref{Del-sym}) leads to
\begin{eqnarray}
\label{P-k-b-3}
_{(b)}P^{AB}_{ab}(k) & = & \frac{1}{2} \, g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^3p}{(2\pi )^3} \frac{4n_g({\bf p}) - 2n_s({\bf p}) + 1}{E_p}
\end{eqnarray}
within the Hard Loop Approximation.
The contributions coming from the gluon tadpole shown in Fig.~\ref{fig-scalar}c and
the scalar tadpole from Fig.~\ref{fig-scalar}d equal, respectively,
\begin{eqnarray}
\label{P-k-c-2}
_{(c)}P^{AB}_{ab}(k) &=& - 2 g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^3p}{(2\pi)^3} \frac{2n_g({\bf p})+1}{E_p},
\\[2mm]
\label{P-k-d-2}
_{(d)}P^{AB}_{ab}(k) &=& - 5g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^3p}{(2\pi)^3} \frac{2n_s({\bf p})+1}{2E_p}.
\end{eqnarray}
In both cases the symmetry factor $1/2$ is included.
Summing up all contributions we obtain the final formula of scalar self-energy
\begin{equation}
\label{P-k-final}
P^{AB}_{ab}(k) = - 2g^2 N_c \delta_{ab} \delta^{AB}
\int \frac{d^3p}{(2\pi)^3} \frac{f({\bf p})}{E_p},
\end{equation}
which depends, as $\Pi$ and $\Sigma$, only on the effective distribution function (\ref{f-def}).
\section{Effective Action}
\label{sec-eff-action}
The Hard Loop Approach can be formulated in an elegant and compact way by introducing the effective action which was first derived for equilibrium plasmas in \cite{Taylor:1990ia,Frenkel:1991ts,Braaten:1991gm} within the thermal field theory. It was also rederived in terms of quasiclassical kinetic theory \cite{Blaizot:1993be,Kelly:1994dh}. Later on a generalization of the action to anisotropic systems was given in \cite{Pisarski:1997cp,Mrowczynski:2004kv}.
A structure of the effective action is constraint by the form of respective self-energies. Since the self-energy of a given field is the second functional derivative of the action with respect to the field, one writes
\begin{eqnarray}
\label{action-A-1}
{\cal L}^{(A_\mu^a)}_2(x) &=&
\frac{1}{2} \int d^4y \; A_\mu^a(x) \Pi_{ab}^{\mu \nu}(x-y) A_\nu^b(y) ,
\\ [2mm]
\label{action-Psi-1}
{\cal L}^{(\Psi_i^a)}_2(x) &=&
\int d^4y \; \bar{\Psi}_i^a(x) \Sigma^{ij}_{ab} (x-y) \Psi_j^b(y) ,
\\ [2mm]
\label{action-Phi-1}
{\cal L}^{(\Phi_A^a)}_2(x) &=&
\int d^4y \; \Phi_A^a(x) P^{AB}_{ab}(x-y) \Phi_B^b(y) ,
\end{eqnarray}
where the self-energies are given by the formulas (\ref{Pi-k-final}, \ref{Si-k-final}, \ref{P-k-final}), respectively. The subscript `2' indicates that the above effective actions generate only two-point functions. To generate $n$-point functions these actions need to be modified to a gauge invariant form. In the nonAbelian gauge theory studied here, the actions (\ref{action-A-1}, \ref{action-Psi-1}, \ref{action-Phi-1}) require a simple change - the ordinary derivatives should be replaced by the covariant ones in the final expressions. Repeating the calculations described in detail in \cite{Mrowczynski:2004kv}, one finds the Hard Loop effective action of the ${\cal N}=4$ super Yang-Mills as
\begin{eqnarray}
{\cal L}_{\rm HL}
&=&
-\frac{1}{4}F^{\mu \nu}_a F_{\mu \nu}^a
+\frac{i}{2}\bar \Psi_i^a (D\!\!\!\!/ \, \Psi_i)^a
+\frac{1}{2}(D_\mu \Phi_A)_a (D^\mu \Phi_A)_a
\\ \nn
&& + \; {\cal L}^{(A_\mu^a)}_{\rm HL} +{\cal L}^{(\Psi_i^a)}_{\rm HL}
+{\cal L}^{(\Phi_A^a)}_{\rm HL} ,
\end{eqnarray}
where
\begin{eqnarray}
\label{action-A-2}
{\cal L}^{A}_{\rm HL} &=& g^2 N_c \int \frac{d^3p}{(2\pi )^3} \,
\frac{f({\bf p})}{E_p} \,
F_{\mu \nu}^a (x) \bigg({p^\nu p^\rho \over (p \cdot D)^2}\bigg)_{ab} F_\rho^{b \;\mu} (x) ,
\\ [2mm]
\label{action-Psi-2}
{\cal L}^{\Psi}_{\rm HL} &=& g^2 N_c
\int \frac{d^3p}{(2\pi )^3} \, \frac{ f({\bf p})}{E_p} \,
\bar{\Psi}^a_i(x) \bigg( {p \cdot \gamma \over p\cdot D}\bigg)_{ab} \Psi^b_i(x) ,
\\ [2mm]
\label{action-Phi-2}
{\cal L}^{\Phi}_{\rm HL} &=& - 2g^2 N_c
\int \frac{d^3p}{(2\pi )^3} \, \frac{f({\bf p})}{E_p} \;
\Phi_A^a(x) \Phi_A^a(x) .
\end{eqnarray}
where $f({\bf p})$ is, as previously, the effective distribution function of plasma constituents.
The actions (\ref{action-A-2}, \ref{action-Psi-2}, \ref{action-Phi-2}) are obtained from the self-energies but the reasoning can be turned around. As argued in \cite{Frenkel:1991ts,Braaten:1991gm}, the actions of gauge bosons (\ref{action-A-2}), fermions (\ref{action-Psi-2}), and scalars (\ref{action-Phi-2}) are of unique gauge invariant form. Therefore, the structures of hard-loop self-energies of gauge bosons, fermions and scalars are unique. Consequently, the self-energies computed in the previous section and those corresponding to the fundamental ${\cal N} = 2$ hypermultiplet can be inferred from the known QED and QCD results with some help of supersymmetry arguments. However, explicit computations, as those presented in Sec.~\ref{sec-self-energies}, seem to be still needed to determine, at least, numerical coefficients.
\section{Collective modes}
\label{sec-modes}
When the self-energies computed in Sec.~\ref{sec-self-energies} are substituted into the dispersion equations presented in Sec.~\ref{sec-dis-eqs}, collective modes can be found as solutions of the equations. Below we briefly discuss the gluon, fermion, and scalar excitations.
\begin{itemize}
\item
The structure of polarization tensor (\ref{Pi-k-final}) is such as of gluon polarization tensor in QCD plasma. It has also analogical form as in both usual and supersymmetric QED plasma. Therefore, the spectrum of collective excitations of gauge bosons is in all cases the same. In equilibrium plasma we have the longitudinal (plasmon) mode and the transverse one which are discussed in {\it e.g.} the textbook \cite{lebellac}. When the plasma is out of equilibrium there is a whole variety of possible collective excitations. In particular, there are unstable modes, see {\it e.g.} the review \cite{Mrowczynski:2007hb}, which exponentially grow in time and strongly influence the system's dynamics.
\item
The form of Majorana fermion self-energy (\ref{Si-k-final}) happens to be the same as the quark self-energy in QCD plasma. It also coincides with the electron self-energy in both non-supersymmetric and supersymmetric QED plasma. Therefore, we have identical spectrum of excitations of fermions in all these systems. In equilibrium plasma there are two modes of opposite helicity over chirality ratio, see in {\it e.g.} the textbook \cite{lebellac}. One mode corresponds to the positive energy fermion, another one, sometimes called a plasmino, is a specific medium effect. In non-equilibrium plasma the spectrum of fermion collective excitations changes but no unstable modes have been found even for an extremely anisotropic momentum distribution \cite{Mrowczynski:2001az,Schenke:2006fz}.
\item
The scalar self-energy (\ref{P-k-final}) is independent of momentum, it is negative and real. Therefore, $P(k)$ can be written as $P(k) = - m^2_{\rm eff}$ where $m_{\rm eff}$ is the effective scalar mass. Then, the solutions of dispersion equation (\ref{dis-eq-selectron}) are $E_p = \pm \sqrt{m^2_{\rm eff} + {\bf p}^2}$.
\end{itemize}
We conclude this section by saying that the gauge boson and fermion excitations of SYMP are the same as in ultrarelativistic QED and QCD plasma. The scalar excitations are of the form of free massive relativistic particle.
\section{Collisional characteristics}
\label{sec-collisions}
We consider here characteristics of the ${\cal N} = 4$ super Yang-Mills plasma which are driven by collisions of plasma constituents. We start with a review of elementary processes and then we discuss transport coefficients.
\subsection{Elementary processes}
The elementary processes, which occur at the lowest nontrivial order of the coupling constant $g$, are binary interactions, the cross sections of which are proportional to $g^4$. Table~\ref{table-collisions} gives the respective matrix elements squared summed over all internal degrees of freedom of interacting particles. $G, F, S$ denote a gluon, fermion and scalar, respectively. The matrix elements, which were first computed in \cite{Huot:2006ys}, are expressed through the Mandelstam invariants $s,t$ and $u$ defined in the standard way. For a process symbolically denoted as $1+2 \longrightarrow 3+4$, we have
\begin{equation}
s \equiv (p_1 +p_2)^2 , \;\;\;\;\;\; t \equiv (p_1 - p_3)^2 , \;\;\;\;\;\; u \equiv (p_1 - p_4)^2,
\end{equation}
where $p_1, p_2, p_3, p_3$ are the four-momenta of particles $1, 2, 3, 4$, respectively. For a given process, the differential cross section, which is summed over the internal degrees of freedom of final state particles and averaged over the internal degrees of freedom of initial state particles, is expressed through the matrix element squared from Table~\ref{table-collisions} as
\begin{equation}
\frac{d \sigma}{d t} = \frac{1}{16 \pi s^2} \frac{1}{N^{\rm dof}_1} \frac{1}{N^{\rm dof}_2}\sum |M|^2,
\end{equation}
where $N^{\rm dof}_1$ and $N^{\rm dof}_2$ are the numbers of internal degrees of freedom of initial state particles given in Table~\ref{table-field-content}. The collisional processes listed in the Table~\ref{table-collisions} determine transport properties of the plasma.
\begin{table}[b]
\caption{\label{table-collisions} Elementary processes in ${\cal N} =4$ super Yang-Mills plasma.}
\begin{ruledtabular}
\begin{tabular}{ccc}
$n^0$ & Process & $\frac{1}{g^4} \frac{1}{N_c^2(N_c^2 -1)}\sum |M|^2$
\\[4mm]
1& $GG \leftrightarrow GG$ & $8 \big(\frac{s^2 + u^2}{t^2} + \frac{u^2 + t^2}{s^2}+\frac{t^2 + s^2}{u^2} +3 \big)$
\\[2mm]
2 & $GF \leftrightarrow GF$ & $32 \big(\frac{s^2 + u^2}{t^2} - \frac{u}{s} - \frac{s}{u} \big)$
\\[2mm]
3 & $GG \leftrightarrow FF$ & $32 \big(\frac{t^2 + u^2}{s^2} - \frac{u}{t} - \frac{t}{u} \big)$
\\[2mm]
4 & $GS \leftrightarrow GS$ & $24 \big(\frac{s^2 + u^2}{t^2} + 1 \big)$
\\[2mm]
5 & $GG \leftrightarrow SS$ & $24 \big(\frac{t^2 + u^2}{s^2} + 1 \big)$
\\[2mm]
6 & $GF \leftrightarrow SF$ & $- 96 \big( \frac{u}{s} + \frac{s}{u} +1 \big)$
\\[2mm]
7 & $GS \leftrightarrow FF$ & $- 96 \big( \frac{u}{t} + \frac{t}{u} +1 \big)$
\\[2mm]
8 & $FS \leftrightarrow FS$ & $- 96 \big[ \frac{2us}{t^2} + 3 \big( \frac{u}{s} + \frac{s}{u} \big) + 1 \big]$
\\[2mm]
9 & $SS \leftrightarrow FF$ & $- 96 \big[ \frac{2ut}{s^2} + 3 \big( \frac{u}{t} + \frac{t}{u} \big) + 1 \big]$
\\[2mm]
10 & $SS \leftrightarrow SS$ & $72 \big(\frac{s^2 + u^2}{t^2} + \frac{u^2 + t^2}{s^2}+\frac{t^2 + s^2}{u^2} +3 \big)$
\\[2mm]
11 & $FF \leftrightarrow FF$ & $128 \big(\frac{s^2 + u^2}{t^2} + \frac{u^2 + t^2}{s^2}+\frac{t^2 + s^2}{u^2} +3 \big)$
\\[2mm]
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Transport coefficients}
Transport coefficients of weakly coupled QGP, which include baryon and strangeness diffusion, electric charge and heat conductivity, shear and bulk viscosity and color conductivity, have been studied in detail, see \cite{Arnold:2000dr,Arnold:2003zc,Arnold:2006fz,Arnold:1998cy} and references therein. The shear viscosity of SYMP has been computed in \cite{Huot:2006ys} and the bulk viscosity is identically zero because of exact conformality of the system. Other transport coefficients of SYMP have not been studied but one expects the coefficients to be qualitatively similar to those of QGP.
Since the temperature is the only dimensional parameter, which characterizes the equilibrium plasma of massless constituents, one finds that, for example, the shear viscosity $\eta$ must be proportional to $T^3$ and the color conductivity $\sigma_c$ to $T$. It appears that the dominant contributions to both transport coefficients of QGP come from the binary collisions driven by a one-gluon exchange which correspond to the matrix elements squared diverging as $t^{-2}$ for $t \rightarrow 0$. The analyses presented in \cite{Arnold:2000dr} and \cite{Arnold:1998cy}, respectively, show that at the leading order $\eta \sim T^3/g^4\ln g^{-1}$ and $\sigma_c \sim T/\ln g^{-1}$. The factor $1/\ln g^{-1}$ appears due to the infrared singularity of the Coulomb-like interaction which is regulated by the gluon self-energy. Actually the physics behind the two formulas is rather different. The viscosity is governed by collisions with the momentum transfer of the order of $gT$ while for the color conductivity the softer collisions with the momentum transfer of the order $g^2T$ play a crucial role.
One expects the same parametric form of $\eta$, $\sigma$ and other transport coefficients in case of SYMP and QGP because, similarly to QGP, there are the Coulomb-like binary interactions for every constituent of SYMP, see Table~\ref{table-collisions}. The analysis \cite{Huot:2006ys} indeed proves that the shear viscosity coefficients of QGP and SYMP differ only by numerical factors which mostly reflect different numbers of degrees of freedom in the two plasmas. The viscosity is strongly dominated by the Coulomb-like interactions, and consequently it does not much matter that the sets of elementary processes in the two plasma systems are different.
In the paper \cite{Czajka:2011zn} we considered two transport characteristics of the ${\cal N} =1$ QED plasma which are not so constrained by dimensional arguments and seemed to strongly depend on elementary process under consideration. Specifically, we computed the collisional energy loss and momentum broadening of a particle traversing the equilibrium plasma. The latter quantity determines a magnitude of radiative energy loss of highly energetic particle in a plasma \cite{Baier:1996sk}. The dimensional argument does not work here because the two quantities depend not only on the plasma temperature but on the energy of test particle as well. We computed the energy loss and momentum broadening due to the processes which, like the Compton scattering on selectrons, are independent of momentum transfer. Such processes are qualitatively different from the Coulomb-like interactions dominated by small momentum transfers. We managed to obtain the exact formulas of the energy loss and momentum broadening due to the momentum-independent scattering. In the limit of high energy of test particle, which is important in the context of jet suppression phenomenology in nucleus-nucleus collisions, the energy loss and momentum broadening appeared to be very similar (at the leading order) to those driven by the Coulomb-like interactions.
The result can be understood as follows. One estimates the energy loss $\frac{dE}{d x}$ as $\langle \Delta E \rangle / \lambda$, where $\langle \Delta E \rangle$ is the typical change of particle's energy in a single collision and $\lambda$ is the particle's mean free path given as $\lambda^{-1} = \rho \, \sigma$ with $\rho \sim T^3$ being the density of scatterers and $\sigma$ denoting the cross section. For the differential cross section, which is independent of momentum transfer, the total cross section is $\sigma \sim e^4/s$. When a highly energetic particle with energy $E$ scatters on massless plasma particle, $s \sim ET$ and consequently $\sigma \sim e^4/(ET)$. The inverse mean free path is thus estimated as $\lambda^{-1} \sim e^4 T^2/E$. When the scattering process is independent of momentum transfer, $\langle \Delta E \rangle$ is of order $E$ and we finally find $-\frac{dE}{d x} \sim e^4 T^2$. In case of Coulomb interaction we have $\langle \Delta E \rangle \sim - e^2 T$, $\lambda^{-1} = e^2 T$ which provide the same estimate of the energy loss. The energy transfer in a single collision is thus much smaller in the Coulomb interaction than in the momentum independent scattering but the cross section is bigger in the same proportion. Consequently, the two interactions corresponding to very different differential cross sections lead to very similar energy losses.
We expect an analogous situation in SYMP. There are various elementary process but the energy loss and momentum broadening of highly energetic particles do not much differ from those in QGP.
\section{Conclusions}
\label{sec-conclusions}
QCD is obviously rather different than ${\cal N} = 4$ super Yang-Mills theory. Nevertheless QGP and SYMP are surprisingly similar in the weak coupling regime (at the leading order). The form of gluon collective excitations is identical and the same is true for the fermion (quark) modes. The scalar modes in SYMP are as of massive relativistic particle. The sets of elementary processes are different in QGP and SYMP but the transport coefficients, which are dominated by the Coulomb-like interactions, are quite similar. The energy loss and momentum broadening of a highly energetic test particle are also rather similar in the two plasma systems. The differences mostly come from different numbers of degrees of freedom in both plasmas which need to be taken into account for a quantitative comparison.
\section*{Acknowledgments}
We are grateful to Simon Caron-Huot for a discussion on conserved charges in SYMP. This work was partially supported by the ESF Human Capital Operational Program and Polish Ministry of Science and Higher Education under grants 6/1/8.2.1/POKL/2009 and 667/N-CERN/2010/0, respectively.
\newpage
|
{
"timestamp": "2012-05-25T02:04:40",
"yymm": "1203",
"arxiv_id": "1203.1856",
"language": "en",
"url": "https://arxiv.org/abs/1203.1856"
}
|
\section{Introduction}
The invention of the laser lead to a giant leap in the field of classical
and quantum optics.
This light source offers unprecedented possibilities regarding features
such as coherence, intensity, and brilliance etc.
Unfortunately, however, it is not easy to transfer this successful concept
beyond the optical or near-optical regime,
cf.~\cite{Baldwin:1997ve,Tkalya:2011dq}.
Free-electron lasers, for example, work at much higher energies -- but their
principle of operation (in their usual design) is more similar to classical
emission instead of stimulated emission.
As a result, their properties (e.g., regarding coherence) are not quite
comparable to optical lasers.
There is another phenomenon in this energy range $\,{\cal O}(\rm keV)$ in which
coherence plays a crucial role -- nuclear excitons known from M\"ossbauer
spectroscopy \cite{Mossbauer:1958fk,Mossbauer:1958kx}.
These coherent excitations of a large number of nuclei
\cite{Hannon:1999fk,Smirnov:2005uq,Habs:2009uq} are analogous to
Dicke states \cite{Dicke:1954kx} (also known as Dicke super-radiance
\cite{Scully:2006fk,Scully:2007fk,Sete:2010fu}) in quantum optics.
The coherence results in constructive interference of the emission
amplitudes from many nuclei \cite{Burnham:1969uq} and is facilitated
by the fact that the
photon recoil is absorbed by the whole lattice
\cite{Mossbauer:1958fk,Mossbauer:1958kx} instead of the individual
nuclei (which would destroy the coherence).
For example, the coherent nature of the propagation of nuclear
excitons through resonant media, showing quantum beats, was observed in
\cite{Frohlich:1991kx,Burck:1999fk}.
Other cooperative effects of coherently excited nuclei have been studied,
such as the collective Lamb shift \cite{Rohlsberger:2010fk},
coherent control of nuclear x-ray pumping \cite{Palffy:2011vn}, and
electromagnetically induced transparency \cite{Rohlsberger:2012uq}.
In the following, we study the possibility of constructing a laser-type
device employing these nuclear excitons,
which is based on stimulated emission \cite{ELIwhitebook}.
Such a device could combine the advantages of the free-electron laser
with the coherence and brilliance of nuclear excitons.
\section{Hamiltonian}
First, we describe a single nucleus as a two-level system with transition
frequency $\omega$ interacting resonantly with a single-mode field.
In rotating-wave and dipole approximation, the Hamiltonian can then
be cast into the standard form ($\hbar = c = \varepsilon_0 = 1$)
\begin{eqnarray}
\label{eq:hsingle}
\hat{H}_{\rm single}
=
\left(
g \hat{a} \sigma_\ell^+ e^{i \fk{\kappa} \cdot \fk{r}_\ell} + {\rm H.c.}
\right)
+ \frac{\omega}{2} \left( \hat{\sigma}^z_\ell + 1 \right)
+ \omega \hat{a}^\dagger \hat{a}
\,.
\end{eqnarray}
As usual, the ladder operators
$\sigma_\ell^{\pm} = ( \sigma_\ell^x \pm i \sigma_\ell^y ) / 2$
and the Pauli matrix $\sigma_\ell^z$ describe the two-level system.
The first term governs the interaction (with coupling constant~$g$)
with the electromagnetic field and
thus contains photonic annihilation and creation operators
$\hat{a} / \hat{a}^\dagger$ and phase factors
$e^{i \fk{\kappa} \cdot \fk{r}_\ell}$ depending on the location
of the nucleus, $\f{r}_\ell$, and the wavenumber $\f{k}=\f{\kappa}$
of the photon mode with $|\f{\kappa}| = \omega$.
The second and third term account for the energy stored in the two-level
nucleus and in the single-mode field, respectively.
When dealing with many $S \gg 1$ two-level nuclei instead of one,
we can sum up the individual-nucleus Hamiltonians and arrive at
\begin{eqnarray}
\label{eq:hmany}
\hat{H}
=
\left( g \hat{a} \hat{\Sigma}^+ + {\rm H.c.} \right)
+ \omega \left( \hat{\Sigma}^z + \frac{S}{2} \right)
+ \omega \hat{a}^\dagger \hat{a}
\,,
\end{eqnarray}
where quasispin-$S$-operators have been introduced
\begin{eqnarray}
\hat{\Sigma}^\pm
=
\sum_{\ell=1}^S \sigma_\ell^\pm \exp\{\pm i \f{\kappa} \cdot \f{r}_\ell\}
\,,\quad
\hat{\Sigma}^z = \frac{1}{2} \sum_{\ell=1}^S \sigma_\ell^z
\,.
\end{eqnarray}
In the interaction picture, the perturbation Hamiltonian,
originating from the first term in Eq.~(\ref{eq:hmany}), reads
\begin{eqnarray}
\label{eq:hinteract}
\hat{V} = g \hat{a} \hat{\Sigma}^+ + {\rm H.c.}
\,.
\end{eqnarray}
The quasispin-$S$-operators
$\hat{\Sigma}^\pm=\hat{\Sigma}^x\pm i\hat{\Sigma}^y$
and
$\hat{\Sigma}^z$ generate an $SU(2)$ algebra \cite{Lipkin:2002fk}.
Thus, the transition matrix elements for collective transitions not
only depend on the number of nuclei involved, but also on the number
of excitations $s$
\begin{eqnarray}
\label{eq:matrix_elements}
\hat{\Sigma}^+\ket{s}&=&\sqrt{(S-s)(s+1)}\ket{s+1}
\,,\nonumber\\
\hat{\Sigma}^-\ket{s}&=&\sqrt{(S-s+1)s}\ket{s-1}
\,,
\end{eqnarray}
where $\ket{s} \propto (\hat{\Sigma}^+)^s \ket{0}$ denotes a coherent
state with $s$ excitons, often referred to as Dicke states
\cite{Dicke:1954kx}.
\section{Coherent emission}
In contrast to the spontaneous decay of a single excited nucleus,
where the resulting photon can be emitted in all directions, exciton
states as in Eq.~(\ref{eq:matrix_elements}) predominantly emit
photons in forward direction $\f{\kappa}$.
Only in this case, all the phases $e^{i \fk{\kappa} \cdot \fk{r}_\ell}$
add up coherently (we assume random locations $\f{r}_\ell$),
see Fig.~\ref{fig:exciton_and_dicke}.
We will now investigate spontaneous and stimulated emission from an
ensemble of $S$ coherently excited nuclei in more detail.
\begin{figure}[h]
\begin{center}
\psfrag{gamma}{$\f{\gamma}$}
\psfrag{kvector}{$\f{k}$}
\psfrag{0}{\scriptsize{$0$}}
\psfrag{1}{\scriptsize{$1$}}
\psfrag{2}{\scriptsize{$2$}}
\psfrag{3}{\scriptsize{$3$}}
\psfrag{S}{\scriptsize{$S$}}
\subfigure[]{\includegraphics[width=0.5\columnwidth]{exciton2}}
\hspace{.5cm}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{dicke2}}
\caption{Sketch of the coherent properties of nuclear excitons.
An incident photon with wave-vector $\f{k}$ is absorbed (a)
by an ensemble of $S\gg1$ nuclei (two-level systems) and thus
generates a Dicke state $\ket{s=1}$.
Then the decay amplitudes of all these nuclei add up coherently
in forward direction such that the absorption is followed by
collective spontaneous emission into the same direction (b).}
\label{fig:exciton_and_dicke}
\end{center}
\end{figure}
\subsection{Spontaneous emission}
We start with the case of collective spontaneous emission
(a.k.a.\ Dicke super-radiance \cite{Scully:2006fk,Scully:2007fk,Sete:2010fu})
from a coherent state $\ket{s}$.
First of all, as the $S$ nuclei are not enclosed by a resonator
or a cavity in our set-up, we have to consider all $\f{k}$-modes.
Thus, the Hamiltonian~(\ref{eq:hinteract}) changes into
\begin{eqnarray}
\label{eq:hmanymodes}
\hat{V}_{\rm sp} \left( \tau \right)
=
\int d^3k\; g_{\fk{k}} \hat{a}_{\fk{k}} \,
e^{-i \left( \omega_{\fk{k}} - \omega \right) \tau} \hat{\Sigma}^+
\left( \f{k} \right) + {\rm H.c.}
\,,
\end{eqnarray}
where $\hat{a}_{\fk{k}}$ is the photonic annihilation operator for the
mode $\f{k}$ with frequency $\omega_{\fk{k}}$ and $g_{\fk{k}}$ the
associated coupling strength.
Note that we neglect polarization effects, i.e., we assume that
the polarization vectors are directed along the same axis as the dipole
moments of the absorbing nuclei.
Furthermore, $\hat{\Sigma}^+\left(\f{k}\right)$ denotes the
quasispin-$S$-operators with the wavenumber $\f{k}$ instead of $\f{\kappa}$.
However, when $\f{k}$ is not close to $\f{\kappa}$, the phase factors
of $\hat{\Sigma}^\pm \left( \f{k} \right)$ and $\ket{s}$ do not match,
and the transition is not coherent, i.e., not enhanced by a factor
$S$ according to Eq.~(\ref{eq:matrix_elements}),
and can thus be neglected.
Note that this is the reason why collectively emitted photons are
directed along (almost) the same axis as previously absorbed
photons \cite{Scully:2006fk,Scully:2007fk,Sete:2010fu},
see also Fig.~\ref{fig:exciton_and_dicke}.
For simplicity, the quasispin-$S$-operators are therefore approximated
by introducing a cut-off function $g \left( \f{\kappa} - \f{k} \right)$
that is only non-zero for small deviations $\f{\kappa} - \f{k}$
of the supported direction $\f{\kappa}$, i.e.,
$\hat{\Sigma}^\pm \left( \f{k} \right) \approx
g \left( \f{\kappa} - \f{k} \right)\,\hat{\Sigma}^\pm$.
For large $S$, we may approximate the quasispin-$S$-operators
classically, i.e.,
$\hat{\Sigma}^- \approx \Sigma^- = \sqrt{\left( S - s +1 \right) s}$,
and thus the effect of the Hamiltonian Eq.~(\ref{eq:hmanymodes}) acting on
the vacuum state can be expressed by a coherent state
\begin{eqnarray}
\hat{U}_{\rm sp} ( t ) \ket{0}
\approx
\exp\left(
\int d^3k \, \alpha_{\fk{k}} \hat{a}_{\fk{k}}^\dagger - {\rm H.c.}
\right)
\ket{0}
\,,
\end{eqnarray}
with the amplitudes
\begin{eqnarray}
i\alpha_{\fk{k}}
=
g_{\fk{k}}^* g ( \f{\kappa} - \f{k} )
\sqrt{\left( S - s +1 \right) s}
\int_{0}^t
d\tau \,
e^{i \left( \omega_{\fk{k}} - \omega \right) \tau}
\,.
\end{eqnarray}
The number of emitted photons per mode is given by $|\alpha_{\fk{k}}|^2$
and the total photon number grows linearly with $t$
\begin{eqnarray}
\mathcal{N}_\gamma
=
\int d^3k \, |\alpha_{\fk{k}}|^2
\approx
2 \pi^2 \left( S - s +1 \right) s |g_{\fk{\kappa}}|^2 \frac{t}{L_\bot^2}
\,,
\end{eqnarray}
where $L_\bot^2$ denotes the transversal cross-section area of the ensemble,
which determines the transverse area in $\f{k}$-space where
$g ( \f{\kappa} - \f{k} )$ is non-zero.
In addition to this spatial resonance condition, the temporal resonance
was incorporated via approximating the squared time-integral by $t^2$ for
$| \omega_k - \omega | < 1/t$ and zero otherwise.
Strictly speaking, this relation is only valid for a fixed number of
excitations $s$, i.e., the time-dependence of $s(t)$ due to the emission
of photons (energy conservation) is neglected.
Assuming that this time-dependence $s(t)$ is slow compared to $\omega$
(i.e., that the coupling strength is small enough), we may take it into
account approximately via defining the instantaneous spontaneous emission
rate
\begin{eqnarray}
\Gamma_{\rm sp}(t)
= \frac{d\mathcal{N}_\gamma}{dt}
= \gamma \left[ S - s(t) +1 \right] s(t)
\,,
\end{eqnarray}
with the abbreviation $\gamma = 2 \pi^2 |g_{\fk{\kappa}}|^2 / L_\bot^2$.
The change of $s(t)$ in the time interval $dt$ is then governed by
$\Gamma_{\rm sp}(t)$
\begin{eqnarray}
\label{eq:dgl}
\frac{ds(t)}{dt}
=
- \Gamma_{\rm sp}(t)
=
-\gamma \left[ S - s(t) + 1 \right] s(t)
\,.
\end{eqnarray}
For the initial condition $s(0) = S/2$ (see below),
the solution for $S \gg 1$ is given by
\begin{eqnarray}
s(t) = \frac{S}{1 + e^{\gamma S t}}
\,.
\end{eqnarray}
This yields the intensity due to spontaneous emission
\begin{eqnarray}
\label{eq:ispon}
I_{\rm sp}(t)
&=&
-\frac{dE}{dt} \frac{1}{L_\bot^2}
=
-\frac{ds(t)}{dt} \frac{\omega}{L_\bot^2}
\nonumber\\
&=&
\frac{1}{4} \gamma S^2 {\rm sech}^2 \left( \gamma S t /2 \right)
\frac{\omega}{L_\bot^2}
\,,
\end{eqnarray}
where $L_\bot^2$ is the cross-section area of the emitted beam,
see also \cite{Rehler:1971fk}.
The time-dependence in the ${\rm sech}$-function can be used to define
an effective time constant via
\begin{eqnarray}
\label{eq:tauspon}
\tau_{\rm sp} = \frac{4}{\gamma S}
\,,
\end{eqnarray}
after which $I_{\rm sp}(t)$ has dropped to 7\% of its initial value.
Let us now briefly compare this time-scale $\tau_{\rm sp}$ for the coherent
spontaneous emission process with the time-scale in the incoherent case.
For incoherent emission from $s$ excited nuclei, we can regard each
nucleus independently.
According to standard Weisskopf-Wigner theory \cite{Scully:1997fk},
the life-time of an excited nucleus is given by
\begin{eqnarray}
\label{eq:tausingle}
\tau_{\rm single}
=
\frac{1}{\Gamma_{\rm single}}
=
\frac{1}{8 \pi^2} \frac{1}{|g_{\fk{\kappa}}|^2 \omega^2}
\,.
\end{eqnarray}
Comparing the two time-scales Eq.~(\ref{eq:tauspon}) and (\ref{eq:tausingle})
\begin{eqnarray}
\frac{\tau_{\rm sp}}{\tau_{\rm single}}
=
64\pi^2 \frac{L_\bot^2}{\lambda^2} \frac{1}{S}
\,,
\end{eqnarray}
we find that for large $S$, the coherent spontaneous emission process is
much faster than the incoherent process
(see also \cite{Junker:2012fk}).
Taking for example $S = 10^{10}$ $^{57}$Fe-nuclei with resonance at
$\Delta E_\gamma = 14.4\;{\rm keV}$, lifetime
$\tau_{\rm single} = 141\;{\rm ns}$ and $L_\bot = 0.1\;\mu{\rm m}$,
the quotient evaluates to $\tau_{\rm sp}/\tau_{\rm single} \approx 0.09$,
i.e. the coherent emission runs over ten times faster than the incoherent
emission.
Note that there are also competing processes, such as decay via
electron conversion -- but they are incoherent and thus can be suppressed
for large $S$, i.e., small $\tau_{\rm sp}$.
\subsection{Stimulated emission}
In order to study stimulated emission from a coherently excited $S$-nuclei
ensemble, we regard the incoming field
$A_{\rm in}(t) = \sqrt{I_{\rm in}(t)}/\omega$ classically.
That is, we use the Hamiltonian Eq.~(\ref{eq:hinteract}), but replace
$g \hat{a}$ by $\tilde{g} A_{\rm in}(t)$.
For simplicity, we assume the transition matrix element $\tilde{g}$
of the nucleus to be real
\begin{eqnarray}
\label{eq:hstim}
\hat{V}_{\rm st}
=
\tilde{g} A_{\rm in}(t) \left( \hat{\Sigma}^+ + \hat{\Sigma}^- \right)
=
2 \tilde{g} A_{\rm in}(t) \hat{\Sigma}^x
\,.
\end{eqnarray}
Applying Heisenberg picture and employing the properties of the
$SU(2)$-algebra yields
\begin{eqnarray}
\label{eq:sigmaz}
\hat{U}_{\rm st}^\dagger (t) \hat{\Sigma}^z \hat{U}_{\rm st} (t)
&=&
\cos \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
\hat{\Sigma}^z
\nonumber\\
&+& \sin \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
\hat{\Sigma}^y
\,.
\end{eqnarray}
As envisaged for laser application (see below), we choose $s(0) = S$ here,
that is all $S$ nuclei are in the coherently excited state.
The time-dependent number of excitations is given by
$\bra{S}\hat{U}_{\rm st}^\dagger(t)\hat{\Sigma}^z\hat{U}_{\rm st}+S/2\ket{S}$
and thus the energy stored in the $S$ nuclei at time $t$ is
\begin{eqnarray}
E(t)
=
\frac{S \omega}{2} \left[ \cos
\left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
+ 1\right]
\,.
\end{eqnarray}
This yields the emitted intensity $I_{\rm st}(t)$ stimulated by the incoming
intensity $I_{\rm in}(t)$
\begin{eqnarray}
\label{eq:istim}
I_{\rm st}(t)
=
\frac{\tilde{g} S}{L_\bot^2}
\sin \left(
\frac{2 \tilde{g}}{\omega} \int_0^t d\tau\, \sqrt{I_{\rm in}(\tau)}
\right)
\sqrt{I_{\rm in}(t)}
\,,
\end{eqnarray}
where we have assumed that both beams have the same cross-section area
$L_\bot^2$.
We define the time-scale of the stimulated emission as the time
$\tau_{\rm st}$, after which all the energy initially stored in the $S$
nuclei has been emitted, i.e.,
\begin{eqnarray}
\label{eq:taustim}
\int_0^{\tau_{\rm st}} d\tau\,\sqrt{I_{\rm in}(\tau)}
=
\frac{\pi \omega}{2 \tilde{g}}
\,.
\end{eqnarray}
Now let us imagine that we have two separate ensembles (e.g., foils)
of coherently excited nuclei, such that the first foil spontaneously
emits the intensity $I_{\rm in}(t)$ as in Eq.~(\ref{eq:ispon}) which
causes stimulated emission according to Eq.~(\ref{eq:istim})
in the second foil.
In this case, we can insert $ I_{\rm in}(t) = I_{\rm sp}(t)$,
and Eq.~(\ref{eq:taustim}) can be solved for $\tau_{\rm st}$
\begin{eqnarray}
\tau_{\rm st}
=
\frac{4}{\gamma S}\,{\rm ArTanh}
\left[ {\rm Tan} \left( \frac{1}{8}
\sqrt{\frac{\pi}{2}} \right) \right]
\approx 0.16\times\tau_{\rm sp}
\,.
\end{eqnarray}
Since both foils contain the same nuclei (with the same coupling strengths),
the time-scale for the stimulated emission of the second foil,
$\tau_{\rm st}$, is completely determined by the time-scale of the
spontaneous emission process of the first foil, $\tau_{\rm sp}$.
\section{Pumping}
After having discussed coherent spontaneous emission as well as coherent
stimulated emission, let us investigate the pumping process for a single
foil of $S$ nuclei, which are initially in the state $s=0$, i.e.,
$\bra{0} \hat{\Sigma}^z \ket{0} = -S/2$.
Note that it is very easy to over- or under-estimate the efficiency of the
pumping process by using too simplified pictures.
On the one hand, one might expect that the number of excitons in the foil
grows linearly with the number of photons incident and thus linearly with
the interaction time $t$.
However, this is only true for pumping with incoherent light
(for further details, see the appendix), but not for coherent pumping,
which is the case considered here.
On the other hand, since the transition matrix elements in
Eq.~(\ref{eq:matrix_elements}) scale with $\sqrt{s}$ and thus the
effective line-width increases with $s$, one might expect a behavior
like $\dot s\propto s$, which would imply an exponential growth
$s(t)\propto e^{\kappa t}$, at least for small $s\ll S$.
This picture is also wrong, since -- in view of the unitarity of the
time-evolution -- not just the absorption rate but also the emission
rate increase with $s$.
Thus, the correct answer is that $s(t)$ grows quadratically
$s(t)\propto t^2$ for small $s$, i.e., somewhere in between
linear and exponential.
To show this, let us consider pumping with one coherent pump-pulse
$A_{\rm pump}(t)$ for the whole interaction time.
We can employ the Hamiltonian Eq.~(\ref{eq:hstim}) again with the sole
difference that the incoming field $A_{\rm in}(t)$ is now given by the
pump-field $A_{\rm in}(t) = A_{\rm pump}(t)$.
Thus, Eq.~(\ref{eq:sigmaz}) again holds, and the exciton number is given by
\begin{eqnarray}
s(t)
&=&
\frac{S}{2}
\left[
1 - \cos \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm pump}(\tau)\right)
\right]
\nonumber\\
&=&
S \tilde{g}^2 \left( \int_0^t d\tau\, A_{\rm pump}(\tau) \right)^2
+ \,{\cal O} \left( \tilde{g}^4 t^4 \right)
\,,
\end{eqnarray}
i.e., the exciton number grows quadratically for small $t$
(for an alternative approach, see the appendix).
We moreover find that a full cycle
(i.e., a sign flip of $\hat{\Sigma}^z\to-\hat{\Sigma}^z$)
occurs after the pump time $\tau_{\rm pump}$ where
\begin{eqnarray}
\label{eq:pump_constraint}
\int_0^{\tau_{\rm pump}} d\tau\, A_{\rm pump}(\tau) = \frac{\pi}{2 \tilde{g}}
\,.
\end{eqnarray}
The simplest example would be a constant pump pulse
$A_{\rm pump}=A_0$ with $\tau_{\rm pump} = \pi / ( 2 \tilde{g} A_0 )$.
In order to see if such a pump-field is feasible in general, we calculate
a rough estimate for the required intensity of the pump-field.
For simplicity, we assume that the intensity is constant over the pulse,
i.e., $A_{\rm pump} (t) = \sqrt{I_{\rm pump}}/\omega$.
Then we find
$I_{\rm pump} = \pi^2 \omega^2 / (4 \tilde{g}^2 \tau_{\rm pump}^2)$.
Now $\tilde{g}$ can be expressed in terms of the (single-nucleus) decay
rate $\Gamma_{\rm single}$ and the frequency $\omega$ of the considered
nuclear excitation (see the appendix).
Moreover, if we replace the coherent pulse-length
$\tau_{\rm pump} = {\mathfrak N} \lambda = 2 \pi {\mathfrak N} / \omega$
by the number ${\mathfrak N}$ of (coherent) wave-cycles, we find
\begin{eqnarray}
\label{eq:ipump}
I_{\rm pump}
=
\frac{1}{32 \pi {\mathfrak N}^2}\,\frac{\omega^5}{\Gamma_{\rm single}}
\,.
\end{eqnarray}
Thus, nuclear resonances with low energies $\omega$ but high decay-rates
$\Gamma_{\rm single}$ require low pump intensities.
Concrete examples will be discussed at the end of this article.
\section{Laser}
Now we have gathered all the tools required to understand the set-up
of the proposed nuclear exciton laser.
The envisaged set-up consists of a series of $N \gg 1$ foils
$n = 1,2,...,N$, each foil containing $S_n$ nuclei
(two-level systems) with a nuclear resonance at frequency $\omega$.
At the beginning, we assume that all $n = 1,2,...,N$ foils are in the
ground state, corresponding to a quasi-spin $\Sigma^z_n = -S_n/2$,
see Fig.~\ref{fig:sequence}(a).
To prepare the emission of a laser pulse, the foils need to be pumped
to suitable coherent states.
Let us distinguish between the first foil and all later foils $n = 2,...,N$.
While the latter all should be pumped to the maximum of $\Sigma^z_n = S_n/2$,
i.e., $s_n = S_n$, the first foil should only be pumped such that half of the
nuclei are in the excited state, i.e., $\Sigma^z_1 = 0$ and $s_1 = S_1/2$.
For simplicity, we envisage the whole pumping process to be achieved by only
one coherent pump-pulse, which goes through all the foils one after another
and is only weakly changed by absorption.
\begin{figure}[h]
\begin{center}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{pump_before}}
\hspace{.5cm}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{pump_after}}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{emission}}
\caption{
Sketch of the operation sequence of the proposed nuclear exciton laser.
Initially, all foils (here $N=3$) are in the ground state
$\Sigma^z_n = -S_n/2$ (a).
The pump-pulse then rotates the quasispin of the first foil to
$\Sigma^z_1 = 0$
and the quasispin of all subsequent foils to $\Sigma^z_n = +S_n/2$ (b).
Then, the ``half-filled'' first foil spontaneously emits a pulse
$I_{\rm sp}(t)$,
which stimulates emission at foils $2$ and $3$, leading to an enhanced
overall intensity $I_{\rm total}^{(3)}(t)$ (c).
}
\label{fig:sequence}
\end{center}
\end{figure}
The pump-pulse should satisfy Eq.~(\ref{eq:pump_constraint}) in order to
rotate the quasispin $\Sigma^z_n$ of each foil from $\Sigma^z_n = -S_n/2$
to $\Sigma^z_n = +S_n/2$.
Additional measures need to be taken to ensure that the first foil is only
pumped to $s_1 = S_1/2$.
One option could be to have a different kind of nuclei in the first foil,
which have the same resonance frequency as those in the other foils, while
the coupling strengths differ by a factor of two (approximately).
Another option could be to switch the first foil
(mechanically or magnetically
\cite{Shvydko:1996fk,Rohlsberger:2000fk,Coussement:2002zr,
Palffy:2009kx,Adams:2011ys})
during the pumping process.
When the set-up is prepared as shown in Fig.~\ref{fig:sequence}(b),
the emission process automatically starts, as the first foil immediately
begins with the spontaneous emission discussed above, Eq.~(\ref{eq:ispon}).
The idea is that, due to the ``half-filled'' coherent state $\Sigma^z_1 = 0$,
the emission process of the first foil happens much faster than the
spontaneous emission of the subsequent foils.
Taking, e.g., the second foil ($s_2 = S_2$), the time-scale for the emission
of a single photon would be $1 / \Gamma_{\rm sp} = 1 / (\gamma S_2)$.
For the first foil, the time-scale for the whole emission process
(of nearly all photons, not only one) is given by
$\tau_{\rm sp} = 4 / (\gamma S_1)$.
So by choosing $S_1 \gg S_2$, e.g., by making the first foil ten times
thicker than the subsequent foils, it is assured that the second foil
is still in the state $\Sigma^z_2 = S_2/2$, when the intensity emitted
from the first foil is incident.
Stimulated emission then occurs at the second foil according to
Eq.~(\ref{eq:istim}) and the second foil has emitted all its energy
after $\tau _{\rm st} \approx 0.16\times\tau_{\rm sp}$, i.e.,
before the stimulating pulse coming from the first foil declines.
After the second foil, the overall intensity thus adds up to
$I_{\rm total}^{(2)}(t) = I_{\rm sp}(t) + I_{\rm st}^{(2)}(t)$.
This overall intensity then causes stimulated emission at the third foil,
resulting in an even bigger intensity
$I_{\rm total}^{(3)}(t) = I_{\rm total}^{(2)}(t) + I_{\rm st}^{(3)}(t)$, etc.
In this way, the intensity of the light pulse grows stepwise with each
passed foil.
Numerical analysis has been done for the case of $N = 50$ foils.
Iteratively, $I_{\rm st}^{(n)} (t)$ was calculated from
$I_{\rm total}^{(n-1)} (t)$, where
$I_{\rm total}^{(n)} (t) = I_{\rm total}^{(n-1)} (t) + I_{\rm st}^{(n)} (t)$,
starting with $I_{\rm total}^{(1)} (t) = I_{\rm sp}(t)$.
It was assumed that the first foil consists of $S_1 = 10^{10}$
$^{57}$Fe-nuclei
(with $\Delta E_\gamma = 14.4\;{\rm keV}$ and $\tau = 141\;{\rm ns}$)
while all other foils are ten times thinner, i.e., $S_n = 10^{9}$.
Transversal dimensions of the foils and the laser beam are chosen as
$L_\bot^2 = (0.1\;\mu{\rm m})^2$.
Note that the useful part of the laser pulse $I_{\rm total}^{(n)}(t)$
is determined by the time after which the last foil has emitted all its
excitations, $\tau^{(n)}_{\rm st}$, because afterwards re-absorption
takes place.
This time $\tau^{(n)}_{\rm st}$ becomes shorter with rising $n$,
as the intensity which causes the stimulated emission grows with $n$.
As a result, the average intensity of the useful part of the laser pulse,
\begin{eqnarray}
\overline{I_{\rm total}^{(n)}}
=
\frac{1}{\tau^{(n)}_{\rm st}}
\int_0^{\tau^{(n)}_{\rm st}} d\tau\,
I_{\rm total}^{(n)} (\tau)
\,,
\end{eqnarray}
increases with a power law. In the concrete example given above,
$\overline{I_{\rm total}^{(n)}}$ grows roughly $\propto n^{3/2}$,
see Fig.~\ref{fig:powerlaw}.
\bigskip
\begin{figure}[h]
\begin{center}
\psfrag{Itotaltau}
{{$\overline{I_{\rm total}^{(n)}}/\overline{I_{\rm total}^{(2)}}$}}
\psfrag{n}{{$n$}}
\includegraphics[width=0.9\columnwidth]{powerlaw}
\caption{Average intensity $\overline{I_{\rm total}^{(n)}}$ over number of
foil $n$ from numerical analysis. In this example,
$\overline{I_{\rm total}^{(n)}}$ roughly increases as a power law
$\propto n^{3/2}$.}
\label{fig:powerlaw}
\end{center}
\end{figure}
\section{Conclusions}
In summary, we described a proposal for a laser in the $\,{\cal O}({\rm keV})$
regime which is based on stimulated emission and works with nuclear
excitons.
The pumping could be achieved with a free-electron laser, for example.
Note that the pump pulse $A_{\rm pump}$ and the generated laser pulse
$A_{\rm laser}$ both correspond to a 180$^\circ$-rotation of the last
foil according to Eq.~(\ref{eq:sigmaz}) and thus are related via
\begin{eqnarray}
\int_0^{\tau_{\rm pump}} d\tau\, A_{\rm pump}(\tau)
=
\int_0^{\tau^{(N)}_{\rm st}} d\tau \, A_{\rm laser}(\tau)
=
\frac{\pi}{2\tilde g}
\,.
\end{eqnarray}
However, the intensity of the pump pulse $\propto|A_{\rm pump}^2|$ is much
larger than that of the laser pulse $\propto|A_{\rm laser}^2|$.
On the other hand, the duration ${\tau^{(N)}_{\rm st}}$ of the laser pulse
is much larger and thus its frequency accuracy is much higher
(see also \cite{Kim:2008qf} for a different approach).
This could be important for spectroscopy etc.
Let us discuss some example data for the required intensity of the
pump-pulse.
First, we consider $^{57}$Fe-nuclei with a resonance at
$\Delta E_\gamma = 14.4\;{\rm keV}$ with a mean lifetime of
$\tau = 141\;{\rm ns}$.
If we assume that the pump pulse consists of ${\mathfrak N} = 10^6$
coherent wave-trains, we would need a pump intensity of
$I_{\rm pump} \approx 8.3 \cdot 10^{20} \;{\rm W}/{\rm cm^2}$
according to Eq.~(\ref{eq:ipump}).
(Comparable or even higher intensities have already been
considered in e.g.\ \cite{Burvenich:2006bh,Palffy:2008uq,Liao:2011cr}.)
This is probably beyond the capabilities of present
free-electron lasers, see, e.g., \cite{felbasics}.
However, future light sources such as seeded free-electron lasers
should achieve improved coherence times and higher intensities
(especially after focussing with X-ray lenses).
On the other hand, when considering other nuclear resonances beyond
the well-known $^{57}$Fe-example, we find that the requirements are
somewhat easier to fulfill.
For example, considering the $^{201}$Hg resonance at
$\Delta E_\gamma = 1.6\;{\rm keV}$ with $\tau = 81\;{\rm ns}$
and again assuming ${\mathfrak N} = 10^6$ coherent wave-trains,
we would ``only'' need a pump intensity of
$I_{\rm pump} \approx 8.0 \cdot 10^{15}\;{\rm W}/{\rm cm^2}$
according to Eq.~(\ref{eq:ipump}).
Unfortunately, this intensity is probably still too large:
After inserting typical values for the absorption cross section of
$1.6\;{\rm keV}$-photons in metals (or other solid materials), we find
that the pump beam deposits enough energy in the foil to evaporate it.
Even though the thermalization dynamics following the illumination
with such a $8.0 \cdot 10^{15}\;{\rm W}/{\rm cm^2}$-beam of
$1.6\;{\rm keV}$-photons is not well studied yet, one would expect that
the foil starts to disintegrate after a few pico-seconds \cite{Siwick:2003fk}
and hence does not survive long enough for our purposes.
In summary, the major difficulty of our set-up is that it requires
extremely large pump intensities.
As we may infer from Eq.~(\ref{eq:ipump}), the pump intensity scales with
the fifth power of the transition energy $\omega$.
Thus, our scheme should be much easier to realize at lower energies.
As one possible example, let us envisage a UV-laser.
In this case, the nuclear transitions could be replaced by
suitable electronic transitions in atoms or molecules.
The pumping could be achieved either directly via a free-electron
laser in the low-energy regime or indirectly via a two-photon transition
generated by two optical lasers, for example.
\begin{figure}[h]
\begin{center}
\psfrag{e1}{$E_2$}
\psfrag{e2}{$E_3$}
\psfrag{e3}{$E_4$}
\psfrag{1s}{2s}
\psfrag{2s}{3d}
\psfrag{3p}{4p}
\psfrag{delta}{$\Delta$}
\psfrag{omega_1}{$\omega_1$}
\psfrag{omega_2}{$\omega_2$}
\psfrag{omega_uv}{$\omega_{\rm uv}$}
\includegraphics[width=0.5\columnwidth]{three_level}
\caption{Sketch (not to scale) of the level scheme.
The pumping process from 2s to 4p is induced by a detuned two-photon
transition, i.e., both photons together are in resonance
$E_4-E_2=\omega_1+\omega_2$ while one-photon absorption is
suppressed by the detuning $\Delta$ where $E_3-E_2=\omega_1-\Delta$.
The laser operates via the one-photon transition from 4p back to 2s and
emits photons of the energy $\omega_{\rm uv}=\omega_1+\omega_2$.}
\label{fig:three_level}
\end{center}
\end{figure}
Let us discuss the latter case using a three-level system as
depicted in Fig.~\ref{fig:three_level}.
Assuming two pump-lasers with optical frequencies $\omega_1$ and
$\omega_2$, the laser could operate in the ultra-violet regime
$\omega_{\rm uv}$.
In this case, the expression $\tilde g A_{\rm pump}$ for one-photon
pumping in Eq.~(\ref{eq:pump_constraint}) should be replaced by
$\tilde g_{23}A^{\rm pump}_1\tilde g_{34}A^{\rm pump}_2/\Delta$.
Note that the coupling constant $\tilde g_{23}$ of the
``dipole-forbidden'' 2s-3d transition
is typically much smaller than $\tilde g_{34}$.
Assuming typical values, such as a dipole coupling length of three Bohr
radii,
we would need pump-laser intensities of about
$I_{\rm pump} = \,{\cal O}(10^{10}\;{\rm W}/{\rm cm}^2)$
over a length of ${\mathfrak N} = 10^4$ coherent wave-trains with a
detuning of $\Delta = \,{\cal O}(10^{13}\;{\rm Hz})$ in order to
prevent unwanted excitations of the middle 3d level.
The condition for dominant coherent emission,
$\tau_{\rm sp}/\tau_{\rm single} \ll 1$,
can be fulfilled for $S_1/L_\bot^2 = \,{\cal O}(10^{5}\;\mu{\rm m}^{-2})$,
which is quite reasonable.
In this scenario, the duration of the laser pulse ${\tau^{(N)}_{\rm st}}$
is comparable to the length of the pump pulse
$\tau_{\rm pump} \approx {\tau^{(N)}_{\rm st}} = \,{\cal O}(10\;{\rm ps})$
and its intensity is well above $\,{\cal O}(10^5\;{\rm W}/{\rm cm}^2)$,
depending on the number of foils.
Again, the main idea would be that the coherent emission is strongly
enhanced for $S\gg1$ in comparison to competing non-coherent decay
channels.
To this end, the two pump lasers must be parallel to ensure the
spatial phase matching.
\section*{Acknowledgements}
R.S.\ acknowledges fruitful discussions, e.g., with Bernhard Adams,
at the Winter Colloquia on the Physics of Quantum Electronics (PQE);
and with Uwe Bovensiepen.
N.t.B.\ would like to thank Sumanta Das for valuable discussions at the
DPG Spring Meeting in Stuttgart.
This work was supported by the DFG (SFB-TR12).
\section*{Appendix}
\subsection{Coherent versus incoherent pumping}
Let us review the pumping process by applying the
Holstein-Primakoff \cite{Holstein:1940vn} transformation
\begin{eqnarray}
\hat{\Sigma}^+
=
\hat{b}^\dagger \sqrt{S - \hat{b}^\dagger \hat{b}}
=
(\hat{\Sigma}^-)^\dagger
\,,\quad
\hat{\Sigma}^z = \hat{b}^\dagger \hat{b} - S/2
\,,
\end{eqnarray}
to the Hamiltonian Eq.~(\ref{eq:hstim}) and considering the limit $S\gg s$,
i.e., the beginning of the pumping process
\begin{eqnarray}
\hat{V}_{\rm st}
\approx
\tilde{g} A_{\rm pump}(t)
\left( \sqrt{S} \hat{b}^\dagger + {\rm H.c.} \right)
\,.
\end{eqnarray}
We first analyze the case of coherent pumping, that is pumping with a
coherent pulse $A_{\rm pump}(t)$, i.e., the same time-evolution operator
for the whole pumping process
\begin{eqnarray}
\hat U_{\rm st}(t) = \exp\left(
\beta(t)\hat b^\dagger-{\rm H.c.}
\right)
\,.
\end{eqnarray}
A time-dependent coherent state of excitons is created
\begin{eqnarray}
\beta(t)=-i \tilde{g} \sqrt{S} \int_0^t d\tau A_{\rm pump}(\tau)
\,,
\end{eqnarray}
whose exciton number grows quadratically with time $t$
\begin{eqnarray}
n(t)=|\beta(t)|^2
=\,{\cal O}\left( \tilde{g}^2 S A_{\rm pump}^2 t^2 \right)
\,.
\end{eqnarray}
An incoherent pump-pulse, in contrast, can be approximated as a
succession of many uncorrelated coherent pulses $A_{\rm pump}^{(i)}(t)$
incident on the target.
The time-evolution operator then is a product of many coherent
displacement operators
\begin{eqnarray}
\hat U_{\rm eff}
\approx
\prod_i
\hat{U}_{\rm st}^{(i)}
=
\exp\left(\sum_i\beta_i \hat b^\dagger-{\rm H.c.}\right)
\,.
\end{eqnarray}
For uncorrelated pulses, the $\beta_i$ have random phases,
such that the sum corresponds to a random walk
\begin{eqnarray}
\beta_{\rm eff}(j)
=\sum_{i=1}^j\beta_i\propto\sqrt{j}\propto\sqrt{t}
\,,
\end{eqnarray}
such that the exciton number $n = |\beta_{\rm eff}|^2$
grows merely linearly with time in this case.
\subsection{Expressing $\tilde{g}$ in terms of $\Gamma_{\rm single}$ and $\omega$}
The coupling constant $|g_{\fk{\kappa}}|$ can be expressed via the decay
rate $\Gamma_{\rm single}$ and frequency $\omega$
\begin{eqnarray}
\Gamma_{\rm single}
=
2 \pi \int d^3k\, |g_{\fk{k}}|^2 \delta
\left( \omega_k - \omega \right)
\approx
8 \pi^2 |g_{\fk{\kappa}}|^2 \omega^2
\,.
\end{eqnarray}
The dimensionless coupling constant $\tilde{g}$ can be obtained via
$\tilde{g} = |g_{\fk{\kappa}}| \sqrt{2 (2 \pi)^3 \omega}$ since our
Hamiltonian contains the classical field $A_{\rm pump}(t)$.
As a result we arrive at
\begin{eqnarray}
\tilde{g} = \sqrt{\frac{2 \pi \Gamma_{\rm single}}{\omega}}
\,.
\end{eqnarray}
\bibliographystyle{apsrev}
|
{
"timestamp": "2012-11-20T02:03:55",
"yymm": "1203",
"arxiv_id": "1203.2135",
"language": "en",
"url": "https://arxiv.org/abs/1203.2135"
}
|
\section*{Introduction}
Given an ascending sequence of positive integers $e_0,\ldots, e_n$
the curve in $\mathbb A^{n+1}$ parameterized by $t\to (t^{e_0}, t^{e_1}\ldots
,t^{e_n})$ is called an affine monomial curve since the
parametrization is by monomials. The minimal number of equations
defining monomial curves and the various structures of monomial
curves have been fascinating algebraists and geometors for a long
time. It is well known that these equations are binomial equations.
In fact, the ideal of a monomial curve in $\mathbb A^{n+1}$ is a weighted
homogeneous binomial prime ideal of height $n$ in the polynomial ring
$R= k[x_0, \ldots x_n]$ . In the plane, they are principal ideals and the
space monomial curves are either complete intersections, generated by
two binomials or determinantal ideals generated by the $2\times 2$
minors of a $2\times 3$ matrix \cite{herzog}. This breaks down even
in dimension 4 because there is no upper bound for the number of
generators for monomial cuves in $\mathbb A^4$ \cite{BH}. However, because
of the structure theorem of Gorenstein ideals in Codimension three as
ideals generated by Pfaffians \cite{BH}, the Gorenstein monomial
curves in dimension three are either complete intersections, generated
by 3 elements or the ideal of $4\times 4$ pfaffians of a $5\times 5$
skew symmetric matrix. Thus, for special classes of monomial curves,
the number of generators are bounded. We partition the monomial curves
in to classes so that two monomial curves are in the same class if their
consecutive parameters have the same differences. That is, if
${\bf m} = \{m_1,\ldots, m_n\}$ is a sequence of positive integers,
$C({\bf m})$ is a class of monomial curves defined by ${\underline{a}} = a_0,
\ldots, a_n$ with $\Delta ({\underline{a}}) = {\bf m}$. Herzog and Srinivasan
conjecture that the minimal number of generators for the ideal
defining the monomial curves in a given class $C({\bf m})$ is bounded. In
fact, they conjecture that this is eventually periodic with period
$\sum_i m_i$. This conjecture is true for monomial curves defined
by arithmetic sequences \cite{GSS2}. In this paper we prove the
conjecture in dimension 3 completely and prove it for complete
intersections in any dimension. We prove that for $a_0\gg 0$, the
complete intersections in the class $C({\bf m})$ occur periodically with
period $\sum_i m_i$. Our proof of the conjecture follows from a
criterion for complete intersection extending the one in \cite{D} for monomial curves with high $a_0$. This generalizes and recovers some results of Adriano
Marzullo, \cite{Ad}.
Now we state the conjecture precisely: Let $\underline{a}= (a_0, \ldots
a_n)$ be a sequence of positive integers and let $j$ be any positive
integer. Let $\underline{a} + (\underline{j})$ denotes sequence
$(j+a_0, j+a_1, j+a_1, \ldots, j+a_n)$. Let $\Gamma_{\underline{a} + (\underline{j})}$
denote the monomial curve corresponding to the sequence $\underline{a} + (\underline{j})$
and $I_{\underline{a}+(\underline{j})}$ denote the defining ideal of $\Gamma_{\underline{a}+(\underline{j})}$.
Then the strong form of
Herzog-Srinivasan conjecture states that the Betti numbers of $I_{\underline{a} +
(\underline{j})}$ are eventually periodic in $j$. Thus, the conjecture says
that within a class of monomial curves associated to increasing
sequences $\underline{a} $ with the same $\Delta {\underline{a}}$, the Betti numbers of
the defining ideals are eventually periodic in $a_0$.
In this paper we prove that for any sequence $\underline{a}$, for large $j$, if
$\underline{a} + (\underline{j})$ is a complete intersection, then, $\underline{a} +
(\underline{j+a_n})$ is a complete intersection. Since we are proving
results for $j \gg 0$, we may as well assume that $a_0 = 0$ and the
sequence $\underline{a} + (\underline{j}) = (j, j+a_1, \ldots, j+a_n)$. To be precise,
let $CI(\underline{a})=\{j~ |~ \Gamma_{\underline{a}+(\underline{j})} \mbox{ is a complete
intersection curve}\}$. We prove:
\vskip 2mm
\noindent
{\bf Theorem \ref{main}} {\em If $\underline{a} = (a_1, \ldots, a_n)$,
then $CI(\underline{a})$ is either finite or eventually periodic with period
$a_n$. If for $j \gg 0$, $\underline{a}+(\underline{j})$ is a complete intersection, then
there exists $1 \leq t \leq n-1$ and $k \in \mathbb{Z}_+$ such that
\begin{enumerate}
\item[(a)] $j = a_nm$ for some $m \in \mathbb{Z}_+$,
\item[(b)] $\operatorname{gcd}(a_1, \ldots, a_{t-1}, a_t+a_{t+1}, a_{t+2}, \ldots,
a_n) = k \neq 1$ and
\item[(c)] $\displaystyle{ a_t \in \left \langle \frac{a_1}{k},
\ldots, \frac{a_{t-1}}{k}, \frac{a_{t+1}}{k}, \ldots,
\frac{a_n}{k} \right\rangle}$.
\end{enumerate}}
\vskip 2mm
\noindent
and
\vskip 2mm
\noindent
{\bf Corollary \ref{cor1}}
{\em For $j \gg 0$, if $\underline{a} + (\underline{j})$ is a complete intersection, then $\underline{a}$
is a complete intersection. }
\vskip 2mm
\noindent
We also give a criterion for these curves to be a complete
intersections in Theorem \ref{crit}.
\section{Monomial curves in $\mathbb{A}^3$}
Let $\underline{a} = (a_0,a_1, \ldots, a_n) \in \mathbb{Z}_+^{n+1}$ with $a_0< a_1 < \cdots <
a_n$ and $R = k[t^{a_0}, \ldots, t^{a_n}]$, where $k$ is a field of
characteristic zero. We say that $\underline{a}$ is a complete intersection
sequence if $R$ is a complete intersection.
For the reason explained in the introduction,
we will assume here that $a_0 = 0$.
We begin by recalling a result characterizing the
complete intersection property of the sequence $\underline{a}$. For any sequence
$\underline{a}$, let $\langle a_1, \ldots, a_n \rangle := \{\sum_{i=1}^n r_ia_i
~ | ~ r_i \in \mathbb{Z}_{\geq 0} \}$ be the semigroup generated by $a_1, \ldots,
a_n$.
\begin{theorem}\label{complete-int}{\em [Proposition 9, \cite{D}]}
The sequence $\underline{a}$ is a complete intersection if and only if $\underline{a}$ can
be written as a disjoint union:
$$
\underline{a} = k_1(b_{i_1}, \ldots, b_{i_r}) \sqcup k_2(b_{i_{r+1}}, \ldots,
b_{i_n}),
$$
where $a_{i_m} = k_1b_{i_m}$ for $m = 1, \ldots, r$, $a_{i_m} = k_2
b_{i_m}$ for $m = r+1, \ldots, n$,
$\operatorname{gcd}(k_1, k_2) = 1$, $k_1 \notin \{b_{i_{r+1}}, \ldots, b_{i_n}\}$,
$k_1 \in \langle b_{i_{r+1}}, \ldots, b_{i_n} \rangle$, $k_2 \notin
\{b_{i_1}, \ldots, b_{i_r}\}$, $k_2 \in \langle \{b_{i_1}, \ldots,
b_{i_r} \rangle$ and both $(b_{i_1}, \ldots, b_{i_r})$ and
$(b_{i_{r+1}}, \ldots, b_{i_n})$ are complete intersection sequences.
\end{theorem}
We say that the sequence $\underline{a}$ is a complete intersection of type $(r,
n-r)$ if it splits as in the above theorem.
\begin{lemma}\label{split-type1}
Suppose $j > a_n^2$. If $(j, j+a_1, j+a_2, \ldots, j+a_n)$ is a
complete intersection of the type $(m, n+1-m)$, then either $m = 1$ or
$m = n$.
\end{lemma}
\begin{proof}
Suppose $1< m < n$, then we have a split of the form
$$
(j, j+a_1, \ldots, j+a_n) = k_1(\alpha_1, \ldots, \alpha_m)
\sqcup k_2(\alpha_{m+1}, \ldots, \alpha_{n+1}),
$$
where $k_1 \in \langle \alpha_{m+1}, \ldots, \alpha_{n+1} \rangle$ and
$k_2 \in \langle \alpha_1, \ldots, \alpha_m \rangle$.
Since $k_i$ divides $j+a_{l}$ for $1 < l < n$, $k_i \leq
a_n$. Since $j > a_n^2$ and $k_2 \leq
a_n$, $\alpha_j > a_n$. This contradicts the
fact that $k_1 \in \langle \alpha_{m+1}, \ldots, \alpha_{n+1}
\rangle$. Therefore $m = 1$ or $m = n$.
\end{proof}
\begin{lemma}\label{split-type2}
Suppose $\underline{a} + (\underline{j})$ is a complete intersection sequence for $j \gg 0$.
Then complete intersection splits of the type
\begin{enumerate}
\item $\underline{a}+(\underline{j}) = k_1\left(\frac{j}{k_1}\right) \sqcup
k_2\left(\frac{j+a_1}{k_2}, \ldots,
\frac{j+a_n}{k_2}\right)$;
\item $\underline{a}+(\underline{j}) = k_1\left(\frac{j}{k_1}, \frac{j+a_1}{k_1}, \ldots,
\frac{j+a_{n-1}}{k_1}\right) \sqcup
k_2\left(\frac{j+a_n}{k_2}\right)$
\end{enumerate}
are not possible.
\end{lemma}
\begin{proof}
First we prove that a split as in $(1)$ is not possible. Suppose $(1)$
is a complete intersection split of $\underline{a}+(\underline{j})$. First note that $k_2$
divides $a_i$ for $i \geq 2$. If $k_1 \neq j$, then by multiplying by
an appropriate factor, we obtain
\begin{eqnarray*}
j & = & \alpha_1 \frac{j+a_1}{k_2} + \cdots + \alpha_n
\frac{j+a_n}{k_2} \\
& = & (\alpha_1 + \cdots + \alpha_n)\frac{j}{k_2} + \alpha_1
\frac{a_1}{k_2} + \alpha_2 \frac{a_2}{k_2} + \cdots + \alpha_n
\frac{a_n}{k_2},
\end{eqnarray*}
where $\alpha_i$'s are non-negative integers.
Therefore,
$$k_2 j = (\sum_{i=1}^n\alpha_i)(j) + \alpha_1a_1 +
\cdots + \alpha_na_n$$
so that
$$
[k_2 - (\alpha_1+\cdots+\alpha_n)]j = \alpha_1a_1 +
\cdots + \alpha_na_n
$$
Since the right hand side consists of linear combination of
non-negative integers, not all of them zero, $k_2 > \sum_{i=1}^n
\alpha_i$. Therefore, we have
\begin{eqnarray*}
0 < [k_2 - (\alpha_1+\cdots+\alpha_n)]j & < &
(\sum_{i=1}^n\alpha_i)(a_n) \leq
k_2 (a_n) \leq a_n^2.
\end{eqnarray*}
This contradicts the fact that $j > a_n^2$. Therefore a
split of the first kind is not possible.
\vskip 2mm \noindent
Now assume that $(2)$ is a complete intersection split for $\underline{a}+ (j)$,
for $j > a_n^2$. If
$k_2 \neq j+\sum_{i=1}^na_i$, then after multiplying with an
appropriate factor we get
$$
j+a_n = \alpha_1 \frac{j}{k_1} + \cdots + \alpha_n
\frac{j+a_{n-1}}{k_1},
$$
where $\alpha_i$'s are non-negative integers. Therefore
\begin{eqnarray}\label{eqn1}
a_n & = &
\left(\sum_{i=1}^n\alpha_i-k_2\right)\frac{j}{k_2} +
\alpha_2\frac{a_1}{k_2}+ \alpha_3 \frac{a_2}{k_2} + \cdots
+\frac{\alpha_n}{k_2} a_{n-1}.
\end{eqnarray}
If $\sum_{i=1}^n \alpha_i < k_2$, then it follows from the above
equality that
\begin{eqnarray*}
a_n & \leq &
\left(\sum_{i=1}^n\alpha_i-k_2\right)\frac{j}{k_2} + \frac{1}{k_2}
a_n\left(\sum_{i=1}^n \alpha_i\right) \\
& \leq & \left(\sum_{i=1}^n\alpha_i-k_2\right)\frac{j}{k_2} +
a_n \leq 0.
\end{eqnarray*}
The last inequality holds since $k_2 \leq \sum_{i=1}^n a_i$ and $j >
a_n^2$. This is a contradiction, since the left hand side of the
inequality is a positive integer.
Now suppose $\sum_{i=1}^n \alpha_i > k_2$. It follows from equation
(\ref{eqn1}) that
\begin{eqnarray*}
k_2a_n & = &
\left(\sum_{i=1}^n\alpha_i-k_2\right)j + \alpha_2 a_1
+ \cdots + \alpha_na_{n-1}.
\end{eqnarray*}
Therefore $j \leq k_2 a_n < a_n^2$, which is a contradiction to the
hypothesis that $j > a_n^2$.
If $\sum_{i=1}^n \alpha_i = k_2$, then we have
\begin{eqnarray*}
a_n & = & \alpha_2 a_1
+ \cdots + \alpha_na_{n-1}\\
& < & \frac{1}{k_2} \left(\sum_{i=1}^n \alpha_i\right)
a_{n-1} = a_{n-1}.
\end{eqnarray*}
This is again a contradiction. Therefore,
all three possibilities lead to contradiction. Hence
a complete intersection split of type (2) is not possible.
\end{proof}
We now prove the periodicity conjecture for monomial curves in
$\mathbb{A}^3$.
Let $a_1 = a$ and $a_2 = b$. We first prove a characterization for
$(j, j+a, j+b)$ to be a complete intersection sequence for $j \gg
0$.
\begin{theorem}\label{complete-int1}
If $j \geq \max\{ab, b(b-a)\}$, then $(j,j+a, j+b)$ defines a
complete intersection ideal if and only if there exist $(j, b) = k
\neq 1$ and non-negative integers $\alpha, \beta$ such that $k(j+a) =
\alpha (j)+\beta (j+b)$. Moreover, in this case, $\alpha +\beta =
k$ and $(a,b-a) = s$ with $b = sk$.
In partiular, if $a$ and $b-a$ are relatively prime, $(j,j+a,j+b)$
is a complete intersection if and only if $b$ divides $j$.
\end{theorem}
\begin{proof}
Let $(j, j+a, j+b)$ be a complete intersection sequence. By Theorem
\ref{complete-int} and Lemma \ref{split-type2}, we can have only one
split possible, namely:
\vskip 2mm \noindent
$$
(j, j+a, j+b) = \frac{j+a}{k'}(k') \sqcup k\left(\frac{j}{k},
\frac{j+b}{k}\right),
$$
where $k' \mid k,$ $\operatorname{gcd}\left(\frac{j+a}{k'}, k\right) = 1$ and
$\frac{j+a}{k'} \in \left\langle \frac{j}{k}, \frac{j+b}{k}
\right\rangle$. Let $\alpha, \beta$ be non-negative integers such that
$k(j+a) = \alpha j+\beta (j+b)$. Since $k\leq b$, we see that
$k(j+a) \le kj+j = j(k+1)$. Therefore $kj+ka = j(\alpha
+\beta)+\beta b$ so that $\alpha +\beta \le k$. If $\alpha+\beta <
k$, then the equation $ka = (\alpha+\beta - k) j + \beta b$ would
imply that $ka < 0$ if $j \gg 0$. Therefore, $\alpha+\beta = k$.
Further, in this even, $\alpha a = \beta (b-a)$. Therefore $b = (\alpha
+\beta )s$ so that $\alpha(b) = (b-a)(\alpha+\beta) = \alpha s (\alpha
+\beta)$. Hence $b-a =\alpha s$ and $a = \beta s$.
If $\operatorname{gcd}(a,b-a)=1$, then $s =1$ and hence $\alpha +\beta = k = b$,
there by establishing that $b$ divides $j$.
The converse is clear.
\end{proof}
We new prove the periodicity conjecture for $n = 2$.
\begin{theorem}
Let $\underline{a} + (\underline{j}) = (j, j+a, j+b)$ and let $I_{\underline{a} + (\underline{j})}$
denote the defining ideal of the monomial curve $(t^j, t^{j+a},
t^{j+b})$. If $j \gg 0$, then the betti numbers of $I_{\underline{a}+(\underline{j})}$
are periodic for with period $b$.
\end{theorem}
\begin{proof}
Since the ideals $I$ in this case are either complete intersections or
height $2$ Cohen-Macaulay ideals generated by $3$ elements, we simply
need to show the periodicity of the number of generators.
By Theorem \ref{complete-int1}, if this is a complete intersection,
then $(j,b) = k,~ k(j+a) = \alpha j+\beta (j+b)$, with $\alpha
+\beta = k$ and $\operatorname{gcd}(j, b) = k$. Thus, $ \alpha (j+b)+\beta
(j+2b) = k(j+a) +(\alpha +\beta)b= k(j+a+b)$. Therefore,
$(j+b, j+a+b, j+2b)$ also defines a complete intersection.
Conversely, Suppose $(j+a+b, j+a+b, j+2b)$ defines a complete
intersection. Since $j\ge \max\{ab,b(b-a)\}$, we have the same
$\alpha, \beta$ giving the equations as before. Therefore, for $j \ge
\max\{ab,b(b-a)\},~ (j+rb,j+a+rb,j+b+rb)$ is a
complete intersection for all $r$ if and only if it is a complete
intersection for $(j, j+a, j+b)$.
Thus the eventual periodicity is true for $d=2$.
\end{proof}
\section{Monomial curves in $\mathbb{A}^n$ for $n \geq 4$}
In this section we prove the periodicity of occurence of complete
intersections in the class $\Gamma_{\underline{a}+(\underline{j})} \subset \mathbb{A}^n$
and characterize complete intersection monomial curves
in $\mathbb{A}^4$.
\begin{theorem}\label{main}
If $\underline{a} = (a_1, \ldots, a_n)$, then $CI(\underline{a})$ is either finite or
eventually periodic with period $a_n$. If for $j \gg 0$, $\underline{a}+(j)$ is
a complete intersection then there exists $1 \leq s \leq n-1$ and $k
\in \mathbb{Z}_+$ such that
\begin{enumerate}
\item[(a)] $j = a_nm$ for some $m \in \mathbb{Z}_+$,
\item[(b)] $\operatorname{gcd}(a_1, \ldots, a_{s-1}, a_s+a_{s+1}, a_{s+2}, \ldots,
a_n) = k \neq 1$ and
\item[(c)] $\displaystyle{ a_s \in \left \langle \frac{a_1}{k},
\ldots, \frac{a_{s-1}}{k}, \frac{a_{s+1}}{k}, \ldots,
\frac{a_n}{k} \right\rangle}$.
\end{enumerate}
\end{theorem}
\begin{proof}
We first assume that $\operatorname{gcd}(a_1, \ldots, a_n) = 1$.
Assume that $CI(\underline{a})$ is not finite.
Assume
that $\underline{a}+(\underline{j})$ is a complete intersection. Therefore it follows from
Lemma \ref{split-type1} and Lemma \ref{split-type2}, that we have the
complete intersection split of the form
\begin{eqnarray*}\label{eqn2}
\underline{a}+(\underline{j}) & = & \frac{j+a_s}{k'} (k') \sqcup k
\left(\frac{j}{k}, \frac{j+a_1}{k}, \ldots,
\frac{j+a_{s-1}}{k}, \frac{j+a_{s+1}}{k},
\ldots, \frac{j+a_n}{k}\right)
\end{eqnarray*}
which satisfies the conditions in Theorem \ref{complete-int}.
Furthermore, we have the following:
\begin{enumerate}
\item $k' \mid k$ and $k' \neq k$.
\item $k \mid \operatorname{gcd}(a_1, \ldots, a_{s-1}, a_{s+1}, \ldots, a_n)$.
\item Since $k' \mid k$, it divides $a_i$ for $i = 1, \ldots,
s-1$ and it divides $j$ as well. Therefore, $k' \mid a_s$ and
hence $k' \mid a_i$ for all $i = 1, \ldots, n$. This implies
that $k' \mid \operatorname{gcd}(a_1, \ldots, a_n) = 1$. Therefore $k' =
1$.
\end{enumerate}
\noindent
Since $\left(\frac{j}{k}, \frac{j+a_1}{k}, \ldots,
\frac{j+a_{s-1}}{k}, \frac{j+a_{s+1}}{k},
\ldots, \frac{j+ a_n}{k}\right)$ is a complete
intersection sequence (associated to a sequence of length $n-1$), it
follows by induction on $n$ that for $j \gg 0$,
$$
\frac{j}{k} = \frac{a_n}{k}m
$$
and hence $j = a_nm$, where $m$ is a positive integer. Since $k' = 1$,
we have
$$j+a_s \in \left\langle
\frac{j}{k}, \frac{j+a_1}{k}, \ldots,
\frac{j+a_{s-1}}{k}, \frac{j+a_{s+1}}{k},
\ldots, \frac{j+ a_n}{k} \right\rangle,$$
and therefore
there exist some non-negative integers $\alpha_1, \ldots, \alpha_n$,
not all zero such that
\begin{eqnarray}\label{eqn3}
j+\sum_{i=1}^s a_i = \alpha_1 \frac{j}{k} + \cdots + \alpha_s
\frac{j+a_{s-1}}{k} + \alpha_{s+1} \frac{j+a_{s+1}}{k} + \cdots + \alpha_n
\frac{j+a_n}{k}.
\end{eqnarray}
\textsc{Claim 1:} $\sum_{i=1}^n \alpha_i = k$.
\vskip 2mm \noindent
\textit{Proof of Claim 1:} From the above equation, we can write
\begin{eqnarray*}
a_s & = & \left(\sum_{i=1}^n \alpha_i - k\right)
\frac{j}{k} + \sum_{l=1}^s \alpha_l\frac{a_{l-1}}{k} + \sum_{l=s+1}^n \alpha_l
\frac{a_l}{k}.
\end{eqnarray*}
Suppose $\sum_{i=1}^n \alpha_i > k$. If $j > a_n^2$,
then $\frac{j}{k} > a_n$ and hence we get
that $a_s > a_n$, a contradiction.
\vskip 2mm \noindent
Suppose $\sum_{i=1}^n \alpha_i < k$. Then we get
\begin{eqnarray*}
a_t & \leq & \left(\sum_{i=1}^n \alpha_i - k\right)
\frac{j}{k} + \frac{1}{k}\left(\sum_{i=1}^n \alpha_i\right)
a_n \\
& < & \left(\sum_{i=1}^n \alpha_i - k\right)
\frac{j}{k} + a_n \leq 0,
\end{eqnarray*}
where the last inequality holds since $\frac{j}{k} \geq \sum_{i=1}^n
a_i$. This is contradiction since $a_s > 0$. Therefore,
we have shown that neither of the cases
\begin{enumerate}
\item[(a)] $\sum_{i=1}^n \alpha_i > k$
\item[(b)] $\sum_{i=1}^n \alpha_i < k$
\end{enumerate}
are not possible. Therefore, $\sum_{i=1}^n \alpha_i = k$. This
completes the proof of the claim.
\vskip 2mm \noindent
\textsc{Claim 2:} $\underline{a} + (\underline{j+a_n}) = \left(j+a_n, j+a_n + a_1, \ldots,
j+ 2a_n \right)$ is a complete intersection sequence.
\vskip 2mm \noindent
\textit{Proof of Claim 2:} We show that this sequence has a complete
intersection split similar to that of $\underline{a}_j$. Choose $\alpha_1,
\ldots, \alpha_n$ as in (\ref{eqn3}). Therefore we have $\sum_{i=1}^n
\alpha_i = k$ so that
\begin{eqnarray*}
\sum_{l=1}^s\alpha_l\left(\frac{j}{k} + \frac{a_n}{k} +
\frac{a_{l-1}}{k}\right) & + &
\sum_{l=s+1}^n \alpha_l \left(\frac{j}{k} + \frac{a_n}{k} +
\frac{a_l}{k}\right) \\
& = & j+a_s + \left(\sum_{i=1}^n \alpha_i \right)
\frac{a_n}{k}\\
& = & j+a_s+ a_n
\end{eqnarray*}
Therefore, $$
j + a_n + a_s \in
\left\langle \frac{j}{k}+\frac{a_n}{k}, \ldots,
\frac{j}{k} +
\frac{a_{s-1}}{k}+ \frac{a_n}{k},
\frac{j}{k} +
\frac{a_{s+1}}{k}+ \frac{a_n}{k},\ldots
\frac{j}{k} +
\frac{a_n}{k}+ \frac{a_n}{k} \right\rangle,
$$
We need to show that this split satisfies
all the properties of Theorem \ref{complete-int}. Since
$\operatorname{gcd}(j+a_s, k) = 1$ and $k \mid a_n$,
$\operatorname{gcd}(j+a_n + a_s, k) = 1$. Note that
$\operatorname{gcd}\left(\frac{a_1}{k}, \ldots, \frac{a_{s-1}}{k},
\frac{a_{s+1}}{k}, \ldots, \frac{a_n}{k}
\right) = 1$. Since
$
\left(\frac{j}{k}, \frac{j+a_1}{k}, \ldots,
\frac{j+a_{s-1}}{k}, \frac{j+a_{s+1}}{k},
\ldots, \frac{j+ a_n}{k}\right)
$
is a complete intersection, by induction on $n$, we get that
$$
\left( \frac{j}{k}+\frac{a_n}{k}, \ldots,
\frac{j}{k} +
\frac{a_{s-1}}{k}+ \frac{a_n}{k},
\frac{j}{k} +
\frac{a_{s+1}}{k}+ \frac{a_n}{k},\ldots,
\frac{j}{k} +
\frac{a_n}{k}+ \frac{a_n}{k} \right)
$$
is a complete intersection sequence. Therefore, if $CI(\underline{a})$ is
infinite, then it is eventually periodic with period $a_n$.
Now let $k' = \operatorname{gcd}(a_1, \ldots, a_n)$.
Assume that $\underline{a} + (\underline{j})$ is a complete intersection.
Then we have a split of the form:
$$
\underline{a} + (\underline{j}) = \frac{j+a_s}{k'} ~ (k') \sqcup k
\left(\frac{j}{k}, \ldots, \frac{j+a_{s-1}}{k},
\frac{j+a_{s+1}}{k}, \ldots, \frac{j+a_n}{k}
\right),
$$
where $k = \operatorname{gcd}(a_1, \ldots, a_{s-1}, a_{s+1}, \ldots,
a_n)$, $k' \mid k$, $k \neq k'$,
$\operatorname{gcd}\left(\frac{j+a_s}{k'}, k\right) = 1$ and \\
$\left(\frac{j}{k}, \ldots, \frac{j+a_{s-1}}{k},
\frac{j+a_{s+1}}{k}, \ldots, \frac{j+a_n}{k}
\right)$ is a complete intersection. Since $\operatorname{gcd}\left(\frac{a_1}{k},
\ldots, \frac{a_{s-1}}{k}, \frac{a_{s+1}}{k},
\ldots, \frac{a_n}{k}\right) = 1$, we can use the Theorem \ref{main}
to conclude that $\frac{j}{k} = \frac{a_n}{k} m$ for some
$m \in \mathbb{Z}_+$. Hence $j = (\sum_{i=1}^n a_i)m$. We also have that\\
$\displaystyle{\left(\frac{j+a_n}{k}, \ldots,
\frac{j+a_n+a_{s-1}}{k},
\frac{j+a_n+a_{s+1}}{k}, \ldots,
\frac{j+a_n+a_n}{k}
\right)}$ is a complete intersection, since the period being
$\frac{a_n}{k}$. This shows that we have a complete
intesection split
$$\underline{a}+(\underline{j+a_n}) =
\frac{j+a_n+a_s}{k'} ~(k') \sqcup
k \left(\frac{j+a_n}{k}, \ldots, \frac{j+a_n+a_{s-1}}{k},
\frac{j+a_n+a_{s+1}}{k}, \ldots,
\frac{j+a_n+a_n}{k}\right).$$
\vskip 2mm \noindent
This implies that $\underline{a} + (\underline{j+a_n})$ is a complete
intersection, proving the periodicity as well.
\end{proof}
As a consequence of the above result, we relate the complete
intersection property of $\underline{a}$ and $\underline{a} + (\underline{j})$ for $j \gg 0$.
\begin{corollary}\label{cor1}
For $j \gg 0$, if $\underline{a} + (\underline{j})$ is a complete intersection, then $\underline{a}$
is a complete intersection.
\end{corollary}
\begin{proof}
First assume that $\operatorname{gcd}(a_1, \ldots, a_n) = 1$.
We prove the first statement by induction on $n$. If $n = 1$, there is
nothing to prove and for $n = 2,$ $(a_1, a_2)$ is always a complete
intersection. Assume now that $\underline{a} = (a_1, \ldots, a_n)$ with $n \geq
3$ and $\underline{a} + (\underline{j})$ is a complete intersection for $j \gg 0$. By
Theorem \ref{main}, there exists a $s$ and $k$ such that
$$
\underline{a} + (\underline{j}) = j+a_s ~ (1) \sqcup k \left(\frac{j}{k},
\frac{j+a_1}{k}, \ldots, \frac{j+a_{s-1}}{k}, \frac{j+a_{s+1}}{k},
\ldots, \frac{j+a_n}{k}\right)
$$
with $\left(\frac{j}{k}, \frac{j+a_1}{k}, \ldots, \frac{j+a_{s-1}}{k},
\frac{j+a_{s+1}}{k}, \ldots, \frac{j+a_n}{k}\right)$ a complete
intersection, $\operatorname{gcd}\left(j+a_s, k\right) = 1$
and $a_s \in \left\langle
\frac{a_1}{k}, \ldots, \frac{a_{s-1}}{k}, \frac{a_{s+1}}{k}, \ldots,
\frac{a_n}{k}\right\rangle$. By induction on $n$, we get that
$\left(\frac{a_1}{k}, \ldots, \frac{a_{s-1}}{k}, \frac{a_{s+1}}{k},
\ldots, \frac{a_n}{k}\right)$ is a complete intersection. Hence we
have a complete intersection split:
$$
\underline{a} = a_s~(1) \sqcup k\left(\frac{a_1}{k}, \ldots,
\frac{a_{s-1}}{k}, \frac{a_{s+1}}{k}, \ldots, \frac{a_n}{k}\right).
$$
Therefore $\underline{a}$ is a complete intersection. Now assume that $\operatorname{gcd}(a_1,
\ldots, a_n) = k'$. Since $\underline{a} + (\underline{j})$ is a complete intersection for
$j \gg 0$, it follows from Theorem \ref{main} that $j = a_nm$ for some
$m \in \mathbb{Z}_+$. Therefore $k' \mid j$. Let $j' = \frac{j}{k'}$ and $a_i
= \frac{a_i}{k'}$. Since $\underline{a} + (j)$ is a complete intersection so is
$(j', j'+a_1', \ldots, j'+a_n')$. By the first part, this implies that
$(a_1', \ldots, a_n')$ is a complete intersection and hence $(a_1,
\ldots, a_n)$ too is a complete intersection.
\end{proof}
We now prove a partial converse of the above corollary. It can be seen
that a converse statement of Corollary \ref{cor1} is not true, cf.
Example \ref{ex2}
\begin{proposition}
If $n \geq 3$ and $\underline{a}$ is a complete
intersection and $k_{i+1}a_i \in \langle a_{i+1}, \ldots, a_n
\rangle$, where $k_i = \operatorname{gcd}(a_i, \ldots, a_n)$, then there exists $j
\gg 0$ such that $\underline{a} + (\underline{j})$ is a complete intersection.
\end{proposition}
\begin{proof}
First assume that $\operatorname{gcd}(a_1, \ldots, a_n) =
1$. We prove the assertion by induction on $n$. Let $n = 3$.
Let
$$
\underline{a} = a_1~(1) \sqcup k\left(\frac{a_2}{k}, \frac{a_3}{k}\right),
$$
where $k = \operatorname{gcd}(a_2, a_3)$. Since $a_1 \in \left\langle \frac{a_2}{k},
\frac{a_3}{k}\right\rangle$, we can write $k a_1 = \beta a_2 + \gamma
a_3$. Since $a_1 < a_i$ for $i = 2, 3$, $k \geq \beta + \gamma$. Let
$\alpha = k - \beta - \gamma$. Then for $j \geq 0$, $(j + a_1) =
\alpha \frac{j}{k} + \beta \frac{j+a_2}{k} + \gamma \frac{j+a_3}{k}$.
By Theorem \ref{complete-int1}, there exists $j \gg 0$, $j = a_3m$
such that $\left(\frac{j}{k}, \frac{j+a_2}{k}, \frac{j+a_3}{k}\right)$
is a complete intersection.
Let $a_1 = \alpha_2 \frac{a_2}{k} + \cdots +
\alpha_n\frac{a_n}{k}$. Since $a_1 < a_i$ for all $i =
2, \ldots, n$, $\sum_{i=2}^n \alpha_i\leq k$. Let $\alpha_1 = k -
\sum_{i=2}^n \alpha_i$. Then for any $j > 0$, we can write
$$
j+a_1 = \alpha_1 \frac{j}{k} + \alpha_2 \frac{j+a_2}{k} +
\cdots + \alpha_n \frac{j+a_n}{k}.
$$
Since $\frac{a_2}{k_2} \in \left\langle \frac{a_3}{k_2k_3}, \ldots,
\frac{a_n}{k_2k_3}\right\rangle$ and $\left(\frac{a_3}{k_2k_3}, \ldots,
\frac{a_n}{k_2k_3}\right)$ is a complete intersection, by induction,
we get that $\left(\frac{j}{k_2}, \frac{j+a_2}{k_2}, \ldots,
\frac{j+a_n}{k_n}\right)$ is a complete intersection for some $j \gg
0$. Therefore by Theorem \ref{complete-int}, $\underline{a} + (\underline{j})$ is a complete
intersection. If $\operatorname{gcd}(a_1, \ldots, a_n) = k_1 \neq 1$, then we can divide
by $k_1$ to get a complete intersection sequence $\underline{a}'$, apply the
first part to obtain a $j'$ such that $\underline{a}' + (\underline{j}')$ is a complete
intersection and then by multiplying by $k_1$ to conclude that
$\underline{a}+(\underline{j})$ is a complete intersection.
\end{proof}
We now characterize the complete intersection sequences when $n = 3$.
It is actually possible to formulate a similar result in
the general case, but it is highly complicated. Therefore, we stick to
the case of $n = 3$.
\begin{theorem}\label{crit}
Let $\underline{a} + (\underline{j}) = (j, j+a, j+b, j+c)$. Then $\underline{a} + (\underline{j})$ is a
complete intersection for $j \gg 0$ if and only if there exist
non-negative integers $m, k, \beta, \gamma$ such that $j = cm$,
$k \neq 1$ and one of the follwing is satisfied:
\begin{enumerate}
\item[(1a)] $\operatorname{gcd}(a, c) = k$ and
\item[(1b)] $ka = \beta b + \gamma c$.
\end{enumerate}
\centerline{OR}
\begin{enumerate}
\item[(2a)] $\operatorname{gcd}(b, c) = k$
\item[(2b)] $kb = \beta a + \gamma(c)$ with $\beta +
\gamma \leq k$.
\end{enumerate}
\end{theorem}
\begin{proof}
If $\underline{a} + (\underline{j})$ is a complete intersection sequence for $j \gg 0$, then
by Theorem \ref{main}, one of the two sets of conditions are
satisfied. We now prove the converse. First assume that (1a) and
(1b) are true. Let $\alpha = k - (\beta + \gamma)$.
Using (1a), we can write $j+b = \alpha
\left(\frac{j}{k}\right) + \beta \left(\frac{j+a}{k}\right) + \gamma
\left(\frac{j+c}{k}\right).$ Note that $\operatorname{gcd}\left(\frac{a}{k},
\frac{c}{k}\right) = 1$. Therefore by Theorem \ref{complete-int1},
we get that $\left(\frac{j}{k}, \frac{j+b}{k},
\frac{j+c}{k}\right)$ is a complete intersection if $j \gg 0$ and
$\frac{j}{k} = \frac{c}{k} m$ for some $m$. Therefore if $j \gg 0$
and $j = cm$, then $\left(\frac{j}{k}, \frac{j+b}{k},
\frac{j+c}{k}\right)$ is a complete intersection. Let $k' =
\operatorname{gcd}(a,b,c) = \operatorname{gcd}(k, a)$. Then we can write
$$
\underline{a} + (\underline{j}) = \frac{j+a}{k'} (k') \sqcup k\left(\frac{j}{k},
\frac{j+b}{k}, \frac{j+c}{k}\right)
$$
with $k' \mid k$, $k' \neq k$, $\operatorname{gcd}\left(\frac{j+a}{k'}, k\right) =
1$ and $\left(\frac{j}{k}, \frac{j+a+b}{k}, \frac{j+a+b+c}{k}\right)$
a complete intersection. Therefore $\underline{a} + (j)$ is a complete
intersection. If we assume the second set of condition, then the proof
can be obtained by interchanging the role of $a$ and $b$.
\end{proof}
\section{Examples}
We conclude the article by giving some examples. In the first example,
we show the periodicity.
\begin{example}
Let $\underline{a} = (11,16,28)$. Let $j = 28m$ for some $m > 1$. Then it can be
seen that $28m + 11 = 2 (7m) + (7m + 4) + (7m + 7)$ and that $(7m,
7m+4, 7m+7)$ is a complete intersection (here we need $m > 1$).
Therefore $(28m, 28m+11, 28m+16, 28m+28)$ is a complete intersection
sequence.
\end{example}
The next example shows that $CI(\underline{a})$ could be non-empty and finite.
\begin{example}\label{ex1}
Let $\underline{a} = (3,8,20)$. For $j = 28$, we have $\underline{a} + (\underline{j}) =
(28,31,36,48)$ and it can be seen that $\underline{a} + (\underline{j}) = 31 ~(1) \sqcup 4
(7,9,12)$ is a complete intersection split. Therefore $\underline{a} + (j)$ is a
complete intersection. Suppose $\underline{a} + (\underline{j})$ is a complete intersection
for $j \gg 0$. Since $\operatorname{gcd}(3, 20) = 1$, the only possible split for $j
\gg 0$ is of the form
$$
\underline{a} + (\underline{j}) = \frac{j+3}{k'} ~(k') \sqcup 4\left(\frac{j}{4},
\frac{j+8}{4}, \frac{j+20}{4}\right),
$$
with $\alpha + \beta + \gamma = 4$. This gives us the equation,
$$
12 = 8 \beta + 20 \gamma.
$$
Since this does not have a non-negative integer solution, we arrive at
a contradiction. Therefore, $CI(\underline{a})$ is finite. This examples also
shows that taking $j > a_n$ is not enough.
\end{example}
The next example shows that converse of Corollary \ref{cor1} is not
always true even for $j > a_n^2$ and $j = a_nm$.
\begin{example}\label{ex2}
Let $\underline{a} = (8,17,18)$. Then $\underline{a}$ is a complete intersection sequence.
For $j \gg 0$ and $j = 18m$, the only possibility of a complete
intersection split is of the form
$$
\underline{a} + (\underline{j}) = j+17 ~(1) \sqcup 2\left(\frac{j}{2}, \frac{j+8}{2},
\frac{j+18}{2}\right)
$$
such that $j+17 = \alpha \frac{j}{2} + \beta \frac{j+8}{2} + \gamma
\frac{j+18}{2}$ with $\alpha + \beta + \gamma = 2$. Therefore, $17 =
4\beta + 9 \gamma = 17$ and $\beta + \gamma \leq 2$. Since this does
not have a non-negative integer solution, this does not occur.
Therefore, $\underline{a} + (\underline{j})$ can not be a complete intersection for $j >
18^2$.
\end{example}
\vskip 2mm
\noindent
\textbf{Acknowledgement:} The work was done during the first author's visit to
University of Missouri. He was funded by the Department of Science and Technology, Government of India. He would like to express sincere thanks to the funding agency and also to the Department of Mathematics at University of Missouri for the great hospitality provided to him.
\bibliographystyle{amsplain}
|
{
"timestamp": "2012-03-20T01:05:23",
"yymm": "1203",
"arxiv_id": "1203.1991",
"language": "en",
"url": "https://arxiv.org/abs/1203.1991"
}
|
\section{Introduction}
The aim of this paper is to extend in a slightly more general
and unified
set up two important steps of the proof of the asymptotic stability
of solitary waves for the Nonlinear Schr\"odinger equation \cite{Cu2,Cu1,bambusi}
and the particular case of Nonlinear Dirac system treated in \cite{boussaidcuccagna}. In both cases
there is a localization at the solitary wave
and a representation of the system in terms of coordinates arising
from the linearization at a solitary wave.
The operators $\mathcal{H}_p$ introduced later play this role.
In general $\mathcal{H}_p$ has both continuous spectrum and non zero eigenvalues. The latter give rise to discrete modes which in the nonlinear problem could produce chaotic Lissaius like motions. It turns out that in
\cite{bambusicuccagna,Cu2,Cu4,boussaidcuccagna,Cu1,bambusi}
discrete modes relax to 0
because of a mechanism
of slow leaking of energy away from the discrete modes into the continuous modes, where energy disperses by linear dispersion. The idea was initiated in special situations in \cite{sigal,BP2,SW3}. We refer to
\cite{Cu2} for more comments and references.
The aim of this paper consists in simplifying two key steps
in the proofs in \cite{Cu1,Cu2,boussaidcuccagna}. The first step
consists in searching Darboux coordinates. This allows to decrease the
number of coordinates in the system and to reduce to the study of
the system at an equilibrium point.
The second step consists in the
implementation of the Birkhoff normal forms, to produce a simple {\textit{effective}} Hamiltonian.
After this, \cite{Cu1,Cu2,boussaidcuccagna} prove the energy
leaking away from the discrete modes. In particular the key step is the proof that certain coefficients
of the discrete modes equations are second powers, the { \it Nonlinear Fermi Golden Rule} (FGR),
which generically are positive and yield discrete mode energy dissipation.
We do not discuss the FGR in this paper limiting ourselves to the search of Darboux coordinates
and to the Birkhoff normal forms argument.
In this paper we avail ourselves with some ideas and notation drawn from early versions of \cite{bambusi} to improve the
presentation in \cite{Cu1}.
\cite{bambusi,Cu1} represent two attempts to
extend the result proved in \cite{Cu2} for standing ground states of the NLS, to the case of moving ground states. A further goal in \cite{bambusi} is to develop the theory in a more abstract set up. Early versions of \cite{bambusi} did not encompass a Birkhoff step extendable to
\cite{boussaidcuccagna}.
\cite{bambusi} is confined (like us here) to systems with
Abelian group of symmetries.
Our present proof was written before the 3rd version of \cite{bambusi} was posted on the Arxiv site.
The 2rd version of \cite{bambusi} contained an incorrect effective Hamiltonian,
see Remark \ref{rem:formal2} later. In the 3rd version of \cite{bambusi} this has been corrected. Still, the discussion in \cite{bambusi} is at times sketchy,
for example in Theorems 3.21 and 5.2
\cite{bambusi}, see Remarks \ref{rem:formal1} and \ref{rem:bamb1} and the discussion below and at the beginning of Sect. \ref{subsec:darboux}
about \eqref{eq:fdarboux}.
We draw from \cite{bambusi}
a better choice of initial coordinates and set up than \cite{Cu1}. Some of it existed also in previous literature, cfr.
the discussion in Sect.6 \cite{RSS}.
We also borrow some notation, i.e. symbols $\resto ^{k,m}$ and $\textbf{S} ^{k,m}$.
Inspired by \cite{bambusi} we simplify the proof of the Birkhoff step in \cite{Cu1}.
Both
here and in \cite{Cu1} we
consider initial data in subsets of $\Sigma _n$ for $n\gg 1$ which are
unbounded in $\Sigma _n$ and invariant for the system.
We require this substantial amount of regularity and spacial decay to 0 for the classes
of solutions of the system, in order to give a rigorous
treatment of the flows and of the pullbacks. \cite{bambusi} suggests that
\cite{Cu1} should prove decay rates in time.
We do not know what is the basis for this statement in \cite{bambusi} since, by the time invariance of the subsets
$\Sigma _n$ considered, the problem considered in
\cite{Cu1} is very similar in this respect to the one with $\Sigma _n$ replaced by $H^1$.
Indeed time decay corresponds to bounds on norms containing time dependent weights.
But if the problem is invariant by translation in time, the only information that can be
derived must be invariant by translation in time, and bounds on time weighted norms do not have this property.
We therefore emphasize that \cite{Cu1} and the present paper are very different from, say, \cite{BP2,SW3}, which consider
initial data in subsets of $H^{k,s}$ which are not invariant by the time evolution. We also point out that
the 1st version of \cite{bambusi} contains a false statement on rate of time decay. The
2nd version of \cite{bambusi} in the acknowledgments, credits us for pointing out this error, although these credits are not any more in the 3rd version.
To find an effective Hamiltonian, we use in a crucial way the regularity
properties of the flows, which in turn depend on the fact that we work in $\Sigma _n$ for $n\gg 1$.
See Theorem \ref{theorem-1.1} where the regularity of the flows is used to prove that the coordinate
changes preserve the system.
To prove for the NLS the same result in $H^1$, where the
coordinate changes are continuous only, one needs to explain how they
preserve the
structure needed to make sense of the NLS. \cite{bambusi} claims the result in $H^1$,
\cite{bambusi} claims the result in $H^1$,
but the proof is not spelled out,
see Remark \ref{rem:bamb1}.
We discuss in some detail
a key formula on the differentiation of the pullback of
a differential form
along a flow, see \eqref{eq:fdarboux}, which is the basis of
Moser's method to find Darboux coordinates.
This formula is simple in
classical set ups, but in our case and in \cite{bambusi} its interpretation and proof are not obvious.
In \cite{bambusi} the formula is stated and used without discussion.
We treat the issue rigorously in
Sect. \ref{subsec:darboux}, regularizing the flow, using
\eqref{eq:fdarboux} for the regularized flow,
and recovering the desired equality between differential forms, by
a limiting argument. Notice that we do not prove formula \eqref{eq:fdarboux} for the non regularized flow.
We end with few remarks on the proofs.
The proof of the Darboux Theorem is a simplification of that in \cite{Cu1} in the part discussing
the vector field. We give in Sect. \ref{subsec:darboux} a detailed proof on the fact that the resulting flow transforms the symplectic
form as desired. See also the introductory remarks in Sect. \ref{sect:symplectic}. Notice that parts of this discussion were skipped in \cite{Cu1}.
The portion of our paper on the Birkhoff normal forms covers from Sect. \ref{sec:pullback}
on and
is quite different from \cite{boussaidcuccagna,Cu1,Cu2}
mainly because the pullback of the terms of the expansion of the Hamiltonian
cannot be treated on a term by term basis,
see Remark \ref{rem:differences}. What is important is to get a general structure
of the pullbacks of the Hamiltonian. This is discussed in Sect. \ref{sec:pullback}.
It is likely
that most of the analysis in Lemmas \ref{lem:back}, \ref{lem:back11} and \ref{lem:ExpH11}, is not necessary to the derivation of the effective
Hamiltonian, which is represented by $H'_2$ and the null terms in
$\mathbf{R}_0$ and $\mathbf{R}_1$ of the expansion in Lemma \ref{lem:ExpH11}, in the final Hamiltonian.
On the other hand, writing the Hamiltonian explicitly should make the arguments transparent and more clearly
applicable to the part on dispersion and Fermi Golden rule.
In Sect. \ref{sec:speccoo} we finally distinguish between discrete and continuous modes.
The present paper treats only equations whose symmetry group is Abelian. This limitation will have to be
overcome to extend the theory to more general systems such for example
the Dirac system without the symmetry constraints of \cite{boussaidcuccagna}.
\section{Set up}
\label{sec:setup}
\begin{itemize}
\item
Given two vectors $ {u},v\in \R ^{2N}$ we denote by $u\cdot v =\sum u_j v_j$
their inner product.
\item We will consider also another quadratic form $|u|_1^2= u\cdot _1 u$
in $ \R ^{2N}$.
\item For any $n \ge 1$ we consider the space $\Sigma _n=\Sigma _n(\R ^3, \R ^{2N})$ defined by
\begin{equation}\label{eq:sigma}
\begin{aligned} &
\| u \| _{\Sigma _n} ^2:=\sum _{|\alpha |\le n} (\| x^\alpha u \| _{L^2(\R ^3, \R ^{2N})} ^2 + \| \partial _x^\alpha u \| _{L^2(\R ^3, \R ^{2N})} ^2 ) <\infty . \end{aligned}\nonumber
\end{equation}
We set $\Sigma _0= L^2(\R ^3, \R ^{2N})$. Equivalently we can define $\Sigma _{r}$ for $r\in \R$ by the norm
\begin{equation}
\begin{aligned} &
\| u \| _{\Sigma _r} := \| ( 1-\Delta +|x|^2) ^{\frac{r}{2}} u \| _{L^2} <\infty . \end{aligned}\nonumber
\end{equation}
For $r\in \N$ the two definitions are equivalent, see \cite{Cu2}. We will not use another
quite natural class of spaces denoted by $H^{k,s}$ and defined by
\begin{equation}
\begin{aligned} &
\| u \| _{H^{k,s}} := \| (1+|x|^2) ^{\frac{s}{2}} ( 1-\Delta ) ^{\frac{k}{2}} u \| _{L^2} <\infty . \end{aligned}\nonumber
\end{equation}
\item ${\mathcal S}(\R ^3, \R ^{2N})=\cap _{n\in \N}\Sigma _n(\R ^3, \R ^{2N})$
is the space of Schwartz functions and the space of tempered distributions is
${\mathcal S}'(\R ^3, \R ^{2N})=\cup _{n\in \N}\Sigma _{-n}(\R ^3, \R ^{2N})$.
\item For $X$ and $Y$ two Banach space, we will
denote by $B(X,Y)$ the Banach space of bounded linear operators from
$X$ to $Y$ and by $B^{\ell}(X,Y)= B ( \prod _{j=1}^\ell X ,Y)$.
\item We denote by $\langle \ , \ \rangle $
the natural inner product in $L^{2}(\R ^3, \R ^{2N})$.
\item $J$ is an invertible antisymmetric matrix in
$\R ^{2N}$. We have also $|J y|_1=| y|_1$ for any $y\in \R ^{2N}$.
In $L^{2}(\R ^3, \R ^{2N})$ we consider the symplectic form $\Omega =\langle J ^{-1}\ , \ \rangle
$.
\item We consider in $ L^2(\R ^3, \R ^{2N})$ a linear selfadjoint elliptic differential operator $\mathcal{D}$
such that $\mathcal{D}\in B(\Sigma _{r},\Sigma _{r-\text{ord}\mathcal{D}} ) $ and $\mathcal{D} \in B(H^{r},H^{r-\text{ord}\mathcal{D}} ) $ for all $r$ and for a fixed integer $\text{ord}\mathcal{D}\ge 1$.
\item
We consider a Hamiltonian of the form
\begin{equation} \label{eq:energyfunctional}\begin{aligned}&
E(U)=E_K(U)+E_P(U)\\&
E_K(U):=\frac{1}{2} \langle {\mathcal D} U, U \rangle \, , \quad
E_P(U):=
\int _{\R ^3}B( |U|_1^2) dx .\end{aligned}
\end{equation}
Here $B\in C^\infty (\R , \R)$, $B( 0)=B'( 0)=0$ and there exists a $p\in(2,6]$ such that for every
$k\ge 0$ there is a fixed $C_k$ with
\begin{equation}\label{eq:growthB} \left| \nabla _\zeta ^k (B ( |\zeta |_1^2) )\right|\le C_k
|\zeta |^{p-k-1} \quad\text{if $|\zeta |\ge 1$ in $\R ^{2N}$.}
\end{equation}
\end{itemize}
Notice that $E_P\in C^5(H ^{1}(\R ^3, \R ^{2N})), \R)$.
Consistently with \cite{boussaidcuccagna,Cu1,Cu2}, we focus only on \textit{semilinear} Hamiltonians. We consider the system
\begin{equation}\label{eq:NLSvectorial} \dot U = J \nabla E (U) \quad , \quad U(0)=U_0\quad
\end{equation}
where for a Frech\'et differentiable function $F$ the gradient $\nabla F (U)$
is defined by
$\langle \nabla F (U), X\rangle = dF(U) (X)$, with $dF(U)$ the exterior differential calculated at $U$.
We assume that
\begin{itemize}
\item[(A1)] there exists $d_0$ such that for $d>d_0$ system \eqref{eq:NLSvectorial} is locally well posed in
$H^d$. Furthermore, the space $\Sigma _d$ is invariant by this motion.
\end{itemize}
\noindent We recall the following definition.
\begin{definition}\label{def:HamField} Given a Frech\'et differentiable function $F$,
the Hamiltonian vectorfield of $F$ with respect to a \textit{strong} symplectic form $\omega $, see \cite{amr} Ch. 9, is the field $X_F $ such that $ \omega (X_F ,Y)= dF (Y)$ for any given
tangent vector $Y.$ For $\omega =\Omega$ we have $X_F=J\nabla F$.
For $F,G $ two scalar Frech\'et differentiable functions,
we consider the Poisson bracket
$
\{ F,G \} := dF (X_G ).
$
If $\mathcal{G} $ has values in a given
Banach space $\mathbb{E}$ and $G$ is a scalar valued function, then we set $
\{ \mathcal{G} ,G \} := \mathcal{G} '(X_G ),
$ for $\mathcal{G}'$ the Frech\'et derivative of $\mathcal{G} $.
\end{definition}
We assume some symmetries in system \eqref{eq:NLSvectorial}. Specifically
we assume what follows.
\begin{itemize}
\item[(A2)] There are selfadjoint differential operators $\Diamond _\ell$
for $\ell =1,..., n_0$ in $L^2$ such that
$ \Diamond _\ell :
\Sigma _{n} \to \Sigma _{n-d_\ell} $ for $\ell =1,..., n_0$.
We set $\textbf{d} =\sup _\ell d_\ell$.
\item[(A3)]We assume
$[\Diamond _\ell , J] =0$ and $[\Diamond _\ell , \Diamond _k] =0$.
\item[(A4)] We assume $\{\Pi _\ell , E_K \} = \{\Pi _\ell , E_P \} =0$ for all $\ell $, where $\Pi _\ell := \frac{1}{2} \langle \Diamond _\ell \ ,
\ \rangle $.
\item[(A5)] Set $
\langle \epsilon \Diamond\rangle ^{ 2}:=1+\sum _j \epsilon ^2 \Diamond ^2_j.$ Then
$ \langle \epsilon \Diamond\rangle ^{ -2} \in B(\Sigma _n ,\Sigma _ n )$ with
\begin{equation} \label{eq:a71} \begin{aligned}
&
\quad \text{ $\| \langle \epsilon \Diamond\rangle ^{ -2} \| _{B(\Sigma _n ,\Sigma _ n ) } \le C_n<\infty$ for any $|\epsilon |\le 1 $ and $n\in \N$.}
\end{aligned}
\end{equation}
Furthermore, for any $n\in \Z$ we have
\begin{equation} \label{eq:a72} \begin{aligned}
& strong-\lim _{\epsilon\to 0} \langle \epsilon \Diamond\rangle ^{ -2} =1 \text{ in $B(\Sigma _n ,\Sigma _ n )$ } \\& \lim _{\epsilon\to 0} \| \langle \epsilon \Diamond\rangle ^{ -2} -1 \| _{B(\Sigma _n ,\Sigma _ {n'} ) }=0
\quad \text{ for any $n'\in \Z$ with $n'<n$. }
\end{aligned}
\end{equation}
\item[(A6)] Consider the
groups
$e^{J \langle \epsilon \Diamond\rangle ^{ -2}\Diamond \cdot \tau } $ defined in $L^2$.
We assume that for any $n\in \N$ these groups leave $ \Sigma _n$ invariant and that for any $n\in \N$ and $c>0$ there a $C$ s.t.
$\| e^{J\langle \epsilon \Diamond\rangle ^{ -2}\Diamond \cdot \tau } \| _{B(\Sigma _n,\Sigma _n)}\le C$ for any $|\tau |\le c$ and any $|\epsilon |\le 1 $ .\end{itemize}
We introduce now our \textit{solitary} waves.
\begin{itemize}
\item[(B1)] We assume that for $ {\mathcal O}$ an open subset of $\R ^{n_0}$
we have a function $p\to \Phi _p \in {\mathcal S}(\R ^3, \R ^{2N}) $ which is in
$C^\infty ( {\mathcal O} , {\mathcal S} )$, with $\Pi _\ell (\Phi _p )=p_\ell$, where the
$\Phi _p $ are constrained critical points of $E$ with associated Lagrange multipliers $\lambda _\ell (p)$
so that
\begin{equation} \label{eq:lagr mult}\begin{aligned}&
\nabla E(\Phi _p )= \lambda (p) \cdot \Diamond \Phi _p \end{aligned}
\end{equation}
\item[(B2)] We will assume that the map $p\to \lambda (p)$ is a diffeomorphism.
In particular this means that the following matrix has rank $n_0 $
\begin{equation} \label{eq:nondegen} \begin{aligned} & \text{rank} \left [ \frac{\partial \lambda _i}{\partial p _j} \right ] _{ \substack{ i\downarrow \ , \ j \rightarrow}}= n_0 .
\end{aligned}
\end{equation}
\end{itemize}
A function $U(t):= e^{J( t \lambda (p) +\tau _0)\cdot \Diamond}\Phi _p $
is a {solitary wave} solution of \eqref{eq:NLSvectorial}
for any fixed vector $\tau _0$.
\subsection{The linearization}
\label{subsec:linearization}
Set ${\mathcal H}_p:= J(\nabla ^2 E(\Phi _p )- \lambda (p) \cdot \Diamond ) $.
Notice that $E (e^{J \tau \cdot \Diamond}U) \equiv E ( U) $ for any $U$ yields
$\nabla E (e^{J \tau \cdot \Diamond}U) = e^{J \tau \cdot \Diamond} \nabla E ( U) $ and
$\nabla ^2 E (e^{J \tau \cdot \Diamond}U) = e^{J \tau \cdot \Diamond} \nabla ^2E ( U) e^{-J \tau \cdot \Diamond} .$
Then
\eqref{eq:lagr mult} implies $\nabla E( e^{J \tau \cdot \Diamond}\Phi _p )= e^{J \tau \cdot \Diamond} \lambda (p) \cdot \Diamond \Phi _p $. So applying $\partial _{ \tau _j}$ we obtain
$ (\nabla ^2 E( \Phi _p ) -\lambda (p) \cdot \Diamond ) J \Diamond _j \Phi _p =0$ and so
\begin{equation} \label{eq:kernel}\begin{aligned}&
{\mathcal H}_p J \Diamond _j \Phi _p =0\end{aligned}
\end{equation}
\begin{itemize}
\item [(C1)] We will assume
\begin{equation} \label{eq:kernel1}\begin{aligned}&\ker
{\mathcal H}_p =\text{Span}\{ J \Diamond _j \Phi _p:j=1,..., n_0 \} .\end{aligned}
\end{equation}
\end{itemize}
\noindent Applying $\partial _{ \lambda _j}$ to \eqref{eq:lagr mult} yields
$ (\nabla ^2 E( \Phi _p ) -\lambda (p) \cdot \Diamond ) \partial _{ \lambda _j} \Phi _p = \Diamond _j\Phi _p $. This yields
\begin{equation} \label{eq:gen kernel}\begin{aligned}&
{\mathcal H}_p \partial _{ \lambda _j} \Phi _p =J \Diamond _j \Phi _p\end{aligned}
\end{equation}
We have
\begin{equation} \label{eq:nondegen1} \begin{aligned} & \langle \partial _{ \lambda _j} \Phi _p , \Diamond _k \Phi _p\rangle =\frac 12 \partial _{ \lambda _j} \langle \Phi _p , \Diamond _k \Phi _p\rangle =\partial _{ \lambda _j}p_k.
\end{aligned}
\end{equation}
Necessarily, by (B2) there exists $j$ such that $\partial _{ \lambda _j}p_k\neq 0.$
This implies that the \textit{generalized kernel} is
\begin{equation} \label{eq:gen kernel1}\begin{aligned}&N_g (
{\mathcal H}_p ) =\text{Span}\{ J \Diamond _j \Phi _p, \partial _{ \lambda _j} \Phi _p :j=1,..., n_0 \} .\end{aligned}
\end{equation}
\noindent The map $(p , \tau ) \to e^{J \tau _0 \cdot \Diamond}\Phi _p $ is in
$C^\infty ( {\mathcal O} \times \R ^{n_0 }, {\mathcal S} )$.
\begin{itemize}
\item [(C2)]
We assume this map is a local embedding
and that the image is a manifold $\mathcal{G}$.
\end{itemize}
At any given point $e^{J \tau \cdot \Diamond}\Phi _{p }$
the tangent space of $\mathcal{G}$ is given by
\begin{equation*} \label{ }\begin{aligned}& T_{e^{J \tau \cdot \Diamond}\Phi _{p }} \mathcal{G} = \text{Span}\{ e^{J \tau \cdot \Diamond}\partial _{p_j}\Phi _{p },
e^{J \tau \cdot \Diamond}\Diamond _{j }\Phi _{p } :j=1,..., n_0\} . \end{aligned}
\end{equation*}
We have $\Omega ( e^{J \tau \cdot \Diamond}\partial _{p_j}\Phi _{p }, e^{J \tau \cdot \Diamond}\partial _{p_k}\Phi _{p } ) = \Omega ( \partial _{p_j}\Phi _{p }, \partial _{p_k}\Phi _{p } )$.
\begin{itemize}
\item [(C3)] We
assume that
\begin{align} \label{eq:gen kernel2} & \Omega ( \partial _{p_j}\Phi _{p }, \partial _{p_k}\Phi _{p } ) = 0
\text{ for all $j$ and $k$} \\& \label{eq:gen kernel20} \Omega ( \partial _{p_j}\Phi _{p }, \Phi _{p } ) = 0
\text{ for all $j$} . \end{align}
\end{itemize}
Notice that \eqref{eq:gen kernel20} is not required in \cite{bambusi} but in any case is true for the applications
in \cite{bambusi,boussaidcuccagna,Cu1,Cu2}. Here we use it in Lemma \ref{lem:1forms}.
We have the following beginning of Jordan block decomposition of $\mathcal{H}_p$.
\begin{lemma}
\label{lem:begspectdec} Consider the operator $\mathcal{H}_p$. We
have
\begin{equation} \label{eq:begspectdec1} \begin{aligned} &
J^{-1}\mathcal{H}_p=-\mathcal{H}_p^{\ast}J^{-1} \, , \quad \mathcal{H}_pJ=-J\mathcal{H}_p^{\ast} .
\end{aligned}\end{equation}
Assume (B1)--(B2) and (C1). Then we have
\begin{align}
\label{eq:begspectdec2}& L^2= N_g(\mathcal{H}_p)\oplus N_g^\perp (\mathcal{H}_p^{\ast}) \ ,
\\& N_g (\mathcal{H}_p^{\ast}) =\text{Span}\{ \Diamond _j \Phi _p, J^{-1}\partial _{ \lambda _j} \Phi _p :j=1,..., n_0 \} . \label{eq:begspectdec3}
\end{align}
\end{lemma}
\proof We have $\mathcal{H}_p=JA$ for a selfadjoint operator $A$ and with
$J$ a bounded antisymmetric operator. Then $\mathcal{H}_p^{\ast}=-AJ$
and \eqref{eq:begspectdec1} follows by direct inspection.
Recall that (B1)--(B2) and (C1) imply \eqref{eq:gen kernel1}.
Then
\eqref{eq:begspectdec1} implies
\eqref{eq:begspectdec3}.
\noindent The map $\psi \to \langle \ , \psi \rangle $ establishes a map $N_g (\mathcal{H}_p^{\ast})
\to B (N_g(\mathcal{H}_p), \R )$. By \eqref{eq:nondegen1}, formulas \eqref{eq:gen kernel1} and \eqref{eq:begspectdec3}
imply that this map is an isomorphism. For any $u \in L^2$ there is exactly one $v\in N_g(\mathcal{H}_p)$ such that $\langle u , \ \rangle$ and $\langle v , \ \rangle$ coincide as elements in $B (N_g (\mathcal{H}_p^{\ast}) , \R )$. Then $ u-v \in N_g^\perp (\mathcal{H}_p^{\ast}) $ and we get
\eqref{eq:begspectdec2}.
\qed
Obviously Lemma \ref{lem:begspectdec} holds true only because our $J$ is very special. For the KdV, where $J=\frac{\partial}{\partial x}$,
\eqref{eq:begspectdec2}--\eqref{eq:begspectdec3} are not true.
\noindent
Denote by $P_{N_g }(p) =P_{N_g(\mathcal{H}_p)}$ the projection onto
$N_g(\mathcal{H}_p) $ associated to \eqref{eq:begspectdec2} and by $P(p):=1-P_{N_g }(p)$ the projection on $N_g^\perp (\mathcal{H}_p^{\ast})$. We have, summing
on repeated indexes,
\begin{equation} \label{eq:projNg} \begin{aligned} & P_{N_g }(p)X= -J\Diamond
_j \Phi _p\ \langle X ,J^{-1} \partial _{p_j}\Phi _p\rangle +\partial
_{p_j}\Phi _p \ \langle X ,\Diamond _j \Phi _p \rangle. \end{aligned}
\end{equation}
\begin{lemma}
\label{lem:projections} Assume (B1)--B(2) and (C1).
Then:
\begin{itemize}
\item[(1)] $P_{N_g }(p)\in B (\mathcal{S}',\mathcal{S}) $ for any $p\in \mathcal{O}$ and
$P_{N_g }(p)\in C^{\infty}
(\mathcal{O}, B (\Sigma _{-k},\Sigma _{ k}))$ for any $k\in \N $.
\item[(2)] For any $p_0\in \mathcal{O}$ and $k $ there exists an $\varepsilon _k>0$
such that for $|p-p_0|< \varepsilon _k $
\begin{equation}\label{eq:projections}
P (p )P ( p_0) :N_g^\perp (\mathcal{H} _{p_0 }^{\ast})\cap \Sigma _{k}\to N_g^\perp (\mathcal{H}_{p }^{\ast})\cap \Sigma _{k}
\end{equation}
is an isomorphism.
\item[(3)] For $h >k$ we have $\varepsilon _h \ge \varepsilon _k $.
\end{itemize}
\end{lemma}
\proof Claim (1) is elementary and we skip the proof.
\noindent Consider the map $P (p )P ( p_0) P(p)= 1 +P (p )(P_{N_g }(p) -P_{N_g }(p_0) ) P(p) $ from
$N_g^\perp (\mathcal{H}_{p }^{\ast})\cap \Sigma _{k }$ into itself. By Claim (1) and by the Fredholm alternative, this is an isomorphism for $|p-p_0|< \varepsilon _k $ with $\varepsilon _k>0$ sufficiently small.
This implies that the $P (p )P ( p_0)$ in \eqref{eq:projections} is onto. For the same reasons also
$P (p_0 )P ( p ) P(p_0) $ is an isomorphism from
$N_g^\perp (\mathcal{H}_{p_0 }^{\ast})\cap \Sigma _{k }$ into itself. Then $P (p )P ( p_0)$ in \eqref{eq:projections} is one to one. This yields Claim (2).
\noindent For $h>k$ we have the commutative diagram
\begin{equation}
\begin{aligned}
N_g^\perp (\mathcal{H} _{p_0 }^{\ast})&\cap \Sigma _{h}
\stackrel{P (p )P ( p_0)}{\rightarrow} &N_g^\perp (\mathcal{H}_{p }^{\ast})\cap \Sigma _{h} \\& \downarrow & \downarrow
\\ N_g^\perp (\mathcal{H} _{p_0 }^{\ast})&\cap \Sigma _{k}\stackrel{P (p )P ( p_0)}{\rightarrow} &N_g^\perp (\mathcal{H}_{p }^{\ast}) \cap \Sigma _{k}
\end{aligned}\nonumber
\end{equation}
with the vertical maps two embedding. This implies that for $|p-p_0|< \varepsilon _k $
we have $\ker P (p )P ( p_0)=0$ in $N_g^\perp (\mathcal{H} _{p_0 }^{\ast}) \cap \Sigma _{h}$.
To complete the proof of Claim (3), we need to show that given $u\in N_g^\perp (\mathcal{H} _{p }^{\ast}) \cap \Sigma _{h}$ and the resulting $v\in N_g^\perp (\mathcal{H} _{p _0 }^{\ast}) \cap \Sigma _{k}$
with $u=P (p )P ( p_0)v$, we have $v\in \Sigma _{h}$.
But this follows immediately from
\begin{equation*} v= u + (P_{N_g }(p) -P_{N_g }(p_0) )v \text{ where $u\in \Sigma _{h}$ and $(P_{N_g }(p) -P_{N_g }(p_0) )v\in \mathcal{S}$.}
\end{equation*}
\qed
We will denote the inverse of \eqref{eq:projections} by
\begin{equation}\label{eq:projections1}
(P (p )P ( p_0))^{-1} :N_g^\perp (\mathcal{H} _{p }^{\ast})\cap \Sigma _{k}\to N_g^\perp (\mathcal{H}_{p_0 }^{\ast})\cap \Sigma _{k}.
\end{equation}
We have the following \textit{Modulation} type lemma.
\begin{lemma}[Modulation]
\label{lem:modulation} Assume (A2), (B.1), (B.2), (C.1) and (C.3). Fix $n \in \Z$, $n\ge 0$
and
fix $\Psi _0=e^{J \tau _0 \cdot \Diamond}\Phi _{p_0}$. Then $\exists$ a neighborhood $\U $ in $ \Sigma _{-n}(\R ^3, \R ^{2N}) $ of $U_0$
and functions $p \in C^\infty (\U , \mathcal{O})$
and $\tau \in C^\infty (\U , \R ^{n_0 })$ s.t. $p(\Psi _0)=p_0 $ and $\tau (\Psi _0) =\tau _0$
and s.t. $\forall U\in \U $\begin{equation}\label{eq:anzatz}\begin{aligned} &
U = e^{J \tau \cdot \Diamond} ( \Phi _{p } +R)
\text{ and $R\in N^{\perp}_g (\mathcal{H}_p ^*)$.}
\end{aligned}\end{equation}
\end{lemma}
\proof Consider the following $2n_0$ functions:
\begin{equation} \label{eq:nals}\begin{aligned} & \mathcal{F}_j (U,p, \tau ):= \Omega (
U -e^{ J \tau \cdot \Diamond} \Phi _ p, e^{ J \tau \cdot \Diamond} \partial _{p_j}\Phi _p )
\\& \mathcal{G}_j (U,p, \tau ) :=\Omega (
U -e^{ J \tau \cdot \Diamond} \Phi _ p, Je^{ J \tau \cdot \Diamond} \Diamond _j \Phi _p ) . \end{aligned}
\end{equation}
These functions belong to
$C^\infty ( \Sigma _{-n} \times \mathcal{O} \times \R ^{n_0}, \R )$.
We introduce the notation $R = e^{ -J \tau \cdot \Diamond}U-\Phi _ p $.
Notice that $R=0$ for $U=\Phi _ p$. Then
\begin{equation*} \label{ }\begin{aligned} & \partial _{\tau _k}\mathcal{F}_j (U,p, \tau ) = \Omega (
e^{ J \tau \cdot \Diamond} R ,e^{ J \tau \cdot \Diamond} J \Diamond _k \partial _{p_j}\Phi _p ) - \Omega ( J \Diamond _k e^{ J \tau \cdot \Diamond} \Phi _p,
e^{ J \tau \cdot \Diamond} \partial _{p_j}\Phi _p )
\\& = - \langle R , \Diamond _k \partial _{p_j}\Phi _p \rangle - \langle \Diamond _k \Phi _p ,
\partial _{p_j}\Phi _p \rangle = - \langle R , \Diamond _k \partial _{p_j}\Phi _p \rangle - \frac{1}{2}\partial _{p_j} \langle \Diamond _k \Phi _p ,
\Phi _p \rangle \\& = - \langle R , \Diamond _k \partial _{p_j}\Phi _p \rangle - \delta _{jk} . \end{aligned}
\end{equation*}
By \eqref{eq:gen kernel2} we have
\begin{equation*} \label{ }\begin{aligned} & \partial _{p _k}\mathcal{F}_j (U,p, \tau ) = \Omega (
e^{ J \tau \cdot \Diamond} R ,e^{ J \tau \cdot \Diamond} \partial _{p_k} \partial _{p_j}\Phi _p ) - \Omega ( J e^{ J \tau \cdot \Diamond} \partial _{p_k} \Phi _p,
e^{ J \tau \cdot \Diamond} \partial _{p_j}\Phi _p )
\\& = \Omega (
R , \partial _{p_k} \partial _{p_j}\Phi _p ) . \end{aligned}
\end{equation*}
By (A3) we have
\begin{equation*} \label{ }\begin{aligned} & \partial _{\tau _k}\mathcal{G}_j = \Omega (
e^{ J \tau \cdot \Diamond} R ,e^{ J \tau \cdot \Diamond} J ^2 \Diamond _k \Diamond _j\Phi _p ) - \Omega ( J \Diamond _k e^{ J \tau \cdot \Diamond} \Phi _p
e^{ J \tau \cdot \Diamond} J \Diamond _j\Phi _p )
\\& = - \langle R , J\Diamond _k \Diamond _j\Phi _p \rangle - \langle J \Diamond _k \Phi _p ,
\Diamond _j\Phi _p \rangle = - \langle R , J\Diamond _k \Diamond _j\Phi _p \rangle , \end{aligned}
\end{equation*}
We have
\begin{equation*} \label{ }\begin{aligned} & \partial _{p _k}\mathcal{G}_j = \Omega (
e^{ J \tau \cdot \Diamond} R ,e^{ J \tau \cdot \Diamond} J \Diamond _j \partial _{p_k} \Phi _p ) - \Omega ( e^{ J \tau \cdot \Diamond} \partial _{p_k} \Phi _p
e^{ J \tau \cdot \Diamond} J \Diamond _j\Phi _p )
\\& = -\langle R , \Diamond _j \partial _{p_k} \Phi _p \rangle +\langle \partial _{p_k} \Phi _p , \Diamond _j \Phi _p \rangle = -\langle R , \Diamond _j \partial _{p_k} \Phi _p \rangle + \delta _{jk}
. \end{aligned}
\end{equation*}
At $U=\Psi _0$, $\tau =\tau _0$ and $p=p_0$ we have $\mathcal{F}_j =\mathcal{G}_j =0$. Since in this case $R=0$
we get the desired result by the Implicit Function Theorem. \qed
\subsection{Spectral coordinates}
\label{subsec:coordinates}
Lemmas \ref{lem:modulation}--\ref{lem:begspectdec} lead to a natural
decomposition of \eqref{eq:NLSvectorial}. To write it we need
further notation.
\noindent We are ready for the natural
coordinates decomposition.
Let $\Pi (U_0)=p_0$.
We consider
for $R\in N_g^\perp (\mathcal{H} _{ p_0 }^{\ast})$ the map
\begin{equation}\label{eq:coordinate}\begin{aligned} &
(\tau , p, R) \to U= e^{ J \tau \cdot \Diamond} ( \Phi _{p } +P(p)R).
\end{aligned} \end{equation}
We have the following formulas,
\begin{equation}\label{eq:vectorfields} \begin{aligned} &
\frac \partial {\partial {\tau _j}} = J \Diamond _jU
\, ,\quad \frac \partial {\partial {p _j}}= e^{ J \tau \cdot \Diamond} ( \partial _ {p _j}\Phi _{p } + \partial _ {p _j}P(p)R) ,\end{aligned}
\end{equation}
with $\frac \partial {\partial {p _j}} \in C^{\infty} (\U \cap \Sigma _{k }, \Sigma _{k' })$
for any pair $(k ,k' )\in \N ^2$, with $\U \subset \Sigma _{-n}$ the neighborhood
of $ e^{J\tau _0\cdot \Diamond }\Phi _{p _0} $ in
Lemma \ref{lem:modulation}. Similarly, $\frac \partial {\partial {\tau _j}} \in C^{0} (\U \cap \Sigma _{k }, \Sigma _{k-d_j })$. We have what follows.
\begin{lemma} \label{lem:gradient R} Consider the $n\ge 0$ and $\U$
in Lemma \ref{lem:modulation} and fix an integer $k\ge -n$.
Then the map
$U\to R(U)=R$ is $ C^{0} (\U \cap \Sigma _{k }, \Sigma _k)$. For $k\ge -n +\textbf{d}$
we have
$R\in C^1(\U \cap \Sigma _{k }, \Sigma _{k -\textbf{d}})$.
For $\U$ sufficiently small in $\Sigma _{-n }$
the
Frech\'et derivative
$R'(U) $ of $R(U)$ is defined by the following formula, summing on the repeated index $j$,
\begin{equation*} \begin{aligned}
R'( U)&= (P (p )P ( p_0))^{-1} P (p )\big [ e^{ - J \tau \cdot \Diamond} \uno-
J \Diamond _j P (p ) R \, d\tau _j- \partial _{p_j} P (p ) R \, dp _j \big ] ,
\end{aligned}
\end{equation*}
where $(P (p )P ( p_0))^{-1} :N_g^\perp (\mathcal{H} _{p }^{\ast})\cap \Sigma _{k -\textbf{d}}\to N_g^\perp (\mathcal{H}_{p_0 }^{\ast})\cap \Sigma _{k -\textbf{d}} $ is well defined by Lemma \ref{lem:projections}.
\end{lemma}
\proof The continuity of $ R(U)$ follows from $R=e^{-J\tau \cdot
\Diamond}U-\Phi _p$ and
\begin{equation*} \begin{aligned} &
R-R'= e^{-J \tau \cdot
\Diamond}U -e^{-J\tau '\cdot
\Diamond}U '+\Phi _{p'} -\Phi _p =\\&\Phi _{p'} -\Phi _p + (e^{-J \tau \cdot
\Diamond} -e^{-J\tau '\cdot
\Diamond} )U +e^{-J\tau '\cdot
\Diamond}(U-U ' ) .
\end{aligned}
\end{equation*}
Then use $p\to \Phi _p\in C^\infty ( {\mathcal O} , {\mathcal S} )$,
the fact that $e^{ J \tau \cdot
\Diamond}$ is strongly continuous in $\Sigma _k$ and locally uniformly bounded therein. The fact that $ R(U)$ has Frech\'et derivative
follows by the chain rule. To get the formula for $R'(U)$ notice that the equalities $R'\frac \partial {\partial {p _j}}=R'\frac \partial {\partial {\tau _j}}=0$ and $R'e^{ J \tau \cdot \Diamond} P (p )P ( p_0)= \uno _{|N_g^\perp (\mathcal{H}_{p_0}^{\ast})} $ characterize $R'$.
We claim we have
\begin{equation}\label{eq:frechr}
R'=\mathbf{a} _j d\tau _j + \mathbf{b} _j dp _j +(P (p )P ( p_0))^{-1} P (p ) e^{ - J \tau \cdot \Diamond}
\end{equation}
for some $\mathbf{a} _j $ and $\mathbf{b} _j$. First of all, by the independence of coordinates $(\tau ,p)$ from
$R\in N_g^\perp (\mathcal{H}_{p_0}^{\ast})$,
\begin{equation*}
d\tau_j \circ e^{ J \tau \cdot \Diamond} P (p )P ( p_0)=dp_j \circ e^{ J \tau \cdot \Diamond} P (p )P ( p_0)=0.
\end{equation*}
Indeed for $g \in N_g^\perp (\mathcal{H}_{p_0}^{\ast}) $ we have for instance
\begin{equation*}\begin{aligned}
0= \frac {d}{dt} \tau _j (u (\tau , p , R+tg)) _{|t=0} &= \frac {d}{dt} \tau _j ( e^{ J \tau \cdot \Diamond} ( \Phi
_p + P(p) P(p_0) (R+tg))) _{|t=0} \\& =
d\tau_j \circ e^{ J \tau \cdot \Diamond} P (p )P ( p_0)g . \end{aligned}
\end{equation*}
Secondarily, by the definition of $(P (p )P ( p_0))^{-1}$,
\begin{equation*}
(P (p )P ( p_0))^{-1} P (p ) e^{ - J \tau \cdot \Diamond} \circ e^{ J \tau \cdot \Diamond} P (p )P ( p_0) =\uno _{N_g^\perp (\mathcal{H}_{p_0}^{\ast})}.
\end{equation*}
Hence we get the claimed equality \eqref{eq:frechr}.
\noindent To get $\mathbf{a} _j $ and $\mathbf{b} _j$ notice that
by $R' \frac \partial {\partial {\tau _j}}=0$ and $P(p)J\Diamond _j \Phi _p
=0$
\begin{equation*} \begin{aligned} &
\mathbf{a} _j= -(P (p )P ( p_0))^{-1} P (p ) e^{ - J \tau \cdot \Diamond}
\frac \partial {\partial {\tau _j}}=\\& -(P (p )P ( p_0))^{-1} P (p ) e^{ - J \tau \cdot \Diamond} e^{ J \tau \cdot \Diamond} J\Diamond _j (\Phi _p+P(p)R)=\\& -(P (p )P ( p_0))^{-1} P (p ) J\Diamond _j P(p)R .
\end{aligned}
\end{equation*}
Similarly by $R'\frac \partial {\partial {p _j}} =0$ and $P(p)\partial _{p_j}\Phi _p
=0$
\begin{equation*} \begin{aligned} &
\mathbf{b} _j= -(P (p )P ( p_0))^{-1} P (p ) e^{ - J \tau \cdot \Diamond}
\frac \partial {\partial {p _j}}=\\& -(P (p )P ( p_0))^{-1} P (p ) (
\partial _{p_j}\Phi _p+\partial _{p_j}P(p)R)=\\& -(P (p )P ( p_0))^{-1} P (p ) \partial _{p_j}P(p)R .
\end{aligned}
\end{equation*}
\qed
\bigskip
A crucial point in the stability proofs in \cite{bambusicuccagna,boussaidcuccagna,Cu1,Cu2}, first realized
and used in \cite{Cu3},
is the importance not to loose track of the Hamiltonian nature of
\eqref{eq:NLSvectorial}, in whichever coordinates the system is written.
Thus we have what follows.
\begin{lemma}\label{lem:HamFor} In the coordinate system \eqref{eq:coordinate}, system \eqref{eq:NLSvectorial} can be written as
\begin{equation} \label{eq:SystPoiss} \begin{aligned} &
\dot p = \{ p , E \} \, , \, \dot \tau = \{ \tau , E \} \, , \, \dot R = \{ R , E \}
. \end{aligned}
\end{equation}
\end{lemma}
\proof
The statement is not standard only for $\dot R = \{ R , E \}$. Notice that \eqref{eq:NLSvectorial} can be written as
\begin{equation}\label{eq:sys1}\begin{aligned} &
\dot U= J \dot \tau \cdot \Diamond U+
e^{ J \tau \cdot \Diamond} \dot p\cdot \nabla _p( \Phi _{p } +P(p)R) +e^{ J \tau \cdot \Diamond} P(p)
\dot R \\& =\sum _j \dot \tau _j \frac{\partial }{\partial \tau _j}+
\dot p_j \frac{\partial }{\partial p _j} +e^{ J \tau \cdot \Diamond} P(p)
\dot R = J \nabla E ( U) .\end{aligned}
\end{equation}
When we apply the derivative $R'(U)$ to \eqref{eq:sys1}, all the terms in the lhs of the last
line cancel except for
\begin{equation}
\begin{aligned} & R'(U)
e^{ J \tau \cdot \Diamond} P(p)
\dot R = R'(U) J
\nabla E ( U) =R'(U)X_E(U)=\{ R,E\} ,
\end{aligned}\nonumber
\end{equation} from
the definition of hamiltonian field and of Poisson bracket.
Finally we use
\begin{equation}
\begin{aligned} & R'(U)
e^{ J \tau \cdot \Diamond} P(p)
\dot R = \frac{d}{ds}_{|_{s=0}}R( U (\tau , p , R+s\dot R) ) =\frac{d}{ds}_{|_{s=0}} (R+s\dot R) = \dot R .
\end{aligned}\nonumber
\end{equation}
\qed
\subsection{Reduction of order of system \eqref{eq:SystPoiss}}
\label{subsec:reduction}
The following Poisson bracket identities are useful.
\begin{lemma}
\label{lem:Involutions} Consider the functions $\Pi _j$. Then
$ X _{\Pi _j }=
\frac{\partial}{\partial \tau _j} .$
In particular
\begin{equation} \label{eq:Ham.VecField1} \{ \Pi _j,\tau _k \} =-\delta _{jk} \ , \quad \{ \Pi _j,p _k \} \equiv 0 \ , \quad \{ R, \Pi _j \} = 0.
\end{equation}
\end{lemma}
\proof \eqref{eq:Ham.VecField1} follows from the first claim, which is a consequence of \eqref{eq:vectorfields}:
\begin{equation}
\begin{aligned} & X _{\Pi _j } (U) = J\nabla \Pi _j (U) = J \Diamond _jU= \frac{\partial}{\partial \tau _j} . \end{aligned}\nonumber
\end{equation}
\qed
\noindent We introduce now a new Hamiltonian:
\begin{equation} \label{eq:K} \begin{aligned} &
{K}(U):=E(U)- E\left ( \Phi _{p _0}\right ) - \lambda _j(p) \left ( \Pi _j (U)- \Pi _j (U_0)\right ) .
\end{aligned}
\end{equation}
Notice that $K(e^{J\tau \cdot \Diamond}U)\equiv K(U)$. Equivalently, $\partial _{\tau _j} {K} \equiv 0$.
We know that for solutions of \eqref{eq:NLSvectorial} we have $\Pi _j (U (t))= \Pi _j (U_0) $ and
\begin{equation*} \begin{aligned} & \{ p _j, K \} = \{ p _j, E \} \, , \, \{ R , K \} = \{ R, E \} \, , \,
\{ \tau _j , K \} = \{ \tau _j , E \}+ \lambda _j(p) .
\end{aligned}
\end{equation*}
By $\partial _{\tau _j} {K} \equiv 0$, the evolution of the variables $p ,R$ is unchanged if we consider the following new
Hamiltonian system:
\begin{equation} \label{eq:SystK} \begin{aligned} &
\dot p_j = \{ p _j, K \} \, , \quad \dot \tau _j = \{ \tau _j , K \} \, , \quad \dot R= \{ R, {K}
\} . \end{aligned}
\end{equation}
It is elementary that the \textit{momenta} $\Pi _j(U)$ are invariants of motion of \eqref{eq:SystK}.
Before exploiting the invariance of $ \Pi _j(U)$ to reduce the order of
the system, we introduce appropriate notation.
First of all we set \begin{equation}\label{eq:PhaseSpace}\begin{aligned} &
{\Ph}^{r }:=\R ^{ n_0}\times (\Sigma _{r } \cap N_g^\perp ({\mathcal H}_{p_0}))=\{(\tau , R)\}
\, , \\&
\widetilde {\Ph}^{r }:= \R ^{ n_0}\times {\Ph}^{r }=\{(\Pi ,\tau , R)\} . \end{aligned}
\end{equation}
We set ${\Ph} ={\Ph}^{0 }$ and $\widetilde {\Ph}=
\widetilde {\Ph}^{0 }$.
\begin{definition}\label{def:scalSymb}
We will say that $F(t,\varrho , R)\in C^{M}(I\times \mathcal{A},\R)$ with
$I$ a neighborhood of 0 in $\R$ and
$\mathcal{A}$ a neighborhood of 0 in
$ {\Ph}^{-K }$
is $\mathcal{R}^{i,j}_{K, M}$ and we will write $F=\mathcal{R}^{i,j}_{ K,M}$, or more specifically $F=\mathcal{R}^{i, j}_{ K,M} (t,\varrho , R)$,
if there exists a $C>0$ and a smaller neighborhood $\mathcal{A}'$ of 0 s.t.
\begin{equation}\label{eq:scalSymb}
|F(t,\varrho , R)|\le C \| R\| _{\Sigma _{-K}}^j (\| R\| _{\Sigma _{-K}}+|\varrho |)^{i} \text{ in $I\times \mathcal{A}'$}.
\end{equation}
We say $F=\mathcal{R}^{i, j} _{K, \infty}$ if $F=\mathcal{R}^{i,j}_{K, m}$ for all $m\ge M$.
We say $F=\mathcal{R}^{i, j}_{\infty, M} $ if for all $k\ge K$ the above $F$ is the restriction of an
$F(t,\varrho , R)\in C^{M}(I\times \mathcal{A}_{k },\R)$ with $\mathcal{A}_k$ a neighborhood of 0 in
$ {\Ph}^{-k }$ and
which is
$F=\mathcal{R}^{i,j}_{k, M}$. Finally we say
$F=\mathcal{R}^{i, j} $ if $F=\mathcal{R}^{i, j} _{K, \infty}$ and $F=\mathcal{R}^{i, j} _{ \infty , M}$.
\end{definition}
\begin{definition}\label{def:opSymb} We will say that an $T(t,\varrho , R)\in C^{M}(I\times \mathcal{A},\Sigma _{K} (\R^3, \R ^{2N}))$, with $I\times \mathcal{A}$ like above,
is $ \mathbf{{S}}^{i,j}_{K,M} $ and we will write $T= \mathbf{{S}}^{i,j}_{K,M}$ or more specifically $T= \mathbf{{S}}^{i,j}_{K,M} (t,\varrho , R)$,
if there exists a $C>0$ and a smaller neighborhood $\mathcal{A}'$ of 0 s.t.
\begin{equation}\label{eq:opSymb}
\|T(t,\varrho , R)\| _{\Sigma _{K}}\le C \| R\| _{\Sigma _{-K}}^j (\| R\| _{\Sigma _{-K}}+|\varrho |)^{i} \text{ in $I\times \mathcal{A}'$}.
\end{equation}
We use notation $T=\mathbf{{S}}^{i, j} $, $T=\mathbf{{S}}^{i,j}_{K,\infty }$ or $T=\mathbf{{S}}^{i,j}_{\infty,M}$ as above.
\end{definition}
These notions will be often used also for functions $F=\mathcal{R}^{i, j}_{ K,M} ( \varrho , R)$ and $T=\mathbf{{S}}^{i, j}_{ K,M} ( \varrho , R)$
independent of $t$.
\begin{remark}
\label{rem:formal1}
We will see later that the coefficients of the vector fields
whose flows are used to change coordinates are symbols as of
Definitions \ref{def:scalSymb} and \ref{def:opSymb}.
The definitions of the symbols
$ \mathcal{R}^{i, j} $ and $ \mathbf{{S}}^{i, j} $
in Def. 3.9 and 3.10 \cite{bambusi} are very restrictive, since they
require for the symbols to be defined in the whole
$I\times \mathcal{S}'$. The proofs in \cite{bambusi} at most prove
that the coefficients of the vector fields in fact are symbols
of the form $ \mathcal{R}^{i, j} _{K,M} $ and $ \mathbf{{S}}^{i, j} _{K,M} $ in our sense.
As an example we refer to Lemmas 3.26 and 5.5 in \cite{bambusi}. In Lemma 3.26 \cite{bambusi}
the fact that the $b_i$ and the $\langle W^l;Y\rangle $ are symbols of the form $ \mathcal{R}^{j, k} $
for some $(j,k)$
in the sense of Def. 3.10 in \cite{bambusi}, requires preliminarily to show
at least that they are functions of $(\varrho ,R)$ for $(\varrho ,R)$ in some neighborhood $\U$ of $(0,0)$ in
$\R ^{n_0}\times \mathcal{S}'$.
This is not addressed in \cite{bambusi} and is far from trivial, since the coefficients of the linear system
right above
formula (3.60) are are unbounded in any such $\U$. The justification that the coefficients
$\Phi _{\mu \nu}(M)$ of $\chi $ in Sect. 5 in \cite{bambusi} are in $\mathcal{S}$ is similarly inconclusive.
The key step should be that the homological equation in
Lemma 5.5 can be solved for all parameters $k$ uniformly in the variable $M\in \R ^n$, at least
for $|M|< a $ for a fixed $a$. But the homological equations involve the perturbation of an operator
and in \cite{bambusi} the perturbation is not fully analyzed. For example there is no discussion of
the norm $\| V_M-V_0\| _{\mathcal{W}^k\to \mathcal{W}^k}$ as $k$ grows
and $|M|< a $. This norm should be expected to grow
and become large, possibly breaking down the proof of $\Phi _{\mu \nu}(M)\in \mathcal{S}$.
In fact it is plausible that $\Phi _{\mu \nu}(M)\in \mathcal{S}$ only for $M=0$.
From the above remarks we can see that no coordinate change in the Birkhoff or in the Darboux steps in \cite{bambusi} is shown to be an \textit{almost smooth} transformation in the sense of Definition 3.15 in \cite{bambusi}. So for instance the proof of the Birkhoff normal forms,
that is Theor. 5.2 \cite{bambusi}, is inconclusive. The proof of the Darboux step, that is Theor. 3.21 \cite{bambusi},
is even sketchier and is similarly inconclusive.
\end{remark}
\bigskip
We proceed now to a reduction of order in \eqref{eq:SystK}. Write
\begin{equation}\label{eq:variables} \begin{aligned} &
\Pi _j(U) =\Pi _j(e^{ J \tau \cdot \Diamond} ( \Phi _{p } +P(p)R))=\Pi _j( \Phi _{p } +P(p)R) \\& = \frac{1}{2}
\langle \Diamond _j( \Phi _{p } +P(p)R) , \Phi _{p } +P(p)R = p_j+ \Pi _j(P(p)R)\\& = p_j+ \Pi _j( R) + \Pi _j( (P(p)-P(p_0)) R) +\langle R, \Diamond _j(P(p)-P(p_0)) R\rangle .
\end{aligned}
\end{equation}
We well move from variables $(\tau , p , R)$ to variables $(\tau , \Pi , R)$. Setting $ \varrho _j=\Pi _j( R)$,
we have
\begin{equation}\label{eq:variables1} \begin{aligned} &
p_j=\Pi _j-\varrho _j+ \widetilde{\Psi} _j (p-p_0, R)
\end{aligned}
\end{equation}
with
$ \widetilde{\Psi} _j =\mathcal{R}^{0, 2}(p-p_0, R)$.
The implicit function theorem yields:
\begin{lemma}
\label{lem:var} There are functions $p_j =p_j ( \Pi ,\Pi ( R) , R)$
defined implicitly by \eqref{eq:variables},or \eqref{eq:variables1},
such that $ p_j=\Pi _j-\varrho _j+{\Psi} _j(\Pi , \varrho ,R)
$ with
${\Psi} (p_0 , \varrho ,R)=
\mathcal{R}^{0, 2} (\varrho , R)
$.
\end{lemma}
We consider now $(\tau , \Pi , R)$ as a new coordinate system.
By $
\frac{\partial}{\partial \tau _k } \Pi _j(U)\equiv 0 $ it follows that the vectorfields $ \frac{\partial}{\partial \tau _k }$ are the same for the two systems of coordinates.
In the new variables, system \eqref{eq:SystK} reduces to the pair
of systems
\begin{align} & \label{eq:SystK1} \dot \tau _j = \{ \tau _j ,K \} \, , \quad {\dot
\Pi _j }= 0 \, , \\& \label{eq:SystK21}
\dot R= \{ R, K
\} .
\end{align}
System \eqref{eq:SystK21} is closed because of $\partial _{\tau _j}K=0$.
\section{Darboux Theorem}
\label{sect:symplectic}
In this section we present one of the two main results of this paper.
We seek to reproduce Moser's proof of the Darboux theorem. Specifically
we look for a vector field ${\mathcal X}^t$ that will produce a flow
as in \eqref{eq:fdarboux} below. The proof of the existence and properties of
${\mathcal X}^t$ is similar to \cite{Cu2}, but influenced
by the choice of coordinates in \cite{bambusi}. We also add material
to justify, once ${\mathcal X}^t$ has been found, the formal formula \eqref{eq:fdarboux}. Notice that for \cite{boussaidcuccagna,Cu2} formula
\eqref{eq:fdarboux} does not require justification because ${\mathcal X}^t$ is a smooth
vectorfield on a given manifold. But the situation in \cite{Cu1,bambusi} is different since now ${\mathcal X}^t$ is not
a standard vectorfield on a manifold and $\Omega $ is not a regular
differential form on the same manifold, so
Lie derivative,
pullbacks, push forwards and the related differentiation formulas, require justification.
Notice that, to be useful in the asymptotic stability theory, the change of variables has to be such that the new Hamiltonian equations is semilinear.
This is why even in \cite{boussaidcuccagna,Cu2}, where
we could apply the standard Darboux theorem for strong symplectic forms
on Banach manifolds, see \cite{amr} Ch. 9, it is important to select ${\mathcal X}^t$ with an {\it ad hoc} process.
\subsection{Search of a vectorfield}
\label{subsec:vector}
Recall that $\Omega =\langle J ^{-1}\ , \ \rangle
$ and consider
\begin{equation} \label{eq:Omega0} \Omega _0 := d \tau _j \wedge
d\Pi _j + \langle J ^{-1}R' , R' \rangle
.
\end{equation}
\begin{lemma}
\label{lem:1forms}
At the points $e^{ J \tau \cdot \Diamond} \Phi _{p_0 }$ for all $\tau \in \R ^{n_0 }$ we have $\Omega _0 =\Omega .$
\noindent Consider the following forms:
\begin{equation} \label{eq:1forms1}\begin{aligned} &
\mathrm{ B} _0 :=\tau _j
d\Pi _j +\frac{1}{2} \langle J ^{-1}R , R' \rangle ; \quad \mathrm{B}:=\mathrm{B_0}+\alpha \text{ for }
\end{aligned}
\end{equation}
\begin{equation} \label{eq:1forms2}\begin{aligned} &
\alpha :=-\beta _j(p,R) d\Pi _j + \left \langle \Gamma (p) R +
\beta _j(p,R) P^*(p) \Diamond _j P(p)R,R'
\right \rangle \ ,
\\&
\Gamma (p):=\frac{1}{2} J ^{-1} \left ( P(p) -P(p_0) \right ) \ ,
\\& \beta _j(p,R) :=\frac{1}{2}\
\frac{ \langle P^*(p) J ^{-1} R, \partial _{p_{ j}}P(p)R \rangle }{ 1+
\langle \Diamond _j P(p)R, \partial _{p_{ j}}P(p)R \rangle } \
.
\end{aligned}
\end{equation}
Then
$ d\mathrm{B_0}=\Omega _0$ and $
d\mathrm{B } =\Omega .$
\end{lemma}
\proof $ d\mathrm{B_0}=\Omega _0$ follows from the definition of exterior differential.
Set $\widetilde{B} :=\frac{1}{2}\langle J ^{-1} U , \ \rangle $. Notice that $d \widetilde{B}=\Omega $.
By \eqref{eq:coordinate} we get:
\begin{equation} \label{eq:beta1} \begin{aligned} &
\widetilde{B} (X)= \frac{1}{2}\langle J ^{-1}e^{ J \tau \cdot \Diamond} \Phi _{ {p} } , X \rangle
+ \frac{1}{2}\langle J ^{-1} P(p)R , e^{ -J \tau \cdot \Diamond} X \rangle .
\end{aligned}
\end{equation}
Set $\psi (U):=\frac{1}{2}\langle J ^{-1} e^{ J \tau \cdot \Diamond} \Phi _{ {p} } ,U
\rangle .$ Then we claim
\begin{equation*} \label{eq:dPsi0} \begin{aligned} & d\psi =\frac{1}{2}\langle J ^{-1} e^{ J \tau \cdot \Diamond} \Phi _{ {p} } ,\
\rangle + p_jd\tau _j,\end{aligned}
\end{equation*}
where in this proof we will sum on repeated indexes.
The last formula implies
\begin{equation} \label{eq:beta11} \begin{aligned} &
\widetilde{B} = d\psi - p_jd\tau _j
+ \frac{1}{2}\langle J ^{-1} P(p)R , e^{ -J \tau \cdot \Diamond} \ \rangle .
\end{aligned}
\end{equation}
The desired formula on $d\psi$ follows by
\begin{equation*} \begin{aligned} & d\psi =\frac{1}{2}\langle J ^{-1} e^{ J \tau \cdot \Diamond} \Phi _{ {p} } ,\
\rangle + \frac{1}{2} \langle e^{ J \tau \cdot \Diamond} \Diamond _j \Phi _{ {p} } ,U
\rangle d\tau _j \\& + \frac{1}{2} \langle e^{ J \tau \cdot \Diamond} J ^{-1} \partial _{ {p}_j }\Phi _{ {p} } ,U
\rangle dp_j = \frac{1}{2}\langle J ^{-1} e^{ J \tau \cdot \Diamond} \Phi _{ {p} } ,\
\rangle +\\& \frac{1}{2} \langle \Diamond _j \Phi _{ {p} } ,\Phi _{ {p} }+P(p)R
\rangle d\tau _j + \frac{1}{2} \langle J ^{-1} \partial _{ {p}_j }\Phi _{ {p} } ,\Phi _{ {p} } +P(p)R
\rangle dp_j \stackrel{\text{by \eqref{eq:begspectdec3}}}{=}
\\& \frac{1}{2}\langle J ^{-1} e^{ J \tau \cdot \Diamond} \Phi _{ {p} } ,\
\rangle+ \underbrace{\frac{1}{2} \langle \Diamond _j \Phi _{ {p} } ,\Phi _{ {p} }
\rangle}_{p_j} d\tau _j + \frac{1}{2} \underbrace{ \langle J ^{-1} \partial _{ {p}_j }\Phi _{ {p} } ,\Phi _{ {p} }
\rangle}_{\qquad \quad 0 \text{ by \eqref{eq:gen kernel20}}} dp_j .
\end{aligned}
\end{equation*}
By Lemma \ref{lem:gradient R} and using $ P(p)^*J ^{-1} =J ^{-1}P(p)$ we have
\begin{equation*} \begin{aligned} & \frac{1}{2}\langle J ^{-1} P(p)R , e^{ -J \tau \cdot \Diamond} \ \rangle =\frac{1}{2}\langle J ^{-1} R ,P(p) R' \ \rangle \\& +\frac{1}{2}\langle J ^{-1} R ,P(p)J \Diamond _jP( {p})R \rangle d\tau _j+\frac{1}{2}\langle J ^{-1} R ,P(p) \partial _{p_j}P( {p})R \rangle dp _j \\& =\frac{1}{2}\langle J ^{-1} R , R' \ \rangle + \frac{1}{2}\langle J ^{-1} R ,(P(p) -P( {p}_0) ) R' \ \rangle \\& -\Pi _j(P( {p})R) d\tau _j+\frac{1}{2}\langle J ^{-1} R ,P(p) \partial _{p_j}P( {p})R \rangle dp _j .
\end{aligned}
\end{equation*}
So by \eqref{eq:beta11} and using $P(p)J =JP^*(p)$ we get
\begin{equation*} \begin{aligned} &
\widetilde{B} - d\psi = - (\overbrace{ p_j+ \Pi _j(P( {p})R)}^{\Pi_j} ) d\tau _j
+ \frac{1}{2}\langle J ^{-1} R , R' \ \rangle \\&+ \frac{1}{2}\langle J ^{-1} R ,(P(p) -P(p_0) ) R' \ \rangle -\frac{1}{2}\langle P^*( {p}) J ^{-1} R , \partial _{p_j}P( {p})R \rangle dp _j .
\end{aligned}
\end{equation*}
Then $d\alpha =\Omega -\Omega _0$ for
\begin{equation*} \label{eq:beta12} \begin{aligned} & \alpha :=
\widetilde{B} - d\psi -B_0 +d(\Pi_j \tau _j) = \\& \frac{1}{2}\langle J ^{-1} R ,(P(p) -P( {p}_0) ) R' \ \rangle -\frac{1}{2}\langle P^*( {p}) J ^{-1} R , \partial _{p_j}P( {p})R \rangle dp _j .
\end{aligned}
\end{equation*}
By $p_j=\Pi _j - \Pi _j(P(p)R)$ we get
\begin{equation*} \label{eq:beta13} \begin{aligned} & dp_j=d\Pi _j -\langle \Diamond _j P(p) R, P(p)R'\rangle - \langle \Diamond _j P(p) R, \partial _{p_j}P(p)R \rangle dp_j.
\end{aligned}
\end{equation*}
Then inserting the next formula in the formula for $\alpha$, we obtain \eqref{eq:1forms2}:
\begin{equation} \label{eq:beta14} \begin{aligned} & dp_j= \frac{ d\Pi _j -\langle \Diamond _j P(p) R, P(p)R'\rangle}{1+\langle \Diamond _j P(p) R, \partial _{p_j}P(p)R \rangle} .
\end{aligned}
\end{equation}
\qed
In the Lemmas \ref{lem:dalpha1}--\ref{lem:vectorfield} we will initially
consider the regularity of the functions in terms of the coordinates
$(\tau , p, R)$.
\begin{lemma}
\label{lem:dalpha1} We have $\beta _j \in C^\infty ( {\mathcal O} \times \Sigma _{-n }, \R )$
for any $n$. For any pair $(n,n' )$ we have $\Gamma \in C^\infty ( {\mathcal O} , B( \Sigma _{-n' }, \Sigma _{ n }) )$.
Summing on repeated indexes, we have
\begin{equation} \label{eq:dalpha1}
\begin{aligned} & d\alpha =-\partial _{p_k}\beta _j dp_k \wedge d\Pi _j - \langle \nabla _R\beta _j,R'
\rangle \wedge d\Pi _j \\& + dp_k \wedge \langle \partial _{p_k}[\Gamma (p) R +
\beta _j(p,R) P^*(p) \Diamond _j P(p)R],R' \rangle
\\& + \langle \nabla _R \beta _j , R'\rangle \wedge \langle P^*(p) \Diamond _j P(p)R,R'\rangle +2\langle \Gamma R',R'\rangle
. \end{aligned}
\end{equation}
\end{lemma}
\proof Follows from a simple computation. In particular, for a $\mathbf{L}\in B(\Sigma _1, L^2)$ fixed, we use the formula
\begin{equation*} \begin{aligned} &
d\langle \mathbf{L}R, R'\rangle (X,Y):=X\langle \mathbf{L}R, R'Y\rangle-Y\langle \mathbf{L}R, R'X\rangle
- \langle \mathbf{L}R, R'[X,Y]\rangle \\&
=
\langle \mathbf{L}R'X, R'Y\rangle -\langle \mathbf{L}R'Y, R'X\rangle . \end{aligned}
\end{equation*}
\qed
\begin{lemma}
\label{lem:dalpha2} Summing on repeated indexes, we have
\begin{equation*}
\begin{aligned} d\alpha &= \widehat{\delta } _ k \partial _{p_k}{\beta } _ {j } d\Pi _j \wedge d\Pi _k + \langle \widehat{\Gamma}_j+
( \widehat{\delta } _ k \partial _{p_k}{\beta } _ {j } -\widehat{\delta } _ j \partial _{p_j}{\beta } _ {k } ) \Diamond _k P(p)R,R'
\rangle \wedge d\Pi _j \\& + 2 \langle \Gamma (p) R ', R'\rangle
+
\langle \widetilde{\beta} _j , R'\rangle \wedge \langle P^*(p) \Diamond _j P(p)R,R' \rangle
\ ,\end{aligned}
\end{equation*}
where we have (this time not summing on repeated indexes)
\begin{equation*}
\begin{aligned}
\widehat{\delta} _k &:= \frac 1{1+\langle \Diamond _k P(p) R, \partial _{p_k}P(p)R \rangle}
\ , \\ \widehat{\Gamma}_j&:=-\nabla _R \beta _j-\widehat{\delta} _j [ \partial _{p_j}\Gamma R +\sum _{i=1}^{n_0}\beta _i\partial _{p_j}
\left ( P^*(p) \Diamond _i P(p) \right )R ]\\&
+ \sum _{k=1}^{n_0}( \widehat{\delta } _ k \partial _{p_k}{\beta } _ {j } -\widehat{\delta } _ j \partial _{p_j}{\beta } _ {k } ) ( P^*(p)-1)\Diamond _k P(p) R
\\
\widetilde{\beta} _j&:= \nabla _R \beta _j+ \widehat{\delta} _j \partial _{p_j} ( \Gamma +\sum _{k=1}^{n_0}
\beta _k P^*(p) \Diamond _k P(p) )R
. \end{aligned}
\end{equation*}
\end{lemma}
\proof Follows by an elementary computation substituting \eqref{eq:beta14} in \eqref{eq:dalpha1}
\qed
\begin{lemma}
\label{lem:vectorfield0} For any fixed large $n$ and for $\varepsilon _0>0$,
consider the set $\U _{\mathbf{d}} \subset \widetilde{\Ph}^{\textbf{d}}=\{ (p,R) \}$ defined by $ \|R\| _{\Sigma _{-n}}\le \varepsilon _0$ and
$| p -p_0|\le \varepsilon _0$.
Then for $ \varepsilon _0 $ small enough
there exists a unique vectorfield $ \mathcal{X}^t :\U _{\mathbf{d}} \to \widetilde{\Ph} $ which solves
$
i_{\mathcal{X}^t} \Omega _t=- \alpha
$, where $\Omega _t:=\Omega _0+t(\Omega -\Omega _0)$.
\end{lemma}
\proof
First of all we consider $Y$ such that $
i_{Y} \Omega _0=- \alpha
$, that is to say
\begin{equation*} \begin{aligned} &
(Y ) _{\tau _j} d\Pi _j - (Y ) _{\Pi _j} d\tau _j +\langle J^{-1} (Y ) _{R} ,R' \rangle \\ &= \beta _j(p,R) d\Pi _j - \left \langle \Gamma (p) R +
\beta _j(p,R) P^*(p) \Diamond _j P(p)R,R'
\right \rangle .
\end{aligned}
\end{equation*}
This yields
\begin{equation} \label{eq:Yvec} \begin{aligned} &
(Y ) _{\tau _j} =\beta _j(p,R)=\mathcal{R}^{0,2}(p,R) \ , \quad (Y ) _{\Pi _j}=0\ , \\ &
(Y ) _{R}= - P(p_0)J \Gamma (p) R -
\beta _j(p,R)P(p_0)J P^*(p) \Diamond _j P(p)R \\& = \mathbf{S}^{1,1} (p-p_0,R) +\mathcal{R}^{0,2}(p,R) P(p_0)P (p)J \Diamond _j P(p)R .
\end{aligned}
\end{equation}
Equation $
i_{\mathcal{X}^t} \Omega _t=- \alpha
$ is equivalent to
\begin{equation} \label{eq:fred1} \begin{aligned}
& (1+ t\mathcal{K})\mathcal{X}^t=Y
\end{aligned}
\end{equation}
where the operator $\mathcal{K}$ is defined by $ i_{X}d \alpha =
i_{\mathcal{K} X} \Omega _0$. In coordinates, \eqref{eq:fred1} becomes
$
(\mathcal{X}^t)_{\Pi _j} =0$ and, for $P=P(p)$,
\begin{align} & (\mathcal{X}^t)_{\tau _j} + t \langle \widehat{\Gamma} _j+
( \widehat{\delta } _ k \partial _{p_k}{\beta } _ {j } -\widehat{\delta } _ j \partial _{p_j}{\beta } _ {k } ) \Diamond _k P R, (\mathcal{X}^t)_{R}
\rangle =-\beta _j , \label{eq:fred20b} \\& (\mathcal{X}^t)_{R} +t{\mathcal L}(\mathcal{X}^t)_{R} =(Y)_{R}
\ ,\text{where for $X\in N_g^\perp (\mathcal{H}_{p_0}^*)$} \label{eq:fred20t}\\& {\mathcal L}X:= P ( {p}_0)J\left [ 2 \Gamma X +
\langle \widetilde{\beta} _j , X\rangle P^* \Diamond _j P R - \langle P^* \Diamond _j P R ,X\rangle \widetilde{\beta} _j \right ] \label{eq:fred20} .\end{align}
\eqref{eq:fred20} implies the following lemma.
\begin{lemma}\label{lem:fred11}
We have, summing on repeated indexes, with $i$ varying in some finite set,
\begin{equation} \label{eq:fred11} \begin{aligned}
&{\mathcal L}X = {\mathcal A}_j(X) J\Diamond _j R +{\mathcal B}_i(X) \Psi _i
\end{aligned}
\end{equation}
where: $\Psi _i = \mathbf{S}^{0,0}(p-p_0,R)$;
for $L= {\mathcal A}_j, {\mathcal B}_i$, we have
$ L \in C^\infty (\mathcal{U}_{\mathbf{d}} , B(L^2 ,\R ))$ with
\begin{equation} \label{eq:fred12} \begin{aligned}
& L(X) = L _{j} \left \langle \Diamond _j R, X \right \rangle + \langle \widetilde{L} ,X \rangle ,
\end{aligned}
\end{equation}
where we have $ \widetilde{L} = \mathbf{S}^{1,0}(p-p_0,R)$ and $ L _{j}\in \resto ^{0,0}(p-p_0,R)$.
\end{lemma}
\proof Schematically, for $\widetilde{L} _i =\mathbf{S}^{0,0}(p-p_0,R)$ and $\Psi _i =\mathbf{S}^{0,0}(p-p_0,R)$ we have
\begin{equation*} \begin{aligned} & P(p) R =R -P_{N_g}(p ) R
=R+\sum _i
\langle \widetilde{L} _i,R \rangle \Psi _i \, , \\& P^*(p) \Diamond _k R =\Diamond _k R-P_{N_g}^*(p) \Diamond _k R = \Diamond _kR+\sum _i
\langle \widetilde{L} _i,R \rangle \Psi _i .\end{aligned}\end{equation*}
Then $(P^*(p) \Diamond _k P(p) -\Diamond _k )R=\mathbf{S}^{0,1}(p-p_0,R)$.
\noindent By the definition of $\widetilde{\beta} _j$ we have
\begin{equation} \begin{aligned} & \widetilde{\beta} _j =\sum _k \widehat{\delta} _j (\partial _{p_j}\beta _k) \Diamond _k R + \widehat{L} \\& \widehat{L} := \nabla _R \beta _j+ \frac 12 J^{-1} \widehat{\delta} _j \partial _{p_j} P(p) R +\sum _{k } \beta _k \partial _{p_j} ( P^*(p) \Diamond _k P(p) )R\\& -\sum _{k }\widehat{\delta} _j \partial _{p_j}\beta _k \left [ P_{N_g}^*(p)\Diamond _k P (p)R + \Diamond _k P_{N_g}(p) R \right ] ,
\end{aligned} \nonumber \end{equation}
where $\widehat{L}= \mathbf{S}^{0,1}(p-p_0,R)$.
\noindent We also have $ \Gamma X =\frac 12 J^{-1} (P_{N_g} (p_0)- P_{N_g} (p ) ) X =\sum _i
\langle \widetilde{L} _i, X \rangle \Psi _i $ with $\widetilde{L} _i= \mathbf{S}^{1,0}(p-p_0,R)$
and $\Psi _i = \mathbf{S}^{0,0}(p-p_0,R)$.
This yields the result.
\qed
\begin{lemma}
\label{lem:vectorfield} System \eqref{eq:fred20b}--\eqref{eq:fred20}
admits exactly one solution $\mathcal{X}^t$.
For $\mathcal{A }_j=\mathcal{R}^{0,2}_{n,\infty}(t,p-p_0,R)$, $\mathcal{D } =\mathbf{S}^{1,1}_{n,\infty}(t,p-p_0,R)$ with $|t|<3$, we have
\begin{equation}\label{eq:quasilin1}
\begin{aligned} &
(\mathcal{X}^t)_R = \mathcal{A }_j J\Diamond _j R + \mathcal{D} . \end{aligned}
\end{equation}
\end{lemma}
\proof Recall $Y$ defined by $i_Y\Omega _0 =-\alpha $. By \eqref{eq:Yvec} with $\widetilde{\mathcal{A }}_j=\mathcal{R}^{0,2}_{n,\infty}(p-p_0,R)$ and $\widetilde{\mathcal{D }} =\mathbf{S}^{1,1}_{n,\infty}(p-p_0,R)$
we have $ ({Y})_{R} = \widetilde{\mathcal{A }}_jJ\Diamond _j R + \widetilde{\mathcal{D} } $. By
$ (\mathcal{X}^t)_{R} +t{\mathcal L}(\mathcal{X}^t)_{R} =(Y)_{R}$ and
Lemma \ref{lem:fred11} this implies for $X=(\mathcal{X}^t)_{R}$
\begin{equation} \begin{aligned} & \langle \Diamond _k R, X\rangle +t\mathcal{B}_i(X)\langle \Diamond _k R, \Psi _i \rangle = \langle \Diamond _k R, (Y)_{R} \rangle \\& \langle \widetilde{L}, X\rangle +t\mathcal{A }_j(X) \langle \widetilde{L}, J\Diamond _j R\rangle +t\mathcal{B}_i(X)\langle \widetilde{L}, \Psi _i \rangle = \langle \widetilde{L}, (Y)_{R} \rangle ,\end{aligned}\nonumber
\end{equation}
as $L$ runs through all the $L= {\mathcal A}_j, {\mathcal B}_i$. Taking appropriate linear combinations of these
equations with the coefficients $L_j$ of $L= {\mathcal A}_j, {\mathcal B}_i$, see Lemma \ref{lem:fred11},
for a matrix $\textbf{R}^{0,1}(p-p_0, R)$ whose coefficients are $\resto^{0,1}(p-p_0, R)$,
we get $$(1+t\textbf{R}^{0,1}(p-p_0, R)) \begin{pmatrix} \mathcal{A }_j((\mathcal{X}^t)_{R}) \\ \mathcal{B}_i((\mathcal{X}^t)_{R})\end{pmatrix} = \begin{pmatrix} \mathcal{A }_j((Y)_{R}) \\ \mathcal{B}_i((Y)_{R})\end{pmatrix}.$$
Then we get
\begin{equation} \label{eq:fredcorr1} \begin{aligned} & \begin{pmatrix} \mathcal{A }_j((\mathcal{X}^t)_{R}) \\ \mathcal{B}_i((\mathcal{X}^t)_{R})\end{pmatrix} =(1+t\textbf{R}^{0,1}(p-p_0, R))^{-1} \begin{pmatrix} \mathcal{A }_j((Y)_{R}) \\ \mathcal{B}_i((Y)_{R})\end{pmatrix}. \end{aligned}
\end{equation}
Using the left hand side of
\eqref{eq:fredcorr1} set
\begin{equation}\label{eq:fredcorr2}
{\mathcal L}(\mathcal{X}^t)_{R} := {\mathcal A}_j((\mathcal{X}^t)_{R}) J\Diamond _j R +{\mathcal B}_i((\mathcal{X}^t)_{R}) \Psi _i.
\end{equation}
The rhs of \eqref{eq:fredcorr2} satisfies the properties stated for the rhs of
\eqref{eq:quasilin1}. Finally set
$ (\mathcal{X}^t)_{R} :=(Y)_{R}- t{\mathcal L}(\mathcal{X}^t)_{R}$.
This is a solution of \eqref{eq:fred20t}. It is elementary to see from the argument that such solution is unique and that it satisfies
the properties of the statement. \qed
Turning to coordinates $(\tau , \Pi , R)$ and by
Lemma \ref{lem:var} we conclude what follows.
\begin{lemma}\label{lem:fred12}
Consider the coordinate system $(\tau ,\Pi ,R)$.
For $G$ any of the $\mathcal{A} _{j}$, $ \mathcal{D} $ in Lemma
\ref{lem:vectorfield},
we have $G=G ( \Pi , \Pi (R),R )$, with $G ( \Pi , \varrho ,R )$
smooth w.r.t. $ (\Pi,\varrho ,R)\in \mathcal{U}_{ \textbf{d}}
$, with $\mathcal{U}_{ \textbf{d}}$ formed by the $ (\Pi,\varrho ,R)\in \R ^{2n_0}
\times (\Sigma _{\textbf{d}}\cap N^\perp _g (\mathcal{H}_{p_0}))$
defined by the inequalities $ \|R\| _{\Sigma _{-n}}\le \varepsilon $,
$|\varrho | \le \varepsilon $ and
$| \Pi -p_0|\le \varepsilon $ for $\varepsilon >0$ small enough.
\end{lemma}
\subsection{Flows}
\label{subsec:ode}
The following lemma is repeatedly used in the sequel, see Lemma 3.24
\cite{bambusi}.
\begin{lemma} \label{lem:ODE} Below we pick $r,M,M_0,s,s',k,l\in \N\cup \{ 0 \}$ with $1\le l\le M$.
Consider a system
\begin{equation} \label{eq:ODE}\begin{aligned} &
\dot \tau _j = T_j (t,\Pi , \Pi (R), R ) \ , \quad \dot \Pi _j =0 \ , \\& \dot R
= \mathcal{A }_j(t,\Pi , \Pi (R), R ) J\Diamond _j R + \mathcal{D}(t,\Pi , \Pi (R), R )
,
\end{aligned} \end{equation}
where we assume what follows.
\begin{itemize}
\item $P_{N _g ( p_0)} (\mathcal{A }_j J\Diamond _j R + \mathcal{D}) \equiv 0$.
\item At $\Pi =p_0$, dropping the dependence on $\Pi$ and for $\U _{-r}$ a neighborhood
of 0 in $\Ph ^{-r}$, we have
$\mathcal{A }(t, \varrho ,R ) \in C^M ((-3,3)\times \U _{-r}, \R ^{n_0} )$ and $ \mathcal{D}(t, \varrho ,R ) \in C^M ( (-3,3)\times\U _{-r}, \Sigma _{r} )$
\item In $ (-3,3)\times \U _{-r}$ for a fixed $i$ in $ \{ 0,1\}$, and a fixed $C_r$, we have:
\begin{equation} \label{eq:symbol} \begin{aligned} & | \mathcal{A }(t, \varrho ,R ) |\le C \| R\| _{\Sigma _{-r}}^{M_0+1} , \\& \| \mathcal{D }(t, \varrho ,R )\| _{\Sigma _{r}} \le C (|\varrho| +\| R\| _{\Sigma _{-r}})^i \| R\| _{\Sigma _{-r}}^{M_0 } .
\end{aligned} \end{equation}
\end{itemize}
Let $k\in \Z\cap [0,r-(l+1)\textbf{d}]$ and set for $s^{\prime \prime}\ge \textbf{d}$ (or $s^{\prime \prime}\ge \textbf{d}/2$ if $\textbf{d}/2\in \N $)
\begin{equation} \label{eq:domain0} \begin{aligned} & \U _{\varepsilon _1,k}^{s^{\prime \prime}} :=
\{ (\tau, \Pi , R) \in \widetilde{ {\Ph}}^{s^{\prime \prime}} \ : \ \Pi =p_0 \ , \ \| R \| _{\Sigma _{-k }} + |\Pi (R)| \le \varepsilon _1\}.
\end{aligned} \end{equation}
Then for $\varepsilon _1>0$ small enough,
the initial value
problem associated to \eqref{eq:ODE} for $\Pi =p_0$
defines a flow $\mathfrak{F} ^t =( \mathfrak{F} ^t _{\tau } , \mathfrak{F} ^t _{R} )$
for $t\in [-2, 2]$ in $\U _{\varepsilon _1,k}^{\textbf{d}}$. In particular
for $\Pi =p_0$, for $R$ in
a neighborhood $ B_{\Sigma _{-k}}$ of
0 in $\Sigma _{-k}$ and $\Pi (R) $ in a neighborhood $ B_{\R ^{ n_0}} $ of
0 in $\R ^{ n_0}$, we have
\begin{equation} \label{eq:ODE1}\begin{aligned} &
\mathfrak{F} ^t _{R} ( \Pi (R), R) = e^{Jq(t, \Pi (R), R)\cdot \Diamond } ( R+ \textbf{S} (t, \Pi (R), R ))
,
\end{aligned} \end{equation}
\begin{equation} \label{eq:ODEpr210}\begin{aligned} \text{with } &
\textbf{S} \in C^l((-2,2)
\times B_{\R ^{ n_0}}\times B_{\Sigma _{-k}} , \Sigma _{r- (l+1)\textbf{d}}
) \\& {q} \in C^l((-2,2)
\times B_{\R ^{ n_0}} \times B_{\Sigma _{-k}}, \R ^{ n_0}
).
\end{aligned} \end{equation}
For fixed $C>0$ we have
\begin{equation} \label{eq:symbol1} \begin{aligned} & | q (t,\varrho ,R ) |\le C \| R\| _{\Sigma _{(l+1)\textbf{d}-r}}^{M_0+1} \, , \\& \| \textbf{S} (t,\varrho ,R ) \| _{\Sigma _{r- (l+1)\textbf{d}}} \le C (|\varrho| +\| R\| _{\Sigma _{(l+1)\textbf{d}-r}})^i \| R\| _{\Sigma _{(l+1)\textbf{d}-r}}^{M_0 } .
\end{aligned} \end{equation}
Furthermore we have $\textbf{S}= \textbf{S}_1+ \textbf{S} _2$ with
\begin{equation} \label{eq:duhamel} \begin{aligned} & \textbf{S}_1 (t, \Pi (R), R ) =
\int _0^t \mathcal{D }(t',\Pi (R (t') ) ,R (t') ) dt'\\& \| \textbf{S}_2 (t, \varrho, R ) \| _{\Sigma _{s}}
\le C\| R \| _{\Sigma _{(l+1)\textbf{d}-r} } ^{2M_0+1} (|\varrho | +\| R \| _{\Sigma _{(l+1)\textbf{d}-r}} )^i .
\end{aligned} \end{equation}
For $r-(l+1)\textbf{d}\ge s' \ge s+l\mathbf{d}\ge l\mathbf{d}$ and $k\in \Z\cap [0,r-(l+1)\textbf{d}]$ and
for $ \varepsilon _1>0$ sufficiently small, we have
\begin{equation} \label{eq:reg1}\begin{aligned} &\mathfrak{F} ^t \in C^l((-2,2)\times \U _{\varepsilon _1,k}^{s'}
, \widetilde{\Ph} ^{s }
)
.
\end{aligned} \end{equation}
Furthermore, there exists $ \varepsilon _2>0$
such that
\begin{equation} \label{eq:main1}\begin{aligned} & \mathfrak{F} ^t( \U _{\varepsilon _2,k}^{s'})\subset \U _{\varepsilon _1,k}^{s'} \text{ for all $|t|\le 2$
.}
\end{aligned} \end{equation}
\noindent We have \begin{equation} \label{eq:ODE11}\begin{aligned} &
\mathfrak{F} ^t (e^{J\tau \cdot \Diamond } U) \equiv e^{J\tau \cdot \Diamond }\mathfrak{F} ^t ( U)
.
\end{aligned} \end{equation}
\end{lemma}
\proof
It is enough to focus on the equation for $R$. Set $S= e^{-Jq \cdot \Diamond }R$ for $q\in \R ^{n_0}$. Then consider the following system:
\begin{equation} \label{eq:ODEpr}\begin{aligned} &
\dot S
= e^{-Jq \cdot \Diamond } \mathcal{D}( t,\varrho ,e^{ Jq \cdot \Diamond }S ) \ ,
\\& \dot q = \mathcal{A } (t,\varrho ,e^{ Jq \cdot \Diamond }S ) \quad , \quad q(0)=0 , \\& \dot \varrho _j= \langle S, e^{-Jq \cdot \Diamond } \Diamond _j\mathcal{D}( t, \varrho , e^{ Jq \cdot \Diamond }S ) \rangle \ .
\end{aligned} \end{equation}
For $ l\le M$ and $ k, s^{\prime \prime}\in [0, r-(l+1)\textbf{d}]$ the field in \eqref{eq:ODEpr} is $C^l( (-3,3)\times \U _{-k}, \Sigma _{ s^{\prime \prime}}
\times \R ^{2n_0})$ with
$\U _{-k}\subset
\Sigma _{-k}
\times \R ^{2n_0} $ a neighborhood of the equilibrium 0.
This follows from the fact that $(q,X)\to e^{ Jq \cdot \Diamond }X$
is in $C^{l} ( \R ^{ n_0} \times \Sigma _{ \ell} ,\Sigma _{ \ell -l\mathbf{d}} )$ for all $\ell \in \Z$ and from the hypotheses on $\mathcal{A}$ and $ \mathcal{D}$. For example
\begin{equation*}\begin{aligned} &
(t,q,\varrho ,S) {\rightarrow} e^{-Jq \cdot \Diamond } \Diamond _j \mathcal{D}( t, \varrho , e^{ Jq \cdot \Diamond }S ) \in C^l ( (-3,3)\times \R ^{2n_0} \times \Sigma _{ l\mathbf{d}-r} , \Sigma _{ r-(l+1)\mathbf{d}}
), \end{aligned}
\end{equation*}
(more precisely for $(q,\varrho ,S)$ in a neighborhood of the origin). So
\begin{equation*}\begin{aligned} &
(t,q,\varrho ,S) {\rightarrow} \langle S, e^{-Jq \cdot \Diamond } \Diamond _j \mathcal{D}( t, \varrho , e^{ Jq \cdot \Diamond }S )\rangle , \end{aligned}
\end{equation*}
is in
$ C^l ( (-3,3)\times \R ^{2n_0} \times \Sigma _{ -k} , \R
)$ for $k\le r - (l+1)\mathbf{d} $ (for $(q,\varrho ,S)$ near origin).
\noindent
For $l\ge 1$ we can apply to \eqref{eq:ODEpr} standard
theory of ODE's to conclude that there are neighborhoods of the origin $B_{\R ^{2n_0}}\subset \R ^{2n_0}$ and $B_{\Sigma _{-k}}\subset \Sigma _{-k} $
such that the flow is of the form
\begin{equation} \label{eq:ODEpr1}\begin{aligned} &
S(t)
= R+ \textbf{S} (t, \varrho , R ) \ , \quad \textbf{S} (0, \varrho ,R) =0\ , \\& q(t)= {q} (t,\varrho , R ) \ , \quad {q}(0, \varrho , R) =0\ , \\& \varrho (t)=\varrho + \overline{\varrho} (t, \varrho , R) \ , \quad \overline{\varrho} (0,\varrho , R ) =0\ ,
\end{aligned} \end{equation}
\begin{equation} \label{eq:ODEpr2}\begin{aligned} \text{with } &
\textbf{S} \in C^l((-2,2)
\times B_{\R ^{ n_0}}\times B_{\Sigma _{-k}} , \Sigma _{r- (l+1)\textbf{d}}
) \\& \overline{\varrho},{q} (t,\varrho , R ) \in C^l((-2,2)
\times B_{\R ^{ n_0}} \times B_{\Sigma _{-k}}, \R ^{ n_0}
).
\end{aligned} \end{equation}
\noindent For $S\in \Sigma _{ {\textbf{d}} }\cap B_{\Sigma _{-k}}$ and $S(0)=S$,
choosing $s^{\prime \prime}\ge \textbf{d}$ we have $S(t)\in\Sigma _ {\textbf{d}} $
with $\Pi (S(t))=\varrho (t)$ for $\varrho (0)=\varrho =\Pi (S)$. Then \eqref{eq:ODEpr2} yields
\eqref{eq:ODEpr210} (we can replace $\Sigma _{ {\textbf{d}} }$ with $\Sigma _{ \frac{{\textbf{d}} }{2} }$ if $ \frac{\textbf{d} }{2}\in \N$). \eqref{eq:ODE1} and \eqref{eq:ODEpr210} yield \eqref{eq:reg1}.
\noindent We have for $R(0)=R$
\begin{equation} \label{eq:duh2}\begin{aligned} & R(t)= e^{ Jq (t) \cdot \Diamond } (R+\int _0^t
e^{- Jq (t') \cdot \Diamond } \mathcal{D}( t', \varrho (t'),R (t') ) dt' ).
\end{aligned} \end{equation}
By (A6), for $\epsilon =0$, and by \eqref{eq:symbol}, for $ |s ^{\prime\prime}|\le r-(l+1)\textbf{d} $ we have
\begin{equation} \label{eq:duh21}\begin{aligned} & \| R(t) \| _{\Sigma _{ s ^{\prime\prime}} }\le C\| R \| _{\Sigma_{ s ^{\prime\prime}} }+C\int _0^t \| \mathcal{D}( t', \varrho (t),R (t') ) \| _{\Sigma _{ r} } dt' \\& \le C\| R \| _{\Sigma _{ s ^{\prime\prime}} }
+C\int _0^t \| R (t')\| _{\Sigma _{-r}}^{ M_0 } (|\varrho (t')| +\| R (t')\| _{\Sigma _{-r}}) ^i dt' \\& \le C\| R \| _{\Sigma _{ s ^{\prime\prime}} }
+C\int _0^t \| R (t')\| _{\Sigma _{ s ^{\prime\prime}} }^{ M_0 } (|\varrho (t')| +\| R (t')\|_{\Sigma _{ s ^{\prime\prime}} }) ^i dt',
\end{aligned} \end{equation}
with the caveat that the second line is purely formal and is used to
get the third line, where the integrand is continuous.
Proceeding similarly, for $\varrho (0) = \varrho$
\begin{equation} \label{eq:duh22}\begin{aligned} & | \varrho (t) - \varrho |
\le \int _0^t | \langle R(t'), \Diamond \mathcal{D}( t', R (t'), \varrho (t') ) \rangle | dt'\\& \le \int _0^t \| R (t')\| _{\Sigma _{(l+1)\textbf{d} -r}} \| \mathcal{D}( t', \varrho (t),R (t') ) \| _{\Sigma _{r-l\mathbf{d}}}
dt'\\&
\le
C\int _0^t \| R (t')\| _{\Sigma _{(l+1)\textbf{d} -r}} ^{ M_0+1 }
(|\varrho (t')| +\| R (t')\| _{\Sigma _{(l+1)\textbf{d} -r}}) ^i dt'.
\end{aligned} \end{equation}
So for $| s ^{\prime\prime}|\le r-(l+1)\textbf{d}$, using the continuity
in $t'$ of the integrals in the last lines of \eqref{eq:duh21} and \eqref{eq:duh22},
by the Gronwall inequality there is a fixed $C$ such that for all $|t|\le 2$ we have
\begin{align} \label{eq:gronwall0} & \| R(t) \| _{\Sigma _{ s ^{\prime\prime}} }\le C\| R \| _{\Sigma _{ s ^{\prime\prime}} } \, , \\& | \varrho (t) - \varrho | \le C \| R \| _{\Sigma _{(l+1)\textbf{d} -r}}^{ M_0+1 }
(|\varrho | +\| R \| _{\Sigma _{(l+1)\textbf{d} -r}}) ^i . \label{eq:gronwall1}
\end{align}
By
\eqref{eq:gronwall0} for $s ^{\prime\prime}=s'$ and $s ^{\prime\prime}=-k$ and by
$ | \varrho (t) - \varrho | \le C \| R \| _{\Sigma _{-k}}^{ M_0+1 }
(|\varrho | +\| R \| _{\Sigma _{-k}}) ^i$, we get $\mathfrak{F} ^t( \U _ {\varepsilon _2,k}^{s^{\prime }})\subset \U _ {\varepsilon _1,k}^{s^{\prime }}$ for all $|t|\le 2$ for $\varepsilon _1 \gg \varepsilon _2,$ that is \eqref{eq:main1}.
\noindent We have \begin{equation*} \begin{aligned} & S(t,\varrho ,R)= \int _0^t
e^{- Jq (t') \cdot \Diamond } \mathcal{D}( t', \varrho (t'),R (t') ) dt' ),
\end{aligned} \end{equation*}
Proceeding as for \eqref{eq:duh21} and using \eqref{eq:gronwall0}--\eqref{eq:gronwall1}
we get the estimate for $\textbf{S}$ in
\eqref{eq:symbol1}. The estimate on $q$ is obtained similarly
integrating the second equation in \eqref{eq:ODEpr1}.
\noindent We have
\begin{equation} \label{eq:duh3}\begin{aligned} & \textbf{S}_2 (t, R, \varrho ) = \int _0^1 dt^{\prime \prime }\int _0^t
e^{- t^{\prime \prime }q (t') \cdot \Diamond } q (t') \cdot \Diamond \mathcal{D}( t', \varrho (t),R (t') ) dt'
\end{aligned} \end{equation}
Then by \eqref{eq:gronwall0}--\eqref{eq:gronwall1} we get
\begin{equation} \label{eq:duh4}\begin{aligned} & \| \textbf{S}_2 (t, R, \varrho ) \| _{\Sigma _{r-\textbf{d}}}\le C ^{\prime \prime} \int _0^t
|q (t') | \| \mathcal{D}(t', \varrho (t),R (t')) \| _{\Sigma _{r-\textbf{d} }} dt' \\& \le C' \int _0^t
\| R (t')\| _{\Sigma _{(l+1)\textbf{d}-r} }^{ 2M_0+1 } (|\varrho (t')| +\| R (t')\| _{\Sigma _{(l+1)\textbf{d}-r} })^i dt' \\&\le C
\| R \| _{\Sigma _{(l+1)\textbf{d}-r} }^{2M_0+1} (|\varrho | +\| R \| _{\Sigma _{(l+1)\textbf{d}-r}} )^i.
\end{aligned} \end{equation}
This yields \eqref{eq:duhamel}. \eqref{eq:reg1} follows by \eqref{eq:ODE1}--\eqref{eq:ODEpr210}.
Finally,
\eqref{eq:ODE11} follows immediately from \eqref{eq:ODE1}.
\qed
\begin{lemma}
\label{lem:ODEdomains} Assume hypotheses and conclusions of Lemma \ref{lem:ODE}.
Consider the flow of system \eqref{eq:ODEpr} for $\Pi =p_0$ . Denote the flow in the space with variables $\{ ( \varrho ,R)\}$ by $\mathfrak{F} ^t =( \mathfrak{F} ^t _{\varrho } , \mathfrak{F} ^t _{R} )$. Then we have
\begin{equation} \label{eq:ODEdomains1} \begin{aligned} &
\mathfrak{F} ^t _{R} ( \varrho ,R ) = e^{Jq(t, \varrho ,R)\cdot \Diamond } ( R+ \textbf{S} (t, \varrho ,R )) \\& \mathfrak{F} ^t _{\varrho} ( \varrho ,R ) = \varrho + \overline{\varrho} (t, \varrho , R)
.
\end{aligned} \end{equation}
Furthermore, the following facts hold.
\begin{itemize}
\item[(1)]
Let $k \in \Z \cap [0,r-(l+1)\textbf{d}]$ and $h\ge \max\{ k +l\textbf{d},(2 l+1)\textbf{d}-r\}$. Then we have
$\mathfrak{F} ^t \in C^l((-2,2)\times \U _{-k }
, \Ph ^{-h }
)
$
for a
neighborhood of the origin
$\U _{-k }\subset \Ph ^{-k }$.
\item[(2)] Let $h$ and $k$ be like above with $h\le r-( l+1)\textbf{d}$. Then given a function $\resto ^{a,b}_{h, l}( \varrho ,R)$, we have $\resto ^{a,b}_{h, l}\circ \mathfrak{F} ^t=\resto ^{a,b}_{k, l}( t, \varrho ,R) $ and given a function $\textbf{S}^{a,b}_{h, l}( \varrho ,R)$, we have $\textbf{S} ^{a,b}_{h, l}\circ \mathfrak{F} ^t=\textbf{S} ^{a,b}_{k, l} ( t,\varrho ,R)$.
\end{itemize}
\end{lemma}
\proof \eqref{eq:ODEdomains1} follows by \eqref{eq:ODEpr1}.
By \eqref{eq:ODEpr2} we have\begin{equation*} \begin{aligned} &
\textbf{S} \in C^l((-2,2)
\times \U _{-k } , \Sigma _{r- (l+1)\textbf{d}}
) \ , \quad q \text{ and }\mathfrak{F} ^t _{\varrho} \in C^l((-2,2)
\times \U _{-k }, \R ^{ n_0}
).
\end{aligned} \end{equation*}
By the above formulas we have $\mathfrak{F} ^t _{R}\in C^l((-2,2)
\times \U _{-k }, \Sigma _{r- (2l+1)\textbf{d}} \cap \Sigma _{-k- l\textbf{d}}
).$ This yields $\mathfrak{F} ^t _{R}\in C^l((-2,2)
\times \U _{-k }, \Sigma _{-h}
) $ and yields Claim (1).
\noindent By Claim (1), $\resto ^{a,b}_{h, l}\circ \mathfrak{F} ^t \in C^l( (-2,2)
\times \U _{-k }, \R ^{ n_0}
)$. Let $(\varrho ^t , R^t)=\mathfrak{F} ^t(\varrho , R )$. Then
\begin{equation*} \begin{aligned} &
|\resto ^{a,b}_{h, l}\circ \mathfrak{F} ^t(\varrho , R )|=| \resto ^{a,b}_{h, l}(\varrho ^t , R^t )| \le C' \| R^t\| _{\Sigma _{-h}}^b (\| R^t\| _{\Sigma _{-h}}+|\varrho ^t|)^{a}\\& \le C \| R\| _{\Sigma _{-h}}^b (\| R\| _{\Sigma _{-h}}+|\varrho |)^{a}\le C \| R\| _{\Sigma _{-k}}^b (\| R\| _{\Sigma _{-k}}+|\varrho |)^{a},
\end{aligned} \end{equation*}
where the first inequality uses Definition \eqref{eq:scalSymb}, the second uses \eqref{eq:gronwall0}--\eqref{eq:gronwall1} for $s^{\prime\prime}=-h$ and the last is obvious. Similarly by Claim (1), $\textbf{S} ^{a,b}_{h, l}\circ \mathfrak{F} ^t \in C^l( (-2,2)
\times \U _{-k }, \Sigma _h
)\subset C^l( (-2,2)
\times \U _{-k }, \Sigma _k )$ and
\begin{equation*} \begin{aligned} &
\| \textbf{S} ^{a,b}_{h, l}(\varrho ^t , R^t )\| _{ \Sigma _k}\le \| \textbf{S} ^{a,b}_{h, l}(\varrho ^t , R^t )\| _{ \Sigma _h} \le C' \| R^t\| _{\Sigma _{-h}}^b (\| R^t\| _{\Sigma _{-h}}+|\varrho ^t|)^{a}\\& \le C \| R\| _{\Sigma _{-h}}^b (\| R\| _{\Sigma _{-h}}+|\varrho |)^{a}\le C \| R\| _{\Sigma _{-k}}^b (\| R\| _{\Sigma _{-k}}+|\varrho |)^{a}.
\end{aligned} \end{equation*}
\qed
\bigskip To prove Theorem \ref{th:main} we will need more information on $(\Pi (R (1)) , R(1)) $. This is provided by the following
lemma.
\begin{lemma} \label{lem:ODEbis} Consider, for $\mathcal{D}$ the function in
\eqref{eq:ODE} at $\Pi =p_0$, the system
\begin{equation} \label{eq:ODEbis}\begin{aligned} & \dot S(t)
= \mathcal{D}( t, \Pi (R_0),S(t) ) \, , \quad S(0)=R_0 .
\end{aligned} \end{equation}
Then for $S'=S(1)$ and for $R'=R(1)$ with $R(t)$ the solution of \eqref{eq:ODE}
with $R(0)=R_0$, we have (same indexes of Lemma \ref{lem:ODE})
\begin{equation} \label{eq:ODE1bis}\begin{aligned} &
\| R'-S'\| _{\Sigma _{-s' }} \le C \| R _0\| _{\Sigma _{ -s }}^{M_0+2} \, , \\&
\Pi (R')- \Pi (S')=\resto ^{i,2M_0+1}_{s , l}
( \Pi (R_0), R_0).
\end{aligned} \end{equation}
\end{lemma}
\proof Recall that for $\varrho =\Pi (R)$ we have $\dot \varrho =\langle R , \Diamond \mathcal{D}( t, \varrho ,R ) \rangle .$ Similarly, for $\sigma =\Pi (S)$ we have $\dot \sigma = \langle S,\Diamond \mathcal{D}(t, \varrho _0 ,S) \rangle $, where $\varrho _0=\Pi (R_0)$. So we have
\begin{equation*} \label{eq:ODE2bis}\begin{aligned} \dot \varrho -\dot \sigma & =
\langle R, \Diamond \mathcal{D}( t,\varrho ,R ) \rangle - \langle S,\Diamond \mathcal{D}(t, \varrho _0,S ) \rangle \\& =\langle R -S, \Diamond \mathcal{D}(t, \varrho ,R ) \rangle +\langle S, \Diamond ( \mathcal{D}(t, \varrho _0 ,S ) -\mathcal{D}(t, \varrho ,R )) \rangle .
\end{aligned} \end{equation*}
By \eqref{eq:symbol} for fixed constants
and using $s'\le r- \mathbf{d}$, we have
\begin{equation*} \label{eq:ODE3bis}\begin{aligned} & |\dot \varrho -\dot \sigma |\lesssim \| R-S\| _{\Sigma _{-s ' }} \| \mathcal{D}( t, \varrho ,R ) \| _{\Sigma _{r }} + \| S\| _{\Sigma _{ -s' }}
\| \mathcal{D}( t, \varrho _0 ,S ) -\mathcal{D}(t, \varrho ,R ) \| _{\Sigma _{r }}
\\& \lesssim \| R-S\| _{\Sigma _{-s' }} \| R \| _{\Sigma _{ -s }}^{M_0} (|\varrho |+ \| R \| _{\Sigma _{ -s ' }}) ^i+| \varrho - \varrho _0 | \ \| S\| _{\Sigma _{ -s ' }} \|(R, S)\| _{\Sigma _{ -s' }}^{M_0}
\\& + \| R-S\| _{\Sigma _{-s' }} \| S\| _{\Sigma _{ -s ' }} \|(R, S)\| _{\Sigma _{ -s'}}^{M_0-1} (| (\varrho , \varrho _0 ) |+ \|(R, S)\| _{\Sigma _{ -s'}})^i.
\end{aligned} \end{equation*}
We have
$ \dot R -\dot S = \mathcal{D}(t, \varrho ,R ) -\mathcal{D}(t, \varrho _0 ,S ) + J \mathcal A(t, \varrho ,R ) (t, \varrho ,R ) \cdot \Diamond R$
and hence for fixed constants we have, using $s\le s'-\mathbf{d}$,
\begin{equation*} \label{eq:ODE4bis}\begin{aligned} & \| R-S\| _{\Sigma _{-s ' }} \le \int _0^t [ \|
\mathcal {D}( \varrho , R) -\mathcal{D}( \varrho _0 ,S ) \| _{\Sigma _{-s' }} + |{\mathcal A} | \| R \| _{\Sigma _{-s }} ]dt' \\& \lesssim \int _0^t \big [
\| R-S\| _{\Sigma _{-s' }} \|(R, S)\| _{\Sigma _{ -s' }}^{M_0-1} (| (\varrho ,\varrho _0 ) |+ \|(R, S)\| _{\Sigma _{ -s'}})^i \\& + | \varrho - \varrho _0 | \ \|(R, S)\| _{\Sigma _{ -s' }}^{M_0}
+ \| R \| _{\Sigma _{ -s }}^{M_0+2} \big ] dt' .
\end{aligned} \end{equation*}
Recall that $ | \varrho - \varrho _0 | \le C \| R _0 \| _{\Sigma _{(l+1)\textbf{d}-r }} ^{M_0+1}(|\varrho _0|+ \| R_0 \| _{\Sigma _{ (l+1)\textbf{d}-r }}) ^i $ by \eqref{eq:gronwall1}, that $s<r-(l+1)\textbf{d}$ and that we have \eqref{eq:gronwall0} for $s^{\prime \prime}=-s,-s'$.
Then by Gronwall inequality, the above inequalities yield
\begin{equation} \label{eq:ODE6bis}\begin{aligned} &
\| R(t)-S(t)\| _{\Sigma _{-s' }} \le C \| R _0\| _{\Sigma _{ -s }}^{M_0+2} \\&
| \varrho (t)- \sigma (t)|\le C \| R _0\| _{\Sigma _{-s }}^{ 2M_0+1 } (|\varrho _0|+ \| R_0 \| _{\Sigma _{ -s }}) ^i .
\end{aligned} \end{equation}
This yields the bounds implicit in \eqref{eq:ODE1bis}. The regularity follows from Lemma \ref{lem:ODE}.
\qed
\subsection{Darboux Theorem: end of the proof}
\label{subsec:darboux}
Formally the proof should follow by $ i_{\mathcal{X} ^t} \Omega _t=-\alpha$, where $ \Omega _t =(1-t)\Omega _0+t
\Omega$, and by
\begin{equation}\label{eq:fdarboux} \begin{aligned} &
\frac{d}{dt}
\left ( \mathfrak{F}_{ t}^*\Omega _t\right ) = \mathfrak{F}_{ t}^*
\left (L_{\mathcal{ X}_t} \Omega _t+\frac{d}{dt}\Omega _t\right ) = \mathfrak{F}_{ t}^*
\left ( d i_{\mathcal{X} ^t} \Omega _t+d\alpha \right ) =0.
\end{aligned}
\end{equation}
But while for \cite{boussaidcuccagna,Cu2} the above formal
computation falls within the classical framework of flows, fields and
differential forms, in the case of \cite{bambusi,Cu1}
this is not rigorous. In order to justify rigorously this computation, we will consider
first a regularization of system \eqref{eq:ODE}.
\begin{lemma} \label{lem:moll} Consider the system
\begin{equation} \label{eq:ODEmoll}\begin{aligned} &
\dot \tau _j = T_j (t,\Pi , \Pi (R), R ) \ , \quad \dot \Pi _j =0 \ , \\& \dot R
= \mathcal{A }_j(t,\Pi , \Pi (R), R ) J\langle \epsilon \Diamond\rangle ^{-2}\Diamond _j R + \mathcal{D}_ \epsilon (t,\Pi , \Pi (R), R )
,
\end{aligned} \end{equation}
where $\mathcal{D}_ \epsilon =\mathcal{D}+\mathcal{A }_j P_{N _g ( p_0)} J\Diamond _j ( 1- \langle \epsilon \Diamond\rangle ^{-2})R . $
\begin{itemize}
\item[(1)]
For $|\epsilon |\le 1$ system \eqref{eq:ODEmoll} satisfies all the conclusions of Lemma \ref{eq:ODEmoll},
if we replace $\Diamond $ in \eqref{eq:ODE1} with $\langle \epsilon \Diamond\rangle ^{-2}\Diamond $
(resp. ${\mathcal D}$ in \eqref{eq:duhamel} with ${\mathcal D} _\epsilon$),
with a fixed
choice of constants $\varepsilon _1 $, $\varepsilon _2 $, $C$,
and with a fixed choice of sets $B_{\R^{n_0}}$, $B_{\Sigma _{-s}}$.
\item[(2)] For ${\mathcal X}^t$ the vector field of \eqref{eq:ODE}, denote by ${\mathcal X}^t_\epsilon$
the vector field of \eqref{eq:ODEmoll}. Let
$ n' > n+\textbf{d}$ with $n,n'\in \N$. Then for $k\in \Z \cap [0,r]$
we have
\begin{equation} \label{eq:reg3}\begin{aligned} & \lim _{\epsilon \to 0}{\mathcal X}^t_\epsilon = {\mathcal X}^t \text{ in } C^{M}((-3,3)\times \U _ {\varepsilon _0,k}^{n'}
, \widetilde{ {\Ph} }^{n }
) \text{ uniformly locally}
,
\end{aligned} \end{equation}
that is uniformly on subsets of $(-3,3)\times \U ^{n'}_{\varepsilon _0,k}$ bounded in $(-3,3)\times \widetilde{ {\Ph} }^{n '}$.
\item[(3)] Denote by $\mathfrak{F} ^t_\epsilon =(\mathfrak{F} ^t_{\epsilon \tau},\mathfrak{F} ^t_{\epsilon R}) $ the flow associated to \eqref{eq:ODEmoll} at $\Pi =p_0$.
Let $s'$,$s$ and $k$ as in the statement of Lemma \ref{lem:ODE}. Then there is a pair $0<\varepsilon _ 1<\varepsilon _0$ such that
\begin{equation} \label{eq:reg2}\begin{aligned} & \lim _{\varepsilon \to 0}\mathfrak{F} ^t_\epsilon = \mathfrak{F} ^t \text{ in } C^{l-1}([-1,1]\times \U _ {\varepsilon _1,k}^{s^{\prime }}
, \U _ {\varepsilon _0,k}^{s }
) \text{ uniformly locally}
.
\end{aligned} \end{equation}
\end{itemize}
\end{lemma}
\proof For claim (1), it is enough to check that ${\mathcal D} _\epsilon$ satisfies
an estimate like the one of ${\mathcal D} $ in \eqref{eq:symbol1} for a fixed $C$ for all $|\epsilon |\le 1$.
Indeed, after this has been checked, the proof of Lemma \ref{eq:ODE} can be repeated verbatim,
exploiting (A6) for $\epsilon \neq 0$ and with $\Diamond$ replaced by $\langle \epsilon \Diamond\rangle ^{-2}\Diamond$.
\noindent The estimate on ${\mathcal D} _\epsilon$ needed for Claim (1)
follows by
the definition of ${\mathcal D} _\epsilon$ , by the estimate on ${\mathcal D} $, by
$P_{N _g ( p_0)}= \textbf{e}_a\langle \textbf{e}^*_a, \ \rangle $ (sum on repeated indexes) for Schwartz functions $\textbf{e}_a$ and $\textbf{e}^*_a$ and, for $n\in \N$ with $n-1\ge s+\textbf{d}$,
and by
\begin{equation}\label{eq:reg4} \begin{aligned} & \| P_{N _g ( p_0)}
J\Diamond _i ( 1- \langle \epsilon \Diamond\rangle ^{-2}) \| _{{B(\Sigma _{-r},\Sigma _{r}) }} \le
\| \textbf{e}_a \langle J\Diamond _i ( 1- \langle \epsilon \Diamond\rangle ^{-2})\textbf{e}^*_a, \ \rangle
\| _{{B(\Sigma _{-r},\Sigma _{r}) }}
\\& \le \| \textbf{e}_a\| _{\Sigma _r}
\| ( 1- \langle \epsilon \Diamond\rangle ^{-2}) \textbf{e}_a^*\| _{\Sigma _{r+\textbf{d}}}
\le C (\epsilon) \| \textbf{e}_a\| _{\Sigma _r} \ \| \textbf{e}_a^*\| _{\Sigma _{r'}}
\end{aligned} \end{equation}
$C (\epsilon)= \|
\Diamond ( 1- \langle \epsilon \Diamond\rangle ^{-2}) \| _{B(\Sigma _{r'},\Sigma _{r+\textbf{d}}) } $ is bounded by \eqref{eq:a71} for $|\epsilon |\le 1$ for any pair $(r',r)$ with $r'>r+\textbf{d}$.
\noindent We consider now Claim (2). We have
\begin{equation*} \begin{aligned}& {\mathcal X}^t -{\mathcal X}^t_\epsilon= \mathcal{A }_j(t,\varrho ,R)\left ( J ( 1- \langle \epsilon \Diamond\rangle ^{-2})\Diamond _j R - P_{N _g ( p_0)} J\Diamond _j ( 1- \langle \epsilon \Diamond\rangle ^{-2})R\right ).
\end{aligned}
\end{equation*}
We have $P_{N _g ( p_0)} J\Diamond _j ( 1- \langle \epsilon \Diamond\rangle ^{-2})R\stackrel{\epsilon\to 0}{\rightarrow} 0$ for $R\in \Sigma_{n'}$ for any $n'\in \Z$
because in fact
$C (\epsilon) \stackrel{\epsilon\to 0}{\rightarrow} 0$ by \eqref{eq:a72}, with $C (\epsilon)$ defined like above for any pair $( r',r)$ with $r'>r+\textbf{d}$.
\noindent Still by \eqref{eq:a72}, for $ n > n'+\textbf{d}$ and for $R\in \Sigma_{n'}$ we have by (A5)
\begin{equation} \label{eq:reg5} \begin{aligned} & \|
J\Diamond ( 1- \langle \epsilon \Diamond\rangle ^{-2}) R \| _{ \Sigma _n } \le
\|
\Diamond ( 1- \langle \epsilon \Diamond\rangle ^{-2}) \| _{B(\Sigma _{n'},\Sigma _n ) } \| R \| _{\Sigma_{n'} } \\& \le C \|
( 1- \langle \epsilon \Diamond\rangle ^{-2}) \| _{B(\Sigma _{n'} ,\Sigma _{n +\textbf{d}}) } \| R \| _{\Sigma_{n'} } \stackrel{\epsilon\to 0}{\rightarrow} 0
.
\end{aligned} \end{equation}
These facts yield \eqref{eq:reg3}.
\noindent We turn now to Claim (3) and to \eqref{eq:reg2}. By the Rellich criterion, the embedding
$\Sigma _a\hookrightarrow\Sigma _b$ for $a>b$ is compact. Hence also $\Ph ^a\hookrightarrow\Ph ^b$
is compact.
Then \eqref{eq:reg2} follows by the Ascoli--Arzela Theorem by a standard argument.
\qed
\begin{corollary}\label{cor:darboux} Consider \eqref{eq:ODE} defined by
the field
${\mathcal X}^t$ and consider indexes and notation
of Lemma \ref{lem:ODE} (in particular we have $M_0=1$ and $i=1$ in
\eqref{eq:symbol} and elsewhere; $r$ and $M$ can be arbitrary).
Consider $s'$,$s$ and $k$ as in \ref{lem:ODE}. Then for the map $\mathfrak{F} ^t \in C^l( \U _ {\varepsilon _1,k}^{s^{\prime }}
, \widetilde{\Ph} ^{s })$
derived
from \eqref{eq:reg1},
we have $ \mathfrak{F} ^{1*}\Omega =\Omega _0$.
\end{corollary}
\proof $\Omega _0$ is constant in the coordinate system $(\tau , \Pi , R)$ where $R\in N^\perp _g({\mathcal H}_{p_0}^*),$
with $\Omega _0=d\tau _j\wedge d\Pi _j +\langle J^{-1} \ , \ \rangle $,
where we apply $\langle J^{-1} \ , \ \rangle$ only to vectors in the $R$ space. Hence $\Omega _0$ is $C^\infty$
in $R\in L^2$, $\tau $ and $\Pi$, with values in
$B^2 (L^2, \R ) . $ From Lemma \ref{lem:dalpha2} we have that $d\alpha$, so also $\Omega $ by
$\Omega =\Omega _0+d\alpha $, belongs to $C^\infty ( \U _ {\varepsilon _0,k}^{s}, B^2 (\widetilde{ \Ph}, \R ))$ for an $\epsilon _0>0$, and so also to $C^\infty ( \U _ {\varepsilon _0,k}^{s }, B^2 (\widetilde{ \Ph}^s, \R ))$. Let now $r-(l+1)\textbf{d}\ge s'\ge s+l\mathbf{d}$ and $k\in \Z \cap [0,r-(l+1)\textbf{d}]$. Then for a fixed $0<\varepsilon _2\ll \varepsilon _1$
and for all $|\epsilon |\le 1$ we have
\begin{equation} \label{eq:darboux2} \begin{aligned} & \mathfrak{{F}}_{\epsilon} ^t \in C^l((-2,2)\times \U _ {\varepsilon _2,k}^{s^{\prime }}
, \U _ {\varepsilon _1,k}^{s }
)
, \quad \mathfrak{F}_{\epsilon} ^t( \U _ {\varepsilon _2,k}^{s^{\prime }} )\subset
\U _ {\varepsilon _1,k}^{s^{ \prime}} \text{ for all $|t |\le 2$}
\end{aligned} \end{equation}
by Lemma \ref{lem:ODE}, for a fixed $l\ge 2$. By Lemma \ref{lem:moll}
we have uniformly locally
\begin{equation} \label{eq:darboux3} \begin{aligned} & \lim _{\varepsilon \to 0}\mathfrak{F} ^t_\epsilon = \mathfrak{F} ^t \text{ in } C^{l }([-1,1]\times \U _ {\varepsilon _2,k}^{s^{\prime }}
, \U _ {\varepsilon _1,k}^{s }
) .
\end{aligned} \end{equation}
\noindent Let us take $0<\varepsilon _3\ll \varepsilon _2$ s.t.
$\mathfrak{F}_{\epsilon} ^t(\U _ {\varepsilon _3,k}^{s^{\prime }} )\subset \U _ {\varepsilon _2,k}^{s^{\prime }}
$ for all $|t |\le 2$ and $|\epsilon |\le 1$.
\noindent In $\U _ {\varepsilon _3,k}^{s^{\prime }} $ the following computation
is valid because $\mathcal{X}_\epsilon ^t$ is a standard vector field in
$\U _ {\varepsilon _1,k}^{s^{\prime }} $ and similarly $\Omega _t$ is a regular
differential form therein:
\begin{equation*} \label{eq:darboux1} \begin{aligned} & \mathfrak{F}_{\epsilon }^{1*}\Omega -\Omega _0
=\int _0^1
\frac{d}{dt}
\left ( \mathfrak{F}_{\epsilon }^{t*}\Omega _t\right ) dt = \int _0^1 \mathfrak{F}_{\epsilon }^{t*}
\left (L_{\mathcal{X}_\epsilon ^t} \Omega _t+\frac{d}{dt}\Omega _t\right ) dt\\&= d
\int _0^1 \mathfrak{F}_{\epsilon }^{t*}
\left ( i_{\mathcal{X}_\epsilon ^t} \Omega _t+ \alpha \right ) dt
,
\end{aligned}
\end{equation*}
where we recall $\Omega _t=\Omega _0+t(\Omega -\Omega _0)$.
\noindent If we consider a ball
$\mathbf{B}$ in $\U _ {\varepsilon _3,k}^{s^{\prime }} $,
in the notation of Lemma \ref{lem:1forms},
for some function $\psi _{\epsilon}\in C^1(\mathbf{B},\R )$ we can write
\begin{equation} \label{eq:darboux11} \begin{aligned} & \mathfrak{F}_{\epsilon }^{1*}(B_0+\alpha ) -B_0 +d\psi _{\epsilon}
=
\int _0^1 \mathfrak{F}_{\epsilon }^{t*}
\left ( i_{\mathcal{X}_\epsilon ^t} \Omega _t+ \alpha \right ) dt
,
\end{aligned}
\end{equation}
\noindent By \eqref{eq:darboux2}--\eqref{eq:darboux3}
we have
\begin{equation*}
\lim _{\epsilon \to 0}(\mathfrak{F}_{\epsilon }^{1*}(B_0+\alpha ) -B_0)=\mathfrak{F} ^{1*}(B_0+\alpha ) -B_0 \text{ in } C^{l -1}( \U _ {\varepsilon _3,k}^{s^{\prime }}
, B (\widetilde{ \Ph}^{s' }, \R )
) .
\end{equation*}
The set $\Gamma :=\{\mathfrak{F}_{\epsilon }^{t}(\mathbf{B}): |t|\le 2, |\epsilon |\le 1 \}$ is a bounded subset in $\U _ {\varepsilon _2,k}^{s^{\prime }} $ because of \eqref{eq:gronwall0}--\eqref{eq:gronwall1}. Then we have
\begin{equation*} \begin{aligned}& \lim _{\epsilon \to 0}{\mathcal X}^t_\epsilon ={\mathcal X} ^t \text{ in } C^{0}((-2,2)\times \Gamma
, \widetilde{{\Ph}} ^{s }
) \text{ uniformly }.
\end{aligned}
\end{equation*}
Hence by $i_{\mathcal{X} ^t} \Omega _t=-\alpha$ we get
\begin{equation*}
\lim _{\epsilon \to 0}\left ( i_{\mathcal{X}_\epsilon ^t} \Omega _t+ \alpha \right )= i_{\mathcal{X} ^t} \Omega _t+ \alpha =0 \text{ in } C^{0}((-2,2)\times \Gamma
, B( \widetilde{{\Ph}} ^{s },\R )
) \text{ uniformly. }
\end{equation*}
This implies \begin{equation*} \begin{aligned}& \lim _{\epsilon \to 0}
\|
\int _0^1 \mathfrak{F}_{\epsilon }^{t*}
\left ( i_{\mathcal{X}_\epsilon ^t} \Omega _t+ \alpha \right ) dt \| _{L^\infty (\mathbf{B} ,B ( \widetilde{{\Ph}} ^{s' },\R ))}\\& \le C \lim _{\epsilon \to 0}\| i_{\mathcal{X}_\epsilon ^t} \Omega _t+ \alpha \| _{L^\infty ([0,1]\times \Gamma ,B ( \widetilde{{\Ph}} ^{s },\R ))}=0,
\end{aligned}
\end{equation*}
for $C$ an upper bound to the norms $\| (\mathfrak{F}_{\epsilon }^{t*}) _{|\mathfrak{F}_{\epsilon }^{t }(\upsilon)} :B ( \widetilde{{\Ph}} ^{s '},\R )\to B ( \widetilde{{\Ph}} ^{s },\R ) \| $ as $\upsilon $ varies in $ \mathbf{B}$. Notice that $C<\infty$
by \eqref{eq:reg2}.
\noindent By \eqref{eq:darboux11} we conclude that uniformly
\begin{equation*}
\lim _{\epsilon \to 0} d\psi _{\epsilon} =B_0-\mathfrak{F} ^{1*}(B_0+\alpha )\text{ in } C^{0}( \mathbf{B}
, B (\widetilde{ \Ph}^{s'}, \R )).
\end{equation*}
Normalizing $\psi _{\epsilon} (\upsilon _0)=0$ at some given $\upsilon _0\in \mathbf{B}$, it follows that
also $\psi _{\epsilon}$ converges locally uniformly to a function
$\psi _{0}$ with $d\psi _{0}= B_0-\mathfrak{F} ^{1*}(B_0+\alpha )$.
Taking the exterior differential, we conclude that
$
\mathfrak{F} ^{1*}\Omega =\Omega _0$ { in } $C^{\infty}( \U _ {\varepsilon _3,k}^{s^{\prime }}
, B^2 (\widetilde{ \Ph}^{s'}, \R )
) .$
\qed
\section{Pullback of the Hamiltonian}
\label{sec:pullback}
In the somewhat abstract set up of this paper it is particularly important to have a general description
of the pullbacks of the Hamiltonian $K$. Our main goal in this section is formula \eqref{eq:back11}. This formula and its related expansion
in Lemma \ref{lem:ExpH11} obtained splitting $R$ in discrete and continuous modes, play a key role in the Birkhoff normal forms argument.
The first and quite general result is the following consequence of
Lemma \ref{lem:ODE}.
\begin{lemma} \label{lem:ODE1} Consider
$\mathfrak{F} =\mathfrak{F}_1 \circ \cdots \circ \mathfrak{F}_L$
with $ \mathfrak{F}_j= \mathfrak{F}_j ^{t=1}$ transformations as of Lemma \ref{lem:ODE}. Suppose that for $j$
we have $M_0=m_j$, with given numbers $1\le m_1\le ...\le m_L$.
Suppose also that all the $j$ we have the same pair $ r$ and $ M $, which we assume sufficiently large. Let $i_j=1$ if $m_j=1$. Fix $0<m'<M$
\begin{itemize}
\item[(1)]
Let $ r > 2L(m'+1) \mathbf{d} + s'_L > 4L(m'+1) \mathbf{d} + s_1$, $s_1\ge \textbf{d}$.
Then, for any $\varepsilon >0$ there exists a $\delta >0$
such that $\mathfrak{F}\in C^{m'} (\U ^{s'_L} _{\delta ,a}, \U ^{s _1} _{\varepsilon ,h})$ for $0\le a\le h $ and
$0\le h < r-(m'+1) \mathbf{d}$.
\item[(2)]
Let $ r > 2L(m'+1) ) \mathbf{d} +h> 4L(m'+1) \mathbf{d} + a$, $a\ge 0$.
The above composition, interpreting the $\mathfrak{F}_j$'s as maps in the $(\varrho , R)$
variables as in Lemma \ref{lem:ODEdomains},
yields also $\mathfrak{F} \in C^{m'}( \U _{-a}
, \Ph ^{-h }
)
$ for $\U_ {-a} $ a sufficiently small neighborhood of the origin in $ \Ph ^{-a} $.
\item[(3)] For $\U_ {-a} \subset \Ph ^{-a} $ like above and for functions
$\mathcal{R}^{i, j} _{a,m'}\in C^{m'} (\U_ {-a} ,\R ) $ and $\mathbf{S}^{i, j} _{a,m'}\in C^{m'} (\U_ {-a} ,\Sigma _{ a}) $, the following formulas hold:
\begin{equation} \label{eq:ODE2}\begin{aligned} & \Pi (R'):= \Pi (R)\circ \mathfrak{F} = \Pi (R)
+\mathcal{R}^{i_1, m_1+1} _{a,m'}(\Pi (R), R) , \\& p':=p\circ \mathfrak{F} = p +\mathcal{R}^{i_1, m_1+1} _{a,m'}(\Pi (R), R),\\& \Phi _{p'} = \Phi _{p} +\mathbf{S}^{i_1, m_1+1} _{a,m'}(\Pi (R), R) .
\end{aligned} \end{equation}
\item[(4)] For a function $F$ such that $F(e^{J\tau \cdot \Diamond }U)\equiv F(U)$ we have
\end{itemize}
\begin{equation*} \label{eq:ODE4}\begin{aligned} &
F\circ \mathfrak{F}(U)= F\left ( \Phi _{p}+P(p) (R+ \mathbf{S}^{i_1,m_1 }_{k' ,m '}) + \textbf{S}^{i_1,m_1 +1 }_{k' ,m'} \right ) \, , \, k'=r- 7L (m'+1) \mathbf{d} .
\end{aligned} \end{equation*}
\end{lemma}
\proof
Recall that by \eqref{eq:reg1} we have $\mathfrak{F}_j\in C^{m'}(\U _{\varepsilon _j ',h}^{s'_j}, \U _{\varepsilon _j ,h}^{s_j} )$ for $r-(m'+1)\textbf{d}> s'_j\ge s_j+m'\textbf{d}$ and appropriate choice of the $0<\varepsilon _j '<\varepsilon _j $ and for $h\in \Z \cap [0,r-(m'+1)\textbf{d}]$.
So for the composition we have $\mathfrak{F} \in C^{m'}(\U _{\varepsilon _L ' ,a }^{\kappa}, \U _{\varepsilon _1,h }^{s_1} )$ for $a\le h$.
The inequalities $ r > 2L(m'+1) \mathbf{d} + s'_L > 4L(m'+1) \mathbf{d} + s_1$, $s_1\ge\textbf{ d} $ can be accommodated
since $r$ is assumed sufficiently large. This yields claim (1).
\noindent By Lemma \ref{lem:ODEdomains} we have $\mathfrak{F}_j\in C^{m'}(\U_ {- h +j m'\textbf{d} } , \Ph ^{- h +(j-1)m'\textbf{d} } )$ with $\U_ {- h +j m'\textbf{d} }\subset \Ph ^{-h+j m'\textbf{d} } $ a neighborhood of the origin. So for the composition we have
$\mathfrak{F} \in C^{m'}( \U _{-a }
, \Ph ^{-h }
)
$ for $a\le h-Lm'\textbf{d}$. The conditions $ r > 2L(m'+1) \mathbf{d} +h$, $h> 4L(m'+1) \mathbf{d} + a$ and $a\ge 0$, can be accommodated
since $r$ is assumed sufficiently large. This yields claim (2).
\noindent We now prove \eqref{eq:ODE2}.
Let first $L=1$.
By \eqref{eq:ODE1} we have $R':=( \mathfrak{F}_1)_R(\Pi (R),R)=e^{Jq _1\cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{r-(m'+1)\mathbf{d},m'} )$, where we use $M> m'$. Here we will omit the variables $(\Pi (R), R)$ in the $\mathbf{S}$'s
and $\resto$'s.
Then we have for $a'=r-(m'+1)\mathbf{d}$
\begin{equation}\label{eq:pi1}
\Pi (R') = \Pi ( R+\textbf{S}^{i_1,m_1 }_{a',m'} ) = \Pi (R)
+\mathcal{R}^{i_1,m_1+1}_{a'-\mathbf{d},m'}.
\end{equation}
Here we have used
\begin{equation*}
\begin{aligned} |\langle R, \Diamond \textbf{S}^{i_1,m_1 }_{a' ,m'}\rangle | \le \| R \| _{\Sigma _{-a'+\mathbf{d}}} \| \textbf{S}^{i_1,m_1 }_{a' ,m'} \| _{\Sigma _{a'}} . \end{aligned}
\end{equation*}
By $ p_j=\Pi _j-\Pi _j(R)+ \mathcal{R}^{0, 2} (\Pi (R) , R)
$ we get
\begin{equation}\label{eq:pi2}\begin{aligned} &
p_j'=\Pi _j-\Pi _j(R')+ \mathcal{R}^{0, 2} (\Pi (R') , R')\\& = \Pi _j-\Pi _j(R )+ \mathcal{R}^{0, 2} (\Pi (R ) , R ) + \mathcal{R}^{i_1,m_1+1}_{a'-\mathbf{d} ,m'}= {p_j}+ \mathcal{R}^{i_1,m_1+1}_{a'-\mathbf{d} ,m'} .
\end{aligned}
\end{equation}
This yields \eqref{eq:ODE2} for $L=1$ since $a\le r-4(m'+1)\textbf{d}<a'-\mathbf{d}$.
We extend the proof to the case $L>1$.
We write here and below $\mathfrak{F} ':= \mathfrak{F}_1 \circ \cdots \circ \mathfrak{F} _{L-1} $.
We suppose that $ \mathfrak{F}'_R (\Pi (R),R)=e^{Jq \cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{a'_{L-1} ,m'} )$ for $a'_{L-1}\le r-2(L-1) m' \textbf{d}$, which is true for $L-1=1$. Then
\begin{equation*}\begin{aligned} &
R'=e^{J(q \circ\mathfrak{F} _{L} ) \cdot \Diamond } \left ( e^{Jq_L \cdot \Diamond } ( R+ \textbf{S}^{i_L,m_L }_{r-(m'+1)\mathbf{d},m'} )+ \textbf{S}^{i_1,m_1 }_{a'_{L-1} ,m'}\circ\mathfrak{F} _{L} \right ) \\& = e^{J ({q\circ\mathfrak{F} _{L}+q_L} ) \cdot \Diamond } \left ( R+ \textbf{S}^{i_L,m_L }_{r-(m'+1)\mathbf{d},m'} )+e^{-Jq_L \cdot \Diamond }\textbf{S}^{i_1,m_1 }_{a'_{L-1}- m' \mathbf{d},m'} \right ),
\end{aligned}
\end{equation*}
where $q_L=\resto ^{0,m_L+1 }_{r-(m'+1)\mathbf{d},m'}$ and where we used the last claim in Lemma \ref{lem:ODEdomains}. Since $e^{-Jq_L \cdot \Diamond }\textbf{S}^{i_1,m_1 }_{a'_{L-1}- m' \mathbf{d},m'} =\textbf{S}^{i_1,m_1 }_{a'_{L-1}-2m'\mathbf{d},m'} $ we conclude that there is an expansion
$ R'=e^{Jq \cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{a'_{L } ,m'} )$
for $a'_{L }\le a'_{L-1}-2m' \mathbf{d}$. Then
\begin{equation}\label{eq:tranR}\begin{aligned} & \mathfrak{F}_R (\Pi (R),R)=e^{Jq \cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{a'_{L } ,m'} ) \, , \quad a'_{L }:=r- 2L m' \textbf{d}.\end{aligned}
\end{equation}
For $a'=a'_{L }$ formulas \eqref{eq:pi1}--\eqref{eq:pi2} continue to hold.
By
$a< a'_{L }-\mathbf{d} $ this yields \eqref{eq:ODE2}.
\noindent We consider the last statement of Lemma \ref{eq:ODE1}. For $a'=r-(m'+1)\mathbf{d}$
we have
\begin{equation*} \begin{aligned} &
F(\mathfrak{F}_1(U))= F(\Phi _{p'}+P(p') e^{Jq _1\cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{a' ,m'} ))=\\& F(\Phi _{p}+P(p)e^{Jq _1\cdot \Diamond } ( R+ \textbf{S}^{i_1,m_1 }_{a' ,m'} )+\mathbf{S}^{i_1,m_1+1 } _{a'+\mathbf{d} ,m'}) =\\& F\left (e^{Jq _1 \cdot \Diamond } \left (\Phi _{p}+P(p) (R+ \mathbf{S}^{i_1,m_1 }_{a' ,m'}) +Y\right ) \right )
\end{aligned} \end{equation*}
with
\begin{equation*} \begin{aligned} &Y =(e^{Jq _1 \cdot \Diamond }-1) \Phi _{p}+ [P(p),e^{Jq _1 \cdot \Diamond }] (R+ \textbf{S}^{i_1,m_1 }_{a' ,m'}) +e^{-Jq _1 \cdot \Diamond } \textbf{S}^{i_1,m_1 +1 }_{a'-\mathbf{d},m'} .
\end{aligned} \end{equation*}
We claim
\begin{equation}\label{eq:Ypull} \begin{aligned} &Y = \textbf{S}^{i_1,m_1 +1 }_{a'-2m'\mathbf{d},m'} .
\end{aligned} \end{equation}
To prove \eqref{eq:Ypull}
we use $(e^{Jq _1 \cdot \Diamond }-1) \Phi _{p} = \mathbf{S}^{i_1,m_1 +1} _{r-(m'+1)\mathbf{d},m'} =\mathbf{S}^{i_1,m_1 +1} _{a' ,m'} $. This follows from $\Phi _{p}\in C^\infty (\mathcal{O},\mathcal{S})$ and
\begin{equation} \label{eq:reg} \begin{aligned} & \left | (e^{Jq _1 \cdot \Diamond }-1) \Phi _{p} \right | _{\Sigma _l}
\le |q_{1j}| \int _0^1 \left | e^{t Jq _1 \cdot \Diamond } \Diamond _j \Phi _{p} \right | _{\Sigma _l} dt\le C_l |q_{1j}| \left | \Diamond _j \Phi _{p} \right | _{\Sigma _l}.
\end{aligned} \end{equation}
\noindent Schematically we have, summing over repeated indexes and for $ \mathbf{e}_j,\mathbf{e}_j^*
\in \mathcal{S}$,
\begin{equation*} \begin{aligned} &[P(p),e^{Jq _1\cdot \Diamond }] = [e^{Jq _1 \cdot \Diamond }, P_{N_{g}}(p) ] = e^{Jq _1 \cdot \Diamond }\mathbf{e}_j
\langle \mathbf{e}_j ^*, \, \rangle -\mathbf{e}_j
\langle e^{-Jq _1\cdot \Diamond }\mathbf{e}_j ^*, \, \rangle \\& =(e^{Jq _1 \cdot \Diamond }-1) \mathbf{e}_j
\langle \mathbf{e}_j ^*, \, \rangle -\mathbf{e}_j
\langle (e^{-Jq _1 \cdot \Diamond }-1)\mathbf{e}_j ^*, \, \rangle \\& =\mathbf{S}^{0,m_1+1 }_{r-(m'+1)\mathbf{d},m'} \langle \mathbf{e}_j ^*, \, \rangle + \mathbf{e}_j\langle \mathbf{S}^{0,m_1+1 }_{r-(m'+1)\mathbf{d},m'}, \, \rangle .
\end{aligned} \end{equation*}
This yields for any $a^{\prime\prime} \le a'=r-(m'+1)\mathbf{d}$
\begin{equation*} [P(p),e^{Jq_1\cdot \Diamond }] (R+ \textbf{S}^{i_1,m_1 }_{a^{\prime\prime},m'})=\textbf{S}^{i_1,m_1 +2 }_{a^{\prime\prime },m'}. \end{equation*}
We have $e^{-Jq _1 \cdot \Diamond } \textbf{S}^{i_1,m_1+1 }_{a'-\mathbf{d},m'} =\textbf{S}^{i_1,m_1+1 }_{a'-(m' +1)\mathbf{d},
m'}$. Then \eqref{eq:Ypull} is proved.
Then
\begin{equation}\label{eq:pi3} \begin{aligned} & F(\mathfrak{F}_1(U))=
F\left ( \Phi _{p}+P(p) (R+ \mathbf{S}^{i_1,m_1 }_{a'-2m'\mathbf{d},m'}) + \textbf{S}^{i_1,m_1 +1 }_{a'-2m'\mathbf{d},m'}\right )
\end{aligned} \end{equation}
for $a'=r-(m'+1)\mathbf{d}$. This proves the last sentence of our lemma for $L=1$.
For $L>1$ set once more $\mathfrak{F} ':= \mathfrak{F}_1 \circ \cdots \circ \mathfrak{F} _{L-1} $.
We assume by induction that $ F(\mathfrak{F}'(U))$ equals the rhs of
\eqref{eq:pi3} for $a'=a'_{L-1}:=r-2(L-1) m' \textbf{d}$.
Then using $\mathbf{S}^{i_1,m_1 }_{l,m '}\circ \mathfrak{F}_L= \mathbf{S}^{i_1,m_1 }_{l- m' \mathbf{d},m '}$ from Lemma
\ref{lem:ODEdomains}, by \eqref{eq:ODE2} for $\mathfrak{F}=\mathfrak{F}_L$ and by \eqref{eq:Ypull} with the index 1 replaced by index $L$, we get
\begin{equation*} \begin{aligned} &
F(\mathfrak{F}(U))= F\big (\Phi _{p'}+\\& P(p') e^{Jq _L\cdot \Diamond } ( R+ \textbf{S}^{i_L,m_L }_{r-(m'+1)\mathbf{d},m'} ) +P(p') \mathbf{S}^{i_1,m_1 }_{a'_{L-1}- m' \mathbf{d},m '}+ \textbf{S}^{i_1,m_1 +1 }_{a'_{L-1}- m' \mathbf{d},m'} \, \big ) \\& =F\left (e^{Jq_L \cdot \Diamond } \left [\Phi _{p}+P(p) (R+ \mathbf{S}^{i_1,m_1 }_{a'_{L-1}- m' \mathbf{d},m '}) + \textbf{S}^{i_1,m_1 +1 }_{a'_{L-1}-2m' \mathbf{d},m'} \right ] \right ) .
\end{aligned} \end{equation*}
\noindent We conclude that $ F(\mathfrak{F}(U))$ equals the rhs of
\eqref{eq:pi3} for $a'_{L}=r-2L m' \textbf{d}$. In particular this proves the last sentence of our lemma for any $L $.
\qed
\begin{lemma}
\label{lem:Taylor ex}
For fixed vectors $\mathbf{u}$ and $\mathbf{v} $ and for $B$ sufficiently regular with $B(0)=0$,
we have
\begin{equation}\label{ExpEP0}\begin{aligned}
& B (|\mathbf{u} +\mathbf{v} |^2_1) =B \left ( |\mathbf{u} |_1^2
\right )+ B (| \mathbf{v} |_1^2)\\& + \sum _{ j =0}^{3}\int _{[0,1]^2} \frac{t^j}{j!} (\partial _t ^{j+1})_{|t=0} \partial _s [B (|s\mathbf{u} +t\mathbf{v} |^2_1) ]\ dt ds \\& +\int _{[0,1]^2} dt ds \int _0^t \partial _\tau ^5 \partial _s [B (|s\mathbf{u} +\tau \mathbf{v} |_1^2)] \frac{(t-\tau )^3}{3!} \ d\tau .
\end{aligned}\end{equation}
\end{lemma} \proof Follows by Taylor expansion in $t$ of
\begin{equation*}\begin{aligned}
& B (|\mathbf{u} +\mathbf{v} |_1^2) =B \left ( |\mathbf{u} |_1^2
\right ) +\int _0^1 \partial _t[ B (| \mathbf{u} +t\mathbf{v} |^2_1)] dt =\\&
B \left ( |\mathbf{u} |^2_1
\right ) + B (| \mathbf{v} |^2_1)+\int _{[0,1]^2} dt ds \ \partial _s \partial _t[ B (|s \mathbf{u} +t\mathbf{v} |_1^2)] .
\end{aligned}\end{equation*} \qed
\begin{lemma}
\label{lem:back} Consider a transformation
$\mathfrak{F} =\mathfrak{F}_1 \circ \cdots \circ \mathfrak{F}_L$
like in Lemma \ref{lem:ODE1} and with $m_1=1$, with same notations, hypotheses and conclusions.
In particular we suppose $r$ and $M$ sufficiently large that the conclusions of Lemma \ref{lem:ODE1}
hold for preassigned sufficiently large $s=s'_L$, $k' $ and $m' $.
Let $k \le k' - \max\{\textbf{d}, \text{ord}(\mathcal{D} )\} $ and $m\le m'$.
Then there are a $ \underline{{{\psi}}} (\varrho) \in C^\infty $ with $\underline{{\psi}} (\varrho) =O(| \varrho |^2)$ near 0
and a small $\varepsilon >0$
such that in
$\U^{s}_{\varepsilon ,k}$ we have the expansion
\begin{align} \label{eq:back1} &
K \circ \mathfrak{F}= \underline{{{\psi}}} (\Pi (R)) + \frac {1}{2}\Omega ( \mathcal{H}_pP(p)R,P(p) R ) + \resto ^{1,2} _{k,m} +E_P(P(p)R)+\textbf{R} ^{\prime \prime}
\\& \textbf{R} ^{\prime \prime }:= \sum _{d=2}^4
\langle B_{d } ( R,\Pi (R) ), (P(p)R) ^{ d} \rangle
+\int _{\mathbb{R}^3}
B_5 (x, R, R(x),\Pi (R) ) (P(p)R)^{ 5}(x) dx \nonumber
\end{align}
with:
\begin{itemize}
\item $ \resto ^{1,2} _{k,m} = \resto ^{1,2} _{k,m} (\Pi (R), R) $;
\item $B_2(0,0 )=0$;
\item $(P(p)R)^d(x)$ represent $d-$products of components of $P(p)R$;
\item
$B_{d
}( \cdot , R,\varrho ) \in C^{m } ( \U _{-k},
\Sigma _k (\mathbb{R}^3, B (
(\mathbb{R}^{2N })^{\otimes d},\mathbb{R} ))) $ for $2\le d \le 4$ with $\U _{-k}\subset \Ph ^{-k}$ a neighborhood of the origin;
\item for
$ \zeta \in \mathbb{R}^{2N }$ with $|\zeta |\le \varepsilon$
and $( \varrho ,R) \in \U _{-k}$
we have for $i\le m$
\begin{equation} \label{eq:B5}\begin{aligned} & \| \nabla _{ R,\zeta, \varrho }
^iB_5( R,\zeta ,\varrho ) \| _{\Sigma _k(\mathbb{R}^3, B (
(\mathbb{R}^{2N })^{\otimes 5},\mathbb{R} )} \le C_i .
\end{aligned} \end{equation}
\end{itemize}
\end{lemma}
\proof Here we will omit the variables $(\Pi (R), R)$ in the $\mathbf{S}$'s
and $\resto$'s.
\noindent By Lemma \ref{lem:ODE1} for $m\le m' \le M$, $k+\max\{\textbf{d}, \text{ord}(\mathcal{D} )\}\le k' \le r-L(m'+2)\textbf{d}$,
we have
\begin{equation} \begin{aligned} &
{K}(\mathfrak{F}(U))= E (\Phi _p+P(p)R+P(p)\mathbf{S}^ {1,1}_{k',m'}+\mathbf{S}^{1,2}_{k',m'})- E\left ( \Phi _{p _0}\right ) \\& -(\lambda _j(p) + \resto ^{1,2} _{k,m}) \left ( \Pi _j (\Phi _p+P(p)R ) + \mathcal{R}^{1,2}_{k,m} -\Pi _j\left ( \Phi _{p _0}\right ) \right ) ,
\end{aligned}
\end{equation}
where, by \eqref{eq:ODE2}, we have used $p':=p\circ \mathfrak{F}= p +\mathcal{R}^{1,2}_{k,m}$ and where by
$k\le k' - \textbf{d} $
\begin{equation*}\begin{aligned} &
\Pi _j (\Phi _p+P(p)R+P(p)\mathbf{S}^{1, 1}_{k',m'} +\mathbf{S}^{1, 2}_{k',m'} )=\Pi _j (\Phi _p+P(p)R ) +\mathcal{R}^{1,2 } _{k,m} .\end{aligned}
\end{equation*}
Set now $\Psi = \Phi _p +P(p)\mathbf{S}^{1,1} _{k',m'} +\mathbf{S}^{1,2}_{k',m'} $. By \eqref{ExpEP0} for $\textbf{u}=\Psi$ and $\textbf{v}=
P(p)R$
\begin{equation} \label{eq:back2} \begin{aligned} & E_P( \Psi +P(p)R )= E_P( \Psi ) + E_P( P(p)R ) \\& +\sum _{ j =0}^{1}\int _{\R ^3}dx \int _{[0,1]^2}
\frac{t^j}{j!} (\partial _t^{j+1})_{|t=0} \partial _s [B (|s \Psi+tP(p)R |^2_1) ] dt ds \\&+ \sum _{ j =2}^{3}\int _{\R ^3}dx\int _{[0,1]^2} \frac{t^j}{j!} (\partial _t^{j+1})_{|t=0} \partial _s [ B (|s \Psi +tP(p)R |_1^2) ] dt ds \\& + \int _{\R ^3}dx \int _{[0,1]^2} dt ds \int _0^t \partial _\tau ^5 \partial _s [B (|s \Psi +\tau P(p)R |_1^2)] \frac{(t-\tau )^3}{3!} d\tau .
\end{aligned}
\end{equation}
The last two lines can be incorporated in $ \textbf{R}^{\prime \prime} $. For example, schematically
we have
\begin{equation*}
\partial _\tau ^5 \partial _s B (|s\Phi _{p } +\tau P(p)R |_1^2)
\sim \widetilde{B} ( s\Phi _{p } +\tau P(p)R) \ \Phi _{p } \ (P(p)R )^5,
\end{equation*}
for some $\widetilde{B}(Y)\in C^\infty (\R ^{2N}, B ^6(\R ^{2N}, \R )).$
This produces a term which can be absorbed in the $B_5$ term of $\mathbf{R} ^{\prime \prime} .$ In particular, \eqref{eq:B5}
follows from \eqref{eq:growthB}. The terms in the third line of \eqref{eq:back2} can be treated similarly yielding terms
which end in the $B_d$ term of $\mathbf{R} ^{\prime \prime} $ with $d=j+1$.
\noindent The second line of \eqref{eq:back2} equals
\begin{equation} \label{eq:back3} \begin{aligned} & \int _{\R ^3}dx \int _{[0,1]^2} dt ds
\sum _{ j =0}^{1}\frac{t^j}{j!}(\partial _t^{j+1})_{|t=0} \partial _s \ \big \{ \ \ B (|s \Phi _p+tP(p)R |^2_1) + \\& + \int _0^1 d\tau \partial _\tau [B (|s (\Phi _p
+\tau (P(p)\mathbf{S}^{1,1}_{k',m'} +\mathbf{S}^{1,2}_{k',m'}) +tP(p)R |^2_1) ] \ \ \big \} .
\end{aligned}
\end{equation}
The contribution from the last line of \eqref{eq:back3} can be incorporated in $\textbf{R}^{\prime \prime}+\resto ^{1,2} _{k,m} $.
\noindent By $k \le k' -\text{ord}(\mathcal{D} )$ we have
\begin{equation*} \label{eq:back31} \begin{aligned} & E_K( \Psi +P(p)R )
= E_K( \Psi ) +\langle {\mathcal D} \Phi _p, P(p)R \rangle \\& + \overbrace{\langle {\mathcal D} (P(p)\mathbf{S}^{1,1}_{k',m'}+\mathbf{S}^{1,2}_{k' ,m'}), P(p)R \rangle}^{\resto ^{1,2}_{k,m}} +E_K( P(p)R ).\end{aligned}
\end{equation*}
Notice that from the
$j=0$ term in the first line of \eqref{eq:back3} we get
\begin{equation*} \begin{aligned} & 2\int _{\R^3}dx \int _{0}^1 ds
\partial _s [B' (|s \Phi _p |^2_1) s \Phi _p\cdot _{1} P(p)R ] =2\int _{\R^3}dx B' (| \Phi _p |^2_1) \Phi _p\cdot _{1} P(p)R \\& =\langle \nabla E_P(\Phi _p), P(p) R\rangle .
\end{aligned}
\end{equation*}
By \eqref{eq:lagr mult} and \eqref{eq:begspectdec2}, that is $\nabla E(\Phi _p )= \lambda (p) \cdot \Diamond \Phi _p \in N_g (\mathcal{H}_p^{\ast})$,
and by $P(p)R\in N_g^\perp (\mathcal{H}_p)$, we have
\begin{equation*}
\langle {\mathcal D} \Phi _p, P(p)R \rangle + \langle \nabla E_P(\Phi _p), P(p) R\rangle =\langle \nabla E(\Phi _p), P(p)R \rangle =0.
\end{equation*}
The
$j=1$ term in the first line of \eqref{eq:back3} is $\frac{1}{2}
\langle \nabla ^2 E_P(\Phi _p)P(p) R, P(p) R\rangle$ which summed
to the $E_K( P(p)R )$ in \eqref{eq:back31} yields the $\frac {1}{2}\Omega ( \mathcal{H}_pP(p)R,P(p) R )$ in \eqref{eq:back1}.
\noindent We have $ E_K ( \Psi )+ E_P ( \Psi )= E ( \Psi )$ and
\begin{equation*} \begin{aligned} & E ( \Psi )
= E ( \Phi _p ) +\overbrace{\langle \nabla E(\Phi _p), P(p)\mathbf{S}^{1,1}_{k',m'}\rangle}^{0}+
\overbrace{\langle \nabla E(\Phi _p), \mathbf{S}^{1,2 } _{k',m'} \rangle }^{ \resto^{1,2 } _{k,m}}+ \resto^{1,2 } _{k,m}.\end{aligned}
\end{equation*}
The last term we need to analyze, for $d(p):=E(\Phi _{p }) - \lambda (p) \cdot \Pi (\Phi _{p })$, is
\begin{equation*} \label{eq:back4} \begin{aligned} &
E(\Phi _{p }) - E(\Phi _{p _0 })-\sum _j\lambda _j(p) (\Pi _j (\Phi _{p }) - \Pi _j (\Phi _{p _0 })) \\& =
d(p)- d(p_0)- \sum _j (\lambda _j(p_0)-\lambda _j(p)) p _{0j} =: \widetilde{\psi} (p,p_0),
\end{aligned}
\end{equation*}
where $ \widetilde{\psi} (p,p_0)=O ((p-p_0)^2)$ by $\partial _{p_j}d(p)=-p\cdot \partial _{p_j} \lambda (p)$.
Notice that $\widetilde{\psi}\in C^\infty (\mathcal{O}^2, \R) $.
Now recall that in the initial system
of coordinates we have $p '=\Pi -\Pi (R') +\resto ^{0,2}(\Pi (R') ,R')$. Substituting
$p'$ and $\Pi (R')$ by means of \eqref{eq:ODE2}, and $R'$ by means of \eqref{eq:tranR}
we conclude that $p=p_0 -\Pi (R)+\resto ^{0,2}_{k',m'} .$
Then $ \widetilde{\psi} (p,p_0)=\underline{{{\psi}}} (\Pi (R))+\resto ^{1,2}_{k,m} $
with
$\underline{{{\psi}}}(\varrho ):= \widetilde{\psi} (p_0-\varrho,p_0)$ a $C^\infty$ function with
$\underline{{{\psi}}}(\varrho )=O(|\varrho |^2)$ for $\varrho$ near $0$.
\qed
\begin{lemma}
\label{lem:back11} Under the hypotheses and notation of Lemma \ref{lem:back},
for an $\textbf{R}'$ like $\textbf{R}^{\prime \prime}$, for a $\psi \in C^\infty $
with $\psi (\varrho )=O(|\varrho|^2)$ near 0, we have
\begin{align} \label{eq:back11} &
K \circ \mathfrak{F}= {{\psi}} (\Pi (R)) + \frac {1}{2}\Omega ( \mathcal{H}_{p _0} R, R )
+ \resto ^{1,2} _{k,m} (\Pi (R),R) +E_P( R)+\textbf{R}',
\\& \textbf{R} ':= \sum _{d=2}^4
\langle B_{d } ( R,\Pi (R) ), R ^{ d} \rangle
+\int _{\mathbb{R}^3}
B_5 (x, R, R(x),\Pi (R) )R^5 ( x) dx, \nonumber
\end{align}
the $B_d$ for $d=2,...,5$ with similar properties of the functions in Lemma \ref{lem:back}.
\end{lemma} \proof
We have
\begin{equation*}
P(p) R=R+(P(p)-P(p_0)) R=R +\textbf{S}^{1,1}(p-p_0,R)=R+\textbf{S}^{1,1}(\Pi (R),R).
\end{equation*}
Substituting $P(p) R= R+\textbf{S}^{1,1}(\Pi (R),R)$
in \eqref{eq:back1} we obtain that $ \resto ^{1,2}_{k,m}+\textbf{R}^{\prime \prime} $ is absorbed in
$ \resto ^{1,2}_{k,m}(\Pi (R),R) +\textbf{R}^{\prime } $.
This is elementary to see for the terms with $d\le 4$. We consider
the case $d=5$.
\begin{equation*} \begin{aligned}&
B_5 (x, R, R(x),\Pi (R) ) R ^{ i}(x) (\textbf{S}^{1,1}) ^{5-i} \\& =
\sum _{j=0}^{5-i} \frac{1}{j!}(\partial _t^j)_{|t=0} [B_5 (x, R, tR(x),\Pi (R) )] R ^{ i}(x) (\textbf{S}^{1,1}) ^{5-i}\\& + \int _0^1 \frac{(1-t)^{4-i}}{(4-i)!}\partial _t ^{5-i} [B_5 (x, R, tR(x),\Pi (R) )] R ^{ i}(x) (\textbf{S}^{1,1}) ^{5-i} \end{aligned}
\end{equation*}
The last term can be absorbed in the $d=5$ term of $\mathbf{R}'$. Similarly, all the other terms either are absorbed in $\mathbf{R}'$
or, like for instance the $i=j=0$ term, they are $ \resto ^{1,2} $.
\noindent We write $E_P(P(p)R)=E_P(R-P_{N_g(p)} R) $ and use \eqref{ExpEP0}
for $\mathbf{u}=R$ and $\mathbf{v}=-P_{N_g(p)} R$.
We get the sum of
$E_P( R)$ with a term which can be absorbed in $ \resto ^{1,2}_{k,m}(\Pi (R),R) +\textbf{R}^{\prime } $.
We finally focus on
\begin{equation} \label{eq:back12} \begin{aligned}& \frac 12 \langle J^{-1} \mathcal{H}_p P(p)R, P(p)R \rangle = \frac 12 \langle \mathcal{D} P(p)R, P(p)R \rangle \\&- \lambda _j(p) \Pi _j(P(p)R) + \frac 12 \langle \nabla ^2 E _P (\Phi _p)P(p)R, P(p)R \rangle .\end{aligned}
\end{equation}
We have
\begin{equation} \label{eq:back13} \begin{aligned} \langle \mathcal{D} P(p)R, P(p)R \rangle
&= \langle \mathcal{D} R, R \rangle +\resto ^{1,2}_{k,m}(\Pi (R),R) \\
\langle \nabla ^2 E _P (\Phi _p)P(p)R, P(p)R \rangle & =
\langle \nabla ^2 E _P (\Phi _{p_0}) R, R \rangle + \resto ^{1,2}_{k,m}(\Pi (R),R) \\&+ \langle ( \nabla ^2 E _P (\Phi _p)-\nabla ^2 E _P (\Phi _{p_0})) R, R \rangle \\
\lambda _j(p) &=\lambda _j(p_0)+ \resto ^{1,0} (\Pi (R)) + \resto ^{1,2}_{k,m}(\Pi (R),R) \\ \Pi _j(P(p)R) &= \Pi _j( R ) +\resto ^{1,2}_{k,m}(\Pi (R),R) .\end{aligned} \nonumber
\end{equation}
Then we conclude that the right hand side of \eqref{eq:back12} is
\begin{equation} \label{eq:back120} \begin{aligned} &
\overbrace{\frac 12 \langle (\mathcal{D} - \lambda (p_0) \cdot \Diamond +\nabla ^2 E _P (\Phi _{p_0})) R, R \rangle }^{\frac 12\langle J^{-1} \mathcal{H} _{p_0} R, R \rangle }+\resto ^{2, 0} (\Pi (R) ) +\resto ^{1,2}_{k,m}(\Pi (R),R)
\\& + \frac 12\langle ( \nabla ^2 E _P (\Phi _p)-\nabla ^2 E _P (\Phi _{p_0})) R, R \rangle \end{aligned}
\end{equation}
where the last term can be absorbed in the $d=2$ term of $\textbf{R}'$.
Setting $\psi (\varrho ) =\underline{\psi }(\varrho ) +\resto ^{2, 0} (\varrho )$ with the $\resto ^{2, 0}$
in \eqref{eq:back120}, we get the desired result.
\qed
\bigskip
We have completed the part of this paper devoted to the Darboux Theorem. The next step consists in the decomposition of $R$ into discrete and
continuous modes, and the search of a new coordinate system by an appropriate Birkhoff normal forms argument.
\section{Spectral coordinates associated to $\mathcal{H} _{p_0}$}
\label{sec:speccoo}
We will consider the operator $\mathcal{H} _{p_0}$, which will be central
in our analysis henceforth. We will list now various hypotheses, starting
with the spectrum of $\mathcal{H} _{p_0}$ thought as an operator in the natural complexification $L^2(\R ^3, \C ^{2N}) $ of $L^2(\R ^3, \R ^{2N}) $.
\begin{itemize}
\item[(L1)] $\sigma _e(\mathcal{H} _{p_0}) $ is a
union of intervals in $\im \R $ with $0\not \in \sigma _e(\mathcal{H} _{p_0})$ and is symmetric with respect to 0.
\item[(L2)] $\sigma _p(\mathcal{H} _{p_0}) $ is finite.
\item[(L3)] For any eigenvalue $\mathbf{e} \in \sigma _p(\mathcal{H} _{p_0}) \backslash \{ 0 \}$ the algebraic and geometric dimensions
coincide and are finite.
\item [(L4)] There is a number $\mathbf{n}\ge 1$ and positive numbers $0<\mathbf{e}
_1' \le \mathbf{e} _2' \le ...\le\mathbf{e} _\mathbf{n} ' $
such that $\sigma _p(\mathcal{H} _{p_0})$ consists exactly of the numbers
$\pm \im \mathbf{e} _j'$ and 0. We assume that
there are fixed integers $\mathbf{n}_0=0< \mathbf{n}_1<...<\mathbf{n}_{l_0}=\textbf{n}$ such that
$\mathbf{e} _j' = \mathbf{e} _i '$ exactly for $i$ and $j$
both in $(\mathbf{n}_l, \mathbf{n}_{l+1}]$ for some $l\le l_0$. In this case $\dim
\ker (\mathcal{H}_{p_0} -\mathbf{e} _j' )=\mathbf{n}_{l+1}-\mathbf{n}_l$.
We assume there exist $N_j\in \mathbb{N}$ such that $ N_j+1 =\inf \{ n\in \N : n \mathbf{e} _j'\in \sigma _e(\mathcal{H} _{p_0}) \} $. We set $\mathbf{N}=\sup _j N_j$. We assume that $\mathbf{e} _j'\not \in \sigma _p(\mathcal{H} _{p_0})$ for all $j$.
\item[(L5)] If $\mathbf{e} _{j_1}'<...<\mathbf{e} _{j_i}'$ are i distinct
$\lambda$'s, and $\mu\in \Z^k$ satisfies
$|\mu| \leq 2N +3$, then we have
$$
\mu _1\mathbf{e} _{j_1}'+\dots +\mu _k\mathbf{e} _{j_i}'=0 \iff \mu=0\ .
$$
\end{itemize}
The following hypothesis holds quite generally.
\begin{itemize}
\item[(L6)] If $\varphi \in \ker (\mathcal{H}_{p_0} -\im \mathbf{e} ) $ for $\im \mathbf{e} \in
\sigma _p (\mathcal{H}_{p_0} ) $ then
$\varphi \in \mathcal{S}(\R ^3, \C ^{2N})$.
\end{itemize}
\bigskip
\noindent
By \eqref{eq:begspectdec1}, $
\mathcal{H}_{p_0} \xi = \mathbf{e} \xi
$ implies $
\mathcal{H}_{p_0} ^* J^{-1} \xi =- \mathbf{e} J^{-1}\xi
$. Then $\sigma _p (\mathcal{H}_{p_0} )=\sigma _p (\mathcal{H}_{p_0} ^*)$. We denote it
by $\sigma _p$.
\noindent By general argument we have:
\begin{lemma}
\label{lem:Specdec} The following spectral decomposition remains determined:
\begin{align} \label{eq:spectraldecomp} &
N_g^\perp (\mathcal{H}_{p_0} ^*)\otimes _\R \C = \big (\oplus _{\mathbf{e} \in
\sigma _p\backslash \{ 0\}} \ker (\mathcal{H}_{p_0} - \mathbf{e}
) \big) \oplus X_c ( {p_0} )\\& \nonumber X_c ( {p_0} ):=
\left\{N_g(\mathcal{H}_{p_0} ^\ast)\oplus \big (\oplus _{\mathbf{e} \in
\sigma _p\backslash \{ 0\}} \ker (\mathcal{H}_{p_0} ^*- \mathbf{e}
) \big)\right\} ^\perp .
\end{align}
\end{lemma}
We denote by $P_c$ the projection on $X_c ( {p_0} )$ associated to \eqref{eq:spectraldecomp}. Set $\mathcal{H}:=\mathcal{H}_{p_0}P_c.$
\noindent The following hypothesis is important to solve the homological
equations in the Birkhoff normal forms argument.
\begin{itemize}
\item[(L7)] We have $R_{\mathcal{H}}\circ \Diamond _j^i \in C^\omega ( \rho (\mathcal{H} ), B(\Sigma _{n },\Sigma _{n } )) $ for any $ n\in \N $, any $j=1,...,n_0$ and for any $i =0,1$, where $\rho (\mathcal{H} ) =\C \backslash \sigma _e (\mathcal{H}_{p_0} )$.
\end{itemize}
\noindent For the examples in Sect. \ref{sec:examples},
(L7) can be checked with standard arguments.
\noindent We discuss now the choice of a good frame of eigenfunctions.
\begin{lemma}
\label{lem:basis} It is possible to choose eigenfunctions $\xi '\in \ker (\mathcal{H}_{p_0} -\im \mathbf{e} _j' )$
so that $ \Omega ( \xi _j ',\overline{\xi} _k')=0$ for $j\neq k$ and $ \Omega ( \xi _j' ,\overline{\xi}' _j)=-\im s_j$ with $s_j \in \{ 1 ,-1 \} $ . We have $ \Omega ( \xi _j ', {\xi} _k')=0$ for all $j$ and $k$.
We have $ \Omega (\xi , f )= 0$ for any eigenfunction
$\xi $ and any $f\in X_c ( {p_0} )$.
\end{lemma}
\proof First of all, if $\lambda ,\mu \in \sigma _p(\mathcal{H}_{p_0} ) $ are two eigenvalues with $\lambda
\neq 0$ and given two associated eigenfunctions $\xi _\mu $ and $\xi _\lambda$
\begin{equation} \label{eq:basis1}
\begin{aligned} \langle J^{-1}\xi _\lambda , \overline{\xi} _\mu \rangle &=
\frac 1 \lambda \langle J^{-1}\mathcal{H}_{p_0}\xi _\lambda , \overline{\xi} _\mu \rangle
=-\frac 1 \lambda \langle \mathcal{H}_{p_0}^* J^{-1}\xi _\lambda , \overline{\xi} _\mu \rangle
\\& =-\frac 1 \lambda \langle J^{-1}\xi _\lambda , \mathcal{H}_{p_0} \overline{\xi} _\mu \rangle
=-\frac {\overline{\mu}} \lambda \langle J^{-1}\xi _\lambda , \overline{\xi} _\mu \rangle ,
\end{aligned}
\end{equation}
where for the second equality we used \eqref{eq:begspectdec1} and for the last one the fact that
$
\mathcal{H}_{p_0} \xi = \mu \xi
$ implies $
\mathcal{H}_{p_0} \overline{\xi } = \overline{\mu } \overline{\xi } .
$ Then, for $\mathbf{e} _j\neq \mathbf{e} _k$ and associated eigenfunctions $\xi _j $ and ${\xi} _k$
we get
$ \Omega ( \xi _j ,\overline{\xi} _k)= 0 $. Notice that by a similar argument we have $ \Omega ( \xi _\lambda , {\xi} _\mu )=-\frac \mu \lambda \Omega ( \xi _\lambda , {\xi} _\mu )$ and so $ \Omega ( \xi _j ', {\xi} _k')\equiv 0$ .
\noindent Since $\mathcal{H}_{p_0} \xi = \mathbf{e} \xi
$ implies $
\mathcal{H}_{p_0} ^* J^{-1} \xi =- \mathbf{e} J^{-1}\xi
$, for any eigenfunction
$\xi $ of $\mathcal{H}_{p_0}$ then $J^{-1}\xi
$ is an eigenfunction
of $\mathcal{H}_{p_0}^*$. By the definition of $X_c ( {p_0} )$ in \eqref{eq:spectraldecomp},
we conclude $ \Omega (\xi , f )= \langle J^{-1}\xi , f\rangle = 0$
for any $f\in X_c ( {p_0} )$.
\noindent Let $\im \textbf{e} \in \im \R\backslash \{ 0\}$ be an eigenvalue. By the above discussion, the Hermitian form $\langle \im J^{-1}\xi , \overline{ \eta }\rangle
$ is non degenerate in $ \ker (\mathcal{H}_{p_0} - \im \mathbf{e}
) $. Then we can find a basis such that $\langle \im J^{-1}\eta _j , \overline{ \eta }_k\rangle =- |a_j|\text{sign}(a_j )
\delta _{jk}$, for appropriate non zero numbers $a_j\in \R$. Then set $\xi '=\sqrt{|a _j|}\eta _j.$
\qed
\bigskip
\noindent We set $\xi _j=\xi '_j$ and $\textbf{e} _j=\textbf{e} '_j$
if $s_j=1$.
\noindent We set $\overline{\xi} _j=\xi '_j$ and $\textbf{e} _j=-\textbf{e} '_j$
if $s_j=-1$.
\noindent Notice that if $f\in X_c ( {p_0} )$ then also $\overline{f}\in X_c ( {p_0} )$. This implies that for $R \in N_g^\perp (\mathcal{H}_{p_0} ^*)\otimes _\R \C$ with real entries, that is if $R=\overline{R}$, then we have
\begin{equation}
\label{eq:decomp2}
R (x) =\sum _{j=1}^{\mathbf{n}}z_j \xi _j (x) +
\sum _{j=1}^{\mathbf{n}}\overline{z}_j\overline{\xi }_j( x )
+ f (x), \quad f \in X_c (p_0).
\end{equation}
with $f=\overline{f}$.
\noindent By Lemma \ref{lem:basis} we have, for the $s_j$ of Lemma \ref{lem:basis},
\begin{equation}
\label{eq:H2}
\frac{1}{2} \Omega (\mathcal{H}_{p_0} R, R ) = \sum _{j=1}^\mathbf{n} \mathbf{e} _j |z_j|^2
+\frac{1}{2} \Omega (\mathcal{H}_{p_0} f, f )=:H_2.
\end{equation}
\noindent Consider the map $R\to (z,f)$ obtained from \eqref{eq:decomp2}. In terms of
the pair $(z,f)$, the Fr\'echet derivative $R'$ can be expressed as
\begin{equation*}
R'=\sum _{j=1}^{\mathbf{n}}(dz_j \xi _j+d\overline{z}_j \overline{\xi} _j) +f'.
\end{equation*}
We have
\begin{equation}
\label{eq:OmegaCoo}
\Omega ( R' , R ') =- \im \sum _{j=1}^{\mathbf{n}} dz_j\wedge d \overline{z}_j
+ \Omega ( f' , f ') .
\end{equation}
For a function $F$ independent of $\tau $ and $\Pi$ let us decompose $X_F$ as of spectral decomposition \eqref{eq:decomp2}:
\begin{equation*}
X_F =\sum _{j=1}^{\mathbf{n}}(X_F)_{z_j} \xi _j (x) +
\sum _{j=1}^{\mathbf{n}}(X_F)_{\overline{z}_j}\overline{\xi }_j( x )
+ (X_F)_{f}, \quad (X_F)_{f} \in X_c (p_0).
\end{equation*}
By $i_{X_F}\Omega =dF$ and by
\begin{equation*} \begin{aligned} &
dF= \partial _{z_j}F dz_j+\partial _{\overline{z}_j}F d\overline{z}_j+ \langle \nabla _fF, f'\ \rangle \\&
i_{X_F}\Omega =- \im (X_F)_{z_j} d\overline{z}_j+ \im (X_F)_{\overline{z}_j} dz_j+ \langle J^{-1} (X_F)_{f}, f'\ \rangle ,
\end{aligned}\end{equation*}
we get
\begin{equation*} \begin{aligned} &
(X_F)_{z_j}=\im \partial _{\overline{z}_j}F\ , \quad (X_F)_{\overline{z}_j}=-\im \partial _{z_j}F\ , \quad (X_F)_{f}=J\nabla _fF.
\end{aligned}\end{equation*}
This implies
\begin{equation} \label{eq:poiss}\begin{aligned} & \{ F,G \} :=dF(X_G) = \im \partial _{z_j}F\partial _{\overline{z}_j}G
-\im \partial _{\overline{z}_j}F\partial _{z_j}G + \langle \nabla _fF, J\nabla _fG \rangle.
\end{aligned}\end{equation}
Hence, for $H_2$ defined in \eqref{eq:H2}, for $z=(z_1,....,z_\mathbf{n})$, using standard multi index notation and by \eqref{eq:begspectdec1},
we have:
\begin{equation} \label{eq:poiss1}\begin{aligned} & \{ H_2,z^\mu \overline{z} ^\nu \} =-\im \mathbf{e} \cdot (\mu - \nu ) z^\mu \overline{z}^\nu \ ; \quad \{ H_2, \langle J ^{-1}\varphi ,f\rangle \} = \langle J ^{-1}\mathcal{H}\varphi ,f\rangle .
\end{aligned}\end{equation}
\subsection{Flows in spectral coordinates}
\label{subsec:flowsspec}
We restate Lemma \ref{lem:ODE} for a special class of transformations.
\begin{lemma}
\label{lem:chi}
Consider
\begin{equation}
\label{eq:chi1}\chi =\sum _{|\mu +\nu |=M_0 +1} b_{\mu\nu} (\Pi (f)) z^{\mu} \overline{z}^{\nu} + \sum _{|\mu +\nu |=M_0 } z^{\mu} \overline{z}^{\nu}
\langle J ^{-1} B_{\mu \nu
}(\Pi (f))
, f \rangle
\end{equation}
with $ b_{\mu\nu}(\varrho)= \resto ^{i,0} _{r,M}(\varrho)$ and $ B_{\mu\nu}(\varrho)= \textbf{S} ^{i,0} _{r,M}(\varrho)$ with $i\in \{ 0,1\}$ fixed and $r,M\in \N$ sufficiently large and with
\begin{equation}
\label{eq:symm} \overline{b}_{\mu\nu} = {b}_{\nu\mu} \ , \quad \overline{B}_{\mu \nu
} =B_{\nu\mu} ,
\end{equation}
(so that $\chi$ is real valued for $f=\overline{f}$).
Then we have what follows.
\begin{itemize}
\item[(1)]
Consider the vectorfield $X_\chi $ defined
with respect to $\Omega _0$.
Then, summing on repeated indexes (with the equalities defining the field $X_\chi ^{st}$), we have:
\begin{equation*} \begin{aligned} &(X_\chi ) _{z_j}= \im \partial _{\overline{z}_j}
\chi =: (X_\chi ^{st} ) _{z_j} \, , \quad (X_\chi ) _{\overline{z}_j}= -\im \partial _{z_j}
\chi =: (X_\chi ^{st} ) _{\overline{z}_j} \, , \\& (X_\chi ) _{f} =\partial _{\Pi _j(f)}\chi \, P_c^*(p_0)
J \Diamond _j f +
(X_\chi ^{st} ) _{f} \text{ where } (X_\chi ^{st} ) _{f}:=z^{\mu} \overline{z}^{\nu}B_{\mu \nu
}(\Pi (f)).
\end{aligned}
\end{equation*}
\item[(2)]
Denote by $\phi ^t$ the flow of $X_\chi$ provided by Lemma \ref{lem:ODE} and
set $(z^t,f^t)= (z,f)\circ \phi ^t$. Then we have
\begin{equation} \label{eq:quasilin51}
\begin{aligned} & z^t = z +
\mathcal{Z}(t) \, \, \quad
& f^t =e^{Jq(t )\cdot \Diamond } ( f+ \textbf{S}(t ) )
\end{aligned}
\end{equation}
where, for $(k,m)$ with $k\in \Z\cap [0,r-(m+1)\textbf{d}]$
and $1\le m \le M$,
for $ B_{\Sigma _{-k}}$
a sufficiently small neighborhood of 0 in $ \Sigma _{-k}\cap X_c(p_0) $ and for $B_{\C ^{\mathbf{n} }} $ (resp.$B_{\R ^{n_0}}$)
a neighborhood of 0 in $\C ^{\mathbf{n} }$ (resp.$ {\R ^{n_0}}$)
\begin{equation} \label{eq:ODEpr21}\begin{aligned} &
\textbf{S} \in C^m((-2,2)\times B_{\C ^{\mathbf{n} }} \times B_{\Sigma _{-k}}
\times B_{\R ^{n_0}} , \Sigma _{k}
) \\& {q} \in C^m((-2,2)\times B_{\C ^{\mathbf{n} }} \times B_{\Sigma _{-k}}
\times B_{\R ^{n_0}} , \R ^{ n_0}
) \\ & \mathcal{Z } \in C^m((-2,2)\times B_{\C ^{\mathbf{n} }} \times B_{\Sigma _{-k}}
\times B_{\R ^{n_0}} , \C ^{ \mathbf{n}}
),
\end{aligned} \end{equation}
with for fixed $C$
\begin{equation} \label{eq:symbol11} \begin{aligned} & | q (t,z,f,\varrho ) |\le C (|z|+\| f\| _{\Sigma _{-k}}) ^{M_0+1} \\&
|\mathcal{Z} (t,z,f,\varrho ) |+ \| \textbf{S} (t,z,f,\varrho ) \| _{\Sigma _{k}} \le C (|z|+\| f\| _{\Sigma _{-k}}) ^{M_0 } .
\end{aligned} \end{equation}
We have $ \textbf{S} (t,z,f,\varrho )= \textbf{S}_1 (t,z,f,\varrho )+ \textbf{S}_2 (t,z,f,\varrho )$ with
\begin{equation} \label{eq:symbol12} \begin{aligned} & \textbf{S}_1 (t,z,f,\varrho ) =\int _0^t
(X_\chi ^{st} ) _{f}\circ \phi ^{t'} dt' \\&
\| \textbf{S}_2 (t,z,f,\varrho ) \| _{\Sigma _{k}} \le C (|z|+\| f\| _{\Sigma _{-k}}) ^{2M_0+1 }(|z|+\| f\| _{\Sigma _{-k}}+|\varrho |)^i .
\end{aligned} \end{equation}
\item[(3)] The flow $\phi ^t$ is canonical: for $s,s',k$ as in Lemma \ref{lem:ODE}, the map $\phi ^t \in C^l( \U ^{s'}_{\varepsilon _1,k}
, \widetilde{\Ph} ^{s })$
satisfies
$\phi ^{t*}\Omega _0=\Omega _0$
{ in } $C^{\infty}( \U ^{s'}_{\varepsilon _2,k}
, B^2 (\widetilde{ \Ph}^{s'}, \R )
) $ for $\varepsilon _2>0$ sufficiently small.
\end{itemize}
\end{lemma}
\proof First of all notice that $\chi$ does not depend on $\tau$ and $\Pi$ so that the only nonzero component
of $X_\chi$ is $(X_\chi )_R=J\nabla _R\chi $. The latter is of the form indicated in claim (1) by a direct
computation. Claim (2) follows now by Lemma \ref{lem:ODE}.
\noindent To prove Claim (3) we need to make rigorous the following formal computation
\begin{equation*}
\frac{d}{dt}\phi ^{t*}\Omega _0=\phi ^{t*}L_{X_\chi}\Omega _0=\phi ^{t*}di_{X_\chi}\Omega _0=\phi ^{t*}d^2\chi =0.
\end{equation*}
To make sense of this we can proceed as in Corollary \ref{cor:darboux}. We skip the proof.
\qed
\begin{lemma}
\label{lem:ExpH11} Consider a transformation
$\mathfrak{F} =\mathfrak{F}_1 \circ \cdots \circ \mathfrak{F}_L$
like in Lemma \ref{lem:ODE1} and with $m_1=2$ and for fixed $r$ and $M$ sufficiently large.
Denote by $(k',m')$ the pair $(k,m)$ of Lemma \ref{lem:back11} and
consider a pair $(k,m)$ with $k\le k'$ and $m\le m' -(2\textbf{N}+5)$.
Set $H':=K\circ \mathfrak{F} $. Consider decomposition
\eqref{eq:decomp2}. Then on a domain $\U^s_{\varepsilon , k} $ like \eqref{eq:domain0} we have
\begin{equation} \label{eq:ExpH11} \begin{aligned} & H '= { \psi} (\Pi (f)) +H_2' +\textbf{R}\ ,
\end{aligned}
\end{equation} for a $\psi \in C^\infty $
with $\psi (\varrho )=O(|\varrho|^2)$ near 0
and with what follows.\begin{itemize}
\item[(1)]
We have
\begin{equation} \label{eq:ExpH2} H_2 '=
\sum _{\substack{ |\mu +\nu |=2\\
\mathbf{e} \cdot (\mu -\nu )=0}}
a_{\mu \nu} ( \Pi (f) ) z^\mu
\overline{z}^\nu + \frac{1}{2} \langle J ^{-1} \mathcal{H}_{p
_0} f, f\rangle .
\end{equation}
\item[(2)] We have $\textbf{R} = {\textbf{R} _{ -1 }} + {\textbf{R} _{0}} + {\textbf{R} _1 } +
{\textbf{R} _2} +\resto ^{1,2}_{k,m+2}(\Pi (f), f)+
{\textbf{R} _3} +
{\textbf{R} _4} $, with:
\begin{equation*} \begin{aligned} &{\textbf{R} _{ -1 }}=
\sum _{\substack{ |\mu +\nu |=2\\
\mathbf{e} \cdot (\mu -\nu )\neq 0 }} a_{\mu \nu } (\Pi (f)
)z^\mu
\overline{z}^\nu +\sum _{|\mu +\nu | = 1} z^\mu
\overline{z}^\nu \langle J ^{-1}G_{\mu \nu }(\Pi (f)
),f\rangle ;\end{aligned}
\end{equation*}
For $\mathbf{N}$ as in (L4) of this section,
\begin{equation*} \begin{aligned} & {\textbf{R} _0}= \sum _{|\mu +\nu |= 3}^{2\mathbf{N}+1} z^\mu
\overline{z}^\nu a_{\mu \nu
}( \Pi (f) ) ;
\end{aligned}
\end{equation*}
\begin{equation*} \begin{aligned} & {\textbf{R} _1}= \sum _{|\mu +\nu
|= 2}^{2\mathbf{N} } z^\mu
\overline{z}^\nu \langle J ^{-1} G_{\mu \nu }( \Pi (f)
), f\rangle ;
\end{aligned}
\end{equation*}
\begin{equation*} \begin{aligned} & {\textbf{R} _2}=
\langle \mathbf{B}_{2 } ( \Pi (f) ), f ^{ 2} \rangle
\text{ with $\mathbf{B}_{2 } (0)=0$}
\end{aligned}\nonumber
\end{equation*}
where $f^d(x)$ represents schematically $d-$products of components of $f$;
\begin{equation*} \begin{aligned} & {\textbf{R} _3}= \sum _{ \substack{ |\mu +\nu |=\\= 2N+2}} z^\mu
\overline{z}^\nu a_{\mu \nu
}( z,f,\Pi (f) ) +\sum _{ \substack{ |\mu +\nu
|=\\= 2N+1}} z^\mu
\overline{z}^\nu \langle J ^{-1} G_{\mu \nu }(z,f, \Pi (f)
), f\rangle ;
\end{aligned}
\end{equation*}
\begin{equation*} \begin{aligned} & {\textbf{R} _4}= \sum _{d=2}^4
\langle B_{d } ( z ,f,\Pi (f) ), f ^{ d} \rangle
+\int _{\mathbb{R}^3}
B_5 (x, z ,f, f(x),\Pi (f) ) f^{ 5}(x) dx\\& + \widehat{\textbf{R}} _2( z ,f,\Pi (f))+ E_P ( f) \text{ with $B_2(0,0,\varrho )=0$.}
\end{aligned}\nonumber
\end{equation*}
\item[(3)] For $\delta _j:=( \delta _{1j}, ..., \delta _{mj}),$
\begin{equation} \label{eq:ExpHcoeff1} \begin{aligned} &
a_{\mu \nu } ( 0 ) =0 \text{ for $|\mu +\nu | = 2$ with $(\mu
, \nu )\neq (\delta _j, \delta _j)$ for all $j$,} \\& a_{\delta _j
\delta _j } ( 0 ) =\lambda _j (\omega _0) ,
\\& G_{\mu \nu }( 0 ) =0 \text{ for $|\mu +\nu | = 1$ }.
\end{aligned}
\end{equation}
These $a_{\mu \nu } ( \varrho )$ and $G_{\mu \nu }( x,\varrho
)$ are $C ^{m }$ in all variables with $G_{\mu \nu }( \cdot ,\varrho )
\in C^m ( \mathrm{U},\Sigma _k (\mathbb{R}^3,\mathbb{C}^{2N }))$,
for a
small neighborhood $\mathrm{U}$ of $( 0,0,0)$ in
$ \mathbb{C}^{ \mathbf{n}}\times (\Sigma _{-k}\cap X_c(p_0))\times \R ^{n_0}$
(the space of the $(z,f,\varrho )$),
and they satisfy
symmetries analogous to \eqref{eq:symm}.
\item[(4)] We have
$a_{\mu \nu
}( z, \varrho ) \in C^{ m }( \mathrm{U},
\mathbb{C} ) $ .
\item[(5)] $G_{\mu \nu
}( \cdot , z, \varrho ) \in C^{ m } ( \mathrm{U},
\Sigma _k (\mathbb{R}^3,\mathbb{C}^{2N }))) $.
\item[(6)] $B_{d
}( \cdot , z,f,\varrho ) \in C^{m } ( \mathrm{U},
\Sigma _k (\mathbb{R}^3, B (
(\mathbb{C}^{2N })^{\otimes d},\mathbb{R} ))) $, for $2\le d \le 4$.
$\mathbf{B}_{2
}( \cdot , \varrho )$ satisfies the same property.
\item[(7)] Let
$ \zeta \in \mathbb{C}^{2N }$. Then for
$B_5(\cdot , z ,f, \zeta ,\varrho )$ we have (the derivatives are not in the holomorphic sense)
\begin{equation} \label{H5power2}\begin{aligned} &\text{for $|l|\le m $ ,
} \| \nabla _{ z ,f,\zeta, \varrho }
^lB_5( z,f,\zeta ,\varrho ) \| _{\Sigma _k(\mathbb{R}^3, B (
(\mathbb{R}^{2N })^{\otimes 5},\mathbb{R} )} \le C_l .
\end{aligned}\nonumber \end{equation}
\item[(8)]
\begin{equation}\label{eq:Rhat0}\begin{aligned} &
\widehat{\textbf{R}} _2
\in C^{m} ( \mathrm{U} ,\C
), \\& | \widehat{\textbf{R}} _2 (z ,f, \varrho
)| \le C (|z|+ \| f \| _{\Sigma _{-k}}) \| f \| _{\Sigma _{-k}}^2;
\end{aligned}\end{equation}
\end{itemize}
\end{lemma}
\proof We need to express $R$ in terms of $(z,f)$
using \eqref{eq:decomp2} inside \eqref{eq:back11}.
\noindent We have $\Pi (R)= \Pi (f) +\resto ^{0,2}(R).$ Then, succinctly,
\begin{equation*}\begin{aligned}
&
\resto ^{1,2}_{k',m' }(\Pi (R),R)=\sum _{a+b=2} ^{2\textbf{N}+1} \frac{1}{a!b!} \langle
\nabla ^a_{ \varrho } \nabla ^b_{ R }\resto ^{1,2}_{k',m' }(\Pi (f),0) , (\resto ^{0,2}(R) )^{a } R ^{b\otimes} \rangle + \\& \sum _{\substack{a+b\\ =2\textbf{N}+2}} \int _0^1
\frac{(1-t)^{2\textbf{N}+1}}{a!b!} \langle \nabla ^a_{ \varrho } \nabla ^b_{ R }
\resto ^{1,2}_{k',m' }(\Pi (f)+t\resto ^{0,2}(R),tR) , (\resto ^{0,2}(R) )^{a } R ^{b\otimes} \rangle dt,
\end{aligned}
\end{equation*}
with $(k',m')$ the pair $(k,m)$ of Lemma \ref{lem:back11}.
We substitute \eqref{eq:decomp2}, that is $R=z\cdot \xi +\overline{z}\cdot \overline{\xi} +f$.
For $m\le m' -(2\textbf{N}+2)$ and $k\le k'$,
the terms from the $R^{b\otimes}$
of degree in $f$ at most 1, go into $\textbf{R}_i$ with $i=-1,0,1,3$ and $H'_2$.
For $m\le m' -(2\textbf{N}+4)$,
the remaining terms are absorbed in $\resto ^{1,2}_{k ',m+2 }(\Pi (f),f)+\widehat{\textbf{R}}_2(z,f,\Pi (f)) $.
\noindent We focus now on the $d=5$ term in \eqref{eq:back11}. We substitute $R=z\cdot \xi +\overline{z}\cdot \overline{\xi} +f$.
This schematically yields, for a $\widetilde{B}_5$ satisfying claim (7) with the pair $(m',k')$,
\begin{equation}\label{eq.b5}\begin{aligned}
& \sum _ {j=0}^5\int _{\mathbb{R}^3}
\widetilde{B}_5 (x, z, f, f(x),\Pi (f) ) (z\cdot \xi +\overline{z}\cdot \overline{\xi} ) ^{5-j} f^j ( x) dx
. \end{aligned}
\end{equation}
For $j=5$ we get a term that can be absorbed in the $B_5$ term in $\textbf{R} _4$. Expand the $j<5$ terms
in \eqref{eq.b5} as
\begin{equation*}\label{eq.b51}\begin{aligned}
& \sum _ {i=0}^{4-j}\int _{\mathbb{R}^3} \frac{1}{i!} (\partial _t^i ) _{|t=0}
\widetilde{B}_5 (x, z, f,t f(x),\Pi (f) ) (z\cdot \xi +\overline{z}\cdot \overline{\xi} ) ^{5-j} f^{i+j } ( x) dx+\\&
\int _{\mathbb{R}^3} \frac{1}{(4-j)!} \int _0^1 \partial _t^{5-j} [\widetilde{B}_5 (x, z, f,t f(x),\Pi (f) )] (z\cdot \xi +\overline{z}\cdot \overline{\xi} ) ^{5-j} f^5 ( x) dx
. \end{aligned}
\end{equation*} go into the $B_d$ term in $\textbf{R} _4$
The last term fits in the $B_5$ term in $\textbf{R} _4$ by $m\le m'-5$. The terms in the first line
go into the $B_d$ of $\textbf{R}_4$
for $d=i+j\ge 2$ . The terms with $ i+j< 2$ can be treated like the $\resto ^{1,2}_{k',m' }(\Pi (R),R)$ for
$m\le m' -(2\textbf{N}+5)$ and $k\le k'$.
\noindent We focus on $E_P(R)=E_P(z\cdot \xi +\overline{z}\cdot \overline{\xi} +f)$. We use Lemma \ref{lem:Taylor ex} for $\textbf{v}=f$ and $ \textbf{u}= z\cdot \xi +\overline{z}\cdot \overline{\xi}.$ Then
\begin{equation*} \begin{aligned} & E_P( R )= E_P( f ) + E_P( z\cdot \xi +\overline{z}\cdot \overline{\xi} ) +\\& \int _{\R ^3}dx
\sum _{ j =0}^{3}\int _{[0,1]^2} \frac{t^j}{j!} (\partial _t^{j+1})_{|t=0} \partial _s [B (|s ( z\cdot \xi +\overline{z}\cdot \overline{\xi} )+tf |^2_1) ] dt ds \\& + \int _{\R ^3}dx \int _{[0,1]^2} dt ds \int _0^t \partial _\tau ^5 \partial _s [B (|s ( z\cdot \xi +\overline{z}\cdot \overline{\xi} ) +\tau f |_1^2)] \frac{(t-\tau )^3}{3!} d\tau .
\end{aligned}
\end{equation*}
By $B(0)=B'(0)=0$, we have $E_P( z\cdot \xi +\overline{z}\cdot \overline{\xi} )=\resto ^{0,4} (R)$.
It is easy to conclude that this term easily fits into $\textbf{R}_0 +\textbf{R}_3$. Similarly, the $j=0$ term
fits in $ \textbf{R}_1+\textbf{R}_3$. The $j\ge 1$ terms
fit in the $B_{j+1}$ term in $ \textbf{R}_4$. The last line fits in the $B_ {5}$ term in $ \textbf{R}_4$.
\noindent The symmetries \eqref{eq:symm} for the coefficients in $H'_2+\textbf{R}_{-1}+\textbf{R}_0+ \textbf{R}_1 $ are an elementary consequence of the fact that $H'$ is real valued.
\qed
\begin{remark}
\label{rem:differences}
Given a Hamiltonian $H'$ expanded as in Lemma \ref{lem:ExpH11} and given
a transformation $\mathfrak{F} $, we cannot obtain the expansion of
Lemma \ref{lem:ExpH11} for $H'\circ\mathfrak{F}$ analysing one by one the terms of the expansion of
$H'$. This works in the set up of \cite{Cu1,Cu2} but not here (see in particular the discussion on the exponential under formula \eqref{eq:key-201}
later).
\end{remark}
\section{ Birkhoff normal forms}
\label{sec:Normal form}
In this section we arrive at the main result of the paper.
\subsection{Homological equations}
\label{subsec:homological}
We consider $a_{\mu \nu}^{(\ell )} ( \varrho )\in C ^{ \widehat{m}} (U,C)$ for $k_0\in \N$ a fixed number and $U$ a neighborhood of 0 in $\R ^{n_0}$.
Then we set
\begin{equation} \label{eq:H2birk} H_2^{(\ell )} (\varrho ):=
\sum _{\substack{ |\mu +\nu |=2\\
\mathbf{e} \cdot (\mu -\nu )=0}}
a_{\mu \nu}^{(\ell )} ( \varrho ) z^\mu
\overline{z}^\nu + \frac{1}{2} \langle J^{-1} \mathcal{H} f, f\rangle .
\end{equation}
\begin{equation} \label{eq:lambda}
\textbf{e} _j ( \varrho ) := a
_{\delta _j\delta _j} ^{(\ell )}(\varrho ), \quad \textbf{e} (\varrho )=
(\lambda _1(\varrho ), \cdots, \lambda _m(\varrho )).\end{equation}
We assume $\textbf{e} _j (0 ) = \textbf{e} _j $ and $a_{\mu \nu}^{(\ell )} ( 0 ) =0 $ if $(\mu ,\nu)\neq (\delta _j,\delta _j)$ for
all $j$, with $\delta _j$ defined in \eqref{eq:ExpHcoeff1}.
\begin{definition}
\label{def:normal form} A function $Z( z,f, \varrho )$ is in normal form if $
Z=Z_0+Z_1
$
where $Z_0$ and $Z_1$ are finite sums of the following type:
\begin{equation}
\label{e.12a}Z_1= \sum _{\mathbf{e} (0) \cdot(\nu-\mu)\in \sigma _e(\mathcal{H} _{p_0})}
z^\mu \overline{z}^\nu \langle J^{-1} G_{\mu \nu}( \varrho ),f\rangle
\end{equation}
with $G_{\mu \nu}( x ,\varrho )\in C^{m} ( U,\Sigma _{k }(\R ^3, \C ^{2N}))$ for fixed $k,m\in \N$ and $U\subseteq \R ^{n_0}$ a neighborhood of 0;
\begin{equation}
\label{e.12c}Z_0= \sum _{ \mathbf{e}(0) \cdot(\mu-\nu)=0} g_{\mu \nu}
( \varrho )z^\mu \overline{z}^\nu
\end{equation}
and $g_{\mu \nu} ( \varrho )\in C^{m} ( U,
\mathbb{C})$.
We assume furthermore that the above coefficients
satisfy the symmetries in \eqref{eq:symm}: that is $\overline{g}_{\mu \nu}=g_{\nu\mu }$ and $\overline{G}_{\mu \nu}=G_{\nu \mu }$.
\end{definition}
\begin{lemma}
\label{lem:NLhom1} We consider $\chi =\chi (b,B)$ with
\begin{equation}
\label{eq:chi}\chi (b,B) =\sum _{|\mu +\nu |=M_0+1} b_{\mu\nu} z^{\mu} \overline{z}^{\nu} + \sum _{|\mu +\nu |=M_0} z^{\mu}
\overline{z}^{\nu}
\langle J^{-1} B_{\mu \nu
}
, f \rangle
\end{equation}
for $b_{\mu\nu} \in \C $ and $B_{\mu \nu
} \in \Sigma _{\widehat{k} }(\R ^3, \C ^{2N})\cap X_c(p_0)$ with $\widehat{k} \in \N$, satisfying the symmetries in \eqref{eq:symm}. Here we interpret
the polynomial $\chi$ as a function with parameters $b=(b_{\mu\nu})$ and $B=(B_{\mu\nu})$. Denote by $X_{\widehat{k} }$ the space of the pairs $(b,B)$.
Let us also consider given polynomials with $K=K( \varrho ) $ and $\widetilde{K}=\widetilde{K}( \varrho ,b,B ) $ where:
\begin{equation}\label{eq:Krho}
K(\varrho ) :=\sum _{|\mu +\nu |=M_0+1} k_{\mu\nu} ( \varrho ) z^{\mu} \overline{z}^{\nu} + \sum _{|\mu +\nu |=M_0} z^{\mu} \overline{z}^{\nu}
\langle J^{-1} K_{\mu \nu
}(\varrho )
, f \rangle ,
\end{equation}
with $k_{\mu\nu} ( \varrho ) \in C^ {\widehat{m} } ( U, \C )$ and $K_{\mu\nu} ( \varrho ) \in C^{\widehat{m} } ( U, \Sigma _{\widehat{k}}(\R ^3, \C ^{2N}) \cap X_c(p_0))$ for $U$ a neighborhood of 0 in $\R ^{n_0}$,
satisfying the symmetries in \eqref{eq:symm};
\begin{equation}\label{eq:tildeKrho} \begin{aligned} &\widetilde{K} ( \varrho ,b,B ) :=
\sum _{|\mu +\nu |=M_0+1} \widetilde{k}_{\mu\nu} ( \varrho , b,B ) z^{\mu} \overline{z}^{\nu} \\&+
\sum _{i=0}^1\sum _{j=1}^{n_0}
\sum _{|\mu +\nu |=M_0} z^{\mu}
\overline{z}^{\nu}
\langle J ^{-1} \Diamond ^i_j K_{j\mu \nu
}^i(\varrho , b,B )
, f \rangle ,\end{aligned}
\end{equation}
with $\widetilde{k}_{\mu\nu} \in C^{\widehat{m}} ( U \times X _{ \widehat{k} }, \R )$ and $\widetilde{K}_{j\mu\nu}^{i} \in C^{\widehat{m}} ( U\times X_{\widehat{ k} }, \Sigma _{ \widehat{k} }(\R ^3, \C ^{2N}) \cap X_c(p_0))$,
satisfying the symmetries in \eqref{eq:symm}.
Suppose also that the sums \eqref{eq:Krho} and \eqref{eq:tildeKrho} do not
contain terms in normal form and that $\widetilde{K}( 0 ,b,B )=0$. Then there exists
a neighborhood $V\subseteq U$ of 0 in $\R ^{n_0}$ and a unique choice of functions
$(b(\varrho ), B(\varrho )) \in C ^{\widehat{m}} (V,X _{\widehat{k}} )$ such that for $\chi (\varrho )
=\chi (b(\varrho ), B(\varrho ))$, $\widetilde{K}(\varrho ) =\widetilde{K} (\varrho ,b(\varrho ),B(\varrho ))$ we have
\begin{equation}
\label{eq:homologicalEq} \left\{ \chi (\varrho ) ,H_2 ^{(\ell )}(\varrho ) \right \} ^{st}= K (\varrho)+\widetilde{K}(\varrho ) +Z(\varrho )
\end{equation}
where $\{
\cdots \} ^{st} $ is the bracket \eqref{eq:poiss} for $\varrho$
fixed and
where $Z(\varrho )$ is in normal form and homogeneous of degree $M_0+1$ in $(z,f)$.
\end{lemma}
\proof Summing on repeated indexes, by \eqref{eq:poiss1} we get \begin{equation} \label{eq:homologicalEq1} \begin{aligned}& \{
H_2^{(\ell )}, \chi \} ^{st} = -\im \mathbf{e} ( \varrho )\cdot (\mu -
\nu) z^{\mu} \overline{z}^{\nu} b_{\mu \nu }(\varrho ) \\&- z^{\mu} \overline{z}^{\nu}\langle f , J ^{-1}( \im \mathbf{e} ( \varrho )\cdot (\mu -
\nu)- \mathcal{H} )B _{\mu \nu }(\varrho )
\rangle + \widehat{ K} (\varrho ,b(\varrho ),B(\varrho )) ,
\end{aligned}
\end{equation}
\begin{equation} \label{eq:homologicalEq2} \begin{aligned} \widehat{ K} (\varrho , b,B) :&= \sum _{\substack{|\mu +\nu |=2\\ (\mu , \nu )\neq (\delta _j , \delta _j ) \ \forall \ j} } a_{\mu \nu}^{(\ell )}(\varrho )
\big [\sum _{|\mu '+\nu '|=M_0 +1}
\{z^\mu
\overline{z}^\nu ,z ^{\mu'}
\overline{z}^{ \nu '} \} b_{\mu ' \nu '} \\& +
\sum _{|\mu '+\nu '|=M_0 }
\{z^\mu
\overline{z}^\nu ,z ^{\mu'}
\overline{z}^{ \nu '} \} \langle J^{-1} B_{\mu ' \nu '}
, f \rangle
\big ] .
\end{aligned}
\end{equation}
$\widehat{ K} $ is a homogeneous polynomial of the same type of the above ones and we have $\widehat{ K} (0 , b,B) =0$. In particular, $\widehat{ K} $ satisfies the symmetries \eqref{eq:symm} by (for $f=\overline{f}$)
\begin{equation*} \begin{aligned} & ( a_{\mu \nu}^{(\ell )} b_{\mu ' \nu '}
\{z^\mu
\overline{z}^\nu ,z ^{\mu'}
\overline{z}^{ \nu '} \} )^* = a_{\nu \mu }^{(\ell )} b_{\nu '\mu ' }
\{z^\nu
\overline{z}^\mu ,z ^{\nu'}
\overline{z}^{ \mu '} \} \,
\\& ( a_{\mu \nu}^{(\ell )} \langle J^{-1} B_{\mu ' \nu '}
, f \rangle
\{z^\mu
\overline{z}^\nu ,z ^{\mu'}
\overline{z}^{ \nu '} \} )^* = a_{\nu \mu }^{(\ell )} \langle J^{-1} B_{\nu ' \mu '}
, f \rangle
\{z^\nu
\overline{z}^\mu ,z ^{\nu'}
\overline{z}^{ \mu '} \}
\end{aligned}
\end{equation*}
which follow by $(\im \partial _{z_j}F\partial _{\overline{z}_j}G
-\im \partial _{\overline{z}_j}F\partial _{z_j}G)^*=\im \partial _{z_j}F^*\partial _{\overline{z}_j}G^*
-\im \partial _{\overline{z}_j}F^*\partial _{z_j}G^*$, where in these formulas $a^*=\overline{a}$, and by the symmetries \eqref{eq:symm} for $\chi$ and for $H_2^{(\ell )}$ .
\noindent Denote by $\widehat{ Z} (\varrho , b,B) $ the sum of
monomials in normal form of $\widetilde{{K}}$ and set $\textbf{K } :=\widetilde{{K}} +\widehat{ K} -\widehat{ Z} $.
We look at \begin{equation} \label{eq:homologicalEq10} \begin{aligned} & - \im \mathbf{e} (\varrho )\cdot (\mu -
\nu) z^{\mu} \overline{z}^{\nu} b_{\mu \nu } -z^{\mu} \overline{z}^{\nu} \langle f , J ^{-1}(\im \mathbf{e} (\varrho ) \cdot (\mu -
\nu)- \mathcal{H} )B _{\mu \nu }
\rangle \\& + \textbf{K }(\varrho ,b ,B ) +K(\varrho ) =0
\end{aligned}
\end{equation}
that is at
\begin{equation} \label{eq:NLhom11}\begin{aligned}&
k_{\mu \nu} (\varrho ) + \textbf{{k}}_{\mu \nu}(\varrho ,b,B) - b_{\mu \nu}(\varrho ) \im \mathbf{e} (\varrho )
\cdot (\mu - \nu)=0 \\& B_{\mu \nu } (\varrho ) =- R_{\mathcal{H}}(\im \mathbf{e} (\varrho ) \cdot (\mu - \nu) )\left [ {K}_{\mu \nu }(\varrho ) + \textbf{{K}} _{\mu \nu}(\varrho ,b,B)\right ]
,
\end{aligned}
\end{equation}
with $\textbf{{k}}_{\mu \nu}$ and $\textbf{{K}} _{\mu \nu}$ the coefficients of $\textbf{K}$.
Notice that when $ \textbf{k}_{\mu \nu}(0 ,b,B) =0$ and $\textbf{K}_{\mu \nu}(0 ,b,B) =0$, for $\varrho = 0$ there is a unique solution $(b,B)\in X_{ \widehat{k} }$ given by
\begin{equation} \label{eq:NLhom12}\begin{aligned}&
b_{\mu \nu}(0)
=\frac{ k_{\mu\nu}(0)}{\im
\mathbf{e} \cdot(\mu-\nu)} \, , \quad B_{\mu \nu } (0 ) =- R _{\mathcal{H}} (\im \mathbf{e} \cdot
(\mu -\nu ) ) K_{\mu \nu
}(0) .
\end{aligned}
\end{equation}
Lemma \ref{lem:NLhom1} is then a consequence of the Implicit Function Theorem by Hypothesis (L7) in Sect. \ref{sec:speccoo}.
\qed
In the particular case $M_0=1$ we need a slight variation of Lemma \ref{lem:NLhom1}.
\begin{lemma}
\label{lem:NLhom2} Suppose now $M_0=1$ and assume the notation of Lemma \ref{lem:NLhom1},
assuming $K(0 )=0$, $\widetilde{K}( 0 ,0,0 )=0$ and $\nabla _{b,B}\widetilde{K}( 0 ,0,0 )=0$. We furthermore
consider function $a_{\mu \nu}^{\mu ' \nu '} \in C^{\widehat{m}} (U\times X_{\widehat{k}}, \C)$
with $| a_{\mu \nu}^{\mu ' \nu '} (\varrho , b, B)|\le C \| (b, B)\| _{X_{\widehat{k}}}$ and we set
\begin{equation}
\label{eq:modop} \begin{aligned} & \left\{ \chi (\varrho ) ,H_2 ^{(\ell )}(\varrho ) \right \} ^{\widetilde{st}}=
\left\{ \chi (\varrho ) ,H_2 ^{(\ell )}(\varrho ) \right \} ^{st} \\& +
\sum _{\substack{|\mu +\nu |=1\\ |\mu '+\nu '|=1}}
a_{\mu \nu}^{\mu ' \nu '} ( \varrho ,b(\varrho ),B(\varrho )) z^{\mu} \overline{z}^{\nu}\langle
\mathcal{H} B_{\mu ' \nu '}(\varrho ), f
\rangle .\end{aligned}
\end{equation}
Then, the same conclusions of Lemma \ref{lem:NLhom1} hold for
\begin{equation}
\label{eq:modhomologicalEq} \left\{ \chi (\varrho ) ,H_2 ^{(\ell )}(\varrho ) \right \} ^{\widetilde{st}}= K (\varrho)+\widetilde{K}(\varrho ) +Z(\varrho ) .
\end{equation}
\end{lemma}
\proof Like above we get to
\begin{equation*} \label{eq:NLhom111}\begin{aligned}&
k_{\mu \nu} (\varrho ) + \textbf{{k}}_{\mu \nu}(\varrho ,b,B) - b_{\mu \nu} \im \mathbf{e} (\varrho )
\cdot (\mu - \nu)=0 \\& B_{\mu \nu } =- R_{\mathcal{H}}(\im \mathbf{e} (\varrho ) \cdot (\mu - \nu) ) [ {K}_{\mu \nu }(\varrho ) + \textbf{{K}} _{\mu \nu}(\varrho ,b,B) + \sum _{\mu ' \nu '}a_{\mu \nu}^{\mu ' \nu '}(\varrho , b ,B ) \mathcal{H} B_{\mu ' \nu '} ]
.
\end{aligned}
\end{equation*}
For $(\varrho , b, B) =(0 ,0,0) $ both sides are 0. Then
Lemma \ref{lem:NLhom2} follows by Implicit Function Theorem.
\qed
\subsection{The Birkhoff normal forms}
\label{subsec:mainbirkhoff}
Our goal in this section is to prove the
following result where $N$ is as of (L4) in Sect. \ref{sec:speccoo}.
\begin{theorem}
\label{th:main} For any integer $2\le \ell \le 2\mathbf{N}+1$
we have transformations $\mathfrak{F} ^{(\ell )} = \mathfrak{F}_1 \circ \phi _2\circ ...\circ \phi _\ell $, with $\mathfrak{F}_1$ the transformation in
Corollary \ref{cor:darboux}
the $\phi _j$'s
like in Lemma \ref{lem:chi}, such that
the conclusions of Lemma \ref{lem:ExpH11} hold,
that is such that we have the following expansion
\begin{equation*}
H ^{(\ell )}:=K\circ \mathfrak{F} ^{(\ell )} ={ \psi} (\Pi (f))+ H_2^{(\ell )} + \resto^{1,2}_{k,m+2}(\Pi (f) , f)+ \sum _{j=-1}^4 \textbf{R}_{j}^{(\ell )},
\end{equation*}
with $H_2^{(\ell )}$ of the form \eqref{eq:ExpH2} and with the following additional properties:
\begin{itemize}
\item[(i)] $\textbf{R} _{-1} ^{(\ell )} =0$;
\item[(ii)] all the nonzero terms in $\textbf{R} _0 ^{(\ell )} $ with $|\mu +\nu |\le \ell $ are in normal form,
that is $\lambda \cdot (\mu -\nu )=0$;
\item[(iii)] all the nonzero terms in $\textbf{R} _1^{(\ell )} $ with $|\mu +\nu |\le \ell -1$ are in normal form,
that is $\lambda \cdot (\mu -\nu )\in \sigma _e(\mathcal{H} _{p_0})$.
\end{itemize}
\end{theorem}
\proof The proof of Theorem \ref{th:main} is by induction. There are two distinct
parts in the proof, \cite{Cu2,Cu1,bambusi}. Here we follow the ordering of
\cite{bambusi}. In the first part
we
assume that for some $\ell \ge 2$ the statement of the theorem is true, and we show
that it continues to be true for $\ell+1$.
The proof of case $\ell =2$, which presents some additional complications, is dealt in the
second part.
In the proof we will get polynomials \eqref{eq:chi1} with $M_0=1,..., 2\textbf{N} $ with decreasing
$(r,M)$ as $M_0$ increases. Nonetheless, in view of the fact that in Lemma \ref{lem:fred12} the $n$ is arbitrarily large and that $(r,M)$ decreases by a fixed amount at each step, these $(r,M)$ are
arbitrarily large. This is exploited in Theorem \ref{theorem-1.1} later.
\bigskip
\noindent {\it The step $\ell +1>2$.} We can assume that
$H^{(\ell)}$ have the desired properties for indexes $(k',m')$ (instead of $(k,m)$) arbitrarily large.
We consider the representation \eqref{eq:ExpH11} for $H^{(\ell)}$ and we set $\mathbf{h}=H^{(\ell)}(z,f,\varrho )$ replacing
$\Pi (f)$ with $\varrho$ in \eqref{eq:ExpH11}. Then $\mathbf{h}=H^{(\ell)}(z,f,\varrho )$ is
$C^{2\mathbf{N}+2}$
near 0 in
$\Ph ^{s _0}=\{ (\varrho , R ) \}$ for $m '\ge 2\mathbf{N}+2 $ for $s_0>\max \{
\text{ord}(\mathcal{H}_{p_0}), 3/2 \} $
by
Lemma \ref{lem:ExpH11}.
So we have equalities
\begin{align}
\label{eq:derivmain1} & a_{\mu \nu}(\varrho )
=\frac{1}{\mu !\nu ! } \partial _z ^\mu \partial _{\overline{z}}^\nu
\mathbf{h}_{|(z,f,\varrho )= (0,0,\varrho )} \ , \quad |\mu +\nu |\le 2\mathbf{N}+1 ,\\& \label{eq:derivmain2}J^{-1}G_{\mu \nu}(\varrho )
=\frac{1}{\mu !\nu ! } \partial _z ^\mu \partial _{\overline{z}}^\nu \nabla _f
\mathbf{h} _{|(z,f,\varrho )= (0,0,\varrho )} \ , \quad |\mu +\nu |\le 2\mathbf{N} .
\end{align}
We consider now a yet unknown $\chi$ as in \eqref{eq:chi1} with $M_0=\ell$,
$i=0$, $M=m'$ and $r=k'$.
Set $\phi :=\phi ^1$, where $\phi ^t$ is the flow
of Lemma \ref{lem:chi}. We are seeking $\chi$ such that $H^{(\ell )}
\circ\phi $ satisfies the conclusions of Theorem \ref{th:main} for $\ell +1$.
\noindent We know that $H^{(\ell )}
\circ\phi $ satisfies the conclusions of Lemma \ref{lem:ExpH11}. Therefore, to prove the induction step,
all we need to do is to check that the expansion of $H^{(\ell )}
\circ\phi $ satisfies $\mathbf{R}_{-1}=0$ and that the only terms in
$\mathbf{R}_{0} $ and $\mathbf{R}_{ 1} $ of degree $\le \ell +1$
are in normal form. We have
\begin{equation} \label{eq:backH} \begin{aligned} & H_2 ^{(\ell )}\circ
\phi = H_2^{(\ell )} + \int _0^1 \{ H_2^{(\ell )} , \chi \} ^{st}\circ \phi ^t dt + \int _0^1 (\partial _{\varrho _j} a_{\mu \nu }z^\mu {\overline{z}}^\nu \{ \Pi _j(f) , \chi \} ) \circ \phi ^t dt.
\end{aligned}
\end{equation}
By \eqref{eq:homologicalEq1}--\eqref{eq:homologicalEq2} we have
for $\varrho =\Pi (f)$
\begin{equation} \label{eq:key0} \begin{aligned} &
\{ H_2 ^{(\ell )} , \chi \} ^{st}= -\im \sum _{|\mu +\nu |=\ell +1}\mathbf{e} ^{(\ell)} ( \varrho )\cdot (\mu -
\nu) z^{\mu} {\overline{z}}^{\nu} b_{\mu \nu } ( \varrho ) \\& - \sum _{|\mu +\nu |=\ell } z^{\mu} {\overline{z}}^{\nu} \langle J^{-1} ( \im \mathbf{e} ^{(\ell)} ( \varrho )\cdot (\mu -
\nu)- \mathcal{H} )B _{\mu \nu }( \varrho ),f
\rangle + \\& \sum _{\substack{|\mu +\nu |=2\\ (\mu , \nu )\neq (\delta _j , \delta _j ) \ \forall \ j} } a_{\mu \nu}^{(\ell )}( \varrho )
\big [\sum _{|\mu '+\nu '|=\ell +1}
\{ z^\mu
{\overline{z}}^\nu , z^{\mu'}
{\overline{z}}^{ \nu '} \} b_{\mu ' \nu '} ( \varrho ) \\& +
\sum _{|\mu '+\nu '|=\ell }
\{ z^\mu
{\overline{z}}^\nu ,z ^{\mu'}
{\overline{z}}^{ \nu '} \} \langle J^{-1} B_{\mu ' \nu '}( \varrho )
, f \rangle
\big ] .
\end{aligned}
\end{equation}
By Lemma \ref{lem:chi} for $M_0=\ell$,
$i=0$, $M=m'$ and $r=k'$ for first and last formula and by the proof
of Lemma \ref{lem:ODE}, in particular by \eqref{eq:gronwall1}, we have
\begin{align}
& z \circ \phi ^t = z +
\resto ^{0, \ell }_{k^{\prime \prime }, m'} (t,\Pi (f) , R) \, , \, \Pi (f) \circ \phi ^t = \Pi (f)
+\resto ^ {0, \ell +1} _{k^{\prime \prime },m'}(t, \Pi (f),R) \, , \nonumber \\& \label{eq:backcoo}
f \circ \phi ^t =e^{J\resto ^{0, \ell +1 }_{k^{\prime \prime }, m'} (t,\Pi (f) , R) \cdot \Diamond } ( f+ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m'}(t,\Pi (f) , R) )
\end{align}
for $ k^{\prime \prime } \le k'-(m'+1)\textbf{d} $. Then, substituting \eqref{eq:backcoo} in \eqref{eq:key0} we get, if $
k\le k^{\prime \prime }-\text{ord}(\mathcal{H}_{p_0})$, where $\text{ord}(\mathcal{H}_{p_0})\le \max \{\text{ord}(\mathcal{D}), \textbf{d}\} $,
for $1\le m \le m'$ and exploiting that an $\resto ^ {0,2\ell } _{k ,m }$ is also an $\resto ^ {0,\ell +2} _{k ,m }$ for $\ell \ge 2$,
\begin{equation} \label{eq:key-11} \begin{aligned} & \int _0^1\{ H_2 ^{(\ell )} , \chi \} ^{st} \circ \phi ^t dt = \{ H_2 ^{(\ell )} , \chi \} ^{st} +\resto ^ {0,\ell +2} _{k ,m }(\Pi (f),R) .
\end{aligned}
\end{equation}
\noindent We have
\begin{equation*} \begin{aligned} & \{ \Pi _j(f) , \chi \} =\sum _{k=1}^{n_0}\{ \Pi _j(f) , \Pi _k(f) \} \partial _{\Pi _k(f)} \chi + \sum _{|\mu '+\nu '|=\ell }z ^{\mu'}
{\overline{z}}^{ \nu '}\langle P_c^*(p_0) \Diamond _j f, B_{\mu ' \nu '} \rangle.
\end{aligned}
\end{equation*}
We have, for $P_d(p_0)=1-P_c(p_0)$ the projection on the direct sum of
$N_g(\mathcal{H}_{p_0})$ and the complement of $X_c(p_0)$ in
\eqref{eq:spectraldecomp}, and using $JP_c^*(p_0)=P_c (p_0)J$ which follows from \eqref{eq:begspectdec1},
\begin{equation} \label{eq:keyPoiss1} \begin{aligned} &
\{ \Pi _i(f), \Pi _j(f) \}= \langle P_c^*(p_0) \Diamond _i f, J P_c^*(p_0) \Diamond _j f\rangle \\& = \langle \Diamond _i f, P_d (p_0)J \Diamond _j f\rangle = \resto ^ {0,2} (f).
\end{aligned}
\end{equation}
Notice also that, for $B_{\mu \nu} \in \Sigma _{k'} $ independent of $\Pi (f)$ and for $|\mu +\nu |=\ell$, we have
\begin{equation} \label{eq:keyPoiss2} \begin{aligned} &
\{ \Pi _i(f), z ^{\mu }
{\overline{z}}^{ \nu } \langle J^{-1} B_{\mu \nu }
, f \rangle \}= z ^{\mu }
{\overline{z}}^{ \nu } \langle P_c^*(p_0) \Diamond _i f, B_{\mu \nu } \rangle = \\&
z ^{\mu }
{\overline{z}}^{ \nu } \langle f, \Diamond _i B_{\mu \nu } \rangle - z ^{\mu }
{\overline{z}}^{ \nu } \langle P_d^*(p_0) \Diamond _i f, B_{\mu \nu } \rangle
= \resto ^ {0,\ell +1} _{k'-\mathbf{d} ,\infty} (R) +\resto ^ {0,\ell +1}(R) .
\end{aligned}
\end{equation}
By \eqref{eq:keyPoiss1}--\eqref{eq:keyPoiss2} we conclude that $\{ \Pi _j(f) , \chi \} =\resto ^ {0,\ell +1} _{k'-\textbf{d} ,m '}(\Pi (f),R) $.
By \eqref{eq:backcoo} we get for $m\le m'$
\begin{equation*} \begin{aligned} & \{ \Pi _j(f) , \chi \} \circ \phi ^t =
\resto ^ {0,\ell +1} _{k'-\textbf{d} ,m'}\left (\Pi (f)+ \resto ^ {0, \ell +1} _{k^{\prime \prime },m'} (t, \Pi (f),R) ,\textsl{S} \right ) ,\\& \text{for } \textsl{S}:=e^{J\resto ^{0, \ell +1 }_{k^{\prime \prime }, m'}
(t,\Pi (f) , R) \cdot \Diamond } \left ( R+ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m'}(t,\Pi (f) , R) \right ).
\end{aligned}
\end{equation*}
Then
\begin{equation} \label{eq:key-21} \begin{aligned} & \{ \Pi _j(f) , \chi \} \circ \phi ^t = \resto ^ {0, \ell +1} _{k^{\prime \prime } -m'\textbf{d},m'}(t, \Pi (f),R) .
\end{aligned}
\end{equation}
By \eqref{eq:backcoo} and \eqref{eq:key-21} the last term in \eqref{eq:backH} is $\resto ^ {0,\ell +2} _{k ,m }(\Pi (f),R)$ for $k\le k^{\prime \prime } -m'\textbf{d}$. This and \eqref{eq:key-11} yield
for $ {k}=\min \{ k^{\prime } -(2m'+1)\textbf{d} ,k^{\prime } -( m'+1)\textbf{d} -\text{ord} (\mathcal {H}_{p_0}) \} $
\begin{equation} \label{eq:key001}\begin{aligned} & H_2^{(\ell )} \circ
\phi = H_2^{(\ell )} + \{ H_2^{(\ell )} , \chi \} ^{st}+ \resto ^ {0,\ell +2}_{\widetilde{k},m}(\Pi (f),R).
\end{aligned}
\end{equation}
A second observation is that $\mathbf{h}=(H^{(\ell)}\circ \phi ) (z,f,\varrho )$ is $C^{2\textbf{N}+2}$
in $\Ph ^{s_0} =\{ (\varrho , R ) \}$ for $m\ge 2\textbf{N}+2$.
We can compute again the corresponding coefficients in \eqref{eq:derivmain1}--\eqref{eq:derivmain2}.
Because of \eqref{eq:symbol11}, for $|\mu +\nu |
\le \ell $ in \eqref{eq:derivmain1} and for $|\mu +\nu |
\le \ell -1$ in \eqref{eq:derivmain2} these coefficients are the same of $\mathbf{h}= H^{(\ell)} (z,f,\varrho )$.
\noindent A third observation is that for $j= 3,4$ we have for $\mathbf{k}=\textbf{R}^{(\ell )} _j\circ \phi$
\begin{equation} \label{eq:derivatives} \begin{aligned} & \partial ^\mu _{z} \partial ^\nu _{{\overline{z}}} \mathbf{k} _{| (0,0, \varrho ) } =0 \text{ for } |\mu |+|\nu |\le \ell +1 \\& \partial ^\mu _{z} \partial ^\nu _{{\overline{z}}} \nabla _f \mathbf{k} _{| (0,0, \varrho ) }=0 \text{ for } |\mu |+|\nu |\le \ell .
\end{aligned}
\end{equation}
By Lemma \ref{lem:ODEbis} for $l=m$, $s=k$ and $r=k'$,
we have for $k\le k'-(2m+1) \textbf{d}$
\begin{equation} \label{eq:key31} \begin{aligned} & \Pi _j(f) \circ \phi = \Pi _j(f) \circ \phi _0
+\resto ^ {0,2\ell +1} _{k,m}(\Pi (f),R),
\end{aligned}
\end{equation}
with $\phi _0=\phi _0^1$ and $ \phi _0^t$ the flow defined as in Lemma
\ref{lem:ODEbis} using the field $X_\chi ^{st}$.
Then we have
\begin{equation} \label{eq:key32back} \begin{aligned} & \Pi _j(f) \circ \phi _0= \Pi _j(f) +\int _0^1
\left \langle \Diamond _j
(X_\chi ^{st} ) _{f} (\Pi (f), R \circ \phi _0^t) , f \circ \phi _0^t \right \rangle dt
.
\end{aligned}
\end{equation}
By the definition of $X_\chi ^{st}$ and by formulas \eqref{eq:backcoo} for $\phi _0^t$,
which are simpler because there are no phase
factors, by $|\mu +\nu | =\ell $ the integrand in \eqref{eq:key32back} is
\begin{equation*} \begin{aligned} & \left (z +
\resto ^{0, \ell }_{k^{\prime \prime }, m} (t,\Pi (f) , R)\right ) ^\mu \left (\overline{z} +
\resto ^{0, \ell }_{k^{\prime \prime }, m} (t,\Pi (f) , R)\right ) ^\nu \\& \times \left \langle \Diamond _j
B_{\mu , \nu } (\Pi (f)), f+ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m}(t,\Pi (f) , R) \right \rangle \\& = z ^\mu \overline{z} ^\nu \langle \Diamond _j
B_{\mu , \nu } (\Pi (f)), f \rangle +\resto ^{0, 2\ell }_{k^{\prime \prime } , m}(t,\Pi (f) , R).
\end{aligned}
\end{equation*}
Then for $k\le k^{\prime \prime } $ we have
\begin{equation} \label{eq:key32} \begin{aligned} & \Pi _j(f) \circ \phi _0= \Pi _j(f) +
\langle \Diamond _j
(X_\chi ^{st} ) _{f} , f \rangle
+\resto ^ {0, 2\ell } _{k,m}(\Pi (f),R).
\end{aligned}
\end{equation}
By $\ell \ge 2$ we have $2\ell \ge \ell +2$ and so $\resto ^ {0, 2\ell } _{k,m} $ is an $\resto ^ {0, \ell +2} _{k,m}$.
\noindent By
$ {\psi} (\varrho)=O(|\varrho|^2)$ near 0, we conclude that
\begin{equation} \label{eq:key-101} \begin{aligned} & {\psi} ( \Pi (f) ) \circ \phi = {\psi} ( \Pi (f) )+\widetilde{K} '+\resto ^{1,\ell +2} _{k,m}(\Pi (f),R),
\end{aligned}
\end{equation}
with $\widetilde{K} '$ a polynomial as in \eqref{eq:tildeKrho} with $M_0=\ell ,$ with $ \widetilde{K} '(0,b,B)=0$
and
$( \widehat{k} , \widehat{m}) =(k',m')$ satisfying. Notice that it was to get the last equality, which follows from \eqref{eq:key32}, that
we introduced the flow
$\phi _0^t$.
\noindent We now focus on $\mathbf{R}_2$. We have by \eqref{eq:backcoo}
\begin{align} \label{eq:key-201} & \mathbf{R}_2 \circ \phi = \langle \mathbf{B}_{2 } ( \Pi (f') ), (f') ^{ 2} \rangle = \\& \langle \mathbf{B}_{2 } \left ( \Pi (f) +\resto ^ {0,\ell +1} _{k,m} (\Pi (f),R)\right ), \left (e^{J\resto ^{0,\ell +1 } _{k^{\prime \prime },m'} (\Pi (f),R) \cdot \Diamond } ( f+ \textbf{S}^{0,\ell } _{k^{\prime \prime },m'}(\Pi (f),R) )\right ) ^{ 2} \rangle
.\nonumber
\end{align}
In our present set up the exponential $e^{J\resto ^{0,\ell +1 } _{k^{\prime \prime },m'} \cdot \Diamond }$ cannot be moved to the
$\mathbf{B}_{2 }$ by a change of variables in the integral
as in \cite{Cu1}. Fortunately we know already that $H^{(\ell )}\circ \phi$ has the expansion of
Lemma \ref{lem:ExpH11} and that all we need to do is to compute some derivatives of $\mathbf{R}_2 \circ \phi $.
\noindent
Using the expansion in \eqref{eq:key-201} and formula
\eqref{eq:symbol12}, for $i=0$ now, we set
\begin{equation}\label{eq:key-301} \begin{aligned} &
\mathfrak{R}_2:= \langle \mathbf{B}_{2 } ( \Pi (f) ), ( f+ \textbf{S}^{i,\ell } _{k^{\prime \prime },m'}( \Pi (f) ,R) ) ^{ 2} \rangle = \\& \left \langle \mathbf{B}_{2 } ( \Pi (f) ), \left [ f+ \int _0^1 (X^{st}_\chi )_f \circ \phi ^t dt + \textbf{S}^{i,2\ell +1 } _{k^{\prime \prime },m'}( \Pi (f) ,R) \right ] ^{ 2} \right \rangle = \\& \langle \mathbf{B}_{2 } ( \Pi (f) ), f ^{ 2} \rangle+ 2 \int _0^1 \langle \mathbf{B}_{2 } ( \Pi (f) ), (X^{st}_\chi )_f \circ \phi ^t \ f \rangle dt + \resto^{i,2\ell } _{k^{\prime \prime } ,m' }( \Pi (f) ,R) . \end{aligned}
\end{equation}
We have that $\mathbf{k}=
\mathbf{R}_2 \circ \phi - \mathfrak{R} _2$ is $C^{\ell +1}$ and satisfies \eqref{eq:derivatives}.
Hence the analysis of $\mathbf{R}_2 \circ \phi$ reduces to that of $\mathfrak{R}_2$.
By \eqref{eq:backcoo}, for $k\le k^{\prime\prime}$, $m\le m'-1$ and $\ell >1$ we have
\begin{equation}\label{eq:key-401} \begin{aligned} & \int _0^1 X^{st}_\chi \circ \phi ^t dt = X^{st}_\chi
+ \textbf{S}_{k^{\prime\prime},m'-1}^{0,2\ell -1}( \Pi (f) ,R)= X^{st}_\chi
+ \textbf{S}_{k,m}^{0, \ell +1}( \Pi (f) ,R) . \end{aligned}
\end{equation}
This implies
\begin{equation}\label{eq:key-501} \begin{aligned} &
\mathfrak{R}_2 = \langle \mathbf{B}_{2 } ( \Pi (f) ), f ^{ 2} \rangle+\widetilde{K} ^{\prime \prime}
+
\resto _{k,m}^{0, \ell +2}(\Pi (f),R) \quad , \\& \widetilde{K}^{\prime \prime } :=2 \langle \mathbf{B}_{2 } ( \Pi (f) ), f (X^{st}_\chi )_f \rangle
. \end{aligned}
\end{equation}
Then $ \widetilde{K}^{\prime \prime } $ is a polynomial like in \eqref{eq:tildeKrho} for the pair $(\widehat{k},\widehat{m})=(k',m')$ satisfying $ \widetilde{K} ^{\prime \prime }(0,b,B)=0$ by $B_2(\varrho )=0$ for $\varrho =0$.
\noindent By \eqref{eq:backcoo} and for the pullback of the term
$\resto _{k',m'+2}^{1, 2 }( \Pi (f) , f )$ in Lemma
\ref{lem:ExpH11} we have for $\varrho =\Pi (f ) $
\begin{equation} \label{eq:key-50110}
\begin{aligned} &\resto _{k',m'+2}^{1, 2 }( \Pi (f') , f ')= \resto _{k',m'+2}^{1, 2 }( \varrho , f ') \\&+
\int _0^1 (\nabla _{\varrho }\resto _{k',m'+2}^{1, 2 })( \varrho +t
\resto _{k^{\prime\prime},m'+2}^{0, \ell +1 }( \varrho , f ) , f ') \cdot \resto _{k^{\prime\prime},m'}^{0, \ell +1 }( \varrho , f ) dt \\& = \resto _{k',m'+2}^{1, 2 }( \varrho , f ') + \resto _{k,m}^{0, \ell +3 } ( \varrho , R )
\end{aligned}
\end{equation}
for $k\le k^{\prime\prime} - m \textbf{d}$ and $m\le m' $, by elementary analysis of the
second line.
\noindent Applying again \eqref{eq:backcoo} we have
\begin{equation} \label{eq:key-50111}
\begin{aligned} \resto _{k',m'+2}^{1, 2 }( \varrho , f ') &= \resto _{k',m'+2}^{1, 2 }\left ( \varrho , e^{J\resto ^{0, \ell +1 }_{k^{\prime \prime }, m' } ( \varrho , R) \cdot \Diamond } \left ( f+ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m'}( \varrho , R)\right ) \right ) \\&= \resto _{k',m'+2}^{1, 2 }\left ( \varrho , f+ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m'}( \varrho , R) \right )+ \resto _{k,m}^{1, \ell +2 } ( \varrho , R )
\end{aligned}
\end{equation}
for $k\le k^{\prime\prime} - m \textbf{d}$ and $m\le m'-1$. Next, by Lemma \ref{lem:chi}, \eqref{eq:symbol12}
and by \eqref{eq:key-401}, we have
$ \textbf{S} ^{0, \ell }_{k^{\prime \prime }, m'}( \varrho , R) =( X^{st}_\chi )_f +\textbf{S}_{k^{\prime\prime},m'-1}^{0, \ell +1}( \varrho ,R) $
and
\begin{equation*} \begin{aligned} & \resto _{k',m'+2}^{1, 2 }\left ( \varrho , f+( X^{st}_\chi )_f +\textbf{S}_{k^{\prime\prime},m }^{0, \ell +1}( \varrho ,R) \right ) =\resto _{k',m'+2}^{1, 2 }( \varrho , f ) + \\& \int _0^1 \langle \nabla _R \resto _{k',m'+2}^{1, 2 }\left (
\varrho , f+t( X^{st}_\chi )_f +t\textbf{S}_{k^{\prime\prime},m }^{0, \ell +1}( \varrho ,R)
\right ) , ( X^{st}_\chi )_f + \textbf{S}_{k^{\prime\prime},m }^{0, \ell +1}
( \varrho ,R) \rangle dt \\& =\resto _{k',m'+2}^{1, 2 }( \varrho , f ) +\langle \nabla _f \resto _{k',m'+2}^{1, 2 }( \varrho , f ) , ( X^{st}_\chi )_f \rangle +\resto _{k ,m }^{1, \ell +2 }( \varrho , R )
\end{aligned}
\end{equation*}
where we have used $\ell \ge 2$, $k\le k^{\prime\prime} \le k^{\prime }$ and $m\le m' -1$.
Notice that we have that $\resto _{k',m' +2}^{1, 2 }( \varrho , f )$ is an $\resto _{k ,m+2}^{1, 2 }( \varrho , f )$.
Finally we have
\begin{equation}\label{eq:key-5013} \begin{aligned} &
\langle \nabla _f \resto _{k',m'+2}^{1, 2 }( \varrho , f ) , ( X^{st}_\chi )_f \rangle =\widetilde{K} ^{\prime \prime \prime }+\overline{\textbf{R}}_2\quad ,
\\& \widetilde{K} ^{\prime \prime \prime }:= \langle \nabla _f ^2\resto _{k',m'+2}^{1, 2 }( \varrho , 0 ) f, ( X^{st}_\chi )_f \rangle ,
\end{aligned}
\end{equation}
with $\overline{\textbf{R}}_2 $ a term we can absorb in $\widehat{\textbf{R}}_2 $ and with $ \widetilde{K} ^{\prime \prime \prime }$ like in \eqref{eq:tildeKrho} for the pair $(\widehat{k},\widehat{m})=(k',m')$ satisfying $ \widetilde{K} ^{\prime \prime \prime }(0,b,B)=0$.
\noindent We set
\begin{equation} \label{eq:key-601} \begin{aligned} & \textbf{R}^{(\ell )} _0 +\textbf{R}^{(\ell )} _1 =Z '+ K + \textbf{R} _{01} \ ,
\end{aligned}
\end{equation}
where: $Z '$ is the sum of the monomials in normal form of degree $\le \ell+1$;
$K$, which is like in \eqref{eq:Krho}, is the sum of the the monomials of degree
equal to $ \ell+1$ not in normal form;
$\textbf{R} _{01}$ is the sum of the monomials of degree $> \ell+1$. By induction there are no monomials not in normal form of degree $\le \ell $ so that each of the monomials of the lhs of \eqref{eq:key-601}
go into exactly one of the three terms of the rhs.
\noindent We define ${Z}^{\prime \prime } $ and $\widetilde{K}$ by setting
\begin{equation} \label{eq:key-701} \begin{aligned} & \widetilde{K}^{\prime }+\widetilde{K}^{\prime \prime }+\widetilde{K} ^{\prime \prime \prime }= {Z}^{\prime \prime } +\widetilde{K},
\end{aligned}
\end{equation}
collecting in ${Z}^{\prime \prime } $ all monomials of the lhs in normal form (all of degree $ \ell+1$) and in $\widetilde{K} $ all monomials of the lhs not in normal form.
Here $\widetilde{K} $ is like in \eqref{eq:tildeKrho} for $(\widehat{k},\widehat{m})=(k',m')$ with $ \widetilde{K} (0,b,B)=0$.
\noindent Applying Lemma \ref{lem:NLhom1} for $(\widehat{k},\widehat{m})=(k',m')$ we
can choose $\chi $ such that for $Z=Z ' +{Z}^{\prime \prime } $ we have
\begin{equation}\label{eq:key1}
\{ H_2^{(\ell )} , \chi \} ^{st}+Z+ K+\widetilde{K}=0.
\end{equation}
Then $H^{(\ell +1)}:=H^{(\ell )}\circ \phi $ satisfies the conclusions of
Theorem \ref{th:main} for $\ell+1$.
\bigskip
{\it The step $\ell +1=2$.}
Set $H^{(1 )}=K\circ \mathfrak{F}_1$. We are seeking a transformation
$\phi$ as in the previous part such that $H^{(2)}:=H^{(1 )}\circ \phi $ has term $\mathbf{R} _{-1}^{(2)}=0 $ in its expansion
in Lemma \ref{lem:ExpH11}.
The argument is similar to the previous one, but this time $\chi $ has degree $\ell +1$ with
$\ell =1$.
So the steps in the previous argument where we exploited $\ell \ge 2 $ need to
be reframed.
\noindent We know that $H^{(1 )}$ satisfies Lemma \ref{lem:ExpH11} for $L=1 $
for some pair that we denote by $(k',m')$ rather than $(k,m)$.
\noindent The proof of \eqref{eq:key-11} is different from the previous one. By \eqref{eq:ODE1bis}
we have for some $(k,m)$ appropriately smaller than $(k',m')$
\begin{equation} \label{eq:keyfirst} \begin{aligned} &
\{ H_2 ^{(1 )} , \chi \} ^{st}\circ \phi ^t= \{ H_2 ^{(1 )} , \chi \} ^{st}\circ \phi ^t _0+ \resto ^{0,4}_{k,m}(\Pi (f),R).
\end{aligned}
\end{equation}
The following linear transformation
\begin{equation*} \label{eq:linop} \begin{aligned} & (Z,\overline{Z},F) \to \begin{pmatrix} \im \nu _j b_{\mu \nu}(\Pi (f)) \frac{Z^\mu \overline{Z}^\nu }{\overline{Z}_j} + \im \nu _j \frac{Z^\mu \overline{Z}^\nu }{\overline{Z}_j} \langle J^{-1}B_{\mu \nu}(\Pi (f)),F\rangle
\\ -\im \mu _j b_{\mu \nu}(\Pi (f)) \frac{ {Z}^\mu \overline{{Z}}^\nu }{ {Z}_j} - \im \mu _j \frac{Z^\mu \overline{Z}^\nu }{ {Z}_j} \langle J^{-1}B_{\mu \nu}(\Pi (f)),F\rangle
\\ B_{\mu \nu}(\Pi (f)) {Z}^\mu \overline{{Z}}^\nu
\end{pmatrix}
\end{aligned}
\end{equation*}
depends linearly on $(b (\varrho),B (\rho))$, for $\varrho =\Pi (f) $.
Then
\begin{equation} \label{eq:linop1} \begin{aligned} & z_j \circ \phi ^t _0=z_j+
a _j(t,b,B)\cdot z + b _j(t,b,B)\cdot \overline{z} + \sum _{\mu \nu} c _{ j\mu \nu} (t,b,B)
\langle J^{-1}B_{\mu \nu} ,f\rangle
\end{aligned}
\end{equation}
for $a _j,b _j\in C^\infty ([0,1]\times X_{k'}, \C^{\textbf{n}})$
with $|a _j| + |b _j| \le C \| (b,B)\| _{ X_{k'}}$
and
$c _{ j\mu \nu} \in C^\infty ([0,1]\times X_{k'}, \C )$.
Similarly
\begin{equation} \label{eq:linop2} \begin{aligned} & f \circ \phi ^t _0=f+
\textbf{a} (t,b,B)\cdot z + \textbf{ b} (t,b,B)\cdot \overline{z} + \sum _{\mu \nu} \textbf{c} _{ \mu \nu} (t,b,B)
\langle J^{-1}B_{\mu \nu} ,f\rangle
\end{aligned}
\end{equation}
with $\textbf{a} ,\textbf{b} \in C^\infty ([0,1]\times X_{k'}, \Sigma _{k'}^{\textbf{n}})$
with $\| \textbf{a} \| _{\Sigma _{k'}^{\textbf{n}}} + \|\textbf{b} \| _{\Sigma _{k'}^{\textbf{n}}} \le C \| (b,B)\| _{ X_{k'}}$
and
$\textbf{c} _{ \mu \nu} \in C^\infty ([0,1]\times X_{k'}, \Sigma _{k'} )$. These coefficients
satisfy appropriate symmetries that ensure $\overline{f \circ \phi ^t _0}=f \circ \phi ^t _0$.
\noindent We have
\begin{equation} \label{eq:key-1310} \begin{aligned} & \{ H_2 ^{(1 )} , \chi \} ^{st}
\circ \phi ^t _0= \{ H_2 ^{(1 )} , \chi \} ^{st}
(\Pi (f), R\circ \phi _0^t) + \resto ^{1,4}_{k,m} (t,\Pi (f), R).
\end{aligned}
\end{equation}
To compute $ \{ H_2 ^{(1 )} , \chi \} ^{st}
(\Pi (f), R\circ \phi _0^t)$ we replace the
$R$ in \eqref{eq:key0} with $R\circ \phi _0^t $. The coordinates of the latter can be expressed in terms
of $R$ by \eqref{eq:linop1}--\eqref{eq:linop2}.
When we substitute $(z,f)$ in \eqref{eq:key0}
using
\eqref{eq:linop1}--\eqref{eq:linop2},
by an elementary
computation we obtain
\begin{equation*} \begin{aligned} & \{ H_2 ^{(1 )} , \chi \} ^{st}
(\varrho, R\circ \phi _0^t)= \{ H_2 ^{(1)} , \chi \} ^{st} (\varrho, R ) \\& + \sum _{\substack{|\mu +\nu |=1\\ |\mu '+\nu '|=1}}
a_{\mu \nu}^{\mu ' \nu '} (t,\varrho ,b(\varrho ),B(\varrho )) z^{\mu} \overline{z}^{\nu}\langle \mathcal{H} B_{\mu \nu}(\varrho ), f
\rangle +A^t + \underline{{\mathbf{R}}}^t.
\end{aligned}
\end{equation*}
Here:
\begin{itemize}
\item $a_{\mu \nu}^{\mu ' \nu '} (t,\varrho ,b ,B ) \in C^{m'}$
with $a_{\mu \nu}^{\mu ' \nu '} (t,0 ,0 ,0 ) =0$;
\item we have \begin{equation*} \begin{aligned} &
A^t=\sum _{|\mu +\nu |=2}
\alpha _{\mu \nu} (t,\varrho ,b(\varrho ),B(\varrho ))z^{\mu} \overline{z}^{\nu} \\& +\sum _{l=0}^1 \sum _{j=1}^{n_0}
\sum _{ |\mu +\nu |=1 }
z^{\mu} \overline{z}^{\nu} \langle \Diamond _j^l A^l_{\mu \nu }(t, \varrho , b(\varrho ),B(\varrho )), f
\rangle ,
\end{aligned}
\end{equation*}
$\alpha _{\mu \nu}(t,\varrho ,b,B)$ and $A^l_{\mu \nu }(t,\varrho ,b,B)$ are $C^{m'}$
with
for $i=2$
\begin{equation}
\label{eq:corr1} |\alpha _{\mu \nu} (t,\varrho ,b,B) | + \| A^l_{\mu \nu }(t,\varrho ,b,B) \| _{\Sigma _{k'}}\le C \| ( b,B)\| _{X_{k'}}^i;
\end{equation}
\item $\underline{\mathbf{R}} ^t( \varrho ,z,f)$ is $C^m$ in $(t,\varrho ,z,f)\in \R ^{n_0+1} \times \C ^{\mathbf{n}} \times \Sigma _{-k}$ with $(\varrho ,z,f)$ near $(0 ,0,0)$, with for $i=2$
\begin{equation}
\label{eq:corr2} | \underline{{\mathbf{R}}}^t | \le C \| ( b,B)\| _{X_{k'}}^2 \| f\| _{\Sigma _{-k}}^2 .
\end{equation}
\end{itemize}
Then, in the notation of Lemma \ref{lem:NLhom2}
\begin{equation} \label{eq:key-110} \begin{aligned} & \int _0^1\{ H_2 ^{(1 )} , \chi \} ^{st}
\circ \phi ^t _0 dt = \{ H_2 ^{(1)} , \chi \} ^{\widetilde{st}} + A
+\underline{{\mathbf{R}}} +\resto ^{1,4}_{k,m}(\Pi (R) , R),
\end{aligned}
\end{equation}
with $A=\int _0^1 A^tdt $ and $\underline{{\mathbf{R}}}=\int _0^1 \underline{{\mathbf{R}}}^tdt $ are like $A^1$ and $\underline{{\mathbf{R}}}^1$.
Then, using also \eqref{eq:keyfirst},
we get the following analogue of \eqref{eq:key001}: \begin{equation} \label{eq:key-001}\begin{aligned} & H_2^{(1 )} \circ
\phi = H_2^{(1 )} + \{ H_2^{(1 )} , \chi \} ^{\widetilde{st}} + A+ \underline{{\mathbf{R}}} + \resto ^ {0,4}_{k,m}(\Pi (f),R).
\end{aligned}
\end{equation}
\eqref{eq:key31} remains true also for $\ell =1$. We consider \eqref{eq:key32back} and expand
\begin{equation*} \begin{aligned} &
\langle \Diamond _j
(X_\chi ^{st} ) _{f} (\Pi (f), R \circ \phi _0^t) , f \circ \phi _0^t \rangle =
\langle \Diamond _j
(X_\chi ^{st} ) _{f} (\Pi (f), R) , f \rangle + A^t + \textbf{R} ^t,
\end{aligned}
\end{equation*}
with $A^t$ and
$\textbf{R} ^t$ like the previous ones but such that \eqref{eq:corr1}--\eqref{eq:corr2}
hold for $i=1$. This yields
\begin{equation} \label{eq:key320} \begin{aligned} & \Pi _j(f) \circ \phi _0= \Pi _j(f)
+A'+\underline{{\mathbf{R}}}'.
\end{aligned}
\end{equation}
Here $ \underline{{\mathbf{R}}}'$ is like $ \underline{{\mathbf{R}}} ^1$ such that \eqref{eq:corr2}
holds for $i=1$.
$A'$ is like $A^1$ such that \eqref{eq:corr1}
holds for $i=1$.
\noindent By $ {\psi} (\varrho)=O(|\varrho|^2)$ near 0 and \eqref{eq:key31} we get the first equality in
\begin{equation} \label{eq:key101} \begin{aligned} & {\psi} ( \Pi (f) ) \circ \phi = {\psi} ( \Pi (f) ) \circ \phi _0 +
\resto ^ {1, 3} _{k,m}(\Pi (f),R)\\&=
{\psi} ( \Pi (f) )+\widetilde{K} '+\resto ^{1,2} _{k',m'}(\Pi (f),f)+\resto ^ {1, 3} _{k,m}(\Pi (f),R),
\end{aligned}
\end{equation}
where $\widetilde{K} '=\resto ^{1,2} _{k',m'}(\Pi (f),R)$ is a polynomial in $R$ as in \eqref{eq:tildeKrho} with $\widetilde{K} ' (0,b,B)=0$. The second line in \eqref{eq:key101} follows by $ {\psi} (\varrho)=O(|\varrho|^2)$, by the fact that $ {\psi} (\varrho)$ is smooth
and by \eqref{eq:key320}. Notice that by choosing $m\le m' -2$ we have $\resto ^{1,2} _{k',m'}(\Pi (f),f)=\resto ^{1,2} _{k ,m+2}(\Pi (f),f).$
\noindent The discussion of $\textbf{R}\circ \phi $ is similar to the previous one after \eqref{eq:key-201} . This time, though,
by \eqref{eq:ODE1bis} we write
\begin{equation}\label{eq:key401} \begin{aligned} & \int _0^1 X^{st}_\chi \circ \phi ^t dt =
\int _0^1 X^{st}_\chi \circ \phi ^t_0 dt + \textbf{S}_{k,m}^{0,3} (\Pi (f),R) . \end{aligned}
\end{equation}
By \eqref{eq:linop1}--\eqref{eq:linop2} we get
\begin{equation}\label{eq:key402} \begin{aligned} & \int _0^1 X^{st}_\chi \circ \phi ^t _0 dt
=
X^{st}_\chi + \mathbf{A} \text{ in $\Ph ^{k'}$} , \end{aligned}
\end{equation}
with $(z,f)\to \mathbf{A}(\varrho ,z,f)$ linear, with $C^{m'} $
dependence in $\varrho$ and with
\begin{equation}\label{eq:key403} \begin{aligned} & \| \mathbf{A} (\varrho , z, f) \|_ { \Ph ^{k'}}\le C \| (b (\varrho ),B(\varrho )) \| _{X_{k'}} (|z|+\| f \| _{\Sigma _{-k'}}). \end{aligned}
\end{equation}
This yields, for $\mathfrak{R}_2$ defined as in \eqref{eq:key-301},
\begin{equation*} \begin{aligned} &
\mathfrak{R}_2 = \left \langle \mathbf{B}_{2 } ( \Pi (f) ), \left [ f+ \int _0^1 (X^{st}_\chi )_f \circ \phi ^t_0 dt \right ] ^{ 2} \right \rangle +\resto ^{1,3 } _{k ,m }( \Pi (f) ,R) =\\& \langle \mathbf{B}_{2 } ( \Pi (f) ), f ^{ 2} \rangle+ 2\langle \mathbf{B}_{2 } ( \Pi (f) ), f \mathbf{A}\rangle + \langle \mathbf{B}_{2 } ( \Pi (f) ), \mathbf{A}^2\rangle +\resto ^{1,3 } _{k ,m }( \Pi (f) ,R) , \end{aligned}
\end{equation*}
where we have used $ \mathbf{B}_{2 } ( 0 )=0$ for the reminder.
\noindent We have
\begin{equation*} \begin{aligned} &
2\langle \mathbf{B}_{2 } ( \Pi (f) ), f \mathbf{A}\rangle + \langle \mathbf{B}_{2 } ( \Pi (f) ), \mathbf{A}^2\rangle = \widetilde{K} ^{\prime \prime}
+ \underline{\mathbf{R}}^{\prime \prime} , \end{aligned}
\end{equation*}
with $\underline{{\mathbf{R}}}^{\prime \prime}$ like $\underline{{\mathbf{R}}}$ and with
$\widetilde{K} ^{\prime \prime}$ like \eqref{eq:tildeKrho}
with $\widetilde{K} ^{\prime \prime}(0,b,B)=0$, by $\mathbf{B}_{2 } ( 0 )=0$, and with $(\widehat{k},\widehat{m})=(k',m')$.
Summing up, we have
\begin{equation}\label{eq:key501} \begin{aligned} &
\mathfrak{R}_2 = \langle \mathbf{B}_{2 } ( \Pi (f) ), f ^{ 2} \rangle+\widetilde{K} ^{\prime \prime}
+ \underline{\mathbf{R}}^{\prime \prime}+\resto^{1,3 } _{k ,m }( \Pi (f) ,R).
\end{aligned}
\end{equation}
Notice that the reduction of $\mathbf{R}_2\circ \phi $ to $ \mathfrak{R}_2$ continues to hold also for $\ell= 1$.
\noindent We consider $\resto _{k',m'+2}^{1, 2 }\circ \phi$ from the $ \resto _{k',m'+2}^{1, 2 }$ term in the expansion of $\mathbf{R}$ in Lemma
\ref{lem:ExpH11}.
Then, by \eqref{eq:key-50110} and by \eqref{eq:key401}--\eqref{eq:key402}, for $\varrho =\Pi (f )$ we have
\begin{equation*} \begin{aligned} &
\resto _{k',m'+2}^{1, 2 }( \Pi (f') , f ')= \resto _{k',m'+2}^{1, 2 }( \varrho , f +( X^{st}_\chi )_f +\mathbf{A} + \textbf{S}_{k,m}^{0,3} ) + \resto _{k,m}^{0, 4 }( \varrho , R )
. \end{aligned}
\end{equation*}
The first term in the rhs can be expanded for $\varrho =\Pi (f )$ as
\begin{equation*} \begin{aligned} &
\resto _{k',m'+2}^{1, 2 }( \varrho , f +( X^{st}_\chi )_f +\mathbf{A} ) +\resto _{k ,m }^{1, 4 }( \varrho , R ).
\end{aligned}
\end{equation*}
We have for $\varrho =\Pi (f )$
\begin{equation*} \begin{aligned} &
\resto _{k',m'+2}^{1, 2 }( \varrho , f +( X^{st}_\chi )_f +\mathbf{A} ) = \mathfrak{B}_2(\varrho ) ( f +( X^{st}_\chi )_f +\mathbf{A} )^2+ \resto _{k ,m }^{1, 3 }( \varrho , R ),
\end{aligned}
\end{equation*}
with $\mathfrak{B}_2(\varrho )$ a $C^{m'}$ function with values
in $B ^2(\Sigma_{-k'},\Sigma_{ k'}) $ with $\mathfrak{B}_2(0 )=0$.
Considering the binomial expansion we get for $\varrho =\Pi (f )$
\begin{equation*} \begin{aligned} &
\resto _{k',m'+2}^{1, 2 }( \Pi (f') , f ') = \mathfrak{B}_2(\varrho ) f ^2 +\widetilde{K}^{\prime\prime\prime} +\underline{\mathbf{R}} ^{\prime\prime\prime}+\resto _{k ,m }^{0, 3 }( \varrho , R ) ,
\end{aligned}
\end{equation*}
with $\underline{\mathbf{R}} ^{\prime\prime\prime}$ like $\underline{\mathbf{R}} $ and with $\widetilde{K}^{\prime\prime\prime}$ like
\eqref{eq:tildeKrho} with $\widetilde{K}^{\prime\prime\prime}(0,b,B)=0$ and $(\widehat{k},\widehat{m})=(k',m')$.
\noindent We now set $K= \textbf{R}^{(1 )} _{-1}$ and with the $A$ of \eqref{eq:key-110} we write
\begin{equation} \label{eq:key701} \begin{aligned} & \widetilde{K}^{\prime }+\widetilde{K}^{\prime \prime }+\widetilde{K} ^{\prime \prime \prime }+A ={Z}^{\prime \prime } +\widetilde{K} ,
\end{aligned}
\end{equation}
where in ${Z}^{\prime \prime }$ we collect the null terms of the lhs
and in $\widetilde{K}$ the other terms.
Now we have $K(0 )=0$, $\widetilde{K}( 0 ,0,0 )=0$ and $\nabla _{b,B}\widetilde{K}( 0 ,0,0 )=0$. By Lemma \ref{lem:NLhom2} for $(\widehat{k},\widehat{m})=(k',m')$ we
can choose $\chi $ such that for we have
\begin{equation} \label{eq:key702}
\{ H_2^{(\ell )} , \chi \} ^{\widetilde{st}}+Z^{\prime \prime }+ K+\widetilde{K}=0.
\end{equation}
Then $H^{(2)}:=H^{(1 )}\circ \phi $ satisfies the conclusions of
Theorem \ref{th:main} for $\ell =2$.
\qed
Summing up, we have proved the following result, whose proof we sketch now.
\begin{theorem}\label{theorem-1.1} For fixed $p_0\in \mathcal{O}$ and for sufficiently large $l\in \N$,
there are a fixed $k\in \N$, an $\epsilon >0$, an $1\ll s' \ll l$ and a $1\ll k \ll k'$ such that for solutions $\widehat{U}(t)$ to
\eqref{eq:NLSvectorial} with $\Pi (U) =p_0$ with $|\Pi (\widehat{R} (t)) |+\| \widehat{R} (t) \| _{\Sigma _{-k}}<\epsilon $ and $\widehat{R} (t) \in \Sigma _{l}$, there
exists a $C^0$ map $\Phi :\U ^l_{\epsilon , k} \to \U ^{s'}_{\epsilon ' , k'} $ such that
\begin{equation} \label{eq:1.11}\begin{aligned} &
R :=\Phi _{R} ( \Pi (\widehat{R} ), \widehat{R} ) = e^{Jq( \Pi (\widehat{R} ), \widehat{R} )\cdot \Diamond } ( \widehat{R} + \textbf{S} ( \Pi (\widehat{R} ), \widehat{R} ))
,
\end{aligned} \end{equation}
\begin{equation} \label{eq:1.12} \begin{aligned} \text{with } &
\textbf{S} \in C^2((-2,2)
\times B_{\R ^{ n_0}}\times B_{\Sigma _{-k}} , \Sigma _{s'}
) \\& {q} \in C^2((-2,2)
\times B_{\R ^{ n_0}} \times B_{\Sigma _{-k}}, \R ^{ n_0}
)
\end{aligned} \end{equation}
such that $ \| \textbf{S} ( \Pi (\widehat{R} ) ,\widehat{R} )
\| _{\Sigma _{ s'} }\le C \epsilon\| \widehat{R} \| _{\Sigma _{ -k}} $
and such that splitting $R(t) $ in spectral coordinates $(z(t), f(t))$ the latter satisfy
\begin{equation} \label{eq:HamSyst}\begin{aligned} & \dot z_j =\im \partial _{\overline{z}_j}H\ , \quad \dot f=J\nabla _fH
\end{aligned}\end{equation}
where $H$ is a given function satisfying the properties of $H^{(2\textbf{N}+1)}$ in Theorem \ref{th:main}.
\end{theorem}
\proof Since
in Lemma \ref{lem:fred12} we can pick arbitrary $n$,
we see by the proof of Theorem \ref{th:main}
that we can suppose that the $2\textbf{N}+1$ transformations $\phi _\ell$ are defined by flows
\eqref{eq:ODE} with pair $(r,M)$ with $r$ and $M$ as large as needed.
\noindent Starting with an appropriate $\U ^{s}_{\varepsilon _0, \kappa _0}$, we know that there is a map
$ \mathfrak{F} : \U ^{s'}_{\varepsilon _1, \kappa '} \to \U ^{s }_{\varepsilon _0, \kappa _0}$ as regular as needed
which satisfies the conclusions of Theorem \ref{th:main}.
In particular here
we have $s'\gg s$ and $ 1\ll \kappa ' \ll \kappa _0$ and in $\U ^{s'}_{\varepsilon _1, \kappa '}$
we get the system \eqref{eq:HamSyst} by pulling back the system which exists in $\U ^{s }_{\varepsilon _0, \kappa _0}$.
\noindent We choose now $l\gg s'$, $1 \ll k\ll \kappa '$ and
sufficiently small $\epsilon $ and $ \delta $ with
$\U ^l_{\delta , k} \subset
\U ^{s }_{\varepsilon _0, \kappa _0}$ and $\U ^l_{\epsilon , k} \subset
\U ^{s'}_{\varepsilon _1, \kappa '} $. Here $l$ and $ \kappa '$ can be as large as we want, thanks to
our freedom to choose $(r,M)$.
\noindent By choosing $\delta $ small we can assume
$\U ^l_{\delta , k} \subset \mathfrak{F} ( \U ^{s'}_{\varepsilon _1, \kappa '})
$. This follows from \eqref{eq:main1} which implies $\mathfrak{F} ^{-1}(\U ^l_{\delta , k})
\subset \U ^l_{\epsilon , k} $. Finally we set $\Phi = \mathfrak{F} ^{-1}$ where
$\mathfrak{F} ^{-1} : \U ^l_{\delta , k} \to \U ^{s'}_{\varepsilon _1, \kappa '}$.
\noindent Formula \eqref{eq:1.11} and the information on $\textbf{S}$ has been proved in the course of the proof of Lemma \ref{lem:ODE1}. The information on the phase function $q$ can be proved by a similar
induction argument, which we skip here.
\qed
\begin{remark}
\label{rem:bamb1} The paper \cite{bambusi}
highlights in the Introduction and states in Theorem 2.2,
that it is able to treat all solutions of the NLS near ground states in $H^1 $. But
in fact, in \cite{bambusi} there is no explicit proof of this. While
\cite{bambusi} does not state the regularity properties
of the maps in Theorem 3.21 and Theorem 5.2 \cite{bambusi},
from the context they appear to be just continuous. Even if we assume that they are \textit{almost smooth} transformations
(but see Remark \ref{rem:formal1} above), nonetheless an explanation
is required on why they preserve the structure needed to make sense of the NLS.
But while pullbacks of the Hamiltonian
are analyzed, the pullbacks of differential forms and the making sense of them,
are not discussed in \cite{bambusi}.
For example, there is no explicit discussion on why $\mathfrak{F} ^{t*} \Omega _t$ makes sense
in formula (3.42) \cite{bambusi}, i.e. \eqref{eq:fdarboux} here.
\end{remark}
\begin{remark}
\label{rem:formal2} In the 2nd version of \cite{bambusi} there is an incorrect effective
Hamiltonian. If we use the correct definition of the symbols $ \mathbf{{S}}^{i, j} $ which we give above,
the functions $\Phi _{\mu \nu }$ used in the normal form expansion in \cite{bambusi} are in $\mathcal{W}^j$ for some large $j$, rather than in $ \cap _{j\ge 0}\mathcal{W}^j$. In pp. 25--27 in the 2nd version of \cite{bambusi}, the
$\mathcal{W}^j$'s are defined using the classical
pair of operators $L_\pm $, see \cite{W2}, and are closed subspaces of $ H^{j-1}(\R^3)$ of finite codimension.
This last fact seems to be unnoticed in \cite{bambusi} and leads to the breakdown of the proof in the 2nd version of \cite{bambusi}, as we explain below.
The space $\mathcal{W}^2$, for example,
is defined by first considering $ \langle L_+u, u\rangle $
for
$u\in \ker ^\perp L_-\cap \ker ^\perp L_+\subset L^2$. Notice that $ \langle L_+u, u\rangle \ge 0 ,$
see Prop. 2.7 \cite{W2} or Lemma 11.12 \cite{RSS}. Proceeding like in Lemma 11.13 \cite{RSS}
it can be shown that for $ u\in \ker ^\perp L_-\cap \ker ^\perp L_+\subset L^2$ with $u\neq 0$ we have
$\| u \| _L^2 :=\langle L_+u, u\rangle >0.$ Then consider
the completion of $\ker ^\perp L_- \cap \ker ^\perp L_+\cap C_0^\infty$ by the norm $\| u \| _L $.
This completion is exactly
$\ker ^\perp L_-\cap \ker ^\perp L_+\cap H^1(\R^3)$. Then $\mathcal{W}^2$ is
a closed subspace of finite codimension of the latter space. Specifically,
$\mathcal{W}^2$ is in the continuous spectrum part
in the spectral decomposition of the operator $L_-L_+$, which is selfadjoint for
$\langle u, v \rangle _L := \langle L^{-1}_- u, v \rangle $ in $\ker ^\perp L_-$.
Notice that, under hypotheses analogous to (L1)--(L6) in Sect. \ref{sec:speccoo}, $L_-L_+$ has finitely many eigenvalues and its eigenfunctions are Schwartz functions.
Likewise, also the other $\mathcal{W}^j $'s are closed subspaces of $ H^{j-1}(\R^3)$ of finite codimension. Later in the 2nd version of \cite{bambusi}, at
p.41, the Strichartz estimates hinge on the false inclusion
of $ \mathcal{W}^j $, or of $ \mathcal{W}^\infty $, in $L^{ \frac{6}5 }(\R ^3, \C )$.
Additional mistakes appear in the justification of the Fermi Golden rule. While formulas
$R^\pm _{L_0}(\rho )\Phi $ in (St.2)--(St.3) on p. 38 of the 2nd version make sense because $ \Phi \in H^{k ,s }$
for $s>0$ appropriate, analogous formulas $R^\pm _{B}(\rho )\Phi $ in (6.50) and elsewhere in Sect. 6.2,
are undefined when we know only that $\Phi \in \mathcal{W}^\infty $. In fact even $R^\pm _{-\Delta }(\rho )\Phi $
is undefined for $\rho \ge 0$ for such $\Phi$'s. So in particular, in the 2nd version of \cite{bambusi},
the discussion of the Fermi Golden rule is purely formal.
The above ones are not simple oversights. Rather, they stem from the fact that,
in the 2nd version of \cite{bambusi}, the
homological equations are solved only in these $\mathcal{W}^j$'s, while it is unclear if they can be solved in spaces
with spacial weights like the $H^{k,n}$ or the $\Sigma _n$ for $n>0$, as we remarked in an early version of \cite{Cu1}.
The 3rd version of \cite{bambusi}
credits our remark for having stimulated
changes in this part of the paper. These changes are classified in the 3rd version of \cite{bambusi}
as plain simplifications.
This might leave the wrong impression that the proof in the 2nd version of \cite{bambusi},
although more complicated than in the 3rd version, is still correct.
\end{remark}
\section{The NLS and the Nonlinear Dirac Equation}
\label{sec:examples}
We give a sketchy discussion of few examples.
\textbf{The Nonlinear Schr\"odinger equation.} We consider the equation
\begin{equation*}
\im U_{t }=-\Delta U + 2B'(|U|^2) U \ .
\end{equation*}
Here $N=1$, ${\mathcal D}=-\Delta $, $|\ | _1=|\ |$, $J=\begin{pmatrix} 0 &
1 \\
-1 & 0
\end{pmatrix}$.
There are four invariants:
\begin{equation*}
\begin{aligned} & Q(U)=\Pi _4(U)=\frac 12\langle U , U \rangle \text{ and }
\Pi _j(U)=\frac 12\langle U , J \frac{\partial }{\partial x_j} U \rangle \text{ for $j\le 3.$}
\end{aligned}
\end{equation*}
For fixed $v\in \R ^3$ we have
\begin{equation*}
\begin{aligned} & Q(e^{-\frac 12 J v\cdot x}U)=Q(U) \ , \
\Pi _j(e^{-\frac 12 J v\cdot x}U)=\Pi _j( U) -\frac{v_j}2 Q(U) \text{ for $j\le 3 $ and} \\&
E(e^{-\frac 12 J v\cdot x}U) = E(U)-\sum _{j=1}^{3} v_j\Pi _j( U) + \frac{v^2}{2}Q(U).
\end{aligned}
\end{equation*}
There is well established theory guaranteeing under appropriate hypotheses existence of open sets $\mathcal{O}\subseteq \R ^+$
and
$(\phi _ {\omega },0)\in C^\infty (\mathcal{O},{ \mathcal S}(\R ^3, \R ^2))$ such that
\begin{equation*}
\Delta \phi _ {\omega } -\omega \phi _ {\omega }+2B'( \phi _ {\omega } ^2)\phi _ {\omega }=0\quad\text{for $x\in \R^3$}.
\end{equation*}
More precisely it is possible to prove exponential decay to 0 of $\phi _ {\omega }(x)$ as $x\to \infty$.
\noindent For $v\in \R ^3$
arbitrary
we get
$\Phi _p (x) =e^{-\frac 12 J v\cdot x} (\phi _ {\omega } (x),0)$ where $p_4= \Pi _4(\phi _ {\omega })$
and $p_j=-\frac 12 v_j p_4$ for $j\le 3.$
We have $\lambda _4 (p)=-\omega - \frac {v^2}4$ and
$\lambda _j (p)=-v_j$ for $j\le 3.$ Notice that for $\frac{d}{d\omega } Q(\phi _ {\omega }) \neq 0$
this yields \eqref{eq:nondegen}. Notice that
\begin{equation*}
\begin{aligned} & \nabla ^2
E(e^{-\frac 12 J v\cdot x}U) = e^{-\frac 12 J v\cdot x}
\left ( \nabla ^2 E(U)- J v\cdot \nabla _x + \frac{v^2}{4} \right ) e^{\frac 12 J v\cdot x}
\end{aligned}
\end{equation*}
and that $v\cdot \nabla _x \circ e^{-\frac 12 J v\cdot x} =e^{-\frac 12 J v\cdot x}\circ (v\cdot \nabla _x -J \frac {v^2}2) $
and
\begin{equation*}
\begin{aligned} & \nabla ^2
E(\Phi _p (x) ) -\lambda (p) \cdot \Diamond = e^{-\frac 12 J v\cdot x}
\left ( \nabla ^2 E((\phi _ {\omega },0))- J v\cdot \nabla _x + \frac{v^2}{4} \right ) e^{\frac 12 J v\cdot x} \\&
+ Jv\cdot \nabla _x e^{-\frac 12 J v\cdot x} e^{\frac 12 J v\cdot x} + (\omega +\frac {v^2}4)e^{-\frac 12 J v\cdot x} e^{\frac 12 J v\cdot x}.
\end{aligned}
\end{equation*}
They imply
\begin{equation}\label{eq:conjNLS}
\begin{aligned} & {\mathcal H}_p =e^{-\frac 12 J v\cdot x}
{\mathcal H}_\omega e^{\frac 12 J v\cdot x} \, , \quad {\mathcal H}_\omega
:=J
(\nabla ^2 E((\phi _ {\omega },0)) +\omega ).
\end{aligned}
\end{equation}
The multiplier operator $e^{-\frac 12 J v\cdot x}$ is an isomorphism in all spaces $\Sigma _n$ so all the information on the spectrum of
${\mathcal H}_p $ is obtained from the spectrum of ${\mathcal H}_\omega .$
We have $ {\mathcal H}_\omega = {\mathcal H}_{0\omega} +V $ where $ {H}_{0\omega}:= J (-\Delta +\omega )$
and
\begin{equation*}
\begin{aligned} & V:= 4 J \begin{pmatrix} - B' (\phi _\omega ^2 )-2 B ^{\prime \prime} (\phi
_\omega ^2) \phi_\omega^2 &
0 \\
0 & - B' (\phi _\omega ^2 )
\end{pmatrix} .
\end{aligned}
\end{equation*}
This yields $\sigma _e( {\mathcal H}_\omega) =\sigma ( {H}_{0\omega}) =(-\infty , -\omega ]\cup [\omega , \infty )$ and that $\sigma _p( {\mathcal H}_\omega)$
is finite with finite multiplicities. The fact that $\sigma _p( {\mathcal H}_\omega) $ is in the complement of
$ \sigma _e(\mathcal{H} _{\omega })$ is expected to be true generically.
Set ${\mathcal H} ={\mathcal H}_\omega P_c(\omega )$ for
$P_c(\omega )$ the projection on $X_c({\mathcal H}_\omega)$.
\begin{lemma} \label{lem:a5} The statement in (A5) is true.
\end{lemma}
\proof Notice that $\Sigma _n$ is invariant by Fourier transform so that \eqref{eq:a71}
is equivalent to the fact that for the following multiplier operator
(that is an operator $\psi (x)$ which maps $u\to (\psi u)(x):=\psi (x)u(x)$)
we have
\begin{equation} \label{eq:a71bis} \begin{aligned}
&
\quad \text{ $\| (1+\epsilon ^2+\epsilon ^2|x| ^2)^{ -2} \| _{B(\Sigma _n ,\Sigma _ n ) } \le C_n<\infty$ $\forall$ $|\epsilon |\le 1 $ and $n\in \N$.}
\end{aligned}
\end{equation}
Similarly \eqref{eq:a72} is equivalent to
\begin{align} \label{eq:a72bis}
& strong-\lim _{\epsilon\to 0} (1+\epsilon ^2+\epsilon ^2|x| ^2)^{ -2} =1 \text{ in $B(\Sigma _n ,\Sigma _ n )$ } \\& \lim _{\epsilon\to 0} \| (1+\epsilon ^2+\epsilon ^2|x| ^2)^{ -2} -1 \| _{B(\Sigma _n ,\Sigma _ {n'} ) }=0
\quad \text{ for any $n'\in \N$ with $n'<n$. } \nonumber
\end{align}
Both \eqref{eq:a71bis}--\eqref{eq:a72bis} are elementary to check
using the first definition of $\Sigma _n$ in Sect.\ref{sec:setup},
computing commutators of the multiplier operators with $\partial ^\alpha _x$ and computing elementary bounds on the derivatives of the multipliers.\qed
\begin{lemma} \label{lem:a6} The statement in (A6) is true.
\end{lemma}
\proof Using the Fourier transformation like in Lemma \ref{lem:a5}, (A6) is equivalent to
the statement that for any $n\in \N$ and $c>0$ there a $C$ s.t.
the following multiplier operator satisfies
\begin{equation*}
\| e^{ (1+\epsilon ^2+\epsilon ^2|x| ^2)^{ -2} J( \tau _4 - \sum _{j=1}^3 x_j \tau _j } ) \| _{B(\Sigma _n,\Sigma _n)}\le C
\end{equation*}
for any $|\tau |\le c$ and any $|\epsilon |\le 1 $. This too is elementary to check.
\begin{lemma} \label{lem:l7} The statement in (L7) is true.
\end{lemma}
\proof From $ \sigma ({\mathcal H} ) =\sigma _e(\mathcal{H} _{\omega })$ we have
$R_{\mathcal H}\in C^\omega (\rho ({\mathcal H} ), B(L^2,L^2))$.
\noindent We have $R _{ {\mathcal H}_{0\omega}}$ and $R _{ {\mathcal H}_{0\omega}}\partial _{x_j }$ are in $ C^\omega (\rho ({\mathcal H} ), B(\Sigma _n,\Sigma _n)) $
for any $n\in \N$. By conjugation by Fourier transform this is equivalent to the statement that for $z\in \rho ({\mathcal H}_{0\omega})$ and $i=0,1$, we have
\begin{equation} \begin{aligned} & \xi _j ^i
\begin{pmatrix} ( |\xi |^2+\omega -z)^{-1} & 0 \\ 0 & - ( |\xi |^2+\omega +z)^{-1} \end{pmatrix}
\in B(\Sigma _n,\Sigma _n).
\end{aligned} \nonumber \end{equation}
This is elementary, using the first definition of $\Sigma _n$ in Sect.\ref{sec:setup}.
\noindent We have for $i=0,1$
\begin{equation}\label{eq:resolv1}
\begin{aligned} & R _{ {\mathcal H } } (z) \partial _{x_j }^i = R _{ {\mathcal H}_{0\omega}} (z) P_c(\omega ) \partial _{x_j }^i -
R _{ {\mathcal H}_{0\omega}}(z) VR _{ {\mathcal H } } (z) \partial _{x_j }^i .
\end{aligned} \end{equation}
From \eqref{eq:resolv1} we derive, for $\| \ \| = \| \ \| _{B(L^2,L^2)}$.
\begin{equation}\label{eq:resolv11}
\begin{aligned} & \| R _{ {\mathcal H } } (z)\partial _{x_j }^i \| \le
\| (1+ R _{ {\mathcal H}_{0\omega}}(z) V) ^{-1}\|
\| R _{ {\mathcal H}_{0\omega}} (z) P_c(\omega ) \partial _{x_j }^i \| ,
\end{aligned}
\end{equation}
which yields the $n=0$ case.
\noindent From \eqref{eq:resolv1} we derive
\begin{equation*}\label{eq:resolv2}
\begin{aligned} & \| R _{ {\mathcal H } } (z)\partial _{x_j }^i \| _{B(\Sigma _n,\Sigma _n)} \le C \| R _{ {\mathcal H}_{0\omega}} (z)\partial _{x_j }^i \| _{B(\Sigma _n,\Sigma _n)}\\& +C
\| R _{ {\mathcal H}_{0\omega}} (z) \| _{B(\Sigma _n,\Sigma _n)} \| \langle x \rangle ^nV \| _{W^{n,\infty}}
\| R _{ {\mathcal H } } (z) \partial _{x_j }^i \| _{B(H^n,H^n)}.
\end{aligned}
\end{equation*}
\noindent The last factor is bounded. Indeed for $\mathbf{v}=R _{ {\mathcal H } } (z)\partial _{x_j }^i\mathbf{u}$ we have
\begin{equation*}\label{eq:resolv3}
\begin{aligned} & \partial ^\alpha _x \mathbf{v}= R _{ {\mathcal H } } (z)\partial ^\alpha _x\partial _{x_j }^i \mathbf{u} +R _{ {\mathcal H } } (z) [V,\partial ^\alpha _x]\partial _{x_j }^i\mathbf{u}
\end{aligned}
\end{equation*}
and induction in $ n$ yields the desired bounds $\| \mathbf{v} \| _{H^n}\le C \| \mathbf{u} \| _{H^n} $.
\qed
\bigskip
\textbf{The Nonlinear Dirac Equation.}
Here the unknown $U$ is $\C^4$-valued, $u^*$ its complex conjugate and for $m>0$
\begin{equation}\label{Eq:NLDE}
\im U_t -D_m U -Vu+2B'(U\cdot \beta {U}^* )\beta U=0
\end{equation}
where we assume for the moment $V=0$ and where
$D_m=-\im \sum _{j=1}^3\alpha_j\partial_{x_j}+m\beta$,
with for $j=1,2,3$
\begin{equation*}
\alpha _j= \begin{pmatrix} 0 &
\sigma _j \\
\sigma_j & 0
\end{pmatrix} , \ \beta = \begin{pmatrix} I _{\C^2} &
0 \\
0 & -I _{\C^2}
\end{pmatrix}
, \end{equation*} \begin{equation*}
\sigma _1=\begin{pmatrix} 0 &
1 \\
1 & 0
\end{pmatrix} \, ,
\sigma _2= \begin{pmatrix} 0 &
\rmi \\
-\rmi & 0
\end{pmatrix} \, ,
\sigma _3=\begin{pmatrix} 1 &
0 \\
0 & -1
\end{pmatrix}.
\end{equation*}
Notice that the symmetry group \eqref{Eq:NLDE} is not Abelian. In \cite {boussaidcuccagna}
there is a symmetry restriction on the solutions considered, by looking only at functions
such that for any $x\in \R ^3 $ we have { $U (-x) =\beta U (
x )$ and $U (-x_1,-x_2,x_3)=S_3U (x_1, x_2,x_3)$} with
$S_3:=\begin{pmatrix}\sigma_3&0\\0&\sigma_3\end{pmatrix}$.
We need to redefine the spaces $\Sigma _n$ in the proof, introducing these symmetries. This does not affect the proof.
\noindent There is a unique invariant $Q(U)=\frac 12 \| u \| _{L^2}.$ In this case $\Diamond _1U=U$ for any $u$. Hence all the changes of variables are diffeomorphism within each space $\Ph ^K$ (or $\widetilde{\Ph} ^K$).
\noindent (A5)--(A6) in this case are elementary. In fact (A5) is unnecessary, (A6) is necessary only for $\epsilon =0$, in which case is trivial. (L7) is necessary only for $i=0$ (given that the only $\Diamond _j$ is the identity)
and can be proved in a way similar to Lemma \ref{lem:l7}.
\textbf{Nonlinear Dirac Equation with a Potential.} Pick $V\in {\mathcal S}(\R ^3, B(\C^4))$ with
$V(x)$ selfadjoint for the scalar product in $\C^4$ for any $x\in \R ^3$. Then generically $\sigma _p(D_m +V) \subset (-m,m)$. Suppose $\sigma _p(D_m +V) =\{ e_0,..., e_{\textbf{n} } \}$ with $ e_0<...< e_{\textbf{n} }$.
Then bifurcation yields corresponding families of small standing waves $e^{-\im \omega t} \phi _\omega (x)$
of \eqref{Eq:NLDE}. For generic $V$ the $e_j$ have multiplicity 1. If we focus on
$e_0$, for generic smooth $B'(r )$
there will be a smooth family $\omega \to \phi _\omega $ in $C^\infty (\mathcal{O}, \Sigma _n)$
for any $n$, with $\mathcal{O}$ an open interval one of whose endpoints is $e_1$. Then it can be shown that for generic $V$
the hypotheses (L1)--(L6) in Sect. are true, as well as all the previous hypotheses. Indeed in this case, taking
$\omega $ sufficiently close to $e_0$,
we have eigenvalues with $\textbf{e}_j'$ arbitrarily close to $e_j-e_0$. Generically this yields (L4)--(L5).
The multiplicity of the $\im \textbf{e}_j'$ is 1. We have $\sigma _e(\mathcal{H}_\omega )= (-\infty , -m +|\omega |]
\cup [m-|\omega |, \infty )$. An eigenvalue $\lambda $ of $\mathcal{H}_\omega$
is either $\lambda =0$, or $\lambda =\pm \im \textbf{e}_j'$ for some $j$.
This in particular yields (L1)--(L3).
|
{
"timestamp": "2012-06-07T02:02:07",
"yymm": "1203",
"arxiv_id": "1203.2120",
"language": "en",
"url": "https://arxiv.org/abs/1203.2120"
}
|
\section{Introduction}
ALICE is a dedicated experiment built to exploit the unique physics
potential of heavy-ion interactions at LHC energies
\cite{Aamodt:2008zz}. Properties of strongly interacting matter at
extreme energy density are explored via a comprehensive studies of
hadron, muon, electron and photon production in the collisions of
heavy nuclei and their comparison with proton-proton collisions.
Presently, the ALICE collaboration consists of about 1600 members from
33 countries. Russian nuclear-physics community takes an active part
in ALICE since the very beginning, now counting 134 members from 12
institutes. Russian institutes contribute in almost every major
sub-detectors of the ALICE experiment, and also take part in physics
analysis of data collected in 2010--2011.
The ALICE experiment has collected a rich sample of data with
proton-proton and lead-lead collisions. In 2010 and beginning of 2011,
about $10^9$~events with the minimum bias trigger were recorded,
corresponding to the integrated luminosity $\int{\cal L}dT =
16~\mbox{nb}^{-1}$. Rare-event triggers on muons, jets and photons
were dominant in data taking with the proton beams at collision energy
$\sqrt{s}=7$~TeV in the second half of 2011, with the delivered
integrated luminosity $\int{\cal L}dT = 4.9~\mbox{pb}^{-1}$. Limited
data samples with the proton beams at collision energies $\sqrt{s}=0.9$
and 2.76~TeV have been also recorded with integrated luminosities
$\int{\cal L}dT = 0.14\mbox{~and~}1.3~\mbox{nb}^{-1}$ respectively.
Among rare-event triggers used in data taking in 2011, one has to
mention the trigger on the MUON detector selecting events with muons
in the high-rapidity range to enrich statistics for $J/\psi$ and
$\Upsilon$ signals (this trigger was in operation since 2010). A
trigger based on the electromagnetic calorimeter (EMCAL) was selecting
events with high-energy photons and jets in the central
barrel. Another ALICE calorimeter, a photon spectrometer PHOS, has
provided a trigger on photons with a moderate energy threshold, to
enhance a data sample for neutral meson and direct photon studies.
The first run with lead-lead beams at collision energy
$\sqrt{s_{_{NN}}}=2.76$~TeV was taken with ALICE in November 2010. The
delivered integrated luminosity was $\int{\cal L}dT =
10~\mu\mbox{b}^{-1}$. The dominant trigger in 2010 was a minimum bias
one. In November 2011, the LHC has delivered 10 times more data, and
the ALICE experiment has restricted the minimum-bias trigger share in
favor of several rare-event triggers with the total life time
80\%. Detector VZERO has deployed triggers on the most central events
with selected centralities $0-10\%$ and semi-central events with
centralities $20-60\%$. A trigger on ultra-peripheral collisions was
realized on SPD and TOF detectors. Other triggers implemented earlier
in pp collisions on EMCAL, PHOS and MUON detectors, were also active
in the PbPb run 2011.
\section{Hadron production in proton-proton collisions}
Measurements of identified hadron spectra are considered as an
important test of various non-perturbative models of hadron production
at high energies, as well as those of perturbative QCD
calculations. ALICE performs extensive studies of hadron production
due to its powerful particle identification capabilities
\cite{Aamodt:2008zz}. Charged particles are identified by several
tracking detectors covering complimentary kinematic ranges. Barrel
tracking detectors are embedded into a solenoidal magnet with magnetic
field of 0.5~T. This is a relatively soft magnetic field which allows
to reconstruct charged tracks at transverse momenta starting from
$p_{\rm t}>50$~MeV/$c$. Inner Tracking System (ITS) and Time
Projection Chamber (TPC) can identify charged particles in the full
$2\pi$ azimuthal angle and pseudorapidity range $|\eta|<0.9$, via
measurements of their ionization loss $dE/dx$. Time-of-flight
measurements, provided by the TOF detector in the same solid
angle as ITS and TPC, can discriminate charged pions, kaons and
protons in a higher momentum range. The limited-acceptance High-Momentum
Particle Identification detector (HMPID) is a Cherenkov detector
covering a solid angle $\Delta\phi=60^\circ$ and $|\eta|<0.6$ is used
to identify charged particles at a higher momentum range, up to
$p=5$~GeV/$c$. Transition Radiation Detector (TRD) is another
barrel detector surrounding TPC, which is designed to identify
electrons and at present covers about a half of the complete azimuthal
angle.
Photons and neutral mesons decaying into photons are detected and
identified by two electromagnetic calorimeters. A precise Photon
Spectrometer (PHOS) is a high-granularity calorimeter built of lead
tungstate crystals (PbWO$_4$). Its small Moli\'ere radius, high density
and high light yield allow to detect photons with the best possible
energy resolution in the energy range up to $E<100$~GeV in the
azimuthal angle range $\Delta\phi=60^\circ$ and $|\eta|<0.13$. Its high
spatial resolution provides measurements of neutral pions via
invariant mass spectrum at transverse momenta $0.6<p_{\rm
t}<50$~GeV/$c$. Another, wide-aperture Electromagnetic Calorimeter
(EMCAL) is a sampling-type calorimeters built of lead-scintillator
modules. Its primary goal is to trigger jets and measure a neutral
component of jets. Dynamic range of EMCAL covers energies up to
250~GeV, and granularity of this calorimeter allows to reconstruct
$\pi^0$ mesons at transverse momenta $1<\pT<20$~GeV/$c$.
Muon identification is provided in ALICE by the muon arm which is
installed in the forward rapidity range $2.5<y<4$. This muon detector
is a magnet spectrometer consisting of a set of proportional chambers
in the dipole magnetic field. Hadronic background is suppressed by
the hadron absorber installed in front of the muon spectrometer.
Using charged hadron identification in ITS, TPC and TOF,
ALICE has measured production spectra $dN/d\pT$ of identified charged
hadron ($\pi^\pm$, $K^\pm$, $p$, $\bar{p})$ in the minimum bias pp
collisions at collision energies $\sqrt{s}=0.9$~\cite{PIDhadron900GeV}
and $7$~TeV~\cite{PIDhadron7TeV} (Fig.\ref{fig:pp-piKp_spectra}).
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\hsize]{figures/pp7TeV_piKp_Tsallis.pdf}
\caption{Transverse momentum spectra of $\pi^-$, $K^-$, $\bar{p}$ in
pp collisions at $\sqrt{s} = 7$~TeV. The lines are the
Levy-Tsallis fits.}
\label{fig:pp-piKp_spectra}
\end{figure}
The spectra were fitted with the Tsallis function~\cite{Tsallis:1987eu}
\begin{equation}
\displaystyle
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
E \frac{{\rm d}^3 \sigma}{{\rm d}p^3} =
\displaystyle
\frac{\sigma_{pp}}{2\pi}\frac{{ \rm d}N}{{\rm d}y}
\frac{c \cdot (n-1)(n-2)}{nC\left[ nC+m(n-2)\right]}
\displaystyle\left(1+\frac{\mT-m}{nC}\right)^{-n},
\label{eq:Tsallis}
\end{equation}
where the fit parameters are ${\rm d}N/{\rm d}y$, $C$ and $n$,
$\sigma_{\rm pp}$ is the proton-proton inelastic cross section, $m$ is
the meson rest mass and $\mT=\sqrt{m^2+\pT^2}$ is the transverse
mass. The integrated yield at $y=0$, defined by the Tsallis parameter
${\rm d}N/{\rm d}y$, was evaluated from the ALICE data, and thus the
total yields of charged pions, kaons and protons was found. The ratios
of integrated yields $K^\pm/\pi^\pm$, $\bar{p}/\pi^-$ and $p/\pi^+$ in
pp collisions at $\sqrt{s}=0.9$ and $7$~TeV were compared with those
measured at lower collision energies, as shown in
Fig.\ref{fig:pp-k2pi-p2pi}.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/pp7TeV_p2pi.pdf}
\hfil
\includegraphics[width=0.48\hsize]{figures/pp7TeV_K2pi.pdf}
\caption{Integrated yield ratio of $K/\pi$ (left) and $\bar{p}/\pi^-$
(right) as a function of collision energy.}
\label{fig:pp-k2pi-p2pi}
\end{figure}
A trend of slight increase of $K^\pm/\pi^\pm$ ratio with $\sqrt{s}$
can be observed. ALICE data also suggest that baryon-antibaryon
asymmetry, observed at RHIC, vanishes at LHC energies, as expected.
Tsallis parameterization allows to find also the mean transverse
momentum $\langle \pT \rangle$ and to observe its evolution with
collision energy (Fig.\ref{fig:pp-meanpt}).
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\hsize]{figures/pp7TeV_meanpt.pdf}
\caption{Mean $p_{\rm t}$ for charged $\pi$, $K$ and $p$ at
different collision energy in pp collisions.}
\label{fig:pp-meanpt}
\end{figure}
Comparison of mean $\pT$ of different hadron species measured at
different collision energies indicates that hadron production spectra
become harder at higher $\sqrt{s}$, and also mean $\pT$ grows with
hadron mass.
ALICE has also measured production spectra of neutral pions and $\eta$
mesons in pp collisions at $\sqrt{s}=0.9$, $2.76$ and 7~TeV, using the
Photon Spectrometer (PHOS) for real photon detection and central
tracking system for converted photon reconstruction
\cite{pp-pi0}. Neutral meson reconstruction, performed via invariant
mass spectra of photon pairs, allowed to measure differential cross
section of $\pi^0$ and $\eta$ in a wide \pT\ range. In particular, the
spectrum of $\pi^0$ production at the three collision energies are
shown of the left plot of Fig.\ref{fig:pp-pi0}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\hsize]{figures/pp7TeV_pi0Spectrum.pdf}
\includegraphics[width=0.48\hsize]{figures/pp7TeV_pi0Ratio.pdf}
\caption{Production spectrum of $\pi^0$ in pp
collisions at $\sqrt{s}=0.9$, $2.76$ and $7$~TeV (left) and
ratio of NLO pQCD calculations to the measured spectra (right).}
\label{fig:pp-pi0}
\end{figure}
Hadron production at high \pT\ can be well calculated in the
next-to-leading orders of perturbative QCD (NLO pQCD). These
calculations are based on parton distribution (PDF) and fragmentation
functions (FF) measured at lower energies. Application of those PDF's
and FF's to the new energy domain delivered by LHC, lead to
extrapolations of those functions to the kinematic region where the
functions have large uncertainties. The ratio of differential cross
sections of $\pi^0$ and $\eta$ mesons in pp collisions, calculated by
NLO pQCD, to the Tsallis fit of the ALICE measurements are shown by
curves on the right plot of Fig.\ref{fig:pp-pi0}. Data points on this
plot represent the ratio of the measured cross section to the Tsallis
fit to the measurement, which demonstrates the quality of the data
description by the Tsallis parameterization. This comparison of
theoretical calculations and experimental measurements demonstrates
that NLO pQCD at the QCD scale $\mu=\pT$ describes well hadron
production in pp collisions at $\sqrt{s}=0.9$~TeV, while significantly
overestimate it at $\sqrt{s}=7$~TeV. No common set of pQCD parameters
can be found to describe equally well the spectra of pion production
at all three collision energies.
Strangeness production is one of the most important observables for
studying the strongly interacting matter produced in heavy-ion
collisions. That is why measurements of complete set of strange
hadrons in pp collisions is necessary as a reference for comparison
with heavy ion collisions. Besides charged kaons mentioned earlier,
ALICE has measured production spectra of many other strange hadrons,
as well as those of mesons with hidden strangeness ($K^*$, $\Lambda$,
$\Sigma$, $\Omega$, $\phi$ and strange resonance baryons). Production
yields of $(\Sigma^*_{} + \bar{\Sigma^*}^-)/2$ and $\phi$ mesons in pp
collisions at $\sqrt{s}=7$~TeV are shown in
Fig.\ref{fig:pp-resonances} and are compared with several MC
predictions.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/pp7TeV_Sigmastar.pdf}
\hfil
\includegraphics[width=0.48\hsize]{figures/pp7TeV_phi.pdf}
\caption{Production spectrum of $(\Sigma^{*+} + \bar{\Sigma^*}^-)/2$ and $\phi$ in pp
collisions at $\sqrt{s}=7$~TeV with Monte Carlo
predictions by different models.}
\label{fig:pp-resonances}
\end{figure}
Identified hadron spectra measured at LHC energies, in conjunction
with spectra measured by previous experiments at lower collision
energies, allow to observe evolution of hadron production properties
with $\sqrt{s}$. Predictions of various phenomenological models, as
well as NLO pQCD calculations were found to be unable to describe all
identified hadron spectra measured by ALICE in pp collisions
\section{Heavy ion collisions}
Analysis of the first heavy-ion data collected in 2010 brought many
results giving an insight into the properties of strongly interacting
matter at the new energy density regime. Observables characterizing
this matter are classified into several groups which will be reviewed
in this section.
\subsection{Global event properties}
As heavy nuclei are extended objects, centrality determination is an
essential point for all heavy-ion measurements. Centrality of the
collision, directly related to the impact parameter and to the number
of nucleons $N_{\rm part}$ participating in the collision, allows to
study particle production versus the density of the colliding system.
In the ALICE experiment, collision centrality can be measured by
several detectors. The best accuracy of centrality measurement is
achieved with the scintillator hodoscope VZERO covering pseudorapidity
ranges $2.8 < \eta < 5.1$ and $-3.7 < \eta < -1.7$. Distribution of
the sum of amplitudes in VZERO in minimum bias Pb-Pb collisions is
shown in Fig.\ref{fig:PbPb-centrality} (left)
\cite{bib:PbPb-dNdy}. Centrality classes were defined by Glauber
model, and the fit of the Glauber model to the data is shown by a
solid line in this plot. Centrality resolution for all the estimators
can be found in Fig.\ref{fig:PbPb-centrality} (right)
\cite{bib:ToiaQM2011} which demonstrates that the best resolution is
achieved with the VZERO detector, and is equal to about 0.5\% in the
most central events, and varies up to 1.5\% in the most peripheral
collisions.
\begin{figure}[ht]
\includegraphics[width=0.51\hsize]{figures/PbPb_v0glauber.pdf}
\hfill
\includegraphics[width=0.46\hsize]{figures/PbPb_centres.pdf}
\caption{Centrality determination in ALICE. Glauber model fit to the
VZERO amplitude with the inset of a zoom of the most peripheral
region (left); Centrality resolution with different detectors
(right).}
\label{fig:PbPb-centrality}
\end{figure}
One of the key observable in heavy ion collision is the charged
particle multiplicity and its dependence on the collision
centrality. The main detector used for this measurements in the
Silicon Pixel Detector (SPD), two innermost layers of the barrel
tracking system covering the pseudorapidity range $|\eta|<1.4$. The
charged particle density, normalized to the average number of
participants in a given centrality class, $dN_{\rm
ch}/d\eta/\left(\langle N_{\rm part} \rangle \right)$ was measured
by ALICE in PbPb collisions at $\sqrt{s_{_{NN}}}=2.76$~TeV and compared with
similar measurements at lower energies at RHIC and SPS
(Fig.\ref{fig:PbPb-dNdeta}, left plot) \cite{bib:ToiaQM2011}.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/PbPb_dNdy.pdf}
\hfill
\includegraphics[width=0.48\hsize]{figures/PbPb_dNdeta_Npart.pdf}
\caption{Charged track density $dN/d\eta$ in pp and AA collisions vs
collision energy (left) and vs the number of participants (right).}
\label{fig:PbPb-dNdeta}
\end{figure}
In the most central events (centrality $0-5\%$) at LHC energy the
charged particle density was found to be $dN_{\rm ch}/d\eta=1601\pm
60$ \cite{bib:PbPb-dNdy} which is, being normalized to the number of
participants, is 2.1 times larger than the charged particle density
measured at RHIC at $\sqrt{s_{_{NN}}}=200$~GeV and 1.9 times larger than that in
pp collisions at $\sqrt{s}=2.36$~TeV. The dependence of $dN_{\rm
ch}/d\eta$ on the number of participants $N_{\rm part}$, shown in the
right plot of Fig.\ref{fig:PbPb-dNdeta}, is very similar at LHC
($\sqrt{s_{_{NN}}}=2.76$~TeV) and RHIC ($\sqrt{s_{_{NN}}}=0.2$~TeV) energies, provided
the RHIC points are scaled by a factor 2.1 to match the LHC points.
Longitudinal and transverse expansion of the highly compressed
strongly-interacting system created in heavy-ion collisions can be
studied experimentally via intensity interferometry, the Bose-Einstein
enhancement of identical bosons emitted close by in phase space, known
as Hanbury Brown-Twiss analysis (HBT). ALICE has measured the HBT
radii and evaluate space-time propertied on the system generated in
Pb-Pb collisions at $\sqrt{s_{_{NN}}}=2.76$~TeV \cite{bib:HBT}. The two-particle
correlation function of the difference $\vec{q}$ of two 3-momenta
$\vec{p_1}$ and $\vec{p_2}$ was measured for like-sign charged pions
which allowed to get the Gaussian HBT radii, $R_{\rm out}$, $R_{\rm
side}$ and $R_{\rm long}$. The product of these 3 radii and
decoupling time extracted from $R_{\rm long}$, measured by
ALICE at LHC energy, together with this value measured at the AGS, SPS
and RHIC, is shown in Fig.\ref{fig:PbPb-HBT} (left) as a function of
charged track density $dN_{\rm ch}/d\eta$.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/PbPb_HBT_R.pdf}
\hfill
\includegraphics[width=0.48\hsize]{figures/PbPb_HBT_tau.pdf}
\caption{System size (left) and lifetime (right).}
\label{fig:PbPb-HBT}
\end{figure}
This measurements indicate that the homogeneity volume in central PbPb
collisions at $\sqrt{s_{_{NN}}}=2.76$~TeV exceeds that measured at RHIC by a
factor of 2. The increase is present in both longitudinal and
transverse radii. The decoupling time for mid-rapidity pions exceeds 10
fm/c which is 40\% larger than at RHIC (Fig.\ref{fig:PbPb-HBT}, right).
\subsection{Collective expansion}
In non-central collision of nuclei, the overlap region, and hence the
initial matter distribution is anisotropic. During evolution of the
matter, the spatial asymmetry of initial state is converted to an
anisotropic momentum distribution. The azimuthal distribution of the
particle yield can be expressed in terms of the angle between the
particle direction $\varphi$ and the reaction place $\Psi_{\rm RP}$:
\begin{eqnarray}
\frac{dN}{d(\varphi-\Psi_{\rm RP})} & \propto
& 1+2\sum_{n=1}^{}v_n \cos\left[n(\varphi-\Psi_{\rm RP})\right], \\
& & v_2 = \langle \cos\left[n(\varphi-\Psi_{\rm RP})\right] \rangle.
\end{eqnarray}
The second coefficient of this Fourier series, $v_2$, is referred to
as elliptic flow. Theoretical models, based on relativistic
hydrodynamics \cite{bib:hydro-v2_Kestin,bib:hydro-v2_Niemi},
successfully described the elliptic flow observed at RHIC
\cite{bib:RHIC_v2} and predict its increase at LHC energies from 10\%
to 30\%.
The first measurements of elliptic flow of charged particles in Pb-Pb
collisions at $\sqrt{s_{_{NN}}}=2.76$~TeV were reported by ALICE in
\cite{bib:ALICE-v2}. Charged tracks were detected and reconstructed in
the central barrel tracking system, consisting of ITS and
TPC. Elliptic flow integrated over $\pT$ range $0.2 < \pT <
5$~GeV/$c$, for the 2- and 4-particle cumulant methods, is shown in
Fig.\ref{fig:PbPb-v2} (left) as a function of centrality.
\begin{figure}[ht]
\includegraphics[width=0.52\hsize]{figures/PbPb_v2_vs_centrality.pdf}
\hfill
\includegraphics[width=0.44\hsize]{figures/PbPb_v2_vs_E.pdf}
\caption{Azimuthal flow $v_2$ of charged particles in Pb-Pb
collisions at $\sqrt{s_{_{NN}}}=2.76$~TeV vs centrality (left) and
$v_2$ vs collision energy (right).}
\label{fig:PbPb-v2}
\end{figure}
It shows that the integrated elliptic flow increases from central to
peripheral collision and reaches the maximum value $v_2 \approx 0.1$
in semi-central collisions in the $40-60\%$ centrality
class. Comparison of the integrated elliptic flow of charged
particles, measured at different center-mass collision energies, shows
a smooth increase of $v_2$ with $\sqrt{s_{_{NN}}}$, and confirms model
expectations that the value of $v_2$ in Pb-Pb collisions at
$\sqrt{s_{_{NN}}}=2.76$~TeV increases by about 30\% with respect to $v_2$ in
Au-Au collisions at $\sqrt{s_{_{NN}}}=0.2$~TeV.
Particle momentum anisotropy is also studied via two-particle
correlations which measure the distributions of azimuthal angles
$\Delta\varphi$ and pseudorapidities $\Delta\eta$ between a
``trigger'' particle at transverse momentum $\pT^t$ and an
``associated'' particle at $\pT^a$. The correlation function
$C(\Delta\varphi,\Delta\eta)$ looks differently in different kinematic
regions. At $\pT^t < 3-4$~GeV/$c$, the shape of the correlation function
reveals the ``bulk-dominated'' regime, where hydrodynamic
modeling has been demonstrated to give a good description of the data
from heavy-ion collisions (see Fig.\ref{fig:PbPb-ridge}, left).
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/PbPb_RidgeSoft.pdf}
\hfill
\includegraphics[width=0.48\hsize]{figures/PbPb_RidgeHard.pdf}
\caption{Di-hadron correlations $C(\Delta\varphi,\Delta\eta)$ in
central Pb-Pb collisions in the ``bult-dominated'' regime (left)
and in the ``jet-dominated'' regime (right).}
\label{fig:PbPb-ridge}
\end{figure}
At high transverse momenta of both particles, jets become dominating,
and the shape of the correlation function in central Pb-Pb collisions
has just a clear near-side peak centered at $\Delta\varphi =
\Delta\eta = 0$ and no evident out-side peak, as shown in
Fig.\ref{fig:PbPb-ridge}, right. Harmonic decomposition of
two-particle correlations \cite{bib:ALICE-harmonic} performed by
ALICE, has shown that in the ``bulk-dominated'' regime a distinct
near-side ridge and a doubly-peaked away-side structure are observed
in the most central events, which reflects a collective response to
anisotropic initial conditions.
The results of global event properties and collective expantion
studied by ALICE, indicate that the fireball formed in nuclear
collisions at the LHC is hotter, lives longer, and expands to a larger
size at freeze-out as compared to lower energies.
\subsection{Strangeness production}
Strange particle production has been considered as a probe of strongly
interacting matter by heavy-ion experiments at AGS, SPS and RHIC. We
have already demonstrated that ALICE, due to its powerful particle
identification technique, has measured strange particle spectra in pp
collisions. Similar analysis was performed on the Pb-Pb data collected
in 2010. Comparison of strange meson and baryon production is
illustrated by the $\Lambda/K^0_S$ ratio measured by ALICE in
different centrality classes (Fig.\ref{fig:PbPb-Lambda_K0s},
left). This ratio in peripheral Pb-Pb collision is similar to that one
measured in pp collisions, but it grows with centrality,
increasing the value of 1.5 in the most central collisions. The
qualitative behaviour of this ratio on $\pT$ at the LHC collision
energy is similar to the ratio measured at RHIC by the STAR experiment
(Fig.\ref{fig:PbPb-Lambda_K0s}, right). An enhancement of strange and
multi-strange baryons ($\Omega^-$, $\bar{\Omega}^+$,
$\Sigma^-$,$\bar{\Sigma}^+$ ) was obsevred in heavy-ion collisions by
experiments at lower energies, and was confirmed by ALICE at LHC
energy \cite{ALICE-Hippolyte}. It was also shown that multi-strange
baryon enhancement scales with the number of participants $N_{\rm
part}$ and decreases with the collision energy.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/PbPb_Lambda_K0s.pdf}
\hfill
\includegraphics[width=0.48\hsize]{figures/PbPb_Lambda_K0s_STAR.pdf}
\caption{Ratio $\Lambda/K^0_S$ in Pb-Pb collisions at
$\sqrt{s_{_{NN}}}=2.76$~TeV in different centralities (left) and
comparison of this ratio at LHC and RHIC in centralities $0-5\%$
and $60-80\%$ (right).}
\label{fig:PbPb-Lambda_K0s}
\end{figure}
\subsection{Parton energy loss in medium}
Experiments at RHIC reported that hadron production at high
transverse momentum in central Au-Au collisions at a center-of-mass
energy per nucleon pair $\sqrt{s_{_{NN}}}=200$~GeV is suppressed by a factor
$4-5$ compared to expectations from an independent superposition of
nucleon-nucleon collisions. This suppression is attributed to energy
loss of hard partons as they propagate through the hot and dence QCD
medium. Therefore, a spectrum suppression of hadron production can be
used as a measure of the properties of the strongly interacting matter.
The strength of suppression of a hadron $h$ is expressed by
the nuclear modification factor $R_{AA}$, defined as a ratio of the
particle spectrum in heavy-ion collision to that in pp, scaled by the
number of binary nucleon-nucleron collisions $N_{\rm coll}$:
\begin{equation}
R_{AA}(\pT) = \frac{(1/N_{AA})d^2N_h^{AA}/d\pT d\eta}
{N_{\rm coll}(1/N_{pp})d^2N_h^{pp}/d\pT d\eta}.
\end{equation}
At the larger LHC energy, the density of the medium is expected
to be higher than at RHIC, leading to a larger energy loss of
high-$\pT$ partons. However, the hadron production spectra are less
steeply falling with $\pT$ at LHC than at RHIC which would reduce the
value of $R_{AA}$ for a given value of the parton energy loss.
ALICE has measured the nuclear modification factor $R_{AA}$ for many
particles. All charged particles, detected in the ALICE central
tracking system (ITS and TPC), show a spectrum suppression
\cite{Otwinowski:2011gq} which is qualitatively similar to that
observed at RHIC (Fig.\ref{fig:PbPb-RAA_charged}). However,
quantitative comparison with RHIC demonstrates that the suppression at
LHC energy is stronger which can be interpreted by a denser medium.
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\hsize]{figures/PbPb_Charged_RAA.pdf}
\caption{Nuclear modification factor $R_{AA}$ of charged particles.}
\label{fig:PbPb-RAA_charged}
\end{figure}
Benefiting from particle identification which has been already mention
earlier in this paper, ALICE has measured suppression of various
identified hadrons, which provides experimental data for studying the
flavor and mass dependence of the spectra suppression.
A nuclear modification factor $R_{AA}$ of charged pion production in
mid-rapidity (Fig.\ref{fig:PbPb-RAA_pions}) has lower values in the
range of moderate transverse momenta ($3<\pT<7-10$~GeV/$c$) than that
of unidentified charged particles, but at higher $\pT$ it coincides with
all charged particles.
\begin{figure}[ht]
\includegraphics[width=0.60\hsize]{figures/RAA_pions.pdf}
\caption{Nuclear modification factor $R_{AA}$ of charged pions.}
\label{fig:PbPb-RAA_pions}
\end{figure}
To the contrary to charged pions, strange hadrons ($K^0_S$, $\Lambda$)
are less suppressed in the most central collisions compared to all
charged particles (Fig.\ref{fig:PbPb-RAA_all}). This is
explained by the fact that strange quark production is enhanced in a
hot nuclear medium, and this strangeness enhancement partially
compensates energy loss of strange quarks, such that the overall
$R_{AA}$ value becomes larger than for pions. Lambda hyperons have no
suppression at $\pT<3-4$~GeV/$c$, which is interpreted by an
additional baryon enhancement in central heavy-ion collisions.
ALICE has reported also the first measurements of $D$ meson
suppression \cite{bib:PbPb-Dmesons} in Pb-Pb collisions in two
centrality classes, $0-20\%$ and $40-80\%$, shown in
Fig.\ref{fig:PbPb-RAA_all}. It was shown that the $R_{AA}$ values for
$D^0$, $D^+$ and $D^{*+}$ are consistent with each other within the
statistical and systematical uncertainties. Although the statistics of
the ALICE run 2010 is marginal for $D$ meson measurement, the obtained
result shows a hint that the $D$ mesons are less suppressed than
charged pions.
\begin{figure}[ht]
\includegraphics[width=0.48\hsize]{figures/2011-Sep-18-raa_central_lin_ch_K0_L_chPion_D.pdf}
\hfill
\includegraphics[width=0.48\hsize]{figures/2011-Sep-18-raa_peri_lin_ch_K0_L_chPion_Pi0_D.pdf}
\caption{Nuclear modification factor $R_{AA}$ of charged particles,
$K^0$, $\Lambda$, $\pi^\pm$, $D^+$, $D^0$, $D^{*+}$ in central
(left) and peripheral (right) collisions.}
\label{fig:PbPb-RAA_all}
\end{figure}
\section{Conclusion}
The ALICE collaboration is running an extensive research program with
proton-proton collisions. The domain where ALICE is competitive with
other LHC experiments, covers event characterization and identified
particle spectra at low and medium transverse momenta. Practically all
measured spectra in pp collisions at $\sqrt{s}=7$~TeV show
statistically significant deviations from models which well described
lower-energy results. Therefore new experimental results from pp
collision allow to tune various phenomenological models and pQCD
calculations.
A plenty of experimental results produced by the ALICE collaboration
from the first Pb-Pb data gives the first insight on strongly
interacting nuclear matter at the highest achievable collision
energy. It is evident that the quark-gluon matter produced in heavy
ion collision at LHC qualitatively has properties similar to what was
observed at RHIC. The matter produced at LHC has about 3 times larger
energy density, twice larger volume of homogeneity and about 20\%
larger lifetime. Like at RHIC, the matter at LHC reveals the
properties on an almost perfect liquid. Particle suppression appeared
to be stronger at LHC than at RHIC which is also an evidence of denser
medium produced at LHC. At the end of 2011, LHC has delivered 10
times more data with Pb-Pb collision at $\sqrt{s_{_{NN}}}=2.76$~TeV, which will
bring more precise results.
|
{
"timestamp": "2012-03-12T01:00:37",
"yymm": "1203",
"arxiv_id": "1203.1984",
"language": "en",
"url": "https://arxiv.org/abs/1203.1984"
}
|
\section{Introduction}
In classical general relativity (GR), in contrast to mathematical differential geometry, it is kind of a dogma that the points of the space-time manifold (S-T) have no real physical individuality. This was already realized in the context of the socalled \tit{Einstein-Hole-Argument} (see for example \cite{Pais},\cite{Stachel},\cite{Norton} or sect.4 in \cite{Rovelli}) and condensed in the mathematical statement
\begin{ob} In classical general relativity all diffeomorphic S-T-manifolds are physically indistinguishable. I.e., with $\Phi$ a diffeomorphism from $M$ to $M'$ (that is, not simply a coordinate transformation), and $g':=\Phi_{\ast}\circ g$ (note that $ \Phi_{\ast}$ denotes the push forward related to the pull back $((\Phi^{-1})^{\ast})$, $(M,g)$ and $(M',g')$ describe the same classical physics. In other words
\begin{equation}S\!-\!T=Riem/Diff \end{equation}
\end{ob}
(cf. e.g. \cite{Hawking})
In the special case $M'=M$ we would have a family of mathematically discernible metrics at the same (coordinate) point, that is
\begin{equation} g(x)\neq \Phi_{\ast}\circ g(x) =g'(x) \end{equation}
However one should note that, physically, we always have given a single metric from the family on the S-T-manifold as, by construction, the metrical properties on the S-T-manifold are given by a concrete measurement prescription which comes in a sense from outside in contrast to mathematics. We will come back to this topic below in the context of spontaneous symmetry breaking (SSB).
The typical way how a metric is introduced in GR exploits the existence of local inertial frames (LIF) in which special relativity holds sway and which allow to perform the usual length- and time-measurements. The \tit{equivalence principle} and \tit{general covariance} then allow to transplant the respective measurement results into arbitrary coordinate systems.
\begin{conclusion}On a given S-T-manifold we can hypothetically envisage several mathematically different but physically equivalent metrical tensors
\begin{equation} g(x)\; ,\; g'(x)=\Phi_{\ast}\circ g(x) \end{equation}
\end{conclusion}
We would like to emphasize that in our context of SSB we always have a large class of mathematically different but physically equivalent metrics (in the above sense) on $M$. Typically, SSB is concerned with the ground state or vacuum state of a system. In our context of GR and/or quantum gravity (QG) this means S-T devoid of macroscopic matter/energy content, i.e. solutions with vanishing \tit{Ricci-curvature}, that is
\begin{equation}R_{\mu\nu}=0\quad\text{or}\quad G_{\mu,\nu}=0 \end{equation}
These vacuum solutions have a large class of diffeomorphisms connecting them. Local deformations of the metric as in the \tit{hole-argument} will for example suffice.
While we will develop the subject matter from a direction starting with a view on SSB as it occurs in systems of many degrees of freedom (DoF) like e.g. condensed matter physics, there has been a chain of reasoning which exploits similarities between GR and classical gauge theories.
A well written early paper, belonging to this class, is for example \cite{Isham}, in which relations between the \tit{tetrad formalism} of GR and socalled \tit{nonlinear realizations} of \tit{gauge groups} on \tit{coset spaces} in high energy physics are established (having been of some prominence at that time). We mention also \cite{Freund}. Written in a similar vein are papers from the Russian school (up to very recent times). To mention a few, \cite{Sarda1},\cite{Sarda2} or \cite{Sarda3}. This approach relies mainly on the fibre bundle framework of socalled \tit{gauge gravity theory} and the reduction theory of principal bundles with respect to the structure groups being used and is of a more formal character. A recent paper, also starting from this group-reduction point of view in connection with SSB is \cite{Tomb}.
It is our aim in the following to unify this more formal group-theoretic approach with a different train of ideas which start from the more concretely given implications of SSB and \tit{gravitons} as \tit{Goldstone modes}, thus emphasizing features which may establish a connection to an underlying bundle of phenomena belonging to the not yet existing field of quantum gravity (QG). Note for example the remark in \cite{Isham}:
\begin{quote}
One does not expect any new development in the notoriously difficult problem of quantizing gravity to result from this modified point of view. However some insight may be gained\ldots
\end{quote}
It is our impression that the observation that gravitons are the Goldstone modes of SSB of diffeomorphism invariance will lead to real physical consequences if one can relate the more formal and abstract aspects on the level of classical gauge theory and fibre bundle reductions to the corresponding physical implications on the deeper levels of quantum space-time physics.
\section{Physical Considerations concerning SSB of Diffeomorphism Invariance}
As the group theoretic aspects of SSB are represented in great detail in the above mentioned literature, we begin our analysis with the development of a more physical point of view concerning the subject matter which makes contact with related phenomena of SSB in systems consisting of many degrees of freedom.
There are two particular points to be mentioned which may shed some light on the scene in GR and QG. We try to elucidate them by briefly discussing two characteristic examples taken from the field of SSB and phase transitions in many-body physics. We will however only stress the points which are of relevance for our corresponding analysis in gravitational physics.
To begin with, we discuss the phenomenon of breaking of translation invariance by crystallization of a continous (quantum) many-body system. In the symmetric unbroken phase the particle density $\rho (x):=<\hat{\rho}>$ is a constant, i.e.
\begin{equation}\rho (x+a)=\rho(x)\quad ,\quad x,a\in \mbb{R}^d \end{equation}
Below some critical point or phase transition line we have instead a periodic dependence of the particle density in the respective \tit{pure phases}, i.e.
\begin{equation}\rho (x+a)\neq\rho(x)\quad\text{in general} \end{equation}
but
\begin{equation}\rho (x+R_i)=\rho(x) \end{equation}
for some discrete subgroup of $\mbb{R}^d$.
\begin{defi}By a pure phase we mean in the above context a crystal having a definite macroscopic position in space. Mixtures may occur if we average over a group of such localized crystals. In a pure phase correlation functions do decay but only slowly due to the existence of collective Goldstone excitations.
\end{defi}
\begin{bem}A method of generating such localized crystal is the method of Bogoliubov quasi-averages (an external localizing field which is switched off in the end after the thermodynamic limit has been taken).
\end{bem}
\begin{ob}If this happens, both in the classical and the quantum regime, long-lived collective excitations do emerge which induce long-range correlations. In the case of a crystal they are called phonons.
\end{ob}
Representations of the Goldstone phenomenon in the quantum regime are so numerous that we mention only very few sources. Almost every textbook about quantum field theory contains a brief discussion (see e.g. \cite{Itzykson}). As to the older literature there is the nice comprehensive review \cite{Guralnik}. A more recent contribution is for example \cite{Requ2}. A detailed development of the Goldstone phenomenon in the regime of classical statistical mechanics can be found in \cite{Requ1}.
There exists an important difference between Goldstone particles in say QFT and in e.g. condensed matter physics and statistical mechanics. In QFT the Goldstone particles are exact mass -zero particles, i.e. they have a sharp excitation branch. On the other hand, in condensed matter physics, or systems having a non-vanishing particle density in general, they aquire an infinite lifetime only for momentum zero, while for non-vanishing momenta they are usually still relatively stable collective excitations but have only a finite lifetime (which typically decreases with increasing momentum) resulting in a smeared dispersion law (cf. e.g. \cite{Requ2}). Furthermore, while in RQFT their spin vanishes due to general principles (see \cite{Reeh}), this is not so in the more general context. The underlying reason is the absence of Lorentz covariance, Einstein causality and the socalled spectrum condition ( energy-momentum concentrated in the forward cone). Furthermore, while in most scenarios we can at least exploit translation invariance and the corresponding Fourier-mode decomposition, this is absent in GR and QG. Therefore, in the following, we will avoid all these concepts and discuss the Goldstone phenomenon in a much broader framework.
The relevant point in our investigation will be the following:
\begin{ob}For a hypothetical observer living inside one of the respective pure phases, i.e. the crystal, being translated by some vector, $a$, the internal physics is the same compared to a corresponding observer in a crystal, being translated by some vector, $a'\neq a$, provided corresponding coordinate systems have been chosen. Only an outside observer is able to discern the various translated pure phases.
\end{ob}
For illustrational purposes we mention another example, i.e. a lattice spin system, being capable of spontaneous magnetization. A pure phase in this scenario is described by a magnetization vector, pointing in a certain direction in configuration space. Again, the internal physics relative to the orientation of this magnetization vector is the same in all the different pure phases. Only an external observer is able to see the different phases (that is, the different directions of magnetization) by using his external reference system.
\begin{bem}All the internal observers are however able to observe the long-range collective Goldstone excitations, that is, phonons or magnons.
\end{bem}
All this now winds up to the observation that all internal observers see essentially the same physics provided they adapt their internal reference systems appropriately. That is, the situation is completely the same as compared to the case of diffeomorphism invariance of vacuum solutions in GR or QG.
\begin{conclusion}By the above observations we feel entitled to attribute to the different members of the class of diffeomorphic realizations of S-T a perhaps less than ephemeral or only formal existence as, by necessity, we are only internal observers in the latter case.
\end{conclusion}
We want to conclude this section with a brief analysis of the character of the goldstone modes under discussion. Phonons are essentially lattice vibrations in the crystal case, magnons are fluctuations of the local magnetization. Phrasing it somewhat differently one can venture to say:
\begin{ob}The Goldstone modes try to locally interpolate between the different potentially coexisting pure phases. I.e., local distortions of the crystal lattice can for example be regarded as local transitions into another slightly shifted crystal configuration. The same holds in the magnon case.
\end{ob}
\begin{conclusion}Exploiting the above correspondence between our examples and diffeomorphism invariance of S-T, one may conclude that the Goldstone modes in the latter case are the gravitons, acting as local distortions of S-T. They interpolate locally between the mathematically different but physically only hypothetically coexisting realizations of S-T as we are living in only one of these possible realizations.
\end{conclusion}
\section{The Conceptual Representation of SSB of Diffeomorphism Invariance in the Context of General Relativity and Quantum Gravity}
In this section we want to analyse the nature of SSB in our context. Note that the different diffeomorphic realisations of S-T can be viewed as an underlying differentiable manifold being equipped with different but diffeomorphic pseudoriemannian metrics. Furthermore, as we are mainly interested in the case of degenerate ground states (in a possibly underlying theory of QG), we assume that the (macroscopic) energy-momentum tensor vanishes.
The cornerstone of GR is the \tit{equivalence principle}, that is, at every point, $P$, of the S-T manifold there exists for a fixed metrical field, $g(\circ,\circ)$, a class of LIF in which the laws of special relativity (SR) hold in an at least infinitesimal neighborhood of the point $P$. Mathematically we can construct such a local coordinate system as follows, while we take at the same time the opportunity to introduce a number of useful concepts and notations (cf. e.g. \cite{Moeller} sect. 9.6. As to the tetrad formalism see also \cite{Synge} sect. I,3 ).
\begin{defi}In a given coordinate system, $x$, a (contravariant) tetrad at $P$ is given by 4 pseudoorthogonal tangent vectors, $e_a=(e_a^{\nu})$ with $a=0,1,2,3$ labelling the 4 vectors and $\nu$ denoting the indices with respect to the local coordinate tangent vectors $\partial_0,\partial_1,\partial_2,\partial_3$. We have
\begin{equation}e_{a\nu}=g_{\nu\mu}e_a^{\mu}\quad ,\quad e_a^{\nu}e_{b\nu}=\eta_{ab} =g(e_a,e_b) \end{equation}
with $\eta_{ab}$ the Minkowski tensor.
\end{defi}
For formal reasons we introduce
\begin{equation}e^{a\nu}:= \eta^ {ab}e_b^{\nu}\quad , \quad e_{\nu}^a:= \eta^ {ab}e_{b\nu} \end{equation}
\begin{ob}
\begin{equation}e_a^{\nu}e^b_{\nu}=\delta_a^b \end{equation}
\end{ob}
\begin{lemma}
\begin{equation}e_a^{\nu}e_{\mu}^a=\delta_{\mu}^{\nu} \end{equation}
\end{lemma}
This follows from the preceding observation. I.e., we have
\begin{equation}(e_a^{\nu}e_{\mu}^a)e_b^{\mu}= e_a^{\nu}(e_{\mu}^ae_b^{\mu})=e_a^{\nu}\delta_b^a=e_b^{\nu}=\delta_{\mu}^{\nu}e_b^{\mu} \end{equation}
\begin{bem}Note that $\nu,\mu$ refer to the covariant coordinate indices and are consequently raised and lowered with the help of $g_{\nu\mu},g^{\nu\mu}$ while $a,b$ as formal indices are raised and lowered with the help of the Minkowski metric.
\end{bem}
\begin{ob}Any two tetrades, $(e_a),(f_b)$, at $P$ are connected by a Lorentz transformation, $L$, i.e.
\begin{equation}e_a^{\nu}=L_a^{\cdot b}f_b^{\nu}\quad\text{or}\quad e_a=L_a^{\cdot b}f_b \end{equation}
\end{ob}
\begin{lemma}
\begin{equation}e_{\nu}^af_b^{\nu}=L_{\cdot b}^a \end{equation}
with
\begin{equation}e_{\nu}^a=L_{\cdot b}^a f_{\nu}^b\quad ,\quad f_b^{\nu}=L_{\cdot b}^a e_a^{\nu} \end{equation}
\end{lemma}
This follows from
\begin{equation}(e_{\nu}^af_b^{\nu})f_{\mu}^b=e_{\nu}^a(f_b^{\nu}f_{\mu}^b)=e_{\nu}^a\delta_{\mu}^{\nu}=e_{\mu}^a \end{equation}
and
\begin{equation}e^a=L_{\cdot b}^af^b\end{equation}
\begin{lemma}We have
\begin{equation}\eta_{ab}=g(e_a,e_b)=g(f_a,f_b) \end{equation}
\end{lemma}
Proof: Under the assumption $g(f^c,f^d)=\eta^{cd}$ we have
\begin{equation}g(e^a,e^b)=g^{\nu\mu}e_{\nu}^ae_{\mu}^b= g^{\nu\mu}f_{\nu}^cf_{\mu}^dL^a_{\cdot c}L^b_{\cdot d}=\eta^{cd} L^a_{\cdot c}L^b_{\cdot d}=\eta^{ab} \end{equation}
As shown in \cite{Moeller}, l.c., one can easily construct a new local coordinate system with the help of the tetrad at $P$:
\begin{equation}(x')^i:=e_{\nu}^i(x^{\nu}-x_P^{\nu})\quad ,\quad x^{\nu}=x_P^{\nu}+e_k^{\nu}(x')^k \end{equation}
which is pseudo-orthogonal at $P$, i.e.
\begin{equation}g'_{ik}(P)=e_i^{\nu}e_k^{\mu}g_{\nu\mu}=\eta_{ik} \end{equation}
Other pseudo-orthogonal coordinate systems can be generated by Lorentz transformations from the $(e^a)$-system, i.e.
\begin{equation}f_a^{\nu}=L_a^{\cdot b}e_b^{\nu}\quad\text{and}\quad y^i=f_{\nu}^i(x^{\nu}-x_P^{\nu}) \end{equation}
It follows
\begin{equation}y^i=L_{\cdot k}^i(x')^k \end{equation}
However, a pseudo-orthogonal coordinate system at $P$ is in general not a LIF. Therefore, an observer at $P$ in such a system still experiences a gravitational field. This can be transformed away by means of a more general coordinate transformation which leads to the
\begin{ob}There exists a coordinate transformation to socalled local Lorentz-coordinates (e.g. Riemann normal coordinates) such that
\begin{equation}g_{\nu\mu}(P)=\eta_{\nu\mu}\quad\text{and}\quad \partial g_{\nu\mu}(P)/\partial x_{\rho}=0\quad\text{or}\quad \Gamma_{\nu\mu}^{\rho}(P)=0 \end{equation}
which is the definition of a LIF.
Further local Lorentz transformations leave the class of LIF invariant while leading, of course, to another system of local Lorentz-coordinates.
\end{ob}
We now will establish the connection to SSB of diffeomorphism invariance. Central in the context of SSB and phase transitions is the notion of \tit{order parameter}. In the above examples, taken from many-body theory, order parameters are certain observable quantities which characterize the pure spontaneously broken phases as e.g. the gradient of the particle density or the magnetization. One can equally well use the corresponding quantum observables, that is, observables whose ground state expectation values vanish in the ordered unbroken phase while being different from zero in the broken phases. Furthermore, the different ground state expectation values are connected by the set of broken symmetry transformations.
More specifically, we have a large group, $G$, of symmetrie (some of which are broken) containing a closed subgroup, $H$, of conserved symmetries. The \tit{configuration manifold} of ground states can be related to and parametrized by the \tit{cosets} of the \tit{homogeneous space} $G/H$. We conclude, that it is important to study the structure of the space $G/H$ with $G$ a Lie group and $H$ a closed subgroup. This problem is in general not trivial and we will deal with the more mathematical aspects of the problem in the following section.
We will now transplant this picture into the more general framework of GR and/or QG. As configuration manifold (of minima of some functional of the occurring fields) we take the set of vacuum solutions of GR. The group $G'$ is the (in general) infinite dimensional group $Diff$. In the following we restrict ourselves, for convenience, to a single orbit under $Diff$, that is, a fixed vacuum solution together with its transforms under $Diff$. In a next step we will shift our point of view to a more local one, i.e. we switch to the action of the respective groups in the corresponding \tit{tangent bundle} of S-T.
\begin{ob}Locally the group $Diff$ is realized by the group $G:=GL(n.\R)$ of general linear transformations in the respective local coordinate systems:
\begin{equation}Diff\:\rightarrow\: (\partial (x')^i/\partial x^j)(P)\: \in GL(n,\R) \end{equation}
\end{ob}
One can equally well consider the local action on the (principal) \tit{frame bundle} over a fixed manifold $M$, i.e. the action of $Diff$ in a local trivialization of the frame bundle.
\begin{defi}As a closed subgroup of $GL(n.\R)$ we take the orthochronous Lorentz group, $L$, (the connected component of the group unit, $e$).
\end{defi}
\begin{bem}In a given coordinate system the local frames are in a one-one relation to elements of $GL(n,\R)$, i.e.
\begin{equation}v_i=v_i^k(\partial/\partial x^k)\quad , \quad i=1,\ldots ,n \end{equation}
with $(v_i^k)\in GL(n.\R)$.
\end{bem}
The field on $M$ we are mainly interested in is the classical metric field $g_{\nu\mu}(x)$ or
\begin{equation}g_{\nu\mu}(x):=< \hat{g}_{\nu\mu}(x)> \end{equation}
with $\hat{g}$ its quantum version (in some framework of semiclassical QG).
\begin{defi}We call $g_{\nu\mu}(x)$ an order parameter field, thus generalizing the notion of order parameter. An order parameter manifold, $(M,g)$, is $M$ plus a fixed metrical field $g_{\nu\mu}(x)$.
\end{defi}
\begin{ob}In a given fixed coordinate patch the group $Diff$ acts on the metrical field $g_{\nu\mu}(x)$ by acting at each point $P$ via $GL(n,\R)$ (in a tensorial way)
\begin{equation}Diff:\: g(x)\:\rightarrow\:g'(x) \end{equation}
\end{ob}
The \tit{hole-problem} was solved by Einstein (cf. \cite{Rovelli} l.c.) by introducing for example four particles $A,B,C,D$ with $A,B$ meeting in point $x_i$ inside the hole and $C,D$ in point $x_j$. He observed that the diffeomorphism now transforms both the metric tensor $g(x)$ and the trajectories of the test particles so that they get shifted as well with their distance becoming equal with respect to the transformed metric compared to their distance relative to the original metric $g(x)$.
\begin{ob} This is exactly the situation we discussed in section 2 in the context of e.g. many-body physics. I.e., physically the different phases become indistinguishable if the respective reference frames are appropriately reoriented. In the hole argument the reference system is provided by the 4 particle trajectories and their intersection points. Put differently, the Einstein hole argument is in fact an illustration of our concept of SSB in general relativity.
\end{ob}
As a last point in this section we discuss what a local observer in a LIF physically experiences concerning the SSB of diffeomorphism invariance. We again emphasize that we have one metric field, $g(x)$, with the other potentially existing fields $g':=\Phi_*\circ g$ being unobservable for the local observer. What is actually at his disposal are \tit{passive} coordinate transformations. He can for example apply Lorentz transformations.
\begin{ob}We learned above that with $(e_a)$ a Lorentz-orthonormal frame, $g(e_a,e_b)=\eta_{ab}$, the same holds for the Lorentz transformed tetrad, $(f_a)$, i.e. $g(f_a,f_b)=\eta_{ab}$.
\end{ob}
That is, if the local observer uses a LIF (local Lorentz coordinates) he observes that gravitational forces are locally absent. The same holds then when he applies the Lorentz group $L_+^{\uparrow}$ (and extending the new frame appropriately to a local Lorentz coordinate system). If, on the other hand, he applies general elements from $GL(n,\R)$ he will observe gravitational forces in the transformed coordinate frame.
\begin{conclusion}In our view this is the local manifestation of the breaking of diffeomorphism invariance.
\end{conclusion}
In the next section we will relate our physical observations with the more abstract formalism of \tit{reduction of pricipal bundles}.
\section{SSB and Reducible Principal Bundles}
A large part of the above mentioned literature is concerned with the reducibility of the \tit{frame bundle} with \tit{structure group} $GL(n,\R)$ to a subgroup, in our case the Lorentz group $L_+^{\uparrow}$ or $SO(n-1,1;\R)$. Frequently this phenomenon is already considered as a case of SSB. In our view reducibility as such is a widespread phenomenon in the field of principal bundles. So we regard it rather as a necessary prerequisite for SSB, not as the phenomenon itself! The mathematical results we are using in the follwing can be found in the following books, \cite{KN},\cite{CB},\cite{CH},\cite{H}. We take some pains of mentioning the places where the results (and more) can be found because exact citations of the sometimes quite nontrivial results are frequently missing in the physical literature.
We denote a general bundle by $(E,B,G,F)$ or simply $E$, $E\rightarrow B$. $E$ is the total space, $B$ the base space (typically in our case the space-time manifold $S-T$), $G$ is the structure group (in physics called the gauge group), $F$ the typical fibre (sometimes $F$ is a vector space). Both $E,B$ are differentiable manifolds. Points in $E,B$ are denoted by $p,x$. $E$ is continuously and surjectively projected by $\pi$ on $B$. With $x\in B$, $\pi^{-1}(x)=E(x)$ or $E_x$ is the fibre over $x$. $G$ acts on $F$ from the left via homeomorphisms or (in case of a vector space) as linear automorphisms. All fibre bundles are assumed to be locally trivial, that is, it exists a covering of $B$ by open sets, $U_i$, so that it exists $\phi_i$ with
\begin{equation}\phi_i:\pi^{-1}(U_i)\simeq U_i\times F \end{equation}
In particular
\begin{equation}\phi_i':\pi^{-1}(x)=E_x\simeq F \end{equation}
with
\begin{equation}\phi_i(p)=(\pi(p),\phi_i'(p) \end{equation}
If $x\in U_i\cap U_j$ the transition maps
\begin{equation}\phi_i'\circ\phi_j'^{-1}:F\rightarrow F \end{equation}
are elements of the structure group.
\begin{defi}A principal bundle, $P$, is a fibre bundle in which the typical fibre $F$ and the structure group $G$ are identical or, equivalently, $E_x$ or $P_x$ are diffeomorphic to $G$ (as Lie groups).
\end{defi}
\begin{ob}It is important that one can define a right action of $G$ on the fibres, $P_x$, i.e.
\begin{equation}R_g\circ p=p\cdot g\quad\text{or}\quad R_g\circ p=\phi_i^{-1}\circ(\phi_i(p)\circ g) \end{equation}
under which $G$ acts freely on $P_x$.
\end{ob}
\begin{bem}Note that left action is in general not! fibre preserving (i.e. independent of the choice of patch $U_i$).
\end{bem}
In our context we are typically concerned with the bundle of frames, $L(M)$, over a manifold $M$. A linear frame at $x\in M$ is an ordered basis of the tangent space $T_x(M)$. The linear group acts on $L(M)$ from the right in the following way.
\begin{equation}Y_j=A_j^ix_i \end{equation}
$(Y_j),(X_i)$ linear frames at $ x\in M$. Taking a coordinate basis $(\partial_{x^i})$ each frame at $x$ can be expressed as
\begin{equation}X_j=X^i_j \partial_{x^i} \end{equation}
for some $(X^i_j)\in GL(n,\R)$. This shows at the same time the local triviality of the frame bundle, i.e.:
\begin{equation}\pi^{-1}(U)\simeq U\times GL(n,\R) \end{equation}
\begin{defi}Let $P(M,G)$ be a principal bundle. Let $P'$ be a submanifold of $P$ and $H$ a Lie-subgroup of $G$ so that $P'(M,H)$ is again a principal bundle with structure group $H$. We call $P'(M,H)$ a reduction of $P(M,G)$. We say, the structure group $G$ is reduced to $H\subset G$.
\end{defi}
\begin{bem}This property is a nontrivial! result. It has a variety of interesting applications (cf. e.g. \cite{KN} or \cite{CB}).
\end{bem}
In the reduction process an important role is played by the \tit{coset space} $G/H$, or in our contex, $GL(n,\R)/L_+^{\uparrow}$. In most of the mathematical literature it is dealt with the case $GL(n,\R)/O(n,\R)$, i.e. $O(n,\R)$ instead of $L_+^{\uparrow}$. The latter situation is technically much simpler (as is the case for Riemannian geometry in general compared to Lorentzian geometry) but one finds frequently the slightly erroneous statement in the physical literature that the former case is more or less equivalent. Therefore we want to briefly comment on this point.
We have the general result (cf. \cite{KN} p.43 or \cite{CB} p.109)
\begin{satz}$G/H$ is an analytic manifold , in particular the projection $G\rightarrow G/H$ is real analytic. Furthermore, $G$ is locally diffeomorphic to $G/H\times H$, i.e. $G(G/H,H)$ with $H$ as structure group is a principal bundle (cf. \cite{KN} p.55).
\end{satz}
In special cases, e.g. if $H$ is a maximal compact subgroup, stronger results hold.
\begin{satz}With $H$ a maximal compact subgroup
\begin{equation}G\simeq H\times \R^m \end{equation}
\end{satz}
This is a result by Iwasara (cf. \cite{CB} p.386 or\cite{CH} p.109ff.
\begin{koro}We have
\begin{equation}GL(n,\R)/O(n)\simeq \R^{n(n+1)/2} \end{equation}
with $dim(O(n))=n(n-1)/2$.
\end{koro}
\begin{bem}One should note that $L_+^{\uparrow}$ is not compact and (to our knowledge) a corresponding result does not hold for $L_+^{\uparrow}$ instead of $O(n)$.
\end{bem}
The reason why such a result holds for $O(n)$ can be understood relatively easily. The wellknown \tit{polar decomposition} tells us that there exists an essentially unique (global!) decomposition.
\begin{satz}[Polar Decomposition]It exists an essentially unique decomposition
\begin{equation}L=O\cdot |L| \end{equation}
with $O$ orthogonal and $L\in GL(n,\R)$, $|L|$ positive semidefinite, more specifically
\begin{equation}|L|=(L^+\cdot L)^{1/2} \end{equation}
and $L^+$ the adjoint of $L$.
\end{satz}
\begin{bem}This important result is much more general and holds also for \tit{closable (unbounded) operators} (cf. e.g. \cite{RS} vol.I)
\end{bem}
It is crucial for such a result to hold that one can exploit the spectral theorem, i.e. that for example $|L|$ is well-defined and, a fortiori, selfadjoint. Nothing in that direction holds (to our knowledge) if $O$ is replaced by an element of $L_+^{\uparrow}$. In the latter case one only has a weaker result of a (in our view) quite different nature (which is however sufficient in our case).
With $G$ a Lie group, $H$ a closed subgroup, let $\hat{G},\hat{H}$ be the respective Lie algebras, $\hat{M}$ some vector subspace of $\hat{G}$ so that $\hat{G}$ is the direct sum, $\hat{G}=\hat{M}+\hat{H}$, (note that $\hat{M}$ is not! unique in general). Let $\exp_{\hat{M}}$ be the restriction of the exponential map to $\hat{M}$ and $\hat{e}:=e\cdot H=H$ the element in $G/H$ under the projection $\pi:G\rightarrow G/H$.
\begin{satz}(cf. e.g. \cite{H} p.113) There exists a neighborhood, $U$, of $0$ in $\hat{M}$ which is mapped homeomorphically under $\exp_{\hat{M}}$ onto $\exp_{\hat{M}}(U)\subset G$ so that $\pi$ maps $\exp_{\hat{M}}(U)$ homeomorphically onto a neighborhood of $\hat{e}$ in $G/H$.
\end{satz}
To prove this important theorem, one needs the following lemma (\cite{H} p.105), which is useful in various contexts.
\begin{lemma}With $\hat{G}=\hat{M}+\hat{H}$ there exist open neighborhoods $U_{\hat{M}},U_{\hat{H}}$ of $0$ in $\hat{M},\hat{H}$ so that the map
\begin{equation}(A,B)\rightarrow \exp(A)\cdot\exp(B) \end{equation}
is a diffeomorphism of $U_{\hat{M}}\times U_{\hat{H}}$ onto an open neighborhood of $e$ in $G$.
\end{lemma}
\begin{bem}It is remarkable that this map generates a full neighborhood of $e$ in $G$ as it is, at first glance, only a product set. It follows from the particular situation in Lie groups. It hinges in particular on the non-vanishing of a certain functional determinant around $e$ in $G$. This guarantees the bijectivity of the map near $e$. But one does not know how large this neighborhood actually is.
\end{bem}
\begin{conclusion}The above theorem says essentially that with the help of $\hat{M}$ we get locally! a transversial submanifold relative to $H$ which yields a parametrization of the fibres around $\hat{e}\in G/H$. Via group multiplication we can then transport this parametrization to a neighborhood of any coset $g\cdot H$.
\end{conclusion}
\begin{satz}In the case of $GL(n,\R),L_+^{\uparrow}(n,\R)$, matrices with
\begin{equation}\eta\cdot A^T=A\cdot\eta \quad\text{(pseudosymmetric)} \end{equation}
span a $n(n+1)/2$-dimensional submanifold being locally transversal to $L_+^{\uparrow}(n,\R)$, i.e. they locally coordinatize $GL(n,\R)/L_+^{\uparrow}(n,\R)$. That is, in a neighborhood of $e\in GL(n,\R)$ each element $L\in GL(n,\R)$ can be written uniquely as
\begin{equation}L=\Lambda\cdot A \end{equation}
with $\Lambda\in L_+^{\uparrow}(n,\R)$ and $A$ pseudosymmetric.
\end{satz}
\begin{bem}Note that e.g. matrices with $A^T=A$ do not have this property while also spanning a $n(n+1)/2$-dimensional submanifold. There exist Lorentz boosts being even positive!. The above local decomposition with $A$ pseudosymmetric was used in \cite{Isham}.
\end{bem}
Proof of theorem: We show that no element in $L_+^{\uparrow}$ different from $e$ is pseudosymmetric. We have with $\Lambda\in L_+^{\uparrow}$:
\begin{equation}\Lambda^T\eta\Lambda=\eta \end{equation}
Assume that
\begin{equation}\Lambda^T=\eta\Lambda\eta \end{equation}
we get
\begin{equation}\eta\Lambda\eta^2\Lambda=\eta\quad\rightarrow\quad \eta\Lambda^2=\eta \end{equation}
hence
\begin{equation}\Lambda^2=e\quad\rightarrow\quad\Lambda=e \end{equation}
which proves the theorem as it holds $n(n-1)/2+n(n+1)/2=n^2= Dim(GL(n,\R)$.
There is an important theorem in the reduction theory of principal bundles (\cite{KN} p.57) saying the following:
\begin{satz}The principal bundle $P(M,G)$ is reducible to $P'(M,H)$ iff $P/H$ admits a cross section.
\end{satz}
\begin{bem}The meaning of $P/H$ is the following. The fibres of $P$ are diffeomorphic to $G$, hence the fibres of the bundle $P/H$ are diffeomorphic to the typical fibre $G/H$. The structure group is again $G$ (multiplication from the left).
\end{bem}
In the case of $O(n)$
\begin{equation}GL(n,\R)/O(n)\simeq\R^{n(n+1)/2} \end{equation}
it is easy to see that local cross sections can be pasted together and extended to a global cross section (cf. \cite{CB} p.385) because $\R^m$ is a vector space and assuming that $M$ is paracompact.
\begin{ob}In the case of $L_+^{\uparrow}$ as subgroup we have local cross sections around every point of $P/H$ or rather $L(M)/L_+^{\uparrow}$ ($L(M)$ the frame bundle) as a consequence of our above results. But (to our knowledge) $GL(n,\R)/L_+^{\uparrow}$ is not diffeomorphic to a vector space.
\end{ob}
We can however proceed as follows. We assume that a Lorentzian metric, $g$, is given on our space-time manifold $M=S-T$.
\begin{bem}In contrast to the Riemannian case (paracompact manifold) this is not automatic (cf. e.g. \cite{CB} p.293).
\end{bem}
With the help of $g$ we generate the set of pseudoorthogonal tetrads $(e_a)$ at every point $x$ of $M$. This set is invariant under $L_+^{\uparrow}$ as we have seen in section 3.
\begin{ob}The subset of $L(M)$ consisting of pseudoorthogonal tetrads at each $x\in M$ is a \tit{reduced subbundle} $Q(M)\subset L(M)$ with structure group $L_+^{\uparrow}$.
\end{ob}
In each fibre of $Q(M)$ $L_+^{\uparrow}$ acts freely from the right. Therefore the projection
\begin{equation}pr:L(M)\rightarrow L(M)/L_+^{\uparrow} \end{equation}
is constant on the fibres of $Q$ as subsets of the fibres of $L(M)$.
\begin{ob}Thus $pr$ induces a mapping
\begin{equation}s:M\rightarrow L(M)/L_+^{\uparrow} \end{equation}
via
\begin{equation}s(x):=pr(I(e_a(x)) \end{equation}
with $\pi_Q(e_a(x))=x$ the projection map in the subbundle $Q(M)$ and $I$ the injection (imbedding) map
\begin{equation}I:Q(M)\rightarrow L(M) \end{equation}
It is obvious that $s(x)$ determines a cross section in $L(M)/L_+^{\uparrow}$.
\end{ob}
\begin{bem}Note that $s(x)$ represents the equivalence class of Lorentz frames in $L(M)/L_+^{\uparrow}$ over $x$. That this is a well defined mapping is obvious. The continuity or differentiability of the cross section is the crucial point. This follows from the properties of the composition of the above maps all of which are differentiable. Furthermore, with the help of this cross section we get back exactly the subbundle $Q(X)$ we started from.
\end{bem}
\section{Conclusion}
We have learned in the preceeding sections that the gravitational field (or, rather, the metrical field), $g$, can be regarded as an \tit{orderparameter field} and the macroscopic, smooth space-time manifold $(S-T;g)$ as an \tit{order parameter manifold}, lying above a presumed microscopic (irregular and erratic) quantum space-time consisting of an array of many interacting DoF.
We have invoked the situation of quantum many-body systems for several times. There the situation is the following. We usually have a symmetric unbroken phase with the order parameter being zero, and, in another region of certain parameters, a phase transition to a set of physically more or less identical spontaneously broken phases characterized by certain non-vanishing values of the order parameter.
Transferring these observations to our field we may venture to say:
\begin{conjecture}We associate the presumed unbroken phase of our (quantum) space-time, that is, the absence of a classical, macroscopic metrical field, $g$, to some \tit{pre-big-bang era}. The emergence of classical space-time is then the result of a phase transition (which may have happened before or near the big-bang).
\end{conjecture}
Another interesting point concerns the nature of the gravitons themselves. In many-body physics the corresponding excitations are the phonons with the ordered phase being the crystal phase and the relatively long-lived lattice phonons as Goldstone particles. On the other hand, phonons do already exist in quantum fluids but they happen to be more strongly damped. The phonons which occur e.g. in fluids can be associated with the SSB of Galilei invariance (cf. e.g. \cite{Requ2}.
\begin{conjecture}We think that in our context gravitons, while being more stable in the ordered phase, i.e. $S-T$ plus a non-vanishing $g$-field, have already existed in the unordered phase (quantum vacuum with vanishing classical $g$). In this phase they represented certain types of more primordial excitations not being related to deformatons of classical $S-T$. They perhaps do reflect, as in the above Galilei-case, the existence of a quantum vacuum as such.
\end{conjecture}
\begin{bem}This may be interesting in connection with string theory. While string theory, at least as a starting point, exploits a classical embedding or target space, gravitons are associated with certain low-lying excitations of closed strings. Our above observations concerning the possible nature of gravitons may perhaps shed some light on a certain connection to string theory and the role of gravitons in this framework.
\end{bem}
The last point we want to address is whether there exists an objective correlate to the different hypothetical phases being described by the different diffeomorphic metrical fields. In SSB of many-body systems all the different phases can really exist while only one is realized in each case. In the corresponding situation of GR we are less accustomed to such a picture. However, in more recent times the general perspective has changed a little bit, given the discussion about the \tit{landscape} in string theory and cosmology or \tit{induced/entropic gravity} which employ more or less openly some microscopic baclground substrate as supposed carrier of the concepts of physics.
|
{
"timestamp": "2012-03-09T02:01:22",
"yymm": "1203",
"arxiv_id": "1203.1702",
"language": "en",
"url": "https://arxiv.org/abs/1203.1702"
}
|
\section{Introduction}
\label{intro}
This paper is a follow-up of the study initiated in \cite{DPTVjsp}, \cite{DPTVpro}, where current reservoirs in the context of stochastic interacting particle systems have been proposed as a method to investigate stationary non-equilibrium states with steady currents produced
by action at the boundary.
Due to the particular difficulties in implementing this new method, we consider the simplest possible particle system.
The bulk dynamics is the symmetric simple exclusion process (SSEP) in the interval $\La_N=[-N,N]\cap \mathbb Z$ ($N$ a positive integer and $N\to \infty$ eventually), namely the state space is $\{0,1\}^{\La_N}$ (at most one particle per site): independently each particle tries to jump at rate $N^2/2$ to each one of its nearest neighbor (n.n.) sites,
the jump then takes place if and only if the chosen site is empty, jumps outside $\La_N$ are suppressed.
To induce a current we send in
particles from the right and take them out from the left, and would like this to happen at rate $Nj/2$,
$j>0$ a fixed parameter independent of $N$. Due to the restrictions imposed by the configurational space, we have to be more precise when defining this dynamics. For this we fix a parameter $K\ge 1$ (an integer) and two intervals
$I_{\pm}$ of length $K$ at the boundaries: $I_+\equiv [N-K+1,N]$ and $I_-\equiv [-N,-N+K-1]$.
At rate $Nj/2$, when $I_+$ is not totally occupied, we create a particle at its rightmost empty site; with the same rate, unless $I_-$ is empty, we take out a particle from its leftmost occupied site. In case $I_+$ is already full, or $I_-$ empty, the corresponding mechanism aborts.
\smallskip
In \cite{DPTVjsp}, \cite{DPTVpro} we have proved that at any time $t>0$ propagation of chaos holds and that in the limit $N\to \infty$ the hydrodynamical equation is the linear heat equation:
\begin{eqnarray}
\nn&&\frac {\partial}{\partial t} \rho(r,t)= \frac 12 \frac {\partial^2}{\partial r^2} \rho(r,t), \qquad r\in (-1,1),\, t>0,
\\&&\rho(r,0)=\rho_0(r),\qquad \rho(\pm 1,t)= u_{\pm}(t),
\label{1.1}
\end{eqnarray}
where $\rho_0(\cdot)$ is given but $u_{\pm}(t)$ are solutions of a nonlinear system of two integral equations, see \eqref{dptv2.4} below.
\smallskip
The goal of this paper is to investigate the limiting density profile (as $N\to \infty$) of the (unique) invariant measure of the process. The main result is Theorem \ref{thm2.2}, which shows that this rescaled limiting profile coincides with the unique stationary solution of \eqref{1.1}. In particular, taking into account the validity of the Fourier law, proven as Theorem 2 in \cite{DPTVjsp}, we see that the effective current in the stationary regime is strictly smaller than its desired maximum value which is $\min\{j/2,1/4\}$, but this value is indeed approached by letting $K\to \infty$.
\vskip1cm
\section{Model and main results}
Particle configurations are elements $\eta$ of $\{0,1\}^{\La_N}$, $\eta(x)=0,1$ being the occupation number at $x\in\La_N$. We consider the Markov process on $\{0,1\}^{\La_N}$ defined via the generator
$$
L_N:=N^{2}\Big(L_0+\frac 1N L_b\Big),
$$
where $L_b= L_{b,+}+ L_{b,-}$ and
\begin{eqnarray}
\nn
&&
L_0 f(\eta):=\frac 12\sum_{x=-N}^{N-1} [f(\eta^{(x,x+1)})-f(\eta)],
\\ \label{1}
\\&& L_{b,\pm} f(\eta):= \frac{j}{2}\sum_{x\in I_\pm}D_{\pm}\eta(x) [f(\eta^{(x)})-f(\eta)\Big],
\nn
\end{eqnarray}
$\eta^{(x)}$ being the configuration obtained from $\eta$ by changing the occupation number at $x$, $\eta^{(x,x+1)}$ by exchanging the occupation numbers at $x,x+1$; for any $u:\La_N\to [0,1]$
\begin{eqnarray}
&& D_+u (x)= [1-u(x)]u(x+1)u(x+2)\dots u(N), \quad x\in I_+
\nn\\
&& D_-u (x)= u(x)[1-u(x-1)][1-u(x-2)]\dots[1-u(-N)], \quad x\in I_-.
\label{6}
\end{eqnarray}
\vskip 0.5cm
Given $\rho_0\in C([-1,1],[0,1])$, let $\nu^{(N)}$ be the product probability measure on $\{0,1\}^{\La_N}$ such that $\dis{\nu^{(N)}(\eta(x))=\rho_0(N^{-1} x)}$ for all $x\in\La_N$. Let $\mathbb{P}_{\nu^{(N)}}$ denote the law of the process with initial distribution $\nu^{(N)}$ and $\E_{\nu^{(N)}}$ the corresponding expectation.\footnote{Omitting the initial profile to avoid too heavy notation.}
\vskip.5cm
The following theorem has been proven (in a stronger form) in \cite{DPTVjsp}, \cite{DPTVpro}. The statement below contains all what is needed in the present paper. In the following, for $n$ a positive integer we write $\Lambda_N^{n,\neq}$ for the set of all sequences $(x_1,...,x_n)$ in $\La_N^n$ such that $x_i\ne x_j$ whenever $i\neq j$.
\vskip.5cm
\begin{thm}
\label{thm2.1}
There exists $\tau>0$ so that for any $\rho_0$ as above and any $n\ge 1$,
\begin{equation}
\label{2.0}
\lim_{N\to \infty} \Big|\E_{\nu^{(N)}}\big(\prod_{i=1}^n \eta(x_i,t)\big)-\prod_{i=1}^n\E_{\nu^{(N)}}\big( \eta(x_i,t)\big)\Big|=0,\qquad \text{for any } t\le \tau\log N.
\end{equation}
Furthermore
\begin{equation}
\label{dptv2.3}
\lim_{N\to \infty} \sup_{x\in \La_N}\sup_{ t \le \tau\log N} \big|\E_{\nu^{(N)}}\big( \eta(x,t) )-\rho( N^{-1}x,t)\big|=0,
\end{equation}
where the function $\rho(r,t)$ solves the heat equation $\frac{\partial \rho}{\partial t}=\frac 12 \frac{\partial^2 \rho}{\partial r^2}$, $r\in (-1,1), t>0$ with initial datum $\rho_0$ and boundary conditions $\rho(\pm 1,t)=u_\pm(t)$, the pair $(u_+(t),u_-(t))$ being the unique solution of the non linear system
\begin{eqnarray}
\nn
&&
\hskip-.3cm
u_\pm(t)= \int_{[-1,1]} P_t(\pm 1,r) \rho_0(r) dr+ \frac j2\int_0^t \Big\{ P_{s}(\pm 1,1)\left(1-u_+(t-s)^K\right)
\\&&\hskip3.8cm
- P_{s}(\pm 1,-1)\left(1- (1-u_-(t-s))^K\right)\Big\}ds,
\label{dptv2.4}
\end{eqnarray}
where $P_t(r,r')$ is the density kernel of the semigroup (also denoted as ${P_t}$) with generator $\Delta/2$, $\Delta$ the laplacian in $[-1,1]$ with reflecting, Neumann, boundary conditions.
The function $\rho(r,t)$ satisfies
\begin{equation}
\label{4.2}
\frac{\partial \rho(r,t)}{\partial r}|_{r=1}=j (1-u_+(t)^K),\quad \frac{\partial \rho(r,t)}{\partial r}|_{r=-1}=j (1-(1-u_-(t))^K).
\end{equation}
\end{thm}
\vskip.5cm
\noindent {\bf Remark.} The following is the integral form of the macroscopic equation:
\begin{eqnarray}
\label{dptv2.4bis}
&& \rho(r,t)= \int_{[-1,1]} P_t(r,r') \rho(r',0) dr' + \frac j2\int_0^t \Big\{ P_{s}(r,1)\left(1-\rho(1,t-s)^K\right)\nn
\\&&\hskip3cm
- P_{s}(r,-1)\left(1- (1-\rho(-1,t-s))^K\right)\Big\}ds.
\end{eqnarray}
It will be convenient to recall the expression for the density kernel $P_t(r,r^\prime)$ in terms of the Gaussian kernel
\begin{equation}
\label{N4.3}
G_t (r,r') = \frac{e^{-(r-r')^2/(2t)}}{\sqrt {2\pi t}},\quad r,r' \in \mathbb{R},
\end{equation}
as
\begin{eqnarray}
\label{N4.10}
P_t (r,r') &=& \sum_{r'':\psi(r'')=r'} G_t(r,r'') \quad \text{for} \; r'\neq \pm 1\nn\\
P_t(r,\pm 1) &=& \sum_{r'':\psi(r'')=\pm 1} 2G_t(r,r''),
\end{eqnarray}
where $\psi:\mathbb R\to [-1,1]$ denotes the usual reflection map:
$\psi (x)= x $ for $x \in [-1,1]$, $\psi(x)=2-x $ for $x \in [1,3]$, $\psi$ extended to the whole line
as periodic of period 4.
\vskip .5cm
\noindent {\bf Notation.} $P_t g(r)=\int P_t(r,r^\prime) g(r^\prime) dr^\prime$, for $g$ a bounded continuous function, $t>0$.
\vskip1cm
\noindent
The main result of this paper is about the density profile of the unique invariant measure $\mu_N$.
\begin{thm}
\label{thm2.2}
For any integer $k\ge 1$ we have
\begin{equation}
\label{2.1-a}
\lim_{N\to\infty} \max_{(x_1,..,x_k)\in \La_N^{k,\ne}}\Big|\mu_N\big(\eta(x_1)\cdots \eta(x_k)\big)-\rho^*(x_1/N)\cdots \rho^*(x_k/N)\Big| = 0
\end{equation}
where $\rho^*(r)$ is the unique stationary solution of the macroscopic equation.
Namely $\rho^*(r)=J\,r +\frac 12$,
\begin{equation}
\label{2.1}
J=j(1-\alpha^K),\quad \text{ with $\alpha$ the solution of}\quad \alpha(1+j\alpha^{K-1})=j+\frac 12.
\end{equation}
\end{thm}
\vskip.5cm
By Theorem \ref{thm2.2} it follows that $\mu_N$ concentrates on a $L^1$-neighborhood of the
limit profile $\rho^*$: let $r\in (0,1)$ and
\begin{equation*}
\rho^{(\ell)}(r;\eta) = \frac1{2\ell+1} \sum_{x\in \La_N: |x-rN| \le \ell} \eta(x)
\end{equation*}
Then for any $a\in (0,1)$
\begin{equation*}
\lim_{N\to \infty} \mu_N \Big( \int_{-1}^1 | \rho^{(N^a)}(r;\eta) -\rho^*(r)|dr\Big)=0
\end{equation*}
Theorem \ref{thm2.2} will follow from $\bullet$\; uniformly on the initial datum $\rho_0$ the solution $\rho(r,t|\rho_0)$ of the macroscopic equation \eqref{dptv2.4bis} converges in sup norm to $\rho^*$ exponentially fast, see Theorem \ref{thm3.3} below; $\bullet$\;
for any integer
$k\ge 1$,
\begin{equation}
\label{2.1-b}
\lim_{t\to \infty}\lim_{N\to\infty}\max_{\eta \in \{0,1\}^{\La_N}}\max_{(x_1,..,x_k)\in \La_N^{k,\ne}}\Big|\mathbb E_\eta \Big(\prod_{i=1}^k\eta(x_i,t)\Big)-\prod_{i=1}^k\rho^*(x_i/N)\Big| = 0
\end{equation}
We are also working on an extension of the theorem where we prove exponential convergence in time to $\mu_N$ uniformly in $N$.
\vskip1cm
\section{Monotonicity properties}
\vskip.5cm
We consider the space $\{0,1\}^{\La_N}$ endowed with the usual partial order, namely we say that $\eta\le \xi$ iff $\eta(x)\le \xi(x)$ for all $x\in \La_N$.
The following proposition is an immediate consequence of general facts on attractive systems, see e.g. \cite{li} (chs. II and III).
\begin{prop}
\label{thm3.1}
Let $\eta_0$ and $\xi_0$ be two particle configurations such that $\eta_0\le \xi_0$, and let $\mathbb{P}_{\eta_0}$, respectively
$\mathbb{P}_{\xi_0}$, be the law of the process starting from $\eta_0$, respectively $\xi_0$. Then there is a coupling $\mathbb{Q}$ of $\mathbb{P}_{\eta_0}$ and $\mathbb{P}_{\xi_0}$ (i.e. $\mathbb{Q}$ is a measure on the product space, with $\mathbb{P}_{\eta_0}$ as its
first marginal, and $\mathbb{P}_{\xi_0}$ as the second one) such that
\begin{equation}
\label{3.1}
\mathbb{Q}\{(\eta,\xi)\colon \eta_t \,\le \,\xi_t\,, \forall t\}=1
\end{equation}
\end{prop}
\noindent {\bf Proof}. Being well known that the process corresponding to $L_0$ is attractive, it suffices to observe that the flip rates $c(x,\eta):=D_\pm \eta (x)$ in $I_\pm$ are attractive in the sense that if $\eta(x)=\xi(x)=0$ and $\eta \le \xi$ then $c(x,\eta) \le c(x,\xi)$, while if $\eta(x)=\xi(x)=1$ and $\eta \le \xi$ then $c(x,\xi) \le c(x,\eta)$. \qed
\vskip.5cm
The analogous monotonicity property holds for the macroscopic equation.
Instead of a direct proof
we derive the result as a consequence of the monotonicity of the particle system and that it converges
to the macroscopic equation.
\vskip.5cm
\begin{thm}
\label{thm3.2}
Let $ \rho_0, \tilde\rho_0$ be bounded measurable functions from $[-1,1]$ to $[0,1]$ such that $\rho_0(r)\le \tilde\rho_0(r)$ for all $r\in[-1,1]$, and let $\rho(r,t)$, respectively $\tilde\rho(r,t)$, be the corresponding solution of \eqref{dptv2.4bis} with initial datum $\rho_0$, respectively $\tilde\rho_0$. Then $\rho(r,t)\le \tilde\rho(r,t)$ for all $r\in[-1,1]$ and $t\ge 0$.
\end{thm}
\noindent {\bf Proof}. Let $\nu^{(N)}$ and $\tilde\nu^{(N)}$ be the product probability measures on $\{0,1\}^{\La_N}$ such that $\nu^{(N)}(\eta(x))=\rho_0(N^{-1} x)$ and $\tilde\nu^{(N)}(\eta(x))=\tilde\rho_0(N^{-1} x)$ for all $x\in\La_N$.
It is well known that a coupling $\la^{(N)}$ of $\nu^{(N)}$ and $\tilde\nu^{(N)}$ such that $\la^{(N)}\{(\eta,\tilde \eta)\colon \eta\le \tilde\eta\}=1$ exists. Using Proposition \ref{thm3.1} and the notation of Theorem \ref{thm2.1} we have
\begin{equation}
\label{3.2-a}
\mathbb{E}_{\nu^{(N)}}(\eta(x,t))\le \mathbb{E}_{\tilde\nu^{(N)}}(\eta(x,t)),\qquad \forall x\in \La_N,\quad \forall t\ge 0.
\end{equation}
From \eqref{dptv2.3} we then have that for all $t\ge 0$ and for all $r\in[-1,1]$, (below $[\cdot]$ denotes the integer part)
\begin{equation}
\label{3.2}
\rho(r,t)=\lim_{N\to\infty}\mathbb{E}_{\nu^{(N)}}\big(\eta([Nr],t)\big)\le \lim_{N\to\infty}\mathbb{E}_{\tilde\nu^{(N)}}\big(\eta([Nr],t)\big)=\tilde\rho(r,t).
\end{equation}
\qed
\vskip1cm
\section{The macroscopic profile}
\vskip.5cm
\nopagebreak
We first prove that the function $\rho^*$ in the statement of Theorem \ref{thm2.2} is a stationary solution to the Dirichlet problem \eqref{1.1} with boundary condition \eqref{dptv2.4} or, equivalently, of the integral equation \eqref{dptv2.4bis}. In fact by requiring that a stationary solution is a linear function we get, due to \eqref{4.2}, that the values of this function at $\pm 1$, denoted with $u_\pm$, must satisfy
\begin{equation*}
j (1-u_+^K) =j (1-(1-u_-)^K).
\end{equation*}
This implies
\begin{equation*}
u_+=(1-u_-),\qquad \text{and }\quad \frac {2u_+-1}2=j(1-u_+^K),\quad u_+=\frac 12 +j(1-u_+^K).
\end{equation*}
Solving we get
\begin{equation*}
u_+(1+ju_+^{K-1}) =j +\frac 12
\end{equation*}
in agreement with \eqref{2.1}.
On the other hand, since $\frac{\partial}{\partial t}P_t(r,r')=\frac12\frac{\partial^2}{\partial(r')^2}P_t(r,r')$ and
it satisfies Neumann boundary conditions at $\pm 1$ we easily see that
\begin{equation*}
\frac{d}{dt} \int_{[-1,1]} P_t(r,r')r'dr'=\frac 12\left(P_t(r,-1)-P_t(r,1)\right).
\end{equation*}
Recalling (from \eqref{2.1}) that $J=j(1-(\rho^*(1))^K)=j(1-(1-\rho^*(-1))^K)$
we see at once that $\rho^*$ satisfies \eqref{dptv2.4bis}, which in this case can be written as:
\begin{equation}
\rho^*(r)= P_t\rho^*(r)+\frac j2 (1-(\rho^*(1))^K)\int_0^t \Big\{ P_{s}(r,1)
- P_{s}(r,-1)\Big\}ds,
\label{4.1}
\end{equation}
for all $t\ge 0$.
\vskip.5cm
We now prove that any solution to the Dirichlet problem converges exponentially fast to $\rho^*$ as $t\to \infty$. In particular, one has uniqueness of the stationary solution.
\vskip.5cm
\begin{thm}
\label{thm3.3}
There exist positive constants $c,c^\prime$ so that for any function $\rho_0$ $\in L^\infty([-1,1],[0,1])$ the solution $\rho(r,t|\rho_0)$ of the
macroscopic equation \eqref{dptv2.4bis} with initial datum $\rho(r,0)=\rho_0(r)$ satisfies
\begin{equation}
\label{t4.1}
\sup_{r\in[-1,1]}|\rho(r,t|\rho_0)-\rho^*(r)|\le c' e^{-ct}.
\end{equation}
\end{thm}
\noindent {\bf Proof}.
Let $\bar \rho(r,t)$ denote the solution with initial datum $\rho\equiv 1$, and $\und \rho(r,t)$ that
corresponding to initial datum $\rho\equiv 0$. From Theorem \ref{thm3.2} we know that
$\und \rho(r,t)\le \rho(r,t|\rho_0)\le \bar \rho (r,t)$, for any initial $\rho_0$. Hence, calling
\begin{equation*}
w(r,t):=\bar\rho(r,t)-\und\rho(r,t) \ge 0,\qquad w(t)=\sup_{r\in[-1,1]}w(r,t)
\end{equation*}
it suffices to show that $w(t)\le c^\prime e^{-ct}$ for suitable positive constants $c,c^\prime$ and all $t>0$.
In the proof below $c, \bar c, \tilde c$ will denote suitable positive constants (that might depend on the model parameter $j$)
whose value may change from line to line.
Let
\begin{equation*}
\bar u_\pm(t):=\bar\rho(\pm 1,t),\quad \und u_\pm(t):=\und\rho(\pm 1,t), \qquad w_\pm(t):=\bar u_\pm(t)-\und u_\pm (t)\ge 0.
\end{equation*}
>From \eqref{dptv2.4bis} we see that for all $r\in[-1,1]$, and all $t\ge t_0\ge 0$,
\begin{equation}
\label{4.3a}
w(r,t)=(P_{t-t_0}w(\cdot,t_0))(r)- \frac j2\int_{t_0}^t f(r,s,t-s) ds,
\end{equation}
where
\begin{eqnarray}
\nn
&&\hskip-.8cm
f(r, s,t-s):= P_{s}(r,1)\left\{\bar u_+(t-s)^K-\und u_+(t-s)^K\right\}
\\&&\hskip1cm +P_{s}( r,-1)\left\{(1-\und u_-(t-s))^K-(1-\bar u_-(t-s))^K\right\}.
\label{4.0a}
\end{eqnarray}
Interchanging particles and holes, one can couple at once the evolutions starting from the configurations $\bar\eta=\und 1$ ({\it all occupied sites}) and $\und \eta =\und 0$ ({\it all empty sites}) so that $\bar \eta(x,t)=1-\und \eta(-x,t)$. Therefore, by the
same argument as in the proof of Theorem \ref{thm3.2} one has $\bar \rho(r,t)=1-\und\rho(-r,t)$ for all $r$ and all $t$. In particular $w(-r,t)=w(r,t)$, $\bar u_{\pm}(t)=1-\und u_{\mp}(t)$ and $w_+(t)=w_-(t)$ for all $t$. (Still from Theorem \ref{thm3.2} we see that
$w(r,t)$ and so also $w(t)$ decrease in $t$.) Of course $w(r,0)=1$ for all $r$.
In particular, we may rewrite \eqref{4.3a} with $t_0=0$ as
\begin{equation}
w(r,t)=1- \frac j2\int_0^{t} [P_{s}(r,1)+P_s(r,-1)]w(1,t-s)h(t-s)ds\label{4.15}
\end{equation}
where
\begin{equation}
h(t-s):=\sum_{\ell=0}^{K-1}\bar u_+(t-s)^{K-1-\ell}\und u_+(t-s)^{\ell}
\label{4.16}
\end{equation}
and where we have used that for any integer $K\ge 1$,
\begin{equation}
\label{4.3a.1}
a^K-b^K=(a-b)\sum_{\ell=0}^{K-1} b^\ell a^{K-1-\ell},\qquad a \ge b \ge 0.
\end{equation}
Also, from \eqref{4.16} and the monotonicity properties we see that
\begin{equation}
\label{ck}
b:={\rho^*(1)}^{K-1}\le h(t) \le b+K-1=: c_K.
\end{equation}
The proof will use local times. To this end we introduce
the kernel operators $K^{(\eps)}_{s}$, $\eps>0$:
\begin{equation*}
K^{(\eps)}_{s} f(r)=\frac 1{\eps}\int_{[-1,-1+\eps]\cup[1-\eps,1]} P_s(r,r')f(r')dr', \quad f \in C([-1,1],\mathbb R).
\end{equation*}
In particular $K^{(\eps)}_{s} f(r)=K^{(\eps)}_{s} f(-r)$ for all $r \in [-1,1]$. Let $w^{(\eps)}$ be the solution to the following
integral equation:
\begin{eqnarray}
\nn
&&\hskip-1cm
w^{(\eps)}(r,t)=1
- \frac j2\int_0^{t} (K^{(\eps)}_{s} w^{(\eps)}(\cdot,t-s))(r)h(t-s)ds.
\label{4.15b}
\end{eqnarray}
We shall next prove that for all $T>0$,
\begin{equation}
\label{4.16'}
\lim_{\eps\to 0}\,\sup_{r\in [-1,1]}\,\sup_{0\le t\le T}\big|w(r,t)- w^{(\eps)}(r,t)\big|=0.
\end{equation}
Calling
\begin{equation}
\label{4.16a}
\psi(r,t)=\big|w(r,t)- w^{(\eps)}(r,t)\big|,\qquad \Psi(t)=\sup_{r\in [-1,1]}\psi(r,t)
\end{equation}
and using \eqref{ck}, we can write
\begin{eqnarray}
\nn
&&\Big|\int_0^{t} \left\{(K^{(\eps)}_{s} w^{(\eps)}(\cdot,t-s))(r)- \{P_{s}(r,1)+P_s(r,-1)\}w(1,t-s)\right\}h(t-s)ds
\Big|\le c\eps\\
&&\nn +c_K
\int_\eps^{t}\left\{\frac 1{\eps}\int_{1-\eps}^1\big|P_s(r,y)-P_s(r, 1)+P_s(-r,y)-P_s(-r,1)\big|dy\right\} \big|w^{(\eps)}(y,t-s)\big|ds \\
&& +c_K\int_\eps^{t} \{P_{ s}(r, 1)+P_s(-r,1)\} \Psi(t-s) ds.
\label{4.17}
\end{eqnarray}
Using that for all $y\in[1-\eps,1], \, r\in [-1,1]$
\begin{equation}
\label{4.18}
|P_s(r,y)-P_s(r, 1)|\le c \frac {1-y}{\sqrt {s^3}}, \qquad \forall s\in [\eps,t]
\end{equation}
we see that the second term on the r.h.s. of \eqref{4.17} is bounded above by
\begin{equation*}
\tilde c \int_\eps^{t} \frac 1{\sqrt {s^3}}ds\frac 1{\eps}\int_{1-\eps}^1(1-y)dy\le c' \sqrt\eps
\end{equation*}
for suitable constants $\tilde c, c'$. We then easily get
\begin{equation}
\psi(r,t)\le c_1 \sqrt\eps+c_2 \int_{0}^t \Psi(s)ds
\label{4.20}
\end{equation}
for suitable constants $c_1,c_2$. By the Gronwall inequality we conclude \eqref{4.16'}.
\vskip.5cm
We now estimate $w^{(\eps)}$. Let $\{B_t\}$ be a standard Brownian motion with reflecting b.c. at $\pm 1$,
with $\mathbb P_r$ denoting its law when $B_0=r$ (and corresponding expectations denoted by $\mathbb E_r$). Then
\begin{equation}
w^{(\eps)}(r,t)= \mathbb E_r\Big( e^{-\int_0^{t} \varphi_\eps(B_s,t-s)ds} w^{(\eps)}(B_{t},0)\Big)
\label{4.21}
\end{equation}
where
\begin{eqnarray}
\varphi_\eps(B,t-s)= \phi_\eps(B)h (t-s),\quad \phi_\eps(r)=\frac {j}{2\eps}\mathbf 1_{[1-\eps,1]}(|r|), \; r\in [-1,1].
\label{4.22}
\end{eqnarray}
By \eqref{ck}
\begin{equation}
w^{(\eps)}(r,t) \le \mathbb E_r\Big( e^{-b \int_0^{t} \phi_\eps(B_s)ds} \Big).
\label{4.21.1}
\end{equation}
For $0<\bar t<t$ we write
\begin{equation}
w^{(\eps)}(r,t) \le \mathbb E_r\Bigg( e^{-b\int_{0}^{t-{\bar t}} \phi_\eps(B_s)ds}\,
\mathbb E_{B_{t-\bar t}}\Big( e^{-b\int_{t-\bar t}^t \phi_\eps(B_s)ds}\Big)\Bigg)
\label{4.23}
\end{equation}
We shall prove below that taking $\bar t$ sufficiently small, we can take $\alpha<1$ so that for all $\eps>0$
\begin{equation}
\sup_{r\in[-1,1]}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)\le 1-\alpha
\label{4.24}
\end{equation}
From \eqref{4.24} and \eqref{4.23} we then get
\begin{equation}
|w^{(\eps)}(r,t)|\le (1-\alpha)^{[t/\bar t]}
\label{4.25}
\end{equation}
($[a]$ the integer part of $a$) which then concludes the proof of the theorem.
\vskip.5cm
{\bf Proof of \eqref{4.24}}. Let $T=\inf\{t \ge 0: |B_t|=1\}$.
We then have
\begin{eqnarray}
\label{4.26}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)&\le& \mathbb E_{r}\Big( \mathbf 1_{\{T\le \bar t/2\}}e^{-b\int_{T}^{\bar t} \phi_\eps(B_s)ds} \Big)+\mathbb P_r(T> {\bar t}/2)
\end{eqnarray}
and write
\begin{eqnarray}
\mathbb E_{r}\Big( \mathbf 1_{\{T\le {\bar t}/2\}}e^{-b\int_{T}^{\bar t} \phi_\eps(B_s)ds} \Big) &\le& \mathbb E_{r}\Big(\mathbf 1_{\{T \le {\bar t}/2\}}\mathbb E_{B_T}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\Big) \nn
\\ &\le& \mathbb{P}_r(T \le {\bar t}/2)\;
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\Big)
\label{a4.26b}
\end{eqnarray}
where we also used that $\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)=\mathbb E_{-1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)$ by symmetry.
By Taylor expansion
\begin{equation}
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s) ds}\big)
\le 1-b\mathbb E_{1}\Big(\int_0^{{\bar t}/2} \phi_\eps (B_s)ds \Big)
+\xi_2
\label{4.27}
\end{equation}
where
\begin{equation}
\xi_2=(\frac{jb}{2\eps})^2\int_0^{{\bar t}/2} dt_1\int_0^{t_1}dt_2
\int_{|y_1|\in [1-\eps,1], |y_2|\in [1-\eps,1]} P_{t_1}(1,y_1)P_{t_2}(y_1,y_2) dy_1dy_2
\label{4.28}
\end{equation}
But, from \eqref{N4.3}--\eqref{N4.10} we see that
\begin{equation}
\label{4.29}
\sup_{x,y\in [-1,1]}P_s(x,y)\le c \frac 1{\sqrt s}
\end{equation}
so that for $\bar t$ small enough we get
\begin{equation}
\label{4.30}
\xi_2\le \bar c {\bar t}/2
\end{equation}
for suitable constant $\bar c$.
Using again \eqref{N4.3}--\eqref{N4.10}, we see at once that a positive constant $c$ can be taken so that for all $\bar t$ small,
and all $\eps>0$
\begin{equation}
\label{4.31}
b\mathbb E_{1}\Big(\int_0^{{\bar t}/2} \phi_\eps(B_s)ds \Big)\ge
c\sqrt {\bar t/2}.
\end{equation}
From \eqref{4.27}, \eqref{4.30} and \eqref{4.31} we then get
for $\bar t$ small (with possibly different constant $c$),
\begin{equation}
\label{4.32}
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\le 1-c\sqrt{{\bar t}/2}.
\end{equation}
By \eqref{4.26} and \eqref{a4.26b} we then have
\begin{equation}
\label{4.41}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)\le \big[1-c\sqrt{{\bar t} /2}\big]\mathbb{P}_r(T\le {\bar t}/2)+\mathbb{P}_r(T>{\bar t}/2)\le 1-\alpha
\end{equation}
with
\begin{equation}
\alpha=\inf_{r\in [-1,1]} \mathbb{P}_r(T\le {\bar t}/2)c\sqrt{{\bar t} /2}.
\end{equation}
\vskip1cm
\section{Proof of Theorem \ref{thm2.2} }
\vskip.5cm
\nopagebreak
The proof is a direct consequence of the following three facts. (i) For any $t>0$ and any integer $k\ge 1$
\begin{equation}
\label{5.0.1}
\lim_{N\to \infty} \max_{ \eta \in \{0,1\}^{\La_N}}\max_{(x_1,..,x_k)\in \La_N^{k,\ne}}
\Big| \mathbb E_{\eta}\big(\prod_{i=1}^k \eta(x_i,t)\big)-\prod_{i=1}^k \mathbb E_{\eta}\big(\eta(x_i,t)\big)\Big|=0
\end{equation}
(ii) For any $t>0$
\begin{equation}
\label{5.0.2}
\lim_{N\to \infty}
\max_{ \eta \in \{0,1\}^{\La_N}}\max_{x\in \La_N}
\Big| \mathbb E_{\eta}\big( \eta(x,t)\big)-\rho(x/N,t| \eta)\Big|=0
\end{equation}
(iii)
\begin{equation}
\label{5.0.3}
\lim_{t\to \infty}
\sup_{ \rho_0\in L^\infty([-1,1];[0,1])}
\| \rho(\cdot,t| \rho_0)-\rho^*(\cdot)\|_\infty=0
\end{equation}
(i) and (ii) are proved in \cite{DPTVpro}--\cite{DPTVjsp}, \eqref{5.0.3} is proved in Theorem \ref{thm3.3}.
\vskip.5cm
{\bf Acknowledgments.}
The work is partially supported by the PRIN project, n.2009TA2595. M.E.V. thanks the Universities of Rome and L'Aquila for kind hospitality. M.E.V. is
partially supported by CNPq grant 302796/2002-9. The research of D.T. has been partially supported by a
Marie Curie Intra European Fellowship within the 7th European
Community Framework Program.
\vskip1cm
\bibliographystyle{amsalpha}
|
{
"timestamp": "2012-03-09T02:02:55",
"yymm": "1203",
"arxiv_id": "1203.1795",
"language": "en",
"url": "https://arxiv.org/abs/1203.1795"
}
|
\section{Introduction}
Primes in arithmetic progression (denoted by AP-$k$, $k \ge 3$) refers
to $k$ prime numbers that are consecutive terms of an
arithmetic progression. For example, 5, 11, 17, 23, 29 is an AP-5, a
five-term arithmetic progression of primes with the common difference 6.
In this example of five primes in arithmetic progression, the primes are
not {\em consecutive} primes. CPAP-$k$ denotes $k$ consecutive primes
in arithmetic progression. An example of CPAP-3 is 47, 53, 59 with the
common difference 6.
Primes in arithmetic progression have been extensively studied both
analytically (see the comprehensive account in~\cite{green-tao})
and numerically (see, \cite{wolfram, wikipedia}). The largest known
sequences contain up to 26 terms, {\it i.e}, AP-26
and 10 consecutive primes {\it i.e}, CPAP-10
(see ~\cite{prime-pages, ap-records} for the AP-$k$ records
and ~\cite{prime-pages, cpap-records} for the CPAP-$k$ records).
The {\it geometric-arithmetic progression} refers to
\begin{equation}\label{ga}
a, \, a r + d, \, a r^2 + 2 d, \, a r^3 + 3 d, \, \dots\,.
\end{equation}
The sequence in~(\ref{ga}) is not be confused with the
{\it arithmetic-geometric progression}, $a$, $(a + d) r$,
$(a + 2 d) r^2$, $(a + 3 d) r^3$, $\dots$, whose terms are composite
by construction. Primes in geometric-arithmetic progression is a set
of $k$ primes (denoted by GAP-$k$) that are the consecutive terms of a
geometric-arithmetic progression in~(\ref{ga}).
For example 3, 17, 79 is a 3-term geometric-arithmetic progression
({\it i.e}, a GAP-3)
with $a = p_1 = 3$, $r = 5$ and $d = 2$. An example of GAP-5 is, 7,
47, 199, 911, 4423, with $p_1 = 7$, $r = 5$ and $d = 12$.
The first term of the GAP-$k$ is called the {\em start},
$r$ the {\em ratio} and $d$ the {\em difference}.
The special case of GAP-2 shall be discussed separately.
For $r = 1$, GAP-$k$ reduces to AP-$k$; in this sense, GAP-$k$ is a
generalization of the AP-$k$. It is possible to generate GAPs with
$p_1 = 1$, in which a case the first term of the sequence is 1 and
has to be excluded when computing the order ($k$) as 1 is excluded
from the set of primes. Example of one such GAP-5 with $p_1 = 1$,
$r = 7$ and $d = 720$ is 1, 727, 1489, 2503, 5281, 20407.
One can also have GAPs with $p_1 = r$; an example for a GAP-5,
is 5, 139, 353, 967, 3581, with $p_1 = r = 5$ and $d = 114$.
There can be GAPs with composite $r$; an example of such a GAP-3 is
7, 107, 1579 with $p_1 = 7$, a composite $r = 15$ and $d = 2$;
and an example for GAP-5 is 11, 919, 14543, 473227, 16509011
with $p_1 = 11$, a composite $r = 35$ and $d = 534$.
Relevant examples are presented in Table-1 and Table-2 respectively.
\section{Results and Analysis}
The following theorem summarizes the conditions on a
geometric-arithmetic progression to be a candidate for GAP-$k$.
\begin{Theorem}\label{theorem-basic}
Let GAP-$k$ denote the set of $k$ primes forming the sequence
$\left\{p_1 r^j + j d \right\}_{j = 0}^{k-1}$, for fixed
$p_1$, $r$ and $d$. Then it is necessary that $d$ is even; $p_1$
is an odd-prime coprime to $d$; $r$ is an odd-number coprime to $d$.
When $p_1 \ne 1$ and $r \ne 1$, the maximum possible order-$k$
of the set is lesser of the two fixed numbers $p_1$ and the smallest
prime factor of $r$.
When $r = 1$, the maximum order of the set is $p_1$.
When $p_1 = 1$, the maximum order of the set is less than the smallest
prime factor of $r$.
\end{Theorem}
When $d$ is odd, the alternate terms of the sequence
$\left\{p_1 r^j + j d \right\}_{j = 0}^{k-1}$, take
even values. Hence, $d$ can not be odd. When $p_1 r^j$ is even
then again the alternate terms of the
sequence are even. So, it is necessary that $d$ is even and $p_1 r^j$
is odd, ensuring that all the terms of the sequence are odd, a
prerequisite for them to be prime. The first term of the sequence is
$p_1$. So, $p_1$ is necessarily an odd-prime. Since, $p_1 r^j$ is odd
it is necessary that $r$ is also odd. For $p_1 r^j + j d$ to be prime it
is necessary that $p_1$ and $r$ are both coprime to $d$. This proves
the theorem except for the part related to the order of the set.
First we consider the scenario $p_1 \ne 1$ and $r \ne 1$.
The $(p_1 + 1)^{\text{th}}$ term of the sequence (obtained for $j = p_1$)
is $p_1 r^{p_1} + p_1 d$, which is composite. Hence, $k \leq p_1$.
Let $r_1$ be the smallest prime factor of $r$. The $(r_1 + 1)^{\text{th}}$
term of the sequence (obtained for $j = r_1$) is
$p_1 r^{r_1} + r_1 d = p_1 r_1^{r_1} (r/r_1)^{r_1} + r_1 d$, which is composite.
Hence $k \leq r_1$.
When $r = 1$ and $p_1 \ne 1$, the sequence simplifies to $\left\{p_1 + j d \right\}$,
whose $(p_1 + 1)^{\text{th}}$ term is $p_1 + p_1 d$, which is composite.
Hence, $k \leq p_1$.
When $p_1 = 1$ and $r \ne 1$, the sequence becomes $r^j + j d$, whose
first term is 1 (for $j = 0$) and the $(r_1 + 1)^{\text{th}}$
term is $r^{r_1} + r_1 d = r_1^{r_1} (r/r_1)^{r_1} + r_1 d$, which is composite.
Since, number 1 is not among the primes, $k \leq (r_1 - 1)$.
The case $p_1 = 1$ and $r = 1$ is trivial and generates only one GAP
(uniquely fixed with $d = 2$), which is the GAP-3, 3, 5, 7.
This completes the proof of the theorem.
For every integer, $n \ge 2$, there exists a prime $p$ such that $n < p < 2n$
(see for instance,~\cite{hardy}). The elements of GAPs ($r \ne 1$) grow faster than $2n$.
Consequently, GAPs can not have consecutive primes as its members.
Hence, we do not have consecutive primes in geometric-arithmetic progression.
Theorem~\ref{theorem-basic} tells us the necessary conditions
on $p_1$, $r$ and $d$, for a geometric-arithmetic progression to be a candidate
for GAP-$k$.
The theorem is {\it nonconstructive}, giving no clues for a recipe to generate the GAP-$k$.
A recipe is required to choose `good' triplets $(p_1, \, r, \, d)$
in order to generate GAP-$k$ with larger values of $k$, and GAPs of a given order with
large number of digits.
GAP-2 is a pair of primes of the form $(p_1, \, p_1 r + d)$ and
consequently structurally much simpler than the larger GAP-$k$.
For GAP-2, theorem~\ref{theorem-basic}
simplifies to the condition that, $p_1 r$ and $d$ are coprime.
For example, with $p_1 = 2$, $r = 6$ and $d = 5$, we have $(2, \, 17)$;
with $p_1 = 3$, $r = 2$ and $d = 1$, $(3, \, 7)$; and
$p_1 = 7$, $r = 100$ and $d = 211$, $(7, \, 911)$ respectively.
In the world of primes, {\em titanic} is $1000+$ digits~\cite{prime-pages}.
Example of a titanic GAP-2 is obtained with $p_1 = M_{4253}$, $r = 7$
and $d = 1422$ as $(M_{4253}, \, 7 M_{4253} + 1422)$.
Here, $M_{4253} = 2^{4253} - 1$ is the 19th Mersenne prime containing 1281 digits.
Mersenne primes were chosen, as they are well-known and easy to express~\cite{mersenne, mersenne-oeis}.
Pairs of primes with specific properties have been extensively studied.
For instance, {\em Sophie Germain primes} have the form $(p, \, 2p + 1)$.
With $r = 1$ the GAP-2 further simplifies to the pair $(p, \, p + d)$. Prime pairs
such as {\em twin primes}, $(p, \, p + 2)$; {\em cousin primes}, $(p, \, p + 4)$;
{\em sexy primes}, $(p, \, p + 6)$, among others have been extensively studied~\cite{prime-pages}.
A GAP-$k$ is said to be {\em minimal} if the minimal start $p_1$ and
the minimal ratio $r$ are equal, {\it i.e}, $p_1 = r = p$, where $p$ is the smallest prime $\ge k$.
Such GAPs have the form $\left\{p*p^j + j d \right\}_{j = 0}^{k-1}$.
Minimal GAPs with different differences, $d$ do exist. For example, the minimal GAP-5
($p_1 = r = 5$) has the
possible differences, 84, 114, 138, 168, $\ldots$ and the minimal
GAP-6 ($p_1 = r = 7$) has the possible differences, 144, 1494, 1740, 2040, $\ldots$.
A minimal GAP-$k$ is further said to be {\em absolutely minimal} if the difference $d$ is minimum.
All the GAPs up to $k = 11$ in Table-1 are absolutely minimal. Computationally obtained lower bounds
of $d$ in search for higher-order minimal GAPs are also presented in Table-1.
From theorem~\ref{theorem-basic}, it is evident that the order, $k$ of any GAP-$k$ does not
exceed both the starting prime $p_1$ and the smallest prime factor of the ratio $r$.
Equipped with this fact and the numerical data, we have the following two conjectures
\begin{Conjecture}[Minimal Start]{}\label{conjecture-one}
The minimal starting prime, $p_1$ in a GAP-$k$ is the smallest prime $\ge k$.
\end{Conjecture}
\begin{Conjecture}[Minimal Start and Minimal Ratio]{}\label{conjecture-two}
The minimal starting prime, $p_1$ and minimal ratio, $r$ in a GAP-$k$ is the
smallest prime $\ge k$ and $p_1 = r$.
\end{Conjecture}
Computational data in Table-1 supports these conjectures up to $k = 11$.
In the context of the absolutely minimal GAPs, it is interesting to note that
the absolutely minimal GAP-9 and the absolutely minimal GAP-10 occur for the same
value of $d = 903030 = 31*971*(5\#) = 30101*(5\#)$, where
$n \#$ is the primorial, $2.3.5. ... p, p \le n$. For example, $10\# = 2.3.5.7 = 210$.
Consequently, GAP-9 is a complete
subset of GAP-10 (in this particular instance, since they have the same $d$).
An individual GAP-9 occurs for a higher $d = 1004250 = (5^2)*13*103*(5\#) = 33475*(5\#)$.
This is analogous to the situation of AP-4 and AP-5 with the minimal start (which is 5).
The corresponding sequence is $\{5 + j d \}$. For $d = 6$ (which is the minimum difference),
the AP-4 and AP-5 are 5, 11, 17, 23, and 5, 11, 17, 23, 29 respectively.
The next AP-4 and AP-5 again occur at $d = 12$. The individual AP-4 occurs only
at $d = 18$, which is 5, 23, 41, 59.
A given pair of start $p_1$ and ratio $r$, in general generates a GAP-$k$ of a certain
order $k$ for different values of the difference $d$. In this note, we shall focus on
the set of differences corresponding to the minimal GAPs.
The minimal GAP-2, $\left\{2*2^j + j d \right\}_{j = 0}^{1}$ is a pair of
primes, $(2, \, 4 + d) \equiv (2, \, p - 4)$, where $p$ is any prime.
Consequently, $d$ belongs to the sequence $\{p - 4 \}$, where $\{p \}$ is the
infinite sequence of primes. Since, the sequence $\{p \}$ is infinite, the
sequence $\{p - 4 \}$ is also infinite.
We shall cite various integer sequences from {\it The On-Line Encyclopedia
of Integer Sequences} (OEIS) created and maintained by Neil Sloane.
For example, the sequence of primes, $\{p \}$ is identified by A000040 in~\cite{Sloane}.
The infinite sequence $\{p - 4 \}$: 1, 3, 7, 9, 13, 15, 19, 25, $\ldots$, is A172367~\cite{GAP-2-d}.
The integer sequences of the differences $d$, corresponding to the minimal GAPs of each
order 3 to 11 are presented in~\cite{GAP-3-d}-\cite{GAP-11-d}.
In general, there are no reasons to believe that the sequence of the differences $d$
corresponding to any GAP (minimal or non-minimal) are finite.
Analogous sequences for the differences also exist for the primes in arithmetic progression.
See~\cite{AP-2-d}-\cite{AP-differences} for the sequences of differences
corresponding to the primes in arithmetic progression with the minimal start.
A study of these integer sequences may provide a pattern, which will potentially guide us in our
search for higher order GAPs and APs.
From the computed sequences, we note that the set of differences for a given
minimal GAP-$k$ have a {\em common} $k$-dependent multiplicative factor.
This factor has been indicated as $(...\#)$ in Table-1. We have included only the
common factor. Individual differences $d$ do have additional factors. For instance,
the first difference for the minimal GAP-11 is $443687580 = 2112798 (7\#) = 14789586(5\#)$.
The second difference is not a multiple of $(7\#)$ and hence we have shown the first
difference as $14789586(5\#)$ in Table-1.
Theorem~\ref{theorem-basic} restricts the values of the differences $d$ for
any GAP-$k$ ($k \ge 3$) to be even ({\it i.e} multiples of 2).
There are additional restrictions on the values of the differences $d$ for
minimal GAPs of a given order, as shown below
\begin{Theorem}[Factors of $d$]{}\label{theorem-primorial} ~~
\begin{enumerate}
{\itemsep -0.103cm
\item
The values of the differences $d$ for each of the minimal GAP-$k$, $k \ge 5$ are
multiples of a $k$-dependent factor denoted along with the order $k$ by $(k:~...\#)$,
where $\#$ is the primorial. They are
$(5:~3\#)$,
$(6-7:~3\#)$,
$(8-11:~5\#)$,
$(12-13:~7\#)$,
$(14-17:~5\#)$,
$(18:~7\#)$,
$(19:~11\#)$,
$(20-23:~11\#)$,
$(24-29:~13\#)$,
$(30-31:~13\#)$,
$(32-37:~19*11\#)$,
$(38-41:~13\#)$,
$(42:~17\#)$,
$(43:~19\#)$,
$(44-47:~23*17\#)$,
$(48-53:~17\#)$,
$(54:~29*13\#)$,
$(55-58:~29*19*13\#)$,
$(59:~29*19\#)$,
$(60-61:~31*19*13\#)$,
$(62-67:~31*19\#)$,
$(68-71:~17\#)$,
$(72-73:~37*23*17\#)$,
$(74-79:~23\#)$,
$(80-83:~41*19\#)$,
$(84-89:~31*23\#)$,
$(90-97:~23\#)$,
$(98-99:~23\#)$,
$(100-101:~31*23\#)$,
$(102-103:~23\#)$ respectively.
\item
The values of the differences $d$ for all minimal GAP-$k$, $k \ge 5$ are multiples of $(3\#)$.
\item
The values of the differences $d$ for all minimal GAP-$k$, $k \ge 8$ are multiples of $(5\#)$.
\item
The values of the differences $d$ for all minimal GAP-$k$, $k \ge 18$ are multiples of $(7\#)$.
\item
The values of the differences $d$ for all minimal GAP-$k$, $k \ge 19$ are multiples of $(11\#)$.
\item
The values of the differences $d$ for all minimal GAP-$k$, $k \ge 38$ are multiples of $(13\#)$.
}
\end{enumerate}
\end{Theorem}
The proof of the theorem~\ref{theorem-primorial} is based on modular arithmetic and is
presented in Appendix-A. Factors up to $k = 103$ are presented in Table-3.
In an arithmetic progression (AP-$k$) with the minimal start, the pattern of the differences
is known to be a multiple of $k\#$, where $k$ is the largest prime $\le k$ (if $k$ is not a prime).
If $k$ is a prime than the common difference is a multiple of $(k - 1)\#$
(see \cite{wolfram, wikipedia}).
Unlike in the case of the AP-$k$ with the minimal start, there is no obvious pattern in the case
of the minimal GAP-$k$. The factor $(...\#)$ is not even monotonic.
In none of the cases, it has been possible to establish the factors containing
the higher powers of 2, 3, 5, or 7. Theorem~\ref{theorem-primorial} only gives the restrictions
on the common difference $d$ in order for the generating sequence to be a candidate for minimal GAP.
The existence of the minimal GAP-$k$, $k \ge 12$ is yet to be established (numerically or otherwise).
The extension of the theorem~\ref{theorem-primorial} to non-minimal GAPs is also discussed in
Appendix-A.
So far, we have considered the GAPs from the sequence $\left\{p_1 r^j + j d \right\}_{j = 0}^{k-1}$.
The sequence, $\left\{p_1 r^j + j d \right\}$ can have sets of primes
for consecutive $j$, not necessarily starting with $j = 0$.
For example, the sequence, $\left\{5*5^j + 4j \right\}_{j = 7}^{j = 9}$ generates the
GAP-3, 390653, 1953157, 9765661.
Another example is the sequence, $\left\{13*13^j + 156497*(11\#)j \right\}_{j = 3}^{j = 12}$
generating the GAP-10,
1084552771, 1446403573, 1812367159, 2231796937, 3346287211, 13496563933, 141112064479,
1795775474737, 23302061711251, 302879444689093.
Such sets, not starting with $j = 0$, can not be put in the form
$\left\{P_1 R^j + j D \right\}_{j = 0}^{j = k-1}$, where $P_1$, $R$ and $D$ are derived from
$p_1$, $r$ and $d$.
Importantly, the orders of such GAPs are restricted, as we shall soon see.
\begin{Theorem}\label{theorem-free}
Let $\left\{p_1 r^j + j d \right\}_{j = 0}^{p^{\prime} - 1}$ be a
GAP-$p^{\prime}$ of order $p^{\prime}$, where $p^{\prime}$ is smaller of the two
primes $p_1$ and $r_1$ the smallest prime factor of $r$.
Then the infinite sequence $\left\{p_1 r^j + j d \right\}_{j = p^{\prime}}^{\infty}$ does not
have any GAPs of order $\ge p^{\prime}$.
\end{Theorem}
Since, theorem~\ref{theorem-basic} requires $p_1$ to be an odd-prime and $r$
to be any odd number, $p^{\prime} \ge 3$. When $r = 1$, the GAP is reduced
to an AP and we have $p^{\prime} = p_1 \ge 3$.
Let us recall that theorem~\ref{theorem-basic} forbids GAPs of orders greater
than $p^{\prime}$ throughout the interval $[j = 0, \, \infty)$.
While proving theorem~\ref{theorem-basic}, we saw that
$(p_1 + 1)^{\text{th}}$ and the $(r_1 + 1)^{\text{th}}$ terms of the
sequence, $\left\{p_1 r^j + j d \right\}_{j = 0}^{\infty}$ are composite.
The $(n p_1 + 1)^{\text{th}}$ term (where, $n = 1, 2, 3, \ldots$) of the sequence
(obtained for $j = n p_1$) is $p_1 r^{n p_1} + n p_1 d$. This term is composite and
belongs to the interval $[j = p_1, \, \infty)$.
There are only $(p_1 - 1)$ terms between any two successive $(n p_1 + 1)^{\text{th}}$
and $(\overline{n+1} p_1 + 1)^{\text{th}}$ terms. So, the interval $[j = p_1, \, \infty)$
can not have any GAPs of order more than $(p_1 - 1)$.
The $(n r_1 + 1)^{\text{th}}$ term (obtained for $j = n r_1$) of the sequence is
$p_1 r^{n r_1} + n r_1 d = p_1 r_1^{n r_1} (r/r_1)^{n r_1} + r_1 d$. This is also composite.
Similar arguments forbid the GAPs of order more than $(r_1 - 1)$ in the
interval $[j = p_1, \, \infty)$. This proves the theorem.
In passing we note that, when $r = 1$, the GAP is reduced to an AP. Then
theorem~\ref{theorem-free} tells us that the
sequence, $\left\{p_1 + j d \right\}_{j = p_1}^{\infty}$ does not have any APs of order $\ge p_1$.
Theorem~\ref{theorem-free} is for the restricted case of GAPs, whose order is any
prime $p^{\prime} \ge 3$
and forbids the existence of GAPs of order $p^{\prime}$ in the infinite interval $[j = p^{\prime}, \, \infty)$.
Moreover, the theorem is silent about the absence (or existence) of GAPs of orders lower than
$(p^{\prime} - 1)$ in the interval $[j = p_1, \, \infty)$. It is interesting to note that
the nine sequences $\left\{p*p^j + j d \right\}_{j = k}^{1000}$, (for $k = 3$ to 11),
for the choice of the absolutely minimal triplets $(p, \, p, \, d)$ in Table-1 do not have
any GAPs of order 3 to $p$ respectively. Some of them do have one or two GAP-2 in the
interval, $[j = k, \, 1000]$ respectively.
This numerical data provides room for extending the theorem~\ref{theorem-free} to cases, when the
order of minimal GAP is not a prime. This leads to the conjecture
\begin{Conjecture}[GAP free]{}\label{conjecture-three}
Whenever the sequence $\left\{p*p^j + j d \right\}_{j = 0}^{k - 1}$ has the minimal GAP-$k$,
the rest of the infinite sequence, $\left\{p*p^j + j d \right\}_{j = k}^{\infty}$
does not have any GAPs of orders $\ge 3$.
\end{Conjecture}
\section{Concluding Remarks}
For a given triplet, $(p_1, \, r, \, d)$, the
sequence, $\left\{p_1 r^j + j d \right\}_{j = 0}^{N}$,
may not always generate very many primes.
For example, the sequence, $\left\{5*3^j + 2j \right\}$, takes prime values for,
$j = 0$, 1, 7, 29, 49, 83, 436, 536, 1274, $\dots$.
The sequence, $\left\{7*13^j + 36j \right\}_{j = 0}^{1000}$, has only a
single pair ({\it i.e}, a GAP-$2$) for $j = 0, 1$, which is $(7, \, 127)$.
Similar is the situation for
a wide range of $(p_1, \, r, \, d)$, making it very hard to find GAPs.
GAP-3 and GAP-4 with 159 digits were obtained using the Mersenne primes.
Numerical data in this article was computed initially
(up to GAP-6 in Table-1), using the Microsoft {\it EXCEL}~\cite{EXCEL}.
The primality of the numbers generated by EXCEL was checked using the database of
primes at {\it The Prime Pages}~\cite{prime-pages} and the {\em Sequence A000040}, from
{\it The On-Line Encyclopedia of Integer Sequences} (OEIS), created and
maintained by Neil Sloane~\cite{Sloane}.
For higher orders, we are using the {\it MATHEMATICA}~\cite{MATHEMATICA}.
Search for GAPs with ever larger $k$ and
geometric-arithmetic progressions containing larger primes is in progress.
We end this note with several open questions, similar to the ones, which exist for the
primes in arithmetic progression~\cite{green-tao}.
Are there arbitrarily long geometric-arithmetic progressions of primes?
Are there infinitely many $k$-term geometric-arithmetic progressions consisting of $k$ primes?
Do the prime numbers contain infinitely many geometric-arithmetic progressions of length $k$ for all $k$?
Are there are infinitely many GAP-$k$ for any $k$?
We conjecture that the answer to all the above questions is in the affirmative.
\newpage
\begin{center}
{\Large\bf
Appendix-A: \\ Proof of Theorem~\ref{theorem-primorial}}
\end{center}
Theorem~\ref{theorem-basic} states that the common factor of all differences of
any GAP-$k$, $k\ge 3$ is 2 ({\it i.e}, $d$ is even).
Theorem~\ref{theorem-primorial} states additional factors of all the differences
$d$ of a given minimal GAP-$k$, $k \ge 5$. The theorem essentially consists of two parts:
part-1 is for the specific GAP-$k$ with $k$ up to 103; part-2 has global statements giving
the common factor of all differences of any minimal GAP, whose order exceeds a particular
number. Its proof is based on modular arithmetic. As we shall soon see, it suffices to
demonstrate the procedure of proving the statements in a few specific cases of both part-1
and part-2 respectively. Rest of the statements in the theorem for higher orders can be
proved closely following the procedures established for lower orders. In fact the
methodology presented can be used to derive results and extend the theorem to still higher
orders and importantly to the non-minimal GAPs.
The minimal GAP-5 is defined by the sequence $\left\{5*5^j + j d \right\}_{j = 0}^{4}$ and
the common difference $d$ is restricted in such a way that the defining sequence has 5 primes.
The first four terms of this sequence belong to GAP-4.
The residues of this sequence $\pmod {3}$ are $\{2, \, 1 + d, \, 2 + 2 d, \, 1, \, 2 + d \}$.
Primality requires the second residue $1 + d$ such that $d \not\equiv 2 \pmod {3}$ and the
fifth residue $2 + d$ such that $d \not\equiv 1 \pmod {3}$. Consequently, $d \equiv 0 \pmod {3}$.
Otherwise, the second and fifth terms in the defining sequence would be multiples of 3 and not prime.
Hence, $d$ is {\em necessarily} a multiple of 3.
Theorem~\ref{theorem-basic} restricts the values of the differences $d$ of any GAP-$k$, $k\ge 3$ to
be multiples of 2. Consequently, the values of the differences for GAP-5 are restricted to be multiples
of $(3\#)$.
The fifth residue does not belong to GAP-4 and hence the result is not applicable to GAP-4.
The third residue $2 + 2 d$ is degenerate as it gives the same information as the second residue.
Among the five residues, the first was {\em numeric} ({\it i.e}, free of $d$) and the third was degenerate.
The defining sequence for the minimal GAP-7 is $\left\{7*7^j + j d \right\}_{j = 0}^{6}$.
The corresponding residues $\pmod {3}$ are $\{1, \, 1 + d, \, 1 + 2 d, \, 1, \, 1 + d, \, 1 + 2 d, \, 1 \}$.
Since, GAP-6 is defined by the same sequence except for the index, its residues are the same
as the first six residues for GAP-7. The second and third residues are sufficient to establish
that $d$ is a multiple of 3 for both GAP-6 and GAP-7.
Consequently, the values of the differences for GAP-6 and GAP-7 are restricted to be multiples
of $(3\#)$.
The residues $\pmod {5}$ are $\{2, \, 4 + d, \, 3 + 2 d, \, 1 + 3 d, \, 2 + 4 d, \, 4, \, 3 + d \}$.
The first and sixth residues are numeric.
The second, fourth and fifth residues require $d \not\equiv 1 \pmod {5}$, $d \not\equiv 3 \pmod {5}$,
and $d \not\equiv 2 \pmod {5}$ respectively. The third and seventh residues are degenerate.
The case, $4 \pmod {5}$ remains unaddressed and hence $d$ need not be multiple of 5.
The presence of numeric and degenerate residues of a given defining sequence hinders the
larger factors.
Rest of the results in part-1 of the theorem are proved closely following the procedure
used for GAP-4 to GAP-7. The procedure is straightforward but becomes laborious
as the order-$k$ grows. We have used the MATHEMATICA to compute the residues~\cite{MATHEMATICA}.
Following are the residues for GAP-7 $\pmod {5}$
{\tt
\begin{verbatim}
In[1]:= Clear[p];
p = 7;
PolynomialMod[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d,
p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d}, 5]
Out[3]= {2, 4 + d, 3 + 2 d, 1 + 3 d, 2 + 4 d, 4, 3 + d}
\end{verbatim}
}
In part-1 of the theorem, the results are for individual GAP-$k$, $k \le 103$. We have
demonstrated the procedure up to $k = 7$ and it is straightforward to extend it to higher
orders. Part-2 onwards the statements are global and the procedure is as follows.
The differences for the minimal GAP-5 are divisible by 3. Since, $5 \equiv 2 \pmod {3}$
the result is applicable to all those primes $> 5$, whose residues $\pmod {3}$ are 2.
The differences for GAP-7 are divisible by 3 and $7 \equiv 1 \pmod {3}$. The result is
again applicable to all those primes $> 7$, whose residues $\pmod {3}$ are 1. The
result was individually proved for GAP-6 in part-1.
The non-zero residues of 3 are $\{1, 2 \}$. Consequently the differences for
all GAP-$k$, $k \ge 5$ are divisible by $(3\#)$ with the factor 2 coming from
theorem~\ref{theorem-basic}.
The non-zero residues of 5 are $\{1, 2, 3, 4 \}$, and the corresponding 4 primes
with these residues are $\{11, 17, 13, 19 \}$. Note, that $7 \equiv 2 \pmod {5}$
but its differences are not divisible by 5 as seen in part-1.
We now individually examine the four generating sequences for GAP-11, GAP-17, GAP-13
and GAP-19 respectively and conclude that the differences for each of them are multiples of 5.
The factor of $(3\#)$ is already established, so we conclude that all GAP-$k$, $k\ge 19$ have
their differences as multiples of $(5\#)$. The inequality $k \ge 19$, is refined by using the
results in part-1. The factor of $(5\#)$ was established for the lower order
GAP-$k$, $k = 8$ to $k = 18$ in part-1.
Hence, the values of the differences $d$ for all minimal GAP-$k$, $k \ge 8$ are multiples of $(5\#)$.
The set $\{11, 17, 13, 19 \}$ was only a {\em candidate} set. Had the differences for any of
the GAP-11, GAP-17, GAP-13 or GAP-19 failed to be divisible by 5, we would have examined the
GAPs of higher orders, corresponding to that residue.
The non-zero residues of 7 are $\{1, 2, 3, 4, 5, 6 \}$ and the corresponding 6 primes
are $\{29, 23, 31, 53, 19, 13 \}$. The primes $11 \equiv 4 \pmod {7}$ and $17 \equiv 3 \pmod {7}$
are not relevant in view of the results in part-1. Following the procedure used for establishing
the factors $(3\#)$ and $(5\#)$, we conclude that the differences $d$ for all
minimal GAP-$k$, $k \ge 18$ are multiples of $(7\#)$.
The candidate set of 10 primes corresponding to the non-zero residues of 11
is $\{23, 79, 47, 37, 71, 61, 29, 19, 31, 43 \}$. The differences $d$ for each of the
GAPs of these orders are divisible by $(11\#)$.
The largest prime in this set is 79 and a spontaneous result is that all GAP-$k$, $k \ge 79$
have their differences as multiples of $(11\#)$.
Using the results from part-1, we refine the inequality and conclude that
the differences $d$ for all minimal GAP-$k$, $k \ge 19$ are multiples of $(11\#)$.
The candidate set of 12 primes corresponding to the non-zero residues of 13
is $\{53, 41, 29, 43, 31, 71, 59, 47, 61, 101, 89, 103 \}$. The largest prime in
this set is 103. Hence, the results in part-1 are up to $k = 103$. The candidate set
successfully works and we conclude that the differences $d$ for all
minimal GAP-$k$, $k \ge 38$ are multiples of $(13\#)$.
The candidate set of 16 primes corresponding to the non-zero residues of 17
is $\{103, 53, 71, 89, 73, 193, 109, 127, 43, 163, 79, 97, 47, 167, 83, 101 \}$.
The largest prime in this set is 193.
The candidate set of 18 primes corresponding to the non-zero residues of 19
is $\{191, 59, 79, 61, 43, 101, 83, 103, 199, 67, 163, 107, 89, 109, 167, 149,
131, 37 \}$.
The largest prime in this set is 199.
The procedure of proving this theorem can be applied to the non-minimal GAPs.
The common factor $...\#$ of the differences $d$ of a GAP-$k$ with the start $p_1$,
and ratio $r$ shall be denoted by $(k: p_1, \, r, \, ...\#)$.
The examples are
$(3: 5, 7, 3\#)$m
$(3: 2^{521} - 1, 19, 3\#)$,
$(4: 11, 35, 2\#)$,
$(4: 2^{521} - 1, 5, 2\#)$,
$(5: 47, 2^{31} - 1, 3\#)$,
$(7: 7, 11, 5\#)$,
$(7: 7, 13, 3\#)$,
$(7: 7, 17, 3\#)$,
$(7: 7, 19, 3\#)$,
$(7: 11, 7, 3\#)$,
$(7: 11, 13, 3\#)$,
$(7: 11, 17, 3\#)$,
$(7: 13, 7, 3\#)$,
$(7: 17, 7, \#)$,
$(7: 19, 7, 3\#)$,
$(7: 19, 23, 3\#)$,
$(7: 23, 19, 3\#)$,
$(11: 11, 13, 7\#)$ and
$(11: 13, 11, 5\#)$.
The choice of the $(k: p_1, \, r, \, ...\#)$ in the above examples includes the cases
covered in Table-2.
\newpage
\begin{center}
{\Large\bf
Appendix-B: \\ MATHEMATICA Codes}
\end{center}
Most of the data in this article was computed using the versatile package {\it MATHEMATICA}~\cite{MATHEMATICA}.
The following program searches for the values of the differences $d$ for the minimal GAP-5, in the
range $[0, 1000]$.
{\tt
\begin{verbatim}
In[1]:= Clear[p]; p = 5;
gapset5d = {};
Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d,
p*p^4 + 4*d}] == {True, True, True, True, True},
AppendTo[gapset5d, d]], {d, 0, 10^3}]; gapset5d // Timing
Out[4]= {6.50521*10^-19, {84, 114, 138, 168, 258, 324, 348,
462, 552, 588, 684, 714, 744, 798, 882, 894, 972}}
\end{verbatim}
}
The output is the set of 17 values of $d$: $\{84, 114, 138, \dots, 972 \}$.
The above program (christened {\em runner}) picks the values of $d$ but skips
the finer details. The following program (christened {\em walker}) gives the
complete GAP sets corresponding to each $d$.
{\tt
\begin{verbatim}
In[5]:= f[n_, m_] := (5)*(5)^n + n*m;
Column[Table[{m,
Cases[Table[{f[n, m], f[n + 1, m], f[n + 2, m], f[n + 3, m],
f[n + 4, m]}, {n, 0, 5}], {a1_, a2_, a3_, a4_, a5_} /;
PrimeQ[{a1, a2, a3, a4, a5}] == {True, True, True, True,
True}]}, {m, 114, 114}]] // Timing
Out[6]= {4.33681*10^-19, {114, {{5, 139, 353, 967, 3581}}}
\end{verbatim}
}
In the above program we choose the difference 114 and obtained the corresponding GAP-5:
$\{5, 139, 353, 967, 3581 \}$. As the name goes the walker is much slower than the
runner and hence, not suitable for generating the sequence of differences $d$.
The runner can be made into an {\em accelerator} by replacing $\{d, 0, 10^{\wedge} 3 \}$
with $\{d, 0, 10^{\wedge} 3, \, {\mbox{\boldmath $2$}} \}$ and confining the search to
multiples of 2 (as restricted by theorem~\ref{theorem-basic}). It can be further
accelerated by the replacement of $\{d, 0, 10^{\wedge} 3, \, {\mbox{\boldmath $2$}} \}$
with $\{d, 0, 10^{\wedge} 3, \, {\mbox{\boldmath $6$}} \}$ and refining the search to
multiple of $(3\#) = 6$ (as restricted by theorem~\ref{theorem-primorial}). Such
replacements are relevant as the numbers grow. It is straightforward to extend the
above programs (for GAP-5) to higher orders.
\newpage
\begin{landscape}
\noindent
{\bf Table-1:}
Primes in Geometric-Arithmetic Progression with
{\em minimal start}, $p_1$, {\em minimal ratio}, $r$ and the {\em minimal difference}, $d$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$k$ & $p_1$ & $r$ & $d$ &
$\begin{array}{c} {\rm Primes~ of~ the~ form}, \\
p_1 *r^n + n d,
{\rm for} ~$n = 0$ ~{\rm to} ~k - 1
\end{array}$ & $\begin{array}{c} {\rm Digits}\\ {\rm of~First}\end{array}$
& $\begin{array}{c} {\rm Digits}\\ {\rm of~Last}\end{array}$ \\
\hline
2 & 2 & 2 & 1 & $2*2^n + n$ & 1 & 1 \\
\hline
3 & 3 & 3 & 2 & $3*3^n + 2n$ & 1 & 2 \\
\hline
4 & 5 & 5 & $3(2\#)$ & $5*5^n + 3(2\#)n$ & 1 & 3 \\
\hline
5 & 5 & 5 & $14(3\#)$ & $5*5^n + 14(3\#)n$ & 1 & 4 \\
\hline
6 & 7 & 7 & $24(3\#)$ & $7*7^n + 24(3\#)n$ & 1 & 6 \\
\hline
7 & 7 & 7 & $554(3\#)$ & $7*7^n + 554(3\#)n$ & 1 & 6 \\
\hline
8 & 11 & 11 & $2087(5\#)$ & $11*11^n + 2087(5\#)n$ & 2 & 9 \\
\hline
9 & 11 & 11 & $30101(5\#)$ & $11*11^n + 30101(5\#)n$ & 2 & 10 \\
\hline
10 & 11 & 11 & $30101(5\#)$ & $11*11^n + 30101(5\#)n$ & 2 & 11 \\
\hline
11 & 11 & 11 & $14789586(5\#)$ & $11*11^n + 14789586(5\#)n$ & 2 & 12 \\
\hline
$12-13$ & 13 & 13 & $> {25*10^7} \times (7\#)$ & & & \\
\hline
$14-17$ & 17 & 17 & $> {6*10^7} \times (5\#) $ & & & \\
\hline
$18$ & 19 & 19 & $> {10*10^7} \times (7\#) $ & & & \\
\hline
$19$ & 19 & 19 & $> {10*10^7} \times (11\#) $ & & & \\
\hline
$20-23$ & 23 & 23 & $> {5*10^7} \times (11\#)$ & & & \\
\hline
$24-29$ & 29 & 29 & $> {6*10^7} \times (13\#)$ & & & \\
\hline
$30-31$ & 31 & 31 & $> {11*10^7} \times (13\#)$ & & & \\
\hline
\end{tabular}
\end{center}
\noindent
$n \#$ is the primorial, $2.3.5. ... p, p \le n$. For example, $10\# = 2.3.5.7 = 210$. \\
The symbol $>$ indicates the computationally obtained lower bound of
the {\em minimal difference}, $d$ in search for the GAPs of the corresponding orders.
\newpage
\noindent
{\bf Table-2:} Miscellaneous examples of Primes in Geometric-Arithmetic Progression
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$k$ & $p_1$ & $r$ & $d$ &
$\begin{array}{c} {\rm Primes~ of~ the~ form}, \\
p_1 *r^n + n d,
{\rm for} ~$n = 0$ ~{\rm to} ~k - 1
\end{array}$ & $\begin{array}{c} {\rm Digits}\\ {\rm of~First}\end{array}$
& $\begin{array}{c} {\rm Digits}\\ {\rm of~Last}\end{array}$ \\
\hline
2 & 2 & 5 & 7 & $2*5^n + 7$ & 1 & 2 \\
\hline
2 & 13 & 80 & 53 & $13*80^n + 53$ & 1 & 4 \\
\hline
2 & $2^{4423} - 1$ & 7 & 802 & $(2^{4423} - 1)*7^n + 802n$ & 1332 & 1333 \\
\hline
3 & 5 & 7 & $4(3\#)$ & $5*7^n + 4(3\#)n$ & 1 & 3 \\
\hline
3 & $2^{127} - 1$ & 3 & 7390 & $(2^{127} - 1)*3^n + 7390n$ & 39 & 40 \\
\hline
3 & $2^{521} - 1$ & 3 & 1106 & $(2^{521} - 1)*3^n + 1106n$ & 157 & 158 \\
\hline
3 & $2^{521} - 1$ & 19 & $4365(3\#)(2^{127} - 1)$ & $(2^{521} - 1)*19^n + 4365(3\#)(2^{127} - 1)n$ & 157 & 160 \\
\hline
4 & 11 & 35 & $12(2\#)$ & $11*35^n + 12(2\#)n$ & 1 & 15 \\
\hline
4 & $2^{521} - 1$ & 5 & $33936(2\#)$ & $(2^{521} - 1)*5^n + 33936(2\#)n$ & 157 & 159 \\
\hline
5 & 47 & $2^{31} - 1$ & $13554(3\#)$ & $47*(2^{31} - 1)^n + 13554(3\#)n$ & 2 & 39 \\
\hline
10 & 17 & 17 & 7700(5\#) & $17*17^n + 7700(5\#)n$ & 2 & 13 \\
\hline
11 & 13 & 11 & $129262(5\#)$ & $13*11^n + 129262(5\#)n$ & 2 & 12 \\
\hline
11 & 13 & 13 & $983234(7\#)$ & $13*13^n + 983234(7\#)n$ & 2 & 13 \\
\hline
11 & 103 & 103 & 2900641(23\#) & $103*103^n + 2900641(23\#)n$ & 3 & 23 \\
\hline
12 & 19 & 19 & $ 9345011(11\#)$ & $19*19^n + 9345011(11\#)n$ & 2 & 16 \\
\hline
12 & 31 & 31 & $70167912(13\#)$ & $31*31^n + 70167912(13\#)n$ & 2 & 18 \\
\hline
13 & 19 & 19 & $ 9345011(11\#)$ & $19*19^n + 9345011(11\#)n$ & 2 & 17 \\
\hline
\end{tabular}
\end{center}
\noindent
$M_{31} = 2^{31} - 1$, $M_{127} = 2^{127} - 1$, $M_{521} = 2^{521} - 1$
and $M_{4423} = 2^{4423} - 1$ are Mersenne primes~\cite{mersenne, mersenne-oeis}.
\end{landscape}
\newpage
\noindent
{\bf Table-3:}
The differences $d$ for the minimal GAP of each order are multiples of a common factor
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Order-$k$ & Generating Prime $p$ & Common Factor \\
\hline
2 & 2 & 1 \\
\hline
3 & 3 & 2 \\
\hline
4 & 5 & 2 \\
\hline
5 & 5 & $3\#$ \\
\hline
6--7 & 7 & $3\#$ \\
\hline
8--11 & 11 & $5\#$ \\
\hline
12--13 & 13 & $7\#$ \\
\hline
14--17 & 17 & $5\#$ \\
\hline
18 & 19 & $7\#$ \\
\hline
19 & 19 & $11\#$ \\
\hline
20--23 & 23 & $11\#$ \\
\hline
24--29 & 29 & $13\#$ \\
\hline
30--31 & 31 & $13\#$ \\
\hline
32--37 & 37 & $19*11\#$ \\
\hline
38--41 & 41 & $13\#$ \\
\hline
42 & 43 & $17\#$ \\
\hline
43 & 43 & $19\#$ \\
\hline
44--47 & 47 & $23*17\#$ \\
\hline
48--53 & 53 & $17\#$ \\
\hline
54 & 59 & $29*13\#$ \\
\hline
55--58 & 59 & $29*19*13\#$ \\
\hline
59 & 59 & $29*19\#$ \\
\hline
60--61 & 61 & $31*19*13\#$ \\
\hline
62--67 & 67 & $31*19\#$ \\
\hline
68--71 & 71 & $17\#$ \\
\hline
72--73 & 73 & $37*23*17\#$ \\
\hline
74--79 & 79 & $23\#$ \\
\hline
80--83 & 83 & $41*19\#$ \\
\hline
84--89 & 89 & $31*23\#$ \\
\hline
90--97 & 97 & $23\#$ \\
\hline
98--99 & 101 & $23\#$ \\
\hline
100--101 & 101 & $31*23\#$ \\
\hline
102--103 & 103 & $23\#$ \\
\hline
\end{tabular}
\end{center}
\newpage
|
{
"timestamp": "2012-03-12T01:01:36",
"yymm": "1203",
"arxiv_id": "1203.2083",
"language": "en",
"url": "https://arxiv.org/abs/1203.2083"
}
|
\section{Introduction}\label{intro}
Analogous to classical principal components analysis (PCA), the
projection-pursuit approach to robust PCA is based on finding projections
of the data which have maximal dispersion. Instead of using the
variance as a measure of dispersion, a robust scale estimator $s_{n}$
is used for the maximization problem. This approach was introduced by
\citet{li}, who proposed estimators based on maximizing (or
minimizing) a~robust scale. In this way, given a sample $\bx_i\in
\real
^d$, $1\le i\le n$, the first robust principal component vector is
defined as
\[
\wba= \argmax_{\{\ba\in\real^d\dvtx \ba\trasp\ba=1\}} s_{n}(\ba
\trasp\bx
_{1},\ldots,\ba\trasp\bx_{n}) .
\]
The subsequent principal component vectors are obtained by imposing
orthogonality conditions. In the multivariate setting, \citet{li}
argue that the breakdown point for this projection-pursuit based
procedure is the same as that of the scale estimator $s_{n}$. Later on,
\citet{crouruiz2} derived the influence functions of the resulting
principal components, while their asymptotic distribution was studied
in \citet{cui}. A maximization algorithm for obtaining $\wba$
was proposed in \citet{crouruiz} and adapted for
high-dimensional data in \citet{crfil}.
The aim of this paper is to adapt the projection pursuit approach to
the functional data setting. We focus on functional data that are
recorded over a period of time and regarded as realizations
of a stochastic process, often assumed to be in the $L^2$ space on a
real interval. Various choices of robust scales, including the median
of the absolute deviation {about the median} (\textsc{mad}) and
$M$-estimates of scale are considered and compared.
Classical functional PCA uses the eigenvalues and eigenfunctions of the
sample covariance operator. \citet{dpr} have studied the
asymptotic properties of these sample functional principal components.
\citet{ris} proposed to smooth the principal components
by imposing an additive roughness penalty to the sample variance. The
consistency of this method was subsequently studied by \citet{pez}. Another approach to smoothing the principal
components, proposed in \citet{s} and reviewed in \citet{rasi2},
is based on penalizing the norm rather than the
sample variance, while \citet{bo} considered a
kernel-based approach. More recent work on estimation of the principal
components and the covariance function includes \citet{ger}, \citet{hall},
\citet{hallmulwa} and
\citet{yao}.
The literature on robust principal components in the functional data
setting, though, is rather sparse. To our knowledge, the first attempt
to provide estimators of the principal components that are less
sensitive to anomalous observations was due to \citet{loc},
although their approach is multivariate in nature. \citet{ger08}
studied a fully functional approach to robust estimation of the
principal components by considering a functional version of the
spherical principal components defined in \citet{loc}.
\citet{hyn} give an application of a
robust projection-pursuit approach, applied to smoothed trajectories,
but did not study the properties of their method in detail.
In this paper, we introduce several robust estimators of the principal
components in the functional data setting. Our approach uses a robust
projection-pursuit combined with various smoothing methods. A primary
focus of this paper is to provide a rigorous theoretical foundation for
this approach to robust functional PCA. In particular, we establish
under very general conditions the strong consistency of the our
proposed estimators.
In Section \ref{prop}, the robust estimators of the principal
components, based on both the raw and smoothed approaches, are
introduced. Consistency results and the asymptotic robustness of the
procedure are established in Section~\ref{consist}, while
Fisher-consistency of the related functionals is studied in Section~\ref {fisher}.
Section~\ref{appen} provides conditions under which one
of the crucial assumptions hold. Selection of the smoothing parameters
for the smooth principal components is discussed in Section~\ref{smoothpar}.
The results of a Monte Carlo study are reported in Section~\ref{monte}.
Finally, Section~\ref{concl} contains some concluding
remarks. Most proofs are relegated to the \hyperref[appenA]{Appendix}
and to the technical supplementary material available online; see
\citet{bali3a}. We begin the next section with notation and a review of
some basic concepts which are utilized in later sections.
\section{Preliminaries}\label{prelim}
\subsection{Functional principal components analysis}\label{fpca}
$\!\!\!$Principal components ana\-lysis, which was originally developed for
multivariate data, has been successfully extended to accommodate
functional data, and is
usually referred to as functional PCA. Principal components analysis
for general Hilbert spaces can be described as follows.
Let $X\in\mathcal{H}$ be a random element of a Hilbert space
$\mathcal
H$ defined in $(\Omega,\mathcal{A},P)$. Denote by $\langle\cdot
,\cdot
\rangle$ the inner product in $\mathcal H$ and by $\|\alpha\|
^2=\langle
\alpha,\alpha\rangle$. Assume that~$X$ has finite second moment, that
is, $\esp(\|X\|^2)<\infty$. The bilinear operator $a_X\dvtx\mathcal{H}
\times\mathcal{H} \to\real$ defined as $a_X(\alpha,\beta) =
\cov(\langle\alpha,X\rangle, \langle\beta,X\rangle)$ leads to a
continuous operator. The Riesz representation theorem then implies that
there exists a bounded operator, $\bGa_X\dvtx\mathcal{H}\to\mathcal{H}$,
such that $a_X(\alpha,\beta) = \langle\alpha, \bGa_X \beta\rangle$.
The operator $\bGa_X$ is called the covariance operator of $X$ and is
linear, self-adjoint and continuous.
Although the general situation in which $X\in\mathcal{H}$ is treated
in this paper, to help simplify the
basic ideas, we first consider the case $X\in L^2(\mathcal{I})$ where
$\mathcal{I}\subset\real$ is a finite interval. We take the usual
inner product for
$L^2(\mathcal{I})$, that is, $\langle\alpha, \beta\rangle= \int
_\mathcal{I} \alpha(t) \beta(t)\,dt$ and denote the covariance function
of $X$ by
$\gamma_X(t,s) = \cov(X(t),X(s))$. The corresponding
covariance operator $\bGa_X\dvtx L^2(\mathcal{I})\to L^2(\mathcal{I})$ is
such that $\bGa_X(\alpha)(t)=\int_\mathcal{I} \gamma_X(t,s) \alpha(s)
\,ds$. It is assumed the covariance function satisfies
$\int_\mathcal{I} \int_\mathcal{I} \gamma_X^2(t,s) \,dt \,ds
<\infty
$. Consequently, $\bGa_X$ is a Hilbert--Schmidt operator.
A Hilbert--Schmidt operator has a countable number of eigenvalues, all
of which are real. $\mathcal{F}$ will stand for the Hilbert space of
such operators with inner product defined by $\langle\bGa_1, \bGa
_2\rangle_{\mathcal{F}} =\sum_{j=1}^\infty\langle\bGa_1 u_j, \bGa_2
u_j\rangle$, where $\{u_j \dvtx j\ge1\}$ is any orthonormal basis of
$L^2(\mathcal{I})$. Furthermore, since the covariance operator~$\bGa_X$
is also positive semi-definite, its eigenvalues are nonnegative. As
with symmetric matrices in finite-dimensional Euclidean spaces, one can
choose the eigenfunctions of a Hilbert--Schmidt operator so that they
form an orthonormal basis for $L^2(\mathcal{I})$. Let $\{\phi_j\dvtx j\ge
1\}$ and
$\{\lambda_j\dvtx j\ge1\}$ be respectively an orthonormal basis of
eigenfunctions and their corresponding eigenvalues for the covariance
operator~$\bGa_X$,
with $\lambda_j\ge\lambda_{j+1}$. The spectral value decomposition for
$\bGa_X$ can then be expressed as $\bGa_X = \sum_{j=1}^\infty
\lambda_j
\phi_j \otimes\phi_j$,
with $\otimes$ being the tensor product, or equivalently $\gamma_X(t,s)
= \sum_{j=1}^\infty\lambda_j \phi_j(t)\phi_j(s)$, with\vspace*{1pt}
$ \sum_{j=1}^\infty\lambda^2_j = \|\bGa_X\|_\mathcal{F}^2 = \int
_\mathcal{I}\int_\mathcal{I} \gamma_X^2(t,s)\,dt \,ds$. The $j$th principal
component variable is then defined as
$Z_j = \langle\phi_j, X \rangle$, which leads to the Karhunen--Lo\`
{e}ve expansion $X(t) =\mu(t)+ \sum_{j = 1}^{\infty} Z_j \phi_j(t)$,
with $\mu(t)=\esp(X(t))$ and the $Z_j$'s being uncorrelated and having
variances $\lambda_j$ in descending order.
In general, for $Y = \langle\alpha,X\rangle$, which is a linear
function of the process $\{X(s)\}$, we have $\var(Y) = \langle\alpha
,\bGa_X \alpha\rangle$. An important optimality
property of the first principal component variable is that it can be
defined as the variable $Z_1 = \langle\alpha_1,X\rangle$ such that
\begin{equation}\label{MAX}
\var(Z_1)=\sup_{\{\alpha\dvtx \|\alpha\|=1\}}
\var(\langle\alpha,X\rangle)=\sup_{\{\alpha\dvtx \|
\alpha\|=1\}}
\langle\alpha,\bGa_X \alpha\rangle.
\end{equation}
Any solution to (\ref{MAX}), that is, any $\alpha$ for which the
supremum is obtained, corresponds
to an eigenfunction associated with the largest eigenvalue of the
covariance operator $\bGa_X$, that is, $\alpha_1=\phi_1$ and $\var
(Z_1)=\lambda_1$. If
$\lambda_1 > \lambda_2$, then~$\alpha_1$ is unique up to a sign change.
As in the multivariate setting, the other principal components can be
obtained successively
via (\ref{MAX}), but under the orthogonality condition that $\langle
\alpha_j, \alpha_k \rangle=0$ for $j < k$.
\subsection{Scale functionals and estimates} \label{scale}
The basic idea underlying our approach is to view principal components
as in (\ref{MAX}), but with the variance replaced by a robust scale functional.
We first recall the definition of a~scale functional. Denote by
$\mathcal{G}$ the set of all univariate distributions. A scale
functional $\sigma_{\rob}\dvtx\mathcal{G} \to[0,+\infty)$
is one which is location invariant and scale equivariant, that is, if
$G_{a,b}$ stands for the distribution of $aY+b$ when $Y\sim G$, then,
$\sigma_{\rob}(G_{a,b})=|a|\sigma_{\rob}(G)$,
for all real numbers $a$ and $b$. Two well-known examples of scale
functionals are the standard deviation, $\mbox{\textsc{sd}}(G)=\{\esp
(Y-\esp(Y))^2\}^{1/2}$, where $Y\sim G$,
and the median absolute deviation about the median, $\mbox{\textsc{mad}}(G)= c
\median(|Y-\median(Y)| )$. The normalization constant $c$,
used in the \textsc{mad}, can be chosen so that its
empirical or sample version is consistent for a scale parameter of
interest. Typically, one chooses $c=1/\Phi^{-1}(0.75)$ so that the
\textsc{mad} equals the standard deviation at a normal distribution.
The breakdown points, a measure of global robustness, for the standard
deviation and the \textsc{mad} are $0$ and $1/2$, respectively. The
\textsc{mad}, however, has a discontinuous influence function, which
reflects some local instability. Furthermore, the empirical version of
the \textsc{mad} is known to be fairly inefficient at the normal and
other distributions; see \citet{hub}.
In the finite-dimensional setting, as reported in \citet{cui}
the impact of a discontinuous influence function on the efficiency of
the estimators of the principal directions is even more dramatic
covariance function.
A broader class of robust scale functionals, which includes as special
cases both the \textsc{sd} and the \textsc{mad}, are the $M$-scale
functionals. An $M$-scale
functional with a bounded and continuous score function can have both a
high breakdown point and a continuous and bounded influence function.
Also, their empirical versions,
the $M$-estimates of scale, can be tuned to have good efficiency over a
broad range of distributions. Given a location parameter $\mu$, an
$M$-scale functionals $\sigma_M(G)$
with a continuous score function $\chi\dvtx\real\to\real$ can be defined
to be a solution to the equation
\begin{equation}\label{sigmarob}
\esp\biggl[ \chi\biggl( \frac{ Y-\mu}{\sigma_{\rob}(G)} \biggr) \biggr]
=\delta.
\end{equation}
Given a location statistic $\widehat{\mu}_n$, the corresponding
$M$-estimate of scale is then a solution $\widehat{\sigma}_n$ to the
$M$-estimating equation
\begin{equation}\label{snrob}
\frac{1}{n}\sum_{i=1}^{n}\chi\biggl( \frac{Y_{i} -\widehat{\mu
}_n}{\widehat{\sigma}_{ n} } \biggr) = \delta.
\end{equation}
If the score function is discontinuous, as is the case with the \textsc
{mad}, then a slight modification to (\ref{sigmarob}) and (\ref{snrob})
is needed; see \citet{mart}.
Typically, the score function $\chi$ is even with $\chi(0) = 0$,
nondecreasing on $\real_+$ and with $0< \sup_{x\in\real}\chi
(x)=\chi
(+\infty)=\lim_{x\to+\infty} \chi(x)$.
When $\chi(+\infty)=2\delta$, the $M$-estimate of scale has a 50\%
breakdown point, and by choosing $\chi$ properly one can also obtain a
highly efficient estimate; see \citet{croux}.
One such popular choice, and the one we use in our Monte Carlo study,
is the score function introduced by \citet{beatuk}, namely
\begin{equation}\label{funcionBT}
\chi_c( y) = \min\bigl(3 ( y/c )^2 - 3 ( y/c)^4
+ ( y/c )^6, 1\bigr)
\end{equation}
with $c$ being a tuning constant chosen so that the corresponding
$M$-estimator of scale is consistent for a scale parameter of interest.
For example, the choice $c = 1.56$ when $\delta=1/2$ ensures
that the $M$-scale functional is Fisher-consistent at the normal
distribution and has a 50\% breakdown point.
For continuous and nondecreasing score functions $\chi$, the solutions
to (\ref{sigmarob}) and (\ref{snrob}) are unique, and the simple
re-weighting algorithm
\[
\bigl\{\widehat{\sigma}_{ n}^{(k+1)}\bigr\}^2=\frac{1}{n \delta}\sum
_{i=1}^{n} w
\biggl( \frac{Y_{i} -\widehat{\mu}}{\widehat{\sigma}_{n}^{(k)} }
\biggr)
(Y_{i} -\widehat{\mu})^2,
\]
where $w(y)=\chi(y)/y^2$ for $y\ne0$ and $w(0)=\chi^{\prime\prime
}(0)$, is known to always converge to the unique solution of (\ref
{snrob}) regardless of the initial value $\widehat{\sigma}_{ n}^{(0)}$.
In practice, the initial value $\widehat{\sigma}_{ n}^{(0)}$ is usually
taken to be the \textsc{mad}. A discussion on the convergence of the
algorithm can be found in \citet{maro}.
For a bounded score function $\chi$, if the solution $\sigma_{\rob
}(G_0)$ of (\ref{sigmarob}) is unique, as it is the case when $\chi$ is
continuous and nondecreasing, then the functional~$\sigma_{\rob}$ is
weakly continuous at $G_0$. Weakly continuity of $\sigma_{\rob}$ at
$G_0$, that is, continuity with respect to the weak topology in
$\mathcal G$ which is given by the Prohorov metric, and consistency in
a neighborhood of $G_0$ entails robustness at $G_0$. For details, see
\citet{hub} and \citet{ham}.
\section{The estimators}\label{prop}
We consider several robust approaches in this section and define them
on a separable Hilbert space $\mathcal H$, keeping in mind that the
main application will be $\mathcal{H}=L^2(\mathcal{I})$.
From now on and throughout the paper, $\{X_i\dvtx 1 \leq i \leq n\}$ denote
realizations of the stochastic process $X\sim P$ in a separable Hilbert
space~$\mathcal{H}$. Thus, $X_{i} \sim P$ are independent stochastic
processes that follow the same law. This independence condition could
be relaxed, since we only need the strong law of large numbers to hold
in order to guarantee the results in this paper.
\subsection{Raw robust projection-pursuit approach}\label{rawpp}
Based on property (\ref{MAX}) of the first principal component and
given $\sigma_{\rob}(F)$ a robust scale functional, the raw (meaning
unsmoothed) robust functional principal component directions are
defined as
\begin{equation}\label{MAXROB}
\cases{
\displaystyle \phi_{\rob,1}(P) =\argmax_{\|\alpha\|=1}\sigma_{\rob}
(P[\alpha
]),\cr
\displaystyle \phi_{\rob,m}(P)= \argmax_{\|\alpha\|=1, \alpha\in\mathcal{B}_m}
\sigma_{\rob}(P[\alpha]),\qquad 2 \leq m,}
\end{equation}
where $P[\alpha]$ stands for the distribution of $\langle\alpha
,X\rangle
$ when $X\sim P$ and $\mathcal{B}_m=\{\alpha\in\mathcal{H}\dvtx
\langle
\alpha, \phi_{\rob,j}(P) \rangle=0, 1\le j\le m-1\}$. We will denote
the $m$th largest principal value by
\begin{equation}\label{lamrob}
\lambda_{\rob,m}(P)=\sigma_{\rob}^2(P[\phi_{\rob,m}]) = \max_{\|
\alpha\| =1, \alpha\in\mathcal{B}_m} \sigma^2_{\rob}(P[\alpha] ) .
\end{equation}
Since the unit ball is weakly compact, the maximum above is attained if
the scale functional $\sigma_{\rob}$ is (weakly) continuous.
Next, denote by $s^2_{n}\dvtx\mathcal{H}\to\real$ the function
$s^2_{n}(\alpha)=\sigma_{\rob}^2(P_n[\alpha])$, where\break
$\sigma
_{\rob}(P_n[\alpha])$ stands for the functional $\sigma
_{\rob}$
computed at the empirical distribu\-tion of $\langle\alpha,X_1\rangle,
\ldots, \langle\alpha,X_n\rangle$. Analogously, the mapping $\sigma
\dvtx\mathcal{H}\to\real$ stands for $\sigma(\alpha)=\sigma_{\rob
}(P[{\alpha}])$. The components in (\ref{MAXROB}) will be estimated
empirically by
\begin{equation}\label{estFPC}
\cases{
\wphi_{\raw,1} =\displaystyle \argmax_{\|\alpha\|=1}s_n(\alpha),\cr
\wphi_{\raw,m}= \displaystyle \argmax_{\alpha\in\widehat{\mathcal{B}}_m}
s_n(\alpha),\qquad 2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_m = \{ \alpha\in\mathcal{H}\dvtx \|
\alpha\|
=1, \langle\alpha, \wphi_{\raw,j} \rangle=0 , \forall 1\le
j \le
m-1\}$. The
estimators of the principal values are then computed as
\begin{equation}\label{esttamanno}
\wlam_{\raw,m}= s^2_{n}(\wphi_{\raw,m}),\qquad 1 \leq m .
\end{equation}
\subsection{Smoothed robust principal components}\label{smoothpp}
Sometimes instead of raw functional principal components, smoothed ones
are of interest. The advantages of smoothed functional PCA are well
documented; see, for instance, \citet{ris} and \citet{rasi2}. One compelling argument is that smoothing is a
regularization tool that might reveal more interpretable and
interesting features of the modes of variation for functional data. As
noted in the \hyperref[intro]{Introduction}, \citet{ris} and \citet{s} proposed two smoothing approaches by penalizing the variance and
the norm, respectively.
To be more specific, \citet{ris} estimate the first principal
component by maximizing over $\|\alpha\|=1$, the objective function
$\widehat{\var}(\langle\alpha,X\rangle)- \rho
\lceil\alpha,\alpha\rceil$, where $\widehat{\var}$ stands for the
sample variance and $\lceil \alpha
,\beta\rceil\!=\!\int_0^1\!\alpha^{\prime\prime}(t)\beta^{\prime \prime
}(t)\,dt$. Consistency for these estimators was established by
\citet{pez}.
Another regularization method proposed by \citet{s} is to penalize
the roughness through a norm defined via the penalized inner product,
$\langle\alpha,\beta
\rangle_{\tau}=\langle\alpha,\beta\rangle+\tau\lceil\alpha ,\beta
\rceil$. The smoothed first direction $\wphi_1$ is then the one that
maximizes $\widehat{\var}(\langle\alpha,X\rangle)$ over $\|
\alpha\|_{\tau}=1$. Consistency of these estimators is also established
in \citet{s} under the assumption that $\phi_j$ has finite
roughness, that is, $\lceil\phi_j,\phi_j \rceil<\infty$.
Clearly the smoothing parameters $\rho$ and $\tau$ need to converge to
$0$ in order to get consistency results.
Let us consider $\mathcal{H}_{\smooth}$, the subset of ``smooth
elements'' of $\mathcal{H}$. In order to obtain consistency results, we
will assume that $\phi_{\rob,j}(P)\in\mathcal {H}_{\smooth }$. Let
$D\dvtx\mathcal{H}_{\smooth}\,{\rightarrow}\,\mathcal{H}$~be a linear
operator, which we will refer to as the ``differentiator.'' Using $D$,
we define the symmetric positive semidefinite bilinear form \mbox{$
\lceil\cdot, \cdot\rceil\dvtx\mathcal{H}_{\smooth}\times\mathcal
{H}_{\smooth} \rightarrow\real$}, where $\lceil\alpha, \beta\rceil=
\langle D\alpha, D\beta\rangle$. The ``penalization operator'' is then
defined as $\Psi\dvtx \mathcal{H}_{\smooth} \rightarrow \mathbb{R}$,
$\Psi(\alpha) = \lceil\alpha,\alpha\rceil$, and the penalized inner
product as $\langle\alpha,\beta\rangle_{\tau }=\langle
\alpha,\beta\rangle+\tau\lceil\alpha,\beta\rceil$. Therefore,
$\|\alpha\|_{\tau}^2=\|\alpha\|^2+\tau\Psi(\alpha)$. As in
\citet{pez}, we will assume that the bilinear form is closable.
\begin{remark}\label{remark31}
The most common setting for functional data is to choose $\mathcal{H} =
L^2(\mathcal{I})$, $ \mathcal{H}_{\smooth} = \{ \alpha\in
L^2(\mathcal{I}), \alpha$ is twice\vspace*{1pt} differentiable, and $
\int_\mathcal{I} (\alpha^{\prime\prime }(t))^2\,dt<\infty\} $,
$D\alpha= \alpha^{\prime\prime}$ and $ \lceil \alpha, \beta\rceil=
\int_\mathcal{I} \alpha^{\prime\prime }(t)\beta ^{\prime\prime}(t) \,dt
$ so that $\Psi(\alpha) = \int_\mathcal{I} (\alpha^{\prime\prime}(t))^2
\,dt$.
\end{remark}
Let $\sigma_{\rob}(F)$ be a robust scale functional and define
$s_{n}(\alpha)$ and $\sigma(\alpha)$ as in Section~\ref{rawpp}.
Then we
can adapt the classical procedure by defining the smoothed robust
functional principal direction estimators either:
\begin{longlist}[(a)]
\item[(a)]
by penalizing the norm as
\begin{equation}\label{estFPCsmooth2}
\cases{
\displaystyle \wphi_{\smoothn, 1} =\argmax_{\|\alpha\|_{\tau
}=1}s_n^2(\alpha
)=\argmax_{\beta\ne0}
\frac{s_n^2(\beta)}
{\langle\beta,\beta\rangle+\tau\Psi(\beta)}
\vspace*{2pt}\cr
\displaystyle \wphi_{\smoothn, m}= \argmax_{\alpha\in\widehat{\mathcal{B}}_{m,
\tau,\smoothn}} s_n^2(\alpha),\qquad 2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_{m, \tau, \smoothn} = \{ \alpha\in
\mathcal{H}\dvtx\|\alpha\|_{\tau}=1, \langle\alpha, \wphi
_{\smoothn, j}
\rangle_{\tau}=0 , \forall 1\le j \le m-1\}$,
\item[(b)] or by penalizing the scale as
\begin{equation}\label{estFPCsmooth1}
\cases{\displaystyle
\wphi_{\smooths, 1}=\argmax_{\|\alpha\|=1}\{s_n^2(\alpha)-
\rho\Psi(\alpha)\},\vspace*{2pt}\cr
\displaystyle \wphi_{\smooths, m}=\argmax_{\alpha\in\widehat{\mathcal
{B}}_{m,\smooths}} \{s_n^2(\alpha)- \rho\Psi(\alpha)\},\qquad
2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_{m,\smooths} = \{ \alpha\in\mathcal
{H}\dvtx\|\alpha\|=1, \langle\alpha, \wphi_{\smooths, j} \rangle=0
,
\forall 1\le j \le m-1\}$.
\end{longlist}
The corresponding principal value estimators are respectively defined as
\begin{equation}\label{lamFPCsmooth}
\wlam_{\smooths,m} = s^2_{n}(\wphi_{\smooths,m}) \quad\mbox{and}\quad
\wlam_{\smoothn,m} = s^2_{n}(\wphi_{\smoothn,m}).
\end{equation}
\subsection{Sieve approach for robust functional principal components}
\label{sievepp}
Another approach, motivated by using $B$-splines as a smoothing tool,
is to consider the method of sieves. The method of sieves involves
approximating an infinite-dimensional parameter space $\Theta$ by a
sequence of finite-dimensional parameter spaces $\Theta_n$, which
depend on the sample size $n$, and then estimate the parameters on the
spaces $\Theta_n$ rather than $\Theta$.
Let $\{\delta_i\}_{i\ge1}$ be a basis of $\mathcal{H}$ and define
$\mathcal{H}_{p_n}$ as the linear space spanned by $\delta_1,\ldots,
\delta_{p_n}$ and by
$\mathcal{S}_{p_n}=\{\alpha\in\mathcal{H}_{p_n}\dvtx
\|\alpha\|=1\} $, that is,
\[
\mathcal{H}_{p_n}=\Biggl\{\alpha\in\mathcal{H}\dvtx\alpha=\sum
_{j=1}^{p_n}a_j \delta_j \Biggr\}
\]
and $
\mathcal{S}_{p_n}=\{\alpha\in\mathcal{H}\dvtx\alpha=\sum
_{j=1}^{p_n}a_j \delta_j$, such that
$\|\alpha\|^2=\sum
_{j=1}^{p_n}\sum
_{s=1}^{p_n}a_j a_s \langle\delta_j, \delta_s\rangle=1\}$.
Note that $\mathcal{S}_{p_n}$ approximates the unit sphere $\mathcal
{S}=\{\alpha\in\mathcal{H}\dvtx\|\alpha\|=1\}$. When $\{\delta_i\}
_{i\ge
1}$ is an orthonormal basis, $\|\alpha\|^2=\sum_{j=1}^{p_n}a_j^2=\ba
\trasp\ba$ where $\ba=(a_1,\ldots,a_{p_n})\trasp$, hence, the norm of
$\alpha$ equals the Euclidean norm of the vector~$\ba$. Define the
robust sieve estimators of the principal components as
\begin{equation}\label{sieveFPC}
\cases{\displaystyle \wphi_{\sieve, 1} =\argmax_{\alpha\in\mathcal
{S}_{p_n}}s_n(\alpha
),\vspace*{2pt}\cr
\displaystyle \wphi_{\sieve, m}= \argmax_{\alpha\in\widehat{\mathcal
{B}}_{n,m}} s_n(\alpha),\qquad 2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_{n,m} = \{ \alpha\in\mathcal
{S}_{p_n}\dvtx
\langle\alpha, \wphi_{\sieve,j} \rangle=0 , \forall 1\le j
\le
m-1\}$, and let the principal value estimators be
\begin{equation}\label{sievelamFPC}
\wlam_{\sieve,m}=s_n^2(\wphi_{\sieve,m}).\vadjust{\goodbreak}
\end{equation}
Some of the frequently used bases for functional data are the Fourier,
polynomial, spline and wavelet bases; see, for instance,
\citet{rasi2}.
To the best of our knowledge, the above sieve approach is new to
functional principal component analysis, even if one considers the
classical sieve estimators,
that is, when $\sigma_{\rob}$ in (\ref{sieveFPC}) is the standard deviation.
\subsection{A unified formulation for the robust projection pursuit approaches}
To help formulate a unified approach to the different estimators
considered in Sections \ref{smoothpp}, \ref{smoothpp} and \ref
{sievepp}, let the products $\rho\Psi(\alpha)$ or $\tau\Psi(\alpha
)$ be
defined as $0$ when $\rho=0$ or $\tau=0$, respectively, even when
$\alpha\notin\mathcal{H}_{\smooth}$ for which case $\Psi(\alpha
)=\infty$. Moreover, when $p_n=\infty$, define $\mathcal{H}_{p_n}=
\mathcal{H}$.
All the projection pursuit estimators considered in the previous
subsections then can be viewed as special cases of the following
general class of estimators:
\begin{equation}\label{estFPCgeneral}
\cases{\displaystyle
\wphi_{1}=\argmax_{\alpha\in\mathcal{H}_{p_n},\|\alpha\|
_{\tau
}=1}\{s_n^2(\alpha)- \rho\Psi(\alpha)\},
\vspace*{2pt}\cr
\displaystyle \wphi_{m}=\argmax_{\alpha\in\widehat{\mathcal{B}}_{m, \tau}}
\{s_n^2(\alpha)- \rho\Psi(\alpha)\},\qquad 2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_{m, \tau} = \{ \alpha\in\mathcal
{H}_{p_n}\dvtx\|\alpha\|_{\tau}=1, \langle\alpha, \wphi_{ j}
\rangle
_{\tau}=0 , \forall 1\le j \le m-1\}$.
With this\vspace*{1pt} definition and by taking $p_n=\infty$, the raw estimators
are obtained when $\rho=\tau=0$, while $ \wphi_{\smoothn, m}$ and $
\wphi_{\smooths, m}$ correspond to $\rho=0$ and $\tau=0$, respectively.
On the other hand, the sieve estimators correspond a~finite choice for
$p_n$ and $\tau=\rho=0$.
\section{Consistency results}\label{consist}
In this section, we show that under mild conditions the functionals
$\phi_{\rob,m}(P)$ and $\lambda_{\rob,m}(P)$ defined through (\ref
{MAXROB}) and (\ref{lamrob}) are weakly continuous. Moreover, we state
conditions that guarantee the consistency of the estimators defined in
Section \ref{prop}. Proofs for this section can be found in the
\hyperref[appenA]{Appendix} and in the supplemental article
[\citet{bali3a}].
To derive the consistency of the proposed estimators, we need the
following assumptions:
\begin{longlist}[(S0)]
\item[(S0)] For some $q\ge2$ and $1 \leq j \leq q$, $\phi
_{\rob,j}(P)$ are unique up to a sign change.
\item[(S1)] $\sigma\dvtx\mathcal{H}\to\real$ is a weakly
continuous function, that is, continuous with respect to the weak
topology in $\mathcal H$.
\item[(S2)] ${\sup_{\| \alpha\| = 1}}|s_n^2(\alpha) -
\sigma^2(\alpha)|\convpp0$.
\end{longlist}
Note that condition (S0) holds if and only if $\lambda_{\rob
,1}(P)> \cdots> \lambda_{\rob, q+1}(P)$.\vspace*{8pt}
\textit{Some remarks.}
(i) (S1) holds when the scale functional $\sigma_{\rob}$ is a
continuous functional (with respect to the weak topology under the
Prohorov distance).
This is because $\alpha_k$ converges weakly to $\alpha$, which implies
$\langle\alpha_k ,X\rangle\convweak\langle\alpha,X\rangle$ and
hence $\sigma_{\rob}(P[\alpha_k])\to\sigma_{\rob}(P[\alpha])$.
For the
case when the scale functional is the standard deviation,
and the underlying probability $P$ has a compact covariance operator
$\bGa_{X}$, we see from the relationship $\sigma^2(\alpha) = \langle
\alpha, \bGa_{X} \alpha\rangle$ that condition (S1) holds, even
though the standard deviation itself is not a~weakly continuous functional.
(ii) Since there exists a metric $d$ generating the weak topology in
$\mathcal H$ and that the closed ball $\mathcal{V}_r=\{\alpha\dvtx \|
\alpha
\|\le r\}$ is weakly compact, we see that (S1) implies that
$\sigma(\alpha)$ is uniformly continuous with respect to the metric $d$
and hence, with respect to the weak topology, over $\mathcal{V}_r$.
(iii) Assumption (S2) holds for the classical estimators based
on the sample variance since the empirical covariance operator,
$\widehat{\bGa}$, is consistent in the unit ball. Indeed, as shown in
\citet{dpr}, $\| \widehat{\bGa} - \bGa_{X}\| \convpp0$,
which entails that
$
{\sup_{\| \alpha\| = 1}}|s_n^2(\alpha) - \sigma^2(\alpha
)|
\leq\| \widehat{\bGa} - \bGa_{X} \| \convpp0.
$
However, this assumption may seem harder to verify for other scale
functionals since the unit sphere $\mathcal{S}=\{\|\alpha\|=1\}$ is not
compact, and $s_n^2(\alpha)$ is usually not defined through a
covariance operator estimator. To be more precise, even under some
conditions to be considered in Section \ref{fisher}, there exists a
compact operator $\bGa$ such that $\sigma(\alpha)=\langle\alpha,
\bGa
\alpha\rangle$, $s_n^2(\alpha)$ cannot be expressed as $\langle
\alpha,
\bGa_n\alpha\rangle$ for some consistent estimator $\bGa_n$ of
$\bGa$.
Corollary \ref{corollary61} in Section \ref{appen} establishes that
(S2) holds for any scale functional $\sigma_{\rob}$ that is continuous
with respect to the weak topology.
The following lemma, whose proof can be found in Section B
of the technical supplemental article given in
\citet{bali3a}, is useful for deriving the consistency and continuity of
the principal direction estimators. In this lemma and in the subsequent
theorems, it should be noted that $\langle\wphi,\phi\rangle^2
\rightarrow1$ implies,\vspace*{1pt} under the same mode of convergence, that the
sign of $\wphi$ can be chosen so that $\wphi\rightarrow\phi$.
For the sake of simplicity, denote by $\lambda_{\rob,j}=\lambda
_{\rob
,j}(P)$ and $\phi_{\rob,j}=\phi_{\rob,j}(P)$.
\begin{lemma}\label{lemma41}
Let $\wphi_m\in\mathcal
{S}$ be such that $\langle\wphi_m, \wphi_j\rangle\stackrel{\mathit{a.s.}}{\longrightarrow}0$ for
$j\ne
m$ and assume that \textup{(S0)} and \textup{(S1)} hold. Then:
\begin{longlist}[(b)]
\item[(a)] If $\sigma^2(\wphi_1)\stackrel{\mathit{a.s.}}{\longrightarrow}\sigma^2(\phi_{\rob,1})$, then
$\langle\wphi_1,\phi_{\rob,1}\rangle^2\stackrel{\mathit{a.s.}}{\longrightarrow}1$.
\item[(b)] Given $2 \leq m \leq q$, if $\sigma^2(\wphi_m)\stackrel{\mathit{a.s.}}{\longrightarrow}
\sigma
^2(\phi_{\rob,m})$ and $\wphi_s \stackrel{\mathit{a.s.}}{\longrightarrow}\phi_{\rob,s}$, for $1\le
s\le
m-1$, then $\langle\wphi_m,\phi_{\rob,m}\rangle^2\stackrel{\mathit{a.s.}}{\longrightarrow}1$.
\end{longlist}
\end{lemma}
Let $d_{\pr}(P,Q)$ stand for the Prohorov distance between the
probability measures $P$ and $Q$. Thus, $P_n \convweak P$ if and only
if $d_{\pr}(P_n, P) \to0$. Theorem~\ref{theorem41} below establishes
the continuity of the functionals defined in (\ref{MAXROB}) and~(\ref
{lamrob}), and hence the asymptotic robustness of the estimators
derived from them, as defined in \citet{ham}. This can be seen just
by replacing almost sure convergence by convergence in its statement.
As it will be shown in Section \ref{appen}, the uniform convergence
required in assumption (ii) below is satisfied, for instance, if
$\sigma
_{\rob}$ is a continuous scale functional when $P_n \convweak P$.
To accommodate data driven smoothing parameters a more general
framework is considered in Theorem \ref{theorem41}, which allows for
the smoothing parameters $\tau_n$ and $\rho_n$ to be random, such that
$\tau_n\convpp0$ and $\rho_n\convpp0$.
\begin{theorem}\label{theorem41}
Let $P_n$ be a sequence
of probability measures and $\tau=\tau_n\ge0$, $\rho=\rho_n\ge0$ be
random smoothing parameters. Denote\vspace*{1pt} by $\sigma_n^2(\alpha)=\sigma
_{\rob
}^2(P_{n}[\alpha])$ and define $ \wlam_{ m}=\sigma_n^2(\wphi_{m})$ with
\begin{equation}\label{estFPC2}
\cases{\displaystyle \wphi_{1} =\argmax_{\|\alpha\|_{\tau}=1} \{
\sigma_n^2(\alpha
)- \rho\Psi(\alpha)\},
\vspace*{2pt}\cr
\displaystyle \wphi_{ m}=\argmax_{\alpha\in\widehat{\mathcal{B}}_{m, \tau}}
\{\sigma_n^2(\alpha)- \rho\Psi(\alpha)\},\qquad 2 \leq m,}
\end{equation}
where $\widehat{\mathcal{B}}_{m, \tau} = \{ \alpha\in\mathcal
{H}\dvtx \|
\alpha\|_{\tau}=1, \langle\alpha, \wphi_{j} \rangle_{\tau}=0
,
\forall 1\le j \le m-1\}$. Let $P$ be a~probability measure
satisfying \textup{(S0)}. Assume that:
\begin{longlist}
\item \textup{(S1)} holds;
\item ${\sup_{\|\alpha\|=1}}|\sigma_n^2(\alpha) - \sigma
_{\rob
}^2(P[\alpha])|\convppp0$;\vspace*{1pt}
\item $\tau_n\convppp0$ and $ \rho_n \convppp0$;
\item moreover, if $\tau_n>0 $ or $ \rho_n> 0 $, for all $n\ge
n_0$, assume that $\phi_{\rob,j} \in\mathcal{H}_{\smooth}$, that is,
$\Psi( \phi_{\rob,j})<\infty$, for all $ 1 \le j\le q$.
\end{longlist}
Then:
\begin{longlist}[(d)]
\item[(a)] $ \wlam_{1}\convppp\lambda_{\rob,1}$ and $\sigma
^2(\wphi
_{1})\convppp\sigma^2(\phi_{\rob,1})$. Moreover, $\rho\Psi(\wphi_{1})
\convppp0$ and $\tau\lceil\wphi_{1},\wphi_{1} \rceil\convppp0$, and
so $\|\wphi_{1}\| \convppp1$;\vspace*{1pt}
\item[(b)] $\langle\wphi_{1},\phi_{\rob,1}\rangle^2\convppp1$;
\item[(c)] for any $2 \leq m \leq q$, if $\wphi_{\ell} \convppp\phi
_{\rob
,\ell}$, $\tau\Psi(\wphi_{ \ell}) \convppp0$ and $\rho\Psi(\wphi_{
\ell}) \convppp0$ for $1\le\ell\le m-1$, then $ \wlam_{m}\convppp{
\sigma^2(\phi_{\rob,m})}$ and $\sigma^2(\wphi_{m})\convppp\sigma
^2(\phi
_{\rob,m})$. Moreover, $\rho\Psi(\wphi_{m}) \convppp0$, $\tau\Psi
(\wphi_{ m}) \convppp0$ and so, $\|\wphi_{m}\| \convppp1$;
\item[(d)] for $1\leq m \leq q$, $\langle\wphi_{m},\phi_{\rob
,m}\rangle
^2\convppp1$.
\end{longlist}
\end{theorem}
Note that assumption (ii) corresponds to (S2) when $P_n$ is
the empirical probability measure. On the other hand, when $\sigma
_{\rob
}(\cdot)$ is a continuous scale functional, Theorem \ref{theorem62}
implies that (ii) holds whenever \mbox{$d_{\pr}(P_n,P)\convpp0$}. Moreover,
if $\sigma_{\rob}(\cdot)$ is a continuous scale functional and $P$
satisfies (S0), Theorem \ref{theorem41} entails the continuity
of the functionals $\phi_{\rob,j}(\cdot)$ and $\lambda_{\rob
,j}(\cdot)$
at~$P$, for $1\le j\le q$, and so the proposed estimators are
qualitatively robust and consistent. In particular, the estimators are
robust at any elliptical distribution $\mathcal{E}(\mu, \bGa)$, as
defined in Section \ref{fisher}, such that the largest $q+1$
eigenvalues of the operator $\bGa$ are all distinct.
Theorem \ref{theorem41} establishes the consistency of the raw
estimators of the principal components under (S0) to (S2)
by taking $\rho=\tau=0$. It also shows that proposals (\ref
{estFPCsmooth2}) and (\ref{estFPCsmooth1}) give consistent estimators
if $\phi_{\rob,j} \in\mathcal{H}_{\smooth}$, $1 \le j \le q$.
In\vspace*{1pt}
\citet{bali2}, it is shown that the estimators $\wphi_{\smoothn,m}$ and
$\wlam_{\smoothn,m}$ defined in (\ref{estFPCsmooth2}) and (\ref
{lamFPCsmooth}) are still consistent if $\phi_{\rob,j} \in\overline
{\mathcal{H}}_{\smooth}$, $1 \le j \le q$, where $\overline{\mathcal
{H}}_{\smooth}$ stands for the closure of $\mathcal{H}_{\smooth}$. The
condition $\phi_{\rob,j} \in\overline{\mathcal{H}}_{\smooth}$
generalizes the assumption of smoothness, $\phi_{\rob,j} \in\mathcal
{H}_{\smooth}$, required in \citet{s} and holds, for example,
when $\mathcal{H}_{\smooth}$ is a dense subset of $\mathcal{H}$.
Theorem \ref{theorem42} establishes the consistency of the estimators
of the principal directions defined through the sieve approach given in
(\ref{sieveFPC}). Below we give a
separate statement for the consistency of the sieve estimators to avoid
imposing additional burdensome assumptions, such as smoothness
conditions for the basis elements, whenever either
$\tau\ne0$ or $\rho\ne0$ in (\ref{estFPCgeneral}). Its proof is
relegated to the Section C of the technical supplement
[\citet{bali3a}].
\begin{theorem}\label{theorem42}
Let $\wphi_{\sieve,m}$
and $\wlam_{\sieve,m}$ be the estimators defined in (\ref{sieveFPC})
and (\ref{sievelamFPC}), respectively. Under \textup{(S0)} to
\textup{(S2)}, if $p_n\to\infty$, then:
\begin{longlist}[(c)]
\item[(a)] $ \wlam_{\sieve,1}\convppp{ \sigma^2(\phi_{\rob,1})}$ and
$\sigma^2(\wphi_{\sieve,1})\convppp\sigma^2(\phi_{\rob,1})$.
\item[(b)] Given $2 \leq m \leq q$, if $\wphi_{\sieve,\ell} \convppp
\phi
_{\rob,\ell}$, for $1\le\ell\le m-1$, then $ \wlam_{\sieve
,m}\convppp{
\sigma^2(\phi_{\rob, m})}$ and $\sigma^2(\wphi_{\sieve,m})\convppp
\sigma
^2(\phi_{\rob,m})$.
\item[(c)] For $1\leq m \leq q$, $\langle\wphi_{\sieve,m},\phi
_{\rob
,m}\rangle^2\convppp1$.
\end{longlist}
\end{theorem}
\section{Fisher-consistency under elliptical distributions}\label{fisher}
The results in Section~\ref{consist} ensure that, under mild
conditions, the estimates of the principal directions converge almost
surely to $\phi_{\rob,m}$ defined in (\ref{MAXROB}). An important point
to highlight is what the functions $\phi_{\rob,m}$ represent, at least
in some particular situations. This section focuses on showing that,
for the functional elliptical families defined in \citet{bali},
the functionals~$\phi_{\rob,m}(P)$ and~$\lambda_{\rob,m}(P)$
have a simple interpretation. In particular, our results hold for the
functional elliptical family, but are not restricted to it. We recall
here their definition for the sake of completeness.
Let $X$ be a random element in a separable Hilbert space $\mathcal H$
and $\mu\in\mathcal{H}$. Let $\bGa\dvtx\mathcal{H} \rightarrow
\mathcal
{H}$ be a self-adjoint, positive semidefinite and compact operator. We
will say that $X$ has an elliptical distribution with parameters $(\mu,
\bGa)$, denoted as $X\sim\mathcal{E}(\mu, \bGa)$, if for any linear
and bounded operator $A\dvtx\mathcal{H} \rightarrow\real^d$, $AX$~has a
multivariate elliptical distribution with parameters $A\mu$ and $A\bGa
A^* $, that is, $AX\sim\mathcal{E}_{d}(A\mu, A\bGa A^*)$, where
$A^*\dvtx
\mathbb{R}^p\rightarrow\mathcal{H}$ stands for the adjoint operator of
$A$. As in the finite-dimensional setting, if the covariance operator,
$\bGa_X$, of~$X$ exists, then $\bGa_X = {a} \bGa$, for some $a\in
\real$.
The elliptical distributions in $\mathcal{H}$ include the Gaussian
distributions. Other elliptical distributions can
be obtained from the following construction. Let~$V_1$ be a Gaussian
element in $\mathcal{H}$ with zero mean and covariance operator $\bGa
_{V_1}$, and
let $Z$ be a random variable independent of $V_1$. Given $\mu\in
\mathcal{H}$, define $X=\mu+Z V_1$. Then, $X$ has an elliptical
distribution $\mathcal{E}(\mu, \bGa)$
with the operator $\bGa$ being proportional to $\bGa_{V_1}$. Note that
$\bGa$ exist even if the second moment of~$X$ do not exist.
For random elements which admit a finite Karhunen--Lo\`eve expansion,
that is, $X(t) = \mu(t) + \sum_{j=1}^q \lambda_j^{1/2} U_j \phi_j(t)$,
the assumption that $X$ has an elliptical distribution is analogous to
assuming that $\bU=(U_1,\ldots,U_q)\trasp$ has a spherical distribution.
Lemma \ref{lemma51} below states the Fisher-consistency of the
functionals defined through (\ref{MAXROB}) under the following
assumption:
\begin{longlist}[(S3)]
\item[(S3)] There exists a constant $c>0$ and a self-adjoint,
positive semidefinite and compact operator $\bGa_0$, such that for any
$\alpha\in\mathcal{H}$, we have $\sigma^2(\alpha)={c}\langle
\alpha,
\bGa_0\alpha\rangle$.
\end{longlist}
Its proof follows immediately and is thus omitted. Note that
(S3) entails that the function $\sigma\dvtx \mathcal{H} \rightarrow\real$
defined as $\sigma(\alpha) = \sigma_{\rob}(P[\alpha] )$ is weakly
continuous, hence (S1) holds. Besides, as a consequence of Lemma
\ref{lemma51}, (S0) holds under~(S3) if the $q$ largest
eigenvalues of $\bGa_0$ are distinct.
\begin{lemma}\label{lemma51}
Let $\phi_{\rob,m}$ and $\lambda_{\rob,m}$ be the functionals defined
in (\ref{MAXROB}) and~(\ref{lamrob}), respectively. Let $X\sim P$ be a
random element such that \textup{(S3)} holds. Denote by
$\lambda_1\ge\lambda_2\ge\cdots$ the eigenvalues of $\bGa_0$ and by
$\phi_j$ the eigenfunction of $\bGa_0$ associated to~$\lambda_j$.
Assume that for some $q\ge2$, and for all $1\le j\le q$, $\lambda
_1>\lambda_2>\cdots>\lambda_q>\lambda_{q+1}$. Then, we have that
$\phi
_{\rob,j}(P) = \phi_j$ and $\lambda_{\rob,j}(P) = c \lambda_j$.
\end{lemma}
For any distribution possessing finite second moments, if the scale
functional is taken to be the standard deviation, then (S3)
holds with $\bGa_0=\bGa_X$. When considering a robust scale functional,
(S3) holds if $X$ has an elliptical distribution $\mathcal{E}(\mu
, \bGa)$ taking $\bGa_0=\bGa$, and so Lemma \ref{lemma51} entails that
the functionals $\phi_{\rob,j}(P)$ defined through (\ref{MAXROB}) are
Fisher-consistent. As in the finite-dimensional setting, the scale
functional $\sigma_{\rob}$ can be calibrated to attain
Fisher-consistency of the principal values.
Assumption (S3) ensures that we are estimating the target
directions. It may seem restrictive since it is difficult to verify outside
the family of elliptical distributions except when the scale is taken
to be the standard deviation. However, even in the finite-dimensional case,
asymptotic properties have been derived only under similar restrictions
when considering projection-pursuit estimators. For instance, both
\citet{li} and \citet{crouruiz2} assume an underlying
elliptical distribution in order to obtain consistency results and
influence functions, respectively. Also, in \citet{cui} the
influence function of the projected data is assumed to be of the form
$h(\bx,\ba)=2 \sigma(F[\ba])\operatorname{IF}(\bx, \sigma_{\ba} ;F_0)$, where
$F[\ba]$ stands for the distribution of $\ba\trasp\bx$ when $\bx
\sim
F$. This condition, though, primarily holds when the distribution is elliptical.
\begin{remark}\label{remark51}
An alternative to the robust
projection pursuit approach for functional principal components is to
consider the spectral value decomposition of a robust covariance or
scatter operator. The spherical principal components, noted in the
\hyperref[intro]{Introduction}, which were proposed by \citet{loc} and
further developed by \citet{ger08}, apply this approach using the
spatial covariance operator. The spatial covariance operator is defined
to be
\[
\bV= \esp\bigl({(X - \eta) \otimes(X- \eta)}/{\|X - \eta\|^2}\bigr)
\]
with $\eta$ being the spatial median, that is,
\begin{equation}\label{spatialm}
\eta= \argmin_{\theta\in\mathcal{H}}\esp( \|X - \theta\| - \| X \| ).
\end{equation}
The spatial median is sometimes referred to as the multivariate $L^1$
median, but this is a misnomer since the the norm in (\ref{spatialm})
is the $L^2$ norm. Note that when the norm is replaced by the square of
the norm in (\ref{spatialm}), the resulting parameter is the mean.
\citet{ger08} proved the Fisher-consistency of the eigenfunctions of
the spatial covariance operator, but under the additional assumption
that $X$ has a finite Karhunen--Lo\`eve expansion so that $\bV$ has
only a finite number of nonzero eigenvalues, which is
essentially the multivariate setting. Unlike the projection pursuit
approach, though, under an elliptical model the eigenvalues of $\bV$
are not proportional to the eigenvalues of the shape parameter~$\bGa$.
Consequently, as discussed, for example, in \citet{mar}, \citet{bofr}
and \citet{vis}, this implies that even if the
second moments exist, the amount of variance explained by a~principal
component variable is not equivalent to the ratio of the eigenvalue to
the sum of all the eigenvalues. That is, $\wtlam_k/\sum_{j=1}^{\infty}
\wtlam_j$ is not the same as the explained proportion $\lambda_k/\sum
_{j=1}^{\infty} \lambda_j$, where
$\wtlam_k$ and $\lambda_k$ are the $k$th largest eigenvalue of~$\bV$
and $\bGa$ respectively.
In the multivariate setting, it is also known that the eigenvectors
obtained from the sample spatial covariance matrix can be extremely
inefficient estimates whenever the eigenvalues differ greatly; see
\citet{croux99}. Intuitively, the reason for this inefficiency is that the
spatial covariance matrix down-weights observations according to their
Euclidean distance from the center. This seems reasonable when the
distribution is close to being spherical, but less so when there are
strong dependencies in the variables. In some sense, this is the
antithesis of PCA, since one is usually interested in principal
components when one suspects the latter.
As noted in \citet{maro}, there is a vast literature on
robust estimates for covariance matrices, such as $M$-estimates,
$S$-estimates and the $\mathit{MCD}$, among others. These estimates downweight
observations relative to the shape of the data cloud. It may seem
reasonable then to try to extend these estimates to the functional
setting. An important feature of these estimates, though, is that they
are affine equivariant, and as shown in \citet{tyler}, this implies
that, when the sample size is no greater than the dimension plus one,
such estimates are simply proportional to the sample covariance matrix.
In the functional data setting, the sample size is always less than the
dimension, which is infinite. Thus, at this time, we view the robust
projection-pursuit approach as more viable.
\end{remark}
\section{Some uniform convergence results}\label{appen}
In this section, we show that when the scale functional is continuous
with respect to the Prohorov distance, (S2) and more generally,
condition (ii) in Theorem \ref{theorem41} hold whenever $P_n\convweak
P$. To derive these results, we will first state some properties
regarding the weak convergence of empirical measures that hold not only
in $L^2(\mathcal{I})$ but in any complete and separable metric space.
These properties may be useful in other settings. The proofs for the
theorems in this section are relegated to Section D of the
supplemental article [\citet{bali3a}].
Let $\mathcal M$ be a complete and separable metric space (Polish
space) and $\mathcal{B}$ the Borel $\sigma$-algebra of $\mathcal M$.
Lemma \ref{lemma61}, which is a restatement of Theorem 3 in
\citet{varad}, ensures that the empirical measures converge
weakly almost surely on a Polish space to the probability measure
generating the observations.
\begin{lemma}\label{lemma61}
Let $(\Omega,\mathcal{A},\prob)$ be a probability space and
$X_n\dvtx\Omega
\rightarrow\mathcal{M}$, $n \in\mathbb{N}$, be a sequence of
independent and identically distributed random elements such that
$X_i\sim P$. Assume that $\mathcal M$ is a Polish space, and denote by
$P_n$ the the empirical probability measure, that is, $P_n(A)=
({1}/{n}) \sum_{i=1}^n I_A(X_i)$ with $I_A(X_i)=1$ if $X_i\in A$ and 0
elsewhere. Then, $P_n \convweak P$ almost surely, that is, $d_{\pr
}(P_n,P)\convppp0$.
\end{lemma}
Let $P$ be a probability measure in $\mathcal M$, a separable Banach
space, and let $\mathcal{M}^*$ denote the dual space. For a given $f
\in\mathcal{M}^*$, define $P[f]$ as the real measure of the random
variable $f(X)$, with $X \sim P$.
\begin{theorem}\label{theorem61}
Let $\{P_n\}_{n \in\mathbb{N}}$ and $P$ be probability measures
defined on~$\mathcal M$ such that $d_{\pr}(P_n, P) \to0$. Then, $\sup
_{\| f \|_{*} = 1} d_{\pr}(P_n[f], P[f]) \to0$.
\end{theorem}
When the Banach space $\mathcal M$ above is a separable Hilbert space
$\mathcal{H}$, the Riesz representation theorem implies that for $f
\in
\mathcal{H}^*$ with $\|f \|_* =1$, there exists $\alpha\in\mathcal
{H}$ such that $f(X)=\langle\alpha, X\rangle$. The following result
states that when~$\sigma_{\rob}$ is a continuous scale functional,
uniform convergence can be attained and so, assumption (ii) in Theorem
\ref{theorem41} is satisfied.
\begin{theorem}\label{theorem62}
Let $\{P_n\}_{n \in
\mathbb{N}}$ and $P$ be probability measures defined on a~separable
Hilbert space $\mathcal{H }$, such that $d_{\pr}(P_n, P) \to0$. Let
$\sigma_{\rob}$ be a continuous scale functional. Then,
$\sup_{\| \alpha\| = 1 } |\sigma_{\rob}(P_n[\alpha]) - \sigma
_{\rob
}(P[\alpha])| \longrightarrow0$.
\end{theorem}
Using Lemma \ref{lemma61} and Theorem \ref{theorem62}, we get the
following result that shows that (S2) holds if $\sigma_{\rob}$
is a continuous scale functional.
\begin{corollary}\label{corollary61}
Let $P$ be a probability measure in a separable Hilbert space $\mathcal
{H }$, $P_n$ be the empirical measure of a random sample $X_1, \ldots,
X_n$ with $X_i \sim P$, and $\sigma_{\rob}$ be a continuous scale
functional. Then we have that\break
$
{\sup_{\| \alpha\| = 1 }} |\sigma_{\rob}(P_n[\alpha]) - \sigma
_{\rob
}(P[\alpha])| \convppp0.
$
\end{corollary}
\section{Selection of the smoothing parameters}\label{smoothpar}
The selection of the smoothing parameters is an important practical
issue. The most popular general approach to address such a selection problem
is to use the cross-validation methods. In nonparametric regression,
the sensitivity of $L^2$ cross-validation methods to outliers
has been pointed out by \citet{wang} and by \citet{can}, among others. The latter also proposed more robust
alternatives to
$L^2$ cross-validation. The idea of robust cross-validation can be
adapted to the present situation. Assume for the moment that we are
interested in
a fixed number, $\ell$, of components. We propose to proceed as follows:
\begin{longlist}[(CV1)]
\item[(CV1)]
Center the data. That is, define $\widetilde
{X}_{i}=X_{i}-\widehat
{\mu}$ where $\widehat{\mu}$ is a robust location estimator, such as
the functional spatial median defined in \citet{ger08}.
\item[(CV2)] For the penalized roughness approaches and for each $m$ in the
range $1\le m\le\ell$ and ${\tau> 0}$, let $\wphi_{m,\tau}^{(-j)}$
denote the robust estimator of the $m$th principal direction computed
without the $j$th observation.
\item[(CV3)] Define $ X_{j}^{\bot}(\tau)= \widetilde{X}_{j}-\pi(\widetilde
{X}_{j};{\widehat{\mathcal{L}}_\ell^{(-j)}})$, for $1\le j \le n$,
where $\pi(X;\mathcal{L})$ stands for the orthogonal projection of $X$
onto the linear (closed) space $\mathcal{L}$, and $\widehat{\mathcal
{L}}_\ell^{(-j)}$ stands for the linear space spanned by $\wphi
_{1,\tau
}^{(-j)}, \ldots, \wphi_{\ell,\tau}^{(-j)}$.
\item[(CV4)] Let $\mathrm{RCV}_\ell(\tau)= \sigma_n^2( \|X_{1}^{\bot}(\tau)\|,
\ldots, \|
X_{n}^{\bot}(\tau)\| )$, where $\sigma_n$ is a robust measure of scale
about zero. We then choose $\tau_n$ to be the value of $\tau$ which
minimizes $\mathrm{RCV}_\ell(\tau)$.
\end{longlist}
By a robust measure of scale about zero, we mean that no location
estimator is applied to center the data. For instance, in the classical
setting, one takes $\sigma_n^2(z_1,\ldots,z_n)=(1/n)\sum_{i=1}^n z_i^2$,
while in the robust situation, one might choose $\sigma_n(z_1,\ldots
,z_n)=\median(|z_1|,\ldots,|z_n|)$ or to be
an $M$-estimator satisfying equation (\ref{snrob}) when setting
$\widehat{\mu}_n =0$.
For large sample sizes, it is well understood that cross-validation
methods can be computationally prohibitive. In such cases, $K$-fold
cross-validation
provides a useful alternative. In the following, we briefly describe a
robust $K$-fold cross-validation procedure suitable for our proposed estimates.
\begin{longlist}[(K1)]
\item[(K1)]
First center the data as above, using $\widetilde
{X}_{i}=X_{i}-\widehat{\mu}$.
\item[(K2)] Partition the centered data set $\{\widetilde{X}_{i}\}$ randomly
into $K$ disjoint subsets of approximately equal sizes with the $j$th
subset having size \mbox{$n_j\ge2$}, $\sum_{j=1}^K n_j =n$. Let $\{
\widetilde
{X}_i^{(j)}\}_{1\le i\le n_j}$ be the elements of the $j$th subset,
and\break
$\{\widetilde{X}_i^{(-j)}\}_{1\le i\le n-n_j}$ denote the elements in
the complement of the $j$th subset. The set $\{\widetilde{X}_i^{(-j)}\}
_{1\le i\le n-n_j}$ will be the training set and $\{\widetilde
{X}_i^{(j)}\}_{1\le i\le n_j}$ the validation set.\vspace*{1pt}
\item[(K3)] Similar to step (CV2) but with leaving the $j$th validation subset
$\{
\widetilde{X}_i^{(j)}\}_{1\le i\le n_j}$ out instead of the $j$th observation.
\item[(K4)] Define $ X_{j}^{(j)\bot}(\tau)$ the same way as in step (CV3), using
the validation set. For instance, $ X_{i}^{(j)\bot}(\tau)= \widetilde
{X}_i^{(j)}-\pi(\widetilde{X}_i^{(j)};{\widehat{\mathcal{L}}_\ell
^{(-j)}})$, $1\le i\le n_j$, where $\widehat{\mathcal{L}}_\ell^{(-j)}$
stands for the linear space spanned by $\wphi_{1,\tau}^{(-j)}, \ldots,
\wphi_{\ell,\tau}^{(-j)}$.
\item[(K5)] Let\vspace*{1pt} $\mathrm{RCV}_{\ell, \KCV}(\tau)= \sum_{j=1}^K
\sigma_n^2( \| X_{1}^{(j)\bot}(\tau)\|,\ldots,
\|X_{n_j}^{(j)\bot}(\tau)\| )$, and choo\-se~$\tau_n$ to be the value of
$\tau$ which minimizes $\mathrm{RCV}_{\ell, \KCV}(\tau)$.
\end{longlist}
A similar approach can be developed to choose $p_n$ for the sieve
estimators.
\section{Monte Carlo study}\label{monte}
The results of Section \ref{consist} established under general
conditions the consistency of the various robust projection pursuit
approaches to functional principal components analysis. The classical
approach to functional principal components analysis also yields
consistent estimates, provided second moment exists. A study of the
influence functions and the asymptotic distributions of the various
procedures would be useful to compare them. We leave these important
and challenging theoretical problems for future research. For now, to
help illuminate possible differences in the various approaches, we
present in this section the results of a Monte Carlo study.
\subsection{Algorithms}\label{algo}
All the computational methods to be considered here are modifications
of the basic \textsc{cr} algorithm proposed by \citet{crouruiz} for the computation of principal components using
projection-pursuit. The basic algorithm applies to multivariate data,
say $m$-dimensional, and requires a search over projections in $\real
^m$. The \textsc{grid} algorithm described in \citet{crfil} can\vadjust{\goodbreak}
also be considered, in particular, when the number of variables $m$ is
larger than the sample size $n$. For the sake of completeness, we
briefly recall the \textsc{cr} algorithm.
Let $\bY= ({\mathbf{y}}_1, \ldots, {\mathbf{y}}_n)$ be the sample in $\real
^m$, and let
$\widehat{\bmu}_n(\bY)$ be a location estimate computed from this
sample. Let $1\le q\le m$ be the desired number of components to be
computed and
denote by $\xi_n$ the univariate projection index to be maximized. In
the multivariate setting, the index $\xi_n$ corresponds to a robust
univariate scale statistic.
\begin{longlist}[(CR1)]
\item[(CR1)]
For $k = 1$, set ${\mathbf{y}}^{(1)}_i = {\mathbf{y}}_i- \widehat{\bmu
}_n(\bY)$. Let
the set of candidate directions for the first
principal direction be $\mathcal{A}_{n,1}(\bY) = \{{\mathbf{y}}^{(1)}_i/\nu_i,
1\le i\le n\}$ where $\nu_i^2={{\mathbf{y}}^{(1)}_i}\trasp{\mathbf{y}}^{(1)}_i$. We then define
${\mathbf{v}}_{1} = \arg\max_{\ba\in\mathcal{A}_{n,1}(\bY)} \xi
_n(\ba\trasp
{\mathbf{y}}_1,\ldots, \ba\trasp{\mathbf{y}}_n)$.
\item[(CR2)] For $2\le k\le q$, define recursively $z^{(k-1)}_i={
\mathbf{v}}_{k-1}\trasp
{\mathbf{y}}_i$ and ${\mathbf{y}}^{(k)}_i =\break{\mathbf{y}}^{(k-1)}_i- z^{(k-1)}_i
{\mathbf{v}}_{k-1} ={\mathbf{y}}
^{(1)}_i-\pi_{\mathcal{V}_{k-1}}({\mathbf{y}}^{(1)}_i)$,
where $\pi_{\mathcal{V}_{k-1}}({\mathbf{y}})$ stands for the orthogonal
projection of ${\mathbf{y}}$ over the linear space $\mathcal{V}_{k-1}$ spanned
by ${\mathbf{v}}_{1},\ldots, {\mathbf{v}}_{k-1}$. Let the set of candidate
directions for
the $k$th principal direction be\break
$\mathcal{A}_{n,k}(\bY) = \{{\mathbf{y}}^{(k)}_i/\nu_i, 1\le i\le n\}
$ where
$\nu_i^2={{\mathbf{y}}^{(1k)}_i}\trasp{\mathbf{y}}^{(k)}_i$, and define
${\mathbf{v}}_{k} =\break\arg\max_{\ba\in\mathcal{A}_{n,k}(\bY)} \xi
_n(\ba\trasp
{\mathbf{y}}_1,\ldots, \ba\trasp{\mathbf{y}}_n)$.
\end{longlist}
The vectors ${\mathbf{v}}_{k}$ then yield approximations to the $k$th
principal direction, for $1\le k \le q$, and approximate scores for the
$k$th principal variable are given by $z^{(k)}_i={\mathbf{v}}_{k}\trasp
{\mathbf{y}}
_i$, for $1\le i\le n$. As mentioned in \citet{crouruiz},
the \textsc{cr} algorithm makes no smoothness assumptions on the index
$\xi_n$, is simple and fast, and requires only $O(n)$ computing space.
To apply the algorithm to functional data when considering either the
raw or a penalized approach, we first discretize the domain of the
observed function over $m$ equally spaced points in $\mathcal{I}=[
-1,1]$. The resulting multivariate observations are then ${\mathbf{y}}
_i=(X_i(t_1),\ldots, X_i(t_m))\trasp$, where $t_0=-1<t_1<\cdots
<t_m<t_{m+1}=1$. The index $\xi_n$ in the algorithm depends on the
approach being used. For instance, for the raw robust estimate and for
those penalizing the norm the index is a robust scale. On the other
hand, for the robust penalized scale approach the index is the square
of the robust scale plus the penalization term. Also, for the penalized
norm approach the orthogonal projection $\pi_{\mathcal{V}_{k-1}}({\mathbf{y}})$
in step (CR2) is with respect to the inner product $\langle\cdot, \cdot
\rangle_{\tau}$ so, the finite-dimensional inner product is modified as
in \citet{s}. The resulting directions ${\mathbf{v}}_k$ then give
numerical approximations for $\{\widehat{\phi}_k(t_1),\ldots,
\widehat
{\phi}_k(t_m) \}$. One can then interpolate or use smoothing methods to
obtain $\widehat{\phi}_k(t)$ for
$t \in\mathcal{I}$.
For the sieve approach, let $\widetilde{\delta}_1,\ldots, \widetilde
{\delta}_{p_n}$ be an orthonormal basis for $\mathcal{H}_{p_n}$, which
can be obtained by using Gramm--Schmidt on the original basis. For
$\alpha\in\mathcal{H}_{p_n}$, we then have $\langle\alpha, X_i
\rangle= \ba\trasp{\mathbf{y}}_i $, where $\alpha= \sum_{j=1}^{p_n} a_j
\widetilde{\delta}_j$, $\ba= (a_1,\ldots,\allowbreak a_{p_n})\trasp$, ${\mathbf{y}}
_i=(\langle X_i, \widetilde{\delta}_1\rangle,\ldots, \langle X_i,
\widetilde{\delta}_{p_n}\rangle)\trasp$. Consequently, we can take
$m=p_n$ and apply the \textsc{cr} algorithm to the inner scores ${\mathbf{y}}
_i$. The inner scores are computed numerically by approximating the
integrals over a grid of $50$ points.\vspace*{1pt} A numerical approximation for
$\widehat{\phi}_k(t)$ is then given by $\sum_{j=1}^{p_n} v_{k,j}
\widetilde{\delta}_j(t)$ with ${\mathbf{v}}_k=(v_{k,1},\ldots
,v_{k,p_n})\trasp$.
\subsection{The estimators}\label{estimators}
There are three main characteristics that distinguish the different
estimators: the method of
centering in the first step of the \textsc{cr} algorithm, the scale
function being used and the type of smoothing method.\vspace*{8pt}
\textit{Centering}: For classical procedures, that is, those based on
the standard deviation, we use the sample mean as the centering point
for the trajectories. For the robust procedures, that is, those based
on \textsc{mad} or \textsc{$M$-scale}, we center the data by using
either (i) the component-wise sample median, that is, the median at
each time point, or (ii) the sample spatial median; see~(\ref{spatialm}).
It turns out that the two robust centering methods produced similar
results, so only the results for the spatial median are reported.\vspace*{8pt}
\textit{Scale function}: Three scale estimators are considered here:
the classical standard deviation (\textsc{sd}), the median absolute
deviation (\textsc{mad}) and an $M$-estimator of scale (\textsc
{$M$-scale}). For the $M$-estimator, we use the score function (\ref
{funcionBT}) introduced by \citet{beatuk}, as discussed in
Section~\ref{scale}, with $c=1.56$, $\delta= 1/2$.\vspace*{8pt}
\textit{Smoothing parameters $\tau$ and $\rho$}: For both the classical
and robust procedures defined in Section \ref{smoothpp}, a penalization
depending on the $L^2$ norm of the second derivative, multiplied by a
smoothing factor, is included, that is, $\Psi(\alpha)=\int_{-1}^1
(\alpha^{\prime\prime}(t))^2\,dt$. Again the integral is computed over
the same grid of points $t_1,\ldots, t_m$, and the second derivative of
$\alpha$ at $t_i$ is approximated by $\{\alpha(t_{i+1})-2
\alpha
(t_{i})+\alpha(t_{i-1}) \}/(t_{i+1}-t_i)^2$, since we choose an
equidistant grid of points. Note that when $\rho=\tau=0$, the raw
estimators defined in Section~\ref{rawpp} are obtained.\vspace*{8pt}
\textit{Sieve}: Two different sieve basis are considered: the Fourier
basis obtained by taking $\delta_j$ to be the Fourier basis functions,
and the cubic $B$-spline basis functions. The Fourier basis used in the
sieve method is the same basis used to generate the data.\vspace*{8pt}
In all tables, the estimators corresponding to each scale choice are
labeled as \textsc{sd}, \textsc{mad}, \textsc{$M$-scale}. For each
scale, we consider four estimators, the raw estimators where no
smoothing is used, the estimators obtained by penalizing the scale
function defined in (\ref{estFPCsmooth1}), those obtained by penalizing
the norm defined in (\ref{estFPCsmooth2}) and the sieve estimators
defined in (\ref{sieveFPC}). In all tables, as in\vspace*{1pt} Section
\ref{prop}, the $j$th principal direction estimators related to each
method are labeled as $\wphi_{\raw,j}$, $\wphi_{\smooths,j}$, $\wphi
_{\smoothn,j}$ and $\wphi_{\sieve,j}$, respectively.
When using the penalized estimators, several values for the penalizing
parameters $\tau$ and $\rho$ were chosen. Since large values of the
smoothing parameters make the penalizing term the dominant component
regardless of the amount of contamination considered, we choose $\tau$
and $\rho$ equal to $a n^{-\alpha}$ for $\alpha=3$ and $4$ and $a$
equal to $0.05$, $0.10$, $0.25$, $0.5$, $0.75$, $1$, $1.5$ and $2$.
For the sieve estimators based on the Fourier basis, ordered as
$\{1, \cos( \pi x)$, $\sin( \pi x),\ldots, \cos( q_n \pi x), \sin(
q_n \pi x), \ldots\}$, the values $p_n=2 q_n$ with $q_n=5$, $10$ and
$15$ are used, while for the sieve estimators based on the $B$-splines,
the dimension of the linear space considered is selected as $p_n= 10$,
$20$ and $50$. The basis for the $B$-splines is generated from the R
function $\mathit{cSplineDes}$, with the knots being equally spaced in the
interval $[-1,1]$ and
the number of knots equal to $p_n+1$.
To conserve space, we only report here the results corresponding to
$p_n=30$ and $p_n=50$ for the Fourier and $B$-spline basis,
respectively. More extensive simulation results are listed in the
technical report [\citet{bali2}].
\subsection{Simulation settings}\label{setting}
The sample was generated using a finite Karhu\-nen--Lo\`eve expansion
with the functions, $\phi_i\dvtx[-1,1] \rightarrow\mathbb{R}$,
$i=1,2,3$, where
$ \phi_1(x)= \sin( 4 \pi x)$, $\phi_2(x) = \cos( 7 \pi x)$ and
$\phi
_3(x)= \cos( 15 \pi x)$.
It is worth noticing that, when considering the sieve estimators based
on the Fourier basis, the third component cannot be detected when $q_n
< 15$, since in this case~$\phi_3(x)$ is orthogonal to the estimating
space. Likewise, the second component cannot be detected when $q_n < 7$.
We performed $NR=1\mbox{,}000$ replications generating independent samples $\{
X_i\}_{i=1}^n$ of size $n= 100$ following the model $X_i = Z_{i1} \phi
_1 + Z_{i2} \phi_2 + Z_{i3} \phi_3$, where $\bZ
_{i}=(Z_{i1},Z_{i2},Z_{i3})\trasp$ are independent vectors whose
distribution will depend on the situation to be considered. The central
model, denoted $C_0$, corresponds to Gaussian samples. We also consider
four contaminations of the central model, labeled $C_{2}$, $C_{3,a}$,
$C_{3,b}$ and $C_{23}$ depending on the components to be contaminated.
Contamination models are commonly considered in robust statistics since
they tend be the more difficult models to be robust against and are the
basis for the concept of bias robustness, see \citet{maro} for
further discussion. In all these situations $Z_{i1}, Z_{i2}$ and
$Z_{i3}$ are also independent. For each of the models, we took $\sigma
_1=4$, $\sigma_2=2$ and $\sigma_3=1$. The central model and the
contaminations can be described as follows:\vspace*{8pt}
${C_{0}}$: $Z_{i1} \sim N( 0,\sigma_1^2)$, $Z_{i2} \sim N(0,\sigma
_2^2)$ and $Z_{i3} \sim N(0,\sigma_3^2)$.
${C_{2}}$: $Z_{i2}$ are independent and identically distributed as $
0.8 N(0,\sigma_2^2)+\break 0.2 N(10,\allowbreak 0.01)$, while
$Z_{i1} \sim N(0,\sigma_1^2)$ and $Z_{i3} \sim N(0,\sigma_3^2)$. This
contamination corresponds to a strong contamination on the second
component and changes the mean value of the generated data $Z_{i2}$ and
also the first principal component. Note that $\var(Z_{i2})=19.202$.
${C_{3,a}}$: $Z_{i1} \sim N( 0,\sigma_1^2)$, $Z_{i2} \sim N(0,\sigma
_2^2)$ and $Z_{i3} \sim0.8 N(0,\sigma_3^2)+ 0.2
N(15$, $0.01)$. This contamination corresponds to a strong
contamination on the third component. Note that $\var(Z_{i3})=36.802$.
${C_{3,b}}$: $Z_{i1} \sim N( 0,\sigma_1^2)$, $Z_{i2} \sim N(0,\sigma
_2^2)$ and $Z_{i3} \sim0.8 N(0,\sigma_3^2)+ 0.2
N(6, 0.01)$. This contamination corresponds to a strong
contamination on the third component. Note that $\var(Z_{i3})=6.562$.
${C_{23}}$: $Z_{ij}$ are independent and such that $Z_{i1} \sim
N(0,\sigma_1^2)$, $Z_{i2} \sim0.9 N(0,\sigma_2^2) + 0.1
N(15, 0.01)$ and $Z_{i3} \sim0.9 N(0,\sigma_3^2) +
0.1 N(20, 0.01)$. This contamination corresponds to a mild
contamination on the last two components. Note that $\var
(Z_{i2})=23.851$ and $\var(Z_{i3})=36.901$.\vspace*{8pt}
We also include a long-tailed model, namely a Cauchy model,
labe-\break led~${C_{\mathrm{Cauchy}}}$,~which is defined by taking $\bZ_i\,{\sim}\,\mathcal
{C}_3(0,\bSi)$ with $\bSi\,{=}\,\operatorname{diag}(\sigma_1^2,\sigma
_2^2,\sigma
_3^2)$, where $\mathcal{C}_p(0,\bSi)$ stands for the $p$-dimensional
elliptical Cauchy distribution centered at $0$ with scatter matrix
$\bSi
$. For this situation, the covariance operator does not exist, and thus
the classical principal components are not defined.
It is worth noting that the directions $\phi_1$, $\phi_2$ and $\phi_3$
correspond to the classical principal components for the case $C_0$,
but not
necessarily for the other cases. For instance, when $\sigma_{\rob
}^2=\operatorname{VAR}$, $C_{3,a}$ interchanges the order between~$\phi_1$
and $\phi_3$, that is, $\phi_3=\phi_{\rob,1}(C_{3,a})$ as defined in
(\ref{MAXROB}), and so it corresponds to the first principal component
of the covariance operator, while $\phi_1$ is the second and $\phi_2$
is the third one.
\subsection{Simulation results}\label{resultados}
For each situation, we compute the estimators of the first three
principal directions and the square distance between the true and the
estimated direction (normalized to have $L^2$ norm 1), that is,
\[
D_j=\biggl\|\frac{\wphi_j}{\| \wphi_j\|}-\phi_j\biggr\|^2 .
\]
Note that all the estimators except those penalizing the norm, are such
that \mbox{$\| \wphi_j\|=1$}. Tables 4 to
9 of the supplementary
material [\citet{bali3b}] report the means of $D_j$ over replications,
which hereafter is referred to as mean square error. To help understand
the influence of the grid size $m$ on the estimators, Tables
3, 4 and 5 give the mean squared errors
for $m=50, 100, 150, 250$ and $250$, under $C_0$ for various values of
the penalizing parameters. As can be seen, for the first two components
some slight improvement is observed when using $m=250$ as opposed to
$m=50$ points, but at the expense of increasing the computing time
about 2.6 fold. On the other hand, for the third principal direction,
taking $m=100$ compared to $m=50$ reduces the mean square error by at
least a half for the penalized estimators,
while the gain is not so prominent for the raw estimates. The size
$m=50$ is selected for presentation in the remainder of our study since it
provides a reasonable a compromise between the performance of the
estimators and the computational time.
To help understand the effect of penalization, consider Table
6. This table shows results for the raw and
penalized estimators under $C_0$ for different choices of the
penalizing parameters. From this table, we see that a better
performance is achieved in most cases when $\alpha=3$ is used. To be
more precise, the results in Table~6 show
that the best choice for $\wphi_{\smooths,j}$ is $\rho=2 n^{-3}$ for
all $j$. Note that $\rho=1.5 n^{-3}$ gives fairly similar results when
using the $M$-scale, reducing the mean squared error relative to the
raw estimate by about a half and a third for $j=2$ and $3$, respectively.
For the norm penalized estimators, $\wphi_{\smoothn,j}$, the best
choice for the penalizing parameter seems to depend upon both the
component to be estimated and the scale estimator used. For instance,
when using the standard deviation, the best choice is $\tau=0.10
n^{-3}$, for $j=1$ and $2$ while for $j=3$ a smaller order is needed to
obtain an improvement over the raw estimators, with the value $\tau
=0.75 n^{-4}$ leading to a small gain over the raw estimators. For the
robust procedures, larger values are needed to see an advantage to
using the penalized norm approach relative to the raw estimators. For
example, for $j=1$, the largest reduction is observed when $\tau=2
n^{-3}$ while for $j=2$, the best choices correspond to $\tau=0.5
n^{-3}$ and $\tau= 0.25 n^{-3}$ when using the \textsc{mad} and
$M$-scale, respectively. When using the $M$-scale, a good compromise is
to choose $\tau=0.75 n^{-3} $, which gives a reduction of around 30\%
and 50\% for the first and second principal directions, respectively,
although smaller values of~$\tau$ are again better for estimating the
third component.
Based upon\vspace*{1pt} the above observations, we report here only the results
corresponding to $\rho=1.5 n^{-3}$ and $\tau=0.75 n^{-3}$ for the
penalized estimators $\wphi_{\smoothn,j}$ and $\wphi_{\smooths,j}$,
respectively, under the contamination models and the Cauchy model.
Results for other choices of $\rho$ and $\tau$ are given in
\citet{bali2}.
The simulation study confirms the expected inadequate behavior of the
classical estimators in the presence of outliers. Under contamination,
the classical estimators of the principal directions do not estimate
the target directions very accurately. This is also the case when
considering the Cauchy distribution. Curiously, though, the principal
directions, under the Cauchy model, do not seem to be totally arbitrary
and they partially recover $\phi_1$, $\phi_2$ and $\phi_3$ when the
standard deviation is used, although not as well as when using a robust
scale, even though the covariance operator does not exist, nor do the
population principal directions as defined in~(\ref{MAX}).
The robust estimators of the first principal directions are not heavily
affected by any of the contaminations, while the estimates of the
second and third principal directions appear to be most affected under
model $C_{3,a}$. In particular, for the third direction, the
projection-pursuit estimators based on an the \textsc{mad} seems to be
most affected by this type of contamination when penalizing the norm,
although much less so than the classical methods. With respect to the
contamination model~$C_{3,a}$, the estimators $\wphi_{\smoothn,j}$,
which are the robust penalized norm estimators, tend to have the best
performance among all the robust competitors for the first two
components, and, in particular, when using the $M$-scale; see Table
8. It is worth noting that the
classical estimators of the first component are not affected by this
contamination when penalizing the norm since the penalization dominates
the contaminated variances. The same phenomena is observed under
$C_{3,b}$ when using the classical estimators for the selected amount
of penalization. For the raw estimators, the sensitivity of the
classical estimators under this contamination can be observed in Table
7. We refer to \citet{bali2}
for the behavior when other values of the smoothing parameters
are chosen.
As noted in \citet{s}, for the classical estimators, some
degree of smoothing in the procedure based on penalizing the norm will
give a better estimation of $\phi_j$ in the $L^2$ sense under mild
conditions. In particular, both the procedure penalizing the norm and
the scale provide some improvement with respect to the raw estimators
if $\Psi(\phi_j)<\Psi(\phi_\ell)$, when $j<\ell$. This means that the
principal directions are rougher as the eigenvalues decrease [see
\citet{pez} and \citet{s}], which is also
reflected in our simulation study. The advantages of the smooth
projection pursuit procedures are most striking when estimating $\phi
_2$ and $\phi_3$ with an $M$-scale and using the penalized scale approach.
As expected, when using the sieve estimators, the Fourier basis gives
the best performance over all the methods under $C_0$, since our data
were generated from this basis; see Table 9. The choice of the $B$-spline
basis give similar results quite to those obtained with $\wphi
_{\smooths
,j}$ when estimating the first direction, except under $C_{\mathrm{Cauchy}}$
where the penalized estimators show a~better performance. For the
second and third components, the estimators obtained with the
$B$-spline basis show larger the mean square errors than the raw or
penalized estimators.
\subsection{$K$th fold simulation}\label{kfold}
Table \ref{tabtiempos} reports the computing times in minutes for 1,000
replications and for a fixed value of $\tau$ or $\rho$, run on a
computer Core Quad I7~930 (2.80 GHz) with 8 Gb of Ram memory. We also
report the computing times when using the sieve approach with the
Fourier basis and a~fixed value of $p_n$. This suggests that the
leave-one-out cross-validation may be difficult to perform, and so a
$K$-fold approach is adopted instead. It is worth noticing that the
robust procedures based on the \textsc{mad} are much faster than those
based on the $M$-scale, so they may be preferred in terms of computing
time. However, as mentioned in Section~\ref{prelim}, the main
disadvantage of the \textsc{mad} is its low efficiency and lack of
smoothness, which is related to the discontinuity of its influence
function. This is particularly important when estimating principal
components in the finite-dimensional setting, since as it was pointed
out by \citet{cui} and \citet{crouruiz2} the
variances of some elements of the estimated principal directions may
blow up when using the \textsc{mad} leading to highly inefficient
estimators. As expected and mentioned in Section \ref{resultados},
Table 6 reveals a high loss of efficiency
for the \textsc{mad}, in our functional setting, for any choice of the
smoothing parameter.
\begin{table}
\tablewidth=250pt
\caption{Computing times in minutes for 1,000
replications and a fixed value of $\tau$, $\rho$ or $p_n$ (when using
the Fourier basis)}\label{tabtiempos}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccc@{}}
\hline
& \textbf{\textsc{sd}} & \textbf{\textsc{mad}} & \textbf{\textsc{$\bolds{M}$-scale}}\\
\hline
$\wphi_{\raw}$ & \hphantom{0}5.62 & \hphantom{0}6.98 & 17.56\\[2pt]
$\wphi_{\smooths}$ & \hphantom{0}7.75 & \hphantom{0}9.00 & 20.18\\[2pt]
$\wphi_{\smoothn}$ & 31.87 & 33.21 & 44.04\\[2pt]
$\wphi_{\sieve}$ & \hphantom{0}0.5\hphantom{0} & \hphantom{0}5.22 & 16.08 \\
\hline
\end{tabular*}
\end{table}
For the procedure which penalizes the scale or the norm, the smoothing
parameters $\rho$ and $\tau$ are selected using the procedure described
in Section \ref{smoothpar} with $K=4$ and $\ell=1$. Due to the
extensive computing time, we have only performed $500$ replications.
The simulation results when penalizing the scale function, that is, for
the estimators defined through (\ref{estFPCsmooth1}), are reported in
Table \ref{tabtablameanL2squarenormKfold}. Under $C_0$, when estimating
the second and third principal directions, the robust estimators based
on the $M$-scale combined with a penalization in the scale clearly have
smaller mean square error than the raw estimators, while those
penalizing the norm improve the performance of the raw estimators and
also that of $\wphi_{\smooths,j}$, on the first and second
directions.\looseness=1
\begin{table}
\caption{Mean values of $\|\wphi_j/\|\wphi_j\|-\phi_j\|^2$
when the penalizing parameter is selected using
$K$-fold cross-validation}\label{tabtablameanL2squarenormKfold}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccccccc@{}}
\hline
& & \multicolumn{3}{c}{$\bolds{\wphi_{\smooths,j}}$} & \multicolumn
{3}{c}{$\bolds{\wphi_{\smoothn,j}}$} \\[-4pt]
& & \multicolumn{3}{c}{\hrulefill} & \multicolumn
{3}{c@{}}{\hrulefill}\\
\textbf{Model} & \textbf{Scale estimator}
& $\bolds{j=1}$ & $\bolds{j=2}$ & $\bolds{j=3}$ & $\bolds{j=1}$
& $\bolds{j=2}$ & $\bolds{j=3}$ \\
\hline
$C_0$ & \textsc{SD} & 0.0073 & 0.0094 & 0.0078 & 0.0075 & 0.0094 & 0.0360 \\
& \textsc{mad} & 0.0662 & 0.0993 & 0.0634 & 0.0497 & 0.0660 &
0.2573 \\
& $M$-scale & 0.0225 & 0.0311 & 0.0172 & 0.0208 & 0.0271 & 0.0839 \\
[4pt]
$C_2$ & \textsc{SD} & 1.2840 & 1.2837 & 0.0043 & 1.2076 & 1.2232 & 0.0301 \\
& \textsc{mad} & 0.3731 & 0.3915 & 0.0504 & 0.3360 & 0.3770 &
0.2832 \\
& $M$-scale & 0.4261 & 0.4286 & 0.0153 & 0.3679 & 0.4049 & 0.1607 \\
[4pt]
$C_{3,a}$ & \textsc{SD} & 1.7840 & 1.8901 & 1.9122 & 1.7795 & 1.8861 & 1.9134 \\
& \textsc{mad} & 0.2271 & 0.5227 & 0.5450 & 0.0573 & 0.2289 &
0.9540 \\
& $M$-scale & 0.2176 & 0.4873 & 0.5437 & 0.0257 & 0.1187 & 0.8710 \\
[4pt]
$C_{3,b}$ & \textsc{SD} & 0.0192 & 0.8350 & 0.8525 & 0.0173 & 0.5902 & 0.7502 \\
& \textsc{mad} & 0.0986 & 0.3930 & 0.3820 & 0.0553 & 0.1417 &
0.5167 \\
& $M$-scale & 0.0404 & 0.2251 & 0.2285 & 0.0241 & 0.1080 & 0.3174 \\
[4pt]
$C_{23}$ & \textsc{SD} & 1.7645 & 0.5438 & 1.6380 & 1.7537 & 0.6496 & 1.4305 \\
& \textsc{mad} & 0.2407 & 0.3443 & 0.2064 & 0.1414 & 0.2214 &
0.6824 \\
& $M$-scale & 0.2613 & 0.3707 & 0.2174 & 0.1313 & 0.1870 & 0.5901 \\
[4pt]
$C_{\mathrm{Cauchy}}$ & \textsc{SD} & 0.3580 & 0.4835 & 0.2287 & 0.2862 & 0.3525 & 0.3435 \\
& \textsc{mad} & 0.0788 & 0.1511 & 0.1082 & 0.0613 &
0.0855 & 0.3147 \\
& $M$-scale & 0.0444 & 0.0707 & 0.0434 & 0.0349 & 0.0463 & 0.1465 \\
\hline
\end{tabular*}
\end{table}
From the results in Table \ref{tabtablameanL2squarenormKfold} we
observe that the classical estimators are sensitive to the
contaminations in the simulation settings, and, except for
contaminations in the third component, the robust counterpart shows a~clear advantage. Note that $C_{3,b}$ affects more the classical
estimators when the smoothing parameter is selected by the robust
$K$-fold cross-validation method than for the fixed values studied in
the previous section. This can be explained by the fact that
contamination $C_{3,b}$ is a mild contamination in the third component
which has a large $\|\phi^{\prime\prime}_3\|^2$, and so the classical
estimators are more sensitive to it, just as the raw estimators, if
smaller values of the smoothing parameter are chosen. It is worth
noticing that the penalized robust estimators based on the $M$-scale
improve the performance of the raw estimators based on the $M$-scale,
even under contaminations, when the penalizing parameter is selected
using the $K$-fold approach. This advantage is more striking when
penalizing the norm and when the two first principal components are considered.
Note that we choose $\ell=1$, and so our focus was on the first
principal component. To improve the observed performance, a different
approach should be considered, maybe by selecting a different smoothing
parameter for each principal direction.
\section{Concluding remarks}\label{concl}
In this paper, we propose robust principal component analysis for
functional data based on a projection-pursuit approach. The different
procedures correspond to robust versions of the unsmoothed principal
component estimators, to the estimators obtained penalizing the scale
and to those obtained by penalizing the norm. A sieve approach based on
approximating the elements of the unit ball by elements over
finite-dimensional spaces is also considered. In particular, the
procedures based on smoothing and sieves are new. A robust
cross-validation procedure is introduced to select the smoothing parameters.
Consistency results are derived for the four type of estimators.
Finally, a simulation study confirms the expected inadequate behavior
of the classical
estimators in the presence of outliers, with the robust procedures
performing significantly better. In particular, the procedure\vadjust{\goodbreak} based on
an $M$-scale combined with a penalization in the norm, where the
smoothing parameter is selected via a robust $K$-fold cross-validation,
is recommended.
Among other contributions we highlight the following:\vspace*{8pt}
(a) We obtain the continuity of the principal directions and
eigenvalue functionals, which implies the asymptotic qualitative
robustness of the corresponding estimators. This extends the results of
\citet{li} from Euclidean spaces to infinite-dimensional
Hilbert spaces, where the unit ball is not compact. Noncompactness
poses technical challenges which we overcome with tools from functional
analysis.
(b) Our results not only include the finite-dimensional case but also
improve upon some of the results obtained in that situation for the
projection pursuit estimators. For example, the assumptions in
\citet{li} regarding the robust scale functional are stronger than
ours. Also, to derive the consistency of the raw estimators, we only
require uniform convergence over the unit ball of $s_n(\alpha)$
to~$\sigma(\alpha)$, which holds if the scale functional
$\sigma_{\rob}$ is continuous. This improves upon the consistency
results given in \citet{cui}, who require a uniform Bahadur
expansion for $s_n(\alpha )$ over the unit ball.
(c) A key step in proving the continuity of the projection pursuit
functional is to show that weak convergence of probability measures
over a~Hilbert space implies uniform convergence of the laws of the
projections of the stochastic processes, that is, Theorem
\ref{theorem62}. This uniform convergence result can be useful in other
statistical problems where projection methods are considered.
(d) The proofs for the penalized estimators include, as particular
cases, the estimators defined by \citet{ris} and studied
by \citet{pez}, and those considered by \citet{s}. Extending the results to scale estimators other than the
standard deviation required more challenging arguments since, unlike
the classical setting, the projection-pursuit index $s^2_n(\alpha)$
cannot be expressed in the simple form $\langle\alpha, \bGa_n \alpha
\rangle$ for some compact operator $\bGa_n$.
\begin{appendix}\label{appenA}
\section*{Appendix}
In this Appendix, we give the proofs of the results stated in Section
\ref{consist}. Some technicalities are omitted, and we refer to the
technical report [\citet{bali2}] for details.
Before presenting the proof, some additional notation is needed.
Denote by $\mathcal{L}_{m-1}$ the linear space spanned by $\{\phi
_{\rob
,1},\ldots,\phi_{\rob, m-1}\}$, and let~$\widehat{\mathcal
{L}}_{m-1}$ be
the linear space spanned by the first $m-1$ estimated principal
directions, that is, by $\{\wphi_{\sieve,1},\ldots,\wphi_{\sieve
,m-1}\}$
or $\{\wphi_{1},\ldots,\wphi_{m-1}\}$, where it will be clear in each
case which linear space we are considering. The latter includes the
situation of the linear spaces spanned by $\{\wphi_{\raw,1},\ldots
,\wphi
_{\raw, m-1}\}$, $\{\wphi_{\smooths,1},\ldots,\wphi_{\smooths
,m-1}\}$
and $\{\wphi_{\smoothn,1},\ldots,\wphi_{\smoothn,m-1}\}$. Finally, for
any linear space~$\mathcal L$, $\pi_\mathcal{L}\dvtx\mathcal{H}\to
\mathcal
{L}$ stands for the orthogonal projection onto the linear space
$\mathcal L$, which exists if $\mathcal L$ is a closed linear space. In
particular, $\pi_{\mathcal{L}_{m-1}}$, $\pi_{\widehat{\mathcal
{L}}_{m-1}}$ and~$\pi_{\mathcal{H}_{p_n}}$ are well defined.
Moreover, for the sake of simplicity, denote by $\mathcal{T}_k =
\mathcal{L}_{k}^{\bot}$ the linear space orthogonal to $\phi_1,\ldots
,\phi_k$ and by $\pi_k = \pi_{\mathcal{T}_k}$ the orthogonal projection
with respect to the inner product defined in $\mathcal H$. On the other
hand, let $\widehat{\pi}_{\tau,k}$ be the projection onto the linear
space orthogonal to $\wphi_{1},\ldots, \wphi_{k}$ in the space
$\mathcal
{H}_{\smooth}$ in the inner product $\langle\cdot, \cdot\rangle
_{\tau
}$, that is, for any $\alpha\in\mathcal{H}_{\smooth}$,
$\widehat{\pi}_{\tau,k} (\alpha) = \alpha- \sum_{j=1}^k \langle
\alpha,
\wphi_{ j} \rangle_{\tau}\wphi_{ j}$.
Moreover, let $\widehat{\mathcal{T}}_{\tau,k}$ stand for the linear
space orthogonal to $\widehat{\mathcal{L}}_k$ with the inner product
$\langle\cdot, \cdot\rangle_{\tau}$. Thus, $ \widehat{\pi}_{\tau,k}$
is the orthogonal projection onto $\widehat{\mathcal{T}}_{\tau,k}$ with
respect to this inner product.
\begin{pf*}{Proof of Theorem \ref{theorem41}}
First note that the
fact that $\sigma_{\rob}$ is a scale functional entails that $\sigma
_n(\alpha)=\|\alpha\| \sigma_n(\alpha/\|\alpha\| )$. Thus from
assumption (ii) and the fact that $\|\alpha\| \leq\|\alpha\|_\tau$, we
get that
\begin{equation}\label{S1enmenorque1}\quad
\sup_{\|\alpha\|\le1}|\sigma_n^2(\alpha) - \sigma^2(\alpha)
|\convpp0 \quad\mbox{and}\quad \sup_{\|\alpha\|_{\tau}\le1}
|\sigma_n^2(\alpha) - \sigma^2(\alpha)|\convpp0 .
\end{equation}
(a) To prove that $\wlam_1 \convpp\sigma^2(\phi_{\rob,1})$ it is
enough to show that
\begin{eqnarray}
\label{SMdesi1}
\sigma^2(\phi_{\rob,1}) &\geq& \wlam_1 + o_{\mathrm{a.s.}}(1) ,
\\
\label{SMdesi2}
\sigma^2(\phi_{\rob,1}) &\leq& \wlam_1 + o_{\mathrm{a.s.}}(1),
\end{eqnarray}
where $ o_{\mathrm{a.s.}}(1)$ stands for a term converging to $0$
almost surely.
Note that from (\ref{S1enmenorque1}), we get that $a_{n,1}=\sigma
_n^2(\wphi_1)-\sigma^2(\wphi_1)\convpp0$ and $b_{n,1}=\sigma
_n^2(\phi
_{\rob,1})-\sigma^2(\phi_{\rob,1})\convpp0$.
Using that $\sigma$ is a scale functional and that\break $\sigma^2(\phi
_{\rob
,1}) = \sup_{\alpha\in\mathcal{S}} \sigma^2(\alpha)$, we obtain
easily that
\[
\sigma^2(\phi_{\rob,1}) \geq\sigma^2 \biggl( \frac{\wphi_1}{\|\wphi
_1\|
}\biggr)=\frac{\sigma^2 (\wphi_1)}{\|\wphi_1\|^2}\geq\sigma^2(\wphi_1)=
\sigma_n^2(\wphi_1) - a_{n,1}= \wlam_1 + o_{\mathrm{a.s.}}(1)
\]
concluding the proof of (\ref{SMdesi1}).
To derive (\ref{SMdesi2}), note that since $\phi_{\rob,1}\in
\mathcal
{H}_{\smooth}$, $\| \phi_{\rob,1} \|_{\tau}<\infty$ and $\|\phi
_{\rob
,1}\|_{\tau} \geq\|\phi_{\rob,1}\|=1$,
then, defining\vspace*{1pt} $\beta_1=\phi_{\rob,1}/ \|\phi_{\rob,1}\|_{\tau}$, we
have that $\|\beta_1\|_{\tau}=1$, which implies that $
\wlam_1 = \sigma_n^2(\wphi_1) \geq\sigma_n^2(\wphi_1)-\rho\Psi
(\wphi
_1)\ge\sigma_n^2(\beta_1)-\rho\Psi(\beta_1)$.
Hence, using that $\sigma_{\rob}$ is a scale functional and that
$\Psi
(a\alpha)=a^2\Psi(\alpha)$, for any $a\in\real$, we get
\begin{eqnarray*}
\wlam_1 &\ge& \sigma_n^2(\beta_1)-\rho\Psi(\beta_1)= \frac
{\sigma
_n^2(\phi_{\rob,1})-\rho\Psi(\phi_{\rob,1})}{\|\phi_{\rob,1}\|
_{\tau}^2}
= \frac{\sigma^2(\phi_{\rob,1})+b_{n,1}-\rho\Psi(\phi_{\rob
,1})}{\|\phi
_{\rob,1}\|_{\tau}^2}.
\end{eqnarray*}
When $\rho=0$, we have defined $\rho\Psi(\phi_{\rob,1})= 0$ and
similarly when $\tau=0$. So from now on, we will assume that $\tau_n>0$
and $\rho_n>0$.
Since\vadjust{\goodbreak} $b_{n,1}=o_{\mathrm{a.s.}}(1)$, $\rho\convpp0$ and $\tau
\convpp0$, we have that $\rho\Psi(\phi_{\rob,1})\convpp0$ and $\|
\phi
_{\rob,1}\|_{\tau} \convpp\|\phi_{\rob,1}\|= 1$, concluding the proof
of (\ref{SMdesi2}). Hence, $\wlam_1 \convpp\sigma^2(\phi_{\rob,1})$.
From (\ref{S1enmenorque1}) and the fact that $\|\wphi_1\| \leq1$, we
obtain that $\wlam_1-\sigma^2(\wphi_1)=\sigma_n^2(\wphi_1)-\sigma
^2(\wphi_1)\convpp0$. Therefore, using that $\wlam_1 \convpp\sigma
^2(\phi_{\rob,1})$, we get that
\begin{equation}\label{sigmawphi}
\sigma^2(\wphi_1) \convpp\sigma^2(\phi_{\rob,1}) .
\end{equation}
Moreover, the inequalities $\sigma^2(\phi_{\rob,1}) \geq\sigma^2 (
{\wphi_1}/{\|\wphi_1\|})\geq\sigma^2(\wphi_1)$
obtained above also imply that
\begin{equation}\label{sigmawphi2}
\sigma^2 ( {\wphi_1}/{\|\wphi_1\|}) \convpp\sigma^2(\phi_{\rob ,1}) .
\end{equation}
Using that $\|\wphi_1\|_{\tau}=1$, we get that $\tau\Psi(\wphi_1)
=1-\|\wphi_1\|^2=1-{\sigma^2(\wphi_1)}/\sigma^2(\wphi_1 /\allowbreak\|\wphi_1\|)$.
Hence, (\ref{sigmawphi}) and (\ref{sigmawphi2}) entail that
$\tau
\Psi(\wphi_1)\convpp0$.
It only remains to show that $\rho\Psi(\wphi_1)\convpp0$, which
follows easily from the fact that $\wlam_1\convpp\sigma^2(\phi
_{\rob
,1})$, $\sigma_n^2(\phi_{\rob,1})\convpp\sigma^2(\phi_{\rob
,1})$, $\rho
\convpp0$\vspace*{1pt} and $\|\phi_{\rob,1}\|_{\tau} \convpp1$ since
$\wlam_1 \geq\sigma_n^2(\wphi_1) - \rho\lceil\wphi_1, \wphi_1
\rceil\geq
(\sigma_n^2(\phi_{\rob,1}) - \rho\lceil\phi_{\rob,1}, \phi
_{\rob,1}
\rceil)/\|\phi_{\rob,1}\|_{\tau}^2$.
Note that we have not used the weak continuity of $\sigma$ as a
function of $\alpha$ to derive~(a).
(b) Note that since $\|\wphi_{1}\|_{\tau}=1$, we have that $\|
\wphi
_{1}\|\le1$. Moreover, from~(a), $\|\wphi_{1}\| \convpp1$. Let
$\wtphi
_1=\wphi_1/\|\wphi_1\|$, then $\wtphi_1\in\mathcal{S}$ and $\sigma
(\wtphi_1)=\sigma(\wphi_1)/\|\wphi_1\|$. Using that $\sigma
^2(\wphi
_{1})\convpp\sigma^2(\phi_{\rob,1})$ and $\|\wphi_{1}\| \convpp
1$, we
obtain that $\sigma^2(\wtphi_{1})\convpp\sigma^2(\phi_{\rob,1})$, and
thus the proof follows using Lemma \ref{lemma41}.
(c) Let us show that $\wlam_m \convpp\sigma^2(\phi_{\rob,m})$.
The proof will be done in several steps by showing
\begin{eqnarray}
\label{lema}
&\displaystyle {\sup_{\|\alpha\|_{\tau} \leq1} }|\sigma^2(\pi_{m-1} \alpha) -
\sigma
_n^2(\widehat{\pi}_{\tau,m-1} \alpha)|
\convpp0,&
\\
\label{desilamm1}
&\sigma^2(\phi_{\rob,m}) \geq\wlam_m + o_{\mathrm{a.s.}}(1),&
\\
\label{desigenm}
&\sigma^2(\phi_{\rob,m}) \leq\wlam_m + o_{\mathrm{a.s.}}(1) .&
\end{eqnarray}
Note that (\ref{lema}) corresponds to an extension of assumption (ii)
while (\ref{desilamm1}) and (\ref{desigenm}) are analogous to (\ref
{SMdesi1}) and (\ref{SMdesi2}).
We begin by proving (\ref{lema}). Note that $ {\sup_{\|\alpha\|_{\tau}
\leq1}} |\sigma_n^2(\pi_{m-1} \alpha) -\sigma
^2(\widehat{\pi}_{\tau,m-1} \alpha)| \leq{\sup_{\|\alpha\|_{\tau}
\leq1}} |\sigma^2(\pi_{m-1} \alpha) - \sigma^2(\widehat{\pi}_{\tau,m-1}
\alpha)| +\vspace*{1pt} {\sup_{\|\alpha\|_{\tau} \leq1}}
|\sigma_n^2(\widehat{\pi}_{\tau,m-1} \alpha) -
\sigma^2(\widehat{\pi}_{\tau,m-1} \alpha)|$. Using
(\ref{S1enmenorque1}) and the fact that if $\|\alpha\|_{\tau} \leq1$,
then $\|\widehat{\pi}_{\tau,m-1} \alpha\|_{\tau} \leq1$, we get that
the second term on the right-hand side converges to 0 almost surely. To
complete the proof of (\ref{lema}), it remains to show that
\begin{equation}\label{falta}
{\sup_{\|\alpha\|_{\tau} \leq1}} |\sigma^2(\pi_{m-1} \alpha) -
\sigma
^2(\widehat{\pi}_{\tau,m-1} \alpha)|\convpp0 .
\end{equation}
Using that $\wphi_j \convpp\phi_{\rob,j}$ and that $\tau\Psi(
\wphi
_j)=\tau\lceil\wphi_j,\wphi_j \rceil\convpp0$, for $1\le j\le m-1$
and arguing as in \citet{s} [see \citet{bali2} for
details], we get that,\vadjust{\goodbreak} for $1\le j \le m-1$,
$
{\sup_{\|\alpha\|_{\tau} \leq1}} \|\langle\alpha, \phi_{\rob
,j}\rangle
\phi_{\rob,j}-\langle\alpha, \wphi_j\rangle_{\tau}\wphi_j\|
\convpp0
$,
entailing that ${\sup_{\|\alpha\|_{\tau} \leq1}} \|\widehat{\pi
}_{\tau
,m-1} \alpha-\pi_{m-1} \alpha\|\convpp0$.
Therefore, using that $\sigma$ is weakly uniformly continuous over the
unit ball, we get easily that (\ref{falta}) holds, concluding the proof
of (\ref{lema}).
As in (a), we will next show that (\ref{desilamm1}) holds. Using again
that $\sigma$ is a scale functional, we get easily that
$\sup_{\alpha\in\mathcal{S}\cap\mathcal{T}_{m-1}} \sigma^2(\alpha) =
\sup_{\alpha\in\mathcal{S}} \sigma^2(\pi_{m-1} \alpha) $, so using
again that $\|\wphi_m\|\le\|\wphi_m\|_\tau=1$, we obtain\vspace*{1pt} that
$\sigma^2(\phi_{\rob,m}) =\break\sup_{\alpha\in\mathcal{S}} \sigma^2(\pi
_{m-1} \alpha) \geq\sigma^2(\pi_{m-1}
{\wphi_m}/{\|\wphi_m\|})\ge\sigma^2 (\pi_{m-1} {\wphi_m})$. From
(\ref{lema}) and the fact that $\|\wphi_m\|_{\tau}=1$, we get that
$b_m=\sigma^2(\pi_{m-1} \wphi_m)-\sigma_n^2 (\widehat{\pi}_{\tau,m-1}
\wphi_m)\convpp0$, and so since $\widehat{\pi}_{\tau,m-1}
{\wphi_m}=\wphi_m$ and $\|\wphi_m\|\le1$, we get that $
\sigma^2(\phi_{\rob,m}) \geq\sigma^2(\pi_{m-1} \wphi_m)= \sigma_n^2
(\widehat{\pi}_{\tau,m-1} \wphi_m) + o_{\mathrm{a.s.}}(1) = \wlam_m +
o_{\mathrm{a.s.}}(1)$, completing the proof of (\ref{desilamm1}).
We will show now (\ref{desigenm}). Note that
$\phi_{\rob,m}\in\mathcal{H}_{\smooth}$, so that $\| \phi_{\rob,m}
\|_{\tau}<\infty$ and $\| \phi_{\rob,m}\|_{\tau}\to\| \phi_{\rob,m}
\|=1$. Using\vspace*{2pt} that $\sigma_{\rob}$ is a scale functional, the fact that
$\wlam_m = \sigma_n^2(\wphi_m)\ge\sigma_n^2(\wphi_m) -\rho\Psi(\wphi
_m)=\sup_{\|\alpha\|_{\tau}=1, \alpha\in\widehat{\mathcal{T}}_{\tau
,m-1}}\{ \sigma_n^2(\alpha) -\rho\Psi(\alpha)\}$ and that for any
$\alpha\in\mathcal{H}_{\smooth}$ such that $\|\alpha\|_{\tau}=1$ we
have that $\| \widehat{\pi}_{\tau,m-1} \alpha\|_{\tau}\le1$, we get
easily that $\wlam_m \ge
\sup_{\|\alpha\|_{\tau}=1}\{\sigma_n^2(\widehat{\pi}_{\tau,m-1} \alpha)
-\rho\Psi(\widehat{\pi}_{\tau,m-1} \alpha)\}$, and so $\wlam_m
\geq({\sigma_n^2( \widehat{\pi}_{\tau,m-1}\phi_{\rob,m} )- \rho\Psi(
\widehat{\pi}_{\tau,m-1} \phi_{\rob,m})}
)/{\|\phi_{\rob,m}\|_{\tau}^2}$. From (\ref{lema}) we obtain that
$d_m=\sigma_n^2(\widehat{\pi}_{\tau,m-1}
\phi_{\rob,m})-\sigma^2(\pi_{m-1} \phi_{\rob,m})\convpp0$. Moreover,
the fact that $\tau\convpp0$ entails that
$\|\phi_{\rob,m}\|_{\tau}\convpp\|\phi_{\rob,m}\|=1$. On the other
hand, using that $\rho\Psi(\wphi_{\ell})\convpp0$, $1\le\ell\le m-1$,
and the fact that $\rho\convpp0$ implies that $\rho\Psi(\phi_{\rob,m})
= o_{\mathrm{a.s.}}(1)$, analogous arguments to those considered in
\citet{pez} allow us to show that $
\rho\Psi(\wpi_{m-1}\phi_{\rob,m})=\break\rho\lceil\wpi_{m-1}\phi_{\rob,m},
\wpi_{m-1}\phi_{\rob,m} \rceil\convpp0$. Hence, we get that
\begin{eqnarray*}
\wlam_m
&\geq&\frac{\sigma^2( {\pi}_{ m-1}\phi_{\rob,m}
)+d_m- \rho
\Psi( \widehat{\pi}_{\tau,m-1} \phi_{\rob,m})}{1+o(1)}
\\
&\geq&\frac{\sigma^2( {\pi}_{ m-1}\phi_{\rob,m})+d_m-
o_{\mathrm{a.s.}}(1)}{1+o(1)}\\
&=&\sigma^2( \phi_{\rob,m}
)+o_{\mathrm{a.s.}}(1) ,
\end{eqnarray*}
where the last equality follows from the fact that ${\pi}_{ m-1}\phi
_{\rob,m}=\phi_{\rob,m}$.
Therefore, from (\ref{desilamm1}) and (\ref{desigenm}), we obtain that
$\wlam_m \convpp\sigma^2(\phi_{\rob,m} )$.
On the other hand, (\ref{lema}) entails that $\wlam_m- \sigma
^2(\wphi
_m)=\sigma_n^2(\wphi_m) - \sigma^2(\wphi_m)\convpp0$, which together
with $\wlam_m\convpp\sigma^2(\phi_{\rob,m})$ implies that $ \sigma
^2(\wphi_m)\convpp\sigma^2(\phi_{\rob,m})$.
To complete the proof of (c), it remains to show that $\tau\Psi(\wphi
_m) \convpp0$ and $\rho\Psi(\wphi_m) \convpp0$. As in (a), we have
that the following inequalities converge to equalities:
\begin{equation}
\label{cotasigmaphim}\qquad
\sigma^2(\phi_{\rob,m}) \geq\sigma^2\biggl(\pi_{m-1} \frac{\wphi
_m}{\|
\wphi_m\|}\biggr)\geq\sigma^2(\pi_{m-1} \wphi_m)=
\wlam_m +
o_{\mathrm{a.s.}}(1) .\vadjust{\goodbreak}
\end{equation}
Using\vspace*{1pt} that $\sigma$ is a scale estimator and that
$\|\wphi_m\|_\tau=1$, we get that $\tau\Psi(\wphi_m) = 1 -
\|\wphi_m\|^2 = 1 -{\sigma^2(\pi _{m-1}\wphi
_m)}/{\sigma^2(\pi_{m-1}\wphi_m /\|\wphi_m\|)} $, which together with
(\ref{cotasigmaphim}) entails that the second term on the right-hand
side is $1+ o_{\mathrm{a.s.}}(1)$ and so, $\tau \lceil\wphi_m,\wphi_m
\rceil\convpp0$, entailing that $\|\wphi _m\| \convpp1$.
On the other hand, we also have that
\begin{equation}
\label{cotasigmawphim}
\wlam_m=\sigma_n^2(\wphi_{m}) \geq\sigma_n^2(\wphi_m
)-\rho
\Psi(\wphi_m)\geq\sigma^2(\phi_{\rob,m}) + o_{\mathrm{a.s.}}(1) ,
\end{equation}
so using that $\wlam_m=\sigma_n^2(\wphi_{m})\convpp\sigma^2
(\phi
_{\rob,m})$, we obtain that $\rho\Psi(\wphi_m)\convpp0$,
concluding the proof of (c).
(d) We have already proved that when $m=1$ the result holds. We
proceed by induction and assume\vspace*{1pt} that $\langle\wphi_\ell, \phi_{\rob
,\ell} \rangle^2 \to1$, $\tau\Psi(\wphi_{ \ell}) \convpp0$ and
$\rho
\Psi(\wphi_{ \ell}) \convpp0$ for $1\le\ell\le m-1$, to show that
$\langle\wphi_m, \phi_{\rob,m} \rangle^2 \to1$. Without loss of
generality, we can assume that $\wphi_\ell\convpp\phi_{\rob,\ell}$,
for $1\le\ell\le m-1$. Denote by $\wtphi_j={\wphi_j}/{\|\wphi_j\|}$.
Then, for $1 \le\ell\le m-1$, $\|\wphi_\ell\|\to1$, and so $\wtphi
_\ell\convpp\phi_{\rob,\ell}$. It suffices to show that $\langle
\phi
_{\rob,m}, \wtphi_m\rangle^2\convpp1$.
Using (c) we have that $\sigma^2(\wphi_m) \convpp\sigma^2(\phi
_{\rob,
m})$ and that $\|\wphi_m\|\convpp1$, and so $\sigma^2(\wtphi_m)
\convpp\sigma^2(\phi_{\rob, m})$. The proof follows now from Lemma
\ref{lemma41} if we show that $\langle\wtphi_m, \wtphi_\ell\rangle
\convpp
0$, $1\le\ell\le m-1$.
Using that $\tau\Psi(\wphi_\ell) \convpp0$, for $1\le\ell\le m-1$,
and that from (c) $\tau\Psi(\wphi_m) \convpp0$ we get that $\tau
\lceil\wphi_\ell,\wphi_m \rceil\convpp0$ for $1\le\ell\le m-1$.
Therefore, the fact that $\langle\wphi_m, \wphi_\ell\rangle_\tau=0$
entails that
$\langle\wphi_m, \wphi_\ell\rangle=\langle\wphi_m, \wphi_\ell
\rangle
_\tau- \tau\lceil\wphi_\ell,\wphi_m \rceil\convpp0$, and so
$\langle
\wtphi_m, \wtphi_\ell\rangle\convpp0$, concluding the proof.
\end{pf*}
\end{appendix}
\section*{Acknowledgments}
We wish to thank the Associate Editor and three anonymous referees for
valuable comments which led to an improved version of the original
paper.
\begin{supplement}[id=suppA]
\sname{Supplement A}
\stitle{Robust functional principal components\\}
\slink[doi]{10.1214/11-AOS923SUPPA}
\sdatatype{.pdf}
\sfilename{aos923\_suppa.pdf}
\sdescription{In this Supplement, we give the proof of some of the
results stated in Sections \ref{consist} and \ref{appen}.}
\end{supplement}
\begin{supplement}[id=suppB]
\sname{Supplement B}
\stitle{Robust functional principal components\\}
\slink[doi]{10.1214/11-AOS923SUPPB}
\sdatatype{.pdf}
\sfilename{aos923\_suppb.pdf}
\sdescription{In this Supplement, we report the results obtained
in the Monte Carlo study for the raw estimators and for the penalized
ones when the smoothing parameters are fixed.}
\end{supplement}
|
{
"timestamp": "2012-03-12T01:01:05",
"yymm": "1203",
"arxiv_id": "1203.2027",
"language": "en",
"url": "https://arxiv.org/abs/1203.2027"
}
|
\section{Introduction}
\label{sec:intro}
In telecommunication, radar and other fields of electronic engineering, it is usually of great
concern about multiband signals, which consist of several narrowband signals modulated by different
carrier frequencies. To acquire such a signal, a common practical method \cite{ref1} is to
demodulate the narrowband signals by their carrier frequencies to baseband, respectively. After
low-pass filtered to reject frequencies from other bands, the narrowband signals are converted to
digital samples at rate corresponding to their actual bandwidths. Thus each band is acquired
separately and the total sampling rate is the sum of the bandwidths. This method can reach the
minimal sampling rate while the carrier frequencies must be known a priori.
When the carrier frequencies are unknown, efficient sampling of the multiband signal is a
challenging task because the Nyquist frequency of the signal may exceed the capability of the
up-to-date analog-to-digital converters (ADCs). In this scenario, several sub-Nyquist sampling
strategies have been proposed to handle this challenge. In \cite{ref2} and \cite{ref3}, a scheme
called multicoset sampling is proposed to perform analog-to-digital conversion at low rate, but
knowledge of the frequency support must be known for recovery. Periodic nonuniform sampling is
another sub-Nyquist sampling strategy where several ADCs are applied to form a high sampling rate
for the multiband signal \cite{ref5}.
In recent years, Compressive Sensing (CS) has been raised and developed to recover sparse signals
from far fewer samples \cite{ref6}. Due to the fact that multiband signal is sparse in frequency domain,
several novel analog-to-digital architectures have been designed based on CS theory. In Random
Demodulator \cite{ref7, ref8}, the original signal is multiplied by a pseudorandom sequence,
integrated and sampled at sub-Nyquist rate. In the recovery stage, CS algorithms are performed to
recover the original signal from these samples.
The Modulated Wideband Converter (MWC) \cite{ref9, ref11, ref12} is another CS-based
sub-Nyquist sampling system. It also consists of two stages: sampling, as depicted in
Fig.~\ref{fig1}, and reconstruction. In the sampling stage, modulated sampling is conducted by
mixers and lowpass filters in multiple channels. In the reconstruction stage, sparsity constraint
algorithms are applied to recover the original signal from the multiple observations. The sampling
rate $f_s$ and the frequency $f_p$ of the periodic mixing function $p_i(t)$ satisfies $f_s=qf_p$
with odd $q$, which is a strategy collapsing $q$ channels to a single one. The strategy allows
designers to trade off between sampling rate and the number of hardware devices. Recent researches
focused on this system include a calibration system with simple structure and low computational
complexity to obtain actual measurement matrix \cite{ref13}.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{MWC.pdf}
\caption{A block diagram of the sampling stage of the Modulated Wideband Converter}\label{fig1}
\end{figure}
In order to fully use the accuracy of an ADC, the quantization reference voltage is an essential parameter to consider about. The reference voltage of an conventional ADC simply equals the maximum value of the input analog signal. In the MWC, however, the quantization takes place at the end of the sampling stage and the maximum of the samples is sensitive to several parameters, especially the randomness of the mixing functions. Thus the reference voltage can not be derived from the input signal intuitively. Moreover, simulation results exploit that the maximum of the samples varies widely even when the input signal is fixed. Therefore, an improper reference voltage may have a severe impact on the performance of the whole system. Though important in practice, few researches have been focused on this problem and the reference voltage in practical implementation is obtained by experiments.
This paper aims to theoretically study the reference voltage of quantizer in the MWC. A reasonable method to acquire reference voltage is proposed. By analyzing the input signal under several assumptions, we draw the conclusion that the reference voltage is proportional to the square root of the collapse parameter $q=f_s/f_p$. Further discussions show
that the assumptions made in the analysis are acceptable and that the conclusion is valid for
arbitrary multiband signals.
This paper is organized as follows. Section 2 briefly introduces the structure of the MWC and related mathematical expressions. Section 3 studies the reference voltage and come to the conclusion under rational assumptions. The theoretical results in Section 3 are numerically validated in Section 4.
\section{The Modulated Wideband Converter}
\label{sec:MWC}
As can be seen from Fig.~\ref{fig1}, the input signal $x(t)$ enters $m$ channels simultaneously. In
the $i$-th channel, it is multiplied by a $T_p$-periodic pseudorandom sequence $p_i(t)$, whose
frequency is $f_p=1/T_p$. Specifically, $p_i(t)$ is chosen as a piecewise constant function that
randomly alternates between the levels $\pm 1$ for each of $M$ equal time intervals. The mixed
signal is then truncated by an ideal lowpass filter with cutoff $1/(2T_s)$. Finally, the filtered
signal $y_i(t)$ is sampled at rate $f_s=1/T_s$. According to \cite{ref11}, the parameters $f_s$ and
$f_p$ satisfy $f_s=qf_p$ with odd $q$. Further assume that $x(t)$ is bandlimited to $[-f_{\rm
NYQ}/2,f_{\rm NYQ}/2]$, where $f_{\rm NYQ}$ is much larger than the total sampling rate of the
system.
Formally, the mathematical description of $p_i(t)$ is
\begin{align}
p_i(t)=\alpha_{ik},\quad &k\frac {T_p} M \le t < (k+1) \frac {T_p} M, \nonumber \\
&0\le k \le M-1, \nonumber
\end{align}
with $\alpha_{ik}\in\{+1,-1\}$ and $p_i(t+nT_p)=p_i(t)$ for every $n\in\mathbb Z$. Since $p_i(t)$
is $T_p$-periodic, it has the Fourier expansion
$$
p_i(t)=\sum_{l=-\infty}^{+\infty}c_{il}{\rm e}^{{\rm j}\frac {2\pi} {T_p} lt},
\vspace{-1em}
$$
where $\{c_{il}\}$ are Fourier coefficients. Define $\phi={\rm e}^{-{\rm j}2\pi/M}$, thus
\begin{equation}\label{eq12}
c_{il}=d_l\sum_{k=0}^{M-1}\alpha_{ik}\phi^{lk},
\vspace{-1em}
\end{equation}
where
$$
d_l=\frac 1 {T_p} \int_0^{\frac{T_p} M}{\rm e}^{-{\rm j}\frac {2\pi} {T_p} lt}\,{\rm d}t.
$$
In \cite{ref11}, it has been proved that the Fourier transform of $y_i(t)$ is expanded as
\begin{equation}\label{eq1}
Y_i(f)=\sum_{l=-L_0}^{L_0}c_{il}X(f-lf_p),\quad f\in\left[-f_s/2,f_s/2\right],
\vspace{-1em}
\end{equation}
where
$$
L_0=\left\lceil \frac {f_{\rm NYQ}+f_s} {2f_p} \right \rceil-1.
$$
\section{Reference Voltage of Quantization}
\label{REF}
\subsection{Preliminary}
We first consider about a basic scenario where the signal is only composed of one modulated
narrowband signal, and then discuss about arbitrary multiband signals. In the sequel, the
Fourier transform of $x(t)$, denoted by $X(f)$, consists of two symmetric bands centered at $\pm
l_0f_p$ respectively, where $(l_0-1/2)f_p\ge f_s/2$. The width of both bands satisfies $B\le f_p$. Formally,
\begin{equation}\label{eq2}
X(f)=X_0(f-l_0f_p)+X_0(f+l_0f_p),
\end{equation}
where
$$
X_0(f)=0,\quad f\notin\left[-f_p/2,f_p/2 \right].
$$
Notice that the structure of each channel is identical, and that the
maximum sample of $y_i[n]$ is no larger than the maximum value of
analog signal $y_i(t)$. As a result, the sampling stage can be
simplified to Fig.~\ref{fig2} when reference voltage is being
analyzed. For convenience, in the sequel, the subscript $i$ which denotes the channel number is omitted.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{Fig2.pdf}
\caption{The simplified sampling stage}\label{fig2}
\end{figure}
For the basic scenario (\ref{eq2}), the series on the right side of
(\ref{eq1}) have only $2q$ nonzero terms. Then
\begin{align}\label{eq3}
Y(f)=\sum_{l=l_1}^{l_1+q-1} \left[c_l X(f-l f_p)+c_{-l}X(f+l f_p)\right],\\
\quad f\in\left[-f_s/2, f_s/2\right], \nonumber
\end{align}
where $l_1=l_0-(q-1)/2$. Fig.~\ref{fig3} exploits the simplification
from (\ref{eq1}) to (\ref{eq3}) intuitively.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{Fig3.pdf}
\caption{The simplification from (\ref{eq1}) to (\ref{eq3})
(Parameters: $q=3$, $l_0=3$, $l_1=2$).}\label{fig3}
\end{figure}
Considering (\ref{eq3}) in time domain, one has
\begin{align}\label{eq4}
y(t)=&\sum_{l=l_1}^{l_1+q-1}c_l\int_{-f_s/2}^{f_s/2}X(f-lf_p){\rm e}^{{\rm j}2\pi
ft}\,{\rm d}t \nonumber\\
&+\sum_{l=l_1}^{l_1+q-1}c_{-l}\int_{-f_s/2}^{f_s/2}X(f+lf_p){\rm e}^{{\rm j}2\pi
ft}\,{\rm d}t.
\end{align}
Since $[-f_s/2,f_s/2]$ contains only one sideband of $X(f-lf_p)$,
Hilbert transform can be applied to represent such single sideband
signal. Setting ${\hat x}(t)=x(t)* \left\{ 1/(\pi t) \right\}$ as Hilbert
transform of $x(t)$, ${\hat X}(f)$ denotes the Fourier transform of
${\hat x}(t)$. According to the definition of Hilbert transform,
$[X(f)-{\rm j}{\hat X}(f)]/2$ represents the negative-frequency band of
$X(f)$. Hence
\begin{align}\label{eq5}
&\sum_{l=l_1}^{l_1+q-1}c_l \int_{-f_s/2}^{f_s/2} X(f-l f_p){\rm e}^{{\rm j}2\pi ft}\,{\rm d}t \nonumber \\
=&\sum_{l=l_1}^{l_1+q-1}c_l {\rm e}^{{\rm j}2\pi lf_pt} \int_{-\infty}^{+\infty} \frac {X(f)-{\rm j}{\hat X}(f)} 2 {\rm e}^{{\rm j}2\pi ft}\,{\rm d}t \nonumber \\
=&\sum_{l=l_1}^{l_1+q-1}c_l {\rm e}^{{\rm j}2\pi lf_pt} \frac {{\tilde x}^*(t)} 2,
\end{align}
where ${\tilde x}(t)=x(t)+{\rm j}{\hat x}(t)$ is the analytic
signal of $x(t)$.
Dealing with the second term on the right side of (\ref{eq4}) in the
same way, it can be derived that
\begin{align}\label{eq6}
\left|y(t)\right|=&\left|\sum_{l=l_1}^{l_1+q-1}\Re\left\{c_l{\rm e}^{{\rm j}2\pi lf_pt}{\tilde x}^*(t)\right\}\right| \nonumber\\
=&\left|{\tilde x}(t)\right| \left|\sum_{l=l_1}^{l_1+q-1} \left|c_l\right|\cos\left(\theta_l+2\pi lf_pt-\theta_{{\tilde x}}(t)\right)\right|,
\end{align}
where $c_l=\left|c_l\right|{\rm e}^{{\rm j}\theta_l}$, ${\tilde
x}(t)=\left|{\tilde x}(t)\right|{\rm e}^{{\rm j}\theta_{\tilde x}(t)}$ and
$\Re(\cdot)$ denotes the real part of its argument. The reference
voltage is determined by the maximum value of $\left|y(t)\right|$.
Define $\left|y\right|_{\max}$ as the maximum value of
$\left|y(t)\right|$. $|y|_{\rm max}$ is a random variable, due to the fact that the mixing function $p(t)$ randomly
alternates between $\pm 1$. Mathematically, (\ref{eq6}) indicates
that the randomness of $|y|_{\rm max}$ results from the
randomness of $\{c_l\}$, which are Fourier coefficients of $p(t)$. The threshold of $\left|y\right|_{\max}$ is
chosen such that the probability
\begin{equation}\label{eq7}
{\rm P}\left\{\left|y\right|_{\rm max}\le \left|y\right|_{\rm
th}\right\}= P_0,
\end{equation}
where $P_0$ is a constant. It is reasonable to use
$\left|y\right|_{\rm th}$ as reference voltage with a proper $P_0$,
say $99\%$.
\subsection{Main Contribution}
The following proposition reveals the property of the reference
voltage defined as (\ref{eq7}).
\begin{Proposition}\label{Pro1}
In the MWC mentioned above, consider fixed input signal (\ref{eq2}).
Further assume\\
\noindent(a) $\left|y(t)\right|$ and $\left|{\tilde
x}(t)\right|$ reach their maximum at the same time;\\
\noindent(b) $\{\theta_l\}$ are i.i.d. random variables with uniform
distribution on $[-\pi,\pi]$, and $\{\left|c_l\right|\}$ are i.i.d.
random variables independent from $\{\theta_l\}$.
Then the reference voltage $\left|y\right|_{\rm th}$ defined by
(\ref{eq7}) is proportional to $\sqrt q=\sqrt{f_s/f_p}$.
Specifically,
\begin{equation}\label{eq8}
\left|y\right|_{\rm th}=\sqrt{q}\left|{\tilde x}(t_0)\right|\sigma Y_{\rm th},
\end{equation}
where $t_0={\rm argmax}_t\left\{{\tilde x}(t)\right\}$,
$\sigma^2={\rm var}\left\{ |c_l|\cos \theta_l\right\}={\rm
var}\left\{ \Re \{c_l\}\right\}$, and $Y_{\rm th}$ satisfies
$$
2\Phi(Y_{\rm th})-1=\frac 1 {\sqrt{2\pi}} \int_{-Y_{\rm th}}^{Y_{\rm th}} {\rm e}^{-t^2/2}\,{\rm d}t=P_0.
$$
where $\Phi(x)$ is the cumulative distribution function of the standard normal distribution.
\end{Proposition}
\begin{proof}
According to assumption (a), denoting $\theta_l+2\pi lf_pt_0-\theta_{{\tilde
x}}(t_0)$ by $\hat {\theta}_l$, one has
$$
|y|_{\rm max}=\left|{\tilde x}(t_0)\right|
\left|\sum_{l=l_1}^{l_1+q-1} \left|c_l\right|\cos\left(\hat {\theta}_l\right)\right|.
$$
Notice that $\hat {\theta}_l$ is also uniform distributed on $[-\pi,\pi]$ as $\theta_l$.
Setting $z_l=|c_l|\cos \hat{\theta}_l$, the equivalent problem is to
find $Z_{\rm th}=|y|_{\rm th}/\left|{\tilde x}(t_0)\right|$ such
that
\begin{equation}\label{eq9}
{\rm P}\left\{ \left| \sum_{l=l_1}^{l_1+q-1} z_l \right|\le Z_{\rm th} \right\}= P_0.
\end{equation}
Since ${\rm E}\{\cos \hat{\theta}_l\}$ equals $0$ when $\hat{\theta_l}$ is uniform distributed on $[-\pi,\pi]$, $\{z_l\}$ are zero-mean i.i.d. random variables.
According to Central Limit Theorem \cite{ref14}, the variable
$$Z=\sum_{l=l_1}^{l_1+q-1} \frac{z_l}{\sigma \sqrt q}$$ satisfies
standard normal distribution when $q$ approaches infinity. Hence
\begin{align}\label{eq10}
{\rm P}\left\{ \left| \sum_{l=l_1}^{l_1+q-1} z_l \right|\le Z_{\rm th} \right\}
&={\rm P}\left\{ -\frac {Z_{\rm th}} {\sigma \sqrt{q}} \le Z \le \frac {Z_{\rm th}} {\sigma \sqrt{q}} \right\} \nonumber \\
&=\frac 1 {\sqrt{2\pi}} \int_{-Y_{\rm th}}^{Y_{\rm th}} {\rm e}^{-t^2/2}\,{\rm d}t,
\end{align}
where $Y_{\rm th}=Z_{\rm th}/\left(\sigma\sqrt{q}\right)$. Combining
(\ref{eq9}) and (\ref{eq10}), it is obvious that $Y_{\rm th}$ is a
constant satisfying
$$
\frac 1 {\sqrt{2\pi}} \int_{-Y_{\rm th}}^{Y_{\rm th}} {\rm e}^{-t^2/2}\,{\rm d}t=P_0.
$$
Thus
$$
|y|_{\rm th}=\left|{\tilde x}(t_0)\right|Z_{\rm th}=\sqrt{q}\left|{\tilde x}(t_0)\right|\sigma Y_{\rm th},
$$
and this completes the proof.
\end{proof}
\subsection{Discussion}
Assumption (a) is drawn from the mechanism of the system. Mapping (\ref{eq2}) to time domain, one has
$x(t)=x_0(t)\cos \left(2\pi l_0f_p t\right)$.
Thus
\begin{align}\label{eq11}
\left|{\tilde x}(t)\right|&=\left|x(t)+{\rm j}{\hat x}(t)\right| \nonumber \\
&=\left|x_0(t)\cos \left(2\pi l_0f_p t\right) + {\rm j}x_0(t)\sin \left( 2\pi l_0f_p t \right)\right| \nonumber \\
&=\left|x_0(t)\right|.
\end{align}
Equation (\ref{eq11}) shows that $\left|{\tilde x}(t)\right|$ and $\left|x_0(t)\right|$ reach the maximum at the same time. On the other hand, $y(t)$ is obtained through three steps from
$x_0(t)$: modulated by $\cos(2 \pi l_0f_p t)$ with large $l_0$
typically, multiplied by $p(t)$ and lowpass filtered. In the first step,
modulation with the carrier at high frequency can hardly change the instant when the signal reaches the maximum. The second step does not make any change because $|p(t)|=1$. For the last step, since $f_s\ge f_p$, the lowpass
filter does not affect the ``envelope'' of the signal. Based on the
analysis above, it is convincing that $|y(t)|$ and $|x_0(t)|$ also
reach the maximum at the same time, which leads to assumption (a).
Assumption (b) is based on the observation of $\{c_l,l_l\le l\le
l_1+q-1\}$. Considering (\ref{eq12}), since $l_1$ is far larger than $q$, one has $l_1+q-1\approx l_1$, which results in the approximation that
$\{c_l,l_1\le l\le l_1+q-1\}$ are i.i.d. Furthermore, the
distribution of $\theta_l$ is determined mainly by the sum of $\alpha_k\phi^{lk}, 0\le k\le M-1$. In the complex plane,
$\{\phi^{lk}, 0\le k\le M-1\}$ represents $M$ points located
uniformly on the unit circle and $M$ is typically very large. Therefore
assume that the argument of their linear combination satisfies
uniform distribution is an acceptable way to simplify the analysis.
Now we discuss about arbitrary multiband signals. Consider a multiband signal $y(t)$ which consists of $N$ basic signals denoted by (\ref{eq2}) with the same maximum value. If there is no overlap in time domain among these $N$ signals, the problem can be reduced to study one input signal. When $N$ signals reach their maximum at the same time, say $t_0$, the following analysis intuitively shows that the reference voltage is proportional to $\sqrt{Nq}$. In frequency domain, according to (\ref{eq3}), $[-f_s/2,f_s/2]$ has $q$ shift copy of each band. Thus, observing the multiband signal within a short time interval around $t_0$, the energy of $Y(f)$ is $Nq$ times the energy of each basic signal. Hence the amplitude $|y(t_0)|$ is $\sqrt{Nq}$ times the maximum of each basic signal.
\section{Simulation}
\label{sec:simu}
The following simulation calculates the reference voltage $\left|y\right|_{\rm th}$ numerically with $P_0=99\%$. For specific parameter settings, the experiment is conducted 5000 trials and the value which is larger than $99\%$ of the results is chosen as $\left|y\right|_{\rm th}$.
The number of the channels is $m=30$ and the frequency of the mixing function is $f_p\approx 51.3{\rm MHz}$. The input $x(t)$ is a multiband signal consisting of $N$ pairs of bands, defined as
\begin{equation}\label{eq13}
x(t) = \sum_{i=1}^{N}\sqrt{EB}{\rm sinc}\left(B\left(t-\tau\right)\right)\cos\left(2\pi f_i \left(t-\tau\right)\right)
\end{equation}
where $E=10$, $B=50 {\rm MHz}$, $\tau = 0.4 \rm{\mu s}$, $f_i=\left\{25if_p\right\}$. The reference voltage is calculated under the parameter choice $N=\{1,2,3\}$ and $\{q=2q'+1,\quad q'=0,1,...,9\}$.
The results are plotted in Fig.~\ref{fig4}. It is shown that $\left|y\right|_{\rm th}/ \sqrt{q}$ is almost constant when $q \le 3$. Though (\ref{eq10}) is obtained under the condition that $q$ approaches infinity, the results indicate that $q$ need not be very large. Furthermore, for each fixed $N$ respectively, calculate arithmetic mean of $\left|y\right|_{\rm th}/ \sqrt{q}$. The ratio of the results is $1:1.37:1.63$, which is close to $1:\sqrt{2}:\sqrt{3}$. The results above correspond to the theoretical analysis and the discussions.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{Result.pdf}
\caption{Reference voltage with respect to collapse parameter $q$ and the number of signals $N$}\label{fig4}
\end{figure}
\section{Conclusion}
\label{sec:conclu}
In this paper, we propose a method to acquire the reference voltage of the MWC. Then we
theoretically analyze the reference voltage under rational assumptions of the input signal and the mixing functions. The conclusion is drawn that the quantization reference voltage is proportional to the square root of collapse parameter $q=f_s/f_p$. Furthermore, discussions and simulation results show that the conclusion is valid for arbitrary multiband signals. Specifically, when the multiband signal consists of $N$ pairs of bands, the reference voltage is proportional to $\sqrt{Nq}$.
|
{
"timestamp": "2012-03-09T02:01:33",
"yymm": "1203",
"arxiv_id": "1203.1711",
"language": "en",
"url": "https://arxiv.org/abs/1203.1711"
}
|
\section{Introduction}
\label{sec1}
Quantum metrology is a research field that examines the characteristic fundamental properties of measurements under the laws of quantum mechanics \cite{NOON05,NOON01}. The ultimate goal of this is to achieve measurements at the information theoretical bounds allowed by the laws of quantum mechanics, far beyond their classical counterparts. For optical system the classic and extremely well studied example are NOON states \cite{NOON05,NOON01,NOON02,NOON03,NOON04,NOON06} whose performance allows them to measure linear phase shifts at the Heisenberg limit. Several
theoretical studies have recently investigated the role of non-linearity to help improve the limits of phase enhancement in linear systems \cite{gerry07,Nonlinear01,Nonlinear02,Nonlinear03,Mitchell10} and the first demonstration of this so-called super-Heisenberg scaling has been shown \cite{Demo11}. The particle-loss and decoherence mechanisms are, however, not fully explored in combined linear and non-linear interferometers, even theoretically \cite{WalmsleyNature10,WalmsleyPRL09,WalmsleyPRA09,Escher11}.
It is not only NOON states that allow linear phase measurements at the Heisenberg limit. Entangled coherent states (ECSs) \cite{Barryarxiv,gerry95,Barryarxiv1,Barryarxiv3,Enk01,Jeong01,munro02,Peter05} are also able to do this \cite{Gerry02,Gerry01,Gerry03} and can outperform that of NOON states in the region of very modest particle numbers with a linear phase operation \cite{Joo11,NewArxiv12}. An important case of entangled coherent states are the two-mode path entangled states, a state analogous to a NOON state, but with one of two modes containing a coherent state rather than a Fock state \cite{Gerry01}. This particular ECS can be represented as a superposition of NOON states with different
photon numbers \cite{Gerry02}. Using linear optical elements, the phase sensitivity of ECSs outperforms that of NOON and bat states \cite{Jess10}, both without\cite{Gerry02,Gerry01,Gerry03} and with losses \cite{Joo11}, because coherent states maintain their properties in the presence of loss. Given all this recent work, a natural question arises regarding comparison of the phase enhancement for ECSs, compared to NOON and other states, in the case of non-linear phase shifts \cite{gerry07}.
In this paper we are going to investigate the non-linear phase enhancement resulting from a generalized non-linearity characterized by an exponent $k$ ($k>0$) on four quantum states: NOON, even and odd ECSs and an approximated ECS (AECS). The AECS is created from a photon subtracted squeezed state and experimentally feasible to realize \cite{photon_sub01,photon_sub02}. Potential enhancements will be quantified with the quantum Fisher information \cite{Braunstein94,Braunstein96}. To begin we will consider a two-mode pure state $|\psi \rangle_{12}$ and a generalized non-linear phase shifter $U(\phi,k)$ given by
\begin{eqnarray}
U(\phi,k) &=& {\rm e}^{i \phi (a^{\dag}_2 a_2)^k},
\label{eq:Phase01}
\end{eqnarray}
where $a^{\dag}_i$ ($a_i$) is a creation (annihilation) operator in spatial mode $i$ \cite{BillPRA2010} (see the details in Section \ref{sec2-1}). The exponent $k$ represents the order of the non-linearity. For example, $k = 1$ corresponds to a linear phase shift on the state, $k =2$ a Kerr phase shift and $k \neq 2$ gives a more general non-linear effect in a phase operation. When the generalized phase operation $U(\phi,k)$ is applied to mode 2 of $|\psi \rangle_{12}$, the resultant state is equal to
\begin{eqnarray}
&& |\psi^k (\phi) \rangle_{12} = (\openone \otimes U(\phi,k) )
|\psi \rangle_{12}.
\end{eqnarray}
Now according to phase estimation theory \cite{Braunstein94,Braunstein96}, the phase uncertainty is bounded by the quantum Fisher information
\begin{eqnarray}
\delta \phi & \ge & {1\over \sqrt{F^Q}},
\label{eq:Fisher01}
\end{eqnarray}
where the value of $F^Q$ for pure states is simply given by
\begin{eqnarray}
&&F_{}^{Q} = 4\Big[ \langle \tilde{\psi}^k |\tilde{\psi}^k \rangle - |\langle \tilde{\psi}^k |\psi^k (\phi) \rangle |^2 \Big] = 4 (\Delta n^k)^2,
\label{eq:Fisher02}
\end{eqnarray}
with $| \tilde{\psi}^k \rangle = \partial |\psi^k (\phi) \rangle/\partial \phi$ and $(\Delta n^k )^2 = \langle (n^k)^2 \rangle - \langle n^k \rangle^2$ ($\langle n^k \rangle= {}_{12} \langle \psi | ( a^{\dag}_2 a_2 )^k |\psi\rangle_{12}$). It is important to note that $\langle n^1 \rangle$ denotes an average (or mean) photon number.
To allow a fair comparison of the phase sensitivity among the various different quantum states under consideration, we will use the same average photon number in one of two modes as the physical resource count for the states \cite{Gerry02,BillPRA2010,gerry2000,gerry2002}. For pure states, we shall demonstrate an inequality for the sensitivity among the three states: NOON (least sensitive), odd ECS and even ECS (most sensitive). Furthermore we will show that in the limit of large $\langle n^1 \rangle$ the AECS slightly better than the other three states. We shall also consider a small amount of loss for the dispersive and non-linear interferometer arm, because the non-linear medium providing the phase operation will generally also provide a scattering effect (e.g., particle losses) \cite{ref1:keq2}. We shall demonstrate that, analogous to the linear case ($k=1$) \cite{Joo11,NewArxiv12}, the phase enhancement of ECSs still outperforms that of NOON states, even for non-linear cases ($k\neq1$).
For a physical realization, it is known that the ideal ECS can be generated by mixing through a 50:50 beam-splitter (BS) a coherent state and a coherent state
superposition (CSS) \cite{Peter05}, given by
\begin{eqnarray}
|CSS_{\pm} (\alpha) \rangle = {N}^{\pm}_{\alpha} (|\alpha \rangle \pm |{\rm -} \alpha \rangle)
\end{eqnarray}
where $|\alpha \rangle$ is a coherent state with amplitude $\alpha$ and ${N}^{\pm}_{\alpha} = 1 / \sqrt{2(1 \pm {\rm e}^{-2|\alpha|^2})}$. Since the CSS with small $\alpha$ has been already demonstrated in experiments \cite{Grangier09,Polzik06,Furusawa08,Knill10}, the scheme of building AECSs with a modest photon number may be experimentally feasible with very high fidelity in the near future \cite{LundPRA2004}. Several experiments have demonstrated that non-linear phase operations can be realized in various set-ups. For example, self-Kerr phase modulation ($k=2$) has been measured as a function of electric field amplitudes in waters, fibers, Nitrobenzene, Rydberg states, etc. \cite{ref1:keq2,ref2:keq2,ref3:keq2}. Notably, the phase shift dependent upon the applied field clearly follows theoretical predictions in the case of a Rydberg electro-magnetically induced transparency medium (see in Fig.~3 in Ref.~\cite{ref1:keq2}).
The remainder of this paper is organized as follows. In Sec. \ref{sec2-1}, the mathematical formalism for generalized phase shifters is described. The phase enhancements of pure ECSs is discussed in the case of ideal preparation in Sec. \ref{sec2-2}. A feasible approach to implementing the AECS is given in Sec.~\ref{sec2-3}. In Sec.~\ref{sec3} we also investigate the effects of loss on the phase enhancement behavior. Finally, in Sec.~\ref{sec4} we summarize and discuss our results.
\section{Optimal phase estimation using nonlinearity in pure states}
\label{sec2-1}
Let us first discuss the validity of Eq.~(\ref{eq:Phase01}) for general $k$.
The generalised phase operation is formed by
\begin{eqnarray}
U(\phi) = \exp \Big[ i \hat{H} (\phi) \,t / \hbar \Big],
\end{eqnarray}
where the total Hamiltonian is equal to
\begin{eqnarray}
\hat{H} (\phi) &=& \hat{H}_0 + \hat{H}_{int} (\phi),
\label{Hamiltonian0}
\end{eqnarray}
consisting of the unperturbed Hamiltonian represented by
\begin{eqnarray}
\hat{H}_0 &=& \int d^3 r \Big[ \frac{1}{2\mu_0} |\hat{B}|^2 + \frac{\epsilon_0 }{2} |\hat{E}|^2 \Big] = \hbar \omega \Big( a^{\dag} a + \frac{1}{2}\Big), ~~\label{Hamiltonian00}
\end{eqnarray}
for mode frequency $\omega$ and an interaction Hamiltonian given by expanding the polarization of the non-linear medium
\begin{eqnarray}
\hat{H}_{int} (\phi) &=& \int d^3 r \, \Big[ \hat{E} \cdot \hat{P} \Big] = \int d^3 r \, \hat{E} \Big[ \sum_{j=1}^{\infty} \frac{\chi^{(j)}}{j+1} (\hat{E})^j \Big], ~~~~ \label{Hamiltonian01}
\end{eqnarray}
where $\chi^{(j)}$ is the $j$-th order susceptibility of the medium \cite{Drummond}. A single-mode electric field is given by
\begin{eqnarray}
\hat{E} = i \sqrt{\frac{\hbar \omega}{2 \epsilon_0}} \Big( a u(r) - a^{\dag} u^*(r) \Big), \label{E01}
\end{eqnarray}
where $u(r)$ is the mode function. Due to the lack of phase matching, the single-mode assumption and the rotating wave approximation, we may
neglect the terms in $\chi^{(2x)}$ ($x$: positive integer) \cite{Drummond}, and then,
\begin{eqnarray}
U (\phi,k) = \exp \Big[ i \omega t (a^{\dag} a + \frac{1}{2})\Big] \prod_{k=1}^{\infty} \exp \Big[ i \phi^{(k)} \Big( a^{\dag} a \Big)^k \Big],~~~~~
\end{eqnarray}
where the phase parameter of the non-linearity $k$ is
\begin{eqnarray}
\phi^{(k)} &=& t \int d^3 r \sum_{x=1}^{\infty} {\cal{F}} \Big( \chi^{(2x-1)} \Big) .
\label{phik}
\end{eqnarray}
We note that $ {\cal{F}} ( \chi^{(2x-1)} )$ is a function of $\chi^{(2x-1)}$. Therefore, the expression of the non-linear phase operation in Eq.~(\ref{eq:Phase01}) is appropriate for fixed $k$.
A well-known example of a non-linear phase operation is given by the Kerr interaction for $k=2$ \cite{Gerry02,Gerry01,Haus89}. In an interaction picture that removes the linear dynamical phase, the non-linear component is
\begin{eqnarray}
U(\phi,2) = \exp \Big[ i \phi^{(2)} \Big( a^{\dag} a \Big)^2 \Big],
\end{eqnarray}
with
\begin{eqnarray}
\phi^{(2)} &=& t \int d^3 r \Big( \frac{3}{2} \chi^{(3)}+ 5 \chi^{(5)} \Big),
\end{eqnarray}
where the interaction Hamiltonian is truncated after the fifth-order susceptibility $\chi^{(5)}$.
\subsection{Ideal (theoretical) cases}
\label{sec2-2}
\begin{figure}[t]
\center
\includegraphics[width=270px]{Allin_0112.eps}
\caption{ (color online). The plots show the inequality $\delta
\phi_{N^k} \ge \delta \phi_{E^k_{-}} \ge \delta \phi_{E^k_{+}}$
with respect to $N=2\langle n^1_N \rangle =\langle n^1_{E_{\pm}}
\rangle$ ($k=1,2,3$). \label{fig:Ideal01} }
\end{figure}
We now need to review and calculate the phase enhancements using the quantum
Fisher information for pure (no loss) cases of the NOON state and ECSs. The NOON state is defined by \cite{NOON03}
\begin{eqnarray}
|\psi_{N}\rangle_{12} = \frac{1}{\sqrt{2}} (|N\rangle_1 |0\rangle_2 + |0\rangle_1 |N\rangle_2),
\end{eqnarray}
where $|N\rangle$ is a number state with photon number $N$. After a generalized phase shifter $U(\phi,k)$ is applied in mode 2, the resulting state is given by $|\psi_{N}^{k} \rangle_{12} =\Big( \openone \otimes U(\phi,k) \Big)|\psi_{N} \rangle_{12}$. From Eq.~(\ref{eq:Fisher02}), the quantum Fisher information of the pure NOON states with a non-linearity of order $k$ is given by
$F_{N^{k}}^{Q}=1/N^{2k}$ and
\begin{eqnarray}
\delta \phi_{N^k} > {1 \over N^{k}}.
\label{eq:FINOON01}
\end{eqnarray}
Similarly for the even and odd ECSs defined by
\begin{eqnarray}
&&|ECS_{\pm} (\alpha_{\pm}) \rangle = {\cal{N}}^{\pm}_{\alpha_{\pm}} \Big[|\alpha_{\pm} \rangle_{1}
|0\rangle_2 \pm |0 \rangle_1 | \alpha_{\pm} \rangle_2 \Big],
\label{eq:Coherent01}
\end{eqnarray}
with amplitude $\alpha_{+}$ ($\alpha_{-}$) for even (odd) ECS and ${\cal{N}}^{\pm}_{\alpha_{\pm}} = 1 / \sqrt{2(1 \pm {\rm e}^{-|\alpha_{\pm}|^2})}$. After the phase shifter $U(\phi,k)$ is performed in mode 2, we find that the resulting state is given by
\begin{eqnarray}
|ECS^{k}_{\pm} (\alpha_{\pm}, \phi) \rangle_{12} = \Big(\openone \otimes U(\phi,k) \Big) |ECS_{\pm} (\alpha_{\pm}) \rangle_{12}.
\end{eqnarray}
The quantum Fisher information is then given by
\begin{eqnarray}
&&F_{E^{k}_{\pm}}^{Q} = 4 f_{\alpha_{\pm}} \left[ \sum_{n=0}^{\infty} { n^{2k} (\alpha_{\pm})^{2n} \over n!} - f_{\alpha_{\pm}} \left(
\sum_{n=0}^{\infty} { n^k (\alpha_{\pm})^{2n} \over n!} \right)^2 \right]. \nonumber \\
\label{eq:QFECS}
\end{eqnarray}
for $f_{\alpha_{\pm}} = {\rm e}^{- {|\alpha_{\pm}|^2 }} ({\cal{N}}^{\pm}_{\alpha_{\pm}})^2$ and
\begin{eqnarray}
\delta \phi_{E^{k}_{\pm}} > {1 \over \sqrt{F_{E^{k}_{\pm}}^{Q}}}.
\label{eq:FIECS01}
\end{eqnarray}
In order to compare the phase sensitivity of the different states, we take into account the same {\it average} particle
number of the states \cite{Gerry02,BillPRA2010,gerry2000,gerry2002} in an arm
\begin{eqnarray}
\langle n^1_N \rangle = \langle n^1_{E_{\pm}} \rangle = {N \over 2} = \big({\cal{N}}^{\pm}_{\alpha_{\pm}}\big)^2 \cdot |\alpha_{\pm}|^2, \label{APN01}
\end{eqnarray}
where $\langle n^1_{E_{\pm}} \rangle= \langle ECS_{\pm} (\alpha_{\pm}) | a^{\dag}_2 a_2 |ECS_{\pm} (\alpha_{\pm}) \rangle$ and in general $\alpha_{+} \neq \alpha_{-}$. In Fig.~\ref{fig:Ideal01}, the values of optimal phase estimation are plotted for the three quantum states and satisfied with the inequality
\begin{eqnarray}
\delta \phi_{N^k} & \ge & \delta \phi_{E^k_{-}} \ge \delta \phi_{E^k_{+}}
\label{inequality01}
\end{eqnarray}
for any $N$ and $k$. The first inequality $\delta \phi_{N^k} \ge \delta \phi_{E^k_{-}}$ shows the pattern of the difference $\delta \phi_{N^k} - \delta \phi_{E^k_{-}}$ for $k=1,2,3$ in Fig.~\ref{fig:Delta_keq123}. Note that $\delta \phi_{E^k_{\pm}}$ approaches to $\delta \phi_{N^k}$ because of $|ECS_{\pm} (\alpha) \rangle \approx |\psi_{N}\rangle$ for larger $N$ and that $\delta \phi_{E^k_{\pm}}$ is a continuous value while $\delta \phi_{N^k}$ exists discretely due to integer $N$.
\begin{figure}[b]
\centering
\includegraphics[width=220px]{Delta_keq123.eps}
\caption{ (color online). Difference of the optimal phase estimation between $|\psi_N \rangle$ and $|ECS^{k}_{-} (\alpha_-) \rangle$ such as $\delta \phi_{N^k} - \delta \phi_{E^k_-}$ for $k=1,2,3$.} \label{fig:Delta_keq123}
\end{figure}
\subsection{Preparation of an approximate ECS}
\label{sec2-3}
We now present an optical set-ups to create an AECS with a very high fidelity to an odd ECS, based on current optical technology methods. We also compare the phase enhancement of the AECS with the other states. As shown in Fig.~\ref{fig:Prepare01}, two steps are required for AECS preparation. First, we create a photon-subtracted quantum state, with high fidelity to an odd CSS, given by the scheme in Ref.~\cite{LundPRA2004}. The generation of squeezed vacuum states can
be given by degenerate parametric down-conversion utilizing non-linearity \cite{Peter05} and a series of experimental results shows that modest strengths of squeezing through second-harmonic generation is achievable with the current technology \cite{Schnabel10,Vahlbruch,Chelkowski}. The scheme of single-photon subtraction through an unbalanced BS from a squeezed vacuum has been already demonstrated \cite{photon_sub01,photon_sub02}. Finally, generation of the odd ECS follows from the well known technique of mixing a traveling CSS with a controlled coherent state through a 50:50 beam-splitter (BS).
\begin{figure}[t]
\hspace{-1.7cm}
\includegraphics[angle=-90,width=210px]{PreparationFig1.eps}
\vspace{-0.7cm} \caption{ (Color online) Schematics of generating
$|AECS (r, \alpha_A) \rangle$ from a squeezed vacuum $S(r)|0\rangle$ and a coherent state $|\alpha \rangle$. After single-photon subtraction at stage 1, the state $\ket{ ACSS (r) }$ is very similar to $|CSS_- (\alpha) \rangle$. The coherent state in stage 2 has the same amplitude of the old CSS ($BS^{\eta}$ and
$D$ are an unbalanced BS with transmission rate $\eta$ and a single-photon detector).
\label{fig:Prepare01}}
\end{figure}
In Fig.~\ref{fig:Prepare01}, the schematics shows how to generate $|AECS (r, \alpha_A) \rangle$ from a squeezed vacuum $S(r)|0\rangle$ and a coherent state $|\alpha \rangle$ ($S(r)=\exp\left[-{r\over 2} \left( a^2-(a^{\dag})^2\right)\right]$ and $r$ is a squeezing parameter). It was shown that the fidelity between a squeezed single photon state $S(r)\ket{1}$ and an odd CSS $|CSS_{-} (\alpha) \rangle$ with small $\alpha$ is extremely high \cite{LundPRA2004} and that a photon-subtracted squeezed vacuum state $a S(r)\ket{0}$ is identical to $S(r)\ket{1}$ \cite{Jeong05}. We begin by preparing a squeezed vacuum $S(r)|0\rangle$ and then performing single-photon subtraction by using $BS^{\eta}$ ($\eta$: transmission rate) and a single-photon detector. The resultant state is called an ACSS possessing a very high fidelity compared with the ideal odd CSS. In detail, when a single photon is detected, the resultant state $\ket{ ACSS (r_0)}= a S(r_0)\ket{0}$ ($r_0 = {\rm arcsinh} [2\alpha_0 /3]$) is given by
\begin{eqnarray}
\label{eq:normalized} \ket{ ACSS (r_0) } = f_r \sum_{k=0}^{\infty}\frac{\sqrt{(2k+1)!}}{2^k \cdot k!}(\tanh r_0)^k\ket{2k+1},~~~~~
\end{eqnarray}
for $f_r =(1-\tanh^2 r_0)^{3/4}$ with the maximized fidelity between $\ket{ ACSS (r_0) }$ and $|CSS_{-}(\alpha_0) \rangle$.
In the second stage, the odd AECS can be built with the generated ACSS $\ket{ ACSS (r_0)}_1$ mixed with an extra coherent state $|\alpha_0 \rangle_2$ through an 50:50 BS. The state is written as
\begin{eqnarray}
\label{eq:AECS01}
\ket{{AECS} (r_0,\alpha_A)}&=&\sum_{m=1}^\infty \sum_{m'=0}^{m-1} \Big[ H_{m,m'} (\ket{m}\ket{m'} - \ket{m'}\ket{m}) \Big], \nonumber \\
&\approx& \ket{ECS_- (\alpha_-)},
\end{eqnarray}
where $H_{m,m'}$ is the coefficient of the state in Fock basis ($\alpha_A = \sqrt{2} \alpha_0$). As shown in Fig.~\ref{fig:Hmn}, the resultant state $\ket{{AECS} (r_0,\alpha_A)}$ is approximately equivalent to the desired odd ECS $|ECS_{-} (\alpha_-) \rangle$ with high fidelity ($\approx 0.975$) if $\alpha_- = \alpha_A \approx 1.9807$. The state $\ket{{AECS} (r_0,\alpha_A)}$ consists of the ideal ECS ($m'=0$) and the unbalanced photon states called $m$ and $m'$ states ($m' \neq 0$) \cite{mandmstate}. In other words, the outcome state contains a superposition of NOON states for $m'=0$ while it also includes the states possessing unbalanced photon numbers in both modes for $m' \neq 0$. Then, after the generalized phase shifter $U(\phi,k)$ in mode 2, we can estimate the phase enhancement of the final state given by $\ket{{AECS}^{k} (r_0, \alpha_A,\phi)} = (\openone \otimes U(\phi,k) )\ket{{AECS} (r_0,\alpha_A)}$.
\begin{figure}[b]
\hspace{-0.5cm}
\includegraphics[width=250px]{Hmn4.eps}
\caption{ (color online). The coefficient $H(m,m')$ of $|AECS (r_0,\alpha_A)\rangle$ ($\alpha_A \approx 1.9807$). It clearly shows that the major contribution of the NOON state and the minor of $m$ and $m'$ states. The blue (red) colour indicates a positive (negative) value.
\label{fig:Hmn} }
\end{figure}
For the phase enhancement of the AECS compared with ideal ECSs, the value of $\langle n^1_{E_{A}} \rangle = \bra{AECS } a^{\dag}_2 a_2 \ket{AECS }$ should be $N/2$ in
Eq.~(\ref{APN01}). As shown in Fig.~\ref{fig:Ideal01}, it is true that the results from AECS are very close to those from ideal odd ECSs for $k=1,2$ but slightly better than those of the other states at $k=3$, for the modest number of $N$ (see thin black lines in Fig.~\ref{fig:Ideal01}). This is because the detailed shape of the AECS is slightly different from the ideal odd ECS. In Fig.~\ref{fig:NOONF}, this advantage of phase enhancement can be explained by the fact that the distribution of $H(m,0)$ is narrower but has a longer tail in the AECS, compared to a coherent state $|2.0\rangle$, indicating the photon distribution of $|ECS_{\pm} (\alpha_\pm) \rangle$, for the same average photon number. The $m$ and $m'$ states might provide
a minor contribution of phase enhancement in the AECS.
\begin{figure}[htb]
\centering
\includegraphics[width=250px]{NOON_factor02.eps}
\caption{ (color online). The amplitude of $H_{m,0} (\alpha_A)$ for $\alpha_A =2.0$ and that of coherent state $|2.0\rangle$ with respect to photon number $m$. Both states contain the same average photon number. It implies that the AECS reaches the peak amplitude in smaller $m$ and has a longer tail than the coherent state.
\label{fig:NOONF} }
\end{figure}
\section{For small loss cases in a non-linear phase operation}
\label{sec3}
The phenomenon of imperfect phase operations may lead to small particle losses in the arm. For the comparison of phase enhancement in lossy states, we choose the fixed average
photon number of the states such as
\begin{eqnarray}
\label{eq:Fixed_Ave01} \langle n^1_{N} \rangle = \langle n^1_{E_{+}} \rangle = \langle n^1_{E_{-}} \rangle = \langle n^1_{E_{A}} \rangle = 2.0,
\end{eqnarray}
which implies $N = 4$. For example, $\ket{{AECS} (r_0,2.0}$ and $\ket{{ECS}_- (1.9807)}$ have the same average photon number such as $\langle n^1_{E_{-}} \rangle = \langle n^1_{E_{A}} \rangle = 2$.
Adding a BS with vacuum input can mimic this lossy condition in the dispersive interferometer arm after the non-linear phase shift ($T$: transmission rate of the BS) \cite{Escher11}. Here, we examine the phase enhancement of mixed states by generalising the results of Ref.~\cite{Escher11} such as
\begin{eqnarray}
\label{eq:Fixed_Ave02} && \delta \phi \ge \frac{1}{\sqrt{F^{Q}}} \ge \frac{1}{\sqrt{C^{Q}_k}} \,,
\end{eqnarray}
where $C^{Q}_k = 4 \Big( \langle H^k_1 \rangle - \langle H^k_2 \rangle^2 \Big)$ for any $k$ \cite{CQ}. In particular, this equation shows an excellent match with the exact value of the quantum Fisher information in the small loss region ($T\approx 1$) \cite{Escher11}. For $k=1$, the bound $C^Q$ is given by
\begin{eqnarray}
\label{eq:CQkEq1} C^Q_1 = 4 \Big[ T^2 \Big( \langle n^2 \rangle - \langle n \rangle^2 \Big) +T(1-T) \langle n \rangle \Big],
\end{eqnarray}
and for $k=2$,
\begin{widetext}
\begin{eqnarray}
\label{eq:CQkEq2}
C^Q_2 &=& 4\Big[ T^4 \,\langle n^4 \rangle + 6 T^3 (1-T) \, \langle n^3\rangle + T^2 (1-T)(3-11 T) \, \langle n^2 \rangle + T(1-T)(1- 6T + 6 T^2) \langle n \rangle \nonumber \\
&& - \Big( T^4 \langle n^2 \rangle^2 + 2 T^3 (1-T) \langle n \rangle \langle n^2 \rangle + T^2 (1-T)^2 \langle n \rangle^2 \Big) \Big].
\end{eqnarray}
\end{widetext}
\begin{figure}[htb]
\hspace{-0.7cm}
\includegraphics[width=270px]{Modify_Final_Lossy02.eps}
\caption{ The phase sensitivity with small lossy conditions for $k=1,2$ ($T$: transmission rate of the BS mimicking photon losses). The thick solid line is for NOON states and the thin solid line is for AECS which is very similar to dashed lines for even (short-dashed) and odd (long-dashed) ECSs.
\label{fig:Lossy} }
\end{figure}
As shown in Fig. \ref{fig:Lossy}, the ECSs including AECS significantly outperform NOON states (thick solid lines) for $k=1,2$ in a small photon-loss window. For NOON, even and odd ECS and AECSs, the expectation values in Eqs.~(\ref{eq:CQkEq1}) and (\ref{eq:CQkEq2}) are given by
\begin{eqnarray}
\langle n^k \rangle &=& {N^{k}\big/ 2},\\
\langle n^k_{E_{\pm}} \rangle &=& f_{\alpha_{\pm}}
\sum_{n=0}^{\infty} { n^k (\alpha_{\pm})^{2n} \over n!},\\
\langle n^k_{E_{A}} \rangle &=& \bra{AECS (r,\alpha_A)} (a^{\dag}_2 a_2)^k \ket{AECS (r,\alpha_A)}. \nonumber \\
&=& \sum_{m=1}^\infty \sum_{m'=0}^{m-1} \Big[ \Big( H_{m,m'}\Big)^2 \Big( m^{k} + (m')^{k} \Big) \Big].
\end{eqnarray}
Therefore, these results show that ECSs still outperform the phase enhancement achieved by NOON states in the region of small losses after the non-linear phase operation ($k=2$).
\section{Summary and Remarks}
\label{sec4}
In summary, we have analyzed phase enhancement of ECSs for non-linear phase shifts, using quantum Fisher information to quantify the results. As shown in linear optical elements \cite{Joo11,NewArxiv12}, the phase sensitivity of ECSs outperforms that of NOON states for modest average photon numbers, converging to the limit of NOON states for large average photon numbers. We have presented the form of generalized (non-linear) phase operations, in terms of the power of number operators, and obtained an inequality for the phase enhancement of NOON states and odd and even ECSs, all with respect to the same average photon number as a physical resource. We have also investigated the feasibility of creating AECS in optical set-ups based on current technology, and examined its phase sensitivity. Finally, we have shown that the behavior of the phase sensitivity for ECSs significantly outperforms that for NOON states, for the $k=2$ non-linear example in the presence of small losses.
The fidelity between the odd ECS $|ECS_{-} (\alpha_-) \rangle$ and AECS $\ket{{AECS} (r_0,\alpha_A)}$ is very high, but in general the AECS has more degrees of freedom with $r$ and $\alpha$ (see Fig. \ref{fig:Prepare01}). Thus, there is an opportunity to generate other useful quantum states, different from ECSs, by tuning $r$ and $\alpha$ which could give fourther improvements. Similarly, one may consider using approximate even CSS generated in Ref.~\cite{photon_sub02}, instead of using odd CSS obtained by subtracting a single photon \cite{photon_sub01}. In addition, even though it is known that the self Kerr-type non-linear interaction can be performed with small losses in tunable three- or four-level systems \cite{smallLOSS}, it is still an open question as to whether useful higher order non-linear phase operations can also be performed minimizing loss mechanisms.
\vskip 2truecm
\begin{acknowledgements}
We acknowledge J. Dunningham for useful discussions and financial support from the European Commission of the European Union under the FP7 Integrated Project Q-ESSENCE, the FIRST program in Japan, the NRF grant funded by the Korea government (MEST) (No. 3348- 20100018) and the World Class University program.
\end{acknowledgements}
|
{
"timestamp": "2012-05-14T02:03:08",
"yymm": "1203",
"arxiv_id": "1203.2099",
"language": "en",
"url": "https://arxiv.org/abs/1203.2099"
}
|
\section{Introduction}
Observations of the soft gamma-ray repeaters and anomalous X-ray pulsars showed that these objects can be associated with the strongly magnetized
neutron stars (NSs) with the magnetic field exceeding the Schwinger critical value of $B_{\rm cr} = m_{\rm e}^2 c^3/e\hbar = 4.412\times 10^{13}$ G \cite{Mere02,WT06}.
This has revived the interest in theoretical studies of the interaction processes
between radiation and matter in such fields \cite{HL06}.
Compton scattering is an important process shaping the radiation spectra of the NS atmospheres. Its properties
in the magnetic field differs substantially from the case when the magnetic field is absent.
Even the classical non-relativistic limit of the scattering cross section has a resonance at the energy related to the Lorentz frequency
and is strongly dependent on the photon energy, polarization and the B-field strength \cite{Can1971, Ven1979}.
While the classical description has been useful for understanding the approximate effects of energy, angle and polarization dependence of the cross section in the magnetic field, it does not include the possibility of the electron excitation to a higher Landau state
corresponding to the resonances at higher harmonics, which required fully relativistic treatment.
In the relativistic regime the recoil of the electron is important and the natural line width of the cyclotron resonances depends on the spin of the electron. The relativistic scattering cross section
for the simplest case of ground-to-ground state scattering in the magnetic field was derived in \cite{Her1979}.
These results were extended to a more general case of scattering
to arbitrary Landau states in \cite{DH1986, BAM1986} and discussed further in \cite{Mes1992}.
The derived expressions have been applied to modeling the cyclotron line formation in accreting neutron star atmospheres,
but only for the case of one-dimensional thermal electron distribution because of the complexity of the expressions
\cite{AlMesz1989,AlMesz1991,HarDaugh1991,ArHar1996,ArHar1999}.
When the incident photons propagate along the magnetic field, the resonance appears only at the fundamental frequency
and scattering to the higher Landau levels can effectively be neglected. This allows to simplify
the expressions for the relativistic cross sections and to approximate them by analytical formulae \cite{GHBCM2000}.
The transport of photons through the atmosphere involves multiple scattering, which have to be
considered either by the Monte Carlo methods or using the kinetic equations.
The former approach was used for a qualitative study of the line formation process in Her X-1 \cite{Yah1979, Yah1980},
but it becomes impractical for a large optical depth and when the induced scattering has to be accounted for, and
therefore has a limited field of applications.
In the cold plasma approximation, assuming the coherent scattering, the radiative transfer equation
can be formulated as a set of coupled equations for two normal polarization modes \cite{GP1974}.
The influence of the electron temperature on the radiation transport can be accounted by the Fokker-Planck
approximation, for example, by modifying the Kompaneets equation \cite{K1956}
to allow for the resonances in the scattering cross section \cite{BHP1979}.
Such a treatment, however, does not account for the effects of the photon angular distribution and polarization.
Photon polarization, however, influences the photon redistribution over the energy \cite{Nag1981,PSM1989}.
In a sufficiently strong magnetic field, owing to the large Faraday depolarization, the radiation can be described in terms of two polarization modes.
Under certain conditions (depending on the field strength, photon energy and propagation direction), however,
the vacuum resonance is accompanied by the phenomenon of mode collapse and the breakdown of Faraday depolarization \cite{Zhel1983,LH2003, PSh1979}. In this case the two-mode description fails and instead the kinetic equations have to written in terms of the Stokes parameters or the coherency matrix.
In the case when the induced scattering needs to be accounted for, the situation complicates further as
there is no intuitive way to get such an equation.
The aim of this paper is to derive from first principles a general kinetic equation for Compton scattering in any magnetic field
accounting simultaneously for photon polarization in terms of the Stokes parameters, for the induced scattering and the Pauli exclusion
principle for electrons.
We use methods of quantum statistics and follow an approach similar to that used for derivation of the kinetic equation
for Compton scattering without magnetic field \cite{NP2001}.
The resulting equations are valid for any photon and electron energies,
and for the magnetic field strength below about $10^{16}$ G.
In the most general case, the electron polarization is also taken into account.
We also consider several special cases and derive the kinetic equations when
the electron gas is non-polarized and rarefied as well as when the radiation can be presented via two polarization modes.
The derived equations provide the basis for construction of the models of radiation transport
in atmospheres and magnetospheres of strongly magnetized neutron stars.
\section{Description of the electron and photon gases}
We use the system of units where $\hbar=c=m_{\rm e}=1$. We
assume that the magnetic field is locally homogeneous, which is justified, because
the scales of changes of the B-field are orders of magnitude larger than the microscopic
magnetic scale for conditions in atmospheres of NS and even the geometrical depth of the atmosphere.
The magnetic field is described by the dimensionless parameter $b=B/B_{\rm cr}$.
We choose the reference frame in any space-point so that the $z$-axis coincides with the magnetic field direction.
The following assumptions about the time scales are used:
\begin {enumerate}
\item The typical time scale on which the distribution function changes (for electrons and photons) is much larger than the typical time scale between the interactions.
\item The plasma is sufficiently rarified, so that we can use a generalization of
the Bogolyubov method for the case of quantum statistics to derive the kinetic equation.
\item The typical time scale of a single interaction is much smaller than the typical time scale between the interactions.
\end {enumerate}
\subsection{Descriptions of single particles}
The electron states are described by the wave-functions $\Psi_{n\sigma}(\unl {r},Y,Z)$. Its arguments are the space-time coordinates,
the momentum projection $Z$ on the direction of the magnetic field,
$Y$ describing location of the center of electron gyro-orbit (its $y$-coordinate),
the Landau level $n$, and the spin projection $\sigma$ on the magnetic field direction ($\sigma=\pm 1$ in $\hbar/2$ units).
Dimensionless energy of an electron in this case,
\be \label{eq:RnZ}
R_n(Z)=\sqrt {1+Z^2+2bn},
\ee
is independent of $Y$.
The energy levels are degenerate with the spin projection on the magnetic field direction,
except for the ground Landau level with $n=0$, where $\sigma$ can have only one value $-1$.
The full electron wave function is presented through the partial solutions of the Dirac equation for the electron in the magnetic field:
\be \label{eq:psiurfull}
\psi(\unl {r})=\sum_{n,\sigma}\int\frac{\d Y\d Z}{R_n(Z)}
\Psi_{n\sigma}(\unl {r},Y,Z)b_{n\sigma}(Y,Z),
\ee
where $b_{n\sigma}$ are coefficients.
The photon state is described by four parameters: the wavenumber $k$, the two angles $\theta$ and $\varphi$, which define the direction of the photon momentum, and the polarization state $s=1,2$. The 3-dimensional photon momentum can be represented as
\be \label{eq:vkphoton}
\textit{\textbf{k}}=(k_x,k_y,k_z)=k(\sin\theta\cos\varphi,\sin\theta\sin\varphi,
\cos\theta).
\ee
The corresponding photon 4-momentum is $\unl {k}=\{k,\textit{\textbf{k}}\},\,k=|\textit{\textbf{k}}|$.
Photon polarization is described by the polarization basis. It consists of two unit vectors, which are orthogonal to the photon momentum $\textit{\textbf{k}}$:
\be \label{eq:ve1ve2orts}
\textit{\textbf{e}}_1=(\sin\varphi,-\cos\varphi,0),\quad
\textit{\textbf{e}}_2=(\cos\theta\cos\varphi,\cos\theta\sin\varphi,-\sin\theta).
\ee
The 4-vector potential can be defined as:
\be \label{eq:uAsur}
\unl {A}_s(\unl {r})=\unl {e}_se^{-i\unl {k}\cdot\unl {r}},\quad\unl {e}_s=\{
0,\textit{\textbf{e}}_s\},\quad s=1,2.
\ee
We note that the photons are described in the same manner as in the case when the magnetic field is absent.
We assume that the dispersion relation for the photons in magnetized vacuum does not differ
from the dispersion relation in the case when the magnetic field is absent.
This approximation constrains the strength of the field and energies of photons.
For estimations one needs to know vacuum dielectric tensor and the inverse permeability tensor
for the case of magnetized vacuum \cite{Adler1971,PLShH2004}.
It is known that the indices of refraction differ from unity by more than $10\%$
only for the fields with strength $b>300$ \cite{ShHL1999}. This restricts
application of the developed formalism to $B\lesssim 10^{16}$ G.
\subsection{Description of the particle ensembles}
\subsubsection{Wave functions}
We describe particle ensembles by density matrix using the rules of quantum statistics. Let us define the wave functions for the case of limited number of particles. These functions will be used for construction of the density matrix.
The wave function for a limited number of particles with defined characteristics of each of them, can be found from the vacuum wave function by applying the operators of creation and annihilation.
Let $\bar{a}_{(s)}(\textit{\textbf{k}})$ and $a_{(s)}(\textit{\textbf{k}})$ be the creation and annihilation operators of a photon in the state with polarization $s$ and 3-momentum $\textit{\textbf{k}}$. According to the methods of second quantization \cite{BS1959}, these operators satisfy the relation
\be \label{eq:aoacommut}
a_{(s)}(\textit{\textbf{k}})\ovl {a}_{(s')}(\textit{\textbf{k}}')-\ovl {a}_{(s')}(\textit{\textbf{k}}')a_{(s)}(\textit{\textbf{k}})=k
\delta(\textit{\textbf{k}}-\textit{\textbf{k}}')\delta_{s}^{s'}.
\ee
Let $b^\dag_{n\sigma}(Y,Z)$ and $b_{n\sigma}(Y,Z)$ be creation and annihilation operators of an electron on the Landau level $n$ in polarization state $\sigma$ with momentum projections $Y$ and $Z$. These operators satisfy the following relation
\be \label{eq:bdbcommut}
b_{n\sigma}(Y,Z)b^\dag_{n'\sigma'}(Y',Z')+b^\dag_{n'\sigma'}(Y',Z')
b_{n\sigma}(Y,Z)=R_n(Z)\delta(Y-Y')\delta(Z-Z')\delta_{n}^{n'}
\delta_{\sigma}^{\sigma'}.
\ee
The system of $N$ photons with fixed parameters $\{s_1,\textit{\textbf{k}}_1;...;s_N,\textit{\textbf{k}}_N\}$ is described by the wave function
\be \label{eq:PsisvkN}
\Psi_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)=\frac{1}{\sqrt {N!}}\bar {a}_{(s_1)}
(\textit{\textbf{k}}_1)...\bar {a}_{(s_N)}(\textit{\textbf{k}}_N)\Psi_0,
\ee
where $\Psi_0$ is the vacuum wave function of the photon gas.
Analogously, the system of $N$ electrons with fixed parameters $\{n_1,\sigma_1,Y_1,Z_1;...;n_N,\sigma_N,Y_N,Z_N\}$ is described by the wave function
\be \label{eq:PhisigYZ}
\Phi^{\sigma_1...\sigma_N}_{n_1...n_N}(Y_1,Z_1,...,Y_N,Z_N)=
\frac{1}{\sqrt {N!}}b^{\dag}_{n_1\sigma_1}(Y_1,Z_1)...b^{\dag}_{n_N
\sigma_N}(Y_N,Z_N)\Phi_0.
\ee
where $\Phi_0$ is the vacuum wave function of the electron-positron gas.
The wave function for arbitrary state of particles can be presented as a sum of the wave functions with fixed particle parameters. For example, the state of $N$ photons is described by the function
\be \label{eq:PsiNdef}
\Psi_N=\int\prod_{j=1}^N\frac{\d\textit{\textbf{k}}_j}{k_j}c_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)
\Psi_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N),
\ee
where $c_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)$ are the weight coefficients.
The wave function for an arbitrary state of $N$ electrons can be represented as
\be \label{eq:PhiNdef}
\Phi_N=\int\prod_{j=1}^N\frac{\d Y_j\d Z_j}{R_{n_j}(Z_j)}
c_{n_1...n_N}^{\sigma_1...\sigma_N}(Y_1,Z_1,...,Y_N,Z_N)
\Phi^{\sigma_1...\sigma_N}_{n_1...n_N}(Y_1,Z_1,...,Y_N,Z_N).
\ee
The wave function for state with $N$ photons and $N_{+}$ electrons can be written as
\beq \label{eq:PsiNNpdef}
\strut\displaystyle \Psi_{N,N_{+}} & = & \int\prod_{i=1}^N\frac{\d\textit{\textbf{k}}_i}{k_i}
\prod_{j=1}^{N_{+}}\frac{\d Y_j\d Z_j}{R_{n_j}(Z_j)}c_{n_1...n_{N_{+}},s_1...
s_N}^{\sigma_1...\sigma_{N_{+}}}(Y_1,Z_1,...,Y_{N_{+}},Z_{N_{+}},\textit{\textbf{k}}_1,...,
\textit{\textbf{k}}_N)\nonumber \\
\strut\displaystyle & \times& \Psi_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)\Phi^{\sigma_1...
\sigma_{N_{+}}}_{n_1...n_{N_{+}}}(Y_1,Z_1,...,Y_{N_{+}},Z_{N_{+}}).
\eeq
\subsubsection{Density matrix}
The density matrix is defined as an averaged dyad product of the state vector with its conjugate.
For the system consisting of $N$ photons and $N_{+}$ electrons it can be written in the form
\beq \label{eq:rhoNNpdef}
& \strut\displaystyle \rho_{N,N_{+}}=\frac{1}{N!N_{+}!}\int\prod_{i=1}^N
\frac{\d\textit{\textbf{k}}_i}{k_i}\frac{\d\textit{\textbf{k}}'_i}{k'_i}\int\prod_{j=1}^{N_{+}}
\frac{\d Y_j\d Z_j}{R_{n_j}(Z_j)}\frac{\d Y'_j\d Z'_j}{R_{n'_j}(Z'_j)}
& \nonumber \\
& \strut\displaystyle \times \left\langle c{^*}{_{n'_1...n'_{N_{+}},s'_1...s'_N}
^{\sigma'_1...\sigma'_{N_{+}}}}(Y'_1,Z'_1,...,Y'_{N_{+}},Z'_{N_{+}},
\textit{\textbf{k}}'_1,...,\textit{\textbf{k}}'_N)c_{n_1...n_{N_{+}},s_1...s_N}^{\sigma_1...\sigma_{N_{+}}}
(Y_1,Z_1,...,Y_{N_{+}}Z_{N_{+}},\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)\right\rangle
& \nonumber \\
& \strut\displaystyle
\times\Psi_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)\ovl {\Psi}_{s'_1...s'_N}
(\textit{\textbf{k}}'_1,...,\textit{\textbf{k}}'_N)\Phi^{\sigma_1...\sigma_{N_{+}}}_{n_1...n_{N_{+}}}(Y_1,Z_1,
...,Y_{N_{+}},Z_{N_{+}})\ovl {\Phi}^{\sigma'_1...\sigma'_{N_{+}}}_{n'_1...n'_
{N_{+}}}(Y'_1,Z'_1,...,Y'_{N_{+}},Z'_{N_{+}}). &
\eeq
The expressions in the triangle brackets are the elements of the density matrix kernel.
\subsubsection{Algebra of the density matrix kernels}
From now on we will operate only with density matrix kernels. All the equations and the final results are written through these kernels.
It is easy to make a transformation from the simplest kernels to the distribution functions or to the coherency matrix.
Let us write the density matrix kernel for the system of $N$ photons and $N_{+}$ electrons
\beq \label{eq:kernelrhodef}
& \strut\displaystyle \rho_{s_1...s_N,\sigma_1...\sigma_{N_{+}},n_1...n_{N_{+}}}^{s'_1
...s'_N,\sigma'_1...\sigma'_{N_{+}},n'_1...n'_{N_{+}}}\left({{\textit{\textbf{k}}'_1...
\textit{\textbf{k}}'_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}}\Biggl|{{Y'_1...Y'_{N_{+}}\;Z'_1...Z'_{N_{+}}}
\atop {Y_1...Y_{N_{+}}\;Z_1...Z_{N_{+}}}}\right)\equiv
& \nonumber \\
& \strut\displaystyle
\equiv\left\langle c{^*}{_{n'_1...n'_{N_{+}},s'_1...s'_N}^{\sigma'_1...
\sigma'_{N_{+}}}}(Y'_1,Z'_1,...,Y'_{N_{+}}Z'_{N_{+}},\textit{\textbf{k}}'_1,...,\textit{\textbf{k}}'_N)
c_{n_1...n_{N_{+}},s_1...s_N}^{\sigma_1...\sigma_{N_{+}}}(Y_1,Z_1,...,
Y_{N_{+}}Z_{N_{+}},\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)\right\rangle. &
\eeq
Further let us write some useful relations for the kernels. For the sake of simplicity we consider only the photon gas.
These relations can be generalized trivially to the case of the electron-photon gas.
The kernel for the system of $N$ photons can be written through the density matrix:
\be \label{eq:matrix2kernel}
\rho_{s_1...s_N}^{s'_1...s'_N}\left({\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_N}
\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}\right)=N!\ \ovl {\Psi}_{s_1...s_N}(\textit{\textbf{k}}_1,...,\textit{\textbf{k}}_N)
\rho_N\Psi_{s'_1...s'_N}(\textit{\textbf{k}}'_1,...,\textit{\textbf{k}}'_N).
\ee
Hereinafter we call it the $N$-particle kernel. It is normalized to unity:
\be \label{eq:normrhoN1}
\int\prod_{i=1}^N\frac{\d\textit{\textbf{k}}_i}{k_i}\rho_{s_1...s_N}^{s_1...s_N}\left(
{\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}\right)=\texttt {Sp} (\rho)=1.
\ee
For any $m\leq N$ the $m$-particle kernel can be calculated as
\be \label{eq:rhomdef}
\rho_{s_1...s_m}^{s'_1...s'_m}\left({\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_m}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_m}
\right)=\frac{1}{(N-m)!}\int\prod_{i=m+1}^N\frac{\d\textit{\textbf{k}}_i}{k_i}\rho_{s_1...
s_m,s_{m+1}...s_N}^{s'_1...s'_m,s_{m+1}...s_N}\left({\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_m \textit{\textbf{k}}_{m+1}
...\textit{\textbf{k}}_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_m \textit{\textbf{k}}_{m+1}...\textit{\textbf{k}}_N}\right).
\ee
The 1-particle kernel can be expressed through the $N$-particle kernel as
\be \label{eq:rhosskk}
\rho_s^{s'}\left({\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}\right)=\frac{1}{(N-1)!}
\int\prod_{i=2}^N\frac{\d\textit{\textbf{k}}_i}{k_i}\rho{_s^{s'}}{_{s_2...s_N}^{s_2...s_N}}\left(
{\textit{\textbf{k}}'\;\textit{\textbf{k}}_2...\textit{\textbf{k}}_N}\atop {\textit{\textbf{k}}\;\textit{\textbf{k}}_2...\textit{\textbf{k}}_N}\right).
\ee
It is normalized to the total number of the particles:
\be \label{eq:normrho1}
\int\frac{\d^3k}{k}\rho_s^s\left({\textit{\textbf{k}}}\atop {\textit{\textbf{k}}}\right)=N.
\ee
The diagonal elements of 1-particle kernel compose the coherency matrix in the case of photon gas.
\subsubsection{Transformation from the simplest kernel to the distribution functions}
The transformation from 1-particle density matrix to the distribution function in the case of electrons
in the case of field-free space can be made using the Wigner function. The Wigner function is defined as
\be \label{eq:rhovpvrdef}
\rho(\textit{\textbf{p}},\textit{\textbf{r}})=\int \d\textit{\textbf{v}}\exp\left(i\textit{\textbf{p}}\cdot\textit{\textbf{v}}\right)\rho_s\left(
\textit{\textbf{r}}+\textit{\textbf{v}}/2,\textit{\textbf{r}}-\textit{\textbf{v}}/2\right),
\ee
where $\rho_s\left(\textit{\textbf{r}}+\textit{\textbf{v}}/2,\textit{\textbf{r}}-\textit{\textbf{v}}/2\right)$ is a 1-particle density matrix in the coordinate representation. The momentum and coordinate representations are connected through the Fourier transforms:
\be \label{eq:rhosvrvr}
\rho_s(\textit{\textbf{r}},\textit{\textbf{r}}\,')=\frac{1}{(2\pi)^6}\int\frac{\d\textit{\textbf{p}}\ \d\textit{\textbf{p}}'}
{\sqrt {p_0\ p'_0}}\exp\left(-i(\textit{\textbf{p}}\cdot\textit{\textbf{r}}-\textit{\textbf{p}}\,'\cdot \textit{\textbf{r}}\,')\right)
\rho\left({{\textit{\textbf{p}}'}\atop {\textit{\textbf{p}}}}\right).
\ee
Then one can rewrite the Wigner function using the density matrix in the momentum representation:
\be \label{eq:Wignerfunc}
\rho(\textit{\textbf{p}},\textit{\textbf{r}})=\frac{1}{(2\pi)^6}\int\frac{\d\textit{\textbf{p}}_1\d\textit{\textbf{p}}'_1}
{\sqrt {p_{01}p'_{01}}}\exp\left(-i(\textit{\textbf{p}}_1-\textit{\textbf{p}}_1{\!\!'})\cdot\textit{\textbf{r}}\right)
\delta\left(\textit{\textbf{p}}-\frac{\textit{\textbf{p}}_1+\textit{\textbf{p}}_1{\!\!'}}{2}\right)\rho\left(
{{\textit{\textbf{p}}_1{\!\!'}}\atop {\textit{\textbf{p}}_1}}\right).
\ee
The inverse transformation from the Wigner function to the density matrix in momentum representation reads
\be \label{eq:Wignerrev}
\rho\left({{\textit{\textbf{p}}_1{\!\!'}}\atop {\textit{\textbf{p}}_1}}\right)=\frac{\sqrt {p_{01}p'_{01}}}
{(2\pi)^3}\int\d\textit{\textbf{p}}\ \d\textit{\textbf{r}}\ \exp\left(i(\textit{\textbf{p}}_1-\textit{\textbf{p}}_1{\!\!'})\cdot\textit{\textbf{r}}\right)
\delta\left(\textit{\textbf{p}}-\frac{\textit{\textbf{p}}_1+\textit{\textbf{p}}_1{\!\!'}}{2}\right)\rho(\textit{\textbf{p}},\textit{\textbf{r}}).
\ee
The time scale of the electron-photon interaction is much smaller than the time scale of noticeable changes of the distribution functions. Therefore, in the last equation one can assume that the Wigner function does not depend on the space variables. In this case, the integration could be made easily and we can write
\be \label{eq:nucl2df1}
\rho\left({{\textit{\textbf{p}}_1{\!\!'}}\atop {\textit{\textbf{p}}_1}}\right)=p_{01}\delta(\textit{\textbf{p}}_1{\!\!'}-\textit{\textbf{p}}_1)
\rho(\textit{\textbf{p}}_1).
\ee
Then one can convert the 1-particle density matrix kernel to the distribution function in the case of spinless particles or to the coherency matrices in the case of particles with non-zero spin. In the last case a trivial generalization is used:
\be \label{eq:nucl2df2}
\rho_{\tau_1}^{\tau'_1}\left({{\textit{\textbf{p}}_1{\!\!'}}\atop {\textit{\textbf{p}}_1}}\right)=p_{01}
\delta(\textit{\textbf{p}}_1{\!\!'}-\textit{\textbf{p}}_1)\rho_{\tau_1}^{\tau'_1}(\textit{\textbf{p}}_1).
\ee
Equations (\ref{eq:nucl2df1}) and (\ref{eq:nucl2df2}) can be written immediately from the physical meaning of the
density matrix and the assumption that the time scale of the interaction and the time between interactions are mush
smaller than the time scale of noticeable changes of the distribution function.
One can also write similar relation for the case of electrons in the external magnetic field.
If we consider a non-interacting electron in the B-field
not accounting for cyclotron radiation (which should be described by another kinetic equation),
then the electron should conserve its $z$-momentum and the Landau level.
This means that the kernel should be diagonal over both $Z$ and $n$, because it is not possible
to have mixed states corresponding to different values of the $z$-projection of momentum or the Landau level.
Non-diagonal elements in the kernel can appear only if one accounts for interactions between particles,
but because of the smallness of the interaction time scale
the kernel should be diagonal over $z$-projection of momentum and the Landau levels.
In this case the relation will have the following form (a detailed derivation is given in Appendix \ref{Ritus}):
\be \label{eq:rhoRnZ}
\rho_{\sigma n}^{\sigma' n'}\left({{Y' Z'}\atop {Y Z}}\right)=
R_{n}(Z)\delta_{n}^{n'}\delta(Y'-Y)\delta(Z'-Z)\rho_{\sigma n}^{\sigma'}(Y,Z),
\ee
where $\sigma$ and $\sigma'$ describe the electron spin-states,
$n$ and $n'$ are the Landau levels, $Z$ and $Z'$ are the momentum projections and
$R_{n}(Z)$ is the electron energy given by equation (\ref{eq:RnZ}).
A transformation from the 1-particle density matrix to the distribution function in momentum space is trivial,
but one must again assume that the
typical time scale of changes of the distribution function is much larger than the typical time scales of interaction between the particles.
\subsection{Description of the interaction}
\subsubsection{Description of the single interaction}
Let us mark parameters of the particles before the interaction with the subscript "i" and particles after interaction with the subscript "f". There are three conservation laws for Compton scattering in the magnetic field. They are the energy conservation, the conservation of the momentum along the magnetic field and the conservation of the transversal momentum:
\be \label{eq:conservlaws}
R_{\rm i}+k_{\rm i}=R_{\rm f}+k_{\rm f},\quad
Z_{\rm i}+k_{\rm i}\cos\theta_{\rm i}=Z_{\rm f}+k_{\rm f}\cos\theta_{\rm f},\quad
Y_{\rm i}+k_{\rm i}\sin\theta_{\rm i}\sin\varphi_{\rm i}=Y_{\rm f}+k_{\rm f}\sin\theta_{\rm f}\sin\varphi_{\rm f}.
\ee
Let us use special designation for product of $\delta$-functions which are describing these conservation laws:
\beq& \strut\displaystyle
\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)\equiv
& \nonumber \\
& \strut\displaystyle
\equiv\delta(R_n(Z)+k-R_{n'}(Z')-k')
\delta(Z+k\cos\theta-Z'-k'\cos\theta')
\delta(Y+k\sin\theta\sin\varphi-Y'- k'\sin\theta'\sin\varphi')&.
\eeq
A single interaction can be described by the $S$-matrix. The elements of the $S$-matrix can be calculated using methods of quantum electrodynamics.
In the simplest case the elements of the $S$-matrix can be obtained using second-order perturbation theory. In this case Compton scattering can be represented by two Feynman diagrams with two vertices in both of them and one can write an expression for the $S$-matrix elements:
\be \label{eq:Sfi}
S_{{\rm f}\,{\rm i}}=-4\pi i\alpha\int\d^4r_1\d^4r_2\ovl {\Psi}_{\rm f}(\unl {r}_2)\left\{
\left[\unl {\gamma}\unl {A}^\dag_{\rm f}(\unl {r}_2)\right]G(\unl {r}_2,\unl {r}_1)\left[
\unl {\gamma}\unl {A}_{\rm i}(\unl {r}_1)\right]+\left[\unl {\gamma}\unl {A}_{\rm i}(\unl {r}_2)\right]G(\unl {r}_2,\unl {r}_1)
\left[\unl {\gamma}\unl {A}^\dag_{\rm f}(\unl {r}_1)\right]\right\}\Psi_{\rm i}(\unl {r}_1),
\ee
where $\unl {\gamma}\unl {A}$ is the Dirac inner product of a 4-vector and $\unl {\gamma}$-matrix, and $G(\unl {r}_2,\unl {r}_1)$ is a relativistic electronic propagator in the presence of a constant magnetic field, $\Psi_{\rm i}(\unl {r})$ and $\Psi_{\rm f}(\unl {r})$ are the electron wave-functions in coordinate representation, and
$\alpha=e^2$ is the fine-structure constant.
The $S$-matrix elements and the cross-sections for Compton scattering in magnetic field contain resonances which have to be
regularized \cite{NK1993}. The calculations are not trivial and have been performed only in special cases \cite{Her1979,DH1986,BAM1986,GHBCM2000}.
\subsubsection{Evolution of the density matrix}
The evolution of the density matrix can be described by equation:
\be \label{eq:drhodt}
i\Dr {\rho(t)}{t}=H(t)\rho(t)-\rho(t)H(t),
\ee
where the Hamiltonian is
\be \label{eq:Hamiltonian}
H(t)=-e\int\d\textit{\textbf{r}}\ \ovl {\psi}(\unl {r})\unl {\gamma}\unl {A}(\unl {r})\psi(\unl {r}).
\ee
Equation (\ref{eq:drhodt}) is written here in non-covariant form, but it can be transformed to the explicitly covariant form using the Tomonaga-Schwinger equation \cite{BS1959}. It means that the form of the equation is covariant for the longitudinal Lorentz transformations (along the magnetic field direction).
The solution of equation (\ref{eq:drhodt}) can be presented by the operator of evolution $U(x,y)$:
\be \label{eq:drhodtsol}
\int\limits_{t_0}^t\Dr {\rho(t')}{t'}\d t'=\rho(t)-\rho(t_{0})=U(t,t_0)\rho(t_0)-
\rho(t_0)U(t,t_0).
\ee
On the other hand the operator $U(t,t_{0})$ can be represented in the following form:
\be \label{eq:Utt0}
U(t,t_0)=\int\limits_{t_0}^t\d t'\int\limits_{t_0}^t\d t''\int\limits_V\d\textit{\textbf{r}}'
\int\limits_V\d\textit{\textbf{r}}''{\mathcal S}(\unl {r}',\unl {r}'')=\int\limits_{\mathcal V}\d^4r'
\int\limits_{{\mathcal V}}\d^{4}r''{\mathcal S}(\unl {r}',\unl {r}''),
\ee
where ${\mathcal V}=[t_0,t]\times V$ is the volume in Minkowski space and
\be \label{eq:cSurur}
{\mathcal S}(\unl {r}',\unl {r}'')=i\frac{e^{2}}{(2\pi)^{9}}\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}
\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}b^\dagger_{n'\sigma'}(Z')b_{n\sigma}(Z)
\bar {a}_{(s')}(\textit{\textbf{k}}')a_{(s)}(\textit{\textbf{k}}){\mathcal N}{_n^{n'}}{_\sigma^{\sigma'}}{_s^{s'}}
\left(\left.{{\textit{\textbf{k}}'}\atop{\textit{\textbf{k}}}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right|{{\unl {r}'}\atop {\unl {r}''}}\right).
\ee
The space integral of ${\mathcal N}$ can be represented through the elements of the scattering $M$-matrix:
\be \label{eq:intd4rcN}
\int\limits_{{\mathcal V}}\d^4r'\int\limits_{{\mathcal V}}\d^4r''{\mathcal N}{_n^{n'}}{_\sigma^{\sigma'}}{_s^{s'}}
\left(\left.{{\textit{\textbf{k}}'}\atop{\textit{\textbf{k}}}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right|{{\unl {r}'}\atop {\unl {r}''}}\right)=
(2\pi)^{8}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
M^{\sigma's'}_{\sigma s}\left(\left.{{n'Y'Z'}\atop {n Y Z}}\right|{{\textit{\textbf{k}}'} \atop
{\textit{\textbf{k}}}}\right),
\ee
then $U(t,t_0)$ can be rewritten in the following form
\be \label{eq:UfM}
U(t,t_0)=\frac{i}{2\pi}\alpha
\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}
\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}b^\dagger_{n'\sigma'}(Z')b_{n\sigma}(Z)
\bar {a}_{(s')}(\textit{\textbf{k}}')a_{(s)}(\textit{\textbf{k}})M^{\sigma's'}_{\sigma s}
\left(\left.{{n'Y'Z'}\atop {n Y Z}}\right|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right).
\ee
Let us assume that the typical time scale of the density matrix changes is much larger than the typical time scales of a single interaction.
In that case changes in the distribution on a macroscopically small times scale can be represented througth the $S$-matrix because
$M$-matrix can be considered as the scattering $S$-matrix divided by the fine-structure constant:
\be
M^{\sigma_{\rm f} s_{\rm f}}_{\sigma_{\rm i} s_{\rm i}}\left(\left.{{n_{\rm f} Y_{\rm f} Z_{\rm f}}\atop {n_{\rm i} Y_{\rm i} Z_{\rm i}}}\right|{{\textit{\textbf{k}}_{\rm f}} \atop{\textit{\textbf{k}}_{\rm i}}}\right)\equiv
M_{\rm fi}=\frac{S_{\rm fi}}{\alpha},
\ee
and the time interval $[t_0, t_0 +t]$ is considered as a macroscopically small time.
Equations (\ref{eq:drhodtsol}) and (\ref{eq:UfM}) determine the solution formulated through the elements of the scattering matrix.
We reformulate these equations below in terms of the kernels of the density matrix.
\section{Derivation of the kinetic equation for the photon gas}
\subsection{Methodology of the kinetic equation derivation}
\subsubsection{Summary of our assumptions and the Bogolyubov method}
We derive kinetic equation using a generalization of the Bogolyubov method (for the case of quantum statistics). At the first step we formulate Liouville's theorem in terms of the kernels of density matrix. One can derive the equations for kernels of different orders (1-particle, 2-particle and other) by integrating over the parameters of different numbers of particles. We use this method to obtain the system of kinetic equations. If the full ensemble contains $N$ particles, the system of equations contains $N$ equations.
In the case of the rarefied gas one can use only a few first equations from the Bogolyubov hierarchy. The criterion of rarefaction can be formulated
through the ``gaseous parameter'', which depends on the concentration of the particles and the cross sections of their interaction:
\be \label{eq:alphagas}
\alpha_{\rm gas} \equiv
\sqrt {\sigma_{\rm T}}(n_{\rm e}n_{{\rm ph}})^{1/6} \ll 1,
\ee
where $\sigma_{\rm T}$ is the Thomson cross section, $n_{\rm e}$ and $n_{\rm ph}$ are the electron and the photon concentrations, correspondingly.
According to the principle of weakening of correlations, which is satisfied for sufficiently rarefied gases, the correlations are accounted for only in the equation for the 1-particle matrix via kernel by entering the right-hand side ({rhs}) of the aforementioned equation. This kernel is assumed to characterize the electron and photon states after the interaction. It can be represented via the same kernel before the interaction and the correlation function.
To derive the kinetic equation for the typical conditions in the neutron star atmospheres, it is enough to use only the first and the second equation from the Bogolyubov hierarchy.
\subsubsection{Formulation of Liouville's theorem and the equations of Bogolyubov hierarchy}
We use the following notations: $R\equiv R_n(Z),\quad R'\equiv R_{n'}(Z')$, etc.
There are $N$ photons and $N_+$ electrons in the system. The equation, describing the change of $(N+N_{+})$-particle kernel during macroscopically small time $T_{0}$ is written as
\beq \label{eq:comptLiuvTh}
& \strut\displaystyle \rho_{s_1...s_N \sigma_1...\sigma_{N_{+}} n_1...n_{N_{+}}}
^{s'_1..s'_N \sigma'_1...\sigma'_{N_{+}} n'_1...n'_{N_{+}}}
\left({{\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}}\left|{{Y'_1...Y'_{N_{+}} Z'_1
... Z'_{N_{+}}}\atop {Y_1...Y_{N_{+}} Z_1...Z_{N_{+}}}}\right|
\frac {T_0}{2}\right)
& \nonumber \\
& \strut\displaystyle
=\rho_{s_1..s_N \sigma_1...\sigma_{N_{+}} n_1...n_{N_{+}}}^{s'_1
...s'_N \sigma'_1...\sigma'_{N_{+}} n'_1...n'_{N_{+}}}\left({{\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_N}
\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_N}}\left|{{Y'_1...Y'_{N_{+}} Z'_1...Z'_{N_{+}}}
\atop {Y_1...Y_{N_{+}} Z_1...Z_{N_{+}}}}\right|-\frac {T_0}{2}\right)+i
\frac {\alpha}{2\pi}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)\
b^\dag_{n'\sigma'}(Y',Z')b_{n\sigma}(Y,Z)\bar {a}_{s'}(\textit{\textbf{k}}')a_s(\textit{\textbf{k}})
M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\left|{{\textit{\textbf{k}}'}
\atop {\textit{\textbf{k}}}}\right) \right.
& \nonumber \\
& \strut\displaystyle
\times\sum^N_{i=1}\sum^{N_{+}}_{i_{+}=1}\left[
\delta^{s'}_{s_i'}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_i)\delta^{n'}_{n'_{i_{+}}}
\delta^{\sigma'}_{\sigma'_{i_{+}}}\delta(Y'-Y'_{i_{+}})\delta(Z'-Z'_{i_{+}})
\right. & \nonumber \\
& \strut\displaystyle \times\rho
_{s_1...s_i...s_N \sigma_1...\sigma_{i_{+}}...\sigma_{N_{+}} n_1...n_{i_{+}}...n_{N_{+}}}
^{s'_1...s...s'_N \sigma'_1...\sigma...\sigma'_{N_{+}} n'_1...n...n'_{N_{+}}}
\left({{\textit{\textbf{k}}'_1...\textit{\textbf{k}}...
\textit{\textbf{k}}'_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}_i...\textit{\textbf{k}}_N}}\right|{{Y'_1...Y...Y'_{N_{+}} Z'_1...
Z...Z'_{N_{+}}}\atop {Y_1...Y_{i_{+}}...Y_{N_{+}} Z_1...Z_{i_{+}}...
Z_{N_{+}}}}\left|-\frac {T_0}{2}\right)
& \nonumber \\
& \strut\displaystyle
-\delta^s_{s_i}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i)\delta^n_{n_{i_{+}}}
\delta^\sigma_{\sigma_{i_{+}}}\delta(Y-Y_{i_{+}})\delta(Z-Z_{i_{+}})
& \nonumber \\
& \strut\displaystyle
\times\rho_{s_1...s'...s_N \sigma_1...\sigma'...\sigma_{N_{+}}
n_1...n'...n_{N_{+}}}^{s'_1...s'_i...s'_N \sigma'_1...\sigma'_{i_{+}}...
\sigma'_{N_{+}} n'_1...n'_{i_{+}}...n'_{N_{+}}}\left(\left.{{\textit{\textbf{k}}'_1...\textit{\textbf{k}}'_{i}
...\textit{\textbf{k}}'_N}\atop {\textit{\textbf{k}}_1...\textit{\textbf{k}}'...\textit{\textbf{k}}_N}}\right|{{Y'_1...Y'_{i_{+}}...
Y'_{N_{+}} Z'_1...Z'_{i_{+}}...Z'_{N_{+}}}\atop {Y_1...Y'...Y_{N_{+}} Z_1
...Z'... Z_{N_{+}}}}\left|-\frac {T_0}{2}\right)\right]. &
\eeq
This equation is a formulation of Liouville's theorem in terms of the density matrix kernels.
It can be used to find the equation describing the evolution of 1-particle photon kernels on the time interval $\left[-T_0/2,T_0/2\right]$,
which is the first equation of the Bogolyubov hierarchy.
We integrate equation (\ref{eq:comptLiuvTh}) over the parameters of $(N-1)$ photons and $N_{+}$ electrons:
\beq
& \strut\displaystyle \rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|
\frac{T_0}{2}\right)-\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right|
-\frac{T_0}{2}\right)=\frac {1}{(N-1)!N_{+}!}i
\frac{\alpha}{2\pi}\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}
\frac {\d\textit{\textbf{k}}'}{k'}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
&\\
& \strut\displaystyle
\times b^\dag_{n'\sigma'}(Y',Z')b_{n\sigma}(Y,Z)\bar {a}_{s'}
(\textit{\textbf{k}}')a_s(\textit{\textbf{k}})
M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right.\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right)
\sum_{i_{+}=1}^{N_{+}}\int\prod_{i=2}^N
\left(\frac {\d\textit{\textbf{k}}_i}{k_i}\frac {\d\textit{\textbf{k}}'_i}{k'_i}\right)\prod_{i_{+}=1}^{N_{+}}
\left(\frac {\d Y_{i_{+}}\d Z_{i_{+}}}{R_{i_{+}}}\frac{\d Y'_{i_{+}}\d Z'_{i_{+}}}{R'_{i_{+}}}
\right)
& \nonumber \\
& \strut\displaystyle
\times \left\{
\delta^{n'}_{n'_{i_{+}}}\delta^{\sigma'}_{\sigma'_{i_{+}}}\delta(Y'-Y'_{i_{+}})
\delta(Z'-Z'_{i_{+}})\right.
\left[\delta ^{s'}_{s_{1}'}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_{1})
P^{s}_{s_1}\left(\left.{{\textit{\textbf{k}}}\atop{\textit{\textbf{k}}_1}}\right|1,1,\textit{J},\textit{G}\right)\right.
\left.+\sum_{i=2}^{N}\delta^{s'}_{s_i}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}_i)
P^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J},\textit{G}\right)
\right]
& \nonumber \\
& \strut\displaystyle -\delta^n_{n_{i_{+}}}\delta^\sigma_{\sigma_{i_{+}}}
\delta(Y-Y_{i_{+}})\delta(Z-Z_{i_{+}})
\left[\delta ^s_{s_1}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)
R^{s'_1}_{s'}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}'}}\right|1,1,\textit{J'},\textit{G'}\right)
\right.
\left.\left.+\sum_{i=2}^{N}\delta ^{s}_{s_{i}}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_{i})
R^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J'},\textit{G'}\right)
\right]
\right\} , &\nonumber
\eeq
where the kernels under the integral in the rhs of the equation correspond to time $t=-T_0/2$ and where
we used special designations for kernels describing the system of $N$ photons and $N_+$ electrons:
$$P^{s_a}_{s_b}\left(\left.{{\textit{\textbf{k}}_c}\atop{\textit{\textbf{k}}_d}}\right|i,j,\textit{J},\textit{G}\right)\equiv \rho{^{s_{a}s_{2}...}_{s_{b}s_{2}...}}{^{s ...}_{s_i ...}}^{s_N}_{s_N}
{^{\sigma_{1}\sigma_{2}...\sigma...\sigma_{N+}}_{\sigma_{1}\sigma_{2}...\sigma_j...\sigma_{N+}}}
{^{n_{1}n_{2}...n...n_{N+}}_{n_{1}n_{2}...n_j...n_{N+}}}
\left({{\textit{\textbf{k}}_c \textit{\textbf{k}}_2 ...}\atop{\textit{\textbf{k}}_d \textit{\textbf{k}}_2 ...}}
{{\textit{\textbf{k}}...}\atop{\textit{\textbf{k}}_i ...}}{{\textit{\textbf{k}}_N}\atop{\textit{\textbf{k}}_N}}\left|
{{Y_1 Y_2 ... Y ... Y_{N_{+}}}\atop{Y_1 Y_2 ... Y_{j} ... Y_{N_{+}}}}
{{Z_1 Z_2 ... Z ... Z_{N_{+}}}\atop{Z_1 Z_2 ... Z_{j} ... Z_{N_{+}}}}
\right)\right.,
$$
$$R^{s_a}_{s_b}\left(\left.{{\textit{\textbf{k}}_c}\atop{\textit{\textbf{k}}_d}}\right|i,j,\textit{J},\textit{G}\right)\equiv \rho{^{s_{a}s_{2}...}_{s_{b}s_{2}...}}{^{s_i ...}_{s ...}}^{s_N}_{s_N}
{^{\sigma_{1}\sigma_{2}...\sigma_j...\sigma_{N+}}_{\sigma_{1}\sigma_{2}...\sigma...\sigma_{N+}}}
{^{n_{1}n_{2}...n_j...n_{N+}}_{n_{1}n_{2}...n...n_{N+}}}
\left({{\textit{\textbf{k}}_c \textit{\textbf{k}}_2 ...}\atop{\textit{\textbf{k}}_d \textit{\textbf{k}}_2 ...}}
{{\textit{\textbf{k}}_i ...}\atop{\textit{\textbf{k}} ...}}{{\textit{\textbf{k}}_N}\atop{\textit{\textbf{k}}_N}}\left|
{{Y_1 Y_2 ... Y_j ... Y_{N_{+}}}\atop{Y_1 Y_2 ... Y ... Y_{N_{+}}}}
{{Z_1 Z_2 ... Z_j ... Z_{N_{+}}}\atop{Z_1 Z_2 ... Z ... Z_{N_{+}}}}
\right)\right.,
$$
where $\textit{J}=(s,\textit{\textbf{k}})$ and $\textit{G}=(\sigma,n,Y,Z)$ are parameters of photons and electrons respectively.
Let us denote the terms under the sum $\sum\limits_{i=2}^{N}$ by $\Xi$. One can transform it:
\beq
& \strut\displaystyle \Xi=\frac{1}{(N-1)!N_{+}!}i
\frac {\alpha}{2\pi}
\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
& \nonumber \\
& \strut\displaystyle
\times b^\dag_{n'\sigma'}(Y',Z')b_{n\sigma}(Y,Z)\bar {a}_{s'}
(\textit{\textbf{k}}')a_s(\textit{\textbf{k}})
M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right.\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right)
\sum_{i_{+}=1}^{N_{+}}\int\prod_{i=2}^{N}
\left(\frac {\d\textit{\textbf{k}}_i}{k_i}\frac {\d\textit{\textbf{k}}'_i}{k'_i}\right)\prod_{i_{+}=1}^{N_{+}}
\left(\frac{\d Y_{i_{+}}\d Z_{i_{+}}}{R_{i_{+}}}\frac{\d Y'_{i_{+}}\d Z'_{i_{+}}}{R'_{i_{+}}}\right)
& \nonumber \\
& \strut\displaystyle \times\left\{
\delta^{n'}_{n'_{i_{+}}}\delta^{\sigma'}_{\sigma'_{i_{+}}}\delta(Y'-Y'_{i_{+}})
\delta(Z'-Z'_{i_{+}}) \right.
\sum_{i=2}^N\delta ^{s'}_{s_i}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}_i)
P^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J},\textit{G}\right)
& \nonumber \\
& \strut\displaystyle
-\delta^n_{n_{i_{+}}}\delta^{\sigma}_{\sigma_{i_{+}}}
\delta(Y-Y_{i_{+}})\delta(Z-Z_{i_{+}})
\left.\sum_{i=2}^{N}\delta^s_{s_{i}}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i)
R^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J'},\textit{G'}\right)
\right\}
& \nonumber \\
& \strut\displaystyle
=\frac {1}{(N-1)!N_{+}!}i
\frac {\alpha}{2\pi}
\int\d Y\d Z\d Y'\d Z'\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
& \nonumber \\
& \strut\displaystyle
\times\sum\limits_{i_{+}=1}^{N_{+}}\int\prod_{i=2}^{N}\left(\frac {\d\textit{\textbf{k}}_i}{k_i}
\frac {\d\textit{\textbf{k}}'_i}{k'_i}\right)\prod_{i_{+}=1}^{N_{+}}\left(\frac {\d Y_{i_{+}}
\d Z_{i_{+}}}{R_{i_{+}}}\frac {\d Y'_{i_{+}}\d Z'_{i_{+}}}{R'_{i_{+}}}\right)
\left\{
\sum_{i=2}^{N}
P^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J'},\textit{G'}\right)
\right.
\left. -\sum_{i=2}^{N}
P^{s'_1}_{s_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J'},\textit{G'}\right)\right\}.
& \nonumber
\eeq
Creation and annihilation operators with the elements of scattering matrix were placed under the integral in the last transformation. Indices in the round
brackets indicate the positions of pairs of parameters ($s$--$s'$), ($n$--$n'$), ($Y$--$Y'$), ($Z$--$Z'$), ($\sigma$--$\sigma'$) and ($\textit{\textbf{k}}$--$\textit{\textbf{k}}$'). One notices that all the terms in the sum $\Xi$ cancel out, giving $\Xi=0$.
The remaining part of the equation can be rewritten as
\beq \label{eq:ker1}
& \strut\displaystyle
\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right.\left|
\frac {T_0}{2}\right)-\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|-
\frac {T_0}{2}\right)=i
\frac {\alpha}{2\pi}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)\right.
& \nonumber \\
& \strut\displaystyle
\times M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right.\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right)
\left[\delta^{s'}_{s'_1}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)
\rho_{s_1 \sigma' n'}^{s \sigma n}\left(\left.{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right|
{{Y Z}\atop {Y' Z'}}\right)-\delta^s_{s_1}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)
\rho_{s' \sigma' n'}^{s'_1 \sigma n}\left(\left.{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}'}}\right|
{{Y Z}\atop {Y' Z'}}\right)\right]. &
\eeq
Thus, we have obtained the first equation of the Bogolyubov hierarchy describing the evolution of 1-particle density matrix kernel through the 2-particle density matrix kernel.
Let us now obtain the second equation of the Bogolyubov hierarchy for the 2-particle kernels. We proceed with the integration and the summation over the parameters of $(N-1)$ photons and $(N_{+}-1)$ electrons in equation (\ref{eq:comptLiuvTh}):
\beq
& \strut\displaystyle \rho_{s_1 \sigma_1 n_1}^{s'_1 \sigma'_1 n'_1}\left({{\textit{\textbf{k}}'_1 Y'_1
Z'_1}\atop {\textit{\textbf{k}}_1 Y_1 Z_1}}\left|\frac {T_0}{2}\right)-
\rho_{s_1 \sigma_1 n_1}^{s'_1 \sigma'_1 n'_1}\left({{\textit{\textbf{k}}'_1 Y'_1 Z'_1}\atop
{\textit{\textbf{k}}_1 Y_1 Z_1}}\right|-\frac {T_0}{2}\right)
=\frac {1}{(N-1)!(N_{+}-1)!}
i\frac {\alpha}{2\pi}
& \nonumber \\
& \strut\displaystyle
\times\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)b^\dag_{n'\sigma'}(Y',Z')b_{n\sigma}(Y,Z)\bar {a}_{s'}(\textit{\textbf{k}}')a_{s}(\textit{\textbf{k}})
M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right.\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right) & \nonumber \\
& \strut\displaystyle \times\int\prod_{i=2}^{N}\left(\frac {\d\textit{\textbf{k}}_i}{k_i}
\frac {\d\textit{\textbf{k}}'_i}{k'_i}\right)\prod_{i_{+}=2}^{N_{+}}\left(\frac {\d Y_{i_{+}}
\d Z_{i_{+}}}{R_{i_{+}}}\frac {\d Y'_{i_{+}}\d Z'_{i_{+}}}{R'_{i_{+}}}\right)
\times\!
\left\{\!
\left[\delta^{n'}_{n'_1}\delta^{\sigma'}_{\sigma'_1}\delta^{s'}_{s'_1}
\delta(Y'\!\!-\!Y'_1)\delta(Z'\!\!-\!Z'_1)\delta(\textit{\textbf{k}}'\!\!-\!\textit{\textbf{k}}'_1)\
P^{s}_{s_1}\left(\left.{{\textit{\textbf{k}}}\atop{\textit{\textbf{k}}_1}}\right|1,1,\textit{J},\textit{G}\right)
\right.\right. & \nonumber \\
& \strut\displaystyle
+\delta^{s'}_{s'_1}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\sum_{i_{+}=2}^{N_{+}}
\delta_{\sigma_{i_+}}^{\sigma'}\delta_{n_{i_+}}^{n'}\delta(Y_{i_+}-Y')
\delta(Z_{i_+}-Z')\
P^{s}_{s_1}\left(\left.{{\textit{\textbf{k}}}\atop{\textit{\textbf{k}}_1}}\right|1,i_{+},\textit{J},\textit{G}\right)
& \nonumber \\
& \strut\displaystyle +\delta^{n'}_{n'_1}\delta^{\sigma'}_{\sigma'_{1}}\delta(Y'-Y'_1)
\delta(Z'-Z'_1)\sum_{i=2}^N\delta_{s_i}^{s'}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i)\
P^{s_1}_{s_1}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}_1}}\right|i,1,\textit{J},\textit{G}\right)
& \nonumber \\
& \strut\displaystyle +\sum_{i=2}^{N}\sum_{i_{+}=2}^{N_{+}}
\delta_{\sigma_{i_+}}^{\sigma'}\delta_{n_{i_+}}^{n'}\delta(Y_{i_+}-Y')
\delta(Z_{i_+}-Z')\delta_{s_i}^{s'}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i)
\left.\
P^{s_1}_{s_1}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J},\textit{G}\right)
\right] & \nonumber \\
& \strut\displaystyle -\left[\delta_s^{s_1}\delta_n^{n_1}\delta_{\sigma}^{\sigma_1}
\delta(\textit{\textbf{k}}\!-\!\textit{\textbf{k}}_1)\delta(Y\!-\!Y_1)\delta(Z\!-\!Z_1)\
R^{s_1}_{s'}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}'}}\right|1,1,\textit{J'},\textit{G'}\right)\right.
& \nonumber \\
& \strut\displaystyle +\delta_s^{s_1}\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\sum_{i_{+}=2}^{N_{+}}
\delta_{n_{i_+}}^n\delta_{\sigma_{i_+}}^{\sigma}\delta(Y-Y_{i_+})
\delta(Z-Z_{i_+}) \
R^{s_1}_{s'}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}'}}\right|1,i_{+},\textit{J'},\textit{G'}\right)
& \nonumber \\
& \strut\displaystyle +\delta_n^{n_1}\delta_\sigma^{\sigma_1}\delta(Y-Y_1)\delta(Z-Z_1)
\sum_{i=2}^N\delta_{s_i}^s\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i) \
R^{s_1}_{s_1}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}_1}}\right|i,1,\textit{J'},\textit{G'}\right)
& \nonumber \\
& \strut\displaystyle +\sum_{i=2}^N\sum_{i_{+}=2}^{N_{+}}\delta_{s_i}^s
\delta_{n_{i_+}}^n\delta_{\sigma_{i_+}}^\sigma\delta(Y-Y_{i_+})
\delta(Z-Z_{i_+})\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_i)
\left.\left.\
R^{s_1}_{s_1}\left(\left.{{\textit{\textbf{k}}_1}\atop{\textit{\textbf{k}}_1}}\right|i,i_{+},\textit{J'},\textit{G'}\right)
\right]\right\}. &
\eeq
The terms with a double sum $\displaystyle\sum_{i=2}^N\sum_{i_{+}=2}^{N_{+}}$ cancel each other.
Other terms are transformed into the form containing 2- and 3-particle kernels.
As a result, we obtain the equation for the 2-particle density matrix kernel:
\beq \label{eq:ker2}
& \strut\displaystyle \rho_{s_1 \sigma_1 n_1}^{s'_1 \sigma'_1 n'_1}\left(\left.
{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right|{{Y'_1 Z'_1}\atop {Y_1 Z_1}}\left|
\frac {T_0}{2}\right)-\rho_{s_1 \sigma_1 n_1}^{s'_1 \sigma'_1 n'_1}\left(\left.
{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right|{{Y'_1 Z'_1}\atop {Y_1 Z_1}}\right|-
\frac {T_0}{2}\right)
& \nonumber \\
& \strut\displaystyle
=i\frac{\alpha}{2\pi}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
M^{\sigma's'}_{\sigma s}\left({{n'}\atop{n}}{{Y'}\atop{Y}}{{Z'}\atop{Z}}\right.\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right) & \nonumber \\
& \strut\displaystyle \times\left[\delta_{s'_1}^{s'}\delta_{n'_1}^{n'}
\delta_{\sigma'_1}^{\sigma'}\delta(Y'-Y'_1)\delta(Z'-Z'_1)\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\
\rho_{s_1 \sigma_1 n_1}^{s \sigma n}\left(\left.{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right|
{{Y Z}\atop {Y_1 Z_1}}\right)\right. & \nonumber \\
& \strut\displaystyle
-\delta_{s_1}^s\delta_{n_1}^n\delta_{\sigma_1}^\sigma
\delta(Y-Y_1)\delta(Z-Z_1)\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\ \rho_{s' \sigma' n'}^{s'_1
\sigma'_1 n'_1}\left(\left.{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}'}}\right|{{Y'_1 Z'_1}\atop
{Y' Z'}}\right)
& \nonumber \\
& \strut\displaystyle
+\delta_{n'_1}^{n'}\delta_{\sigma'_1}^{\sigma'}
\delta(Y'\!-\!Y'_1)\delta(Z'\!-\!Z'_1)\ \rho_{s_1 s' \sigma_1 n_1}^{s'_1 s
\sigma n}\left(\left.{{\textit{\textbf{k}}'_1 \textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1 \textit{\textbf{k}}'}}\right|{{Y Z}\atop
{Y_1 Z_1}}\right)\!-\!\delta_{n_1}^n\delta_{\sigma_1}^\sigma\delta(Y\!-\!Y_1)
\delta(Z\!-\!Z_1)\ \rho_{s_1 s' \sigma'_1 n'_1}^{s'_1 s \sigma' n'}\left(\left.
{{\textit{\textbf{k}}'_1 \textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1 \textit{\textbf{k}}'}}\right|{{Y' Z'}\atop {Y'_1 Z'_1}}\right)
& \nonumber \\
& \strut\displaystyle
\left. +\delta_{s'_1}^{s'}\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\ \rho_{s_1 \sigma_1
n_1 \sigma' n'}^{s \sigma'_1 n'_1 \sigma n}\left(\left.{{\textit{\textbf{k}}}\atop
{\textit{\textbf{k}}_1}}\right|{{Y'_1 Z'_1 Y Z}\atop {Y_1 Z_1 Y' Z'}}\right)-
\delta_{s_1}^s\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\ \rho_{s' \sigma_1 n_1 \sigma' n'}^{s'_1
\sigma'_1 n'_1 \sigma n}\left(\left.{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}'}}\right|
{{Y'_1 Z'_1 Y Z}\atop {Y_1 Z_1 Y' Z'}}\right)\right]. &
\eeq
Thus, we have derived the equations for kernels of 1-particle density matrix of photons (\ref{eq:ker1}) and for the
2-particle density matrix (\ref{eq:ker2}), which includes both the photon and the electron parameters.
\subsection{Completion of the derivation}
\subsubsection{Expression for the 2-particle kernels through the 1-particle kernels}
Equations (\ref{eq:ker1}) and (\ref{eq:ker2}) are the first two equations of the Bogolyubov hierarchy. The hierarchy can be continued, but using the principle of weakening of correlations, we have stopped at the first two equations. To obtain an equation describing the evolution of the 1-particle kernel through the 1-particle kernels, one must use these two equations. We use the ``molecular chaos'' approximation, according to which there is no correlation between the distribution functions of photons and electrons before interaction. This approximation works better in the case, when the typical time between the interactions is much larger than the typical time of an interaction. The independence of the photons and electrons distributions can be expressed through the following equation:
\be \label{eq:m_chaos}
\rho_{\sigma n s}^{\sigma'n's'}\left({{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\left|{{Y'Z'}\atop {Y Z}}
\right)=\rho_{\sigma n}^{\sigma'n'}\left({Y'Z'}\atop {Y Z}\right)\rho_{s}^{s'}
\left({{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right). \right.
\ee
The 2-particle kernels of the photon gas and the 2-particle kernels of the electron gas are presented through 1-particle kernels according to the properties of symmetry and anti-symmetry of bosonic and fermionic wave functions:
\beq \label{eq:exch_f}
& \strut\displaystyle \rho_{s_1s_2}^{s'_1s'_2}\left({{\textit{\textbf{k}}'_1\textit{\textbf{k}}'_2}\atop {\textit{\textbf{k}}_1\textit{\textbf{k}}_2}}
\right)=\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s_2}^{s'_2}
\left({{\textit{\textbf{k}}'_2}\atop {\textit{\textbf{k}}_2}}\right)+\rho_{s_1}^{s'_2}\left({{\textit{\textbf{k}}'_2}\atop
{\textit{\textbf{k}}_1}}\right)\rho_{s_2}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_2}}\right),
& \\ \label{eq:exch_e}
& \strut\displaystyle \rho_{\sigma_1n_1\sigma_2n_2}^{\sigma'_1n'_1\sigma'_2n'_2}
\left({{Y'_1 Z'_1 Y'_2 Z'_2}\atop {Y_1 Z_1 Y_2 Z_2}}\right)=\rho_{\sigma_1n_1}^{\sigma'_1n'_1}
\left({{Y'_1 Z'_1}\atop {Y_1 Z_1}}\right)\rho_{\sigma_2n_2}^{\sigma'_2n'_2}
\left({{Y'_2 Z'_2}\atop {Y_2 Z_2}}\right)-\rho_{\sigma_1n_1}^{\sigma'_2n'_2}
\left({{Y'_2 Z'_2}\atop {Y_1 Z_1}}\right)\rho_{\sigma_2n_2}^{\sigma'_1n'_1}
\left({{Y'_1 Z'_1}\atop {Y_2 Z_2}}\right). &
\eeq
These equations become more accurate, when photons and electrons gases are sufficiently rarefied.
Transformations of the 3-particle kernels in equations (\ref{eq:ker1}) and (\ref{eq:ker2}) are simple because there are no pure photon or pure electron kernels among them and one can rewrite them easily through the 1- and 2-particle kernels. Then we use equations (\ref{eq:m_chaos}) and (\ref{eq:exch_f}) to complete the transformation.
\subsubsection{Closure of the Bogolyubov hierarchy}
Substituting equation (\ref{eq:ker2}) to equation (\ref{eq:ker1}), we get:
\beq
& \strut\displaystyle \rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|
\frac {T_0}{2}\right)-\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right|-
\frac {T_0}{2}\right)=i
\frac{\alpha}{2\pi}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
M_{\sigma s}^{\sigma's'}\left({{n'Y'Z'}\atop {nYZ}}\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}
\right) \right. & \nonumber \\
& \strut\displaystyle
\times\left[k'\delta_{s'_1}^{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\rho_{s_1\sigma'n'}^
{s\sigma n}\left({{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\left|{{YZ}\atop {Y'Z'}}\right)-
k\delta_{s_1}^s(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\rho_{s'\sigma'n'}^{s'_1\sigma n}\left({{\textit{\textbf{k}}'_1}\atop
{\textit{\textbf{k}}'}}\right|{{YZ}\atop {Y'Z'}}\right)\right] & \nonumber \\
& \strut\displaystyle -
\frac {\alpha^2}{(2\pi)^2}
\sum_{n,n',n'',n'''}\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}
\frac {\d Y''\d Z''}{R''}\frac {\d Y'''\d Z'''}{R'''}\frac {\d\textit{\textbf{k}}}{k}
\frac {\d\textit{\textbf{k}}'}{k'}\frac {\d\textit{\textbf{k}}''}{k''}\frac {\d\textit{\textbf{k}}'''}{k'''}
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
\delta\left(n'',Y'',Z'',\textit{\textbf{k}}''\;|\;n''',Y''',Z''',\textit{\textbf{k}}'''\right)
&\nonumber \\
& \strut\displaystyle
\times M_{\sigma s}^{\sigma's'}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right)M_{\sigma''s''}^{\sigma'''s'''}\left({{n'''Y'''Z'''}\atop
{n''Y''Z''}}\right|{{\textit{\textbf{k}}'''}\atop {\textit{\textbf{k}}''}}\right)
& \nonumber \\
& \strut\displaystyle
\times\left\{
\delta_{s'_1}^{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\left[k'k'''R'''\delta_{sn\sigma}^
{s'''n'''\sigma'''}(\textit{\textbf{k}}-\textit{\textbf{k}}''')\delta(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')\rho_{s_1\sigma' n'}^
{s''\sigma''n''}\left({{\textit{\textbf{k}}''}\atop {\textit{\textbf{k}}_1}}\left|{{Y''Z''}\atop {Y'Z'}}\right)
\right.\right.\right.
& \nonumber \\
& \strut\displaystyle
-k'k''R''\delta_{s_1n'\sigma'}^{s''n''\sigma''}(\textit{\textbf{k}}_1-\textit{\textbf{k}}'')
\delta(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s'''\sigma'''n'''}^{s\sigma n}\left({{\textit{\textbf{k}}}\atop
{\textit{\textbf{k}}'''}}\left|{{YZ}\atop {Y'''Z'''}}\right) \right. & \nonumber \\
& \strut\displaystyle +k'R'''\delta_{n\sigma}^{n'''\sigma'''}(\textit{\textbf{p}}_{YZ}'''-\textit{\textbf{p}}_{YZ})
\rho_{s_1s'''\sigma' n'}^{s s''\sigma''n''}\left({{\textit{\textbf{k}}\vk''}\atop
{\textit{\textbf{k}}_1\textit{\textbf{k}}'''}}\left|{{Y''Z''}\atop {Y'Z'}}\right)+k'k'''\delta_s^{s'''}
(\textit{\textbf{k}}-\textit{\textbf{k}}''')\rho_{s_1\sigma' n'\sigma'''n'''}^{s''\sigma n\sigma''n''}
\left({{\textit{\textbf{k}}''}\atop {\textit{\textbf{k}}_1}}\right|{{YZY''Z''}\atop {Y'Z'Y'''Z'''}}\right)
& \nonumber \\
& \strut\displaystyle
\left. -k'k''\delta_{s_1}^{s''}(\textit{\textbf{k}}_1-\textit{\textbf{k}}'')
\rho_{s'''\sigma'n'\sigma'''n'''}^{s\sigma n\sigma''n''}\left({{\textit{\textbf{k}}}\atop
{\textit{\textbf{k}}'''}}\left|{{YZY''Z''}\atop {Y'Z'Y'''Z'''}}\right)-k'R''
\delta_{n'\sigma'}^{n''\sigma''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s_1s'''\sigma n}^
{ss''\sigma'''n'''}\left({{\textit{\textbf{k}}\vk''}\atop {\textit{\textbf{k}}_1\textit{\textbf{k}}'''}}\right|{{Y'''Z'''}\atop
{YZ}}\right)\right]
& \nonumber \\
& \strut\displaystyle
+\delta_{s_1}^s(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\left[-kk'''R'''\delta_{s_1'n\sigma}^
{s'''n'''\sigma'''}(\textit{\textbf{k}}'''-\textit{\textbf{k}}_1')\delta(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')\rho_{s'\sigma' n'}^
{s''\sigma''n''}\left({{\textit{\textbf{k}}''}\atop {\textit{\textbf{k}}'}}\left|{{Y''Z''}\atop {Y'Z'}}\right)
\right.\right.
& \nonumber \\
& \strut\displaystyle
+kk''R''\delta_{s'}^{s''}(\textit{\textbf{k}}'-\textit{\textbf{k}}'')\delta_(n'\sigma')^
{n''\sigma''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s'''\sigma''' n'''}^{s_1'\sigma n}
\left({{\textit{\textbf{k}}_1'}\atop {\textit{\textbf{k}}'''}}\left|{{YZ}\atop {Y'''Z'''}}\right) \right.
& \nonumber \\
& \strut\displaystyle
-kR'''\delta_{n\sigma}^{n'''\sigma'''}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')
\rho_{s's'''\sigma'n'}^{s_1's''\sigma''n''}\left({{\textit{\textbf{k}}_1'\textit{\textbf{k}}''}\atop
{\textit{\textbf{k}}'\textit{\textbf{k}}'''}}\left|{{Y''Z''}\atop {Y'Z'}}\right)+kR''\delta_{n'\sigma'}^
{n''\sigma''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s's'''\sigma n}^{s_1's''\sigma''' n'''}
\left({{\textit{\textbf{k}}_1'\textit{\textbf{k}}''}\atop {\textit{\textbf{k}}'\textit{\textbf{k}}'''}}\right|{{Y'''Z'''}\atop {YZ}}\right)
& \nonumber \\
& \strut\displaystyle
\left.\left. -kk'''\delta_{s_1'}^{s'''}(\textit{\textbf{k}}_1'-\textit{\textbf{k}}''')
\rho_{s\sigma n}^{s'\sigma' n'}\left({{\textit{\textbf{k}}''}\atop {\textit{\textbf{k}}'}}\left|{{Y Z Y'' Z''}
\atop {Y'Z'Y'''Z'''}}\right)+kk''\delta_{s'}^{s''}(\textit{\textbf{k}}'-\textit{\textbf{k}}'')
\rho_{s'''\sigma'n'\sigma'''n'''}^{s_1'\sigma n\sigma''n''}\left({{\textit{\textbf{k}}_1'}\atop
{\textit{\textbf{k}}'''}}\right|{{YZY''Z''}\atop {Y'Z'Y'''Z'''}}\right)\right]\right\} , &
\eeq
where we introduced a symbol for the product of several Kronecker's deltas:
$\delta_{\alpha_1...\alpha_N}^{\beta_1...\beta_N}\equiv\prod_{i=1}^N
\delta_{\alpha_i}^{\beta_i},$
a symbol for the product of Kronecker's deltas and a $\delta$-function:
$\delta_{\alpha_1...\alpha_N}^{\beta_1...\beta_N}(a)\equiv\delta_{\alpha_1...
\alpha_N}^{\beta_1...\beta_N}\delta(a)$,
and $\textit{\textbf{p}}_{YZ}\equiv(0,Y,Z)$ is the electron momentum.
Using equations (\ref{eq:m_chaos}), (\ref{eq:exch_f}) and (\ref{eq:exch_e}), one can transform 2- and 3-particle kernels in the {rhs} of the equation to the 1-particle kernels. Then using equation (\ref{eq:nucl2df2}), the equation can be represented in terms of the coherency matrix. After some algebra we get:
\beq \label{eq:ku_1}
& \strut\displaystyle \rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|
\frac {T_0}{2}\right)-\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\right|-
\frac {T_0}{2}\right) & \nonumber \\
& \strut\displaystyle =i
\frac {\alpha}{2\pi}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\frac {\d\textit{\textbf{k}}'}{k'}
\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
{\mathcal A}_1
& \nonumber \\
& \strut\displaystyle
+
\frac {\alpha^2}{4\pi^2}
\sum_{n,n',n'',n'''}\int\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}
\frac {\d Y''\d Z''}{R''}\frac {\d Y'''\d Z'''}{R'''}\frac {\d\textit{\textbf{k}}}{k}
\frac {\d\textit{\textbf{k}}'}{k'}\frac {\d\textit{\textbf{k}}''}{k''}\frac {\d\textit{\textbf{k}}'''}{k'''}
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
M_{\sigma s}^{\sigma's'}\left({{n'Y'Z'}\atop {nYZ}}\left|{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right) \right.
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n'',Y'',Z'',\textit{\textbf{k}}''\;|\;n''',Y''',Z''',\textit{\textbf{k}}'''\right)
M_{\sigma''s''}^{\sigma'''s'''}\left({{n'''Y'''Z'''}\atop{n''Y''Z''}}\left|{{\textit{\textbf{k}}'''}\atop {\textit{\textbf{k}}''}}\right) \right.
& \nonumber \\
& \strut\displaystyle \times\left\{
{\mathcal B}_1+{\mathcal B}_2+{\mathcal B}_3+{\mathcal B}_4 +{\mathcal B}_5+{\mathcal B}_6\right\}, &
\eeq
where
\beq
& \strut\displaystyle {\mathcal A}_1=M_{\sigma s}^{\sigma' s'}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}'}\atop {\textit{\textbf{k}}}}\right)R'\delta_n^{n'}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}')\rho_{\sigma'n'}^{\sigma}
(\textit{\textbf{p}}_{YZ})\left[k'\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\delta(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\left(k_1
\delta_{s_1'}^{s_1}\rho_{s_1}^{s}(\textit{\textbf{k}}_1)-k\delta_{s_1}^s\rho_{s'}^{s'_1}
(\textit{\textbf{k}}')\right)\right], \right. & \\
& \strut\displaystyle {\mathcal B}_1=k'k'''\delta(\textit{\textbf{k}}''-\textit{\textbf{k}}''')\delta_n^{n'''}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')
\delta_{n'}^{n''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s'''}^{s''}(\textit{\textbf{k}}''')\left(R'R'''
\delta_\sigma^{\sigma'''}\rho_{\sigma'n'}^{\sigma''}(\textit{\textbf{p}}_{YZ}')-RR''
\delta_{\sigma'}^{\sigma''}\rho_{\sigma n}^{\sigma'''}(\textit{\textbf{p}}_{YZ})\right)
& \nonumber \\
& \strut\displaystyle \times\left(k\delta(\textit{\textbf{k}}_1'-\textit{\textbf{k}}')\delta_{s_1}^s(\textit{\textbf{k}}-\textit{\textbf{k}}_1)
\rho_{s'}^{s'_1}(\textit{\textbf{k}}')-k_1\delta(\textit{\textbf{k}}_1-\textit{\textbf{k}})\delta_{s'_1}^{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)
\rho_{s_1}^{s}(\textit{\textbf{k}}_{1})\right), & \\
& \strut\displaystyle {\mathcal B}_2=R'\delta_n^{n'}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}')\rho_{\sigma'n'}^{\sigma}(\textit{\textbf{p}}_{YZ}')
R'''\delta_{n'''}^{n''}(\textit{\textbf{p}}_{YZ}'''-\textit{\textbf{p}}_{YZ}'')\rho_{\sigma'''n'''}^{\sigma''}(\textit{\textbf{p}}_{YZ}''')
& \nonumber \\
&\strut\displaystyle \times\left[\delta_{s'_1}^{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\left(k'k''k'''
\delta_{s_1}^{s''}(\textit{\textbf{k}}_1-\textit{\textbf{k}}'')\delta(\textit{\textbf{k}}-\textit{\textbf{k}}''')\rho_{s'''}^{s}(\textit{\textbf{k}}''')-
k_1k'k'''\delta_s^{s'''}(\textit{\textbf{k}}-\textit{\textbf{k}}''')\delta(\textit{\textbf{k}}''-\textit{\textbf{k}}_1)\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)
\right) \right. & \nonumber \\
& \strut\displaystyle \left. +\delta_{s_1}^s(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\left(kk'k''\delta_{s'_1}^
{s'''}(\textit{\textbf{k}}'_1-\textit{\textbf{k}}''')\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'')\rho_{s'}^{s''}(\textit{\textbf{k}}')-kk''k'''
\delta_{s'}^{s''}(\textit{\textbf{k}}'-\textit{\textbf{k}}'')\delta(\textit{\textbf{k}}'_1-\textit{\textbf{k}}''')\rho_{s'''}^{s'_{1}}(\textit{\textbf{k}}''')
\right)\right], & \\
& \strut\displaystyle {\mathcal B}_3=\delta_{s_1}^s(\textit{\textbf{k}}-\textit{\textbf{k}}_1)kk'R'R'''\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'')
\delta(\textit{\textbf{k}}'_1-\textit{\textbf{k}}''')\delta_{n'}^{n''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\delta_n^{n'''}
(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')\rho_{s'}^{s''}(\textit{\textbf{k}}')\rho_{\sigma'n'}^{\sigma''}(\textit{\textbf{p}}_{YZ}')
& \nonumber \\
& \strut\displaystyle \times\left[k'''\delta_{\sigma}^{\sigma'''}\rho_{s'''}^{s'_1}
(\textit{\textbf{k}}''')+k'''\delta_{s'_1}^{s'''}\left(\delta_\sigma^{\sigma'''}-
\rho_{\sigma'''n'''}^\sigma(\textit{\textbf{p}}_{YZ}''')\right)\right], & \\
& \strut\displaystyle {\mathcal B}_4=-k_1k'k'''R'R'''\delta_{s'_1}^{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)
\delta(\textit{\textbf{k}}-\textit{\textbf{k}}''')\delta(\textit{\textbf{k}}_1-\textit{\textbf{k}}'')\delta_n^{n'''}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')
\delta_{n'}^{n''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)\rho_{\sigma'n'}^
{\sigma''}(\textit{\textbf{p}}_{YZ}') & \nonumber \\
& \strut\displaystyle \times\left[\delta_\sigma^{\sigma'''}\rho_{s'''}^s(\textit{\textbf{k}}''')+
\delta_s^{s'''}\left(\delta_\sigma^{\sigma'''}-\rho_{\sigma'''n'''}^{\sigma}
(\textit{\textbf{p}}_{YZ}''')\right)\right], & \\
& \strut\displaystyle {\mathcal B}_5=k'k'''\delta(\textit{\textbf{k}}-\textit{\textbf{k}}''')\delta(\textit{\textbf{k}}_1-\textit{\textbf{k}}'')\delta_{s'_1}^
{s'}(\textit{\textbf{k}}'-\textit{\textbf{k}}'_1)\delta_{n}^{n'''}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')\delta_{n'}^{n''}(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')
\rho_{s'''}^s(\textit{\textbf{k}}''') & \nonumber \\
& \strut\displaystyle \times\left[RR''k_{1}\delta_{\sigma'}^{\sigma''}\rho_{s_1}^{s''}
(\textit{\textbf{k}}_1)\rho_{\sigma n}^{\sigma'''}(\textit{\textbf{p}}_{YZ})+k''R'''\delta_{s_1}^{s''}
\rho_{\sigma'''n'''}^{\sigma}(\textit{\textbf{p}}_{YZ}''')\left(R''\delta_{\sigma'}^{\sigma''}-R'
\rho_{\sigma'n'}^{\sigma''}(\textit{\textbf{p}}_{YZ}')\right)\right], & \\
& \strut\displaystyle {\mathcal B}_6=-kk'''\delta(\textit{\textbf{k}}_1'-\textit{\textbf{k}}''')\delta(\textit{\textbf{k}}'-\textit{\textbf{k}}'')
\delta_{s_1}^{s}(\textit{\textbf{k}}-\textit{\textbf{k}}_1)\delta_n^{n'''}(\textit{\textbf{p}}_{YZ}-\textit{\textbf{p}}_{YZ}''')\delta_{n'}^{n''}
(\textit{\textbf{p}}_{YZ}'-\textit{\textbf{p}}_{YZ}'')\rho_{s'''}^{s'_1}(\textit{\textbf{k}}''') & \nonumber \\
& \strut\displaystyle \times\left[RR''k'\delta_{\sigma'}^{\sigma''}\rho_{s'}^{s''}
(\textit{\textbf{k}}')\rho_{\sigma n}^{\sigma'''}(\textit{\textbf{p}}_{YZ})+k''R'''\delta_{s'}^{s''}
\rho_{\sigma'''n'''}^\sigma(\textit{\textbf{p}}_{YZ}''')\left(R''\delta_{\sigma'}^{\sigma''}-R'
\rho_{\sigma'n'}^{\sigma''}(\textit{\textbf{p}}_{YZ}')\right)\right]. &
\eeq
\subsubsection{Simplification of equation (\ref{eq:ku_1})}
The presence of the $\delta$-functions under the integrals in equation (\ref{eq:ku_1}) allows us to reduce a number of integrations. Let us define two singular measures $\mu_1$ and $\mu_2$:
\beq
& \strut\displaystyle \d\mu_1=\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}
\frac {\d\textit{\textbf{k}}'}{k'}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right), & \\
& \strut\displaystyle \d\mu_2=\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}
\frac {\d Y''\d Z''}{R''}\frac {\d Y'''\d Z'''}{R'''}\frac {\d\textit{\textbf{k}}}{k}
\frac {\d\textit{\textbf{k}}'}{k'}\frac {\d\textit{\textbf{k}}''}{k''}\frac {\d\textit{\textbf{k}}'''}{k'''}\times
& \nonumber \\
& \strut\displaystyle
\times\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}'\right)
\delta\left(n'',Y'',Z'',\textit{\textbf{k}}''\;|\;n''',Y''',Z''',\textit{\textbf{k}}'''\right). &
\eeq
The terms from the {rhs} of equation (\ref{eq:ku_1}) can be written as:
\beq \label{eq:a1}
& \strut\displaystyle \sum_{n,n'}\int\d\mu_1{\mathcal A}_1=\sum_n\int\frac {\d Y\d Z}{R}
M_{\sigma s}^{\sigma' s'}\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}
\right)\rho_{\sigma'n}^\sigma(Z)\left[\delta_{s'_1}^{s'}\rho_{s_1}^{s}(\textit{\textbf{k}}_1)-
\delta_{s_1}^{s}\rho_{s'}^{s'_1}(\textit{\textbf{k}}_1)\right] \right. & \\ \label{eq:b1}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_1=\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R'-R)M_{\sigma''s''}^
{\sigma''' s'''}\left({{nYZ}\atop {n'Y'Z'}}\left|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}}}\right)
M_{\sigma s}^{\sigma' s'}\left({{n'Y'Z'}\atop {nYZ}}\right|{{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}
\right)\rho_{s'''}^{s''}(\textit{\textbf{k}}) & \nonumber \\
& \strut\displaystyle \times\left[\delta_\sigma^{\sigma'''}\rho_{\sigma'n'}^{\sigma''}
(Z')-\delta_{\sigma'}^{\sigma''}\rho_{\sigma n}^{\sigma'''}(Z)\right]
\left[\delta_{s_1}^s\rho_{s'}^{s'_1}(\textit{\textbf{k}}'_1)-\delta_{s'_1}^{s'}\rho_{s_1}^{s}
(\textit{\textbf{k}}_1)\right], &
\eeq
\beq \label{eq:b2}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_2=\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(\textit{\textbf{k}}_1-\textit{\textbf{k}})
\rho_{\sigma'n}^\sigma(Z)\rho_{\sigma'''n'}^{\sigma''}(Z')\left[
M_{\sigma s}^{\sigma's'_1}\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}
\right)M_{\sigma''s''}^{\sigma'''s'''}\left({{n'Y'Z'}\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}
\atop {\textit{\textbf{k}}_1}}\right) \right. & \nonumber \\
& \strut\displaystyle \left. \times\left(\delta_{s_1}^{s''}\rho_{s'''}^s(\textit{\textbf{k}})-
\delta_s^{s'''}\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)\right)+M_{\sigma s_1}^{\sigma's'}
\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma''s''}^
{\sigma'''s'''}\left({{n'Y'Z'}\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)
\left(\delta_{s'_1}^{s'''}\rho_{s'}^{s''}(\textit{\textbf{k}})-\delta_{s'}^{s''}\rho_{s'''}^
{s'_1}(\textit{\textbf{k}}_1)\right)\right], &
\eeq
\beq\label{eq:b3}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_3=\sum_{n,n'}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta\left(n,Y,Z,\textit{\textbf{k}}_1\;|\;n',Y',Z',\textit{\textbf{k}}\right)
& \nonumber \\
& \strut\displaystyle \times M_{\sigma s_1}^{\sigma's'}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma''s''}^{\sigma''' s'''}\left({{nYZ}\atop
{n'Y'Z'}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)\rho_{s'}^{s''}(\textit{\textbf{k}})\rho_{\sigma'n'}^
{\sigma''}(Z')\left[\delta_{\sigma}^{\sigma'''}\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)+
\delta_{s'_1}^{s'''}\left(\delta_\sigma^{\sigma'''}-\rho_{\sigma'''n}^{\sigma}
(Z)\right)\right], &
\eeq
\beq\label{eq:b4}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_4=-\sum_{n,n'}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}_{1}\right)
& \nonumber \\
& \strut\displaystyle
\times M_{\sigma s}^{\sigma' s'_1}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma'' s''}^{\sigma'''s'''}\left({{nYZ}\atop
{n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)
\rho_{\sigma'n'}^{\sigma''}(Z')\left[\delta_{\sigma}^{\sigma'''}\rho_{s'''}^s
(\textit{\textbf{k}})+\delta_s^{s'''}\left(\delta_\sigma^{\sigma'''}-\rho_{\sigma'''n}^{\sigma}
(Z)\right)\right], &
\eeq
\beq\label{eq:b5}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_5=\sum_{n,n'}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}_{1}\right)
& \nonumber \\
& \strut\displaystyle
\times M_{\sigma s}^{\sigma's'_1}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma'' s''}^{\sigma''' s'''}\left({{nYZ}\atop
{n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s'''}^{s}(\textit{\textbf{k}})\left[
\delta_{\sigma'}^{\sigma''}\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)\rho_{\sigma n}^{\sigma'''}
(Z)+\delta_{s_1}^{s''}\rho_{\sigma'''n}^{\sigma}(Z)\left(\delta_{\sigma'}^
{\sigma''}-\rho_{\sigma'n'}^{\sigma''}(Z')\right)\right], &
\eeq
\beq\label{eq:b6}
& \strut\displaystyle \sum_{n,...,n'''}\int\d\mu_2{\mathcal B}_6=-\sum_{n,n'}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta\left(n,Y,Z,\textit{\textbf{k}}\;|\;n',Y',Z',\textit{\textbf{k}}_{1}\right)
& \nonumber \\
& \strut\displaystyle
\times M_{\sigma s_1}^{\sigma' s'}\left({{nYZ}\atop {n'Y'Z'}}\left|
{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma''s''}^{\sigma''' s'''}\left({{n'Y'Z'}\atop
{nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)\right[
\delta_{\sigma'}^{\sigma''}\rho_{s'}^{s''}(\textit{\textbf{k}})\rho_{\sigma n'}^{\sigma'''}(Z')
+\delta_{s'}^{s''}\rho_{\sigma'''n'}^{\sigma}(Z')\left(\delta_{\sigma'}^
{\sigma''}-\rho_{\sigma'n}^{\sigma''}(Z)\right)\right]. &
\eeq
We use the {rhs} of equations (\ref{eq:a1})--(\ref{eq:b6}) in the {rhs} of the final kinetic equation.
One can notice that the {rhs} of expression (\ref{eq:b1}) vanishes after summation over the electrons spin states and the photons polarizations.
\subsubsection{Transformation of the left-hand size of equation (\ref{eq:ku_1})}
It is necessary to rewrite the {lhs} of equation (\ref{eq:ku_1}) in terms of the distribution function. Then one can rewrite {lhs} through the differential operator. The later transformation is similar to the one, which is made in the quantum field theory for transition from the limited space-box to the infinite space. Finally we get:
\be
\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|
\frac {T_0}{2}\right)-\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop {\textit{\textbf{k}}_1}}\left|-
\frac {T_0}{2}\right)=T_0\DR {}{t}\rho_{s_1}^{s'_1}\left({{\textit{\textbf{k}}'_1}\atop
{\textit{\textbf{k}}_1}}\right|t\right)=T_0k_1\delta(\textit{\textbf{k}}'_1-\textit{\textbf{k}}_1)\DR {}{t}\rho_{s_1}^{s'_1}
(\textit{\textbf{k}}_{1})=k_1\delta(\unl {k}'_1-\unl {k}_{1})\frac {2\pi}{c}\frac{\d r}{\d t}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1), \right.
\ee
where in the later transition the relation $\delta(k_{1}-k'_{1})=cT_0/(2\pi)$ is used. Then one can restore the dependence of the photon matrix on time and space coordinates. After rewriting the derivative over the line of sight as the full derivative, the
{lhs} of the equation takes the covariant form
\be
k_1\delta(\unl {k}'_1-\unl {k}_{1})\frac {2\pi}{c}\frac{\d r}{\d t}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1)
\longmapsto
2\pi\delta(\unl {k}_{1}-\unl {k}'_{1})
\left(\frac{k_{1}}{c}\frac{\partial}{\partial t}+\textit{\textbf{k}}_{1}\right)\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t)\equiv
2\pi\delta(\unl {k}_{1}-\unl {k}'_{1})\unl {k}_{1}\underline{\nabla}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t).
\ee
\section{Different forms of the kinetic equation}
\subsection{Kinetic equation for coherency matrix}
\subsubsection{The general form of the kinetic equation}
Now one can write the final form of the kinetic equation. In the most general case, we formulate it for the coherency matrix, where the polarization of electrons is taken into account (i.e. for the situation when there can be non-trivial spin-distribution of the electron gas).
The equation for the case of polarized electrons is:
\beq \label{eq:ku_e_pol}
& \strut\displaystyle \unl {k}_1\unl {\nabla}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t)=I_{1}+I_{2}+I_{3},&
\eeq
where
\beq
&I_{1}=i\alpha \strut\displaystyle
\frac {1}{(2\pi)^2}\sum_n\int\frac {\d Y\d Z}{R}\rho_{\sigma'n}^\sigma(Z)
\left[\rho_{s_1}^s(\textit{\textbf{k}}_1)M_{\sigma s}^{\sigma's'_1}
\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)-
\rho_{s'}^{s'_1}(\textit{\textbf{k}}_1)M_{\sigma s_1}^{\sigma's'}\left({{nYZ}\atop {nYZ}}\right|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)\right],&
\eeq
\beq& \strut\displaystyle
I_{2}=\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(k-k_1)
\delta(k\cos\theta-k_1\cos\theta_1)\rho_{\sigma'n}^{\sigma}(Z)
\rho_{\sigma'''n'}^{\sigma''}(Z')
& \nonumber \\
& \strut\displaystyle
\times\left[2M_{\sigma s}^{\sigma's'_{1}}\left({{nYZ}\atop {nYZ}}
\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma''s_1}^{\sigma'''s'''}\left({{n'Y'Z'}
\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s'''}^s(\textit{\textbf{k}})-
M_{\sigma s}^{\sigma's'_1}\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}
\right)M_{\sigma''s''}^{\sigma'''s}\left({{n'Y'Z'}\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop
{\textit{\textbf{k}}_1}}\right)\rho_{s_1}^{s''}(\textit{\textbf{k}}_1) \right.
& \nonumber \\
& \strut\displaystyle
\left. -M_{\sigma''s'}^{\sigma'''s'''}\left({{n'Y'Z'}\atop {n'Y'Z'}}
\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma s_1}^{\sigma's'}\left({{nYZ}\atop
{nYZ}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)\right],
&\eeq
\beq& \strut\displaystyle
I_{3}=
\frac{\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac{\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_1)
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1)
& \nonumber \\
& \strut\displaystyle
\times\left\{
\left[\delta_\sigma^{\sigma'''}\rho_{\sigma'n}^{\sigma''}(Z)-
\delta_{\sigma'}^{\sigma''}\rho_{\sigma n'}^{\sigma'''}(Z')\right]
\left[M_{\sigma s_1}^{\sigma's'}\left({{nYZ}\atop {n'Y'Z'}}
\left|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma'' s''}^{\sigma''' s'''}
\left({{n'Y'Z'}\atop {nYZ}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)\rho_{s'''}^{s'_1}
(\textit{\textbf{k}}_1) \right. \right.
&\\
& \strut\displaystyle
\left.+M_{\sigma' s''}^{\sigma s'_1}\left({{n'Y'Z'}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}
\atop {\textit{\textbf{k}}}}\right)M_{\sigma'''s'''}^{\sigma''s'}\left({{nYZ}\atop {n'Y'Z'}}
\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s_1}^{s'''}(\textit{\textbf{k}}_1)\right]
\rho_{s'}^{s''}(\textit{\textbf{k}})
& \nonumber \\
& \strut\displaystyle
+2M_{\sigma s_1}^{\sigma's'}\left({{nYZ}\atop {n'Y'Z'}}\left|{{\textit{\textbf{k}}}
\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma''s''}^{\sigma''' s'_1}\left({{n'Y'Z'}\atop {nYZ}}
\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)\rho_{s'}^{s''}(\textit{\textbf{k}})
\rho_{\sigma'n}^{\sigma''}(Z)\left[\delta_\sigma^{\sigma'''}-
\rho_{\sigma'''n'}^\sigma(Z')\right]
& \nonumber \\
& \strut\displaystyle
-\rho_{\sigma'''n'}^\sigma(Z')\left[\delta_{\sigma'}^{\sigma''}-
\rho_{\sigma'n}^{\sigma''}(Z)\right]
& \nonumber \\
& \strut\displaystyle
\left.\times\left[M_{\sigma s_1}^{\sigma' s''}\left({{nYZ}\atop
{n'Y'Z'}}\left|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)M_{\sigma'' s''}^{\sigma''' s'''}
\left({{n'Y'Z'}\atop {nYZ}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)\rho_{s'''}^{s'_1}
(\textit{\textbf{k}}_1)+M_{\sigma's}^{\sigma'''s'_1}\left({{n'Y'Z'}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop
{\textit{\textbf{k}}}}\right)M_{\sigma s''}^{\sigma''s}\left({{nYZ}\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}
\atop {\textit{\textbf{k}}_1}}\right)\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)\right]\right\}.
&\nonumber\eeq
There are three terms in the {rhs} of equations (\ref{eq:ku_e_pol}).
The first and the second terms describes redistribution only over polarization.
The last one describes the general redistribution of the photons over quantum states (energies, momentum directions and polarization).
\subsubsection{Equation for the case of non-polarized electrons}
The kinetic equation in the case non-polarized electrons one can deduced from equation (\ref{eq:ku_e_pol}) by averaging over the spin states of the electrons. There is a relation between the distribution function of the electrons $f_{n}(Z)$ and the diagonal elements of the electron coherency matrix:
\be
f_{n}(Z)=\rho_{1n}^{1}(Z)+\rho_{2n}^{2}(Z),
\ee
where
$\rho_{1n}^{1}(Z)=\rho_{2n}^{2}(Z).$ The distribution function is normalized to the total number of the electrons:
\be
\sum_{n}\int\d Z f_n(Z)=N_{{\rm e}}.
\ee
Then equation for the case of non-polarized electrons is:
\beq \label{eq:ku_e_nopol}
& \strut\displaystyle
\unl {k}_1\unl {\nabla}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t)
=
J_{1}+J_{2}+J_{3}, &
\eeq
where
\beq& \strut\displaystyle
J_{1}=
i\frac{\alpha}
{(2\pi)^2}\sum_n\int\frac {\d Y\d Z}{R}\frac{f_n(Z)}{2}\left[
\rho_{s_1}^{s}(\textit{\textbf{k}}_1)M_{\sigma s}^{\sigma s'_{1}}\left({{nYZ}\atop {nYZ}}\left|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)-\rho_{s'}^{s'_1}(\textit{\textbf{k}}_1)M_{\sigma
s_1}^{\sigma s'}\left({{nYZ}\atop {nYZ}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)\right],
&\eeq
\beq& \strut\displaystyle
J_{2}=
\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(k-k_1)
\delta(k\cos\theta-k_1\cos\theta_1)\frac{f_n(Z)}{2}\frac{f_{n'}(Z')}{2}
& \nonumber \\
& \strut\displaystyle \times\left[2M_{\sigma s}^{\sigma s'_{1}}\left({{nYZ}\atop {nYZ}}
\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma''s_1}^{\sigma''s'''}\left({{n'Y'Z'}
\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s'''}^s(\textit{\textbf{k}})-
M_{\sigma s}^{\sigma s'_1}\left({{nYZ}\atop {nYZ}}\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}
\right)M_{\sigma''s''}^{\sigma''s}\left({{n'Y'Z'}\atop {n'Y'Z'}}\right|{{\textit{\textbf{k}}}\atop
{\textit{\textbf{k}}_1}}\right)\rho_{s_1}^{s''}(\textit{\textbf{k}}_1) \right. & \nonumber \\
& \strut\displaystyle \left. -M_{\sigma''s'}^{\sigma''s'''}\left({{n'Y'Z'}\atop {n'Y'Z'}}
\left|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma s_1}^{\sigma s'}\left({{nYZ}\atop
{nYZ}}\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right)\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)\right],
&\eeq
\beq& \strut\displaystyle
J_{3}=
\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_1)
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1) & \nonumber \\
& \strut\displaystyle \times
\frac{1}{2}\left\{ \left[f_n(Z)-f_{n'}(Z') \right]
\left[T_{s''s_1}^{s'''s'}\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)+T_{s''s'''}^{s'_1s'}
\rho_{s_1}^{s'''}(\textit{\textbf{k}}_{1})\right]\rho_{s'}^{s''}(\textit{\textbf{k}})
+2T_{s''s_1}^{s'_{1}s'}\rho_{s'}^{s''}(\textit{\textbf{k}})f_n(Z)\left[1-\frac{f_n'(Z')}{2} \right] \right.
& \nonumber \\
& \strut\displaystyle \left. -f_{n'}(Z')\left[1-\frac{f_n(Z)}{2}\right]\left[T_{s''s_1}^{s'''s''}
\rho_{s'''}^{s'_1}(\textit{\textbf{k}}_1)+T_{ss''}^{s'_1s}\rho_{s_1}^{s''}(\textit{\textbf{k}}_1)\right]
\right\} ,
&\eeq
where we use a notation for the product of pairs of the scattering matrix elements:
\be\label{T-matrix_def}
T_{jm}^{ik}\equiv M_{\sigma'j}^{\sigma i}\left({{n'Y'Z'}\atop {nYZ}}\left|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}}}\right)M_{\sigma m}^{\sigma'k}\left({{nYZ}\atop {n'Y'Z'}}
\right|{{\textit{\textbf{k}}}\atop {\textit{\textbf{k}}_1}}\right).
\ee
After averaging over the spin states of the electrons the last term in the {rhs} of the kinetic equation simplifies.
The $\delta$-functions under the integrals in the {rhs} of equations (\ref{eq:ku_e_pol}) and (\ref{eq:ku_e_nopol})
can be use to reduce the number of integrations and to simplify the sum.
\subsubsection{Equation for the case of non-polarized rarefied electron gas}
Another form of the kinetic equations can be obtained in the case of rarefied electron gas.
Neglecting in equation (\ref{eq:ku_e_nopol}) the terms containing squares of the electron distribution function, we gets
\beq \label{eq:ku_e_nopol_nodeg}
& \strut\displaystyle \unl {k}_1\unl {\nabla}\rho_{s_1}^{s'_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t)=
K_{1}+K_{2}+K_{3}, &
\eeq
where
\beq&\strut\displaystyle
K_{1}=i\frac {\alpha}
{(2\pi)^2}\sum_n\int\frac {\d Y\d Z}{R}\frac{f_n(Z)}{2}\left[
\rho_{s_1}^{s}(\textit{\textbf{k}}_1)M_{\sigma s}^{\sigma s'_{1}}\left({{nYZ}\atop {nYZ}}\left|
{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)-
\rho_{s'}^{s'_1}(\textit{\textbf{k}}_1)M_{\sigma
s_1}^{\sigma s'}\left({{nYZ}\atop {nYZ}}\right|{{\textit{\textbf{k}}_1}\atop {\textit{\textbf{k}}_1}}\right)\right],
&\eeq
\beq&
K_{2}=0,
&\eeq
\beq& \strut\displaystyle
K_{3}=\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_1)
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1)\times & \nonumber \\
& \strut\displaystyle \times\frac{1}{2}\left\{
f_{n}(Z)\rho_{s'}^{s''}(\textit{\textbf{k}})\left[ 2T_{s''s_{1}}^{s'_{1}s'}+T_{s''s_{1}}^{s'''s'}\rho_{s'''}^{s'_{1}}(\textit{\textbf{k}}_{1})
+T_{s''s'''}^{s'_{1}s'}\rho_{s_{1}}^{s'''}(\textit{\textbf{k}}_{1})\right]-\right.
& \nonumber \\
& \strut\displaystyle
\left. -f_{n'}(Z')\left[ \rho_{s'''}^{s'_{1}}(\textit{\textbf{k}}_{1})\left( T_{s''s_{1}}^{s'''s''}+T_{s''s_{1}}^{s'''s'}\rho_{s'}^{s''}(\textit{\textbf{k}})\right)+
\rho_{s_{1}}^{s'''}(\textit{\textbf{k}}_{1})\left(T{^{s'_{1}}_{s}}{^{s}_{s'''}}+T_{s''s'''}^{s'_{1}s'}\rho_{s'}^{s''}(\textit{\textbf{k}})\right)
\right]
\right\}. &
\eeq
We notice that the first term of equation (\ref{eq:ku_e_nopol}) has not changed, while the second term has now disappeared.
\subsubsection{Kinetic equation in terms of two polarization modes}
In the case of non-polarized rarefied electron gas, equation (\ref{eq:ku_e_nopol_nodeg}) can be simplified further
if one assumes the absence of correlations between the two linear polarization modes:
$\rho_{1}^{2}(\textit{\textbf{k}})=\rho_{2}^{1}(\textit{\textbf{k}})=0.$
Then one can use only one polarization index for the diagonal elements of the coherency matrix:
$\rho_{i}(\textit{\textbf{k}})\equiv\rho_{i}^{i}(\textit{\textbf{k}}).$
The kinetic equation then get the form:
\beq \label{eq:ku_e_nopol_nodeg_2m}
& \strut\displaystyle \unl {k}_1\unl {\nabla}\rho_{s_1}(\textit{\textbf{k}}_1,\textit{\textbf{r}}_1,t)=
L_{1}+L_{2}+L_{3},
&\eeq
where $L_{1}=0$ and $L_{2}=0$, and
\beq& \strut\displaystyle
L_{3}=\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_1)
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1) & \nonumber \\
& \strut\displaystyle \times
T{^{s_1}_{s}}{^{s}_{s_1}}\left\lbrace f_{n}(Z)\rho_{s}(\textit{\textbf{k}})\left[ 1+\rho_{s_{1}}(\textit{\textbf{k}}_{1})\right]-
f_{n'}(Z')\rho_{s_{1}}(\textit{\textbf{k}}_{1})\left[ 1+\rho_{s}(\textit{\textbf{k}})\right]
\right\rbrace.
&\eeq
This form of the equation is obvious and can be written immediately using physical arguments \cite{PSM1989}.
In this case the two modes are considered independently and a possibility of correlation between their phases is not taken into account.
\subsection{Kinetic equation in terms of Stokes parameters}
Equations (\ref{eq:ku_e_pol}) and (\ref{eq:ku_e_nopol}) can be rewritten in terms of Stokes parameters. Transformation to this form can be done using trivial linear transformation. Elements of the coherency matrix $\{\rho_j^i(\textit{\textbf{k}})\}$ and the Stokes vector $\textit{\textbf{N}}=(n_{\rm I},n_{\rm Q},n_{\rm U},n_{\rm V})^{T}$ are connected by relations:
\beq
& \strut\displaystyle n_{\rm I}=(\rho_1^1+\rho_2^2)/2,\quad n_{\rm Q}=(\rho_1^1-\rho_2^2)/2,\quad
n_{\rm U}=(\rho_1^2+\rho_2^1)/2,\quad n_{\rm V}=i(\rho_1^2-\rho_2^1)/2, & \\
& \strut\displaystyle \rho_1^1=n_{\rm I}+n_{\rm Q},\quad \rho_2^2=n_{\rm I}-n_{\rm Q},\quad
\rho_1^2=n_{\rm U}-in_{\rm V},\quad\rho_2^1=n_{\rm U}+in_{\rm V}. &
\eeq
\subsubsection{General equation for the case of polarized electrons}
\label{sec:genpol1}
Using equation (\ref{eq:ku_e_pol}), one can find the kinetic equation in the case of polarized electrons in terms of Stokes parameters:
\beq \label{eq:Sp_p}
& \strut\displaystyle \unl {k}_1\unl {\nabla}\textit{\textbf{N}}_1=
I^{P}_{1}+I^{P}_{2}+I^{P}_{3}, &
\eeq
where
\beq \label{eq:Sp_p_I1}
& \strut\displaystyle
I^{P}_{1}=
i\frac {\alpha}{(2\pi)^2}\sum_n\int
\frac {\d Y\d Z}{R}f^\sigma_{\sigma'n}(Z)\hat{{\mathcal F}}\textit{\textbf{N}}_1,
&\eeq
\beq& \strut\displaystyle
I^{P}_{2}=
\frac {1}{2}\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(k-k_1)
\delta(k\cos\theta-k_1\cos\theta_1)f^\sigma_{\sigma'n}(Z)
f^{\sigma''}_{\sigma'''n'}(Z')\left[\hat {{\mathcal F}}'\textit{\textbf{N}}+\hat{{\mathcal F}}''\textit{\textbf{N}}_1\right],
&\eeq
\beq \label{eq:Sp_p_I3}
& \strut\displaystyle
I^{P}_{3}=
\frac {1}{2}\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_1)
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1) & \\
& \strut\displaystyle \times\left\{\left[\delta^{\sigma'''}_{\sigma}f^{\sigma''}_{\sigma'n}
(Z)-\delta^{\sigma''}_{\sigma'}f^{\sigma'''}_{\sigma n'}(Z') \right]
\hat{{\mathcal R}}\textit{\textbf{N}}_{1}+2f^{\sigma''}_{\sigma'n}(Z)\left[\delta^{\sigma'''}_{\sigma}-
f^\sigma_{\sigma'''n'}(Z') \right] \hat {{\mathcal R}}'\textit{\textbf{N}}-2f^\sigma_{\sigma'''n'}(Z')
\left[\delta^{\sigma''}_{\sigma'}-f^{\sigma''}_{\sigma'n}(Z) \right] \hat{{\mathcal R}}''\textit{\textbf{N}}_1
\right\}, \nonumber
&\eeq
where $\textit{\textbf{N}}=\textit{\textbf{N}}(\textit{\textbf{k}})$ and $\textit{\textbf{N}}_{1}=\textit{\textbf{N}}(\textit{\textbf{k}}_{1})$ are the Stokes vectors, $\hat{{\mathcal F}}$, $\hat{{\mathcal F}}'$, $\hat{{\mathcal F}}''$, $\hat{{\mathcal R}}$, $\hat{{\mathcal R}}'$ and $\hat{{\mathcal R}}''$ are $4\times4$ complex matrices acting like linear operators from the real 4-dimensional space to the real 4-dimensional space.
The expressions for these matrices can be found in Appendix \ref{app_st1}.
\subsubsection{Equation for the case of non-polarized electrons}
\label{sec:nonpol2}
The equation for the case of non-polarized electrons can be derived from equation (\ref{eq:ku_e_nopol}):
\beq \label{eq:Sp_np}
& \strut\displaystyle \unl {k}_1\underline{\nabla}\textit{\textbf{N}}_1=
J^{P}_{1}+J^{P}_{2}+J^{P}_{3}, &
\eeq
where
\beq \label{eq:Sp_np_J1}
& \strut\displaystyle
J^{P}_{1}=i\frac {\alpha}{(2\pi)^2}
\sum_n\int\frac {\d Y\d Z}{R}\frac{f_n(Z)}{2}\hat {{\mathcal F}}\textit{\textbf{N}}_1,
&\eeq
\beq& \strut\displaystyle
J^{P}_{2}=\frac {1}{2}\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int\frac {\d Y\d Z}{R}
\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(k-k_1)\delta(k\cos\theta-k_1
\cos\theta_1)\frac{f_{n}(Z)}{2}\frac{f_{n'}(Z')}{2}\left[\hat {{\mathcal F}}'\textit{\textbf{N}}+\hat{{\mathcal F}}''\textit{\textbf{N}}_1\right],
&\eeq
\beq \label{eq:Sp_np_J3}
& \strut\displaystyle
J^{P}_{3}=\frac{1}{2}\frac{\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_{1})
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1) & \nonumber \\
& \strut\displaystyle \times\left\{\left[f_n(Z)-f_{n'}(Z')\right]\hat {{\mathcal R}}\textit{\textbf{N}}_1+f_n(Z)\left[1-\frac{f_{n'}(Z')}{2} \right]
\hat {{\mathcal R}}'\textit{\textbf{N}}-f_{n'}(Z')\left[1-\frac{f_n(Z)}{2} \right] \hat {{\mathcal R}}''\textit{\textbf{N}}_1\right\},
&\eeq
where $\hat{{\mathcal F}}$, $\hat{{\mathcal F}}'$, $\hat{{\mathcal F}}''$, $\hat{{\mathcal R}}$, $\hat{{\mathcal R}}'$ and $\hat{{\mathcal R}}''$ are $4\times4$ complex matrices
that can be found in Appendix \ref{app_st2}.
\subsubsection{Equation for the case of non-polarized rarefied electron gas}
We derive the equation for the case of rarefied electron gas by neglecting terms containing squared electron distribution functions in equation (\ref{eq:Sp_np}). The equation takes the following form:
\beq \label{eq:Sp_npr}
& \strut\displaystyle \unl {k}_1\underline{\nabla}\textit{\textbf{N}}_1=
K^{P}_{1}+K^{P}_{2}+K^{P}_{3}, &
\eeq
where
\beq& \strut\displaystyle
K^{P}_{1}=i\frac {\alpha}{(2\pi)^2}
\sum_n\int\frac {\d Y\d Z}{R}\frac{f_n(Z)}{2}\hat {{\mathcal F}}\textit{\textbf{N}}_1,
&\eeq
\beq&
K^{P}_{2}=0,
&\eeq
\beq& \strut\displaystyle
K^{P}_{3}=\frac{1}{2}\frac {\alpha^2}{(2\pi)^3}\sum_{n,n'}\int
\frac {\d Y\d Z}{R}\frac {\d Y'\d Z'}{R'}\frac {\d\textit{\textbf{k}}}{k}\delta(R+k-R'-k_{1})
\delta(Z+k\cos\theta-Z'-k_1\cos\theta_1)\times & \nonumber \\
& \strut\displaystyle \times\left[\left((f_{n}(Z)-f_{n'}(Z'))\hat{{\mathcal R}}-f_{n'}(Z')\hat{{\mathcal R}}''\right)\textit{\textbf{N}}_{1}+f_{n}(Z)\hat{{\mathcal R}}'\textit{\textbf{N}}
\right],
&\eeq
where $\hat{{\mathcal F}}$, $\hat{{\mathcal R}}$, $\hat{{\mathcal R}}'$ and $\hat{{\mathcal R}}''$ are the same $4\times4$ complex matrices as in equations (\ref{eq:Sp_np})--(\ref{eq:Sp_np_J3}) and presented in Appendix \ref{app_st2}.
\section{Summary}
We have deduced a kinetic equation for Compton scattering of polarized radiation in magnetic field. Polarizations of photons and spin states of electrons, the induced scattering and the Pauli exclusion principle were taken into account. The equations are written for both the coherency matrix and the Stokes parameters. Additional forms of the equations valid for the two polarization mode description of radiation is also derived.
The equations for both polarized and non-polarized electrons were obtained.
There are no significant (for the conditions in neutron stars atmospheres) limitations on the energies and the concentrations of the electrons and the photons. The assumptions made are usual for the kinetic theory and related to the typical time scales of the problem and do not limit
significantly the applicability range.
The equations describe the interaction of radiation and electrons in strong magnetic field up to about $10^{16}$ G.
There is no low limit on the B-field strength.
At the same time, the derived equations become rather cumbersome in the case of weak magnetic field, because the electrons can occupy high Landau levels and there will be many terms in the {rhs} of the equation, where the summation over the Landau levels is carried out.
On that other hand, in the case of strong magnetic field in neutron star atmospheres electrons typically occupy only ground Landau level
(or only a few low levels), because of a rather low electron temperature and absence of high-energy photons.
Therefore, the sums over $n$ and $n'$ has only a few terms, which simplifies the equations significantly.
The most general form of the kinetic equation (\ref{eq:ku_e_pol}) has three terms. The second and the third terms there contain
the products of two elements of the scattering matrix. These terms can be rewritten through the interaction cross sections. On the contrary, the first term in the {rhs} of equation (\ref{eq:ku_e_pol}) contains single elements of the scattering matrix.
This term describes changes of the photon polarization with no corresponding changes in energy and the momentum direction.
The polarization change term has the following form: the changes of the diagonal elements of the coherency matrix depend only on the non-diagonal elements and changes of the non-diagonal elements depend only on the diagonal ones.
This term describes the rotation of the polarization plane when radiation can be well described only by the Stokes parameters, and it disappears, if the kinetic equation is reformulated in terms of two polarization modes. It is possible that this term provides correction to the depolarization in the region of vacuum resonance, which most likely will not be large because of a small optical depth of this region.
The second term in the {rhs} of equation (\ref{eq:ku_e_pol}) describes redistribution of photons with only changes in polarization.
It contains the products of the electron distribution functions and disappears in the equation (\ref{eq:ku_e_nopol_nodeg}) for rarefied electron gas.
The last term in the {rhs} of equation (\ref{eq:ku_e_pol}) describes the general redistribution of photons over energy, directions and polarizations. This term can be simplified significantly for the cases of non-polarized electrons (\ref{eq:ku_e_nopol}), rarefied electron gas (\ref{eq:ku_e_nopol_nodeg}) and for the two-polarization mode description of radiation (\ref{eq:ku_e_nopol_nodeg_2m}).
In the latter case this term is the only term in the {rhs} of the kinetic equation and coincides with previously known expressions.
The derived equations form the basis for the construction of models of the radiative transfer in
strongly magnetized neutron stars atmospheres and magnetospheres.
\acknowledgments
This study was supported by the CIMO grant TM-10-7326 (A.M.) and the Academy of
Finland grant 127512 (J.P.).
We are grateful to Dmitry Yakovlev, Yuri Shibanov, Dmitry Rumyantsev, Valery Suleimanov,
and an anonymous referee for a number of useful comments.
|
{
"timestamp": "2012-03-12T01:01:21",
"yymm": "1203",
"arxiv_id": "1203.2055",
"language": "en",
"url": "https://arxiv.org/abs/1203.2055"
}
|
\section{Introduction}
\label{sec:introduction}
Let $X$ be a
smooth projective variety over a number field $k$ and
$\Br(X)$ its Brauer group.
The quotient $\Br(X)/\Br(k)$
plays an important role in the study of arithmetic properties of $X$.
Its effective computation
is possible in certain cases, for example:
when $X$ is a geometrically rational surface (see, e.g., \cite{kst}),
a Fano variety of dimension at most 3 (\cite{kteff}),
or a diagonal $K3$ surface over ${\mathbb Q}$ (\cite{ISZ}, \cite{kt}, \cite{swinnertondyer}).
Skorobogatov and Zarhin proved the finiteness of
$\Br(X)/\Br(k)$ when $X$ is a $K3$ surface \cite{skorobogatovzarhin}.
Letting $X_{\bar k}$ denote $X\times_{\Spec k}{\Spec \bar k}$,
where $\bar k$ is an algebraic closure of $k$,
there is the natural map
\begin{equation}
\label{XtoXbar}
\Br(X)\to\Br(X_{\bar k}).
\end{equation}
Its kernel is known as algebraic part of the
Brauer group.
This is a finite group, and may be identified with the Galois cohomology
\begin{equation}
\label{h1}
H^1(\Gal(\bar k/k),\Pic(X_{\bar k})).
\end{equation}
Knowledge of $\Pic(X_{\bar k})$ is essential to its computation.
A first goal of this paper is the effective computation of
$\Pic(X_{\bar k})$ when $X$ is a $K3$ surface of degree 2 over a number field.
Special cases and examples have been treated previously by, e.g.,
van Luijk \cite{vl}, while a more general treatment, that is however
conditional on the Hodge conjecture, appears in \cite{charles}.
The image of \eqref{XtoXbar} is contained in the invariant
subgroup
\[\Br(X_{\bar k})^{\Gal(\bar k/k)}.\]
The finiteness of this invariant subgroup is
one of the main results of \cite{skorobogatovzarhin}, yet the proof does
not yield an effective bound.
In this paper we give an \emph{effective} bound
for the order of the group $\Br(X)/\Br(k)$.
Combined with the results in \cite{kt}, this permits the effective
computation of the subset
\[
X(\mathbb{A}_k)^{\Br(X)}\subseteq X(\mathbb{A}_k)
\]
of Brauer-Manin unobstructed adelic points of $X$.
Examples of computations of Brauer-Manin obstructions on $K3$ surfaces can
be found in
\cite{bright}, \cite{hvv}, \cite{ieronymou}, \cite{ssd}, \cite{wittenberg}.
The results here,
combined with results in \cite{ctskoro},
imply as well an effective
bound for the order of $\Br(X_{\bar k})^{\Gal(\bar k/k)}$.
The finiteness results of \cite{skorobogatovzarhin} are based on
the Kuga-Satake construction, which associates an abelian variety of
dimension $2^{19}$ to a given $K3$ surface
and relates their cohomology,
together with the Tate conjecture for abelian varieties,
proved by Faltings \cite{faltings}.
The Kuga-Satake correspondence is \emph{conjectured} to be given by an
algebraic correspondence,
but is proved only in some special cases, e.g.,
\cite{vangeemen}, \cite{inose}, \cite{shiodainose}.
Assuming this (in some effective form), one could apply
effective versions of Faltings' results, obtained by
Masser and W\"ustholz \cite{mw1}, \cite{mw2}, \cite{mw3}, \cite{mw4}, \cite{mw5}
(see also \cite{bost}).
Lacking this, we treat the transcendental construction directly, showing
that computations to bounded precision can replace an algebraic
correspondence, in practice.
The construction proceeds in several steps.
First of all, rigidity allows us to construct the Kuga-Satake morphism between
moduli spaces (of $K3$ surfaces with polarization and level structure on
one side and polarized abelian varieties with level structure on the other)
algebraically over a number field, at least up to an explicit finite
list of possibilities.
This allows us to identify an abelian variety corresponding to a $K3$ surface,
and its field of definition.
For the computation of the induced map on homology we work with
integer coefficients and simplicial complexes, and computations up to bounded
precision suffice to determine all the necessary maps.
Then we follow the proof in Section 4 of \cite{skorobogatovzarhin}
and obtain the following result.
\begin{theo}
\label{mainresult}
Let $k$ be a number field and $X$ a $K3$ surface of degree $2$ over $k$,
given by an explicit equation.
Then there is an effective bound
on the order of $\Br(X)/\Br(k)$.
\end{theo}
In fact, we provide an effective bound on $\Br(X)/\Br(k)$
when $X$ has an ample line bundle of arbitrary degree $2d$
provided that there is an effective construction of the moduli space of
primitively quasi-polarized $K3$ surfaces of degree $2d$
(see Definition \ref{def:pola}).
For $d=1$ this is known, via effective geometric invariant
theory (see Remark \ref{whenknowDmodGamma}).
The first step of the proof is, as mentioned above, the
effective computation of the Galois module $\Pic(X_{\bar k})$.
Since this is finitely generated and torsion-free,
this permits the effective computation of the Galois cohomology
group \eqref{h1}.
Hence the proof of Theorem \ref{mainresult} is
reduced to effectively bounding the image of \eqref{XtoXbar}.
\medskip
\noindent\textbf{Acknowledgements.}
The first author was supported by NSF grants 0901645 and 0968349.
The second author was supported by the SNF.
The third author was supported by NSF grants 0739380, 0901777, and 0968349.
The authors benefited from helpful discussions with
N. Katz and G. W\"ustholz.
\section{Effective algebraic geometry}
\label{sec:effalggeo}
We work over an algebraic number field $k$ and denote by $\bar k$ its
algebraic closure.
The term \emph{variety} refers to geometrically integral separated
scheme of finite type over $k$.
We say that a quasiprojective variety or scheme $X$
is given by explicit equations
if homogeneous equations are supplied defining a scheme in a projective space
${\mathbb P}^M$ for some $M$
and a closed subscheme whose complement is $X$.
By convention $\mathcal{O}_X(1)$ will denote the restriction of
$\mathcal{O}_{{\mathbb P}^M}(1)$ to $X$.
The base-change of $X$ to a field extension $k'$ of $k$ will be denoted
$X_{k'}$.
\begin{lemm}
\label{finitemorlem}
Let $X$ be a quasiprojective scheme, given by explicit equations.
Let $f:Y\to X$ be a finite morphism, given by explicit equations
on affine charts.
Then we may effectively determine integers $n$ and $N$ and an embedding
$Y\to X\times {\mathbb P}^N$, such that $f$ is the composite of projection to $X$
with the embedding and the pullback of $\mathcal{O}_{{\mathbb P}^N}(1)$ to $X\times {\mathbb P}^N$
restricts to $f^*\mathcal{O}_X(n)$.
In particular, $f^*\mathcal{O}_X(n+1)$ is very ample on $Y$ and
we may obtain explicit equations for $Y$ as a quasiprojective scheme.
\end{lemm}
\begin{proof}
The morphism $f$ may be presented on affine patches by
finitely many new indeterminates adjoined
to the coordinate rings of the patches (with additional
relations).
We may determine, effectively, an integer $n$ such that
each extends to a section of $\mathcal{O}_X(n)$.
Then for suitable $N$ we have an embedding
$Y\to X\times {\mathbb P}^N$ satisfying the desired conditions.
\end{proof}
We collect effectivity results that we will be using freely.
\begin{itemize}
\item \emph{Effective normalization in a finite function field extension}:
Given a quasiprojective variety $X$
over $k$, presented by means of explicit equations, and another algebraic
variety $V$ with generically finite morphism $V\to X$ given by
explicit equations on affine charts,
to compute effectively
the normalization $Y$ of $X$ in $k(V)$, with finite morphism $Y\to X$.
See \cite{mesnager} and references therein.
\item \emph{A form of effective resolution of singularities}:
Given a nonsingular quasiprojective variety $X^\circ$ over $k$,
to produce a nonsingular projective variety $X$ and
open immersion $X^\circ\to X$, such that
$X\smallsetminus X^\circ$ is a simple normal crossings divisor.
This follows by standard formulations of Hironaka resolution theorems,
for which effective versions are available; see, e.g., \cite{bgmw}.
\item \emph{Effective invariant theory for actions of projective varieties}:
Given a projective variety $X$
and a linearized action of a reductive algebraic group $G$ on $X$ for
$L=\mathcal{O}_X(1)$, to
compute effectively the subsets $X^{ss}(L)$ and $X^s(L)$ of
geometric invariant theory,
the projective variety $X//G$, open subset $U\subset X//G$
corresponding to $X^s(L)$, and quotient morphisms
$X^{ss}(L)\to X//G$ and $X^s(L)\to U$.
This is standard, using effective computation of invariants
$k[V]^G$ for a finitely generated $G$-module $V$
\cite{derksen}, \cite{kempf}, \cite{popov}.
\end{itemize}
\begin{lemm}
\label{hilbschemeargument}
Let $X$ be a quasiprojective normal variety over $k$,
given by explicit equations, and let $U\subset X$ be a nonempty subvariety.
Given $d\in{\mathbb N}$, there is an effective procedure to produce a finite
extension $k'$ of $k$ and
a finite collection of normal quasiprojective varieties
$Y^{(1)}$, $\ldots$, $Y^{(m)}$ over $k'$, with finite morphisms
$f_i:Y^{(i)}\to X_{k'}$,
such that for each $i$ the restriction of $f_i$ to $f_i^{-1}(U_{k'})$
is \'etale of degree $d$,
and such that the $Y^{(i)}_{\bar k}\to X_{\bar k}$ ($i=1$, $\ldots$, $m$)
are up to isomorphism all the degree $d$ coverings by
normal quasiprojective varieties over $\bar k$ which are
\'etale on the pre-image of $U_{\bar k}$.
\end{lemm}
\begin{proof}
Let $N=\dim X$.
Shrinking $U$ if necessary we may suppose that there is morphism
$U\to {\mathbb P}^{N-1}$, given by a suitable linear projection, such that
the generic fiber is smooth and one-dimensional, i.e., the restriction
to the generic point $\eta=\Spec(k(x_1,\ldots,x_{N-1}))$ is a
nonsingular quasi-projective curve $C_{\eta}$.
We may effectively compute a nonsingular projective compactification
$\overline{C}_{\eta}$.
The genus $g=g(\overline{C}_{\eta})$ and degree
$e=\deg(\overline{C}_{\eta}\smallsetminus C_{\eta})$, together with $d$,
determine by the Riemann-Hurwitz formula an upper bound $g_{\mathrm{max}}$
on the genus of $Y\times_X \eta$.
Using effective Hilbert scheme techniques we construct parameter spaces
containing every isomorphism class of genus $g'\le g_{\mathrm{max}}$ curve
equipped with a morphism of degree $d$ to $\overline{C}_{\eta}$.
That the ramification divisor in contained in the
scheme-theoretic pre-image of $\overline{C}_{\eta}\smallsetminus C_{\eta}$
is a closed condition and one that may be implemented effectively
(cf.\ \cite[Corollary 3.14]{mochizuki}) and determines a finite field extension
$k'$ of $k$ and a finite set of
candidates for
$Y\times_X \eta$, over $k'$.
Applying effective normalization to each of these candidates and
eliminating those which are not \'etale over $U_{k'}$,
we obtain the $Y^{(i)}$.
\end{proof}
\section{Baily-Borel compactifications}
\label{sec:bb}
Let $\mathbb{D}=G/K$ be a bounded symmetric domain and
$\Gamma$ an arithmetic subgroup of $G$.
It is known that $\Gamma\backslash \mathbb{D}$ admits a canonical compactification
$(\Gamma\backslash \mathbb{D})^*$,
the Baily-Borel compactification, which is a normal projective variety
\cite{bb};
however, the construction of this variety does not supply algebraic equations.
Let $\widetilde{\Gamma}<\Gamma$ be a finite-index subgroup, and assume that
$\widetilde{\Gamma}$ is neat.
(Recall that an arithmetic subgroup is called neat if, for every element,
the subgroup of ${\mathbb C}^*$ generated by its eigenvalues is torsion-free.)
In this section, we show that if we know
$\Gamma\backslash \mathbb{D}$ as a quasiprojective variety (with explicit
equations over some number field),
we can effectively construct $\widetilde{\Gamma}\backslash \mathbb{D}$ as
a quasiprojective variety (as one of finitely many candidates),
together with its Baily-Borel compactification.
In the following, we let $k$ denote a number field.
\begin{lemm}
\label{neatlem}
Let $G$ be a linear algebraic group over $k$ and
$\Gamma$ an arithmetic subgroup.
There is an effective procedure to construct a neat subgroup of
finite index in $\Gamma$.
\end{lemm}
\begin{proof}
It suffices to establish the result for a discrete subgroup of
$GL_n({\mathbb Z})$.
Fix a prime $\ell$.
There is a finite extension $\mathbf{k}$ of ${\mathbb Q}_\ell$
over which every polynomial of degree $n$ with coefficients in ${\mathbb Q}_\ell$
factors completely
(see \cite{krasner},
effectively computed in \cite{pr}).
The structure of $\mathbf{k}^*$ is known as a direct sum of
${\mathbb Z}$, a finite group, and
${\mathbb Z}_\ell^N$ for some $N$.
Then there is an
$\varepsilon>0$ such that ball of radius $\varepsilon$ around $1$
is contained in the free part, so an $n\times n$
integer matrix sufficiently
close $\ell$-adically to the identity matrix has its eigenvalues not
more than $\varepsilon$ away from $1$.
Hence they generate a
torsion free subgroup of $\mathbf{k}^*$, and also of ${\mathbb C}^*$.
\end{proof}
We fix an embedding $k\hookrightarrow {\mathbb C}$.
\begin{prop}
\label{gammacover}
Let $\mathbb{D}=G/K$ be a bounded symmetric domain,
$\Gamma$ an arithmetic subgroup of $G$,
and $\widetilde{\Gamma}$ a finite-index subgroup of $\Gamma$ which is neat.
Let $X^\circ$ be a quasiprojective variety over $k$,
given by explicit equations, and let $U\subset X^\circ$ be a nonempty
open subscheme, also explicitly given.
Suppose that there exists an isomorphism
$X^\circ_{{\mathbb C}}\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to \Gamma\backslash \mathbb{D}$ such that
the map $\mathbb{D}\to \Gamma\backslash \mathbb{D}$ is unramified over
the image of $U_{\mathbb C}$.
Then there is an effective procedure to produce a finite extension $k'$
of $k$ with compatible embedding in ${\mathbb C}$ and a finite collection
of nonsingular quasiprojective varieties $Y^{(1)}$, $\ldots$, $Y^{(m)}$
defined over $k'$ with morphisms
$f_i:Y^{(i)}\to X^\circ_{k'}$, such that, for some $i$,
setting $\widetilde{X}^\circ:=Y^{(i)}$ there exists
an isomorphism
$\widetilde{X}^\circ_{{\mathbb C}}\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to \widetilde{\Gamma}\backslash \mathbb{D}$
fitting into a commutative diagram
\[
\xymatrix{
\widetilde{X}^\circ_{{\mathbb C}} \ar[r]^\sim \ar[d] &
\widetilde{\Gamma}\backslash \mathbb{D} \ar[d] \\
X^\circ_{{\mathbb C}} \ar[r]^\sim & \Gamma\backslash \mathbb{D}
}
\]
\end{prop}
\begin{proof}
Since $\widetilde{\Gamma}\backslash \mathbb{D}\to \Gamma\backslash \mathbb{D}$
ramifies over the same set of points as
$\mathbb{D}\to \Gamma\backslash \mathbb{D}$,
this follows directly from Lemma \ref{hilbschemeargument}.
\end{proof}
\begin{prop}
\label{neatbb}
Let $\mathbb{D}=G/K$ be a bounded symmetric domain,
$\Gamma$ a neat arithmetic subgroup of $G$, and
$X^\circ$ a quasiprojective variety over $k$ given by explicit equations
such that $X^\circ_{{\mathbb C}}$ is isomorphic to $\Gamma\backslash \mathbb{D}$.
Assume that $PGL_2$ is not a quotient of $G$.
Then there is an effective procedure to construct
a projective variety $X$ over $k$, together with open immersion
$X^\circ\to X$, such that
$X_{{\mathbb C}}$ is isomorphic to the
Baily-Borel compactification $(\Gamma\backslash \mathbb{D})^*$.
\end{prop}
The first ingredient in the proof of
Proposition \ref{neatbb}
is a result of Alexeev \cite[\S 3]{alexeev}, building on
earlier work of Mumford \cite{mumford}:
\begin{theo}
\label{alexeev}
Let $\mathbb{D}$ be a bounded Hermitian symmetric domain and $\Gamma$ a neat
arithmetic subgroup acting on $\mathbb{D}$. Let $X^{\circ}=\Gamma\backslash \mathbb{D}$
with Baily-Borel compactification $X$ and boundary $\Delta$.
Then $(X,\Delta)$ is log canonical, with the automorphic factor
coinciding with the log canonical divisor $K_{X}+\Delta$.
\end{theo}
We will also use a result of Fujino \cite{fujino}:
\begin{theo}
\label{fujino2}
Let $(X,\Delta)$ be a projective log canonical pair and $M$ a line bundle on $X$.
Assume that $M\equiv K_X+\Delta+N$, where $N$ is an ample ${\mathbb Q}$-divisor on $X$.
Let $x_1,x_2 \in X$ be closed points and assume there there are positive numbers $c(k)$
with the following properties:
\begin{enumerate}
\item{If $Z\subset X$ is an irreducible (positive-dimensional) subvariety which contains
$x_1$ or $x_2$ then
$$(N^{\dim(Z)}\cdot Z) > c(\dim(Z))^{\dim(Z)}.$$ }
\item{The numbers $c(k)$ satisfy the inequality
$$\sum_{k=1}^{\dim(X)} \sqrt[k]{2} \frac{k}{c(k)} \le 1.$$
}
\end{enumerate}
Then the global sections of $M$ separate $x_1$ and $x_2$.
\end{theo}
\begin{proof}[Proof of Proposition \ref{neatbb}]
The hypotheses guarantee that the complement of $\Gamma\backslash \mathbb{D}$
in the Baily-Borel compactification has codimension $\ge 2$.
If we define
\[X=\Proj\Big(\bigoplus_{d\ge 0} H^0(X^\circ, dK_{X^\circ})\Big).\]
then $X$ satisfies the conditions of the proposition.
It remains to show that we can construct $X$ effectively.
Effective resolution of singularities as in Section \ref{sec:effalggeo} allows us
to construct a nonsingular compactification $\widetilde{X}$ of $X^\circ$,
projective, such that $\widetilde{X}\smallsetminus X^\circ$ is a
simple normal crossings divisor
$D_1\cup\cdots\cup D_m$.
Now the Borel extension property \cite[Thm.\ A]{borel} implies that
the inclusion $X^\circ\to X$ extends to a
birational morphism $\pi:\widetilde{X}\to X$.
By Theorem \ref{alexeev}, $X$ has at worst log canonical singularities.
Hence there are integers $c_i\le 1$ such that
\[\pi^*K_X=K_{\widetilde{X}}+\sum c_iD_i.\]
By the chain of inclusions
\[H^0(\widetilde{X},d(\pi^*K_X))
\subseteq
H^0(\widetilde{X},d(K_{\widetilde{X}}+\sum D_i))
\subseteq
H^0(X^\circ,dK_{X^\circ})\]
we deduce that
\begin{equation}
\label{KplussumD}
H^0(X,dK_X)=H^0(\widetilde{X},d(K_{\widetilde{X}}+\sum D_i)).
\end{equation}
Theorem \ref{fujino2} supplies a universal constant $n$ depending only on
the dimension of $X$, such that for any $d\ge n$, the linear system
$|dK_X|$ separates points on $X$.
The image $X'$ of $|dK_X|$ may be effectively computed using
\eqref{KplussumD}.
The normalization of $X'$, which may also be computed effectively,
is then isomorphic to $X$.
\end{proof}
\begin{rema}
\label{whenknowDmodGamma}
There are examples in the literature in which $X^\circ$ as in
Proposition \ref{gammacover} has been constructed.
\begin{itemize}
\item Abelian varieties
\begin{itemize}
\item Polarized, with
symmetric theta structure \cite{mumfordequationsiii}, \cite{mumfordtataiii}.
\item Polarized, with level $n$ structure ($n\ge 3$):
a construction based on Hilbert scheme and geometric invariant
theory, presented in \cite[\S7.3]{git}, can be carried out effectively
by the techniques mentioned in Section \ref{sec:effalggeo}.
\end{itemize}
\item $K3$ surfaces
\begin{itemize}
\item
A six-dimensional ball quotient as a moduli space of
$K3$ surfaces which are cyclic degree $4$ covers of ${\mathbb P}^2$ ramified
along a quartic \cite{artebani}, \cite{kondocrelle}.
\item
A nine-dimensional ball quotient coming from $K3$ surfaces which are
cyclic triple covers of ${\mathbb P}^1\times {\mathbb P}^1$ with branch curve
of bidegree $(3,3)$ \cite{kondo}.
\item
$K3$ surfaces of degree $2$, described by Horikawa \cite{horikawa} and
Shah \cite{shahdegree2}; see also \cite{looijengavancouver}.
\end{itemize}
\item
A four-, resp.\ ten-dimensional ball quotient arising
from the moduli space of cubic surfaces \cite{ACT1},
resp.\ threefolds \cite{ACT2}.
\end{itemize}
Mumford's construction in the abelian variety setting yields
explicit equations for the moduli space together with a universal family.
In each of the other examples, explicit GIT constructions are given,
and from these we may obtain explicit equations as mentioned in
Section \ref{sec:effalggeo}.
This allows us,
e.g., to compute the point in $X^\circ$ corresponding to
a given $K3$ surface of degree $2$ presented as a double cover of
the plane branched along an explicitly given sextic curve.
\end{rema}
\begin{rema}
$K3$ surfaces of degree $4$ are analyzed in
\cite{looijengaamsterdam} and \cite{shahdegree4},
and degrees up to $8$ in \cite{looijengaduke}, via the GIT of
quartic surfaces, respectively complete intersections.
The analysis yields a factorization of the rational map from the
Baily-Borel compactification to the GIT quotient.
It would be interesting to use this to
give an effective construction of $X^\circ$ for these degrees
generalizing the one for degree $2$, which is based on
an explicit weighted Kirwan blowup
of the GIT quotient of plane sextics \cite{kirwanlee}.
\end{rema}
\begin{rema}
\label{igusa}
An effective construction of a Baily-Borel compactification is
tantamount to effectively bounding degrees of generators of the corresponding
ring of automorphic functions.
The technique in \cite{bb}
for proving the \emph{existence} of projective compactifications is
\emph{not effective} as it relies on a compactness argument.
In some examples these rings have been computed explicitly:
Igusa \cite{Ig62} shows that the ring of Siegel modular forms
for principally polarized abelian surfaces are generated
by Eisenstein series of weights $4,6,10$, and $12$.
The case of threefolds is explored by Tsuyumine \cite{tsu}, who shows
that $34$ modular forms, of weights ranging from $4$ to $48$, suffice.
Not all of these may be expressed in terms of Eisenstein series.
The case of fourfolds is addressed in Freitag-Oura \cite{FO}, who
introduce some specific relations and dimension formula.
Additional work in this direction was done by Oura-Poor-Yuen
\cite{opy}.
\end{rema}
\begin{prop}
\label{rigidity}
Let $k$ be a number field with a given embedding in ${\mathbb C}$.
Let $X$ and $X'$ be projective varieties over $k$
satisfying $X_{{\mathbb C}}\cong (\Gamma\backslash \mathbb{D})^*$ and
$X'_{{\mathbb C}}\cong (\Gamma'\backslash \mathbb{D}')^*$,
and suppose that $\Gamma'$ is neat.
Fix an integer $d$.
Then there is an effective procedure to produce a finite extension $k'$ of
$k$ with compatible embedding in ${\mathbb C}$ and morphisms
$f_1$, $\ldots$, $f_m:X_{k'}\to X'_{k'}$ such that
$(f_1)_{{\mathbb C}}$, $\ldots$, $(f_m)_{{\mathbb C}}$ are all the morphisms
$X_{{\mathbb C}}\to X'_{{\mathbb C}}$ under which the pullback of $K_{X'_{{\mathbb C}}}$ is
isomorphic to $dK_{X_{{\mathbb C}}}$.
\end{prop}
\begin{proof}
The Hilbert scheme representing morphisms of the given degree from
$X$ to $X'$ may be constructed effectively and by
rigidity (\cite{mok}) has dimension zero.
\end{proof}
\section{Abelian varieties}
\label{sec:av}
Let $A$ be an abelian variety over a number field $k$.
\begin{lemm}
\label{lem:semistable}
There is an effective way to produce a finite extension $k'$ of $k$ such that
$A$ acquires semistable reduction after base change to $k'$.
\end{lemm}
\begin{proof}
This is done by Proposition 4.7 of \cite{SGA7IX}.
\end{proof}
We recall two notions of heights of abelian varieties.
The Faltings height is computed using a semistable model.
Let $K$ be a finite extension of $k$ and
$\mathcal{A}$ a semi-stable model over $\mathfrak{o}_K$.
Then the Faltings height is the arithmetic degree of a particular
metrized canonical sheaf on $\mathcal{A}$
\[h_F(A)=\frac{1}{[K:{\mathbb Q}]}\widehat{\deg} \overline{\omega}_{\mathcal{A}/\Spec(\mathfrak{o}_K)}.\]
This is idependent of the choices of $K$ and $\mathcal{A}$.
For details see, e.g., \cite{bost} \S 2.1.3.
Alternatively, the theta height is defined purely algebraically, in terms
of a principal polarization.
In the following two effectivity results,
the complexity is bounded explicitly in terms of
$[k:{\mathbb Q}]$, $\dim A$, and $h_F(A)$.
Effective comparison results between $h_F(A)$ and the theta height are well known;
see, e.g., \cite{pazuki}.
\begin{prop}
\label{endav}
Let $A$ be a polarized abelian variety over $k$ defined by explicit equations.
Then there is an effective procedure to compute:
\begin{itemize}
\item
A finite extension $k'$ of $k$ for which we have
\[\End_{k'} (A) = \End_{\bar k} (A);\]
\item
Generators of the ${\mathbb Z}$-module $\End_{k'} (A)$;
\item
Generators of the group $\mathrm{NS}(A_{k'})=\mathrm{NS}(\overline{A})$
\end{itemize}
\end{prop}
\begin{proof}
First we reduce to the case when $A$ has a semi-stable model over $k$
by effective semi-stable reduction (Lemma \ref{lem:semistable}).
There is an effective bound on $[k':k]$ from Lemma 2.1 of \cite{mw1}.
The minimal $k'$ is unramified over $k$ by Theorem 1.3 of \cite{ribet}.
The main result of \cite{mw3}
(see also \cite{bost})
bounds the discriminant of the ring of
endomorphisms, which by positive-definiteness bounds the degree of
the elements in a ${\mathbb Z}$-basis.
So they can be found effectively.
See Lemma 2.3 of \cite{mw1}, also 5.2 of \cite{bl}, for the
(standard) identification of $\mathrm{NS}(A)\otimes{\mathbb Q}$
with $\End^{sym}(A)\otimes{\mathbb Q}$.
Pulling back the given polarization by these endomorphisms we get
generators of $\mathrm{NS}(A_{k'})\otimes{\mathbb Q}$.
Possibly after a further field extension, we can find representatives of
generators of $\mathrm{NS}(A_{k'})$.
\end{proof}
\begin{prop}[\cite{mw5}, Theorem 1]
\label{endmodl}
Let $A$ be an abelian variety over $k$.
Then there exists an effective $M\in{\mathbb N}$ such that for any $m$,
\[\End(A)\to \End_\Gamma(A_m)\]
has cokernel annihilated by $M$.
In particular, for a prime $\ell \nmid M$ the natural homomorphism
\[\End(A)/\ell\to \End_\Gamma(A_\ell)\]
is an isomorphism.
\end{prop}
\section{Effective Kuga-Satake construction}
\label{sec:kugasatake}
Let $k$ be a number field with an embedding in ${\mathbb C}$
and $d$ a positive integer.
\begin{defi}
\label{def:pola}
A polarization (resp.\ quasi-polarization) of degree $2d$ on a $K3$ surface $S$
over $k$ a Galois-invariant
class in $\Pic(S_{\bar k})$ which is ample (resp.\ nef) and
has self-intersection $2d$.
A primitive polarization (or quasi-polarization) is one that is not a nontrivial
multiple of another polarization (or quasi-polarization).
\end{defi}
\begin{rema}
Suppose $S$ is given by explicit equations.
These determine a very ample line bundle $L=\mathcal{O}_S(1)$.
We can effectively determine whether the polarization $L$
is primitive, and when it is not, we can produce explicitly
a finite extension $k'$ of $k$ and a primitive polarization represented
by a line bundle $L'$ on $S_{k'}$.
If we assume, further,
that $S(k_v)\ne \emptyset$ for all places $v$ of $k$,
then a standard descent argument
(see, e.g., \S4 of \cite{kteff}) produces effectively a line bundle
on $S$ whose base change to $S_{k'}$ is isomorphic to $L'$.
\end{rema}
For the remainder of the paper we make the following assumptions.
\begin{assu}
\label{assume}
We assume there is an effective construction of
$X^\circ$ over $k$ with $X^\circ_{{\mathbb C}}$ isomorphic to the period space
$\Gamma\backslash \mathbb{D}$ of
primitively quasi-polarized $K3$ surfaces of degree $2d$.
Given a $K3$ surface $S$ over $k$ with explicit equations and
supplied with an explicitly given ample
polarizing class of degree $2d$,
we assume
we can effectively produce the corresponding point in $X^\circ$.
\end{assu}
Let $n$ be a positive integer, greater than or equal to $3$.
The Kuga-Satake construction has been treated in
\cite{deligne}, \cite{vangeemen}, \cite{kugasatake}, and
\cite{rizovcrelle}.
Here we follow the treatment in \cite{rizovcrelle}, where the
relevant level structures are described explicitly and the result
is the existence of morphisms
\[
f^{ks}_{d,a,n,\gamma}:
\mathcal{F}_{2d,n^{\mathrm{sp}}}\to \mathcal{A}_{g,d',n}
\]
of moduli spaces defined over an explicit number field.
Here, there is a standard quadratic form $Q$ on the primitive $H^2$ lattice
$P$ of the $K3$ surface, whose even Clifford algebra will be denoted
$C^+(P)$, and
$a$ is an element of the opposite algebra $C^+(P)^{\mathrm{op}}$
satisfying certain conditions.
Then $d'$ depends explicitly on $a$ and $d$, and
$\gamma$ belongs to a nonempty finite index set.
We suppose these choices are fixed.
The morphism extends to a morphism of Baily-Borel compactifications.
The compactified
source and target spaces can be constructed as projective varieties
over an explicit number field (up to
finitely many candidates) using
Propositions \ref{gammacover} and \ref{neatbb} by the
observations of Remark \ref{whenknowDmodGamma}.
Then (again up to finitely many candidates) Proposition \ref{rigidity}
produces $f^{ks}_{d,a,n,\gamma}$.
The Kuga-Satake abelian variety $A$
associated to the polarized $K3$ surface $S$ has the following
characterization (cf.\ \cite{vangeemen}).
Let $e_1$, $\ldots$, $e_{21}$ be linearly independent
vectors in $P$
diagonalizing the quadratic form so that the span of $e_1$ and $e_2$ is
negative-definite and the span of $e_3$, $\ldots$, $e_{21}$ is
positive-definite.
Let $f_1$, $f_2\in P_{\mathbb R}$ satisfy
$f_1+if_2\in P^{2,0}$ and $Q(f_1)=-1$.
Then $f_1$ and $f_2$ determine an element
\[J:=f_1f_2\in C^+(P)_{\mathbb R}.\]
The element $J$ is independent of the choice of $f_1$ and $f_2$, and
determines a complex structure on $C^+(P)_{\mathbb R}$.
The ${\mathbb C}^*$-action on $C^+(P)_{\mathbb R}$
\[(a+bi)\cdot x:=(a-bJ)x\]
determines a Hodge structure of weight $1$ on $C^+(P)$.
For a suitable choice of sign $\pm$,
the element $\alpha:=\pm e_1e_2\in C^+(P)$ and anti-involution
$\iota:C^+(P)\to C^+(P)$,
\[\iota(e_{i_1}\cdots e_{i_m}):=e_{e_m}\cdots e_{e_1},\qquad
(i_1<\cdots<i_m),\]
determine a polarization
\[E:C^+(P)\times C^+(P)\to {\mathbb Z},\qquad
(v,w)\mapsto tr(\alpha\iota(v)w)\]
where $tr(c)$ denote the trace of the map $x\mapsto cx$.
Then the Kuga-Satake abelian variety associated with the polarized $K3$
surface $S$ is
\[A:=(C^+(P)_{{\mathbb R}},J)/C^+(P),\]
which is a complex torus with
polarized Hodge structure, i.e., a polarized abelian variety over ${\mathbb C}$.
The following properties hold.
There is an injective ring homomorphism
\begin{equation}
\label{morphismu}
u:C^+(P)\to \End(H^1(A,{\mathbb Z}))
\end{equation}
compactible with the weight \emph{zero} Hodge structures on source and target.
The abelian variety $A$ will be defined over a number field,
and the homomorphism of ${\mathbb Z}_\ell$-modules obtained
from $u$ is a homomorphism of Galois modules.
\section{Computing the Picard group of a $K3$ surface}
\label{sec:picard}
Continuing with the assumptions and notation of the previous section, we have
\[P\to\End(C^+(P))\]
sending $v$ to $y\mapsto vye_1$.
This is an injective map of Hodge structures.
From $u$ we get
an injective homomorphism of Hodge structures
\[\End(C^+(P))\to H^2(A\times A,{\mathbb Z})\]
The intersection of the $(0,0)$-part of $\End(C^+(P))_{\mathbb C}$ with
$\End(C^+(P))$ may be effectively computed by identifying
$\End(C^+(P))_{\mathbb Q}$ with a direct summand of
$H^2(A\times A,{\mathbb Q})$, thereby reducing the computation to the determination
of the N\'eron-Severi group of a polarized
abelian variety.
\begin{prop}
\label{effectivePicK3}
Let $S$ be a $K3$ surface as in Assumption \ref{assume}.
Then there is an effective procedure to compute $\Pic(S_{\bar k})$
by means of generators with explicit equations over a finite extension of $k$.
\end{prop}
\begin{rema}
\label{computetopology}
We are interested in the computation of
$H_*(X({\mathbb C}),{\mathbb Z})$, where $X$ is a smooth projective variety over $k$.
This can be done effectively, as explained in \cite{whitney}, by embedding
$X({\mathbb C})$ in a Euclidean space, subdividing the Euclidean space into cubes,
and intersecting with $X({\mathbb C})$.
When $\dim X=2$ this has been treated in \cite{krexp}.
When $X({\mathbb C})$ is isomorphic to
a quotient ${\mathbb C}^N/\Lambda$ for some $N$ and lattice
$\Lambda\subset{\mathbb C}^N$ this is known and standard.
\end{rema}
\begin{proof}[Proof of Proposition \ref{effectivePicK3}]
By the assumptions, we may choose a lift in
$\mathcal{F}_{2d,n^{\mathrm{sp}}}$ of the moduli point of $S$,
and hence obtain a finite set of candidates for the
Kuga-Satake abelian variety.
We compute $P\subset H^2(S({\mathbb C}),{\mathbb Z})$ and $E:C^+(P)\times C^+(P)\to {\mathbb Z}$ exactly
and $J\in C^+(P)_{\mathbb R}$ to high precision.
Evaluating theta functions to high precision, we may identify the
correct image point of the Kuga-Satake morphism, let us say $A^{ks}$ defined
over a finite extension of $k$,
and we may obtain an
analytic map ${\mathbb C}^{2^{19}}\to A^{ks}({{\mathbb C}})$ giving rise to
$A\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to A^{ks}({\mathbb C})$, to arbitrarily high precision.
Using Proposition \ref{endav} we compute representative cycles for generators
of $NS(A^{ks}_{{\mathbb C}}\times A^{ks}_{{\mathbb C}})$.
Computation to sufficiently high precision determines their classes in
$H^2(A\times A,{\mathbb Z})$.
This determines the classes of type $(0,0)$ in
$\End(C^+(P))$, and therefore in $P$.
For a choice of generators we have a degree bound, and by
a Hilbert scheme argument we obtain algebraic representatives of generators,
defined over a finite extension of $k$.
\end{proof}
\section{Proof for good primes}
\label{sec:good}
We keep the notation of the previous section and let
$\Gamma=\Gal(\bar k/k)$.
\begin{prop}
\label{prop:good}
Let $S$ be a $K3$ surface as in Assumption \ref{assume}.
Then there exists, effectively, an $\ell_0$ such that for all primes
$\ell > \ell_0$ we have $\Br(S_{\bar k})^\Gamma_\ell=0$.
\end{prop}
The rest of this section is devoted to the proof.
Proposition \ref{effectivePicK3} tells us that after suitably
extending $k$ we may suppose that $\Pic(S_{\bar k})$ is defined over $k$
and is known explicitly.
We suppose also that we have obtained the Kuga-Satake abelian variety
$A^{ks}$ with analytic map $A\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to A^{ks}({\mathbb C})$ that can be computed to
arbitrarily high precision, and that
$\End(A^{ks}_{\bar k})$ is defined over $k$ and is known explicitly.
Then we may suppose that the subalgebra of
$\End(H_1(A,{\mathbb Z}))$ corresponding to endomorphisms of $A^{ks}$ has been
identified, i.e., that we have computed
\begin{equation}
\label{endosub}
\End(A)\subset \End(H_1(A({\mathbb C}),{\mathbb Z})).
\end{equation}
We have the exact sequence (cf.\ equation (5) of \cite{skorobogatovzarhin})
\begin{align*}
0 \to\Pic(S)/\ell^n \to
H^2&(S_{\bar k},\mu_{\ell^n})^\Gamma \to
\Br(S_{\bar k})_{\ell^n}^\Gamma \\
&{}\to H^1(k,\Pic(S_{\bar k})/\ell^n) \to
H^1(k,H^2(S_{\bar k},\mu_{\ell^n})).
\end{align*}
Let $\delta$ be the (absolute value of the) discriminant of the N\'eron-Severi group.
Then there is an exact sequence
\[0\to \Pic(S)\oplus T_S\to H^2(S({\mathbb C}),{\mathbb Z})\to K\to 0\]
where the cokernel $K$ is finite, of order $\delta$.
After tensoring with ${\mathbb Z}_\ell$ and using a comparison theorem this becomes
an exact sequence of Galois modules
\begin{equation}
\label{eqn:Kell}
0\to (\Pic(S)\otimes {\mathbb Z}_\ell)\oplus T_{S,\ell}\to
H^2(S_{\bar k},{\mathbb Z}_\ell(1))\to K_\ell\to 0
\end{equation}
for some finite $K_\ell$,
where $T_{S,\ell}$ is the submodule of $H^2(S_{\bar k},{\mathbb Z}_\ell(1))$
orthogonal to $\Pic(S)$; note that as abelian groups,
$T_{S,\ell}\cong T_S\otimes {\mathbb Z}_\ell$ and
$K_\ell\cong K\otimes {\mathbb Z}_\ell$.
In particular, for $\ell \nmid \delta$ and any $n$ we have
\[H^1(k,\Pic(S)/\ell^n)\hookrightarrow H^1(k,H^2(S_{\bar k},\mu_{\ell^n})),\]
and the 5-term exact sequence reduces to an isomorphism
\[(T_{S,\ell}/\ell^n)^\Gamma\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to \Br(S_{\bar k})_{\ell^n}^\Gamma.\]
Applying transpose to
the homomorphism $u$ of \eqref{morphismu} we get a homomorphism
\[tu:C^+(P)\to \End(H_1(A,{\mathbb Z})).\]
If $\Pic(S)$ has rank at least $2$, then we let $m\in P$ be an algebraic
class and construct
\[m\wedge T_S\subset \End(H_1(A({\mathbb C}),{\mathbb Z})).\]
By consideration of Hodge type, $\End(A)$ and $m\wedge T_S$ are
disjoint in $\End(H_1(A({\mathbb C}),{\mathbb Z}))$.
Now, outside of finitely many $\ell$ (effectively) we have
an injective map
\[\End(A)/\ell\to \End(H_1(A({\mathbb C}),{\mathbb Z}))/\ell\]
coming from \eqref{endosub} and a pair of injective maps
\[T_S/\ell\to (m\wedge T_S)/\ell\to \End(H_1(A({\mathbb C}),{\mathbb Z}))/\ell,\]
such that the images in $\End(H_1(A({\mathbb C}),{\mathbb Z}))/\ell$ are disjoint.
We use the natural isomorphism of Galois modules
\[A^{ks}_\ell\cong \Hom(H^1(A^{ks},{\mathbb Z}/\ell),{\mathbb Z}/\ell)\]
(cf.\ \cite[\S4.1]{skorobogatovzarhin}) and view
$(m\wedge T_{S,\ell})/\ell$ as a subgroup of $\End(A^{ks}_\ell)$.
So, we have an injective homomorphism of Galois modules
\[T_{S,\ell}/\ell\to \End(A^{ks}_\ell).\]
Applying Proposition \ref{endmodl} we have, away from an effectively
determined finite set of primes $\ell$, an isomorphism
\[\End(A^{ks})/\ell\stackrel{\lower1.5pt\hbox{$\scriptstyle\sim\,$}}\to \End_\Gamma(A^{ks}_\ell).\]
We conclude, outside of an effectively determined finite set of primes $\ell$,
we have
\[(T_{S,\ell}/\ell)^\Gamma=0.\]
If $\Pic(S)$ has rank one, then we have $T_S=P$, and we repeat the
above argument using $\wedge^{20}T_S$ in place of
$m\wedge T_S$ and an identification of $T_S/\ell$ with
$(\wedge^{20}T_S)/\ell$ coming from $\wedge^{21}T_S\cong {\mathbb Z}$.
\section{Bad primes}
\label{sec:bad}
Here we refine the arguments of Section \ref{sec:good} to get an effective
bound on $\Br(S)/\Br(k)$.
We treat the primes excluded from consideration in Section \ref{sec:good}
one at a time, obtaining for each such prime $\ell$ an
effective bound on the order of the
$\ell$-primary subgroup of the image in $\Br(S_{\bar k})$ of
$\Br(S)$.
As in the previous section, we extend $k$ and assume that
$\Pic(S_{\bar k})$ is defined over $k$ and
the Kuga-Satake abelian variety together with its full
ring of geometric endomorphisms is defined over $k$.
We let $m$ be an integer such that the group
$K_\ell$ of \eqref{eqn:Kell} is $\ell^m$-torsion
and further extend $k$ so that
the group $\Br(S_{\bar k})_{\ell^m}$ is defined over $k$.
By \cite{kt} such a field extension may be produced effectively.
To obtain an effective bound on the order of the
$\ell$-primary subgroup of $\Br(S)$ in $\Br(S_{\bar k})$
it suffices
to produce an effective bound on
the order of the cokernel of
\[\Pic(S)/\ell^n\to
H^2(S_{\bar k},\mu_{\ell^n})^{\Gal(\bar k/k')}\]
that is independent of $n$.
The analysis of the previous section, in a refined form,
yields an effective bound for
$(T_{S,\ell}/\ell^n)^\Gamma$, independent of $n$.
Indeed, Proposition \ref{endmodl} yields the effective
annihilation of the cokernel of
$\End(A^{ks})\to \End_\Gamma(A^{ks}_{\ell^n})$.
In the portions of the argument where an injective homomorphism
of finitely generated abelian groups is tensored with
${\mathbb Z}/\ell$, we obtain bounds independent of $n$
on the kernel of the homomorphism
tensored with ${\mathbb Z}/\ell^n$, rather than injective homomorphisms.
This suffices for the analysis.
Equation (1) of \cite{skorobogatovzarhin} yields an exact sequence
of Galois modules
\[0\to \Pic(S)/\ell^m\to H^2(S_{\bar k},\mu_{\ell^m})\to \Br(S_{\bar k})_{\ell^m}\to 0,\]
and therefore $\Gal(\bar k/k')$ acts trivially on
$H^2(S_{\bar k},\mu_{\ell^m})$.
By considering the sequence \eqref{eqn:Kell} tensored by
${\mathbb Z}/\ell^m{\mathbb Z}$ it follows that $\Gal(\bar k/k')$ acts trivially on
$K_\ell$.
We consider $n\ge m$ in what follows.
Tensoring \eqref{eqn:Kell} with ${\mathbb Z}/\ell^n$ yields a four-term
exact sequence of Galois modules with one Tor term:
\begin{equation}
\label{eqn:fourterm}
0\to K_\ell\to
\Pic(S)/\ell^n\oplus T_{S,\ell}/\ell^n\to
H^2(S_{\bar k},\mu_{\ell^n})\to
K_\ell\to 0.
\end{equation}
Since $T_{S,\ell}/\ell^n\to H^2(S_{\bar k},\mu_{\ell^n})$ is
injective, it follows that
\begin{equation}
\label{eqn:isinjective}
K_\ell\to \Pic(S)/\ell^n
\end{equation}
is injective.
We split the exact sequence \eqref{eqn:fourterm} into two
short exact sequences
\begin{gather*}
0\to K_\ell\to
\Pic(S)/\ell^n\oplus T_{S,\ell}/\ell^n\to C\to 0,\\
0\to C\to
H^2(S_{\bar k},\mu_{\ell^n})\to
K_\ell\to 0.
\end{gather*}
This gives the long exact sequences of Galois cohomology
\begin{gather*}
K_\ell\hookrightarrow
\Pic(S)/\ell^n\oplus ( T_{S,\ell}/\ell^n )^\Gamma\to
C^\Gamma
\to H^1(\Gamma,K_\ell)
\to
H^1(\Gamma,\Pic(S)/\ell^n\oplus T_{S,\ell}/\ell^n), \\
0\to
C^\Gamma\to
H^2(S_{\bar k},\mu_{\ell^n})^\Gamma
\to
K_\ell \to H^1(\Gamma, C).
\end{gather*}
Since \eqref{eqn:isinjective} is an injective homomorphism of
trivial Galois modules,
the first three terms of the top sequence split off as a short
exact sequence
\[
0\to K_\ell\to
\Pic(S)/\ell^n\oplus ( T_{S,\ell}/\ell^n )^\Gamma\to
C^\Gamma\to 0.
\]
We conclude by calculating that
\[
\frac{|H^2(S_{\bar k},\mu_{\ell^n})^\Gamma|}
{|\Pic(S)/\ell^n|}
\le
\frac{|K_\ell|\cdot |C^\Gamma|}
{|\Pic(S)/\ell^n|}=
|(T_{S,\ell}/\ell^n)^\Gamma|,
\]
which is bounded as explained above.
|
{
"timestamp": "2012-03-13T01:00:36",
"yymm": "1203",
"arxiv_id": "1203.2214",
"language": "en",
"url": "https://arxiv.org/abs/1203.2214"
}
|
\section{Introduction}
In the rapidly developing field of scattering amplitudes in maximally
supersymmetric Yang-Mills (SYM) theory it sometimes happens that a
new result obtained at enormous cost in time and effort can surprisingly
soon thereafter be rederived by a different
method, or at least understood in terms
of a new framework, as progress marches inexorably on.
This is in no way an indication of wasted effort, as
it is precisely the availability of expensive but valuable new
`theoretical data' which is crucial for inspiring the insights which
in turn allow the next generation of data to be collected.
This bootstrap paradigm suggests that it is never inappropriate to
reexamine previously obtained calculational results to look for
signs of underlying structure or properties which could provide hints
for helping to guide the journey forward. Our focus in this paper
is on integrands for planar multi-loop MHV amplitudes in SYM theory.
In principle there is no conceptual obstacle to constructing these objects
using a variety of reasonably efficient techniques, including generalized
unitarity~\cite{Bern:1994zx,Bern:1994cg,Buchbinder:2005wp,Bern:2007ct,Cachazo:2008dx},
integrand-level recursion
relations~\cite{CaronHuot:2010zt,ArkaniHamed:2010kv,Boels:2010nw}, or
the equivalence between SYM integrands and
supersymmetric Wilson
loops~\cite{Mason:2010yk,CaronHuot:2010ek,Belitsky:2011zm,Adamo:2011pv}
or correlation
functions~\cite{Alday:2010zy,Eden:2010zz,Eden:2010ce,Eden:2011yp,Eden:2011ku,Eden:2011we,Eden:2012tu}.
Specifically our focus is to investigate the extent to which the results
of~\cite{Bourjaily:2011hi} generalize to $n>4$-particle amplitudes.
In that paper it was
shown through seven loops, and conjectured to be true to all loop order,
that the
4-particle planar integrand of the logarithm in SYM theory
has mild behavior (the leading divergence vanishes) in a collinear limit.
In this paper we reformulate the condition of~\cite{Bourjaily:2011hi}
in a very simple way (see equation~(\ref{eq:colliconst}) below) which no longer makes
any reference to the logarithm, and we verify that it holds for
two and three-loop MHV integrands for $n>4$.
Furthermore in~\cite{Bourjaily:2011hi} it was conjectured (and again, shown
through seven loops) that the 4-particle planar integrand
is uniquely determined by simultaneously
imposing dual conformal invariance, dihedral symmetry, and the mild
collinear behavior of the logarithm.
For $n>4$ it is almost immediately seen that this collinear limit by itself cannot
be enough to determine integrands (and is actually a relatively weak constraint
for large $n$), so we will add to our arsenal a second constraint: that the
$n$-particle integrand must reduce smoothly to the $n-1$-particle integrand
in the soft limit, i.e. as the momentum of particle $n$ is taken to zero.
This condition, which was pointed out in~\cite{Drummond:2010mb}, was not discussed
in~\cite{Bourjaily:2011hi} as it is trivial for four-particle
integrands. Here we find that the soft constraint is interestingly
complementary to the collinear constraint, becoming more powerful at larger $n$.
Before proceeding too far let us address an immediate comment one might have on our
approach. Certainly there are various well-known but non-trivial integrat{\it ed}
quantities, such as for $n=6$
the MHV remainder function~\cite{Bern:2005iz} or the NMHV ratio
function~\cite{Drummond:2008vq}
(both of which have been explicitly evaluated,
see~\cite{DelDuca:2009au,Goncharov:2010jf}
and~\cite{Dixon:2011nj} respectively), which
vanish in any soft or collinear limit. However let us emphasize that the `collinear'
limit we discuss here is one in which a loop integration variable, not an external
momentum, becomes collinear with another external momentum. This type of collinear
limit can therefore only be probed at the level of an integrand, not on an
integrated quantity. Moreover the MHV remainder function $R_n^{(L)}$
has no known integrand---that is,
there is no known rational function of external data and
$L$ loop integration variables which, when integrated, gives precisely
$R_n^{(L)}$.
It would be extremely interesting if such an objected could be
found, or even better an algorithm given showing how to determine it for any $n$ and
$L$ in terms of the relevant amplitude integrands (which are explicitly known
at two and three loops~\cite{ArkaniHamed:2010gh}).
In order to apply the soft and collinear constraints towards the
generation of integrands it is necessary to first choose a basis of objects in terms
of which one believes a certain integrand should be expressible.
The soft+collinear constraints then become a system of linear equations
on the coefficients in that basis.
The analysis carried out in~\cite{Bourjaily:2011hi} for $n=4$ benefitted greatly
from an especially manageable basis,
the collection of dual conformally invariant
four-point diagrams.
In particular it seems
(and was proven through seven loops) that this collection is linearly independent
at the level of the integrand,
which is certainly not true of the collection of
natural $n>4$-point dual conformally invariant diagrams which one can easily write
down.
At two loops, the construction of a basis $\mathcal{B}_2$
for general planar Feynman integrals
has been discussed in~\cite{Gluza:2010ws}. It would be fascinating
to study the strength of the collinear and soft limits in the subset
$\mathcal{D}_2
\subset \mathcal{B}_2$ of dual conformally invariant integrals, but we feel
this very general analysis is still beyond our reach at the moment.
Instead
we will take a `Goldilocks' approach to the basis:
first we try small bases at two and three loops which are just
barely large enough
to be known~\cite{ArkaniHamed:2010gh} to be able to express the MHV
amplitude for all $n$. Encouragingly we find that in these cases it is
true that the coefficient of every term in the integrand is uniquely determined
by the soft+collinear constraints.
Then at two loops and for relatively small $n$ we investigate the strength of
the soft+collinear constraints in two other bases: a `too large' basis composed
of arbitrary dual conformally invariant
rational functions of momentum-twistor four-brackets, where
not surprisingly the constraints are too weak,
and finally a `just right' subset
of the latter in which we find that the soft+collinear constraints appears
precisely
powerful to uniquely determine integrands for all $n$.
\section{Setup}
\subsection{Conventions}
We begin by making a potentially jarring notational change: for the rest of this paper, we will refer to ``the number of loops" as $\l$ instead of $L$. We make this change in order to denote the planar $n$-particle $\l$-loop MHV integrand\footnote{Our convention for the overall
normalization of integrands agrees with that of~\cite{ArkaniHamed:2010gh}
and differs slightly from that of~\cite{Bourjaily:2011hi}: the latter
$\l$-loop integrands are a factor of $\l!$ larger than the former.} by $M^{(\l)}_n$, and the corresponding integrand of the logarithm by $L^{(\l)}_n$.
More details about these quantities, and the exact relation between them,
are reviewed in~\cite{Bourjaily:2011hi}.
They are rational functions of the momentum-twistor
four-brackets~\cite{Hodges:2009hk}
involving external and loop integration dual momentum variables.
The former are denoted by $Z_i$ for $i=1,\ldots,n$ while the latter, being
off-shell, need to be described
by a pair of momentum twistors which we denote by $Z_{A_i}, Z_{B_i}$ for
$i=1,\ldots,\l$.
Occasionally we will use the notation $M^{(\l)}_n(1,2,\ldots,n)$ or
$M^{(\l)}_n[1,\ldots,\ell]$ when it is necessary to make explicit reference
to the dependence of an integrand on the external or internal variables
respectively. In examples through three loops we will typically use
the loop integration variables $Z_A,Z_B,Z_C,Z_D,Z_E$ and $Z_F$.
\subsection{Constraints}
Constraints define how the integrand $M^{(\l)}_n$ behaves when the external and/or internal variables are taken to certain limits. In this paper we will be probing the behavior of integrands under two different limits: soft and collinear.
The soft limit involves taking the limit where one of the external momenta vanishes: $p_n\to 0$. In the language of momentum twistors, the soft limit takes the form
\begin{equation}\label{eq:softlimit}
Z_n\to \alpha\, Z_1 + \beta\, Z_{n-1}
\end{equation}
for arbitrary $\alpha,\, \beta$. It was first noted in~\cite{Drummond:2010mb} that the integrand behaves very nicely under the soft limit, in particular the $n$-particle integrand reduces directly to the $(n-1)$-particle integrand when $p_n \to 0$. For the purpose of this paper we will formalize this as the {\it soft constraint}:\\ \\
\noindent{\bf \underline{Soft Constraint}}\\ \noindent{\it The integrand of the $n$-particle $\l$-loop amplitude
in planar SYM theory behaves as}
\begin{equation}
M^{(\l)}_n(1,2,\ldots,n-1,n)\,\to\,M^{(\l)}_{n-1}(1,2,\ldots,n-1)
\end{equation}
{\it in the limit~(\ref{eq:softlimit}) for all $n$ and $\l$.}\\ \\
Of course it is to be understood that $M^{(\l)}_n = 0$ for $n<4$, $\l>0$.
The collinear constraint is based on the observation that the {\it logarithm} of an amplitude has softer IR-divergences than expected when an internal momenta becomes collinear with an external momenta. This softness is well-understood at the level of the integral, but has only been recently discovered to be a property of integrands~\cite{Bourjaily:2011hi}. We emphasize that this was an empirical discovery, and there is no rigorous proof that the property must hold in general. The collinear limit we will be considering in this paper is (see~\cite{Bourjaily:2011hi} for more discussion)
\begin{equation}\label{eq:collinearlimit1}
Z_{A_1} \to Z_2+\O(\epsilon), \qquad Z_{B_1} \to Z_1 + Z_3\,+\O(\epsilon)\,.
\end{equation}
Naturally this limit will produce infrared divergences, and at the level of the integrand these manifest as $1/\epsilon^2$ and $1/\epsilon$ poles in the limit $\epsilon\to0$. However, we conjecture that the $1/\epsilon^2$ pole in $L^{(\l)}_n$ vanishes. We phrase this explicitly as the {\it collinear constraint}: \\ \\
\noindent{\bf \underline{Collinear Constraint}}\\
\noindent
{\it The integrand of the logarithm of the $n$-particle $\l$-loop amplitude
in planar SYM theory behaves as}
\begin{equation}
\label{eq:logcoll}
L^{(\l)}_n[1,2,\ldots,n-1,n]\,\to\,\frac{0}{\epsilon^2} + \O(1/\epsilon)
\end{equation}
{\it in the limit~(\ref{eq:collinearlimit1}) for all $\l>1$}.\\ \\
We have verified this conjecture by explicit
calculation through seventeen points at two loops and eleven
points at three loops.
\subsection{Collinear Redux}
The collinear constraint in the previous section describes how the integrand of the logarithm behaves under the collinear limit. However, there is a more direct way of framing the collinear constraint in terms of the integrand of the amplitude itself. First, let us briefly reintroduce the collinear limit in a more explicit way:
\begin{equation}\label{eq:collinearlimit}
Z_{A_1} \to Z_2+\epsilon Z_X, \qquad Z_{B_1} \to \alpha Z_1 + \beta Z_2 +\gamma Z_3,
\end{equation}
where $Z_X$ is an arbitrary four-vector describing the path taken as $\epsilon\to0$ and the $\alpha,\beta,\gamma$ terms, with $\alpha$ and $\beta$ non-zero, are included in the interest of complete generality. To understand how integrands behave under the collinear limit, we will first consider the $n$-particle one-loop integrand
\begin{equation}\label{eq:oneloop}
M^{(1)}_n =\sum_{i<j} \frac{\ab{AB\,(i\smm1\,i\,i\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(j\smm1\,j\,j\smp1)}\ab{ijn1}}{\ab{AB\,i\smm1\,i}\ab{AB\,i\,i\smp1}\ab{AB\,j\smm1\,j}\ab{AB\,j\,j\smp1}\ab{ABn1}}.
\end{equation}
Under the collinear limit~(\ref{eq:collinearlimit}), the only $\O(1/\epsilon^2)$ pole comes from the $i=2, j=3$ term. This can be easily evaluated to give
\begin{equation}\label{eq:oneloopresult}
\left(M^{(1)}_n\right)_{\rm collinear}=-\frac{1}{(\epsilon\alpha\gamma\ab{123X})^2} + \O(1/\epsilon).
\end{equation}
With this explicit example worked out, we will drop future factors of $\alpha, \gamma$, and $\ab{123X}$ for notational clarity. They can be restored in any formula by replacing $\epsilon^2$ with $(\epsilon\alpha\gamma\ab{123X})^2$.
Direct manipulation of the general $n$-particle two-loop integrand is more difficult, however, we can simplify matters considerably by employing the collinear constraint on the integrand of the logarithm~(\ref{eq:colliconst1}). The two-loop integrand of the logarithm is given in terms of the one- and two-loop integrands by
\begin{equation}
L^{(2)}_n[1,2] = M^{(2)}_n[1,2]-\frac{1}{2}M^{(1)}_n[1]M^{(1)}_n[2].
\end{equation}
Taking the collinear limit and employing~(\ref{eq:oneloopresult}) gives
\begin{equation}
L^{(2)}_n[1,2] \to \left(M^{(2)}_n[1,2]\right)_{\rm collinear}+\frac{1}{2\epsilon^2}M^{(1)}_n[2]+\O(1/\epsilon).
\end{equation}
Imposing the collinear constraint then requires that
\begin{equation}
\left(M^{(2)}_n[1,2]\right)_{\rm collinear}=-\frac{1}{2\epsilon^2}M^{(1)}_n[2] +\O(1/\epsilon).
\end{equation}
A similar calculation, worked out in detail in appendix A, shows that imposing the collinear constraint at three loops is equivalent to requiring
\begin{equation}
\left(M^{(2)}_n[1,2,3]\right)_{\rm collinear}=-\frac{1}{3\epsilon^2}M^{(2)}_n[2,3] +\O(1/\epsilon).
\end{equation}
By now a likely pattern has emerged,
and indeed we prove in appendix A that the logarithm-level collinear
constraint~(\ref{eq:logcoll}) is exactly equivalent, to all loop order,
to the following amplitude-level {\it collinear constraint}:\\ \\
\noindent{\bf \underline{Collinear Constraint} (redux)}\\
\noindent{\it The integrand of the $n$-particle $\l$-loop amplitude
in planar SYM theory behaves as
\begin{equation}\label{eq:colliconst}
M^{(\l)}_n[1,2,\ldots,\l]\,\to\,-\frac{1}{\l} M^{(\l-1)}_{n}[2,3,\ldots,\l]
\end{equation} at $\O(1/\epsilon^2)$ in the limit~(\ref{eq:collinearlimit1}) for all $\l>0$.}\\ \\
We make no attempt here to prove that this is indeed a necessary property of SYM theory amplitudes (a very nice argument in this direction has recently been discovered by~\cite{CHH}). Instead, we conjecture that it is true in general and have verified it to be true at three loops through eleven particles. For the rest of the paper the phrase ``collinear constraint" will refer to~(\ref{eq:colliconst}). The benefit of defining the collinear constraint in this way is not that it is stronger or more informative than the previous definition. Rather, it is computationally and conceptually more simple while containing the same information. Furthermore, this new definition highlights the fact that the collinear and soft limits induce complementary behavior on the same object $M_n^{(\l)}$.
\section{Procedure}
Now that we have defined our constraints, we will describe in detail how they can be used to construct integrands. We begin by postulating a suitable basis of integrands, which will ideally be general enough to describe our desired amplitude while also being small enough to be computationally feasible. We then try to find a linear combination of these basis integrands that satisfy both the collinear and soft constraints.
\subsection{Defining a Basis}\label{sec:basis}
Constructing a basis for the $n$-particle $\l$-loop integrand is a multi-step process, briefly outlined here:
\begin{enumerate}
\item Select a {\it seed} for the basis.
\item Use the seed to generate a {\it pre}-basis.
\item Impose symmetries and linear independency on the pre-basis to determine the basis.
\end{enumerate}
The underlying theme here is to consider as large of a basis as possible, with the hope that the power of our constraints will cast aside all but the small sub-set of desired basis terms.
For example, in a later section, we will consider a basis of general rational functions of momentum-twistor four-brackets, abandoning the notion of a ``diagram" completely.
The choice of seed determines what type of rational functions will enter into our final basis. For example, at two-loops we will first consider the seed
\vspace{-0.2cm}
\begin{equation}
\nonumber\hspace{-0.6cm}\figBox{0}{-1.82}{0.55}{two_loop_integrand.pdf}\!\!\!\!\!=\left\{\begin{array}{l}
\displaystyle\frac{\ab{AB\,(i\smm1\,i\,i\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(j\smm1\,j\,j\smp1)}\ab{i\,j\,k\,l}}{\ab{AB\,i\smm1}\ab{AB\,i\,i\smp1}\ab{AB\,j\smm1\,j}\ab{AB\,j\,j\smp1}\ab{AB\,CD}}\\\displaystyle\times\frac{\ab{CD\,(k\smm1\,k\,k\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(l\smm1\,l\,l\smp1)}}{\ab{CD\,k\smm1k}\ab{CD\,k\,k\smp1}\ab{CD\,l\smm1\,l}\ab{CD\,l\,l\smp1}}\end{array}\right\}\label{MHVInt}
\end{equation}
This particular diagram will be discussed in more depth in the two-loop section, for now we are just using it as an example. We define the pre-basis as the collection of {\it all} iterations of the seed. When dealing with diagrams we also of course restrict ourselves to only planar diagrams. For more general, non-diagrammatic bases considered in later sections we will have no notion of ``planar" and ``non-planar." For this particular seed, we will consider $\{i,j,k,l\}$ ranging over all possible planar orderings for a given $n$, so there will be a total of $4 {n \choose 4}$ elements in the pre-basis.
It is well-known that the amplitude must obey a dihedral symmetry $D_n$, composed of cyclic and mirror symmetries, in the external variables:
\begin{equation}
\mathcal{A}(1,2,\ldots,n) = \mathcal{A}(n,1,2,\ldots,n-1)=\mathcal{A}(n,n-1,\ldots,2,1)
\end{equation}
as well as an $\mathfrak{s}_{\l}$ permutation symmetry in the $\l$-integration variables. We can forcibly impose these symmetries by summing each member of the pre-basis over all of its $D_n \times \mathfrak{s}_{\l}$ images, keeping of course each uniquely generated sum only once. This will give
\begin{equation}\label{eq:prebasis}
\text{pre-basis}=\left\{{\cal I}_1,{\cal I}_2,\ldots,{\cal I}_m\right\}
\end{equation}
where the ${\cal I}_i$ are each $D_n \times \mathfrak{s}_\l$-invariant sums over pre-basis terms. At this stage we must also take into account any linear dependencies, such as Schouten identities, that may exist between elements of the pre-basis. We therefore want to eliminate elements in our pre-basis that can be decomposed into a linear combination of other elements. Computationally, this can be achieved by evaluating~(\ref{eq:prebasis}) over $m$ sets of random values $\{r_i\}$ of the kinematic and loop variables to generate an $m\times m$ matrix:
\begin{equation}
\begin{pmatrix}
{\cal I}_1(r_1) &{\cal I}_2(r_1) & \cdots &{\cal I}_m(r_1) \\
{\cal I}_1(r_2) &{\cal I}_2(r_2) & \cdots &{\cal I}_m(r_2) \\
\vdots & \vdots & \ddots & \vdots \\
{\cal I}_1(r_m) &{\cal I}_2(r_m) & \cdots &{\cal I}_m(r_m) \\
\end{pmatrix}
\end{equation}
This can then be row-reduced to expose a set of $k\le m$ linearly independent basis elements.
(Of course there is always the danger that some of the randomly selected values may
accidentally indicate some apparent linear dependencies which are not truly present.
To guard against this one can in practice choose far more than only $m$ sets
of different random values.)
We then define our basis as the collection of linearly independent, $D_n \times \mathfrak{s}_{\l}$-symmetric rational functions resulting from this procedure:
\begin{equation}\label{eq:prebasis2}
\text{basis}=\left\{{\cal I}_1,{\cal I}_2,\ldots,{\cal I}_k\right\}.
\end{equation}
{}From such a basis one can then construct an ansatz
\begin{equation}\label{eq:ansatz}
M_n^{(\l)}=\sum_i c_i \,{\cal I}_i.
\end{equation}
Our goal is now to determine the collection $\{c_i\}$ of numerical constants which will give us the correct integrand.
\subsection{Imposing Constraints}
Once we have a fully symmetrized and linearly independent basis, the procedure for obtaining the $\l$-loop integrand $M_n^{(\l)}$ from the collinear constraint can be carried out as follows:
\begin{enumerate}
\item Compute the collinear limit of the $n$-point $\l$-loop ansatz~(\ref{eq:ansatz}), and collect from each ${\cal I}$ the $\O(1/\epsilon^2)$ contribution; call these contributions $c_i({\cal I}_i)_{\mathrm{collinear}}$;
\item We now wish to determine the collection of numerical constants $\{c_i\}$ such that \[ \sum_i c_i({\cal I}_i)_{\mathrm{collinear}}=-\frac{1}{\l}M^{(\l-1)}_n.\] By repeatedly evaluating this equation at sufficiently many random independent values of the remaining variables (i.e., $(Z_{A_2},Z_{B_2}), \ldots, (Z_{A_\ell},Z_{B_\ell})$
as well as $Z_1,\ldots,Z_n)$, this equation can be turned into a linear system on the $c_i$ with constant coefficients.
\end{enumerate}
The soft constraint can be imposed in an exactly analogous way, except we are now determining $c_i$ that satisfy
\[\sum_i c_i({\cal I}_i)_{\mathrm{soft}}=M_{n-1}^{(\l)}.\]
and the remaining variables are $(Z_{A_1},Z_{B_1}),\ldots,(Z_{A_\ell},Z_{B_\ell})$ as well as
$Z_1,\ldots,Z_{n-1})$.
It should be noted that, depending on the choice of basis, $({\cal I}_i)_{\mathrm{collinear}}$ and $({\cal I}_i)_{\mathrm{soft}}$ may vanish for certain choices of $i$. If, for example, $({\cal I}_1)_{\mathrm{collinear}}=0$ , the collinear limit will be blind to ${\cal I}_1$ and will provide no constraint on the coefficient $c_1$. Throughout the rest of this paper we will be essentially be asking the same questions: for a given $n$-particle, $\l$-loop ansatz,
\begin{itemize}
\item how many of the $c_i$ are fixed by the collinear constraint alone?
\item how many of the $c_i$ are fixed by the soft constraint alone?
\item how many of the $c_i$ are fixed by the collinear and soft constraints in conjunction?
\end{itemize}
Answering these questions over a range of bases will give us some insight into how strong these constraints are at various $n$ and $\l$.
\section{Two-Loops}\label{sec:twoloop}
An important first step in employing the collinear and soft constraints is to verify that they reproduce previously known integrands. In~\cite{ArkaniHamed:2010gh}, the $n$-particle MHV integrand was given at two- and three-loops. These representations are built upon a special set of integrands which are chiral and have unit leading singularities. Our first goal, then, is to use these integrands to generate a basis using these integrands and see if the collinear and soft constraints can reproduce the known integrand. At two-loops, the $n$-particle integrand is given by
\vspace{-0.4cm}
\begin{equation}\label{eq:twoloopintegrand}
M_{n}^{(2)}=\displaystyle\underset{\substack{i<j<k<l<i}}{\frac{1}{2}\text{{\Huge$\sum$}}\phantom{\frac{1}{2}\!\!}}\hspace{-0.2cm}\raisebox{-1.6cm}{\includegraphics[scale=0.5]{two_loop_integrand.pdf}}
\end{equation}
with
\vspace{-0.1cm}
\begin{equation}\label{eq:twoloopbasis}
\hspace{-1cm}\figBox{0}{-1.82}{0.5}{two_loop_integrand.pdf}\!\!\!\!\!=\left\{\begin{array}{l}
\displaystyle\frac{\ab{AB\,(i\smm1\,i\,i\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(j\smm1\,j\,j\smp1)}\ab{i\,j\,k\,l}}{\ab{AB\,i\smm1}\ab{AB\,i\,i\smp1}\ab{AB\,j\smm1\,j}\ab{AB\,j\,j\smp1}\ab{AB\,CD}}\\\displaystyle\times\frac{\ab{CD\,(k\smm1\,k\,k\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(l\smm1\,l\,l\smp1)}}{\ab{CD\,k\smm1k}\ab{CD\,k\,k\smp1}\ab{CD\,l\smm1\,l}\ab{CD\,l\,l\smp1}}\end{array}\right\}
\end{equation}
We will start by taking~(\ref{eq:twoloopbasis}) as the seed of our basis, and then impose constraints in the hope of reproducing~(\ref{eq:twoloopintegrand}).
Boundary terms occur when $j=i+1$ and/or $l=k+1$, in which case one or both of the wavy-line numerators cancels out a propagator, resulting in a box rather than pentagon topology. As we will see, these boundary terms will be of fundamental importance with regards to the collinear limit, so we write them down explicitly here:
\begin{equation}
\hspace{-1.85cm}\figBox{0}{-1.7}{0.45}{two_loop_boundary_1.pdf}\;\figBox{0}{-1.7}{0.45}{two_loop_boundary_2.pdf}\;\figBox{0}{-1.15}{0.45}{two_loop_boundary_3.pdf}\hspace{-2cm}\label{MHVInt2}
\end{equation}
\subsection{Five- and Six-Point Examples}
With the two-loop basis, imposing the dihedral and loop-variable symmetries is equivalent to selecting topologically distinct diagrams. At $n=5$, there are only two non-isomorphic diagrams:
\begin{equation}
\hspace{-1.85cm}\figBox{0}{-1.7}{0.45}{two_loop_five_point_1.pdf}\;\figBox{0}{-1.15}{0.45}{two_loop_five_point_2.pdf}\hspace{-2cm}
\end{equation}
We therefore have the ansatz
\begin{equation}
M_5^{(2)}= a\,{\cal I}_1^{(2)}+b\,{\cal I}_2^{(2)}
\end{equation}
where each ${\cal I}^{(2)}_i$ is a sum of ten diagrams related by symmetry:
\begin{equation}
\begin{split}
&\hspace{-.2cm}{\cal I}_1^{(2)}=\frac{\ab{1235}^2 (\ab{1245}\ab{CD34}-\ab{1345} \ab{CD24})}{\ab{AB12}\ab{AB23}\ab{AB51} \ab{ABCD} \ab{CD23}\ab{CD34} \ab{CD45}\ab{CD51}}\,+\,\text{sym.} \\ \\
&\hspace{-.2cm}{\cal I}_2^{(2)}=\frac{\ab{1235}\ab{1245}\ab{1345}}{\ab{AB12}\ab{AB23}\ab{AB51}\ab{ABCD}\ab{CD34}\ab{CD45}\ab{CD51}}\,+\,\text{sym.}
\end{split}
\end{equation}
Both ${\cal I}_1$ and ${\cal I}_2$ clearly have non-zero $\O(1/\epsilon^2)$ contributions under the collinear limit. We can therefore assemble the equation
\begin{equation}
a\left({\cal I}_1^{(2)}\right)_{\rm collinear}+b\left({\cal I}_2^{(2)}\right)_{\rm collinear} = -\frac{1}{2}M^{(1)}_5.
\end{equation}
After generating a system of equations on $a$ and $b$ by evaluating this equation at random kinematical values, we see that the only solution is $a = b = 1/2$, in accordance with the known value for the two-loop five-particle integrand.
A similar story occurs with the soft limit: both diagrams give non-zero, linearly independent contributions under the soft limit, and so the soft limit constraint fixes $a=b=1/2$.
At $n=6$ there are five topologically distinct diagrams:
\begin{equation}
\begin{split}
\hspace{-1.85cm}\figBox{0}{-1.7}{0.5}{two_loop_six_point_1.pdf}\;\figBox{0}{-1.7}{0.45}{two_loop_six_point_2.pdf}\;\figBox{0}{-1.7}{0.45}{two_loop_six_point_3.pdf}\hspace{-2cm}\\
\hspace{-3cm}\figBox{0}{-1.15}{0.45}{two_loop_six_point_4.pdf}\;\figBox{0}{-1.15}{0.45}{two_loop_six_point_5.pdf}
\end{split}
\end{equation}
Under the collinear limit, the double-pentagon topology does {\it not} have a contribution at $\O(1/\epsilon^2)$. The other topologies, all featuring at least one box, have $\O(1/\epsilon^2)$ divergences under the collinear limit. Furthermore, the $\O(1/\epsilon^2)$-poles from each diagram are linearly independent rational functions. Therefore, the collinear limit constraint determines four of the five coefficients in the ansatz.
All of the topologies survive the soft limit and are linearly independent, so the integrand is completely determined by the soft limit constraint.
\subsection{General Results}
The size of the basis and the number of coefficients fixed by the collinear constraint are summarized through $n=17$ in table~\ref{table:twoloop}.
The soft constraint is not included in table~\ref{table:twoloop} because it fixes all of the coefficients through $n=17$, and we conjecture that it fixes all of the coefficients for general $n$. This is because the basis generated by~(\ref{eq:twoloopbasis}) is very well-behaved under the soft limit. Specifically, the $n$-particle integrand directly becomes the $(n-1)$-particle integrand term-by-term under the soft limit:
\begin{displaymath}
\{i,j,k,l\}_n \xrightarrow{p_n~ \rightarrow~ 0~} \left\{
\begin{array}{lr}
\{i,j,k,l\}_{n-1} & : n \notin \{i,j,k,l\}\\
0 & : n \in \{i,j,k,l\}
\end{array}
\right.
\end{displaymath}
(where the notation $\{i,j,k,l\}_n$ describes a particular diagram in the $n$-particle basis). Therefore, each basis term will give a non-zero contribution under the soft limit. As we have explicitly verified through $n=17$, these contributions remain linearly independent and so all are fixed by the soft constraint.
\begin{table}[h]
\begin{center}
\begin{tabular}{ c|c|c|}
\cline{2-3}
& \multicolumn{2}{|c|}{{\bf \# of unfixed coefficients}}\\
\hline
\multicolumn{1}{|c|}{${\mathbf n}$} & Symmetrized basis & After collinear constraint \\ \hline
\multicolumn{1}{|c|}{5}& 2 & 0 \\ \hline
\multicolumn{1}{|c|}{6} & 5 & 1 \\ \hline
\multicolumn{1}{|c|}{7} & 8 & 2 \\ \hline
\multicolumn{1}{|c|}{8} & 14 & 5 \\ \hline
\multicolumn{1}{|c|}{9} & 20 & 8 \\ \hline
\multicolumn{1}{|c|}{10} & 30 & 14 \\ \hline
\multicolumn{1}{|c|}{11} & 40 & 20 \\ \hline
\multicolumn{1}{|c|}{12} & 55 & 30 \\ \hline
\multicolumn{1}{|c|}{13} & 70 & 40 \\ \hline
\multicolumn{1}{|c|}{14} & 91 & 55 \\ \hline
\multicolumn{1}{|c|}{15} & 112 & 70 \\ \hline
\multicolumn{1}{|c|}{16} & 140 & 91 \\ \hline
\multicolumn{1}{|c|}{17} & 168 & 112 \\ \hline
\end{tabular}\caption{Results of the collinear constraint on the two-loop basis~(\ref{eq:twoloopbasis})}\label{table:twoloop}
\end{center}
\end{table}
The collinear limit only fixes the coefficients of diagrams involving $\O(1/\epsilon^2)$ divergences under the collinear limit. An interesting feature of the wavy-line numerator term of this basis is that it dampens IR divergences. In the collinear limit, diagrams that feature wavy-line numerators across both loops, i..e two pentagons (rather than two boxes or a box and a pentagon), will only have $\O(1/\epsilon)$ divergences under the collinear limit and will not be fixed by the collinear constraint. Put more succinctly, the diagrams left unfixed by the collinear constraint are those without boxes. The question ``how many coefficients are left unfixed after imposing the collinear constraint" is equivalent to ``how many diagrams (up to isomorphism) involve two pentagons". In other words, imposing the collinear constraint is, topologically, equivalent to symmetrization with the added stipulation that there has to be at least one leg present at $x$ and $y$:
\vspace{-0.4cm}
\begin{equation}
\figBox{0}{-1.7}{0.5}{two_loop_fix_legs.pdf}\vspace{-0.4cm}
\end{equation}
This leads to the relationship
\begin{equation}
\begin{split}
\text{\{\# of coefficients left unfixed after imposing the collinear constraint\}}(n)&\\
=\text{\{\# of topologies with only pentagons\}}(n)&\\
=\text{\{\# of topologies with pentagons and boxes\}}(n-2)&.
\end{split}
\end{equation}
The pattern that emerges in table~\ref{table:twoloop}, $\{0,1,2,5,8,14,\ldots\}$, has a number of possible interpretations, though none seem particularly insightful. The pattern can be interpreted as~\cite{OEIS}
\begin{itemize}
\item the number of aperiodic necklaces (Lyndon words) with 4 black beads and $n-4$ white beads.
\item the maximum number of squares that can be formed from $n$ lines.
\item partial sums of $\{1, 1, 3, 3, 6, 6, 10, 10,\ldots\}$.
\end{itemize}
It is interesting to note that, assuming this pattern holds for all $n$, the collinear constraint will asymptotically become useless at two-loops for large $n$. For odd $n$ the number of unconstrained coefficients left after imposing the collinear constraint goes as
\begin{equation}
\text{\{\# of unfixed coeff. after the collinear const.\}}(n_{{\rm odd}}) = \frac{(n-5)(n-3)(n-1)}{24}
\end{equation}
Therefore,
\begin{equation}
\frac{\text{\{\# of unfixed coefficients after the collinear const.\}}}{\text{\{\# of unfixed coefficients in the basis\}}}(n_{{\rm odd}})=\frac{n-5}{n+1}
\end{equation}
We can thus conclude that the percentage of coefficients fixed by the collinear limit in this particular basis at two-loops asymptotically approaches zero as $n$ goes to infinity.
\section{Three-Loops}\label{sec:threeloop}
The three-loop integrand given in~\cite{ArkaniHamed:2010gh} involves the pentagon and hexagon integrands with wavy-line numerators. The general form is given by
\vspace{-0.2cm}
\begin{equation}\label{eq:threeloopintegrand}
\hspace{-2cm}M_n^{(3)}=\!\!\!\!\displaystyle\underset{\substack{i_1\leq i_2<j_1\leq\\\leq j_2<k_1\leq k_2<i_1}}{\frac{1}{3}\!\text{{\Huge$\sum$}}\phantom{\frac{1}{2}\!\!}}\!\!\!\figBox{0}{-2.15}{0.55}{three_loop_integrand_1.pdf}+\!\!\displaystyle\underset{\substack{i_1\leq j_1< k_1<\\< k_2\leq j_2< i_2<i_1}}{\frac{1}{2}\!\text{{\Huge$\sum$}}\phantom{\frac{1}{4}\!\!}}\!\!\!\!\figBox{0}{-1.8}{0.55}{three_loop_integrand_2.pdf}
\end{equation}
While the two-loop basis had simple boundary terms, (\ref{eq:threeloopintegrand}) hides a considerable amount of complexity that occurs at boundaries (such as $i_1 = i_2$ or $i_1=j_1$). These boundary terms are too numerous and complicated to be discussed in detail here (and can be found in the appendix of~\cite{ArkaniHamed:2010gh}), but as in the two-loop case we find that they play a crucial role in the effectiveness of the collinear, and also soft, constraints. We will therefore construct our $n$-particle three-loop basis by generating and then symmetrizing over all planar iterations of the diagrams in~(\ref{eq:threeloopintegrand}) (including the boundary terms). The goal is then to use the collinear and soft constraints to reproduce the relevant factors of 1/3 and 1/2 in the known integrand.
The size of the basis and the number of coefficients fixed by the collinear constraint are summarized through $n=11$ in table~\ref{table:threeloop}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{ c|c|c|c|}
\cline{2-4}
& \multicolumn{3}{|c|}{{\bf \# of unfixed coefficients}}\\
\hline
\multicolumn{1}{|c|}{{\bf n}} & Symmetrized basis & After collinear & After soft\\ \hline
\multicolumn{1}{|c|}{5} & 6 & 0 & 1 \\ \hline
\multicolumn{1}{|c|}{6} & 17 & 1 & 2 \\ \hline
\multicolumn{1}{|c|}{7} & 33 & 2 & 0 \\ \hline
\multicolumn{1}{|c|}{8} & 63 & 5 & 0 \\ \hline
\multicolumn{1}{|c|}{9} & 109 & 9 & 0 \\ \hline
\multicolumn{1}{|c|}{10} & 178 & 16 & 0 \\ \hline
\multicolumn{1}{|c|}{11} & 277 & 26 & 0 \\ \hline
\end{tabular}\caption{Results of the collinear and soft constraints on the three-loop basis~(\ref{eq:threeloopintegrand})}\label{table:threeloop}
\end{center}
\end{table}
The three-loop basis is not as well-behaved as the two-loop basis under the soft limit due to the large number of boundary terms. This leads to the unfixed coefficients at five- and six-points. It is also worth noting that the collinear constraint, when applied in conjunction with the soft constraint, fixes all coefficients at $n=6$.
The collinear constraint is more effective at three-loops than at two-loops because more of the diagrams involve boxes. This points towards an interesting, though perhaps not entirely surprising feature of the collinear constraint: it grows stronger at higher loop order. At two-loops, the efficacy of the collinear constraint decreased dramatically as $n$ increased. The data so far indicates that this is not the case at three-loops. Instead, we see that the percentage of coefficients fixed by the collinear constraint is asymptotically approaching a value somewhere near $90\%$. This is due to the large number of boundary terms involved in the three-loop basis, all of which involve boxes and are thus sufficiently divergent to be detected by the collinear limit. The total number of boundary terms at three-loops increases considerably when $n$ increases.
\section{Expanding the Basis}
The arguments of the previous section indicate that the collinear and soft constraints are useful tools for determining integrands given a suitable choice of basis. The natural question that arises, then, is what if we didn't {\it a priori} know, or at least knew very little about, the form of the integrand we wanted to construct? In other words, the chiral integrands with unit leading singularities that we used to generate bases in sections~\ref{sec:twoloop} and~\ref{sec:threeloop} are already very refined objects. If we consider a more general basis, will the collinear and soft constraints still determine the integrand?
In order to answer these questions, we will start by determining the five-point two-loop integrand from a very general basis of rational functions. We will then generalize to arbitrary $n$, using the collinear and soft constraints along with symmetry considerations to essentially re-derive the $n$-particle two-loop integrand~(\ref{eq:twoloopintegrand}).
\subsection{Five-Point Two-Loop Integrand}\label{sec:5pt2lp}
In the interest of complete generality, we will abandon diagrams and deal solely with rational functions. This means we will also have no concepts of planar or non-planar terms, which will also expand our basis considerably.
To develop a basis of rational functions, we will conjecture that they all share the common denominator
\begin{equation*}
\ab{AB12}\ab{AB23}\ab{AB34}\ab{AB45}\ab{AB51}\ab{ABCD}\ab{CD12}\ab{CD23}\ab{CD34}\ab{CD45}\ab{CD51}
\end{equation*}
We expect possible numerator terms to look like
\begin{equation}\label{eq:general} \hspace{-.5cm}
\text{\{a product of three $\ab{}$'s involving $Z_1,\ldots,Z_5$\}$\,\ab{AB\,W}\ab{AB\,X}\ab{CD\,Y}\ab{CD\,Z}$}
\end{equation}
where $W,X,Y,Z$ are bi-twistors composed out of $Z_1,\ldots,Z_5$. Taking this rational function as our seed, we iterate over all possible numerators of this form, with the only restriction being that they have the correct conformal weight.
After symmetrization, we are left with 17 linearly independent collections of these terms. Imposing the collinear and soft limits leaves us with four unfixed coefficients in addition to a constant (inhomogeneous) term. The four unfixed terms necessarily obey an interesting set of properties since they escaped detection by our methods. In particular they are all
\begin{itemize}
\item linearly independent
\item fully $D_5\times\mathfrak{s}_2$ symmetric
\item vanish under the soft limit
\item have only $\O(1/\epsilon)$ divergences under the collinear limit.
\end{itemize}
One such object is
\begin{equation} \hspace{-.5cm}
\frac{\makebox[1cm][c]{\small $\ab{1345}\,N + \{AB \leftrightarrow CD\}+\{\mathrm{cyc.}\}$}}{\makebox[14.5cm][c]{\small $\ab{AB12}\ab{AB23}\ab{AB34}\ab{AB45}\ab{AB51}\ab{ABCD}\ab{CD12}\ab{CD23}\ab{CD34}\ab{CD45}\ab{CD51}$}}
\end{equation}
where
\begin{equation}
\begin{split}
N=-&\,\ab{2345}\ab{CD51}\ab{CD23}\ab{AB24}\ab{AB\,(123){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(145)}\\
+&\,\ab{1245}\ab{CD51}\ab{CD23}\ab{AB24}\ab{AB\,(123){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(345)}\\
+&\,\ab{2345}\ab{CD13}\ab{CD24}\ab{AB12}\ab{AB\,(125){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(345)}\\
+&\,\ab{1345}\ab{CD12}\ab{CD24}\ab{AB25}\ab{AB\,(123){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(345)}\\
+&\,\ab{1245}\ab{2345}\ab{CD23}\ab{CD13}\ab{AB12}\ab{AB45}\\
+&\,\ab{1245}\ab{2345}\ab{CD23}\ab{CD51}\ab{AB23}\ab{AB14}
\end{split}
\end{equation}
It is unclear what these objects might represent. They are clearly related in some way to non-planar diagrams, but the exact (function $\leftrightarrow$ diagram) correspondence is non-obvious.
\subsection{$n$-point Two-Loop Integrand}
For $n>5$, it is computationally prohibitive to generate a basis in the same manner as described in section~\ref{sec:5pt2lp}, so we must make some restrictions. After looking at the results for $n=4$ and 5 at two-loops, a plausible conjecture would be that two-loop integrands can be described in terms of double-pentagon integrands, i.e., rational functions with the denominator
\vspace{.4cm}\begin{equation*}
\makebox[1cm][c]{\small $\ab{AB\,i{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,i}\ab{AB\,i\,i\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\ab{AB\,j{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,j}\ab{AB\,j\,j\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\ab{ABCD}\ab{CD\,k{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,k}\ab{CD\,k\,k\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\ab{CD\,l{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,l}\ab{CD\,l\,l\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}$}\vspace{.2cm}
\end{equation*}
Our task is now to determine the possible numerators for the double-pentagon integrand. In other words, we can ask: is the wavy-line numerator structure in~(\ref{eq:twoloopintegrand}) the unique numerator that satisfies the soft and collinear constraints? To answer this questions, we consider the general numerator
\begin{equation}\label{eq:conjecture}
\begin{split}
\frac{\ab{ijkl}}{\ab{ABCD}}\Big(&\frac{f(A,B,i,j)}{\ab{AB\,i{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,i}\ab{AB\,i\,i\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\ab{AB\,j{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,j}\ab{AB\,j\,j\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}}\\ &\hspace{3cm}\times\frac{g(C,D,k,l)}{\ab{CD\,k{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,k}\ab{CD\,k\,k\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\ab{CD\,l{\rm\rule[2.4pt]{6pt}{0.65pt}} 1\,l}\ab{CD\,l\,l\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}}\Big).
\end{split}
\end{equation}\\
Due to the symmetry of~(\ref{eq:conjecture}), we can focus our investigations on $f$:
\begin{itemize}
\item it has either odd or even parity under $\{i\leftrightarrow j\}$
\item it has odd parity under $\{A\leftrightarrow B\}$
\item it is a product of 2 $\ab{}$'s, involving one instance each of $A,B,i{\rm\rule[2.4pt]{6pt}{0.65pt}} 1,i,i\smp1,j\smm1,j,j\smp1$
\end{itemize}
The equivalent statements are true about $g$ after swapping $\{A,B,i,j\}\to\{C,D,k,l\}$. Because the product $f(A,B,i,j)g(C,D,k,l)$ must be invariant under the transformation $\{i\leftrightarrow j,k\leftrightarrow l\}$, $f$ and $g$ must have the same parity under a flip of the external indices.
To determine $f$, we start with the $70={8 \choose 4}$ possible products of $\ab{}$'s. Imposing the above symmetries leaves six linearly independent possible numerator terms that have odd parity under $\{i\leftrightarrow j\}$\footnote{One might wonder why $\ab{AB\,i\smm1\,j\smm1}\ab{i\,i\smp1\,j\,j\smp1}$ and $\ab{AB\,i\,j}\ab{i\smm1\,i\smp1\,j\smm1\,j\smp1}$ are included, but $\ab{AB\,i\smp1\,j\smp1}\ab{i\smm1\,i\,j\smm1\,j}$ is not. This is because it can be expressed as a linear combination of $o_2,o_5,$ and $o_6$.}:
\begin{equation}\label{eq:odd}
\begin{split}
f_{{\rm odd}}\,=\phantom{+}o_1 &\ab{AB\,i\,j}\ab{i\smm1\,i\smp1\,j\smm1\,j\smp1}\\
+\,o_2 &\ab{AB\,i\smm1\,j\smm1}\ab{i\,i\smp1\,j\,j\smp1}\\
+\,o_3 &\ab{AB\,(i\smm1\,i\,i\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(j\smm1\,j\,j\smp1)}\\
+\,o_4 &\ab{AB\,(i\smm1\,j\,i\smp1){\small\mathrm{\raisebox{0.95pt}{{$\,\bigcap\,$}}}}(j\smm1\,i\,j\smp1)}\\
+\,o_5 &\big(\ab{AB\,i\,j\smm1}\ab{i\smm1\,i\smp1\,j\,j\smp1}+\ab{AB\,i\smm1\,j}\ab{j\smm1\,j\smp1\,i\,i\smp1}\big)\\
+\,o_6 &\big(\ab{AB\,i\smp1\,j\smm1}\ab{i\smm1\,i\,j\,j\smp1}+\ab{AB\,i\smm1\,j}\ab{i\,i\smp1\,j\smm1\,j\smp1}-\\
&\phantom{\big(}\ab{AB\,i\smp1\,i\smm1\,}\ab{i\,j\smm1\,j\,j\smp1}-\ab{AB\,i\smm1\,i}\ab{i\smp1\,j\smm1\,j\,j\smp1}\big)
\end{split}
\end{equation}
There are three linearly independent possible numerator terms that have even parity:
\begin{equation}\label{eq:even}
\begin{split}\hspace{-1cm}
f_{{\rm even}}\,=\phantom{+}e_1 &\big(\ab{AB\,i\,j\smm1}\ab{i\smm1\,i\smp1\,j\,j\smp1}-\ab{AB\,i\smm1\,j}\ab{j\smm1\,j\smp1\,i\,i\smp1}\big)\\
+\,e_2 &\big(\ab{AB\, i \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1\,j{\rm\rule[2.4pt]{6pt}{0.65pt}} 1} \ab{i\smm1\, i\, j\, j \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1} -
\ab{AB\, i {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, j} \ab{i\, i \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1\, j {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, j \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1} + \ab{AB\, i {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, j {\rm\rule[2.4pt]{6pt}{0.65pt}} 1} \times\\
& \ab{i\, i \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1\, j\, j \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1} -
\ab{AB\, i {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, i \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1} \ab{i\, j {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, j\, j \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1} +
\ab{AB\, i {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, i} \ab{i \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1\, j {\rm\rule[2.4pt]{6pt}{0.65pt}} 1\, j\, j \hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt} 1}\big)\\
+\,e_3 &\big(\ab{AB\, i\smp1\, j} \ab{i\smm1\, i\, j\smm1\, j\smp1} - \ab{AB\, i\, j} \ab{i\smm1\, i\smp1\, j\smm1\, j\smp1} - \ab{AB\, i\, i\smp1}\times\\
& \ab{i\smm1\, j\smm1\, j\, j\smp1} + \ab{AB\, i\smm1\, j} \ab{i\, i\smp1\, j\smm1\, j\smp1} -
\ab{AB\, i\smm1\, i} \ab{i\smp1\, j\smm1\, j\, j\smp1}\big)\end{split}
\end{equation}
Again, the equivalent statements are true about $g$ after swapping $\{A,B,i,j\}\to\{C,D,k,l\}$. We thus have two different possible bases generated by~(\ref{eq:conjecture}), one involving $(f_{{\rm even}})(g_{{\rm even}})$ and the other with $(f_{{\rm odd}})(g_{{\rm odd}})$.
\subsubsection*{Even-Even Parity}
The even-even parity basis involves 6 possible $\mathfrak{s}_2$-symmetric terms generated by multiplying~(\ref{eq:even}) by the corresponding $\big(\{A,B,i,j\}\to\{C,D,k,l\}\big)$ set. Note that when constructing the basis, as outlined in section~\ref{sec:basis}, we can then absorb the $e_i$ of~(\ref{eq:even}) into the $c_i$ of~(\ref{eq:ansatz}).
Imposing the soft and collinear constraints on this basis, we find that there is no choice of coefficients able to satisfy the collinear constraint for any $n$ (checked explicitly through $n=12$). The soft constraint is possible to satisfy at $n=5$ but at no other points (again, checked explicitly through $n=12$). The even-even basis is thus not a possible basis for two-loop MHV integrands.
\subsubsection*{Odd-Odd Parity}
The odd-odd parity basis involves 21 possible $\mathfrak{s}_2$-symmetric terms generated by multiplying~(\ref{eq:odd}) by the corresponding $\big(\{A,B,i,j\}\to\{C,D,k,l\}\big)$ set. We again absorb the $o_i$ of~(\ref{eq:odd}) into the $c_i$ of~(\ref{eq:ansatz}). The size of this expanded basis and the number of coefficients fixed by the collinear constraint are summarized through $n=10$ in table~\ref{table:twoloopexp}.
\begin{table}[h]
\begin{center}
\begin{tabular}{ c|c|c|c|}
\cline{2-4}
& \multicolumn{3}{|c|}{{\bf \# of unfixed coefficients}}\\
\hline
\multicolumn{1}{|c|}{{\bf n}} & Symmetrized basis & After collinear & After soft\\ \hline
\multicolumn{1}{|c|}{5} & 3 & 0 & 1 \\ \hline
\multicolumn{1}{|c|}{6} & 13 & 1 & 1 \\ \hline
\multicolumn{1}{|c|}{7} & 33 & 3 & 0 \\ \hline
\multicolumn{1}{|c|}{8} & 80 & 9 & 0 \\ \hline
\multicolumn{1}{|c|}{9} & 152 & 17 & 0 \\ \hline
\multicolumn{1}{|c|}{10} & 272 & 32 & 0 \\ \hline
\end{tabular}\caption{Results of the collinear and soft constraints on the expanded two-loop basis~(\ref{eq:odd})}\label{table:twoloopexp}
\end{center}
\end{table}
At $n=6$, imposing both the soft and collinear constraints fixes all coefficients. With this data we see that the odd-odd basis, when combined with the soft and collinear constraints, is able to produce all two-loop MHV integrands. And at all $n$, the completely fixed integrand involves setting $o_3 =1$ and $o_{i\ne3}=0$ in~(\ref{eq:odd}) and then taking all planar diagrams with equal weight, reproducing the two-loop integrand~(\ref{eq:twoloopintegrand}). Furthermore, a familiar story emerges: as $n$ increases, the soft limit grows stronger while the collinear limit grows weaker. We can also compare this data with the smaller two-loop basis summarized in table~\ref{table:twoloop}. The results are intriguing: at $n=8$, for example, the collinear result constrained 64\% of coefficients with the known-numerator, but constrained 89\% of coefficients with the (much) more general numerator.
\section{Multi-Collinear Limits}
We conclude by briefly describing the behavior of integrands under multi-collinear limits, such as the double-collinear limit
\begin{equation}\label{eq:multi2}
\begin{split}
&Z_{A_1} \rightarrow Z_2 + \O(\epsilon) ,~~~~~~~~Z_{B_1}\rightarrow Z_1 + Z_3 + \O(\epsilon)\\
&Z_{A_2} \rightarrow Z_3 + \O(\epsilon) ,~~~~~~~~Z_{B_2}\rightarrow Z_2 + Z_4 + \O(\epsilon).
\end{split}
\end{equation}
As one might expect given the form of the collinear constraint~(\ref{eq:colliconst}), under this limit we have
\begin{equation}
M^{(\l)}_n\to\frac{1}{\l(\l-1)\epsilon^4}M^{(\l-2)}_n +\O(1/\epsilon^3).
\end{equation}
We extend this to the full $\l$-collinear limit, defined as
\begin{equation}\label{eq:multicollilim}
\begin{split}
Z_{A_1}& \rightarrow Z_{2} + \O(\epsilon) ,\qquad Z_{B_1}\rightarrow Z_1 + Z_{3} + \O(\epsilon)\\
&\vdots\hspace{4.75cm}\vdots\\
Z_{A_{\l-1}}& \rightarrow Z_{n} + \O(\epsilon) ,\qquad Z_{B_{\l-1}}\rightarrow Z_{n-1} + Z_1 + \O(\epsilon)\\
\end{split}
\end{equation}
Under this limit, we have
\begin{equation}
M^{(\l)}_n\to\frac{(-1)^\l}{\epsilon^{2\l}\l!}+ \O(1/\epsilon^{2\l-1}).
\end{equation}
The collinear constraint on the integrand of the logarithm can also be generalized in this manner. Specifically, $L^{(\l)}_n$ behaves as $\O(1/\epsilon^\ell)$ in the limit~(\ref{eq:multicollilim}) for $\ell>1$. (One would na\"ively expect $\O(1/\epsilon^{2\ell})$ divergences.)
These multi-collinear constraints are interesting from a purely conceptual standpoint, but are less useful than the single-collinear limit in terms of actually constraining a conjectured integrand basis. This is because fewer terms in the basis are sufficiently divergent to be captured by the multi-collinear constraint. This is analogous to the notion of a ``multi-soft" limit, where taking successive soft limits of external particles is essentially equivalent to removing data that would otherwise be used to construct the integrand.
\section*{Acknowledgments}
It is a pleasure to acknowledge A.~Volovich for many helpful insights.
Also we are grateful to J.~Bourjaily for his helpful insights and for the use of his figures in our paper.
This work was supported in part by US Department of Energy under contract
DE-FG02-91ER40688.
|
{
"timestamp": "2012-04-16T02:00:35",
"yymm": "1203",
"arxiv_id": "1203.1915",
"language": "en",
"url": "https://arxiv.org/abs/1203.1915"
}
|
\section{Introduction}
The question of the microscopic origin of well established Fourier's law of heat conduction has remained unsolved for many decades \cite{giulio,ford,bonetto,lepri,david,dhar08}. The main issue is to find the necessary and sufficient conditions for normal transport. Therefore, it is important to understand basic, simple models, that are expected to embrace important features of complicated, more realistic models. It is believed that for solid systems disorder and anharmonicity play a key role in determining the transport properties. Focusing on the quantum of quantum mechanical systems, this encouraged the development of different formalisms for treating large non-equilibrium quantum systems, such as the Keldysh technique \cite {keldysh}, Landauer-Butikker formalism \cite{landauer}, quantum Langevin equation \cite{qle}, quantization in the Fock space of operators \cite{njp,njp2,pro10,prosel10} giving insight into the region far from equilibrium.
The simplest, but nevertheless non-trivial model in the context of heat transport properties, is the coupled one-dimensional harmonic oscillator chain. Many different methods on modeling the heat reservoirs have been presented. Much effort has been given to solve the classical many-body Langevin equation, or the corresponding master (Liouville) equation. For the homogeneous harmonic oscillator chain size independent heat flux and vanishing temperature gradient have been proven \cite{lieb}. On the other hand, a model dependent scaling of the heat flux has been found for the disordered chain \cite{dharprl}. Dhar and Shastry have derived a generalized quantum Langevin equation by using the Ford-Kac-Mazur formalism \cite{dhar}, and have got similar results as in the classical case. Further, it has been argued in detail that the heat and particle transport properties of the model depend on the spectral properties of the heat bath \cite{dhar08} and boundary conditions of the model, in the case of a gapless harmonic oscillator chain, i.e. with vanishing on-site potential. In Ref. \cite{gaubut07} ordered and disordered one-dimensional harmonic chains within the quantum mechanical Langevin equation have been analyzed. No clear statement on scaling of the heat current in the disordered chain has been given. The connection to the entanglement has been observed as well.
In this paper we consider a different, effective approach of open quantum systems \cite{breuer} in the many-body context, which in the past lead us to interesting and fruitful results in the realm of one-dimensional fermionic models \cite{njp,njp2} and can be used to investigate heat transport properties via the Redfield master equation \cite{saito,satami00}. It has been shown \cite{prosel10} that the Lindblad master equation, a special case of the Redfield master equation, can be solved exactly for quadratic bosonic models. We shall see that similar formalism can be used to get the non-equilibrium stationary state (NESS) of the Redfield master equation for a general quadratic boson Hamiltonian with linear bath operators. We further apply the method to one-dimensional harmonic oscillator chain, which serves as a well known model and can be used to test our approach and possibly give an alternative description of heat baths.
This paper is organized as follows. In section \ref{sec2.0} we review the main concepts of the quantization in the Fock space of operators for bosonic systems \cite{prosel10}. We show that this method can be used to solve the Redfield master equation and derive the Lyapunov equation for the covariance matrix, which determines the NESS. In section \ref{sec3.0} we consider the one-dimensional harmonic oscillator chain model in the context of the Redfield master equation. First, in subsection 3.1 we describe a general method of solution of the open harmonic chain and in subsections 3.2, 3.3 provide two explicit solutions for the homogeneous chain, namely a compact solution for the case of symmetric (left/right) system-bath coupling and ohmic bath correlation function and a recursive perturbative (in the coupling) solution, again for symmetric coupling strengths and a general bath correlation function. In both cases simple expressions for the heat conductivity are obtained. Further, in
subsection 3.4 we introduce disorder and find power law scaling of heat current, that is independent of the fine reservoir properties, i.e. the spectral function of the bath, and the coupling strength.
In section 4 we summarize and conclude.
This paper mainly presents original research results which have not been published before, however we review some material from \cite{prosel10} and \cite{njp,njp2} in order to make the manuscript more smoothly readable and self-contained.
\section{Exact solution of the Redfield equation for quadratic boson system}
\label{sec2.0}
Our aim in this section is to extend operator Fock space quantization for bosonic open quantum system. We focus on the Redfield master equation since the Lindblad case can be considered as a limit of equal-time bath correlation functions similar as in Ref. \cite{njp2}. We shall provide an exact solution of the Liouville master equation in the Redfield form
\begin{eqnarray}
\label{eq:master}
&\frac{{\rm d}}{{\rm d}t}\rho(t)=\hat{\mathcal{L}}\rho(t),\\ \label{eq:Liouville}
&\hat{\mathcal{L}}\rho={\rm i}[\rho,H]+\hat{\mathcal{D}}\rho,\\ \label{eq:dissipator}
&\hat{\mathcal{D}}\rho=\sum_{\mu\nu}\int_0^\infty\mathrm{d}t\Gamma_{\nu,\mu}(\tau)[X_\mu(-\tau)\rho,X_\nu]+{\rm h.c.}
\end{eqnarray}
The Liouvillean (\ref{eq:Liouville}) includes two parts. The first part is the generator of the unitary evolution and is determined by the Hamiltonian $H$. The second part, the dissipator, introduces effectively the influence of infinite large environments (reservoirs or baths) with the correlation functions $\Gamma_{\nu,\mu}(t)$.
We assume a general quadratic Hamiltonian and linear coupling operators
\begin{eqnarray}
\label{eq:H_X_def}
&H=\underline{p}\cdot{\bf P}\underline{p}+\underline{q}\cdot{\bf Q}\underline{q}+\underline{p}\cdot{\bf R}\underline{q}+\underline{q}\cdot{\bf R}\underline{p},\\ \nonumber
&X_{\nu}=\underline{x}_{\nu}^{\rm q}\cdot\underline{q}+\underline{x}_{\nu}^{\rm p}\cdot\underline{p},
\end{eqnarray}
where ${\bf P, Q, R}$ are real, symmetric matrices, $p_j, q_j$ are momentum and coordinate operators, respectively, and $x^{\rm p}_{\nu,j}, x^{\rm q}_{\nu,j}$ are real numbers (coupling amplitudes), which determine the coupling operators $X_\nu$. Greek subscript letters are used to denote different reservoirs and latin subscript letters denote the position in the system (site index $j=1,2,\ldots n$) of size $n$. We shall use this notation throughout the paper, as well as bold latin letters for matrices (${\bf A}$) and the hat for super-operators ($\hat{a}$). In the equation (\ref{eq:H_X_def}) we introduced the following notation of a column vector $\underline{x}=(x_1, x_2,\ldots x_n)^{\rm T}$, where $x$ can represent a number, an operator or a super-operator, which is evident from the context, as well as a dot product that assigns only transposition and {\it not} the complex conjugate of the left operand $\underline{x}\cdot\underline{y}=\sum_{j=1}^n x_jy_j$.
The problem could equally well be formulated in the second quantization picture using the creation and annihilation operators $a^\dag_j, a_j$. It is obvious that the two representations are equivalent, however, using the Hermitian operators results in a real Lyapunov equation for the stationary correlation matrix. This simplifies the algebra and the numerical calculations.
\subsection{Coordinate-momentum picture super-operators}
We conform to Ref. \cite{prosel10}, where the solution of the Lindblad master equation has been found exploiting the algebra in a pair of dual vector spaces $\mathcal{K}$ and $\mathcal{K}'$, where $\mathcal{K}$ contains trace class operators (density matrices) and $\mathcal{K}'$ contains unbounded operators (physical observables) \footnote{For details see Ref. \cite{prosel10}.}. We will adopt the Dirac notation and write an element of $\mathcal{K}'$ as {\it ket} $|\rho\rangle$ and an element of $\mathcal{K}$ as {\it bra} $(A|$, so that their contractions give the expectation value of $A$ for the state $\rho$
\begin{equation}
(A|\rho\rangle={\rm tr}\{ A\rho\}.
\label{eq:inner_prod}
\end{equation}
The basic tool used in this section are left and right multiplication maps over $\mathcal{K}$ defined as
\begin{equation}
\label{eq:mult_maps}
\hat{b}^{\rm L}|\rho\rangle=|b\rho\rangle, \quad \hat{b}^{\rm R}=|\rho b\rangle.
\end{equation}
From the cyclicity of the trace and the definition of the inner product (\ref{eq:inner_prod}) we deduce the action of the maps (\ref{eq:mult_maps}) on the elements of the adjoint space
\begin{equation}
(A|\hat{b}^{\rm L}=(Ab|,\quad (A|\hat{b}^{\rm R}=(bA|.
\end{equation}
Further we define $4n$ maps $\hat{c}_{\nu,j},\hat{c}_{\nu,j}'$ for $j=1,2,\ldots,n$ and $\nu=0,1$
\begin{eqnarray}
\label{eq:def_c}
&\hat{c}_{0,j}=\hat{p}_j^{\rm L}, \quad \hat{c}_{0,j}'={\rm i}(\hat{q}_j^{\rm L}-\hat{q}_j^{\rm R}),\\ \nonumber
&\hat{c}_{1,j}=\hat{q}_j^{\rm R}, \quad \hat{c}_{1,j}'={\rm i}(\hat{p}_j^{\rm R}-\hat{p}_j^{\rm L})
\end{eqnarray}
satisfying almost canonical commutation relations
\begin{eqnarray}
[\hat{c}_{\mu,j},\hat{c}_{\nu,k}]=[\hat{c}_{\mu,j}',\hat{c}_{\nu,k}']=0,\quad [\hat{c}_{\mu,j},\hat{c}_{\nu,k}']=\delta_{j,k}\delta_{\mu,\nu}.
\end{eqnarray}
Note that $\hat{c}_{\mu,j}'\neq\hat{c}_{\mu,j}^\dag$. Another important property is that the maps $\hat{c}'_{\nu,j}$ left-annihilate the identity operator $(1|\hat{c}'_{\nu,j}=1$. We shall now formulate the Liouvillean as a quadratic form in the maps $\hat{c}_{\nu,j}, \hat{c}_{\nu,j}'$ using the inverse relations
\begin{eqnarray}
\label{eq:inv_rel3}
\hat{p}_j^{\rm L}=\hat{c}_{0,j},& \quad \hat{q}_j^{\rm L}=\hat{c}_{1,j}-{\rm i}\hat{c}_{0,j}',\\ \nonumber
\hat{q}_j^{\rm R}=\hat{c}_{1,j},& \quad \hat{p}_j^{\rm R}=\hat{c}_{0,j}-{\rm i}\hat{c}_{1,j}'.
\end{eqnarray}
Let us first rewrite the contribution of the Hamiltonian to the Liouvillean
\begin{eqnarray}
\label{eq:liouv_unit}
&{\rm i}(\hat{H}^{\rm R}-\hat{H}^{\rm L})={\rm i}(\underline{\hat{p}}^{\rm R}\cdot{\bf P}\underline{\hat{p}}^{\rm R}+\underline{\hat{q}}^{\rm R}\cdot{\bf Q}\underline{\hat{q}}^{\rm R}+\underline{\hat{q}}^{\rm R}\cdot{\bf R}\underline{\hat{p}}^{\rm R}+\underline{\hat{p}}^{\rm R}\cdot{\bf R}\underline{\hat{q}}^{\rm R}\\ \nonumber
&~-\underline{\hat{p}}^{\rm L}\cdot{\bf P}\underline{\hat{p}}^{\rm L}-\underline{\hat{q}}^{\rm L}\cdot{\bf Q}\underline{\hat{q}}^{\rm L}-\underline{\hat{p}}^{\rm L}\cdot{\bf R}\underline{\hat{q}}^{\rm L}-\underline{\hat{q}}^{\rm L}\cdot{\bf R}\underline{\hat{p}}^{\rm L})\\ \nonumber
&=2(\underline{\hat{c}}_{1}'\cdot{\bf P}\underline{\hat{c}}_{0}-\underline{\hat{c}}_{0}'\cdot{\bf Q}\underline{\hat{c}}_{1}+\underline{\hat{c}}_{1}'\cdot{\bf R}\underline{\hat{c}}_{1}-\underline{\hat{c}}_{0}'\cdot{\bf R}\underline{\hat{c}}_{0})+{\rm i}(-\underline{\hat{c}}_{1}'\cdot{\bf P}\underline{\hat{c}}_{1}'+\underline{\hat{c}}_{0}'\cdot{\bf Q}\underline{\hat{c}}_{0}').
\end{eqnarray}
Above, anti-Hermitian part of the Liouvillean (\ref{eq:liouv_unit}) left-annihilates the identity operator and is quadratic in the maps $\hat{c}_{\nu,j}, \hat{c}_{\nu,j}'$. We shall shortly see that these characteristics are satisfied for the second, dissipative part of the Liouvillean (\ref{eq:dissipator}) as well. The dissipator can be represented in the following, useful form
\begin{equation}
\label{eq:dissip_def}
\hat{\mathcal{D}}=\sum_{\mu,\nu}\sum_{l,m}^{\{{\rm p,q}\}}\sum_{j,k=1}^nx^m_{\nu,k}\int_0^\infty{\rm d}\tau f^l_{\mu,j}(-\tau)\left( \Gamma^\beta_{\nu,\mu}(\tau)\hat{\mathcal{L}}^{l,m}_{j,k}+\Gamma^{\beta*}_{\nu,\mu}(\tau)\hat{\mathcal{R}}^{l,m}_{j,k}\right).
\end{equation}
We introduced the Heisenberg propagator
\begin{equation}
\label{eq:heiss_prop_def}
\underline{x}_\nu=(\underline{x}_\nu^{\rm p},\underline{x}_\nu^{\rm q}),\quad
\underline{f}_\nu^{\rm a}(t)=\underline{x}_\nu^{\rm a}\cdot\exp({-{\rm i~ad}\,H}t),\quad {\rm a\in\{p,q\},}
\end{equation}
and the fundamental dissipators
\begin{eqnarray}
\hat{\mathcal{L}}^{\rm a,b}_{j,k}|\rho\rangle=|[a_j,\rho b_k]\rangle,\quad \hat{\mathcal{R}}^{\rm a,b}_{j,k}|\rho\rangle=|[b_k\rho, a_j]\rangle,
\quad j,k=1,\ldots n,
\end{eqnarray}
where $a_j = p_j$ if $\rm a = p$ or $a_j = q_j$ if $\rm a = q$ and
$b_j = p_j$ if $\rm b = p$ or $b_j = q_j$ if $\rm b = q$. Expressing the relations (\ref{eq:inv_rel3}) we obtain after some algebra a compact form of the fundamental dissipators
\begin{eqnarray}
\label{eq:fund_dissip}
&\hat{\mathcal{L}}^{\rm q,q}_{j,k}={\rm i}\hat{c}_{0,k}'(\hat{c}_{1,j}-{\rm i}\hat{c}_{0,j}'),\qquad\hat{\mathcal{R}}^{\rm q,q}_{j,k}=-{\rm i}\hat{c}_{0,k}'\hat{c}_{1,j},\\ \nonumber
&\hat{\mathcal{L}}^{\rm p,p}_{j,k}=-{\rm i}\hat{c}_{1,k}'\hat{c}_{0,j},\qquad\hat{\mathcal{R}}^{\rm p,p}_{j,k}={\rm i}\hat{c}_{1,k}'(\hat{c}_{0,j}-{\rm i}\hat{c}_{1,j}'),\\ \nonumber
&\hat{\mathcal{L}}^{\rm p,q}_{j,k}={\rm i}\hat{c}_{0,k}'\hat{c}_{0,j},\qquad\hat{\mathcal{R}}^{\rm p,q}_{j,k}=-{\rm i}\hat{c}_{0,k}'(\hat{c}_{0,j}-{\rm i}\hat{c}_{1,j}'),\\ \nonumber
&\hat{\mathcal{L}}^{\rm q,p}_{j,k}=-{\rm i}\hat{c}_{1,k}'(\hat{c}_{1,j}-{\rm i}\hat{c}_{0,j}'),\qquad\hat{\mathcal{R}}^{\rm q,p}_{j,k}={\rm i}\hat{c}_{1,k}'\hat{c}_{1,j}.
\end{eqnarray}
Let us now rearrange and group the terms in the dissipator (\ref{eq:dissip_def}) according to their meaning. All information about the baths can be encoded in the bath matrices
\begin{eqnarray}
\mathbf{M}^{m,l}=\sum_{\mu,\nu}\int_0^\infty{\rm d}\tau \left(\underline{x}^m_{\nu}\otimes\underline{f}_{\mu}^l(-\tau)\right)\Gamma^\beta_{\nu,\mu}(\tau), \quad l,m\in\{{\rm p,q}\},
\label{eq:bath_matrix_def}
\end{eqnarray}
which simplifies the equation (\ref{eq:dissip_def}) for the dissipator
\begin{eqnarray}
\label{disip_full:eq}
\hat{\mathcal{D}}&=\sum_{l,m}^{\{\rm{p,q}\}}\sum_{j,k=1}^n\bigg( ~M^{m,l}_{k,j}\hat{\mathcal{L}}^{l,m}_{j,k}+M^{m,l*}_{k,j}\hat{\mathcal{R}}^{l,m}_{j,k}\bigg).
\end{eqnarray}
By $(\bullet)^*$ we denote the complex-conjugation.
Plugging the relations (\ref{eq:fund_dissip}) in the equation (\ref{disip_full:eq}) we obtain an apprehensible, quadratic form of the Redfield dissipator
\begin{eqnarray}
\label{eq:dissipator_simpl}
\hat{\mathcal{D}}&=2(-\underline{\hat{c}}_0'\cdot{\bf M}^{\rm q,p}_{\rm i}\underline{\hat{c}}_0-2\underline{\hat{c}}_0'\cdot{\bf M}^{\rm q,q}_{\rm i}\underline{\hat{c}}_1+\underline{\hat{c}}_1'\cdot{\bf M}^{\rm p,p}_{\rm i}\underline{\hat{c}}_0+\underline{\hat{c}}_1'\cdot{\bf M}^{\rm p,q}_{\rm i}\underline{\hat{c}}_1)\\ \nonumber
&~~\,+\underline{\hat{c}}_0'\cdot{\bf M}^{\rm q,q}\underline{\hat{c}}_0'-\underline{\hat{c}}_0'\cdot({\bf M}^{\rm q,p})^*\underline{\hat{c}}_1'-\underline{\hat{c}}_1'\cdot{\bf M}^{\rm p,q}\underline{\hat{c}}_0'+\underline{\hat{c}}_1'\cdot({\bf M}^{\rm p,p})^*\underline{\hat{c}}_1'.
\end{eqnarray}
We shall use the following notation ${\bf M}^{\rm a,b}_{\rm r}={\rm Re}({\bf M}^{\rm a,b})$ and ${\bf M}^{\rm a,b}_{\rm i}={\rm Im}({\bf M}^{\rm a,b})$ for real and imaginary part of the matrices ${\bf M}^{\rm a,b}$, ${\rm a,b}\in \{{\rm p,q}\}$. Inserting now the forms (\ref{eq:liouv_unit}) and (\ref{eq:dissipator_simpl}) in the definition (\ref{eq:Liouville}) and defining the super-operator vector $\underline{\hat{b}}=(\underline{\hat{c}}_0,\underline{\hat{c}}_1,\underline{\hat{c}}_0',\underline{\hat{c}}_1')^{\rm T}$ we arrive at the compact form of the complete Liouvillean
\begin{eqnarray}
\label{eq:lyap}
&\hat{\mathcal{L}}=\hat{\underline{b}}\cdot {\bf S} \hat{\underline{b}}-S_0\hat{\mathds{1}},\\ \nonumber
&{\bf S}=\left(
\begin{array}{cc}
{\bf 0}&{\bf -X} \\
{\bf -X}^{\rm T}&{\bf Y}\\
\end{array}
\right) ,\\ \nonumber
&{\bf X}^{\rm T}=\left(
\begin{array}{cc}
{\bf R}+{\bf M}^{\rm q,p}_{\rm i},&{\bf Q}+{\bf M}^{\rm q,q}_{\rm i}\\
-{\bf P}-{\bf M}^{\rm p,p}_{\rm i},&-{\bf R}-{\bf M}^{\rm p,q}_{\rm i}\\
\end{array}
\right) ,\\ \nonumber
&{\bf Y}=\frac{1}{2}\left(
\begin{array}{cc}
2{\rm i}{\bf Q}+{\bf M}^{\rm q,q}+({\bf M}^{\rm q,q})^{\rm T}, & -({\bf M}^{\rm q,p})^*-({\bf M}^{\rm p,q})^{\rm T}\\
-({\bf M}^{\rm q,p})^\dag-({\bf M}^{\rm p,q}), &-2{\rm i}{\bf P}+ ({\bf M}^{\rm p,p})^*+({\bf M}^{\rm p,p})^{\dag}\\
\end{array}
\right)={\bf Y}^{\rm T} ,\\ \nonumber
&S_0=2{\rm Im}({\rm Tr}\left( {\bf M}^{\rm p,q}-{\bf M}^{\rm q,p} \right)).
\end{eqnarray}
We name the part of the matrix ${{\bf X}}$ arising from the dissipator the coupling matrix and the matrix ${\bf Y}$ the driving matrix. The left stationary state of the Liouville equation (\ref{eq:master}) is simply the identity operator, $(1|\hat{\cal L} = 0$, whereas the right stationary state determines the density matrix
$\rho_{\rm NESS}$ of the non-equilibrium stationary state, $\hat{\cal L}\rho_{\rm NESS}=0$. We now employ the main theorem of Ref. \cite{prosel10}. It states, that the right stationary state of a Liouvillean, which may be represented as a quadratic form in the creation ($\hat{c}_j'$) and annihilation ($\hat{c}_j$) super-operators, that satisfy canonical commutation relations, is determined by the covariance matrix (two point correlation function)
\begin{equation}
\label{eq:corr_def}
Z_{j,k}=(1|\hat{c}_j\hat{c}_k|{\rm NESS}\rangle.
\end{equation}
This correlation function is calculated from the continuous Lyapunov equation
\begin{equation}
\label{eq:lyap_def}
{\bf X}^{\rm T}\bf{Z}+{\bf ZX}={\bf Y}.
\end{equation}
The NESS is a Gaussian state, hence all expectation values can be calculated from the two point correlation function (\ref{eq:corr_def}) using the Wick theorem. In our case the correlation matrix ${\bf Z}$ gives us the momentum-coordinate correlations
\begin{eqnarray}
{\bf Z}=\left(\begin{array}{cc}
{\bf Z}^{\rm p,p},&({\bf Z}^{\rm q,p})^{\rm T}\\
{\bf Z}^{\rm q,p},&{\bf Z}^{\rm q,q}
\end{array}\right), \quad Z^{\rm a,b}_{j,k}={\rm tr}\{a_jb_k\rho_{\rm NESS}\},\quad {\rm a,b\in \{p,q\}}.
\end{eqnarray}
The correlation matrix ${\bf Z}$ is made real by adding the matrix
${\bf Z}_0:=\frac{\rm i}{2}\sigma^{\rm x}\otimes\mathds{1}_n$. The map ${\bf Z}\rightarrow{\bf Z}+{\bf Z}_0$ implies ${\bf Y}\rightarrow{\bf Y}-{\bf X}^{\rm T}{\bf Z}_0-{\bf Z}_0{\bf X}$, therefore we have
\begin{eqnarray}
{\bf Z}=\left(\begin{array}{cc}
{\bf Z}^{\rm p,p},&(\tilde{\bf Z}^{\rm q,p})^{\rm T}\\
\tilde{\bf Z}^{\rm q,p},&{\bf Z}^{\rm q,q}
\end{array}\right),\quad \tilde{\bf Z}^{\rm q,p}={\bf Z}^{\rm q,p}+\frac{\rm i}{2}\mathds{1}_n,\\ \nonumber
{\bf Y}=\frac{1}{2}\left(
\begin{array}{cc}
{\bf M}_{\rm r}^{\rm q,q}+({\bf M}_{\rm r}^{\rm q,q})^{\rm T}, & -{\bf M}^{\rm q,p}_{\rm r}-({\bf M}^{\rm p,q}_{\rm r})^{\rm T}\\
-({\bf M}^{\rm q,p}_{\rm r})^{\rm T}-{\bf M}^{\rm p,q}_{\rm r}, &{\bf M}^{\rm p,p}_{\rm r}+({\bf M}^{\rm p,p}_{\rm r})^{\dag}\\
\end{array}
\right).
\end{eqnarray}
The coupling and the driving matrices are connected to the real and imaginary part of the bath matrices (\ref{eq:bath_matrix_def}), respectively. The Liouville super-operator in Redfield form does not necessary conserve positivity, therefore one should check if the obtained covariance matrix determines a bona fide density matrix. This is done by verifying the positivity of the covariance matrix $C_{j,k}={\rm tr}\{b_jb_k\rho_{\rm NESS}\}$, which is simply an unusual form of the Heisenberg uncertainty relation expressed for the covariance matrix \cite{simon94}. We introduced an operator vector $\underline{b}=(\underline{p},\underline{q})$.
\section{Harmonic oscillator chain}
\label{sec3.0}
In this section we use the derived theory to solve the example of harmonic oscillator chain coupled to heat baths at the ends of the chain. The classical and quantum version of this problem have been studied in the literature \cite{dhar08,gaubut07} using the generalized quantum Langevin equation, derived in Ref. \cite{dhar} utilizing the Ford-Kac-Mazur formalism. A boundary condition dependent scaling of the heat flux for the gapless disordered chain has been observed. In Ref. \cite{gaubut07} the connection between heat current and entanglement has been analyzed. Here we propose a different approach to introduce the effect of the heat baths. We use the slightly modified Redfield dissipator, namely, we extend the lower integration limit in the equation dissipator (\ref{eq:dissip_def}) to $-\infty$, which is the same as neglecting the principal value of the integral. This further simplification is necessary for the Gibbs state to be the stationary state, when all baths have equal temperature. It is not clear under which conditions this approximation is valid, however it is necessary to obtain physically meaningful heat reservoirs (see Ref. \cite{njp2} for more discussion on this issue).
The Hamiltonian of a harmonic chain with nearest neighbor interaction may be expressed as a sum of local Hamiltonians and some boundary Hamiltonian
\begin{equation}
H=\sum_{j=1}^{n-1}H_j+H_{\rm B}.
\end{equation}
The local Hamiltonians $H_j$ contain the contribution from the kinetic energy, on-site potential, and the interaction energy, whereas the boundary Hamiltonian consists of the kinetic and the potential part on the ends of the chain and the boundary condition term
\begin{eqnarray}
&H_j=\frac{1}{4}\left(p_j^{2}+p_{j+1}^{2}+\omega_j^2q_j^{2}+\omega_{j+1}^2q_{j+1}^{2}\right)+\frac{k_j}{2}\left(\frac{q_j}{\sqrt{m_j}}-\frac{q_{j+1}}{\sqrt{m_{j+1}}}\right)^{2},\\ \nonumber
&H_{\rm B}=\frac{1}{4}\left(p_1^{2}+p_{n}^{2}+\omega_1^2q_1^{2}+\omega_{n}^2q_{n}^{2}\right)+\frac{k'_1}{2m_1}q_1^{2}+\frac{k'_n}{2m_n}q_n^{2}.
\end{eqnarray}
Employing a compact matrix notation we get
\begin{eqnarray}
&H=\frac{1}{2}\underline{p}\cdot\underline{p}+\frac{1}{2}\underline{q}\cdot{\bf Q}\underline{q},\\ \nonumber
&{\bf Q}=\left( \begin{array}{cccc}
\frac{k_1'+k_1}{m_1}+\omega_1^2&-\frac{k_1}{\sqrt{m_1m_2}},&\cdots,&0\\
-\frac{k_1}{\sqrt{m_1m_2}},&\frac{k_1+k_2}{m_2}+\omega_2^2,&\ddots,&\vdots\\
\vdots,&\ddots,&\ddots,&-\frac{k_{n-1}}{\sqrt{m_{n-1}m_n}}\\
0,&\cdots,&-\frac{k_{n-1}}{\sqrt{m_{n-1}m_n}},&\frac{k_{n-1}+k_n'}{m_n}+\omega_n^2\\
\end{array}\right).
\end{eqnarray}
We consider a natural choice of the coupling operators
\begin{eqnarray}
\!\!\!\!\!\!\!\!\!\!\!\!X_{\rm L}=\sqrt{\epsilon_{\rm L}}q_1,\quad X_{\rm R}=\sqrt{\epsilon_{\rm R}}q_n, \quad \epsilon_{\rm R}=\frac{\epsilon'_{\rm R}}{m_n},\quad \epsilon_{\rm R}=\frac{\epsilon'_{\rm R}}{m_n}
\end{eqnarray}
and two baths with possible different temperatures and general spectral functions
\begin{equation}
\Gamma^{\beta_j}_{j,k}(\omega)=\delta_{j,k}\frac{{\rm sign}({\omega})|\omega|^\nu}{\exp(\omega\beta_j)-1}.
\end{equation}
Three different types of the heat bath according to the parameter $\nu$ are possible: a) sub-ohmic bath ($\nu<1$), b) ohmic bath ($\nu=1$), and c) super-ohmic bath ($\nu>1$). We use the values $\nu=\frac{1}{2}$ and $\nu=2$ for the sub-ohmic and the super-ohmic case, respectively. The main goal of this section is to examine the behavior of heat current defined by the continuity equation in the bulk \footnote{The reasoning leading to equation (\ref{eq:heat_current}) for the heat current is the same as in \cite{njp2}.}
\begin{eqnarray}
\label{eq:heat_current}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
&0=\frac{\rm d}{{\rm d}t}{\rm tr} (H_j \rho_{\rm NESS})
={\rm tr}\{{\rm i}[H,H_j]\rho_{\rm NESS}\}=J_{j-1,j}+J_{j+1,j},\\ \nonumber
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
&J_{j-1,j}={\rm tr}\left\{{\rm i}[H_{j-1},H_j]\rho_{\rm NESS}\right\}=-\frac{k_{j-1}}{m_{j}}{\rm tr}\left\{\left(p_{j}q_{j}-\frac{\rm i}{2}-p_{j}q_{j-1}\sqrt{\frac{m_j}{m_{j-1}}}\right)\rho_{\rm NESS}\right\},
\end{eqnarray}
where $J_{j-1,j}$ denotes the current from site $j-1$ to site $j$. From the antisymmetry constraint for the correlation matrix $\tilde{\bf Z}^{\rm qp}$, which we shall derive shortly, follow vanishing expectation values $\langle p_jq_j+q_jp_j\rangle=0$. Therefore the equation for the heat current simplifies to
\begin{eqnarray}
J=J_{j-1,j}=\frac{k_{j-1}}{\sqrt{m_{j}m_{j-1}}}{\rm tr}\left\{p_{j}q_{j-1}\rho_{\rm NESS}\right\}.
\end{eqnarray}
In the remaining part of this paper we investigate the problem of heat transport through the harmonic oscillator chain in the framework of third quantization for bosonic Redfield master equation. First, we recover some results for the homogeneous chains, then we focus on the scaling law of the heat current in the disordered chain.
\subsection{Homogeneous chain}
\label{sec3.1}
In this section we study the homogeneous chain, where all coupling constants, masses and on-site potentials are equal; $k_j=k,~m_j=m,~\omega_j=\omega_0$ for all $j$ . Thus, only two parameters determine the behavior of the system, namely the on site potential $\omega_0^2$ and the coupling frequency $\omega_{\rm c}=\sqrt{\frac{k}{m}}$. The Hamiltonian is
\begin{eqnarray}
&H=\sum_{j=1}^n\left(\frac{p_j^2}{2}+\frac{\omega_0^2q_j^2}{2}\right)+\sum_{j=1}^{n-1}\frac{k}{2m}(q_{j}-q_{j+1})^2+\frac{k}{2m}(q_1^2+q_n^2), \\ \nonumber
&H=\frac{1}{2}\underline{p}\cdot\underline{p}+\frac{1}{2}\underline{q}\cdot{\bf Q}\underline{q},\\ \nonumber
&{\bf Q}=\omega^2\left( \begin{array}{cccc}
2&-1,&\cdots,&0\\
-1,&2,&\ddots,&\vdots\\
\vdots,&\ddots,&\ddots,&-1\\
0,&\cdots,&-1,&2\\
\end{array}\right)+\omega_0^2\mathds{I}_n.
\end{eqnarray}
In this section we restrict ourselves to fixed boundary conditions $k'_1=k'_n=k=1$, whereas in later discussion of the disordered chain we refer also to free boundary conditions, $k'_1=k'_n=0$.
The coordinate part of the Hamiltonian is diagonalizable by sine-Fourier transformation
\begin{eqnarray}
&{\bf Q}={\bf U} \Omega {\bf U}^\dag,\\ \nonumber
&U_{i,j}=\sqrt{\frac{2}{n+1}}\sin\left(\frac{ij\pi}{n+1}\right),\quad\Omega_{i,j}=\left(\omega_0^2+2\omega^2-2\omega^2\cos\left(\frac{i\pi}{n+1}\right)\right)\delta_{i,j}.
\end{eqnarray}
Normal coordinate and momentum operators are obtained by the transformations $\underline{q}'=\underline{q}\cdot{\bf U}$, $\underline{p}'=\underline{p}\cdot{\bf U}$. The Hamiltonian is then diagonalized via the usual transformation to creation and annihilation operators $q'_j=(a_j^\dag+a_j)/\sqrt{2\lambda_j}, ~\,p'_j={\rm i}\sqrt{\lambda_j}(a_j^\dag-a_j)/\sqrt{2}.$ We can make this transformations for a general harmonic chain where the columns of the matrix ${\bf U}$ are the right eigenvectors and $\lambda_j$ are the corresponding eigenvalues of the matrix ${\bf Q}$. For the homogeneous chain we have ${\bf U}^\dag={\bf U}^{\rm T}={\bf U}$ and $\lambda_j=\Omega_{j,j}$. The Heisenberg propagator in the normal basis contains a coordinate and a momentum part
\begin{eqnarray}
& q_j'(t)=f^{\rm q'}_j(t)q_j'+f^{\rm p'}_j(t)p_j'\\ \nonumber
& f^{\rm q'}_j=\frac{1}{2}\left({\rm e}^{{\rm i}\omega_j t}+{\rm e}^{-{\rm i}\omega_j t} \right) , \qquad f^{\rm p'}_j=\frac{\rm i}{\omega_j}\left({\rm e}^{-{\rm i}\omega_j t}-{\rm e}^{{\rm i}\omega_j t} \right).
\end{eqnarray}
The time dependent coupling operators in the normal basis are
\begin{eqnarray}
x_j^{\rm L}&=\sqrt{\frac{2\epsilon_{\rm L}}{n+1}}\sin\left(\frac{j\pi}{n+1}\right),\quad
X_{\rm L}(t)&=\underline{x}^{\rm L}\cdot ({\rm diag}(\underline{f}^{\rm q'}(t))\underline{q}'+{\rm diag}(\underline{f}^{\rm p'}(t))\underline{p}'),\\ \nonumber
x_j^{\rm R}&=\sqrt{\frac{2\epsilon_{\rm R}}{n+1}}\sin\left(\frac{j n \pi}{n+1}\right),\quad
X_{\rm R}(t)&=\underline{x}^{\rm R}\cdot ({\rm diag}(\underline{f}^{\rm q'}(t))\underline{q}'+{\rm diag}(\underline{f}^{\rm p'}(t))\underline{p}').
\end{eqnarray}
The problem now is to solve the continuous Lyapunov equation
\begin{eqnarray}
\label{eq:lyapun_chain}
&{\bf X}^{\rm T}{\bf Z}+{\bf ZX}={\bf Y},\\ \nonumber
&{\bf X}^{\rm T}=\left(\begin{array}{cc}
{\bf M}^{\rm q'p'}_{\rm i}, & \frac{{\Omega}}{2}\\
-\frac{\mathds{1}_n}{2},&{\bf 0}
\end{array}\right),\quad
{\bf Y}=\frac{1}{2}\left(\begin{array}{cc}
{\bf M}^{\rm q'q'}_{\rm r}+({\bf M}^{\rm q'q'}_{\rm r})^{\rm T}, & {\bf 0}\\
{\bf 0},&{\bf 0}
\end{array}\right).
\end{eqnarray}
Bath matrices ${\bf M}^{\rm q'q'}, {\bf M}^{\rm q'p'}$ are obtained from the equation (\ref{eq:bath_matrix_def}) applying the Fourier transformation $\Gamma^\beta(\omega)=\frac{1} {2\pi}\int_{-\infty}^{\infty}{\rm d}t\exp(-{\rm i}\omega t)\Gamma^\beta(t)$ and the Kubo-Martin-Schwinger (KMS) condition $\Gamma^{\beta}(-\omega)=\exp(\beta\omega)\Gamma^{\beta}(\omega)$
\small
\begin{eqnarray}
&{\bf M}^{\rm q'q'}=\frac{\pi}{2}\underline{x}^{\rm L}\otimes\underline{x}^{\rm L}{\rm diag}\left(\left(1+{\rm e}^{\beta_{\rm L}\lambda_j} \right)\Gamma^{\beta_{\rm L}}(\lambda_j)\right)+\frac{\pi}{2}\underline{x}^{\rm R}\otimes\underline{x}^{\rm R}{\rm diag}\left(\left(1+{\rm e}^{\beta_{\rm R}\lambda_j} \right)\Gamma^{\beta_{\rm R}}(\lambda_j)\right),\\ \nonumber
&{\bf M}^{\rm q'p'}=\frac{{\rm i}\pi}{2}\underline{x}^{\rm L}\otimes\underline{x}^{\rm L}{\rm diag}\left(\left( \frac{{\rm e}^{\beta_{\rm L}\lambda_j}-1}{\lambda_j}\right)\Gamma^{\beta_{\rm L}}(\lambda_j) \right)+\frac{{\rm i}\pi}{2}\underline{x}^{\rm R}\otimes\underline{x}^{\rm R}{\rm diag}\left(\left( \frac{{\rm e}^{\beta_{\rm R}\lambda_j}-1}{\lambda_j}\right)\Gamma^{\beta_{\rm R}}(\lambda_j) \right).
\end{eqnarray}
\normalsize
In the case of equal temperatures of the baths the Gibbs state should be a stationary state of the modified Redfield master equation. Therefore, it is a good exercise and consistency check to calculate the equilibrium solution (for $\beta_{\rm L}=\beta_{\rm R}=\beta$) of the Lyapunov equation (\ref{eq:lyapun_chain})
\small
\begin{eqnarray}
Z^{\rm p'p'}_{i,j}=\delta_{i,j}\frac{\lambda_j\left(1+{\rm e}^{\beta\lambda_j} \right)}{2\left({\rm e}^{\beta\lambda_j} -1\right)},\quad Z^{\rm q'q'}_{i,j}=\delta_{i,j}\frac{\left(1+{\rm e}^{\beta\lambda_j} \right)}{2\lambda_j\left({\rm e}^{\beta\lambda_j} -1\right)},\quad {\bf Z}^{\rm q'p'}=-\frac{\rm i}{2}\mathds{1}_n,\quad{\bf Z}^{\rm p'q'}=\frac{\rm i}{2}\mathds{1}_n.
\label{eq:gibbs_ness}
\end{eqnarray}
\normalsize
Transforming the last result to the annihilation-creation operator correlation matrix we indeed obtain a well known formula for the occupation number of a harmonic oscillator in the equilibrium (thermal) state at inverse temperature $\beta$
\begin{eqnarray}
Z^{\rm a^\dag a}_{j,k}&=\frac{1}{2}\left( \lambda_j Z^{\rm q'q'}_{j,k}+\frac{Z^{\rm p'p'}_{j,k}}{\lambda_j}+{\rm i}Z^{\rm p'q'}_{j,k}-{\rm i}Z^{\rm q'p'}_{j,k}\right)\\ \nonumber
&=\frac{\delta_{j,k}}{2}\left(\frac{\left(1+{\rm e}^{\beta\lambda_j} \right)}{\left({\rm e}^{\beta\lambda_j} -1\right)}-1\right)=\frac{\delta_{j,k}}{{\rm e}^{\beta\lambda_j} -1}.
\end{eqnarray}
This solution is independent of the coupling operators $X_{\rm L}, X_{\rm R}$ and detailed structure of the bath correlation functions, however, they should satisfy the KMS condition. Therefore, the same correlation matrix is obtained for a general harmonic oscillator chain with heat baths of equal temperatures. We prove that by expanding the Lyapunov equation (\ref{eq:lyapun_chain})
\begin{eqnarray}
\label{eq:lyap2}
&{\bf M}^{\rm q'q'}+({\bf M}^{\rm q'q'})^{\rm T}={\bf M}_{\rm i}^{\rm q'p'}{\bf Z}^{\rm p'p'}+\frac{1}{2}{\Omega}(\tilde{\bf Z}^{\rm q'p'})^{T}+{\bf Z}^{\rm p'p'}({\bf M}^{\rm q'p'}_{\rm i})^{T}+\frac{1}{2}\tilde{\bf Z}^{\rm q'p'}{\Omega},\\ \nonumber
&{\bf 0}={\bf M}^{\rm q'p'}_{\rm i}\tilde{\bf Z}^{\rm q'p'}+\frac{1}{2}{\Omega}{\bf Z}^{\rm q'q'}-\frac{1}{2}{\bf Z}^{\rm p'p'},\\ \nonumber
&{\bf 0}=\tilde{\bf Z}^{\rm q'p'}+(\tilde{\bf Z}^{\rm q'p'})^{\rm T}.
\end{eqnarray}
These equations must be solved consistently with the symmetry conditions $({\bf Z}^{\rm q'q'})^{\rm T}={\bf Z}^{\rm q'q'}$ and $ ({\bf Z}^{\rm p'p'})^{\rm T}={\bf Z}^{\rm p'p'}$. From the last equation (\ref{eq:lyap2}) follows the antisymmetry of the matrix $\tilde{\bf Z}^{\rm q'p'}={\bf Z}^{\rm q'p'}+\frac{\rm i}{2}\mathds{1}$. Further, by simplifying the bath matrices for the case of equal temperatures
\begin{eqnarray}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\bf M}^{\rm q'q'}&=\frac{\pi}{2}\left(\underline{x}^{\rm L}\otimes\underline{x}^{\rm L}+\underline{x}^{\rm R}\otimes\underline{x}^{\rm R}\right){\rm diag}\left(\left(1+{\rm e}^{\beta\lambda_j} \right)\Gamma^{\beta}(\lambda_j)\right),\\ \nonumber
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\bf M}^{\rm q'p'}&=\frac{{\rm i}\pi}{2}\left(\underline{x}^{\rm L}\otimes\underline{x}^{\rm L}+\underline{x}^{\rm R}\otimes\underline{x}^{\rm R}\right){\rm diag}\left(\left( \frac{{\rm e}^{\beta_{\rm R}\lambda_j}-1}{\lambda_j}\right)\Gamma^{\beta_{\rm R}}(\lambda_j) \right)
\end{eqnarray}
it is straightforward to see that correlations (\ref{eq:gibbs_ness}) solve the equation (\ref{eq:lyap2}).
\begin{figure}[h!!!]
\includegraphics[scale=0.67]{prevodnost1.pdf}
\includegraphics[scale=0.67]{prevodnost.pdf}
\includegraphics[scale=0.67]{prevodnost2.pdf}
\caption{Comparison of numerically calculated thermal conductance and analytic solutions in the high temperature (\ref{eq:prevodnost_high}), and the low temperature (\ref{eq:prevodnost_low}) limits and the small coupling limit (\ref{eq:prevodnost_small_eps}) for a homogeneous chain. We remark that analytic solutions for high and low temperature limits are asymptotically correct for all coupling strengths, which can be seen from the middle graph ($\epsilon'=5\times10^{-2}$ and ohmic bath), whereas the small coupling solution is valid for all correlation functions, as shown on the right graph ($\epsilon'=10^{-2}$, sub-ohmic bath). The left graph is plotted for $\epsilon'=10^{-2}$ and ohmic bath. Other model parameters are: $n=20,~ (T_{\rm L} - T_{\rm R})/(T_{\rm L}+T_{\rm R})=0.01,~ \omega_0=10,~ \omega_{\rm c}=2/3,~ \epsilon_{\rm L}'=\epsilon_{\rm R}'=\epsilon$.}
\label{fig:prevodnost}
\end{figure}
\subsection{Homogeneous chain - general solution}
\label{sec3.2}
In this section we provide a general solution for the homogeneous oscillator chain with ohmic bath correlation function $\Gamma^{\beta}(\omega)=\frac{\omega}{\exp(\beta\omega)-1}$. In the case of fixed boundary conditions and equal, symmetric coupling $\epsilon_{\rm L}=\epsilon_{\rm R}=\epsilon$ it is possible to find an exact explicit expression for the correlation matrix. Straightforward but tedious calculation, in which we exploit the particular form of the bath matrices, shows that the solution of equations (\ref{eq:lyap2}), rewritten in real space coordinates, can be sought in the form of the ansatz
\small
\begin{eqnarray}
\label{eq:corr_sol_ohmic}
&\tilde{Z}^{\rm qp}_{i,j}={\rm sign}(i-j)z_{|i-j|},\quad z_n=z_0=0,\\ \nonumber
&Z^{\rm pp}_{i,j}= [{\bf U}{\rm diag}(\underline{\tilde{\Gamma}}^+){\bf U}]_{i,j}+\epsilon \left\{
\begin{array}{cc}
z_{|i+j-2|}-az_{|i+j-1|}+z_{|i+j|}, & \quad \mbox{if $i+j<n+1$}\\
-z_{|2n-i-j|}+az_{|2n-i-j-1|}-z_{|2n-i-j-2|}, & \quad \mbox{if $i+j>n+1$}\\
0,& \quad \mbox{otherwise}\\
\end{array} \right.
\end{eqnarray}
\normalsize
We defined the bath functions
\begin{equation}
\label{eq:ansatz}
\tilde{\Gamma}_j^\pm=\frac{1}{2}\left(\frac{\lambda_j\left(1+{\rm e}^{\beta_{\rm L}\lambda_j} \right)}{\exp(\beta_{\rm L}\lambda_j)-1}\pm\frac{\lambda_j\left(1+{\rm e}^{\beta_{\rm R}\lambda_j} \right)}{\exp(\beta_{\rm R}\lambda_j)-1}\right)
\end{equation}
and the constant $a=2+\omega_0^2/\omega_{\rm c}^2.$ The first part of the correlation matrix ${\bf Z}^{\rm pp}$ in equation (\ref{eq:ansatz}) has the same form as in the equilibrium case, except the temperatures can be different. On the other hand, the second part is calculated from the equations (\ref{eq:lyap2}) and the ansatz for $\tilde{\bf Z}^{qp}$. The Lyapunov equation is now simplified to a linear system of $n-1$ equations for the unknown coefficients $z_1,\ldots z_{n-1}.$
For a large number of oscillators $n\rightarrow\infty$ the covariance matrix is connected to a solution of the continuous fraction equation
\begin{eqnarray}
x=\frac{\tilde{a}+\sqrt{\tilde{a}^2-4}}{2},\quad \tilde{a}=\frac{\omega^2_{\rm c}}{\epsilon^2}+a.
\end{eqnarray}
The heat current $J=\omega^2_{\rm c}z_1$ is then given by
\begin{equation}
z_1=\frac{1}{\epsilon^2x^2}(2 x \phi_1+\phi_2+\phi_3+\ldots \phi_{n-1})=\frac{2(x \phi_1-\phi_1)}{\epsilon^2x^2}+\frac{1}{\epsilon^2x^2}\sum_{k=1}^{n-1}\phi_k,
\end{equation}
where the values $\phi_1,\phi_2,\ldots \phi_n$ are calculated as follows
\begin{eqnarray}
\phi_k=\sum_{j=1}^n\frac{2\epsilon\tilde{\Gamma}^-_j}{n+1}\sin\left(\frac{\pi j}{n+1}\right)\sin\left(\frac{k\pi j}{n+1}\right).
\end{eqnarray}
In the high-temperature limit the expression for the heat current is simplified to
\begin{equation}
J_{\rm high}\approx\frac{\omega_{\rm c}^2}{2x\epsilon}(T_{\rm L}-T_{\rm R}).
\label{eq:prevodnost_low}
\end{equation}
This is an expected behavior since for high temperatures the thermal conductance $G_{\rm th}=J/(T_{\rm L}-T_{\rm R})$ should be independent of the temperature, which follows from classical mechanical considerations. In the low temperature limit we expect that thermal conductance will go to zero. We can simplify the thermal conductance for large on-site potential ($\omega_0\gg\omega_{\rm c}$) and small relative temperature difference $|T_{\rm R}-T_{\rm L}|/(T_{\rm L}+T_{\rm R}) \ll 1$, to
\begin{eqnarray}
G_{\rm th}\approx\frac{2\omega^2_{\rm c}\omega^2_0}{\epsilon x}\frac{{\rm e}^{-2\omega_0/(T_{\rm L}+T_{\rm R})}}{(T_{\rm L}+T_{\rm R})^2}.
\label{eq:prevodnost_high}
\end{eqnarray}
\subsection{Recursive solution}
For a general spectral function of the form $\Gamma^{\beta}(\omega)={\rm sign}(\omega)\frac{|\omega|^{\nu}}{\exp(\beta\omega)-1}$ and equal couplings it is possible to find a recursive solution for the correlation matrix in terms of a perturbative ansatz
\begin{equation}
{\bf Z}^{\rm p'p'}=\sum_{j=0}^\infty\epsilon^{2j}{\bf Z}^{{\rm p'p'}(2j)},\quad {\bf Z}^{\rm q'q'}=\sum_{j=0}^\infty\epsilon^{2j}{\bf Z}^{{\rm q'q'}(2j)},\quad{\bf Z}^{\rm p'p'}=\sum_{j=0}^\infty\epsilon^{2j+1}{\bf Z}^{{\rm p'p'}(2j+1)}.
\end{equation}
The form of the ansatz can be guessed if we rewrite the equations $(\ref{eq:lyap2})$ order by order. Starting from the lowest order we have:
\begin{eqnarray}
&Z^{\rm p'p'(0)}_{j,k}=\frac{\delta_{j,k}}{4|\lambda|^{\nu-1}}\left(\left(1+{\rm e}^{\beta_{\rm L}\lambda_j} \right)\Gamma^{\beta_{\rm L}}(\lambda_j)+\left(1+{\rm e}^{\beta_{\rm R}\lambda_j} \right)\Gamma^{\beta_{\rm R}}(\lambda_j)\right)=\frac{\tilde{\Gamma}^+_j}{2},\\ \nonumber
&Z^{\rm q'q'(0)}_{j,k}=\frac{\delta_{j,k}}{4|\lambda|^{\nu+1}}\left(\left(1+{\rm e}^{\beta_{\rm L}\lambda_j} \right)\Gamma^{\beta_{\rm L}}(\lambda_j)+\left(1+{\rm e}^{\beta_{\rm R}\lambda_j} \right)\Gamma^{\beta_{\rm R}}(\lambda_j)\right)=\frac{\tilde{\Gamma}^+_j}{2\lambda_j^2},\\ \nonumber
&\tilde{Z}^{\rm q'p'(1)}_{j,k}=(1-(-1)^{k+j})\frac{(M^{\rm qq}_{j,k}+M^{\rm qq}_{k,j})}{\lambda^2_j-\lambda^2_k}.
\end{eqnarray}
Each subsequent order is then calculated from the previous one using the recursion relations
\begin{eqnarray}
\label{eq:recursive_solution}
&Z^{\rm q'q'(n)}_{j,k}=\frac{2}{\lambda_k^2-\lambda_j^2}\left\{[{\bf M}^{\rm q'p'}\tilde{\bf Z}^{{\rm q'p'}(n-1)}]_{j,k}-[{\bf M}^{\rm q'p'}\tilde{\bf Z}^{{\rm q'p'}(n-1)}]_{k,j}\right\}, \\ \nonumber
&{\bf Z}^{\rm p'p'(n)}=2{\bf M}^{\rm q'p'}\tilde{\bf Z}^{{\rm q'p'}(n-1)}+{\Omega}{\bf Z}^{{\rm q'q'}(n-1)},\\ \nonumber
&\tilde{Z}_{j,k}^{\rm q'p' (n+1)}=\frac{1-(-1)^{j+k}}{\lambda_j^2-\lambda_k^2} \left\{[{\bf M}^{\rm q'p'}{\bf Z}^{{\rm p'p'}(n)}]_{j,k}+[{\bf M}^{\rm q'p'}{\bf Z}^{{\rm p'p'}(n)}]_{k,j}\right\}.
\end{eqnarray}
We can show that in each order for arbitrary initial condition $\tilde{\bf Z}^{{\rm qp}(n-1)}$ respecting the antisymmetry condition one can calculate the next order of of the correlations ${\bf Z}^{{\rm pp}(n)}$ and ${\bf Z}^{{\rm qq}(n)}$, which also satisfy the symmetry conditions and from which the next correction for the coordinate-momentum part of the correlation matrix can be calculated uniquely. There might be problems on the diagonal due to the difference of the eigenvalues, but the special form of the bath matrix $M^{\rm qp}_{j,k}$, which is nonzero only when the sum $j+k$ is odd, ensures that all diagonal corrections apart from ${\bf Z}^{{\rm pp}(0)}$ and ${\bf Z}^{{\rm qq}(0)}$ vanish. We emphasize that the solution (\ref{eq:recursive_solution}) is usefull only for $\epsilon_{\rm L}=\epsilon_{\rm R}=\epsilon<1$.
In the first order in the coupling $\epsilon$ it is possible to find simple explicit results for the kinetic energy profile and the heat current. We connect the local kinetic energy to the local temperature at the same site
\begin{equation}
T_j=Z^{\rm p,p}_{j,j},
\label{eq:temp}
\end{equation}
which is in principle justified only if the system obeys the condition of local thermal equilibrium. This is definitely not the case for harmonic systems, so our ``temperature" field $T_j$ which is simply defined by (\ref{eq:temp}) should be understood only as a technical term.
We find constant temperature profile for long chains and large on site potential $\omega_0\gg\omega$
\begin{equation}
\!\!\!\!\!\!\!\!\!\!\!\!Z^{\rm p'p'}_{k,k}\approx\frac{\tilde{\Gamma}^+_{n/2}}{4}\approx\frac{1}{4}\left(\frac{\omega_0\left(1+{\rm e}^{\beta_{\rm L}\omega_0} \right)}{\exp(\beta_{\rm L}\omega_0)-1}+\frac{\omega_0\left(1+{\rm e}^{\beta_{\rm R}\omega_0} \right)}{\exp(\beta_{\rm R}\omega_0)-1}\right).
\end{equation}
We get the same form of the matrix $\tilde{\bf Z}^{\rm q'p'}$ as in the ohmic bath case, hence, the heat current in the weak coupling limit is
\begin{equation}
J=\epsilon\sum_{j=1}^n\sin^2\left(\frac{j\pi}{n+1}\right)|\lambda_j|^{\nu-1}\tilde{\Gamma}^-_j.
\end{equation}
In the large on-site potential limit ($\omega_0\gg\omega_{\rm c}$) the current is simplified to
\begin{equation}
J\approx\frac{\epsilon}{2}|\lambda_{n/2}|^{\nu-1}\tilde{\Gamma}^-_{n/2}\approx\frac{\epsilon\omega_0^\nu}{4}\left(\frac{1+{\rm e}^{\beta_{\rm L}\omega_0} }{\exp(\beta_{\rm L}\omega_0)-1}-\frac{1+{\rm e}^{\beta_{\rm R}\omega_0} }{\exp(\beta_{\rm R}\omega_0)-1}\right).
\label{eq:prevodnost_small_eps}
\end{equation}
Comparison of analytical solutions in the low-temperature (\ref{eq:prevodnost_low}) and high-temperature (\ref{eq:prevodnost_high}) limits and small-coupling limit (\ref{eq:prevodnost_small_eps}) with the numerical results for the thermal conductance is shown in Fig. (\ref{fig:prevodnost}).
\begin{figure}[!htb]
\vbox{
\includegraphics[scale=0.66]{scaling_disorder_o10.pdf}
\includegraphics[scale=0.66]{scaling_disorder_o1.pdf}
\includegraphics[scale=0.66]{scaling_disorder_o10_-6.pdf}
}
\caption{Averaged heat current scaling for different bath spectral functions (ohmic, Sub-ohmic, Super-ohmic) and boundary conditions ($k'=0,1$). The first panel (top) is for $\omega_0=10$, the middle panel for $\omega_0=1$, and the bottom panel for $\omega_0=10^{-6}$ (almost gapless). The thermodynamic behavior for nonzero $\omega$ does not depend on the properties of the baths and varying model parameters ($\omega_0, k'$). The estimated asymptotic behavior is $\sim n^{-1.4}$. The statistical error is approximately the size of the symbols. Model parameters: $\delta=0.8,~\omega=2/3,~\epsilon_{\rm L}'=0.3,~\epsilon_{\rm R}'=0.1,~\beta_{\rm L}=0.1,~\beta_{\rm R}=1.0.$}
\label{fig:current_scal_dis}
\end{figure}
\begin{figure}[!htb]
\includegraphics[scale=0.75]{T_profil_disorder.pdf}
\caption{Averaged temperature profile (\ref{eq:temp}) for different temperatures and on site potential and boundary conditions. Average the is made over 1000 realizations.
Model parameters: $\delta=0.3,\omega=2/3,~\epsilon_{\rm L}'=0.3,~\epsilon_{\rm R}'=0.1,~\beta_{\rm R}=1.0$ and ohmic bath spectral function.
}
\label{fig:t_profile}
\end{figure}
\subsection{Disordered chain}
\label{subs:dis_chain}
The homogeneous harmonic chains exhibit ballistic transport. Either one of the two different effects is needed for a sub-ballistic transport, say of a finite (non-zero) temperature gradient and finite (non-infinite) heat conductivity in the thermodynamic limit, namely: the interaction, which causes the dissipation of the phonons, or the disorder, which localizes the phonons. As interaction represents a whole new spectrum of problems and phenomena (see e.g. \cite{prozni10}), we shall focus in this subsection to disorder in harmonic chains. However disorder often leads even to a sub-diffusive (insulating) behavior in the thermodynamic limit.
Disorder can be introduced by (i) randomly changing the masses, (ii) the on-site frequencies, or (iii) the coupling constants. We shall confine ourselves to the first type (i), sometimes referred to as isotopic disorder. The masses are chosen randomly $m_j=m_0(1+\delta \xi_j)$, where $m_0$ denotes the mean mass, $\delta\in[0,1]$ denotes the relative width of the deviation, and $\xi_j$ is a random number uniformly distributed in the interval [-1,1]. In the literature both, the isotopic disorder \cite{dhar08} and the disorder in the coupling constants \cite{gaubut07}, have been studied in the context of quantum Langevin equation. A finite temperature gradient and thermal conductivity in the thermodynamic limit have been observed \cite{gaubut07}.
Here, by employing our formalism, we show some new numerical results which differ from previous numerical conjectures \cite{dhar, dhar08,gaubut07}. Investigating the disordered harmonic oscillator Hamiltonian with free boundary conditions by using the quantum Langevin equation has been found that the thermodynamic behavior of the system depends on the properties of the heat baths \cite{dhar}.
However, using our Redfield operator space formalism we find heat current scaling independent of the bath spectral functions and coupling parameters. Numerical calculation of the heat current for different bath correlation functions, coupling strengths and boundary conditions is presented in Fig. (\ref{fig:current_scal_dis}). Note that in order to obtain reliable data the heat current and the temperature profile are calculated as an average over many realizations of disorder. For the gapped cases, $\omega_0 \neq 0$, we find asymptotically, as $n\to \infty$, anomalous sub-diffusive power law scaling
\begin{equation}
J(n) \propto n^{-\alpha}, \quad {\rm where} \quad \alpha = 1.4.
\end{equation}
On the other hand, data seem less conclusive for the gapless case (in order to avoid numerical instabilities with our method we have chosen $\omega_0 = 10^{-6}$) where the scaling $J(n)$ is either sensitive to the bath correlation function or the accessible system sizes $n$ are still too small, and the behaviors $J(n)$ are not yet asymptotic.
In Fig.~ (\ref{fig:t_profile}) we plot the temperature profiles for the ohmic baths and different on-site petentials and boundary conditions. It is perhaps remarkable to observe non-monotonicities in the profiles which are possible due to absence of local thermal equilibrium.
\section{Summary}
We generalized the method of the bosonic quantization in the Fock space of operators (third quantization) to a general markovian master equation. Lyapunov equation for the coordinate-momentum correlation function (the covariance matrix), which fully determines the non-equilibrium stationary state, has been obtained using the concept of left and right multiplication maps in the operator space.
The formalism that we had derived was used to investigate the heat transport in the one-dimensional quantum harmonic oscillator chain with modified Redfield dissipators. An additional approximation turned out to be necessary in order to obtain consistent heat baths, which yield -- in the case of equal temperatures of the reservoirs -- the correct Gibbs state as the steady state solution of the master equation. For the homogeneous harmonic chain two analytical results for the covariance matrix were derived. The first one, given by equations (\ref{eq:corr_sol_ohmic}), is valid for the ohmic bath and equal, possibly large, strengths of the coupling. The second is given in a perturbative, recursive form (\ref{eq:recursive_solution}) and is valid for general spectral function and again equal couplings. This solution involves complicated expressions for higher orders, therefore it is useful only in the small coupling limit, where already the first correction is adequate. Simplified expressions for the heat conductance and the temperature profile were derived. As expected from the classical results and previous quantum calculations \cite{dhar08} for the ordered chain we obtained vanishing temperature gradient and constant heat current in the thermodynamic limit. In the more interesting disordered case we find heat current scaling independent of the coupling, the bath properties, the on-site potential (as long as the phonon spectrum is gapped), and of the boundary conditions. The asymptotical decrease is faster than $1/n$ (Fourier law), in fact it seems asymptotically $\propto n^{-1.4}$, which corresponds to an insulator.
\section*{Acknowledgements}
We acknowledge useful discussions with Daniel Kosov and financial support by the Programme P1-0044, and the Grant J1-2208, of the Slovenian Research Agency (ARRS).
|
{
"timestamp": "2012-03-19T01:01:16",
"yymm": "1203",
"arxiv_id": "1203.2139",
"language": "en",
"url": "https://arxiv.org/abs/1203.2139"
}
|
\section{Introduction}
It is well known that if $f$ is a convex function on the interval $I=\left[
a,b\right] $ with $a<b$, then
\begin{equation*}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int\limits_{a}^{b}f\left(
x\right) dx\leq \frac{f\left( a\right) +f\left( b\right) }{2},
\end{equation*
which is known as the Hermite-Hadamard inequality for the convex functions.
In \cite{pec2} Pearce et. al. generalized this inequality to $r$-convex
positive function $f$ which defined on an interval $[a,b]$, for all $x,y\in
\lbrack a,b]$ and $t\in \lbrack 0,1]
\begin{equation*}
f\left( tx+(1-t)y\right) \leq \left\{
\begin{array}{ll}
\left( t\left[ f\left( x\right) \right] ^{r}+\left( 1-t\right) \left[
f\left( y\right) \right] ^{r}\right) ^{\frac{1}{r}}, & \text{if }r\neq 0 \\
\left[ f\left( x\right) \right] ^{t}\left[ f\left( y\right) \right] ^{1-t},
& \text{if }r=0
\end{array
\right.
\end{equation*
We have that $0$-convex functions are simply $\log $-convex functions and $1
-convex functions are ordinary convex functions.
Recently, the generalizations of the Hermite-Hadamard's inequality to the
integral power mean of a positive convex function on an interval $[a,b]$,
and to that of a positive $r$-convex function on an interval $[a,b]$ are
obtained by Pearce and Pecaric, and others (see \cite{pec1}-\cite{ngoc}).
For some results related to this classical results, (see\cite{dragomir1}
\cite{dragomir2},\cite{pach},\cite{pec1}$)$ and the references therein.
Dragomir and Mond \cite{dragomir1} proved the following Hermite-Hadamard
type inequalities for the $log$-convex functions:
\begin{eqnarray}
f\left( \frac{a+b}{2}\right) &\leq &\exp \left[ \frac{1}{b-a
\int\limits_{a}^{b}\ln \left[ f\left( x\right) \right] dx\right] \label{z2}
\\
&\leq &\frac{1}{b-a}\int\limits_{a}^{b}G\left( f\left( x\right) ,f\left(
a+b-x\right) \right) dx \notag \\
&\leq &\frac{1}{b-a}\int\limits_{a}^{b}f\left( x\right) dx \notag \\
&\leq &L\left( f\left( a\right) ,f\left( b\right) \right) \notag \\
&\leq &\frac{f\left( a\right) +f\left( b\right) }{2}, \notag
\end{eqnarray
where $G\left( p,q\right) =\sqrt{pq}$ is the geometric mean and $L\left(
p,q\right) =\frac{p-q}{\ln p-\ln q}$ $\left( p\neq q\right) $ is the
logarithmic mean of the positive real numbers $p,q$ $\left( \text{for }p=q
\text{ we put }L\left( p,q\right) =p\right) $.
This paper, except for the introduction, is divided into two sections. In
Section 1, we give the some definitions of $\varphi -$convex functions given
by Noor in \cite{noor1} and \cite{noor5} and we will give a new definition.
By using the new definition is defined in Section 1, we will give the proof
of main theorems in Section 2.
\section{Definitions}
Let $K$ be a nonempty closed set in
\mathbb{R}
^{n}$. Let $f,\varphi :K\rightarrow
\mathbb{R}
$ be continuous functions. First of all, we recall the following well know
results and concepts, which are mainly due to Noor and Noor \cite{noor5} and
Noor \cite{noor1}. In \cite{noor1} and \cite{noor5}, the following new class
of functions are defined by Noor:
\begin{definition}
\label{d1} Let $u\in K$. Then the set $K$ is said to be $\varphi -convex$ at
$u$ with respect to $\varphi $, if
\end{definition}
\begin{equation*}
u+te^{i\varphi }\left( v-u\right) \in K,\text{ }\forall u,v\in K,\text{
t\in \left[ 0,1\right] .
\end{equation*}
\begin{remark}
\label{r1} We would like to mention that the (\ref{d1}) of a $\varphi
-convex $ set has a clear geometric interpretation. This definition
essentially says that there is a path starting from a point $u$ which is
contained in $K$. We do not require that the point $v$ should be one of the
end points of the path. This observation plays an important role in our
analysis. Note that, if we demand that $v$ should be an end point of the
path for every pair of points, $u,v\in K$, then $e^{i\varphi }\left(
v-u\right) =v-u$ if and only if, $\varphi =0$, and consequently $\varphi
-convexity$ reduces to convexity. Thus, it is true that every convex set is
also an $\varphi -convex $ set, but the converse is not necessarily true,
see \cite{noor1},\cite{noor5} and the references therein.
\end{remark}
\begin{definition}
\label{d2} The function $f$ on the $\varphi -convex$ set $K$ is said to be
\varphi -convex$ with respect to $\varphi $, i
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left( 1-t\right)
f\left( u\right) +tf\left( v\right) ,\text{ }\forall u,v\in K,\text{ }t\in
\left[ 0,1\right] .
\end{equation*}
\end{definition}
The function $f$ is said to be $\varphi -concave$ if and only if $-f$ is
\varphi -convex$. Note that every convex function is a $\varphi -convex$
function, but the converse is not true.
\begin{definition}
\label{d3} The function $f$ on the $\varphi -convex$ set $K$ is said to be
logarithmic $\varphi -convex$ with respect to $\varphi $\ , such tha
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left( f\left(
u\right) \right) ^{1-t}\left( f\left( v\right) \right) ^{t},\text{ }u,v\in K
\text{ }t\in \left[ 0,1\right] ,
\end{equation*}
\end{definition}
where $f\left( .\right) >0\left( \text{\cite{noor1},\cite{noor3},\cite{noor5
}\right) .$
Now, we will define a new definition for $\varphi -r-convex$ fonctions as
follows:
\begin{definition}
\label{d4} The positive function $f$ on the $\varphi -r-convex$ set $K$ is
said to be $\varphi -r-convex$ with respect to $\varphi $, i
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left\{
\begin{array}{ll}
\left( \left( 1-t\right) \left[ f\left( u\right) \right] ^{r}+t\left[
f\left( v\right) \right] ^{r}\right) ^{\frac{1}{r}}, & r\neq 0 \\
\left[ f\left( u\right) \right] ^{1-t}\left[ f\left( v\right) \right] ^{t},
& r=0
\end{array
\right.
\end{equation*}
\end{definition}
We have that $\varphi -0-$convex functions are simply logarithmic $\varphi -
convex functions and $\varphi -1-$convex functions are $\varphi -$convex
functions.
From the above definitions, we hav
\begin{eqnarray*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) &\leq &\left( f\left(
u\right) \right) ^{1-t}\left( f\left( v\right) \right) ^{t} \\
&\leq &\left( 1-t\right) f\left( u\right) +tf\left( v\right) \\
&\leq &\max \left\{ f\left( u\right) ,f\left( v\right) \right\} .
\end{eqnarray*
In \cite{noor3}, Noor established following theorem for $\varphi -$convex
functions:
\begin{theorem}
\label{tt1} Let $f:K=\left[ a,a+e^{i\varphi }\left( b-a\right) \right]
\rightarrow \left( 0,\infty \right) $ be a $\varphi -convex$\ function on
the interval of real numbers $K^{0}$ (the interior of $K$) and $a,b\in K^{0}$
with $a<a+e^{i\varphi }\left( b-a\right) $ and $0\leq \varphi \leq \frac{\pi
}{2}$. The
\begin{equation}
f\left( \frac{2a+e^{i\varphi }\left( b-a\right) }{2}\right) \leq \frac{1}
e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi }\left(
b-a\right) }f\left( x\right) dx\leq \frac{f\left( a\right) +f\left( b\right)
}{2}. \label{z3}
\end{equation}
\end{theorem}
The main purpose of this note is to establish new integral inequalities
Hadamard type involving product of two $\varphi -r-convex$ fonctions. Two
refinements of Hadamard's integral inequality for $r$-convex functions
recently established by Ngoc et. al. are shown to be recaptured as special
instances. The method employed in our analysis is based on the basic
properties of logarithms and the application of the well known H\"{o}lder's
integral inequality and Minkowski's integral inequality.
\section{Main Results}
Now, we start with the following our main theorem.
\begin{theorem}
\label{tt2} Let $f:K=\left[ a,a+e^{i\varphi }\left( b-a\right) \right]
\rightarrow \left( 0,\infty \right) $ be $\varphi -r-convex$ functions on
the interval of real numbers $K^{0}$ (the interior of $K$) and $a,b\in K^{0}$
with $a<a+e^{i\varphi }\left( b-a\right) $ and $0\leq \varphi \leq \frac{\pi
}{2}$. Then
\end{theorem}
\begin{equation}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) dx\leq \left( \frac{r}{r+1}\right) ^
\frac{1}{r}}\left( \left[ f\left( a\right) \right] ^{r}+\left[ f\left(
b\right) \right] ^{r}\right) ^{\frac{1}{r}}. \label{z4}
\end{equation}
\begin{proof}
Since $f$ is $\varphi -r-convex$ function and $r\neq 0$, we hav
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left( \left(
1-t\right) \left[ f\left( u\right) \right] ^{r}+t\left[ f\left( v\right)
\right] ^{r}\right) ^{\frac{1}{r}},r\neq 0
\end{equation*
for all $t\in \left[ 0,1\right] $. It is easy to observe tha
\begin{eqnarray}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) dx &=&\int\limits_{0}^{1}f\left(
a+te^{i\varphi }\left( b-a\right) \right) dt \label{z5} \\
&\leq &\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right)
\right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}}dt.
\notag
\end{eqnarray
Using Minkowski's inequality (\ref{z5}), we hav
\begin{eqnarray*}
\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right) \right]
^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}}dt &\leq
\left[ \left( \int\limits_{0}^{1}\left( 1-t\right) ^{\frac{1}{r}}f\left(
a\right) dt\right) ^{r}+\left( \int\limits_{0}^{1}t^{\frac{1}{r}}f\left(
b\right) dt\right) ^{r}\right] ^{\frac{1}{r}} \\
&=&\left( \left( \frac{r}{r+1}\right) \left[ f\left( a\right) \right]
^{r}+\left( \frac{r}{r+1}\right) \left[ f\left( b\right) \right] ^{r}\right)
^{\frac{1}{r}} \\
&=&\left( \frac{r}{r+1}\right) ^{\frac{1}{r}}\left( \left[ f\left( a\right)
\right] ^{r}+\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}}.
\end{eqnarray*
Thus, it is the required inequality in (\ref{z4}). This proof is complete.
\end{proof}
\begin{corollary}
\label{c1} Under the asumptions of Theorem \ref{tt2} with $r=1$, the
following inequality holds:
\end{corollary}
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) dx\leq \frac{f\left( a\right) +f\left(
b\right) }{2}.
\end{equation*}
\begin{theorem}
\label{tt4} Let $f,g:K=\left[ a,a+e^{i\varphi }\left( b-a\right) \right]
\rightarrow \left( 0,\infty \right) $ be $\varphi -r-convex$ and $\varphi
-s-convex$\ functions on the interval of real numbers $K^{0}$ (the interior
of $K$) and $a,b\in K^{0}$ with $a<a+e^{i\varphi }\left( b-a\right) $ and
0\leq \varphi \leq \frac{\pi }{2}$. The
\begin{eqnarray}
&&\frac{2}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx \notag \\
&& \label{16} \\
&\leq &\left( \frac{r}{r+2}\right) \left( \left[ f\left( a\right) \right]
^{r}+\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{2}{r}}+\left( \frac
s}{s+2}\right) \left( \left[ g\left( a\right) \right] ^{s}+\left[ g\left(
b\right) \right] ^{s}\right) ^{\frac{2}{s}} \notag
\end{eqnarray
an
\begin{eqnarray}
&&\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx \notag \\
&& \label{160} \\
&\leq &\left( \frac{rs}{\left( r+2\right) \left( s+2\right) }\right) ^{\frac
1}{2}}\left( \left[ f\left( a\right) \right] ^{r}+\left[ f\left( b\right)
\right] ^{r}\right) ^{\frac{2}{r}}\left( \left[ g\left( a\right) \right]
^{s}+\left[ g\left( b\right) \right] ^{s}\right) ^{\frac{2}{s}}. \notag
\end{eqnarray}
\end{theorem}
\begin{proof}
Since $f$ is $\varphi -r-convex$ function and $g$ is $\varphi -s-convex$
function ($r>0,s>0$), then we hav
\begin{equation}
f\left( a+te^{i\varphi }\left( b-a\right) \right) \leq \left( \left(
1-t\right) \left[ f\left( a\right) \right] ^{r}+t\left[ f\left( b\right)
\right] ^{r}\right) ^{\frac{1}{r}} \label{17}
\end{equation}
\begin{equation}
g\left( a+te^{i\varphi }\left( b-a\right) \right) \leq \left( \left(
1-t\right) \left[ f\left( a\right) \right] ^{r}+t\left[ f\left( b\right)
\right] ^{r}\right) ^{\frac{1}{r}}. \label{18}
\end{equation
Multiplying both sides of (\ref{17}) by (\ref{18}), it follows tha
\begin{eqnarray}
&&f\left( a+te^{i\varphi }\left( b-a\right) \right) g\left( a+te^{i\varphi
}\left( b-a\right) \right) \label{19} \\
&\leq &\left( \left( 1-t\right) \left[ f\left( a\right) \right] ^{r}+t\left[
f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}}\left( \left( 1-t\right)
\left[ g\left( a\right) \right] ^{s}+t\left[ g\left( b\right) \right]
^{s}\right) ^{\frac{1}{s}}. \notag
\end{eqnarray
Integrating the inequality (\ref{19}) with respect to $t$ over $\left[ 0,
\right] $,\ we obtai
\begin{eqnarray}
&&\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx \label{20} \\
&\leq &\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right)
\right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r
}\left( \left( 1-t\right) \left[ g\left( a\right) \right] ^{s}+t\left[
g\left( b\right) \right] ^{s}\right) ^{\frac{1}{s}}dt. \notag
\end{eqnarray
Using Cauchy Swartz's inequality, we hav
\begin{eqnarray}
&&\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right) \right]
^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}}\left(
\left( 1-t\right) \left[ g\left( a\right) \right] ^{s}+t\left[ g\left(
b\right) \right] ^{s}\right) ^{\frac{1}{s}}dt \label{21} \\
&\leq &\left( \int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left(
a\right) \right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{
}{r}}dt\right) ^{\frac{1}{2}}\left( \int\limits_{0}^{1}\left( \left(
1-t\right) \left[ g\left( a\right) \right] ^{s}+t\left[ g\left( b\right)
\right] ^{s}\right) ^{\frac{1}{s}}dt\right) ^{\frac{1}{2}} \notag
\end{eqnarray
Using Young's inequality($2ab\leq a^{2}+b^{2}$) for right-hand side of the
inequality (\ref{21}), we hav
\begin{eqnarray}
&&\left( \int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right)
\right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{2}{r
}dt\right) ^{\frac{1}{2}}\left( \int\limits_{0}^{1}\left( \left( 1-t\right)
\left[ g\left( a\right) \right] ^{s}+t\left[ g\left( b\right) \right]
^{s}\right) ^{\frac{1}{s}}dt\right) ^{\frac{1}{2}} \label{22} \\
&\leq &\frac{1}{2}\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left(
a\right) \right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{
}{r}}dt+\frac{1}{2}\int\limits_{0}^{1}\left( \left( 1-t\right) \left[
g\left( a\right) \right] ^{s}+t\left[ g\left( b\right) \right] ^{s}\right) ^
\frac{2}{s}}dt. \notag
\end{eqnarray
Using Minkowski's inequality right-hand side of the inequality (\ref{22}),
we hav
\begin{eqnarray}
&&\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right) \right]
^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{2}{r}}dt
\label{23} \\
&\leq &\left[ \left( \int\limits_{0}^{1}\left( 1-t\right) ^{\frac{2}{r}
\left[ f\left( a\right) \right] ^{2}dt\right) ^{\frac{r}{2}}+\left(
\int\limits_{0}^{1}t^{\frac{2}{r}}\left[ f\left( b\right) \right]
^{2}dt\right) ^{\frac{r}{2}}\right] ^{\frac{2}{r}} \notag \\
&=&\left( \frac{r}{r+2}\right) \left( \left[ f\left( a\right) \right] ^{r}
\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{2}{r}}. \notag
\end{eqnarray
Similarly we have
\begin{equation}
\int\limits_{0}^{1}\left( \left( 1-t\right) \left[ g\left( a\right) \right]
^{s}+t\left[ g\left( b\right) \right] ^{s}\right) ^{\frac{2}{s}}dt\leq
\left( \frac{s}{s+2}\right) \left( \left[ g\left( a\right) \right] ^{s}
\left[ g\left( b\right) \right] ^{s}\right) ^{\frac{2}{s}}. \label{240}
\end{equation
Adding (\ref{23}) and (\ref{240}) and rewriting (\ref{20}), we obtain (\re
{16}).
Now, using Minkowski's inequality for right-hand side of the inequality (\re
{21}), we hav
\begin{eqnarray}
&&\left( \int\limits_{0}^{1}\left( \left( 1-t\right) \left[ f\left( a\right)
\right] ^{r}+t\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{2}{r
}dt\right) ^{\frac{1}{2}} \label{25} \\
&\leq &\left[ \left( \int\limits_{0}^{1}\left( 1-t\right) ^{\frac{2}{r}
\left[ f\left( a\right) \right] ^{2}dt\right) ^{\frac{r}{2}}+\left(
\int\limits_{0}^{1}t^{\frac{2}{r}}\left[ f\left( b\right) \right]
^{2}dt\right) ^{\frac{r}{2}}\right] ^{\frac{1}{r}} \notag \\
&=&\left( \frac{r}{r+2}\right) ^{\frac{1}{2}}\left( \left[ f\left( a\right)
\right] ^{r}+\left[ f\left( b\right) \right] ^{r}\right) ^{\frac{1}{r}},
\notag
\end{eqnarray
and similarl
\begin{equation}
\left( \int\limits_{0}^{1}\left( \left( 1-t\right) \left[ g\left( a\right)
\right] ^{s}+t\left[ g\left( b\right) \right] ^{s}\right) ^{\frac{1}{s
}dt\right) ^{\frac{1}{2}}\leq \left( \frac{s}{s+2}\right) ^{\frac{1}{2
}\left( \left[ g\left( a\right) \right] ^{s}+\left[ g\left( b\right) \right]
^{s}\right) ^{\frac{1}{s}}. \label{26}
\end{equation
Writing (\ref{25}) and (\ref{26}) in (\ref{21}), and rewriting (\ref{20}),
we get the desired inequality in (\ref{160}). The proof is complete.
\end{proof}
\begin{corollary}
\label{c2} Under the asumptions of Theorem \ref{tt4} and with $s=r=1$ we hav
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx\leq \frac{\left(
\left[ f\left( a\right) \right] +\left[ f\left( b\right) \right] \right)
+\left( \left[ g\left( a\right) \right] +\left[ g\left( b\right) \right]
\right) }{6}
\end{equation*
an
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx\leq \frac{\left(
\left[ f\left( a\right) \right] +\left[ f\left( b\right) \right] \right)
^{2}\left( \left[ g\left( a\right) \right] +\left[ g\left( b\right) \right]
\right) ^{2}}{3}.
\end{equation*}
\end{corollary}
\begin{corollary}
\label{c3} Under the asumptions of Theorem \ref{tt4} and with $s=r$ and
f\left( x\right) =g\left( x\right) $, we hav
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f^{2}\left( x\right) dx\leq \left( \frac{r}{r+2}\right)
\left( \left[ f\left( a\right) \right] ^{r}+\left[ f\left( b\right) \right]
^{r}\right) ^{\frac{2}{r}}
\end{equation*
an
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f^{2}\left( x\right) dx\leq \left( \frac{r}{r+2}\right)
\left( \left[ f\left( a\right) \right] ^{r}+\left[ f\left( b\right) \right]
^{r}\right) ^{\frac{4}{r}}.
\end{equation*}
\end{corollary}
\begin{remark}
If we take $g\left( x\right) =1$ in Corollary \ref{c2} we hav
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) dx\leq \frac{\left( \left[ f\left(
a\right) \right] +\left[ f\left( b\right) \right] \right) +2}{6}
\end{equation*
an
\begin{equation*}
\frac{1}{e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi
}\left( b-a\right) }f\left( x\right) g\left( x\right) dx\leq \frac{4\left(
\left[ f\left( a\right) \right] +\left[ f\left( b\right) \right] \right) ^{2
}{3}.
\end{equation*}
\end{remark}
|
{
"timestamp": "2012-03-13T01:01:29",
"yymm": "1203",
"arxiv_id": "1203.2278",
"language": "en",
"url": "https://arxiv.org/abs/1203.2278"
}
|
\section{Introduction}
The r$\hat{\mathrm{o}}$le of the $\Lambda$ hyperon in stabilizing nuclear
cores was pointed out long ago by Dalitz and Levi Setti \cite{dalitz_setti} as
part of a discussion focusing on light hypernuclei with large neutron excess.
This property is demonstrated by the observation of $^{6}_{\Lambda}$He,
$^{7}_{\Lambda}$Be, $^{8}_{\Lambda}$He, $^{9}_{\Lambda}$Be and
$^{10}_{~\Lambda}$B hypernuclei in emulsion experiments \cite{juric}.
No unstable-core hydrogen $\Lambda$ hypernuclei have been established so far,
although the existence of the lightest possible one $^{6}_{\Lambda}$H was
predicted by Dalitz and Levi Setti \cite{dalitz_setti} and subsequently
reinforced in estimates by Majling \cite{majling}. The neutral-baryon
excess in $^{6}_{\Lambda}$H, in particular, would be $(N+Y)/Z=5$, with $Y=1$
for a $\Lambda$ hyperon, larger than the maximal value in light nuclei,
$N/Z=3$ for $^{8}$He \cite{tilley04}. Neutron-rich light hypernuclei could
thus go beyond the neutron drip line for ordinary nuclear systems.
Two-body reactions in which neutron rich hypernuclei could be produced
are the following double charge-exchange reactions:
\begin{equation}
K^{-} + {^{A}Z} \rightarrow {^{A}_{\Lambda}(Z-2)} + \pi^{+},
\label{prodK}
\end{equation}
induced on nuclear targets by stopped $K^{-}$ mesons or in flight, and
\begin{equation}
\pi^{-} + {^{A}Z} \rightarrow {^{A}_{\Lambda}(Z-2)} + K^{+}
\label{prodpi}
\end{equation}
with $\pi^{-}$ mesons in flight ($p_{\pi^{-}} > 0.89$ GeV/c).
The simplest description of the above reactions is a two-step process on
two different protons of the same nucleus, converting them into a neutron
and a $\Lambda$, with the additional condition that the final nuclear system
is bound. For (\ref{prodK}) it amounts to $K^{-}p\rightarrow\Lambda\pi^{0}$
reaction followed by $\pi^{0}p\rightarrow n\pi^{+}$ or $K^{-}p\rightarrow
{\bar K^0}n$ followed by ${\bar K^0}p\rightarrow\Lambda\pi^{+}$,
for (\ref{prodpi}) to a $\pi^{-}p\rightarrow n\pi^{0}$ reaction followed by
$\pi^{0}p \rightarrow K^{+}\Lambda$ or $\pi^{-}p\rightarrow K^{0}\Lambda$
followed by $K^{0}p\rightarrow K^{+}n$. Another mechanism is a single-step
double charge exchange $m_i^{-}p\rightarrow\Sigma^{-}m_f^{+}$ (where $m$
stands for meson) feeding the $\Sigma$ component coherently admixed into the
final $\Lambda$ hypernuclear state. Such admixtures are essentially equivalent
to invoking a second step of $\Sigma^{-}p\rightarrow\Lambda n$ conversion.
These two-step processes are expected to occur at a much lower rate (reduction
factor $\leq 10^{-2}$ \cite{chrien}) than the production of normal $\Lambda$
hypernuclei by means of the corresponding single-step two-body reactions
($K^{-},\pi^{-}$) and ($\pi^{+},K^{+}$).
The first experimental attempt to produce neutron-rich hypernuclei by
the reaction (\ref{prodK}) with $K^{-}$ at rest was carried out at
KEK \cite{kubota}. Upper limits were obtained for the production of
$^{9}_{\Lambda}$He, $^{12}_{~\Lambda}$Be and $^{16}_{~\Lambda}$C hypernuclei
(on $^{9}$Be, $^{12}$C and $^{16}$O targets respectively) in the range of
$(0.6-2.0)\cdot 10^{-4}/K^{-}_{\rm stop}$, while the theoretical predictions
for $^{12}_{~\Lambda}$Be and $^{16}_{~\Lambda}$C \cite{tretyak01} lie in the
interval $(10^{-6}-10^{-7})/K^{-}_{\rm stop}$, which is at least one order
of magnitude lower than the experimental upper limits and three orders of
magnitude smaller than the standard one-step ($K^{-}_{\rm stop},\pi^{-}$)
reaction rates on the same targets ($10^{-3}/K^{-}_{\rm stop}$).
Another KEK experiment \cite{saha} reported the observation of
$^{10}_{~\Lambda}$Li in the ($\pi^{-},K^{+}$) reaction on a $^{10}$B target
with a 1.2 GeV/c $\pi^{-}$ beam. A production cross section of $11.3\pm 1.9$
nb/sr was evaluated; the result, however, is not directly comparable with
theoretical calculations \cite{tretyak03} since no discrete structure was
observed and the production cross section was integrated over the whole bound
region ($0<B_{\Lambda}<20$ MeV).
A further attempt to observe neutron-rich hypernuclei by means of the
reaction (\ref{prodK}), with $K^{-}$ at rest, was made at the DA$\Phi$NE
collider at LNF by the FINUDA experiment \cite{nrich1}, on $^{6}$Li and
$^{7}$Li targets. The limited data sample collected during the first run
period of the experiment was used to estimate the production rates per
stopped $K^{-}$ of $^{6}_{\Lambda}$H and $^{7}_{\Lambda}$H. The inclusive
$\pi^{+}$ spectra from $^{6}$Li and $^{7}$Li targets were analyzed in
momentum regions corresponding, through momentum and energy conservation,
to $B_{\Lambda}$ values discussed in the literature. Because of the dominant
contribution of the reactions
\begin{eqnarray}
K^{-}_{\rm stop} + p & \rightarrow & \Sigma^{+} + \pi^{-} \nonumber \\
& & \hookrightarrow n + \pi^{+} \ \ \
(\sim 130 < p_{\pi^{+}} < 250\ {\rm MeV/c})
\label{Kp}
\end{eqnarray}
and
\begin{eqnarray}
K^{-}_{\rm stop} + p p & \rightarrow & \Sigma^{+} + n \nonumber \\
& & \hookrightarrow n + \pi^{+} \ \ \
(\sim 100 < p_{\pi^{+}} < 320\ {\rm MeV/c}),
\label{Kpp}
\end{eqnarray}
which give the main component of the inclusive $\pi^{+}$ spectra for
absorption of stopped $K^{-}$ mesons on nuclei, and owing to a limited
statistics, only upper limits could be evaluated for $\Lambda$ hypernuclear
production:
\begin{eqnarray}
R_{\pi^{+}}({^{6}_{\Lambda}{\rm H}}) & < & (2.5 \pm
{0.4_{\rm stat}}^{+0.4}_{-0.1{\rm syst}})\cdot 10^{-5}/K^{-}_{\rm stop},
\label{Rpi+6} \\
R_{\pi^{+}}({^{7}_{\Lambda}{\rm H}}) & < &(4.5 \pm
{0.9_{\rm stat}}^{+0.4}_{-0.1{\rm syst}})\cdot 10^{-5}/K^{-}_{\rm stop},
\label{Rpi+7} \\
\nonumber
\end{eqnarray}
in addition to an upper limit determined in $^{12}$C:
\begin{eqnarray}
R_{\pi^{+}}({^{12}_{~\Lambda}{\rm Be}}) & < & (2.0 \pm
{0.4_{\rm stat}}^{+0.3}_{-0.1{\rm syst}})\cdot 10^{-5}/K^{-}_{\rm stop},
\label{Rpi+12}
\end{eqnarray}
lowering by a factor $\sim 3$ the previous KEK determination \cite{kubota}.
In this article we present the analysis of the total data sample
of the FINUDA experiment, collected from 2003 to 2007 and corresponding to
a total integrated luminosity of 1156 pb$^{-1}$, aiming at assessing the
existence of $^{6}_{\Lambda}$H and determining the production rate by means
of the ($K^{-}_{\rm stop},\pi^{+}$) reaction on $^{6}$Li targets.
A preliminary account of the results, reporting three clear events of
$^{6}_{\Lambda}$H, appeared in \cite{PRL}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=90mm]{schema_BL_NPA3.pdf}
\caption{Left: Binding energy scheme for a system of one proton, four neutrons
and one $\Lambda$ relative to the summed mass of $^{5}{\rm H}+\Lambda$,
$M=5805.44$ MeV, with the (blue in the web version) hatched box denoting the width of $^{5}$H
\cite{korshen}. The two lowest horizontal lines stand for predictions from
Refs.~\cite{dalitz_setti,akaishi}. Right: mean value of the
$^{6}_{\Lambda}{\rm H}$ g.s. mass obtained in the present analysis jointly
from production and decay; the (red in the web version) shaded box represents the error on the
mass mean value obtained from the three $^{6}_{\Lambda}{\rm H}$ events
reported here.}
\label{fig6}
\end{center}
\end{figure}
The binding energy of $^{6}_{\Lambda}$H with respect to the unstable $^{5}$H
core was estimated in Refs.~\cite{dalitz_setti,majling} as $B_{\Lambda}=4.2$
MeV, making $^{6}_{\Lambda}$H particle stable with respect to its
$^{4}_{\Lambda}{\rm H}+2n$ lowest threshold, as shown in Fig.~\ref{fig6}.
We recall that the binding energy $B_{\Lambda}$ of hypernucleus
$^{A}_{\Lambda}$Z is defined as:
\begin{equation}
B_{\Lambda} = M_{\rm core} + M_{\Lambda} - M_{^{A}_{\Lambda}{\rm Z}},
\label{blambda}
\end{equation}
where $M_{\rm core}$ is the mass of the $^{(A-1)}$Z core nucleus in its
ground state (g.s.), as deduced from the atomic mass tables \cite{wapstra}.
The $^{5}$H nuclear core, colloquially termed ``superheavy hydrogen", was
observed as a broad resonance (1.9 MeV FWHM) at energy about 1.7 MeV above
the $^{3}{\rm H}+2n$ threshold \cite{korshen}. A substantially stronger
binding, $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})=5.8$ MeV, was predicted by
Akaishi et al. \cite{akaishi} for the $0^+$ g.s. on the basis of a coherent
$\Lambda N - \Sigma N$ mixing model originally practised for the
$^{4}_{\Lambda}$H cluster \cite{akaishi00}. This coherent $\Lambda N-\Sigma N$
mixing induces a spin-dependent $\Lambda NN$ three-body interaction which
affects primarily the $0^+$ g.s., increasing thus the $\approx$ 1 MeV $1^+$
excitation expected from $^{4}_{\Lambda}$H to 2.4 MeV in $^{6}_{\Lambda}$H.
If this prediction is respected by Nature, it could imply far-reaching
consequences to strange dense stellar matter.
In the next sections we describe briefly the FINUDA experimental apparatus,
and the analysis technique applied to the data collected on $^{6}$Li targets.
We then report on three $^{6}_{\Lambda}$H candidate events found by observing
$\pi^+$ mesons from production and $\pi^-$ mesons from decay in coincidence.
These events prove robust against varying the cuts selected in the analysis,
and give evidence for a particle stable $^{6}_{\Lambda}$H. The measurement
background is evaluated and the production rate of $^{6}_{\Lambda}$H is
estimated. We end with a brief discussion of the $^{6}_{\Lambda}$H excitation
spectrum as constrained by the three candidate events.
\section{Experimental apparatus}
FINUDA was a hypernuclear physics experiment installed at one of the two
interaction regions of the DA$\Phi$NE $e^{+}e^{-}$ collider, the INFN-LNF
$\Phi$(1020)-factory. A detailed description of the experimental apparatus
can be found in Ref.~\cite{fnd}. The layout figured a cylindrical symmetry
arrangement; here we briefly sketch its main components moving outwards
from the beam axis: the {\it interaction/target region}, composed by a barrel
of 12 thin scintillator slabs (TOFINO), surrounded by an octagonal array of
$\mathrm{Si}$ microstrips (ISIM) facing eight target tiles; the {\it tracking
device}, consisting of four layers of position sensitive detectors
(a decagonal array of $\mathrm{Si}$ microstrips (OSIM), two octagonal layers
of low mass drift chambers (LMDC) and a stereo system of straw tubes (ST))
arranged in coaxial geometry; the {\it external time of flight detector}
(TOFONE), a barrel of 72 scintillator slabs. The whole apparatus was placed
inside a uniform 1.0 T solenoidal magnetic field; the tracking volume was
immersed in $\mathrm{He}$ atmosphere to minimize the multiple scattering
effect.
The main features of the apparatus were the thinness of the target
materials needed to stop the low energy ($\sim$ 16 MeV) $K^{-}$'s from
the $\Phi\rightarrow K^{-}K^{+}$ decay channel, the high transparency of
the FINUDA tracker and the very large solid angle ($\sim 2\pi$ sr) covered
by the detector ensemble; accordingly, the FINUDA apparatus was suitable
to study simultaneously the formation and the decay of $\Lambda$ hypernuclei
by means of high resolution magnetic spectroscopy of the emitted charged
particles.
In particular, for $\pi^{+}$ with momentum $\sim 250$ MeV/c the resolution
of the tracker can be evaluated by measuring the width of the momentum
distribution of the monochromatic (235.6 MeV/c) $\mu^{+}$ coming from the
$K_{\mu 2}$ decay channel; for reactions occurring in the apparatus sector
where $^{6}$Li targets were located, it is $\sigma _{p} = (1.1 \pm 0.1)$
MeV/c \cite{spectrFND}; the precision on the absolute momentum calibration,
obtained from the mean value of the same distribution, is better than 0.12
MeV/c for the $^{6}$Li targets, which corresponds to a maximum systematic
uncertainty in the kinetic energy $\sigma_{T sys}(\pi^{+}) = 0.1$ MeV.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=90mm]{pim_mom_133_3.pdf}
\caption{Distribution of low momentum $\pi^{-}$ from $^{6}$Li targets.
The continuous black curve represents a fit the the spectrum given by
the sum of a second degree polynomial [dashed (red in the web version) curve] and a gaussian
function [dot-dashed (blue in the web version) curve]. For more details, see text.}
\label{fig1}
\end{center}
\end{figure}
For $\pi^{-}$ with momentum $\sim 130$ MeV/c the resolution and absolute
calibration can be evaluated from the momentum distribution of the
monochromatic $\pi^{-}$ coming from the two-body mesonic weak decay of
$^{4}_{\Lambda}$H, produced as hyperfragment with a formation probability
of the order of $10^{-3}-10^{-2}$ per stopped $K^{-}$ \cite{tamura_fr}.
Figure~\ref{fig1} shows the distribution for low momentum $\pi^{-}$ from
$^{6}$Li targets, before acceptance correction; the spectrum is fitted in
the 120--140 MeV/c momentum range (continuous black curve) with the sum of
a second degree polynomial, representing the background from quasi-free
$\Lambda$ decay and quasi-free $\Sigma^{+}$ production (dashed (red in the web version) curve
in the figure), and a gaussian function representing the $^{4}_{\Lambda}$H
mesonic decay contribution (dot-dashed (blue in the web version) curve); the fit gives
a $\chi^{2}/{\rm ndf}=79.1/74$, a mean $\mu_{p}=(132.6\pm 0.1)$ MeV/c and
a standard deviation $\sigma _{p}= (1.2 \pm 0.1)$ MeV/c for the gaussian
function, directly measuring the experimental resolution.
For comparison, $p_{\pi^-}=(132.80\pm 0.08)$ MeV/c from
$B_{\Lambda}({_{\Lambda}^{4}{\rm H}})=2.04\pm 0.04$ MeV,
as determined from emulsion studies \cite{juric}; hence the absolute uncertainty is 0.2 MeV/c.
and the corresponding systematic uncertainty in the kinetic energy is
then $\sigma_{T sys}(\pi^{-})=0.14$ MeV.
To perform particle identification, the information of the specific energy
loss in both OSIM and the LMDC's is used; the mass identification from the
time of flight system (TOFINO-TOFONE) for high momentum tracks is also used.
The final selection is performed by requiring the same identification from
at least two different detectors.
\section{Analysis technique}
In the second data taking the statistics collected with $^{6}$Li targets was
improved by a factor 5 with respect to the first run. However, even with the
improved statistics, we could not observe in the inclusive $\pi^{+}$ spectra
clear peaks that could be attributed to the two-body reaction:
\begin{equation}
K^{-}_{\rm stop}+{^{6}{\rm Li}}\rightarrow {^{6}_{\Lambda}{\rm H}}+\pi^{+}\ \ \
(p_{\pi^{+}}\sim 252\ \mathrm{MeV/c}).
\label{nrich6LH}
\end{equation}
Exploiting the increased statistics, we tried then to reduce the background
overwhelming the events from reaction (\ref{nrich6LH}) by examining the
spectra of $\pi^{+}$ in coincidence with the $\pi^{-}$ coming from the mesonic
decay of $^{6}_{\Lambda}$H:
\begin{equation}
{^{6}_{\Lambda}{\rm H}} \rightarrow {^{6}{\rm He}} + \pi^{-} \ \ \
(p_{\pi^{-}}\sim 130-140\ \mathrm{MeV/c}).
\label{6LHmwd}
\end{equation}
The branching ratio for (\ref{6LHmwd}) is expected to be about 50$\%$
taking into account the value measured for the analogous decay
${^{4}_{\Lambda}{\rm H}}\rightarrow {^{4}{\rm He}}+\pi^{-}$ \cite{tamura_fr}.
($\pi^{+}$, $\pi^{-}$) coincidence events, associated with $K^{-}$'s
stopped in the $^{6}$Li targets, were thus considered; only reaction
(\ref{Kp}) contributes to the background of this sample.
We examined thus the two-dimensional raw spectrum of $\pi^{+}$ versus
$\pi^{-}$ momentum, shown in Fig.~\ref{fig2}, in order to recognize
possible enhancements due to occurrence of the reactions (\ref{nrich6LH})
and (\ref{6LHmwd}) in sequence. The low statistics and the strong background
prevented us from finding statistically significant accumulations of events
in the plot arising from a bound $^{6}_{\Lambda}$H.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=90mm]{pim_pip_2d_cut_tot_A.pdf}
\caption{$\pi^{+}$ momentum vs $\pi^{-}$ momentum for $^{6}$Li targets.
The (red in the web version) band dots stand for events with $T(\pi^{+})+T(\pi^{-})=202-204$ MeV.
See text for more details.}
\label{fig2}
\end{center}
\end{figure}
In order to isolate the events due to the possible formation of a bound
$^{6}_{\Lambda}$H, we considered energy conservation for both reactions
(\ref{nrich6LH}) and (\ref{6LHmwd}). Momentum conservation is automatically
ensured by the fact that both reactions of formation (\ref{nrich6LH}) and
decay (\ref{6LHmwd}) occur at rest. The stopping time of $^{6}_{\Lambda}$H
in the material is indeed shorter than its lifetime.
For (\ref{nrich6LH}) we may write explicitly:
\begin{equation}
M(K^{-})+3M(p)+3M(n)-B(^{6}{\rm Li})=M({^{6}_{\Lambda}{\rm H}})+
T({^{6}_{\Lambda}{\rm H}})+M(\pi^{+})+T(\pi^{+}),
\label{Eform}
\end{equation}
in which, in obvious notation, $M$ stands for a particle mass, $T$ -- its
kinetic energy, and $B({^{6}{\rm Li}})$ -- the binding energy of $^{6}$Li.
For (\ref{6LHmwd}) we may write:
\begin{equation}
M({^{6}_{\Lambda}{\rm H}})=2M(p)+4M(n)-B({^{6}{\rm He}})+T({^{6}{\rm He}})+
M(\pi^{-})+T(\pi^{-}),
\label{Edecay}
\end{equation}
in the same notation as above. Combining Eqs.~(\ref{Eform}) and (\ref{Edecay})
in order to eliminate $M({^{6}_{\Lambda}{\rm H}})$, we get the following
equation:
\begin{eqnarray}
T(\pi^{+}) + T(\pi^{-}) & = & M(K^{-}) + M(p) - M(n) - 2M(\pi) \nonumber \\
& & - B({^{6}{\rm Li}}) + B({^{6}{\rm He}}) -T({^{6}{\rm He}})-
T({^{6}_{\Lambda}{\rm H}}).
\label{Ebal}
\end{eqnarray}
All the terms on the right-hand side are either known constants or quantities
that can be evaluated from momentum and energy conservation, except for
$T({^{6}_{\Lambda}{\rm H}})$ ($T({^{6}{\rm He}})$) that depends explicitly
(implicitly) on the unknown value of $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})$.
A variation of $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})$ between 0 and 6 MeV
introduces a change of $\sim$ 0.3 MeV in the kinetic energy
$T({^{6}_{\Lambda}{\rm H}})$ in (\ref{Eform}), corresponding to a sensitivity
of 50 keV per MeV of $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})$, and a change of
$\sim 0.2$ MeV in $T(\pi^{+})+T(\pi^{-})$ in (\ref{Ebal}), corresponding to
a sensitivity of 30 keV per MeV of $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})$.
These variations are much lower than the experimental energy resolutions for
$\pi^{+}(250\ \mathrm{MeV/c})$ and $\pi^{-}(130\ \mathrm{MeV/c})$:
$\sigma_{T}(\pi^{+})=0.96$ MeV and $\sigma_{T}(\pi^{-})=0.84$ MeV. The FINUDA
energy resolution for a ($\pi^{+},\pi^{-}$) pair in coincidence is therefore
$\sigma_{T}=\sqrt{\sigma_{T{\rm exp}}^{2}+\sigma_{T{\rm sys}}^{2}}=1.3$ MeV,
where $\sigma_{T{\rm exp}}=\sqrt{0.96^{2}+0.84^{2}}=1.3$ MeV is
the total experimental energy resolution and $\sigma_{T{\rm sys}}=
\sqrt{\sigma_{T{\rm sys}}(\pi^{+})^{2}+\sigma_{T{\rm sys}}(\pi^{-})^{2}}=0.17$
MeV is the total systematic error on energy. To be definite, we assume
a value of $B_{\Lambda}({^{6}_{\Lambda}{\rm H}})=5$ MeV, halfway between
the conservative estimate of 4.2 MeV \cite{dalitz_setti,majling} and Akaishi's
prediction of 5.8 MeV \cite{akaishi}. The r.h.s. of Eq.~(\ref{Ebal}) assumes
then a value of $T(\pi^{+})+T(\pi^{-})=203.0\pm 1.3$ MeV.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=80mm]{Tpip_Tpim_A.pdf}
\caption{Distribution of raw total kinetic energy $T(\pi^{+})+T(\pi^{-})$
for ($\pi^{+},\pi^{-}$) coincidence events from $^{6}$Li targets. The (red in the web version)
shaded vertical bar represents the cut $T(\pi^{+})+T(\pi^{-})=202-204$ MeV.}
\label{fig3}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=80mm]{Ppip_fit_mc3.pdf}
\hspace{2mm}
\includegraphics[width=80mm]{Ppim_fit_mc3.pdf}
\caption{Continuous histograms: distribution of $\pi^{+}$
(upper part) and $\pi^{-}$ (lower part) momenta for ($\pi^{+},\pi^{-}$)
coincidence events with $T(\pi^{+})+T(\pi^{-})=202-204$ MeV from $^{6}$Li
targets before acceptance correction. The (blue in the web version) shaded vertical bars
indicate the final selection regions. Dashed (red in the web version) histograms represent the
$\Sigma^{+}$ background spectra: see text for more details.}
\label{fig4}
\end{center}
\end{figure}
We considered then the raw spectrum of the total kinetic energy,
$T(\pi^{+})+T(\pi^{-})$, for the coincidence events, shown in Fig.~\ref{fig3}.
Events in the summed energy distribution were selected in the region
($203\pm 1$) MeV, indicated by the (red in the web version) filling in the figure.
The half-width of the interval corresponds to $\sim 77\%$ of the FINUDA
total energy resolution; the value was chosen as a compromise between the
strong requirement of reducing the contamination from background reactions,
as will be discussed in more detail in the following, and the plight for
reasonable statistics, leading to application of a selection narrower than
the experimental resolution. The selected events are represented by red dots
in Fig.~\ref{fig2}.
The raw distributions of $p_{\pi^{+}}$ and $p_{\pi^{-}}$ for the events
selected are shown in Fig.~\ref{fig4} by the continuous line histogram,
falling off to zero at $p_{\pi^{+}}=245$ MeV/c in the higher momentum region,
and at $p_{\pi^{-}}=145$ MeV/c in the lower momentum region. These limiting
values, when inserted in Eqs.~(\ref{nrich6LH}) and (\ref{6LHmwd}) for
two-body kinematics $^{6}_{\Lambda}$H production from rest and decay at rest,
yield $^{6}_{\Lambda}$H mass values higher than the total mass of both
($\Lambda+{^{3}\mathrm{H}}+2n$) and ($\Lambda+{^{5}\mathrm{H}}$) thresholds
marked in Fig.~\ref{fig6}. A $^{6}_{\Lambda}$H mass equal to the mass of its
lowest particle stability threshold ${^{4}_{\Lambda}{\rm H}}+2n$ corresponds
to values of $p_{\pi^{+}}=251.9$ MeV/c and $p_{\pi^{-}}=135.6$ MeV/c.
A genuinely bound $^{6}_{\Lambda}$H system, therefore, requires that pion
momenta satisfying $p_{\pi^{+}}>251.9$ MeV/c and $p_{\pi^{-}}<135.6$ MeV/c
are selected. The cuts actually applied in the analysis of the data,
$p_{\pi^{+}}=(250-255)$ MeV/c and $p_{\pi^{-}}=(130-137)$ MeV/c, as marked by
the (blue in the web version) shaded vertical bars in Fig.~\ref{fig4}, allow for a wide range of
$^{6}_{\Lambda}$H masses from the ($\Lambda+{^{3}\mathrm{H}}+2n$) threshold,
about 2 MeV in the $^{6}_{\Lambda}$H continuum, down to
$B_{\Lambda}({^{6}_{\Lambda}{\rm H}})\geq 6$ MeV, somewhat below the
mass predicted by Akaishi \cite{akaishi}. These cuts do not exclude
completely an eventual contribution from the production and decay of
${^{4}_{\Lambda}\mathrm{H}} + 2n$, of a weight which is anyway negligible,
as discussed in the next section.
\section{Results}
Three events, out of a total number of $\sim 2.7\cdot 10^7$ $K^{-}$
detected at stop on the $^{6}$Li targets, satisfy the final requirements,
$T(\pi^{+})+T(\pi^{-})=202-204$ MeV, $p_{\pi^{+}}=250-255$ MeV/c and
$p_{\pi^{-}}=130-137$ MeV/c. These events, within the (red in the web version) shaded rectangle
on the l.h.s. of Fig.~\ref{fig5}, are candidates for $^{6}_{\Lambda}$H.
The $\pi^{+}$ momenta which this rectangle encompasses go up from a value
corresponding to the ($\Lambda+{^{3}\mathrm{H}}+2n$) threshold to a value
corresponding to the binding energy predicted by Akaishi, whereas the
$\pi^{-}$ momenta which the rectangle encompasses go down from a value
corresponding to the same ($\Lambda+{^{3}\mathrm{H}}+2n$) threshold to about
$2\sigma(p_{\pi^{-}})$ below the value predicted by Akaishi \cite{akaishi}.
\begin{figure}[htbp]
\vspace{-5mm}
\begin{center}
\includegraphics[width=140mm]{pim_pip_2d_cut_3D3.pdf}
\caption{$\pi^{+}$ momentum vs $\pi^{-}$ momentum for $^{6}$Li target
events with $T(\pi^{+})+T(\pi^{-})=202-204$ MeV (l.h.s.) and with
$T(\pi^{+})+T(\pi^{-})=200-206$ MeV (r.h.s.). The shaded (red in the web version) rectangles
on each side consist of a subset of events with $p_{\pi^{+}}=250-255$ MeV/c
and $p_{\pi^{-}}=130-137$ MeV/c. The hatched (blue in the web version) rectangles on each side
are symmetric subsets of events to those in the shaded rectangles.}
\label{fig5}
\end{center}
\end{figure}
Different choices of $T(\pi^{+})+T(\pi^{-})$ interval widths ($2-6$ MeV)
and position (center in $202-204$ MeV) and of $p_{\pi^{+}}/p_{\pi^{-}}$
interval widths ($5-10$ and $8-15$ MeV/c) with fixed limits at 250 and
137 MeV/c respectively to exclude the unbound region, affect the populations
of the corresponding single spectra but not the coincidence spectrum.
As an example, in Fig.~\ref{fig5} a comparison is made between the
($p_{\pi^{+}},p_{\pi^{-}}$) plots satisfying the actual selection
$T(\pi^{+})+T(\pi^{-})=202-204$ MeV (l.h.s.), and similar plots admitting
a wider selection range $T(\pi^{+})+T(\pi^{-})=200-206$ MeV (r.h.s.). The
global population increases for the wider cut, as expected, but the events
that satisfy simultaneously also the separate selections imposed on
$p_{\pi^{+}}$ and $p_{\pi^{-}}$ (shaded rectangles in the upper left part of
the plots) remain the same. A similar stability is {\it not} observed in the
opposite corner of the plots where, on top of the events already there on the
left plot, five additional events appear on the right plot upon extending the
cut. Quantitatively, fitting the projected $\pi^{\pm}$ distributions of the
l.h.s. of Fig.~\ref{fig5} by gaussians, an excess of three events in both
$p_{\pi^{\pm}}$ distributions is invariably found, corresponding to the shaded
(red in the web version) rectangle. The probability for the three events to belong to the fitted
gaussian (background) distribution is less than $0.5\%$ in both cases.
It is possible, moreover, to see directly from the two dimesional plots that
variations of the independent momentum selections do not produce any effect.
Systematic errors due to the applied analysis selection are thus ruled out.
It is also worth noticing that the tight momentum cuts imposed on the
($\pi^{+},\pi^{-}$) coincidence events allow to eliminate completely any
contamination due to possible $\pi^{-}/e^{-}$ misidentification. Furthermore,
$\mu^{+}$'s from $K^{+}_{\mu 2}$ decay are clearly separated from $\pi^{+}$'s
coming from the opposite $K^{-}$ interaction vertex. Figure~\ref{fig6b} shows
a front view of one of the three events, as reconstructed by FINUDA.
\begin{figure}[h]
\begin{center}
\includegraphics[width=70mm]{run11874_10663_tracker3.pdf}
\hspace{2mm}
\includegraphics[width=40mm]{run11874_10663_vertex3.pdf}
\caption{Left: front view of one of the $^{6}_{\Lambda}$H candidate events
reconstructed by FINUDA where a ($\pi^{+},\pi^{-}$) pair emerges from a
$^{6}$Li target and crosses the spectrometer. Right: expanded view of the
target region for the same event where the $K^{-}$ track stops in a $^{6}$Li
target.}
\label{fig6b}
\end{center}
\end{figure}
\begin{table}[h]
\begin{center}
\caption{Kinematical properties and $^{6}_{\Lambda}$H mass,
$M({^{6}_{\Lambda}{\rm H}})$, of the three $^{6}_{\Lambda}$H candidate
events from production (\ref{nrich6LH}) and decay (\ref{6LHmwd}) reactions.
Listed in the last two columns are the mean and the difference of the
production and decay masses. $T_{\rm tot}$ indicates the total $\pi^{\pm}$
kinetic energy $T(\pi^{+})+T(\pi^{-})$.}
\label{tab1}
\vspace{2mm}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$T_{\rm tot}$ & $p_{\pi^{+}}$ & $p_{\pi^{-}}$ & $M({^{6}_{\Lambda}{\rm H}})$ &
$M({^{6}_{\Lambda}{\rm H}})$ & $M({^{6}_{\Lambda}{\rm H}})$ &
$\Delta M({^{6}_{\Lambda}{\rm H}})$ \\
(MeV) & (MeV/c) & (MeV/c) & prod. (MeV) & decay (MeV) & mean (MeV) & (MeV) \\
\hline
202.6$\pm$1.3 & 251.3$\pm$1.1 & 135.1$\pm$1.2 & 5802.33$\pm$0.96 &
5801.41$\pm$0.84 & 5801.87$\pm$0.96 & 0.92$\pm$1.28 \\
202.7$\pm$1.3 & 250.1$\pm$1.1 & 136.9$\pm$1.2 & 5803.45$\pm$0.96 &
5802.73$\pm$0.84 & 5803.09$\pm$0.96 & 0.72$\pm$1.28 \\
202.1$\pm$1.3 & 253.8$\pm$1.1 & 131.2$\pm$1.2 & 5799.97$\pm$0.96 &
5798.66$\pm$0.84 & 5799.32$\pm$0.96 & 1.31$\pm$1.28 \\
\hline
\end{tabular}
\end{center}
\end{table}
By evaluating event by event the corresponding $^{6}_{\Lambda}$H mass from
both production (\ref{nrich6LH}) and decay (\ref{6LHmwd}) reactions, the mass
values listed in Table~\ref{tab1} are obtained. A mean value for each event
jointly from production and decay was also evaluated, with an error given by
the highest of the two uncertainties, 0.96 MeV. For the global mean mass value
of the three events we then find
${\bar M}({^{6}_{\Lambda}{\rm H}})=(5801.43\pm 0.55)$ MeV,
where the uncertainty $0.55=0.96/\sqrt{3}$ MeV reflects the uncertainty
assigned to each event. This global mass uncertainty, however, is considerably
smaller than the mean-mass spread of the three events. We therefore decided to
relax the assigned uncertainty by calculating it from the spread of the three
mean mass values, which yields
\begin{equation}
{\bar M}({^{6}_{\Lambda}{\rm H}}) = 5801.4\pm 1.1~{\rm MeV},
\label{finalmass}
\end{equation}
with uncertainty larger than the 0.96 MeV and 0.84 MeV measurement
uncertainties in production and decay respectively. The standard deviation
of this uncertainty is 0.55 MeV which together with $\sigma=1.1$ MeV is still
short of the 2.11 MeV deviation of the third event mean mass from the mean
mass value. This observation could indicate some irregularity in the
reconstruction of the third event. To regain confidence, each one of the
three events was checked visually for irregularities but none was found.
The third event, in particular, is shown in Fig.~\ref{fig6b}.
Listed in the last column of Table~\ref{tab1} are values of
$\Delta M({^{6}_{\Lambda}{\rm H}})$, defined as the difference between the
$^{6}_{\Lambda}$H mass values obtained from production and from decay. The
mass values obtained from production are systematically higher than those
from decay by
\begin{equation}
\Delta M({^{6}_{\Lambda}{\rm H}}) = 0.98 \pm 0.74~{\rm MeV},
\label{finaldeltamass}
\end{equation}
where the uncertainty is evaluated from the 1.3 MeV uncertainty for
$T(\pi^{+})+T(\pi^{-})$ from which each of the mass differences is directly
determined. Unlike the mean $M({^{6}_{\Lambda}{\rm H}})$ mass value, the
spread of the production vs decay mass differences is well within $1\sigma$.
A possible physical origin of the $\Delta M({^{6}_{\Lambda}{\rm H}})$
systematics is discussed in a subsequent section.
The mean mass value (\ref{finalmass}) corresponds to a $^{6}_{\Lambda}$H
binding energy $B_{\Lambda}=(4.0\pm1.1)$ MeV with respect to the
$(\Lambda + {^{5}{\rm H}})$ threshold, and to $B_{\Lambda}=(0.3\pm1.1)$ MeV
with respect to the lowest threshold $({_{\Lambda}^{4}{\rm H}}+2n)$.
The $^{6}_{\Lambda}$H mean mass value and its uncertainty are indicated on
the r.h.s. of Fig.~\ref{fig6} with respect to the various thresholds and
predictions shown on the l.h.s. of the figure.
\section{Background estimation}
Before discussing the physical interpretation of the above results, it is
mandatory to check carefully that the three observed events do not arise
from physical or instrumental backgrounds that could affect the data.
Concerning the physical backgrounds, a complete simulation has been performed
of possible $K^{-}_{\rm stop}$ absorption reactions on both single nucleons
and pairs of strongly correlated nucleons that lead to the formation and decay
of $\Lambda$ and $\Sigma$ hyperons. Of these reactions, only the following
chain leads to ($\pi^{+},\pi^{-}$) coincidences in the same momentum ranges
corresponding to the production and mesonic decay of $^{6}_{\Lambda}$H and
which are respected by the three candidate events:
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow & \Sigma^{+}+{^{4}{\rm He}}+n+
\pi^{-} \ \ \ \ (p_{\pi^{-}}\leq 190\ \mathrm{MeV/c}) \nonumber \\
& & \hookrightarrow n+\pi^{+} \ \ \ \ (p_{\pi^{+}}\leq 282\ \mathrm{MeV/c}).
\label{bgd_S_1}
\end{eqnarray}
This reaction chain has been studied by means of the FINUDA simulation program
fully reproducing the apparatus geometry, detection efficiency and the trigger
efficiency. The interaction of $K^{-}$ with the target nucleus has been
simulated with two different approaches. In the first approach, the quasi-free
approximation was adopted for the interaction of the $K^{-}$ with a proton of
the target nucleus, $K^{-}_{\rm stop}+p\to\Sigma^{+}+\pi^{-}$, taking into
account the nucleon Fermi motion; the residual nucleus was considered as
a spectator and the notation ``$^{4}$He + n" is just a label to indicate that
the system is highly particle unstable. Pions arising from (\ref{bgd_S_1})
were processed by the pattern recognition and reconstruction programs of
FINUDA as real data. In common with all simulated reaction chains, the
simulated events were then submitted to the same quality cuts and to the same
selections criteria applied in the data analysis. Three events were found out
of a total of $2.2\cdot 10^{7}$ $K^{-}$ mesons simulated to stop on $^{6}$Li
targets and forced to undergo the (\ref{bgd_S_1}) ``quasi-free" reaction chain
with a probability of 1. Taking into account the number of actual $K^{-}$
mesons stopped on $^6$Li targets, the branching fraction for the
$K^{-}_{\rm stop}+p\to\Sigma^{+}+\pi^{-}$ reaction on nuclei measured on
$^{12}$C \cite{vander} and on $^{4}$He \cite{katz}, $(0.159\pm 0.012)$
evaluated as a weighted mean, the $\Sigma^{+}+n\rightarrow \Lambda+p$
conversion probability \cite{outa}, $(0.45\pm 0.04)$, and the
$\Sigma^{+}\rightarrow n+\pi^{+}$ decay branching ratio, $(0.483\pm 0.003)$,
an expected $\Sigma^{+}$ background of $0.15\pm 0.09$ events on $^{6}$Li
targets is obtained.
In a second approach, the interaction of $K^{-}$ mesons with the target
nucleus as a whole was considered, applying directly the 4-body kinematics to
(\ref{bgd_S_1}). Five events were found out of a total of $2.7\cdot 10^{7}$
$K^{-}$ mesons simulated to stop on $^{6}$Li targets and forced to undergo the
(\ref{bgd_S_1}) ``4-body" reaction chain with a probability of 1. Taking into
account the same normalization factors used for the ``quasi-free" approach,
an expected $\Sigma^{+}$ reaction chain background of $0.20\pm 0.11$ events
on $^{6}$Li targets was obtained under the hypothesis that $\Sigma^{+}$
production on $^{6}$Li in this approach always gives a recoiling $^{4}$He
nucleus. Final states corresponding to further fragmentation of the $^{6}$Li
target nucleus, such as $K^{-}_{\rm stop}+{^{6}{\rm Li}}\rightarrow\Sigma^{+}+
{^{3}{\rm He}}+n+n+\pi^{-}$, give weaker background contribution, owing to the
requirements imposed on $T(\pi^{+})+T(\pi^{-})$ ($<180$ MeV for final states
of the $\Sigma^{+}$ production reaction with more than 4 bodies) and on the
$\pi^{+}$ and $\pi^{-}$ momenta.
We also considered the distortion of Eq.~(\ref{bgd_S_1}) reaction chain
spectra due to the $^{4}{\rm He}+n$ final state interaction leading to $^5$He,
a resonance centered at $\sim 0.8$ MeV above the $^{4}{\rm He}+n$ threshold
with $\Gamma=1.36$ MeV \cite{tilley02}. To this end we required that once the
$^{4}$He and neutron momenta generated by the 4-body phase space simulation
corresponded to the formation of the $^{5}$He resonance, the momenta of the
remaining particles, $\Sigma^{+}$ and $\pi^{-}$, should be modified
accordingly. We passed then these modified phase space distributions
through the selection criteria described above and found variation of less
than $1\%$ in the background value evaluated for a $100\%$ $^{4}{\rm He}+n$
final state.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=90mm]{Fit_mc_final.pdf}
\caption{$T(\pi^{+})+T(\pi^{-})$ distributions; (red in the web version) points: experimental
data; (blue in the web version) dashed histogram: ``quasi-free" simulation; (violet in the web version) dotted
histogram: ``4-body" simulation; (black) continuous histogram: best fit to the
data with fractions of the simulated templates. The simulated distributions
have been normalized to the experimental distribution area. For more details
see text.}
\label{fig7}
\end{center}
\end{figure}
In Fig.~\ref{fig7} the experimental $T(\pi^{+})+T(\pi^{-})$ spectrum is shown
together with spectra obtained from the ``quasi-free" and ``4-body"
simulations of the (\ref{bgd_S_1}) process: the simulated spectra were
normalized to the area of the experimental distribution. As may be seen, the
``quasi-free" spectrum (dashed (blue in the web version) histogram) reproduces the experimental
distribution better than the ``4-body" (dotted (violet in the web version) histogram), but
exhibits a too sharp decrease in the 200--210 MeV region and underestimates
the low energy tail. To obtain a satisfactory description, a fit of the
experimental spectrum was performed with fractions of the two simulated
templates; a standard likelihood fitting method, using Poisson statistics,
was applied in which both data and Monte Carlo statistical uncertainties were
taken into account \cite{barlow}. Particular care was devoted to the
description of the 200--210 MeV slope, where the selection of the
$^{6}_{\Lambda}$H candidate events is made. The continuous (black in the web version) histogram
in Fig.~\ref{fig7} represents the best fit to the 180--220 MeV region;
the resulting fractions are $0.743\pm 0.019$ and $0.257\pm 0.017$ for the
``quasi-free" and ``4-body" templates respectively, with a $\chi^{2}$/ndf =
40.0/39. We note that varying the width of the fit region from 180--220 MeV
to 130--220 MeV spoils the fit, increasing the $\chi^{2}$/ndf value by a
factor of $\sim 3.8$, while the fractions of the two templates change by less
than 0.025, corresponding to $1.3-1.5~\sigma$.
Back in Fig.~\ref{fig4}, the dashed (red) histograms represent the separate
$p_{\pi^{+}}$ and $p_{\pi^{-}}$ distributions obtained by adopting the above
fractions to the events successfully simulated within the two approaches
discussed above and satisfying the cut $T(\pi^{+})+T(\pi^{-})=202-204$ MeV.
Only a qualitative agreement is reached with the experimental, low statistics
distributions. The estimated background spectra are shifted from the
experimental ones toward higher momentum for the $\pi^+$ spectrum and toward
lower momentum for the $\pi^-$ spectrum. The difference is significant over
the statistical fluctuation. However, these shifts cause overestimates of the
background counts when the tails of the simulated distributions are used for
an estimate. It is thus possible to conclude that the background estimate
is safe in spite of the slight deviations noted here. In particular, in the
$^{6}_{\Lambda}$H selected regions, indicated by shaded (blue in the web version) vertical bars,
the contribution due to the $\Sigma^{+}$ background for events satisfying also
the cuts on $p_{\pi^{+}}$ and $p_{\pi^{-}}$ in coincidence corresponds to
$0.16\pm 0.07$ events on $^{6}$Li targets (BGD1).
\vspace{3mm}
Another reaction chain capable of providing background events is
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow & {^{4}_{\Lambda}{\rm H}}+2n+
\pi^{+}\ \ \ \ (p_{\pi^{+}} \leq 252\ \mathrm{MeV/c}) \nonumber \\
& & \hookrightarrow {^{4}{\rm He}}+\pi^{-} \ \ \ \ (p_{\pi^{-}} \sim 132.8\
\mathrm{MeV/c}).
\label{bgd_4LH_1}
\end{eqnarray}
The momentum of the $^{4}_{\Lambda}$H decay $\pi^{-}$ is close to the
momentum of the $\pi^{-}$ from the two-body decay of $^{6}_{\Lambda}$H,
$p_{\pi^{-}}\sim 134$ MeV/c, evaluated assuming $B_{\Lambda}=5$ MeV which is
halfway between the two theoretical estimates exhibited in Fig.~\ref{fig6}.
The probability of having background contribution from this reaction chain was
evaluated taking into account the phase space fraction of the reaction
(\ref{bgd_4LH_1}) available for $\pi^{+}$'s satisfying the momentum selection
$p_{\pi^{+}}=250-255$ MeV/c, $4\cdot 10^{-6}$, and the probability for a
$K^{-}_{\rm stop}$ to produce a $^{4}_{\Lambda}$H accompanied by a $\pi^{+}$
on a $^{6}$Li target. In Ref.~\cite{tamura_fr} the probability of producing
$^{4}_{\Lambda}$H on $^{7}$Li targets was reported to be $(3.0\pm 0.4)\cdot
10^{-2}/K^{-}_{\rm stop}$ and, furthermore, the probability of producing it
together with a charged pion was indicated to be $0.49\pm 0.08$. Using these
values, the formation probability of ${^{4}_{\Lambda}{\rm H}}+\pi^{\pm}$
on $^{6}$Li target, the closest isotope of $^{7}$Li, was evaluated to be
$(1.47\pm 0.15)\cdot 10^{-2}/K^{-}_{\rm stop}$. In addition, a branching
ratio 0.49 \cite{tamura_fr} for the two-body decay
$^{4}_{\Lambda}{\rm H}\to {^{4}{\rm He}}+\pi^{-}$ has to be included.
A total probability of $(2.8\pm 0.5)\cdot 10^{-8}$ is obtained.
From this value, taking into account the global efficiency of FINUDA
(acceptance, reconstruction and analysis cuts) it is possible to evaluate a
background of $0.05\pm 0.01$ reconstructed events for the $2.7\cdot 10^{7}$ $K^{-}$
mesons stopped on the $^{6}$Li targets. It should be noted that this value
overestimates by far the actual contribution from (\ref{bgd_4LH_1}) since the
analysis of Ref.~\cite{tamura_fr} is incapable of separating the contribution
of $^{4}_{\Lambda}{\rm H}+\pi^{+}$ out of the global $^{4}_{\Lambda}{\rm H}+
\pi^{\pm}$ fraction. Note that a microscopic reaction evaluation of
(\ref{bgd_4LH_1}) requires a theoretical model which takes into account
different channels ($\Lambda$ production, $\Sigma$ production, compound
nucleus formation) the weights of which are not experimentally known.
We chose to avoid relying on a model and to assume, instead, an overly
conservative evaluation of the background from the chain (\ref{bgd_4LH_1}).
The estimated level of this background is negligible with respect to the
BGD1 background from the chain (\ref{bgd_S_1}).
Other reaction chains have been considered, such as:
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow &
\Sigma^{+}+{^{3}_{\Lambda}{\rm H}}+d+\pi^{-}\ \ \ (p_{\pi^{-}}\leq 165\
{\rm MeV/c}) \nonumber \\
& & \hookrightarrow n + \pi^{+} \ \ \ (p_{\pi^{+}} < 250\ {\rm MeV/c})
\label{bgd_m1}
\end{eqnarray}
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow &
{^{3}_{\Lambda}{\rm H}}+3n+\pi^{+}\ \ \ (p_{\pi^{+}}\leq 242\
{\rm MeV/c}) \nonumber \\
& & \hookrightarrow {^{3}{\rm He}}+\pi^{-}\ \ \ (p_{\pi^{-}}\sim 115\
{\rm MeV/c})
\label{bgd_m2}
\end{eqnarray}
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow &
\Lambda + {^{3}{\rm H}}+2n+\pi^{+}\ \ \ (p_{\pi^{+}}\leq 247\
{\rm MeV/c}) \nonumber \\
& & \hookrightarrow p+\pi^{-}\ \ \ ( p_{\pi^{-}} < 195\ {\rm MeV/c})
\label{bgd_m3}
\end{eqnarray}
These reaction chains may be safely discarded since all of them involve too
low values of $T(\pi^{+})+T(\pi^{-})$, less than 190 MeV, which are way
outside the cut applied on $T_{\rm tot}(\pi)$; the chain (\ref{bgd_m2}) may
be discarded in addition by the cut imposed on $p_{\pi^{-}}$.
Finally, another mechanism which could produce a ($\pi^{+},\pi^{-}$) pair
in the final state is
\begin{eqnarray}
K^{-}_{\rm stop}+{^{6}{\rm Li}} & \rightarrow &
{^{6}_{\Lambda}{\rm He}}+\pi^{0}\ \ \ (p_{\pi^{0}}\sim 280\ {\rm MeV/c})
\nonumber \\
& & \hookrightarrow {^{6}{\rm Li}}+\pi^{-}\ \ \ (p_{\pi^{-}} \sim 108\
{\rm MeV/c}),
\label{bgd_2pi0}
\end{eqnarray}
followed by a reaction on another $^{6}$Li nucleus:
\begin{equation}
\pi^{0}+{^{6}{\rm Li}} \rightarrow {^{6}{\rm He}}+\pi^{+}\ \ \
(p_{\pi^{+}}\sim 280\ \rm{MeV/c\ in\ the\ forward\ direction}).
\label{bgd_2pi0_3}
\end{equation}
However, the kinematics of (\ref{bgd_2pi0}) rules out a contribution from this
reaction chain when applying the cuts on $\pi^{-}$ momenta. Moreover, the mean
free path of a $\pi^{0}$ with momentum $\sim 280$ MeV/c,
$l_{\rm free} < 0.1~\mu$m, strongly reduces the probability of the second
reaction of the chain with respect to $\pi^{0}$ decay.
\vspace{3mm}
The main source of instrumental background could be the presence of fake
tracks, due to fake signals from the detectors, misidentified as true events
by the track reconstruction algorithms. For this purpose we analyzed the
events with a $\pi^{+}$ and a $\pi^{-}$ emitted in coincidence with a $K^{-}$
stopping in a given nuclear target under the following criteria: \\
\begin{enumerate}
\item events relative to target nuclei other than $^{6}$Li ($^{7}$Li,
$^{9}$Be, $^{13}$C, $^{16}$O) were selected with the same selection criteria
$T(\pi^{+})+T(\pi^{-})=202-204$ MeV, $p_{\pi^{+}}=250-255$ MeV/c and
$p_{\pi^{-}}=130-137$ MeV/c, as for $^{6}$Li. Incidentally, for events coming
from the (\ref{bgd_S_1}) reaction chain on nuclear targets heavier than
$^{6}$Li, $T(\pi^{+})+T(\pi^{-})<202$ MeV by at least $2\sigma_{T}$ so that
this criterion actually selects the instrumental background exclusively.
Only one event was found, coming from $^{9}$Be target; \\
\item events relative to the $^{6}$Li targets were selected with values of
$T(\pi^{+})+T(\pi^{-})=193-199$ MeV so as to search for neutron-rich
hypernuclei produced on the other targets. No events were found.
\end{enumerate}
\vspace{2mm}
Taking into account the number of $K^{-}_{\rm stop}$ detected in $^{6}$Li
targets and in all the other targets, we concluded that $0.27\pm 0.27$
fake events should be expected from $^{6}$Li due to the instrumental
background (BGD2).
Combining together the expected number of events arising from physical and
instrumental backgrounds that affect our selected data, $0.16\pm 0.07$ (BGD1)
and $0.27\pm 0.27$ (BGD2), a total of $0.43\pm 0.28$ background events has
been established.
From this value, following Poisson statistics, we may state that the three
observed $^{6}_{\Lambda}$H candidate events do not belong to the background
distribution with a confidence level of $99\%$; the difference between the
measured yield and the total expected background can thus be safely considered
a $^{6}_{\Lambda}$H signal.
The probability of observing three or more events from the background fluctuation
following Poisson distribution with $\mu=0.43$ (BGD1+BGD2) or $\mu=0.16$ (only BGD1) is
0.0096 or 0.0006, respectively. In terms of a {\it statistical significance} $S$ defined by
$S=C/\sqrt{\rm BGD}$, with a signal $C=3-{\rm BGD}$, the statistical significance of the signal
is 3.9 or 7.1, respectively.
\section{Production rate evaluation}
Using the background estimates of the last section, it is possible to evaluate
the product $R \cdot {\rm BR}(\pi^{-})$, where $R$ is the $^{6}_{\Lambda}$H
production rate per stopped $K^{-}$ and ${\rm BR}(\pi^{-})$ is the branching
ratio for the two-body weak decay
$^{6}_{\Lambda}{\rm H}\rightarrow {^{6}{\rm He}}+\pi^{-}$:
\begin{equation}
R \cdot {\rm BR}(\pi^{-}) = \frac{3 - {\rm BGD1} - {\rm BGD2}}
{\epsilon(\pi^{+})\ \epsilon(\pi^{-})\ K^{-}_{\rm stop}({^{6}{\rm Li}})} =
(1.3 \pm 0.9)\cdot 10^{-6} /K^{-}_{\rm stop}.
\label{rate}
\end{equation}
In Eq.~(\ref{rate}), $\epsilon(\pi^{+})$ and $\epsilon(\pi^{-})$ indicate
the global efficiencies for $\pi^{+}$ and $\pi^{-}$, respectively, including
detection efficiency, geometrical and trigger acceptances and pattern
recognition, reconstruction and selection efficiencies, all of which have
been evaluated by means of the full FINUDA simulation code, well tested in
calculations for other reactions in similar momentum ranges
\cite{spectrFND,NPA775,mwd}. $K^{-}_{\rm stop}({^{6}{\rm Li}})$ is the number of
$K^{-}$ detected at stop in $^{6}$Li targets.
The value (\ref{rate}) has to be corrected for the purity of the $^{6}$Li
targets used, 90$\%$, for the 0.77 $\sigma_{T}$ cut applied to $T(\pi^{+})+
T(\pi^{-})$, and for the fraction of $^{6}_{\Lambda}$H decaying in flight.
In Ref.~\cite{tamura_fr} a contribution of $20\%$ is reported for the decay
in flight of $^{4}_{\Lambda}$H produced on a $^{7}$Li target; extending this
value also to $^{6}_{\Lambda}$H and considering that the cut applied to
$p_{\pi^{-}}$, 130--137 MeV/c, allows to accept about one half of the pions
emitted in flight, a correction factor of $10\%$ is evaluated. The corrected
result is
\begin{equation}
[R \cdot {\rm BR}(\pi^{-})]_{\rm corr.}=(2.9 \pm 2.0)\cdot
10^{-6}/K^{-}_{\rm stop}.
\label{finalrate}
\end{equation}
By assuming ${\rm BR}(\pi^{-})=49\%$, in analogy to the weak decay
$^{4}_{\Lambda}{\rm H}\rightarrow {^{4}{\rm He}}+\pi^{-}$ \cite{tamura_fr},
we find $R =(5.9 \pm 4.0)\cdot 10^{-6}/K^{-}_{\rm stop}$, fully consistent
with the upper limit (\ref{Rpi+6}) obtained previously by FINUDA
\cite{nrich1}. Although no theoretical calculation of this capture rate has
been reported to provide direct comparison, the order of magnitude of the
rate determined here is compatible with the interval of values calculated
for production of heavier neutron-rich hypernuclei \cite{tretyak01} with
stopped $K^{-}$ mesons and, as expected, is approximately three orders of
magnitude lower than the capture rate for the production of ordinary
particle-stable $\Lambda$ hypernuclei.
\section{Discussion}
The binding energy deduced from the three measured events listed
in Table~\ref{tab1} was recorded in Eq.~(\ref{finalmass}):
$B_{\Lambda}({^{6}_{\Lambda}{\rm H}})=4.0\pm 1.1$ MeV with respect to
$^{5}{\rm H}+\Lambda$, close to the value
$B_{\Lambda}({_{\Lambda}^{6}{\rm He}})=4.18\pm 0.10$ MeV with respect to
$^{5}{\rm He}+\Lambda$ for the other known $A=6$ hypernucleus \cite{juric}.
It is in good accord with the estimate 4.2 MeV made originally by Dalitz
and Levi Setti \cite{dalitz_setti} and confirmed by Majling \cite{majling}.
It is lower by 1.8 MeV than the value 5.8 MeV suggested by Akaishi et
al. \cite{akaishi}, leaving little room for an attractive contribution from
a $\Lambda NN$ three-body force of a similar magnitude, 1.4 MeV, which in
Akaishi's calculations arises from a coherent $\Lambda N-\Sigma N$ mixing
model. This is consistent with a substantial weakening of $\Lambda N-\Sigma N$
mixing contributions for the excess $p$ shell neutrons in
$_{\Lambda}^{6}{\rm H}$ with respect to the strong effect calculated in the
$s$-shell hypernucleus $_{\Lambda}^{4}{\rm H}$ \cite{akaishi00}. Indeed,
recent shell-model calculations by Millener indicate that $\Lambda N-\Sigma N$
mixing contributions to $B_{\Lambda}$ and to doublet spin splittings in the
$p$ shell are rather small, about $(10\pm 5)\%$ of their contribution
in $_{\Lambda}^{4}{\rm H}$ \cite{millener}. Nevertheless, given the
measurement uncertainty of 1.1 MeV, one may not conclude that this
$\Lambda NN$ force contribution is negligible, but only that its influence
appears considerably lower than predicted. For illustration, see
Fig.~\ref{fig6}.
It is possible to avoid considering explicitly the $\Lambda N-\Sigma N$
mixing effect in the evaluation of $B_{\Lambda}({_{\Lambda}^{6}{\rm H}})$
by updating the shell-model (SM) argument used in Ref.~\cite{dalitz_setti}.
We adopt a cluster model for $_{\Lambda}^{6}$H in terms of $_{\Lambda}^{4}$H
plus two $p$-shell neutrons coupled to $J^{\pi}=0^{+}$ as in $^{6}$He g.s.
The interaction of the $\Lambda$ hyperon with this dineutron cluster,
including any $\Lambda nn$ force arising from $\Lambda N-\Sigma N$ mixing,
may be deduced from $_{\Lambda}^{7}$He which consists of an $\alpha$ cluster
plus precisely the same $\Lambda nn$ configuration under consideration in
$_{\Lambda}^{6}$H. Subtracting $B_{\Lambda}({_{\Lambda}^{5}{\rm He}})=3.12\pm
0.02$ MeV from $B_{\Lambda}({_{\Lambda}^{7}{\rm He}})$, with a value
$B_{\Lambda}({_{\Lambda}^{7}{\rm He}})= 5.36\pm 0.09$ MeV obtained by
extrapolating linearly from the known binding energies of the other members of
the $A=7$ hypernuclear $T=1$ isotriplet (see Fig.~3, Ref.~\cite{hashimoto}),
we obtain $2.24\pm 0.09$ MeV for the $\Lambda nn$ sum of two-body and
three-body interactions involving the $\Lambda$ hyperon. The value of
$B_{\Lambda}({_{\Lambda}^{6}{\rm H}})$ is then obtained adding this 2.24 MeV
to $B_{\Lambda}({_{\Lambda}^{4}{\rm H}})=2.04\pm 0.04$ MeV \cite{juric},
so that $B_{\Lambda}^{\rm SM}({_{\Lambda}^{6}{\rm H}})=4.28\pm 0.10$
MeV.{\footnote{We thank Dr. D.J. Millener for alerting us to this estimate.
A somewhat higher value,
$B_{\Lambda}^{\rm SM}({_{\Lambda}^{6}{\rm H}})=4.60\pm 0.24$ MeV, is obtained
if the preliminary value $B_{\Lambda}({_{\Lambda}^{7}{\rm He}})=5.68\pm
0.03({\rm stat})\pm 0.22({\rm syst})$ MeV from the $(e,e'K^+)$ reaction in the
JLab E01-011 experiment is used \cite{hashimoto}.}} We have thus recovered the
estimate originally made by Dalitz and Levi Setti \cite{dalitz_setti}.
As mentioned in the discussion of Table~\ref{tab1}, the $^{6}_{\Lambda}$H
mass values obtained from production are systematically higher than
the corresponding values obtained from decay, leading to a mass
difference of $\Delta M({^{6}_{\Lambda}{\rm H}})=0.98\pm 0.74$ MeV,
see Eq.~(\ref{finaldeltamass}). This suggests that $^{6}_{\Lambda}$H
is produced in an excited state, while decaying from its ground state.
We recall that Pauli spin is conserved in capture at rest.
For $K^{-}_{\rm stop}+{^{6}{\rm Li}}\to {^{6}_{\Lambda}{\rm H}}+\pi^{+}$
production, since $^{6}$Li is very well approximated (about $98\%$) by a
$L=0,~S=1$ configuration \cite{millener}, $^{6}_{\Lambda}$H is dominantly
produced in its $1^{+}$ first excited state, decaying then by a fast magnetic
dipole transition to the $0^{+}$ ground state from which the mesonic weak
decay occurs. In this situation, the pion kinetic energies $T(\pi^{+})$ and
$T(\pi^{-})$, directly measured by the FINUDA spectrometer, should reflect
this systematic difference between production and decay. The mass of the
$0^{+}$ ground state should be calculated from the decay reaction only,
giving a mean value $M({^{6}_{\Lambda}{\rm H}_{\rm g.s.}})=(5800.9\pm 1.2)$
MeV, corresponding to a binding energy of $4.5\pm 1.2$ MeV with respect
to ($\Lambda+{^{5}{\rm H}}$) and of $0.8\pm 1.2$ MeV with respect to
(${^{4}_{\Lambda}{\rm H}}+2n$). For the $1^{+}$ excited state it should be
possible to evaluate a mean mass $M({^{6}_{\Lambda}{\rm H}^{\ast}})=(5801.9\pm
1.0)$ MeV. Although nominally unstable by $0.2\pm 1.0$ MeV, the low $Q$ value
for two-neutron emission plus the associated $\Delta S=1$ spin flip required
in the decay to $^{4}_{\Lambda}{\rm H}(0^{+}_{\rm g.s.})+2n$ are likely to
make the $^{6}_{\Lambda}{\rm H}(1^{+}_{\rm exc}\to 0^{+}_{\rm g.s.})$ $M1$
$\gamma$-ray transition competitive with the strong decay of the
$1^{+}_{\rm exc}$ level.
It is worth noting that the uncertainty placed on the excitation
energy $\Delta E (0^{+}\to 1^{+})$, identifying this $\Delta E$ with
$\Delta M({^{6}_{\Lambda}{\rm H}})$, is considerably smaller than
the uncertainty of each one of the $0^{+}$ and the $1^{+}$ levels because
$\Delta M$ has been determined directly from the sum of kinetic energies
$T(\pi^{+})+T(\pi^{-})$ and its associated uncertainty. The value determined
in the present experiment for $\Delta E$ is smaller by $2\sigma$ than
the value $\Delta E_{\rm akaishi}(0^{+}\to 1^{+})=2.4$ MeV predicted in
Ref.~\cite{akaishi}. This is in line with the conclusion drawn from the
absolute energy location of the $0^{+}$ g.s., casting doubts on the
applicability of the model developed by Akaishi et al.~\cite{akaishi}. The
value $\Delta E({^{6}_{\Lambda}{\rm H}}:0^{+}\to 1^{+})=0.98\pm 0.74$ MeV is
in good accord with $\Delta E({^{4}_{\Lambda}{\rm H}}:0^{+}\to 1^{+})=1.04\pm
0.03$ MeV \cite{juric}, consistently with a weak-coupling picture for the two
`halo' $p$-shell neutrons in $^{6}_{\Lambda}{\rm H}$ outside the $s$-shell
cluster of $^{4}_{\Lambda}{\rm H}$.
It is also worth noting that the width of the selected $T(\pi^{+})+T(\pi^{-})$
interval (2 MeV) allows to include both production and decay pions within the
experimental resolution of the mass determination at a $1\sigma$ level, thus
extending the validity of the working assumption on which the analysis method
was based, namely that the masses of the produced and decaying hypernucleus
are equal. A variation of the binding energy from the value of 5 MeV used to
fix the selection on $T(\pi^{+})+T(\pi^{-})$ to 4 MeV, the average binding
energy of the $0^+$ and $1^+$ levels, produces a completely negligible
variation of the accepted fraction of events due to the selection criteria
with respect to the errors. Finally, the absence of systematics arising
from the particular choice of both width and position of the selected
$T(\pi^{+})+T(\pi^{-})$ interval indicates that the difference between the
$^{6}_{\Lambda}$H formation and decay masses is not influenced by the cut
itself. It is important, however, to realize that the above deductions on
the $^{6}_{\Lambda}$H excitation spectrum rely on very scarce statistics and,
therefore, have to be considered as indication, even if quite solid.
Before closing we wish to discuss briefly another scenario for the excitation
spectrum of $^{6}_{\Lambda}{\rm H}$ motivated by the somewhat large spread
among the three $^{6}_{\Lambda}{\rm H}$ candidate events. Apart from the $0^+$
g.s. and $1^+$ spin-flip excited state as in $_{\Lambda}^{4}{\rm H}$, a $2^+$
excited state as for the $p$-shell dineutron system in $^{6}{\rm He}$ (1.80
MeV) is expected at about 2 MeV excitation in $^{6}_{\Lambda}{\rm H}$.
Furthermore, a triplet of spin-flip excitations $1^+,2^+,3^+,$ built on the
$2^+$ dineutron excitation is expected 1 MeV higher, at about 3 MeV excitation
in $^{6}_{\Lambda}{\rm H}$. It is then not unreasonable to assign event 3 in
Table~\ref{tab1} to formation and decay of $^{6}_{\Lambda}{\rm H}$ involving
the $0^+$ g.s. and its $1^+$ spin-flip excited state, as considered above,
whereas the other two events which are relatively close to each other
correspond to formation of one of the $1^+,2^+,3^+$ levels and to decay
from the $2^+$ dineutron excitation. This scenario generates an additional
excitation scale to confront the $\approx 3$ MeV separation between the first
two events of Table~\ref{tab1} and the third one. This results in the
assignment of $^{6}_{\Lambda}{\rm H}$ levels listed in Table~\ref{tab2}.
\begin{table}[h]
\begin{center}
\caption{Masses and $B_{\Lambda}$ values (in MeV) of $^{6}_{\Lambda}$H levels
assuming that event 3 in Table~\ref{tab1} corresponds to the lowest two
levels and events 1 and 2 correspond to higher levels.}
\label{tab2}
\vspace{2mm}
\begin{tabular}{ccccc}
\hline
$^{6}_{\Lambda}$H & $0^+$ & $1^+$ & $2^+$ & $(1^+,2^+,3^+)$ \\
\hline
$M$ \ & \ 5798.66$\pm$0.84 \ & \ 5799.97$\pm$0.96 \ & \ 5802.07$\pm$0.59 \ &
\ 5802.89$\pm$0.68 \\
$B_{\Lambda}$ \ & \ 6.78$\pm$0.84 \ & \ 5.47$\pm$0.96 \ & \ 3.37$\pm$0.59 \ &
\ 2.55$\pm$0.68 \\
\hline
\end{tabular}
\end{center}
\end{table}
The table exhibits that the g.s. of $^{6}_{\Lambda}$H is bound in this
scenario much stronger than the SM estimate outlined above, and in fact it
is even more bound than predicted by Akaishi \cite{akaishi}, although the
excitation energy of the spin-flip $1^+$ level appears considerably smaller
than in his prediction. The excitation energy of the $2^+$ level comes
out about 3.4 MeV, too high with respect to the simple SM consideration.
We conclude that this scenario is unlikely, but this conclusion does not
derive from any model-independent experimental observation.
Future experiments will tell.
\section {Conclusions}
We have reported the first observation of the hyper superheavy hydrogen
$^{6}_{\Lambda}$H, based on detecting 3 candidate events that cannot be
attributed to pure instrumental or physical backgrounds. The resulting binding
energy of $^{6}_{\Lambda}$H, $B_{\Lambda}=4.0\pm 1.1$ MeV, agrees with simple
shell-model estimates initiated by Dalitz and Levi Setti \cite{dalitz_setti},
but disagrees with the prediction made by Akaishi \cite{akaishi} based on
a strongly attractive $\Lambda NN$ interaction within a coherent
$\Lambda N - \Sigma N$ mixing model. It was suggested that the excitation
energy of the $1^+$ spin-flip state with respect to the $0^+$ g.s. be
identified with the systematic difference $\Delta M=0.98\pm 0.74$ MeV between
values of $^{6}_{\Lambda}$H mass derived separately from production and from
decay. This value is consistent with the 1.04 MeV for the analogous spin-flip
excitation in $^{4}_{\Lambda}$H, confirming again the applicability of the
shell-model estimates. An experiment to produce $^{6}_{\Lambda}$H via the
($\pi^{-}, K^{+}$) reaction on $^{6}$Li at 1.2 GeV/c was recently approved
at J-PARC \cite{P10} and should run soon. The expected energy resolution is
2.5 MeV FWHM, and the expected statistics about 1--2 orders of magnitude
higher than previous KEK experiments.
\section*{Aknowledgements}
We dedicate this article to the memory of our colleague Ambrogio Pantaleo,
prematurely passed away. Dr. Pantaleo participated actively since the
beginning to the FINUDA experiment and started the study of neutron-rich
hypernuclei production in FINUDA, first results of which have been published
in \cite{nrich1}; this article marks the completion of his dedicated
contribution.
|
{
"timestamp": "2012-05-16T02:03:02",
"yymm": "1203",
"arxiv_id": "1203.1954",
"language": "en",
"url": "https://arxiv.org/abs/1203.1954"
}
|
\section{Auditing Scheme}
\label{sec:audit-auditing}
\subsection{Definitions and Auditing Framework}
\label{audit-subsec:frame_work}
We follow the literature of integrity checking of remote data \cite{Ateniese2007, Juels2007, Shacham2008, Bowers2009, CWang2010} and adapt the proposed framework to our privacy-preserving auditing system. In particular, we consider an auditing scheme which consists of four algorithms:
\begin{itemize}
\item $\ensuremath{\mathsf{KeyGen}}\xspace(1^\lambda) \rightarrow (k_v, k_e)$\quad is a key generation algorithm that is run by the user to setup the scheme. It takes a security parameter $\lambda$ as input and outputs two different private keys: $k_v$ used to generate {\em verification} metadata, and $k_e$ used to {\em encrypt} the possession proof.
\item $\ensuremath{\mathsf{TagGen}}\xspace(\vct{e}, k_v) \rightarrow t$\quad is an algorithm run by the user to generate the verification metadata. It takes as input a coded block, $\vct{e}$, a private key, $k_v$, and outputs a verification tag of $\vct{e}$, $t$.
\item $\ensuremath{\mathsf{GenProof}}\xspace(k_e, (\vct{e}_1, \cdots, \vct{e}_M), (t_{\vct{e}_1}, \cdots, t_{\vct{e}_M}), \ensuremath{\mathsf{chal}}\xspace) \rightarrow V$\quad is run by the storage node to generate a proof of possession. It takes as input a private key, $k_e$; coded blocks stored at the node, $\vct{e}_1, \cdots, \vct{e}_M$; their corresponding verification metadata, $t_{\vct{e}_1}, \cdots, t_{\vct{e}_M}$; and a challenge, \ensuremath{\mathsf{chal}}\xspace, which includes block indices and coding coefficients. It outputs a proof of possession, $V$, for the coded blocks determined by \ensuremath{\mathsf{chal}}\xspace.
\item $\ensuremath{\mathsf{VerifyProof}}\xspace(k_v, \ensuremath{\mathsf{chal}}\xspace, V) \rightarrow \{1, 0\}$\quad is run by the TPA in order to validate a proof of possession. It takes as inputs a private key, $k_v$, a challenge, $\ensuremath{\mathsf{chal}}\xspace$, and a proof of possession $V$. It returns 1 (success) if $V$ is the correct proof of possession for the blocks determined by $\ensuremath{\mathsf{chal}}\xspace$ and 0 (failure) otherwise.
\end{itemize}
An auditing system can be constructed from the above algorithms and consists of two phases:
\begin{itemize}
\item {\em Setup}: The user initializes the security parameters of the system by running \ensuremath{\mathsf{KeyGen}}\xspace. The encoded blocks are prepared as previously described in Section \ref{audit-subsec:system_model}. The user then runs {\ensuremath{\mathsf{TagGen}}\xspace} to generate verification metadata for each encoded block. Afterwards, both the encoded blocks and verification metadata are uploaded to the storage node. The encoded blocks are then deleted from the user's local storage. Finally, the user sends metadata needed to perform the audit to the TPA.
\item {\em Audit}: The TPA issues an audit message, {\em i.e.}\xspace, a {\ensuremath{\mathsf{chal}}\xspace}, to the storage node to make sure that the node correctly stores its assigned coded blocks. The node generates a proof of possession for the blocks specified in {\ensuremath{\mathsf{chal}}\xspace} by running {\ensuremath{\mathsf{GenProof}}\xspace}, and it sends the possession proof back to the TPA. Finally, the TPA runs {\ensuremath{\mathsf{VerifyProof}}\xspace} to verify the possession proof it receives.
\end{itemize}
\subsection{Basic Scheme and Key Techniques}
\label{audit-subsec:basic_scheme}
Here we describe the most basic scheme that supports remote data checking and show that it does not provide the desired properties. This basic scheme is also described in \cite{Ateniese2007}. Afterwards, we describe how we improve this basic scheme to arrive at our proposed scheme.
{\flushleft \bf The Basic Scheme.} During the {\em Setup} phase, the user precomputes a Message Authentication Code (MAC) tag, $t_i$, for each coded block, $\vct{e}_i$, using a secret key, $k_v$, and a standard MAC scheme, {\em e.g.}\xspace, \ensuremath{\mathsf{HMAC}}. The user then uploads both the tags and the coded blocks to the storage node and sends $k_v$ to the TPA. During the {\em Audit} phase, to verify that the node stores $\vct{e}_i$ correctly, the TPA issues a request for $\vct{e}_i$. The node then sends $\vct{e}_i$ and its tag $t_i$ to the TPA. The TPA can use $k_v$ and $t_i$ to check for the integrity of $\vct{e}_i$.
Although providing the possession checking, this scheme suffers from many drawbacks:
\begin{itemize}
\item It is inefficient in both computation and communication since the computation and bandwidth overhead increases linearly in the number of checked blocks.
\item It does not efficiently support {\em node repair} \cite{Dimakis2011, Dimakis2007}: It requires the user to download all the blocks necessary to compute the new (recovering) blocks. The user then computes verification tags for all the new blocks, essentially re-setting up the storage node.
\item It violates privacy because the TPA learns about the blocks. A straightforward way to provide privacy is to encrypt the response block using a standard encryption scheme, {\em e.g.}\xspace, \ensuremath{\mathsf{AES}}. However, in this case, the TPA will not be able to verify the integrity of the original block because the provided tag is not computed on the encrypted block but on the original block.
\end{itemize}
{\flushleft \bf Key Techniques.} We improve the basic scheme to arrive at our proposed scheme by leveraging a novel combination of (i) a homomorphic MAC scheme and (ii) a novel encryption scheme that exploits properties of linear network coding.
In detail, we adopt \ensuremath{\mathsf{SpaceMac}}, a homomorphic MAC scheme that we previously designed specifically for network coding \cite{LeLocate2010, LeJSAC2011}. We use {\ensuremath{\mathsf{SpaceMac}}} to generate verification tags. With \ensuremath{\mathsf{SpaceMac}}, the integrity of multiple blocks can be verified with the computation and communication cost of a single block verification, thanks to the ability to combine blocks and tags. {\ensuremath{\mathsf{SpaceMac}}} also facilitates repair as verification metadata at a newly constructed node can be computed efficiently from existing metadata at healthy nodes.
We custom design a novel encryption scheme, called \ensuremath{\mathsf{NCrypt}}\xspace, to protect the privacy of the response blocks. {\ensuremath{\mathsf{NCrypt}}\xspace} is constructed in a way that preserves the correctness of {\ensuremath{\mathsf{SpaceMac}}}: A response block, even when encrypted, can be used by the TPA for the integrity check. We stress that it is not possible to use other standard encryption schemes, such as {\ensuremath{\mathsf{AES}}}, in place of \ensuremath{\mathsf{NCrypt}}\xspace, because they will break the {\ensuremath{\mathsf{SpaceMac}}} integrity verification. The reason is that in general, a MAC tag computed on a data block can only be used to verify the integrity of the block upon the reception of the tag and the data block, but it cannot be used when the encrypted data block is received instead of the original block.
Formally, let $(\ensuremath{\mathsf{Enc}}\xspace, \ensuremath{\mathsf{Dec}}\xspace)$ denote a symmetric-key encryption scheme and $(\ensuremath{\mathsf{Mac}}, \ensuremath{\mathsf{Verify}})$ denote a MAC scheme. Let $\vct{e}$ be an (encoded) data block, and $k_e$ and $k_v$ be the keys for the encryption and MAC schemes. Let $\vct{c} = \ensuremath{\mathsf{Enc}}\xspace(k_e, \vct{e})$ and $t = \ensuremath{\mathsf{Mac}}(k_v, \vct{e})$. The encryption and MAC schemes are compatible with each other when $\ensuremath{\mathsf{Verify}}(k_v, \vct{c}, t)$ outputs 1 if and only if $\vct{c} = \ensuremath{\mathsf{Enc}}\xspace(k_e, \vct{e})$ and outputs 0 otherwise.
The main novelty of {\ensuremath{\mathsf{NCrypt}}\xspace} lies in its compatibility with \ensuremath{\mathsf{SpaceMac}}: It is carefully designed to maintains both the correctness of {\ensuremath{\mathsf{SpaceMac}}} (Theorem \ref{thm:correctness}) as well as the security of {\ensuremath{\mathsf{SpaceMac}}} (Theorem \ref{thm:PossessionSpaceMac}). {\ensuremath{\mathsf{NCrypt}}\xspace} employs the random linear combination technique of network coding and is semantically secure under a chosen-plaintext attack (CPA-secure). Next, we describe how we use {\ensuremath{\mathsf{SpaceMac}}} and {\ensuremath{\mathsf{NCrypt}}\xspace} in detail.
\subsection{The Homomorphic MAC: \ensuremath{\mathsf{SpaceMac}}}
\label{audit-subsec:spacemac}
In prior work, we originally designed $\ensuremath{\mathsf{SpaceMac}}$ and used it to combat pollution attacks in network coding \cite{LeLocate2010, LeJSAC2011, LeTESLA2011, LeInter2012}. {\ensuremath{\mathsf{SpaceMac}}} was inspired by and an improvement of another homomorphic MAC scheme, \ensuremath{\mathsf{HomMac}}, proposed by Agrawal and Boneh \cite{Agrawal2009}. The novelty of {\ensuremath{\mathsf{SpaceMac}}} and a detailed comparison between the two schemes can be found in \cite{LeLocate2010, LeJSAC2011}. Here, we adopt $\ensuremath{\mathsf{SpaceMac}}$ to support the aggregation of file blocks and tags to allow for efficient auditing (similar to \cite{Shacham2008, Bowers2009Hail}). Furthermore, as we show in Section \ref{sec:audit-repair}, {\ensuremath{\mathsf{SpaceMac}}} also facilitates efficient node repairs.
{\flushleft \bf Definition.} A ($q, n, m$) homomorphic MAC scheme is defined by three probabilistic, polynomial-time algorithms: $\ensuremath{\mathsf{Mac}}$, $\ensuremath{\mathsf{Combine}}$, and $\ensuremath{\mathsf{Verify}}$. The $\ensuremath{\mathsf{Mac}}$ algorithm generates a tag for a given block; the $\ensuremath{\mathsf{Combine}}$ algorithm computes a tag for a linear combination of some given blocks; and the $\ensuremath{\mathsf{Verify}}$ algorithm verifies whether a tag is a valid tag of a given block.
\begin{itemize}
\item $\ensuremath{\mathsf{Mac}}(k, \text{id}, \mathbf{e})$:
\begin{itemize}
\item Input: A secret key, $k$, the identifier, $\text{id}$, of the file, and a source block or encoded block, $\mathbf{e} \in \mathbb{F}^{n+m}_q$.
\item Output: Tag $t$ for $\mathbf{e}$.
\end{itemize}
\item $\ensuremath{\mathsf{Combine}}((\mathbf{e}_1,t_1,\alpha_1), \cdots, (\mathbf{e}_\ell,t_\ell,\alpha_\ell))$:
\begin{itemize}
\item Input: $\ell$ blocks, $\vct{e}_1, \cdots, \vct{e}_\ell$, their tags, $t_1, \cdots, t_\ell$, under key $k$, and their coefficients, $\alpha_1, \cdots, \alpha_\ell \in \mathbb{F}_q$.
\item Output: Tag $t$ for block $\vct{e} \overset{\text{def}}{=} \sum_{i=1}^\ell \alpha_i\,\vct{e}_i$.
\end{itemize}
\item $\ensuremath{\mathsf{Verify}}(k, \text{id}, \vct{e}, t)$:
\begin{itemize}
\item Input: A secret key, $k$, the identifier, $\text{id}$, of the file, a block, $\mathbf{e} \in \mathbb{F}^{n+m}_q$, and its tag, $t$.
\item Output: 0 (reject) or 1 (accept).
\end{itemize}
\end{itemize}
Also, the scheme must satisfy the following correctness requirement:\\
Let $t = \ensuremath{\mathsf{Combine}}((\mathbf{e}_1,t_1,\alpha_1), \cdots, (\mathbf{e}_\ell,t_\ell,\alpha_\ell)) $, then $\ensuremath{\mathsf{Verify}} \left( k, \text{id}, \sum_{i=1}^\ell \alpha_i \vct{e}_i, t \right) = 1$.
Note that the homomorphic property of the MAC scheme, or the existence of $\ensuremath{\mathsf{Combine}}$, which does not exist in regular MAC schemes, such as $\ensuremath{\mathsf{HMAC}}$, ensures that multiple blocks can be audit at the bandwidth and verification computation cost of a single block.
{\flushleft \bf Construction.} $\ensuremath{\mathsf{SpaceMac}}$ consists of a triplet of algorithms: $\ensuremath{\mathsf{Mac}}$, $\ensuremath{\mathsf{Combine}}$, and $\ensuremath{\mathsf{Verify}}$. The construction of $\ensuremath{\mathsf{SpaceMac}}$ uses a pseudo-random function (PRF) $F_1: \mathcal{K}_1 \times (\mathcal{I} \times [1,n+m]) \rightarrow \mathbb{F}_q$, where $\mathcal{K}_1$ is the PRF key domain and $\mathcal{I}$ is the file identifier domain.
\begin{itemize}
\item $\ensuremath{\mathsf{Mac}}(k, \ensuremath{\mathsf{id}}, \vct{e}) \rightarrow t$: The MAC tag $t \in \eff{}{q}$ of a source block or encoded block, denoted by $\vct{e} \in \eff{n+m}{q}$, under key $k$, can be computed by the following steps:\\
-- $\vct{r} \leftarrow (F_1 (k, \ensuremath{\mathsf{id}}, 1), \cdots, F_1 (k, \ensuremath{\mathsf{id}}, n+m))$ .\\
-- $t \leftarrow \vct{e} \cdot \vct{r} \in \mathbb{F}_q$ .
\item $\ensuremath{\mathsf{Combine}}((\mathbf{e}_1,t_1,\alpha_1), \cdots, (\mathbf{e}_\ell,t_\ell,\alpha_\ell)) \rightarrow t$: The tag $t \in \eff{}{q}$ of $\vct{e} \triangleq \sum_{i=1}^\ell \alpha_i \, \vct{e}_i \in \eff{n+m}{q}$ is computed as follows:\\
-- $t \leftarrow \sum_{i=1}^\ell \alpha_i \, t_i \in \mathbb{F}_q$ .
\item $\ensuremath{\mathsf{Verify}}(k, \ensuremath{\mathsf{id}}, \vct{e}, t) \rightarrow \{0, 1\}$: To verify if $t$ is a valid tag of $\vct{e}$ under key $k$, we do the following:\\
-- $\vct{r} \leftarrow (F_1 (k, \ensuremath{\mathsf{id}}, 1), \cdots, F_1 (k, \ensuremath{\mathsf{id}}, n+m))$ .\\
-- $t' \leftarrow \vct{e} \cdot \vct{r}$ .\\
-- If $t' = t$, output 1 (accept); otherwise, output 0 (reject).
\end{itemize}
\begin{lemma}[Theorem 1 in \cite{LeLocate2010}]\label{lemma:SpaceMac}
Assume that $F_1$ is a secure PRF. For any fixed $q$, $n$, $m$, {\ensuremath{\mathsf{SpaceMac}}} is a secure $(q, n, m)$ homomorphic MAC scheme.
\end{lemma}
We refer the reader to \cite{LeLocate2010} for the security game and proof of \ensuremath{\mathsf{SpaceMac}}. We provide security proof of $\ensuremath{\mathsf{SpaceMac}}$ when used in {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} in Section \ref{audit-subsec:possession}. If the user computes the verification tags for the source blocks using the {\ensuremath{\mathsf{Mac}}} algorithm of {\ensuremath{\mathsf{SpaceMac}}}, then the storage node can compute a valid MAC tag for any encoded block using the {\ensuremath{\mathsf{Combine}}} algorithm. The security of {\ensuremath{\mathsf{SpaceMac}}} guarantees that if a block, $\vct{e}'$, is not a linear combination of the source blocks, then the storage node can only forge a valid MAC tag for $\vct{e}'$ with probability $\frac{1}{q}$. The security when using $\ell$ tags is improved to $\frac{1}{q^\ell}$. For clarity, we focus on a single file $\mathcal{F}$ and thus omit the file identifier {\ensuremath{\mathsf{id}}} used by the above three algorithms in our subsequent discussion.
\subsection{The Random Linear Encryption: \ensuremath{\mathsf{NCrypt}}\xspace}
\label{audit-subsec:ncrypt}
To protect the privacy of the response file block, we need to encrypt it. The encryption, however, needs to still allow for the verification of the block. To this end, we design a novel encryption scheme that is compatible with {\ensuremath{\mathsf{SpaceMac}}}, called {\ensuremath{\mathsf{NCrypt}}\xspace}. In particular, {\ensuremath{\mathsf{NCrypt}}\xspace} will protect $n-2$ elements of the response block while still allowing {\ensuremath{\mathsf{SpaceMac}}} integrity checking. The remaining 2 elements are random padded elements. These 2 elements are needed to guarantee the security of the schemes, as we will show in the construction and proofs of {\ensuremath{\mathsf{NCrypt}}\xspace} and {\ensuremath{\mathsf{SpaceMac}}}\footnote{In particular, the 2 random padded elements is to control the number of equations in the system of equations $\Pi_1$ and $\Pi_2$ described in the proofs of Theorems \ref{thm:NCrypt} and \ref{thm:PossessionSpaceMac}, respectively. Intuitively, these 2 random elements are needed to compensate for the extra information learned by the adversary in {\ensuremath{\mathsf{NCrypt}}\xspace} (the element $p$ as part of the ciphertext) and in {\ensuremath{\mathsf{SpaceMac}}} (the equations related to $\bar{\vct{r}}$).}.
Let $\bar{\vct{x}}$ denote the vector formed by the first $n-2$ elements of a vector $\vct{x}$. The construction of $\ensuremath{\mathsf{NCrypt}}\xspace$ uses two PRFs: $F_2: \mathcal{K}_2 \times ([1,n-1] \times [1,n-2]) \rightarrow \mathbb{F}_q$ and $F_3: \mathcal{K}_2 \times (\{0,1\}^\lambda \times [1,n-1]) \rightarrow \mathbb{F}_q$, where $\mathcal{K}_2$ is a PRF key domain. {\ensuremath{\mathsf{NCrypt}}\xspace} consists of three probabilistic, polynomial time algorithms:
\begin{itemize}
\item $\ensuremath{\mathsf{Setup}}(k, \bar{\vct{r}}) \rightarrow (p_1, \cdots, p_{n-1})$: This algorithm is run by the user to setup the encryption scheme. It takes as input a secret key $k$ and a vector $\bar{\vct{r}} \neq \vct{0}, \bar{\vct{r}} \in \eff{n-2}{q}$. It outputs $n-1$ elements in $\eff{}{q}\,$, which are called {\em auxiliary elements} and are used by the encryption. The details are as follows:\\
-- Compute $\bar{\vct{p}}_i \leftarrow ( F_2 (k, i, 1), \cdots, F_2 (k, i, n-2) ) \in \eff{n-2}{q}$, for $i \in [1,n-1]$.\\
-- Compute $p_i \leftarrow \bar{\vct{r}} \cdot \bar{\vct{p}}_i \in \eff{}{q}$, for $i \in [1,n-1]$.
\item $\ensuremath{\mathsf{Enc}}\xspace(k, \bar{\vct{e}}, (p_1, \cdots, p_{n-1}) ) \rightarrow \langle \bar{\vct{c}}, (r, p) \rangle$: This algorithm is run by the storage node to encrypt the $n-2$ first elements of the aggregated response block. It takes as input a secret key, $k$, vector formed by the first $n-2$ elements of the response block, $\bar{\vct{e}}$, and the auxiliary elements, $p_1, \cdots, p_{n-1}$. It computes the encryption, $\langle \bar{\vct{c}}, (r, p) \rangle$, of $\bar{\vct{e}}$ as follows:\\
-- Compute $\bar{\vct{p}}_i, i \in [1,n-1]$, using key $k$ as in \ensuremath{\mathsf{Setup}}.\\
-- Choose $r$ uniformly at random: $r \overset{R}{\leftarrow} \{0, 1\}^\lambda$.\\
-- Compute the {\em masking coefficients}: $\beta_i \leftarrow F_3 (k, r, i) \in \eff{}{q}, \text{for } i \in [1,n-1]\,.$\\
-- Compute the {\em masking vector}: $\bar{\vct{m}} \leftarrow \sum_{i=1}^{n-1} \beta_{i}\,\bar{\vct{p}}_i \in \eff{n-2}{q}\,.$\\
-- Compute $\bar{\vct{c}} \leftarrow \bar{\vct{e}} + \bar{\vct{m}} \in \eff{n-2}{q}$.\\
-- Compute $p \leftarrow \sum_{i=1}^{n-1} \beta_i \, p_i \in \eff{}{q}\,.$
In essence, the data is masked with a randomly chosen vector $\bar{\vct{m}} \in \lspan{ \bar{\vct{p}}_1, \cdots, \bar{\vct{p}}_{n-1} }$.
\item $\ensuremath{\mathsf{Dec}}\xspace(k, \langle \bar{\vct{c}}, (r, p) \rangle) \rightarrow \bar{\vct{e}}$: This algorithm takes as input a secret key, $k$, and the cipher text, $\langle \bar{\vct{c}}, (r, p) \rangle$. The decryption is done as follows:\\
-- Compute $\bar{\vct{p}}_i, i \in [1,n-1]$, using key $k$ as in \ensuremath{\mathsf{Setup}}.\\
-- Compute $\beta_i \leftarrow F_3 (k, r, i) \in \eff{}{q}$, for $i \in [1,n-1]$.\\
-- Compute $\bar{\vct{m}} \leftarrow \sum_{i=1}^{n-1} \beta_{i}\,\bar{\vct{p}}_i \in \eff{n-2}{q}$.\\
-- Compute $\bar{\vct{e}} \leftarrow \bar{\vct{c}} - \bar{\vct{m}} \in \eff{n-2}{q}$.
\end{itemize}
\begin{theorem}\label{thm:NCrypt}
Assume that $F_2$ and $F_3$ are secure PRFs, then {\ensuremath{\mathsf{NCrypt}}\xspace} is a fixed-length private-key encryption scheme for messages of length $(n-2) \times \log_2 q$ that has indistinguishable encryptions under a chosen-plaintext attack.
\end{theorem}
\begin{IEEEproof}
Intuitively, the security of {\ensuremath{\mathsf{NCrypt}}\xspace} holds because $\bar{\vct{m}}$ looks completely random to an adversary who observes a ciphertext $ \langle \bar{\vct{c}}, (r, p) \rangle $ since it is computationally difficult for the adversary to compute $\beta_i$'s without knowing the secret key $k$.
The proof follows a textbook technique used to prove the security of Construction 3.24 in \cite{Katz2007}. We follow the notation in \cite{Katz2007}. Denote the CPA security experiment of an encryption scheme $\Pi = (\ensuremath{\mathsf{Setup}}, \ensuremath{\mathsf{Enc}}\xspace, \ensuremath{\mathsf{Dec}}\xspace)$ and an adversary $\mathcal{A}$ by $\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi}$. The game is as follows:
\begin{itemize}
\item A key $k$ is chosen uniformly at random from $\{0,1\}^\lambda$.
\item The adversary $\mathcal{A}$ is given $\bar{\vct{r}}, p_1, \cdots, p_{n-1}$, and oracle access to $\ensuremath{\mathsf{Enc}}\xspace_k$. $\mathcal{A}$ outputs a pair of messages $\bar{\vct{e}}_0$ and $\bar{\vct{e}}_1$, both are in $\eff{n-2}{q}$.
\item A random bit $b \leftarrow \{0,1\}$ is chosen, and then a ciphertext $c \leftarrow \ensuremath{\mathsf{Enc}}\xspace(k, \bar{\vct{e}}_b, (p_1, \cdots, p_{n-1}))$ is computed and given to $\mathcal{A}$. We call $c$ the challenge ciphertext.
\item The adversary $\mathcal{A}$ continues to have oracle access to $\ensuremath{\mathsf{Enc}}\xspace_k$, and outputs a bit $b'$.
\item The output of the experiment is defined to be 1 if $b' = b$, and 0 otherwise. In case $\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi} = 1$, we say that $\mathcal{A}$ succeeded.
\end{itemize}
Let $\Pi_1$ be an encryption scheme that is exactly the same as $\Pi$ except that a truly random function $f_2$ is used in place of $F_2$. Let Adv$[\mathcal{B}, F_2]$ be the probability of an adversary $\mathcal{B}$ with similar runtime to $\mathcal{A}$ winning the PRF security game (can tell a pseudo-random function $F_2$ from a truly random function $f_2$). By the security of PRF, we have that $\text{Adv}[\mathcal{B}, F_2]$ is negligible in $\lambda$ and it can be shown that (details are provided in the proof of Construction 3.24 in \cite{Katz2007})
\begin{equation}\label{audit-eq:PRF2}
\text{Adv}[\mathcal{B}, F_2] = | \text{Pr}[\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi} = 1] - \text{Pr}[\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi_1} = 1] |\,.
\end{equation}
Similarly, let $\Pi_2$ be an encryption scheme that is exactly the same as $\Pi_1$ except that a truly random function $f_3$ is used in place of $F_3$. Let Adv$[\mathcal{C}, F_3]$ be the probability of an adversary $\mathcal{C}$ with similar runtime to $\mathcal{A}$ winning the PRF security game. Similar to the above, by the security of PRF, we have that $\text{Adv}[\mathcal{B}, F_3]$ is negligible in $\lambda$ and
\begin{equation}\label{audit-eq:PRF3}
\text{Adv}[\mathcal{C}, F_3] = | \text{Pr}[\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi_1} = 1] - \text{Pr}[\mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi_2} = 1] |\,.
\end{equation}
We claim that for every adversary $\mathcal{A}$ that makes at most $g(\lambda)$ queries to its encryption oracle, where $g$ is a polynomial function, we have
\begin{align}{\label{audit-eq:CPA}}
\text{Pr}\left[ \mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi_2} = 1 \right] \leq \frac{1}{2} + \frac{g(\lambda)}{2^\lambda}\,.
\end{align}
Let $r_c$ denote the random string used when generating the challenge ciphertext, which is of the form $\langle \bar{\vct{c}}, (r_c, p) \rangle$ (by encrypting $\bar{\vct{e}}_b$). There are two cases:
{\em (a) $r_c$ is never used by the oracle in the encryption algorithm to produce ciphertext to answer any of $\mathcal{A}$'s queries:} In the following, we will show that each element of {\em any} plaintext $\bar{\vct{e}}$ is masked with a uniformly random value, thus the adversary will not be able to tell which message ($\bar{\vct{e}}_0$ or $\bar{\vct{e}}_1$) was encrypted, as in the case of one-time pad.
Parse $\bar{\vct{e}}$ as $(e^{(1)}, \cdots, e^{(n-2)})$, $\bar{\vct{m}}$ as $(m^{(1)}, \cdots, m^{(n-2)})$, and $\bar{\vct{p}}_i$ as $(p^{(1)}_i, \cdots, p^{(n-2)}_i)$. From a ciphertext returned from an oracle query of $\bar{\vct{e}}$, the adversary can construct the following system of equations $\Pi_1$ by subtracting the query plaintext from the ciphertext:
\[
(\Pi_1) \quad
\begin{cases}
\beta_1 \, p^{(1)}_1 + \cdots + \beta_{n-1} \, p^{(1)}_{n-1} &= m^{(1)}\\
\cdots\\
\beta_1 \, p^{(n-2)}_1 + \cdots + \beta_{n-1} \, p^{(n-2)}_{n-1} &= m^{(n-2)}\\
\beta_1 p_1 + \cdots + \beta_{n-1} p_{n-1} &= p
\end{cases}
\,.
\]
Note that $p^{(j)}_i$ are not all zeros w.h.p. since they are chosen uniformly at random from $\eff{}{q}$ by $f_2$. Let $\beta_i$ be unknowns, $i \in [1,n-1]$. The above system of $n-1$ linear equations is consistent regardless of the values of $m^{(j)}$'s since the rank of the coefficient matrix is at most $n-1$, which is the number of unknowns. Let $s$ be the rank of the coefficient matrix. Now for any $w \in [1,n-2]$, assume that all $m^{(j)}, j \neq w, j \in [1,n-2]$, are fixed. Then $m^{(w)}$ still can take any value in \eff{}{q} equally likely because (i) for any value of $m^{(w)}$, there is the same number of solutions, which is $q^{n - 1 - s}$, and (ii) $\beta_j$ are chosen uniformly at random from $\eff{}{q}$ (as a truly random function $f_3$ is used in place of $F_3$). Thus, each element of the plaintext, $e^{(w)}$, is masked with a uniformly random value, $m^{(w)}$, independent of other masking elements $m^{j \neq w}, j \in [1, n-2]$. Therefore, the probability that $\mathcal{A}$ outputs $b' = b$ is exactly 1/2, as in the case of the one-time pad.
{\em (b) $r_c$ is used by the oracle to answer at least one of $\mathcal{A}$'s queries:} In this case, $\mathcal{A}$ may easily determine which of its messages was encrypted. This is because whenever the oracle returns a ciphertext, $\langle \bar{\vct{c}}, (r, p) \rangle$, it learns the masking vector $\bar{\vct{m}}$ associated with $r$ as $\bar{\vct{m}} = \bar{\vct{c}} - \bar{\vct{e}}$. Thus, by leveraging the corresponding $\bar{\vct{m}}$ of $r_c$, the adversary can tell if $\bar{\vct{e}}_0$ or $\bar{\vct{e}}_1$ was encrypted by actually decrypting the challenge response. Since $\mathcal{A}$ makes at most $g(\lambda)$ queries, and $r$ is chosen uniformly at random, the probability of this event is at most $g(\lambda) / 2^\lambda$.
Equation (\ref{audit-eq:CPA}) follows from (a) and (b). Equations (\ref{audit-eq:PRF2}), (\ref{audit-eq:PRF3}), and (\ref{audit-eq:CPA}) show that \[\text{Pr}\left[ \mathsf{PrivK}^\mathsf{cpa}_{\mathcal{A}, \Pi} = 1 \right] \leq \frac{1}{2} + \frac{g(\lambda)}{2^\lambda} + \epsilon(\lambda)\,,\]
where $\epsilon$ is a cryptographically negligible function in $\lambda$. This completes the proof.
\end{IEEEproof}
\subsection{The Privacy-Preserving Auditing Scheme: \ensuremath{\mathsf{NC\text{-}Audit}}\xspace}
\label{audit-subsec:audit-auditing}
Now we are ready to describe our symmetric-key based auditing protocol, called \ensuremath{\mathsf{NC\text{-}Audit}}\xspace. In particular, \ensuremath{\mathsf{NC\text{-}Audit}}\xspace is built from a novel combination of {\ensuremath{\mathsf{SpaceMac}}} and {\ensuremath{\mathsf{NCrypt}}\xspace} as follows:
{\flushleft \em Setup phase:}
\begin{itemize}
\item The user divides the file into $m$ blocks of size $n-2$ instead of $n$ and pads to each block two random elements in $\eff{}{q}$. This is necessary as {\ensuremath{\mathsf{NCrypt}}\xspace} encrypts only the first $n-2$ elements. We still denote each padded block with its coding coefficients by $\vct{b}_i, i \in [1,m]$.
\item The user runs {\ensuremath{\mathsf{KeyGen}}\xspace} to generate MAC verification key, $k_v$, and encryption key, $k_e$:\\
-- $\ensuremath{\mathsf{KeyGen}}\xspace(1^\lambda) \rightarrow (k_e, k_v)$: $k_e \overset{R}{\leftarrow} \{0, 1\}^\lambda, k_v \overset{R}{\leftarrow} \{0, 1\}^\lambda$.
\item The user then setups the encryption scheme by computing the auxiliary elements, $p_1, \cdots, p_{n-1}$:\\
-- $\bar{\vct{r}} \leftarrow (F_1 (k_v, 1), \cdots, F_1 (k_v, n-2))$.\\
-- $(p_1, \cdots, p_{n-1}) \leftarrow \ensuremath{\mathsf{Setup}}(k_e,\bar{\vct{r}})$.
\item Afterward, the user computes a tag for each source block $\vct{b}_i$ using the {\ensuremath{\mathsf{Mac}}} algorithm of {\ensuremath{\mathsf{SpaceMac}}}:\\
-- $t_{\vct{b}_i} = \ensuremath{\mathsf{Mac}}(k_v, \vct{b}_i)$.
\item The user computes MAC tags of encoded blocks using the {\ensuremath{\mathsf{Combine}}} algorithm of {\ensuremath{\mathsf{SpaceMac}}}. Assume $\vct{e} = \sum_{i=1}^{m} \alpha_i \, \vct{b}_i$, then its tag is computed as follows:\\
-- $\ensuremath{\mathsf{TagGen}}\xspace(\vct{e}, k_v) \rightarrow t_{\vct{e}} = \sum_{i=1}^{m} \alpha_i \, t_{\vct{b}_i}$.
\item Finally, the user sends the encoded blocks, $\vct{e}_1, \cdots, \vct{e}_M$, their tags, $t_{\vct{e}_1}, \cdots, t_{\vct{e}_M}$, the auxiliary elements, $p_1, \cdots, p_{n-1}$, and the encryption key, $k_e$, to the storage node. The user also sends the coding coefficients, $\aug{\vct{e}_1}, \cdots, \aug{\vct{e}_M}$, and the MAC key, $k_v$, to the TPA. We assume that the user uses private and authentic channels to send $k_v$ and $k_e$\footnote{Exchanging secret keys, in particular, and establishing secure and authentic channels, in general, could be done with the support of a public key infrastructure (PKI). This is an important, well studied problem in the cryptography community and is orthogonal to this work.}.
The user then keeps the coding coefficients and the keys but delete all other data.
\end{itemize}
Note that maintaining coding coefficients is necessary for the repair process and is an inherent characteristic of NC storage systems. The overhead of storing the coefficients is negligible compared to the outsource data and could be constant for practical purposes (see Section \ref{audit-subsec:storageOverhead}). If the user outsources the management of the nodes to a third party, such as a proxy as in NCCloud \cite{Hu2012}, then he/she does not need to store the coding coefficients. However, in this case, the proxy must be trusted.
{\flushleft \em Audit phase:}
\begin{itemize}
\item The TPA chooses a set of indexes of blocks to be audited, $\mathcal{I} \subseteq [1,M]$, and chooses the coefficients for these blocks uniformly at random: $\alpha_i \overset{R}{\leftarrow} \eff{}{q}, i \in \mathcal{I}$. The challenge includes the indexes of the blocks and their corresponding coefficients:\\
-- Prepare $\ensuremath{\mathsf{chal}}\xspace = \{ (i, \alpha_i) \,|\, i \in \mathcal{I} \}$.
\item {\ensuremath{\mathsf{GenProof}}\xspace} run by the storage node to generate the proof of storage, $V$, is implemented as follows:\\
-- Compute the aggregated block: $\hat{\vct{e}} = \sum_{i \in \mathcal{I}} \alpha_i \, \hat{\vct{e}}_i$. Parse $\hat{\vct{e}}$ as $(\bar{\vct{e}}, e^{(n-1)}, e^{(n)})$.\\
-- Compute the aggregated tag: $t = \sum_{i= \in \mathcal{I}} \alpha_i \, t_{\vct{e}_i}$.\\
-- Encrypt the response block: $\langle \bar{\vct{c}}, (r, p) \rangle \leftarrow \ensuremath{\mathsf{Enc}}\xspace(k_e, \bar{\vct{e}}, (p_1, \cdots, p_{n-1}))$.\\
The node then sends $V = (\langle \bar{\vct{c}}, (r, p) \rangle, e^{(n-1)}, e^{(n)}, t)$ back to the TPA.
\item $\ensuremath{\mathsf{VerifyProof}}\xspace$ run by the TPA to verify the proof $V$ is implemented as follows:\\
-- Compute coefficients of $\hat{\vct{e}}$: $\aug{\vct{e}} = \sum_{i \in \mathcal{I}} \alpha_i \, \aug{\vct{e}_i}$.\\
-- Let $\vct{c} = (\bar{\vct{c}} \, | \, e^{(n-1)} \,|\, e^{(n)} \, | \, \aug{\vct{e}})$, where ``$|$'' denotes augmentation.\\
\hspace*{2mm} Return result of $\ensuremath{\mathsf{Verify}}(k_v, \vct{c}, t + p)$.
\end{itemize}
{\flushleft \bf Correctness.} The correctness of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, {\em i.e.}\xspace, if the file is correct then the algorithm will accept the proof, is guaranteed by the following Lemma \ref{thm:correctness}. And its security, {\em i.e.}\xspace, if there is corruption then the algorithm will reject the proof, is proved in Section \ref{sec:audit-security}.
\begin{lemma}\label{thm:correctness}
If the storage node follows {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} and computes the aggregated response block using uncorrupted blocks, then the TPA will accept the proof.
\end{lemma}
\begin{IEEEproof}
Let $\vct{r} = (F_1 (k_v, 1), \cdots, F_1 (k_v, n+m))$.
Note that
\begin{align*}
\vct{c} &= (\bar{\vct{c}}\,|\, e^{(n-1)} \,|\, e^{(n)}\,|\, \aug{\vct{e}})\\
~ &= ((\bar{\vct{e}} + \bar{\vct{m}})\,|\, e^{(n-1)} \,|\, e^{(n)} \,|\, \aug{\vct{e}}) = \vct{e} + (\bar{\vct{m}}\,|\, 0, \cdots, 0)\,.
\end{align*}
Thus, in the {\ensuremath{\mathsf{Verify}}},
\begin{align*}
t' &= \vct{c} \cdot \vct{r} = \vct{e} \cdot \vct{r} + \bar{\vct{m}} \cdot \bar{\vct{r}}\\
~ &= t + \sum_{i=1}^{n-1} \beta_i \, \bar{\vct{p}}_i \cdot \bar{\vct{r}} = t + \sum_{i=1}^{n-1} \beta_i \, p_i = t + p\,.
\end{align*}
Therefore, {\ensuremath{\mathsf{Verify}}} returns 1. Hence, the TPA accepts the proof.
\end{IEEEproof}
\section{Conclusion}
\label{sec:audit-conclusion}
In this work, we propose {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, a cryptography-based remote data integrity checking scheme for NC-based storage systems. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is based on a novel combination of an existing MAC scheme custom made for network coding, $\ensuremath{\mathsf{SpaceMac}}$, and a novel CPA-secure encryption scheme, {\ensuremath{\mathsf{NCrypt}}\xspace}, which we carefully design in this work to work in synergy with $\ensuremath{\mathsf{SpaceMac}}$. To the best of our knowledge, {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is the first scheme that efficiently supports auditing for NC storage systems. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} also provides protection against leakage of the outsourced data when the audit is done by a third party. Our evaluation results based on a real implementation in Java demonstrate that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is significantly more efficient than the state-of-the-art schemes.
\section{Performance Evaluation}
\label{sec:audit-evaluation}
\subsection{Client Storage Overhead}
\label{audit-subsec:storageOverhead}
{\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} requires the user and the TPA to store the coding coefficients, which is in $O (m M N)$ space. The user needs the coefficients to carry out repairs while the TPA needs the coefficients to carry out audits. In any case, the overhead of $O (m M N)$ is orders of magnitude less than the outsourced data, which is in $O ( (n+m) M N )$ space; this is because $n \gg m$ for NC-based storage systems. In fact, in a practical NC storage cloud, the space necessary for storing the coding coefficients could be kept less than 160 B ({\em i.e.}\xspace, constant storage) while being able to support arbitrary file size (by increasing the block size $n$, see Section 5.1 of NCCloud \cite{Hu2012}). Table \ref{tab:comparison} compares client storage overhead of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} and other state-of-the-art schemes \cite{QWang2009, CWang2010, Chen2010}.
\subsection{Bandwidth Overhead}
\label{audit-subsec:bandwidthOverhead}
{\flushleft \bf Integrity Checking.}\quad For each audit round, the major communication cost is the cost of sending the proof of possession from the storage node to the TPA, which is dominated by the size of the (encrypted) data bock. Thanks to homormophic property of {\ensuremath{\mathsf{SpaceMac}}}, blocks in the challenge can be aggregated. We achieve similar bandwidth overhead compared to prior schemes for integrity checking of cloud data \cite{Shacham2008, QWang2009, Chen2010, CWang2010}. In particular, the proof of possession for multiple blocks contains only a single block (of size varying from 4 KB \cite{Ateniese2007} to 1.6 MB \cite{Chen2010}).
{\flushleft \bf Repairing.}\quad As discussed in Section \ref{sec:audit-repair}, when using {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, the user does not need to download any data block to repair a failed node. This stands in stark contrast with the state-of-the-art scheme for NC storage systems \cite{Chen2010}, where the user needs to download an amount of data equal to the repair bandwidth to setup integrity metadata for the new coded blocks him/herself.
{\flushleft \bf Encryption.}\quad The amount of additional bandwidth to support encryption is small. In particular, {\ensuremath{\mathsf{NCrypt}}\xspace} requires the storage node to send with the encrypted block, $\bar{\vct{c}}$; the random value, $r$, of size $\lambda$ (typically 80 bits \cite{Ateniese2007}); the auxiliary tag, $p$, and the random padding elements, $e^{(n-1)}, e^{(n)}$, which are of size $\log_2 q$. These are negligible compared to the block size: $n \log_2 q$, {\em e.g.}\xspace, 0.3\% for $q = 2^8, n=4 \times 2^{10}$ (4 KB block).
The bandwidth overhead of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} when compared to other schemes \cite{QWang2009, CWang2010, Chen2010} are summarized in Table \ref{tab:comparison}.
\subsection{Computational Overhead}
\label{audit-subsec:computationOverhead}
\begin{table*}[t]
\begin{center}
{\footnotesize
\begin{tabular}{|l|l|l|l|l|l|}
\hline
~ & ~ & {\bf Wang 2009 \cite{QWang2009}} & {\bf Wang 2010 \cite{CWang2010}} & {\bf Chen 2010 \cite{Chen2010}} & {\bf \ensuremath{\mathsf{NC\text{-}Audit}}\xspace}\\
\hline
\multirow{3}{*}{ \bf Features } & ~ & Public-Key Audit & Public-Key Audit & Private-Key Audit & Private-Key Audit\\
\cline{3-6}
& ~ & {\em No} NC Repair & {\em No} NC Repair & NC Repair & {\em Efficient} NC Repair\\
\cline{3-6}
& ~ & {\em No} Audit Privacy & Audit Privacy & {\em No} Audit Privacy & Audit Privacy \\
\hline
\multirow{2}{*}{ \bf Client Storage } & Audit Overhead & $O(1)$ & $O(1)$ & $O(1)$ & $O(mMN)$\\
\cline{2-6}
& Repair Overhead & N/A & N/A & $O(mMN)$ & $O(mMN)$\\
\hline
\multirow{3}{*}{ \bf Bandwidth } & Audit Overhead & 1 block & 1 block & 1 block & 1 block\\
\cline{2-6}
& Repair Overhead & N/A & N/A & repair bandwidth & 0*\\
\cline{2-6}
& Enc. Overhead & N/A & 0* & N/A & 0*\\
\hline
\multirow{4}{*}{ \bf Computation } & Security & \multicolumn{4}{|c|}{80-bit}\\
\cline{2-6}
& Parameters & \multicolumn{4}{|c|}{300 blocks per challenge, 4 KB block size}\\
\cline{2-6}
& Testbed Config. & \multicolumn{2}{|c|}{1.86 Ghz CPU, 2GB RAM} & \multicolumn{2}{|c|}{2.8 Ghz CPU, 32 GB RAM}\\
\cline{2-6}
& Server Overhead & 270 ms & 273 ms & 3.19 ms & 4.69 ms\\
\cline{2-6}
& Auditor Overhead & 491 ms & 493 ms & 2.76 s & 0.73 ms\\
\hline
\end{tabular}
\end{center}
\caption{Comparison of different remote data integrity checking schemes. 0* indicates no data block needs to be downloaded by the user to support the feature. N/A means not applicable due to the lack of support.}
\label{tab:comparison}
\vspace*{-20pt}
\end{table*}
We first analyze the cost of each operation in {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} by the number of finite field multiplications involved, which is the dominating cost factor. We then present the cost of each operation from our real implementation in Java. We omit the cost of computing PRF values that do not take as input random seeds since they can be precomputed.
{\flushleft \bf Integrity Checking with Encryption:}
{\flushleft \em 1. Storage Node Overhead:} In {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, the cost to compute a proof of possession includes the cost to compute (i) the aggregated response block, $\bar{\vct{e}}$, (ii) the response tag, $t$, (iii) the masking vector, $\bar{\vct{m}}$, and the auxiliary element, $p$. The total cost is dominated by the cost to compute $\bar{\vct{e}}$ and $\bar{\vct{m}}$. $\bar{\vct{m}}$ can be precomputed in advance as it is independent of the challenge. Let $C$ be the average number of blocks specified in a challenge. The average cost to compute a response per challenge is $C \times n$ multiplications with a precomputation of $\bar{\vct{m}}$ and $C\times n + (n-2) \times (n-1)$ without.
{\flushleft \em 2. TPA Overhead:} In {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, verifying a proof of possession can be done very efficiently. In particular, the cost to verify include the time to (i) compute the coefficients of the response block and (ii) run the $\ensuremath{\mathsf{Verify}}$ of $\ensuremath{\mathsf{SpaceMac}}$. Let $\ell$ be the number of tags used (to increase the security to $1/q^\ell$). The total cost is $C \times m + \ell \times (n+m)$ multiplications.
{\flushleft \bf Repairing:}
As described in Section \ref{sec:audit-repair}, repairing a failed node does not incur any computation cost at the user side to maintain the security metadata of the auditing.
{\flushleft \bf Implementation:}
We implement {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} in Java to compare its performance with the state-of-the-art schemes \cite{QWang2009, CWang2010, Chen2010}. For a fair of comparison with \cite{QWang2009, CWang2010}, we use $q = 2^8$ and $\ell = 10$ to provide 80-bit security, and we also set block size to 4 KB ($n = 4 \times 2^{10}$), $m=500$, and the number of blocks indicated by a challenge to $C = 300$. We stress that the choice of parameters may be different in a practical NC storage system, {\em e.g.}\xspace, in \cite{Hu2012}, a block size could be as big as 4 MB while the storage space taken by the coefficients could be kept below 160 B. We implement finite field multiplications in $\eff{}{2^8}$ by table look-ups and additions using XORs.
We also precomputed values that do not depend on the challenges.
Table \ref{tab:comparison} compares the computational overhead of different remote data integrity checking schemes. The reported numbers for \cite{QWang2009} and \cite{CWang2010} are taken from \cite{CWang2010}. (The overhead of the scheme in \cite{QWang2009} is similar to the public-key based scheme in \cite{Shacham2008}.) We refer the reader to \cite{CWang2010} for the detailed setup. We implement the checking scheme in \cite{Chen2010} ourselves. For this scheme, we use AES with CBC mode from the Java {\em crypto} library to decrypt coding coefficients. We refer the reader to Appendix A in \cite{Chen2010} for the detailed description of this scheme. The number reported for {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} and the scheme in \cite{Chen2010} are the average of 100 runs on a computer with 2.8 Ghz CPU and 32 GB RAM. We note that among the three schemes under comparison \cite{QWang2009, CWang2010, Chen2010}, the scheme in \cite{Chen2010} is the only one specifically designed for NC storage systems and thus supports NC repair.
Table \ref{tab:comparison} shows that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} manages to achieve very modest computational overhead. The computational overhead of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is orders of magnitude smaller than those of \cite{QWang2009} and \cite{CWang2010}. This is due to the fact that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is symmetric-key based while the schemes in \cite{CWang2010} and \cite{QWang2009} are public-key based and make heavily use of expensive bilinear mapping operations\footnote{Due to the fundamental difference: the use of expensive bilinear mapping operations in \cite{CWang2010, QWang2009}, we expect a similar gap (in order of magnitude) between the computational overhead of \cite{CWang2010, QWang2009} and that of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} when we run them on the same hardware.}. The scheme in \cite{Chen2010} achieves similar storage node's computational overhead to {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} as it is also symmetric-key based. However, due to the cost of executing $C \times m = 150,000$ numbers of decryption for the coefficients, the computational overhead of the TPA of \cite{Chen2010} is much larger than that of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, in the order of seconds as opposed to milliseconds.
\section{Problem Formulation}
\label{sec:audit-formulation}
\subsection{System Model and Operations}
\label{audit-subsec:system_model}
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{service.pdf}
\vspace{-10pt}
\caption{Parties and Steps Involved in {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}.}
\label{fig:CSP}
\vspace{-15pt}
\end{figure}
Fig. \ref{fig:CSP} illustrates an overview of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}. We consider a cloud storage service that involves three entities: a user, NC-based storage nodes, which make up the storage cloud, and a third party auditor (TPA). The user distributes his/her data on the storage nodes. The user resorts to a TPA to check for the integrity of the data stored at each node; at the same time, he/she does not want the TPA to learn about the outsourced data. We assume that {\em the user is responsible for repairing of a failed node.} The user here acts as a proxy that manages the storage nodes as in the case of NCCloud \cite{Hu2012}. Our work is also applicable to scenarios where there is a cloud service provider, who is independent from the user and acts as the proxy.
The user follows the following basic steps to store his/her data on the storage cloud. We adopt the notation used in \cite{LeTESLA2011}. Denote the original file by $\mathcal{F}$. The user first divides $\mathcal{F}$ into $m$ blocks, $\hat{\vct{b}}_1, \cdots, \hat{\vct{b}}_m$. Each block is a vector in an $n$-dimensional linear space $\eff{n}{q}$, where $\eff{}{}$ is a finite field of size $q$. To facilitate the decoding, the user then augments each block $\hat{\vct{b}}_i$ with its $m$ {\em global coding coefficients}. The resulting blocks, $\vct{b}_i$, have the following form:
\[
\vct{b}_i = (\, \overbrace{\textrm{---}\vct{\hat{b}}_i\textrm{---}}^n, \overbrace{\underbrace{0, \cdots, 0, 1}_i, 0, \cdots, 0}^m)\,\in \mathbb{F}^{n+m}_q\,.
\]
We call $\vct{b}_i$ {\em source blocks} and the space spanned by them {\em source space}, denoted by $\Pi$. We use \aug{\vct{b}_i} to denote the coefficients of $\vct{b}_i$. Typically, $n \gg m$, and this presentation is also called an $n$-extended version of a storage code \cite{Dikaliotis2010}.
The user then creates a number of encoded blocks using an appropriate linear coding scheme for the desired reliability, {\em e.g.}\xspace, an array MDS Evenodd code is used in Fig. \ref{fig:repair}. Each encoded block is a linear combination of the source blocks. Note that if an encoded block $\vct{e}$ equals $\sum_{i=1}^m \alpha_i \, \vct{b}_i$, then the last $m$ coordinates of $\vct{e}$ are exactly the coding coefficients $\alpha_i$'s. These encoded blocks are then distributed across the $N$ storage nodes of the storage cloud. Let $M$ be the number of encoded blocks stored at a storage node, $P$ be the number of healthy nodes that need to send the (encoded) repair blocks, and $Q$ be the number of repair blocks each healthy node needs to send to the new node. In the example given in Fig. \ref{fig:repair}, $m=4$, $N=4$, $M=2$, $P=3$, and $Q=1$.
\subsection{Threat Model}
\label{audit-subsec:threat_model}
We adopt the threat model considered in \cite{CWang2010} and \cite{Yu2010}. In particular, we consider semi-trusted storage nodes that behave properly and do not deviate from the prescribed auditing protocol. However, for their own benefits, they may deliberately delete rarely accessed, archival user's data to reduce operational cost; they may also decide to hide data corruptions, caused by either internal or external factors to maintain reputation. For clarity, we focus our discussion on a single storage node except when discussing the repair process.
We assume that the TPA, who is in the business of auditing, is reliable and independent. We assume that the TPA does {\em not collude} with the storage node during the auditing process to hide data corruption. This is a standard assumption when relying on a TPA for integrity checking \cite{QWang2009, CWang2010, Wang2013}. The TPA, however, must not be able to learn any information about the user's data through the auditing process, aside from the metadata needed for the auditing, as in \cite{CWang2010}. In order words, the auditing protocol should not introduce a data leakage vulnerability. Similar to standard applications of cryptographic protocols, we assume that both the node and the TPA are fully aware of all the cryptographic constructions and protocols used; however, their runtime is polynomial in the security parameter.
\section{Introduction}
\label{sec:audit-introduction}
\IEEEPARstart{T}{raditional} distributed storage architectures provide reliability through block replication, whose major disadvantage is the large storage overhead. As the amount of stored data is growing faster than hardware infrastructure, this becomes a major cost bottleneck. In contrast, coding techniques achieve higher data reliability with considerably smaller storage overhead~\cite{Ford2010}. For that reason, coding techniques are under investigation for different distributed storage systems. Specifically, novel storage codes are currently being deployed in production cloud storage systems, such as Windows Azure~\cite{Huang2012}, analytics clusters ({\em e.g.}\xspace, Facebook Analytics Hadoop clusters~\cite{Sathiamoorthy2013}), archival storage systems, and peer-to-peer storage systems like Cleversafe and Wuala \cite{cleversafe, wuala}.
Distributed storage codes operate by splitting files into blocks and creating additional parity blocks that provide fault tolerance. If the original file consists of $K$ blocks, an $(N, K)$ maximum distance separable (MDS) code is typically used to produce $N$ blocks to be stored individually on $N$ storage nodes, thus tolerating up to $(N-K)$ node failures. A well-known problem of classical erasure codes, like Reed-Solomon, is the so-called repair problem: when a single node fails, typically one block is lost from the file; however, the reconstruction of that single block requires reading and transferring $K$ blocks from other nodes.
Novel storage codes that use network coding (NC) were recently developed to reduce this {\em repair bandwidth}. These distributed storage codes require significantly less than $K$ blocks to repair a single node failure and rely on network coding to perform in-network processing \cite{Dimakis2011, Dimakis2007}. Key ingredients of NC-based distributed storage codes include (i) storing {\em coded blocks}, {\em i.e.}\xspace, linear combinations of original blocks that form the original data, and (ii) {\em block mixing} when repairing. An example is shown in Fig. \ref{fig:repair}. The repair bandwidth, however, is only one aspect of cloud storage.
\input{repair.tex}
Another practical aspect of cloud storage, besides the repair bandwidth, is data integrity checking. Integrity checking is extremely important for distributed storage systems, especially when data is stored with untrusted cloud providers. Data can be lost or corrupted for various reasons while users may remain completely unaware of for long periods of time. For example, storage errors, such as torn writes \cite{Krioukov2008} and latent errors \cite{Schroeder2010}, may damage data in a way that remains undetected. Cloud storage providers may also have incentives to misbehave, {\em e.g.}\xspace, misreport data loss incidents in order to maintain their reputation \cite{Ateniese2007, Shah2007, Wang2009}. This problem is further exacerbated in systems that use coding because corrupted data can propagate to multiple nodes during repair re-encoding \cite{Chen2010}. Therefore, it is important for the user to be able to audit the integrity of the data stored on the cloud.
Another complication is that frequent integrity checking of large data sets may be out of the ability or budget of users with limited resources \cite{Wang2009, CloudSecurity}. As a result, users often resort to a third party to perform audits on their behalf \cite{Ateniese2007, Wang2009, Shacham2008, Juels2007}. In this latter case, it is important that the auditing protocols are privacy-preserving, {\em i.e.}\xspace, do not leak information to the third party \cite{Wang2009, CWang2010}. Indeed, users can leverage data encryption to protect their data before outsourcing it \cite{Juels2007}. However, data encryption should be complementary and orthogonal to integrity checking protocols. In other words, the auditing protocol should not introduce new vulnerabilities of unauthorized data leakage. Furthermore, the users may want to outsource unencrypted instead of encrypted data to support more efficient and complex computations.
As a result, auditing for distributed systems that use modern NC-based storage codes is an important emerging problem. Despite the rich literature on auditing protocols for general distributed and cloud storage~\cite{Juels2007, Ateniese2007, Shah2007, Ateniese2008, Shacham2008, Bowers2009, Bowers2009Hail, Erway2009, Wang2009, QWang2009, CWang2010, Yu2010, Wang2011}, there have been very few auditing protocols for NC-based distributed storage systems \cite{Chen2010, Dikaliotis2010}. These protocols, however, are generic in the sense that they do not specifically exploit coding properties for efficient integrity checking \cite{Chen2010}. Moreover, they do not prevent data leakage \cite{Chen2010, Dikaliotis2010}. Most importantly, they do not efficiently support repair, which is the main advantage of NC-based storage systems when compared to other storage systems.
In this work, we propose a symmetric key-based cryptographic protocol, called {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, to check for the integrity of data stored on an NC-based distributed storage system. To the best of our knowledge, this is the first scheme proposed for NC-based systems that possesses all the following desired properties:
\begin{enumerate}[(i)]
\item {\bf Efficient Integrity Checking}: The integrity check incurs a small bandwidth and computational overhead (on the order of milliseconds). It guarantees that, with high probability, the storage provider passes the integrity check if and only if it possesses the data. The proposed protocol also supports unlimited number of checks.
\item {\bf Efficient Support for Repair}: The repair of failed nodes require negligible bandwidth (no data download) as well as computation for maintaining the metadata used by the integrity checking.
\item {\bf Efficient Privacy Protection}: A third party auditor cannot learn any information about the user data through the checking protocol (except for the metadata used by the integrity checking). This privacy preserving property incurs a small bandwidth ($<$ 1\%) and computational overhead (on the order of milliseconds).
\end{enumerate}
We would like to emphasize that, independently of (iii), properties (i) and (ii) together are already useful to users who could and prefer to audit the data themselves. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is the first protocol that possesses (i) and (ii) at the same time. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} achieves these properties by fully exploiting network coding in its design. The main novelty of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} come from a careful combination of {\ensuremath{\mathsf{SpaceMac}}} -- a homomorphic message authentication code (MAC) that was previously specifically designed for network coding \cite{LeLocate2010, LeInter2012}, and {\ensuremath{\mathsf{NCrypt}}\xspace} -- a novel chosen-plaintext attack (CPA) secure encryption scheme that we custom designed, in this work, to operate in synergy with and preserve the correctness of {\ensuremath{\mathsf{SpaceMac}}}.
We implemented {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} in Java, utilizing our previous implementation of {\ensuremath{\mathsf{SpaceMac}}} \cite{LeInter2012}. Our evaluation of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} shows that it has very low computational overhead. In particular, when performing an audit, both the storage node and the auditor only need to spend a few milliseconds. Furthermore, the auditor's overhead is much less than that of the state-of-the-art approach for NC-based storage systems \cite{Chen2010}, which is on the order of seconds.
The rest of the paper is organized as follows. In Section \ref{sec:audit-related_work}, we discuss related work. In Section \ref{sec:audit-formulation}, we formulate the problem and describe the threat model. In Section \ref{sec:audit-auditing}, we describe the auditing framework and the key building blocks of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, namely {\ensuremath{\mathsf{SpaceMac}}} and {\ensuremath{\mathsf{NCrypt}}\xspace}, before presenting {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} itself. In Section \ref{sec:audit-repair}, we show how {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} efficiently supports repair. In Section \ref{sec:audit-security}, we analyze the security of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}. In Section \ref{sec:audit-evaluation}, we evaluate its storage, bandwidth, and computational efficiency. In Section \ref{sec:audit-conclusion}, we conclude the paper.
\section{Related Work}
\label{sec:audit-related_work}
\subsection{Integrity Checking for Remote Data}
\label{subsec:related-checking}
There has been a rich body of work on integrity checking for remote data \cite{Juels2007, Ateniese2007, Shah2007, Ateniese2008, Shacham2008, Bowers2009, Erway2009, Wang2009, QWang2009, CWang2010, Yu2010, Wang2011}, commonly known as {\em Proof of Retrievability} and {\em Proof of Data Possession}.
{\flushleft \bf Proof of Retrievability (POR).}\quad In \cite{Juels2007}, Juels and Kaliski introduced the notion of POR, where a POR enables a client (verifier) to determine that the server (prover) possesses a file or data object. Furthermore, a successful execution of POR would allow a verifier to extract the file from the proof. The main POR scheme presented there uses {\em sentinels}, {\em i.e.}\xspace, small check blocks, that are inserted into the outsourced data to guard against large file corruption. At the same time, it also utilizes error correcting codes to protect against small file corruption. This scheme can only handle a limited number of queries, which has to be fixed a priori. In contrast, {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} does not use sentinels and supports unlimited number of queries.
In \cite{Shacham2008}, Shacham and Waters proposed two POR schemes with full proofs of security and extract-ability. The first one, built on Boneh-Lynn-Shacham (BLS) signatures, provides public verifiability. The second one, built on pseudorandom functions (PRFs), provides private verifiability. Recently, Bowers {\em et al.}\xspace \cite{Bowers2009Hail} proposed HAIL, an improvement of existing POR schemes that allows for performing data integrity checking with multiple servers against stronger, mobile adversaries.
These schemes \cite{Shacham2008, Bowers2009Hail} exploit homomorphic properties to aggregate authenticator values to improve the audit efficiency. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} also exploits homomorphic properties (of \ensuremath{\mathsf{SpaceMac}}) and provides private verifiability. In terms of extract-ability, {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is different from existing approaches, {\em e.g.}\xspace, \cite{Shacham2008}, in that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} exploits the inherent embedded coding coefficients in the stored blocks to perform the extraction. Meanwhile, \cite{Shacham2008} relies on additional erasure codes (pre-applied to the data) for the extraction.
{\flushleft \bf Proof of Data Possession (PDP).}\quad The notion of PDP was introduced by Ateniese {\em et al.}\xspace \cite{Ateniese2007}. The PDP scheme in \cite{Ateniese2007} uses homomorphic RSA signatures to generate verification tags. The data possession guarantee provided by this scheme is under the RSA and KEA1 assumptions in the random oracle model. Earlier in \cite{Schwarz2006}, Schwarz and Miller proposed using a combination of both erasure-correcting coding and algebraic signatures (homomorphic hashes) to perform integrity checking for remote data. As discussed in \cite{Shacham2008}, the notion of PDP is considered to be weaker than POR. This is because in POR, a successful audit guarantees that all the data can be extracted while in PDP, only a certain percentage of the data ({\em e.g.}\xspace, 90\%) is guaranteed to be available. Integrity checking for groups with efficient user revocation was recently introduced in \cite{Wang2013}. We will show that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} provides the stronger data possession checking with data extraction as in POR (Section \ref{audit-subsec:possession}).
{\flushleft \bf Data Modification.}\quad In \cite{Ateniese2008}, Ateniese {\em et al.}\xspace proposed a symmetric-key based checking scheme that supports data modification. This scheme is built on regular PRFs, hash functions, and encryptions. It provides private verifiability and supports a limited number of queries. In \cite{Erway2009}, Erway {\em et al.}\xspace proposed an auditing scheme built on rank-based authenticated skip lists and requires the storage server to maintain the lists for verification. In \cite{QWang2009}, Wang {\em et al.}\xspace~proposed a public auditing scheme that uses a combination of the BLS-based scheme in \cite{Shacham2008} and Merkle Hash Tree (MHT).
In practice, most current deployments of distributed storage codes \cite{Sathiamoorthy2013, Huang2012} initially set all files to replication mode. When certain files become \textit{cold} ({\em i.e.}\xspace, rarely accessed and modified) the replicated blocks are deleted and corresponding parity blocks are created. This dynamic switching of files from replication to coding allows distributed storage systems to benefit from the high performance of replication for \textit{hot} files and the storage benefits of coding for \textit{cold} files. Interestingly, in most analytics clusters and cloud storage systems, the vast majority of data seem to be \textit{cold} \cite{Sathiamoorthy2013, Huang2012}. Therefore, we do not expect data modification to be a critical operation for encoded data. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} provides some preliminary support for data modification, and the details can be found in the Appendix.
{\flushleft \bf Privacy Preserving.}\quad In \cite{Shah2007}, Shah {\em et al.}\xspace~proposed an auditing protocol that is privacy preserving. This protocol first encrypts the data and then sends a number of message authentication code (MAC) tags of the encrypted data to the auditor. The auditor verifies both the outsourced data and the outsourced encryption key. This approach only works on encrypted files. It also requires the auditor to maintain states and supports only limited number of audits. In \cite{CWang2010}, Wang {\em et al.}\xspace~ proposed a privacy preserving auditing protocol that has public verifiability. This protocol can be considered an extension of the BLS-based protocol in \cite{Shacham2008}. In this approach, the aggregated (proving) block sent by the storage server is masked with a random element to protect the privacy of the block. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is explicitly designed to provide privacy preserving-auditing (Section \ref{audit-subsec:audit-auditing} and \ref{audit-subsec:privacy}). Different from \cite{CWang2010}, {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} relies on symmetric-key cryptographic primitives instead of public-key ones, and thus it provides private instead of public auditing.
Finally, we stress that none of the schemes described above was customized for NC-based storage. In particular, they do not provide efficient support for node repair. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} was designed to achieve all the above properties: providing proof of retrievability and privacy-preserving auditing while efficiently supporting node repair.
\subsection{Integrity Checking for NC-based Storage Systems}
\label{audit-subsec:nc_based_check}
{\flushleft \bf NC-based Storage Systems.}\quad The benefits of network coding for distributed storage were first formalized by the work of Dimakis {\em et al.}\xspace \cite{Dimakis2007}. In particular, in \cite{Dimakis2007}, the authors proposed the notion of {\em regenerating codes} and show that they can significantly reduce the repair bandwidth. This work showed the fundamental tradeoff between node storage and repair bandwidth and proposed regenerating codes that can achieve any point on the optimal tradeoff curve. A survey on recent advances in NC-based storage system can be found at \cite{Dimakis2011}. A wiki on NC-based storage cloud is maintained at \cite{DimakisWiki}. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} is designed to {\em fully support regenerating codes}.
An NC-based distributed file system (NCFS) is proposed in \cite{Hu2011}. One of the first implementations of NC-based storage cloud is NCCloud by Hu {\em et al.}\xspace \cite{Hu2012}. In particular, NCCloud is a proxy-based system for multiple-cloud storage. It utilizes a functional minimum-storage regenerating code to provide cost-effective repair for a permanent single-cloud failure. This efficient repair is achieved without the cost of storage or redundancy level. NCCloud prototype was deployed on top of Windows Azure Storage.
{\flushleft \bf Integrity Checking Schemes for NC-Based Storage Systems.}\quad There have been only a few number of work that provide remote data checking for NC-based storage. In \cite{Dikaliotis2010}, Dikialotis {\em et al.}\xspace~proposed an integrity checking scheme that utilizes the error-correction capabilities of the storage system. This scheme aims to detect errors with a very small amount of bandwidth. The key technique for reducing the bandwidth is to project data blocks onto a small random vector. This checking scheme is inherently different from {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} as it relies on the communication between the auditor and multiple nodes to perform a single check while {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} does not. Moreover, this scheme is information-theory based while {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} leverages cryptographic primitives to provide the checking.
A more recent integrity checking scheme for NC-based storage was proposed in \cite{Chen2010}. In this work, Chen {\em et al.}\xspace~adopted the symmetric-key based scheme that Shacham and Waters proposed for regular cloud storage \cite{Shacham2008} with minor modification. In particular, based on the symmetric-key based scheme in \cite{Shacham2008}, the scheme in \cite{Chen2010} proposed to encrypt the coding coefficients of the outsourced encoded blocks to prevent {\em replay attacks}, where a malicious storage node may store old (incorrect) encoded blocks instead of the new (correct) encoded blocks as required by the repair \cite{Chen2010}. {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} overcomes this attack by requiring the user/auditor to store the coding coefficients, which is also needed for the repair process and only occupies a negligible amount of storage (see Section \ref{audit-subsec:storageOverhead}).
What really sets {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} apart from \cite{Chen2010} is that {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} fully exploits network coding for integrity checking. In particular, the scheme proposed in \cite{Chen2010} relies on two independent logical representation of file blocks for two different purposes: data possession checking and network coding operation. Because of this, during the repair process, the user has to download blocks from the remaining healthy nodes to compute the integrity checking data for the new coded blocks (to be stored at the recovery node). This approach puts heavy bandwidth and computational overhead on the user. In contrast, {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} uses a {\em single} representation for both purposes and thereby achieving integrity checking while eliminating the heavy user's bandwidth and computational overhead. Details of how {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} support efficient repair are provided in Section \ref{audit-subsec:repair}. Furthermore, the scheme in \cite{Chen2010} does not support privacy-preserving auditing while {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} does. We provide detailed performance comparison between {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} and \cite{Chen2010} in Section \ref{sec:audit-evaluation}.
Finally, a recent work by Cao {\em et al.}\xspace \cite{Cao2012} proposed an LT codes-based storage system with an integrity checking and an exact repair schemes; however, it neither supports functional repair \cite{Dimakis2007} (discussed in Section \ref{audit-subsec:repair}) nor privacy-preserving auditing.
{\flushleft \bf Other Security Issues.}\quad
Other security problems for NC-based storage include protecting the privacy and integrity of the blocks while repairing. The work in \cite{Pawar2010} and \cite{Rouayheb2010} prevents eavesdroppers from accessing/decoding all the data. In \cite{Pawar2010}, Pawar {\em et al.}\xspace~provide an explicit code construction that achieves the secrecy capacity for the bandwidth-limited regime of the storage systems under repair dynamics. \cite{Rouayheb2010} analyzes the effects of interaction between the storage nodes on the amount of data revealed to the eavesdroppers. The work in \cite{Pawar2011a} provides upper bounds on the maximum amount of information that can be stored safely when there are malicious nodes.
In \cite{Pawar2011a} and \cite{Buttyan2010}, the authors provide protection against pollution attacks during the repair. In \cite{Buttyan2010}, Buttyan {\em et al.}\xspace~provide a lightweight, pollution-resilient decoding algorithm that is capable of finding adversarial blocks. The scheme in \cite{Chen2010} also protects the repair phase against pollution attacks, {\em i.e.}\xspace, preventing remaining nodes from sending corrupted data to the new (recovering) node. Dealing with pollution attacks is out of the scope of this work. We refer the reader to the rich literature, including our previous work, that deal with pollution attacks \cite{Gkantsidis2006, Agrawal2009, Li2010, Boneh2009, Zhang2011, LeTESLA2011, LeJSAC2011}.
\subsection{This Work in Perspective}
A preliminary version of this work has appeared in NetCod 2012 \cite{Le2012}. In this paper, we provide the following revisions and extensions of the previous version: We revise and provide complete proofs of all lemmas and theorems; we described in detail a repair process; we discuss and compare our storage overhead to prior work \cite{QWang2009, CWang2010, Chen2010}; finally, we provide a comprehensive discussion of related literature.
\section{Support for Node Repair}
\label{sec:audit-repair}
\label{audit-subsec:repair}
When there is a node failure, the user creates a new node to replace this node. Based on the coding coefficients of the coded blocks at the remaining healthy nodes, the user instructs the healthy nodes to send appropriate coded blocks to the new node. The new node then linearly combines them, according to the user instruction, to construct its own coded blocks. This new node may construct the same coded blocks that the failed node had ({\em exact repair}), or completely different coded blocks that still preserve the same level of reliability ({\em functional repair}) \cite{Dimakis2011}. In the example given in Fig. \ref{fig:repair}, the user instructs the first three storage nodes to send coded blocks to exactly repair the fourth node.
Formally, for each healthy node, $N_i, i = 1, \cdots, P$, recall that it needs to send $Q$ encoded repair blocks to the new node. Let $(\vct{e}_{i,1}, \cdots, \vct{e}_{i,M})$ be the encoded blocks currently stored on $N_i$. For $j = 1, \cdots, Q$, the user sends a set of {\em repair coding coefficients} $(\gamma_{i,j,1}, \cdots, \gamma_{i,j,M})$ to $N_i$. This node then uses these coefficients to compute the repair blocks, $\vct{g}_{i,j} = \sum_{k=1}^M \gamma_{i,j,k} \, \vct{e}_{i,k}$, to send to the new node. The new node will receive $P \times Q$ repair blocks, $\vct{g}_{i,j}$, from the healthy nodes. It uses them to reconstruct the encoded blocks, $\vct{h}_1, \cdots, \vct{h}_M$, that it needs to store. For $k = 1, \cdots, M$, the user sends a set of $P \times Q$ {\em reconstruction coding coefficients}, $(\theta_{i,j,k}, \cdots \theta_{P,Q,k})$, to the new node to instruct its reconstruction. The new node then reconstructs $\vct{h}_k = \sum_{i=1}^{P} \sum_{j=1}^{Q} \theta_{i,j,k} \, \vct{g}_{i,j}$. Note that the coding coefficients $\gamma$'s and $\theta$'s are dependent on the repairing scheme.
Using {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, the verification tags of the newly constructed blocks, $\vct{h}_k$, at the new node do not need to be computed by the user. In particular, the healthy nodes can send along the verification tags of the repair blocks, $\vct{g}_{i,j}$, that they send to the new node, where the tags of $\vct{g}_{i,j}$ can be computed using the $\ensuremath{\mathsf{Combine}}$ algorithm of $\ensuremath{\mathsf{SpaceMac}}$ on the tags of $\vct{e}_{i,k}$. The new node then can also use {\ensuremath{\mathsf{Combine}}} on the tags of $\vct{g}_{i,j}$ to generate tags of $\vct{h}_k$. Finally, the user sends the coding coefficients of the coded blocks at the newly constructed node, $\aug{\vct{h}_k}$ (dependent on the repair scheme), to the TPA so that it can audit this new node.
Consequently, with {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, there is negligible cost to the user when repairing a failed node, in terms of both bandwidth and computation of verification metadata. In particular, the user does not need to download data, {\em i.e.}\xspace, $\vct{e}_{i,k}$, and the user also does not need to compute the tags, {\em i.e.}\xspace, runs {\ensuremath{\mathsf{Mac}}} on $\vct{h}_k$. This stands in stark contrast with the prior integrity checking scheme for NC-based storage \cite{Chen2010}, which requires the user to download many data blocks (equal to the repair bandwidth) and compute security metadata for the newly coded blocks him/herself.
Last but not least, since the TPA audits the new node based on the new set of coefficients, a malicious node cannot carry out a {\em replay attack} \cite{Chen2010} (discussed in Section \ref{audit-subsec:nc_based_check}); otherwise, it will not pass the audit because the {\ensuremath{\mathsf{SpaceMac}}} tags are computed on both the data and coefficients. Here we assume that the healthy remaining nodes send valid data and tags to the new node. If there is a malicious node that sends corrupted data or tags, the storage systems is considered polluted. Dealing with pollution attacks is out of the scope of this paper; we refer the reader to previous work, including our own, which explicitly combats pollution attacks \cite{Gkantsidis2006, Agrawal2009, Boneh2009, Agrawal2010, Li2010, LeJSAC2011, LeTESLA2011, LeInter2012, Buttyan2010}.
\section{Security Analysis}
\label{sec:audit-security}
\subsection{Data Possession Guarantee}
\label{audit-subsec:possession}
When using $\ensuremath{\mathsf{SpaceMac}}$ in $\ensuremath{\mathsf{NC\text{-}Audit}}\xspace$, some information about the vector $\vct{r}$ in the $\ensuremath{\mathsf{SpaceMac}}$ construction is available to the adversary. In particular, the storage node knows the following $n-1$ equations: $\bar{\vct{p}}_i \cdot \bar{\vct{r}} = p_i\,, i \in [1,n-1]$. The following theorem states that even when these $n-1$ equations are exposed, $\ensuremath{\mathsf{SpaceMac}}$ is still a secure homomorphic MAC, {\em i.e.}\xspace, any corruption will be detected w.h.p.
\begin{theorem}\label{thm:PossessionSpaceMac}
Assume that $F_1$ is a secure PRF. For any fixed $q$, $n$, $m$, assume that a probabilistic polynomial time adversary $\mathcal{A}$ knows any $n-1$ linearly independent vectors, $\bar{\vct{p}}_1, \cdots, \bar{\vct{p}}_{n-1}$, and any $n-1$ constants, $p_1, \cdots, p_{n-1}$, such that $\bar{\vct{p}}_i \cdot \bar{\vct{r}} = p_i$, where $\vct{r}$ is used in the construction of $\ensuremath{\mathsf{SpaceMac}}$. The probability that $\mathcal{A}$ wins the $\ensuremath{\mathsf{SpaceMac}}$ security game, denoted by $\text{\em Adv}[\mathcal{A}, \ensuremath{\mathsf{SpaceMac}}]$, is at most
\[ \text{\em PRF-Adv}[\mathcal{B}, F_1] + \frac{1}{q}\,,\]
where {\em PRF-Adv}$[\mathcal{B}, F_1]$ is the probability of an adversary $\mathcal{B}$ with similar runtime to $\mathcal{A}$ winning the PRF security game.
\end{theorem}
\begin{IEEEproof}
The security game, called the Attack Game 1, of $\ensuremath{\mathsf{SpaceMac}}$ involves a challenger $\mathcal{C}$ and an adversary $\mathcal{A}$, and is as follows:
\begin{itemize}
\item {\em Setup.} $\mathcal{C}$ generates a random key $k \overset{\text{R}}{\leftarrow} \mathcal{K}$
\item {\em Queries.} $\mathcal{A}$ adaptively queries $\mathcal{C}$, where each query is of the form $(\text{id}, \vct{y})$. For each query, $\mathcal{C}$ replies to $\mathcal{A}$ with the corresponding tag $t \leftarrow \mathsf{Mac}(k, \text{id}, \vct{y})$.
\item {\em Output.} $\mathcal{A}$ eventually outputs a tuple $(\text{id}^*, \vct{y}^*, t^*)$.
\end{itemize}
Up to the time $\mathcal{A}$ outputs, it has queried $\mathcal{C}$ multiple times. Let $l$ denote the number of times $\mathcal{A}$ queried $\mathcal{C}$ using $\text{id}^*$ and get tags for $l$ vectors, $\vct{y}^*_1, \cdots, \vct{y}^*_l$, of these queries. We consider that the adversary wins the security game if and only if
\begin{itemize}
\item $(y_*^{(n+1)}, \cdots, y_*^{(n+m)}) \neq \vct{0}$ (trivial forge otherwise),
\item $\mathsf{Verify} (k, \text{id}^*, \vct{y}^*, t^*) = 1$, and
\item $\vct{y}^* \notin \mathsf{span}(\vct{y}^*_1, \cdots, \vct{y}^*_l).$
\end{itemize}
Here, we prove Theorem \ref{thm:PossessionSpaceMac} with respect to a slightly different security game, called Attack Game 2. This Attack Game 2 is similar to Attack Game 1, except that in the {\em Queries} phase, for each distinct id, the space spanned by the vectors used in the queries has dimension at most $m$. This Attack Game 2 is stricter but better fits the reality: since the dimension of the source space $\Pi$ is only $m$, the adversary must only learn tags of vectors in spaces having dimensions at most $m$.
Now the proof is done by using a sequence of games denoted Game 0 and Game 1. Let $W_0$ and $W_1$ denote the events that $\mathcal{A}$ wins the homomorphic MAC security in Game 0 and Game 1, respectively. Game 0 is identical to Attack Game 2 applied to the scheme $\ensuremath{\mathsf{SpaceMac}}$. Hence,
\begin{equation}\label{audit-eq:1}
\text{Pr}[W_0] = \text{Adv}[\mathcal{A}, \ensuremath{\mathsf{SpaceMac}}]
\end{equation}
Game 1 is identical to Game 0 except that the challenger $\mathcal{C}$ computes $\vct{r} \leftarrow (r_1, \cdots, r_{n+m})$, where $r_i$ is chosen uniformly at random from $\eff{}{q}$: $r_i \overset{\text{R}}{\leftarrow} \mathbb{F}_q$ instead of $r_i \leftarrow F(k, \text{id}, i)$, and everything else remains the same. Then, there exists a PRF adversary $\mathcal{B}$ such that
\begin{equation}\label{audit-eq:2}
|\text{Pr}[W_0] - \text{Pr}[W_1]| = \text{PRF-Adv}[\mathcal{B}, F]
\end{equation}
The complete challenger in Game 1 works as follows:
{\flushleft \emph{Queries.}} $\mathcal{A}$ adaptively queries $\mathcal{C}$, where each query is of the form $(\text{id}, \vct{y})$. If id is already used in $m$ previous query, $\mathcal{C}$ discards the query. Otherwise, $\mathcal{C}$ replies to query $i$ of $\mathcal{A}$ as follows:\\
\hspace*{0.3 cm} if id is never used in any of the previous queries:\\
\hspace*{0.6 cm} $\vct{r}_i := (r^i_1, \cdots, r^i_{n+m})$, where $r^i_j \overset{\text{R}}{\leftarrow} \mathbb{F}_q, j \in [1,n+m]$\\
\hspace*{0.3 cm} else:\\
\hspace*{0.6 cm} $\vct{r}_i$ := the one used in the previous response\\
\hspace*{0.3 cm} send $t := \vct{y}_i \cdot \vct{r}_i$ to $\mathcal{A}$
{\flushleft \emph{Output.}} $\mathcal{A}$ eventually outputs a tuple $(\text{id}^*, \vct{y}^*, t^*)$. When $\vct{y}^*$ does not equal $\vct{0}$, to determine if $\mathcal{A}$ wins the game, we compute\\
\hspace*{0.3 cm} if $\text{id}^* = \text{id}_i$ (for some $i$) then \hspace*{0.5 cm} // case (i)\\
\hspace*{0.6 cm} set $\vct{r}^* := \vct{r}_i$\\
\hspace*{0.3 cm} else \hspace*{5 cm} // case (ii)\\
\hspace*{0.6 cm} set $\vct{r}^* := (r^*_1, \cdots, r^*_{n+m})$, where $r^*_i \overset{\text{R}}{\leftarrow} \mathbb{F}_q, i \in [1,n+m]$\\
Let $l$ denote the number of times $\mathcal{A}$ queried $\mathcal{C}$ using $\text{id}^*$ and get tags for $l$ vectors, $\vct{y}^*_1, \cdots, \vct{y}^*_l$, of these queries. The adversary wins the game, \emph{i.e.}, event $W_1$ happens, if and only if
\begin{align}
~&t^* = \vct{y}^* \cdot \vct{r}^*\,,\text{ and}\label{audit-eq:3}\\
~&\vct{y}^* \notin \mathsf{span}(\vct{y}^*_1, \cdots, \vct{y}^*_l)\label{audit-eq:4}\,.
\end{align}
Subsequently, we will show that Pr[$W_1$] = $\frac{1}{q}$. Let $T$ be the event that $\mathcal{A}$ outputs a tuple with a completely new $\text{id}^*$, \emph{i.e.}, $\mathcal{A}$ never made queries using $\text{id}^*$ before.
$\bullet$ When T happens, \emph{i.e.}, in case (ii), since $r^*_i\,$'s are indistinguishable from random values and $(y_*^{(n+1)}, \cdots, y_*^{(n+m)}) \neq \vct{0}$, the right hand side of equation (\ref{audit-eq:3}) is a completely random value in $\mathbb{F}_q$. Thus,
\begin{equation}\label{audit-eq:5}
\text{Pr}[W_1 \wedge T] = \frac{1}{q}\,\text{Pr}[T]\,.
\end{equation}
$\bullet$ When T does not happen, \emph{i.e.}, in case (i): $\vct{r}^*$ of equation (\ref{audit-eq:3}) equals $\vct{r}_i$ for some $i$, and $\vct{r}^*$ has been used to generate tags for vectors $\vct{y}^*_1, \cdots, \vct{y}^*_l$. In this case, we proceed by showing that for a fixed $\vct{y}^*$, $t^*$ looks indistinguishable from a random value in $\mathbb{F}_q$. The given prior knowledge, the queries, and the output form the following system of linear equations $\Pi_2$:
\[
(\Pi_2) \quad
\begin{cases}
\bar{\vct{p}}_1 \cdot \bar{\vct{r}}^* = p_1\\
\cdots~\\
\bar{\vct{p}}_{n-1} \cdot \bar{\vct{r}}^* = p_{n-1}\\
\vct{y}^*_1 \cdot \vct{r}^* = t_{\vct{y}^*_1}\\
\cdots~\\
\vct{y}^*_l \cdot \vct{r}^* = t_{\vct{y}^*_l}\\
\vct{y}^* \cdot \vct{r}^* = t^*
\end{cases}
\,.
\]
Let the elements $r^*_i, i \in [1,n+m]$, of $\vct{r}^*$ be the unknowns of the system. The above system is consistent regardless of the value of $t^*$ because the coefficient matrix has rank at most $n+m$, which equals the number of unknowns. Let $d$ be the rank of the coefficient matrix, $d\leq n+m$. For a fixed $\vct{y}^*$, its valid tag $t^*$ could be any value in $\mathbb{F}_q$ equally likely because (i) for any value $t^*$, the solution space always has the same size $q^{n+m-d}$, and (ii) $r^*_i$'s are chosen uniformly at random from $\mathbb{F}_q$. As a result, the probability that the adversary chooses a correct $t^*$ is $1/q$. Thus,
\begin{equation}\label{audit-eq:6}
\text{Pr}[W_1 \wedge \neg T] = \frac{1}{q}\,\text{Pr}[\neg T]\,.
\end{equation}
$\bullet$ From equations (\ref{audit-eq:5}) and (\ref{audit-eq:6}), we have
\begin{equation}\label{audit-eq:7}
\text{Pr}[W_1] = \text{Pr}[W_1 \wedge T] + \text{Pr}[W_1 \wedge \neg T] = \frac{1}{q}\,.
\end{equation}
Equations (\ref{audit-eq:1}), (\ref{audit-eq:2}), and (\ref{audit-eq:7}) together prove the theorem.
\end{IEEEproof}
Now, we are ready to prove the data possession guarantee of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}.
\begin{lemma}\label{thm:possession}
With probability at least $1 - \frac{2}{q}$, the storage node can pass a check if and only if it possesses the blocks specified in the challenge of the check.
\end{lemma}
\begin{IEEEproof}
Lemma \ref{thm:correctness} shows that if the storage node possesses the data then it can pass the check. It remains to show that if the node passes the check then it possesses the corresponding blocks w.h.p. Let us prove the converse, {\em i.e.}\xspace, if there are corrupted or missing blocks, the node will fail the check w.h.p. For simplicity, we assume that when responding to a challenge involving a block that no longer exists in the storage, the node replaces it with a block chosen uniformly at random in $\eff{n+m}{q}$.
{\em Case (a) - The storage node is able to compute a correct response block even when some blocks are missing or corrupted:} Denote the correct, unencrypted aggregated block by $\vct{e}$, {\em i.e.}\xspace, $\vct{e} = \sum_{i \in \mathcal{I}} \alpha_i \, \vct{e}_i $. Denote the data of the response block actually computed by the storage node by $\hat{\vct{a}}$ and denote $(\hat{\vct{a}} \, | \, \aug{\vct{e}} )$ by $\vct{a}$. If there is at least one error in the data of one of the block or there is at least one missing block, then $\Prob{\hat{\vct{a}} = \hat{\vct{e}}} \leq \frac{1}{q}$ because $\alpha$'s are chosen uniformly at random from $\eff{}{q}$. Note that $\vct{e}$ is in the source space: $\vct{e} \in \Pi$, thus if $\hat{\vct{a}} \neq \hat{\vct{e}}$ then $\vct{a} \notin \Pi$. Therefore, $\Prob{\vct{a} \in \Pi} = \Prob{\vct{a} = \vct{e}} \leq \frac{1}{q}$.
{\em Case (b) - The storage node responds with an incorrect block:} The security of $\ensuremath{\mathsf{SpaceMac}}$ from Theorem \ref{thm:PossessionSpaceMac} guarantees that the node can provide a valid tag of $\vct{a} \notin \Pi$ with probability at most $\frac{1}{q}$. Without loss of generality, we can ignore the encryption because if the node already knows a valid tag of $\vct{a}$, it can provide the correct encryption to pass the check. Meanwhile, if the node does not know a valid tag of $\vct{a}$, its chance of forging a valid tag for the cipher text $\vct{c}$ is still bounded by the security guarantee of $\ensuremath{\mathsf{SpaceMac}}$, which is at most $\frac{1}{q}$.
As a result, from cases (a) and (b), the probability of passing the check when there is error or missing block is at most $\frac{2}{q}$.
\end{IEEEproof}
Not only does {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace} provide detection in the presence of corrupted or missing blocks, it also ensures that the user can extract the data stored on the storage node just by collecting responses of the node from the checking protocol. This is also known as the {\em retrievability} property. We provide the proof of retrievability based on the theoretical framework of \cite{Bowers2009}, which is derived from \cite{Shacham2008} and \cite{Juels2007}.
\begin{lemma}
Assume that the storage node responds correctly to a fraction, $1-\epsilon$, of the challenges uniformly, where $\epsilon < \frac{1}{2}$. The user can extract the encoded blocks stored on the node, $\vct{e}_1, \cdots, \vct{e}_M$, by performing $\gamma$ challenge-response interactions with the storage node with high probability (depending on $\gamma$, $\epsilon$, and $q$).
\end{lemma}
\begin{IEEEproof}
Lemma \ref{thm:possession} implies that if a node responds correctly to a fraction of challenge, then with probability at least $1-\frac{2}{q}$, the response block is a correct linear combination of the blocks stored at the node. For a challenge coefficient vector $(\alpha_1, \cdots, \alpha_M)$, the user can challenge the node using a number of constant-multiples of the vector, {\em e.g.}\xspace, $(c\,\alpha_1, \cdots, c\,\alpha_M)$ for some constant $c$, to learn the responses (including incorrect responses), and then use majority decoding to learn the correct equation $\sum_{i=1}^M \alpha_i \vct{e}_i = \vct{d}$, where $\vct{d}$ is some constant vector. By collecting $M$ linearly independent equations of this form, the user can solve for $\vct{e}_1, \cdots, \vct{e}_M$ using Gaussian elimination.
Note that for a fixed $\epsilon < \frac{1}{2}$, the probability of learning one correct equation depends on both $q$ and the number of queries made using the multiples of the corresponding coefficient vector. For a fixed $q$, this probability can be made arbitrarily high by increasing the number of queries.
\end{IEEEproof}
\subsection{Privacy-Preserving Guarantee}
\label{audit-subsec:privacy}
{\ensuremath{\mathsf{NCrypt}}\xspace} provides the privacy guarantee of {\ensuremath{\mathsf{NC\text{-}Audit}}\xspace}, which we stated in the following lemma.
\begin{lemma}
From the responses of the storage node, the TPA does not learn any information about the outsourced data, except for the information that could be derived from the MAC tag.
\end{lemma}
\begin{IEEEproof}
The claim is a direct consequence of Theorem \ref{thm:NCrypt} and the fact that the padding elements are chosen randomly.
\end{IEEEproof}
\section{Acknowledgement}
\input{references.bbl}
\input{audit-appendix}
\end{document}
|
{
"timestamp": "2014-05-14T02:02:30",
"yymm": "1203",
"arxiv_id": "1203.1730",
"language": "en",
"url": "https://arxiv.org/abs/1203.1730"
}
|
\section{Introduction}
Non-perturbative effects play a very important r\^ole in the study of low-energy implications of string theory. In the past few years a very good understanding of these effects has been achieved despite the absence of a fully non-perturbative formulation of the theory. Much progress has been made on the origin, structure and computability of these terms in the effective field theory at low energy and weak coupling. D-branes and their Euclidean counterparts generalise the r\^ole of non-perturbative effects (gaugino condensation and instantons) in field theories.
In IIB string compactifications to four dimensions with $\mathcal{N}=1$ supersymmetry and O3/O7-planes,
there are two known ways to generate non-perturbative effects \cite{NPreview}:
\begin{itemize}
\item by gaugino condensation in the field theory living on D7-branes or Euclidean D3-brane (E3) instantons wrapped on geometric four-cycles;
\item by gaugino condensation on spacetime-filling D3-branes or Euclidean D(-1)-brane (E(-1)) instantons at singularities.
\end{itemize}
The corresponding gauge kinetic function is determined in the first case by the K\"ahler modulus $T$ controlling the size of the wrapped four-cycle whereas in the second case by the dilaton field $S$ and the blow-up mode $Q$ resolving the singularity.
Despite being very much suppressed at weak coupling,
non-perturbative effects are nevertheless relevant as corrections to the superpotential $W$ due to non-renormalisation theorems that keep nominally higher-order contributions from contributing. For instance, symmetries forbid the appearance of the K\"ahler moduli in $W$ at any order in perturbation theory and so their appearance in $W$ is necessarily purely non-perturbative. For the simpler constructions normally considered the non-perturbative superpotential coming from these two types of mechanisms can be written as:
\begin{equation} \label{vanillaWnp}
W_{\rm np} =\sum_i A_i\, e^{-\,a_i \, \left( T_i+ h_i(\mathcal{F}) S\right)} + B\, e^{-\, b \, \left(S+ h_j\, Q_j\right)}\,,
\end{equation}
where the prefactors $A_i$ and $B$ are in general functions of the complex structure moduli and the open string moduli (D7-deformation and D3-position moduli). The constants $a_i$ and $b$ are given by $a_i=2\pi/N_i$ and $b=2\pi/N_b$ where, in the case of gaugino condensation, $N_i=\beta_i/3$ and $N_b=\beta_b/3$ with $\beta_i$ and $\beta_b$ the one-loop beta function coefficients of the condensing gauge theories, while, in the case of instantons, $N_i=N_b=1$.
The quantities $h_i(\mathcal{F})$ and $h_j$ are non-zero in the presence of a non-vanishing magnetic flux $\mathcal{F}$
on the world-volume of a D7-brane or if a stack of D3-branes is placed at a singularity. In this way chiral matter is generated in the low-energy theory.
More precisely, for flux on a D7-brane the K\"ahler moduli get charged under the anomalous $U(1)$ and the gauge kinetic function $f_i$ gets a flux-dependent shift $h_i(\mathcal{F})$ that introduces a new dependence on the dilaton, modifying it from $f_i=T_i$ to $f_i=T_i+h(\mathcal{F}_i) S$. On the other hand, for D3-branes at a singularity the blow-up modes $Q_j$ resolving the singularity get charged under the anomalous $U(1)$ and the gauge kinetic function receives extra contributions of the form $f=S + h_j Q_j$ where $h_j$ are constants proportional to the $U(1)$-charge of the blow-up modes. In both cases the coefficients $A_i$ and $B$ must also depend on charged matter fields in order for $W$ to be gauge invariant.
To date only the $T_i$ dependent part in $W_{\rm np}$ has been used for modulus stabilisation. However, there is no reason to assume $B=0$ and therefore also the $S$-dependent term in $W_{\rm np}$ has to be included unless it can be argued to be subdominant. One of the main points of this article is to explore the implications of these non-perturbative effects for modulus stabilisation.
We here study their inclusion within the context of the LARGE Volume Scenario (LVS) \cite{LVS}, which provides a concrete and controlled realisation of modulus stabilisation in flux compactifications of IIB string theory. The mechanism used in the LVS is very generic \cite{ccq2} and shows that the natural minimum of the leading contributions to the scalar potential corresponds to (non-supersymmetric) anti de Sitter (AdS) 4D spacetime with exponentially large volume. As for other stabilisation scenarios it is necessary to seek uplifting mechanisms that can lift the minimum to flat or de Sitter (dS) vacua, and it has been assumed that this can be realised in a similar manner as in the KKLT scenario \cite{KKLT}, where an anti D3-brane is located at the tip of a highly warped throat in the extra dimensions. The anti-brane provides a positive-definite contribution to the scalar potential and the warping can be adjusted to explicitly tune the minimum
to have a small positive vacuum energy.
The use of anti-branes in this way raises standard concerns about control over the approximations used to compute the low-energy effective field theory, due to the explicit breaking of supersymmetry by the anti-brane. For the LVS there is also an additional challenge regarding whether an exponentially large volume is consistent with a large warping. We address this issue when we describe LVS models in the first part of this article, and identify the domain of validity for which both are consistent.
A second uplifting proposal obtains de Sitter vacua from D-terms produced by magnetised branes \cite{Dterm,CMQS,kq}. We review this proposal to restate the criteria required to obtain a viable uplifting term from a localised magnetic flux turned on a stack of branes wrapping a large four-cycle. Moreover we also show that fibred Calabi-Yau manifolds open up the possibility
of achieving de Sitter vacua from the interplay of D-terms and string loop corrections to the K\"ahler potential
(and possibly warping).
Our main result is to introduce a novel fully supersymmetric mechanism for obtaining dS solutions in the LVS. This mechanism is based on the presence of the dilaton-dependent non-perturbative effects discussed above, either from gaugino condensation on D3-branes or E(-1)-instantons at singularities.
We show that the corresponding non-perturbative superpotential can give rise to a positive contribution to the scalar potential which differs from the one arising from K\"ahler moduli-dependent non-perturbative effects, as previously considered in KKLT and LVS. These dilaton-dependent non-perturbative effects have never been considered in the context of modulus stabilisation even though non-perturbative superpotentials at a singular locus have been discussed in the past as potential string theory realisations of dynamical supersymmetry breaking and gauge mediation \cite{quiverSUSYbreaking,Romans,Florea,k1}.
We find that combining such a positive contribution with the standard LVS scalar potential gives an uplifting term very similar to the one produced by warped anti D3-brane tension, thereby giving rise to dS vacua in a fully supersymmetric set-up. The fields whose F-terms are responsible
for the realisation of dS vacua are the blow-up modes resolving the singularity of the non-perturbative quiver. Moreover these $S$-dependent non-perturbative effects, when combined to string loop corrections to the K\"ahler potential, can provide a realisation of the LVS also for manifolds with zero or positive Euler number.
This paper is organised as follows. In section \ref{Review} we review the LVS framework and discuss various ways to obtain de Sitter vacua in this setting: uplifting by anti-branes in highly warped throats and D-terms. Readers familiar with the LVS can safely skip this section. Then in section \ref{dSquiver} we analyse the effect of dilaton-dependent non-perturbative effects which provide a novel way to realise de Sitter solutions. This is the main result of the paper. Furthermore, we show that these new effects can also give rise to new LVS for Calabi-Yau
three-folds with positive Euler number due to their interplay with string loop corrections to the K\"ahler potential. In section \ref{Implications} we discuss phenomenological and cosmological implications of this new scenario, and finally conclude in section \ref{Conclusions}. Appendix \ref{AppFieldRedef} is devoted to the study of the effect of loop induced field redefinitions on the form of the scalar potential while in appendix \ref{OnlyNPquiver} we analyse the
subcase with only non-perturbative effects at the singularity.
\section{Review of LVS de Sitter vacua}
\label{Review}
In this section we give a brief review of the basics of the LVS whose main feature is the emergence of a non-supersymmetric AdS minimum at exponentially large value of the internal volume. We will then discuss the main ideas that have been put forward
to obtain de Sitter solutions within the LVS framework working out the conditions under which this can achieved either via anti-branes in highly warped throats or D-terms.
\subsection{LVS in a nutshell}
\label{LVSreview}
The LARGE Volume Scenario corresponds to a concrete mechanism of modulus stabilisation on orientifolded Calabi-Yau compactifications of type IIB string theory. The dilaton and complex structure moduli are fixed by turning on fluxes
of the complex three-form $G_3=F_3+iS H_3$ where $F_3,H_3$ are the Ramond-Ramond and Neveu-Schwarz three-forms and $S=e^{-\phi}+i C_0$ with $\phi$, $C_0$ the dilaton and axion fields of the 10D theory \cite{gkp}. The quantisation of the fluxes gives rise to the discrete landscape of minima.
The K\"ahler moduli are stabilised by a combination of perturbative and non-perturbative effects. The perturbative effects correspond to the leading order term in the expansion in inverse overall volume ${\cal V}$ of the $\alpha'$ corrections which compete with the leading order non-perturbative effect in a blow-up modulus $\tau_s$, leading to the stabilisation at
${\cal V}\sim e^{a\tau_s}$. The constant $a$ is fixed by the nature of the non-perturbative effect that can be either gaugino condensation on D7-branes or a Euclidean D3-instanton.
It was long thought that this kind of stabilisation was impossible, due to the obstacle identified long ago by Dine and Seiberg \cite{ds} that states that the natural vacuum cannot be analysed in weak coupling since the potential then arises as a series in the coupling constant, which is a field (the dilaton). But then dilaton stabilisation requires a competition among different orders in the weak-coupling expansion, and so cannot be understood within such an expansion.
This objection is overcome in the LVS by realising that in string theory there are actually a number of independent expansions at play, instead of only the one implicitly assumed in \cite{ds}. Besides the string coupling, there are also expansions in inverse powers of all of the various large moduli of the background geometry, of which there can be very many. Even though the minimum identified in the LVS is very generic \cite{ccq2,CMV}, the simplest realisation is for an orientifold of the $\mathbb{P}^4_{[1,1,1,6,9]}$ Calabi-Yau having two K\"ahler moduli $\tau_b$, $\tau_s$ and volume ${\cal V} = \tau_b^{3/2} - \tau_s^{3/2}$. The simplest set-up then gives rise to an effective field theory determined by a K\"ahler potential of the type:
\begin{equation}
K=-2\ln\left({\cal V}+\frac{\hat\zeta}{2}\right)=-2\ln\left(\tau_b^{3/2}-\tau_s^{3/2}+\frac{\hat\zeta}{2}\right)\,,
\label{KLVS}
\end{equation}
with $\hat\zeta=\zeta/g_s^{3/2}$, and a superpotential:
\begin{equation}
W=W_0+ A \,e^{-a T_s}\,.
\label{WLVS}
\end{equation}
Here $T_s = \tau_s + i \psi_s$ and $T_b = \tau_b + i \psi_b$ are the complex moduli for $\tau_s$ and $\tau_b$.
After stabilising the axion-like fields, $\psi_s$ and $\psi_b$, the corresponding scalar potential takes the form:
\begin{equation}
V_{\rm LVS}= c_1 \frac{\sqrt{\tau_s}\,e^{-2a\tau_s}}{{\cal V}} -c_2\frac{ W_0 \tau_s\,e^{-a\tau_s}}{{\cal V}^2} + c_3\frac{W_0^2}{{\cal V}^3}\,,
\label{Vlvs}
\end{equation}
where the $c_i$ are $\mathcal{O}(1-10)$ constants.
Notice that the three terms of this scalar potential conspire to give a non trivial minimum. The negative sign in the second term comes from the minimisation of the axion component of $T_s$
and drives the minimum to negative values of the vacuum energy. The last term dominates at large values of $\tau_s$ guaranteeing that the potential asymptotes to zero from the positive side (this requires $\zeta>0$). The minimum of this scalar potential is at $\tau_s\sim 1/g_s$ and ${\cal V} \sim W_0\,e^{a\tau_s}$ where the three terms of $V_{\rm LVS}$ are of the same order, giving rise to exponentially large volumes in a natural way. The minimum of this potential corresponds to a non-supersymmetric AdS vacuum.
\subsection{De Sitter LVS}
Several proposals have been put forward to realise a dS solution at exponentially large volume. The most promising so far rely on the inclusion of anti D3-branes at the tip of a warped throat
and D-terms from magnetised D7-branes. Let us briefly review both of these constructions discussing the pros and cons involved.
\subsubsection{Anti-branes and warping}
The simplest proposal to achieve dS space follows KKLT by adding an anti D3-brane at the tip of
a highly warped region somewhere within in the Calabi-Yau space \cite{KKLT}. This contributes a positive term to the scalar potential of the order:
\begin{equation}
V_{\rm up}= \frac{\nu}{{\cal V}^{\gamma}}\,,
\label{up}
\end{equation}
where $\nu$ can be order unity if $\gamma = 3$, but $\nu$ must be parametrically small if $\gamma\leq 3$. This condition on $\nu$ arises from the requirement that this term can compete with the negative potential generated by the LVS mechanism (in order to uplift it), but not be so large as to dominates at large distances and destabilise the minimum. A small size for the coefficient $\nu$ is plausible if the anti-brane is in a strongly warped region, since then $\nu$ is suppressed by a power of the warp factor, which can be very small.
The analysis of \cite{KKLT} treated the anti-brane in a probe approximation. The effects of the brane were included in the effective field theory by simply considering a potential term equal to the brane tension. The configuration is metastable
with the possibility of decay involving brane flux annihilation \cite{KPV}. There has been much recent work studying the back-reaction of the brane on the geometry \cite{Gr,Gro,Grt,Dym} to develop an understanding of the system which does not rely on the probe approximation.
One finds that the presence of the brane modifies the geometry in the ultraviolet, the associated mode is normalisable supporting the interpretation of the state as a configuration in the same theory.
\subsubsection*{LARGE volume and large warping}
It is very interesting that the two proposals for generating a large hierarchy from extra dimensions --- {\em i.e.} exponentially large volume and large warping --- are realised in a natural way in type IIB string compactifications. Since both can be exponentially large, care must be taken to ensure there are no problems with the validity of the effective field theories which are the main tools for analysis.
We now therefore pause to establish the consistency conditions for having exponentially large volume and large warping at the same time. We identify a common domain of validity for both, and so show that large warping can be possible within the LVS.
We start with a warped ten-dimensional metric of the form:
\begin{equation}
ds^2= e^{2A} \eta_{\mu\nu} dx^\mu dx^\nu + e^{-2A} g_{mn} dy^mdy^n\,,
\end{equation}
with warp factor $w=e^{-2A}$ and $g_{mn}$ the metric for the Calabi-Yau manifold. The 10D field equations of type IIB supergravity have as solutions $e^{-4A} = e^{-4A_0} + c$ where $c \simeq {\cal O}({\cal V}^{2/3})$ is an otherwise arbitrary constant and $A_0$ is a profile that depends on the underlying Calabi-Yau geometry. The condition for having a throat that dominates over the volume in part of the geometry is $e^{-4A_0} \gg c$.
But we must also be sure that the warping is not so strong as to invalidate the 4D effective field theory at energies below the warped Kaluza-Klein scale. This requires in particular that the moduli masses must be much smaller than the warped KK scale. Requiring this to be true in a region where warping dominates large-volume in the geometry therefore leads to the twin conditions \cite{warping}:
\begin{equation}
e^{-4A_0}>{\cal V}^{2/3} > e^{-A_0}\,.
\label{constraint}
\end{equation}
Here $A_0$ is considered at its maximum value (tip of the throat). The first of these is the condition that the effective warp factor is dominated by $A_0$ rather than $c$: $e^{-4A} \simeq e^{-4A_0}$ with $c\sim {\cal V}^{2/3}$. The second condition is the condition that modulus masses are not higher than the warped KK scale, which requires $e^{A_0} > {\cal V}^{-2/3}$ \cite{warping}.
For the present purposes, what is important is that we must check if any candidate warped uplifting term --- as in KKLT and LVS scenarios --- is consistent with these constraints. That is, uplifting potentials usually have the form:
\begin{equation}
V_{\rm up} \sim \frac{e^{4A_0}}{{\cal V}^{\alpha-2/3}}\,,
\label{uplift1}
\end{equation}
where the unwarped contribution would go like $1/{\cal V}^{\alpha}$ and for an anti-D3 brane $\alpha=2$. In the LVS case, for tuning the cosmological constant, the uplifting term (\ref{uplift1}) has to be comparable in size to the rest of the potential (\ref{Vlvs}), and so we need:
\begin{equation}
\frac{e^{4 A_0}}{\langle {\cal V} \rangle^{\alpha -2/3} } \sim \frac{1}{\langle {\cal V} \rangle^3}
\quad \Leftrightarrow \quad
e^{- A_0} \sim \langle {\cal V} \rangle^{11/12-\alpha/4}\,.
\label{eap}
\end{equation}
This satisfies the above constraint (\ref{constraint}) for $\alpha$ in the range:
\begin{equation}
1<\alpha<3\,,
\end{equation}
which is satisfied in particular by anti-D3 branes, for which $\alpha=2$. Therefore it can be consistent to realise de Sitter uplifting via anti-D3 branes at the tip of a warped throat in the LARGE Volume Scenario.
\subsubsection{D-terms from magnetised branes revisited}
Another interesting mechanism to achieve dS vacua relies on D-terms from magnetised D7-branes since they give rise to a positive contribution to the scalar potential of the form (\ref{up}). Concrete examples have been worked out in \cite{Dterm,CMQS,CGJR}. Notice that in KKLT scenarios D-terms are not appropriate for uplifting since the AdS minimum is supersymmetric and vanishing F-terms imply vanishing D-terms also. On the other hand, the situation is much better in LVS since the AdS minimum is non supersymmetric and therefore D-terms can be non-zero.
The D-term potential associated to the diagonal $U(1)$ factor
of a stack of D7-branes wrapping the Calabi-Yau divisor $D_i$,
looks like \cite{HKLVZ}:
\begin{equation}
V_D = \frac{g_i^2}{2} \left( \sum_j c_{ij} \hat{K}_j\varphi_j -\xi_i\right)^2\,,
\label{VD}
\end{equation}
where $\hat{K}$ is the matter K\"ahler potential.
In the previous expression the gauge coupling is given by $g_i^{-2}= {\rm Vol}(D_i) /(4\pi) =\tau_i/(4\pi)$,
$\varphi_j$ are matter fields (open string modes) with $U(1)$ charges $c_{ij}$, while $\xi_i$ is
the Fayet-Iliopoulos term. This term is generated by turning on a magnetic flux $\mathcal{F}_i=\tilde{f}_i^k \hat{D}_k$
on the D7-brane stack wrapping $D_i$:
\begin{equation}
\xi_i=\frac{1}{4\pi{\cal V}}\int_X \hat{D}_i \wedge J\wedge \mathcal{F}_i
=-\frac{q_{ij}}{4\pi}\,\frac{\partial K}{\partial T_j}\,,
\label{FI}
\end{equation}
where $q_{ij}$ are the $U(1)$ charges of the K\"ahler moduli which can be expressed in terms of
the triple intersection numbers $k_{ijk}$ of the Calabi-Yau as:
\begin{equation}
q_{ij}= \tilde{f}^k k_{ijk}\,.
\label{triple}
\end{equation}
Ref. \cite{CMQS} studied the interplay between F- and D-terms in the LVS in order to obtain dS vacua, finding that an attractive candidate can be found when the following three criteria are satisfied:
\begin{itemize}
\item The magnetised D7-brane stack wraps the large four-cycle;
\item The K\"ahler modulus $T_b$ (described in section \ref{LVSreview}) is charged under the $U(1)$;
\item One of the open string modes, stretching between the D7-brane with magnetic flux and its orientifold image, is tachyonic at the origin of field space. Such a situation is fairly generic and can be shown to arise as a result of relationships between the charges of various open string fields connected to anomaly cancellation conditions.
\end{itemize}
Under such a situation, ref. \cite{CMQS} found that, upon minimisation, the combined F- and D-term potential yields a contribution which scales as ${\cal V}^{-8/3}$ which is good since $1<8/3<3$.
However, such a contribution can uplift the AdS vacuum to dS
in a controlled fashion only if the magnetic flux on the brane is localised in a warped region. This requires that a two-cycle of the large four-cycle be deep in a warped throat which is not impossible but not straightforward to achieve, as we now see.
To see why it can be hard to have localised magnetic flux which can generate the uplift term of \cite{CMQS}, note first that
we require that the brane wraps the large cycle and also that the K\"ahler modulus $T_b$ is charged under the $U(1)$, i.e:
\begin{equation}
q_{bb} = \tilde{f}^i k_{i b b} \neq 0\,.
\label{bcharge}
\end{equation}
In order for the magnetic flux to be localised, no flux can thread the large cycle since such a contribution would
have support in the entire Calabi-Yau, and so it is difficult to conceive that the
associated energy can be lowered by warping. Thus we would like to set $\tilde{f}^b =0$,
and have non trivial flux threading a small cycle, i.e $\tilde{f}^s \neq 0$. Then the condition (\ref{bcharge}) implies that:
\begin{equation}
k_{sbb} \neq 0\,.
\end{equation}
However ref. \cite{CKM} pointed out that in order
to find a LVS the small four-cycle supporting the non-perturbative effects has to be a `diagonal' del Pezzo divisor which is characterised by the fact that there exists a basis of toric divisors where the only non-vanishing intersection number is $k_{sss}\neq 0$. Hence this four-cycle enters the overall volume in a completely diagonal
way, showing that it can be shrunk to zero size without affecting the Calabi-Yau geometry. Therefore in this case the condition $k_{sbb} \neq 0$ cannot be realised.
However this condition might still be realised if the number of K\"ahler moduli is greater than two and there are other small four-cycles which do not support non-perturbative effects
as in \cite{CMV}. These additional small four-cycles support a GUT- or MSSM-like visible sector and, in order to avoid the shrinking induced by D-terms, have to be `non-diagonal' rigid
but not del Pezzo divisors, according to the classification of \cite{CKM}. Then one can indeed satisfy the condition $k_{sbb} \neq 0$ \cite{CMV}.
\subsubsection*{D-term uplift for fibred Calabi-Yau manifolds}
An alternative way to achieve dS vacua via D-terms without having to deal with localised magnetic flux,
has been proposed in \cite{CGJR} for Calabi-Yau three-folds where the overall volume is controlled by two
four-cycles $\tau_1$ and $\tau_2$:
\begin{equation}
{\cal V}=\sqrt{\tau_1}\tau_2-\tau_s\,.
\end{equation}
The sum of F- and D-term potential coming from a magnetised D7-brane stack wrapping $\tau_1$ is:
\begin{equation}
V=V_D + V_F = \frac{\pi}{\tau_1} \left(
q_{\phi} |\phi_c|^2-\xi_1 \right)^2+k \frac{W_0^2}{{\cal V}^2}\,|\phi_c|^2+V_{\rm LVS}\,,\quad\text{with}\quad\xi_1=p\frac{\sqrt{\tau_1}}{{\cal V}}\,,
\label{NewVtotale}
\end{equation}
where $\phi_c$ is a canonically normalised matter field while $k$, $q_{\phi}$ and $p$ are all $\mathcal{O}(1)$ numbers. The LVS potential is as described above, and depends only on the moduli $\tau_s$ and ${\cal V}$.
The minimum for $\phi_c$ is at:
\begin{equation}
\langle|\phi_c|^2\rangle= \frac{\xi_1}{q_{\phi}}-\frac{k W_0^2 \,\tau_1}{2\pi q_{\phi}^2 {\cal V}^2}\simeq \frac{\xi_1}{q_{\phi}}\,,
\end{equation}
where the approximate equality uses $\xi_1 \simeq \sqrt{\tau_1}/{\cal V}$ and ${\cal V} \gg 1$. Eq.~(\ref{NewVtotale}) then reduces to:
\begin{equation}
V = \lambda W_0^2 \frac{\sqrt{\tau_1}}{{\cal V}^3}+V_{\rm LVS}\,,\quad\text{with}\quad\lambda=\frac{k p}{q_{\phi}}\,.
\label{NewVsu}
\end{equation}
It is now a problem that the potential $V_{\rm LVS}$ (\ref{Vlvs}) does not depend on $\tau_1$, because the modulus $\tau_1$ cannot be fixed. However we know that another term in $V$ depends on $\tau_1$ once string-loop corrections to $K$ are included, coming from loops of open strings living on the D7-branes wrapped around $\tau_1$ \cite{ccq1}. Including also this $\tau_1$-dependent part of the scalar potential we find:
\begin{equation}
V =\left(\lambda\frac{\sqrt{\tau_1}}{{\cal V}}+\frac{g_s^2 c_{\rm loop}^2}{\tau_1^2}\right)\frac{W_0^2}{{\cal V}^2}\,,
\label{Vcombined}
\end{equation}
where $c_{\rm loop}$ is a loop coefficient that depends on the complex structure moduli. The potential (\ref{Vcombined}) has a minimum for $\tau_1$ at:
\begin{equation}
\tau_1 =\left(4 g_s^2 c_{\rm loop}^2/\lambda\right)^{2/5}{\cal V}^{2/5}\,,
\label{tau1VEV}
\end{equation}
which substituting back in (\ref{Vcombined}) gives:
\begin{equation}
V =\mu\frac{W_0^2}{{\cal V}^{14/5}}+V_{\rm LVS}\,,
\quad\text{with}\quad\mu=5\left(\frac{\lambda}{4}\right)^{4/5}\left(g_s c_{\rm loop}\right)^{2/5}.
\label{A}
\end{equation}
We see that in order to get a Minkowski vacuum, the coefficient $c_{\rm loop}$ must be tuned such that \cite{CGJR}:
\begin{equation}
\mu \sim \mathcal{O}(1) \,g_s^{-1/2}\langle{\cal V} \rangle^{-1/5}\qquad\Leftrightarrow\qquad c_{\rm loop}\sim \mathcal{O}(1)\, g_s^{-9/4}\langle{\cal V} \rangle^{-1/2}\,.
\end{equation}
Substituting this tuning back in (\ref{tau1VEV}) we realise that:
\begin{equation}
\langle\tau_1\rangle \sim g_s^{-1}\sim \mathcal{O}(10)\,.
\end{equation}
Notice that the tuning of the loop coefficient includes the value of the volume
at the minimum which may not be natural for very large volumes. However there is always
the possibility that the corresponding cycle lies within a warped region for which the coefficient $\lambda$
is warped and volume dependent and therefore the tuning is similar to the
antibrane uplifting in KKLT.
We finally mention that another mechanism to achieve dS vacua that has been proposed, relies on F-terms from a hidden sector with metastable susy breaking \cite{ss}. However this mechanism has not yet been implemented within the LVS. It would be very interesting to have a concrete realisation
of this mechanism in the LVS.
\section{De Sitter vacua from non-perturbative effects}
\label{dSquiver}
Non-perturbative effects play a major r\^ole in stabilising the K\"ahler moduli $T_i$ of type IIB Calabi-Yau flux compactifications. In fact, the starting point of KKLT and LVS scenarios is the inclusion of $T_i$-dependent
non-perturbative corrections to the superpotential of the form:
\begin{equation}
W_{\rm np} =\sum_i A_i\, e^{-\,a_i \, f_i}\,,
\label{WnpinT}
\end{equation}
where the prefactors $A_i$ are in general functions of the complex structure moduli $U$
and the open string moduli (D7-deformation and D3-position moduli), the $a_i$ are constants and
the $f_i$ are the corresponding gauge kinetic functions which are given by:
\begin{equation}
f_i = T_i + h_i(\mathcal{F}) S\,,
\label{gkfinT}
\end{equation}
where $T_i$ is the K\"ahler modulus controlling the size of the divisor $\Sigma_i$
wrapped by either a stack of D7-branes or a Euclidean D3-instanton,
while $h$ is a function of the world-volume magnetic flux $\mathcal{F}$.
Let us briefly describe the two cases for $\mathcal{F}=0$ and $\mathcal{F}\neq 0$ which give rise to
two physically different situations.
\begin{itemize}
\item {\it Vanishing magnetic flux: $\mathcal{F}=0$}. In this case $h_i=0$ and
the non-perturbative effects have three different microscopic realisations:
\begin{itemize}
\item {\it Divisor $\Sigma_i$ wrapped just by a stack of D7-branes}:
In this case the superpotential (\ref{WnpinT}) is generated by gaugino condensation.
The simplest realisation of a pure $\mathcal{N}=1$ super Yang-Mills theory that undergoes
gaugino condensation is via a stack of D7-branes wrapping an orientifold-invariant four-cycle without any
magnetic flux on the D7-branes. The constants $a_i$ are given by $a_i=6\pi/\beta_i$ where $\beta_i$ are
the one-loop beta function coefficients of the condensing gauge theories. For example,
in the case of a pure $SU(N)$ theory, $\beta_i= 3 N$ while the in case of a pure $Sp(2N)$ theory,
$\beta_i = 3 (N+1)$.
\item {\it Divisor $\Sigma_i$ wrapped just by a Euclidean D3-brane instanton}:
In this case the superpotential (\ref{WnpinT}) is generated by a so-called `stringy' instanton.
The simplest realisation which yields the right fermionic zero mode structure,
involves a rigid four-cycle which is transversally invariant under the orientifolds and
a vanishing magnetic flux on the instanton. In this case the constants $a_i$ are given by $a_i=2\pi$.
\item {\it Divisor $\Sigma_i$ wrapped by both a stack of D7-branes and a Euclidean D3-brane instanton}:
In this case the main contribution to the superpotential (\ref{WnpinT}) comes from gaugino condensation since
the contribution due to E3-instantons is more suppressed due to the different behaviour of the constants $a_i$.
\end{itemize}
\item {\it Non-vanishing magnetic flux: $\mathcal{F}\neq 0$}. In this case $h_i\neq 0$ and
the non-perturbative effects can have again three different microscopic realisations:
\begin{itemize}
\item {\it Divisor $\Sigma_i$ wrapped just by a stack of D7-branes}:
In the presence of a non-vanishing magnetic flux $\mathcal{F}$ on the world-volume of a D7-brane stack,
chiral matter gets generated, and so the theory undergoes gaugino condensation only for particular configurations.
For examples an $SU(N_c)$ theory with $N_f$ flavours, undergoes gaugino condensation only for $N_f<N_c-1$.
Moreover, in this case the K\"ahler moduli get charged under the anomalous $U(1)$, and so the prefactors
$A_i$ have to depend also on charged matter fields whose $U(1)$ transformation has to compensate the shift transformation of the $T_i$ fields
in order to render the superpotential (\ref{WnpinT}) gauge invariant.
\item {\it Divisor $\Sigma_i$ wrapped just by a Euclidean D3-brane instanton}: Given that $\mathcal{F}$ is anti-invariant
under the orientifold and in order to have a non-vanishing contribution to the superpotential the instanton configuration
has to be invariant under the orientifold, we can have $\mathcal{F}\neq 0$ only if the magnetic flux is purely odd,
i.e. $\mathcal{F} \in H^{1,1}_-(\Sigma_i)$ \cite{GKPW}. This is possible only for orientifold projections such that $h_{1,1}^-\neq 0$.
In this case the gauge kinetic functions acquire a dependence on the $G$-moduli associated with the $h_{1,1}^-$ odd four-cycles,
$f_i = T_i + h_{ij} (\mathcal{F}) G_j$.
\item {\it Divisor $\Sigma_i$ wrapped by both a stack of D7-branes and a Euclidean D3-brane instanton}:
In this case the superpotential (\ref{WnpinT}) is generated by a so-called `gauge' instanton corresponding to the case $N_f=N_c-1$
and the prefactors $A_i$ have again to depend on charged matter field in order to guarantee the gauge invariance of the superpotential.
\end{itemize}
\end{itemize}
Scenarios with vanishing magnetic flux are simpler for modulus stabilisation.
The case with $\mathcal{F}\neq 0$ includes more general expressions but also constraints.
In particular, it complicates the stabilisation of the four-cycle supporting the Standard Model (SM)
by these non-perturbative effects since requiring non-vanishing coefficients $A_i$
would generically break the SM symmetry at high energies by the non-vanishing values of the charged matter fields \cite{Tension}.
In those cases D-terms or perturbative, rather than non-perturbative, effects are preferred to fix the SM cycle \cite{CMV}.
The non-perturbative effects described above are the only ones considered so far for modulus stabilisation in type IIB flux
compactifications. However, there are further dilaton-dependent non-perturbative effects generated by
either gaugino condensation on spacetime-filling D3-branes or Euclidean D(-1)-brane (E(-1)) instantons.
Given that the associated branes do not wrap any internal cycle,
the corresponding gauge kinetic function is determined by the dilaton field $S$:
\begin{equation}
f_i = S +h_i \,Q_i\,,
\label{gkfinS}
\end{equation}
where, in analogy with (\ref{gkfinT}), we have included a shift proportional to $h$. As we have seen,
in the case of D7-branes and E3-instantons, $h$ is non-zero only in the presence of a magnetic flux.
In the present case, this is equivalent to place the D3-branes and/or the E(-1)-instantons at a singularity.
In fact, as a magnetic flux is responsible for the emergence of chirality in the geometric case,
here chiral matter gets generated by the presence of the singularity. Hence the $h_i$ in (\ref{gkfinS})
are constants proportional to the $U(1)$-charge of the blow-up modes $Q_i$ resolving the singularity.
In parallel with the previous discussion for the geometric case, we have two cases for $h_i=0$
(branes at smooth points and no chiral matter)
and $h_i\neq 0$ (branes at singularities and chiral matter).
For each case we have again three different microscopic realisations corresponding to:
\begin{itemize}
\item {\it Stack of spacetime-filling D3-branes}:
In this case the superpotential (\ref{WnpinT}) is generated by gaugino condensation.
The simplest realisation of a pure $\mathcal{N}=1$ super Yang-Mills theory that undergoes
gaugino condensation is via a stack of spacetime-filling D3-branes at a smooth point on top of the O-plane.
If the D3-branes are located at a singularity chiral matter gets generated,
and so the theory undergoes gaugino condensation only for particular configurations.
Moreover, in this case the blow-up modes $Q_i$ get charged under the anomalous $U(1)$, and so the prefactors
$A_i$ have to depend also on charged matter fields whose $U(1)$ transformation has to compensate the shift of the $Q_i$ fields
in order to render the superpotential gauge invariant.
We finally point out that the location of the D3-branes can be determined by minimising the potential
for the D3-position moduli $\zeta_i$ which is generated by different effects ($\zeta_i$-dependence of the K\"ahler potential and
the prefactors $A_i$ plus D-terms) \cite{malda}.
\item {\it Euclidean D(-1)-brane instanton}:
In this case the superpotential (\ref{WnpinT}) is generated by a so-called `stringy' instanton.
\item {\it Euclidean D(-1)-brane instanton on top of a stack of spacetime-filling D3-branes}:
When the branes are located at smooth points, the main contribution to the superpotential (\ref{WnpinT}) comes from gaugino condensation since
the contribution due to E(-1)-instantons is more suppressed due to the different behaviour of the constants $a_i$.
On the other hand, for branes at singularities, the superpotential (\ref{WnpinT}) is generated by a so-called `gauge' instanton
corresponding to the case $N_f=N_c-1$ and the prefactors $A_i$ have again to depend on charged matter fields in order to guarantee the gauge invariance of the superpotential.
\end{itemize}
In this section we will explore a combination of both classes of non-perturbative effects
within the LVS framework.
We shall focus, without loss of generality,
on the case with a single K\"ahler moduli-dependent and a single dilaton-dependent contribution
to the non-perturbative superpotential:
\begin{equation}
W_{\rm np} = A\, e^{-\,a \, T } + B\, e^{-\, b \, \left(S+ h Q\right)}\,,
\label{TotalWnp}
\end{equation}
where we will consider just one blow-up mode $Q$.
We will find that the dilaton-dependent non-perturbative effect proportional to $B$
can give rise to de Sitter vacua only for $h\neq0$. In fact, in the case $h=0$, the new $S$-dependent
non-perturbative effect can be included in a redefinition of $W_0$ $\to$ $W_0'\equiv W_0 + B\,e^{-b\,S}$,
showing that this term cannot give rise to any new interesting dynamics.
We shall find that for $h\neq 0$ it is the F-term of the blow-up mode $Q$ which is responsible for uplifting
since it gives rise to
a positive definite term in the scalar potential of the form $e^{-2\,b {\rm Re}(S)}/{\cal V}$.
Given that ${\rm Re}(S)$ is fixed by ratios of flux integers,
this term has a similar effect as the standard warped anti D3-brane contribution
with the advantage that it comes from a manifestly supersymmetric effective action.
As is well known, the LVS applies for Calabi-Yau compactifications with negative Euler number ($h^{1,2}>h^{1,1}$).
However in this section we shall also present another interesting implication of adding an $S$-dependent $W_{\rm np}$:
the possibility to obtain exponentially large volumes and de Sitter space also for positive and vanishing Euler numbers.
This requires the introduction of string loop contributions to the effective action which are less understood.
In appendix \ref{OnlyNPquiver} we will discuss briefly a simplified case with non-perturbative effects only at the singularity regime.
\subsection{De Sitter LVS from D3/E(-1) non-perturbative effects at singularities}
\label{NPsing}
Let us study how the LVS stabilisation mechanism gets modified by the inclusion of
dilaton-dependent non-perturbative effects. We shall analyse the case for $h\neq 0$ corresponding to D3-branes and/or E(-1)-instantons at singularities
since, as we have pointed out above, the case for $h=0$ does not give rise to any new result.
As reviewed in section \ref{LVSreview}, the standard LVS involves at least two K\"ahler moduli,
$T_b=\tau_b+{\rm i}\,\psi_b$ and $T_s=\tau_s + {\rm i}\,\psi_s$, which can be fixed by the interplay of the leading order $\alpha'$
correction to the K\"ahler potential and a single $T_s$-dependent non-perturbative superpotential.
Hence we shall extend the superpotential (\ref{WLVS}) as in (\ref{TotalWnp}) obtaining:
\begin{equation}
W=W_0(S)+ A\, e^{-a T_s}+B \,e^{-b \left(S+h Q\right)}\,,
\label{Wnon-pert}
\end{equation}
where $Q=\rho + {\rm i}\,\psi_{\rho}$ is the modulus that blows up the singularity ($\rho=0$ corresponds to a collapsed four-cycle) and $h\neq 0$. \footnote{This extra sector can also be considered for the KKLT scenario for which there will be a non-perturbative superpotential for the overall K\"ahler modulus $T$ and also for the blow-up mode $Q$. However since the original minimum was supersymmetric, $D_TW=0$, as long as there is a solution to $D_QW=0$ the vacuum remains supersymmetric AdS; although such a solution is likely to be away from the singular locus.}
The K\"ahler potential (\ref{KLVS}) of the effective supersymmetric field theory has to be supplemented with the term for the blow-up mode $\rho$,
and so it takes the more general form (writing $s = {\rm Re}(S)$):
\begin{equation}
K=-\ln(2s)-2\ln \left({\cal V}+\frac{\zeta \,s^{3/2}}{2}\right)+\alpha \frac{\rho^2}{{\cal V}}\,,
\label{KahlerPot}
\end{equation}
with:
\begin{equation}
{\cal V}= \tau_b^{3/2}-\tau_s^{3/2}\,.
\end{equation}
Notice that, even if the dilaton is fixed at leading semi-classical order by imposing $D_S W=0$,
both in the superpotential (\ref{Wnon-pert}) and in the K\"ahler potential (\ref{KahlerPot})
we kept the dilaton dependence manifest in order to have control over higher order contributions to the scalar potential
which could play an important r\^ole for achieving dS vacua.
In the large volume limit $\tau_b\gg \tau_s$, the K\"ahler potential (\ref{KahlerPot}) can be expanded as:
\begin{equation}
K\simeq -\ln(2s)-3\ln \tau_b +2 \,\epsilon_{\tau_s} - \frac{\zeta \,s^{3/2}}{\tau_b^{3/2}}\left(1+\epsilon_{\tau_s} \right)
+\alpha \frac{\rho^2}{\tau_b^{3/2}}\left(1+\epsilon_{\tau_s} \right)\,,
\label{KPot}
\end{equation}
where
\begin{equation}
\epsilon_{\tau_s} \equiv \left(\frac{\tau_s}{\tau_b}\right)^{3/2}\sim \mathcal{O}({\cal V}^{-1})\ll 1\,.
\end{equation}
We point that the moduli $T_b$ and $T_s$ are both
larger than the string scale, while $Q$ is at the singular locus. It is worth stressing that
the effective field theory can be trusted not just in the case when the moduli are in the geometric regime,
but also in the regime close to a singularity thanks to the detailed description
of strings at orbifold singularities. The field $Q$ can indeed be shown to be fixed
at the singular locus as a result of
D-term stabilisation. If we turn on a flux on a two-cycle inside the four-cycle whose volume is given by the blow-up mode $\rho$,
the D-term potential takes the same form as in (\ref{VD}):
\begin{equation}
V_D = \frac{g^2}{2} \left( \sum_i c_i \hat{K}_i\varphi_i -\xi\right)^2\,,
\end{equation}
where, according to (\ref{FI}), the FI-term $\xi$ becomes:
\begin{equation}
\xi=-\frac{q_{\rho}}{4\pi} \frac{\partial K}{\partial Q}=-\frac{\alpha q_{\rho}}{4\pi} \frac{\rho}{{\cal V}}\,.
\end{equation}
Here $q_{\rho}$ is the $U(1)$ charge of the blow-up mode $\rho$
that depends on the magnetic flux and its triple self intersection number (see (\ref{triple})).
In this section we shall proceed assuming that the D-term potential can be minimised
with non-zero VEVs for some charged matter fields such that $\langle\sum_i c_i \hat{K}_i\varphi_i\rangle=0$
together with a non-vanishing prefactor $B$
of the non-perturbative superpotential ($B\neq 0$). \footnote{The situation is similar to the tension between moduli
stabilisation by non-perturbative effects and chirality discussed in \cite{Tension};
although in our case this results into a much weaker constraint as the open string fields belong to a hidden sector,
and so can get non-zero VEVs.}
The details of such settings will be
elaborated in section \ref{generation}. Thus the D-term scalar potential reduces to:
\begin{equation}
V_D= \frac{g^2}{2}\,\xi^2=\frac{g^2}{2}\left(\frac{\alpha q_{\rho}}{4\pi}\right)^2 \frac{\rho^2}{{\cal V}^2}
\simeq \frac{(\alpha q_{\rho})^2}{8\pi}
\frac{\rho^2}{(s+h\rho) \tau_b^3}\left(1+2\epsilon_{\tau_s}\right),
\end{equation}
which uses $4\pi g^{-2} = s + h\rho$ and ${\cal V}^{-2} \simeq \tau_b^{-3}(1 + 2 \epsilon_{\tau_s})$.
We now turn to the computation of the F-term scalar potential:
\begin{equation}
V_F =e^K\left(K^{I\bar{J}}D_I W D\overline{W}_{\bar{J}}-3|W|^2\right),
\end{equation}
where the K\"ahler covariant derivative is $D_I W= W_I+W K_I$.
The K\"ahler metric is a symmetric matrix which reads (with the leading order correction in a large volume expansion):
\begin{equation}
K_{I\bar{J}}=\left(
\begin{array}{cccc}
\frac{1}{4 s^2}\left(1-\frac{3\epsilon_s}{4} \right) &
\frac{9 \sqrt{s}\, \zeta }{16 \tau_b^{5/2}}(1+2\epsilon_{\tau_s}) & -\frac{9 \sqrt{s\,\tau_s} \,\zeta
}{16 \tau_b^3} & 0 \\
K_{1\bar{2}} & \frac{3}{4
\tau_b^2}\left(1+\frac{ 10 \epsilon_{\tau_s} + 5\epsilon_{\rho}-5\epsilon_s}{4} \right) & -\frac{9 \sqrt{\tau_s}}{8 \tau_b^{5/2}}(1+\epsilon_{\rho}-\epsilon_s) & -\frac{3 \alpha \, \rho}{4 \tau_b^{5/2}}(1+2\epsilon_{\tau_s}) \\
K_{1\bar{3}} & K_{2\bar{3}} & \frac{3 }{8
\tau_b^{3/2} \sqrt{\tau_s}} \left(1+\frac{\epsilon_{\rho}-\epsilon_s}{2} \right)& \frac{3 \alpha \,\rho \sqrt{\tau_s}}{4 \tau_b^3} \\
K_{1\bar{4}} & K_{2\bar{4}} & K_{3\bar{4}} & \frac{\alpha }{2 \tau_b^{3/2}}(1+\epsilon_{\tau_s})
\end{array}
\right), \nonumber
\end{equation}
where:
\begin{equation}
\epsilon_{\rho}\equiv \frac{\alpha \, \rho^2}{\tau_b^{3/2}}\sim \mathcal{O}({\cal V}^{-1})\ll 1
\qquad\text{and}\qquad\epsilon_s\equiv \zeta\left(\frac{s}{\tau_b}\right)^{3/2}\sim \mathcal{O}({\cal V}^{-1})\ll 1\,.
\end{equation}
The inverse K\"ahler metric with the leading order correction in a large volume expansion reads:
\begin{equation}
\left(
\begin{array}{cccc}
4 s^2 \left(1+\frac{3\epsilon_s}{4} \right) &
-\frac{3 s^{5/2}\, \zeta}{\sqrt{\tau_b}}\left(1+\frac{4\epsilon_{\tau_s} + 8\epsilon_s + \epsilon_{\rho}}{4} \right) &
-\frac{3 s^{5/2} \, \zeta \tau_s }{\tau_b^{3/2}}\left(1+ \frac{12 \epsilon_{\tau_s} + 8 \epsilon_s + \epsilon_{\rho}}{4} \right) &
-\frac{9 s^{5/2} \,\zeta \rho}{2 \tau_b^{3/2}}\left(1+\frac{4\epsilon_{\tau_s}+8\epsilon_s+ \epsilon_{\rho}}{4}\right) \\
K^{1\bar{2}} &
\frac{4}{3} \tau_b^2 \left(1+\frac{8\epsilon_{\tau_s}+ 5\epsilon_s+\epsilon_{\rho}}{4} \right) &
4 \tau_b \tau_s \left(1+\frac{8\epsilon_{\tau_s}+ 3\epsilon_s-\epsilon_{\rho}}{4} \right) &
2 \,\rho \tau_b \left(1+\frac{5\epsilon_s+\epsilon_{\rho}}{4} \right) \\
K^{1\bar{3}} & K^{2\bar{3}} &
\frac 83 \tau_b^{3/2} \sqrt{\tau_s} \left(1+\frac{9\epsilon_{\tau_s} + \epsilon_s - \epsilon_{\rho}}{2} \right) &
2 \,\rho \tau_s\left(1+\frac{8\epsilon_{\tau_s}+ 5\epsilon_s + \epsilon_{\rho}}{4} \right) \\
K^{1\bar{4}} & K^{2\bar{4}} & K^{3\bar{4}} & \frac{2\tau_b^{3/2}}{\alpha}\left(1+\frac{3\epsilon_{\rho} - 2\epsilon_{\tau_s}}{2}\right)
\end{array}
\right). \nonumber
\end{equation}
Therefore the F-term scalar potential is given by:
\begin{eqnarray}
V_F&=&\frac{(1+\epsilon_{\rho}) (1-\epsilon_s) (1+2\,\epsilon_{\tau_s})}{2 s \tau_b^3} \left\{ K^{S\bar{S}} D_{\bar{S}}\overline{W} D_S W
-a A K^{S \bar{T_s}} \left(D_S W e^{{\rm i} a \psi_s}+D_{\bar{S}} \overline{W} e^{-{\rm i} a \psi_s}\right) e^{-a \tau_s}
\right. \nonumber \\
&& \left. -h b B K^{S\bar{Q}} \left(D_S W e^{+ {\rm i }b (C_0 + h \psi_{\rho})}
+D_{\bar{S}} \overline{W} e^{- {\rm i} b (C_0 + h \psi_{\rho})}\right)
e^{-b (s+ h \rho)}
+ (\overline{W} D_S W+W D_{\bar{S}}\overline{W}) K^{Si} K_i \right. \nonumber \\
&& \left. + (a A)^2 K^{T_s \bar{T_s}} e^{-2 a \tau_s}-2 a A W \cos\left(a \psi_s\right)
K^{T_s i} K_i\, e^{-a \tau_s} \right. \nonumber \\
&& \left. + (h b B)^2 K^{Q \bar{Q}} e^{-2 b (s+ h \rho)}
-2 h b B W \cos\left[ b \left(C_0 +h \psi_{\rho}\right)\right]
K^{Qi} K_i \,e^{-b (s +h \rho)} \right. \nonumber \\
&& \left. + 2 h a A b B K^{T_S \bar{Q}} \cos\left[a \psi_s + b \left(C_0 + h \psi_{\rho}\right)\right]
e^{- a \tau_s - b (s + h \rho)}
+W^2 \left(K^{ij} K_i K_j-3\right)\right\}.
\end{eqnarray}
Substituting the values of the elements of the inverse K\"ahler metric we obtain:
\begin{eqnarray}
V_F&=&\frac{(1+\epsilon_{\rho}) (1-\epsilon_s) (1+2\,\epsilon_{\tau_s})}{2 s \tau_b^3} \left\{ 4 s^2 D_{\bar{S}}\overline{W} D_S W
\left(1+\frac{3\epsilon_s}{4} \right) \right. \nonumber \\
&& \left. +3 a A \zeta\,s^{5/2} \,\tau_s \left(D_S W e^{{\rm i} a \psi_s}+D_{\bar{S}} \overline{W} e^{-{\rm i} a \psi_s}\right)
\frac{e^{-a \tau_s}}{\tau_b^{3/2}}\left(1+ \frac{12 \epsilon_{\tau_s} + 8 \epsilon_s +\epsilon_{\rho}}{4} \right)
\right. \nonumber \\
&& \left. + \frac 92 h b B \zeta\,s^{5/2} \, \rho \left(D_S W e^{+ {\rm i }b (C_0 + h \psi_{\rho})}
+D_{\bar{S}} \overline{W} e^{- {\rm i} b (C_0 + h \psi_{\rho})}\right) \frac{e^{-b (s+ h \rho)}}{\tau_b^{3/2}}
\left(1+\frac{4\epsilon_{\tau_s}+8\epsilon_s+\epsilon_{\rho}}{4}\right)\right. \nonumber \\
&& \left. + \frac 92 \zeta (\overline{W} D_S W+ W D_{\bar{S}}\overline{W}) \frac{s^{5/2}}{\tau_b^{3/2}}
\left(1+ \frac{4\epsilon_{\tau_s}+6 \epsilon_s-\epsilon_{\rho}}{4}\right)
+W^2 \left(3+\frac{3 \epsilon_s- \epsilon_{\rho}}{4}-3\right)\right. \nonumber \\
&& \left. + \frac 83 (a A)^2 \sqrt{\tau_s}\, \tau_b^{3/2}\,e^{-2 a \tau_s}
\left(1+\frac{9\epsilon_{\tau_s} + \epsilon_s - \epsilon_{\rho}}{2} \right)
+ 4 a A W \cos\left(a \psi_s\right) \tau_s\, e^{-a \tau_s}
\left(1+\frac{3\epsilon_s-\epsilon_{\rho}}{4}\right) \right. \nonumber \\
&& \left. + \frac{2}{\alpha}(h b B)^2 \tau_b^{3/2} e^{-2 b (s+ h \rho)}
\left(1+\frac{3 \epsilon_{\rho} - 2\epsilon_{\tau_s}}{2}\right)
+2 h b B W \cos\left[ b \left(C_0 +h \psi_{\rho}\right)\right] \rho \,e^{-b (s +h \rho)}
\left(1+\frac{9\epsilon_s - 3 \epsilon_{\rho}}{4}\right) \right. \nonumber \\
&& \left. + 4 h a A b B \,\rho \tau_s \cos\left[a \psi_s + b \left(C_0 + h \psi_{\rho}\right)\right]
e^{- a \tau_s - b (s + h \rho)}\left(1+\frac{8\epsilon_{\tau_s}+5 \epsilon_s +\epsilon_{\rho}}{4} \right)
\right\}. \nonumber
\end{eqnarray}
Notice that the K\"ahler covariant derivative with respect to the dilaton scales as:
\begin{equation}
D_S W = \underset{\mathcal{O}(1)}{\underbrace{D_S W_0|_{\zeta=0}}}
- \underset{\mathcal{O}({\cal V}^{-1})}{\underbrace{\left[\frac 34 \frac{\epsilon_s}{s} W_0(S)+\left(b+\frac{1}{2\,s}\right) B e^{-b(S+hQ)}\right]}}
-\underset{\mathcal{O}({\cal V}^{-2})}{\underbrace{\frac 34 \frac{\epsilon_s}{s} B e^{-b(S+hQ)}}}\,,
\label{DSW}
\end{equation}
where:
\begin{equation}
D_S W_0|_{\zeta=0}=\left(\frac{\partial_s W_0}{2}-\frac{W_0}{2s}\right)\,.
\end{equation}
The volume scaling of (\ref{DSW}) follows from the fact that we shall be interested in studying the
scalar potential in the large volume limit $a \tau_s\sim b s\sim \ln {\cal V}\gg 1$.
The total scalar potential $V=V_D+V_F$ receives contributions at different orders in a large volume expansion.
It is therefore convenient to study its behaviour order by order in $1/{\cal V}$ writing:
\begin{equation}
V= V_{\mathcal{O}({\cal V}^{-2})}+V_{\mathcal{O}({\cal V}^{-3})}+V_{\mathcal{O}({\cal V}^{-4})}+\dots \,,
\end{equation}
where:
\begin{equation}
V_{\mathcal{O}({\cal V}^{-2})}=\frac{1}{\tau_b^3} \left[ 2 s | D_S W_0|_{\zeta=0}|^2
+\frac{(\alpha q_{\rho})^2}{8\pi}\frac{\rho^2}{\left(s+h\rho\right)}\right].
\end{equation}
and:
\begin{eqnarray}
V_{\mathcal{O}({\cal V}^{-3})}&=&\frac{1}{\tau_b^3}
\left\{2 s |D_S W_0|_{\zeta=0}|^2 (\epsilon_{\rho}-\frac{\epsilon_s}{4}+2\,\epsilon_{\tau_s})
+\frac{(\alpha q_{\rho})^2}{4\pi}\frac{\rho^2}{(s+h\rho) }\, \epsilon_{\tau_s}\right. \nonumber \\
&& \left.
-2 s (b+1/(2s)) B \left[ \cos[b(C_0+h \psi_{\rho})] {\rm Re}(D_S W_0|_{\zeta=0})
-\sin [b(C_0+h \psi_{\rho})] {\rm Im}(D_S W_0|_{\zeta=0})\right]e^{-b (s+h\rho)}\right. \nonumber \\
&& \left. + \frac 32 \left[{\rm Re}(W_0) {\rm Re}(D_S W_0|_{\zeta=0})+{\rm Im}(W_0) {\rm Im}(D_S W_0|_{\zeta=0})\right] \epsilon_s\right. \nonumber \\
&& \left. + \frac{1}{2s}\left[\frac 83 (a A)^2 \sqrt{\tau_s}\, \tau_b^{3/2}\,e^{-2 a \tau_s}
+ 4 a A W_0 \cos\left(a \psi_s\right) \tau_s\, e^{-a \tau_s} +W_0^2 \frac{3 \epsilon_s- \epsilon_{\rho}}{4} \right.\right. \nonumber \\
&& \left.\left. + \frac{2}{\alpha}(h b B)^2 \tau_b^{3/2} e^{-2 b (s+ h \rho)}
+2 h b B W_0 \cos\left[ b \left(C_0 +h \psi_{\rho}\right)\right] \rho \,e^{-b (s +h \rho)}\right] \right\}.
\label{VO3}
\end{eqnarray}
The potential at order ${\cal V}^{-2}$ depends on four fields: $\tau_b$, $s$, $C_0$ and $\rho$.
However, the minimisation with respect to $S$ and $\rho$ implies that:
\begin{equation}
\langle D_S W|_{\zeta=0}\rangle=0\qquad\text{and}\qquad\langle\rho\rangle=0\,,
\end{equation}
leaving a flat potential for $\tau_b$ and justifying our expansion of the
K\"ahler potential around the singularity obtained by shrinking the blow-up mode $\rho$.
The potential at order ${\cal V}^{-3}$ then reads:
\begin{equation}
V_{\mathcal{O}({\cal V}^{-3})}=\frac{1}{2 \langle s\rangle}
\left[\frac 83 (a A)^2 \sqrt{\tau_s}\, \frac{e^{-2 a \tau_s}}{\tau_b^{3/2}}
- 4 a A W_0 \tau_s\frac{e^{-a \tau_s}}{\tau_b^3} + \frac{3}{4} \frac{\zeta \langle s\rangle ^{3/2}}{\tau_b^{9/2}} W_0^2
+ \frac{2}{\alpha}(h b B)^2 \frac{e^{-2 b \langle s\rangle}}{\tau_b^{3/2}}\right] .
\label{VO3s}
\end{equation}
where we have already minimised with respect of $\psi_s$ and we have set $s=\langle s\rangle = 1/g_s$.
Notice that the leading order stabilisation of the blow-up mode $\rho$ at $\langle\rho\rangle=0$
eliminates the $\rho$ dependence in the exponentials and the last term in (\ref{VO3}).
This is important because this is the only extra term that could give a negative contribution to the scalar potential.
Considering the scalar potential for $\rho$ by adding the $\rho$-dependent terms in (\ref{VO3})
to the D-term, the VEV of $\rho$ is slightly moved away from the singularity
but the minimum is at a value $\langle\rho\rangle\sim 1/{\cal V}$
inducing a much suppressed contribution to the scalar potential of order
$\delta V_F\sim 1/{\cal V}^4$, and therefore can be safely neglected.
Similarly for $(D_S W_0|_{\zeta=0})$, this quantity has been fixed
at $\langle D_S W|_{\zeta=0}\rangle=0$ only focusing on the potential at order ${\cal V}^{-2}$.
The leading order correction to this result comes from considering also the
dilaton dependent terms in (\ref{VO3}). They slightly move the minimum to
$\langle D_S W|_{\zeta=0}\rangle\sim 1/{\cal V}$ giving rise again to contributions of the order
$\delta V_F\sim 1/{\cal V}^4$, which can therefore be safely neglected.
What we are left with then, is a potential of the standard LVS form (\ref{Vlvs})
plus an additional positive definite term coming from the non-perturbative effects at the singularity:
\begin{equation}
V = V_{\rm LVS} + V_{\rm up}\,,
\end{equation}
where:
\begin{equation}
V_{\rm up} \propto h^2\,\frac{e^{-2b\langle s\rangle}}{{\cal V}}\,,
\label{uplift2}
\end{equation}
with positive proportionality factor. This is precisely of the form (\ref{uplift1}), with $\alpha=5/3$ and the warp factor substituted by a very similar expression now as an exponential of the dilaton field which is fixed by fluxes. Therefore the effect of this term is identical to a large warping for weak coupling strings. Notice also that the uplifting term (\ref{uplift2}) is proportional to $h$,
and so it goes to zero for $h=0$.
Let us now minimise the scalar potential with respect to $\tau_s$ by solving $\partial V/\partial \tau_s=0$.
In the limit $a\tau_s\gg 1$, we find:
\begin{equation}
e^{-a \tau_s}= \frac{3\sqrt{\tau_s}}{4 a A}\frac{W_0}{\tau_b^{3/2}}\sim\mathcal{O}({\cal V}^{-1})
\quad\Rightarrow\quad a \tau_s =\ln\left(\frac{4 a A}{3\sqrt{\tau_s}}\right)+\ln\left(\frac{\tau_b^{3/2}}{W_0}\right)
\simeq\ln\left(\frac{\tau_b^{3/2}}{W_0}\right)\,.
\label{tsVEV}
\end{equation}
Substituting this result back in (\ref{VO3s}) we end up with the following effective potential:
\begin{equation}
V_{\mathcal{O}({\cal V}^{-3})}=\frac{3}{4\langle s\rangle} \frac{W_0^2}{\tau_b^{9/2}} \left\{\frac{\zeta \langle s\rangle ^{3/2}}{2}
-\left[\frac{\ln\left(\tau_b^{3/2}/W_0\right)}{a}\right]^{3/2}
+ \frac{4}{3\alpha}\left(\frac{h b B}{W_0}\right)^2 e^{-2 b \langle s\rangle}\tau_b^3 \right\}.
\label{VO3tb}
\end{equation}
The solution of $\partial V/\partial \tau_b=0$ is then:
\begin{equation}
\frac{\zeta \langle s\rangle ^{3/2}}{2}=\left[\frac{\ln\left(\tau_b^{3/2}/W_0\right)}{a}\right]^{3/2}
\left(1-\frac{1}{2\ln\left(\tau_b^{3/2}/W_0\right)}\right)-\frac{4}{9\alpha}
\left(\frac{h b B}{W_0}\right)^2 e^{-2 b \langle s\rangle}\tau_b^3\,.
\label{tbVEV}
\end{equation}
This implicitly fixes the volume ${\cal V}$ or $\tau_b$ as a function of $\langle s \rangle, W_0, B$ and the constants $h,a, b,\alpha$. Substituting this result back in (\ref{VO3tb}) we find that the value of the potential at the
minimum is: \footnote{Explicit string calculations at one-loop order have led to the need to redefine the K\"ahler modulus that corresponds to the proper chiral superfield in the supergravity action \cite{Redef}. This field redefinition is model dependent but has been found to be needed for blow-up modes for orientifolds and when D3 and D7-branes are present at a singularity. It is then important to re-analyse our results of this section taking into account that the blow-up field is subject to a field redefinition. This analysis is carried out in appendix \ref{AppFieldRedef} where we show that a field redefinition does not qualitatively change our results.}
\begin{equation}
\langle V_{\mathcal{O}({\cal V}^{-3})}\rangle=\frac{3}{4\langle s\rangle}\frac{W_0^2}{\tau_b^{9/2}}
\left\{-\frac{\left[\ln\left(\tau_b^{3/2}/W_0\right)\right]^{1/2}}{2 a^{3/2}}
+\frac{8}{9\alpha}
\left(\frac{h b B}{W_0}\right)^2 e^{-2 b \langle s\rangle}\tau_b^3\right\}.
\label{VO3min}
\end{equation}
Notice that in the absence of the non-perturbative quiver superpotential ($B=0$) this is negative giving rise to the standard AdS vacuum.
For $B\neq 0$ the second term, being positive, lifts the minimum allowing the possibility of having a de Sitter minimum and even a destabilisation of the minimum for large enough values.
\subsubsection{`Tuning' of the cosmological constant}
Given that the parameters $\langle s\rangle, W_0, B$ are determined by the fluxes and
the VEVs of hidden sector matter fields,
they can be adjusted to cancel the vacuum energy up to this order.
In order to find a Minkowski vacuum we need therefore to perform the following tuning:
\begin{equation}
\frac{\left[\ln\left(\tau_b^{3/2}/W_0\right)\right]^{1/2}}{2 a^{3/2}}=
\frac{8}{9\alpha}
\left(\frac{h b B}{W_0}\right)^2 e^{-2 b \langle s\rangle}\tau_b^3\,.
\label{tuning}
\end{equation}
Let us now try to estimate the amount of tuning needed to get a Minkowski vacuum,
by considering all the underlying parameters fixed except for $B$.
Hence we have
three equations, (\ref{tsVEV}), (\ref{tbVEV}) and (\ref{tuning}), in three unknowns,
$\tau_s$, $\tau_b\simeq {\cal V}^{2/3}$ and $B$ whose solution is:
\begin{equation}
\langle\tau_s\rangle\simeq \langle s \rangle \left(\frac{\zeta}{2}\right)^{2/3},
\quad\langle{\cal V}\rangle \simeq W_0 \,e^{a\langle\tau_s\rangle},
\quad B\simeq\underset{\mathcal{O}(1)}{\underbrace{\left(\frac{3\langle\tau_s\rangle^{1/4}}{4 h b}\sqrt{\frac{\alpha}{a}}\right)}}
\, \left(\frac{\langle{\cal V}\rangle}{W_0}\right)^{\frac{b}{a} \left(\frac{2}{\zeta}\right)^{2/3}-1}\,.
\label{sol}
\end{equation}
Hence the order of magnitude of $B$ depends crucially on $\zeta$ (and so the Euler number of the underlying Calabi-Yau) and the choice of parameters $a$ and $b$.
Given a Minkowski vacuum, one can obtain a discretuum of de Sitter vacua with small cosmological constant by varying $W_0$ which depends on all flux quanta.
This can be seen by considering a small deviation $\epsilon$ in the relation (\ref{tuning}),
which would result in a vacuum energy of the form:
\begin{equation}
\langle V_{\mathcal{O}({\cal V}^{-3})}\rangle=\frac{3}{4\langle s\rangle}\frac{W_0^2}{\tau_b^{9/2}}\,\epsilon\,.
\end{equation}
We have then achieved in the LVS the same effect as anti D3-branes. In fact,
we showed that D3/E(-1) non-perturbative effects at singularities provide an uplifting term that
allows an almost continuum tuning of the vacuum energy by small changes in the fluxes. This is in contrast with proposals for D-term uplift in which the needed substantial warping was not explicitly achieved and may require topological conditions on the internal manifold difficult to satisfy \cite{Dterm}. The main advantage over anti D3-brane constructions is that in this case the effective field theory is under control especially since it is manifestly supersymmetric.
\subsubsection{Generation of non-perturbative superpotentials at singularities}
\label{generation}
We now discuss how to obtain a non-zero prefactor
of the dilaton-dependent non-perturbative superpotential in a way compatible with the stabilisation
of the blow-up mode $\rho$ at the singular locus by imposing the vanishing of the Fayet-Iliopoulos term.
We can envisage three different scenarios:
\begin{itemize}
\item In the presence of just one anomalous $U(1)$, one would require
matter fields with opposite charges to have VEVs such that their contribution to the D-term vanishes
consistent with a non-zero value of the prefactor $B$.
\item In the more general case with more than one anomalous $U(1)$ and multiple blow-up modes,
a non-vanishing prefactor can be obtained without necessarily requiring cancellation of
matter field contributions to all the D-terms.
An explicit example of this was studied by the Romans \cite{Romans}. The construction involved
a $\mathbb{Z}_5$ quiver with gauge group $SU(5)\times U(1)_5 \times U(1)_1$
(and so two anomalous $U(1)$'s) and two blow-up modes $Q_1$ and $Q_2$.
The non-perturbative superpotential at the singularity is given by:
\begin{equation}
W_{\rm np} = B \, e^{-b (S+a_1 Q_1+a_2 Q_2)}\,,
\end{equation}
with the prefactor $B$ depending on open string fields $B= \phi/Z^2$,
where $\phi$ is charged only under $U(1)_1$ and $Z$ is a composite field
built out of fields charged under both anomalous $U(1)$'s.
Given that the combination of blow-ups entering in $W_{\rm np}$ is the same
as that entering in the FI-term of $U(1)_5$, $\xi_5 \propto a_1 \rho_1+a_2 \rho_2$,
the D-term associated with this $U(1)$
can fix exactly this combination to zero (in accord with our result in the previous section)
without having to require cancellations between matter fields VEVs.
On the other hand, the D-term of $U(1)_1$ fixes
\begin{equation}
|\phi|^2 = \xi_1 \propto b_1 \rho_1+b_2 \rho_2 = m^2 \neq 0\,.
\end{equation}
Notice that this stabilisation would leave a flat direction which could then be
fixed at non-zero value by the inclusion of a tachyonic
susy-breaking mass term for $\phi$ (justifying our assumption of $m \neq 0$).
Hence the prefactor $B$ is non-zero since it turns out to be $B=m/Z_{\rm light}^2$
where in the denominator $Z_{\rm light}$ depends only on the fields which remain light.
The composite field $Z_{\rm light}$ then corresponds to the standard run-away
direction of ADS-like superpotentials in the global supersymmetry case where the potential is proportional to $|\partial W/\partial Z|^2 $ \cite{seiberg}.
This direction could be fixed at non-zero value by
supergravity effects if there is a finite solution to $D_ZW=0$. For a discussion along these lines see for instance
\cite{Florea}. Furthermore soft breaking terms will also provide contributions to the the $Z$ potential and generically lifting the runaway behaviour.
\item The prefactor $B$ might not depend at all on matter fields for orientifold projections
such that $h^{1,1}_-\neq 0$ as discussed in \cite{GKPW}.
\end{itemize}
\subsection{De Sitter LVS for positive Euler numbers}
It is well known that the standard realisation of the LARGE Volume Scenario
with non-perturbative effects in the geometric regime and $\alpha'$ corrections
relies on the fact that the Calabi-Yau manifold has negative Euler number $\chi$
(or positive coefficient $\zeta \propto -\chi$).
This means that the number of complex structure moduli ($h^{1,2}$)
is larger than the number of K\"ahler moduli ($h^{1,1}$).
This amounts to essentially half of all Calabi-Yau manifolds because of mirror symmetry.
However ref. \cite{CMV} opens up the possibility to obtain LVS also for $\chi\geq 0$
(or equivalently $\zeta\leq 0$). In fact, the authors of \cite{CMV} pointed
out that, in order to avoid the shrinking of the four-cycle supporting chiral matter,
one has to wrap the visible sector branes on intersecting rigid divisors.
In this way, in the absence of singlets which can get non-zero VEVs without
breaking any of the visible sector gauge groups, one can
perform an appropriate choice of brane set-up and world-volume fluxes which
leads to the D-term stabilisation of all these rigid divisors except for
one. This remaining flat direction, which we shall denote $\tau_{\rm vs}$
since its size is constrained to be small by the requirement of obtaining a
visible sector gauge coupling of the correct size, can be fixed by the
inclusion of string loop corrections \footnote{For a similar mechanism
to fix the visible sector four-cycle via string loops but in the
presence of non-vanishing singlet VEVs see \cite{ccq2}.} to be proportional to $\tau_s$:
$\langle\tau_{\rm vs}\rangle\sim\langle\tau_s\rangle$.
The final contribution to the scalar potential looks like \cite{CMV}:
\begin{equation}
V_{\rm loop}^{({\rm s})}\simeq\frac{c_{\rm loop}^{({\rm s})} W_0^2}{{\cal V}^3 \sqrt{\tau_s}}\,,
\label{loop}
\end{equation}
where $c_{\rm loop}^{({\rm s})}$ is an unknown coefficient which depends on the complex structure moduli fixed at tree level.
Given that $\tau_s \sim \mathcal{O}(10)$ this term scales as $1/{\cal V}^3$, and so for $c_{\rm loop}^{({\rm s})}>0$,
it has the potentiality to give rise to an AdS minimum at exponentially large volume even if $\zeta\leq 0$.
We need however to check that the tuning of $c_{\rm loop}^{({\rm s})}$ needed to obtain such a minimum
does not lead us to a regime where we cannot trust the perturbative expansion anymore.
As explained in \cite{ccq1}, the parameter that controls this expansion is:
\begin{equation}
\epsilon_{\rm loop}^{({\rm s})}\equiv\frac{c_{\rm loop}^{({\rm s})}}{\tau_s}\ll 1\,,
\label{expansionparams}
\end{equation}
which comes from the expansion of the inverse one-loop corrected K\"ahler metric.
Notice that we expect two-loop contributions to the scalar potential
to be suppressed by additional loop factors of $(16\pi^2)$, and so
these higher-loop corrections can be neglected if $\epsilon_{\rm loop}^{({\rm s})}\ll 1$.
On top of these loop corrections, we have also $g_s$ effects coming from loops
of open string states living on the large cycle $\tau_b$:
\begin{equation}
V_{\rm loop}^{({\rm b})}\simeq\frac{c_{\rm loop}^{({\rm b})} W_0^2}{{\cal V}^{10/3}}\,,
\label{Looplarge}
\end{equation}
and in this case the parameter that controls the perturbative expansion is:
\begin{equation}
\epsilon_{\rm loop}^{({\rm b})}\equiv\frac{c_{\rm loop}^{({\rm b})}}{\tau_b}\simeq \frac{c_{\rm loop}^{({\rm b})}}{{\cal V}^{2/3}}\ll 1\,.
\label{expansionparamb}
\end{equation}
Let us start by neglecting the string loop correction (\ref{Looplarge}) due to their volume suppression.
The scalar potential takes the form (neglecting the prefactor):
\begin{equation}
V=\mu_1 \sqrt{\tau_s}\, \frac{e^{-2 a \tau_s}}{{\cal V}}
- \mu_2 W_0 \tau_s\frac{e^{-a \tau_s}}{{\cal V}^2} - \mu_3 \frac{W_0^2}{{\cal V}^3}
+ \frac{c_{\rm loop}^{({\rm s})} W_0^2}{{\cal V}^3 \sqrt{\tau_s}} .
\label{VO3geom}
\end{equation}
where:
\begin{equation}
\mu_1\equiv\frac 83 (a A)^2>0\,,\qquad \mu_2 \equiv 4 a A>0 \qquad\text{and}\qquad
\mu_3\equiv \frac 34 \frac{|\zeta|}{g_s^{3/2}}\geq 0\,.
\end{equation}
Solving $\partial V/\partial \tau_s=0$ in the limit $a\tau_s\gg 1$ gives:
\begin{equation}
e^{-a\tau_s}\simeq\frac{3\sqrt{\tau_s}}{8 a A}\frac{W_0}{{\cal V}}\left[1\pm \left(1-\frac{ c_{\rm loop}^{({\rm s})}}{3 a\tau_s^3}\right)\right]
\quad\Rightarrow\quad \tau_s \simeq\frac{\ln {\cal V}}{a}\quad \text{for}\quad W_0\sim \mathcal{O}(1)\,.
\label{compare}
\end{equation}
A careful analysis considering also the second derivative with respect to $\tau_s$
shows that in order to get a minimum we need to take the solution with the positive sign.
Notice that in the case with $c_{\rm loop}^{({\rm s})}=0$ this solution reduces to (\ref{tsVEV}).
Substituting this result back in (\ref{VO3geom}) we find (again in the limit $a\tau_s\gg 1$):
\begin{equation}
V\simeq \frac 32 \frac{W_0^2}{{\cal V}^3}\left[
- \frac{\left(\ln {\cal V}\right)^{3/2}}{a^{3/2}}
- \frac{|\zeta|}{2 g_s^{3/2}}
+ \frac{2 \sqrt{a} \,c_{\rm loop}^{({\rm s})}}{3\sqrt{\ln {\cal V}}} \right] .
\label{VO3geoms}
\end{equation}
The solution of $\partial V/\partial {\cal V}=0$ is then:
\begin{equation}
\frac{\ln{\cal V}}{a}= \sqrt{\frac{2\,c_{\rm loop}^{({\rm s})}}{3}} \left(1-\epsilon\right)^{1/2}\simeq \sqrt{\frac{2\,c_{\rm loop}^{({\rm s})}}{3}}\,,
\label{tbVEVgeom}
\end{equation}
where:
\begin{equation}
\epsilon \equiv \frac{3 |\zeta| g_s^{-3/2}}{4 c_{\rm loop}^{({\rm s})}}\sqrt{\frac{\ln{\cal V}}{a}}
\simeq \frac{|\zeta|}{2} \left(\frac 32\right)^{3/4}\left(\frac{g_s^{-2}}{c_{\rm loop}^{({\rm s})}}\right)^{3/4}\ll 1
\quad \text{for}\quad c_{\rm loop}^{({\rm s})}\gg g_s^{-2}\,.
\end{equation}
Taking this value of $c_{\rm loop}^{({\rm s})}$ and comparing (\ref{tbVEVgeom}) with (\ref{compare}),
we realise that the minimum is at:
\begin{equation}
\langle\tau_s\rangle\simeq \sqrt{\frac{2\,c_{\rm loop}^{({\rm s})}}{3}}\sim g_s^{-1}\sim\mathcal{O}(10)
\qquad\text{and}\qquad\langle{\cal V}\rangle\sim W_0 \,e^{a \langle\tau_s\rangle}\gg 1\,.
\end{equation}
However this minimum is not in a regime where we can trust the perturbative expansion
since the parameter (\ref{expansionparams}) becomes:
\begin{equation}
\epsilon_{\rm loop}^{({\rm s})}\sim \sqrt{c_{\rm loop}^{({\rm s})}}\sim g_s^{-1}\gg 1\qquad\text{for}\qquad
g_s\ll 1\,.
\end{equation}
This implies that the string loop correction (\ref{loop}) can never beat the $\alpha'$
correction and yield a minimum in a regime where we can trust the effective field theory.
However this is not the case for the loop correction (\ref{Looplarge}) coming from the large cycle.
In fact, setting $c_{\rm loop}^{({\rm s})}=0$ without loss of generality,
the effective potential in terms of ${\cal V}$ reads:
\begin{equation}
V\simeq \frac 32 \frac{W_0^2}{{\cal V}^3}\left[
- \frac{\left(\ln {\cal V}\right)^{3/2}}{a^{3/2}}
- \frac{|\zeta|}{2 g_s^{3/2}}
+ \frac{2\,c_{\rm loop}^{({\rm b})}}{3{\cal V}^{1/3}} \right] .
\label{VO3geomsnew}
\end{equation}
The solution of $\partial V/\partial {\cal V}=0$ is then:
\begin{equation}
\frac{\ln{\cal V}}{a}= \left(\frac{20}{27}\right)^{2/3}\left(\frac{c_{\rm loop}^{({\rm b})}}{{\cal V}^{1/3}}\right)^{2/3}
\left(1-\delta\right)^{2/3}\simeq \left(\frac{20}{27}\right)^{2/3}\left(\frac{c_{\rm loop}^{({\rm b})}}{{\cal V}^{1/3}}\right)^{2/3}\,,
\label{tbVEVgeomnew}
\end{equation}
where:
\begin{equation}
\delta \equiv \frac{27 |\zeta| }{40 }\frac{g_s^{-3/2} {\cal V}^{1/3}}{c_{\rm loop}^{({\rm b})}} \ll 1
\qquad \text{for}\qquad c_{\rm loop}^{({\rm b})}> g_s^{-3/2}{\cal V}^{1/3}\,.
\end{equation}
Given that all the known Calabi-Yau three-folds have $1/2\lesssim|\zeta|\lesssim 3/2$,
the condition $\delta\ll 1$ is well satisfied by taking $c_{\rm loop}^{({\rm b})}\simeq 4\, g_s^{-3/2}{\cal V}^{1/3}$
since it gives $0.08\lesssim\delta\lesssim 0.25$.
In this case the parameter (\ref{expansionparamb}) that controls the perturbative expansion
is still smaller than unity since it turns out to be volume suppressed:
\begin{equation}
\epsilon_{\rm loop}^{({\rm b})}\sim \frac{4\, g_s^{-3/2}}{{\cal V}^{1/3}}\ll 1\qquad \text{for}\qquad
{\cal V}\gg 1\,.
\end{equation}
This minimum is AdS but it can be turned into a dS vacuum due to the
positive contribution (\ref{uplift2}) coming from D3/E(-1) non-perturbative effects at singularities
in the same way described in the previous sections.
\section{Phenomenological implications}
\label{Implications}
Even though the mechanism we have used for getting de Sitter vacua is very different from
anti D3-branes at a warped throat, once uplifting has been achieved,
the main physical implications are very similar to the standard LVS.
However, they can provide a more robust origin to the different implications of LVS.
\begin{itemize}
\item{} {\it Inflation:} Models of K\"ahler moduli inflation \cite{KahlerInflation} are based on the LVS and
depend very much on the positive uplift term to get early Universe accelerated slow roll. A criticism to these scenarios could be that the most important
contribution to the almost de Sitter expansion relies in the uplifting mechanism that generically requires the introduction of anti D3-branes.
However, if this term comes from a manifestly supersymmetric theory, as we have here, instead of a non linear realisation of supersymmetry, it
makes the status of these models more robust. This is what can be obtained from our mechanism since the whole dynamics is identical to the existing models of inflation except for the origin of the
`uplift' term. \footnote{We want to emphasise that uplift term is not a proper terminology in our mechanism since the term appears on equal footing to the other terms in the scalar potential and we have explicitly found the minimum of the full potential.}
\item{} {\it Supersymmetry breaking:}
The new mechanism for obtaining dS vacua can play an important r\^ole in the study of soft supersymmetry breaking. There are several realisations of the soft breaking terms within the LVS. This depends on whether the cycle supporting the Standard Model brane participates or not in the breaking of supersymmetry. If it does, the soft terms are of order the gravitino mass $m_{3/2}\propto 1/{\cal V}$ \cite{cqs}. In the general expression for the soft scalar masses there appears a contribution from the vacuum energy $m_0^2= V_0 +\cdots$. Since the uplifting term is of order $1/{\cal V}^3$ its contribution to scalar masses is much smaller than the gravitino mass and therefore does not play any r\^ole on the soft breaking Lagrangian.
However if the Standard Model cycle does not contribute to supersymmetry breaking, then the soft terms can be hierarchically smaller than the gravitino mass with scalar masses of order $m_0\sim 1/{\cal V}^{3/2}, 1/{\cal V}^2$ \cite{bckmq} (see however \cite{joepedro} when field redefinitions are needed). In this case the uplifting term is crucial for determining the soft scalar masses \cite{shanta1}. Given that our scenario provides an explicit supersymmetric source for obtaining dS vacua, the contribution to the scalar masses can be computed in a reliable way. In particular what we have is an `F-term uplift'
realised via the F-term of a blow-up mode whose contribution to the soft masses in the Standard Model brane is negligible since the visible sector is
localised at a different singularity. Hence the leading order contribution to the soft masses comes from the F-term of the dilaton which is
of order $1/{\cal V}^2$, allowing for a realisation of the large gravitino mass scenario of \cite{bckmq,shanta2} (see also \cite{joe}).
\item{} {\it An AdS/CFT dual description?:}
Our uplift potential takes the form:
\begin{equation}
V_{\rm up}\simeq \frac{e^{-2 b s}}{ {\cal V}^{\alpha}},
\label{Vup}
\end{equation}
with $\alpha=1$ for the case with $h\neq0$ and $\alpha=2$ for $h=0$ and $s$ fixed by ratios of integers from fluxes. This has a very similar structure as the anti D3-brane potential with $e^{-2bs}$ playing the r\^ole of the minimal value of the warp factor.
The term (\ref{Vup}) can arise from a non-perturbatively generated superpotential of a gauge theory; one thus might ask whether there is a relation to a dual theory along the lines of \cite{k1}.
On the other hand there seem to be some key differences between our system and a warped throat with an anti-brane. Firstly, supersymmetry is broken by a bulk modulus; unlike the anti-brane case where the brane is the source of supersymmetry breaking. Furthermore, the value at which $s$ is fixed is sensitive to the flux quanta in all the three-cycles in the Calabi-Yau; this makes it difficult to associate $e^{-2bs}$ with the infrared scale of a throat. It may be interesting to have a proper understanding of this in terms of AdS/CFT duality.
\end{itemize}
\section{Conclusions}
\label{Conclusions}
Obtaining de Sitter space solutions from string theory is hard but very relevant for phenomenology. Despite several attempts, so far the most convincing proposals have involved anti D3-branes which are required to provide a positive source of potential energy that combines with a concrete mechanism of modulus stabilisation to lift an AdS minimum
to positive values. Moreover the amount of this lifting is controlled in an almost continuous manner given the fact that this contribution comes with an exponential warped factor depending on ratios of integers determined by fluxes.
The fact that this mechanism is not explicitly supersymmetric has raised criticism and doubts on how reliable and stringy this mechanism is. Even though these doubts may not be fully justified, it is reassuring to identify alternative mechanisms that achieve the same results but coming from a fully supersymmetric effective action. We regard this to be the main result of this article.
We achieved de Sitter vacua from a combination of non-perturbative effects. The first is the standard one coming from either gaugino condensation on D7-branes or Euclidean D3 instantons wrapping rigid four-cycles in the geometric regime. The new element is to include dilaton-dependent non-perturbative effects arising from either gaugino condensation on space-time filling D3-branes or E(-1)-instantons at singularities. A combination of both is what gives rise to a tunable (as in the anti D3-branes) positive vacuum energy. This makes possible fully supersymmetric treatments of inflation and soft supersymmetry breaking in the LVS making the calculations more controllable and the scenarios more robust.
Moreover these dilaton-dependent non-perturbative superpotentials, when combined with string loop effects,
open up the possibility of realising new LVS for manifolds with zero or positive Euler number.
Open questions are legion: a detailed calculation of soft terms for different scenarios of supersymmetry breaking, explicit realisations of this scenario in compact Calabi-Yau models following the recent constructions in \cite{CMV}, explicit calculations of the next order corrections to the scalar potential (of order $1/{\cal V}^4$), etc. We hope to report on some of these issues in the near future.
\section*{Acknowledgments}
We thank Matteo Bertolini, Massimo Bianchi, Shanta de Alwis, Thomas Grimm, Sven Krippendorf,
Liam McAllister, Jose F. Morales, Marco Serone, Gary Shiu and in particular Joe Conlon and Roberto Valandro for useful discussions.
We acknowledge DAMTP-CTC, Cambridge and ICTP, Trieste where part of this work was done.
We thank the organisers of PASCOS 2011, Cambridge for
providing a good atmosphere to develop this project. AM is funded by the EU and the University of Cambridge.
CB's research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information (MRI).
FQ also wishes to thank the hospitality of KITPC during the final stages of this project.
This research was partly supported by the Project of Knowledge Innovation
Program (PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10.
|
{
"timestamp": "2012-03-20T01:05:42",
"yymm": "1203",
"arxiv_id": "1203.1750",
"language": "en",
"url": "https://arxiv.org/abs/1203.1750"
}
|
\section{Zero coupon bond prices and yield curves}
\label{section introduction}
Insurance cash flows are valued using the risk-free yield curve.
First, today's yield curve needs to be estimated
from government bonds, swap rates and corporate bonds and, second, future
yield curves then need to be predicted. This prediction is a complex task because, in general, it involves
the forecast of infinite dimensional random vectors and/or random functions. In the
present paper we tackle the problem of yield curve prediction using a non-parametric approach, which is based on ideas
presented in Ortega et al.~\cite{Ortega}. In contrast to \cite{Ortega} we are heading for long term predictions as needed in insurance industry. Assume $t\ge 0$ denotes time in years. Choose $T \geq t$ and denote, at time $t$, the price of the (default-free) zero coupon bond (ZCB) that pays one unit of currency at maturity date $T$ by $P(t,T)$. The yield curve at time $t$ for maturity dates $T \geq t$ is then given by the continuously-compounded spot
rate defined by \begin{equation*}
Y(t,T) =- \frac{1}{T-t}~\log P(t,T).
\end{equation*}
{\bf Aim and scope.}
Model stochastically the yield curves $T \mapsto Y(t,T)$ for future dates
$t\in (0,T)$ such that:
(i) the model is free of arbitrage;
(ii) explains past yield curve observations;
(iii) allows to predict the future yield curve development.
In contrast to standard literature on prediction of yield curves we insist that models should be free of arbitrage. This requirement is crucial when it comes to the prediction of highly correlated prices as it is the case for interest rates. Otherwise it is possible to ``artificially'' shift P\&L distributions. More precisely, if a prediction model admits arbitrage then implementing this arbitrage portfolio yields an always positive P\&L. In practice adding such an arbitrage portfolio can then be used to shift P\&L distributions of general portfolios, which is an undesired effect from the point of view of
valuation and risk management, see Figure \ref{arbitrage}
and Section \ref{sec.arbitrage}.
~
{\bf Organization of the paper.}
The remainder of the paper is organized as follows: in Section 2 we propose our discrete time model for (discretized) yield curve evolution. In Section 3 we describe the ubiquitous no arbitrage conditions for our modeling setup. In Section 4 we describe the actual calibration procedure and in Section 5 we present a concrete calibration to real market data.
\section{Model proposal on a discrete time grid}
Choose a fixed grid size $\Delta = 1/n$ for $n\in \N$. We consider
the discrete time points $t \in \Delta \N_0=\{0,\Delta, 2\Delta, 3\Delta, \ldots\}$ and the maturity dates $T\in t+\Delta \N$. For example, the choice
$n=1$ corresponds to a yearly grid, $n=4$ to a quarterly grid,
$n=12$ to a monthly grid, $n=52$ to a weekly grid
and $n=250$ to a business days grid.
~
The filtered probability space is denoted by $(\Omega, {\cal F}, \p, \F)$
with real world probability measure $\p$ and (discrete time)
filtration $\F=({\cal F}_t)_{t \in \Delta \N_0}$.
~
We assume that the ZCBs
exist at all time points $t \in \Delta \N_0$ for all maturity
dates $T=t+m$ with
times to maturity $m\in \Delta \N$. Thus, we can consider
the discrete time yield curves
\begin{equation*}
\mathbf{Y}_t=( Y(t,t+m))'_{m \in \Delta \N}
\end{equation*}
for all time points $t\in \Delta \N_0$. Assume that $(\mathbf{Y}_t)_{t
\in \Delta \N_0}$ is $\F$-adapted, that is,
$(\mathbf{Y}_s)_{s\le t}$
is observable at time $t$ and this information is contained
in the $\sigma$-field
${\cal F}_t$. Our aim is (as described above) to model and predict
$(\mathbf{Y}_t)_{t \in \Delta \N_0}$. We assume that there exists
an equivalent martingale measure $\p^\ast \sim \p$ for the bank
account numeraire discount $(B_t^{-1})_{t\in \Delta \N_0}$
and, in a first step,
we describe $(\mathbf{Y}_t)_{t \in \Delta \N_0}$
directly under this equivalent martingale measure $\p^\ast$. Notice here that the bank account
numeraire is actually a discrete
time roll-over portfolio, as will be seen in the next section.
~
{\bf Remark.}
The assumption that the yield curve is given at any moment $ t \in \Delta \N_0 $ for sufficiently many maturities is a very strong one. In practice the yield curve is inter- and extrapolated every day from quite different traded quantities like coupon bearing bonds, swap rates, etc.
This inter- and extrapolation allows for a lot of freedom, often
parametric families are used, e.g.~the Nelson-Siegel \cite{NelsonSiegel}
or the Svensson \cite{Svensson1, Svensson2}
family, but also non-parametric approaches such as
splines are applied (see Filipovi\'c \cite{Damir}).
~
\section{Stochastic yield curve modeling and no-arbitrage}\label{modelling_na}
Assume the initial yield curve
$\mathbf{Y}_0=(Y(0,m))_{m\in \Delta \N}$
at time $t=0$ is given.
For $t,m\in \Delta\N$ we make the following model assumptions:
assume there exist deterministic functions
$\alpha_\Delta(\cdot, \cdot, \cdot)$ and
$\mathbf{v}_\Delta(\cdot, \cdot, \cdot)$
such that the yield curve has the following stochastic
representation
\begin{eqnarray}
m~Y(t,t+m)&=&(m+\Delta)~Y(t-\Delta,t+m)-\Delta ~Y(t-\Delta,t)
\label{yield curve dynamics}
\\&&
+~{\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
+\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})~
\boldsymbol{\varepsilon}^\ast_t,\nonumber
\end{eqnarray}
where the innovations $\boldsymbol{\varepsilon}^\ast_t$
are ${\cal F}_t$-measurable and independent of ${\cal F}_{t-\Delta}$
under $\p^\ast$. In general, the innovations $\boldsymbol{\varepsilon}^\ast_t$
are multivariate
random vectors and the last product in \eqref{yield curve dynamics}
needs to be understood in the inner product sense.
~
{\bf Remark.}
The first two terms on the right-hand side of \eqref{yield curve dynamics}
will exactly correspond to the no-arbitrage condition in
a deterministic interest rate model (see (2.2) in Filipovi\'c \cite{Damir}).
The fourth term on the right-hand side of \eqref{yield curve dynamics}
described by $\mathbf{v}_\Delta
(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
\boldsymbol{\varepsilon}^\ast_t$
adds the stochastic part to the future yield curve
development. Finally, the third term
${\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})$
will be recognized as a Heath-Jarrow-Morton \cite{HJM}
(HJM) term that makes the stochastic model free of arbitrage. This term is going to be analyzed in detail in Lemma \ref{HJM yield condition}
below. This approach allows us to separate conceptually the task of estimating volatilities, i.e.~estimating $ v_{\Delta} $, and estimating the market price of risk, i.e.~the difference of $ \p $ and $ \p^\ast $.
~
Assumption \eqref{yield curve dynamics}
implies for the price of the ZCB
at time $t$ with time to maturity $m$
\begin{equation*}
P(t,t+m)=
\frac{P(t-\Delta,t+m)}{P(t-\Delta,t)}~
\exp \left\{
-{\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})-
\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})~
\boldsymbol{\varepsilon}^\ast_t \right\}.
\end{equation*}
In order to determine the HJM term
${\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})$
we define the discrete time bank account value for an initial
investment of 1 as follows:
$B_0=1$ and for $t\in \Delta \N$
\begin{equation*}
B_t = \prod_{s=0}^{t/\Delta-1} P(\Delta s, \Delta (s+1))^{-1}
= \exp \left\{\Delta \sum_{s=0}^{t/\Delta-1}
Y(\Delta s, \Delta (s+1))\right\}~>~0.
\end{equation*}
The process $\mathbf{B}=(B_t)_{t\in \Delta \N_0}$ considers the
roll over of an initial
investment 1 into the (discrete time) bank account with grid
size $\Delta$.
Note that $\mathbf{B}$ is previsible, i.e.~$B_t$
is ${\cal F}_{t-\Delta}$-measurable for all $t\in \Delta \N$.
~
Absence of arbitrage is now expressed in terms of the following
$(\p^\ast,\F)$-martingale property (under the assumption that all the
conditional
expectations exist). We require for all $t,m\in \Delta \N$
\begin{equation}\label{no arbitrage condition}
\E^\ast \left[\left.B_{t}^{-1}~ P(t,t+m)
\right|{\cal F}_{t-\Delta} \right]~\stackrel{!}{=}~
B_{t-\Delta}^{-1}~ P(t-\Delta,t+m).
\end{equation}
The necessity of such a martingale property is due to the fundamental theorem
of asset pricing (FTAP) derived in Delbaen-Schachermayer
\cite{DS}.
For notational convenience we set $\E^\ast_t\left[\cdot\right]
= \E^\ast\left[\left.\cdot \right|{\cal F}_t\right]$ for $t\in \Delta \N_0$.
The no-arbitrage condition \eqref{no arbitrage condition} immediately provides the following lemma.
\begin{lemma}\label{HJM yield condition}
Under the above assumptions the absence of arbitrage
condition \eqref{no arbitrage condition}
implies
\begin{equation*}
{\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
=\log~
\E^\ast_{t-\Delta} \left[
\exp \left\{-
\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})~
\boldsymbol{\varepsilon}^\ast_t \right\} \right].
\end{equation*}
\end{lemma}
This solves item (i) of the aim and scope list.
~
{\footnotesize
{\bf Proof of Lemma \ref{HJM yield condition}.}
We rewrite \eqref{no
arbitrage condition} as follows
(where we use assumption \eqref{yield curve dynamics}
of the yield curve development and the appropriate
measurability properties)
\begin{eqnarray*}
&&\hspace{-.51cm}
\exp \left\{- \Delta~
Y( t-\Delta, t)\right\}~
\E^\ast_{t-\Delta} \left[ P(t,t+m) \right]
~=~
P( t-\Delta, t)~
\E^\ast_{t-\Delta} \left[ P(t,t+m) \right]
\\
&&=~P(t-\Delta,t+m)~\exp \left\{-
{\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
\right\}
\E^\ast_{t-\Delta} \left[
\exp \left\{-
\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})~
\boldsymbol{\varepsilon}^\ast_t \right\} \right]\\
&&\stackrel{!}{=}~P(t-\Delta,t+m).
\end{eqnarray*}
Solving this requirement proves the claim of Lemma \ref{HJM yield condition}.
{\begin{flushright}\vspace{-2mm}$\Box$\end{flushright}}}
\section{Modeling aspects and calibration}
We need to discuss the choices
$\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})$ and
$\boldsymbol{\varepsilon}^\ast_t$ as well as the description
of the equivalent martingale measure $\p^\ast\sim \p$.
Then, the model and the prediction
is fully specified through Lemma \ref{HJM yield
condition}.
\subsection{Data and explicit model choice}
Assume we would like to study a finite set ${\cal M}\subset \Delta \N$ of
times to maturity.
We specify below necessary properties of ${\cal M}$ for yield curve prediction.
For these times to maturity choices we define for $t\in \Delta \N$
\begin{equation*}
\mathbf{Y}_{t, +}
=\left(Y(t,t+m)\right)_{m\in {\cal M}}'
\qquad \text{ and }
\qquad
\mathbf{Y}_{t, -}
=\left(Y(t-\Delta,t+m)\right)_{m\in {\cal M}}',
\end{equation*}
that is, in contrast to $\mathbf{Y}_{t}$
the random vectors $\mathbf{Y}_{t, +}$
and $\mathbf{Y}_{t, -}$ only
consider the times to maturity $m$ and $m+\Delta$
for $m\in {\cal M}$.
Note that
$\mathbf{Y}_{t, -}$ is ${\cal F}_{t-\Delta}$-measurable
and $\mathbf{Y}_{t, +}$ is ${\cal F}_{t}$-measurable.
Our aim is to model the change from $\mathbf{Y}_{t, -}$ to
$\mathbf{Y}_{t, +}$. In view of \eqref{yield curve dynamics}
we define the vector
\begin{equation*}
\boldsymbol{\Upsilon}_t
~=~(\Upsilon_{t,m})'_{m\in {\cal M}}
~=~\left(m~Y(t,t+m)-(m+\Delta)~Y(t-\Delta,t+m)\right)_{m\in {\cal M}}'.
\end{equation*}
We set the dimension $d=|{\cal M}|$. For
$\boldsymbol{\varepsilon}^\ast_t|_{{\cal F}_{t-\Delta}}$
we then choose
a $d$-dimensional standard Gaussian distribution with
independent components under the equivalent
martingale measure $\p^\ast$.
~
{\bf Remark.}
We are aware that the choice of multivariate Gaussian innovations
$\boldsymbol{\varepsilon}^\ast_t$ is only a first step towards more
realistic innovation processes. However, we believe that already in this
model, with suitably chosen estimations of the instantaneous covariance
structure, the results are quite convincing -- additionally chosen jump
structures might even improve the situation. The independence assumption
with respect to the martingale measure is an additional strong assumption
which could be weakened.
~
Thus, we re-scale
the volatility term with the grid size $\Delta$ and assume that at time $t$
it only
depends on the last observation $\mathbf{Y}_{t,-}$: define
$\mathbf{v}_\Delta(\cdot, \cdot, \cdot)$ by
\begin{equation*}
\mathbf{v}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
=\sqrt{\Delta}~\boldsymbol{\sigma}(t,m,\mathbf{Y}_{ t,-}),
\end{equation*}
where the function $\boldsymbol{\sigma}(\cdot, \cdot, \cdot)$
does not depend on the grid size $\Delta$.
Lemma \ref{HJM yield condition}
implies for these choices for the HJM term
\begin{equation*}
{\alpha}_\Delta(t,m,(\mathbf{Y}_s)_{s\le t-\Delta})
=\log~
\E^\ast_{t-\Delta} \left[
\exp \left\{-\sqrt{\Delta}~
\boldsymbol{\sigma}(t,m,\mathbf{Y}_{t,-})~
\boldsymbol{\varepsilon}^\ast_t \right\} \right]
=
\frac{\Delta}{2}~
\left\| \boldsymbol{\sigma}(t,m,\mathbf{Y}_{ t,-})
\right\|^2
.
\end{equation*}
From \eqref{yield curve dynamics}
we then obtain for $t\in \Delta \N$ and $m\in {\cal M}$
under $\p^\ast$
\begin{equation}
\Upsilon_{t,m}
\label{yield curve dynamics 2}
= \Delta\left[
-Y(t-\Delta,t)
+\frac{1}{2}
\left\| \boldsymbol{\sigma}(t,m,\mathbf{Y}_{t,-})
\right\|^2\right]
+\sqrt{\Delta}~\boldsymbol{\sigma}(t,m,\mathbf{Y}_{ t,-})~
\boldsymbol{\varepsilon}^\ast_t.
\end{equation}
Note that $(\boldsymbol{\Upsilon}_t)_{t\in \Delta \N}$
is a $d$-dimensional process, thus,
we need a $d$-dimensional Gaussian random vector
$\boldsymbol{\varepsilon}^\ast_t|_{{\cal F}_{t-\Delta}}$ for
obtaining full rank and no singularities.
Next, we specify explicitly the $d$-dimensional function
$\boldsymbol{\sigma}(\cdot, \cdot, \cdot)$.
We proceed similar to
Ortega et al.~\cite{Ortega}, i.e.~we directly model volatilities
and return directions.
Assume that for every $\mathbf{y}\in \R^d$ there exists
an invertible and linear map
\begin{equation}\label{invertible map}
\varsigma(\mathbf{y}):\R^d \to \R^d, \qquad
\boldsymbol{\lambda} \mapsto
\varsigma(\mathbf{y})(\boldsymbol{\lambda}).
\end{equation}
In the sequel we identify the linear map $\varsigma(\mathbf{y})(\cdot)$
with the corresponding (invertible)
matrix $\varsigma(\mathbf{y})\in \R^{d\times d}$
which generates this linear map, i.e.~$\varsigma(\mathbf{y})(\boldsymbol{\lambda})=\varsigma(\mathbf{y})~\boldsymbol{\lambda}$.
In the next step, we choose vectors
$\boldsymbol{\lambda}_1, \ldots, \boldsymbol{\lambda}_d \in \R^d$
and define the matrix
${\Lambda}=
\left[
\boldsymbol{\lambda}_1, \ldots, \boldsymbol{\lambda}_d\right]\in \R^{d\times
d}$. Moreover, for $\mathbf{y}\in \R^d$ we set
\begin{eqnarray*}
\Sigma_{{\Lambda}} (\mathbf{y})
&=& \varsigma(\mathbf{y})~
\Lambda ~\Lambda' ~\varsigma'(\mathbf{y})
\in \R^{d\times d}.
\end{eqnarray*}
Using vector form
we make the following model
specification for \eqref{yield curve dynamics 2}:
\begin{model ass} \label{Model Assumptions 1}
We choose the following model
for the yield curve at time $t\in \Delta \N$ with
time to maturity dates ${\cal M}$:
\begin{equation*}
\boldsymbol{\Upsilon}_t
= \Delta\left[-\mathbf{Y}(t-\Delta,t)+\frac{1}{2}
~{\rm sp}
(\Sigma_{{\Lambda}}
({\mathbf{Y}_{t, -}}))
\right]+ \sqrt{\Delta}~
\varsigma({\mathbf{Y}_{t, -}})~{\Lambda}
~ \boldsymbol{\varepsilon}_{t}^\ast,
\end{equation*}
with $\mathbf{Y}(t-\Delta,t)=({Y}(t-\Delta,t),\ldots, {Y}(t-\Delta,t))'
\in \R^d$ and ${\rm sp}(\Sigma_{{\Lambda}})$
denotes the $d$-dimensional vector that
contains the diagonal elements of the matrix $\Sigma_{{\Lambda}}
\in \R^{d\times d}$.
\end{model ass}
For the
$j$-th maturity $m_j \in {\cal M}$
we have done the following choice
\begin{equation*}
\boldsymbol{\sigma}(t,m_j,\mathbf{Y}_{t,-})~
\boldsymbol{\varepsilon}^\ast_t
=\sum_{i=1}^d
\boldsymbol{\sigma}_i(t,m_j,\mathbf{Y}_{t,-})~
{\varepsilon}^\ast_{t,i}
=\sum_{i=1}^d
\left[\varsigma(\mathbf{Y}_{t, -})
~ \boldsymbol{\lambda}_i\right]_j~
{\varepsilon}^\ast_{t,i}.
\end{equation*}
The linear map $\varsigma(\cdot)$ describes the
{\it volatility scaling factors},
$\boldsymbol{\lambda}_1, \ldots, \boldsymbol{\lambda}_d \in \R^d$
specify the {\it return directions}, and the volatility choice
does not depend on the grid size $\Delta$.
Our aim is to calibrate these terms.
~
{\bf Remark.}\label{remark1}
The volatility scaling factors $\varsigma(\cdot)$ mimic how volatility for different maturities scales with the level of yield at this maturity. Several approaches have been discussed in the literature. The choice of a square-root dependence seems to be quite robust over different maturities and interest rate regimes, but for small rates -- as we face it for the Swiss currency CHF -- linear dependence seems to be a good choice, too, see choice \eqref{choice sigma}.
\begin{lemma} \label{Lemma 3.2}
Under Model Assumptions \ref{Model Assumptions 1}, the random
vector
$\boldsymbol{\Upsilon}_t|_{{\cal F}_{t-\Delta}}$ has
a $d$-dimensional conditional Gaussian distribution
with the first two conditional moments given by
\begin{eqnarray*}
\E^\ast_{t-\Delta} \left[
\boldsymbol{\Upsilon}_t\right]&=&
\Delta\left[-\mathbf{Y}(t-\Delta,t)+\frac{1}{2}
~{\rm sp}(\Sigma_{\Lambda} ({\mathbf{Y}_{t, -}}))
\right],\\
{\rm Cov}^\ast_{t-\Delta } \left(
\boldsymbol{\Upsilon}_t\right)&=&
\Delta ~
\Sigma_{\Lambda}
({\mathbf{Y}_{t, -}}).
\end{eqnarray*}
\end{lemma}
\subsection{Calibration procedure}
In order to calibrate our model
we need to choose the volatility scaling factors $\varsigma(\cdot)$
and we need to
specify the return directions
$\boldsymbol{\lambda}_1, \ldots, \boldsymbol{\lambda}_d \in \R^d$
which provide the matrix $\Lambda$. In fact we do not need to specify the direction $\boldsymbol{\lambda}_1, \ldots, \boldsymbol{\lambda}_d \in \R^d$
themselves, but rather $ \Sigma_{\Lambda} $, which we shall do in the sequel.
Assume we have observations
$(\boldsymbol{\Upsilon}_t)_{t=\Delta,\ldots, \Delta K}$,
$(Y(t-\Delta,t))_{t=\Delta,\ldots \Delta(K+1)}$,
and $(\mathbf{Y}_{t,-})_{t=\Delta,\ldots, \Delta (K+1)}$. We
use these observations to predict/approximate the
random vector $\boldsymbol{\Upsilon}_{\Delta (K+1)}$
at time $\Delta K$.
For $\mathbf{y}\in \R^d$ we define the matrices
\begin{eqnarray*}
C_{(K)} &=&\frac{1}{\sqrt{K}} \left(\left[
\varsigma(\mathbf{Y}_{\Delta k,-})^{-1}~
\boldsymbol{\Upsilon}_{\Delta k}\right]_j
\right)_{j=1,\ldots, d; ~k=1, \ldots, K}\in \R^{d\times K},\\
S_{(K)}(\mathbf{y})&=&
\varsigma(\mathbf{y})~
C_{(K)}~C'_{(K)}~\varsigma'(\mathbf{y})\in \R^{d\times d}.
\end{eqnarray*}
Choose $t=\Delta(K+1)$.
Note that $C_{(K)}$ is ${\cal F}_{t-\Delta}$-measurable.
For $\mathbf{x},\mathbf{y}\in \R^d$
we define the $d$-dimensional random vector
\begin{eqnarray}\label{model kappa}
\boldsymbol{\kappa}_t~=~\boldsymbol{\kappa}_t(\mathbf{x},\mathbf{y})
&=& - \Delta~\mathbf{x}+\frac{1}{2}~
{\rm sp}\left(S_{(K)}(\mathbf{y})\right)
+ \varsigma(\mathbf{y})~
C_{(K)}~ \mathbf{W}^\ast_{t},
\end{eqnarray}
with $\mathbf{W}^\ast_{t}$ is independent
of ${\cal F}_{t-\Delta}$, ${\cal F}_t$-measurable, independent
of $\boldsymbol{\varepsilon}^\ast_t$ and a $K$-dimensional
standard Gaussian random vector
with independent components under $\p^\ast$.
\begin{lemma} \label{Lemma 3.3} The random
vector
$\boldsymbol{\kappa}_t|_{{\cal F}_{t-\Delta}}$ has
a $d$-dimensional Gaussian distribution
with the first two conditional moments given by
\begin{eqnarray*}
\E^\ast_{t-\Delta} \left[
\boldsymbol{\kappa}_t\right]&=&
- \Delta~\mathbf{x}+\frac{1}{2}~
{\rm sp}\left(S_{(K)}(\mathbf{y})\right),\\
{\rm Cov}^\ast_{t-\Delta} \left(
\boldsymbol{\kappa}_t\right)&=&
S_{(K)}(\mathbf{y}).
\end{eqnarray*}
\end{lemma}
Our aim is to show that the matrix $S_{(K)}(\mathbf{y})$
is an appropriate estimator for
$\Delta
\Sigma_{\Lambda}
(\mathbf{y})$ and then Lemmas \ref{Lemma 3.2} and \ref{Lemma 3.3}
say that $\boldsymbol{\kappa}_t$ is an appropriate stochastic
approximation to $\boldsymbol{\Upsilon}_t$, conditionally
given ${\cal F}_{t-\Delta}$.
~
{\bf Remark.} The random vector $\boldsymbol{\kappa}_t$ can be seen
as a filtered historical simulation where $\mathbf{W}^\ast_{t}$
re-simulates the $K$ observations which are appropriately
historically scaled through the matrix $C_{(K)}$.
~
We calculate the expected value of $S_{(K)}(\mathbf{y})$ under $\p^\ast$.
Choose $\mathbf{z},\mathbf{y} \in \R^d$ and define the function
\begin{equation*}
f_\Lambda(\mathbf{z},\mathbf{y})
= \varsigma(\mathbf{y})^{-1}~
\left[-\mathbf{z}+\frac{1}{2}{\rm sp}\left(
\Sigma_{\Lambda} ({\mathbf{y}})
\right)
\right]\left[-\mathbf{z}+\frac{1}{2}{\rm sp}\left(
\Sigma_{\Lambda} ({\mathbf{y}})
\right)
\right]'\left(\varsigma(\mathbf{y})^{-1}\right)'.
\end{equation*}
Note that this function does {\it not} depend on the grid size $\Delta$.
Lemma \ref{Lemma 3.2} then implies
that
\begin{equation}\label{function f analysis}
f_\Lambda(\mathbf{Y}(t-\Delta,t),\mathbf{Y}_{t, -})~=~
\Delta^{-2}~
\varsigma(\mathbf{Y}_{t, -})^{-1}~
\E^\ast_{t-\Delta} \left[
\boldsymbol{\Upsilon}_t\right]~\E^\ast_{t-\Delta} \left[
\boldsymbol{\Upsilon}_t\right]'
\left(\varsigma(\mathbf{Y}_{t, -})^{-1}\right)',
\end{equation}
where the left-hand side only depends on $\Delta$ through the fact
that the yield curve $\mathbf{Y}_{t-\Delta}$ is observed at time $t-\Delta$, however otherwise
it does not depend on $\Delta$ (as a scaling factor).
\begin{theo}\label{theorem unbiased}
Under Model Assumptions \ref{Model Assumptions 1} we obtain
for all $K\in \N$ and $\mathbf{y}\in \R^d$
\begin{equation*}
\E^\ast_{0}\left[S_{(K)}(\mathbf{y})\right]
=\Delta~
\Sigma_{\Lambda}
(\mathbf{y})+\Delta^2~
\varsigma(\mathbf{y}) \left(\frac{1}{K}\sum_{k=1}^K
\E^\ast_{0}\left[
f_\Lambda(\mathbf{Y}(\Delta(k-1),\Delta k),\mathbf{Y}_{\Delta k, -})\right]
\right)
\varsigma(\mathbf{y})'.
\end{equation*}
\end{theo}
{\bf Interpretation.}
Using $S_{(K)}(\mathbf{y})$ as estimator for $\Delta
\Sigma_{\Lambda} (\mathbf{y})$ provides, under $\p^\ast_0$, a bias given by
\begin{equation*}
\Delta^2~
\varsigma(\mathbf{y}) \left(\frac{1}{K}\sum_{k=1}^K
\E^\ast_{0}\left[
f_\Lambda(\mathbf{Y}(\Delta(k-1),\Delta k),\mathbf{Y}_{\Delta k, -})\right]
\right)
\varsigma(\mathbf{y})'.
\end{equation*}
If we choose $t=\Delta K$ fixed and assume that the
term in the bracket is uniformly bounded for $\Delta \to 0$ then we see that
\begin{equation}\label{asymptotic result}
\E^\ast_{0}\left[S_{(K)}(\mathbf{y})\right]
=\Delta~
\Sigma_{\Lambda}
(\mathbf{y})+\Delta^2 ~O(\mathbf{1}),\qquad \text{ for }\Delta\to 0.
\end{equation}
That is, for small grid size $\Delta$ the second term
should become negligible.
~
The only term that still needs to be chosen is the
invertible and linear map $\varsigma(\mathbf{y})$, i.e.~the
volatility scaling factors.
For $\vartheta \ge 0$ we define that function
\begin{equation}\label{h function}
h:\R_+ \to \R_+, \qquad y \mapsto
h(y)=\vartheta^{-1/2}~y 1_{\{y \le \vartheta \}}+y^{1/2} 1_{\{y > \vartheta \}}.
\end{equation}
As already remarked in Subsection \ref{remark1}
in the literature one often finds
the square-root scaling, however for small rates a linear scaling
can also be appropriate. For the Swiss currency CHF it turns out below
that the linear scaling is appropriate for a threshold
of $\vartheta=2.5\%$. In addition, we define the
function $h(\cdot)$ as above to guarantee that the processes
do not explode for large volatilities and small grid sizes.
Assume that there exist constants $\sigma_j>0$, then
we set for $\mathbf{y}=(y_1,\ldots, y_d)\in \R^d$
\begin{eqnarray}
\varsigma(\mathbf{y})&=& {\rm diag}(\sigma_1 h(y_1) ,\ldots,
\sigma_d h(y_d))~
=~
{\rm diag}(\sigma_1,\ldots,
\sigma_d)~
{\rm diag}(h(y_1),\ldots, h(y_d)).
\nonumber
\end{eqnarray}
Basically, volatility is scaled according to the actual
observation $\mathbf{y}$. This choice implies
\begin{eqnarray*}
\varsigma(\mathbf{y})~C_{(K)} &=&\frac{1}{\sqrt{K}}~\varsigma(\mathbf{y})~ \left(\left[
\varsigma(\mathbf{Y}_{\Delta k,-})^{-1}~
\boldsymbol{\Upsilon}_{\Delta k}\right]_j
\right)_{j=1,\ldots, d; ~k=1, \ldots, K}\\
&=&\frac{1}{\sqrt{K}}~ {\rm diag}(h(y_1),\ldots,
h(y_d))~ \left(\left[
{\rm diag}(h(\mathbf{Y}_{\Delta k,-}))^{-1}~
\boldsymbol{\Upsilon}_{\Delta k}\right]_j
\right)_{j=1,\ldots, d; ~k=1, \ldots, K},
\end{eqnarray*}
thus, the constants $\sigma_j>0$ do not need to be estimated because they are
already (implicitly) contained in the observations and, hence, in $\Lambda$.
Therefore, we set them to 1 and we choose
\begin{eqnarray}
\varsigma(\mathbf{y})&=&
{\rm diag}(h(y_1),\ldots, h(y_d)).
\label{choice sigma}
\end{eqnarray}
These assumptions now allow to directly
analyze the bias term given in \eqref{asymptotic result}.
Therefore, we need to evaluate the function $f_\Lambda$
in Theorem \ref{theorem unbiased}.
However, to this end we would need to know $\Sigma_\Lambda$,
i.e.~we obtain from Theorem \ref{theorem unbiased} an implicit
solution (quadratic form) that can be solved for $\Sigma_\Lambda$.
We set $\mathbf{y}=\mathbf{1}$ and then obtain
from Theorem \ref{theorem unbiased}
\begin{eqnarray*}
{\Delta}^{-1}~
\E^\ast_{0}\left[S_{(K)}(\mathbf{1})\right]
&=&
\Sigma_{\Lambda}
(\mathbf{1})+\Delta
\left(\frac{1}{K}\sum_{k=1}^K
\E^\ast_{0}\left[
f_\Lambda(\mathbf{Y}(\Delta(k-1),\Delta k),\mathbf{Y}_{\Delta k, -})\right]
\right).
\end{eqnarray*}
Note that $\Sigma_{\Lambda} (\mathbf{y})
=\varsigma(\mathbf{y})\Lambda\Lambda'\varsigma(\mathbf{y})$, thus
under \eqref{choice sigma} its elements are
given by $h(y_i)h(y_j)s_{ij}$, $i,j=1,\ldots, d$, where we have defined
$\Lambda\Lambda'=(s_{ij})_{i,j=1,\ldots, d}$.
Let us first concentrate on the diagonal elements, i.e.~$i=j$, and
assume that time to maturity $m_i$ corresponds to index $i$.
\begin{eqnarray*}
\Delta^{-1}\left(
\E^\ast_{0}\left[{S_{(K)}(\mathbf{1})}\right]\right)_{ii}
&=&s_{ii}+
\frac{\Delta}{K}\sum_{k=1}^K
\Bigg(
\E^\ast_{0}\left[
\left(\frac{Y(\Delta(k-1),\Delta k)}
{h(Y(\Delta(k-1),\Delta k+m_i))}\right)^2\right]
\\
&&
+~\frac{1}{4}
\E^\ast_{0}\left[h(Y(\Delta(k-1),\Delta k+m_i))^2\right]s^2_{ii}
-\E^\ast_{0}\left[Y(\Delta(k-1),\Delta k)\right]s_{ii}\Bigg).
\end{eqnarray*}
This is a quadratic equation that can be solved for $s_{ii}$.
Define
\begin{eqnarray}
\label{a_i calculation}
a_i&=&\frac{\Delta}{4K}\sum_{k=1}^K
\E^\ast_{0}\left[h(Y(\Delta(k-1),\Delta k+m_i))^2\right],\\
\label{b calculation}
b&=&
1-
\frac{\Delta}{K}\sum_{k=1}^K\E^\ast_{0}\left[Y(\Delta(k-1),\Delta k)\right],
\\
\label{c_i calculation}
c_i&=&
-\Delta^{-1}\left(
\E^\ast_{0}\left[{S_{(K)}(\mathbf{1})}\right]\right)_{ii}
+\frac{\Delta}{K}\sum_{k=1}^K
\E^\ast_{0}\left[
\left(\frac{Y(\Delta(k-1),\Delta k)}
{h(Y(\Delta(k-1),\Delta k+m_i))}\right)^2\right],
\end{eqnarray}
then we have $a_i s_{ii}^2+bs_{ii}+c_i=0$ which provides the solution
\begin{equation}
\label{s_ii calculation}
s_{ii}= \frac{-b + \sqrt{b^2-4a_ic_i}}{2a_i}.
\end{equation}
Thus, the bias terms of the diagonal elements are given by
\begin{equation*}
\beta_{ii}=
\Delta^{-1}\left(
\E^\ast_{0}\left[{S_{(K)}(\mathbf{1})}\right]\right)_{ii}
-s_{ii},
\end{equation*}
which we are going to analyze below for the different maturities $m_i\in
{\cal M}$. For the off-diagonals $i\neq j$ and the corresponding
maturities $m_i$ and $m_j$ we obtain
{\small
\begin{eqnarray}
\nonumber
\Delta^{-1}\left(
\E^\ast_{0}\left[{S_{(K)}(\mathbf{1})}\right]\right)_{ij}
&=&s_{ij}+
\frac{\Delta}{K}\sum_{k=1}^K
\Bigg(
\E^\ast_{0}\left[
\frac{Y(\Delta(k-1),\Delta k)}
{h(Y(\Delta(k-1),\Delta k+m_i))}
\frac{Y(\Delta(k-1),\Delta k)}
{h(Y(\Delta(k-1),\Delta k+m_j))}
\right]
\\\nonumber
&&\qquad
+\frac{1}{4}
\E^\ast_{0}\left[h(Y(\Delta(k-1),\Delta k+m_i))
h(Y(\Delta(k-1),\Delta k+m_j))\right]
s_{ii}s_{jj}\\&&\label{s_ij calculation}
\qquad
-\frac{1}{2}\E^\ast_{0}\left[Y(\Delta(k-1),\Delta k)
\frac{Y(\Delta(k-1),\Delta k+m_i)}
{h(Y(\Delta(k-1),\Delta k+m_j))}\right]s_{ii}
\\&&\nonumber
\qquad
-\frac{1}{2}\E^\ast_{0}\left[Y(\Delta(k-1),\Delta k)
\frac{Y(\Delta(k-1),\Delta k+m_j)}
{h(Y(\Delta(k-1),\Delta k+m_i))}
\right]s_{jj}
\Bigg).
\end{eqnarray}}
This can easily be solved for $s_{ij}$ for given $s_{ii}$ and $s_{jj}$.
\section{Calibration to real data}
\subsection{Calibration}
For the time-being we assume that
$\p=\p^\ast$, i.e.~we assume that the market price of risk is identical
equal to 0. This simplifies the calibration and as a consequence we can
directly work on the observed data. The choice of the drift
term will be discussed below.
The first difficulty is the choice of the data. The reason therefore
is that risk-free ZCBs do {\it not} exist and, thus, the risk-free
yield curve needs to be estimated from data that has different spreads
such as a credit spread, a liquidity spread, a long-term premium, etc.
We calibrate the model to the Swiss currency CHF.
For short times to maturity (below one year) one typically chooses
either the LIBOR (London InterBank Offered Rate)
or the SAR (Swiss Average Rate), see Jordan \cite{Jordan}, as (almost)
risk-free financial instruments. The LIBOR is the rate at which
highly-credit banks borrow and lend money at the inter-bank market.
The SAR is a rate determined by the Swiss National Bank at which
highly-credited institutions borrow and lend money with securization.
We display the yields of these two financial time series for instruments of a
time to maturity of 3 months, see Figure \ref{Figure 1}. We see that
the SAR yield typically lies below the LIBOR yield
(due to securization). Therefore,
we consider the SAR to be less risky and we choose it as approximation
to a risk-free financial instrument with short time to maturity.
For long times to maturity (above one year)
one either chooses government bonds
(of sufficiently highly rated countries) or swap rates.
In Figure \ref{Figure 2} we give the time series of the Swiss government
bond and the CHF swap yields both for a time to maturity of 5 years.
We see that the rate of the Swiss government bond is below the swap rate
(due to lower credit risk and maybe an illiquidity premium coming from a
high demand) and
therefore we choose Swiss government bonds as approximation to the
risk-free yield curve data for long times to maturity.
We mention that these short terms and long terms data are not completely
compatible which may give some difficulties in the calibration. We will
also see this in the correlation matrices below.
Thus, for our analysis we choose the SAR for times to maturity
$m \in \{1/52, 1/26, 1/12, 1/4 \}$ and the Swiss government bond
for times to maturity $m \in \{1,2,3,4,5,6,7,8,9,10,15, 20, 30 \}$.
We choose time grid $\Delta=1/52$ (i.e.~a weekly time grid) and then
we calculate $\boldsymbol{\Upsilon}_t$ for our observations.
Note that we cannot directly calculate $\Upsilon_{t,m}=
m~Y(t,t+m)-(m+\Delta)~Y(t-\Delta,t+m)$ for all $m\in {\cal M}$ because
we have only a limited set of observed times to maturity. Therefore,
we make the following interpolation:
assume $m+\Delta \in (m, \widetilde{m}]$ for $m,\widetilde{m} \in {\cal M}$,
then approximate
\begin{equation*}
Y(t-\Delta,t+m)~\approx~
\frac{\widetilde{m}-(m+\Delta)}{\widetilde{m}-m} ~Y(t-\Delta, t+m-\Delta) +
\frac{\Delta}{\widetilde{m}-m}~
Y(t-\Delta, t+\widetilde{m}-\Delta).
\end{equation*}
In Figure \ref{Figure 3} we give the time series of these
estimated $(\boldsymbol{\Upsilon}_t)_t$ and in Figure \ref{Figure 4}
we give the component-wise ordered time series
obtained from $(\boldsymbol{\Upsilon}_{t})_t$.
We observe that the volatility is increasing in the time to maturity due to
scaling with time to maturity. Using \eqref{choice sigma}
we calculate
\begin{equation*}
\sqrt{K} ~ C_{(K)} = \left(\left[
\varsigma(\mathbf{Y}_{\Delta k,-})^{-1}~
\boldsymbol{\Upsilon}_{\Delta k}\right]_j
\right)_{j=1,\ldots, d; ~k=1, \ldots, K}\in \R^{d\times K}
\end{equation*}
for our observations. In Figures \ref{Figure 5} and \ref{Figure 6}
we plot the time series $\Upsilon_{t,m}$ and
$[\sqrt{K} ~ C_{(K)}]_m=
\Upsilon_{t,m}/h(Y(t-\Delta, t+m))$
for illustrative purposes only for maturities $m=1/52$ and $m=5$.
We observe that the scaling $\varsigma(\mathbf{Y}_{t,-})^{-1}$
gives more stationarity for short times to maturity, however in financial
stress periods it substantially increases the volatility of the observations,
see Figure \ref{Figure 5}.
For longer times to maturity one might discuss or even question the scaling
because it is less obvious whether it
is needed, see Figure \ref{Figure 6}. Next figures will show
that this scaling is also needed for longer times to maturity.
We then calculate the observed matrix
\begin{equation*}
\left(\widehat{s}_{ij}^{\rm bias}(K)\right)_{i,j=1,\ldots, d}=
\Delta^{-1}S_{(K)}(\mathbf{1})
\end{equation*}
as a function of the number of observations
$K$ (we set $\mathbf{1}=(1,\ldots, 1)'\in \R^d$).
Moreover, we calculate the bias correction terms
given in \eqref{a_i calculation}-\eqref{c_i calculation}
where we simply replace the expected values on the right-hand
sides by the observations. Formulas
\eqref{s_ii calculation}-\eqref{s_ij calculation} then provide
the estimates $\widehat{s}_{ij}(K)$ for $s_{ij}$ as a function
of the number of observations $K$. The bias correction term
is estimated by
\begin{equation*}
\widehat{\beta}_{ij}(K)=
\widehat{s}_{ij}^{\rm bias}(K)
-\widehat{s}_{ij}(K).
\end{equation*}
We expect that for short times to maturity the bias correction term is larger due to more
dramatic drifts. The results for selected times to maturity
$m\in \{1/52, 1/4, 1, 5, 20\}$
are presented in Figures \ref{Figure 7}-\ref{Figure 11}. Let us
comment these figures:
\begin{itemize}
\item Times to maturity in the set ${\cal M}_1=\{1/52,1/26, 1/12\}$
look similar to $m=1/52$ (Figure \ref{Figure 7});
${\cal M}_2=\{1/4\}$ corresponds to Figure \ref{Figure 8};
times to maturity in the set ${\cal M}_3=
\{1,2,3,4,5,6,7,8,9,10,15\} $ look similar to $m=1,5$
(Figures \ref{Figure 9}-\ref{Figure 10});
times to maturity $m\in {\cal M}_4=\{20,30\}$ look similar to $m=20$
(Figure \ref{Figure 11}).
\item Times to maturity in ${\cal M}_1 \cup {\cal M}_3$ seem to have converged,
for ${\cal M}_2$ the convergence picture is distorted by the
last financial crisis, where volatilities relative to yields have
substantially increased, see also Figure \ref{Figure 5}. One might
ask whether during financial crisis we should apply a different
scaling (similar to regime switching models). For ${\cal M}_4$ the
convergence picture suggest that we should probably study longer time
series (or scaling should be done differently).
Concluding, this supports the choice of the function $h$
in \eqref{h function}. Only long times to maturity
$m \in {\cal M}_4$ might suggest a different scaling.
\item For times to maturities in ${\cal M}_3 \cup {\cal M}_4$ we observe that
the bias term given in \eqref{asymptotic result} is negligible,
see Figures \ref{Figure 9}-\ref{Figure 11},
that is, $\Delta
=1/52$ is sufficiently small for times to maturity $m\ge 1$. For
times to maturities in ${\cal M}_1 \cup {\cal M}_2$ it is however
essential that we do a bias correction, see Figure
\ref{Figure 7}-\ref{Figure 8}. This comes from the fact that for
small times to maturity the bias term is driven by
$\mathbf{z}$ in $f_\Lambda(\mathbf{z},\mathbf{y})$ which then is
of similar order as $s_{ii}$.
\end{itemize}
In Table \ref{Table 1} we present the resulting
estimated matrix $\widehat{\Sigma}_\Lambda(\mathbf{1})
=(\widehat{s}_{ij}(K))_{i,j=1,\ldots, d}$ which is based on all
observations in $ \{01/2000,\ldots, 05/2011\}$. We observe that
the diagonal $\widehat{s}_{ii}(K)$ is an increasing function
in the time to maturity $m_i$.
Therefore, in order to further analyze this
matrix, we normalize it as follows (as a correlation matrix)
\begin{equation*}
\widehat{\Xi}=(\widehat{\rho}_{ij})_{i,j=1,\ldots, d}
=\left(\frac{\widehat{s}_{ij}(K)}{\sqrt{\widehat{s}_{ii}(K)}
\sqrt{\widehat{s}_{jj}(K)}}\right)_{i,j=1,\ldots, d}.
\end{equation*}
Now all the entries $\widehat{\rho}_{ij}$ live on the same
scale and the result is presented in Figure \ref{Figure 12}.
We observe two different structures, one for times to maturity
less than 1 year, i.e.~$m\in \widetilde{\cal M}_1
={\cal M}_1 \cup {\cal M}_2$,
and one for times to maturity $m\in
\widetilde{\cal M}_2=
{\cal M}_3 \cup {\cal M}_4$.
The former times to maturity $m\in \widetilde{\cal M}_1$
were modeled using the observations
from the SAR, the latter $m\in \widetilde{\cal M}_2$
with observations from the Swiss
government bond. This separation shows that these two data
sets are not completely compatible which gives some
,,additional independence'' (diversification) between
$\widetilde{\cal M}_1$ and $\widetilde{\cal M}_2$.
If we calculate the eigenvalues of $\widehat{\Xi}$
we observe that the first 5 eigenvalues explain 95\% of the
total observed cross-sectional volatility (we have a $d=17$ dimensional
space). Thus, a principal component analysis says that
we should at least choose a 5-factor model. These are more factors
than typically stated in the literature (see Brigo-Mercurio \cite{BM},
Section 4.1). The reason therefore
is again that the short end $\widetilde{\cal M}_1$
and the long end $\widetilde{\cal M}_2$ of the estimated yield curve
behave more independently due to different choices of the data
(see also Figure \ref{Figure 12}). If we restrict this principal
component analysis to $\widetilde{\cal M}_2$ we find the classical
result that a 3-factor model explains 95\% of the observed
cross-sectional volatility.
In the next step we analyze the assumption
of the independence of $\Sigma_\Lambda(\mathbf{1})=\Lambda \Lambda'
=(s_{ij})_{i,j=1,\ldots, d}$ from the grid size $\Delta$. Similar
to the analysis above we estimate $\Sigma_\Lambda(\mathbf{1})$ for the
grid sizes $\Delta=1/52, 1/26, 1/13, 1/4$ (weekly, bi-weekly,
4-weekly, quarterly grid size). The first observation is that the
bias increases with increasing $\Delta$ (for illustrative purposes
one should compare Figure \ref{Figure 9} with $m=1$ and $\Delta=1/52$
and Figure \ref{Figure 9_3} with $m=1$ and $\Delta=1/4$). Of course,
this is exactly the result expected.
In Table \ref{Table 2} we give the differences between
the estimated matrices $\widehat{\Sigma}_\Lambda(\mathbf{1})
=(\widehat{s}_{ij}(K))_{i,j=1,\ldots, d}$ on the
weekly grid $\Delta=1/52$ versus the
estimates on a quarterly grid $\Delta =1/4$ (relative to the
estimated values on the quarterly grid). Of course, we can only
display these differences for times to maturity $m\in {\cal M}_2
\cup \widetilde{\cal M}_2$ because in the latter model the times
to maturity in ${\cal M}_1$ do not exist. We observe rather small
differences within $\widetilde{\cal M}_2$ which supports the
independence assumption from the choice of $\Delta$ within
the Swiss government bond yields. For the SAR in ${\cal M}_2$ this
picture does not entirely hold true which has also to do with
the fact that the model does not completely fit to the data,
see Figure \ref{Figure 8}. Thus, we only observe larger difference
for covariances that have a bigger difference in times to maturity
compared. The pictures for $\Delta=1/26, 1/13$
are quite similar which justify our independence choice.
\begin{conclusions}~\label{conclusions}\end{conclusions}
We conclude that the independence assumption
of $\Sigma_\Lambda(\mathbf{1})$ from $\Delta$
is not violated by our observations
and that the bias terms $\widehat{\beta}_{i,j}(K)$ are negligible
for maturities $m_i,m_j \in \widetilde{\cal M}_2$ and time grids
$\Delta=1/52, 1/16, 1/13$, therefore we can directly work with
model \eqref{model kappa} to predict future yields for
times to maturity in $\widetilde{\cal M}_2$.
\subsection{Back-testing and market price of risk}
\label{subsection back-testing}
In this subsection we back-test our model against the
observations. We therefore choose a fixed-term annuity with nominal
payments of size 1 at maturity dates $m\in {\cal M}_3$. The present
value of this annuity at time $t$ is given by
\begin{equation*}
\pi_t = \sum_{m \in {\cal M}_3} P(t,t+m)=
\sum_{m \in {\cal M}_3} \exp \left\{- m~ Y(t,t+m) \right\}
\approx \sum_{m \in {\cal M}_3} 1- m ~Y(t,t+m)~ \stackrel{\rm def.}{=}~
\widetilde{\pi}_t.
\end{equation*}
Our back-testing setup is such that we try to predict $\widetilde{\pi}_{t}$ based on the observations ${\cal F}_{t-\Delta }$ and then (one period later) we compare this forecast with the realization
of $\widetilde{\pi}_{t}$. In view of Conclusions \ref{conclusions} we
directly work with $C_{(K)}$ for small time grids $\Delta$
(for $t=\Delta(K+1)$). Moreover, the Taylor approximation $\widetilde{\pi}_t$ to
${\pi}_t$ is used in order to avoid (time-consuming) simulations. Here a first order Taylor expansion is sufficient since the portfolio's variance will be -- due to high positive correlation -- quite large in comparison to possible second order -- drift like -- correction terms. Such an approximation does not work for short-long portfolios.
For the approximation (under $\p^\ast$)
\begin{equation*}
\boldsymbol{\Upsilon}_{t}|_{{\cal F}_{t-\Delta}}
~\stackrel{(d)}{\approx}~
\boldsymbol{\kappa}_{t}(\mathbf{Y}(t-\Delta, t),
\mathbf{Y}_{t,-})|_{{\cal F}_{t-\Delta}},
\end{equation*}
we obtain an approximate forecast to $\widetilde{\pi}_t$ given by
(denote the cardinality of ${\cal M}_3$ by $d_3$)
\begin{eqnarray}\nonumber
\widetilde{\widetilde{\pi}}_t|_{{\cal F}_{t-\Delta}}
&=&d_3-
\sum_{m \in {\cal M}_3} (m+\Delta)~Y(t-\Delta,t+m)
+d_3 ~\Delta Y(t-\Delta, t)\\\label{life portfolio}
&&\qquad
-\frac{1}{2}~\mathbf{1}_{{\cal M}_3}'~{\rm sp}\left(S_{(K)}
(\mathbf{Y}_{t,-})\right)
-\mathbf{1}_{{\cal M}_3}'~\varsigma(\mathbf{Y}_{t,-})C_{(k)}\mathbf{W}^\ast_t~
\big|_{{\cal F}_{t-\Delta}},
\end{eqnarray}
where $\mathbf{1}_{{\cal M}_3}=(1_{\{1\in {\cal M}_3\}},\ldots,
1_{\{d\in {\cal M}_3\}})'\in \R^d$.
Thus, the conditional distribution of $\widetilde{\widetilde{\pi}}_t$
under $\p^\ast$, given ${\cal F}_{t-\Delta}$, is a Gaussian distribution
with conditional mean and conditional variance given by
\begin{eqnarray*}
\mu^\ast_{t-\Delta}&=&
d_3-
\sum_{m \in {\cal M}_3} (m+\Delta)~Y(t-\Delta,t+m)
+d_3 ~\Delta Y(t-\Delta, t)
-\frac{1}{2}~\mathbf{1}_{{\cal M}_3}'~{\rm sp}\left(S_{(K)}
(\mathbf{Y}_{t,-})\right),\\
\tau_{t-\Delta}^2&=&
\mathbf{1}_{{\cal M}_3}'~
S_{(K)}
(\mathbf{Y}_{t,-})~\mathbf{1}_{{\cal M}_3}.
\end{eqnarray*}
We calculate these conditional moments for
$t \in \{01/2005, \ldots, 05/2011 \}$ based on the $\sigma$-fields
${\cal F}_{t-\Delta}$
generated by the data in
$\{01/2000, \ldots, t-\Delta \}$, for $\Delta=1/52, 1/12$ (weekly and
monthly grid). From these we can calculate the observable residuals
\begin{equation*}
z^\ast_t = \frac{\widetilde{\pi}_t-\mu^\ast_{t-\Delta}}{\tau_{t-\Delta}}.
\end{equation*}
The sequence of these observable residuals should
approximately look like an i.i.d.~standard Gaussian distributed
sequence. The result for $\Delta=1/52$ is given in Figure \ref{residuals 1}
and for $\Delta=1/12$ in Figure \ref{residuals 2}.
At the first sight this sequence $(z^\ast_t)_t$ seems to fulfill
these requirements, thus the out-of-sample back-testing provides the
required results. In Figure \ref{QQ Plot} we also provide
the Q-Q-plot for the residuals $(z^\ast_t)_t$
against the standard Gaussian distribution for
$\Delta =1/52$. Also in this plot we observe a good fit, except
for the tails of the distribution. This suggests that one may
relax the Gaussian assumption on $\boldsymbol{\varepsilon}_t^\ast$
by a more heavy-tailed model (this can also be seen
in Figure \ref{residuals 1} where we a few outliers). We have already
mentioned this in Section \ref{modelling_na} but for this exposition we keep
the Gaussian assumption.
If we calculate the auto-correlation for time lag $\Delta$
between the residuals $z^\ast_t$
we obtain 5\% which is a convincingly small value.
This supports the assumption having independent residuals. The
same holds true if we consider the
auto-correlation for time lag $\Delta$ between the
absolute values $|z^\ast_t|$ of the residuals resulting in 11\%.
The only observation which may contradict the i.i.d.~assumption
is that we observe slight clustering in Figure \ref{residuals 1}.
This non-stationarity might have to do with that we calculate the
residuals under the equivalent martingale measure $\p^\ast$,
however we make the observations under the real world probability
measure $\p$. If these measures coincide the statements are the same.
The classical approach is that one assumes that the two probability
measures are equivalent, i.e.~$\p^\ast \sim \p$, with density process
\begin{equation}\label{market-price of risk}
\xi_{t} = \prod_{s=1}^{t/\Delta} \exp \left\{ -\frac{1}{2}\left\|
\boldsymbol{\lambda}_{\Delta s} \right\|^2+
\boldsymbol{\lambda}_{\Delta s}
~ \boldsymbol{\varepsilon}_{\Delta s} \right\},
\end{equation}
with $\boldsymbol{\varepsilon}_t$ is independent
of ${\cal F}_{t-\Delta}$, ${\cal F}_{t}$-measurable
and a $t/\Delta$-dimensional
standard Gaussian random vector
with independent components under $\p$.
Moreover, it is assumed that $\boldsymbol{\lambda}_{t}$ is
$d$-dimensional and previsible, i.e.~${\cal F}_{t-\Delta}$-measurable.
Note that this density process
$(\xi_t)_t$ is a strictly positive and normalized $(\p,\F)$-martingale.
For any $\p^\ast$-integrable and ${\cal F}_t$-measurable random variable
$X_t$ we have, $\p$-a.s.,
\begin{equation*}
\E^\ast_{t-\Delta} \left[X_t\right]=
\frac{1}{\xi_{t-\Delta}}~\E_{t-\Delta} \left[\xi_t X_t\right].
\end{equation*}
This implies that
\begin{equation*}
\boldsymbol{\varepsilon}_{t}-\boldsymbol{\lambda}_{t}
~\stackrel{(d)}{=}~\boldsymbol{\varepsilon}^\ast_t
\quad \text{under $\p^\ast_{t-\Delta}$.}
\end{equation*}
$\boldsymbol{\lambda}_{t}$ is called market price
of risk at time $t$ and reflects
the difference between $\p^\ast_{t-\Delta}$
and $\p_{t-\Delta}$. Under Model Assumptions \ref{Model Assumptions 1}
we then obtain under the real world probability measure $\p$
\begin{equation*}
\boldsymbol{\Upsilon}_t
= \Delta\left[-\mathbf{Y}(t-\Delta,t)+\frac{1}{2}
~{\rm sp}
(\Sigma_{{\Lambda}}
({\mathbf{Y}_{t, -}}))
\right]+ \sqrt{\Delta}~
\varsigma({\mathbf{Y}_{t, -}})~{\Lambda}
~ \boldsymbol{\lambda}_{t}
+ \sqrt{\Delta}~
\varsigma({\mathbf{Y}_{t, -}})~{\Lambda}
~ \boldsymbol{\varepsilon}_{t},
\end{equation*}
i.e.~we have a change of drift given by $
\sqrt{\Delta}~
\varsigma({\mathbf{Y}_{t, -}})~{\Lambda}
~ \boldsymbol{\lambda}_{t}$.
Thus, under the (conditional) real world probability
measure $\p_{t-\Delta}$ the approximate forecast
$\widetilde{\widetilde{\pi}}_t$ has a Gaussian distribution
with conditional mean and conditional covariance given by
\begin{equation*}
\mu_{t-\Delta}=
\mu^\ast_{t-\Delta}
-\sqrt{\Delta}~\mathbf{1}_{{\cal M}_3}'~\varsigma
(\mathbf{Y}_{t,-})~\Lambda~\boldsymbol{\lambda}_t\qquad
\text{ and } \qquad
\tau_{t-\Delta}^2=
\mathbf{1}_{{\cal M}_3}'~
S_{(K)}
(\mathbf{Y}_{t,-})~\mathbf{1}_{{\cal M}_3}.
\end{equation*}
For an appropriate choice of the market price of risk
$\boldsymbol{\lambda}_{t}$ we obtain residuals
\begin{equation*}
z_t = \frac{\widetilde{\pi}_t-\mu_{t-\Delta}}{\tau_{t-\Delta}},
\end{equation*}
which should then form an i.i.d.~standard Gaussian distributed
sequence under the real world probability measure $\p$.
In order to detect the market price of risk term,
we look at residuals for
individual times to maturity $m\in {\cal M}$, i.e.~we replace the indicators
$\mathbf{1}_{{\cal M}_3}$ in \eqref{life portfolio} by indicators
$\mathbf{1}_{\{m\}}$. We denote the resulting residuals
by $z_{m,t}^\ast$ and the corresponding volatilities by
$\tau_{m,t-\Delta}$. In Figures \ref{maturity 1}, \ref{maturity 5}
and \ref{maturity 10} we show the results for $m=1,5,10$.
The picture is similar to Figure \ref{residuals 1}, i.e.~we observe
clustering but not a well-defined drift. This implies that
we suggest to set the market price of risk
$\boldsymbol{\lambda}_{t}=0$ for the prediction of future
yield curves (we come back to this in Section \ref{Section Vasicek}).
\subsection{Comparison to the Vasi\v{c}ek model}
\label{Section Vasicek}
We compare our findings to the results in the Vasi\v{c}ek model \cite{Vasicek}.
The Vasi\v{c}ek model is
the simplest short rate model that
provides an affine term structure for interest rates (see also
Filipovi\'c \cite{Damir}), and hence a closed-form solution
for ZCB prices. The price of the ZCB in the Vasi\v{c}ek model
takes the following form
\begin{equation*}
P(t,t+m)= \exp \left\{ A(m)-r_t~ B(m) \right\},
\end{equation*}
where the short rate process $(r_t)_t$ evolves as an Ornstein-Uhlenbeck
process under $\p^\ast$,
and $A(m)$ and $B(m)$ are constants only depending on the
time to maturity $m$ and the model parameters
$\kappa^\ast$, $\theta^\ast$ and $g$
(see for instance (3.8) in Brigo-Mercurio \cite{BM}).
The short rate $r_t$ is then
under $\p^\ast_{t-\Delta}$ normally distributed
with conditional mean and conditional variance given by
\begin{eqnarray*}
\E^\ast_{t-\Delta}[r_t]
&=&r_{t-\Delta}~e^{-\Delta \kappa^\ast}
+\theta^\ast\left(1-e^{-\Delta \kappa^\ast}\right),\\
{\rm Var}^\ast_{t-\Delta}(r_t)&=&
\frac{g^2}{2\kappa^\ast}\left[1-e^{-2\kappa^\ast\Delta}\right].
\end{eqnarray*}
Thus, the approximation $\widetilde{\pi}_t$ has under $\p^\ast_{t-\Delta}$
a normal distribution with conditional mean
\begin{equation*}
\E^\ast_{t-\Delta}[\widetilde{\pi}_t]
= \sum_{m\in {\cal M}_3} \left(1 + A(m) - \E^\ast_{t-\Delta}[r_t] ~B(m)\right),
\end{equation*}
and conditional variance
\begin{equation*}
{\rm Var}^\ast_{t-\Delta}(\widetilde{\pi}_t)
= {\rm Var}^\ast_{t-\Delta}(r_t)
\left(\sum_{m\in {\cal M}_3} B(m)\right)^2.
\end{equation*}
As in the previous section we
assume $\p^\ast=\p$, i.e.~we set the market price of risk
$\boldsymbol{\lambda}_{t}=0$:
(i) this allows to estimate the model parameters
$\kappa^\ast$, $\theta^\ast$
and $g$, for instance, using maximum likelihood
methods (see (3.14)-(3.16) in Brigo-Mercurio \cite{BM}); (ii) makes
the model comparable to the calibration of our model. We will comment on this ``comparability'' below.
Thus we estimate these
parameters and obtain parameter estimates $\widehat{\kappa}^\ast$,
$\widehat{\theta}^\ast$ and $\widehat{g}$ from which
we get the estimated
functions
$\widehat{A}(\cdot)$ and $ \widehat{B}(\cdot)$.
This then allows to estimate the conditional mean and variance of
$\widetilde{\pi}_t$, given ${\cal F}_{t-\Delta}$.
From these we calculate the observable residuals
\begin{equation*}
v^\ast_t = \frac{\widetilde{\pi}_t-\widehat{\E}^\ast_{t-\Delta}[\widetilde{\pi}_t]}{
\widehat{{\rm Var}}^\ast_{t-\Delta}(\widetilde{\pi}_t)^{1/2}}.
\end{equation*}
In Figure \ref{residuals_3} we plot the time series
$z^\ast_t$ and $v^\ast_t$ for $t \in \{01/2005,\ldots, 05/2011\}$.
The observation is that $v^\ast_t$
is far too small! The explanation for this observation lies in the assumption
$\p^\ast=\p$, i.e.~$\boldsymbol{\lambda}_{t}=0$. Since the Vasi\v{c}ek
prices are calculated by conditional expectations of the
{\it entire} future
development of the short rate $r_t$ until expiry of the ZCB, the
choice of the market price of risk $\boldsymbol{\lambda}_{t}$
has a huge influence on the
resulting ZCB price in the Vasi\v{c}ek model. Thus, the calibration of
$\widehat{A}(\cdot)$ and $\widehat{B}(\cdot)$ is completely
wrong if we set $\boldsymbol{\lambda}_{t}=0$.
Compare
\begin{eqnarray}
\log P(t,t+m)&=& - m~ Y(t,t+m),\label{ZCB 1}\\
\log P(t,t+m)&=& A(m) - r_t ~B(m) \label{ZCB 2}.
\end{eqnarray}
Conditionally, given ${\cal F}_{t-\Delta}$, we model the development
from $Y(t-\Delta, t+m)$ to $Y(t, t+m)$ for the study of
\eqref{ZCB 1}. That is, we model a change of the yield curve
$\mathbf{Y}_{t-\Delta}$ at time $t-\Delta$ to
$\mathbf{Y}_{t}$ at time $t$. Since the yield curve $\mathbf{Y}_{t-\Delta}$
already corresponds to market prices it already contains the actual
market risk aversion, and thus the market price of risk
$\boldsymbol{\lambda}_t$ in \eqref{market-price of risk}
only influences one single period in our consideration.
The (pricing) functions $A(\cdot)$ and $B(\cdot)$ in \eqref{ZCB 2}, however,
are calculated completely within the Vasi\v{c}ek model by a forward projection
of $r_t$ until maturity date $t+m$. If this forward projection is done
under the wrong measure $\p$, then these pricing components
completely miss the
market risk aversion and hence are not appropriate. Thus, in general,
we should have $A(m)=A(m,\boldsymbol{\lambda}_{t})$
and $B(m)=B(m,\boldsymbol{\lambda}_{t})$ which
requires a detailed knowledge of the market price of risk $\boldsymbol{\lambda}_{t}$
and, thus, the Vasi\v{c}ek model reacts much more sensitively to non-appropriately
calibrated equivalent martingale measures $\p^\ast$. Note that this
is true for all models where ZCB prices are entirely determined
by the short rate process $(r_t)_t$.
\begin{conclusions}~\end{conclusions}
\begin{itemize}
\item
We conclude that the HJM models (similar to Model Assumptions
\ref{Model Assumptions 1})
are much more robust against inappropriate choices of the market price
of risk compared to short rate models, because in the former we only need
to choose the market price of risk for the one-step ahead for the prediction
of the ZCB prices at the end of the period (i.e.~from $t-\Delta$ to $t$)
whereas for short rate models we need to choose the market price of
risk appropriately for the entire life time of the ZCB (i.e.~from $t-\Delta$
to $t+m$).
\item Our HJM model (Model Assumptions \ref{Model Assumptions 1})
always captures the actual yield curve, whereas this is not
necessarily the case for short rate models.
\end{itemize}
\subsection{Forward projection of yield curves and arbitrage}
\label{sec.arbitrage}
For the calibration of the model and for yield curve prediction we
have chosen a restricted set ${\cal M}$ of times to maturity.
In most applied cases one has to stay within such a restricted
set because there do not exist observations for all times to
maturity. We propose that we predict future yield
curves within these families ${\cal M}$ and then approximate
the remaining times to maturity using a parametric family
like the Nelson-Siegel \cite{NelsonSiegel}
or the Svensson \cite{Svensson1, Svensson2}
family, see also Filipovi\'c \cite{Damir}.
~
Finally, we demonstrate the absence of arbitrage condition given
in Lemma \ref{HJM yield condition}. At the end of
Section \ref{section introduction} we have emphasized the importance
of the no-arbitrage property of the prediction model. Let us choose an
asset portfolio $w_t P(t,t+m_1)-P(t,t+m_2)$ for two different times
to maturity $m_1$ and $m_2$. We approximate this portfolio
by a Taylor expansion up to order $2$ and set
{\small
\begin{equation*}
\widetilde{\pi}_t=
w_t \left(1-m_1Y(t,t+m_1)+\frac{{(m_1Y(t,t+m_1))}^2}{2} \right)-\left(1-m_2Y(t,t+m_2)+\frac{{(m_2Y(t,t+m_2))}^2}{2}\right).
\end{equation*}}
Under our model assumptions, the returns of both terms
$m_iY(t,t+m_i)$
in portfolio
$\widetilde{\pi}_t$ have, conditionally given ${\cal F}_{t-\Delta}$,
a Gaussian distribution term with standard deviations given by
\begin{equation*}
\tau_{t-\Delta}^{(i)}=
\sqrt{
\mathbf{1}_{\{m_i\}}'~
S_{(K)}
(\mathbf{Y}_{t,-})~\mathbf{1}_{\{m_i\}}}
\qquad \text{ for } i=1,2.
\end{equation*}
If we choose $w_t=\tau_{t-\Delta}^{(2)}/\tau_{t-\Delta}^{(1)}$
then the returns of the Gaussian parts of both terms in portfolio $\widetilde{\pi}_t$ have the same
variance and, thus, under the Gaussian assumption have the same
marginal distributions. Since the conditional expectation of the second order term in the Taylor expansion cancels the no-arbitrage drift term (up to a small short rate correction) we see that the returns of the portfolio $ \widetilde{\pi}_t $ should provide zero returns conditionally.
In Figure \ref{arbitrage} we give an example for times to
maturity $m_1=10$ and $m_2=20$. The correlation between the
prices of these ZCBs is high, about 85\%, i.e.~their prices tend
to move simultaneously. The resulting weights $w_t$ are in the
range between 1.4 and 1.9. In Figure \ref{arbitrage} we plot the aggregated
realized gains of the portfolio $ \widetilde{\pi} $ minus their prognosis including
and excluding the HJM correction term. Recall that the predicted
gains should be zero conditionally on the current information. We observe that the model without the HJM term
clearly drifts away from zero, which opens the possibility of arbitrage. Therefore, we insist on a prediction model that is free of arbitrage.
|
{
"timestamp": "2012-03-12T01:00:59",
"yymm": "1203",
"arxiv_id": "1203.2017",
"language": "en",
"url": "https://arxiv.org/abs/1203.2017"
}
|
\section{Introduction}
States living on the boundary of Fractional Quantum Hall (FQH) systems represent one of the more intriguing examples of
one-dimensional interacting electron gas.\cite{DasSarma} The general theory describing these edge states involves the idea of Chiral Luttinger
Liquid ($\chi$LL).\cite{Wen95, Wenbook}
In particular, for the simple Laughlin sequence \cite{Laughlin83} at filling factor $\nu=1/(2n+1)$,
with $n \in \mathbb{N}$, all the properties of the system, including the fractional charge and statistics of edge excitations, are described
in terms of a single chiral bosonic field. More involved is the description of states belonging to the Jain sequence \cite{Jain89} at
filling factor $\nu = p/ (2n p + 1)$, being $n \in \mathbb{N}$ and $p \in \mathbb{Z}$, where the introduction of a charge bosonic field and
additional neutral bosonic modes are required by the proposed hierarchical theories leading to an hidden SU($|p|$)
symmetry.\cite{Wen91,Kane95}
Recently, great interest was dedicated to more exotic states, such as \cite{Willett87} $\nu=5/2$ where different models were proposed,
with excitations supporting both Abelian\cite{Halperin83,Halperin93,Wen90} or
non-Abelian\cite{Moore91,Wen91b, Fendley07, Levin07, Lee07, Bishara08, Carrega12} statistics. To date, different experimental
observations\cite{Radu08,Bid10, Dolev08, Venkatachalam11} suggest the non-Abelian anti-Pfaffian model as the proper
candidate for $\nu=5/2$ even if the debate is still open.\cite{Lin12}
In the effective field theories the peculiar non-Abelian properties are encoded in an additional conformal field, which belongs to the
Ising sector.\cite{ Nayak08}
The non-Abelian nature of the
excitations of this state arose interest in perspective of possible applications to topologically protected quantum computation.\cite{Nayak08}
The simpler experimental test for all these models is the study of transport properties in a quantum point contact (QPC) geometry.\cite{Chang03}
In absence of interactions between edges and external degrees of freedom, the power-law behavior
of the transport properties in the QPC geometry, as a function of bias or temperature, directly reflects the universal exponents of the $\chi$LL theory.
Unfortunately, sometimes strong discrepancies between the predictions of such theories and experimental observations are reported. For example,
even for the simple Laughlin sequence \cite{Roddaro03, Roddaro04} the behavior of the differential conductance, as a function of the voltage, is in
qualitative agreement with predictions only at high temperature showing a peak at zero bias. However, decreasing temperature, the observed peak
turns into a completely unexpected dip.
Anomalous current/voltage characteristics have been also measured for other filling factors such as $\nu=2/5$ in the Jain sequence.\cite{Chung03}
Furthermore, renormalizations of the $\chi$LL exponents are sometimes crucial to fully explain the measured crossover of the tunneling
charges at low temperatures.\cite{Chung03, Bid09, Dolev10, Ferraro08, Ferraro10a, Ferraro10b, Ferraro10c, Carrega11}
Possible explanations for these disagreements have been traced back to the inhomogeneity of the filling factor below the QPC due to the action
of the electrostatic gates\cite{Roddaro04, Lal08}, or to an energy dependent tunneling amplitude caused by the extended nature of the
contact.\cite{Overbosch09, Chevallier10}
Alternatively, various mechanisms leading to the renormalization
of the Luttinger parameters through coupling with external environments have been proposed. They range from
the coupling with one dimensional phonons \cite{Rosenow02, Khlebnikov06}, edge reconstruction induced by the smoothness of the confinement
potential \cite{Yang03}, possible Coulomb interaction between the different edges \cite{Mandal02, Papa04}, to interactions with a compressible
component of a composite fermions liquid with very small longitudinal conductivity.\cite{Shytov98,Levitov01}
Many of these approaches have focused on the Laughlin case and cannot be easily extended to composite edge states
where anomalous behaviors are usually observed.
In particular, many of the above mechanisms are not robust against the disorder induced intra-edge electron tunneling, an unavoidable effect in real
samples responsible for the equilibration of the different channels. This is a crucial ingredient in explaining the universal quantization of the
conductance in presence of counter-propagating modes.\cite{Kane94, Kane95, Levin07, Lee07, Bishara08, Carrega12}
Recently, Dalla Torre \emph{et al.}\cite{DallaTorre10, DallaTorre11} observed that the interplay between the $1/f$ noise, generated by
the external environment, and the dissipation induced by the cooling setup
could lead to the renormalization of the Luttinger parameter for one dimensional systems of cold atoms.
In this paper we will apply this idea to the case of the edge states in the FQH effect, taking in account the peculiar chiral
nature of the $\chi$LL theories and investigating the effects of an external noisy environment. The $1/f$ noise, a
quite universal and unavoidable perturbation in any electronic circuitry, can be indeed generated by trapped
charges in the semiconductor substrate.\cite{Paladino02, Muller06}
These sources of noise, with $1/f$ spectrum, drive stochastically the system into an out-of-equilibrium condition and the stationary condition is
recovered, by the dissipation mechanism.
For one dimensional electron systems and $\chi$LL, has been considered in literature different mechanisms: from the coupling
with metallic gates used to confine
the electron gas \cite{Cazalilla06} to the coupling with electromagnetic environment \cite{Sassetti94,CastroNeto97,Safi04} or with other
systems.\cite{Shytov98, Levitov01}
In this paper we will discuss the consequences caused by the joint presence of $1/f$ noise and dissipation mainly because
those effects are unavoidable in order to get a realistic description of the physical systems.
A relevant advantage of the proposed mechanism
is its robustness to the disorder dominated phase, a key feature in order to apply the model to edge states with counter-propagating channels,
such as $\nu=2/3$ and $\nu=5/2$. The aim of this paper is to present a detailed analysis of this fact and its applicability to real cases.
The paper is organized as follow. In Sec.\ref{laughlin} we consider the Laughlin sequence.
Using this paradigmatic example we introduce the notations and the general methods that we will use later for the composite edge case, which is the
main issue of the paper. The effects of the renormalization induced by the noisy environments and the possible
consequences on the QPC transport are discussed. In Sec.\ref{Jainseq} we analyze the effect of the noisy environment on the Jain
sequence limiting for simplicity only to the two-modes cases $\nu=2/5,2/3$. In particular for the co-propagating case $\nu=2/5$
we investigate how the scaling becomes non-universal and also dependent on the strength of the intra-edge Coulomb interaction
when the external noise is present. This is strongly different from the standard hierarchical result where the scaling
dimension is predicted to be independent from interaction between the modes.
Discussing the case of counter-propagating modes (i.e. $\nu=2/3$), we exploit this condition to get the disorder dominated phase in presence of a
noisy environment.
In Sec.\ref{fivehalf}
the properties of the anti-Pfaffian model for $\nu=5/2$
as a function of the strength of the $1/f$ noise are analyzed, showing the regions of the parameters space where the
elementary excitation with non-Abelian nature could dominate. We finally discuss the counterintuitive result that the external noise
could help the manipulation of non-Abelian excitation in the QPC geometry. Conclusions are summarized in Sec.\ref{Conclusions}.
\section{Laughlin sequence}
\label{laughlin}
Let us consider the edge states of a quantum Hall fluid, described in terms of $\chi$LL theories, and investigate the effect induced by the
joint presence of an external environment, $1/f$ noise and dissipation. We stress that due to the presence of $1/f$ noise we have to face
with an out-of-equilibrium problem, therefore in the following we will employ proper techniques, i.e. the Keldysh contour formalism.\cite{Schwinger61,Keldysh64, Rammerbook, Kamenev09}
\subsection{Model}
We start our analysis considering the Laughlin sequence \cite{Laughlin83} with filling factor $\nu=1/(2n+1)$, being $n \in \mathbb{N}$.
The Lagrangian density of the $\chi$LL for an infinite edge is described in terms of a single bosonic mode
\begin{equation}
\label{free_Laughlin}
\mathcal{L}_0=\frac{1}{4 \pi \nu} \partial_{x}\varphi ( -\partial_{t} -v \partial_{x} ) \varphi,
\end{equation}
where $\varphi$ is a right-moving field along the edge with propagation velocity $v$.
In view of dealing with an out-of-equilibrium system, we treat the problem in the Keldysh contour
formalism. According to the standard path integral
formulation, the non-equilibrium problem
on the doubled time contour is easily encoded
by introducing the bosonic field in the forward/backward time branch $\varphi^{\mathrm{f}/\mathrm{b}}$.
We refer the reader to Ref.~\onlinecite{Rammerbook} and
Ref.~\onlinecite{Kamenev09} for the general treatment of these issues, and to Ref.~\onlinecite{Martin05} for the applications of these
methods to the edge states of the quantum Hall effect.
It is useful to write $\varphi^{\mathrm{f}/\mathrm{b}}= (\varphi^{\mathrm{cl}}\pm \varphi^{\mathrm{q}})/\sqrt{2}$ where
$\varphi ^\mathrm{cl}$ ($\varphi ^\mathrm{q}$) represents the so called classical (quantum) component of the field.\cite{Kamenev09} In terms of
the classical-quantum basis the bosonic Green's functions (GFs) are enclosed in the matrix
\begin{equation}
\mathcal{G}_{a,b}(x, t)=-i\langle \varphi^{a}(x,t)\varphi^{b}(0,0)\rangle=\left(
\begin{array}{ccc}
G^{K}(x, t)& G^{R}(x, t)\\
G^{A}(x, t)& 0\\
\end{array}
\right),
\label{KeldyshGF}
\end{equation}
where $a, b=\mathrm{cl, q}$. With $G^{R}$, $G^{A}$ and $G^{K}$
the retarded, advanced and Keldysh GFs respectively.\cite{Kamenev09}
In terms of these fields and in Fourier transform, defined as
$\varphi^{\mathrm{cl/q}} (q, \omega)=\int dx dt\ e^{i(\omega t-q x)}\ \varphi^{\mathrm{cl/q}} (x,t)$, the free Keldysh action,
deduced from Eq. (\ref{free_Laughlin}), reads
\begin{equation}
\label{KeldyshAction}
\mathcal{S}_{0}= \frac{1}{2}\sum_{q\neq0,\omega}\
(\Phi^*(q, \omega))^T\cdot \mathcal{G}_{0}^{-1}(q, \omega)\cdot\Phi(q, \omega),
\end{equation}
with the vector $\Phi=(\varphi^{\mathrm{cl}}, \varphi^{\mathrm{q}})^T$.
The matrix kernel of the action
is\cite{Kamenev09}
\begin{equation}
\label{chiLLGF}
\mathcal{G}^{-1}_{0}(q, \omega)=\frac{q}{2\pi\nu}\left[
\begin{array}{cccc}
0&(\omega-i\epsilon)- vq\\
(\omega+i\epsilon)- vq & 2iq\epsilon \mathrm{sgn}(\omega)
\end{array}
\right],
\end{equation}
where $\epsilon\to0$ is the standard regularization factor\cite{Note1} and the top-left $0$ component corresponds to the standard
continuum limit of the
Keldysh action.\cite{Kamenev09} Inverting the Kernel matrix and taking the cl-q component $(\mathcal{G}_{0})_{\mathrm{cl},\mathrm{q}}$
we get, according to Eq. (\ref{KeldyshGF}), the retarded GF for the free bosonic fields
\begin{equation}
\label{ChiralRetardedGF}
G_{0}^{\mathrm{R}}(q, \omega)=\left(\frac{2\pi\nu}{q}\right)\frac{1}{(\omega+i\epsilon)- vq}
\end{equation}
and analogously for the advanced GF, $G_{0}^{\mathrm{A}}=(\mathcal{G}_{0})_{\mathrm{q},\mathrm{cl}}$. From the linear response theory
one can show that the current along the Hall bar is given by $I=\nu \mathfrak{g}_0 V_H$
with $V_H$ the Hall potential and $\mathfrak{g}_0=e^2/h$ the quantum of conductance.\cite{Kane95}
We discuss now the influence
of $1/f$ noise term at low energies.\cite{Paladino02, Muller06} This contribution can be described in terms of a classical stochastic external
potential $f(x,t)$
that describes the effective interaction of the edge with the localized trapped
charges. The correlation function of the external force
is $K(q, \omega)=\langle f^{*}(q, \omega) f(q, \omega) \rangle=F/|\omega |$ where $F$ is the strength of the noise. For simplicity the noise
is assumed $\delta$-correlated in space, a natural assumption for short-range impurities in the low energy/long wavelength limit. Note
that the presence of this time dependent external force brings the system out-of-equilibrium.
The external gaussian random force $f(x,t)$ couples directly with the electron density
$\rho= \partial_{x}\varphi/(2\pi)$
of the $\chi$LL.
In the Keldysh formalism this interaction is \cite{Note1b}
\begin{equation}
\mathcal{S}_{f,\varphi}=\!\!\int\!\! dt \left(\mathcal{L}_{f,\varphi^{\mathrm{f}}}-\mathcal{L}_{f,\varphi^{\mathrm{b}}}\right)
=\sqrt{2} \sum_{q, \omega} (i q)
f^*(q, \omega) \varphi^\mathrm{q}(q, \omega)
\end{equation}
with $\mathcal{L}_{f,\varphi}\propto f(x,t)\partial_{x} \varphi(x,t)$ and where, in the first identity, we write the Keldysh action in terms of the
fields $\varphi^{\mathrm{f/b}}$ and, in the second one, in terms of
quantum component $\varphi^{\mathrm{q}}$ only. This result is standard in the Keldysh formalism and comes directly from the fact that
a purely classical external force couples \textit{only} with the quantum component $\varphi^{\mathrm{q}}$ of the field.\cite{Kamenev09}
The total $1/f$ effective Keldysh action\cite{Rammerbook} $\mathcal{S}_{1/f}$ for the bosonic field $ \varphi$ derives
from the functional integration
\begin{equation}
e^{i\mathcal{S}_{1/f}}=\int\!\! Df\ e^{-\frac{1}{2} \sum_{q, \omega} K^{-1}(q, \omega)|f(q, \omega)|^2}\ e^{i\mathcal{S}_{f,\varphi}}
\end{equation}
which averages on the disorder realizations of the noise potential $f(x,t)$.
The averaged effective Keldysh action $\mathcal{S}_{1/f}$ can be written in a form similar to Eq.(\ref{KeldyshAction}) with kernel\cite{DallaTorre10}
\begin{equation}
\label{G1f}
\mathcal{G}_{1/f}^{-1}(q, \omega)=\left[
\begin{array}{cccc}
0&0\\
0&+ 2 i q^{2}F/|\omega|
\end{array}
\right].
\end{equation}
Here, only the Keldysh q-q component is different from zero. This is a direct consequence of the fact that $1/f$ noise brings the system
out-of-equilibrium.
In this case the usual relations between retarded, advanced and Keldysh GFs, dictated by the fluctuation-dissipation
theorem, are no more valid.
The system, under the external driving force ($1/f$ noise), will reach a stationary condition only in presence of a dissipative mechanism
that drains the energy accumulated in the system.
Various mechanisms may introduce dissipation in the edge states.\cite{Cazalilla06,Sassetti94,CastroNeto97,Safi04,Shytov98, Levitov01} Here we limit
to consider the most general assumption with a dissipative term, induced by the external bath,
generalizing the Caldeira-Leggett approach to the $\chi$LL.\cite{Caldeira83, CastroNeto97}
The one dimensional edge mode can be coupled with oscillators through
the current density $j\propto\partial_t\varphi$ or the charge density $\rho\propto\partial_x\varphi$.
Hereafter, we will discuss the Keldysh action for a generic spectral function of the bath. Later on
we will focus only on the ohmic behavior.
The general Lagrangian density, which couples edge and harmonic oscillator modes, is
$\mathcal{L}_{\xi, \varphi}\propto \xi (x, \textbf{x}_{\perp}=0,t) \partial_\mu \varphi(x,t) $
where $\mu=t$ ($\mu=x$) describes the coupling with the current (charge) density.
The field $\xi(x,\mathbf{x}_\perp,t)$ represents a bath of oscillators with extra spatial degrees of freedom $\mathbf{x}_{\perp}$ orthogonal
to the 1D system.\cite{CastroNeto97}
Integrating out the harmonic bath degrees of
freedom it is easy to obtain the usual Matsubara euclidean effective action\cite{Kamenev09} ($\beta=(k_B T)^{-1}$)
\begin{equation}
S^{\mathrm{E}}_{\mathrm{diss}}=\frac{1}{2\beta}\sum_{q, i\omega_n} \mathcal{D}^{-1}(q, i\omega_n) |\varphi(q, i\omega_n)|^2
\label{Mats}
\end{equation}
with $\omega_n=2\pi n/\beta$. The spectral function $\mathcal{D}^{-1}(q, i\omega_n)$ encodes all the dynamical information about the external
bath and the coupling mechanism.\cite{CastroNeto97} In the Keldysh contour formalism this dissipative
contribution $\mathcal{S}_{\mathrm{diss}}$
can be written as in Eq.(\ref{KeldyshAction}) with kernel \cite{Kamenev09}
\begin{equation}
\mathcal{G}_{\mathrm{diss}}^{-1}(q, \omega)=
\left[
\begin{array}{cccc}
0&[\mathcal{D}^{A}(q, \omega)]^{-1}\\\
[\mathcal{D}^{R}(q, \omega)]^{-1}& [\mathcal{D}^{-1}(q, \omega)]^K
\end{array}
\right],
\label{Gdiss}
\end{equation}
$[\mathcal{D}^{R/A}(\omega,q)]^{-1}$ being the retarded/advanced analytic continuation of the spectral function $\mathcal{D}^{-1}(q, i\omega_n)$.
In this case the Keldysh component of the dissipation is computed by using the fluctuation-dissipation theorem\cite{Kamenev09}
\begin{equation}
[\mathcal{D}^{-1}]^K=([\mathcal{D}^{R}]^{-1}-[\mathcal{D}^{A}]^{-1})\mathrm{coth}(\beta\omega/2)
\end{equation}
that must be satisfied by a bath in thermal equilibrium.\cite{Kamenev09, DallaTorre10}
As stated before, in this paper we will only consider a specific type of dissipation, the ohmic one. The form of the bath spectral function
for such case is $\mathcal{D}^{-1}(q, i\omega_n)=\gamma|\omega_n|$, with $\gamma$ the friction coefficient.
At zero temperature the Keldysh kernel becomes
\begin{equation}
\label{Gdiss}
\mathcal{G}_{\mathrm{diss}}^{-1}(q, \omega)=\left[
\begin{array}{cccc}
0&-i\gamma\omega\\
+i\gamma\omega&+2i\gamma |\omega|
\end{array}
\right].
\end{equation}
Finally, the total Keldysh action is
$\mathcal{S}_{\mathrm{tot}}=\mathcal{S}_{0}+\mathcal{S}_{1/f}+\mathcal{S}_{\mathrm{diss}}$
with total kernel for $\varphi$
\begin{equation}
\mathcal{G}^{-1}=\mathcal{G}_{0}^{-1}+\mathcal{G}_{1/f}^{-1}+\mathcal{G}_{\mathrm{diss}}^{-1}
\end{equation}
(cf. Eq.(\ref{chiLLGF}), Eq.(\ref{G1f}) and Eq.(\ref{Gdiss})).
Inverting the kernel, the non-equilibrium Keldysh GFs at zero temperatures read
\begin{equation}
\label{Gtot}
\frac{\mathcal{G}}{2\pi\nu}=\left[
\begin{array}{cccc}
-\frac{2 i \tilde{\gamma} |\omega| (1+ (\tilde{F}/\tilde{\gamma}) q^2/\omega^2)}{q^{2}(\omega- vq)^{2} +\tilde{\gamma}^{2} \omega^{2}} &(q(\omega- vq)+ i \tilde{\gamma} \omega)^{-1}\\
(q(\omega- vq) - i \tilde{\gamma} \omega)^{-1}&0
\end{array}
\right]
\end{equation}
where $\tilde{F}=2\pi\nu F$ is the rescaled strength of the noise and $\tilde{\gamma}=2\pi\nu\gamma$ the friction coefficient. The regularization
factor $\epsilon$ in Eq.({\ref{chiLLGF}) is suppressed because the causal structure is already guaranteed by the dissipative contribution.\cite{Note2}
It is worth to underline that both the $1/f$ noise and the dissipation terms are relevant perturbations in the Renormalization Group (RG) sense with
massive coupling constants, namely $\mathrm{dim}[\tilde{F}]=\mathrm{dim}[\tilde{\gamma}]=1$ ($\mathrm{dim}[...] $ indicates the canonical
mass dimension).
The relevance of these terms will completely spoil the scale invariance property that characterize the standard $\chi$LL theory
for the edge states. Fortunately, one can demonstrate that for a noisy environment only \emph{weakly} coupled with the
edge, i.e. $\tilde{\gamma}, \tilde{F}\rightarrow 0$, but with the ratio $\tilde{F}/\tilde{\gamma}$ constant \cite{DallaTorre10} the scale invariance is
preserved. Indeed in this case, the combined action of the two environmental effects leads only to a {\emph {marginal}}
perturbation of the
theory. Consequently the conductance in the Hall bar will be quantized. Indeed, in the limit discussed before,
with $\tilde{\gamma}\to0$, it is easy to verify that the retarded (advanced) GF $G^R(t)$ ($G^A(t)$), namely the
anti-transform of the off-diagonal entry
$\mathcal{G}_{\mathrm{cl,q}}$ ($\mathcal{G}_{\mathrm{q,cl}}$) of the matrix in Eq.(\ref{Gtot}), coincides with the results obtained from the
retarded (advanced) GFs of the
free theory\cite{Wen95,Wenbook} given Eq.(\ref{ChiralRetardedGF}). Therefore the linear response shows that a weakly coupled noisy
environment does not modify the
conductance of the system with respect to the free $\chi$LL theory.
From the Keldysh GF $G^K=(\mathcal{G})_{\mathrm{cl,cl}}$ (anti-transform in time of the left-top entry of Eq.(\ref{Gtot})), we can define
the bosonic correlation function $\tilde{G}^{K}(t)=G^{K}(t)-G^{K}(0)$ with
\begin{equation}
\tilde{G}^{K}(t)=i\nu g \ln \left[1+ \omega_{\mathrm{c}}^2t^2\right]
\label{GK}
\end{equation}
where $\omega_{\mathrm{c}}=v/a$, with $a$ a finite length cut-off, and
\begin{equation}
g=\left(1+ \frac{\tilde{F}}{v^2\tilde{\gamma}}\right).
\label{g}
\end{equation}
Comparing Eq.(\ref{GK}) with the same
quantity calculated from the free $\chi$LL theory described in Eq.(\ref{chiLLGF}), i.e.
$\tilde{G}_0^{K}(t)= i\nu \ln \left[1+ \omega_{\mathrm{c}}^2t^2\right]$, we see that the functional dependence remains exactly the same, but
with an additional renormalization factor $g$.
\subsection{Scaling dimension renormalizations}
The above result leads to extremely important physical consequences. As a remarkable example we can consider a generic
$m$-agglomerate quasiparticle (qp) annihilation operator in the bosonized form \cite{Wen95}
\begin{equation}
\Psi^{(m)}(x)= \frac{e^{i m \varphi(x)}}{\sqrt{2\pi a}}
\end{equation}
and the two point greater/lesser GFs
\begin{equation}
\label{gtrlessGF}
\mathsf{C}_{m}^{>}(t)=\langle \Psi^{(m)}(t)\Psi^{(m)\dagger}(0)\rangle=-\mathsf{C}_{m}^{<}(-t).\\
\end{equation}
These quantities determine the tunneling densities of states of edges and consequently the
transport properties in a QPC geometry. They can be expressed in terms of the bosonic correlation
function \cite{Gutman10} $\tilde{G}^{>}(t)=G^{>}(t)-G^{>}(0)$ with\cite{Rammerbook} $G^{>}(t)=(G^{K}(t)+G^{R}(t)-G^{A}(t))/2$ and retarded,
advanced and Keldysh GFs
obtained from Eq.(\ref{Gtot}). At zero temperature one then has\cite{Gutman10,Mitra11}
\begin{equation}
\label{greaterGF}
\mathsf{C}_{m}^{>}(g,t)=e^{im^2\tilde{G}^{>}(t)}=
\left[\frac{1}{1+\omega_c^2 t^2}\right]^{\frac{m^2\nu g}{2}} e^{-im^2\nu\phi(t)}
\end{equation}
where
\begin{equation}
\phi(t)=\tan^{-1}\left[\frac{\omega_c t}{\sqrt{1+\omega_c^2 t^2}}\right]\underset{\omega_c\to\infty}\to\frac{\pi}{2}\mathrm{sgn}(t).
\end{equation}
From the comparison of the previous expressions with the results obtained for the free $\chi$LL,
it is possible to see that the renormalization factor $g$ only influences
the absolute value of the GF. We explicitly indicate the peculiar functional dependence on $g$ in left hand term of Eq.(\ref{greaterGF}).
The phase instead, as expected, is not affected being related to the universal statistical properties of the excitations.
We define now the scaling dimension $\Delta(m)$
of the $m$-agglomerate operator $\Psi^{(m)}(x)$ as the long-time behavior of the
two-point GF $|\mathsf{C}^{\gtrless}_{m}(t)|\underset{t\to\infty}\approx|t|^{-2\Delta(m)}$.
This quantity is
\begin{equation}
\label{ScalingLaughlin}
\Delta(m)=g\Delta_0(m)=g\nu\frac{m^{2}}{2},
\end{equation}
note that the scaling of the raw theory $\Delta_0(m)$ is renormalized by the factor $g$.
This result induces a modification of the power-law behavior of the transport properties, with respect to the free (unrenormalized) case.
For simplicity we calculate only the single quasiparticle (single-qp) contribution to the back-scattering current, the most dominant one in the Laughlin
sequence, for the weak-backscattering regime. Notice that, for the Laughlin model, the renormalization mechanism cannot affect the relevance
of the excitations. We will see that for the models with composite edges this won't be in general the case.
We model the QPC in terms of a local tunneling term at $x=0$ between the right- ($R$) and left-($L$) moving edges
such as $H_T=\mathbf{t} \Psi_R^{(1)} \Psi_L^{(1)\dagger}+\mathrm{H.c}$.\cite{Sassetti94,Cuniberti96,Braggio01,Cavaliere04} We also assume that
the edge are affected by different environments and consequently they may have different renormalization parameters $g_{R/L}$ for the right-moving
edge ($R$) and the left-moving one ($L$).
The average current at zero temperature, at the lowest order in the tunneling, reads ($\hbar=1$)
\begin{equation}
\label{IB}
\langle I_B\rangle=e^*\left(\frac{|\mathbf{t}|}{2\pi a}\right)^2\int_{-\infty}^{+\infty}\!\!\!d t\ e^{i E t} \mathsf{C}^>(g_R,t)\mathsf{C}^<(g_L,-t)
\end{equation}
where $E=e^* V$ is the energy involved in the tunneling, with $V$ the bias and $e^*=\nu e$ the single-qp charge.
From Eq.(\ref{greaterGF}) one has
\begin{equation}
\mathsf{C}^>(g_R,t)\mathsf{C}^<(g_L,-t)=\left(\frac{1}{1-i\omega_c t}\right)^{\nu(\bar{g}-1)}\left(\frac{1}{1+i\omega_c t}\right)^{\nu(\bar{g}+1)}
\end{equation}
with $\bar{g}=(g_R+g_L)/2$. From this result one can calculate the expression
of the current at zero temperature in Eq.(\ref{IB}) obtaining
\begin{equation}
\langle I_B\rangle=e^*\theta(E)\left(\frac{|\mathbf{t}|^2}{a^2\omega_c}\right)\frac{(E/\omega_c)^{2\nu\bar{g}-1}}{\Gamma[\nu( \bar{g}-1)]
\Gamma[\nu(\bar{g}+1)]}\ \mathcal{N}
\end{equation}
where $\Gamma[x]$ is the Gamma function and $\mathcal{N}=\ _2F_1[\nu(\bar{g}-1),1-\nu(\bar{g}+1),1+\nu(\bar{g}-1),-1]$ is a constant with
$_2F_1[a,b,c,z]$ the hypergeometric function.\cite{Chang03}
The power-law behavior of the back-scattering current at zero temperature is therefore $\langle I_B\rangle\propto V^{2\nu \bar{g}-1}$
with renormalized exponent $\nu \bar{g} $.
In the following we will always assume that the renormalization phenomena affects identically the right and left edges.
A generalization to the case of different couplings can be done straightforwardly.
Note that in Eq.(\ref{g}) the strength of the renormalization can take any value $g\geq 1$. The
same formula suggests that, for fixed environmental contribution ($\tilde{F}/\tilde{\gamma}$ constant), the renormalization
would be typically stronger for slow propagating modes, due to the explicit dependence on the inverse of the squared mode velocity
in the expression.
This renormalization could reach also high values with important modifications of the power-law behavior of transport
properties.\cite{Chung03, Roddaro03, Roddaro04}
As we have mentioned before, other mechanisms could explain the same renormalization of the
exponents.\cite{Rosenow02,Khlebnikov06,Yang03,Mandal02,Papa04,Lal08,Overbosch09,Chevallier10}
However, some of these
mechanism (such as the
coupling with phonons) contains intrinsic limitation on the strength renormalization, differently from our
model where the only real limitation is the request that $g \geq 1$. We will see, in the next sections, that this model can be simply
generalized to more complex fractional quantum Hall states, such as composite edge states and, more importantly, it also reveals robust
to the presence of disorder.
\section{Composite edges: Jain sequence}
\label{Jainseq}
\subsection{Model}
\label{modelJain}
Here, we focus on the effects of the out-of-equilibrium noise source in the case of multichannel edge states. The prototype of these Hall states is
represented by the Jain sequence \cite{Jain89} with filling factor $\nu = p/ (2n p + 1)$, being $n \in \mathbb{N}$ and $p \in \mathbb{Z}$.
Following the hierarchical construction \cite{Wenbook}, one has one charged bosonic mode, analogous to the one described for the Laughlin
sequence, and $|p|-1$ additional neutral modes which propagate either in the same direction ($p > 0$) or in opposite one ($p < 0$). For simplicity
we restrict the discussion only to the case of two edge modes ($|p|=2$), underlying the differences
between co-propagating modes ($p > 0$, $\nu = 2/5$) and counter-propagating ones ($p < 0$, $\nu = 2/3$).\cite{Wenbook}
The edge states in the former case are described in terms of two co-propagating bosonic charged fields with different filling
factors $\nu_1 = 1/3$ and $\nu_2 = 1/15$, such as $\nu= \nu_1 + \nu_2 = 2/5$, while in the latter case the bosonic fields, with $\nu_1 = 1$ and
$\nu_2 = 1/3$ respectively, propagate in opposite directions leading to $\nu= \nu_1 - \nu_2 = 2/3$.
The Lagrangian densities are
\begin{equation}
\mathcal{L}_\zeta= \sum_{j=1,2}\frac{-1}{4 \pi \nu_j} \partial_{x} \varphi_{j} \left((\zeta)^{j+1}\partial_{t} \varphi_{j}+v_{j} \partial_{x} \varphi_{j}\right)
\label{Phi0}
\end{equation}
where $\zeta = \pm$ indicates the co-propagating ($\zeta=+$) or counter-propagating ($\zeta=-$) case, and $v_1$, $v_2$ are the velocities
of the modes and effectively contain the information on intra-edge interactions. The fields commutation
relations are $[\varphi_j(x),\varphi_k(x')]=i\delta_{jk} \eta_k\nu_k\mathrm{sgn}(x-x')$
where $\eta_k=(\zeta)^{k+1}$ is related to the direction of propagation of the fields ($j,k = 1, 2$).
The two modes are close to each other and interact via the density-density coupling (inter-edge interaction)
\begin{equation}
\label{coupling}
\mathcal{L}_{12}=\frac{v_{12}}{2\pi\nu_{12}} \partial_{x} \varphi_{1} \partial_{x} \varphi_{2},
\end{equation}
with strength $v_{12}$, where $\nu_{12}=\sqrt{\nu_1 \nu_2}$.
Notice that, $v_1$, $v_2$ and $v_{12}$ are non-universal parameters related to the intra- and inter-channel interaction strengths.
Under the reasonable assumption that the confinement potential is sufficiently smooth the two modes are localized in slightly different
positions. Therefore one reasonably assumes that they effectively "feel" different noisy environments. Indeed, in general, the trapped charges in
the substrate beneath the Hall bar affect the $\varphi_1$ and $\varphi_2$ modes in a different way. One can introduce two distinct
$1/f$ noise fields $f_{1/2}(x,t)$ operating on the edge, with noise strength $F_{1/2}$ and spectrum $K_{1/2}(\omega,q)=F_{1/2}/|\omega|$
respectively. We will also consider two different ohmic dissipations with friction coefficients $\gamma_{1/2}$. Also in this case the total Keldysh action
can be written in a form analogous to Eq.(\ref{KeldyshAction}), but in terms of the four components vector
$\Phi=(\varphi_1^{\mathrm{cl}},\varphi_2^{\mathrm{cl}}, \varphi_1^{\mathrm{q}}, \varphi_2^{\mathrm{q}})^T$, due to the presence of the two fields $\varphi_{1/2}$.
The Keldysh kernel now reads
\begin{widetext}
\begin{equation}
\label{KernelComposite}
\mathcal{G}^{-1}_{\zeta}=\frac{1}{2\pi}\left[
\begin{array}{cccc}
0&0&\nu_{1}^{-1} \left[q\left(\omega-v_{1} q\right)-i \tilde{\gamma}_1 \omega\right]&\nu_{12}^{-1} v_{12} q^{2}\\
0&0&\nu_{12}^{-1} v_{12} q^{2} &\nu_{2}^{-1}\left[ q\left(\zeta\omega-v_{2}q\right)-i \tilde{\gamma}_2 \omega\right]\\
\nu_{1}^{-1} \left[q\left(\omega-v_{1} q\right)+i \tilde{\gamma}_1 \omega\right] &\nu_{12}^{-1} v_{12} q^{2} & +2 i \nu_{1}^{-1} (\tilde{\gamma}_1 |\omega| +\tilde{F}_1 q^2/ |\omega|)&0\\
\nu_{12}^{-1} v_{12} q^{2}&\nu_{2}^{-1}\left[q\left(\zeta\omega-v_{2}q\right) +i \tilde{\gamma}_2 \omega\right]&0&+2 i \nu_2^{-1}(\tilde{\gamma}_2 |\omega|+\tilde{F}_2 q^2/|\omega|)
\end{array}
\right]
\end{equation}
\end{widetext}
where $\tilde{\gamma}_{i}=2\pi\nu_{i}\gamma_{i}$ and $\tilde{F}_{i}=2\pi\nu_{i}F_{i}$ with $i=1,2$.
\subsection{Interaction effects}
We will discuss now how the presence of a noisy environment could affect
the renormalization of the $\chi$LL exponents in composite edge systems.
We will start this analysis by considering the presence of interaction between the channels described in Eq.(\ref{coupling}). Let us start
to discuss the case of co-propagating channels such as
$\nu=2/5$. For such cases, indeed, the clean system, i.e. with no static disorder along the edge, properly describes
the physics of the edge states. In the next subsection we will investigate the renormalization effects for $\nu=2/3$ (counter-propagating modes)
assuming instead the presence of the static disorder, which is crucial in the equilibration process between counter-propagating modes.
The Keldysh action kernel for $\nu=2/5$ is given by $\mathcal{G}^{-1}_{+}$ in Eq.(\ref{KernelComposite}) with $\nu_1 =1/3$ and $\nu_2=1/15$.
To better analyze the problem it is useful to make a rescaling $\varSigma$ and a
rotation $\mathcal{R}(\theta)$ of the fields $\varphi_{1/2}$. In the new basis $\varphi'_{1/2}$ we have
\begin{equation}
\label{rotation}
\binom{\varphi_1}{\varphi_2}
=\underbrace{\left[\begin{array}{cc}\sqrt{\nu_1}&0\\0&\sqrt{\nu_2}\end{array}\right]}_{\varSigma}\cdot\underbrace{\left[\begin{array}{cc}\cos(\theta)&\sin(\theta)\\-\sin(\theta)&\cos(\theta)\end{array}\right]}_{\mathcal{R}(\theta)}
\cdot\binom{\varphi'_1}{\varphi'_2}
\end{equation}
where the angle $\theta$ satisfies $\tan(2\theta)=2v_{12}/(v_1-v_2)$. The new fields eigenmodes $\varphi'_{1/2}$ are
decoupled, with respect to the density-density interaction, and have different velocities
\begin{equation}
\label{newvelocities}
v'_{1,2}=\left(\frac{v_1+v_2\pm\sqrt{(v_1-v_2)^{2}+4v_{12}^2}}{2}\right).
\end{equation}
From the above relations we get a criterium, called \emph{stability}, which requires these velocities to be always positive.\cite{Kane95}
This is reflected in a constraint $v_{12}^2\leq v_1 v_2$ between the intra- and inter-mode couplings.\\
At the same time the dissipative and $1/f$ terms acquire off-diagonal contributions due to the transformation in Eq.(\ref{rotation}).
In particular, we can see what happen to those terms by focusing at
the $2\times2$ bottom left block-matrix, that coincides with the q-cl component of the Kernel in Eq.(\ref{KernelComposite}).
In the new $\varphi'$ basis it becomes
\begin{equation}
(\mathcal{G}^{\prime -1})_{\mathrm{q,cl}}
=
\frac{q}{2\pi}\left[\begin{array}{cc} \omega-v'_{1} q &
0
\\
0
& \omega-v'_{2}q \end{array}\right]
+(\mathcal{G}'^{-1}_{\mathrm{diss}})_{\mathrm{q,cl}}
\end{equation}
where the dissipative contribution is no more diagonal and reads
\begin{equation}
\label{rotgamma}
(\mathcal{G}'^{-1}_{\mathrm{diss}})_{\mathrm{q,cl}}=
\mathcal{R}^{T}(\theta)\cdot\left[\begin{array}{cc} i \tilde{\gamma}_1\omega &0\\0&i \tilde{\gamma}_2\omega\end{array}\right]\cdot\mathcal{R}(\theta)
\end{equation}
with the rotation $\mathcal{R}(\theta)$ defined in Eq.(\ref{rotation}).
Notice that the advanced component ($2\times2$ top right block-matrix) $(\mathcal{G}^{\prime -1})_{\mathrm{cl,q}}$ can be easily
derived from the previous result by
complex conjugation, $(\mathcal{G}^{\prime -1}(\omega,q))_{\mathrm{cl,q}}=\left((\mathcal{G}^{\prime -1}(\omega,q))_{\mathrm{q,cl}}\right)^*$.
The Keldysh component $(\mathcal{G}'^{-1})_{\mathrm{q,q}}$ of Eq.(\ref{KernelComposite}) ($2\times 2$ bottom right
block-matrix) transforms in the new basis according to Eq.(\ref{rotgamma})
\begin{equation}
\label{1fprime}
\frac{(\mathcal{G}'^{-1})_{q,q}}{2i|\omega|}=
\mathcal{R}^{T}(\theta)\cdot\left[\begin{array}{cc}
\tilde{\gamma}_1 +\tilde{F}_2 \frac{q^2}{|\omega|^2}&0\\0&
\tilde{\gamma}_2+\tilde{F}_2 \frac{q^2}{|\omega|^2}
\end{array}\right]\cdot\mathcal{R}(\theta)
\end{equation}
where the linear dependence on the coefficients
$\tilde{F}_1$, $\tilde{F}_2$ is now explicit.
In the limit of weak contribution of noise and dissipation (see Sec.\ref{laughlin}), i.e. $\tilde{\gamma}_{i},\tilde{F}_{i}\to0$ but keeping the ratios
$\tilde{F}_{i}/\tilde{\gamma}_{i}$ constant, it is possible to calculate all the Keldysh GFs following the same approach used for the Laughlin case.\\
To simplify the discussion we consider only the case where the effective friction coefficients of the dissipative contributions are the
same, $\tilde{\gamma}_1=\tilde{\gamma}_2=\tilde{\gamma}$, allowing only different strengths for the $1/f$ noise.\cite{Note5} This assumption
greatly simplify our discussion without affecting the key results.
For $\tilde{\gamma} \to 0$ one recovers again the standard form of the retarded/advanced GFs for composite edges. A direct consequence of
this fact is that the argument discussed in Ref.~\onlinecite{Kane95}, where the conductance of a multichannel edge of interacting
co-propagating modes is calculated using the retarded GFs, is still valid here. Therefore, also for the multichannel edge the Hall bar current is
correctly quantized $I=\nu \mathfrak{g}_0 V_H$, independent of the value of the inter-edge
interaction $v_{12}$.\cite{Note6}
The presence of an external noisy environment modifies the scaling dimension of the excitations. Now we analyze this point,
taking care of the presence of inter-edge coupling $v_{12}$. In particular
we will show that the scaling properties are no more universal and depend, in general, on inter-edge coupling. This result differentiates
from the standard theory \cite{Kane95} that predicts only universal scaling properties in the co-propagating case. It is important to stress
that this is a \emph{direct consequence} of the presence of a
noisy external environment. To better demonstrate this fact, we will consider now all the possible excitations and their scaling dimensions.
In the bosonized form a generic excitation is written in terms of a linear combination of the bosonic field $\varphi_{1/2}$ as
\begin{equation}
\label{Psialpha}
\Psi^{(\alpha_1,\alpha_2)}(x)\propto e^{i\left[\alpha_1\varphi_1(x)+\alpha_2\varphi_2(x)\right]}
\end{equation}
with $\alpha_{1/2}$ coefficients that determine the considered excitation.\cite{Ferraro08, Ferraro10a, Ferraro10c} Here, we just recall that
the charges of all the excitations are integer multiples of the fundamental single qp, e.g. for $\nu=2/5$, the fundamental charge is
$e^* = e/5$ ($e$ the
electron charge). The statistical properties of the excitations are directly connected to the values of $\alpha_{1}$ and $\alpha_2$
and the commutation
relations of the $\varphi_{1/2}$ fields\cite{Ferraro10a,Ferraro10c}, while the scaling dimensions depends additionally, as we will see, on the
presence of a noisy environment. The two-point correlation function of the operator is\cite{Gutman10}
\begin{equation}
\label{CompositeGreaterGF}
\mathsf{C}^>_{\alpha_1,\alpha_2}(t)=\langle \Psi^{(\alpha_1,\alpha_2)}(t) {\Psi^{(\alpha_1,\alpha_2)}}^\dagger(0)\rangle=e^{\sum_{i,j=1,2}\alpha_i\tilde{G}^>_{ij}(t)\alpha_j}
\end{equation}
where, in the second equality, we introduced the greater GFs $\tilde{G}^>_{jk}(t)=G^>_{jk}(t)-G^>_{jk}(0)$ such that
$G^>_{jk}(t)=-i\langle\varphi_j(t)\varphi_k(0)\rangle$ for the $\varphi_j$ fields with $j,k=1,2$.
In the new basis $\varphi'_{1/2}$, taking the limit of weak coupling with the environment $\tilde{\gamma},\tilde{F}_1,\tilde{F}_2\to 0$, the
Keldysh GFs read ($j,k=1,2$)
\begin{equation}
\label{Gkvarphiprime}
\tilde{G}'^K_{jk}(t)=i\delta_{jk}g'_j \ln [1+\omega'^2_{c,j}t^2]
\end{equation}
where the cut-offs are $\omega'_{c,j}=v'_j/a$ with $j=1,2$. The "mixed" terms $\tilde{G}'^K_{jk}(t)$ with $j\neq k$ vanish due to the
assumption $\tilde{\gamma}_1=\tilde{\gamma}_2=\tilde{\gamma}$ on the dissipative friction coefficients. The renormalization
coefficients for the two normal modes
are now
\begin{equation}
\label{gprime}
g'_j=\left(1+\frac{\tilde{F}_+-(-)^{j}\tilde{F}_-\cos(2\theta)}{2 v_j'^2 \tilde{\gamma}}\right)
\end{equation}
with $\tilde{F}_\pm=\tilde{F}_1\pm\tilde{F}_2$ and the mode velocities $v_j'^2$ given in Eq.(\ref{newvelocities}).
Notice that the renormalization parameters depend on the coupling strength $v_{12}$ and the
mode velocities, through the angle $\theta$. This appears a quite natural generalization of the result given in Eq.(\ref{g}).
From this result one can calculate $\tilde{G}'^>_{jk}(t)$ with the same procedure used for the Laughlin sequence, in the
form of Eq.(\ref{greaterGF}), after the proper replacement of the renormalization parameter $\nu g \to g'_1,g'_2$ and of the
cut-offs $\omega_c \to v'_{1}/a,v'_{2}/a$. The greater GF in Eq.(\ref{CompositeGreaterGF}) can be expressed as
\begin{equation}
\tilde{\mathbf{G}}^>(t)=\varSigma\cdot\mathcal{R}(\theta)\cdot\tilde{\mathbf{G}}'^>(t)\cdot\mathcal{R}^T(\theta)\cdot\varSigma
\end{equation}
where we used the compact matrix notation with $\tilde{G}^>_{jk}(t)=(\tilde{\mathbf{G}}^>(t))_{jk}$ with the rescaling matrix $\Sigma$ and
the rotation matrix $\mathcal{R}(\theta)$ defined in Eq.(\ref{rotation}).
From the long-time behavior of the two-point correlation function of Eq.(\ref{CompositeGreaterGF}) one can calculate the scaling
dimension $\Delta^{(\alpha_1,\alpha_2)}$ of the $\Psi^{(\alpha_1,\alpha_2)}(x)$ operator
\begin{eqnarray}
\Delta^{(\alpha_1,\alpha_2)}=&\frac{1}{2}\left\{\nu_1\ \alpha_1^2\left[g'_1\cos^2(\theta)+g'_2\sin^2(\theta)\right]\nonumber\right.\\
&+\nu_2\ \alpha_2^2\left[g'_2\cos^2(\theta)+g'_1\sin^2(\theta)\right]
\nonumber
\\
&\left.+\sqrt{\nu_1\nu_2}\ \alpha_1\alpha_2\sin(2\theta)\left[g'_1-g'_2\right]\right\}
\end{eqnarray}
where we see an explicit dependence on the coupling $v_{12}$ and the mode velocities $v_{1,2}$, via the angle $\theta$. Note that, in the
absence of environmental
effects ($g'_{1}=g'_{2}=1$), the scaling dimensions reduce to the standard
$\Delta_0^{(\alpha_1,\alpha_2)}=(\nu_1\alpha_1^2+\nu_2\alpha_2^2)/2$ obtained for hierarchical theories.
Furthermore, if the renormalizations of the normal modes are exactly the same $g'_1=g'_2=g$ the scaling dimension become independent of the
angle $\theta$ (i.e. the coupling $v_{12}$) with $\Delta^{(\alpha_1,\alpha_2)}=g\Delta_0^{(\alpha_1,\alpha_2)}$.
In conclusion, we showed that the scaling of a generic excitation in presence of a noisy environment is influenced by the strength of the
coupling $v_{12}$ even for co-propagating modes. This strongly differs from
the standard result\cite{Kane95} where the
scaling is \emph{independent} from the coupling. This fact shows that scaling dimension in the presence of $1/f$ noise are, in general, \emph{no
more universal}, i.e. determined only by the coefficients $(\alpha_1,\alpha_2)$, the filling factor $\nu$ and the model of composite edge we consider.
As a remarkable consequence, in the presence of environmental effects, the relevance between the excitations could differ from
the raw theory. We have already discussed this phenomenology in relations to experimental observations.\cite{Ferraro08,Ferraro10a}}
We will see now that the most important advantage of the presented model is its robustness with respect to static
impurity disorder along the edge. Indeed, all the results up to now are
essentially based on the assumption of a clean edge, without any contribution coming from static disorder. We will also see that including such
contribution the role of inter-mode interactions will be less important but the scaling dimensions will be still affected by renormalizations
effects due to the noisy environment.
\subsection{Disorder effects}
A more realistic discussion of edge states in real samples requires the inclusion of static disorder along the edge.\cite{Note4}
The disorder plays a fundamental role to recover the proper quantization of the Hall
conductance for counter-propagating modes. We refer the reader to the seminal paper
by Kane and Fisher in Ref.~\onlinecite{Kane95} for a detailed discussion of the disorder dominated phase in the hierarchical theories.
In the following we will analyze the case of $\nu=2/3$ where the two channels are counter-propagating. The discussions about the disorder
effects in the composite edge, presented here, can be also generalized
to the whole Jain sequence. In the next section we adapt the
argument even to the $\nu=5/2$ state.
The Keldysh action for the multichannel edge state at $\nu=2/3$ under the influence of $1/f$ noise and dissipation is given by the kernel
$\mathcal{G}_-^{-1}$ of Eq.(\ref{KernelComposite}) with $\nu_1=1$ and $\nu_2=1/3$. The effect of static disorder on the $\chi$LL channels
can be naturally described by adding two more terms to the action.
The first one describes the coupling of two static disorder potential profiles $V_i(x)$
with the charge densities $\rho_i(x)=\partial_x\varphi_i(x)/(2\pi)$ of the two channel $\varphi_i$ composing the edge
$\nu=2/3$ with $i=1,2$. The Lagrangian of this term, which affects locally the two channels is $\mathcal{L}_{V, \varphi}\propto \sum_{i}V_i(x)\partial_{x} \varphi_i(x)$. These forward scattering terms can be easily eliminated from the action
by a simple redefinition of the $\varphi_i(x)$ fields and will be neglected in the following.\cite{Kane95}
The second term describes the effect of the disorder in terms of impurity scattering, i.e. how the disorder potential mediates
the electron transfer between the two
counter-propagating modes. These tunneling terms have the main effect to equilibrate the two channels when they are at different potentials restoring the proper value of the quantized conductance.\cite{Kane95}
This random tunneling term is
\begin{equation}
\mathcal{L}_{\rm{rdm}}= \xi(x) e^{i\left[\varphi_{1}(x)+3\varphi_{2}(x)\right]}+h.c.,
\label{Lagrangian_random}
\end{equation}
with $\xi(x)$ a complex random tunneling amplitude. This process leads to the destruction of an electron into the $\nu=1$ channel ($e^{i\varphi_1(x)}$) and its creation ($e^{i3\varphi_{2}(x)}$) into the $\nu=1/3$ one and viceversa.\cite{Kane95, Kane94} For simplicity it is assumed $\xi(x)$ as a Gaussian random variable $\delta$-correlated in space satisfying
\begin{equation}
\label{disorderLagrangian}
\langle\xi^{*}(x)\xi(y)\rangle_{\rm{ens}}=W \delta(x-y),
\end{equation}
where $\langle...\rangle_{\rm{ens}}$ indicates the ensemble average over the realizations of disorder.
To further analyze the disorder terms, it is convenient to express the system in the basis $\varphi'_{1/2}(x)$
that diagonalizes the problem with respect to inter-edge coupling. We follow essentially the same
procedure used for the $\nu=2/5$ case. The transformation now is given by the composition of a rescaling $\varSigma$ and a Lorentz boost
$\mathcal{B}(\chi)$ with rapidity $\chi$, instead of the standard rotation used for co-propagating modes.
The relation between the old fields $\varphi_{1,2}$ with the new ones $\varphi'_{1,2}$ is
\begin{equation}
\label{Lorentzboost}
\binom{\varphi_1}{\varphi_2}
=\underbrace{\left[\begin{array}{cc}\sqrt{\nu_1}&0\\0&\sqrt{\nu_2}\end{array}\right]}_{\varSigma}\cdot\underbrace{\left[\begin{array}{cc}\cosh(\chi)&\sinh(\chi)\\\sinh(\chi)&\cosh(\chi)\end{array}\right]}_{\mathcal{B}(\chi)}
\cdot\binom{\varphi'_1}{\varphi'_2}
\end{equation}
where $\tanh(2\chi)=-2v_{12}/(v_1+v_2)$.\\
Note that we can calculate the scaling dimension for a generic qp operator $\Psi^{(\alpha_1,\alpha_2)}(x)$ defined in Eq.(\ref{Psialpha})
following the same steps as in the previous section. It is also useful, without loosing generality, to assume $\tilde{\gamma}_1=\tilde{\gamma}_2=\tilde{\gamma}$ but keeping as free parameters the strengths of the $1/f$ noise $\tilde{F}_{1/2}$.
We find the scaling dimension
\begin{eqnarray}
\Delta^{(\alpha_1,\alpha_2)}=&\frac{1}{2}\left\{\nu_1\ \alpha_1^2\left[g'_1\cosh^2(\chi)+g'_2\sinh^2(\chi)\right]\right.\nonumber\\
&+\nu_2\ \alpha_2^2\left[g'_2\cosh^2(\chi)+g'_1\sinh^2(\chi)\right]
\nonumber
\\
&\left.+\sqrt{\nu_1\nu_2}\ \alpha_1\alpha_2\sinh(2\chi)\left[g'_1+g'_2\right]\right\}
\label{Deltaa1a2}
\end{eqnarray}
where the contribution of the Lorentz boost is explicit in the terms $\cosh^2(\chi)$ and $\sinh^2(\chi)$. The renormalization factors of the new modes $g'_i$ are now ($j=1,2$)
\begin{equation}
\label{gboost}
g'_j=\left(1+\frac{\tilde{F}_+-(-)^{j}\tilde{F}_- \mathrm{sech}(2\chi)}{2 v_j'^2 \tilde{\gamma}}\right),
\end{equation}
where the eigenmode velocities $v'_i$ are in Eq.(\ref{newvelocities}) and $\tilde{F}_\pm$ are defined after Eq.(\ref{gprime}).
Note that a direct comparison with the co-propagating expression shows that, in the counter-propagating case, the $\mathrm{sech}(2\chi)$ takes the
role of the $\cos(2\theta)$ of Eq.(\ref{gprime}), but the form of the renormalization factors remains essentially the same.
We address now the role of the disorder terms, starting with the inspection of the scaling $\Delta_{\mathcal{O}}=\Delta^{(1,3)}$ of the tunneling
disorder operator $\mathcal{O}\propto e^{i\left[\varphi_{1}(x)+3\varphi_{2}(x)\right]}$ introduced in Eq.(\ref{Lagrangian_random}).
One can demonstrate that the flow equation for the inter-edge disorder strength $W$ in Eq.(\ref{disorderLagrangian})
is \cite{Kane94, Kane95, Giamarchi88}
\begin{equation}
\label{RGflow}
\frac{d W}{d l}=(3-2 \Delta_{\mathcal{O}})W.
\end{equation}
In particular, if $\Delta_{\mathcal{O}}<3/2$ the disorder is a relevant contribution driving the system in the so called disorder-dominated phase.
For such phase the Hall bar conductance $\mathfrak{g}$ is universal (i.e. independent of the environment and the intra- and
inter-mode couplings) and is properly quantized at the value $\mathfrak{g}=\nu \mathfrak{g}_0$.\cite{Kane94}
All the discussions of quantum Hall states with counter-propagating modes are typically done assuming the system being exactly in this phase. \\
On the contrary, if the scaling of the electron tunneling is $\Delta_{\mathcal{O}}>3/2$, the disorder is irrelevant and, at the fixed
point, the conductance is no more universal depending on the intra- and inter-edge interactions.\cite{Kane94}
In general the environmental
effects, through the parameters
$\tilde{F}_{i}/\tilde{\gamma}_{i}$ and the intra- and inter-mode couplings, could affect the scaling $\Delta_{\mathcal{O}}$ making the
discussion quite cumbersome. As a simple check, we first need to recover the standard result obtained in absence of any environmental effects,
namely the ratios $\tilde{F}_1/\tilde{\gamma},\tilde{F}_2/\tilde{\gamma}\to 0$. From Eq.(\ref{Deltaa1a2}) and the definition of the
operator $\mathcal{O}$ one gets
\begin{equation}
\label{Delta023}
\Delta^0_{\mathcal{O}}=\lim_{\tilde{F}_i\to 0}\Delta^{(1,3)}=2\frac{v_1+v_2-\sqrt{3} v_{12}}{\sqrt{(v_1+v_2)^2-4 v^2_{12}}}
\end{equation}
which coincides with the result of Kane, Fisher and Polchinski.\cite{Kane94}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth,]{figure1.ps}
\caption{Colored areas represent the disorder dominated phase for $\nu=2/3$ in the parameters space $(v_2/v_1,v_{12}/v_1)$ for different
strength of the noise. In light blue the case without the noisy environment \cite{Kane94}, i.e. $\Delta^0_{\mathcal{O}}<3/2$ (see in the text).
This area is limited by the \emph{stability} criterium (solid line) and the
two dashed lines representing respectively the maximum/minimum value of the ratio $v_{12}/v_1$ to get the disordered phase
$\Delta^0_{\mathcal{O}}<3/2$.
Other colored areas represent $\Delta_{\mathcal{O}}<3/2$ with successive reduction of the disordered phase due to the increasing of noise
strength $\tilde{F}_1/(v^2_1\tilde{\gamma})=0$ (lighter blue), $ 0.1, 0.5, 1$ (darker blue) with $\tilde{F}_2=0$. }
\label{fig:Fig1}
\end{figure}
It is convenient now to measure the velocities in units of $v_1$. For the values of the couplings where the \emph{stability} criterium
$v_{12}^2\leq v_1 v_2$ is satisfied
we can determine when $\Delta^0_{\mathcal{O}}<3/2$. In Fig.~\ref{fig:Fig1} we report, in the plane
$(v_2/v_1,v_{12}/v_1)$, the regions where this condition is fulfilled. The area is delimited by the \emph{stability} (solid black) curve and
two (dashed black) lines representing respectively the maximum/minimum value of $v_{12}/v_1$ compatible with the disorder dominated phase. These
two lines are given by $v_{12}/v_1=(4\sqrt{3}/21\pm\sqrt{5}/14)(1+v_2/v_1)$.\\
Now we can evaluate how this area changes under the presence of a noisy environment. One could expect, in analogy with the
renormalizations induced by coupling with phonons\cite{Rosenow02}, that the effects of an external environment
lead always to an enhancement of the scaling dimension
$\Delta_{\mathcal{O}}\geq \Delta^0_{\mathcal{O}}$.
A direct consequence of this fact would be the progressive reduction of the region of existence of the disorder dominated phase. This is
explicit in the figure
where we calculated the regions where $\Delta_{\mathcal{O}}<3/2$, using Eq.(\ref{Deltaa1a2})
varying $\tilde{F}_i/(v_1^2\tilde{\gamma})=0, 0.1, 0.2, 0.3$ for a fixed ratio $\tilde{F}_2/(v_1^2\tilde{\gamma})=0$. Notice that for very strong
noise the disordered dominated phase could be completely washed out.
We conclude this discussion observing that, for moderate noise strength, the disorder dominated phase is still present even if the condition
on the inter- and intra-mode coupling are modified.
Our analysis generalizes some of the results of Ref.~\onlinecite{Kane94} in presence of a noisy environment.\\
The robustness of the proposed model for the renormalization of the exponents in
presence of disorder along the edge is one of the most important result of this paper.
We will discuss now quantitatively how renormalizations are affected by noise intensity. In the disorder dominated phase the system
naturally decouples in charged and neutral contributions.\cite{Kane94, Kane95} Therefore, it is convenient to change the basis from
the original $\varphi_{1/2}$
to the charged $\varphi_\rho$ and neutral fields $\varphi_{\sigma}$
\begin{equation}
\binom{\varphi_{\rho}}{\varphi_{\sigma}}=\frac{1}{\sqrt{2}}\binom{\sqrt{3}(\varphi_{1}+\varphi_{2})}{3\varphi_{2}+\varphi_{1}}
\label{charged_neutral}
\end{equation}
as obtained from the transformation in Eq.(\ref{Lorentzboost}) with $\tanh(\chi^*)=-\sqrt{1/3}$.\cite{Note7}
The action is expressed in the form of Eq.(\ref{KernelComposite}), but with the propagation velocities $v_i$ with index $i=\rho,\sigma,\rho\sigma$ given by
\begin{equation}
\left[\begin{array}{cc}v_\rho&v_{\rho\sigma}\\v_{\rho\sigma}&v_\sigma\end{array}\right]=\mathcal{B}(\chi^*)^T\cdot\left[\begin{array}{cc}v_1&v_{12}\\v_{12}&v_2\end{array}\right]\cdot\mathcal{B}(\chi^*)
\end{equation}
with $\mathcal{B}(\chi^*)$ the Lorentz boost in Eq.(\ref{Lorentzboost}). The same transformation defines the noise strengths $\tilde{F}_i$, in the
new basis as
\begin{equation}
\label{Frhosigma}
\left[\begin{array}{cc}\tilde{F}_\rho&\tilde{F}_{\rho\sigma}\\\tilde{F}_{\rho\sigma}&\tilde{F}_\sigma\end{array}\right]
=\frac{1}{2}\left[\begin{array}{cc}3\tilde{F}_1+\tilde{F}_2&\sqrt{3}(\tilde{F}_1+\tilde{F}_2)\\\sqrt{3}(\tilde{F}_1+\tilde{F}_2) &\tilde{F}_1+3\tilde{F}_2\end{array}\right]
\end{equation}
in terms of the coefficients $\tilde{F}_{1/2}$ of Eq.(\ref{KernelComposite}). An equivalent transformation can be written for the friction
coefficients of the dissipative bath, with the introduction of the quantities
$\tilde{\gamma}_{\rho},\tilde{\gamma}_\sigma,\tilde{\gamma}_{\rho\sigma}$, as linear combinations of $\tilde{\gamma}_{1/2}$.
The off-diagonal terms of the action containing
$v_{\rho\sigma}$ are irrelevant in the RG sense and can be neglected at the fixed point. This was clearly shown in
Ref.~\onlinecite{Kane95}. In the limit of weak coupling such that $\tilde{\gamma}_i,\tilde{F}_i\to0$, but keeping constant
the ratios $\tilde{F}_i/\tilde{\gamma}_i$, with $i=1,2$, the environmental contributions are marginal in the RG sense.\cite{DallaTorre11}
This shows that, at the fixed point of the disorder
dominated phase, we could safely neglect the residual coupling between charged and neutral modes but we \emph{have to} include the
noisy environmental contributions. In the following we will take
$v_{\rho\sigma}=0$ keeping explicitly the dissipative and $1/f$ noise terms in account.
We observe that, in the case of the coupling with 1D phonon $\varphi_{\mathrm{ph}}$ modes\cite{Rosenow02}, one has also a
term analogous to the one proportional to $v_{\rho \sigma}$ discussed above. The canonical mass dimension of the phonons in $1+1$ dimensions is the same
of a chiral bosonic field $\mathrm{dim}[\varphi_{\mathrm{ph}}]=\mathrm{dim}[\varphi_{1,2}]$. As a natural consequence, this coupling term
becomes RG irrelevant in the disordered dominated fixed point as already discussed. This shows that, even
if the coupling with phonons could in principle generate renormalizations of the scaling exponent, in the disorder dominated phase
the phonons are effectively decoupled from the system and their renormalization effects do not survive against disorder. This indicates that our
model is qualitatively different and
presents concrete advantages in comparison with other mechanisms especially for all those cases where counter-propagating modes are
present and, consequently, the disorder dominated phase has to be considered.
We can now evaluate the GFs along the same line followed in the previous section for
$\nu=2/5$. Also in this case we assume $\tilde{\gamma}_1=\tilde{\gamma}_2=\tilde{\gamma}$ and consider the strengths of the $1/f$ noises $\tilde{F}_{1/2}$ as free parameters.
In the limit of $\tilde{\gamma}\to 0$ the retarded/advanced GFs are exactly the same as the ones in Ref.~\onlinecite{Kane95} and consequently
the edge conductance returns the appropriate
quantized value of $\mathfrak{g}=\nu \mathfrak{g}_0$.
For the Keldysh GF contributions of the charged and neutral fields we get, in the disordered phase, a result identical to Eq.(\ref{Gkvarphiprime}),
where the only non-zero GF are the
$\tilde{G}^K_{\rho\rho}$ and $\tilde{G}^K_{\sigma\sigma}$. These are characterized by the cut-off energies $\omega_i=v_i/a$
with $i=\rho,\sigma$ and by
renormalization parameters
\begin{equation}
\label{renormalizNC}
g_{\rho}=\left(1+\frac{\tilde{F}_{\rho}}{2v^2_\rho\tilde{\gamma}}\right)\qquad g_{\sigma}=
\left(1+\frac{\tilde{F}_{\sigma}}{2v^2_\sigma\tilde{\gamma}}\right).
\end{equation}
They coincide with Eq.(\ref{gboost}) choosing $\chi=\chi^*$ and using the definition of Eq.(\ref{Frhosigma}).
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figure2.ps}
\caption{Renormalization parameters $g_\rho$ (black curves) and $g_\sigma$ (gray curves) as a function of the ratio $v_\sigma/v_\rho$. Different
linestyles correspond to different noise strength $\tilde{F}_1/(v_1^2 \tilde{\gamma})=1$ (dashed), $2$ (dot-dashed), $3$ (dotted) having kept fixed
the value of $\tilde{F}_2/(v_1^2 \tilde{\gamma})=0.1$}
\label{fig:Fig2}
\end{figure}
Their values depend on the noise strength $\tilde{F}_i$, the dissipation $\tilde{\gamma}$ and
on the neutral and charged mode velocities $v_i$. The Fig.~\ref{fig:Fig2} shows how the charged renormalization parameter (black curves) and the
neutral one (gray curves) depend on the ratio $v_\sigma/v_\rho$ for different values of the $1/f$ noise strengths. At fixed noise
strength and increasing the ratio $v_\sigma/v_\rho$ the charged mode renormalization rises while the
neutral one decreases.
This behavior is directly connected to the dependence on the inverse of the squared mode velocities of Eq.(\ref{renormalizNC}).
When the modes velocities become small the renormalization parameters increase rapidly.
Interestingly, it is also possible to obtain the counterintuitive condition $g_\sigma>g_\rho$. Following physical intuition indeed, it appears
natural to assume that neutral bosonic modes are less coupled with the environment with respect to the charged ones.
Nevertheless, this intuition failsbecause the neutral bosonic modes, in the composite edges, derives from a particle-hole combination
between the two
modes and are strongly affected by the differences in the noisy environments (differential mode) where the charge
modes instead are affected
by the common mode only.
In conclusion the dependence of the renormalization parameters on the noise strengths
$\tilde{F}_{1/2}$ (see Eq.(\ref{Frhosigma})) guarantees the possibility to get very high renormalization values for
almost any values of the velocities ratio $v_\rho/v_\sigma$. Note that these high values are sometimes
necessary to fully explain the experimental observations.\cite{Ferraro10b}
We conclude this section showing that, rescaling the two fields $\varphi_c=\sqrt{2/3}\varphi_\rho$ and $\varphi_n=\sqrt{2}\varphi_\sigma$,
it is possible to write all qp operators as\cite{Kane94, Ferraro10b}
\begin{equation}
\Psi^{(m,l)}\propto e^{i[ (m/2) \varphi_{c}+(l/2) \varphi_{n}]},
\end{equation}
with the coefficients $m,l\in \mathbb{Z}$ and with the same parity. These operators destroy an $m$-agglomerate, namely an excitation
with charge $m e^{*}$ being $e^{*}=e/3$ the minimal charge allowed by the model. Their scaling dimensions become
\begin{equation}
\Delta(m,l)=
\frac{1}{2}\left[\left(\frac{2}{3}\right) g_{\rho}\left(\frac{m}{2}\right)^2 +2g_{\sigma} \left(\frac{l}{2}\right)^{2}\right],
\end{equation}
where the $g_\rho$ and $g_\sigma$ renormalize the charge and neutral sectors of the excitation separately. Obviously we recover the scaling
dimension reported in the literature \cite{Wen95, Ferraro10b} for $g_{\rho}=g_{\sigma}=1$. The last formula shows that, in the
disorder dominated phase, the presence of a noisy environment naturally leads to different
renormalizations for the neutral and charged modes. Consequence of this fact is the possibility to change the relevance of the
excitations and, indeed when $g_\rho,g_\sigma\neq1$, this could happen due to environmental effects we are discussing. This phenomenology
could have a deep impact on transport properties of the QPC especially in the
weak backscattering regime where the dominant excitations are different from the electrons. The possibilities opened by our model
for composite edges could therefore explain the extremely rich phenomenology observed in QPC transport at low temperatures
for these systems. In Ref.~\onlinecite{Ferraro10a} and
Ref.~\onlinecite{Ferraro10b}
we have discussed in detail the experiments on noise and transport in QPC for $\nu=2/3$. To fully match the theory with the data the presence of
the renormalization parameters $g_\rho, g_\sigma\geq1$ was sufficient. Here, we have shown that a noisy environment can be considered a
proper renormalization mechanisms, robust to unavoidable disorder effects.
\section{Composite edges: the $\nu=5/2$ case}
\label{fivehalf}
\subsection{Anti-Pfaffian model}
Another relevant example of composite edge state is represented by $\nu=5/2$. Possible descriptions have been proposed for this state
predicting both Abelian \cite{Halperin93} and non-Abelian \cite{Moore91, Fendley07, Lee07, Levin07} statistical properties for the elementary
excitations. Particularly interesting is the so called anti-Pfaffian model \cite{Lee07, Levin07}, supporting non-Abelian statistics,
that seems to be indicated by experimental
evidences as a proper description for this state.\cite{Radu08, Bid10} According to this model, the edge states are described as a narrow
region at $\nu=3$ with nearby a Pfaffian edge of holes with $\nu=1/2$.\cite{Lee07} Assuming the second Landau level as the ``vacuum'', the edge is
modeled in terms of a single $\nu=1$ bosonic branch $\varphi_1$ and a counter-propagating $\nu=1/2$ Pfaffian branch \cite{Fendley07}, composed
by a bosonic mode $\varphi_2$ and a Majorana fermion $\mbox{\boldmath $\psi$}$.
The Lagrangian for the free system is $\mathcal{L}_0=\mathcal{L}_-+\mathcal{L}_{12}+\mathcal{L}_{\psi}$
where the bosonic contribution $\mathcal{L}_-$ and $\mathcal{L}_{12}$ are given in Eq.(\ref{Phi0}) and Eq.(\ref{coupling}) respectively
with $\nu_1=1$ and $\nu_2=1/2$. The Lagrangian describing the free evolution of the Majorana fermion \mbox{\boldmath $\psi$}\ in the Ising sector is
\begin{equation}
\mathcal{L}_{\psi}= i \mbox{\boldmath $\psi$} \left( -\partial_{t} +v_{\psi} \partial_{x}\right) \mbox{\boldmath $\psi$},
\end{equation}
with propagation velocity $v_{\psi}$. In addition to the free theory we have the coupling of the bosonic modes $\varphi_{1/2}$ with the
different noisy environments ($1/f$ noise and the dissipative ohmic bath) surrounding them. Also in this case we consider coupling with
the $1/f$ noise strengths $\tilde{F}_i$ and friction coefficients of the dissipative baths $\tilde{\gamma}_1=\tilde{\gamma}_2=\tilde{\gamma}$. Note that the noise
and the dissipation couple electrostatically with $\varphi_{1/2}$ but not with the neutral Ising sector of the theory that is decoupled
from electromagnetic environment. The total Keldysh bosonic action coupled with the noisy environment has the
kernel $\mathcal{G}_-^{-1}$ of Eq.(\ref{KernelComposite}) with $\nu_1=1$ and $\nu_2=1/2$. The Lagrangian density is completed
by the addition of the disorder term
\begin{equation}
\mathcal{L}_{rdm}= \xi(x) \mbox{\boldmath $\psi$}(x)\ e^{i \left[\varphi_{1}(x)+2 \varphi_{2}(x)\right]}+h.c.
\end{equation}
that describes the random electron tunneling processes
which equilibrate the two branches, in fully analogy with $\nu=2/3$.
The complex random coefficients $\xi(x)$, Gaussian distributed, satisfy also Eq.(\ref{disorderLagrangian}).
This unavoidable contribution guarantees that the appropriate value of the Hall resistance is recovered in the disorder dominated phase.
The RG flow equation for the disorder term $W$ is the same of Eq.(\ref{RGflow}), with $\Delta_{\mathcal{O}}$ the scaling dimension
of the tunneling operator $\mathcal{O}\propto \mbox{\boldmath $\psi$} e^{i \left[\varphi_{1}+2 \varphi_{2}\right]}$. Consequently, investigating when
$\Delta_{\mathcal{O}}< 3/2$, it identifies the conditions for a disorder dominated phase of $\nu=5/2$.\cite{Lee07, Levin07}
We firstly identify the conditions of the existence of the disorder
dominated phase as a function of the couplings $v_i$ and the noisy environment. The
scaling is
\begin{equation}
\Delta_{\mathcal{O}}=\frac{1}{2}+\Delta^{(1,2)},
\end{equation}
where the first term in the sum represents the contribution of the Ising sector (Majorana fermion) and the second one
is the bosonic contribution of Eq.(\ref{Deltaa1a2}) with $\nu_1=1$ and $\nu_2=1/2$. The bosonic contribution can
be indeed derived
following exactly the same steps considered for $\nu=2/3$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figure3.ps}
\caption{Colored areas represent the disorder dominated phase for $\nu=5/2$ in the
parameter space $(v_2/v_1,v_{12}/v_1)$ for different strength of the noise. In light blue the case without the noisy
environment \cite{Kane94}, $\Delta^0_{\mathcal{O}}<3/2$ (see in the text).
This area is limited by the \emph{stability} condition (solid line) and the
two dashed lines representing respectively the maximum/minimum value of the ratio $v_{12}/v_1$ to get the disordered phase. Other colored area
represents the successive reduction of the disordered phase due the increasing of noise
strength $\tilde{F}_1/v^2_1\tilde{\gamma}= 0$(lighter blue), $0.1, 0.5, 1$(darker blue) keeping fixed $\tilde{F}_2=0$.}
\label{fig:Fig3}
\end{figure}
The scaling dimension in general depends on the renormalization
parameters $g'_{i}$, defined in Eq.(\ref{gboost}).
Without the noisy environment $g'_{1}=g'_{2}=1$ we then recover
\begin{equation}
\label{Delta052}
\Delta^0_{\mathcal{O}}=\lim_{\tilde{F}_i\to0} \Delta_{\mathcal{O}}=\frac{1}{2}\left(1+\frac{3 (v_1+v_2)-4\sqrt{2} v_{12}}{\sqrt{(v_1+v_2)^2-4 v^2_{12}}}
\right)
\end{equation}
that is the scaling dimension of the intra-edge electron tunneling reported in the literature.\cite{Levin07}
The region of existence of the disorder dominated phase ($\Delta_{\mathcal{O}}< 3/2$) is represented in Fig.~\ref{fig:Fig3} for
different velocities of the modes
$(v_2/v_1,v_{12}/v_1)$. The lines delimiting the area are the same discussed in the previous section: the
\emph{stability} condition (black solid curve) and the two lines (dashed black) that limit the values of $v_{12}/v_1$, i.e.
$v_{12}/v_1=(3\sqrt{2}\pm\sqrt{3})(1+v_2/v_1)/12$.
The discussion hereafter goes in parallel with what we have done for $\nu=2/3$. The noisy
environment will further restrict the set of values of intra- and inter-mode couplings where
the system is dominated by the disordered phase. In
the figure this is represented by the progressive reduction of the colored area: from the lighter blue to the darker blue when the
environmental noise increases. If the noise
becomes strong enough the disorder dominated phase could even disappear.
Also for $\nu=5/2$, at the fixed point of the disorder dominated phase, the system naturally decouples into a charged bosonic
mode $\varphi_{c}=\varphi_{1}+\varphi_{2}$ with velocity $v_{\mathrm{\rho}}$ and a neutral counter-propagating sector (one
bosonic mode $\varphi_{n}=\varphi_{1}+2 \varphi_{2}$ and one Majorana fermion $\mbox{\boldmath $\psi$}$ with the same velocity
$v_{\mathrm{\sigma}}$).\cite{Levin07,Lee07} It is again natural
to introduce the charged $g_\rho$ and neutral $g_\sigma$ renormalization parameters, according to Eq. (\ref{renormalizNC}).
The renormalizations can be very strong, for realistic values of the ratio $v_\sigma/v_\rho$, and satisfying the condition $g_\sigma>g_\rho$
as we anticipated for $\nu=2/3$. In conclusion our model explains the values of the renormalizations proposed in Ref.~\onlinecite{Carrega11} for $\nu=5/2$.\cite{Dolev10}
\subsection{Agglomerate dominance}
Here, we will discuss the effects of noisy environment on the relevance of excitations in the anti-Pfaffian model.
Using the charged and neutral modes basis one
can express the more general qp operator as \cite{Levin07, Carrega11}
\begin{equation}
\Psi^{(\mbox{\scriptsize \boldmath $\chi$},m,l)}\propto \mbox{ \boldmath $\chi$}(x) e^{i \left[(m/2) \varphi_{c}+(l/2) \varphi_{n}\right]},
\end{equation}
where the integer coefficients $m,l$ and the Ising field operator \mbox{ \boldmath $\chi$}$(x)$ define the admissible excitations. In the Ising sector
\mbox{ \boldmath $\chi$}$(x)$ can be $\mathbf{I}$
(identity operator), \mbox{ \boldmath $\psi$}$(x)$ (Majorana fermion) or \mbox{ \boldmath $\sigma$}$(x)$ (spin operator).
The monodromy condition force $m$, $l$ to be even integers for \mbox{ \boldmath $\chi$}$=\mathbf{I},$\mbox{\boldmath $\psi$} and odd integers
for \mbox{ \boldmath $\chi$}$=$\mbox{ \boldmath $\sigma$}. The charge associated to the above operator is $(m/4)e$ depending on the
charged mode contribution only, while its scaling dimension is
\begin{equation}
\Delta(\mbox{\boldmath $\chi$}, m, l)=\frac{1}{2}\delta_{\mbox{\scriptsize \boldmath $\chi$}}+\frac{g_{\mathrm{\rho}}}{16} m^{2}+\frac{g_{\sigma}}{8}l^{2}\,,
\label{Delta}
\end{equation}
with $\delta_{\mathbf{I}}=0$, $\delta_{\mbox{\scriptsize \boldmath $\psi$}}=1$ and $\delta_{\mbox{\scriptsize \boldmath $\sigma$}}=1/8$.
Note that, as stated before, the contribution of the Ising sector to the scaling
dimension is not affected by any renormalization.
We adopted the previous formula to predict the scaling dimension and the transport properties in the
experiment done by the Heiblum group at Weizmann.\cite{Carrega11}. We found a
good agreement with the experiment where, at the lowest temperatures, the dominant excitation is the $2$-agglomerate $2 e^*=e/2$, that is
described by the operator $\Psi^{(\mathbf{I},2,0)}$.
Our explanation clarifies why the anomalous
increasing of the effective charge is observed at extremely low temperatures.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{figure4.ps}
\caption{Three dimensional picture of the region (colored) where the inequality $g_\rho<(1+2g_\sigma)/3$ is fulfilled, namely where the $2$-agglomerate dominates with respect to the single-qp. On the vertical axis is reported the ratio $v_{\sigma}/v_{\rho}$, while in the plane the renormalization factors $\tilde{F}_1/(\tilde{\gamma}v_1^2)$ and $\tilde{F}_2/(\tilde{\gamma}v_1^2)$. The plane $v_{\sigma}/v_{\rho}=1$ is highlighted with a thick line.}
\label{fig:Fig4}
\end{figure}
Let's see now when the noise environmental parameters determine the dominance of the $2$-agglomerate. In general,
the excitation with the lowest scaling dimension dominates the properties in the low energy sector.
Without any renormalization ($g_\rho=g_\sigma=1$) the scaling dimensions
are exactly the same: $\Delta(\mathbf{I}, 2, 0)=\Delta(\mbox{\boldmath $\sigma$}, 1, \pm1)=1/4$.
So \emph{only} the presence of environmental renormalization will determine the dominance of an excitation over the other.
The effect of renormalizations
are indeed crucial to make the single-qp excitation - described by the operator $\Psi^{(\mbox{\scriptsize \boldmath $\sigma$},1,\pm1)}$
with charge $e^*=e/4$ - less relevant than the agglomerate.\cite{Note8}
The agglomerate with charge $e/2$ will be
dominant over the single-qp if $\Delta(\mathbf{I}, 2, 0)<\Delta(\mbox{\boldmath $\sigma$}, 1, \pm1)$ so we get the
inequalities\cite{Carrega11}
\begin{equation}
g_\rho<\frac{1+2g_\sigma}{3}.
\end{equation}
In Fig.~\ref{fig:Fig4} we show the domain where the agglomerate $\Psi^{(\mathbf{I},2,0)}$ is a dominant
over the single-qp $\Psi^{(\mbox{\scriptsize \boldmath $\sigma$},1,\pm1)}$.
We see that agglomerates are more easily dominant for $v_\sigma/v_\rho<1$ - the regime probably valid in the real samples.
Conversely, when $v_\sigma/v_\rho\gtrsim 2$,
the dominance of agglomerate is possible only at very small values of noise strength as show by the peak in the figure.
Note that, for small values of $\tilde{F}_2$ and strong enough $\tilde{F}_1$, it is also possible to have the dominance of the
single-qp for $v_\sigma/v_\rho<1$.
In the last figure this corresponds to the volume underneath the plane identified by the thick line that coincide with $v_\sigma/v_\rho=1$.
In conclusion the dominance of the agglomerate is quite common and only in the case of neutral modes velocity similar to the charged modes
\emph{and} in the presence of a noisy environment the the single-qp could be \emph{more} dominant. Anyway, we want to mention that the
excitation that dominates at very low
energy, potentially, couldn't be also the
dominant at higher energies, i.e. by increasing bias or temperature. This explains why the single-qp seems to be the
dominant charge carriers in measurements carried out at higher
temperatures\cite{Radu08,Dolev08}, as we discussed in more details in Ref.~\onlinecite{Carrega11}.
We conclude this section commenting on the need of a correct identification of the dominant excitations at low energy.
We recall that one of the most important properties of anti-Pfaffian (Pfaffian) states
for $\nu=5/2$ is the possibility to support excitations which satisfying non-Abelian statistics. Indeed, the single-qp is represented by the operator $\Psi^{(\mbox{\scriptsize \boldmath $\sigma$},1,\pm1)}$ that, due to the peculiar fusion rule in the Ising sector
\mbox{\boldmath $\sigma$} $\times$ \mbox{\boldmath $\sigma$}$=\mathbf{I}+$\mbox{\boldmath $\psi$} is intrinsically non-Abelian.
On the other hand the agglomerate is Abelian, being represented in term of the operator $\Psi^{(\mathbf{I},2,0)}$, i.e. with an identity operator $\mathbf{I}$ on the Ising sector.
Therefore, the dominance of the agglomerates with respect to the single-qp could have important consequence on the real possibility to manipulate
non-Abelian excitations with the help of QPC setups. The hope to encode topological protected quantum computation protocols in this system
may be potentially affected by this issue. Counterintuitively, given the previous analysis, a noisy environment could become a helpful resource
leading, in some regions of the parameter space, to the dominance of the non-Abelian single-qp.
In perspective we like to mention that our approach and also many of the discussed results could be recovered also
for other models, such as the Pfaffian or the Abelian 331.\cite{Halperin93,Lin12} This shows that, for an large class of models of edges states, the renormalization phenomena induced
by the noisy environment can play an important role influencing the physics in the low energy regime.
\section{Conclusions}
\label{Conclusions}
We have presented a renormalization mechanism of the tunneling exponent in the $\chi$LL theories for edge states, based
on the joint effects of the weak coupling with out-of-equilibrium $1/f$ noise and dissipation. The model is very general and can be applied
to many different states, such as in the Jain sequence or even the anti-Pfaffian model for $\nu=5/2$.
Considering the paradigmatic case of the Laughlin sequence, we showed how a noisy environment could modify the Luttinger exponents.
The direct consequences of this renormalization are derived for the QPC current in the weak-backscattering regime, mainly focusing on
the effects on the power-law behavior as a function of bias.
In the Jain sequence, and in particular for $\nu=2/5$, we have investigated how the scaling dimensions of the excitations are affected by the interplay between the inter-channel couplings and the noisy environment. Here, the possibility of a change in the dominance of the excitations is reported.
The rich phenomenology induced by these facts was already considered by us\cite{Ferraro08, Ferraro10a, Ferraro10c}, and we found
a good match with the experimental observations.\\
The case of counter-propagating modes has been analyzed in detail. We investigated how the noisy environment modifies the conditions of
stability of the disorder dominated phase. We demonstrated that, for moderate noise strength, the renormalization mechanism is robust
against disorder and remains valid at the fixed point of the disorder dominated phase. This is a crucial result, because of all the quantum Hall
edge theories with counter-propagating modes require the presence of static disorder to guarantee the equilibration along the edge and the
proper universal value of the quantum resistance experimentally observed.
This robustness makes our model a good candidate for a realistic renormalization mechanism of the Luttinger exponent while other models,
such as the coupling with 1D phonons or other bosonic baths, might not survive in presence of disorder.
In the last part of the paper we discussed the $\nu=5/2$ case considering the non-Abelian anti-Pfaffian model for the edge states. In analogy
with the previous analysis we studied the effect of external environments and their role on the disorder dominated phase.
Our proposal for the renormalization mechanism seems to be applicable to a plethora of cases giving a convincing and rather simple unified perspective.
Our results suggest that the values of the Luttinger exponents, that typically in the literature are related to universal features of the adopted theoretical models, have to be taken with care due to the presence of unavoidable noisy environments that can modify, even consistently, some of the predictions.
\section*{Acknowledgements.} We thank E. G. Dalla Torre for valuable discussions and acknowledge the support of the CNR STM 2010 program and the EU-FP7 via ITN-2008-234970 NANOCTM.
|
{
"timestamp": "2012-03-09T02:04:45",
"yymm": "1203",
"arxiv_id": "1203.1906",
"language": "en",
"url": "https://arxiv.org/abs/1203.1906"
}
|
\section{Introduction}
We take the one-dimensional moving singular sources equation
\begin{align}
\label{eq:heat_eq}
& u_t - u_{xx} = \sum_{i=0}^{q-1}F_i(t,x,u) \delta(x-\alpha_i(t)), \quad -\infty < x < \infty,~ t>0, \\
\label{eq:init_cond}
& u(x,0) = u_0(x), \quad -\infty <x<\infty,\\
\label{eq:bound_cond}
& u(x,t) \to 0 \quad \text{as} \quad |x| \to \infty, ~t>0.
\end{align}
as the model problem in this paper. Here $q>0$ is the number of singular
sources. The initial value $u_0(x)$ is taken to be continuous and
compatible with the boundary conditions, i.e. $u_0(x)\to 0$ as $|x|\to
\infty$. The local source functions $F_i(t, x, u)$
$(i=0,1,\ldots,q-1)$ might be given a priori or can be determined from
some additional constraints on the solution. The traveling sources
are located at $\alpha_i(t)$, $i=0,1,\ldots,q-1$. In general, their
velocities can be described by several ordinary differential equations
\begin{align}
\label{eq:source_eq}
\frac{\mathrm{d} \alpha_i}{\mathrm{d} t} = \psi_i ( t, \alpha_i(t), u ), \quad i = 0, 1, \ldots, q-1,
\end{align}
which are coupled with the solution $u$. We assume that the sources do
not intersect with each other during the time in consideration. This
model arises in many areas such as laser beams traveling problems
where $u$ is the temperature of the material \cite{kirk2002blow}, or
free-boundary solidification problems where $\alpha_i(t)$,
$i=0,1,\ldots,q-1$, are the moving interfaces between different phases
\cite{beyer1992analysis}.
It is well known that the solution of the model is continuous and
piecewise smooth \cite{li1997immersed}. However, the derivative of
the solution has a jump at each source due to the delta function
singularity on it, and the jump is given by \cite{ma2009moving}
\begin{align}
\label{eq:jump_ux}
[u_x]_{(\alpha_i(t), t)} = -F_i(t, \alpha_i(t), u(\alpha_i(t),t)), \quad \quad i = 0, 1, \ldots, q-1.
\end{align}
This leads the standard numerical methods, either finite difference
method or finite element method, might fail when crossing the
time-dependent source positions. Various approaches, have been used to
deal with the delta function singularity, such as the immerse boundary
(IB) method and the immerse interface method (IIM)
\cite{smereka2006numerical,beyer1992analysis,leveque1994immersed,
li1997immersed, kandilarov2003immersed, yang2009smoothing}. For the
IB method originally proposed by \cite{peskin1977numerical}, the delta
function is approximated by an appropriately chosen discrete delta
function. Beyer and LeVeque \cite{beyer1992analysis} studied various
cases of the model \eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} for the
IB method with $q=1$, and the source position $\alpha_i(t)$ being
priori specified. In contrast, the IIM first introduced by LeVeque
and Li \cite{leveque1994immersed} incorporates the known jumps of
solution or its derivatives into the finite difference scheme to
obtain a modified discretization scheme. Li \cite{li1997immersed}
developed an IIM numerical algorithm on the uniform mesh for the model
\eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} and \eqref{eq:source_eq}
with $q=1$.
The model \eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} and
\eqref{eq:source_eq} is more difficult to be solved when the source
function $F_i(t,x,u)$ is high nonlinearity. In this case, the
solution is always blow-up in some finite time $T>0$ if the sources
are stationary or move at sufficiently low speed, while blow-up will
be avoided if the sources move at sufficiently high speed (see
e.g. \cite{olmstead1997critical,kirk2000influence,kirk2002blow}). As
the solution evolves singularity, the uniform mesh method always
become computationally prohibitive. Hence, moving mesh method has to
be employed,
which is one of the most popular adaptive methods and have been
successfully used to investigate the blow-up phenomenon
\cite{budd1996moving,huang2008study}. In MMPDE's approaches, the
movement of the mesh is controlled by the moving mesh partial
differential equations (MMPDEs) based on the equidistribution
principle \cite{huang1994MMPDE}. Among these MMPDEs, MMPDE4, MMPDE5
and MMPDE6 are popular to use. Readers interested in the moving mesh
method and its applications can refer to the books
\cite{tang2007adaptive,Huang2011book}.
Recently, several papers have been devoted to moving mesh method for
the model \eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} with a priori
specified source position $\alpha_i(t)$ for $q=1$
\cite{zhu2010numerical,ma2009moving} and for $q>1$, in which the
sources move with the same speed \cite{zhu_moving,hu2011moving}. This
paper is a further study of \cite{hu2011moving} and \cite{hu2012zh}
for the model \eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} with general
movement of the sources, which do not intersect with each other during
time evolution. First, we choose a finite observed domain containing
all sources with appropriate boundary conditions, and divide it into
$q+1$ subdomains by the $q$ sources. Obviously, the sizes of the
subdomains are changed as the sources traveling. MMPDEs are applied on
each subdomain to obtain a local equidistributed mesh on it. Then the
underlying PDE \eqref{eq:heat_eq} is solved on the whole observed
domain with the mesh composed of the local mesh on each
subdomain. Taking the advantages of domain decomposition
\cite{Toselli2005book}, MMPDEs could be solved efficiently by parallel
computing. Moreover, It can be found that the computation of
$[\dot{u}]$ is avoided, thus the discretization scheme for the
underlying PDE becomes very simple. In addition, our method has an
expected second-order convergence in space.
The organization of the paper is as follows. In section
\ref{sec:DDMMM}, we introduce the moving mesh method in conjunction
with domain decomposition for the model problem. In section
\ref{sec:discretization}, the discretization schemes for the physical
problem will be derived in detail. In section \ref{sec:example},
several numerical examples are given to demonstrate the numerical
efficiency and accuracy of our method. The conclusions are presented
in the last section
\section{Moving mesh method in conjunction with domain decomposition}\label{sec:DDMMM}
In the last decade, moving mesh method in conjunction with a Schwarz
domain decomposition has been developed by Haynes and his co-workers
(see i.e. \cite{haynes2007schwarz, gander2012domain} and references
therein). And in this section, we will introduce a slight different
method, that is, moving mesh method in conjunction with a
non-overlapping domain decomposition.
Denote the observed domain by $[x_l,x_r]$ and assume it containing all
sources, that is,
$x_l<\alpha_0(t)<\alpha_1(t)<\cdots<\alpha_{q-1}(t)<x_r$. Here $x_l$,
$x_r$ are either constants or variables of $t$. Then, the observed
domain is divided into $q+1$ subdomains $[\alpha_{i-1},\alpha_{i}]$
$(i=0,1,\ldots,q)$ with $\alpha_{-1}=x_l$, $\alpha_{q}=x_r$, by the
$q$ sources respectively. Obviously, the sizes of the subdomains are
variables of $t$ too.
Let $x$ and $\xi$ denote physical and computational coordinates,
respectively. Without loss of generality we assume the computational
domain is $[0,1]$. Then an one-to-one coordinate transformation
between the observed domain $[x_l,x_r]$ and the computational domain
$[0,1]$ is defined by
\begin{equation}\label{eq:logical_map}
x = x(\xi,t), \quad \xi \in [0,1],
\end{equation}
with
\begin{equation*}
x(0,t)=x_l, \quad x(1,t) = x_r.
\end{equation*}
For a given uniform mesh, $\xi_j=\frac{j}{N}$, $j=0,1,\ldots,N$,
on the computational domain, the corresponding mesh on the
observed domain $[x_l,x_r]$ is
\begin{equation*}
x_l=x_0(t)<x_1(t)<\cdots<x_{N-1}(t)<x_N(t)=x_r.
\end{equation*}
In our method, the coordinate transformation
\eqref{eq:logical_map} is determined as a piecewise smooth
function. On each subdomain $[\alpha_{i-1},\alpha_{i}]$,
$i=0,1,\ldots,q$, it is the solution of an MMPDE which is derived
from the equidistribution principle. In the literature, the
following MMPDEs
\begin{equation}\label{eq:MMPDE4}
\frac{\partial}{\partial\xi} \bigg(M \frac{\partial{\dot{x}}}{\partial\xi}\bigg)
=-\frac{1}{\tau}\frac{\partial}{\partial\xi}
\bigg(M\frac{\partial x}{\partial \xi}\bigg),
\end{equation}
\begin{equation}\label{eq:MMPDE5}
-\dot{x}=-\frac{1}{\tau}\frac{\partial}{\partial\xi}
\bigg(M\frac{\partial x}{\partial \xi}\bigg),
\end{equation}
\begin{equation}\label{eq:MMPDE6}
\frac{\partial^2\dot{x}}{\partial\xi^2}=-\frac{1}{\tau}\frac{\partial}{\partial\xi}
\bigg(M\frac{\partial x}{\partial \xi}\bigg),
\end{equation}
which known as MMPDE4, MMPDE5 and MMPDE6, respectively, are
popularly used after they were originally established and analyzed
in \cite{huang1994MMPDE}. Here $M=M(x,t)$ is the monitor function
giving some measure of the solution error on the physical domain
and $\tau>0$ is a parameter representing a timescale for adjusting
the mesh toward equidistribution. In the asymptotic case
$t\to\infty$, the solution of MMPDE4, MMPDE5 and MMPDE6 would
satisfy the equidistribution principle, which is stated that
\cite{huang1994MMPDE}
\begin{align}
\label{eq:EP}
\frac{\partial}{\partial\xi}
\bigg(M\frac{\partial x}{\partial \xi}\bigg) = 0.
\end{align}
For more details about MMPDE, one can refer to
\cite{huang1994MMPDE} or the recent book \cite{Huang2011book}. In
this paper, MMPDE6 \eqref{eq:MMPDE6} with the boundary condition
\begin{align}
\label{eq:BV_MMPDE}
x(\xi_{j_{i-1}^s},t) = \alpha_{i-1}(t), \quad x(\xi_{j_i^s},t) = \alpha_i(t), \qquad i=0,1,\ldots,q,
\end{align}
is employed as an example to describe our moving mesh strategy in
conjunction with domain decomposition. Here $j_i^s$ is some fixed
index satisfying $0<j^s_i<N$. The resulting mesh, used to solve the
model problem on $[x_l,x_r]$, satisfies the property that a fixed mesh
point is located on each source during the time in consideration,
i.e. $x_{j^s_i}\equiv \alpha_i(t)$.
\begin{figure}[!htbp]
\centering
\small
\begin{tikzpicture}[scale = 0.75]
\matrix[row sep=5mm,column sep=5mm, top color=white, bottom color=white, scale=0.75]{
\node[startstop, text width=0.5\textwidth, scale=0.75, text centered] (start) {Given the old mesh $x_j^n$ on the observed domain $[x_l,x_r]$ and the corresponding solution on the mesh.};\\
\node[passprocess, scale=0.75] (monitor) {Compute the monitor function $M$ on the mesh.};\\
\node[passprocess, text width=0.5\textwidth, scale=0.75] (MMPDE) {Solve MMPDE6 \eqref{eq:MMPDE6} with the boundary condition \eqref{eq:BV_MMPDE} on each subdomain $[\alpha_{i-1},\alpha_i]$, $i=0,1,\ldots,q$.};\\
\node[startstop, text width=0.5\textwidth, scale=0.75, text centered] (end) {Combining the local equidistributed mesh on each subdomain to give the new mesh $x_j^{n+1}$ on $[x_l,x_r]$.}; \\
};
\begin{scope}[every path/.style=line]
\path (start) -- (monitor);
\path (monitor) -- (MMPDE);
\path (MMPDE) -- (end);
\end{scope}
\end{tikzpicture}
\caption{The moving mesh strategy in conjunction with domain
decomposition.}
\label{fig:mesh_strategy}
\end{figure}
Figure \ref{fig:mesh_strategy} shows the moving mesh strategy in
conjunction with domain decomposition. Here the computation of the
monitor function will be presented in section \ref{sec:example}. And
MMPDE6 \eqref{eq:MMPDE6} is solved by the following finite difference
scheme
\begin{multline}\label{eq:MMPDE6_dis}
\frac{\big(x_{j+1}^{n+1}-2x_j^{n+1}+x_{j-1}^{n+1}\big) -
\big(x_{j+1}^n-2x_j^n+x_{j-1}^n\big)}{\Delta t_n} \\
=
-\frac{1}{\tau}\bigg(M_{j+\frac{1}{2}}\big(x_{j+1}^{n+1}-x_j^{n+1}\big)
-
M_{j-\frac{1}{2}}\big(x_{j}^{n+1}-x_{j-1}^{n+1}\big)\bigg)
\end{multline}
in our numerical examples, where $\Delta t_n = t_{n+1}-t_{n}$ and
$M_{j+\frac{1}{2}}=(M_{j+1}+M_{j})/2$.
The new mesh could be obtained very efficiently by parallel
computing based on domain decomposition methods
\cite{Toselli2005book}. And it is best in the sense of
equidistribution on each subdomain. On the other hand, we will
found in the next section that the computation of the jump
$[\dot{u}]$ is avoided, hence the discretization scheme for
the physical PDE \eqref{eq:heat_eq} becomes very simple.
\section{Model discretization and final algorithm}\label{sec:discretization}
In this section, we derive the discretization schemes for the physical
model problem \eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} and
\eqref{eq:source_eq} on the observed domain $[x_l,x_r]$ with
appropriate boundary conditions. Then present a full algorithm of
moving mesh method for the model problem.
\subsection{Discretization schemes}
For an arbitrary function $f=f(x,t)=f(x(\xi,t),t)$, we have
\begin{equation*}
\dot{f}=\frac{\partial{f}}{\partial{t}}(x(\xi,t),t)
\bigg|_{\xi~\text{fixed}}
= f_{t} + f_{x}\dot{x}.
\end{equation*}
Through the coordinate transformation \eqref{eq:logical_map}, we
can rewrite equation \eqref{eq:heat_eq} on the computational
coordinates as
\begin{align}
\label{eq:heat_logical}
\dot{u} - u_x \dot{x} - u_{xx} = \sum_{i=0}^{q-1}F_i(t,x,u) \delta(x-\alpha_i(t)).
\end{align}
Since the right-hand side of \eqref{eq:heat_logical} vanishes when
$x\neq \alpha_i(t)$, that is,
\begin{equation}\label{eq:reduce_logical}
\dot{u} - u_x \dot{x} - u_{xx} = 0,
\end{equation}
we conduct the discretization scheme for \eqref{eq:heat_logical}
on the above equation as \cite{hu2011moving}, with each term on
the left-hand side of \eqref{eq:reduce_logical} containing the
information of jumps when they cross the sources. Physically, the
value $u$ of $i$th source changes smoothly as time evolution,
which means the jump of the directional derivative of $u(x,t)$
along the vector $(\alpha_i'(t),1)$ is zero \cite{ma2009moving,
hu2011moving}, i.e.,
\begin{equation}\label{eq:jump_u_t}
[u_t]_{(\alpha_i(t),t)}+[u_x]_{(\alpha_i(t),t)}\alpha_i'(t)=0, \quad i=0,1,\ldots,q-1.
\end{equation}
Recalling that $x_{j^s_i}\equiv \alpha_i(t)$, it follows that
\begin{equation}\label{eq:jump_u_dot}
[\dot{u}]_{(\alpha_i(t),t)} =
[u_{t}+u_{x}\dot{x}]_{(\alpha_i(t),t)}=
[u_{t}]_{(\alpha_i(t),t)}+\dot{x}[u_{x}]_{(\alpha_i(t),t)}=0,
\quad i=0,1,\ldots,q-1.
\end{equation}
By using the above equation, we can deduce from
\eqref{eq:reduce_logical} that
\begin{equation}\label{eq:jump_u_xx_tmp}
[u_{xx}]_{(\alpha_i(t),t)} = [\dot{u}-u_x\dot{x}]_{(\alpha_i(t),t)} =
[\dot{u}]_{(\alpha_i(t),t)} - \dot{x} [u_{x}]_{(\alpha_i(t),t)} =
- \alpha_i'(t) [u_{x}]_{(\alpha_i(t),t)},
\end{equation}
$i=0,1,\ldots,q-1$. Then we obtain immediately
\begin{equation}\label{eq:jump_u_xx}
[u_{xx}]_{(\alpha_i(t),t)} = \psi_i(t,\alpha_i(t),u)F_i(t,\alpha_i(t),u(\alpha_i(t),t)), \quad i=0,1,\ldots,q-1,
\end{equation}
by taking \eqref{eq:source_eq} and \eqref{eq:jump_ux} into
\eqref{eq:jump_u_xx_tmp}.
Similarly to \cite{hu2011moving}, the discretization scheme for
\eqref{eq:heat_logical} are divided into two cases due to
$x_{j^s_i}\equiv \alpha_i(t)$ during time integration. For $j\neq
j^s_i$, $i=0,1,\ldots,q-1$, the mesh point not located at the source,
\eqref{eq:heat_logical} is discretized by standard center difference
for spatial variable and backward difference for temporal variable,
that is,
\begin{equation}\label{eq:phy_dis_regular}
\frac{u_j^{n+1}-u_j^{n}}{\Delta t_n} -
\frac{u_{j+1}^{n+1}-u_{j-1}^{n+1}}{h_{j+1}^{n+1}+h_{j}^{n+1}}
\bigg(\frac{x_j^{n+1}-x_j^n}{\Delta t_n}\bigg) - \frac{2}{h_{j+1}^{n+1}+h_{j}^{n+1}} \bigg(
\frac{u_{j+1}^{n+1}-u_j^{n+1}}{h_{j+1}^{n+1}} -
\frac{u_{j}^{n+1}-u_{j-1}^{n+1}}{h_{j}^{n+1}} \bigg) = 0,
\end{equation}
where $h_j^n = x_j^{n}-x_{j-1}^n$. Here $x_j^n$, $u_j^n$ are the
mesh and the solution on it at time step $t_n$, respectively. For
$j=j^s_i$, the mesh point just located at the source, the jump
informations should be incorporated into the discretization
scheme. For this case, the discretization scheme for
\eqref{eq:heat_logical} reads
\begin{align}\label{eq:phy_dis_irregular}
\frac{u_{j^s_i}^{n+1}-u_{j^s_i}^n}{\Delta t_n} -
\frac{u_{j^s_i+1}^{n+1}-u_{j^s_i-1}^{n+1}}
{h_{j^s_i+1}^{n+1}+h_{j^s_i}^{n+1}} \psi_i^{n+1} & -
\frac{2}{h_{j^s_i+1}^{n+1}+h_{j^s_i}^{n+1}} \bigg(
\frac{u_{j^s_i+1}^{n+1}-u_{j^s_i}^{n+1}}{h_{j^s_i+1}^{n+1}} -
\frac{u_{j^s_i}^{n+1}-u_{j^s_i-1}^{n+1}}{h_{j^s_i}^{n+1}} \bigg)\nonumber\\
& - \frac{2}{h_{j^s_i+1}^{n+1}+h_{j^s_i}^{n+1}} F_i(u_{j^s_i}^{n+1})
= 0,
\end{align}
where $\psi_i^{n+1}\approx \psi_i(t_{n+1}, \alpha_i^{n+1}, u^{n+1}) $,
$F_i(u_{j^s_i}^{n+1})\approx F_i(t_{n+1}, \alpha_i^{n+1},
u_{j^s_i}^{n+1})$, $i=0,1,\ldots,q-1$.
In the above schemes, we need the source position $\alpha_i(t_{n+1})$
at time step $t_{n+1}$. For the general movement \eqref{eq:source_eq},
it is computed by the following Crank-Nicolson scheme
\begin{align}
\label{eq:source_CN}
\alpha_i^{n+1} = \alpha_i^n + \frac{\Delta t_n}{2} (\psi_i^{n+1}+\psi_i^{n}), \quad i=0,1,\ldots,q-1,
\end{align}
as in \cite{hu2012zh}. If $\psi_i(t,\alpha_i(t),u)$,
$i=0,1,\ldots,q-1$, are independent of $u$, the source position
$\alpha_i^{n+1}$ and the speed $\psi_i^{n+1}$ can be calculated in
advance before solving the discretization schemes for MMPDE6
\eqref{eq:MMPDE6} and physical PDE \eqref{eq:heat_logical}. Otherwise,
the resulting system would be too complicated to be solved. In this
case, we decouple the discretization system by a predictor-corrector
algorithm. For the predictor step, assume $\psi_i^{n+1}=\psi_i^n$ and
solve \eqref{eq:source_CN} to get an approximate variable $\alpha_i^*$
of $\alpha_i^{n+1}$. Then substituting $\psi_i^{n+1}$ and $\alpha_i^*$
into the discretization schemes for \eqref{eq:MMPDE6} and
\eqref{eq:heat_logical} to obtain an approximate solution $u^*$ of
$u^{n+1}$. For the corrector step, compute
$\psi_i^{n+1}=\psi_i(t_{n+1},\alpha_i^*,u^*)$ and solve
\eqref{eq:source_CN} to get $\alpha_i^{n+1}$, then obtain the solution
$u^{n+1}$ at time step $t_{n+1}$ by the discretization schemes for
\eqref{eq:MMPDE6} and \eqref{eq:heat_logical}.
To complete the discretization schemes, we require an appropriate
condition for $u$ on the boundary of the observed domain $[x_l,x_r]$.
For the observed domain is small enough, we employ a third-order local
absorbing boundary condition (LABC) proposed in
\cite{brunner2010computational}
\begin{equation}\label{eq:LABCs_phy}
3s_0 u_x + u_{xt} \pm s_0\sqrt{s_0}u\pm
3\sqrt{s_0}u_t = 0
\end{equation}
for \eqref{eq:heat_eq} as in \cite{hu2011moving,hu2012zh}. Here $s_0$
is an user-defined parameter, the plus sign in "$\pm$" corresponds to
the LABC at the right boundary $x_r$, and the minus sign corresponds
to the one at the left boundary $x_l$. Under the map
\eqref{eq:logical_map}, we get the LABC for \eqref{eq:heat_logical} as
follows
\begin{equation}\label{eq:LABCs_log}
\dot{u}_{x} \pm 3 \sqrt{s_0} \dot{u} + 3 s_0 u_{x}-
u_{xx}\dot{x} \pm s_0\sqrt{s_0}u \pm \left( - 3\sqrt{s_0}u_{x}\dot{x}\right) = 0,
\end{equation}
where the plus sign in "$\pm$" corresponds to the right boundary, and
the minus sign corresponds to the left boundary. According to
\cite{hu2011moving,hu2012zh}, a finite difference scheme for
\eqref{eq:LABCs_log} is
\begin{align}\label{eq:LABC_left}
& \frac{1}{\Delta t_n}\bigg(\frac{u_1^{n+1}-u_{-1}^{n+1}}{2h_1^{n+1}} -
\frac{u_1^{n}-u_{-1}^{n}}{2h_1^{n}}\bigg) - 3 \sqrt{s_0}
\frac{u_0^{n+1}-u_0^{n}}{\Delta t_n} + 3\left(s_0+\sqrt{s_0}\frac{x_0^{n+1}-x_0^n}{\Delta t_n}\right)
\frac{u_1^{n+1}-u_{-1}^{n+1}}{2h_1^{n+1}} \nonumber \\
& - \left(\frac{x_0^{n+1}-x_0^n}{\Delta t_n}\right)
\frac{u_1^{n+1}-2u_0^{n+1}+u_{-1}^{n+1}}{(h_1^{n+1})^2} - s_0
\sqrt{s_0} u_0^{n+1} = 0,
\end{align}
on the left boundary, and
\begin{align}\label{eq:LABC_right}
& \frac{1}{\Delta t_n}\bigg(\frac{u_{N+1}^{n+1}-u_{N-1}^{n+1}}{2h_{N}^{n+1}} -
\frac{u_{N+1}^{n}-u_{N-1}^{n}}{2h_{N}^{n}}\bigg) + 3 \sqrt{s_0}
\frac{u_N^{n+1}-u_N^{n}}{\Delta t_n} + 3\left(s_0-\sqrt{s_0}\frac{x_N^{n+1}-x_N^n}{\Delta t_n}\right)
\frac{u_{N+1}^{n+1}-u_{N-1}^{n+1}}{2h_N^{n+1}} \nonumber \\
& - \left(\frac{x_N^{n+1}-x_N^n}{\Delta t_n}\right)
\frac{u_{N+1}^{n+1}-2u_N^{n+1}+u_{N-1}^{n+1}}{(h_N^{n+1})^2} + s_0
\sqrt{s_0} u_0^{n+1} = 0.
\end{align}
on the right boundary. Here two ghost points $x_{-1}$ and $x_{N+1}$
are used. On the other hand, if the observed domain is big enough or
else, Dirichlet boundary conditions are employed.
\subsection{Full algorithm}
We close this section with a full algorithm in Figure
\ref{fig:full_algorithm} for the model problem
\eqref{eq:heat_eq}$-$\eqref{eq:bound_cond} and
\eqref{eq:source_eq}. Here the choice of the time step $\Delta t_n$
will be determined in the following concrete examples, and $Tol>0$ is
set to be $10^{-16}$.
\begin{figure}[!htb]
\centering
\small
\begin{tikzpicture}[scale = 0.75]
\matrix[row sep=5mm,column sep=5mm, top color=white, bottom color=white, scale=0.75]{
\node[startstop, text width=0.4\textwidth, scale=0.75, text centered] (start) {Prepare the initial values $x^0$, $u^0$ and the terminate time $T$. Let $n=0$, $t_n=0$.};\\
\node[passprocess, scale=0.75] (timestep) {Determine $\Delta t_n$ and let $t_{n+1} = t_n + \Delta t_n$.};\\
\node[decision, scale=0.75] (branchsource) {$\psi_i$ is dependent of $u$};\\
\node[passprocess, text width=0.4\textwidth, left, scale=0.75, text centered] (pre_step1) {Assume $\psi_i^{n+1} = \psi_i^n$, obtain $\alpha_i^*$ by \eqref{eq:source_CN}.};\\
\node[passprocess, left, text width=0.4\textwidth, scale=0.75, text centered] (preStep2) {Moving mesh strategy in conjunction with domain decomposition (see Figure \ref{fig:mesh_strategy}).};\\
\node[passprocess, left, text width=0.4\textwidth, scale=0.75, text centered] (preStep3) {Solve the discretization schemes for the physical PDE \eqref{eq:heat_logical} to calculate $u^*$.};\\
\node[passprocess, text width=0.4\textwidth, left, scale=0.75] (corstep1) {Compute $\psi_i^{n+1}=\psi_i(t_{n+1}, \alpha_i^*, u^*$) and solve \eqref{eq:source_CN} to get $\alpha_i^{n+1}$.};\\
\node[passprocess, text width=0.4\textwidth, scale=0.75, text centered] (corstep2) {Moving mesh strategy in conjunction with domain decomposition (see Figure \ref{fig:mesh_strategy}).};\\
\node[passprocess, text width=0.4\textwidth, scale=0.75, text centered] (corstep3) {Solve the discretization schemes for the physical PDE \eqref{eq:heat_logical} to calculate $u^{n+1}$.};\\
\node[decision, scale=0.75, text centered] (branchtime) {$t_{n+1}<T$ and $\Delta t_n>Tol$};\\
\node[startstop, scale=0.75] (end) {The computation is finished.}; \\
};
\node[passprocess, scale=0.75] (pdeStep1) at ($(preStep3.north east)+(3.25cm,3mm)$) {Compute $\alpha_i^{n+1}$ and $\psi_i^{n+1}$.};
\begin{scope}[every path/.style=line]
\path (start) -- (timestep);
\path (timestep) -- (branchsource);
\path (branchsource) -| node[near start, above, scale=0.75] {YES} (pre_step1);
\path (branchsource) -| node[near start, above, scale=0.75] {NO} (pdeStep1);
\path (pre_step1) -- (preStep2);
\path (preStep2) -- (preStep3);
\path (preStep3) -- (corstep1);
\path (corstep2) -- (corstep3);
\path (corstep1) |- ($(corstep2.north)+(0,2.5mm)$) -- (corstep2);
\path (corstep3) -- (branchtime);
\path (pdeStep1) |- ($(corstep2.north)+(0,2.5mm)$) -- (corstep2);
\path (branchtime) -- node[midway, right, scale=0.75] {NO} (end);
\path (branchtime) -- node[midway, above, scale=0.75] {YES\quad Let $n:=n+1$} ($(branchtime.east)+(4.5,0)$) |- (timestep);
\end{scope}
\end{tikzpicture}
\caption{Full algorithm for numerical solution of the model problem \eqref{eq:heat_eq}-\eqref{eq:source_eq}.}
\label{fig:full_algorithm}
\end{figure}
\section{Numerical examples}\label{sec:example}
In this section, we present some numerical examples to verify the
convergence rate and illustrate efficiency of the full algorithm in
Figure \ref{fig:full_algorithm}.
\begin{example}
We consider a nonlinear moving interface problem with the following
exact solution
\begin{align}
\label{eq:ex_true_u}
u(x,t)=
\begin{cases}
\sin( \omega_1 x ) e^{- \omega_1^2 t}, \quad & x\leq \alpha_0(t), \\
\sin( \omega_2 (1-x)) e^{- \omega_2^2 t}, \quad & x\geq \alpha_0(t),
\end{cases}
\end{align}
for some choice of $\omega_1$ and $\omega_2$. The interface
$\alpha_0(t)$ is determined by solving the scalar equation
\begin{equation}
\label{ex:interface_eq}
\sin(\omega_1 \alpha_0 ) e^{-\omega_1^2t} = \sin(\omega_2(1-\alpha_0)) e^{-\omega_2^2 t},
\end{equation}
so that $u(x,t)$ is continuous across the interface.
\end{example}
The equation \eqref{ex:interface_eq} has a unique solution on $[0,1]$
if we take, for example, $\pi<\omega_1,~\omega_2<2\pi$. Then we have
the ordinary differential equation for the motion of the interface
\begin{align}
\label{ex:interface_ode}
\frac{\mathrm{d} \alpha_0}{\mathrm{d}t} = \frac{(\omega_1^2-\omega_2^2) u(\alpha_0, t)}{u_x(\alpha^-_0, t) - u_x(\alpha_0^+, t)}.
\end{align}
Based on the jump conditions, the source function $F_0(t,x,u)$ is
\begin{align}
\label{ex:source_fun}
F_0(t,x,u) &= -[u_x]_{\alpha_0} = u_x(\alpha^-_0, t) - u_x(\alpha_0^+, t)\nonumber\\
& = \omega_1\cos(\omega_1\alpha_0)e^{-\omega_1^2t} + \omega_2 \cos(\omega_2(1-\alpha_0))e^{-\omega_2^2t}.
\end{align}
Same as in \cite{beyer1992analysis}, we take $\omega_1=5\pi/4$,
$\omega_2=7\pi/4$. The observed domain is set by $[0,1]$, where the
initial position of the interface is $\alpha_0(0)=0.58333$. Since we
have the exact solution, Dirichlet boundary conditions are
employed. In this example, we simply use the uniform time step,
i.e. $\Delta t_n \equiv const$, and the total number of the time
meshes is $L$. The monitor function for MMPDE6 \eqref{eq:MMPDE6} takes
the form
\begin{equation}\label{ex:monitor}
M(x,t) = (1-\theta) \bigg|\frac{\partial u}{\partial
x}\bigg| + \theta((x-\alpha_0(t))^2+\varepsilon)^{-1/4},
\end{equation}
where $0<\theta<1$, $0<\varepsilon\ll 1$. This is consistent with the
choice in \cite{hu2011moving,hu2012zh}. In practice, smoothing the
monitor function can improve the accuracy of the numerical solution,
and we utilize the smoothing technique proposed in
\cite{huang1994moving}. Here the parameters in MMPDE6
\eqref{eq:MMPDE6} and the monitor function \eqref{ex:monitor} are
given by $\tau=10^{-3}$, $\theta=0.5$, and $\varepsilon=10^3/N^4$.
Since backward Euler scheme is used to solve the physical PDE in this
paper, the truncation error for time discretization is only
first-order. To verify our algorithm has a second-order convergence
rate for space, the number of $L$ should be fourfold when $N$ is
double in the convergence test.
Computational results with different number of $N$ and $L$ at the time
$T=0.1$ are listed in Table \ref{table:err_result}, where the errors
are defined as
\begin{align*}
E_{N,L} = \parallel U - u_e \parallel_\infty,
\quad \tilde{E}_{N,L} = \parallel \tilde{U} - u_e \parallel_\infty,
\quad E_{N,L}^\alpha = \mid \tilde{\alpha}_0 - \alpha_0^* \mid.
\end{align*}
Here, $u_e$ is the true solution, $\alpha_0^*$ used as the exact
interface is the solution of a zero-finding MATLAB function
\emph{fzero} for \eqref{ex:interface_eq}. The numerical solution $U$
is obtained by the algorithm where $\alpha_0(t)$ and $\alpha_0'(t)$
are exactly calculated. And $\tilde{U}$, $\tilde{\alpha}_0$ represent
respectively the numerical solution and interface, solved with the
full predictor-corrector algorithm. The ratios in Table
\ref{table:err_result} are $E_{2N,4L}/E_{N,L}$,
$\tilde{E}_{2N,4L}/\tilde{E}_{N,L}$ and
$E_{2N,4L}^\alpha/E_{N,L}^\alpha$, respectively. It is shown that our
algorithm solves the solution and the interface very well, and has a
second-order convergence rate for space,
i.e. $O(1/N^2)$. Additionally, compared the corresponding results in
\cite{li1997immersed}, our algorithm is better than the method
proposed in \cite{li1997immersed}.
\begin{table}[!htb]
\caption{Error and convergence rates at $T=0.1$.
\label{table:err_result
\centering
\tiny
\colorbox{white}{
\begin{tabular}{lllllll}
\hline
N,~L & $E_{N,L}$ & ratio & $\tilde{E}_{N,L}$ & ratio & $E_{N,L}^\alpha$ & ratio \\
\hline
40, ~40 & 1.0931e-2 & - & 1.3721e-2 & - & 8.2720e-3 & - \\
80, ~160 & 2.6996e-3 & 0.24697 & 3.3908e-3 & 0.24713 & 2.0540e-3 & 0.24830 \\
160, ~640 & 6.6945e-4 & 0.24798 & 8.4144e-4 & 0.24816 & 5.1038e-4 & 0.24848 \\
320, ~2560 & 1.6687e-4 & 0.24927 & 2.0981e-4 & 0.24935 & 1.2732e-4 & 0.24947 \\
640, ~10240 & 4.1678e-5 & 0.24976 & 5.2408e-5 & 0.24978 & 3.1807e-5 & 0.24982 \\
1280, ~40960 & 1.0416e-5 & 0.24992 & 1.3098e-5 & 0.24993 & 7.9500e-6 & 0.24994 \\
2560, ~163840 & 2.6038e-6 & 0.24998 & 3.2743e-6 & 0.24998 & 1.9874e-6 & 0.24998 \\
\hline
\end{tabular}}
\end{table}
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.335\textwidth]{ode_source1_mesh.eps}
{\includegraphics[width=0.33\textwidth]{ode_source1_U_phy.eps}
{\includegraphics[width=0.33\textwidth]{ode_source1_U_c.eps}
\caption{\small Mesh trajectories and the profiles of $u$ in
physical variable and computational variable (from left to right)
as time changes with $N=24$. The solid lines are the computed
solution and the dots are the exact solution on the mesh.}
\label{fig:ex1}
\end{figure}
Figure \ref{fig:ex1} presents the profiles of the solution in physical variable and computational variable and the evolving mesh from $t=0$ to $t=0.1$. The number of the mesh is $N=24$, with half mesh points on each side of the interface. We can see that we get excellent resolution of the example even with a grid as coarse as $N=24$.
\bigskip
The rest examples are from traveling heat sources problems with the solution may be blow-up \cite{zhu2010numerical,ma2009moving,hu2011moving,hu2012zh}. If not specifically pointed out, the initial value is given by
\begin{align}\label{eq:init_value}
u(x,0) = \left\{ \begin{array}{ll}
\cos^2(\pi x/2), & -1<x<1,\\ 0, &
\text{otherwise},
\end{array} \right.
\end{align}
the observed domain is set by $[-10,10]$ with $u(-10,t)=u(10,t)=0$, and the source functions $F_i(t,x,u)$ are simply specified by
\begin{equation}\label{eq:F_source}
F_i(t,x,u) = 1 + u^2, \qquad i=0,1,\ldots,q-1.
\end{equation}
The resulting nonlinear system is solved by Newton iteration with the tolerance $tol = 10^{-8}$.
The monitor function for MMPDE6 \eqref{eq:MMPDE6} takes the form
\begin{equation}\label{eq:monitor}
M(x,t) = \theta_{q+1} u^p + \theta_q \bigg|\frac{\partial u}{\partial
x}\bigg| + \sum_{i=0}^{q-1}\theta_i((x-\alpha_i(t))^2+\varepsilon)^{-1/4},
\end{equation}
where the parameters $0<\theta_i<1$, $\sum_{i=0}^{q+1}\theta_i=1$, $0<\varepsilon\ll 1$, $p>0$ will be determined later. For non-blowup case, the following graded time steps
\cite{ma2009moving,zhu2010numerical}
\begin{equation*}
t_n=\bigg(n\frac{T}{L}\bigg)^2, \quad n = 0, 1, \ldots, L,
\end{equation*}
are used with $[0,T]$ the time integration interval and $L$ the number of time meshes. While for blow-up case, the time step $\Delta t_n = t_{n+1}-t_{n}$ is chosen to be \cite{ma2009moving,budd2001scaling}
\begin{equation*}
\Delta t_n =\min\left \{\mu, \frac{\mu}{\Big(\max_{j}\big\{u_j^n\big\}+\varepsilon\Big)^2}\right \},
\end{equation*}
where $\varepsilon$ is same in the monitor function, $\mu$ is a small positive constant with $\mu=10^{-3}$ in the test.
\begin{example}[Linear moving sources]
We consider that all sources move with a constant velocity $k$, and
the position of the $i$-th source has a constant distance $d_i$ to
the $0$-th source, i.e.,
\begin{equation*}
\alpha_i'(t) = k, \quad \alpha_i(0)=d_i,\qquad i=0,1,\ldots,q-1,
\end{equation*}
where $d_0=0$.
\end{example}
Since our method is trivial for multi-sources case, only $q=1,2$ are considered. Different velocities $k$ are investigated in \cite{zhu2010numerical,ma2009moving,hu2011moving} and we specify $k=2$ here. With this velocity, blow-up would occur for $q=2$, and be avoided for $q=1$. The parameters are set by $\tau=10^{-3}$, $\theta_0=0.9$, $\theta_1=0.1$, and $\varepsilon=10^3/N^4$ for $q=1$, while $\tau=5\times 10^{-4}$, $\theta_0=\theta_1=0.3$, $\theta_3=0.4$, $p=2$, $\varepsilon = 10^{-5}$, and $d_1=2.5$ for $q=2$, respectively.
The profiles of the computed solution in physical variable and computational variable and the evolving mesh are presented in Figure \ref{fig:linear_nonblowup} for $q=1$ and in Figure \ref{fig:linear_blowup} for $q=2$. For simplicity, each subdomain has $50$ mesh points, i.e. $N=100$ for $q=1$ and $N=150$ for $q=2$. The numerical results are coincide with that in \cite{zhu2010numerical,ma2009moving}, and the blow-up time is $2.039708648680643$ at the first source $x=4.079417297361286$, corresponding to the maximum value of $u_{\max}=3.16\times 10^6$.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.33\textwidth]{linear2_source1_mesh.eps}
{\includegraphics[width=0.33\textwidth]{linear2_source1_U_phy.eps}
{\includegraphics[width=0.33\textwidth]{linear2_source1_U_c.eps}
\caption{\small Mesh trajectories and the profiles of $u$ from $t=0$
to $t=1.0$ for one source case with $N=100$.}
\label{fig:linear_nonblowup}
\end{figure}
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.36\textwidth]{linear2_source2_mesh.eps}
{\includegraphics[width=0.32\textwidth]{linear2_source2_U_phy.eps}
{\includegraphics[width=0.32\textwidth]{linear2_source2_U_c.eps}
\caption{\small Mesh trajectories and the profiles of $u$ for two sources case with $N=150$.}
\label{fig:linear_blowup}
\end{figure}
\begin{example}[Sin-type moving sources]
We now consider two sources case, in which the sources move
periodically with the same speed while separated by a constant
distance $d_1=2.5$, that is,
\begin{equation*}
\alpha_0'(t) = \alpha_1'(t) = A \cos(\pi t),\quad \alpha_0(0) = 0.
\end{equation*}
\end{example}
The Blow-up phenomenon is studied in \cite{hu2011moving} for different amplitudes $A$. Here we only give the numerical results for $A=\pi$ (see Figure \ref{fig:sin1_source2}), since all results are similar to those in \cite{hu2011moving}. All parameters are chosen the same as those in last example. The blow-up occurs at $t=1.689611393639939$ on the second source with the maximum value of $u_{\max}=3.16\times 10^6$.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.36\textwidth]{sin1_source2_mesh.eps}
{\includegraphics[width=0.32\textwidth]{sin1_source2_U_phy.eps}
{\includegraphics[width=0.32\textwidth]{sin1_source2_U_c.eps}
\caption{\small Mesh trajectories and the profiles of $u$ for $q=2$, $A=\pi$ with $N=150$.}
\label{fig:sin1_source2}
\end{figure}
\begin{example}[Symmetric periodic moving sources]
We consider the case for two sources, which move periodically and
symmetrically. The motion are described by
\begin{equation*}
\alpha_0'(t) = A \cos(\pi t),\quad \alpha_0(0) = -2.0,
\end{equation*}
and $\alpha_1(t)=-\alpha_0(t)$, with e.g. $A=\pi$.
\end{example}
To our best knowledge, there has no theoretical results for multi-sources with different speeds and this is the first time numerically investigating the phenomenon for this case. It is shown in Figure \ref{fig:sym1_source2} that blow-up occurs on both sources at $t=2.496881990359248$, corresponding to the maximum value of $u_{\max}=3.16\times 10^6$.
\begin{figure}[!t
\centering
{\includegraphics[width=0.36\textwidth]{symmetric1_source2_mesh.eps}
{\includegraphics[width=0.32\textwidth]{symmetric1_source2_U_phy.eps}
{\includegraphics[width=0.32\textwidth]{symmetric1_source2_U_c.eps}
\caption{\small Numerical results for symmetric periodic moving sources with $N=150$.}
\label{fig:sym1_source2}
\end{figure}
If local absorbing boundary conditions \eqref{eq:LABCs_log} are used, the observed domain can be chosen more smaller while the results do not be influenced almost. See Figure \ref{fig:sym1_source2_LABC} as an example, where the observed domain is set by $[\alpha_0(t)-4.0,\alpha_1(t)+4.0]$, changed as time evolution. Now blow-up occurs on both sources at $t=2.496370241342059$ with the maximum value of $u_{\max}=3.16\times 10^6$.
\begin{figure}[!t
\centering
{\includegraphics[width=0.36\textwidth]{symmetric1_source2_ABC_mesh.eps}
{\includegraphics[width=0.32\textwidth]{symmetric1_source2_ABC_U_phy.eps}
{\includegraphics[width=0.32\textwidth]{symmetric1_source2_ABC_U_c.eps}
\caption{\small Numerical results for symmetric periodic moving sources with $[x_l,x_r]=[\alpha_0(t)-4.0,\alpha_1(t)+4.0]$.}
\label{fig:sym1_source2_LABC}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
In this paper, our work focus on the problem of traveling singular sources with different speeds.
A new moving mesh method in conjunction with a non-overlapping domain decomposition is proposed for solving this problems.
The whole domain is splitted into $q+1$ subdomains by the $q$ sources,
whose positions are gotten by a predictor-corrector algorithm.
Taking the advantages of the domain decomposition,
the computation of jump $[\dot{u}]$ is avoided and there are only two different cases discussed
in the discretization of the physical PDE.
Thus, it is easy for the implementation to solve the problems with two traveling sources or more.
Moreover, the moving mesh method of MMPDEs can be applied into each sub-domain respectively.
The second-order of the spatial convergence can be proved for the new method under
a special time marching implementation. The good performance of the new method for the blow-up phenomenon
is demonstrated through a number of examples with two sources.
Furthermore, using the new method, we successfully simulate the solutions of two sources with different speeds.
To our best knowledge, this is the first time investigation for this case.
The case of three sources or more can be implemented similarly.
\section*{Acknowledgment}
This work was partially supported by a grant of key program from
the National Natural Science Foundation of China (No.~10731060, 10801120, 11171305),
National Basic Research Program of China (2011CB309704),
Chinese Universities Scientific Fund No.~2010QNA3019
and Zhejiang Provincial Natural Science Foundation of China under Grant No.~Y6110252.
\bibliographystyle{unsrt}
|
{
"timestamp": "2012-12-27T02:04:28",
"yymm": "1203",
"arxiv_id": "1203.1826",
"language": "en",
"url": "https://arxiv.org/abs/1203.1826"
}
|
\section{Overview}
\label{sec:overview}
The topic of these lectures is the use of renormalization group (RG)
methods in low-energy nuclear systems, which include the full range
of atomic nuclei as well as astrophysical systems such as neutron
stars.
We will examine why the RG has become an increasingly useful tool
for nuclear physics theory over the last ten years and consider
how to apply RG technology both formally and in practice. Of particular emphasis will be flow equation approaches applied to Hamiltonians both in free space and in the medium, which are an accessible but powerful method to make nuclear physics computationally more
like quantum chemistry. We will see how interactions are evolved to increasingly universal form and become more amenable to perturbative methods. A key element in nuclear systems is the role of many-body forces and operators; dealing with their evolution is an important on-going challenge.
The expected background for these lectures is a thorough knowledge of nonrelativistic quantum mechanics, including scattering, the basics of quantum field theory,
and linear algebra (it's all matrices!).
We will not assume a knowledge of nuclear structure or reactions,
or even many-body physics beyond Hartree-Fock.
No advanced computing experience is assumed (although Mathematica or MATLAB
knowledge will be very helpful in exploring simple RG examples).
By necessity, we will only scratch the surface in these lectures.
For a thorough treatment of flow equations for many-body systems
not including nuclei,
see the book by Kehrein~\cite{Kehrein:2006}. For more details on
applications
of flow-equation and similar renormalization group methods to
low-energy nuclear physics, the review article~\cite{Bogner:2009bt}
and references therein are recommended.
\section{Atomic nuclei at low resolution via RG}
\label{sec:resolution}
\subsection{Goals and scope of low-energy nuclear physics}
The playing field for low-energy nuclear physics is the table of the nuclides,
shown in Fig.~\ref{fig:landscape}.
There are several hundred stable nuclei (black squares)
but also several thousand \emph{unstable}
nuclei are known through experimental measurements. The total
number of nuclides is still unknown (see the region marked ``terra incognita''), with theoretical
estimates suggesting it could be as high as ten thousand!
These unstable nuclei are the object of scrutiny for new and planned
experimental facilities around the world.
The challenge for low-energy nuclear theory is to describe their structure and
reactions.
We'll return in the final lecture to discuss the overlapping regions where
the theoretical many-body methods listed in the figure can be applied.
Let's start with some of the questions that drive low-energy nuclear physics
research. These include general questions about the physics of
nuclei~\cite{LRP:2007}:
\bi
\I
How do protons and neutrons make stable nuclei and rare isotopes?
Where are the limits?
\I
What is the equation of state of nucleonic matter?
\I
What is the origin of simple patterns observed in complex nuclei?
\I How do we describe fission, fusion, and other nuclear reactions?
\ei
These topics inform and are in turn illuminated by applications
to other fields, such as astrophysics, where one can ask:
\bi
\I
{ How did the elements from iron to uranium originate? }
\I
{ How do stars explode? }
\I
{ What is the nature of neutron star matter?}
\ei
or of fundamental symmetries:
\bi
\I Why is there now more matter than antimatter in the
universe?
\I What is the nature of the neutrinos,
what are their masses,
and how have they shaped the evolution of the universe?
\ei
Finally, there are applications, for which
we are led to ask: How can our knowledge of nuclei and our ability to produce them benefit humankind? The impact is very broad, encompassing the
Life Sciences, Material Sciences, Nuclear Energy, and National Security.
\begin{figure}[!tbp]
\includegraphics[width=3in]{nuclide_overview}
\caption{The nuclear landscape. A nuclide is specified by the number
of protons and neutrons~\cite{LRP:2007}.}
\label{fig:landscape}
\end{figure}
\begin{figure}[!tbp]
\includegraphics[width=3in]{overview_1cut}
\caption{Hierarchy of nuclear degrees of freedom and associated
energy scales~\cite{LRP:2007}.}
\label{fig:scales}
\end{figure}
In Figure~\ref{fig:scales}, the energy scales of nuclear physics are illustrated. There is an extended hierarchy, which is a challenge, but also an opportunity to make use of effective field theory (EFT) and renormalization group (RG) techniques. The ratio of scales can become
an expansion parameter, leading to a systematic treatment at lower
energies.
The progression from top to bottom can be viewed as a reduction in resolution.
In these lectures, our focus is on the intermediate region only, where protons and neutrons are the relevant degrees of freedom. But even within this
limited scope, the concept of reducing resolution by RG methods is extremely powerful.
\subsection{Lowering the resolution with RG}
\begin{figure}[t!]
\includegraphics[width=3in]{hatsuda_phen-pot_new}
\caption{Some phenomenological potentials that accurately describe
proton-neutron scattering up to laboratory energies
of 300\,MeV~\cite{Aoki:2008hh}.}
\label{fig:phenpots}
\end{figure}
What do we mean by resolution? Even the general public these days
is familiar with the concept of digital resolution for computer screens,
cellphones, televisions. High resolution is associated with more
pixels, which follows because as pixel size becomes small compared to
characteristic scales in an image, greater detail is seen. In our
discussion, we associate resolution with the Fourier transform space and
the phenomenon of diffraction.
Recall the basic physics: if the wavelength of light is comparable
to or larger than an aperture, then diffraction is significant. If there are
two sources, we say we can resolve them if the diffracted images
don't overlap too much. For a fixed angle between sources or details
in the object being observed, we find that the wavelength determines
whether or not we resolve the details.
Being unable to resolve details at long wavelength is generally
considered to be a disadvantage (e.g., for astronomical observations),
but we turn it to an advantage.
A fundamental principle of \emph{any} effective low-energy description
(not restricted to nuclear physics!) is that if a system is probed
at low energies, fine details are not resolved, and one can instead use
low-energy variables for low-energy processes. Renormalization theory
tells us that the short-distance structure can be replaced by something
simpler without distorting low-energy \emph{observables}. The familiar
analog from classical electrodynamics is the replacement of a complicated
charge or current distribution with a truncated multipole expansion.
In the quantum case, the replacement can
be done by constructing a model, or in a systematic way using
effective field theory. We emphasize that while observable quantities
(such as cross sections) do not change, the physics interpretation
can (and generally does) change with resolution.
What if there is no external probe? Then the particles still probe each other
with resolution set by their de Broglie wavelengths. Low-density
nuclear systems would seem to imply low resolution.
But the picture is complicated by the nature of traditional
internucleon potentials.
\begin{figure}[t]
\includegraphics[width=3in]{vnn_1S0_kvnn06_k_surf}
\caption{Momentum space representation of the Argonne $v_{18}$ (AV18)
NN potential in the $^1$S$_0$ channel~\cite{Bogner:2009bt}.}
\label{fig:vnnsurf}
\end{figure}
\begin{figure}[t]
\includegraphics[width=3in]{vnn_1S0_kvnn06_k_cntr}
\caption{Alternative momentum space representation of the AV18 potential
in the $^1$S$_0$ channel~\cite{Bogner:2009bt}.}
\label{fig:vnncntr}
\end{figure}
Figure~\ref{fig:phenpots} shows several phenomenological potentials
that reproduce nucleon-nucleon scattering phase shifts up to about
300\,MeV in lab energy. They are characterized by a long-range
attractive tail from one-pion exchange, intermediate attraction,
and a strongly repulsive short-range ``core''.
For our purposes, the partial-wave momentum-space representation, such as
\beq
\langle k | V_{L=0} | k' \rangle
\propto \int\! r^2\,dr \, j_0(kr)\, V(r)\, j_0(k'r)
\eeq
for S-waves is more useful.
Here $k$ and $k'$ are the relative momenta of the two nucleons.
This is shown for the AV18 potential in Fig.~\ref{fig:vnnsurf}
in the $^1$S$_0$ channel.
(The spin and isospin dependence of the nuclear interactions is very
important, unlike the situation in quantum chemistry.)
In the momentum basis, the potentials in Fig.~\ref{fig:phenpots}
are no longer diagonal, so we
need three-dimensional information, but we generally use the flat
contour representation of the same information, as in
Fig.~\ref{fig:vnncntr}. The RG evolution of potentials will be
visualized as changes in such pictures.
We work in units for which $\hbar = c = 1$. Then the typical
relative momentum in the Fermi sea of any large nucleus
is of order $1\fmi$ or 200\,MeV. However, it is evident from
Figs.~\ref{fig:vnnsurf} and \ref{fig:vnncntr} that there are large matrix elements connecting
such momenta to much larger momenta. This is directly associated with
the repulsive core of the potential. For our discussion,
we will adopt $2\fmi$ as the (arbitrary but reasonable)
dividing line between low and high momentum for nuclei.
The consequences of the coupling to high momentum are readily seen
in the probability density of the only two-body bound state,
the deuteron.
Consider the Argonne $v_{18}$~\cite{Wiringa:1994wb} curve in Fig.~\ref{fig:deutprob}.
The probability at small separations is significantly suppressed
as a result of high-momentum components in the
wave function.
This suppression, called ``short-range correlations'' in this context,
carries over to many-body wave functions and greatly complicates
basis expansions.
For example, in a harmonic oscillator basis, which is frequently
the choice for self-bound nuclei because it readily allows removal
of center-of-mass contamination, convergence is greatly slowed
by the need to accommodate high-momentum components.
The factorial growth of the basis size with the number of
nucleons then greatly limits the reach of calculations.
\begin{figure}
\includegraphics[width=3in]{3s1rspacewfsq_rev2}
\caption{Short-range correlations in the deuteron $^3\mbox{S}_1$
probability density from the AV18 potential (solid line). They
are essentially eliminated by the RG evolution to lower flow
parameter $\lambda$ (see Section~\ref{subsec:floweq}).}
\label{fig:deutprob}
\end{figure}
The underlying problem is that the resolution scale
induced by the potential is mismatched with the basic scale of the
low-energy nuclear states (given, for example, by the Fermi momentum).
A solution is to eliminate the coupling to high momentum.
This is readily accomplished for spatial images (i.e., photographs)
by Fourier transforming
and then applying a low-pass filter---simply set the short wavelength
parts to zero---and then transforming back. Let's try that
for our Hamiltonian by setting to zero all of the matrix elements
in Fig.~\ref{fig:vnncntr} for $k > 2\fmi$.
We test the implications by seeing the effect on scattering phase shifts
in the region of laboratory energies corresponding to $k \leq 2\fmi$.
\begin{figure}[tb!]
\includegraphics[width=2.8in]{srg_1S0_phases_bare_cut_v2}
\caption{Effect of a low-pass filter on observables: the $^1$S$_0$
phase shifts. Note that the AV18 phase shifts reproduce experimentally extracted phase shifts
in this energy range.}
\label{fig:lowpass}
\end{figure}
The result is shown in Fig.~\ref{fig:lowpass}.
It would be unsurprising that our filtered Hamiltonian fails close
to the cut-off, but it is evident that there is a failure at all
energies. What happened? The basic problem is that low momentum
and high momentum are coupled when solving quantum mechanically
for observables. For example, consider perturbation theory for the (tangent)
of the phase shift (represented schematically here):
\beq
\langle k | V | k \rangle
+ \sum_{k'}\frac{\langle k | V | k' \rangle \langle k' | V | k \rangle}
{(k^2 - {k'}^2)/m} + \cdots
\label{eq:ptphase}
\eeq
where a low momentum $k$ is mixed with all other momenta at second order
to a degree based on the size of the off-diagonal matrix elements.
(As a computational aside, although momentum is continuous
in principle, in practice we work on a discrete grid. This means
that Eq.~\eqref{eq:ptphase} becomes a matrix equation. For two-body
potentials, roughly $100\times100$ matrices are sufficient.)
The phase shift even at low energy or $k$ will get large contributions from
high $k'$ if the coupling matrix elements $\langle k | V | k' \rangle$
are large.
How can we fix this? Our solution is to use a (short-distance)
unitary transformation of the Hamiltonian
matrix to \emph{decouple} low and high energies.
That is, insert the operator $U^\dagger U = 1$ repeatedly:
\bea
E_n \ampseq \langle \Psi_n | H | \Psi_n \rangle
= \langle \Psi_n | U^\dagger ) U H U^\dagger
U | \Psi_n \rangle ) \nonumber\\
\amps{\equiv} \langle \widetilde\Psi_n | \widetilde H | \widetilde\Psi_n \rangle \;.
\eea
In doing so we have modified operators \emph{and} wavefunctions but
observables (measurable quantities) are unchanged.
An appropriate choice of the unitary transformation can, in principle,
achieve the
desired decoupling. This general approach has long been used in
nuclear structure physics and for other many-body applications.
The new feature here is the use of renormalization group flow
equations to create the net unitary transformation via a series
of infinitesimal transformations.
The renormalization group is well suited to this purpose.
The common features of RG for critical phenomena and high-energy
scattering are discussed by Steven Weinberg in an essay in
Ref.~\cite{Guth:1984rq}. He summarizes:
\begin{quote}
``The method in its most general form can I think be understood
as a way to arrange in various theories that the degrees of freedom
that you're talking about are the relevant degrees of freedom for the
problem at hand.''
\end{quote}
This is the essence of what is done with the low-momentum interaction
approaches considered here: arrange for the degrees of freedom for nuclear structure
to be the relevant ones. This does not mean that other degrees of
freedom cannot be used, but to again quote
Weinberg~\cite{Guth:1984rq}: ``You can use any degrees of freedom you
want, but if you use the wrong ones, you'll be sorry.''
The consequences of using RG for high-energy (particle) physics include improving perturbation
theory, e.g., in QCD.
A mismatch of energy scales can generate large logarithms
that ruins perturbative convergence even when couplings by
themselves are small.
The RG shifts strength between loop integrals and coupling constants
to reduce these logs. For critical phenomena in condensed matter
systems, the RG reveals the
nature of observed
universal behavior by filtering out short-distance degrees
of freedom. We will see both these aspects in our calculations
of nuclear structure and reactions. The end result can be said
to make nuclear physics look more like quantum chemistry, opening
the door to a wider variety of techniques (such as many-body perturbation
theory) and simplifying calculations (e.g., by improving convergence
of basis expansions).
\subsection{Summary points}
Low-energy nuclear physics
is made difficult by a mismatch
of energy scales inherent in conventional phenomenological potentials
and those of nuclear structure and reactions.
The renormalization group offers a way out by lowering the resolution,
which means decoupling low- from high-momentum degrees of freedom.
\section{Overview of flow equations}
\label{sec:flow}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=2.2in]{vlowk_schematic}
\hspace*{.5in}
\includegraphics[width=2.2in]{srg_schematic}
\caption{Schematic $\vlowk$ RG evolution (left) and
flow equation RG evolution (right).}
\label{fig:vlowkschematic}
\end{center}
\end{figure*}
In Fig.~\ref{fig:vlowkschematic}, we show
schematically two options for how the RG can be used to decouple
a Hamiltonian matrix. The more conventional approach on the left in
Fig.~\ref{fig:vlowkschematic} lowers a cutoff $\Lambda$ in momentum
in small steps, with the matrix elements adjusted by requiring some
quantity such as the on-shell T-matrix to be invariant.
(In practice this may be carried out by enforcing that the half-on-shell
T-matrix is independent of $\Lambda$.)
Matrix elements above $\Lambda$ are zero and are therefore trivially
decoupled.
When adapted to low-energy nuclear physics, this approach is typically
referred to as ``$\vlowk$'' \cite{Bogner:2003wn,Bogner:2009bt}.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=5.8in]{tiled_s_3S1_kvnn06_reg_0_3_0_ksq}
\caption{SRG flow of the AV18 NN potential in the $^3$S$_1$ channel
at selected values of the flow parameter $\lambda$~\cite{Bogner:2009bt}.}
\label{fig:srgAV183S1}
\end{center}
\end{figure*}
The approach we will focus on is illustrated in Fig.~\ref{fig:vlowkschematic}
on the right,
in which the matrix is driven toward band-diagonal form, achieving
decoupling again but without truncating the matrix.
The corresponding RG was developed in the early 1990's
by Wegner~\cite{Wegner:1994,Wegner:2000gi,Wegner:2006zz} for condensed matter applications
under the name ``flow equations'' and independently by
Glazek and Wilson~\cite{Glazek:1993rc} for solving quantum chromodynamics
in light-front formalism under the name ``similarity renormalization group''
(SRG).
Only in the last five years was it realized that the band-diagonal
approach is
particular well suited for low-energy nuclear physics, where
it is technically simpler and more versatile than other methods~\cite{Bogner:2006pc,Bogner:2009bt}.
We will apply formalism closer to the flow equation formulation,
but generally use the simple abbreviation SRG.
We have introduced a cutoff-like parameter $\lambda$ that serves
as a momentum decoupling scale. Elsewhere we will also use the natural
flow parameter $s = 1/\lambda^4$.
\subsection{Flow equation basics}
\label{subsec:floweq}
The basic flow equation is a set of coupled differential equations
for the (discrete) matrix elements
of the Hamiltonian matrix (or the potential in practice, because
we fix the kinetic energy matrix by construction).
Let's see an example in action before stepping back and considering details.
For the two-body
potential in a partial-wave momentum basis, the flow equation takes the form
\bea
&& \frac{dV_\lambda}{d\lambda}(k,k') \propto
- (\epsilon_k - \epsilon_{k'})^2 V_\lambda(k,k') \nonumber \\
&& \quad \null + \sum_q (\epsilon_k + \epsilon_{k'} - 2\epsilon_q)
V_\lambda(k,q) V_\lambda(q,k')
\;,
\eea
where $k$ and $k'$ are the relative momenta and
$\epsilon_k \equiv \hbar^2 k^2/M$ and we have omitted an inessential constant.
The evolution is continuous, but snapshots at selected $\lambda$ values
are shown in Fig.~\ref{fig:srgAV183S1}.
(Note that the axes are the kinetic energy $k^2$.)
The evolution toward diagonalization is evident, with the width
of the band in $k^2$ roughly given by $\lambda^2$.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=4.8in]{low_pass_works}
\caption{Nucleon-nucleon (NN)
phase shifts showing the effect of a low-pass filter
at $k_{\rm max} = 2.2\fmi$ in various partial-wave
channels~\cite{Bogner:2007jb,Jurgenson:2007td}. In each
panel, the solid line is for the original AV18 potential while
the dotted line is the result when the potential is set to zero
for $\{k,k'\}>k_{\rm max}$. The dashed line is the results
when the potential is evolved first and then cut at $k_{\rm max}$.}
\label{fig:lowpassworks}
\end{center}
\end{figure*}
It is evident that the off-diagonal matrix elements are driven toward
zero, so we expect that a low pass filter will now be effective.
Indeed it is, as shown in Fig.~\ref{fig:lowpassworks}, where NN phase
shifts for the AV18 potential in a variety of channels are compared
with the results from applying a low-pass filter to the original
(dotted) and evolved (dashed) potentials. The evolved result agrees
up to the low-pass cutoff. Note that if this cutoff is not applied, the phase
shifts for the evolved Hamiltonian agree precisely \emph{at all energies},
because the transformation is unitary in the two-body system
(and this is preserved to high numerical precision).
If we now revisit the consequence of a repulsive core for the deuteron
probability density, we see in Fig.~\ref{fig:deutprob} that the
``wound'' in the wave function at small $r$ is filled in
as the core is transformed away.
(Note: it may look like the normalization is not conserved, but
if we multiplied by $r^2$, we would see that the area under the curves
is the same, as are the large-$r$ tails.)
Thus the short-range correlations in the wave function are
drastically reduced. This means that the physics interpretation of
various phenomena is altered, even though the observable quantities
such as energies and cross sections are unchanged.
(The long-range part of the wave function \emph{is} preserved;
this is related to the asymptotic normalization constants, which can
be extracted from experiment.)
We cannot immediately visualize the changes in the potential in coordinate
space in a conventional plot like Fig.~\ref{fig:phenpots}, however,
because it is now \emph{non-local}, which is to say it is not diagonal
in coordinate representation:
\beq
V({\bf r})\psi({\bf r}) \longrightarrow
\int\! d^3{\bf r'}\, V({\bf r},{\bf r'})\psi({\bf r'})
\;.
\eeq
This is a technical problem for certain quantum many-body methods (such
as Green's function Monte Carlo) but not for methods using harmonic
oscillator matrix elements.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=5.8in]{LP_Side_pot_Swave}
\caption{Local projection of AV18 and N3LO(500\,MeV) potentials
in $^3$S$_1$ channel
at different resolutions~\cite{Wendt:2011private}.}
\label{fig:LPsideSwave}
\end{center}
\end{figure*}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=5.8in]{LP_Side_pot_Dwave_scaled}
\caption{Local projection of AV18 and N3LO(500\,MeV) potentials in mixed
$^3$S$_1$--$^3$D$_1$ channel
at different resolutions~\cite{Wendt:2011private}.}
\label{fig:LPsideDwave}
\end{center}
\end{figure*}
We can visualize the evolution \emph{approximately} by considering a local
projection of the potential:
\beq
\overline V_\lambda(r) = \int\! d^3r'\, V_\lambda(r,r')
\eeq
which leaves a local $V(r)$ unchanged.
For a non-local potential, this roughly gives the action of
the potential on long-wavelength nucleons.
This is shown in Figs.~\ref{fig:LPsideSwave} and \ref{fig:LPsideDwave},
where in the former we see the central potential dissolving and
in the latter similar effects on the tensor part of the potential~\cite{Wendt:2011private}.
Also evident is the flow of the two potentials, initially quite
different (potentials are not observables!), toward a universal
flow at the lower values of $\lambda$.
Very recent work suggests that such a local projection may
capture most of the physics of the full evolved potential, with
the effects of the residual potential calculable in perturbation
theory. This may open the door to using RG-evolved potential
with quantum Monte Carlo methods.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{He4_vs_nmax_highlight_after}
\caption{Convergence of NCFC calculations of the ${^4}$He ground state
energy with basis size at SRG different resolutions. The initial
potential includes both two- and three-body components~\cite{Jurgenson:2009qs,Jurgenson:2010wy}.}
\label{fig:He4vsNmax}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.55in]{Li6_vs_nmax_highlight_after}
\caption{Convergence of NCFC calculations of the ${^6}$Li ground state
energy with basis size at SRG different resolutions. The initial
potential includes both two- and three-body components~\cite{Jurgenson:2009qs,Jurgenson:2010wy}.}
\label{fig:Li6vsNmax}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.3in]{ncsm_matrix_dimension_vs_Nmax}
\caption{Dimension of the Hamiltonian matrix for NCFC/NCSM calculations
as a function of the harmonic oscillator basis size ($\Nmax$ shells)
for selected nuclei.}
\label{fig:ncsmsize}
\end{center}
\end{figure}
The consequences of low-momentum potentials
for harmonic-oscillator-basis calculations are
illustrated in Figs.~\ref{fig:He4vsNmax} and \ref{fig:Li6vsNmax}.
(We'll refer to the method used, which is a direct diagonalization of
the Hamiltonian matrix and therefore is variational, as no-core full configuration, or NCFC.)
The original potential in this case is already soft (that is, there
is much
less coupling to high momentum than in the AV18 potential), but
convergence in a harmonic oscillator basis with $\Nmax$ shells for
excitation is slow (these are the ``Original'' curves).
Note that the matrix dimension grows rapidly with $\Nmax$ \emph{and}
the number of nucleons $A$;
for example $\Nmax = 8$ has dimension about 50,000 for $^4$He
but over one million for $^6$Li (see Fig.~\ref{fig:ncsmsize} for
other examples).
But with SRG evolution, there is vastly improved convergence.
The convergence is also smooth, which makes it possible to
reliably extrapolate
partly converged results to the $\Nmax \rightarrow \infty$ limit.
Nevertheless, the rapid growth of the basis with $A$ is still
a major hindrance to calculating larger nuclei. One solution
being explored by Roth and collaborators~\cite{Roth:2007sv,Roth:2011ar} is
to use importance sampling of matrix elements, in which only
a fraction of the full matrix is used. This technique
is enabled by the RG softening of the potential, which allows the
importance to be evaluated perturbatively.
Let's now return to the basics of SRG flow equations.
We wish to transform an initial hamiltonian, $ H = T + V$
in a series of steps, labeled by the flow parameter $s$:
\beq
H_\flow = U_\flow H U^\dagger_\flow \equiv T + V_\flow
\eeq
with
\beq
\quad U^\dagger_\flow U_\flow =
U_\flow U^\dagger_\flow = 1 \;.
\eeq
Note that the kinetic energy $T$ is taken to be independent of $s$. Differentiating
with respect to $s$:
\bea
\frac{dH_\flow}{d\flow}
\amps{=} \frac{dU_\flow}{d\flow} U^\dagger_\flow U_\flow H
U^\dagger_\flow + U_\flow H U^\dagger_\flow U_\flow
\frac{dU^\dagger_\flow}{ds} \nonumber \\
\amps{=} [\eta_\flow,H_\flow]
\eea
with
\beq
\eta_\flow \equiv \frac{dU_\flow}{d\flow} U^\dagger_\flow
= -\eta^\dagger_\flow
\ .
\eeq
The anti-Hermitian generator $\eta_\flow$ can be specified by a commutator
of $H_\flow$ with
a Hermitian operator $G_s$:
\beq
\eta_\flow = [G_s, H_\flow] \ ,
\eeq
which yields the flow equation (with $T$ held fixed!),
\beq
\frac{dH_\flow}{d\flow}
= \frac{dV_s}{d\flow}
= [ [G_s, H_\flow], H_\flow] \ .
\eeq
The operator $G_s$ determines the flow and there are many choices
one can consider.
\begin{figure}[b!]
\begin{center}
\includegraphics[width=2.55in]{srg_E_zero_1p0_beta_2p00_lam_1p0_2body}
\caption{SRG flow for a simple two-state system (see text).}
\label{fig:two-state}
\end{center}
\end{figure}
\begin{figure*}[thb!]
\begin{center}
\includegraphics[width=1.5in]{srg_rhs_1S0_kvnn_10_lam2p0_reg_0_3_0_pot}
\includegraphics[width=1.13in]{srg_rhs_1S0_kvnn_10_lam2p0_reg_0_3_0_rhs1}
\includegraphics[width=1.5in]{srg_rhs_1S0_kvnn_10_lam2p0_reg_0_3_0_rhs2}
\includegraphics[width=1.5in]{srg_rhs_1S0_kvnn_10_lam1p5_reg_0_3_0_pot}
\caption{SRG flow in stages. On the far left and far right are potentials
in the $^1$S$_0$ channel evolved to $\lambda = 2.0\fmi$ and $\lambda=1.5\fmi$, respectively. The middle panels show the first (left) and second (right)
terms on the right side of Eq.~\eqref{eq:srgeq2}.}
\label{fig:srgstages}
\end{center}
\end{figure*}
Probably the simplest example we can consider is just a two-state
system~\cite{Bogner:2007qb}. Let $H = T + V$, where
\beq
T|i\rangle = \epsilon_i | i\rangle
\quad\mbox{and}\quad
V_{ij} \equiv \langle i|V|j\rangle
\;.
\eeq
Then we can choose $G_s = T$ and
\beq
\frac{dH_s}{ds} = [[T,H_s],H_s]
\eeq
becomes (with the $s$ dependence implicit)
\bea
\frac{d}{ds} V_{ij} \ampseq
-(\epsilon_i-\epsilon_j)^2 V_{ij}
\nonumber \\ \amps {} \null
+ \sum_k (\epsilon_i+\epsilon_j-2 \epsilon_k) V_{ik} V_{kj}
\;.
\eea
For a two-level system with $i = \{a,b\}$, we can express the
flowing Hamiltonian in terms of Pauli matrices:
\beq
T = \frac12(\epsilon_a + \epsilon_b) I + \frac12(\epsilon_a - \epsilon_b)
\sigma_z
\eeq
and
\bea
V_s \ampseq \frac12 (V_{aa}+V_{bb})I
\nonumber \\ \amps{} \null + \frac12 (V_{aa}-V_{bb})\sigma_z
+ V_{ab}\sigma_x \;.
\eea
The solution to the SRG flow equation is easily found.
It is convenient to parametrize
the result in terms of $\theta(s)$ with constant $\omega$:
\beq
\frac{d\theta}{ds} = -2(\epsilon_a-\epsilon_b) \omega \sin\theta(s)
\eeq
where
\beq
\theta(s) = 2 \tan^{-1}[\tan(\theta(0)/2) e^{-2(\epsilon_a-\epsilon_b)
\omega s}]
\;,
\eeq
\beq \omega \cos\theta = (\epsilon_a-\epsilon_b+ V_{aa}-V_{bb})/2
\;,
\eeq
and
\beq
\omega \sin\theta = V_{ab} \;.
\eeq
The resulting flow is plotted for sample energies $\epsilon_a$ and $\epsilon_b$ in Fig.~\ref{fig:two-state}. We clearly see the off-diagonal matrix
element $V_{ab}$ driven to zero. (Try reproducing
this in Mathematica!)
For a nuclear two-body (NN) potential
in a partial-wave momentum basis with $\eta_s = [T,H_\flow]$,
we project on relative momentum states $|k\rangle$
using $1 = \frac{2}{\pi}\int_0^\infty|q\rangle q^2\,dq \langle q |$
with $\hbar^2/M = 1$. The flow equation reduces to:
\beq
\frac{dV_s}{d\flow}
= [ [\Hzero, V_\flow], H_\flow]
\quad \mbox{with} \quad
\Hzero | k \rangle = \epsilon_k | k \rangle
\eeq
and $\lambda^2 = 1/\sqrt{s}$.
$\Trel$ is the relative kinetic energy of the nucleons.
Then
\bea
{\frac{dV_\lambda}{d\lambda}(k,k')} \amps{\propto}
{ - (\epsilon_k - \epsilon_{k'})^2 V_\lambda(k,k') }
\nonumber \\ \amps{\null} \hspace*{-.4in}
\null + { \sum_q (\epsilon_k + \epsilon_{k'} - 2\epsilon_q)
V_\lambda(k,q) V_\lambda(q,k')} \;.
\label{eq:srgeq2}
\eea
This particular equation is for $A=2$, but the results
are generic if one lets $k$ represent a set of Jacobi momenta.
The first term in Eq.~\eqref{eq:srgeq2} drives $V_\lambda$ toward the diagonal:
\beq
V_\lambda(k,k') = V_{\lambda=\infty}(k,k')
\, e^{-[(\epsilon_k-\epsilon_{k'})/\lambda^2]^2} + \cdots
\;, \label{eq:diagonalize}
\eeq
which can be visualized in Fig.~\ref{fig:srgstages}.
These panels represent a sequence from $\lambda = 2.0$ to $\lambda = 1.5$.
The potentials at the beginning and the end are on the outside, while the
two middle panels are the first and second term of Eq.~\eqref{eq:srgeq2}.
For off-diagonal matrix elements, the first term is numerically dominant
and each element is driven to zero
according to Eq.~\eqref{eq:diagonalize}, with further off-diagonal
elements changing more rapidly.
Note that the width of the diagonal is given roughly by $\lambda^2$,
in accord with Eq.~\eqref{eq:diagonalize}.
A more general proof follows if we use the generator advocated by
Wegner, which includes the diagonal part of the Hamiltonian, $H_d$.
Call the diagonal elements $H_{ii} = e_i$, then (with $\eta_s = [H_d,H_s]$),
\bea
\frac{dH_{ij}}{ds} \ampseq \langle i | [[H_d,H_s],H_s] | j \rangle
\nonumber \\ \ampseq \sum_k (e_i + e_j - 2 e_k) H_{ik}H_{kj}
\nonumber \\ \ampseq 2\sum_k (e_i - e_k) |H_{ik}|^2
\;.
\eea
But
\bea
\frac{d}{ds} \sum_i |H_{ii}|^2
\ampseq 2\sum_i H_{ii}\frac{dH_{ii}}{ds} \nonumber\\
\ampseq 4 \sum_{i\neq k} e_i(e_i-e_k)|H_{ik}|^2 \nonumber\\
\ampseq 2\sum_{i\neq k}(e_i-e_k)^2 |H_{ik}|^2 \geq 0
\;.
\eea
Now use:
\beq
\Tr H_s^2 = \mbox{const.} = \sum_{ij} |H_{ij}|^2 =
\sum_{i} |H_{ii}|^2 + \sum_{i\neq j} |H_{ij}|^2
\;,
\eeq
to obtain
\bea
\frac{d}{ds}\sum_{i\neq j}|H_{ij}|^2
\ampseq -\frac{d}{ds}\sum_{i}|H_{ii}|^2 \nonumber \\
\ampseq -2\sum_{i\neq k}(e_i-e_k)^2 |H_{ik}|^2 \leq 0 \;.
\eea
Thus, in the absence of degeneracies, the off-diagonal matrix
elements will decrease (or at least remain unchanged)~\cite{Kehrein:2006}.
This feature of the diagonal generator is desirable, but we also
note that for nuclear Hamiltonians in a momentum basis, the
diagonal is completely dominated by the kinetic energy, so
$H_d \approx \Trel$ is a very good approximation.
Can this break down? Glazek and Perry~\cite{Glazek:2008pg}
showed that it can (see also Wendt et al.~\cite{Wendt:2011qj}).
Reconsider the proof of diagonalization, but now with $G_s = \Hzero$.
Now we have $H_{ii} = e_i$ and
$\Hzero | i\rangle = \epsilon_i | i \rangle$, so that
\beq
\frac{d H_{ii}}{ds} = 2\sum_k (\epsilon_i - \epsilon_k) |H_{ik}|^2
\;.
\eeq
So consider
\beq
\frac{d}{ds} \sum_i |H_{ii}|^2
= 2\sum_i H_{ii}\frac{dH_{ii}}{ds}
= 4 \sum_{i\neq k} e_i(\epsilon_i-\epsilon_k)|H_{ik}|^2
\;,
\eeq
from which we conclude
\beq
\frac{d}{ds}\sum_{i\neq j}|H_{ij}|^2
= -2\sum_{i\neq k}(e_i-e_k)(\epsilon_i-\epsilon_k) |H_{ik}|^2
\;.
\eeq
Thus the off-diagonal decrease depends on
$ e_i - e_k \approx \epsilon_i - \epsilon_k $.
But there is the possibility of this not being true,
e.g., if there are spurious bound states as in large-cutoff EFT~\cite{Wendt:2011qj}.
\subsection{Alternative generators}
Other choices of generator are also possible. Recent work by
Shirley Li while an undergraduate physics major at Ohio State
explored choices designed to accelerate evolution.
In particular, one can choose $G_s$ as
\beq
G_s = -\frac{\Lambda^2}{1+\Trel/\Lambda^2} \approx c + \Trel
+ \cdots
\eeq
or
\beq
G_s = -{\Lambda^2}e^{-\Trel/\Lambda^2} \approx c + \Trel
+ \cdots
\eeq
The expansions show that these reduce to the conventional $\Trel$
for momenta that are small compared to the cutoff parameter $\Lambda$
(not to be confused with $\lambda$). For $\Lambda = 2\fmi$, the low
energy part of the potential is still decoupled but there is much
less evolution at high energy, which makes it computationally much faster
and allows evolution to low $\lambda$. See Ref.~\cite{Li:2011sr} for details.
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{fig_LSvlowk}
\caption{Schematic version of the Lippmann-Schwinger equation
for the T-matrix, with cutoff $\Lambda$ on the intermediate
states. The $\vlowk$ potential $V_{\Lambda}$ is determined
by requiring half-on-shell matrix elements of this equation
to be invariant under changes in
$\Lambda$\cite{Bogner:2003wn,Bogner:2009bt}.}
\label{fig:LSvlowk}
\end{center}
\end{figure}
The usual approach to $\vlowk$ RG evolution
(left diagram in Fig.~\ref{fig:vlowkschematic}) is based on the Lippmann-Schwinger
equation for the half-on-shell T-matrix illustrated in Fig.~\ref{fig:LSvlowk}.
A cutoff $\Lambda$ is imposed on the integral in the second term
and we demand that matrix elements of T are invariant with an
infinitesimal reduction of $\Lambda$. That is, we require
$dT(k,k';E_k)/d\Lambda = 0$, which establishes an RG equation
for $V_\Lambda$. In contrast to the SRG equation, which is second
order in the running potential, the $\vlowk$ RG equation has the
T-matrix on the right side. Thus
\bea
T(k',k;k^{2}) \ampseq \vnn(k',k) \nonumber \\
\amps{} \hspace*{-.3in} \null +\frac{2}{\pi} \, \mathcal{P}
\int_{0}^{\lm_\infty}
\frac{\vnn(k',p) \, T(p,k;k^{2})}{k^{2}-p^{2}} \, p^{2} dp \nonumber \\
\ampseq
\vlowk^\lm(k',k) \nonumber \\
\amps{} \hspace*{-.3in} \null + \frac{2}{\pi} \, \mathcal{P}
\int_{0}^{\lm} \frac{\vlowk^\lm(k',p) \, T(p,k;k^{2})}{k^{2}-p^{2}} \,
p^{2} dp
\nonumber \\
\eea
for all $k,k' < \Lambda$.
(Note: we are using standing-wave boundary conditions for numerical
reasons; this is often called the K-matrix.)
From $dT/d\Lambda = 0$, we get the $\vlowk$ RG equation:
\beq
\frac{d}{d \lm} \vlowk^\lm(k',k) = \frac{2}{\pi} \frac{\vlowk^\lm(k',\lm) \,
T^\lm(\lm,k;\lm^{2})}{1-(k / \lm)^{2}} \;.
\eeq
Note that the full T~matrix appears on the right side,
in contrast to the partial-wave SRG flow equation (with $G_s = \Hzero$),
\bea
{\frac{d}{d\lambda}V_\lambda(k,k')} \amps{\propto}
{ - (\epsilon_k - \epsilon_{k'})^2 V_\lambda(k,k') } \nonumber \\
&&
\hspace*{-.6in}
\null + { \sum_q (\epsilon_k + \epsilon_{k'} - 2\epsilon_q)
V_\lambda(k,q) V_\lambda(q,k')}
\;,
\eea
which is only second-order in the potential.
We can also define smooth regulators for $\vlowk$ as in
Ref.~\cite{Bogner:2006vp}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=5.6in]{tiled_s_3S1_kvnn06_reg_1_1_2_ksq}
\includegraphics[width=5.6in]{tiled_s_3S1_kvnn06_reg_2_2_2_8_ksq}
\vspace*{-.1in}
\caption{$\vlowk$ flow of AV18 in the $^3$S$_1$ channel for a sharp
(top) and smooth (bottom) regulator~\cite{Bogner:2006vp}.}
\label{fig:vlowkAV183S1}
\end{center}
\end{figure*}
The evolution of NN potentials using the $\vlowk$ method is illustrated
for a sharp and smooth regulator in Fig.~\ref{fig:vlowkAV183S1}.
Comparing the $\vlowk$ flow in these figures to the
SRG flow in Fig.~\ref{fig:srgAV183S1}, we see the same decoupling
of low and high momentum.
Other non-RG unitary transformations (which perform the transformation
all at once, rather than in steps) also decouple;
an example is the UCOM method~\cite{Feldmeier:1997zh},
which has close
connections to the SRG (see Refs.~\cite{Hergert:2007wp,Roth:2008km}).
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=5.8in]{srg_pot_eta_BD_sharp_Lam2p0_1P1_series}
\vspace*{-.1in}
\caption{SRG sharp block-diagonal flow of AV18 in
the $^1$P$_1$ channel at $\lambda = 4$, $3$, $2$, and $1\fmi$~\cite{Anderson:2008mu}.
The initial N$^3$LO potential is from Ref.~\cite{Entem:2003ft}
and $\Lambda = 2\fmi$. The axes are in units of $k^2$ from 0 to 11\,fm$^{-2}$
and the color scale is from $-0.5$ to $+0.5\,$fm.}
\label{fig:srgbd1P1}
\end{center}
\end{figure*}
It is also possible to choose a generator that reproduces the
block diagonal (as opposed to band diagonal) form of the $\vlowk$ RG
shown schematically in Fig.~\ref{fig:vlowkschematic}, except that
the transformation will be unitary.
In particular, we can use
\beq
\frac{dH_\flow}{d\flow} = [ [G_s, H_\flow], H_\flow]
\label{eq:srgbd3}
\eeq
with
\beq
G_s = \left(
\begin{array}{cc}
PH_{s}P & 0 \\
0 & QH_{s}Q
\end{array}
\right)
\;,
\eeq
where projection operators $P$ and $Q = 1 - P$ are simply step functions
at a given $\Lambda$ in partial-wave momentum representation.
An example of the subsequent
flow is shown in Fig.~\ref{fig:srgbd1P1} for the
$^1$P$_1$ channel~\cite{Anderson:2008mu}. To get the fully block-diagonal form, one would
have to evolve to $\lambda = \infty$. But in practice, going to
$\lambda = 1\fmi$ is sufficient for essentially complete decoupling
at $\Lambda = 2\fmi$.
The proof of block diagonalization (see Gubankova et al.~\cite{Gubankova:1997ha,Gubankova:1997mq}) goes
as follow.
The generator $\eta_s = [G_s,H_s]$ is non-zero only where $G_s$ is zero,
which means in the off-diagonal blocks.
This will then evolve the potential in this same pattern (this is
generically true, if one desires a different pattern~\cite{Anderson:2008mu}).
A measure of off-diagonal coupling of $H_s$ is
\beq
{\rm Tr}[(Q H_s P)^{\dagger} (Q H_s P)]
= {\rm Tr}[P H_s Q H_s P] \geqslant 0
\;.
\eeq
Now we can calculate the derivative of this expression
by applying the SRG equation \eqref{eq:srgbd3}:
\bea
\frac{d}{ds} \, {\rm Tr}[P H_s Q H_s P]
\ampseq
\nonumber \\
\amps{} \hspace*{-.55in}
{\rm Tr}[P\eta_s Q(Q H_s Q H_s P - Q H_s P H_s P)]
\nonumber \\
\amps{} \hspace*{-.55in}
\null + {\rm Tr}[(P H_s P H_s Q - P H_s Q H_s Q) Q\eta_s P]
\nonumber \\
\amps{} \hspace*{-.55in} =
-2 \, {\rm Tr} [(Q \eta_s P)^{\dagger} (Q \eta_s P)] \leqslant 0
\;.
\eea
Thus the off-diagonal $Q H_s P$ block will decrease as $s$
increases.
The low-momentum block of this SRG is found to be remarkably similar
to the corresponding $\vlowk$ RG potential.
Examples are shown in Figs.~\ref{fig:srgbd2} and \ref{fig:srgbd3}.
However, \emph{proving} that these different RG approaches should
yield the same potential remains an open problem.
\begin{figure}[ht!]
\begin{center} \includegraphics[width=2.9in]{vlowk_vsrg_sharp_bd_3S1_kvnn_10_lam0p5_Lam2p0_contour_k}
\vspace*{-.1in}
\caption{Comparison of momentum-space (a) $\vlowk$ and (b) SRG
block-diagonal $^3$S$_1$ potentials with $\Lambda = 2\fmi$~\cite{Anderson:2008mu}. }
\label{fig:srgbd2}
\vspace*{.2in}
\includegraphics[width=2.9in]{vlowk_vsrg_sharp_bd_3S1_kvnn_10_lam0p5_Lam2p0_surface_k}
\vspace*{-.1in}
\caption{Same as Fig.~\ref{fig:srgbd2}, only a surface plot.}.
\label{fig:srgbd3}
\end{center}
\end{figure}
\subsection{Many-body forces}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.7in]{three_body_energies}
\caption{Convergence of three-body energies for two potentials
and several sets of $\lambda$'s.}
\label{fig:threebodyenergies}
\end{center}
\end{figure}
\begin{figure}[bh!]
\begin{center}
\includegraphics[width=2.7in]{steiner_be_vs_density}
\vspace*{-.1in}
\caption{Binding energies of nuclear and neutron
matter from Akmal et al.\ for several equations of state~\cite{Akmal:1998cf}.}
\label{fig:matter}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.7in]{nm_vlowk_chiral_NNcompare_improved}
\vspace*{-.1in}
\caption{Nuclear matter at third order in perturbation theory
in the particle-particle channel (this appears sufficient for
convergence) using a $\vlowk$ RG-evolved
potential~\cite{Hebeler:2010xb}. The ``NN only'' curves include
no three-body interactions while the ``NN + 3N'' curves include
a 3NF fit to the triton binding energy and the alpha particle radius.}
\label{fig:nm_vlowk}
\end{center}
\end{figure}
In Fig.~\ref{fig:threebodyenergies}, the convergence of the
triton ground-state energy with harmonic oscillator basis
size ($\Nmax\hbar\omega$ excitations)
is shown for two chiral EFT NN potentials and for the corresponding
SRG-evolved potentials at $\lambda$'s from $4\fmi$ to $1\fmi$.
In accord with our previous discussion, we see
increasingly rapid convergence as $\lambda$ decreases.
(Note that there is softening already at $\lambda = 3\fmi$ for
the N$^3$LO EFT with $\Lambda = 600\,$MeV, which corresponds to
$3\fmi$. The moral is that it is not sufficient to simply compare the cutoff numerically
to the decoupling scale $\lambda$.)
However, we also see that the converged (or extrapolated) binding energies are different for each $\lambda$!
How could this be, if
the SRG is supposed to generate unitary transformations?
There are further signs of trouble, such as nuclear matter failing
to saturate after the two-body potential is softened by
RG evolution.
Let's review the facts about uniform nuclear matter.
Figure~\ref{fig:matter} shows the binding energy per particle
for pure neutron matter ($Z=0$) and symmetric nuclear matter ($N=Z=A/2$) from calculations
adjusted to be consistent with extrapolations from nuclei.
Neutron matter has positive pressure, while symmetric
nuclear matter \emph{saturates}; that is, there is
a minimum at a density of about $0.16\,\mbox{fm}^3$ with a binding energy
of about $-16\,$MeV/$A$.
Reproducing this minimum with microscopic interactions fit only
to few-body data is one of the holy grails of nuclear
structure theory.
But if we evolve an NN potential with either the $\vlowk$ RG or
the SRG, we find that nuclear matter does not converge. This is shown
for $\vlowk$ by the ``NN only'' curves in Fig.~\ref{fig:nm_vlowk} and
similar behavior is found for SRG.
The failure of softened low-momentum potentials to reproduce nuclear matter
saturation should sound familiar to anyone who knows the long
history of low-energy nuclear theory.
There were active attempts to transform away hard cores and soften
the tensor interaction in the late sixties and early seventies.
But the requiem for soft potentials was given by Bethe in 1971~\cite{Bethe:1971xm}:
``Very soft potentials must be excluded because they do not give
saturation; they give too much binding and too high density. In
particular, a substantial tensor force is required.''
The next thirty-five years were spent
struggling to solve accurately with such ``hard'' potentials.
But the story is not complete: the three-nucleon forces (3NF)
were not properly accounted for!
Three-body forces between protons and neutrons have a classical analog
in tidal forces: the gravitational force on the
Earth is not just the pairwise sum of Earth-Moon and Earth-Sun forces.
Quantum mechanically, an analog is with the three-body force
between atoms and molecules,
which is called the Axilrod-Teller term and dates from 1943~\cite{Axilrod:1943aa}.
The origin is from triple-dipole mutual polarization.
It is a third-order perturbation correction, so the weakness of
the fine structure constant means that these forces are usually
negligible in metals and semiconductors. However, in solids bound
by van der Waals potentials it can be significant; for example, it is
ten percent of the binding energy in solid xenon~\cite{Bell:1976aa}.
This is the same relative size as needed in the triton.
In general, three-body forces arise from eliminating degrees of
freedom. In the nuclear case, this can mean eliminating excited
states of the nucleon ($N^\ast$ or $\Delta$) or
from relativistic effects;
see Fig.~\ref{fig:3N_diags_left} for diagrammatic representations.
If the intermediate states are not included in the low-energy degrees
of freedom, we have irreducible vertices with three nucleons in and
three nucleons out.
But 3NF's also result from
the decoupling of high-momentum intermediate states, whether
they are eliminated explicitly by a cutoff (as with the $\vlowk$ RG)
or the degree of coupling modified (as with the SRG).
Omitting three-body forces leads to model dependence: observables
will depend on the decoupling scale, whether it is the $\vlowk$ $\Lambda$
or the SRG $\lambda$.
This dependence also becomes a tool, because it is a diagnostic
for errors (more on this later).
To eliminate this dependence,
the 3NF at different $\Lambda$ or $\lambda$ must be either fit or
evolved.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.4in]{3N_diags_left}
\vspace*{-0.05in}
\caption{Sources of internucleon three-body forces. On the
left, the intermediate state includes an excitation of the
nucleon ($\Delta$ or $N^\ast$). On the right, the intermediate
state includes a virtual anti-nucleon ($\overline N$). When
these degrees of freedom are integrated out, the resulting
nucleons-only vertex is a 3NF.}
\label{fig:3N_diags_left}
\end{center}
\end{figure}
It is easy to see that RG flow equations lead to many-body operators.
Consider the SRG operator flow equation written with
second-quantized $a$'s and $a^\dagger$'s:
\bea
\frac{dV_s}{ds} \ampseq
\Bigl[ \Bigl[\sum\underbrace{a^\dagger a}_{G_s},
\sum \underbrace{a^\dagger a^\dagger a a}_{\mbox{2-body}} \Bigr],
\sum \underbrace{a^\dagger a^\dagger a a}_{\mbox{2-body}} \Bigr]
\nonumber \\
\ampseq \cdots + {\sum \underbrace{a^\dagger a^\dagger a^\dagger a a a}_{\mbox{3-body!}}}
+ \cdots
\label{eq:2ndquant}
\eea
where the second equality reflects that even if the initial
Hamiltonian is two-body, the commutators give rise to
three-body terms. (For future reference, recall
that the creation and destruction operators are always defined
with respect to a single-particle basis and a reference state, which
in this case is the vacuum.)
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.4in]{3N_diags_right}
\vspace*{-0.05in}
\caption{Leading three-body forces from chiral EFT. These contributions
represent three different ranges: long-range 2-pion exchange, short-range
contact with one-pion exchange, and pure contact interaction.}
\label{fig:3N_diags_right}
\end{center}
\end{figure}
As the evolution continues,
there inevitably will be $A$-body forces (and operators) generated.
Is this a problem?
Not if these ``induced'' many-body forces are the same size
as those that naturally occur.
Indeed, nuclear three-body forces are already needed in
almost all potentials in common use to get even the triton binding
energy correct.
In fact, low-energy effective theory
tell us generalized diagrams such as those in Fig.~\ref{fig:3N_diags_left}
with four or more legs imply that
there are $A$-body forces (and operators) initially!
However, there is a natural hierarchy predicted from chiral EFT, whose leading
contributions are given in Fig.~\ref{fig:3N_diags_right}
(we'll return to this in Section~\ref{subsec:chiralEFT} and supply additional details).
If we stop the flow equations before induced $A$-body forces are unnaturally
large
or if we can
tailor the SRG $G_s$ to suppress their growth, we will be ok.
(Another option is to choose a non-vacuum reference state,
which is what is done with in-medium SRG, to be discussed later.)
Note that analytic bounds on $A$-body growth have not been derived,
so we need to explicitly monitor the contribution in different
systems.
But the bottom line that makes the SRG attractive as a method
to soften nuclear Hamlitonians is that
it is a tractable method to evolve many-body operators.
To include the 3NF using SRG with normal-ordering in the vacuum,
we start with the SRG flow equation
$dH_s/ds = [[G_s,H_s],H_s]$ (e.g., with $G_s = \Hzero$).
The right side is evaluated without
solving bound-state or scattering equations, unlike the
situation with $\vlowk$, so the SRG
can be applied directly in the three-particle space.
The key observation is that for normal-ordering in the vacuum,
$A$-body operators are completely fixed in the $A$-particle subspace.
Thus we can first solve for the evolution of the two-body potential
in the $A=2$ space, with no mention of the 3NF (either initial
or induced), and then use this NN potential in the equations
applied to $A=3$.
What about spectator nucleons?
There is a decoupling of the 3NF part.
We can see this from the first-quantized version of the SRG
flow equation,
\bea
\frac{dV_s}{ds} \ampseq
\frac{d\Vtwo12}{ds} + \frac{d\Vtwo13}{ds} + \frac{d\Vtwo23}{ds}
+ \frac{d\Vthree}{ds}
\nonumber \\
\ampseq [[\Trel, V_s], H_s] \,,
\eea
where we isolate the contributions from each pair and the 3NF.
Using each SRG equation for the two-body derivatives,
we can cancel them against terms on the right side.
The result is~\cite{Bogner:2006pc}:
\bea
\frac{d\Vthree}{ds} \ampseq
[[\Ttwo12,\Vtwo12], (T_3 + \Vtwo13 + \Vtwo23 + \Vthree)]
\nonumber \\
&& \null
+ \{123\rightarrow132\}
+ \{123\rightarrow231\} \nonumber \\
&& + [[\Trel,\Vthree],H_s] \;.
\eea
The key is that there are no ``multi-valued'' two-body interactions
remaining
(i.e., dependence on the excitation energy of unlinked spectators);
all the terms are connected.
An implementation of these equations in a momentum basis would
be very useful and has very recently been achieved by Hebeler~\cite{Kai}.
But an alternative approach has also succeeded:
a direct solution in a discrete basis~\cite{Jurgenson:2008jp,Jurgenson:2009qs,Jurgenson:2010wy}.
The idea is that the SRG flow equation is an operator equation,
and thus we can choose to evolve in any basis. If one chooses
a discrete basis, than a separate evolution of the three-body
part is not needed. This was first done for nuclei by Jurgenson and
collaborators in 2009 using an anti-symmetrized Jacobi harmonic
oscillator (HO) basis~\cite{Jurgenson:2009qs}.
The technology for working with such a basis had already been
well established for applications to the no-core shell model (NCSM)~\cite{Navratil:1999pw}.
This approach leads to SRG-evolved matrix elements of the potential
directly in the HO basis, which is just what is needed for many-body
applications such as NCFC or coupled cluster.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{H3_Ebind_vs_lambda_nmax36_hw20}
\caption{Triton binding energy during SRG evolution. The three curves
are for an initial potential with only NN components where the
induced 3NF is not kept (``NN-only''), for the same initial NN
potential but keeping the induced 3NF (``NN + NNN-induced''), and
with an initial NNN included as well (``NN + NNN'').}
\label{fig:H3_Ebind_lambda}
\end{center}
\end{figure}
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=2.5in]{He4_vs_lam_nmaxbig28_nmax18_hw36}
\caption{Alpha particle binding energy during SRG evolution.
The curves correspond to those in Fig.~\ref{fig:H3_Ebind_lambda}.}
\label{fig:He4_Ebind_lambda}
\end{center}
\end{figure}
In Fig.~\ref{fig:H3_Ebind_lambda}, the comparison of two-body-only
to full two-plus-three-body evolution is shown for the
triton ($^3$H). The NN-only curve
uses the evolved two-body potential. The change in energy with
$\lambda$ reflects the violation of unitarity by omission of the
induced three-body force. When this induced 3NF is included
(``NN + NNN--induced''), the energy is independent of $\lambda$
for $A=3$. If we now turn to the alpha particle ($^4$He) in
Fig.~\ref{fig:He4_Ebind_lambda}, we see similar behavior, except
now the inclusion of the induced
3NF does not lead to a completely flat
curve at the lowest $\lambda$ values. If there is sufficient
convergence, this is a signal of missing induced 4NF.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{tjon_line_v6}
\caption{Correlation plot of the binding energies of the alpha
particle and the triton. The dotted line connects (approximately)
the locus of
points found for phenomenological potentials, and is
known as the Tjon line~\cite{Nogga:2000uu}.}
\label{fig:tjon_line}
\end{center}
\end{figure}
In both cases, it is evident that starting with an initial NN-only
interaction (in this case, an \nthreelo(500\,MeV) interaction~\cite{Entem:2003ft}), does
not reproduce experiment. The third line in each plot
of Figs.~\ref{fig:H3_Ebind_lambda} and \ref{fig:He4_Ebind_lambda} shows
that an initial 3NF (labeled NNN) contribution leads to a good
reproduction of experiment. The triton energy is part of the
fit of this initial force, but the alpha particle energy
is a prediction.
Note that the magnitude of the NN-only variation is comparable
to the initial 3NF needed. This is an example of the natural
size of the 3NF being manifested by the running of the potential
(which is, in effect, the beta function).
The nature of the evolution is illustrated in Fig.~\ref{fig:tjon_line},
which is a correlation plot of the binding energies in each nucleus.
The dotted line is known as the Tjon line for NN-only phenomenological
potentials. It was found that different potentials that fit NN
scattering data gave different binding energies, but that they
clustered around this line. With the SRG evolution starting with
just an NN potential, the path follows
the line, passing fairly close to the experimental point.
With an initial NNN force and keeping the induced 3-body part,
the trajectory is greatly reduced (see inset), at least until
$\lambda$ is small.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{Expect_vals_H3_NNN_A3nmax32_nmax18_hw28}
\caption{Contributions to the triton binding energy during SRG evolution.
Plotted are the expectation values of the kinetic energy, the two-body
potential, and the three-body potential~\cite{Jurgenson:2010wy}.}
\label{fig:H3_Ebind_contributions}
\end{center}
\end{figure}
\begin{figure}[tbh!]
\begin{center}
\includegraphics[width=2.5in]{Expect_vals_He4_NNN_A3nmax32_nmax18_hw28}
\caption{Contributions to the alpha particle binding energy during SRG evolution. Plotted are the expectation values of the kinetic energy, the two-body
potential, the three-body potential, and the four-body potential~\cite{Jurgenson:2010wy}.}
\label{fig:He4_Ebind_contributions}
\end{center}
\end{figure}
Figures~\ref{fig:H3_Ebind_contributions} and \ref{fig:He4_Ebind_contributions}
show individual contributions to the energy in the form of ground-state
matrix elements of the kinetic energy, two-body, three-body, and (implied)
four-body potentials. The hierarchy of contributions is quite clear
but the graphs also manifest the strong cancellations between the NN
and kinetic energy contributions. These cancellations magnify the impact
of higher-body forces.
Even so, it appears that a truncation including the NNN but omitting
higher-body forces is workable, particularly with $\lambda > 1.5\fmi$.
But what about the $A$ dependence of the 4NF (and beyond)?
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{Li6_running_srg_extrap4-8_v4b}
\caption{SRG running of the $^6$Li ground-state energy. The error
bars are estimates of the errors from extrapolating the results
to $N_{\rm max} = \infty$. The gray shaded region shows the
uncertainty from NSCM calculations using a Lee-Suzuki effective
interaction~\cite{Jurgenson:2010wy}.}
\label{fig:Li6_Ebind_running}
\end{center}
\end{figure}
This $A$ dependence is the topic of current research.
In Fig.~\ref{fig:Li6_Ebind_running}, results for $^6$Li
are shown~\cite{Jurgenson:2010wy}.
Assessing these results is made difficult because of insufficient
convergence of comparison calculations (the shaded areas) and of calculations
at the large $\lambda$ values. Note, however, that the variations are
with the 1\,MeV level; nevertheless we expect to do better.
Roth and collaborators have since used importance truncated NCSM
(IT-NCSM) to extend these calculations to much higher $\Nmax$ and
all the way to $^{16}$O~\cite{Roth:2011ar}.
They find that the SRG with initial NN-only
but including the induced NNN shows only small running with $\lambda$.
On the other hand, with increasing $A$ they find significant deviations.
This has been traced to the influence of the initial long-range NNN
interaction.
If they lower the cutoff of this part of the interaction, then approximate
SRG unitarity is restored and with coupled-cluster methods
they find reasonable results even
for medium-size nuclei (although not yet with fully consistent Hamiltonians)~\cite{Roth:2011vt}.
\subsection{Summary points}
Renormalization group flow equations dramatically reduce correlation
in many-body wave functions, leading to faster convergence of
many-body calculations.
Flow equations (SRG) achieve this lower resolution by decoupling
via a series of unitary transformations, which leave observables
invariant (if no approximations are made) but alter the physics
interpretation.
Few-body forces are inevitable, but the flow-equation approach allows
the evolution of vacuum interactions.
\section{Features of SRG applied to nuclear problems}
\label{sec:features}
\subsection{Chiral EFT, many-body forces, and the SRG}
\label{subsec:chiralEFT}
Before continuing with the SRG for nuclear systems, let's say a bit
more about chiral effective field theory (EFT). In the SRG flow
equations, the input interaction is merely an initial condition;
the equations are the same whether we start with an EFT potential
or a more phenomenological potential such as Argonne $v_{18}$.
However, increasingly nuclear theorists are moving toward using
EFT interactions because they promise a more systematic construction
of many-body forces and consistent operators.
There are three fundamental ingredients of an effective field theory
(e.g., see Ref.~\cite{Beane:2000fx}). The first is to use the most
general lagrangian with low-energy degrees of freedom consistent with
the global and local symmetries of the underlying theory. For nuclei,
the underlying theory is quantum chromodynamics (QCD).
We can identify a hierarchy of nuclear QCD scales:
\bi
\I $M_{\rm QCD} \sim 1\,$GeV [$M_{\rm hadrons}$, $4\pi f_\pi$]
\I $M_{\rm nuc} \sim 100\,$MeV [$\kf$, $f_\pi$, $m_\pi$,
$\delta_{\Delta N}$]
\I $M_{\rm nuc}^2/M_{\rm QCD} \sim 10\,$MeV [nuclear binding energy
per nucleon]
\ei
For all but the lightest nuclei, the EFT of choice at present draws
a line between the first two levels to define the high-energy and
low-energy scales. This is the chiral EFT, with degrees of freedom
consisting of the nucleon (proton and neutron) and the pion~\cite{Epelbaum:2008ga}. In the
near future, the $\Delta$ resonance will be included as well because
of the small mass difference with the nucleon, $\delta_{\Delta N}$.
Besides the usual space-time symmetries,
terms in the chiral EFT lagrangian are constrained by the
requirements of spontaneously broken (as well as explicitly broken)
chiral symmetry~\cite{Epelbaum:2008ga}.
The other two ingredients are the declaration of a regularization and
renormalization scheme, and the identification of a well-defined
power counting based on well-defined expansion parameters.
The separation of scales provides the expansion parameter as
ratios of $Q/M_{\rm QCD}$, where $Q$ is one of the quantities
lumped together above as $M_{\rm nuc}$.
The chiral EFT potentials used here are derived
using a momentum cutoff and what is called ``Weinberg counting'',
in which the counting is done at the level of an irreducible
potential that is summed non-perturbatively with the Lippmann-Schwinger
equation.
This scheme has been criticized because it does not allow
systematic renormalization (meaning, in this context, the removal of
cutoff dependence at each order), which limits the range of cutoffs used.
This in turn hinders the validation of the EFT, because
the sensitivity to the cutoff is used as a measure of uncertainties,
See Ref.~\cite{Epelbaum:2008ga} for a thorough overview of
chiral EFT for nuclei and the status of the power counting and renormalization
controversies.
With any scheme, however, the power counting implies a hierarchy
of many-body contributions. Weinberg counting associates a power
$\nu$ of $Q/M_{\rm QCD}$ with diagrams for the potential,
where
\beq
\nu = -4 + 2N + 2L + \sum_i (d_i + n_i - 2) \;.
\eeq
The definitions and details of its implementation can be found in
Ref.~\cite{Epelbaum:2008ga}. For our purposes, the relevant
term is ``$2N$'', which says that adding a nucleon to go from
an $A$--body potential to an $A+1$--body potential generally suppresses
the contribution by $Q^2/M_{\rm QCD}$.
In the theory without $\Delta$'s, the suppression of the leading 3NF
compared to the leading NN interaction is actually $(Q/M_{\rm QCD})^3$,
and a four-body force first appears at order $(Q/M_{\rm QCD})^4$
\cite{Epelbaum:2008ga}.
It is this hierarchy that we want to preserve as we run our SRG
flow equations.
As noted, the flow equation technology discussed here does not rely
on a particular implementation of chiral EFT, except that the SRG
is inherently non-perturbative. This apparently
excludes alternative renormalization
and power counting schemes that require a perturbative treatment
beyond leading order~\cite{Beane:2001bc,Bedaque:2002mn,Nogga:2005hy}. The current belief is that the two approaches
should give comparable results as long as the EFT cutoff is taken
to be of order $M_{\rm QCD}$, but the issue is far from settled~\cite{Epelbaum:2008ga}.
Another consideration is whether one could bypass the SRG by
simply applying chiral EFT with a lower cutoff.
Indeed, there exists low-cutoff implementations on the market
that display similar characteristics to low-momentum RG-evolved
interactions. However, the lower cutoff also means the effective
expansion parameter is smaller, and therefore the truncation error
is reduced. With the RG, one preserves the truncation error from
the cutoff of order $M_{\rm QCD}$. An additional advantage of
the RG is the controlled variation of the decoupling scale, which
provides a tool for assessing errors from the Hamiltonian truncation
and many-body approximations. But this is also not a settled issue
and further study would be welcome.
\subsection{More perturbative nuclear systems flow equations}
Earlier we mentioned the role of RG in high-energy physics in improving
perturbation theory.
Much of low-energy nuclear physics is intrinsically non-perturbative
because of large scattering lengths and bound states, so how do
we quantify ``perturbativeness''?
We study the convergence of the Born series for scattering
to see how this can be done.
Consider whether the Born series for the T-matrix operator
at a given (complex) $z$,
\beq
T(z) = V + V \frac{1}{z-H_0}V + V\frac{1}{z-H_0}V\frac{1}{z-H_0}V +
\cdots
\eeq
converges.
This is something like a geometric series, for which we know clearly
the convergence criterion: $1 + w + w^2 + \cdots$ diverges if
$|w| \geq 1$.
We get a clue for how to use this if we consider a bound state
$|b\rangle$ and the special value $z = E_b$, which is the bound-state
energy. Then we can rearrange the Schr\"odinger equation
\beq
(H_0 +V)|b\rangle = E_b | b\rangle
\eeq
to the form
\beq
\frac{1}{E_b-H_0}V|b\rangle = |b\rangle
\label{eq:bswein}
\eeq
and look at $T(E_b)|b\rangle$. Using Eq.~\eqref{eq:bswein} repeatedly,
the divergence is manifest, i.e., we get $V(1+1+1+\cdots)|b\rangle$.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{hatsuda_phen-pot_new_flipped3}
\vspace*{-0.25in}
\caption{The potentials of Fig.~\ref{fig:phenpots} inverted as
part of the Weinberg eigenvalue analysis (see text).}
\label{fig:weinberg1}
\end{center}
\vspace*{-0.1in}
\end{figure}
\begin{figure}[b!]
\begin{center}
\includegraphics[width=2.8in]{3s1weinberg_-2MeV}
\vspace*{-0.05in}
\caption{$\vlowk$ RG evolution of the largest positive and negative Weinberg
eigenvalue at fixed energy (corresponding to the deuteron binding
energy) for AV18 in the $^3$S$_1$ channel~\cite{Bogner:2006tw}. }
\label{fig:weinberg2}
\end{center}
\end{figure}
Now we see that we can generalize Eq.~\eqref{eq:bswein} for fixed $E$
by looking for eigenstates of $(E-H_0)^{-1}V$ with
eigenvalue $\eta_\nu(E)$,
\beq
\frac{1}{E-H_0}V|\Gamma_\nu\rangle
= \eta_\nu(E) |\Gamma_\nu\rangle \;.
\label{eq:eigwein}
\eeq
Then with $T$ applied to these eigenstates, there is manifestly a divergence
for $|\eta_\nu(E)| \geq 1$:
\beq
T(E)|\Gamma_\nu\rangle
= V|\Gamma_\nu\rangle
(1 + \eta_\nu + \eta_\nu^2 + \cdots)
\;.
\eeq
So we characterize the perturbativeness of a potential at energy
$E$ by the $\eta_\nu(E)$ with the largest magnitude, which dictates
convergence of the T~matrix.
This analysis follows work in the early 1960's by Weinberg~\cite{Weinberg:1963zz},
so we call $\eta_\nu$ a ``Weinberg eigenvalue'',
although others have made similar treatments of the convergence of
the Lippmann-Schwinger series.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.4in]{complex_vs_energy_1s0}
\includegraphics[width=2.4in]{complex_vs_energy_3S1}
\vspace*{-.05in}
\caption{Trajectories of Weinberg eigenvalues in the complex
plane for several realistic NN potentials in two channels. The symbols
go from 0\,MeV on the axis to 25, 66, 100, and 150\,MeV~\cite{Bogner:2006tw}.}
\label{fig:weinberg3}
\end{center}
\vspace*{-.2in}
\end{figure}
If we compare Eq.~\eqref{eq:eigwein} to Eq.~\eqref{eq:bswein},
we can express the convergence criterion for $E < 0$ as saying that
$T(E)$ diverges if there exists a bound state at $E$ for $V/\eta_\nu$
with $|\eta_\nu| \geq 1$. Or, in other words, what is the
largest $\eta_\nu$ for which $V/\eta_\nu$ supports a bound state at $E$?
We have convergence only for $\eta_\nu < 1$.
This means we'll have two types of eigenvalues, because $\eta_\nu$ could
be negative (``repulsive'') as well as positive (``attractive'').
The negative eigenvalue corresponds to looking for bound states with
a scaled ``flipped'' potential, as in Fig.~\ref{fig:weinberg1}.
We see that the repulsive core becomes a deep attractive well,
implying that we will have a large negative eigenvalue in these
cases. But then we expect that RG evolution to a softened form,
which eliminates the core, should result in decreased eigenvalues.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=2.5in]{complex_vs_energy_1s0_repulsive_N3LO}
\hspace*{.2in}
\includegraphics[width=2.5in]{complex_vs_energy_3s1_repulsive_N3LO}
\vspace*{-.05in}
\caption{Trajectories of Weinberg eigenvalues in the complex
plane for the AV18 NN potential in two channels at various
states in a $\vlowk$ RG evolution (labeled by $\Lambda$). The symbols
go from 0\,MeV on the axis to 25, 66, 100, and 150\,MeV~\cite{Bogner:2006tw}.}
\label{fig:weinberg4}
\end{center}
\begin{center}
\includegraphics[width=2.5in]{weinberg_1s0_n3lo_v2}
\hspace*{.2in}
\includegraphics[width=2.5in]{weinberg_3s1_n3lo_v2}
\vspace*{-.05in}
\caption{Repulsive (negative) Weinberg eigenvalues at $E=0$
for several N$^3$LO chiral EFT potentials as a function
of $\vlowk$ $\Lambda$~\cite{Bogner:2006tw}.}
\label{fig:weinberg5}
\end{center}
\end{figure*}
These expectations hold in practice, as illustrated in
Fig.~\ref{fig:weinberg2}, which shows the evolution of the largest
positive and negative Weinberg eigenvalues as a function of the
$\vlowk$ cutoff $\Lambda$. (Very similar behavior is observed for
the SRG with $\lambda$ replacing $\Lambda$.)
In free space, the largest attractive eigenvalue is unity at all
$\Lambda$'s, corresponding to the deuteron bound state. But the
negative eigenvalue starts very large (because of the repulsive
core) and drops dramatically as the potential is evolved, ending
up well less than unity, indicating it is perturbative (but the
positive eigenvalue still makes this channel nonperturbative).
The situation is even more dramatic in the medium. The other
curves in the figure show the result from considering the T~matrix
in the nuclear medium, where Pauli blocking effects are included
in the intermediate states.
We see a further reduction of the negative eigenvalue and now the
positive eigenvalue is perturbative as well: the deuteron has dissolved!
Starting from negative energies,
we can follow the evolution of
the largest eigenvalue into the complex plane as we increase $E$ to
positive values. Figures~\ref{fig:weinberg3} and
\ref{fig:weinberg4} show paths in the
complex plane for Weinberg eigenvalues, with the symbols indicating
energies of 0, 25, 66, 100, and 150\,MeV.
The shaded region is the unit circle; the potential is perturbative
for energies where both eigenvalues lie inside.
The softening effect of the RG evolution is manifest in
Fig.~\ref{fig:weinberg4} and the eigenvalues provide a quantitative
measure of the perturbativeness.
It's also clear from Fig.~\ref{fig:weinberg3} that at least
the particular chiral potential
used there is already quite soft.
However,
Fig.~\ref{fig:weinberg5} shows that significant \emph{additional} softening
is possible with RG evolution. In all of these plots,
the differences between
the $^1$S$_0$ channel and the $^3$S$_0$--$^3$D$_0$ coupled channel
stem from the latter having an additional source of nonperturbative
behavior: the short-range tensor force.
The increasing ``perturbativeness'' at finite density is documented again
in Fig.~\ref{fig:weinberg6}. At typical nuclear densities with
$1\fmi \leq \kf \leq 1.3\fmi$, both positive and negative eigenvalues
are small at the lowest $\Lambda$'s. This implies that nuclear
matter may actually be perturbative!
(Note that at the Fermi surface, pairing as a nonperturbative
phenomenon is revealed by $|\eta_\nu|>1$~\cite{Ramanan:2007bb}.)
We can understand how this
happens from Fig.~\ref{fig:weinberg7}, which shows the phase space
available to two nucleons that scatter in the medium. Pauli blocking
means they must go outside the two Fermi spheres, but the volume
is increasingly restricted with decreasing $\Lambda$. In addition,
the magnitudes of the matrix elements that scatter such particles
decrease as well~\cite{Bogner:2005sn}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.8in]{paper_3s1weinberg_medium}
\caption{Dependence of the largest Weinberg eigenvalues on
density for several evolved $\vlowk$ potentials~\cite{Bogner:2006tw}.}
\label{fig:weinberg6}
\end{center}
\end{figure}
\begin{figure}[th!]
\begin{center}
\includegraphics[width=1.9in]{fermispheres}
\caption{Overlapping Fermi spheres showing available phase space
for two nucleons excited above the Fermi surface~\cite{Bogner:2005sn}.}
\label{fig:weinberg7}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.8in]{pert2nf}
\caption{Many-body perturbation theory for symmetric nuclear
matter up to third order in the particle-particle channel~\cite{Hebeler:2010xb}.}
\label{fig:weinberg8}
\end{center}
\end{figure}
Perturbation theory in the particle-particle channel is shown
in Fig.~\ref{fig:weinberg8} for the high-resolution Argonne $v_{18}$ potential
initially and after evolution by the $\vlowk$ RG to low resolution ($\Lambda=1.9\fmi$).
Whereas many-body perturbation theory (MBPT) manifestly diverges for the
original potential, it converges for the low-momentum interaction
at second order
(at least in this channel; more complete many-body approximations
must be studied to be more definite).
It is also evident that there is no saturation. But adding a
3NF fit only to few-body properties, as in Fig.~\ref{fig:weinberg9}, shows that the empirical saturation
point can be reproduced with an uncertainty of about 2--3\,MeV/particle.
On-going work to improve this result includes the development of SRG
evolution for the 3NF in momentum space~\cite{Kai}
and of coupled cluster methods
for infinite matter with 3NF's to provide a high-order resummation
of perturbation theory to test convergence.
\begin{figure*}[th!]
\begin{center}
\includegraphics[width=5.2in]{nuclear_matter_trio_new}
\caption{Many-body perturbation theory calculations of symmetric
nuclear matter with $\vlowk$ NN potentials at four different
cutoffs and the leading-order 3NF from chiral EFT~\cite{Hebeler:2010xb}.
The two
free parameters in the 3NF are fit at each density to the triton binding
energy and the alpha particle radius.}
\label{fig:weinberg9}
\end{center}
\end{figure*}
\subsection{Universality from flow equations}
Another general aspect of RG flows known from the study of critical
phenomena is the appearance of universal behavior. In the application
of RG to nuclear interactions, the universality we observe is that
distinct initial NN potentials that reproduce the experimental low-energy
scattering phase shifts, are found to collapse to a single universal
potential. We've already seen an indication of this, but here we
document it in more detail.
We focus on the SRG, but very similar conclusions are found
for $\vlowk$ evolution. In Figs.~\ref{fig:srguniv1} and \ref{fig:srguniv2},
S-wave N$^3$LO chiral EFT potentials from Refs.~\cite{Entem:2003ft} and ~\cite{Epelbaum:2004fk} are evolved with the SRG.
Although the level of truncation is the same and the cutoffs
approximately equal, the methods of regulating the potential
differ (particularly for the two-pion exchange part).
The result is very different looking initial interactions. This is not
a concern, because the NN potential is not an observable.
We also observe there is significant off-diagonal strength coupling
low and medium momenta in the initial potentials.
As the potentials are evolved, we see the characteristic driving
toward the diagonal, with the diagonal width in $k^2$ given roughly by
$\lambda^2$.
At the end of the evolution shown, the interactions still look quite
different at first glance. However, if we focus on the low-momentum
region, where $k^2 < 2\,\mbox{fm}^{-2}$, they appear much more
similar. We can quantify this by taking a slice along the edge
(i.e., $V(k,0)$ and along the diagonal
(i.e., $V(k,k)$) and plot these quantities for these potentials and two
additional ones.
This is done in Fig.~\ref{fig:srguniv3}.
We see a dramatic collapse of the interaction between $\lambda = 5\fmi$
and $\lambda = 2\fmi$ for the region of $k$ below $\lambda$
(or maybe 3/4 $\lambda$).
An open question under active investigation is whether evolved
3NF interactions will be universal.
\begin{figure*}[p!]
\begin{center}
\includegraphics[width=5.3in]{tiled_s_1S0_kvnn10_reg_0_3_0_ksq}
\vspace*{0in}
\includegraphics[width=5.3in]{tiled_s_1S0_kvnn32_reg_0_3_0_ksq}
\vspace*{-.1in}
\caption{SRG evolution of two chiral EFT potentials in
the $^1$S$_0$ channel~\cite{Bogner:2009bt}.}
\label{fig:srguniv1}
\vspace*{.1in}
\includegraphics[width=5.3in]{tiled_s_3S1_kvnn10_reg_0_3_0_ksq}
\vspace*{0in}
\includegraphics[width=5.3in]{tiled_s_3S1_kvnn32_reg_0_3_0_ksq}
\vspace*{-.1in}
\caption{SRG evolution of two chiral EFT potentials
in the $^3$S$_1$ channel~\cite{Bogner:2009bt}.}
\label{fig:srguniv2}
\vspace*{.14in}
\includegraphics[width=2.3in]{vsrg_collapse_1s0_n3lo_5to2}
\hspace*{.2in}
\includegraphics[width=2.3in]{vsrg_collapse_1s0_n3lo_5to2d}
\vspace*{-.1in}
\caption{SRG flow toward universality for several chiral EFT
potentials~\cite{Bogner:2009bt}.}
\label{fig:srguniv3}
\end{center}
\end{figure*}
\subsection{Operator evolution via flow equations}
We have focused almost exclusively on the evolution of the
Hamiltonian, but an RG transformation will also change the
operators associated with measurable quantities.
It is essential to be able to start with operators consistent
with the Hamiltonian
and then to evolve them maintaining this consistency.
The first step is a prime motivation for using EFT; we will
assume that we have consistent initial operators in hand.
The second step can be technically difficult, especially
since we will inevitably induce many-body operators as we
evolve (for the same arguments as we made for the Hamiltonian).
This is where the SRG is particularly advantageous, because
it is technically feasible to evolve operators along with the
Hamiltonian.
The SRG evolution with $\flow$ (recall $s = 1/\lambda^4$)
of \emph{any} operator $O$
is given by:
\beq
O_\flow = U_\flow O U^\dagger_\flow \;,
\label{eq:Oflow}
\eeq
so $O_\flow$ evolves via
\beq
\frac{dO_\flow}{d\flow}
= [ [G_\flow,H_\flow], O_\flow] \;,
\eeq
where we use the same $G_\flow$ to evolve the Hamiltonian
and all other operators.
While we can directly evolve any operator like this in parallel
to the evolution of the Hamiltonian, in practice it is more
efficient and numerically robust to either evolve the unitary
transformation $U_s$ itself:
\beq
\frac{dU_\flow}{d\flow}
= \eta_s U_s = [G_s,H_s] U_s
\;,
\eeq
with initial value $U_{s=0} = 1$, or calculate it directly
from the eigenvectors of $H_{s=0}$ and $H_s$:
\beq
U_s = \sum_i |\psi_i(s)\ra \la \psi_i(0)| \;.
\eeq
Then any operator is directly evolved to the desired $\flow$
by applying Eq.~\eqref{eq:Oflow} as a matrix multiplication.
The second method works well in practice.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{deuteron_md_bare_v4}
\caption{Momentum distribution of the deuteron for the
AV18 potential and CD-Bonn potentials, compared to
results from SRG potentials evolved to $\lambda=2.0\fmi$
and $1.5\fmi$.}
\label{fig:deuteron_md_1}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{deuteron_md_N3LO_bare_dotted}
\caption{Momentum distribution for an unevolved chiral N$^3$LO
potential compared to results from SRG potentials evolved to $\lambda=2.0\fmi$
and $1.5\fmi$~\cite{Anderson:2010aq}.}
\label{fig:deuteron_md_2}
\end{center}
\end{figure}
To simplify our study of operator evolution, we consider the simplest
possible operator: the momentum number operator $a^\dagger_q a_q$.
In Fig.~\ref{fig:deuteron_md_1}, we show the momentum distribution
in the deuteron, i.e., $<\psi_d| a^\dagger_q a_q |\psi_d> $
for two different realistic potentials, AV18 and CD-Bonn.
As implied by the $y$-axis label, the momentum distribution is just the
square of the deuteron wave function in momentum space.
The results for the two potentials agree up to about $2\fmi$ and then
are different.
If we evolve the AV18 potential \emph{and} the momentum operator, then
the matrix element
$<\psi_d^\lambda| O_\lambda |\psi_d^\lambda> $
will be the same for any $\lambda$ and the curve will exactly reproduce
the AV18 deuteron momentum distribution.
However, if we
calculate $<\psi_d^\lambda| a^\dagger_q a_q |\psi_d^\lambda> $ for
$\lambda = 2\fmi$ and $1.5\fmi$, that is, we use the evolved wave function
but the unevolved operator, then we get the other curves.
Besides directly illustrating that the high-$q$ part of the momentum
distribution is not an observable, since we can change it at will
by unitary transformations, this manifests that the high-momentum
part of the wave function is removed.
The latter is potentially disturbing, because if the evolved operator
is supposed to reproduce a high momentum result when the evolved
wave function has a vanishingly small component at that momentum, this
may be because the operator is becoming pathological. To explore
this further, we consider the momentum distribution in Fig.~\ref{fig:deuteron_md_2} at low ($q=0.34\fmi$) and moderately
high ($q=3.0\fmi$) values of the momentum~\cite{Anderson:2010aq}.
In Fig.~\ref{fig:contourmd} we plot the \emph{integrand} of
\beq
\left\langle\psi_{d}^\lambda \right \vert (Ua^{\dagger}_{q}a_{q}U^{\dagger})\left\vert \psi_d^\lambda \right\rangle
\;,
\eeq
at each of these two $q$ values. The full integral is the momentum
distribution at those $q$'s, so the plots tell us where the strength
of the operator lies.
For the low-momentum operator, there is little renormalization, but
the nature of the high-momentum operator changes completely.
Originally, the integral comes entirely from the region of $q=3.0\fmi$,
but the evolution of the operator shifts its strength entirely
to low momentum. This result is similar for other operators, such
as electromagnetic form factors~\cite{Anderson:2010aq}.
As we move to $A \geq 3$, the operator evolution and extraction process
becomes more involved. A flowchart for the procedure is given in
Fig.~\ref{fig:operator_embed}.
Imagine we initially have a one-body operator and we want to evolve
and then evaluate
it in an $A$-particle nucleus.
The difficulty is that $n$-body components are induced as we evolve
and these must be separated out so we can correctly embed them
in the nucleus. In particular, to embed an $n$-body operator in an anti-symmetrized $A$-particle nucleus, we need an embedding factor of $\binom{A}{n}$, so we need to isolate the components first.
With the usual SRG generators, there is no evolution of one-body
operators. This is easiest to see in second quantization, where the
commutators on the right side yield operators that have at least
four creation/destruction operators, meaning it is two- and higher-body.
To isolate the two-body part, we first evolve the Hamiltonian
in the two-particle basis to find the unitary operator $U_s^{(2)}$
and use it to evolve our operator. Then if we subtract the embedded
one-body operator, we will have our two-body part. The one- and two-body
parts are then embedded in the 3-particle basis and subtracted from
the evolved operator in that basis. And so on until we reach the
desired level of truncation, at which point we embed in the $A$-particle
basis and perform the $A$-particle calculation.
More details and examples can be found about operator evolution in Ref.~\cite{Anderson:2010aq}.
\begin{figure*}[p!]
\begin{center}
\includegraphics[width=5.2in]{movie_vsrg_3S1_kvnn10_reg_0_3_0_Mg4p4_Deut_mom_dist_integrand_L0p34_k_cntr_straight_6p0_3p0_2p0_1p5_Rep0}
\includegraphics[width=5.2in]{movie_vsrg_3S1_kvnn10_reg_0_3_0_Mg4p4_Deut_mom_dist_integrand_L3p01_k_cntr_straight_6p0_3p0_2p0_1p5_Re-3}
\caption{Integrand of momentum distribution operator~\cite{Anderson:2010aq}.}
\label{fig:contourmd}
\vspace*{.3in}
\includegraphics[width=5.8in]{process_boost}
\caption{Schematic of the SRG operator evolution and embedding
process~\cite{Anderson:2011private}.}
\label{fig:operator_embed}
\end{center}
\end{figure*}
\subsection{Computational aspects}
Before concluding this lecture, we'd like to make some brief comments
on the computational aspects of the calculations behind the figures
we've seen.
As noted earlier, the continuous momentum is discretized into a finite
number of momentum points.
The subsequent discretization of integrals leads directly to matrices,
and most of the manipulations are efficiently cast in this language.
For example, momentum-space flow equations have integrals like:
\beq
I(p,q) \equiv \int\!dk\, k^2\, V(p,k) V(k,q) \;.
\eeq
The usual choice of discretization is to use gaussian quadrature to
accurately evaluate integrals with a minimum of points. (The size
is not an issue for two-body operators, but becomes critical for
higher-body operators.)
We introduce gaussian nodes and weights $\{k_n,w_n\}$ with ($n = 1,N$)
to reduce integrals to finite sums:
\beq
\quad \int\!dk\, f(k) \approx
\sum_{n} w_n\, f(k_n) \;.
\eeq
(Note: these sets of nodes and weights are generally a combination
of separate smaller rules over adjacent intervals with a \emph{total} of $N$ points.)
Then $I(p,q) \rightarrow I_{ij}$, where $p = k_i$
and $q = k_j$, and
\beq
I_{ij} = \sum_n k_n^2 w_n\, V_{in} V_{nj}
\rightarrow \sum_n \wt V_{in} \wt V_{nj}
\eeq
where
\beq
\wt V_{ij} \equiv \sqrt{w_i}k_i\, V_{ij}\, k_j \sqrt{w_j} \;.
\eeq
This allows us to solve SRG equations and integral equations
for phase shifts,
Schr\"odinger equation in momentum representation.
In practice, $N\approx 100$ gauss points is adequate
for accurate nucleon-nucleon partial waves.
A computer code that carries out SRG evolution can be remarkable
simple.
Here is a possible pseudocode that is suitable:
\be
\I Set up basis (e.g., momentum grid with gaussian quadrature
or HO wave functions with $N_{\rm max}$)
\I Calculate (or input) the initial Hamiltonian and $G_s$
matrix elements (including any weight factors)
\I Reshape the right side $[[G_s,H_s],H_s]$ to a vector
and pass it to a coupled differential equation solver
\I Integrate $V_s$ to desired $s$ (or $\lambda = s^{-1/4}$)
\I Diagonalize $H_s$ with standard symmetric eigensolver
\Lra energies and eigenvectors
\I Form $U = \sum_i |\psi^{(i)}_s\rangle
\langle \psi^{(i)}_{s=0} |$
from the eigenvectors
\I Output or plot or calculate observables
\ee
Such a code has been implemented in
MATLAB, Mathematica, Python, C++, and Fortran-90.
While any basis can be used, so far only discretized momentum
and harmonic oscillator bases have been implemented.
Note that the same procedure (and even the same code in some cases)
is relevant for a many-particle basis, but the number of differential
equations will grow rapidly. For accurate results in the two-body
evolution, $100^2 = 10,000$ matrix elements are needed, so there are
that many differential equations. For an accurate 3NF evolution
in a harmonic oscillator basis, at least 20~million coupled differential
equations need to be solved. This sounds intimidating, but is well
within reach of MATLAB (for example) on a machine with a moderate
amount of memory.
\subsection{Summary points}
Chiral EFT establishes a hierarchy of many-body forces.
Using flow equations to run to low resolution makes many-body
calculations more perturbative and interactions flow
to universal form (at least for NN; this is not yet
established for the 3NF).
Operators can be evolved consistently with interaction.
Long-distance operators change very little, while
short-distance operators renormalize significantly,
accompanied in some cases by a change in physics interpretation.
The basic SRG flow equations in a partial wave momentum basis
can be cast in a form that
involves
just matrix manipulations and the solution of
ordinary first-order, coupled differential equations.
\section{Nuclear applications and open questions}
\label{sec:applications}
In this final lecture, we take a look at the in-medium
similarity renormalization group (IM-SRG) and make a broad
survey of nuclear applications. We conclude with a summary
of open problems and new challenges.
\subsection{In-medium similarity renormalization group}
\label{subsec:imsrg}
We start with a review of Hartree-Fock.
The Hartree-Fock wave function is the
best single Slater determinant
\beq
| \Psi_{\rm HF} \rangle = \det\{\phi_i(\xvec), i=1\cdots A \}
\,, \quad \xvec = ({\bf r}, \sigma, \tau)
\eeq
in the variational sense.
The $\phi_i(\xvec)$ single-particle wave functions
satisfy \emph{non-local} Schr\"odinger equations:
\beq
-\frac{\bm{\nabla}^2}{2M}\phi_i(\xvec)
+ V_{\rm H}(\xvec)
\phi_i(\xvec)
+ \int\! d\yvec\, V_{\rm E}(\xvec,\yvec) \phi_i(\yvec)
= \epsilon_i \phi_i(\xvec)
\eeq
with direct
\beq
V_{\rm H}(\xvec) = \int\! d\yvec \sum_{j=1}^A |\phi_j(\bf y)|^2
v(\xvec,\yvec)
\eeq
and exchange
\beq
V_{\rm E}(\xvec,\yvec) = - v(\xvec,\yvec) \sum_{j=1}^A
\phi_j(\xvec) \phi_j^\ast(\yvec)
\eeq
potentials~\cite{FETTER71,Ring:2005}.
The direct and exchange potentials are shown diagrammatically
in Fig.~\ref{fig:HartreeFock}, with the rightmost diagram an
abbreviated form called a Hugenholtz diagram~\cite{Negele:1988vy}.
We must solve self-consistently using occupied orbitals for $V_H$ and $V_E$.
Then Slater determinants from \emph{all} orbitals form an $A$-body basis.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=2.8in]{fig_effa2p}
\caption{Feynman and Hugenholtz diagrams for Hartree-Fock.}
\label{fig:HartreeFock}
\end{center}
\end{figure}
The in-medium SRG (IM-SRG)
for nuclei, developed recently by Tsukiyama, Bogner, and
Schwenk~\cite{Tsukiyama:2010rj}, applies the flow equations in an $A$--body system using
a different reference state than the vacuum.
For example, we can choose the Hartree-Fock ground state as a
reference state.
The appealing consequence is that,
unlike the free-space SRG evolution, the in-medium
SRG can approximately evolve
$3,...,A$-body operators using only two-body machinery.
However, also in contrast to the free-space SRG,
the in-medium evolution must be repeated for each nucleus or
density.
The key to
the IM-SRG
simplification is the use of \emph{normal-ordering} with respect to a
finite-density reference state.
That is, starting from the
second-quantized Hamiltonian with two- and three-body interactions,
\bea
H \ampseq \sum_{12} T_{12} \ad_1 a_2 + \frac{1}{(2!)^2} \sum_{1234}
\, \langle12|V|34\rangle \ad_1\ad_2a_4a_3
\nonumber \\
\amps{}
\hspace*{-.2in}
\null + \frac{1}{(3!)^2} \, \sum_{123456} \langle123|V^{(3)}|456\rangle
\ad_1\ad_2\ad_3a_6a_5a_4 \,,
\label{eq:Ham}
\eea
all operators are normal-ordered with respect to a finite-density
Fermi vacuum $|\Phi\rangle$ (for example, the Hartree-Fock ground
state or the non-interacting Fermi sea in nuclear matter), as
opposed to the zero-particle vacuum. Wick's theorem can then be
used to {\it exactly} write $H$ as
\bea
H \ampseq E_0 + \sum_{12} f_{12} \{\ad_1a_2\}
\nonumber \\
\amps{}
\hspace*{-.25in}
\null
+ \frac{1}{(2!)^2} \sum_{1234} \, \langle12|\Gamma|34\rangle
\{\ad_1\ad_2a_4a_3\}
\nonumber \\
\amps{}
\hspace*{-.25in}
\null
+ \frac{1}{(3!)^2} \sum_{123456} \, \langle123|\Gamma^{(3)}|456\rangle
\{\ad_1\ad_2\ad_3a_6a_5a_4\} \,,
\label{eq:NorderedHam}
\eea
where the zero-, one-, and two-body normal-ordered terms are given by
\bea
E_0 \ampseq \langle \Phi | H | \Phi \rangle = \sum_1 T_{11} n_1
\nonumber \\ \amps{} \null
+ \frac{1}{2} \sum_{12} \, \langle12|V|12\rangle n_1 n_2
\nonumber \\ \amps{} \null
+ \frac{1}{3!} \sum_{123} \, \langle123|V^{(3)}|123\rangle n_1 n_2 n_3
\,, \label{eq:NorderedCoeff1} \\[10pt]
f_{12} \ampseq T_{12} + \sum_i \langle1i|V|2i\rangle n_i
\nonumber \\ \amps{} \null
+ \frac{1}{2} \, \sum_{ij} \langle1ij|V^{(3)}|2ij\rangle n_i n_j \,,
\label{eq:NorderedCoeff2} \\[10pt]
\langle12|\Gamma|34\rangle \ampseq \langle12|V|34\rangle
+ \sum_i \langle12i|V^{(3)}|34i\rangle n_i \,,
\label{eq:NorderedCoeff3}
\eea
and $n_i = \theta(\varepsilon_{\rm F} - \varepsilon_i)$ denotes the
sharp occupation numbers in the reference state, with Fermi level or
Fermi energy $\varepsilon_{\rm F}$.
By construction, the
normal-ordered strings of creation and annihilation operators obey
$\langle\Phi|\{a^{\dagger}_1\cdots a_n\}|\Phi\rangle = 0$. It is
evident from Eqs.~\eqref{eq:NorderedCoeff1}--\eqref{eq:NorderedCoeff3}
that the coefficients of the normal-ordered zero-, one-, and two-body
terms include contributions from the three-body interaction $V^{(3)}$
through sums over the occupied single-particle states in the reference
state $|\Phi\rangle$. Therefore, truncating the in-medium SRG
equations to two-body {\it normal-ordered} operators will
(approximately) evolve induced three- and higher-body interactions
through the density-dependent coefficients of the zero-, one-, and
two-body operators in Eq.~(\ref{eq:NorderedHam}).
The in-medium SRG flow equations at the normal-ordered two-body level
are obtained by evaluating $dH/ds = [\eta,H]$ with the normal-ordered
Hamiltonian $H = E_0 + f + \Gamma$ and the SRG generator $\eta =
\eta^{1b} + \eta^{2b}$ (with one- and two-body terms) and neglecting
three- and higher-body normal-ordered terms. For infinite matter,
a natural generator choice is $\eta = [f,\Gamma]$ in analogy
with the free-space SRG. In this case, the explicit form of the SRG
equations simplifies because $\eta^{1b}=0$ and $f_{ij} = f_i \,
\delta_{ij}$. This leads to
\bea
\frac{dE_0}{ds} \ampseq
\frac{1}{2} \sum_{1234} \, (f_{12} - f_{34})
|\Gamma_{1234}|^2 \, n_1 n_2 \bar{n}_3 \bar{n}_4
\,, \label{eq:NorderedSRG0} \\[10pt]
\frac{df_1}{ds} \ampseq \sum_{234} \, (f_{41} - f_{23})
|\Gamma_{4123}|^2
\nonumber \\ \amps{} \qquad \null \times
(\bar{n}_2\bar{n}_3n_4 + n_2n_3\bar{n}_4)
\,, \label{eq:NorderedSRG1} \\[10pt]
\frac{d\Gamma_{1234}}{ds} \ampseq -(f_{12} - f_{34})^2 \,
\Gamma_{1234}
\nonumber \\ \amps{} \null
+ \frac{1}{2} \sum_{ab} \,
(f_{12} + f_{34} - 2f_{ab})
\Gamma_{12ab} \Gamma_{ab34}
\nonumber \\ \amps{} \qquad\qquad\null \times
(1 - n_a - n_b)
\nonumber \\ \amps{} \null
+ \sum_{ab} \, (n_a - n_b) \nonumber \\
\amps{} \hspace*{-.25in} \null\times \Bigl\{ \Gamma_{a1b3} \Gamma_{b2a4}
\bigl[(f_{a1}-f_{b3}) - (f_{b2} - f_{a4})\bigr]
\nonumber \\ \amps{} \hspace*{-.25in} \null
- \Gamma_{a2b3}\Gamma_{b1a4}
\bigl[(f_{a2}-f_{b3}) - (f_{b1} - f_{a4})\bigr] \Bigr\} \,,
\label{eq:NorderedSRG2}
\eea
where the single-particle indices refer to momentum states and include
spin and isospin labels.
While the in-medium SRG equations are of second
order in the interactions, the flow equations build up
non-perturbative physics via the successive interference between the
particle-particle and the two particle-hole channels in the SRG
equation for $\Gamma$, Eq.~(\ref{eq:NorderedSRG2}), and between the
two-particle--one-hole and two-hole--one-particle channels for $f$,
Eq.~(\ref{eq:NorderedSRG1}). In terms of diagrams, one can imagine
iterating the SRG equations in increments of $\delta s$. At each
additional increment $\delta s$, the interactions from the previous
step are inserted back into the right side of the SRG equations.
Iterating this procedure, one sees that the SRG accumulates
complicated particle-particle and particle-hole correlations to all orders.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=2.5in]{SNMinmedium}
\caption{Symmetric nuclear matter energy/particle at $\kf = 1.4\fmi$
as evolved in the IM-SRG~\cite{Tsukiyama:2010rj}.}
\label{fig:SNMinmedium}
\end{center}
\end{figure}
\begin{figure}[th!]
\begin{center}
\includegraphics[width=2.5in]{PNMinmedium}
\caption{Neutron matter energy/particle at $\kf = 1.35\fmi$
as evolved in the IM-SRG~\cite{Tsukiyama:2010rj}.}
\label{fig:PNMinmedium}
\end{center}
\end{figure}
\begin{figure*}[p!]
\begin{center}
\includegraphics[width=5.3in]{in_medium_srg_schwenk_2}
\caption{``Off-diagonal'' terms (e.g., $2p2h$ sectors) driven to zero
as $s$ increases,
decoupling them from the Hartree-Fock reference state, which becomes exact as $s\rightarrow \infty$~\cite{Tsukiyama:2010rj}. }
\label{fig:imsrg1}
\vspace*{.1in}
\vfill
\includegraphics[width=4.in]{imsrg_flow}
\vspace*{-.1in}
\caption{IM-SRG flow for the energy of the alpha
particle~\cite{Tsukiyama:2010rj}. }
\label{fig:imsrg2}
\vspace*{.1in}
\vfill
\includegraphics[width=5.3in]{imsrg_finite}
\vspace*{-.1in}
\caption{IM-SRG convergence in finite nuclei compared
to coupled cluster CCSD and CCSD(T) calculations~\cite{Tsukiyama:2010rj}. }
\label{fig:imsrg3}
\end{center}
\end{figure*}
With the choice of generator $\eta = [f,\Gamma]$, the Hamiltonian is
driven towards the diagonal. This means that Hartree-Fock becomes
increasingly dominant with the off-diagonal $\Gamma$ matrix elements
being driven to zero. As with the free-space SRG, it is convenient for
momentum-space evolution to switch to the flow variable $\lambda
\equiv s^{-1/4}$, which is a measure of the resulting band-diagonal
width of $\Gamma$. In the limit $\lambda \rightarrow 0$,
Hartree-Fock becomes exact for the evolved Hamiltonian; the zero-body
term, $E_0$, approaches the interacting ground-state energy, $f$
approaches fully dressed single-particle energies, and the remaining
diagonal matrix elements of $\Gamma$ approach a generalization of the
quasiparticle interaction in Landau's theory of Fermi
liquids~\cite{Kehrein:2006}.
An approximate
solution of the $E_0$ flow equation for
symmetric nuclear matter and neutron matter as a function of $\lambda$
is shown in Figs.~\ref{fig:SNMinmedium} and \ref{fig:PNMinmedium} for two different Fermi momenta
$k_F$ (corresponding to different densities). As
expected, the in-medium SRG drives the Hamiltonian to a form where
Hartree-Fock becomes exact in the limit $\lambda \rightarrow 0$. In
contrast to the ladder approximation based on NN-only SRG interactions
evolved in free space, the same many-body calculation using
interactions evolved with the in-medium SRG at the two-body level
gives energies that are approximately independent of $\lambda$. This
indicates that truncations based on normal-ordering efficiently
include the dominant induced many-body interactions via the
density-dependent zero-, one-, and two-body normal-ordered terms.
In a similar manner, the in-medium SRG can be used as an ab initio
method for finite nuclei. Figure~\ref{fig:imsrg1} shows the off-diagonal normal-ordered two-body matrix elements $\Gamma$ being
driven to zero with the IM-SRG evolution. Figure~\ref{fig:imsrg2} shows
the evolution of the ground-state energy of
$^4$He~\cite{Tsukiyama:2010rj}. As the flow parameter $s$ increases,
the $E_0$ flow and
second-order (in $\Gamma$) many-body perturbation theory
contributions approach each other, as was the case for the infinite
matter results in Fig.~\ref{fig:SNMinmedium}. In addition, the
convergence behavior with increasing harmonic-oscillator spaces in
Fig.~\ref{fig:imsrg3} for $^4$He and $^{40}$ Ca is very promising.
Based on these
calculations, the in-medium SRG truncated at the
normal-ordered two-body level appears to give accuracies comparable to
coupled-cluster calculations truncated at the singles and doubles
(CCSD) level. Finally, we note that the
in-medium SRG is a promising method for non-perturbative calculations
of valence shell-model effective interactions and operators.
\subsection{Implications of RG for nuclear calculations}
There are many on-going and potential applications of low resolution
methods for calculations of nuclear structure and reactions. In many
cases RG techniques are used explicitly but there are also examples
of low resolution being achieved by other means. Here we'll survey
some of both types.
This will be far from a comprehensive list because it focuses largely
on developments
associated entirely or in part with a project called UNEDF, which stands for
Universal Nuclear Energy Density Functional.
UNEDF is a collaboration
of more than fifty physicists, applied mathematicians, and computer
scientists in the United States plus many international
collaborators, funded through the U.S.\ Department of Energy's
SciDAC program.
The long-term vision of the project is to arrive at a comprehensive and quantitative description of nuclei
and their reactions. The focused mission is to construct, optimize, validate,
and apply energy density functionals for structure and reactions,
but to carry out this mission the team has developed many
crosscutting physics collaborations where none existed previously between
the main physics areas: ab initio structure, ab initio
functionals, DFT applications, DFT extensions, reactions.
These interconnections are indicated schematically in the UNEDF
strategy diagram in Fig.~\ref{fig:UNEDF}.
This type of large-scale collaboration
represents a transformation in how low-energy nuclear theory
is done.
UNEDF has been very productive, with over 200 publications to date, including 11 Physical Review Letters and a Science article in the 2011 calendar year
alone. Further background, references, and scientific highlights can be found
at unedf.org, the project website.
\begin{figure*}[p!]
\begin{center}
\includegraphics[width=3.9in]{UNEDF-strategy2a}
\vspace*{-.1in}
\caption{Strategy diagram for UNEDF.}
\label{fig:UNEDF}
\includegraphics[width=2.6in]{MGT_14Cto14N}
\hspace*{.3in}
\raisebox{.5in}{\includegraphics[width=2.6in]{F14_spectrum_measured3}}
\vspace*{-.1in}
\caption{Left: Matrix elements from Carbon-14 lifetime calculation~\cite{Maris:2011as}.
Right: Fluorine-14 spectrum predicted by NCSM and measured
experimentally\cite{Maris:2009bx,Goldberg:2010zz}. }
\label{fig:F14C14}
\end{center}
\end{figure*}
One of the important tools for nuclear structure enabled by low-resolution
interactions is the diagonalization of enormous but very sparse
Hamiltonian matrices, usually in a harmonic
oscillator basis to permit center-of-mass effects to be excluded. This
is referred to as no-core full configuration (NCFC) or no-core shell
model (NCSM), depending on the context.
Two recent examples of what is enabled are represented in Fig.~\ref{fig:F14C14}. On the left are Gamov-Teller matrix elements from a large-scale calculation
of Carbon-14 using a soft chiral EFT potential~\cite{Maris:2011as}.
They highlight the critical role of the 3NF in suppressing
the beta decay rate, which explains the anomalously long lifetime
of $^{14}$C (which is used to great advantage for dating artifacts!).
On the right is the low-lying spectrum of Fluorine-14, which is unstable to
proton decay. This spectrum was predicted in advance of the experimental measurements (not a common occurrence until now!), which meant solving a Hamiltonian matrix of dimension 2 billion using 30,000 cores with a soft interaction (derived from inverse scattering rather than RG, but with similar characteristics)~\cite{Maris:2009bx,Goldberg:2010zz}. The predictions and measurement agree within the combined experimental and theoretical uncertainties.
These calculations would not be possible with the potentials of Fig.~\ref{fig:phenpots}.
RG-softened interactions will allow many more of these confrontations
of experiment with theory in the future.
One of the principal aims of the UNEDF project is to calculate reliable reaction cross sections for astrophysics, nuclear energy, and national security, for which extensions of standard phenomenology is insufficient. The interplay of structure and reactions is essential for a successful description of exotic nuclei as well. Such interplay is characteristic of the ab initio no-core shell model/resonating-group method (NCSM/RGM), which treats bound and scattering states within a unified framework using fundamental interactions between all nucleons. A quantitative proof-of-principle calculation of
this approach is shown in Fig.~\ref{fig:ncsmrgm1}~\cite{Navratil:2010ey,Navratil:2010jn}. A wide range of applications is now possible including $^3$H$(d,n){}^4$He fusion and
the $^7$Be$(p,\gamma){}^8$B reaction important for solar neutrino physics, and many more to come.
In Fig.~\ref{fig:ncsmrgm2}, the first-ever ab-initio calculation
of the $^7$Be$(p,\gamma){}^8$B
astrophysical S-factor is shown~\cite{Navratil:2011sa}.
This calculation uses NCSM/RGM with an N$^3$LO NN interaction evolved
by the SRG to a fine-tuned value of $\lambda = 1.86\fmi$.
The ab initio theory predicts
both the normalization \emph{and} the shape of $S_{17}$.
This very promising techniques has many additional applications.
The inclusion of an SRG-evolved 3NF is planned for the near future.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.5in]{ncsm_rgm_he4}
\caption{NCSM/RGM calculation of neutron scattering from the alpha particle using SRG-evolved interactions, compared with experimental measurements~\cite{Navratil:2010ey,Navratil:2010jn}.}
\label{fig:ncsmrgm1}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.8in]{ncsm_rgm_7Be8B3}
\caption{First ever ab initio calculations of $^7$Be$(p,\gamma){}^8$B
astrophysical S-factor~\cite{Navratil:2011sa}. Uses SRG-N3LO with $\lambda = 1.86\fmi$.}
\label{fig:ncsmrgm2}
\end{center}
\end{figure}
\begin{figure}[tbh!]
\begin{center}
\includegraphics[width=2.5in]{cc_ni56_convergence}
\caption{Convergence of CCSD for Ni-56 evolved N3LO to $\lambda = 2.5\fmi$.}
\label{fig:cc_ni56}
\end{center}
\end{figure}
\begin{figure}[bh!]
\begin{center}
\includegraphics[width=2.5in]{oxygen_3NF_shellmodel1}
\caption{Influence of a three-body force on valence neutrons
in oxygen isotopes.}
\label{fig:oxygen_sm2}
\end{center}
\end{figure}
\begin{figure*}[tbh!]
\begin{center}
\includegraphics[width=4.0in]{oxygen_3NF_shellmodel3}
\hspace*{.1in}
\includegraphics[width=2.0in]{oxygen_3NF_shellmodel2}
\caption{Three-body force impact on oxygen single-particle levels.
Left: NN-only, middle: phenomenological forces, right: NN + 3NF.}
\label{fig:oxygen_sm}
\end{center}
\end{figure*}
UNEDF members have demonstrated that the powerful coupled-cluster (CC) ab initio method, which is a workhorse in quantum chemistry, can be used to accurately calculate closed-shell medium-mass nuclei such as $^{40}$C, $^{48}$C,
$^{56}$Ni with chiral EFT two-body interactions or the RG-softened versions~\cite{Hagen:2008iw,Hagen:2010gd}, as well as proton halo nuclei like $^{19}$F~\cite{Hagen:2010zz}.
A proof-of-principle convergence curve (ground-state energy versus the
size of the orbital space) for $^{56}$Ni
with an SRG-evolved potential is shown in
Fig.~\ref{fig:cc_ni56}.
The CC formalism has been extended to include NNN forces and their inclusion in calculations of the heavier nuclei will break new barriers.
The in-medium SRG diagonalization of closed-shell nuclei such as
$^{40}$C~\cite{Tsukiyama:2010rj}, discussed in Section~\ref{subsec:imsrg} is a complementary approach to CC and is one of several advances
in our understanding of the phenomenological nuclear shell model enabled
by softened potentials.
Another is the direct use of MBPT to examine the effect of the 3NF
on the location of the neutron dripline: the limits of nuclear existence
where an added neutron is no longer bound---it ``drips'' away.
The new physics is indicated schematically
in Fig.~\ref{fig:oxygen_sm2}.
It's been established experimentally that as you add neutrons to
stable oxygen-16, the neutrons stay bound until $^{24}$O. But adding
one more proton to get fluorine extends the dripline all the
way to $^{31}$F. This result is \emph{not} predicted by previous
microscopic calculations using NN interactions, because the single-particle
neutron energy levels that get filled are predicted to be bound
(leftmost panel in Fig.~\ref{fig:oxygen_sm}), leading to
$^{28}$O as the calculated dripline.
The phenomenological shell model, in which matrix elements of the
Hamiltonian are fit to nearby nuclei, has a very different pattern
(e.g., compare the $d3/2$ single-particle energies in the middle
and left panels of Fig.~\ref{fig:oxygen_sm})
and predicts the correct dripline. However, recent calculations with a $\vlowk$ RG force and fitted 3NF yield the right panel of Fig.~\ref{fig:oxygen_sm}. When the 3NF effect is added~\cite{Otsuka:2009cs}, the interaction of
valence neutrons with a core neutron, as in Fig.~\ref{fig:oxygen_sm2}, is repulsive, pushing up the
$d3/2$ level so that the dripline is at $^{24}$O.
\begin{figure}[b!]
\begin{center}
\includegraphics[width=3.0in]{2010_ect_rroth_43_cropped}
\caption{Second-order MBPT applied to closed shell nuclei
with SRG-evolved NN interactions, including the induced 3NF~\cite{Roth:2009private}.
Several different values of the flow parameter are shown
(note that $\alpha$ here is the same as $s=1/\lambda^4$).}
\label{fig:rothmbpt}
\end{center}
\end{figure}
The apparent success of MBPT with low-momentum potentials has been
tested by Roth and collaborators~\cite{Roth:2009private}, who have
done calculations of closed shell nuclei across the mass table
in second-order perturbation theory (first-order is Hartree-Fock).
Results for SRG evolved interactions from an initial NN chiral EFT
and including the induced 3NF show excellent independence of the
flow parameter $\lambda$ (see Fig.~\ref{fig:rothmbpt}) and both the energies
and radii are in good agreement with coupled cluster results~\cite{Hagen:2008iw,Hagen:2010gd}.
However, adding a 3NF leads to large $\lambda$ dependence.
Very recent results from Roth et al.\ that use importance truncated NCSM as well as coupled cluster calculations show that the long-range 3NF is the source to apparent large 4NF contributions for oxygen and heavier nuclei, causing
a strong dependence on the flow parameter.
However, by using a lower cutoff for the initial 3NF, remarkable agreement with experimental binding energies is achieved with fits only to
few-body properties~\cite{Roth:2011ar,Roth:2011vt}.
Work is in progress to identify SRG generators to better control the
RG evolution of the initial 3NF.
We have already seen the convergence of MBPT for symmetric
infinite nuclear matter. Perturbation theory is even more
controlled for pure neutron matter, as illustrated in
Figs.~\ref{fig:PNMachim1} and \ref{fig:PNMachim2}, where
$\vlowk$ RG-evolved interactions and fitted 3NF's are
used in calculations of the neutron matter energy per nucleon
as a function of the density~\cite{Hebeler:2009iv}.
The cutoff dependence of the result is used to estimate the many-body uncertainty.
Figure~\ref{fig:PNMachim1}
shows that the 3NF contribution is important (compare to
the NN-only curves) but that the dominant theoretical uncertainty
is the value of the coupling constants for the long-range part
of the N$^2$LO chiral EFT 3NF.
In Fig.~\ref{fig:PNMachim2}, comparisons with non-perturbative
calculations demonstrate the consistency of the much easier
MBPT calculations. The results for the neutron matter equation
of state have been used by Hebeler et al.\ to provide tight constraints
on neutron star masses and radii~\cite{Hebeler:2010jx}.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=2.5in]{neutron_matter_schwenk_1}
\vspace*{-.1in}
\caption{Neutron matter energy/particle versus density calculated using a $\vlowk$ RG-evolved NN interaction plus fit 3NF. The arrow shows the importance of the 3NF and the width of the band indicates the uncertainties due to 3NF couplings~\cite{Hebeler:2009iv}.}
\label{fig:PNMachim1}
\end{center}
\end{figure}
\begin{figure}[th!]
\begin{center}
\includegraphics[width=2.5in]{neutron_matter_schwenk_2}
\vspace*{-.1in}
\caption{Neutron matter energy/particle versus density as in Fig.~\ref{fig:PNMachim1} compared to other calculations~\cite{Hebeler:2009iv}.}
\label{fig:PNMachim2}
\end{center}
\end{figure}
The MBPT results for infinite matter are also valuable input for
work to develop a microscopic nuclear energy density functional (EDF).
A key tool to incorporate microscopic input into nuclear
EDF's is the density matrix expansion (DME) originally proposed
by Negele and Vautherin, which has been revived
and improved in the UNEDF project.
The DME provides a route to an EDF based on microscopic nuclear
interactions through a quasi-local expansion of the energy in
terms of various local densities and currents, including resummations that can
treat long-range one- and two-pion exchange interactions given by chiral EFT.
With sufficiently soft microscopic interactions,
many-body perturbation theory (MBPT) for nuclei is a quantitative
framework for implementing the DME.
The formal development of DME with MBPT is on-going,
but there are already hybrid formulations between purely ab initio and
phenomenological functionals~\cite{Stoitsov:2010ha},
which allow improvements to be made
while more systematic functionals are developed.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=1.7in]{kohn_sham1a_new}
\caption{Schematic representation of interacting neutrons in a
(theoretical) trapping potential.}
\label{fig:KS}
\end{center}
\end{figure}
\begin{figure}[b!]
\begin{center}
\includegraphics[width=2.7in]{Etot_scaled_neutron_drops_minnesota_all_v2}
\caption{Calculations of the ground-state energies and radii
of trapped systems of twenty (top) and eight (bottom) neutron drops~\cite{Bogner:2011kp}.
The different clusters of points are for different harmonic oscillator
trap frequencies. The NCFC results are exact within the error bars.
They are compared to various approximate microscopic functions,
with the BHF and ``fit'' calculations expected to be
the best approximations.}
\label{fig:neutron-drops}
\end{center}
\end{figure}
Several DME implementations strategies have been developed, with
the first tests recently made
against ab-initio calculations using a semi-realistic
interaction (Minnesota) in trapped neutron drops~\cite{Bogner:2011kp}.
Neutron drops are a powerful theoretical laboratory for improving existing
nuclear energy functionals, with particular value in providing
microscopic input needed for neutron-rich nuclei, where there
are fewer constraints from experiment.
The necessity of an external potential (because the untrapped system
is unbound, with positive pressure) is turned into a virtue by
allowing external control over the environment
(see Fig.~\ref{fig:KS} for a schematic of the trapped neutrons).
Density functional theory, which provides the theoretical underpinning
for the microscopic EDF's, dictates that the same functional applies
for any external potential, which can therefore be varied to
probe and isolate different aspects of the EDF.
Results summarized in Fig.~\ref{fig:neutron-drops} show promising agreement
with the DME functionals and the (essentially exact) ab initio
results using NCFC.
Many more developments in this line will be forthcoming.
\subsection{Summary and survey of open problems}
In these lectures, we've made a whirlwind tool of
atomic nuclei at low resolution. With the renormalization
group (RG), the strategy has been to lower the resolution
and track dependence on it. We've seen how high resolution
leads to coupling of low momenta to high momenta, which hinders solutions
of the many-body problem for low-energy properties.
With RG evolution, correlations
in wave functions are reduced dramatically, leading to faster
convergence of many-body methods. A consequence is that non-local
potentials and many-body operators are induced, so these must
be accommodated.
Flow equations (SRG) achieve low resolution by \emph{decoupling}.
This can be in the form of band or block diagonalization
of the Hamiltonian matrix. The flow equations implement a
series of unitary
transformations, in which observables (measurable quantities)
are not altered but the physics interpretation can change!
In the nuclear case, the usual plan is to evolve until
few-body forces start to grow rapidly, or to use an in-medium
version of the SRG.
With the RG, cutoff dependence becomes a new tool in low-energy
nuclear physics.
The basic idea is that, in principle, observables should be unchanged
with RG evolution. In practice, there are approximations in the
RG implementation and in calculating nuclear observables.
These come from truncation or approximation of ``induced'' many-body
forces/operators and from many-body approximations. For nuclei
there can be dramatic changes even with apparently
small changes in the resolution scale. We can use
these changes as diagnostics of approximations and to estimate
theoretical errors.
Some specific applications of cutoff dependence include:
\bi
\I using cutoff dependence at different orders in an EFT expansion,
which carries over to the corresponding RG-evolved interactions;
\I using the running of ground-state energies with cutoff in
few-body systems to estimate errors and identify
correlations (e.g., the Tjon line);
\I in nuclear matter calculations, validating MBPT convergence
and setting lower bounds on the errors from
uncertainties in many-body interactions;
\I in calculations of finite nuclei,
diagnosing missing many-body forces;
\I identifying and characterizing scheme-dependent observables,
such as spectroscopic factors.
\ei
The possibilities have really only begun to be explored.
We have seen glimpses of the
many promising applications of RG methods to nuclei.
Configuration interaction and coupled cluster approaches
using softened interactions
converge faster, opening up new possibilities and allowing the
limits of computational feasibility to be extended.
Ground-breaking ab-initio reaction calculations are now possible.
Applications of low-momentum interactions
to microscopic shell model calculations bring new understanding
to phenomenological results, highlighting the role of three-body
forces. Because many-body perturbation theory (MBPT) is feasible with
the evolved interactions,
the door is opened to constructive nuclear density functional
theory.
There are also many open questions and difficult problems in applying
RG to low-energy nuclear physics. Here is a subset:
\bi
\I Power counting for evolved many-body operators.
That is, how do we anticipate the size of contributions from
induced many-body interactions and other operators?
This is essential if we are to have reliable estimates of
theoretical errors, because truncations are unavoidable.
We need both analytic estimates to guide us as
well as more extensive numerical tests.
Many of the same issues apply to chiral EFT; can the additional
information available from SRG flow parameter dependence help with
analyzing or even constructing EFT's?
\I Only a few possibilities for SRG generators have been
considered for nuclear systems.
Can other choices for the SRG $G_s$ operator help to control
the growth of many-body forces? Can convergence be improved in the
harmonic oscillator basis, which is limited by an infrared
cutoff as well as an ultraviolet cutoff?
Can a generator be found to drive non-local potentials to
local form, so they can be used with quantum Monte Carlo methods?
Or can the SRG equations be formulated to directly produce
a local projection and a perturbative residual interaction?
\I An apparent close connection between the block-diagonal
generator SRG and the ``standard'' $\vlowk$ RG has been established
empirically, but a formal demonstration of the connection and
its limits has not been made.
\I What other bases for SRG evolution would be advantageous?
The need for a momentum-space implementation for evolution
in the $A=3$ space and beyond is foremost. Beyond providing
necessary checks of evolution in the harmonic oscillator
basis, the evolved interactions in this form could be directly
applied to test MBPT in infinite matter and to test nuclear
scaling. Another possibility is to use hyperspherical coordinates,
which combine the advantage of a discrete basis with better
asymptotic behavior
(and which would be useful for visualization of many-body forces).
\I There are many open questions and problems involving operators.
These include formal issues such as the scaling of many-body
operators and technical issues such as how to handle boosts of
operators that are not galilean invariant. And there are simply
many applications that are yet to be made (e.g., electroweak
processes).
\I The flow to universal form exhibited by two-body interactions
has been clear from the beginning of RG applications to nuclei,
but whether this same behavior is expected for many-body interactions
or for other operators is still open.
\I How can we use more of the power of the RG?
\ei
There is no shortage of opportunities and challenges!
\bibliographystyle{elsarticle-num-names}
|
{
"timestamp": "2012-03-09T02:02:43",
"yymm": "1203",
"arxiv_id": "1203.1779",
"language": "en",
"url": "https://arxiv.org/abs/1203.1779"
}
|
\section{Introduction}
There are a variety of theoretical schemes predicting violations of Lorentz symmetry. Attempts to quantize gravity have resulted in theories that
allow for it \cite{Ame05}. Certain string theories envisage the possibility of a spontaneous breaking of it \cite{koste89}.
The possibility of detecting experimental signatures of breakings of Lorentz symmetry \cite{Ame00,Ame05} from an underlying unified theory at the Planck scale has recently raised interest in Lorentz violation; for a recent review of modern tests of Lorentz invariance see, e.g., \cite{Matt05}.
The Standard Model Extension (SME) is a theoretical framework which allows for generic violations of the Lorentz symmetry for both gravity and electromagnetism \cite{koste1,koste2,koste3,koste4}. In general, there are 20 coefficients for Lorentz violation in the gravitational sector; by assuming spontaneous Lorentz-symmetry breaking, the main effects in the weak-field approximation are accounted for by the traceless coefficients $\overline{s}^{\mu\nu}$ \cite{Bail06} containing 9 independent quantities.
According to Bailey \cite{Bail010}, in the weak field and slow motion approximation, a test particle moving with velocity $\bds v$ at distance $r$ from a central, static body of mass $M$ experiences a Lorentz-violating gravitomagnetic acceleration
\eqi \bds A_{\rm GM} =\ton{\rp{\bds v}{c}}\bds\times \bds B_{\rm G}, \lb{accel}\eqf
where
\eqi\bds B_{\rm G}\doteq \rp{2GM}{r^3}\ton{\bds s\bds\times \bds r},\lb{BG}\eqf
with \eqi \bds s\doteq -\overline{s}^{0j},\ j=1,2,3;\eqf $G$ is the Newtonian constant of gravitation, while $c$ is the speed of light in vacuum.
Constraints from Lunar Laser Ranging (LLR) \cite{Dick} are \cite{Batt07}
\begin{align}
\overline{s}^{01} \lb{llr1} & = \ton{-0.8\pm 1.1}\times 10^{-6}, \\ \nonumber \\
\overline{s}^{02} \lb{llr2} & = \ton{-5.2\pm 4.8}\times 10^{-7},
\end{align}
so that
\eqi s_{\rm max}\lesssim 1-2\times 10^{-6}.\lb{smax}\eqf
A general account of the present-day bounds on all the SME Lorentz violating coefficients can be found in \cite{Koste011}.
From simple dimensional considerations, planetary orbital precessions in the field of the Sun, if present, would be as large as
\eqi \left|\dot\Psi\right|\lesssim \rp{A_{\rm G}}{v}\sim \rp{2GMs}{cr^2} = \rp{1.77\times 10^6\ {\rm m^2\ s^{-1}}}{r^2}, \lb{prece}\eqf
where $\Psi$ denotes a generic orbital element, and we used \rfr{smax}.
The present-day level of accuracy in constraining the secular rates of orbital changes of the planets of the solar system are listed in Table \ref{tavolapit} and Table \ref{tavolafie}.
\begin{table*}[ht!]
\caption{Uncertainties in the rates of change of the semimajor axis $a$, the eccentricity $e$, the inclination $I$ to the mean ecliptic at J$2000.0$, the longitude of the ascending node $\Om$, the longitude of perihelion $\varpi\doteq \Om + \omega$, the mean motion $n_{\rm b}\doteq\sqrt{GM a^{-3}}$ and the mean longitude $\lambda\doteq\Om+\omega+\mathcal{M}$ of the planets of the solar system; $\omega$ is the argument of perihelion, while $\mathcal{M}$ is the mean anomaly. They were inferred by us by taking the ratios of the formal, statistical errors in Table 3 of \cite{pitjeva07}, all rescaled by a factor 10, to the data time span $\Delta T=93$ yr (1913-2006) of the EPM2006 ephemerides used by Pitjeva \cite{pitjeva07}. The figures for $n_{\rm b}$ account also for the uncertainty $\sigma_{GM}=10\ {\rm km^3\ s^{-2}}$ in the Sun's gravitational parameter $GM$ retrieved from \cite{konopliv011}. The results for Saturn are relatively inaccurate with respect to those of the inner planets since radiotechnical data from Cassini were not yet processed when Table 3 of \cite{pitjeva07} was produced. Here mas cty$^{-1}$ stands for milliarcseconds per century.
}\label{tavolapit}
\centering
\bigskip
\begin{tabular}{llllllll}
\hline\noalign{\smallskip}
& $\dot a$ $\ton{\rp{\rm m}{\rm cty}}$ & $\dot e$ $\ton{\rp{1}{\rm cty}}$ & $\dot I$ $\ton{\rp{\rm mas}{\rm cty}}$ & $\dot \Om$ $\ton{\rp{\rm mas}{\rm cty}}$ & $\dot \varpi $ $\ton{\rp{\rm mas}{\rm cty}}$ & $n_{\rm b}$ $\ton{\rp{\rm mas}{\rm cty}}$ & $\dot \lambda$ $\ton{\rp{\rm mas}{\rm cty}}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Mercury & $3.6$ & $4\times 10^{-9}$ & $14.8$ & $121$ & $5.4$ & $50.6$ & $3.7$ \\
Venus & $2.3$ & $2\times 10^{-10}$ & $0.3$ & $9.9$ & $5.7$ & $10.2$ & $0.3$\\
Earth & $1.5$ & $5\times 10^{-11}$ & $-$ & $-$ & $0.6$ & $5.2$ & $-$ \\
Mars & $2.8$ & $5\times 10^{-11}$ & $0.03$ & $0.8$ & $0.1$ & $2.8$ & $0.02$\\
Jupiter & $6612.9$ & $2\times 10^{-8}$ & $23.4$ & $1138.9$ & $79.8$ & $131.2$ & $15.7$\\
Saturn & $45763.4$ & $1\times 10^{-7}$ & $42.7$ & $806.7$ & $778.2$ & $196.9$ & $37.6$\\
Uranus & $433269.0$ & $3\times 10^{-7}$ & $66.2$ & $3671.9$ & $836.5$ & $323.7$ & $80.7$\\
Neptune & $4.9818\times 10^6$ & $8\times 10^{-7}$ & $75.3$ & $2284.0$ & $20249.1$ & $1210.7$ & $263.2$ \\
Pluto & $3.66961\times 10^7 $ & $3\times 10^{-6}$ & $167.4$ & $305.4$ & $2713.4$ & $4501.6$ & $468.3$\\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table*}
\begin{table*}[ht!]
\caption{Supplementary advances of
perihelia and nodes of some planets of the solar system estimated by \cite{fienga011} with the INPOP10a ephemerides. Data from Messenger and Cassini were included for Mercury and Saturn. The reference $\{x,y\}$ plane is the mean Earth's equator at J$2000.0$.
}\label{tavolafie}
\centering
\bigskip
\begin{tabular}{lll}
\hline\noalign{\smallskip}
& $\dot \Om$ $\ton{\rp{\rm mas}{\rm cty}}$ & $\dot \varpi $ $\ton{\rp{\rm mas}{\rm cty}}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Mercury & $1.4 \pm 1.8$ & $0.4 \pm 0.6$ \\
Venus & $0.2 \pm 1.5$ & $ 0.2\pm 1.5$ \\
Earth & $0.0\pm 0.9$ & $-0.2\pm 0.9$ \\
Mars & $-0.05\pm 0.13$ & $-0.04\pm 0.15$ \\
Jupiter & $-40\pm 42$ & $-41\pm 42$ \\
Saturn & $-0.1\pm 0.4$ & $0.15\pm 0.65$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table*}
In Table \ref{tavolaSME} we show tentative upper bounds on putative precessional effects according to \rfr{smax} for all the planets of the solar system.
\begin{table*}[ht!]
\caption{Orders of magnitude of the maximum values $\left|\dot \Psi\right|_{\rm max}$ of putative planetary orbital precessions due to \rfr{accel}, with $s = s_{\rm max}\sim 2\times 10^{-6}$. Cfr. with the empirical bounds in Table \ref{tavolapit} and Table \ref{tavolafie}.
}\label{tavolaSME}
\centering
\bigskip
\begin{tabular}{lll}
\hline\noalign{\smallskip}
& $\left|\dot \Psi\right|_{\rm max}$ $\ton{\rp{\rm mas}{\rm cty}}$ & $\left|\dot \Psi\right|_{\rm max} $ $\ton{\rp{1}{\rm cty}}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Mercury & $343.70$ & $1.7\times 10^{-6}$ \\
Venus & $98.43$ & $4.7\times 10^{-7}$ \\
Earth & $51.50$ & $2.5\times 10^{-7}$ \\
Mars & $22.18$ & $1.1\times 10^{-7}$ \\
Jupiter & $1.92$ & $9.3\times 10^{-9}$ \\
Saturn & $0.56$ & $2.7\times 10^{-9}$ \\
Uranus & $0.14$ & $6.7\times 10^{-10}$\\
Neptune & $0.05$ & $2.7\times 10^{-10}$ \\
Pluto & $0.03$ & $1.6\times 10^{-10}$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table*}
A tension between the LLR bounds \cite{Batt07} of \rfr{llr1}-\rfr{llr2} and the resulting expected planetary precessions of Table \ref{tavolaSME} exist, especially for the inner planets.
Thus, a more detailed analysis is worthwhile. Actually, a mere order-of-magnitude analysis based just on \rfr{prece} and Table \ref{tavolaSME} would be insufficient to draw meaningful
conclusions. Indeed, exact calculations of the secular variations of all the osculating Keplerian orbital elements caused by \rfr{accel} must
be performed, with, e.g., standard perturbative techniques, in order to check if it really induces averaged non-zero orbital changes.
Moreover, also in such potentially favorable case caution is still in order. Indeed, it may well happen, in principle, that the resulting
analytical expressions retain multiplicative factors $1/e^k, k = 1, 2, 3,\ldots$ or $e^k, k = 1, 2, 3,\ldots $ which would notably alter the size of the
found non-zero secular rates with respect to the expected values according to \rfr{prece}. Thus, in Section \ref{calcolo}, we will analytically work out the long-term rates of change of all the osculating Keplerian orbital elements of a test particle acted upon by \rfr{accel} by using the Gauss perturbative scheme \cite{roy05}.
\section{Analytical calculation of the orbital precessions}\lb{calcolo}
In the following, we will not make any particular assumption about the orientation of $\bds s$ in space, so that we will adopt a generic reference frame centered in $M$, with respect to which $\bds s=\{s_x,s_y,s_z\}$. Moreover, we will not make any definite choice about the reference $\{x,y\}$ plane, so that our results can be applied to a variety of specific astronomical and astrophysical scenarios.
The\footnote{The radial unit vector $\bds{\hat{R}}$ is directed from $M$ to the test particle, the transverse unit vector $\bds{\hat{T}}$ lies in the osculating orbital plane and is perpendicular to $\bds{\hat{R}}$, while the normal unit vector $\bds{\hat{N}}$ is perpendicular to the orbital plane and is directed along the osculating orbital angular momentum $\bds L$. Such three unit vectors constitute a right-handed orthonormal basis comoving with the test particle.} radial, transverse and normal components $A_R,A_T,A_N$ of \rfr{accel}, evaluated onto the unperturbed Keplerian ellipse, are
\begin{align}
A_R \lb{ar} & = \rp{-2 GM n_{\rm b} \ton{1 + e \cos f}^3 }{ a c \ton{1 -e^2}^{5/2} } \mathcal{A_R}, \\ \nonumber \\
\mathcal{A_R} & = s_z \cos u \sin I + \cos I \cos u \ton{s_y \cos \Om - s_x \sin \Om } - \sin u \ton{s_x \cos \Om + s_y \sin \Om }, \\ \nonumber \\
A_T & = \rp{2 e GM n_{\rm b} \ton{1 + e \cos f}^2 \sin f }{ a c \ton{1 - e^2}^{5/2}}\mathcal{A_T}, \\ \nonumber \\
\mathcal{A_T} & = s_z \cos u\sin I + \cos I \cos u \ton{s_y \cos \Om - s_x
\sin \Om } - \sin u \ton{s_x \cos \Om + s_y \sin \Om }, \\ \nonumber \\
A_N \lb{an} & = \rp{2 e GM n_{\rm b} \ton{1 + e \cos f}^2 \sin f }{a c \ton{1 - e^2}^{5/2}} \mathcal{A_N}, \\ \nonumber \\
\mathcal{A_N} & = s_z \cos I + \sin I\ton{s_x \sin \Om -s_y \cos \Om},
\end{align}
where \cite{roy05} $u\doteq f + \omega$ is the argument of latitude defined in terms of the true anomaly $f$ and the argument of pericenter $\omega$. Notice that $A_T = A_N = 0$ for circular orbits, i.e. for $e\rightarrow 0$.
Inserting \rfr{ar}-\rfr{an} in the right-hand-sides of the Gauss equations for the variation of the elements and averaging them over one orbital revolution yield the following non-vanishing rates of changes of the Keplerian orbital elements
\begin{align}
\ang{\dert a t} \lb{dadt} & = 0, \\ \nonumber \\
\ang{\dert e t} \lb{dedt} & =\rp{ 2GM\sqrt{1-\ee}}{ c a^2 \ton{ 1 + \sqrt{1-\ee} } }\mathcal{E}, \\ \nonumber \\
\mathcal{E} & = \co \ton{s_x \cO + s_y \sO} + \so \qua{s_z \sI + \cI \ton{s_y \cO - s_x \sO}}, \\ \nonumber \\
\ang{\dert I t} \lb{dIdt} & = -\rp{ 2GM\ton{ 1 - \sqrt{1-\ee}}}{ c a^2 e\sqrt{1-\ee} }\mathcal{I},\\ \nonumber \\
\mathcal{I} & = \so\qua{s_z \cI +\sI\ton{s_x\sO - s_y\cO}}, \\ \nonumber \\
\ang{\dert \Om t} \lb{dOdt} & = \rp{ 2GM\ton{ 1 - \sqrt{1-\ee}}}{ c a^2 e\sqrt{1-\ee} }\mathcal{N},\\ \nonumber \\
\mathcal{N} & = \co\csc I\qua{s_z\cI +\sI\ton{s_x\sO - s_y\cO} }, \\ \nonumber \\
\ang{\dert \varpi t} \lb{dodt} & = -\rp{ 2GM}{ c a^2 e^3\ton{1-e^2} }\mathcal{P},\\ \nonumber \\
\mathcal{P} \nonumber & = \ton{1 - e^2} \ton{1 - \sqrt{1 - e^2}} \sin\omega \ton{s_x \cos\Om + s_y \sin\Om} + \\ \nonumber \\
\nonumber & + \cos\omega \grf{s_y \qua{e^2 \ton{-1 + e^2 + \sqrt{1 - e^2}} + \right.\right. \\ \nonumber \\
\nonumber & + \left.\left. \ton{-1 + \sqrt{1 - e^2} - e^2 \ton{-2 + e^2 + 2 \sqrt{1 - e^2}}} \cos I} \cos\Om +\right.\\ \nonumber \\
\nonumber & + \left. \ton{1 - e^2} \ton{1 - \sqrt{1 - e^2}} s_z \sin I - e^2 \sqrt{1 - e^2} s_x \sin\Om - \sqrt{1 - e^2} s_x
\cos I \sin\Om + \right. \\ \nonumber \\
\nonumber & + \left. 2 e^2 \sqrt{1 - e^2} s_x \cos I \sin\Om + \ton{-1 + e^2} s_x \qua{-e^2 + \ton{-1 + e^2} \cos I} \sin\Om -\right.\\ \nonumber \\
& - \left. e^2 \sqrt{1 - e^2} s_z \cos I \tan\ton{\rp{I}{2}} + e^2 \ton{1 - e^2} s_z \cos I \tan\ton{\rp{I}{2}}}, \\ \nonumber \\
\ang{\dert {\mathcal{M}} t} \lb{dMdt} & =-\rp{ 2GM\ton{1 - e^2}}{ c a^2 e\ton{1 - e^2 + \sqrt{1-e^2}} }\mathcal{L},\\ \nonumber \\
\mathcal{L} & = s_z \cos\omega \sin I + \cos I \cos\omega \ton{s_y \cos\Om - s_x \sin\Om} - \sin\omega \ton{s_x \cos\Om + s_y \sin\Om}.
\end{align}
The results of \rfr{dadt}-\rfr{dMdt} are exact in the sense that no a priori simplifying assumptions on either $e$ and $I$ have been assumed.
By expanding in powers of $e$, it turns out that, in the limit of small eccentricities, the inclination and node precessions are of order $\mathcal{O}(e)$, while the precessions of the perihelion and the mean anomaly are not defined when $e\rightarrow 0$ since they are of order $\mathcal{O}(e^{-1})$; on the other hand, the precession of the mean longitude $\lambda\doteq \varpi + \mathcal{M}$ is of order $\mathcal{O}(e)$. For other computation of the orbital effects, see \cite{Bail06}.
The generality of \rfr{dadt}-\rfr{dMdt} allows one to use them for sensitivity analyses in a variety of specific astronomical and astrophysical scenarios, where different $\{x,y\}$ reference planes and orbital configurations of test particles appear.
\section{Results and conclusions}
By using \rfr{dodt} for Mercury, Venus and the Earth and the figures of Table \ref{tavolafie} for their supplementary advances of perihelia, we solve for $s_x,s_y,s_z$ and obtain
\begin{align}
s_x \lb{dsx} & \doteq -\overline{s}^{01} = (0.9\pm 1.5)\times 10^{-8}, \\ \nonumber \\
s_y \lb{dsy} & \doteq -\overline{s}^{02} = (-4 \pm 6)\times 10^{-9}, \\ \nonumber \\
s_z \lb{dsz} & \doteq -\overline{s}^{03} = (0.3\pm 1)\times 10^{-9}.
\end{align}
The forthcoming analysis of more data from the ongoing MErcury Surface, Space ENvironment,
GEochemistry, and Ranging (MESSENGER) mission \cite{messenger} to Mercury should allow one to improve such bounds in a near future. In view of that fact that. according to \rfr{dadt}-\rfr{dMdt}, the strongest signals occur for the closest particles to the central body, it would be desirable that supplementary advances of all the Keplerian orbital elements of Mercury will be produced in future ephemerides.
We stress that \rfr{dsx}-\rfr{dsz} should be regarded as a sort of predicted parameter sensitivity since, actually, Fienga et al. \cite{fienga011} did not model any Lorentz-violating terms in the INPOP010a ephemerides. Actually, in more refined analyses, it would be required, in principle, to explicitly include \rfr{accel} in the dynamical force models usually fitted to planetary observations, and solve for the $\overline{s}^{0j},\ j=1,2,3$ parameters in a dedicated global solution obtained by reprocessing the entire data set with such modified softwares. The use of a similar approach, including also LLR data, was envisaged in \cite{Batt07}; see also the considerations by Nordtvedt \cite{Nord} in a different context.
The Moon yields us a benchmark for testing the degree of reliability of our approach. Indeed, from the small eccentricity limit of \rfr{dodt}, it can naively be posed
\eqi \dot\varpi \sim \rp{GMs}{c a^2 e}, \eqf from which one can infer
\eqi s\lesssim \rp{c a^2 e}{GM}\delta\dot\varpi.\lb{approx}\eqf Since for the lunar perigee it is \cite{Dick,Muller1,Williams,Muller2,Muller3}
\eqi \delta\dot\varpi \sim 0.1 \ {\rm mas\ yr^{-1}},\eqf
\rfr{approx} yields just
\eqi s \sim 10^{-7},\eqf in substantial agreement with the constraints of \rfr{llr1}-\rfr{llr2} obtained by Battat et al. \cite{Batt07} as the outcome of a fit of modified models to LLR data.
Finally, it may be interesting to look at the Earth and the LAGEOS satellites. Indeed, by looking at, say, the nodes of both LAGEOS and LAGEOS II, and the perigee of LAGEOS II it is possible to constrain $\bds s$ as previously done with the perihelia of three planets. Generally speaking, a major source of systematic uncertainty in the knowledge of the orbit of a terrestrial spacecraft is represented by the mismodeling in the first even zonal coefficient $\overline{C}_{2,0}$ accounting for the terrestrial quadrupole mass moment because of the resulting relatively huge secular precessions of the node and the perigee \cite{Capde}. By assuming \cite{IERS} $\delta \overline{C}_{2,0}\sim 10^{-10}$, corresponding to uncertainties in the orbital elements of LAGEOS and LAGEOS II of the order of $\sim 100-200$ mas yr$^{-1}$, it turns out
\begin{align}
s_x \lb{Tdsx} & \doteq -\overline{s}^{01} \lesssim 1.4\times 10^{-3}, \\ \nonumber \\
s_y \lb{Tdsy} & \doteq -\overline{s}^{02} \lesssim 2\times 10^{-4}, \\ \nonumber \\
s_z \lb{Tdsz} & \doteq -\overline{s}^{03} \lesssim 1.3 \times 10^{-3}.
\end{align}
It can be noticed that \rfr{Tdsx}-\rfr{Tdsz} are neither competitive with the planetary bounds of \rfr{dsx}-\rfr{dsz} nor with the lunar ones of \rfr{llr1}-\rfr{llr2}.
|
{
"timestamp": "2012-03-13T01:01:46",
"yymm": "1203",
"arxiv_id": "1203.1859",
"language": "en",
"url": "https://arxiv.org/abs/1203.1859"
}
|
\section{Introduction}
Note that if $M$ is a rank-$(r-c+1)$ flat of $PG(r-1,q)$, then
$|M| = \frac{q^{r-c+1}-1}{q-1}$ and each rank-$m$ flat of $PG(r-1,q)$
intersects $M$ in a flat of rank at least $m-c+1$.
Our main result is the following:
\begin{theorem}[Main Theorem]\label{main}
For each prime-power $q$, all integers $m> c\ge 0$, and
any real number $\epsilon>0$, there is an integer
$R=R_{\ref{main}}(m,q,c,\epsilon)$ such that,
if $n>R$ and $G$ is a set of points in $PG(n-1,q)$ with
$|G|\le (1-\epsilon)\left(\frac{q^{n-c+1}-1}{q-1}\right)$,
then there exists a rank-$m$ flat $F$ of $PG(n-1,q)$
such that $rank(F\cap G)\le m-c$.
\end{theorem}
We were motivated by a problem in extremal matroid theory
posed by Kung~[\ref{kung}]; the matroidal origins
of the problem are reflected in our terminology which
we briefly review below.
Let $\mathbb F$ be a finite field of order $q$ and let $V$
be a rank-$r$ vector space over $\mathbb F$.
A {\em rank-$k$ flat} of $PG(r-1,\mathbb F)$ is
a $(k+1)$-dimensional subspace of $V$;
the {\em points} are the rank-$1$ flats;
the {\em lines} are the rank-$2$ flats; and
the {\em hyperplanes} are the rank-$(r-1)$ flats.
Technically the projective geometry depends on the particular
vector space $V$; to make this explicit, we write
$PG(V)$ for the projective geometry given by $V$.
We refer to a set $H$ of points in $PG(r-1,\mathbb F)$, for some $r$,
as a {\em geometry over $\mathbb F$} and we define
$rank(H)$ to be the rank of the flat spanned by $H$.
If $H$ and $G$ are geometries over $\mathbb F$, then
there are vector spaces $V_1$ and $V_2$ over $\mathbb F$
so that $H$ is a spanning set of points in $PG(V_1)$ and
$G$ is a spanning set of points in $PG(V_2)$.
We say that {\em $H$ is a restriction of $G$}
or that {\em $G$ contains $H$}, if
there is a rank-preserving projective transformation
from $V_2$ to a vector space $V'_2$ containing $V_1$
so that $H$ is contained in the image of $G$.
For a geometry $H$ over $\mathbb F$ and positive integer $n$,
we let $ex_q(H;n)$ denote the maximum number of points
in a rank-$n$ geometry over $\mathbb F$ not containing $H$.
For integers $0\le c \le m$,
let $F$ be a rank-$(m-c)$ flat of $PG(m-1,q)$ and let
$G(m-1,q,c)$ be the geometry obtained by restricting
$PG(m-1,q)$ to the complement of $F$; thus
$G(m-1,q,m) = PG(m-1,q)$ and $G(m-1,q,1) = AG(m-1,q)$, the rank-$m$ affine geometry over $GF(q)$.
The {\em critical exponent} of $H$ over $GF(q)$, written $c(H;q)$, is
the minimum $c$ such that $H$ is contained in $G(r(M)-1,q,c)$.
The critical exponent was introduced by
Crapo and Rota~[\ref{cr}] and is related to the chromatic number
of a graph.
The following result, which is an easy corollary of
Theorem~\ref{main}, was all but conjectured by Kung~[\ref{kung}].
\begin{theorem}\label{cor}
Let $\mathbb F$ be a finite field of order $q$. If $H$ is a geometry
over $\mathbb F$ with with critical exponent $c>0$, then
$$ \lim_{n\rightarrow \infty} \frac{ ex_q(H;n) }{|PG(n-1,q)|}
= 1-q^{1-c}.$$
\end{theorem}
This theorem bears a striking resemblance to the following theorem
of Erd\H os and Stone~[\ref{es}].
For a graph $H$, let $ex(H;n)$ denote the
maximum number of edges in a simple $n$-vertex
graph that does not contain a subgraph isomorphic to $H$.
The {\em chromatic-number}, $\chi(G)$, of a graph $G$ is the
minimum number of colours needed to colour the vertices so that
no two adjacent vertices get the same colour.
\begin{theorem}[Erd\H os-Stone Theorem]\label{esthm}
For any graph $H$ with chromatic-number $\chi\ge 2$,
$$ \lim_{n\rightarrow \infty} \frac{ex(H;n)}{{n\choose 2}} =
1-\frac{1}{\chi-1}.$$
\end{theorem}
\section{old results}
In this section we briefly review related results.
Note that $G(n-1,q,m-1)$ does not contain $PG(m-1,q)$;
Bose and Burton~[\ref{bb}] showed that $G(n-1,q,m-1)$
is extremal among geometries not containing $PG(m-1,q)$.
\begin{theorem}\label{bbthm}
Let $\mathbb F$ be a field of order $q$ and $m$ and $r$ be integers
with $n\ge m\ge 0$. Then
$$ ex_q(PG(m-1,\mathbb F);\, n) = |G(n-1,q,m-1)|.$$
\end{theorem}
Bonin and Qin~[\ref{bq}] determine $ex_q(H;n)$ exactly for
several other interesting families of geometries.
Our main result is an easy application of the
following deep result due to Furstenberg and Katznelson~[\ref{fur}, Theorem 9.10] in 1985.
\begin{theorem}\label{mdhjthm}
For each field $\mathbb F$ of order $q$, integer $ m\ge 2$, and
real number $\epsilon > 0$, there is an integer
$R=R_{\ref{mdhjthm}}(m,q,\epsilon)$ such that,
$$ ex_q(AG(m-1,\mathbb F);\, n) < \epsilon |PG(n-1,q)|$$
for all $n> R$.
\end{theorem}
This result can be obtained as an easy application of the
density version of the multidimensional Hales-Jewett theorem, also proved by Furstenberg and Katznelson~[\ref{fk}], in 1991,
using ergodic theory. An easier proof was later obtained via the
polymath project~[\ref{polymath}].
The ``easier proof'' is still, however, more than 30 pages long.
Bonin and Qin~[\ref{bq}] have a much simpler proof
of Theorem~\ref{mdhjthm} in the case that $q=2$.
\section{New results}
We start with a proof of Theorem~\ref{main}; for convenience we
restate it in a complementary form.
(The equivalence between the two statements is easy and
is left to the reader.)
\begin{theorem}[Reformulation of Theorem~\ref{main}]\label{main2}
For any integers $m> c\ge 1$ and
real number $\epsilon>0$, there is an integer
$R=R_{\ref{main2}}(m,q,c,\epsilon)$ such that,
$$ex_q( G(m-1,q,c); n) < (1-q^{1-c}+\epsilon)|PG(n-1,q)|,$$
for all $n>R$.
\end{theorem}
\begin{proof}
Let $m> c\ge 1$ be integers and let
$\epsilon>0$ be a real number.
The proof is by induction on $c$;
the case that $c=1$ follows directly from Theorem~\ref{mdhjthm}.
Assume that $c>1$ and that the result holds for $c-1$.
Let $r = R_{\ref{mdhjthm}}(m-c+1,q,\epsilon/2)$,
let $t$ be sufficiently large so that
$q^{1-c}(q^r-1) \le \tfrac{\epsilon}{2}(q^n-q^r)$ for all $n > t$,
and define
$$R_{\ref{main2}}(m,q,c,\epsilon)=
\max(t,R_{\ref{main2}}(r,q,c-1,q^{2-c}-q^{1-c})).$$
Now let $n > R_{\ref{main2}}(m,q,c,\epsilon)$ and
let $M$ be a restriction of $PG(n-1,q)$ with
$|M|\ge (1-q^{1-c}+\epsilon)|PG(n-1,q)|$.
By the inductive assumption, $M$ has a $G(r-1,q,c-1)$-restriction.
Thus there are flats $F_0\subseteq F_1$ of $PG(n-1,q)$ such that
$rank(F_1)=r$, $rank(F_0)=r-c+1$, and $F_1-F_0\subseteq M$. Let
$F_0^c\subseteq F_1$ be a rank-$(c-1)$ flat that is disjoint from $F_0$.
Note that, by our definition of $t$,
$$|M\setminus F_1|\ge \left(1-q^{1-c} +\tfrac{\epsilon}{2}\right)
|PG(n-1,q)-F_1|.$$
So by an elementary averaging argument,
there exists a rank-$(r+1)$ flat $F_2$ containing
$F_1$ such that
$$|M\cap(F_2-F_1)| \ge \left(1-q^{1-c} +\tfrac{\epsilon}{2}\right)|F_2-F_1| =
(1-q^{1-c})q^{r} +\tfrac{\epsilon}{2} q^r .$$
We want to find a rank-$m$ flat $F\subseteq F_2$
such that $F^c_0\subseteq F\not\subseteq F_1$ and $F-F_1\subseteq M$.
If $F$ satisfies these conditions, then
$rank(F \cap F_0) = m-c$ and, hence,
the restriction of $M$ to $F$ contains $G(m-1,q,c)$.
Let $S= (F_2-F_1)\cap M$.
For a flat $F$ of $PG(n-1,q)$ and point $e\not\in F$,
we let $F+e$ denote the flat spanned by $F\cup\{e\}$.
Let $e\in F_2-F_1$ and let $Q = (F_0+e)-F_0$.
Now, for each $f\in Q$,
let $S_f = (F_0^c + f)\cap S$.
Note that $(S_f\, : \, f\in Q)$ partitions $S$ and
$|S_f|\le q^{c-1}$. Finally, let $Q_1$ be the set of all $f\in Q$,
such that $|S_f|= q^{c-1}$.
All vectors in $Q-Q_1$
extend to at most $q^{c-1}-1$ elements in $S$, so
\begin{eqnarray*}
(q^{c-1}-1) q^{r-c+1} + |Q_1| &\ge& |S| \\
&\ge & \left( 1 -q^{1-c}+\tfrac{\epsilon}{2} \right) q^r\\
&= & ( q^{c-1}-1) q^{r-c+1}+\tfrac{\epsilon}{2} q^r.
\end{eqnarray*}
Thus $|Q_1|\ge \frac{\epsilon}{2} q^r$.
By Theorem~\ref{mdhjthm},
there is a subset $Q_2$ of $Q_1$ such that
$Q_2 \cong AG(m-c, q)$.
Let $F$ be the flat of $PG(n-1,q)$
spanned by $F_0^c$ and $Q_2$.
Thus $F$ has rank $m$, $F_0^c\subseteq F$, and, since $Q_2\subseteq Q_1$,
$F-F_1\subseteq M$.
So the restriction of $M$ to $F-F_0$ gives $G(m-1,q,c)$.
\end{proof}
We can now prove Theorem~\ref{cor}, which we restate here
for convenience.
\begin{corollary}
Let $\mathbb F$ be a finite field of order $q$. If $H$ is a geometry
over $\mathbb F$ with with critical exponent $c>0$, then
$$ \lim_{n\rightarrow \infty} \frac{ ex_q(H;n) }{|PG(n-1,q)|}
= 1-q^{1-c}.$$
\end{corollary}
\begin{proof}
Observe that $H$ is a restriction of
$G(r(N)-1,q,c)$ but it is not a restriction of
$G(n-1,q,c-1)$.
Then, by Theorem~\ref{main2}, for
all $\epsilon>0$ and all
sufficiently large $n$,
$$
\frac{q^n}{q^n-1}(1-q^{1-c})=
\frac{|G(n-1,q,c-1)|}{|PG(n-1,q)|}\le
\frac{ex_q(H;n)}{|PG(n-1,q)|} \le 1-q^{1-c} +\epsilon,$$
so the result holds.
\end{proof}
The value of $R_{\ref{main2}}(m,q,c,\epsilon)$ provided by Theorem~\ref{main2}
depends on that of $R_{\ref{mdhjthm}}(m,q,\epsilon)$,
for which the bounds in [\ref{polymath}] are Ackermann-like
for all $q > 2$.
In the binary case, however, the main theorem
of [\ref{bq}] implies that the relatively small function
$R_{\ref{mdhjthm}}(m,2,\epsilon) = 2^{m-2}\lceil 1-\log_2 \epsilon\rceil$
will satisfy Theorem~\ref{mdhjthm}. From this,
one can derive from the proof that
$R_{\ref{main2}}(m,2,c,\epsilon) = T_c(m+d)$ will satisfy Theorem~\ref{main2}, where
$d = \lceil\log_2\lceil(2-\log_2 \epsilon)\rceil\rceil$,
and $T_c$ is the tower function recursively defined by $T_0(s) = s$
and $T_i(s) = T_{i-1}(2^s)$ for all $i > 0$.
\section*{References}
\newcounter{refs}
\begin{list}{[\arabic{refs}]}%
{\usecounter{refs}\setlength{\leftmargin}{10mm}\setlength{\itemsep}{0mm}}
\item \label{bq}
J.E. Bonin, H. Qin,
Size functions of subgeometry-closed classes of
representable combinatorial geometries,
Discrete Math. 224, (2000) 37-60.
\item \label{bb}
R.C. Bose, R.C. Burton,
A characterization of flat spaces in a finite
geometry and the uniqueness of the Hamming
and MacDonald codes,
J. Combin. Theory 1, (1966) 96-104.
\item\label{cr}
H.H. Crapo, G.-C. Rota,
On the foundations of combinatorial theory:
Combinatorial geometries, M.I.T. Press, Cambridge, Mass., 1970.
\item \label{es}
P. Erd\H os, A.H. Stone,
On the structure of linear graphs,
Bull. Amer. Math. Soc. 52, (1946) 1087-1091.
\item \label{fur}
H. Furstenberg, Y. Katznelson,
An ergodic Szemeredi theorem for IP-systems and combinatorial theory,
Journal d'Analyse Math\' ematique 45, (1985) 117-168.
\item \label{fk}
H. Furstenberg, Y. Katznelson,
A density version of the Hales-Jewett Theorem,
J. d'Analyse Math\' ematique 57 (1991) 64-119.
\item\label{kung}
J.P.S. Kung,
Extremal matroid theory, in:
Graph Structure Theory, N. Robertson and P.D. Seymour, eds.
(Amer. Math. Soc., Providence RI, 1993), 21-61.
\item\label{polymath}
D.H.J. Polymath,
A new proof of the density Hales-Jewett theorem,
arXiv:0910.3926v2 [math.CO], (2010) 1-34.
\end{list}
\end{document}
|
{
"timestamp": "2012-03-09T02:04:48",
"yymm": "1203",
"arxiv_id": "1203.1911",
"language": "en",
"url": "https://arxiv.org/abs/1203.1911"
}
|
\section{Analytical expression for a scattering force exerted on atoms by
delta-function pulse train}
\subsection{Delta-function-like pulse model}
\label{Sec:Propagators}
As in our previous work \cite{IliDer12} we parametrize the
electric field of the pulse train at a fixed spatial coordinate as
\begin{equation}
\mathbf{E}(t)=\hat{\varepsilon}\,E_{p}\,\sum\limits_{m}\cos(\omega_{c}%
t+\Phi_{m})\,g(t-mT)
\label{Eq:TrainField} \, ,
\end{equation}
where $\hat{\varepsilon}$ is the polarization vector, $E_{p}$ is
the field amplitude, and $\Phi_{m}$ is the phase shift. The
frequency $\omega_{c}$ is the carrier frequency of the laser field
and $g(t)$ is the shape of the pulses. We normalize $g(t)$ so that $\max |g( t)|
\equiv 1$, then $E_{p}$ has the meaning of the peak amplitude.
While typically pulses have identical shapes and
$\Phi_{m}=m\phi$, one may want to install an active optical
element at the output of the cavity that could vary the phase and
the shape of the pulses.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=4.0in]{Fig1.eps}
\end{center}
\caption{ (Color online ) Energy levels of $\Lambda$-type system and positions of frequency comb teeth. The comb is Doppler shifted in the atomic frame moving with velocity $v$. \label{Fig:lambdasystem}}%
\label{Fig:Setup}%
\end{figure}
The $\Lambda$-system, Fig.~\ref{Fig:lambdasystem}, is composed of the excited state
$|e\rangle$ and the ground states $|g_1\rangle$, $|g_2\rangle$ separated by $\Delta_{12}$; the transition frequencies between the excited and each of the ground states are $\omega_{eg_1}$, $\omega_{eg_2}$ correspondingly.
The single pulse area corresponding to a transition $|g_j\rangle\rightarrow|e\rangle$ is
\begin{equation}
\theta_j=\Omega_{j}^{peak}\,\int\limits_{-\infty}^{\infty}g(t)dt \, ,
\end{equation}
where $\Omega^{peak}_j=\frac{E_{p}%
}{\hbar}\langle e|\mathbf{D}\cdot\hat{\varepsilon}|g\rangle$ is the peak Rabi frequency expressed in terms of the dipole matrix element.
As long as the duration of the pulse $\tau_p$ is
much shorter than the repetition time, the atomic system behaves
as if it was a subject to a perturbation by a series of
delta-function-like pulses: $\Omega^{peak}_jg(t) \rightarrow \theta_j \delta(t)$. In this limit, the only relevant
parameter affecting the quantum-mechanical time evolution is the
effective area of the pulse. The optical Bloch equations, in rotating wave approximation, may be written in form:
\begin{eqnarray}
\dot{\rho}_{ee}&=&-\gamma\rho_{ee}-\sum\limits_{n=0}^{N-1}\delta(t-nT)\sum\limits_{j=1}^2(\theta_{j}\label{Eq:DMdeltamodel} Im\left[e^{-i(k_cz(t)-\delta_jt-\Phi_n)}\rho_{eg_j}\right],\\
\dot{\rho}_{eg_j}&=&-\frac{\gamma}2\rho_{eg_j}+\frac{i}2 \sum\limits_{n=0}^{N-1}\delta(t-nT)\sum\limits_{p=1}^{2} \theta_{p}e^{i(k_cz(t)-\delta_pt-\Phi_n)}(\rho_{ee}\delta_{jp}-\rho_{g_pg_j}), \\
\dot{\rho}_{g_jg_{j'}}&=&\delta_{jj'}\gamma_j \rho_{ee}+\frac{i}2\sum\limits_{n=0}^{N-1}\delta(t-nT)(\theta_{j'}e^{i(k_cz(t)-\delta_{j'}t-\Phi_n)}\rho_{g_je}-\theta_{j}e^{-i(k_cz(t)-\delta_jt-\Phi_n)}\rho_{eg_{j'}}),
\end{eqnarray}
where the detunings $\delta_{j}=\omega_c-\omega_{eg_j}$ are the detunings of the carrier frequency from the frequencies of transitions $|g_j\rangle\rightarrow|e\rangle$.
The dynamics of three-level $\Lambda$-type system driven by the coherent train of delta-function like pulses
has been studied in detail in our previous work \cite{IliDer12}.
Here we employ analytical expression for the density matrix in a quasi-steady-state regime from that work. Although general expression for the density matrix was presented there, here we restrict ourself to the case most commonly realized.
If the energy gap between the two ground states is much smaller than the frequency of transition from the ground to excited state, then the ratio of
decay rates $\gamma_1/\gamma_2$ is proportional to the ratio of relevant dipole matrix elements in the same way as the ratio of pulse areas $\theta_1/\theta_2$.
In this case we can use the following parametrization: $\theta_1/\theta_2=\gamma_1/\gamma_2=\tan\chi$. Then the post-pulse excited state population $\left( \rho_{ee}^s \right)_r$ in QSS regime reads
\begin{eqnarray}
\left(\rho_{ee}^s\right)_r&=&8e^{\frac{\gamma T}2 } \sin^2\frac{\Theta}2\sin ^2\pi\kappa/D\nonumber\\
D&=&8 \cos 2\chi\left(4 \sin^4\frac{\Theta }{4}+\sin ^2\frac{\Theta }{2} \cos 2\pi\kappa\right)\sin\pi\kappa\sin \left(\bar{\eta} +\pi\kappa\right)+\nonumber\\&&
+\cos\pi\kappa\cos \left(\bar{\eta} +\pi\kappa\right)\left(4 \cos\frac{\Theta }{2}\left(\cos2\pi\kappa-5\right)+(\cos\Theta+3) (3 \cos2\pi\kappa+1)-\right.\nonumber\\&&
\left. -16 \sin ^4\frac{\Theta }{4}\sin ^2\pi\kappa\cos (4 \chi )\right)-\nonumber\\&&
-4\cosh\left(\frac{\gamma T}2\right)\left(4 \cos^2\frac{\Theta }{4}\cos2\pi\kappa+2 \cos\frac{\Theta }{2}-\cos\Theta-5\right).\label{Eq:rgen}
\end{eqnarray}
In this formula and below we employ the following notation (see also Fig.~\ref{Fig:lambdasystem})
\begin{enumerate}
\item[(i)]{ The effective single-pulse area
\begin{equation}\Theta=\sqrt{\theta_1^2+\theta_2^2},\end{equation}
where $\theta_j$ are the single-pulse areas for the two transitions $|g_j\rangle\rightarrow|e\rangle$, $j=1,2$.
}
\item[(ii)]{ Number of teeth fitting in the energy gap $\hbar\Delta_{12}$ between the two ground states
\begin{equation} \kappa=\Delta_{12}/\omega_{rep} \, .\end{equation}
Notice that $\kappa$ generally is not an integer number. When it is integer, the two-photon resonance condition is satisfied and the system evolves into the dark state.}
\item[(iii)]{ Doppler shifted phase offset between subsequent pulses
\begin{equation}
\overline{\eta}=\eta(t)-\eta(t+T)=\left(k_cv+\delta_1\right)T+\phi.
\end{equation}
Here $v$ is the atomic velocity and $\phi$ is the carrier-envelope phase offset between subsequent pulses, i.e., $\phi=\Phi_{m} - \Phi_{m+1}$ in Eq.~(\ref{Eq:TrainField}). These phase parameter will be used to characterize the spectral profile of the scattering force. As shown below the density matrix of a system and the scattering force are periodic functions of $\overline{\eta}$. }
\item[(iv)]{ Residual detunings $\overline{\delta}_j$, $j=1,2$, between $|g_j\rangle$ levels and the nearest FC modes in the reference frame moving with the atom.
In general,
$\bar{\delta}_1=(\bar{\eta}+2\pi n_1)/T$ and $\overline{\delta}_2=(\bar{\eta}+2\pi\kappa+2\pi n_2)/T$, where integers $n_j$ are chosen to renormalize the
residual detunings to the interval
$-\omega_{rep}/2<\overline{\delta}_j<\omega_{rep}/2$. }
\end{enumerate}
Eq.~(\ref{Eq:rgen}) gives the value of the excited state population just after the pulse. The time evolution between the pulses is described by
($mT<t<\left(m+1\right)T)$)
\begin{eqnarray}\label{decayeq}
\rho_{ee}^s(t) &=&\left(\rho_{ee}^s\right)_{r}e^{-\gamma t}.
\end{eqnarray}
The dependence on the phase offset $\bar{\eta}$ is the result of interference between the elementary responses of a system to subsequent pulses (the persistent ``memory'' of the system).
Particularly, when $\gamma T\rightarrow\infty$, the excited state completely decays between the pulses and the interference factor vanishes (the ``memory'' is erased),
\begin{equation}
\left(\rho_{ee}^s\right)_r\rightarrow \frac{ 4\sin^2\left(\pi\kappa\right)}{\tan^2\frac{\Theta }{4}+\frac{\sin^2\left(\pi\kappa\right)}{\sin^2\frac{\Theta }{4}}}.
\label{Eq:DMQSSbranchlargedecay}
\end{equation}
At equal pulse areas $\theta_1=\theta_2$ and decay rates $\gamma_1=\gamma_2$ ($\chi=\pi/4$) the equation (\ref{Eq:rgen}) can be simplified further
\begin{eqnarray}
\left(\rho_{ee}^{s}\right)_r&=&\frac{e^{\frac{\gamma T}{2}} \sin ^2\left(\pi\kappa\right) \sin^2\frac{\Theta}2}{4D'}, \nonumber\\
D'&=& \left(\cos \left(\pi\kappa\right) \cos
\left(\bar{\eta}+\pi\kappa\right) \left(\cos ^2\left(\pi\kappa\right) \cos^4\left(\frac{\Theta }{4}\right)-\cos\frac{\Theta}2
\right)+\right.\nonumber\\
&& \left.\cosh \left(\frac{\gamma T }{2}\right) \left(\sin^4\left(\frac{\Theta }{4}\right)+\cos^2\left(\frac{\Theta }{4}\right) \sin ^2\left(\pi\kappa\right)\right)\right). \label{Eq:DMQSSsimplea}
\end{eqnarray}
\subsection{Scattering force}
Now we focus on the evaluation of the
cooling force,
\begin{equation}\label{ClFrc}
F_z=\hbar k_c \sum\limits_{j=1}^2\mathrm{Im}[\rho_{eg_j}\Omega_{eg_j}] \, .
\end{equation}
The laser field is present only during the
pulse, so effectively we deal with a sum over instantaneous forces
\begin{equation}
\mathbf{F}(t)=p_{r}\,\sum\limits_{m,j}\theta_j\delta(t-mT)\,\mathrm{Im}%
[e^{-i(k_cz(t)-\delta_jt-\Phi(t))}\rho_{eg_j}(t)]~\mathbf{\hat{k}}_{c}, \label{Eq:Force}%
\end{equation}
where $\mathbf{\hat{k}}_{c}$ is the unit vector along the
direction of the pulse propagation. The change in the
linear momentum of a particle due to the $m$-th pulse is $\Delta\mathbf{p}%
_{m}=\lim_{\varepsilon\rightarrow0^{+}}\int_{mT-\varepsilon}^{mT+\varepsilon
}\mathbf{F}(t)dt$. We find
\begin{equation}
\frac{-\Delta\mathbf{p}_{m}}{p_{r}}=\left( \left(
\rho_{ee}^{m}\right) _{r}-\left( \rho_{ee}^{m}\right)
_{l}\right) ~\mathbf{\hat{k}}_{c},
\label{Eq:DPoverP}%
\end{equation}
where $\left(\rho_{ee}^{m}\right) _{r}=\rho_{ee}(mT+\varepsilon)$, $\left( \rho_{ee}^{m}\right)_{l}=\rho_{ee}(mT-\varepsilon)$ ($\tau\ll\varepsilon\ll T$) are the excited state population values just before and just after the pulse.
This result follows from noticing that $\Delta p_{m}$ is an
integral of a particular combination
$\,\sum\limits_j\delta(t-mT)\theta_j\mathrm{Im}[e^{-i(k_cz(t)+\delta_jt+\Phi(t)}\rho_{eg_j}(t)]$ over time.
This combination enters the r.h.s. of Eq.~(\ref{Eq:DMdeltamodel}).
Then by integrating Eq.~(\ref{Eq:Force}) over time we
immediately arrive at Eq.~(\ref{Eq:DPoverP}).
Several insights may be gained from analyzing Eq.(\ref{Eq:DPoverP}).
\begin{enumerate}
\item[(i)]{ Eq.~(\ref{Eq:DPoverP}) simply states that the averaged over a big number of cycles single laser pulse
fractional momentum kick is equal to a difference of
populations before and after the pulse.
}
\item[(ii)]{ As elucidated earlier for CW laser cooling (see e.g., Ref.~\cite{MetStr99Book})
radiative decay plays a crucial role in maintaining
force directed along the laser beam. In the context of the
pulse-train cooling, Eq.(\ref{Eq:DPoverP}), radiative decay brings down
the excited state population in the time-interval between the
pulses, thus keeping pre- and post-pulse excited
state population difference negative; this leads to a net force
along the direction of the pulse train propagation.
}
\item[(iii)]{ In the regime when two FC modes match both transition frequencies between the excited and ground states,
the system evolves into a ``dark'' superposition of two ground states which is transparent to the pulses.
The population of the excited state in this case and consequently the scattering force are both zero.
}
\end{enumerate}
In the quasi-steady-state regime the value of single pulse fractional momentum kick is
\begin{equation}
\frac{-\Delta p_{s}}{p_{r}}=\left(\rho_{ee}^{s}\right)
_{r}~\times\left(
1-e^{-\gamma T}\right) \, \label{DPoverPQ}%
\end{equation}
and the average scattering force can be represented as
\begin{equation}
F_{sc}=\frac{\Delta p_s}{T}.\label{Eq:forcegeneral}
\end{equation}
In a particular case of equal branching ratios $b_1=b_2=1/2$, the expression for the scattering force reads (this was obtained using Eq. (\ref{Eq:DMQSSsimplea}) )
\begin{eqnarray}\label{Eq:scforce}
F_{sc}&=&\frac{\Delta p}{T}=-\frac{\hbar k_c}{ T}\frac{\sinh{\gamma T /2} \sin ^2\left(\frac{\Theta }{2}\right) \sin ^2(\pi \kappa )}{D'}\end{eqnarray}
\begin{figure}[h]
\begin{center}
\includegraphics*[width=3.2in]{Figure2.eps}
\end{center}
\caption{ (Color online ) The dependence of the fractional momentum kick $\Delta p/(p_r)$ on the phase offset $\bar{\eta}$ at different values of pulse repetition period $T$. Solid purple line $T=4$ $ns$, dashed purple line $T=50$ $ns$. The parameters of the system are: $\gamma=0.05$ $GHz$, $\Theta=\pi/4$, $\kappa=0.12$. \label{Fig:lambda}}%
\label{Fig:Setup}%
\end{figure}
In Fig.~\ref{Fig:lambda} we plot the fractional momentum kick $\Delta p/p_r $ as a function of the phase offset $\bar{\eta}$.
The radiative force (fractional momentum kick) exerted by the train of coherent pulses depends on the atomic
velocity via Doppler shift $\bar{\eta}=(kv+\delta_1)T+\phi$. As velocity is varied across the ensemble, the maxima of the force
would occur at discrete values of velocities
\begin{equation}
v_n = ( \pi \, (2 n-\kappa)- \phi)/(k_c T) ,\, n = \ldots, -2, -1, 0, 1,2, \ldots \label{Eq:VelocityTeeth}
\end{equation}
In other words the fractional momentum kick (scattering force) spectral profile exhibits the periodic structure of the comb (see Fig.~\ref{Fig:lambda}).
As an example, for $T = 5 \, \mathrm{ns}$ and $\lambda_c=600\, \mathrm{nm}$ carrier wavelength, the force peaks are
separated by $v_{n+1}-v_n = 2\pi/(k_c T)= \lambda_c/T = 120 \, \mathrm{m/s}$ in the velocity space.
Depending on the temperature of the ensemble, the comb may have several teeth effectively interacting with the ensemble.
Notice, however, that if $\gamma T \gg 1$ (Fig.~\ref{Fig:lambda}, dashed purple line) the teeth structure of the radiative force washes out
and the atoms experience radiative force even if their velocities are far away from peaks.
In this case the power stored in the pulse is delivered to the entire ensemble.
This is in a contrast with highly-velocity selective CW laser, where the interaction
window in the velocity-space is typically $1 \, \mathrm{m/s}$.
\subsection{Maximum momentum kick}
The scattering force Eq.~(\ref{Eq:forcegeneral}) is linearly proportional to the post-pulse excited state population.
Therefore the discussion of the excited state population dependence on FC parameters in \cite{IliDer12} directly applies to the scattering force too.
In Ref. \cite{IliDer12} we found that the maximum of $(\rho_{ee}^s)_r$ and correspondingly the maximum of fractional momentum kick for the case of equal pulse areas $\theta_1=\theta_2$ and decay rates $\gamma_1=\gamma_2$ is reached at
optimal residual detunings $\bar{\delta}_1=-\bar{\delta}_2=-mod(2\pi\kappa^{opt}, 2\pi)/T$ and optimal parameter $\kappa=\kappa^{opt}$ determined by
\begin{equation}
\kappa^{opt}=\frac1\pi\arccos(x),\label{Eq:kappaopt}
\end{equation}
where $x$ is a root of the following algebraic equation:
\begin{equation}
16 x^4 \cos ^4\frac{\Theta }{4}-32 x \cosh \frac{\gamma T }{2}\sin ^4\frac{\Theta }{4}+16 \cos \frac{\Theta }{2}-2 x^2 \left(4 \cos \frac{\Theta }{2}+3 \cos (\Theta )+9\right)=0. \label{Eq:findkappa}
\end{equation}
One can show that for the general case of non-equal decay rates, $\gamma_1\neq\gamma_2$, and pulse areas $\frac{\theta_1}{\theta_2}=\frac{\gamma_1}{\gamma_2}=\tan\chi\neq1$ and fixed value of parameter $\kappa$,
the optimal residual detunings are determined as
\begin{equation}
\bar{\delta}_1=\left\lbrace \begin{matrix}mod(\bar{\eta}^{opt},2\pi)/T, \qquad |mod(\bar{\eta}^{opt},2\pi)|<\pi\\
(mod(\bar{\eta}^{opt},2\pi)\mp2\pi)/T, \qquad |mod(\bar{\eta}^{opt},2\pi)|>\pi \end{matrix} \right.,
\end{equation}
\begin{equation}
\bar{\delta}_2=\left\lbrace \begin{matrix}mod(\bar{\eta}^{opt}+2\pi\kappa,2\pi)/T, \qquad |mod(\bar{\eta}^{opt}+2\pi\kappa,2\pi)|<\pi\\
(mod(\bar{\eta}^{opt}+2\pi\kappa,2\pi)\mp2\pi)/T, \qquad |mod(\bar{\eta}^{opt}+2\pi\kappa,2\pi)|>\pi\end{matrix} \right. .
\end{equation}
Here
\begin{eqnarray}
\bar{\eta}^{opt}&=&-\arctan\frac{B}A-\pi\kappa+2\pi n,\label{Eq:etta}\nonumber\\
A&=&\cos\pi\kappa( 4\cos\frac{\Theta}2(\cos2\pi\kappa-5)+(\cos\Theta+3) (3\cos2\pi\kappa+1)-16\sin^4\frac{\Theta}4 \sin^2\pi\kappa\cos(4\chi)),\nonumber\\
B&=&8\cos\left(2\chi\right)(4\sin^4\frac{\Theta}4+\sin^2\frac{\Theta}2\cos2\pi\kappa)\sin\pi\kappa.
\end{eqnarray}
At $\chi=\pi/4$ the coefficient $B$ in Eq.~(\ref{Eq:etta}) vanishes and $\bar{\eta}^{opt}=-\kappa/2+2\pi n$, $n=0,1..$.
After substituting (\ref{Eq:etta}) into the equation for the density matrix (\ref{Eq:rgen}) one can find the optimal value of the parameter $\kappa^{opt}$, corresponding to the maximum of the post-pulse excited state population and consequently maximum fractional momentum kick.
The value of the single pulse area $\Theta$, maximizing the value of the fractional momentum kick is equal to $\pi+2\pi n$, $n=0,1..$.
In Fig.~\ref{Fig: maxGSSPop} (a,b) we show the dependencies of the QSS values of the excited state population $\left(\rho_{ee}^s\right)_r$ (at $\theta_1=\theta_2$, $\gamma_1=\gamma_2$) and corresponding single pulse momentum kick $\Delta p/p_r$ on the effective single pulse area $\Theta$. Different curves correspond to different values of parameter $\mu=\gamma T$. The values of $\left(\rho_{ee}^s\right)_r$ and $\Delta p/ p_r$ were calculated at the
optimal value of $\kappa$, determined by Eq. (\ref{Eq:findkappa}) for each $\Theta$ and $\mu=\gamma T$.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=5.2in]{Figure3.eps}
\end{center}
\caption
{ The dependencies of the quasi-steady-state values of the post-pulse excited state population $\left(\rho_{ee}^s\right)_r$ and single pulse momentum kick $\Delta p/(p_r)$ on effective single pulse area $\Theta$ at different values of $\mu=\gamma T$: $\mu=10$ (dashed pink line),
$\mu=1/2$ (dashd blue line), $\mu=1/100$ (solid purple line)
and optimal parameters $\bar{\eta}=-\pi\kappa^{opt}$, where $\kappa^{opt}$ is obtained from Eq. (\ref{Eq:kappaopt}).
\label{Fig: maxGSSPop}} \end{figure}
At $\Theta=\pi$ the optimal value of the parameter $\kappa$ is equal to $1/2$, independently on the ratio of individual pulse areas $\theta_1/\theta_2$,
\begin{equation}
\kappa^{opt}_{\Theta=\pi}=\frac12.
\end{equation}
For $\kappa=1/2$ and $\Theta=\pi$ the excited state population and the fractional momentum kick are:
\begin{eqnarray}
(\rho_{ee}^s)_r(\Theta=\pi,\kappa=\frac12)&=&\frac{1}{3} e^{\gamma T /2} /\cosh(\gamma T/2),\\
\Delta p/(p_r)_{\Theta=\pi,\kappa=\pi,\bar{\eta}=-\pi/2}=\frac23\tanh(\frac{\gamma T}2).\label{Eq:PiForce}
\end{eqnarray}
The spectral resolution of scattering force at $\kappa=\kappa_{opt}$ vanishes as $\Theta\rightarrow\pi$.
As it was shown in \cite{IliDer12} the maximum post-pulse excited state population in three-level $\Lambda$-system (with $b_1=b_2=1/2$, $\theta_1=\theta_2=\sqrt{2}\pi$) is reached at $\gamma T\gg1$ and is equal to $2/3$. Consequently the maximum of the fractional momentum kick is also $2/3$.
This result can be generalized to the case of unequal pulse areas $\theta_1\neq\theta_2$ and branching ratios $\gamma_1\neq \gamma_2$, (in case if $\theta_1\neq \pi n$, $n=0,1$). Here at $\Theta=\pi$ and $\kappa=1/2$, the three-level $\Lambda$-system, which is initially in the ground state $|g_1\rangle$, eventually reaches the QSS with the fractional momentum kick expressed as
\begin{eqnarray}
\left(\rho_{ee}^s\right)_r=\frac{\Delta p_{max}}{p_r}=-\frac{2\sin^2(2\chi)}{(b_2- b_1) \cos (2\chi )+\cos (4\chi )-2}.\label{Eq:kickmax}
\end{eqnarray}
If the decay rates and pulse areas are $\gamma_1=\gamma\sin^2\chi=\frac{\theta_1^2}{\Theta^2}$, $\gamma_2=\gamma\cos^2\chi=\frac{\theta_2^2}{\Theta^2}$, $\theta_1=\Theta\sin^2\chi$, $\theta_2=\Theta\cos^2\chi$, the maximum fractional momentum kick (\ref{Eq:kickmax}) is equal to $2/3$. This limit is independent on the value of $\chi$ ($\theta_1\neq \pi n$ requires $\chi\neq \pi n/2$).
\subsection{Friction coefficient }
\label{Sec:FrictionCoeffBeta}
In general, one would be interested in both slowing down the atomic beam and compressing (i.e., cooling) the velocity distribution.
Cooling would occur if there is a negative velocity gradient of the radiative force $F_{sc}$.
One may introduce a friction coefficient $\beta$ by expanding the force about some velocity $v$, corresponding to a certain value of parameter $\bar{\eta}(v)$,
\begin{equation}
F_{sc}(v + \Delta v ) \approx F_{sc}(v) - \beta(v) \Delta v \, .
\label{Eq:ForceExpansionBeta}
\end{equation}
When the friction coefficient is positive $\beta>0$, one observes the compression of velocity distribution around $v$. Negative values of $\beta$ lead to heating of the ensemble.
In the limiting case when the radiative lifetime is much shorter then the pulse repetition period there is no interference between the action of subsequent pulses on a system
and consequently no velocity dependence of the scattering force. The friction coefficient is thereby $\beta =0$ and while the ensemble slows down, there is no compression of the velocity distribution.
The friction coefficient of Eq.(\ref{Eq:ForceExpansionBeta}) may be directly determined from the analytical expression for the force
(\ref{Eq:forcegeneral}),
\begin{eqnarray}\beta&=&-16\hbar k_c^2 \sinh{\frac{\gamma T}2}\sin ^2\left(\frac{\Theta }{2}\right) \sin ^2(\pi \kappa )\frac{B \cos \left(\bar{\eta }+\pi \kappa \right)-A \sin \left(\bar{\eta }+\pi \kappa \right)}{\left(A \cos \left(\bar{\eta }+\pi \kappa \right)+B \sin \left(\bar{\eta }+\pi \kappa \right)+C\right){}^2}\nonumber\\
C&=&8\cosh\frac{\gamma T}2 \left(\left(\cos \left(\frac{\Theta }{2}\right)+1\right) (1-\cos (2 \pi \kappa ))+\left(1-\cos \left(\frac{\Theta
}{2}\right)\right)^2\right),\label{Eq:bettagen}
\end{eqnarray}
where coefficients $A$, $B$ are defined in (\ref{Eq:etta}).
For the case of equal decay rates and pulse areas ($\chi=\pi/4$), one has
\begin{eqnarray}
\beta(\chi=\pi/4)&=&\frac{\hbar k_c^2}{2D'}\sinh\frac{\gamma T}2\sin ^2\frac{\Theta }{2} \sin ^2(\pi \kappa ) \cos (\pi \kappa ) \sin (\bar{\eta}+\pi \kappa )\times\nonumber\\ &&\left(\cos ^4\frac{\Theta }{4} \cos ^2(\pi \kappa )-\cos \frac{\Theta
}{2}\right).\label{Eq:bta}
\end{eqnarray}
This result depends on the effective pulse area,
$\Theta$, the product $\mu=\gamma T$ and $\kappa=\Delta_{12}/\omega_{rep}$.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=3.2in]{Figure4.eps}
\end{center}
\caption{(Color online)
Dependence of the friction coefficient $\beta/\hbar k_c^2$ on
phase detuning $\bar{\eta}$ at (a)
$\gamma T=1/2$ and (b) $\gamma T=1/100$. Each panel has three curves with different values of pulse area $\Theta$, $\Theta=\pi/10$ (solid purple line), $\Theta=\pi/2$ (dashed purple line), $\Theta=\pi$ (dashed pink line).}%
\label{Fig:Beta}%
\end{figure}
In Fig.~\ref{Fig:Beta} (a,b) we plot the dependence of friction coefficient $\beta$ (at $\theta_1=\theta_2$) (\ref{Eq:bta}) on the phase offset $\bar{\eta}$ at different values of $\gamma T$ and $\Theta$ at $\kappa=\kappa_{opt}$, optimally chosen for each pair of parameters $\gamma T$ and $\Theta$.
It acquires the maximum value at $\bar{\eta} = \bar{\eta}_{\beta}$,
\begin{equation}
\bar{\eta}_{\beta}=-\cos ^{-1}\left(\frac{b-\sqrt{8 a^2+b^2}}{2 a}\right)-\pi\kappa,\label{Eq:etab}
\end{equation}
where
\begin{eqnarray}
a=\cos (\pi \kappa ) \left(\cos ^4\left(\frac{\Theta }{4}\right) \cos ^2(\pi \kappa )-\cos \left(\frac{\Theta }{2}\right)\right),\\
b=\cosh \left(\frac{\mu }{2}\right) \left(\cos ^2\left(\frac{\Theta }{4}\right) \sin ^2(\pi \kappa )+\sin ^4\left(\frac{\Theta
}{4}\right)\right).
\end{eqnarray}
For the case of non-equal pulse areas $\chi\neq\pi/4$)
\begin{eqnarray}
\bar{\eta}_{\beta}&=&-\sec^{-1}\left({\frac{A^2+B^2}{A \left(C-D\right)+\sqrt{2} B \sqrt{C D-2 \left(A^2+B^2\right)-C^2}}}\right),\label{Eq:etabetagen}\nonumber\\
D&=&\sqrt{8 \left(A^2+B^2\right)+C^2},
\end{eqnarray}
where $A$, $B$ and $C$ are defined in (\ref{Eq:etta}, \ref{Eq:bettagen}).
One can see (Fig.\,~\ref{Fig:Beta}) that as the pulse repetition rate grows (smaller $\gamma T$), smaller values of single pulse area $\Theta$ lead to larger values of the friction coefficient $\beta$.
Notice however, that at very small values of $\gamma T\ll1$ the momentum kick per pulse becomes smaller and the number of pulses needed to decelerate the atomic beam is increased.
At very large values of $\gamma T\gg1$, while the friction coefficient $\beta$ vanishes, the momentum kick $\Delta p$ reaches its maximum. If the large value of
$\gamma T\gg1$ is due to the low pulse repetition rate, then the scattering force $F_{sc}=\Delta p/T$ also becomes smaller and the overall cooling time is increased.
For $\pi$-pulse and $\theta_1=\theta_2$ ($\chi=\pi/4$) the Eq. (\ref{Eq:etab}) reduces to
\begin{equation}
\beta_{\pi}=\frac{\hbar k_c^2}2\frac{\sinh\frac{\gamma T}2 \sin ^2(\pi \kappa ) \cos ^3(\pi \kappa ) \sin (\bar{\eta}+\pi \kappa )}{(\cos ^3(\pi \kappa ) \cos (\bar{\eta}+\pi \kappa )-(\cos (2 \pi \kappa )-2) \cosh \left(\frac{\gamma T }{2}\right))^2}.
\end{equation}
At $\kappa=\frac12$ (chosen in order to maximize the scattering force) the friction coefficient $\beta_{\pi}$ vanishes (similar to the case when $\gamma T\gg1$). One can show that the friction coefficient at $\kappa=1/2$ and $\Theta=\pi$ turns to zero for arbitrary finite ratio of individual pulse areas $\theta_1/\theta_2$ and decay rates $\gamma_1/\gamma_2$ ($\chi\neq\pi/4$).
\subsection{Finding the optimal cooling regime}
Before discussing criteria for the optimal choice of FC parameters (single pulse area and pulse repetition rate ) we analyze the dependence of the scattering force profile on parameters $\gamma T$ and $\Theta$ at optimally chosen number of teeth $\kappa$ fitting into the energy gap between the two ground states. It is worth noticing, that the optimal value of $\kappa^{opt}$ is defined with an accuracy up the integer number, that is the values $\kappa^{opt}+n$, $n=0,1..$, where $\kappa^{opt}$ is defined from Eq. (\ref{Eq:kappaopt}), are also optimal. Analysis in this section is carried out assuming the equal individual pulse areas $\theta_1=\theta_2$ and branching ratios $b_1=b_2$ ($\chi=\pi/4$).
In Fig.\,~\ref{Fig:ForceDependence} we study the dependence of the scattering force $F_{sc}$ on phase offset parameter $\bar{\eta}$ at optimally chosen $\kappa$, Eq. (\ref{Eq:kappaopt}), as the single pulse area $\Theta$ and the parameter $\mu$ vary.
\begin{figure}[h]
\begin{center}
\includegraphics*[width=3.2in]{Figure5.eps}
\end{center}
\caption
{The dependence of the scattering force on the Doppler-shifted phase offset $\bar{\eta}$ at the optimal value of $\kappa=\kappa^{opt}$, chosen according to Eq. (\ref{Eq:findkappa}) at fixed value of $\mu=\gamma T$: panel (a)$\mu=1/2$, panel (b) $\mu=1/100$ and different values of
effective single pulse area $\Theta$. Different curves correspond to the distinct values of single pulse areas: $\Theta=\frac{\pi}{10}$ (solid purple line), $\Theta=\frac{\pi}{2}$ (dashed blue line), $\Theta=\pi$ (dashed pink line). \label{Fig:ForceDependence}}\end{figure}
One can see that at small $\mu=\gamma T$ in Fig.~\ref{Fig:ForceDependence} the maxima of the scattering force is nearly independent on the pulse area $\Theta$ as long as $\Theta>\mu$. However, as $\Theta$ is increased the friction coefficient becomes smaller.
As an example, at $\gamma T=1/100$ the amplitudes of scattering force corresponding to $\Theta=\pi/10$ and $\Theta=\pi/2$ are the same, but width of the peaks is smaller at $\Theta=\pi/10$.
At higher values of parameter $\mu$ the scattering force saturates at higher values of the pulse area $\Theta$. But the gradient of the scattering force is decreased.
At very small pulse areas $\Theta\rightarrow 0$ the scattering force vanishes (as well as the momentum kick $\Delta p$) regardless of the parameter $\gamma T$.
To summarize, at lower pulse repetition rates $\gamma T\gg1$ and larger values of pulse area one can obtain larger momentum kick and smaller scattering force and compression rate.
At larger pulse repetition rate ($\gamma T\ll1$) and properly chosen $\Theta$ one can obtain the maximum of compression rate, but smaller momentum kick.
In the first case the cooling time is increased and the compression of the velocity distribution is slow.
In the second case the number of pulses needed to decelerate the beam is increased and the scattering force velocity capture range is decreased.
To find the opimal cooling regime one has to compromise between the fast slowing of the entire ensemble and its velocity distribution compression rate.
In case when the scattering force rapidly vanishes in the vicinity of its maxima only the atoms within narrow groups of velocity are decelerated.
Below we show that this problem can be mitigated.
The spectral dependence of the scattering force can be varied in time,
so that the positions of maxima follow the center of velocity distribution of the decelerating ensemble.
In this case those atoms, which initially were outside of the scattering force velocity capture range and not decelerated will be eventually captured by the force profile being moved in the spectral domain (e.g. by changing the CEPO $\phi$).
However, if the initial atomic beam is too fast and (or) the velocity distribution is too wide, one can be interested initially in slowing down the ensemble, so that the cooling distance will be not too large.
In this case one would prefer to have a broad scattering force profile (wide velocity capture range). The amplitude of the scattering force has to be large enough to mitigate the increase of cooling time and consequently cooling distance. This can be realized at larger pulse areas $\Theta$.
For example, at $\mu\sim 1/2$, $\Theta\sim\pi/2$ for the atoms with the excited state lifetime $\tau\sim 15$ $ns$ at the laser field wavelength $\lambda=589$ $nm$, the velocity capture range $\Delta v_{cptr}$ is estimated by $\Delta v_{cptr}\sim 20$ $m/s$. The scattering force amplitude is quite the same as its maximum value, reached at $\Theta=\pi$.
At $\Theta=\pi$ the velocity capture range can be extended up to $\lambda/\tau_p$, where $\tau_p$ is the duration of pulse. For $\tau_p\sim 1 $ $ps$ and $\lambda=589$ $nm$ the maximum velocity capture range is very broad $\Delta v_{cptr}^{max}\sim 5.89\times 10^5$ $m/s$.
If the intial velocity distribution is already narrow and (or) the central velocity value is not too high, the priority can be given to the
fast velocity distribution compression.
The optimal set values of pulse area and pulse repetition rate can be chosen based on the initial velocity distribution, desired velocity compression rate and the limiting factors such as given cooling length and the laser power.
\section*{Acknowledgments}
This work was supported in part by the NSF and ARO. We would like to thank Mahmoud Ahmad for discussions.
\section{Conclusion}
In this paper we studied Doppler cooling of a three-level $\Lambda$-type system driven by a train of ultra-short laser pulses. Analytical expression for the scattering force was obtained and its dependence on the FC parameters was analyzed.
The scattering force $F_{sc}$ is linearly proportional to the quasi-steady-state post-pulse excited state population.
Its spectral (velocity) dependence exhibits periodic pattern mimicking the spectrum of the frequency comb.
The contrast of the spectral profile of $F_{sc}$ is a function of the ratio between the excited state lifetime and the pulse repetition period, the effective single pulse area and the residual detunings $\bar{\delta_j}$ between the frequencies of individual transitions and nearest FC teeth.
In a particular case when the pulse repetition period is much longer than the lifetime of the excited state, the spectral dependence of the scattering force reflects the broad-band spectral profile of a singe pulse.
The residual detunings $\bar{\delta}_j$ can be optimized to maximize the scattering force. At optimally chosen detunings the maximum of the scattering force is reached at single pulse area equal to $\pi$. However for $\pi$-pulses the spectral dependence of the scattering force is lost and consequently the friction coefficient vanishes. To optimize the cooling process one has to compromise between maximizing the scattering force and its velocity capture range and maintaining the sufficient gradient of the scattering force (friction coefficient).
The spectral profile of the scattering force and consequently the friction coefficient can be varied in time to follow the moving center of the velocity distribution of decelerating ensemble. This can be realized by simply tuning the carrier envelope phase offset. Such manipulation enables sustained velocity distribution compression as the atoms slow down.
As a result, initially smooth velocity distribution of a thermal beam evolves into a series of narrow groups of velocities separated by $\lambda_c/T$,
so called ``velocity comb''.
\section{Evolution of the velocity distribution}
\label{Ssec:slowing}
Now we turn to the dynamics of slowing down and cooling an entire atomic ensemble,
characterized by some velocity distribution $f(v,t)$ (time-dependence is caused by radiative force).
\subsection{No-cooling theorem for fixed FC parameters}
Suppose that the positions of FC teeth remain fixed in the frequency domain during the deceleration.
As the atoms slow down, they come in and out of resonances with different FC teeth.
The gradient of the scattering force changes its sign (see Fig.~\ref{Fig:Beta}) as the Doppler-shifted phase (velocity) vary. As a result the sustained cooling can not be realized if the positions of FC teeth remain fixed in frequency space during deceleration. This can be demonstrated as follows.
Suppose the parameters of the frequency comb remain fixed. As a result of scattering $N$ pulses the atom with initial velocity $v_i$ will be decelerated to the final velocity $v_f$ determined from the implicit equation
\begin{eqnarray}
N v_r&=&\frac{2 \csc ^2\frac{\Theta }{2}\csc ^2\pi \kappa }{ k_c T \sinh\frac{\gamma T}2}\left( k_c T(v_f-v_i) \cosh\frac{\gamma T }{2}\left(\cos ^4\frac{\Theta }{4} \sin ^2\pi \kappa+\sin ^4\frac{\Theta
}{4}\right)+\right. \nonumber \\ &&
\left. \cos \pi \kappa \left(\cos ^4\frac{\Theta }{4}\cos ^2\pi \kappa -\cos \frac{\Theta }{2}\right) (\sin (k_cTv_f+\pi \kappa )-\sin (k_cTv_i+\pi \kappa ))\right).\qquad
\label{Eq:FixedCombDeltaV}
\end{eqnarray}
where $v_r \equiv p_r/M$ is the recoil velocity. This equation was obtained by integrating Eq. (\ref{Eq:scforce}).
Eq.~(\ref{Eq:FixedCombDeltaV}) implies that
the decrement in velocities would vary across the ensemble.
Yet if we fix the change of velocity equal to the spacing between the teeth,
$v_f=v_i-\lambda_c/T$, we find that the required number of pulses $N_0$ (or time $N_0 T$),
\begin{equation}\label{N0}
N_0 = \frac{2\lambda_c \csc ^2\frac{\Theta }{2}\csc ^2\pi \kappa }{T v_r \sinh\frac{\gamma T}2} \cosh\frac{\gamma T }{2}\left(\cos ^4\frac{\Theta }{4}\sin ^2(\pi \kappa )+\sin ^4\frac{\Theta
}{4}\right),
\end{equation}
does not dependent on the initial value $v_i$. This implies that if we start with a certain
velocity distribution $f(v)$,
the entire distribution is uniformly shifted by $-\lambda_c/T$ every $N_0$ pulses:
$f(v) \rightarrow_{N_0} f(v+\lambda_c/T)$.
Thus, the radiative force exerted by FC with fixed parameters does not lead to velocity compression
--- there is no cooling.
Notice that the above analysis has neglected variation of intensity across comb teeth. Also
while there is no compression of the velocity distribution, there is a residual heating due
to atomic recoil (this arises from treatments beyond our model,
see, e.g., Ref.~\cite{MetStr99Book}).
\subsection{Cooling via tuning the FC}
In order to compress the velocity distribution, one has to maintain the positive gradient of the absolute value of the scattering force in the vicinity of the center of velocity distribution. To attain this condition, the scattering force profile has to follow the center of velocity distribution, moving towards the smaller velocities (frequencies) during the process of deceleration. In other words, the FC tooth closest to the atomic transition frequency $\omega_{eg_1}$ (in the reference frame moving with the center of velocity distribution) has to be somewhat red-detuned from $\omega_{eg_1}$. Tuning the positions of the FC teeth and consequently the scattering force profile can be achieved by tuning the phase of pulses during the cooling process \cite{IliAhmDer11} .
Initially, we start with some velocity distribution $f(v,t=0)$.
To optimize the number of cooled atoms, we focus on atoms with
velocities grouped around
the position of the maximum of $f(v,t=0)$, i.e. the most probable velocity $v_{mp}(t=0)$.
Radiative force will cause both the distribution $f(v,t)$ and the
most probable velocity $v_{mp}(t)$ to evolve in time.
To maximize the rate of compression, the friction coefficient needs to be kept at
its maximum value at $v_{mp}(t)$.
We may satisfy this requirement
by tuning the phase offset $\phi(mT)=\Phi((m+1)T)-\Phi(mT)$ according to
\begin{equation}
\phi(t) = -\left( \delta + k_c v_{mp}(t) \right) T - \bar{\eta}_{\beta} \, ,
\end{equation}
where $\bar{\eta}_{\beta}$, Eq.~(\ref{Eq:etabetagen}),
depends only on (time-independent) values of $\gamma T$, $\Theta$ and $\Delta_{12}/\omega_{rep}$. As $v_{mp}(t)$ becomes smaller
due to the radiative force, the offset phase needs to be reduced.
We may find required pulse-to-pulse increment of the phase offset explicitly
\begin{equation}
\Delta \phi_T=\phi((m+1)T)-\phi(mT)=-\frac{k_cT}{M}\Delta p(\bar{\eta}_{\beta})\,.\label{Eq:DeltaPhi}%
\end{equation}
When the phase offset is driven according to (\ref{Eq:DeltaPhi}), there is a dramatic
change in time-evolution of velocities of individual atoms.
As the phase offset is varied over time,
the entire frequency-comb structure shifts towards lower frequencies.
As the teeth sweep through the velocity space, atomic $v(t)$ trajectories are ``snow-plowed''
by teeth, ultimately leading to narrow velocity spikes collected on the teeth.
This emergence of ``velocity comb'' was discussed in Ref. \cite{IliAhmDer11} for two-level system.
Formally, we may separate initial velocities into groups
\begin{equation}
v_{mp}(t=0)+\left( 2\pi (n-1)-\bar{\eta}_{\beta} \right)/k_c T
< v(t=0) < v_{mp}(t=0)+\left(2\pi n- \bar{\eta}_{\beta} \right)/k_c T, n=0,\pm 1 \ldots
\end{equation}
The width of each velocity group is equal to the distance between neighboring teeth in
velocity space, $2\pi/k_c T$. As a result of ``snow-plowing'', the $n^\mathrm{th}$ group
will be piled up at $v_n(t)=v_{mp}(t)+ 2\pi n /k_c T$. The final velocity spread of individual
velocity groups will be limited by the Doppler temperature, $T_D = \hbar \gamma/2 k_B$.
\begin{figure}[h]
\begin{center}
\includegraphics*[scale=0.5]{Rainbowbox2.eps}
\end{center}
\caption{Time-evolution of velocity distribution for a thermal beam subjected to
a coherent train of laser pulses. Pulse-to-pulse phase offset of the train is varied
linearly in time as prescribed by Eq.~(\ref{Eq:DeltaPhi}). $N$ is the number of pulses. (a) Atomic and pulse train parameters are:
$\gamma T=0.25$, $\Theta=\pi/2$. The optimal phase detuning is $\bar{\eta}=-1.23$.
The center of initial velocity distribution is $v_{mp}=500$ m/s.\label{Fig:fvt}}
\end{figure}
To illustrate the train-driven time-evolution for the entire ensemble,
we consider a 1D thermal beam characterized by the initial velocity distribution
\begin{equation}
f(v,t=0)=\frac{v^3}{2\tilde{v_0}^4}\exp(-\frac{v^2}{2\tilde{v_0}^2}).
\end{equation}
The most probable $v_{mp}$, average $v_{ave}$ and the r.m.s. $v_{rms}$ values are expressed in terms of $\widetilde{v}_0$ as
\begin{eqnarray}
v_{mp}=\sqrt{3}\widetilde{v}_0,\nonumber\\
v_{ave}=\sqrt{\frac{9\pi}{8}}\widetilde{v}_0,\nonumber\\
v_{rms}=2\widetilde{v}_0.
\end{eqnarray}
$v_{mp}$ is the most probable velocity at $t=0$.
A typical time-evolution of the velocity distribution is shown in Fig.~\ref{Fig:fvt}. Local compression of velocity distribution happens near
the points $v_c(t)+\lambda_c n/T$, $n=0,\pm1..$,
where $v_c$ is the time-dependent position of velocity distribution center.
Clearly, velocity distribution, while initially being continuous, after a certain number
of pulses develops a comb-like profile.
This is the ``velocity comb'' of sharp peaks separated by equal intervals $\lambda_c/T$ in the velocity space.
\section{Introduction}
Doppler cooling \cite{HanSch75} relies
on radiative force originating from momentum transfer to atoms from a laser field and subsequent spontaneous emission in random directions.
Cooling by CW lasers has been widely studied both theoretically and experimentally within the last several decades \cite{MinLet87Book,MetStr99Book,BerMal10_Book}.
Schemes for cooling the two-level atoms by the trains of ultrashort laser pulses \cite{Hof88, Str89,WatOhmTan96,Kie06, IliAhmDer11} were proposed.
The possibility of stimulated cooling the two-level atoms by the pairs of counter propagating $\pi$-pulses \cite{Kaz74,NoeNoeSch96,GoeBloHau97, SodGriOvc97} and similar idea of cooling by bichromatic standing wave \cite{SodGriOvc97} were studied both theoretically and experimentally. The interest in cooling by the pulse trains is stimulated by the rapid development of a pulsed laser technology and frequency combs (FC) \cite{SchHarYos08,AdlCosTho09,VodSorSor11}. In particular the mechanical action of FC on atoms was observed experimentally in Ref.~\cite{MarStoLaw04}.
In many cases the atom can not be approximated as a two-level system because the excited state may decay to some intermediate sublevels. As an example, group III atoms have no single-frequency closed transition on which the cooling of the ground state could be based, because their ground states are composed of two fine-structure sublevels $nP_{1/2}$ and $nP_{3/2}$.
CW laser cooling of this type of $\Lambda$-systems
in the presence of bichromatic force-assisted velocity-selective coherent population trapping has been studied in \cite{PruAri03}. Other schemes of CW sub-Doppler cooling of three-level atoms based on velocity-selective coherent population trapping have been proposed earlier \cite{AspAriKai88,KasChu92}. There were also proposals for bichromatic force cooling of three-level $\Lambda$-atoms \cite{GupXiePad93,AspDalHei86}.
Here we propose the scheme for decelerating and cooling the three-level atoms with
the ultrafast pulse train. In our scheme both ground states of the $\Lambda$-type system are coupled to the excited state by the same laser field. As a result, the cooling does not require additional repumping of population from the intermediate state.
The exerted scattering force depends on atomic velocity via the Doppler shift. Similar to the case of a two-level system, studied in \cite{IliAhmDer11}, the spectral profile of the scattering force mimics the periodic structure of the frequency comb (FC) spectra.
Since the positions of FC teeth depend on the pulse-to-pulse carrier envelope phase offset, CEPO, the
velocity-dependence of the scattering force can be varied in time by simply changing the phase offset between subsequent pulses. Thereby, continuous compression of velocity distribution in velocity space can be achieved.
During the pulse-train cooling, continuous velocity distributions gravitate toward a series of sharp peaks (typically of the Doppler width) in the velocity space, reflecting the underlying frequency comb structure.
There are several motivations for this work.
Wide spectral coverage of FC allows one to cool the atoms in a broad range of velocities at the same time.
In some cases, FC cooling could be used for reducing number of required lasers.
Cooling setup based on tunable FC can be alternative to Zeeman slowers , whose fields may be detrimental for precision measurements~\cite{ZhuOatHal91}.
The presented analysis is applicable for laser cooling in ion storage rings \cite{SchKleBoo90etal,MieGriGri96etal} where the circulating ions are subjected to
chopped laser field.
We start consideration by deriving analytical expression for the scattering force in the quasi-steady-state regime (QSS), based on the expression for the density matrix obtained in our previous work \cite{IliDer12}.
In the quasi-steady-state regime the radiative decay-induced drop in the excited state population between two pulses is fully restored by the second pulse.
This regime is similar to the saturation regime in a classical system of two kicked coupled damped oscillators.
Based on our analytical expressions, we show that the $\Lambda$-system can be Doppler cooled without additional repumping of population from the intermediate ground state.
We analyze the dependence of the scattering force on the FC parameters. Based on this analysis we propose a principle of choosing FC parameters for optimal cooling of ensemble of $\Lambda$-type three-level atoms.
For the pulse-train-driven $\Lambda$-system there are two major qualitative effects: ``memory'' and ``pathway-interference'' effects.
Both effects play an important role in understanding of the radiative force exerted by the pulse train on the multilevel system.
The system retains the memory of the preceding pulse as long as the population of the excited state does not completely decay between subsequent pulses. This is satisfied for finite values of the product $\gamma T$, $\gamma$ being the excited state radiative decay rate and $T$ being the pulse repetition period. Then the quantum-mechanical amplitudes driven by successive pulses interfere and the response of the system reflects the underlying frequency-comb structure of the pulse train. If we fix the atomic lifetime and increase the period between the pulses, the interference pattern is expected to ``wash out'', with a complete loss of memory in the limit $\gamma T \gg 1$. This memory effect is qualitatively identical to the case of the two-level system, explored in Ref.~\cite{IliAhmDer11}.
The ``pathway-interference'' effect is unique for multilevel systems. The excited-state amplitude arises from simultaneous excitations of the two ground states. The two excitation pathways interfere. The ``pathway-interference'' effect is perhaps most dramatic in the coherent population trapping (CPT) regime \cite{Har97,MorVia11,SoaAra07,SoaAra10} where the ``dark'' superposition of the ground states conspires to interfere destructively, so that there is no population transfer to the excited state at all.
This paper is organized as follows. In Section II we derive analytical expression for the scattering force exerted on atoms by the pulse train in a quasi steady-state regime.
In Section III we study the dependence of scattering force on FC parameters and propose a method of their optimization.
In Section IV we study the process of cooling of thermal beam of three-level $\Lambda$-type atoms by pulse train. We demonstrate that in the optimal cooling regime the initial velocity distribution evolves to a comb-like profile with sharp equidistant maxima, ``velocity comb''. The width of each peak is determined by the Doppler temperature limit. Finally, the conclusions are drawn in Section V.
|
{
"timestamp": "2012-10-02T02:01:49",
"yymm": "1203",
"arxiv_id": "1203.1963",
"language": "en",
"url": "https://arxiv.org/abs/1203.1963"
}
|
\section{Introduction}
\section{\hspace*{-6mm}{\bf .\hspace{2mm}Introduction}}
Consider the class of dissipative phenomena described by the
following model:
\begin{equation} \label {11}
{\cal L} _\varepsilon u_\varepsilon \equiv
(\varepsilon\partial_{xxt} +
c^2 \partial_{xx} - \partial_{tt}) u_\varepsilon = \ f,\ \ \
\end{equation}
\noindent
where $ \varepsilon, c$ are positive constants and the source term $f $
may be linear or not.
In the linear case, when a prefixed boundary-initial problem ${\cal P}
_\varepsilon$ is stated, the knowledge of the related Green function
$G_\varepsilon $ allows to solve explicitly ${\cal P}
_\varepsilon$. When the function $f$ is not linear, then
$G_\varepsilon $ represents the explicit kernel of the integral equation
to which the problem ${\cal P}
_\varepsilon$ can be reduced.
\noindent
Two typical examples in the non linear case are:
i) For $f=a u_t +b \sin u$, the equation (\ref{11}) is the perturbed
Sine-Gordon
\hspace {4mm} equation which models the Josephson effect in
Superconductivity \cite{bp}.
ii) For $f=f(u_x,u_{xt},u_{xx})$,the equation (\ref{11}) is the
Navier-Stokes equation
\hspace{4mm} for a compressible gas with small viscosity
\cite{mm}.
Moreover, as for the artificial viscosity methods, (\ref{11}) represents a
model of wave equations perturbed by viscous terms with a small
parameter $ \varepsilon$ \cite {kl}, \cite{n}. In this framework, the behaviour of
$u_\varepsilon$ as $\varepsilon \rightarrow 0 $ must be examined and the
interaction of pure waves with the diffusion effects caused by $\varepsilon u_{xxt}$
must be estimated.This interaction is meaningful in
the evolution of many dissipative models (viscoelastic liquids or
solids \cite{ta} - \cite{s}, real gases with viscosity \cite{la},
magnetohydrodynamic
fluids \cite{na}.)
For $\varepsilon \equiv 0$, the parabolic equation (\ref{11})
turns into the wave equation
\begin{equation} \label {12}
{\cal L} _ 0 u_0 \equiv
(
c^2 \partial_{xx} - \partial_{tt}) u_0 = \ f,\ \ \
\end{equation}
\noindent
and ${\cal P}
_\varepsilon$ changes into a problem ${\cal P}
_0$ for $u_0$, with the {\em same initial-boundary} conditions of ${\cal P}
_\varepsilon$. So, boundary-layers are missing, but the approximation
of $u_\varepsilon$ by
$u_0$ is rough, because in large time-intervals the diffusion effects are
dominant. As consequence, the asymptotic behaviour of $u_\varepsilon$
should depend on the {\em slow time} $\varepsilon \, t$ typical of
the diffusion; on the contrary, boundary layer control terms characterized
by the {\em fast time} $t/\varepsilon$, should be negligible.
Aim of the paper is to derive and analyze an appropriate functional
relation between
$G_\varepsilon $ and the wave Green function $ G_0$ related to
${\cal P}
_0$
(Th. 3.1). By means of this transformation, a rigorous asymptotic
analysis of $G_\varepsilon$ is achieved (Th. 4.1), and $G_\varepsilon $
is approximated by solutions $v(x,t)$ of the second-order
{\em diffusion-wave} equations
\begin{equation} \label{13}
\frac{\varepsilon}{\mit 2} \, v_{xx}
\ = \ v_t + c v_x, \ \ \ \ \ \ \\ \ \frac{\varepsilon}{\mit 2} \, v_{xx}
\ = \ v_t - c v_x
\end{equation}
\noindent
which correspond to the {\em heat equations} $v_{yy} \, = \,
v_{\theta}$, where the time
- variable
$\theta$ is just the slow - time $(\theta = \varepsilon \, t) $ and the
space-variables $y \,= \, x \pm ct$ are related to traveling or
backword waves.
The physical meaning of the above analysis is clarified by the
explicit solution related to the case
$f=0$ (n.5). Except errors of order $O (\varepsilon /t)$, this relationship
is
\begin{equation} \label{14}
u_\varepsilon(x,t) = \frac{c}{\sqrt{2 \pi \varepsilon t}} \
\int_{- \infty} ^{\infty} e^{-\frac{c^2(\tau-t)^2}{2\varepsilon t}}
u_0(x,\tau) \ d\tau,
\end{equation}
\noindent
and it clearly shows the interaction between diffusion and wave
propagation. In the time - interval $(\varepsilon,
\frac{1}{\varepsilon})$, when $\varepsilon \, t < 1$ , pure waves
propagate almost without perturbation; while as from the instant $ t>1/\varepsilon $,
damped oscillations predominate.
Let us remark, generally, asymptotic theories are developed for models
with first and second order operators as documented in \cite{bs} and
\cite{d}. For these operators, in fact, numerous maximum theorems for
the rigorous estimate of the remainder are well known.
As for the third - order model (\ref{11}), the error of the
approximation has been examinated by means of the functional relation
(\ref{31}).
\section{\hspace*{-6mm}{\bf .\hspace{2mm}Statement of the problem}}
\setcounter{equation}{0}
\hspace{5.1mm}
If $ u_\varepsilon(x,t)$ is a function defined in the strip\vspace{4mm}
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Omega =\{(x,t) : 0 \leq x
\leq
l, \ \ t>0 \}$,
\vspace{3mm}
\noindent
let ${\cal P}_\varepsilon $ the initial- boundary value problem related
to the Eq. (\ref{11}) with conditions
\begin{equation} \label{21}
\left \{
\begin{array}{ll}
& u_\varepsilon (x,0)=f_0(x), \ \ \partial_t u_\varepsilon
(x,0)=f_1(x),
\ \ \ \ x\in [0,l],\vspace{2mm} \\
& u_\varepsilon (0,t)=\varphi(t), \ \ u_\varepsilon
(l,t)=\psi(t), \ \ \ \ \ \ \ \ \ \ \ t>0,
\end{array}
\right.
\end{equation}
\noindent
where $f_0, f_1, \varphi, \psi$ are arbitrary given data.
Boundary conditions $(\ref{21})_2$ represent only an example of the
analysis we are going to apply; {\em flux-boundary} conditions or {\em mixed
boundary} conditions could be considered too. Further, as $f(x,t),
f_0(x), f_1(x)$ are quite arbitrary, it is not restrictive assuming
$\varphi=0, \ \psi=0$; otherwise, it suffices to put
\vspace{3mm}
\begin{equation} \label {22}
\bar{u}_\varepsilon = u_\varepsilon - \frac{x}{l} \psi -
\frac{l-x}{l} \varphi ; \ \ \ \bar{f} = f + \frac{x}{l}
\ddot{\psi} +\frac{l-x}{l} \ddot{\varphi}
\end{equation}
\noindent
and to modify $f_0(x), f_1(x)$ consequently.
Let $\hat{z}(s) $denote the Laplace transform of the function $z(t)$ and
let
\begin{equation} \label{23}
\hat{F}(x,s) \ = \ \hat{f}(x,s) - s f_0 (x) - f_1(x).
\end{equation}
\noindent
Further, if $\sigma (\varepsilon) = s ( \varepsilon s + c^2 )^{-1/2}$ ,
let
\begin{equation} \label {24}
\hat{g}(y,s) = \frac{cosh[(l-y)\sigma]}{2(\varepsilon s +c^2)
\sigma senh (l\sigma)}
\end{equation}
\noindent
and
\begin{equation} \label{25}
\hat{G}_\varepsilon(\xi, x, s) = \hat{g} (|x-\xi|, s) -
\hat{g} (|x+\xi|, s).
\end{equation}
\noindent
Then, the Laplace tranform $\hat{u}_\varepsilon$ of the solution of the
problem ${\cal P}_\varepsilon $ is given by
\begin{equation} \label {26}
\hat{u}_\varepsilon (x,s) = \int_{0}^{l}
[\hat F(\xi,s) +\varepsilon f^{''}_0(\xi)] \hat{G}_\varepsilon
(x,\xi,s) \ d\xi.
\end{equation}
When $\varepsilon =0 $ one has $\sigma(0) = s/c $ and
(\ref{24}),(\ref{25}) turn into
\begin{equation} \label {27}
\hat{g}_0(y,s) = \frac{cosh[(l-y)s/c]}{2 \
c \ s
\ senh (ls/c)},
\end{equation}
\begin{equation} \label{28}
\hat{G}_0(\xi, x, s) = \hat{g}_0 (|x-\xi|, s) -
\hat{g}_0 (|x+\xi|, s).
\end{equation}
\noindent
Problem ${\cal P}_0 $ is defined by (\ref{12}), (\ref{21}) and the
Laplace transform $\hat{u}_0$ of it's solution is:
\begin{equation} \label{29}
\hat{u}_0 (x,s) = \int_{0}^{l}
\hat F(\xi,s) \\\\\\ \hat{G}_0(\xi,x,s) \ d\xi.
\end{equation}
By comparing (\ref{24}),(\ref{25}) with (\ref{27}), (\ref{28}), the
following relationship between $\hat{G}_\varepsilon$ and
$\hat{G}_0$ is pointed out:
\begin{equation} \label{210}
\hat{G}_\varepsilon(s) = \frac {c^2}{\varepsilon s +c^2} \ \ \hat{G}_0
( \frac{cs}{\sqrt{\varepsilon s+c^2}}
).
\end{equation}
In order to estimate $G_\varepsilon (t)$ in terms of
$G_0(t)$
and to achieve the rigorous
asymptotic behaviour of
$ G_\varepsilon(t)$ for $\varepsilon \rightarrow 0$ ,
the corrispondence (\ref{210}) must be
analyzed and its inverse Laplace transform must be established.
\section{\hspace*{-6mm}{\bf .\hspace{2mm} Functional dependence between
$ G_\varepsilon$ and
$ G_0$ }}
\setcounter{equation}{0}
\hspace{5.1mm}
If $I_0$ denotes the modified
Bessel function of first kind and order zero, the inverse
Laplace - transform of (\ref{210}) is fully given by the following
theorem:
\vspace {5mm}
{\bf Theorem 3.1}{\em - The relationship between the Green functions}
$ G_\varepsilon(t)$ {\em and}
$ G_0(t)$ {\em is}:
\begin{equation} \label{31}
G_\varepsilon(t)= \frac{c^3
e^{-c^2t/\varepsilon}}{\varepsilon\sqrt{\pi\varepsilon t}} \ \
\int_0^\infty e^{-\frac{c^2v^2}{4\varepsilon t}} \ dv \int_0^v
G_0(u) I_0 (\frac{2c^2}{\varepsilon} \sqrt{u(v-u)} \ du
\end{equation}
\noindent
{\em and their} ${\cal L}$- {\em transforms satisfy} (\ref{210}) {\em in
the half- plane}
${\it{ Re}}(s) > - c^2/\varepsilon$.
\vspace{6mm}\noindent
{\bf Proof.} For ${\it{ Re}}(\varepsilon s+c^2) > 0$, one has:
\begin{equation} \label{32}
{\cal L} ( \frac{e^{-\frac{c^2t}{\varepsilon}}}{\sqrt{\pi \varepsilon
t}}e^{-\frac{c^2v^2}{4\varepsilon t}}) \ \ = \
\frac{e^{\frac{-c}{\varepsilon} \, \sqrt{\varepsilon s +c^2 } \, v }}{\sqrt{\varepsilon
s +c^2}},
\end{equation}
\noindent
therefore the Laplace tranform of (\ref{31}) is :
\begin{equation} \label{33}
\hat G_\varepsilon(s)= \frac{c^3
}{\varepsilon\sqrt{\varepsilon s +c^2}}
\int_0^\infty e^{-\frac{c}{\varepsilon} \sqrt{\varepsilon s +c^2 } \,
v} dv \int_0^v
G_0(u) I_0 (\frac{2c^2}{\varepsilon} \sqrt{u(v-u)} du
\end{equation}
\hspace* {2cm} \[ = \frac{c^3
}{\varepsilon\sqrt{\varepsilon s +c^2}} \ \
\int_0^\infty G_0(u) du
\int_0^\infty e^{-\frac{c}{\varepsilon} (u+v)
\sqrt{\varepsilon s +c^2 }}
\ I_0 (\frac{2c^2}{\varepsilon} \sqrt{uv}) \ dv . \]
\vspace{8mm}\noindent
Considering that
\begin{equation} \label{34}
\int_0^\infty e^{- \frac{cv}{\varepsilon}
\sqrt{\varepsilon s +c^2}}
\ \ I_0 (\frac{2c^2}{\varepsilon} \sqrt{uv}) \ dv \ =
\frac{\varepsilon}{c} \ \ \frac{e^{\frac{c^3}{\varepsilon} \frac{u}{
\sqrt{\varepsilon s +c^2 }}}}{\sqrt{\varepsilon
s +c^2}}
\end{equation}
\noindent
see (\cite{em}), one has
\begin{equation} \label{35}
\hat G_\varepsilon(s) = \frac{c^2}{\varepsilon s +c^2} \int_0^\infty
e^{-\frac{cs }{\sqrt{\varepsilon s +c^2}} \ u } \ \ \ G_0 (u) du
\end{equation}
\vspace{8 mm}\noindent
therefore the above formula coincides with (\ref{210}).
\hbox{} \hfill \rule {1.85mm}{2.82mm}
\vspace {5mm}
Consider now a first application of Theorem 3.1. By means of the
transformation (\ref{31}), it's possible to deduce explicit
expressions for $
G_\varepsilon$ from known formulae of
$ G_0$. So, for instance,
if
$ G_0$ is represented by its Fourier series
\begin{equation} \label{36}
G_0=\frac{2}{c\pi} \ \ \\ \ \sum_{n=1}^{\infty} \ \ \frac{1}{n} \ \
\sin (\frac{\pi}{l}cnt) \ \sin (\frac{\pi}{l}nx) \sin
(\frac{\pi}{l} n\xi),
\end{equation}
\noindent
then (\ref{31}) allows to make Fourier series of $G_\varepsilon$
explicit. It suffices to evaluate the transform (\ref{31}) of the
time-component $ G_{0,n}$ of $ G_{0}$:
\begin{equation} \label{37}
G_{0,n} (t) \ \ = \ \ \sin(\frac{\pi}{l} \ c \ n \ t).
\end{equation}
Owing to:
\begin{equation} \label{38}
I = \int_{0}^{2v} \sin(ay) I_0(b\sqrt{y(2v-y)})dy =
2 \sin (av) \ \ \frac{ \sin (v \sqrt{a^2-b^2})}{\sqrt{a^2-b^2}},
\end{equation}
\noindent
if one puts:
\begin{equation}
a=\pi c n/l, \ \ \ b=2c^2/\varepsilon , \ \ \ k= 2 c l / (\pi
\varepsilon), \ \ \ \omega_n = \sqrt{1-(n/k)^2},
\end{equation}
the integral transform (\ref{31}) of $ G_{0,n}$
gives
\begin{equation} \label {39}
G_{\varepsilon,n}(t) \ = \
\ e^{-\frac{\pi^2 n^2 }{2 l^2} \, \varepsilon \, t } \
\frac{\sin (a \omega_n t)}
{ \omega_n}.
\end{equation}
\noindent
So, by (\ref{31}), (\ref{36}), (\ref{39}), one obtains
\begin{equation} \label {310}
G_\varepsilon = \frac{2}{\pi c} \sum
_{n=1}^{\infty}e^{-\frac{\pi^2}{2l^2}n^2 \varepsilon t } \sin
(\frac{c \pi n}{l} \omega_n t
) \frac{1}{n \, \omega_n} \sin (\frac{\pi}{l}nx) \sin
(\frac{\pi}{l} n\xi).
\end{equation}
\noindent
and this formula represents the Fourier series of $G_\varepsilon $
previously established in \cite{mda} by
Fourier method.
\section{\hspace*{-6mm}{\bf .\hspace{2mm} Asymptotic behavior of
$G_\varepsilon$}}
\setcounter{equation}{0}
\hspace{5.1mm}
An additional consequence of (\ref{31}) is the asymptotic
analysis of $G_\varepsilon$ when $\varepsilon$ tends to zero. Referring
to (\ref{31}), let
$\tau=t/\varepsilon$ and let
\begin{equation} \label{41}
\Gamma(u,v,\tau)=\frac{(c\sqrt{\tau})^3}{\sqrt{\pi}}
\ G_0(tu) \ I_0(2c^2\tau \sqrt{uv}) e^{
-c^2\tau[1+(u+v)^2/4]}.
\end{equation}
\noindent
Then, the function $G_\varepsilon $ can be given the form
\begin{equation} \label{42}
G_\varepsilon = \ \ \int \! \! \int_{Q} \Gamma(u,v,\tau) du \ \ dv,
\end{equation}
\noindent
where $Q\equiv \{ (u,v)\in [0,\infty]\times[0.\infty] \}.$
So, $G_\varepsilon$ depends on $\varepsilon$ only through the fast time
$\tau$. When $\varepsilon$ is vanishing and $t>\varepsilon$, the parameter
$\tau$ goes to infinity and the asymptotic behaviour of integral
(\ref{42}) can be rigorously obtained by Laplace method \cite{db}
To establish
precise estimates of the remainder terms, let
\begin{equation} \label{43}
\chi(\tau)=\chi_0 \ \ \tau^{-1/3},\ \ \ \ \sigma(\tau)=\sigma_0 \ \
\tau^{-1/3},
\end{equation}
\noindent
where $\chi_0 $ and $\sigma_0$ are arbitrary real positive constants such
that $\chi$ and $\sigma$ are less then one for all $\tau>1.$ Further, let
\begin{equation} \label{44}
Q_0\equiv
\{ (u,v)\in [1-\chi,1+\chi]\times[1-\sigma,1+\sigma] \} \subset Q.
\end{equation}
\noindent
The following Lemma holds:
\vspace{3mm}\noindent
{\bf Lemma 4.1 -} {\em For all } $\tau>1$, {\em it results}:
\begin{equation} \label{45}
|G_\varepsilon - \int \! \! \int_{Q_0} \ \Gamma(u,v,\tau) \, du\, dv| < \ \mu
\ e^{-\lambda^2 \tau^{1/3}},
\end{equation}
\noindent
{\em where the constants} $\lambda^2$, $\mu$ {\em depend only on }$ c, \chi_0, $ $\sigma_0$.
\vspace{6mm}\noindent
{\bf Proof.} For all real and positive $z$ one has $I_o(z)<e^z$, so that
by (\ref{41}) one obtains
\begin{equation} \label{46}
|\Gamma| \leq \frac{|G_0|}{\sqrt{\pi}} \ \ (c \sqrt{\tau})^3 \ \ e^{-c^2\tau
\ h(u,v)}
\end{equation}
\noindent
with
\begin{equation} \label{47}
h(u,v)=1 \ + \ (1/4)(u+v)^2 \ -2\sqrt{uv}.
\end{equation}
\noindent
\vspace{3mm}
Further, for all $(u,v)\in Q$, one has
\begin{equation} \label{48}
h(u,v)=\frac{1}{4}[(u-1)^2+(v-1)^2+(\sqrt u-1)^2 (\sqrt v+1)^2
\end{equation}
\hspace*{2cm} \[
+ (\sqrt v
-1)^2 (\sqrt u +1)^2 ] \ \ \geq \frac{1}{4} [(u-1)^2+ (v-1)^2 ]. \]
\noindent
Therefore, indicating by C a constant independent by $\tau$, one has:
\begin{equation} \label{49}
|\int \! \! \int_{Q-Q_0} \ \Gamma \, du \, dv|\leq C
\int \! \! \int_{Q-Q_0}
e^{- \frac{c^2}{4}[(u-1)^2+(v-1)^2]\tau} \ du \ dv.
\end{equation}
\noindent
Hence, by standard computations, the estimate (\ref{45}) is obtained.
\hbox{} \hfill \rule {1.85mm}{2.82mm}
\vspace{8 mm}
The estimates of
Lemma 4.1 allow us to define the following
function
\begin{equation} \label{410}
H(x,t,\varepsilon ) = \frac{c}{\sqrt{2 \pi \varepsilon t}} \
\int_{- \infty} ^{\infty} e^{-\frac{c^2(\tau-t)^2}{2\varepsilon t}}
G_0(x,\tau) \ d\tau
\end{equation}
\noindent
and to obtain the following result:
\vspace{3mm}\noindent
{\bf Theorem 4.1 -} {\em For all } $(x,t)\in \Omega, $
{\em and} $t>\varepsilon$, {\em one has:}
\begin{equation} \label{411}
G_\varepsilon(x,t) = H(x,t,\varepsilon) \ \ (1+ \rho_1) \ + \rho_2,
\end{equation}
\noindent
{\em where the remainder terms} $ \rho _1 , \rho_2 $ {\em are decreasing functions of
the fast time} $\tau=t/\varepsilon$ {\em such that}
\vspace{3mm}
\begin{equation} \label{412}
|\rho _1| \leq k_1 \ \tau^{-1}; \ \ \ \ \ \rho_2 \leq \ k_2 \ e^{-
\lambda^2 \tau^{1/3}},
\end{equation}
\noindent
{\em with the constants} $k_1$, $k_2$, $ \lambda^2$ {\em depending only on }$ c, \chi_0, $ $\sigma_0$.
\vspace{4mm}
{\bf Proof.} By asymptotic formulae of Bessel functions, for real,
positive and large $z$ one has
(\cite{w}):
\begin{equation} \label{413}
I_0(z)=\ \frac {e^z}{\sqrt{2 \pi z}} \ (1+r_0) \ \ \ \ \\ \ (|r_0| < c_0
\ \ z^{-1})
\end{equation}
\noindent
and so the integral of $\Gamma$ on $Q_0 $ (see (\ref{41}), (\ref{42}))
can be given the form
\begin{equation} \label{414}
\int \! \! \int_{Q_0} \\ \Gamma du dv = \frac{c \tau}{2 \pi} \
\int_{1-\chi}^{1+\chi}
G_0(tu) \, du \int_{1-\sigma}^{1+\sigma} \ \frac{1+r_1}{(uv)^{1/4}} \ \
e^{- c^2 \tau \ h(u,v)} \ dv ,
\end{equation}
\noindent
where $h(u,v)$ is defined in (\ref{47}) and the remainder term $r_1$ is
such that $|r_1| \, \leq c_1 \,( \tau \sqrt{uv})^{-1}$, according to
(\ref{413}).
The function h(u,v) has a point of absolute minimum in
$(u_0,v_0)=(1,1)$, where it vanishes.
Therefore, it results: $h\, =\,h_0 \, +\,h_*$ with
\begin{equation} \label{415}
h_0=\frac{1}{2}[(u-1)^2+(v-1)^2]; \ \ \ \ \ |h_*| \leq c_*
[|u-1|+|v-1|]^3.
\end{equation}
\noindent
According to the Laplace method, \cite{db}, for large $\tau$ , the dominant
contribution is due to a neighbourhood of $(u_0,v_0)$ , so that it's
enough to expand $\Gamma$ near to this point. Further, if $u$ and $v$ are
substituted by
\begin{equation} \label{416}
u - 1 \ = \ \frac{1}{\sqrt{\tau}} \ u_1, \ \ \ v - 1 \ = \
\frac{1}{\sqrt{\tau}} \ v_1,
\end{equation}
\noindent
the terms of third order like $h_*$ are vanishing with order of $\tau
^{-3/2}$ ; besides, the limits of integration become $( -\infty, \
+\infty)$ as: $\chi \sqrt{\tau}=\chi_0 \ \tau^{1/6}$ and $\sigma
\sqrt{\tau}=\sigma_0 \ \tau^{1/6}$.
Then, for $\tau \rightarrow \infty$, one has:
\begin{equation} \label {417}
\int \! \! \int_{Q_0} \Gamma dudv = \frac{c^2 }{2\pi} \int
_{-\infty}^{\infty} e^{-\frac{c^2}{2} u^2} G_0(t+\sqrt{\varepsilon t} \ u) du
\int
_{-\infty}^{\infty} e^{-\frac{c^2}{2} v^2} dv+
O(\frac{1}{\tau}).
\end{equation}
\noindent
The above formula, together with the results of Lemma 4.1
(see(\ref{45}),(\ref{410})), implies (\ref{41}) and the estimate
$(\ref{412})_2$ for $\rho_2$. Finally, by means of routine calculations,
the order of the error $\rho_1$ can be rigorously specified according to
$(\ref{412})_1$.
\hbox{} \hfill \rule {1.85mm}{2.82mm}
\section{\hspace*{-6mm}{\bf .\hspace{2mm} Diffusion and waves}}
\setcounter{equation}{0}
\hspace{5.1mm}
To remark the physical meaning of the results, let consider the simple
case: $ f=0$, $f_0=0, f_1 \neq 0.$ By (\ref{26}) and (\ref{29}) one
has:
\begin{equation} \label{51}
u_\varepsilon = - \int_0^l f_1 (\xi)G_\varepsilon (\xi,x,t) d\xi ; \
\ \ \ u_0 = - \int_0^l f_1(\xi) G_0 (\xi,x,t) d\xi
\end{equation}
\noindent
Consequently, except errors of order $O(\varepsilon /t) $, Theorem 4.1 allows to
obtain (see (\ref{410}),(\ref{411})) the following estimate:
\begin{equation} \label{52}
u_\varepsilon(x,t) = \frac{c}{\sqrt{2 \pi \varepsilon t}} \
\int_{- \infty} ^{\infty} e^{-\frac{c^2(\tau-t)^2}{2\varepsilon t}}
u_0(x,\tau) \ d\tau.
\end{equation}
\noindent
The above explicit corrispondence between $u_0$ and $u_\varepsilon$ clearly
shows the interaction between diffusion and wave propagation.
For instance, when $ f_1= \frac{c\pi}{l} \ \sin (\frac{\pi x}{l})$, one has
the {\em pure wave}
\begin{equation} \label{53}
u_0(x,t)= \sin (\frac{\pi x}{l}) \ \ \sin (\frac{\pi }{l} \ c t ),
\end{equation}
\noindent
while $u_\varepsilon$, owing to (\ref{52}), is given by
\begin{equation} \label{54}
u_\varepsilon(x,t)= e^{-\frac{\pi^2 }{2 l^2} \, \varepsilon \, t }
\\
\ u_0(x,t).
\end{equation}
\vspace{3mm}
\noindent
This means that the dissipation caused by $\varepsilon u_{xxt}$ is
significant by slow
times $\varepsilon \, t$ only; the terms depending on the fast time
$t/\varepsilon$ are fully negligible for all $t >\varepsilon$.
\noindent
As consequence, in the time interval $(\varepsilon,
\frac{1}{\varepsilon})$ where $\varepsilon t <1$, the pure wave (\ref{53})
is propagated nearly undisturbed; when $t>\frac{1}{\varepsilon}$, damped
oscillations predominate.
These aspects are generally valid, whatever the initial data may be,
and for appropriate $f$. In fact,
if one transforms by (\ref{410}) the time-components
$ G_{0,n} (t) = \sin(\frac{n\pi}{l} \, c \, t)$ of $G_0$,
one has \cite{em}:
\begin{equation} \label{55}
\frac{c}{\sqrt{2 \pi \varepsilon t}} \
\int_{- \infty} ^{\infty} e^{-\frac{c^2}{2} \, \frac{(\tau-t)^2}{\varepsilon t}}
\sin (\frac{\pi}{l} \, c \, n \, \tau) \ d\tau = e^{-\frac{\pi ^2
n^2}{2l^2}
\,
\varepsilon \, t} \ \sin (\frac{\pi}{l} \, c \, n \, t).
\end{equation}
\noindent
Therefore, the Fourier series of the function $H$ which approximates
$G_\varepsilon$ is given by
\begin{equation} \label{56}
H =
\frac{2}{c\pi} \ \ \\ \ \sum_{n=1}^{\infty} \ \ \frac{1}{n} \ \
e^{-\frac{1}{2}(\frac{\pi n}{l})^2 \, \varepsilon \, t } \ \sin (\frac{\pi}{l}cnt) \ \sin (\frac{\pi}{l}nx) \sin
(\frac{\pi}{l} n\xi).
\end{equation}
This formula, together with (\ref{26}), (\ref{411}), makes explicit the
asymptotic behavior of the solution $u_\varepsilon $ of the problem $\cal
P_\varepsilon$ related to arbitrary data $f_0,f_1$ and $f$. As (\ref{56})
shows, this behavior is like (\ref{54}), with damped waves which
become vanishing as from the instants $t> 1/\varepsilon$.
It must be remarked that the function $H $ can be given the form:
\noindent
$H=H^--H^+$, with
\begin{equation} \label{57}
H^{\pm} =
\frac{2}{c\pi} \sum_{n=1}^{\infty} \frac{1}{n} \
e^{- \frac{1}{2} (\frac{\pi n}{l})^2 \ \varepsilon t } \ \sin (\frac{\pi n}{l}\xi)
\cos [\frac{\pi n}{l}(x \pm ct)].
\end{equation}
\noindent
Then, the basic variables which are typical of the diffusion and wave
propagation are identified by the following
\vspace{4mm}
\noindent
{\bf Property 5.1} - {\em The function} $H^-$ {\em (or} $H^+${\em ) is solution of
the diffusion-wave equation}
\begin{equation} \label{58}
\frac{\varepsilon}{2} v_{xx}
\ = \ v_t + c v_x \ \ \ \ \ (or : \frac{\varepsilon}{2} v_{xx}
\ = \ v_t - c v_x).
\end{equation}
\noindent
{\em Moreover, if one puts}: $y=x \pm ct, \ \ \ \theta =
\frac{\varepsilon}{2} \,t$, {\em Eq.} (\ref{58}) {\em turns
into the heat equation} $v_{yy} =v_{\theta}$, {\em with the time
-variable}
$\theta$ {\em given just by the slow time and the
space-variable} $y$ {\em related to traveling (or retrograde) waves.}
\vspace {5mm}
Finally, we observe that $H^{\pm}_x $ is given by
Jacobi Theta function $\theta_3(y,\theta)$;
in fact, for $y=x + ct$ or $y=x - ct$, one has:
\begin{equation} \label{59}
H^{\pm}_x = \frac{1}{4cl} [ \theta_3 (\frac{y-\xi}{2l},
\frac{\theta}{2l^2})-
\theta_3 (\frac{y+\xi}{2l
}, \frac{\theta}{2l^2})],
\end{equation}
\noindent
according to well-known formulae related to the heat equation.
\vspace {10mm}
\noindent
\vspace{5mm}
\bf{REFERENCES}
\vspace{3mm}
\begin{enumerate}
{\small
\bibitem {bp} A.Barone, G. Patern\`o, {\small \bf {Physics and Application of the
Josephson Effect} } Wiley, N. Y. 530 (1982)
\vspace{-3mm}
\bibitem {mm} V.P. Maslov, P. P. Mosolov, {\small {\bf Non linear wave
equations perturbed by viscous terms}} Walter deGruyher Berlin N. Y.
329 (2000).
\vspace{-3mm}
\bibitem{kl} A.I. Kozhanov, N. A. Lar`kin, {\small \it Wave equation with
nonlinear dissipation in noncylindrical Domains}, Dokl. Math 62, 2,
17-19 (2000)
\vspace{-3mm}
\bibitem {n} Ali Nayfeh, {\small \it A comparison of perturbation methods for
nonlinear hyperbolic waves} in {\bf Proc. Adv. Sem. Wisconsin} 45, 223-276 (1980).
\vspace{-3mm}
\bibitem {ta} R. I. Tanner, {\small \it Note on the Rayleigh Problem for a
Visco-Elastic Fluid}, ZAMP vol XIII, 575-576 (1962)
\vspace{-3mm}
\bibitem {jrs} D.D Joseph, M. Renardy and J. C. Saut, {\small\it
Hyperbolicity and change of type in the flow of viscoestic fluids},
Arch Rational Mech. Analysis, 87 213-251, (1985).
\vspace{-3mm}
\bibitem {M} J. A. Morrison, {\small \it Wave propagations in rods of Voigt
material and visco-elastic materials with three-parameters models,
}Quart. Appl. Math. 14 153-173, (1956).
\vspace{-3mm}
\bibitem {r2} P. Renno, {\small \it On some viscoelastic models}, Atti Acc.
Lincei Rend. fis.
75 (6) 1-10, (1983).
\vspace{-3mm}
\bibitem {s} Y.Shibata, {\small \it On the rate of decay of solutions to
linear viscoelastic Equation}, Math. Meth. Appl. Sci.,23 203-226 (2000)
\vspace{-3mm}
\bibitem {la} H. Lamb, {\small {\bf Hydrodynamics}}, Dover Publ. Inc., (1932)
\vspace{-3mm}
\bibitem {na} R. Nardini, {\small \it Soluzione di un problema al contorno
della magneto idrodinamica}, Ann. Mat. Pura Appl.,35 269 (1953)
\vspace{-3mm}
\bibitem{bs} L.L.Bonilla and J.S. Solen, {\small \it High field limit of the
Vlasov Poisson Fokker Plank system: A comparison of different
perturbation methods}, Math. Models Meth. Appl. Sci. 11, 1457-1468,
(2001)
\vspace{-3mm}
\bibitem{d} P.Degond, {\small \it An infinite system of diffusion equation
arising in trasport theory: the coupled spherical armonics expansion
model},Math. Models Meth. Appl. Sci. 11. 903-932, (2001)
\vspace{-3mm}
\bibitem {em} Erdelyi, Magnus, Oberhettinger, Tricomi,{\small \bf{ Tables of
integral transforms }} vol. I Mac Graw-Hill Book (1956)
\vspace{-3mm}
\bibitem {db}De Bruijn, {\small \bf{Asymptotic Methods in Analysis}} North-
Holland Publishin (1958)
\vspace{-3mm}
\bibitem {mda} M. De Angelis, {\small \it Asymptotic analysis for the strip
problem related to a parabolic third- order
operator}, Appl.Math.Letters
14 (4),
425-430 (2001)
\vspace{-3mm}
\bibitem {w} G. N. Watson, {\small {\bf Theory of Bessel Function }} Cambridge
p.804 (1944)
\vspace{-3mm}
\bibitem {r1} P. Renno, {\small \it On a Wave Theory for the Operator
$\varepsilon \partial_t(\partial_t^2-c_1^2
\Delta_n)+\partial_t^2-c_0^2\Delta_n$}, Ann. Mat. pura e Appl.,136(4)
355-389 (1984).
}
\end{enumerate}
\end {document}
|
{
"timestamp": "2012-03-13T01:00:21",
"yymm": "1203",
"arxiv_id": "1203.2198",
"language": "en",
"url": "https://arxiv.org/abs/1203.2198"
}
|
\section{Introduction}
It is well known that if $f$ is a convex function on the interval $I=\left[
a,b\right] $ and $a,b\in I$ with $a<b$, the
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int\limits_{a}^{b}f\left(
x\right) dx\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \label{H}
\end{equation
which is known as the Hermite-Hadamard inequality for the convex functions.
Both inequalities hold in the reversed direction if $f$ is concave. We note
that Hadamard's inequality may be regarded as a refinement of the concept of
convexity and it follows easily from Jensen's inequality. Hadamard's
inequality for convex functions has received renewed attention in recent
years and a remarkable variety of refinements and generalizations have been
found (see, for example, \cite{SSDRPA}, \cite{Dragomir2}, \cite{USK}-\cite{K
, \cite{CEMPJP}, \cite{sarikaya}-\cite{yildiz}).
In \cite{USK} some inequalities of Hermite-Hadamard type for differentiable
convex mappings connected with the left part of (\ref{H}) were proved by
using the following lemma:
\begin{lemma}
\label{l1} Let $f:I^{\circ }\subset \mathbb{R}\rightarrow \mathbb{R}$, be a
differentiable mapping on $I^{\circ }$, $a,b\in I^{\circ }$ ($I^{\circ }$ is
the interior of $I$) with $a<b$. If \ $f^{\prime }\in L\left( \left[ a,
\right] \right) $, then we hav
\begin{equation}
\begin{array}{l}
\dfrac{1}{b-a}\int_{a}^{b}f(x)dx-f\left( \dfrac{a+b}{2}\right) \\
\\
\ \ \ \ \ =\left( b-a\right) \left[ \int_{0}^{\frac{1}{2}}tf^{\prime
}(ta+(1-t)b)dt+\int_{\frac{1}{2}}^{1}\left( t-1\right) f^{\prime
}(ta+(1-t)b)dt\right]
\end{array}
\label{HH}
\end{equation}
\end{lemma}
One more general result related to (\ref{HH}) was established in \cit
{USKMEO}. The main result in \cite{USK} is as follows:
\begin{theorem}
\label{t1} Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$, be a
differentiable mapping on $I^{\circ }$, $a,b\in I$ with $a<b$. If the
mapping $\left\vert f^{\prime }\right\vert $ is convex on $\left[ a,b\right]
$, the
\begin{equation}
\left\vert \frac{1}{b-a}\int_{a}^{b}f(x)dx-f\left( \frac{a+b}{2}\right)
\right\vert \leq \frac{b-a}{4}\left( \frac{\left\vert f^{\prime
}(a)\right\vert +\left\vert f^{\prime }(b)\right\vert }{2}\right) .
\label{H1}
\end{equation}
\end{theorem}
In \cite{SSDRPA}, Dragomir and Agarwal established the following results
connected with the right part of (\ref{H}) as well as to apply them for some
elementary inequalities for real numbers and numerical integration:
\begin{theorem}
Let $f:I^{\circ }\subset \mathbb{R}\rightarrow \mathbb{R}$ be a
differentiable mapping on $I^{\circ }$, $a,b\in I^{\circ }$ with $a<b,$ and
f^{\prime }\in L(a,b).$ If the mapping $\left\vert f^{\prime }\right\vert $
is convex on $\left[ a,b\right] $, then the following inequality holds
\begin{equation}
\left\vert \dfrac{f(a)+f(b)}{2}-\dfrac{1}{b-a}\int_{a}^{b}f(x)dx\right\vert
\leq \left( b-a\right) \left( \frac{\left\vert f^{\prime }(a)\right\vert
+\left\vert f^{\prime }(b)\right\vert }{8}\right) . \label{1H}
\end{equation}
\end{theorem}
In \cite{CEMPJP}, Pearce and Pe\v{c}ari\'{c} proved the following theorem:
\begin{theorem}
Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$, be a differentiable
mapping on $I^{\circ }$, $a,b\in I^{\circ }$ with $a<b$. If the mapping
\left\vert f^{\prime }\right\vert ^{q}$ is convex on $\left[ a,b\right] $
for some $q\geq 1$, the
\begin{equation}
\left\vert \dfrac{f(a)+f(b)}{2}-\dfrac{1}{b-a}\int_{a}^{b}f(x)dx\right\vert
\leq \frac{b-a}{4}\left( \frac{\left\vert f^{\prime }(a)\right\vert
^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{2}\right) ^{\frac{1}{q}}
\label{2H}
\end{equation
an
\begin{equation}
\left\vert \frac{1}{b-a}\int_{a}^{b}f(x)dx-f\left( \frac{a+b}{2}\right)
\right\vert \leq \frac{b-a}{4}\left( \frac{\left\vert f^{\prime
}(a)\right\vert ^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{2}\right) ^
\frac{1}{q}}. \label{3H}
\end{equation}
\end{theorem}
We recall that the notion of quasi-convex functions generalizes the notion
of convex functions. More precisely, a function $f:[a,b]\subset \mathbb{R
\rightarrow \mathbb{R}$ is said quasi-convex on $[a,b]$ if
\begin{equation*}
f(tx+(1-t)y)\leq \sup \left\{ f(x),f(y)\right\}
\end{equation*
for all $x,y\in \lbrack a,b]$ and $t\in \left[ 0,1\right] .$ Clearly, any
convex function is a quasi-convex function. Furthermore, there exist
quasi-convex functions which are not convex (see \cite{ion}).
The classical Hermite-Hadamard inequality provides estimates of the mean
value of a continuous convex function $f:[a,b]\rightarrow \mathbb{R}$. Ion
in \cite{ion} presented some estimates of the right hand side of a Hermite-
Hadamard type inequality in which some quasi-convex functions are involved.
The main results of \cite{ion} are given by the following theorems.
\begin{theorem}
Assume $a,b\in \mathbb{R}$ with $a<b$ and $f:[a,b]\rightarrow \mathbb{R}$ is
a differentiable function on $(a,b)$. If $\left\vert f^{\prime }\right\vert $
is quasi-convex on $[a,b],$ then the following inequality hold
\begin{equation*}
\left\vert \dfrac{f(a)+f(b)}{2}-\dfrac{1}{b-a}\int_{a}^{b}f(x)dx\right\vert
\leq \frac{\left( b-a\right) }{4}\sup \left\{ \left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert \right\} .
\end{equation*}
\end{theorem}
\begin{theorem}
Assume $a,b\in \mathbb{R}$ with $a<b$ and $f:[a,b]\rightarrow \mathbb{R}$ is
a differentiable function on $(a,b)$. Assume $p\in
\mathbb{R}
$ with $p>1$. If $\left\vert f^{\prime }\right\vert ^{p/p-1}$ is
quasi-convex on $[a,b],$ then the following inequality hold
\begin{equation*}
\left\vert \dfrac{f(a)+f(b)}{2}-\dfrac{1}{b-a}\int_{a}^{b}f(x)dx\right\vert
\leq \frac{\left( b-a\right) }{2(p+1)^{\frac{1}{p}}}\left[ \sup \left\{
\left\vert f^{\prime }(a)\right\vert ^{p/p-1},\left\vert f^{\prime
}(b)\right\vert ^{p/p-1}\right\} \right] ^{\frac{p-1}{p}}.
\end{equation*}
\end{theorem}
Convexity plays a central and fundamental role in mathematical finance,
economics, engineering, management sciences and optimizastion theory. In
recent years, several extensions and generalizations have been considered
for classical convexity. A significant generalization of convex functions is
that of $\varphi $-convex functions introduced by Noor in \cite{Noor1}. In
\cite{Noor1} and \cite{Noor5}, the authors have studied the basic properties
of the $\varphi $-convex functions. It is well-know that the $\varphi
-convex functions and $\varphi $-sets may not be convex functions and convex
sets. This class of nonconvex functions include the classical convex
functions and its various classes as special cases. For some recent results
related to this nonconvex functions, see the papers \cite{Noor1}-\cite{Noor5}
\section{Preliminaries}
Let $f,\varphi :K\rightarrow
\mathbb{R}
^{n}$, where $K$ is a nonempty closed set in
\mathbb{R}
^{n}$, be continuous functions. First of all, we recall the following well
know results and concepts, which are mainly due to Noor and Noor \cite{Noor1
\ and Noor \cite{Noor5} as follows:
\begin{definition}
\label{d1} Let $u,v\in K$. Then the set $K$ is said to be $\varphi -convex$
at $u$ with respect to $\varphi $, i
\begin{equation*}
u+te^{i\varphi }\left( v-u\right) \in K,\text{ }\forall u,v\in K,\text{
t\in \left[ 0,1\right] .
\end{equation*}
\begin{remark}
\label{r1} We would like to mention that the Definition \ref{d1} of a
\varphi -convex$ set has a clear geometric interpretation. This definition
essentially says that there is a path starting from a point $u$ which is
contained in $K$. We do not require that the point $v$ should be one of the
end points of the path. This observation plays an important role in our
analysis. Note that, if we demand that $v$ should be an end point of the
path for every pair of points, $u,v\in K$, then $e^{i\varphi }\left(
v-u\right) =v-u$ if and only if, $\varphi =0$, and consequently $\varphi
-convexity$ reduces to convexity. Thus, it is true that every convex set is
also an $\varphi -convex$ set, but the converse is not necessarily true, see
\cite{Noor1},\cite{Noor5} and the references therein.
\end{remark}
\end{definition}
\begin{definition}
\label{d2} The function $f$ on the $\varphi -convex$ set $K$ is said to be
\varphi -convex$ with respect to $\varphi $, i
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left( 1-t\right)
f\left( u\right) +tf\left( v\right) ,\text{ }\forall u,v\in K,\text{ }t\in
\left[ 0,1\right] .
\end{equation*
The function $f$ is said to be $\varphi -concave$ if and only if $-f$ is
\varphi -convex$. Note that every convex function is a $\varphi -convex$
function, but the converse is not true. For eample, the function $f:\mathbb
R\rightarrow }\mathbb{R}$, $f(x)=-\left\vert x\right\vert $ is not a convex
function$,$ but $f(x)=-\left\vert x\right\vert $ is a $\varphi -convex$ with
respect to $\varphi $ wher
\begin{equation*}
\varphi (v,u)=\left\{
\begin{array}{c}
2k\pi ,\text{ \ \ \ \ \ }u.v\geq 0,\ k\in \mathbb{Z} \\
k\pi ,\text{ \ \ \ \ \ \ }u.v<0,\ k\in \mathbb{Z}\text{.
\end{array
\right.
\end{equation*}
\end{definition}
\begin{definition}
\label{d3} The function $f$ on the $\varphi -convex$ set $K$ is said to be
logarithmic $\varphi -convex$ with respect to $\varphi $, such tha
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \left( f\left(
u\right) \right) ^{1-t}\left( f\left( v\right) \right) ^{t},\text{ }u,v\in K
\text{ }t\in \left[ 0,1\right]
\end{equation*
where $f\left( .\right) >0$.
\end{definition}
Now we define a new definition for quasi-$\varphi -convex$ functions as
follows:
\begin{definition}
\label{d4} The function $f$ on the quasi $\varphi -convex$ set $K$ is said
to be quasi $\varphi -convex$ with respect to $\varphi $, i
\begin{equation*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) \leq \max \left\{ f\left(
u\right) ,f\left( v\right) \right\} .
\end{equation*}
\end{definition}
From the above definitions, we hav
\begin{eqnarray*}
f\left( u+te^{i\varphi }\left( v-u\right) \right) &\leq &\left( f\left(
u\right) \right) ^{1-t}\left( f\left( v\right) \right) ^{t} \\
&\leq &\left( 1-t\right) f\left( u\right) +tf\left( v\right) \\
&\leq &\max \left\{ f\left( u\right) ,f\left( v\right) \right\} .
\end{eqnarray*
In \cite{Noor3},\ Noor proved the Hermite-Hadamard inequality for the
\varphi -$convex functions as follows:
\begin{theorem}
\label{tt1} Let $f:K=\left[ a,a+e^{i\varphi }\left( b-a\right) \right]
\rightarrow \left( 0,\infty \right) $ be a $\varphi -convex$\ function on
the interval of real numbers $K^{0}$ (the interior of $K$) and $a,b\in K^{0}$
with $a<a+e^{i\varphi }\left( b-a\right) $ and $0\leq \varphi \leq \frac{\pi
}{2}$. Then the following inequality holds
\begin{eqnarray}
f\left( \frac{2a+e^{i\varphi }\left( b-a\right) }{2}\right) &\leq &\frac{1}
e^{i\varphi }\left( b-a\right) }\int\limits_{a}^{a+e^{i\varphi }\left(
b-a\right) }f\left( x\right) dx \label{2} \\
&\leq &\frac{f\left( a\right) +f\left( a+e^{i\varphi }\left( b-a\right)
\right) }{2}\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \notag
\end{eqnarray}
\end{theorem}
This inequality can easily show that using the $\varphi -convex$\ function's
definition and $f\left( a+e^{i\varphi }\left( b-a\right) \right) <f\left(
b\right) .$
In this article, using functions whose derivatives absolute values are
\varphi $-convex and quasi-$\varphi $-convex, we obtained new inequalities
releted to the right and left side of Hermite-Hadamard inequality. In
particular if $\varphi =0$ is taken as, our results obtained reduce to the
Hermite-Hadamard type inequality for classical convex functions.
Throughout this study, we always assume that $K=\left[ a,a+e^{i\varphi }(b-a
\right] $ and $0\leq \varphi \leq \frac{\pi }{2}$ the interval, unless
otherwise specified.
We shall start with the following refinements of the Hermite-Hadamard
inequality for $\varphi -$convex functions. Firstly, we give the following
results connected with the right part of (\ref{2}):
\begin{theorem}
\label{tt2} Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on
K^{0}$. If $\left\vert f^{\prime }\right\vert $ is $\varphi -$convex
function on the interval of real numbers $K^{0}$ (the interior of K) and
a,b\in K$ with $a<a+e^{i\varphi }(b-a)$. Then, the following inequality
holds
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+e^{i\varphi }(b-a))}{2}\right\vert \notag \\
&& \label{3} \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}\left[ \left\vert f^{\prime
}(a)\right\vert +\left\vert f^{\prime }(b)\right\vert \right] . \notag
\end{eqnarray}
\begin{proof}
Since $K$ is $\varphi -$convex with respect to $\varphi $, for every $t\in
\left[ 0,1\right] $, we have $a+te^{i\theta }(b-a)\in K$. Integrating by
parts implies tha
\begin{eqnarray}
&&\int_{0}^{1}(1-2t)f^{\prime }(a+te^{i\varphi }(b-a))dt \notag \\
&& \label{4} \\
&=&\left[ \frac{(1-2t)f(a+te^{i\varphi }(b-a))}{e^{i\varphi }(b-a)}\right]
_{0}^{1}+\frac{2}{e^{i\varphi }(b-a)}\int_{0}^{1}f(a+te^{i\varphi }(b-a))dt
\notag \\
&& \notag \\
&=&-\frac{f(a)+f(a+e^{i\varphi }(b-a))}{e^{i\varphi }(b-a)}+\frac{2}
e^{2i\varphi }(b-a)^{2}}\int_{a}^{a+e^{i\varphi }(b-a)}f(x)dx \notag
\end{eqnarray
By $\varphi -$convexity\ of\ $\left\vert f^{\prime }\right\vert $ and (\re
{4}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x))dx-\frac{f(a)+f(a+e^{i\varphi }(b-a))}{2}\right\vert \\
&=&\frac{e^{i\varphi }(b-a)}{2}\left\vert \int_{0}^{1}(1-2t)f^{\prime
}(a+te^{i\varphi }(b-a))dt\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\int_{0}^{1}\left\vert 1-2t\right\vert
\left[ (1-t)\left\vert f^{\prime }(a)\right\vert +t\left\vert f^{\prime
}(b)\right\vert \right] dt \\
&=&\frac{e^{i\varphi }(b-a)}{8}\left[ \left\vert f^{\prime }(a)\right\vert
+\left\vert f^{\prime }(b)\right\vert \right] .
\end{eqnarray*
which completes the proof.
\end{proof}
\end{theorem}
\begin{theorem}
\label{tt3}Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on
K^{0}$. Assume $p\in
\mathbb{R}
$ with $p>1$. If $\left\vert f^{\prime }\right\vert ^{p/p-1}$ is $\varphi -
convex function on the interval of real numbers $K^{0}$ (the interior of $K
) and $a,b\in K$ with $a<a+e^{i\varphi }(b-a)$. Then, the following
inequality holds
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+e^{i\varphi }(b-a))}{2}\right\vert \notag \\
&& \label{5} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left( \frac
\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}}{2}\right) ^{\frac{p-1}{p}} \notag
\end{eqnarray}
\begin{proof}
From H\"{o}lder's inequality and by using (\ref{4}) in the proof of Theorem
\ref{tt2}, we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x))dx-\frac{f(a)+f(a+e^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\int_{0}^{1}\left\vert 1-2t\right\vert
\left\vert f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\left( \int_{0}^{1}\left\vert
1-2t\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert ^{\frac{p}{p-1}}dt\right) ^
\frac{p-1}{p}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\left( \int_{0}^{1}\left\vert
1-2t\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left[
(1-t)\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+t\left\vert
f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right] dt\right) ^{\frac{p-1}{p}}
\\
&=&\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left( \frac{\left\vert
f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}}{2}\right) ^{\frac{p-1}{p}}.
\end{eqnarray*
This implies ineqyality (\ref{5}).
\end{proof}
\end{theorem}
Now, we give the following results connected with the left part of (\ref{2}):
\begin{theorem}
\label{tt4} Under the assumptions of Theorem \ref{tt2}. Then the following
inequality holds
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert
\notag \\
&& \label{6} \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}\left[ \left\vert f^{\prime
}(a)\right\vert +\left\vert f^{\prime }(b)\right\vert \right] . \notag
\end{eqnarray}
\begin{proof}
Since $K$ is $\varphi -$convex with respect to $\varphi $, for every $t\in
\left[ 0,1\right] $, we have $a+te^{i\theta }(b-a)\in K$. Integrating by
parts implies tha
\begin{eqnarray}
&&\int_{0}^{\frac{1}{2}}tf^{\prime }(a+te^{i\varphi }(b-a))dt+\int_{\frac{1}
2}}^{1}(t-1)f^{\prime }(a+te^{i\varphi }(b-a))dt \notag \\
&& \label{7} \\
&=&\left[ \frac{tf(a+te^{i\varphi }(b-a))}{e^{i\varphi }(b-a)}\right] _{0}^
\frac{1}{2}}+\left[ \frac{(t-1)f(a+te^{i\varphi }(b-a))}{e^{i\varphi }(b-a)
\right] _{\frac{1}{2}}^{1} \notag \\
&&-\frac{1}{e^{i\varphi }(b-a)}\int_{0}^{1}f(a+te^{i\varphi }(b-a))dt \notag
\\
&& \notag \\
&=&\frac{1}{e^{i\varphi }(b-a)}f\left( \frac{2a+e^{i\varphi }(b-a)}{2
\right) -\frac{1}{e^{2i\varphi }(b-a)^{2}}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dt. \notag
\end{eqnarray
By $\varphi -$convexity of $\left\vert f^{\prime }\right\vert $, we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime
}(a+te^{i\varphi }(b-a))\right\vert dt+\int_{\frac{1}{2}}^{1}(1-t)\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt\right] \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left[
(1-t)\left\vert f^{\prime }(a)\right\vert +t\left\vert f^{\prime
}(b)\right\vert \right] dt+\int_{\frac{1}{2}}^{1}(1-t)\left[ (1-t)\left\vert
f^{\prime }(a)\right\vert +t\left\vert f^{\prime }(b)\right\vert \right] d
\right] \\
&\leq &e^{i\varphi }(b-a)\left[ \frac{\left\vert f^{\prime }(a)\right\vert
+\left\vert f^{\prime }(b)\right\vert }{8}\right] .
\end{eqnarray*
The proof is completed.
\end{proof}
\end{theorem}
\begin{theorem}
\label{tt5} Under the assumptions of Theorem \ref{tt3}. Then the following
inequality holds
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert
\notag \\
&& \label{8} \\
&\leq &\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p
}\left[ \left( 3\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1
}+\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{
}}+\left( \left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+3\left\vert
f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\right] .
\notag
\end{eqnarray}
\begin{proof}
From H\"{o}lder's inequality and by using (\ref{7}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f(\frac{2a+e^{i\varphi }(b-a)}{2})\right\vert \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime
}(a+te^{i\varphi }(b-a))\right\vert dt+\int_{\frac{1}{2}}^{1}(1-t)\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt\right] \\
&\leq &e^{i\varphi }(b-a)\left( \int_{0}^{\frac{1}{2}}t^{p}dt\right) ^{\frac
1}{p}}\left( \int_{0}^{\frac{1}{2}}\left\vert f^{\prime }(a+te^{i\varphi
}(b-a))\right\vert ^{\frac{p}{p-1}}dt\right) ^{\frac{p-1}{p}} \\
&&+e^{i\varphi }(b-a)\left( \int_{\frac{1}{2}}^{1}\left( 1-t\right)
^{p}dt\right) ^{\frac{1}{p}}\left( \int_{\frac{1}{2}}^{1}\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert ^{\frac{p}{p-1}}dt\right) ^
\frac{p-1}{p}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2^{1+\frac{1}{p}}(p+1)^{\frac{1}{p}}}\left[
\int_{0}^{\frac{1}{2}}\left[ (1-t)\left\vert f^{\prime }(a)\right\vert ^
\frac{p}{p-1}}+t\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right]
dt\right] ^{\frac{p-1}{p}} \\
&&+\frac{e^{i\varphi }(b-a)}{2^{1+\frac{1}{p}}(p+1)^{\frac{1}{p}}}\left[
\int_{\frac{1}{2}}^{1}\left[ (1-t)\left\vert f^{\prime }(a)\right\vert ^
\frac{p}{p-1}}+t\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right]
dt\right] ^{\frac{p-1}{p}} \\
&=&\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p}
\left[ \left( 3\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1
}+\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{
}}+\left( \left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+3\left\vert
f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\right]
\end{eqnarray*
which completes the proof.
\end{proof}
\end{theorem}
\begin{theorem}
\label{tt6} Under the assumptions of Theorem \ref{tt2}. Then, the following
inequality holds
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f(\frac{2a+e^{i\varphi }(b-a)}{2})\right\vert \notag \\
&& \label{9} \\
&\leq &\frac{e^{i\varphi }(b-a)}{4}\left( \frac{4}{p+1}\right) ^{\frac{1}{p}
\left[ \left\vert f^{\prime }(a)\right\vert +\left\vert f^{\prime
}(b)\right\vert \right] . \notag
\end{eqnarray}
\begin{proof}
We consider the inequality (\ref{8}) i.
\begin{eqnarray}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f(\frac{2a+e^{i\varphi }(b-a)}{2})\right\vert \notag \\
&& \label{10} \\
&\leq &\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p
}\left[ \left( 3\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1
}+\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{
}}+\left( \left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+3\left\vert
f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\right] .
\notag
\end{eqnarray
Let $a_{1}=3\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}$,
b_{1}=\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}$,
a_{2}=\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}$,
b_{2}=3\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}$. Here
0<\left( p-1\right) /p<1$, for $p>1$. Using the fact that
\begin{equation*}
\sum\limits_{k=1}^{n}\left( a_{k}+b_{k}\right) ^{s}\leq
\sum_{k=1}^{n}a_{k}^{s}+\sum_{k=1}^{n}b_{k}^{s},
\end{equation*
for $\left( 0\leq s<1\right) $, $a_{1},a_{2},...,a_{n}\geq 0$,
b_{1},b_{2},...,b_{n}\geq 0$, we obtai
\begin{eqnarray*}
&&\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p}
\left[ \left( 3\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1
}+\left\vert f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{
}}+\left( \left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}}+3\left\vert
f^{\prime }(b)\right\vert ^{\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\right] \\
&\leq &\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p
}(3^{\frac{p-1}{p}}+1)\left[ \left\vert f^{\prime }(a)\right\vert
+\left\vert f^{\prime }(b)\right\vert \right] \\
&\leq &\frac{e^{i\varphi }(b-a)}{16}\left( \frac{4}{p+1}\right) ^{\frac{1}{p
}4\left[ \left\vert f^{\prime }(a)\right\vert +\left\vert f^{\prime
}(b)\right\vert \right]
\end{eqnarray*
which completed proof.
\end{proof}
\end{theorem}
\begin{theorem}
\label{z} Let $f:\rightarrow (0,\infty )$ be a differentiable mapping on
K^{0}$. Assume $q\in
\mathbb{R}
$ with $q\geq 1$. If $\left\vert f^{\prime }\right\vert ^{q}$ is $\varphi -
convex function on the interval of real numbers $K^{0}$ (the interior of $K
) and $a,b\in K$ with $a<a+e^{i\varphi }(b-a)$. Then the following
inequality holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert \\
&& \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}\left[ \left( \frac{2\left\vert f^{\prime
}(a)\right\vert ^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^
\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }(a)\right\vert
^{q}+2\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^{\frac{1}{q}
\right] .
\end{eqnarray*}
\begin{proof}
From H\"{o}lder's inequality and by using (\ref{7}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f(\frac{2a+e^{i\varphi }(b-a)}{2})\right\vert \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime
}(a+te^{i\varphi }(b-a))\right\vert dt+\int_{\frac{1}{2}}^{1}(1-t)\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt\right] \\
&\leq &e^{i\varphi }(b-a)\left( \int_{0}^{\frac{1}{2}}tdt\right) ^{\frac{1}{
}}\left( \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime }(a+te^{i\varphi
}(b-a))\right\vert ^{q}dt\right) ^{\frac{1}{q}} \\
&&+e^{i\varphi }(b-a)\left( \int_{\frac{1}{2}}^{1}\left( 1-t\right)
dt\right) ^{\frac{1}{p}}\left( \int_{\frac{1}{2}}^{1}\left( 1-t\right)
\left\vert f^{\prime }(a+te^{i\varphi }(b-a))\right\vert ^{q}dt\right) ^
\frac{1}{q}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{8^{\frac{1}{p}}}\left[ \int_{0}^{\frac{1}{2
}t\left[ (1-t)\left\vert f^{\prime }(a)\right\vert ^{q}+t\left\vert
f^{\prime }(b)\right\vert ^{q}\right] dt\right] ^{\frac{1}{q}} \\
&&+\frac{e^{i\varphi }(b-a)}{8^{\frac{1}{p}}}\left[ \int_{\frac{1}{2
}^{1}\left( 1-t\right) \left[ (1-t)\left\vert f^{\prime }(a)\right\vert
^{q}+t\left\vert f^{\prime }(b)\right\vert ^{q}\right] dt\right] ^{\frac{1}{
}} \\
&=&\frac{e^{i\varphi }(b-a)}{8}\left[ \left( \frac{2\left\vert f^{\prime
}(a)\right\vert ^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^
\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }(a)\right\vert
^{q}+2\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^{\frac{1}{q}
\right] .
\end{eqnarray*
The proof is completed.
\end{proof}
\end{theorem}
\begin{theorem}
Under the assumptions of Theorem \ref{z}. Then the following inequality
holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert \\
&& \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}(\frac{2^{\frac{1}{q}}+1}{3^{\frac{1}{q}}
)\left[ \left\vert f^{\prime }(a)\right\vert +\left\vert f^{\prime
}(b)\right\vert \right] .
\end{eqnarray*}
\begin{proof}
We consider the inequality (\ref{8}) i.
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f(\frac{2a+e^{i\varphi }(b-a)}{2})\right\vert \\
&& \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}\left[ \left( \frac{2\left\vert f^{\prime
}(a)\right\vert ^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^
\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }(a)\right\vert
^{q}+2\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^{\frac{1}{q}
\right] .
\end{eqnarray*
Let $a_{1}=2\left\vert f^{\prime }(a)\right\vert ^{q}/3$, $b_{1}=\left\vert
f^{\prime }(b)\right\vert ^{q}/3$, $a_{2}=\left\vert f^{\prime
}(a)\right\vert ^{q}/3$, $b_{2}=2\left\vert f^{\prime }(b)\right\vert ^{q}/3
. Here $0<1/q<1$, for $q\geq 1$. Using the fact tha
\begin{equation*}
\sum\limits_{k=1}^{n}\left( a_{k}+b_{k}\right) ^{s}\leq
\sum_{k=1}^{n}a_{k}^{s}+\sum_{k=1}^{n}b_{k}^{s},
\end{equation*
for $\left( 0\leq s<1\right) $, $a_{1},a_{2},...,a_{n}\geq 0$,
b_{1},b_{2},...,b_{n}\geq 0$, we obtai
\begin{eqnarray*}
&&\frac{e^{i\varphi }(b-a)}{8}\left[ \left( \frac{2\left\vert f^{\prime
}(a)\right\vert ^{q}+\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^
\frac{1}{q}}+\left( \frac{\left\vert f^{\prime }(a)\right\vert
^{q}+2\left\vert f^{\prime }(b)\right\vert ^{q}}{3}\right) ^{\frac{1}{q}
\right] \\
&\leq &\frac{e^{i\varphi }(b-a)}{8}(\frac{2^{\frac{1}{q}}+1}{3^{\frac{1}{q}}
)\left[ \left\vert f^{\prime }(a)\right\vert +\left\vert f^{\prime
}(b)\right\vert \right] .
\end{eqnarray*
This concludes the proof.
\end{proof}
\end{theorem}
\section{Hermite-Hadamard type inequalities for quasi $\protect\varphi -
convex functions}
In this section, we prove some new inequalities of Hermite-Hadamard for
quasi $\varphi -$convex function as follows:
\begin{theorem}
Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on $K^{0}$. If
\left\vert f^{\prime }\right\vert $ is quasi $\varphi -$convex function on
the interval of real numbers $K^{0}$ (the interior of $K$) and $a,b\in K$
with $a<$ $a+e^{i\varphi }(b-a)$. Then the following inequality holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&& \\
&\leq &\frac{e^{i\varphi }(b-a)}{4}\max \{\left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert \}.
\end{eqnarray*}
\begin{proof}
By quasi $\varphi -$convexity of $\left\vert f^{\prime }\right\vert $ and by
using (\ref{4}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\int_{0}^{1}\left\vert (1-2t)\right\vert
\left\vert f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\max \{\left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert
\}\int_{0}^{1}\left\vert (1-2t)\right\vert dt \\
&\leq &\frac{e^{i\varphi }(b-a)}{4}\max \{\left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert \}
\end{eqnarray*
which completes the proof.
\end{proof}
\end{theorem}
\begin{theorem}
Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on $K^{0}
.Assume $p\in
\mathbb{R}
$ with $p>1$. If $\left\vert f^{\prime }\right\vert ^{p/p-1}$ is quasi
\varphi -$convex function on the interval of real numbers $K^{0}$ (the
interior of $K$) and $a,b\in K$ with $a<e^{i\varphi }(b-a)$. Then the
following inequality holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left[ \max
\{\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}},\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}\}\right] ^{\frac{p-1}{p}}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
By quasi $\varphi -$convexity of $\left\vert f^{\prime }\right\vert ^{p/p-1}$
and by using (\ref{4}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a)+f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\int_{0}^{1}\left\vert (1-2t)\right\vert
\left\vert f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\left( \int_{0}^{1}\left\vert
(1-2t)\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert ^{\frac{p-1}{p}}dt\right) ^
\frac{p}{p-1}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2}\left( \int_{0}^{1}\left\vert
(1-2t)\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\max
\{\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}},\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}\}dt\right) ^{\frac{p}{p-1}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left[ \max
\{\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}},\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}\}\right] ^{\frac{p-1}{p}}
\end{eqnarray*
which completes the proof.
\end{proof}
\begin{theorem}
Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on $K^{0}$ . If
$\left\vert f^{\prime }\right\vert $ is quasi $\varphi -$convex function on
the interval of real numbers $K^{0}$ (the interior of $K$) and $a,b\in K$
with $a<$ $a+e^{i\varphi }(b-a)$. Then the following inequality holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&& \\
&\leq &\frac{e^{i\varphi }(b-a)}{4}\max \{\left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert \}.
\end{eqnarray*}
\begin{proof}
By quasi $\varphi -$convexity of $\left\vert f^{\prime }\right\vert $ and by
using (\ref{7}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-f\left( \frac{2a+e^{i\varphi }(b-a)}{2}\right) \right\vert \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime
}(a+te^{i\varphi }(b-a))\right\vert dt+\int_{\frac{1}{2}}^{1}(1-t)\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt\right] \\
&\leq &e^{i\varphi }(b-a)\max \{\left\vert f^{\prime }(a)\right\vert
,\left\vert f^{\prime }(b)\right\vert \}\left[ \int_{0}^{\frac{1}{2
}tdt+\int_{\frac{1}{2}}^{1}(1-t)dt\right] \\
&\leq &\frac{e^{i\varphi }(b-a)}{4}\max \{\left\vert f^{\prime
}(a)\right\vert ,\left\vert f^{\prime }(b)\right\vert \}.
\end{eqnarray*
This concludes the proof.
\end{proof}
\end{theorem}
\begin{theorem}
Let $f:K\rightarrow (0,\infty )$ be a differentiable mapping on $K^{0}
.Assume $p\in
\mathbb{R}
$ with $p>1$. If $\left\vert f^{\prime }\right\vert ^{p/p-1}$ is quasi
\varphi -$convex function on the interval of real numbers $K^{0}$ (the
interior of $K$) and $a,b\in K$ with $a<e^{i\varphi }(b-a)$. Then the
following inequality holds
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left[ \max
\{\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}},\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}\}\right] ^{\frac{p-1}{p}}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
By quasi $\varphi -$convexity of $\left\vert f^{\prime }\right\vert ^{p/p-1}$
and by using (\ref{4}), we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{e^{i\varphi }(b-a)}\int_{a}^{a+e^{i\varphi
}(b-a)}f(x)dx-\frac{f(a+te^{i\varphi }(b-a))}{2}\right\vert \\
&\leq &e^{i\varphi }(b-a)\left[ \int_{0}^{\frac{1}{2}}t\left\vert f^{\prime
}(a+te^{i\varphi }(b-a))\right\vert dt+\int_{\frac{1}{2}}^{1}(1-t)\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert dt\right] \\
&\leq &e^{i\varphi }(b-a)\left( \int_{0}^{\frac{1}{2}}t^{p}dt\right) ^{\frac
1}{p}}\left( \int_{0}^{\frac{1}{2}}\left\vert f^{\prime }(a+te^{i\varphi
}(b-a))\right\vert ^{\frac{p}{p-1}}dt\right) ^{\frac{p-1}{p}} \\
&&+e^{i\varphi }(b-a)\left( \int_{\frac{1}{2}}^{1}\left( 1-t\right)
^{p}dt\right) ^{\frac{1}{p}}\left( \int_{\frac{1}{2}}^{1}\left\vert
f^{\prime }(a+te^{i\varphi }(b-a))\right\vert ^{\frac{p}{p-1}}dt\right) ^
\frac{p-1}{p}} \\
&\leq &\frac{e^{i\varphi }(b-a)}{2(p+1)^{\frac{1}{p}}}\left[ \max
\{\left\vert f^{\prime }(a)\right\vert ^{\frac{p}{p-1}},\left\vert f^{\prime
}(b)\right\vert ^{\frac{p}{p-1}}\}\right] ^{\frac{p-1}{p}}
\end{eqnarray*
which completes the proof.
\end{proof}
|
{
"timestamp": "2013-04-03T02:01:46",
"yymm": "1203",
"arxiv_id": "1203.2282",
"language": "en",
"url": "https://arxiv.org/abs/1203.2282"
}
|
\section{Introduction}
Clustering of inertial particles in turbulent flows is relevant for meteorology
and engineering, as well as fundamental research. It is believed to play a
crucial role in rain-drop formation \cite{falkovich_nature02}, as well as in the aggregation of
proto-planetesimals in Keplerian accretion disks \cite{bracco_pof99}. The physical mechanism which
originates such clustering is indeed rather simple: particles heavier than the
fluid in which they are transported experience inertial forces which expel
them from vortices; particles lighter than the fluid are attracted into
vortical structures, for similar reasons \cite{Squires1991, cencini_jot06, Bec2005}. In realistic flows, however,
particles are advected by the small scale vortical structures of turbulent
flows: these have highly non-trivial statistical features, resulting in a
complex clustering process which is still far from being completely understood.
From the point of view of applications, the properties of concentration and
distribution of inertial particles play a crucial role in engineering and
for the design of industrial processes involving combustion and mixing \cite{Warnatz2006,Rouson2001, Sbrizzai2006}. Suspensions of particles in viscoelastic fluids are used in many products of commercial and industrial relevance \cite{barnes2003}.
In this paper we investigate, by means of direct numerical simulations
of a turbulent flow, how the clustering properties of a dilute
suspension of inertial particles can be affected by the addition of
small amounts of polymer additives.
The effects induced by polymers on turbulent flows
are themselves of enormous relevance. It is enough to mention the celebrated
drag reduction effect which occurs in pipe flows \cite{Lumley1969},
or the recently discovered elastic turbulence regime \cite{gs_nature00}.
Polymers have striking effects also on Lagrangian properties of the flow.
In particular it has been shown that polymer addition in turbulent flows
reduces the chaoticity of Lagrangian trajectories \cite{bcm_prl03}
and affects acceleration of fluid tracers \cite{cmxb_njp08}.
Conversely in the elastic turbulence regime polymers are
able to generate Lagrangian chaos in flows at vanishing
Reynolds number, which would be non chaotic in the Newtonian case
\cite{gs_nature01,bcm_prl03}.
Here we show that the addition of polymers in a turbulent flow has
important effects on the statistical properties of inertial particles
which can result in both an increase or a decrease of the clustering.
An example of the effect of polymers on clustering
is shown in Fig.~\ref{fig1} which represents the distribution
of an ensemble of inertial particles in a turbulent flow
before and after the introduction of polymers. It is evident, already
at the qualitative level of Fig.~\ref{fig1}, that polymers are able
to change the statistical distribution of particles.
We show that these effects can be understood and quantified
in terms of the Lyapunov exponents of inertial particles,
which are very sensitive to the presence of polymers.
Previous systematic investigations of inertial particle dynamics
in Newtonian turbulent flows \cite{calzavarini_jfm08}
and stochastic flows \cite{bec_pof03}
have shown that clustering (quantified by means of the Lyapunov Dimension of particle attractor) is maximum when the particle relaxation time
is of the order of the shortest characteristic time of the flow.
\begin{figure}
\includegraphics[width=4.2cm]{fig1a.eps}
\includegraphics[width=4.2cm]{fig1b.eps}
\includegraphics[width=4.2cm]{fig1c.eps}
\includegraphics[width=4.2cm]{fig1d.eps}
\caption{Section on plane $z=0$ of the distribution of heavy particles
with $\tau_S=0.035$ (upper panels) and light particles with $\tau_S=0.03$ (lower panels) in statistically stationary conditions in a Newtonian
flow (left) and a viscoelastic flow at $\rm Wi=1$ (right).
Both flows are forced with the same
forcing ${\bf f}({\bf x},t)$ $\delta$-correlated in time and localized
on large scales. Numerical simulations are done by a pseudo-spectral,
fully dealiased code at resolution $256^3$. For the viscoelastic simulations,
a small diffusive term is added to (\ref{eq:4}) to prevent numerical
instabilities \cite{sb_jnnfm95}.}
\label{fig1}
\end{figure}
We consider the case of a dilute suspension of small inertial particles,
in which the effects of the disturbance flow induced by the particles
can be neglected. The dynamics of the suspension is hence
modeled by an ensemble of non-interacting point particles,
which experience viscous drag and added mass forces.
The equation of motion of each particle reads \cite{maxey_pof83}:
\begin{eqnarray}
{d {\bm x} \over dt}&=& \bm v
\label{eq:1} \\
{d {\bm v} \over dt}&=&-{1 \over \tau_S}\left[\bm v - \bm u(\bm x(t),t)\right]+
\beta {d {\bm u} \over dt}
\label{eq:2}
\end{eqnarray}
where $\tau_S=a^2/(3\beta\nu)$ is the Stokes relaxation time,
$a$ is the particle radius, $\beta=3\rho_f/(\rho_f+2\rho_p)$
($\rho_p$ and $\rho_f$ representing particle and fluid densities
respectively) and $\nu$ is the kinematic viscosity of the fluid
(replaced by the total viscosity $\nu_T$ in a viscoelastic fluid, see below).
Light (heavy) particles correspond to $\beta>1$ ($\beta<1$).
In this work we consider the two extreme cases of
very light particles (e.g. air bubble in water) for which $\beta=3$
and very heavy particles with $\beta=0$.
We define the Stokes number as $\rm St=\tau_S \lambda^ 0_1$, where
$\lambda^0_1$ is the maximum Lyapunov exponent of neutral Lagrangian tracers
(i.e. $\rm St=0$ particles) in the flow. With this definition,
maximum clustering is obtained for $\rm St \simeq 0.1$
\cite{bec_pof03, calzavarini_jfm08}.
The viscoelastic flow ${\bf u}({\bf x},t)$ in which the particles are
suspended can be described by standard viscoelastic models,
such as the Oldroyd-B model or the nonlinear FENE-P model,
which accounts for the finite extensibility of polymers.
In spite of their simplicity, these models are able to reproduce
many relevant properties of dilute polymer solutions, including
turbulent drag reduction~\cite{sbh_pof97,bcm_pre05} and elastic turbulence
phenomenology~\cite{bbbcm_pre08}.
Here we choose the Oldroyd-B model~\cite{bird87}, in which the
coupled dynamics of the velocity field ${\bf u}({\bf x},t)$
and the polymer conformation tensor $\sigma({\bf x},t)$
(which is proportional to local square polymer elongation)
reads:
\begin{eqnarray}
{\partial \bm u \over \partial t}+\bm u\cdot\bm\nabla\bm u &=&
-\bm\nabla p+\nu\nabla^2\bm u+{2\nu\gamma \over \tau_p}\bm\nabla\cdot\sigma
+{\bm f}
\label{eq:3}\\
{\partial \sigma \over \partial t}+\bm u\cdot\nabla\sigma &=&
(\nabla\bm u)^T\cdot\sigma+\sigma\cdot(\nabla\bm u)-
{2 \over \tau_p}(\sigma-\mathbb{I})
\label{eq:4}
\end{eqnarray}
The total viscosity of the solution $\nu_T=\nu (1+\gamma)$
is written in terms of the kinematic viscosity of the solvent $\nu$
and the zero-shear contribution of the polymer $\gamma$ which
is proportional to the polymer concentration.
The polymer time $\tau_p$ represents the longest relaxation time to the equilibrium configuration ($\sigma=\mathbb{I}$ in dimensionless units).
Viscoelasticity of the turbulent flow is parametrized by the
Weissenberg number $\rm Wi$, the ratio between $\tau_p$ and a characteristic
time of the flow. Here we use $\rm Wi=\tau_p \lambda_1^N$
where $\lambda_1^N$ is the Lagrangian Lyapunov exponent of the Newtonian flow,
before the addition of polymers (i.e. (\ref{eq:3}) with $\gamma=0$). We
stress that $\lambda_1^0$ introduced above refers instead to the specific flow
that carries the suspension and it clearly depends on $\rm Wi$.
Therefore $\lambda_1^N\equiv\lambda_1^0|_{Wi=0}$.
\begin{table}[b]
\label{tab:1}
\begin{tabular}{c c c c c}
$\rm Wi$ &$\varepsilon_f$& $\varepsilon_\nu$&$u_{\rm rms}$ &$\lambda^0_1$ \\
\hline
0 & 0.28 & 0.28 & 0.76 & 1.36 \\
0.5 & 0.28 & 0.18 & 0.73 & 1.08 \\
1 & 0.28 & 0.092 & 0.68 & 0.75 \\
\end{tabular}
\caption{Parameters for the Newtonian and viscoelastic simulations.
The Weissenberg number $\rm Wi$, energy input $\varepsilon_f$, viscous
dissipation rate $\varepsilon_\nu$, rms velocity $u_{rms}$ and Lagrangian
Lyapunov exponent $\lambda_1^0$ of the carrier flow are shown.
In both viscoelastic runs an additional dissipative term was added on
polymers (see text), with coefficient $\nu_p=2.3\times 10^3$ }
\label{table1}
\end{table}
In the following we discuss results obtained by integrating numerically
the viscoelastic model (\ref{eq:3}-\ref{eq:4}) at high resolution for different
values of $\rm Wi$ (see Table~\ref{table1}). The flow is sustained by a
stochastic Gaussian forcing ${\bf f}({\bf x},t)$
$\delta$-correlated in time and localized on large scales. Fluid equations were integrated by means of a standard, fully dealiased, pseudo spectral code, on a cubic, triple-periodic domain with 256 grid points per side.
When the flow reaches a turbulent, statistically stationary state, different
families (i.e. with different values of parameters $\beta$ and $\tau_S$)
of inertial particles are injected, with initial homogeneous
distribution in space, and their motion integrated according to
(\ref{eq:1}-\ref{eq:2}). For each value of $\rm Wi$, we
integrated the motion of $1024$ particles for each of $21$ values
of $\tau_S$ and two values of $\beta$, namely very heavy particles with
$\beta=0$ and "bubbles" with $\beta=3$.
As an effect of inertia the distribution of particles does not remain
homogeneous and evolves to a fractal set dynamically evolving with
the flow, such as the examples shown in Fig.~\ref{fig1}. In the language
of dynamical systems, the equations (\ref{eq:1}-\ref{eq:2}) for particle motion
represent a dissipative system whose chaotic trajectories evolve to
a fractal attractor (which evolves in time following the flow).
A quantitative measure of clustering at small scales
is therefore obtained by measuring the fractal dimension of the attractor
(for each family of particles) using the Lyapunov dimension
\cite{bec_pof03,bec_pof06} defined in terms of Lyapunov exponents as
$D_L=K+\sum_{i=1}^{K} \lambda_i/|\lambda_{K+1}|$ where $K$ is the
largest integer for which $\sum_{i=1}^{K} \lambda_i \ge 0$ \cite{ccv2010}.
Since the space distribution of the particles is the projection
of the attractor on the sub-space of particle positions,
the fractal dimension of clusters
is given by $\min(D_L, 3)$~\cite{sy97,hk97},
provided that the projection is generic
(for a discussion on this issue see e.g. \cite{bch07}).
This implies that $D_L<3$ gives fractal
distributions of dimension $D_L$, while $D_L>3$ corresponds to space-filling
configurations, which however can be non-homogeneous.
\begin{figure}
\includegraphics[width=8.0cm]{fig2a.eps}
\includegraphics[width=8.0cm]{fig2b.eps}
\caption{
Lyapunov dimension for light (upper panel) and heavy (lower panel) particles
plotted as a function of $\tau_S$. Different lines correspond
to the different Weissenberg numbers: $\rm Wi=0$ (squares), $\rm Wi=0.5$ (circles) and
$\rm Wi=1.0$ (triangles).
}
\label{fig2}
\end{figure}
In Fig.~\ref{fig2} we plot the fractal dimensions for both heavy
and light particles as a function of $\tau_S$ for the three simulations
at different $\rm Wi$.
It is evident that the addition of polymer changes substantially
the clustering properties of the particles, both increasing
$D_L$ and reducing $D_L$ depending on value of $\tau_S$.
Figure~\ref{fig1} shows examples of clustering reduction, for heavy and light particles respectively. The upper panels refer to heavy particles ($\beta=0$) with $\tau_S=0.035$, while the bottom ones are extracted from a simulation with $\beta=3$ and $\tau_S=0.03$. Both values of Stokes time are, for the Newtonian flow, on the left of the minimum in $D_L$. As a consequence, polymers produce a reduction of clustering. Such effect is more visible for light particles. A possible reason for this difference will be discussed further on.
The mechanism at the basis of this effect is not trivial and
is a consequence of the change induced by the polymers on the
small-scale properties of the turbulent flow.
In Fig.\ref{fig3} we plot the energy spectra for the different $\rm Wi$ numbers.
The effect of polymers is evident in the high-wavenumber range
where velocity fluctuations are clearly suppressed, resulting in a
depletion of the energy spectrum, while large-scale fluctuations are
unaffected.
Indeed one can expect that only the fastest eddies of the flow,
i.e. those whose eddy turn-over time $\tau_\ell$ is shorter that
the polymer relaxation time $\tau_p$,
can produce a significant elongation of polymers.
The elastic feedback therefore affects only small scales $\ell$
with $\tau_\ell < \tau_p$.
Conversely, large scales exhibit the same phenomenology of a
Newtonian flow, characterized by a turbulent cascade with a
constant energy flux equal to the energy input rate $\varepsilon_f$.
The turbulent cascade proceeds almost unaffected by the presence
of polymers down to the Lumley scale $\ell_L$,
whose eddy turn-over time equals the polymer relaxation time.
A dimensional estimate, based on the Kolmogorov scaling for
the typical velocity $u_\ell\sim\varepsilon_f^{1/3}\ell^{1/3}$
and turn-over time $\tau_\ell=\ell/u_\ell \sim\varepsilon_f^{-1/3}\ell^{2/3}$
of an eddy of size $\ell$, gives $\ell_L=\tau_p^{3/2}\varepsilon_f^{1/2}$.
Polymers would therefore affect only the small scales $\ell < \ell_L$.
Our results are in qualitative agreement with this picture: the
$\rm Wi=0.5$ spectrum differs from the Newtonian
one only for $k\gtrsim 8$, while
at $\rm Wi=1$ polymers are active over a larger range of scales.
The reduction of kinetic energy at small scales, due to the transfer of
energy to the polymers, is accompanied by a reduction of the viscous
dissipation $\varepsilon_\nu=\nu\langle (\nabla u)^2\rangle$
at fixed energy input $\varepsilon_f$,
as can be seen from Table~\ref{table1} and in the inset of
Fig.\ref{fig3}.
This phenomenon has been previously observed both in forced and decaying
simulations of statistically homogeneous and isotropic turbulence
(see, e.g., ~\cite{dcbp05,pmp06}).
\begin{figure}
\includegraphics[width=9.0cm]{fig3.eps}
\caption{
Energy spectra for the Newtonian case ${\rm Wi}=0$ (squares) and for the
viscoelastic ones ${\rm Wi}=0.5$ (circles) and ${\rm Wi}=1$ (triangles). The
depletion due to polymer feed-back is evident on large wavenumbers, while the
larger scales are unaffected. The effect of polymers extends at lower
wave-numbers as $\rm Wi$ increases. Inset: viscous energy dissipation $\varepsilon_\nu$ during
a typical time interval in the stationary simulations, for the Newtonian (solid
line), ${\rm Wi}=0.5$ (dashed line) and ${\rm Wi}=1$ (dash-dot) flows. The
decrease in $\varepsilon_\nu$ with ${\rm Wi}$ is evident, as well as the reduction in
fluctuations.
}
\label{fig3}
\end{figure}
The suppression of small-scale motions caused by polymers
has major consequences also on the Lagrangian statistics.
It is responsible of the reduction of chaoticity
of Lagrangian trajectories~\cite{boffetta_prl03}.
Indeed the chaoticity of the flow is directly related to its
stretching efficiency via the Lyapunov exponents.
When polymers are stretched, the elastic stress tensor produces a negative
feed-back on small scale stretching, thus reducing the degree of chaoticity of
the flow \cite{boffetta_prl03,balkovsky_pre01}. This effect is clearly
observable in the decrease of the Lagrangian Lyapunov exponent of the flow
at increasing polymer elasticity (see the inset of Fig.\ref{fig4}).
It is worth to notice that, because of polymers counteraction,
the Lyapunov exponent of the resulting viscoelastic flow is smaller than $\tau_p^{-1}$.
In other words, the $\rm Wi$ number computed a posteriori (i.e. after polymer injection)
is always smaller than unity.
This is not in contrast with the hypothesis that polymers have a strong active effect on the flow
mainly when they are stretched, i.e. above the so-called coil-stretch transition, which is
expected to happen around $\rm Wi\simeq 1$ \cite{balkovsky_prl00}.
Indeed, the Lyapunov exponent simply provides a measure of the
average stretching in a chaotic flow. One should bear in mind that large fluctuations
of the stretching rates (and therefore strong viscoelastic effects)
can occur also when $\rm Wi \lesssim 1$.
\begin{figure}
\includegraphics[clip=true,keepaspectratio,width=8.0cm]{fig4.eps}
\caption{Comparison between the Cram\'er functions of the stretching rate
$\gamma_1$ computed at ${\rm Wi}=0$ (solid line),${\rm Wi}=0.5$ (dashed
line),${\rm Wi}=1$ (dash-dot). Inset: first Lagrangian Lyapunov exponent
$\lambda^0_1$ (circles) and width $\mu$ (squares) of the Cram\'er function (see
text) as a function of $\rm Wi$. The Lyapunov exponents are compared with the
Newtonian value $\lambda_1^N$.
}
\label{fig4}
\end{figure}
Detailed information on the fluctuations of the stretching rates can be
obtained from the statistics of the Finite Time Lyapunov Exponents (FTLE)
$\gamma_i$.
The FTLE are defined via the exponential growth rate during a finite time
$T$ of an infinitesimal $M$-dimensional volume as
$\sum_{i=1}^M\gamma_i=(1/T)\ln[V^M(T)/V^M(0)]$ \cite{ccv2010}.
From the definition of the Lyapunov exponents it follows that
$\lim_{T\rightarrow\infty}\gamma^T_i=\lambda_i$. A large deviation approach
suggests that the probability density function (PDF) of the largest
stretching rate $\gamma_1$ measured over a long time
$T\gg 1/\lambda_1$ takes the asymptotic form
$P_{T}(\gamma_1)\sim N(t)\exp[-H(\gamma_1)T]$ where the Cram\'er function
$H(\gamma_1)$ is convex and obeys the conditions
$H(\lambda_1)=0$, $H^\prime(\lambda_1)=0$.
We computed the Cram\'er function for the Lagrangian FTLE for the Newtonian case
and the two viscoelastic cases.
In the inset of Fig.~\ref{fig4} we plotted the average of the stretching rates
(i.e., the first Lagrangian Lyapunov exponent of the flow $\lambda_1^0$) and
the rescaled variance $\mu=T\langle\gamma_1^2\rangle$, for the three values of
$\rm Wi$ that we considered. The decrease of the Lyapunov exponent (rescaled
with the Newtonian value $\lambda_1^N$ for comparison) gives a measure of the
decrease in the chaoticity of the flow, due to the action of Polymers. On the
other hand, we also observe a decrease in the relative variance
$\mu/\Lambda_1^0$, which implies that polymer feedback induces also a reduction
of the fluctuations of of stretching rates. Inspection of the main panel of
Fig.~\ref{fig4}, however, shows that fluctuations are not reduced uniformly.
Indeed, the shape of $P(\gamma_1)$ changes when polymers are added.
As is evident in Fig.\ref{fig4}, elasticity has the effect of raising the
right branch of the Cram\'er function, while the left one is comparatively
less affected.
Given the definition of $H(\gamma_1)$, this amounts to a relative
suppression of positive fluctuations in the stretching rate: as one could
expect, polymers have a larger (negative) feedback on events of larger
stretching.
\begin{figure}
\includegraphics[width=8.0cm]{fig5a.eps}
\includegraphics[width=8.0cm]{fig5b.eps}
\caption{
Lyapunov dimension for light (upper panel) and heavy (lower panel) particles
plotted as a function of ${\rm St=\tau_S \lambda^0_1}$. Different lines correspond
to the different Weissenberg numbers with symbols as in Fig.~\ref{fig2}.
}
\label{fig5}
\end{figure}
The effect of polymers on Lyapunov exponents and the Lagrangian nature of the latter suggests to introduce the dimensionless Stokes
number defined as $\rm St=\tau_S \lambda^0_1$ which depends on $\rm Wi$ by the
dependence of $\lambda^0_1$ shown in Fig.~\ref{fig4}. Figure~\ref{fig5}
shows the Lyapunov dimension $D_L$ for both heavy and light particles
as a function of $\rm St$.
It is evident that, with respect to Fig.~\ref{fig2},
the collapse of the curves at different $\rm Wi$ is improved.
In particular, the minimum
of the fractal dimension (which corresponds to maximum clustering) occurs
almost for the same $\rm St$ number.
Still, some differences are observable, in particular for small $\rm St$ in the case
of light particles.
This can be understood by the following argument.
Bubbles, at variance with heavy particles,
have tendency to concentrate on filaments of high vorticity.
Indeed, while the minimal dimension for heavy particles is about $2.5$
(at $\rm St \simeq 0.1$), for light particles at maximal clustering
it becomes as small as $1.26$.
Vortex filaments correspond to quasi-one-dimensional
regions of intense stretching, in the direction
longitudinal to the vortex, which give major contributions to the right
tail of the Cram\'er function.
As shown in Fig.~\ref{fig4}, the effects of polymers on the distribution
of Lyapunov exponent is more evident in this region of strong fluctuations,
where the distribution does not rescale with $\lambda^0_1$. It is therefore
not surprising that also the effects on clustering of light particles
cannot be completely absorbed in the rescaling of $\tau_S$ with the mean
stretching rate $\lambda^0_1$.
As the fractal dimension is given by a combination of the Lyapunov exponents,
in order to better understand the differences on light and heavy particles,
in Fig.\ref{fig6} we show the first three Lyapunov exponents as a function
of $\rm St$. The first observation is that bubbles, at variance with heavy
particles, exhibit negative values of $\lambda_2$, consistently with the
lower value of $D_L$ and the tendency of light particles to concentrate
towards vortex filaments.
The first Lyapunov exponent decreases with $\rm Wi$ for any value of $\rm St$,
thus indicating that the phenomenon of chaos reduction, already discussed
for the case of Lagrangian tracers, is generic also for inertial particles.
On the contrary, the second Lyapunov exponent shows a different behavior
for light and heavy particles: it increases for the former but slightly
decreases for the latter. Figure~\ref{fig6} shows that the effect of
polymers is not a simple rescaling of the Lyapunov spectrum, which
would trivially keep the dimension $D_L$ unchanged. From this point
of view, the almost perfect rescaling of the Lyapunov dimensions
shown in Fig.~\ref{fig5} is quite surprising and arises as the result
of compensations of different effects.
\begin{figure}
\includegraphics[width=8.0cm]{fig6a.eps}
\includegraphics[width=8.0cm]{fig6b.eps}
\caption{The first three Lyapunov exponents for light ($\beta=3$, upper panel)
and heavy ($\beta=0$, lower panel) particles, at different $\rm Wi$.
Continuous, dashed and dotted lines represent the first, second and
third Lyapunov exponents, while symbols correspond to different $\rm Wi$
as in Fig.~\ref{fig2}.
}
\label{fig6}
\end{figure}
In conclusion, we investigated the clustering properties of inertial
(heavy and light) particles in a turbulent viscoelastic fluid.
The main effect of polymers on turbulent
flows is to counteract small-scale fluctuations and to reduce its chaoticity.
Quantitatively, this results in a decrease in the first Lyapunov exponent of
the flow, which, in turn, affects clustering of inertial particles.
The latter can be quantified by means of the fractal (Lyapunov) dimension of
particle distributions. Although the effects of polymers on the particle
Lyapunov exponents are complex and qualitatively different for light
and heavy particles, the overall effect on fractal dimension is relatively
simple and can be rephrased in the rescaling of the characteristic
time of the flow.
Indeed, when particle inertia is parametrized by the Stokes number $\rm St$
defined with the Lyapunov time of the flow, one can approximately rescale
the curves $D_L(\rm St)$ at all $\rm Wi$.
In contrast, as polymers do not affect large scale properties of the
flow, a parametrization of particle inertia based on integral time scales
would not show a collapse of the curves $D_L(\rm St)$ at different $\rm Wi$.
As a consequence, any prediction of
particle clustering in turbulent polymeric solutions requires an accurate
estimate of small scale stretching rates.
We acknowledge support from the the EU COST Action MP0806.
\input{paper.bbl}
\end{document}
|
{
"timestamp": "2012-03-09T02:03:34",
"yymm": "1203",
"arxiv_id": "1203.1838",
"language": "en",
"url": "https://arxiv.org/abs/1203.1838"
}
|
\section{Introduction}
{\em Graphs} in this paper are allowed to have loops and multiple edges.
A graph is a {\em minor} of another if the first can be obtained from
a subgraph of the second by contracting edges. An {\em $H$ minor}
is a minor isomorphic to $H$.
A graph $G$ is {\em apex} if it has a vertex $v$ such that
$G\backslash v$ is planar.
(We use $\backslash$ for deletion.)
J\o rgensen~\cite{Jor} made the following beautiful conjecture.
\begin{conjecture}
\label{con:jorgensen}
\showlabel{con:jorgensen}
Every $6$-connected graph with no $K_6$ minor is apex.
\end{conjecture}
\noindent
This is related to Hadwiger's conjecture~\cite{Had}, the following.
\begin{conjecture}
\label{con:hadwiger}
\showlabel{con:hadwiger}
For every integer $t\ge1$, if a loopless graph has no $K_t$ minor, then it
is $(t-1)$-colorable.
\end{conjecture}
Hadwiger's conjecture is known for $t\le6$.
For $t=6$ it has been proven in~\cite{RobSeyThoHad} by showing
that a minimal counterexample to Hadwiger's conjecture for $t=6$
is apex.
The proof uses an earlier result of Mader~\cite{MadHomkrit}
that every minimal counterexample to Conjecture~\ref{con:hadwiger}
is $6$-connected.
Thus Conjecture~\ref{con:jorgensen}, if true, would give more
structural information.
Furthermore, the structure of all graphs with no $K_6$ minor is not known,
and appears complicated and difficult.
On the other hand, Conjecture~\ref{con:jorgensen} provides a nice
and clean statement for $6$-connected graphs.
Unfortunately, it, too, appears to be a difficult problem.
In this paper we prove Conjecture~\ref{con:jorgensen} for all
sufficiently large graphs, as follows.
\begin{theorem}
\label{thm:largejorgensen}
\showlabel{thm:largejorgensen}
There exists an absolute constant $N$ such that every $6$-connected
graph on at least $N$ vertices with no $K_6$ minor is apex.
\end{theorem}
The second and third author recently announced
a generalization~\cite{NorTho}
of Theorem~\ref{thm:largejorgensen}, where $6$ is
replaced by an arbitrary integer $t$.
The result states that for every integer $t$ there exists
an integer $N_t$ such that every $t$-connected graph on at least
$N_t$ vertices with no $K_t$ minor has a set of at most $t-5$ vertices
whose deletion makes the graph planar.
The proof follows a different strategy, but makes use of several
ideas developed in this paper and its companion~\cite{KawNorThoWolbdtw}.
We use a number of results from the Graph Minor series of Robertson
and Seymour, and also three results of our own that are proved
in~\cite{KawNorThoWolbdtw}.
The first of those is a version of Theorem~\ref{thm:largejorgensen}
for graphs of bounded tree-width, the following.
(We will not define tree-width here, because it is sufficiently well-known,
and because we do not need the concept {\it per se},
only several theorems that use it.)
\begin{theorem}
\label{thm:bdedtwjorgensen}
\showlabel{thm:bdedtwjorgensen}
For every integer $w$ there exists an integer $N$
such that every $6$-connected
graph of tree-width at most $w$ on at least $N$ vertices and with no
$K_6$ minor is apex.
\end{theorem}
Theorem~\ref{thm:bdedtwjorgensen} reduces the proof of
Theorem~\ref{thm:largejorgensen} to graphs of large tree-width.
By a result of Robertson and Seymour~\cite{RobSeyGM5} those graphs
have a large grid minor.
However, for our purposes it is more convenient to work with walls
instead.
Let $h\ge2$ be even.
An {\em elementary wall of height $h$} has vertex-set
$$\{(x,y): 0\le x\le 2h+1, 0\le y\le h\}-\{(0,0),(2h+1,h)\}$$
and an edge between any vertices $(x,y)$ and $(x',y')$ if either
\par\hangindent0pt\mytextindent{$\bullet$} $|x-x'|=1$ and $y=y'$, or
\par\hangindent0pt\mytextindent{$\bullet$} $x=x'$, $|y-y'|=1$ and $x$ and $\max\{y,y'\}$
have the same parity.
\noindent
Figure~\ref{fig:wall} shows an elementary wall of height $4$.
A {\em wall of height $h$} is a subdivision of an elementary wall
of height $h$. The result of~\cite{RobSeyGM5}
(see also~\cite{DieGorJenTho,ReedBCC,RobSeyThoQuickPlanar})
can be restated as follows.
\nocite{DieGorJenTho}
\nocite{ReedBCC}
\nocite{RobSeyThoQuickPlanar}
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = 1]{wall.eps}
\end{center}
\caption{
An elementary wall of height $4$.
}
\label{fig:wall}
\end{figure}
\showfiglabel{fig:wall}
\begin{theorem}
\label{thm:grid}
\showlabel{thm:grid}
For every even integer $h\ge2$ there exists an integer $w$ such that
every graph of tree-width at least $w$ has a subgraph isomorphic to
a wall of height $h$.
\end{theorem}
The {\em perimeter} of a wall is the cycle that bounds the infinite face
when the wall is drawn as in Figure~\ref{fig:wall}.
Now let $C$ be the perimeter of a wall $H$ in a graph $G$.
The {\em compass of $H$ in $G$} is the restriction of $G$ to $X$, where
$X$ is the union of $V(C)$ and the vertex-set of the unique component
of $G\backslash V(C)$ that contains a vertex of $H$.
Thus $H$ is a subgraph of its compass, and the compass is connected.
A wall $H$ with perimeter $C$ in a graph $G$ is {\em planar} if its compass
can be drawn in the plane with $C$ bounding the infinite face.
In Section~\ref{sec:planarwall} we prove the following.
\begin{theorem}
\label{thm:plwall}
\showlabel{thm:plwall}
For every even integer $t\ge2$ there exists an even integer $h\ge2$
such that if a $5$-connected graph $G$ with no $K_6$ minor
has a wall of height at least $h$, then either it is apex, or
has a planar wall of height $t$.
\end{theorem}
Actually, in the proof of Theorem~\ref{thm:plwall} we need
Lemma~\ref{lem:pinwheel} that is proved in~\cite{KawNorThoWolbdtw}. The lemma says
that if a $5$-connected graph with no $K_6$ minor has a
subgraph isomorphic to subdivision
of a pinwheel with sufficiently many vanes
(see Figure~\ref{fig:pinwheel}), then it is apex.
By Theorem~\ref{thm:plwall} we may assume that our graph $G$ has
an arbitrarily large planar wall $H$. Let $C$ be the perimeter of
$H$, and let $K$ be the compass of $H$. Then $C$ separates $G$ into
$K$ and another graph, say $J$, such that $K\cup J=G$,
$V(K)\cap V(J)=V(C)$ and $E(K)\cap E(J)=\emptyset$.
Next we study the graph $J$. Since the order of the vertices on $C$
is important, we are lead to the notion of a ``society", introduced
by Robertson and Seymour in~\cite{RobSeyGM9}.
Let $\Omega$ be a cyclic permutation of the elements of some set;
we denote this set by $V(\Omega)$.
A {\em society} is a pair $(G,\Omega)$, where $G$ is a graph, and $\Omega$
is a cyclic permutation with $V(\Omega)\subseteq V(G)$.
Now let $J$ be as above, and let $\Omega$ be one of the cyclic permutations
of $V(C)$ determined by the order of vertices on $C$.
Then $(J,\Omega)$ is a society that is of primary interest to us.
We call it the {\em anticompass society of $H$ in $G$}.
We say that $(G,\Omega,\Omega_0)$ is a {\em neighborhood} if $G$ is a graph
and $\Omega,\Omega_0$ are cyclic permutations, where both $V(\Omega)$
and $V(\Omega_0)$ are subsets of $V(G)$.
Let $\Sigma$ be a plane, with some orientation called ``clockwise."
We say that a neighborhood $(G,\Omega,\Omega_0)$ is {\em rural} if $G$ has
a drawing $\Gamma$ in $\Sigma$ without crossings (so $G$ is planar)
and there are closed discs $\Delta_0\subseteq \Delta\subseteq \Sigma$, such
that
\par\hangindent0pt\mytextindent{(i)} the drawing $\Gamma$ uses no point of $\Sigma$ outside $\Delta$,
and none in the interior of $\Delta_0$, and
\par\hangindent0pt\mytextindent{(ii)} for $v\in V(G)$, the point of $\Sigma$ representing $v$ in the
drawing $\Gamma$ lies in $bd(\Delta)$ (respectively, $bd(\Delta_0)$)
if and only if $v\in V(\Omega)$ (respectively, $v\in V(\Omega_0))$, and
the cyclic permutation of $V(\Omega)$ (respectively, $V(\Omega_0))$
obtained from the clockwise orientation of $bd(\Delta)$ (respectively,
$bd(\Delta_0)$) coincides (in the natural sense) with $\Omega$ (respectively,
$\Omega_0$).
\noindent We call $(\Sigma,\Gamma,\Delta,\Delta_0)$ a {\em presentation}
of $(G,\Omega,\Omega_0)$.
Let $(G_1,\Omega,\Omega_0)$ be a neighborhood, let $(G_0,\Omega_0)$
be a society with $V(G_0)\cap V(G_1)=V(\Omega_0)$, and let $G=G_0\cup G_1$.
Then $(G,\Omega)$ is a society, and we say that $(G,\Omega)$
is the {\em composition} of the society $(G_0,\Omega_0)$ with
the neighborhood $(G_1,\Omega,\Omega_0)$.
If the neighborhood $(G_1,\Omega,\Omega_0)$ is rural, then
we say that $(G_0,\Omega_0)$ is a {\em planar truncation} of $(G,\Omega)$.
We say that a society $(G,\Omega)$ is {\em $k$-cosmopolitan}, where $k\ge 0$
is an integer, if for every planar truncation $(G_0,\Omega_0)$ of
$(G,\Omega)$ at least $k$ vertices in $V(\Omega_0)$ have at least two
neighbors in $V(G_0)$.
At the end of Section~\ref{sec:planarwall} we deduce
\begin{theorem}
\label{cosmopolitan}
\showlabel{cosmopolitan}
For every integer $k\ge 1$ there exists an even integer $t\ge 2$ such that
if $G$ is a simple graph of minimum degree at least six and $H$ is a
planar wall of height $t$ in $G$, then the anticompass society of
$H$ in $G$ is $k$-cosmopolitan.
\end{theorem}
For a fixed presentation $(\Sigma,\Gamma,\Delta,\Delta_0)$
of a neighborhood $(G,\Omega,\Omega_0)$ and an integer $s\ge0$ we define an
{\em $s$-nest} for $(\Sigma,\Gamma,\Delta,\Delta_0)$
to be a sequence $(C_1,C_2,\ldots,C_s)$
of pairwise disjoint cycles of $G$ such that
$\Delta_0\subseteq\Delta_1\subseteq\cdots\subseteq\Delta_s\subseteq\Delta$,
where $\Delta_i$ denotes the closed disk in $\Sigma$ bounded by
the image under $\Gamma$ of $C_i$.
We say that a society $(G,\Omega)$ is {\em $s$-nested} if it
is the composition of a society $(G_1,\Omega_0)$ with a rural
neighborhood $(G_2,\Omega,\Omega_0)$ that has
an $s$-nest for some presentation of $(G_2,\Omega,\Omega_0)$.
Let $\Omega$ be a cyclic permutation.
For $x\in V(\Omega)$ we denote the image of $x$ under $\Omega$
by $\Omega(x)$.
If $X\subseteq V(\Omega)$, then we denote by $\Omega|X$ the restriction
of $\Omega$ to $X$.
That is, $\Omega|X$ is the permutation $\Omega'$ defined by saying
that $V(\Omega')=X$ and $\Omega'(x)$ is the first term of the
sequence $\Omega(x),\Omega(\Omega(x)),\ldots$ which belongs to $X$.
Let $v_1,v_2,\ldots,v_k\in V(\Omega)$ be distinct.
We say that $(v_1,v_2,\ldots,v_k)$ is {\em clockwise} in $\Omega$ (or simply
{\em clockwise} when $\Omega$ is understood from context) if
$\Omega'(v_{i-1})=v_{i}$ for all $i=1,2,\ldots,k$, where $v_0$ means $v_k$
and $\Omega'=\Omega|\{v_1,v_2,\ldots,v_k\}$.
For $u,v\in V(\Omega)$ we define $u\Omega v$ as the set of all
$x\in V(\Omega)$ such that either $x=u$ or $x=v$ or $(u,x,v)$ is clockwise
in $\Omega$.
A separation of a graph is a pair $(A,B)$ such that $A\cup B=V(G)$
and there is no edge with one end in $A-B$ and the other end in $B-A$.
The order of $(A,B)$ is $|A\cap B|$.
We say that a society $(G,\Omega)$ is {\em $k$-connected}
if there is no separation $(A,B)$ of $G$ of order at most $k-1$
with $V(\Omega)\subseteq A$ and $B-A\ne\emptyset$.
A {\em bump} in $(G,\Omega)$ is a path in $G$ with at least one edge,
both ends in $V(\Omega)$ and otherwise disjoint from $V(\Omega)$.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = 0.75]{bumps.eps}
\end{center}
\caption{
(a),(b) A turtle. (c),(d) A gridlet. (e),(f) A separated doublecross.
}
\label{fig:bumps}
\end{figure}
\showfiglabel{fig:bumps}
Let $(G,\Omega)$ be a society and let $(u_1,u_2,v_1,v_2,u_3,v_3)$ be
clockwise in $\Omega$. For $i=1,2$ let $P_i$ be a bump in $G$
with ends $u_i$ and $v_i$,
and let $L$ be either a bump with ends $u_3$ and $v_3$,
or the union of two internally disjoint bumps, one with
ends $u_3$ and $x\in u_3\Omega v_3$ and the other with ends $v_3$ and
$y\in u_3\Omega v_3$. In the former case let $Z=\emptyset$, and in
the latter case let $Z$ be the subinterval of $u_3\Omega v_3$ with
ends $x$ and $y$, including its ends.
Assume that $P_1,P_2,L$ are pairwise disjoint.
Let $q_1,q_2\in V(P_1)\cup V(P_2)\cup v_3\Omega u_3-\{u_3,v_3\}$ be
distinct such that neither of the sets $V(P_1)\cup v_3\Omega u_1$,
$V(P_2)\cup v_2\Omega u_3$ includes both $q_1$ and $q_2$.
Let $Q_1$ and $Q_2$ be two not necessarily disjoint paths with
one end in $u_3\Omega v_3-Z-\{u_3,v_3\}$ and the other end $q_1$ and $q_2$,
respectively, both internally disjoint from $V(P_1\cup P_2\cup L)\cup V(\Omega)$.
In those circumstances we
say that $P_1\cup P_2\cup L\cup Q_1\cup Q_2$ is a {\em turtle}
in $(G,\Omega)$.
We say that $P_1,P_2$ are the {\em legs},
$L$ is the {\em neck}, and $Q_1\cup Q_2$ is the {\em body} of the turtle. (See Figure~\ref{fig:bumps}(a),(b).)
Let $(G,\Omega)$ be a society, let $(u_1,u_2,u_3, v_1,v_2,v_3)$ be
clockwise in $\Omega$, and let $P_1,P_2,P_3$ be disjoint bumps
such that $P_i$ has ends $u_i$ and $v_i$. In those
circumstances we say that $P_1,P_2,P_3$ are {\em three crossed paths}
in $(G,\Omega)$.
Let $(G,\Omega)$ be a society, and let
$u_1,u_2,u_3,u_4,v_1,v_2,v_3,v_4\in V(\Omega)$ be such that either
$(u_1,u_2,u_3,v_2,u_4,v_1,v_4,v_3)$ or
$(u_1,u_2,u_3,u_4,v_2,v_1,v_4,v_3)$ or
$(u_1,u_2,u_3,v_2=u_4,v_1,v_4,v_3)$ is clockwise.
For $i=1,2,3,4$ let $P_i$ be a bump with ends $u_i$ and $v_i$ such that
these bumps are pairwise disjoint, except possibly for $v_2=u_4$.
In those circumstances we say that $P_1,P_2,P_3,P_4$ is
a {\em gridlet}. (See Figure~\ref{fig:bumps}(c),(d).)
Let $(G,\Omega)$ be a society and let $(u_1,u_2,v_1,v_2,u_3,u_4,v_3,v_4)$
be clockwise or counter-clockwise in $\Omega$.
For $i=1,2,3,4$ let $P_i$ be a bump with ends $u_i$ and $v_i$ such that
these bumps are pairwise disjoint, and let $P_5$ be a path with one
end in $V(P_1)\cup v_4\Omega u_2-\{u_2,v_1,v_4\}$, the other
end in $V(P_3)\cup v_2\Omega u_4-\{v_2,v_3,u_4\}$, and otherwise
disjoint from $P_1\cup P_2\cup P_3\cup P_4$.
In those circumstances we say that $P_1,P_2,\ldots,P_5$ is
a {\em separated doublecross}.(See Figure~\ref{fig:bumps}(e),(f).)
A society $(G,\Omega)$ is {\em rural} if $G$ can be drawn in a disk
with $V(\Omega)$ drawn on the boundary of the disk in the order given
by $\Omega$.
A society $(G,\Omega)$ is {\em nearly rural} if there exists a vertex
$v\in V(G)$ such that the society $(G\backslash v,\Omega\backslash v)$
obtained from $(G,\Omega)$ by deleting $v$ is rural.
In Sections~\ref{sec:leap}--\ref{sec:lack} we prove the following.
The proof strategy is explained in Section~\ref{sec:transactions}.
It uses a couple of theorems from~\cite{RobSeyGM9} and
Theorem~\ref{thm:leap} that we prove in Section~\ref{sec:leap}.
\begin{theorem}
\label{thm:society}
\showlabel{thm:society}
There exists an integer $k\ge1$ such that for every
integer $s\ge0$ and every $6$-connected $s$-nested
$k$-cosmopolitan society $(G,\Omega)$ either $(G,\Omega)$ is nearly rural,
or $G$ has a triangle $C$ such that
$(G\backslash E(C),\Omega)$ is rural,
or $(G,\Omega)$ has an $s$-nested planar
truncation that has a turtle, three crossed paths, a gridlet,
or a separated doublecross.
\end{theorem}
Finally, we need to convert a turtle, three crossed paths,
gridlet and a separated double-cross
into a $K_6$ minor.
Let $G$ be a $6$-connected graph, let $H$ be a sufficiently high planar wall
in $G$, and let $(J,\Omega)$ be the anticompass society of $H$ in $G$.
We wish to apply to Theorem~\ref{thm:society} to $(J,\Omega)$. We can,
in fact, assume that $H$ is a subgraph of a larger planar wall $H'$ that
includes $s$ concentric cycles $C_1,C_2,\ldots,C_s$
surrounding $H$ and disjoint from $H$,
for some suitable integer $s$, and hence $(J,\Omega)$ is
$s$-nested.
Theorem~\ref{thm:society} guarantees a turtle
or paths in $(J,\Omega)$ forming three crossed paths, a gridlet,
or a separated double-cross, but it
does not say how the turtle or paths might intersect the cycles $C_i$.
In Section~\ref{sec:nest} we prove a theorem that says that the cycles and
the turtle (or paths) can be changed such that after possibly sacrificing
a lot of
the cycles, the remaining cycles and the new turtle (or paths)
intersect nicely.
Using that information it is then easy to find a $K_6$ minor in $G$.
We complete the proof of Theorem~\ref{thm:largejorgensen} in
Section~\ref{sec:turtle}.
\section{Finding a planar wall}
\label{sec:planarwall}
\showlabel{sec:planarwall}
Let a {\em pinwheel with four vanes} be the graph pictured in
Figure~\ref{fig:pinwheel}. We define a pinwheel with $k$ vanes analogously.
A graph $G$ is {\em internally $4$-connected} if it is simple, $3$-connected, has at least
five vertices, and for every separation $(A,B)$ of $G$ of
order three, one of $A,B$ induces a graph with at most three edges.
\begin{figure}[ht!]
\begin{center}
\leavevmode
\includegraphics[trim = 0mm 5mm 65mm 0mm, clip]{pinwheel2.eps}
\end{center}
\caption{
A pinwheel with four vanes.
}
\label{fig:pinwheel}
\end{figure}
\showfiglabel{fig:pinwheel}
The objective of this section is to prove the following theorem.
\begin{theorem}
\label{plsepwall}
\showlabel{plsepwall}
For every even integer $t\ge2$ there exists an even integer $h$ such that if
$H$ is a wall of height at least $h$ in an internally $4$-connected\ graph $G$, then either
\par\hangindent0pt\mytextindent{(1)}$G$ has a $K_6$ minor, or
\par\hangindent0pt\mytextindent{(2)}$G$ has a subgraph isomorphic to a subdivision of a
pinwheel with $t$ vanes, or
\par\hangindent0pt\mytextindent{(3)}$G$ has a planar wall of height $t$.
\end{theorem}
In the proof we will be using several results from~\cite{RobSeyGM13}.
Their statements require the following terminology:
distance function, $(l,m)$-star over $H$,
external $(l,m)$-star over $H$, subwall, dividing subwall,
flat subwall, cross over a wall.
We refer to~\cite{RobSeyGM13} for precise definitions, but we offer
the following informal descriptions.
The distance of two distinct vertices $s,t$ of a wall is the minimum number
of times a curve in the plane joining $s$ and $t$ intersects the drawing
of the wall, when the wall is drawn as in Figure~\ref{fig:wall}.
An $(l,m)$-star over a wall $H$ in $G$ is a subdivision of a star
with $l$ leaves such that only the leaves and possibly the center
belong to $H$, and the leaves are pairwise at distance at least $m$.
The star is external if the center does not belong to $H$.
A subwall of a wall is dividing if its perimeter separates the subwall from
the rest of the wall. A cross over a wall is a set of two disjoint paths
joining the diagonally opposite pairs of ``corners" of the wall,
the vertices represented by solid circles in Figure~\ref{fig:wall}.
A subwall $H$ is flat in $G$ if there is no cross $P,Q$ over $H$
such that $P\cup Q$ is a subgraph of the compass of $H$ in $G$.
We begin with the following easy lemma. We leave the proof to the reader.
\begin{lemma}
\label{usestar}
\showlabel{usestar}
For every integer $t$ there exist integers $l$ and $m$ such that
if a graph $G$ has a wall $H$ with an external $(l,m)$-star, then
it has a subgraph isomorphic to a subdivision of a pinwheel with $t$ vanes.
\end{lemma}
We need one more lemma, which follows immediately
from~\cite[Theorem~8.6]{RobSeyGM13}.
\begin{lemma}
\label{crossinwall}
\showlabel{crossinwall}
Every flat wall in an internally $4$-connected\ graph is planar.
\end{lemma}
\begin{figure}[ht!]
\begin{center}
\leavevmode
\includegraphics[scale = 1]{gridk6.ps}
\end{center}
\caption{
A $K_6$ minor in a grid with two crosses.
}
\label{fig:gridk6}
\end{figure}
\showfiglabel{fig:pinwheel}
\noindent
{\bf Proof of Theorem~\ref{plsepwall}}.
Let $t\ge1$ be given, let $l,m$ be as in Lemma~\ref{usestar},
let $p=6$, and let $k,r$ be as in~\cite[Theorem 9.2]{RobSeyGM13}.
If $h$ is sufficiently large, then $H$ has
$k+1$ subwalls of height at least $t$, pairwise at distance at least $r$.
If at least $k$ of these subwalls are non-dividing, then
by \cite[Theorem 9.2]{RobSeyGM13} $G$ either has a $K_6$ minor, or
an $(l,m)$-star over $H$, in which case it has a subgraph isomorphic
to a pinwheel with $t$ vanes by Lemma~\ref{usestar}. In either case
the theorem holds, and so we may assume that at least two of the
subwalls, say $H_1$ and $H_2$, are dividing.
We may assume that $H_1$ and $H_2$ are not planar,
for otherwise the theorem holds.
Let $i\in\{1,2\}$.
By Lemma~\ref{crossinwall} the wall $H_i$ is not flat, and hence
its compass has a cross $P_i\cup Q_i$.
Since the subwalls $H_1$ and $H_2$ are dividing, it follows that
the paths $P_1,Q_1,P_2,Q_2$ are pairwise disjoint.
Thus $G$ has a minor isomorphic to the graph shown in Figure~\ref{fig:gridk6},
but that graph has a minor isomorphic to a minor of $K_6$,
as indicated by the numbers in the figure.
Thus $G$ has a $K_6$ minor, and the theorem holds.~$\square$\bigskip
To deduce Theorem~\ref{thm:plwall} we need the following lemma, proved
in~\cite[Lemma~5.3]{KawNorThoWolbdtw}.
\begin{lemma}
\label{lem:pinwheel}
\showlabel{lem:pinwheel}
If a $5$-connected graph $G$ with no $K_6$ minor has a subdivision
isomorphic to a pinwheel with $20$ vanes, then $G$ is apex.
\end{lemma}
\noindent
{\bf Proof of Theorem~\ref{thm:plwall}.}
Let $t\ge2$ be an even integer. We may assume that $t\ge 20$.
Let $h$ be as in Theorem~\ref{plsepwall}, and let $G$ be a $5$-connected
graph with no $K_6$ minor. From Theorem~\ref{plsepwall} we deduce that $G$ either
satisfies the conclusion of Theorem~\ref{thm:plwall}, or has a
subdivision isomorphic to a pinwheel with $t$ vanes.
In the latter case the theorem follows from Lemma~\ref{lem:pinwheel}.~$\square$\bigskip
We need the following theorem of DeVos and Seymour~\cite{DevSeyExt3col}.
\begin{theorem}
\label{devosseymour}
\showlabel{devosseymour}
Let $(G,\Omega)$ be a rural society such that $G$ is a simple graph
and every vertex of $G$ not in $V(\Omega)$ has degree at least six.
Then $|V(G)|\le |V(\Omega)|^2/{12}+|V(\Omega)|/2+1$.
\end{theorem}
\noindent
{\bf Proof of Theorem~\ref{cosmopolitan}.}
Let $k\ge 1$ be an integer, and let $t$ be an even integer such that
if $W$ is the elementary wall of height $t$ and $|V(W)|\le\ell^2/12+\ell/2+1$, then
$\ell >6k-6$. Let $K$ be the compass of $H$ in $G$, let $(J,\Omega)$
be the anticompass society of $H$ in $G$, let $(G_0,\Omega_0)$ be a planar
truncation of $(J,\Omega)$, and let $\ell=|V(\Omega_0)|$.
Thus $(J,\Omega)$ is the composition of $(G_0,\Omega_0)$ with a rural
neighborhood $(G',\Omega,\Omega_0)$. Then $|V(H)|\le\ell^2/12 + \ell/2+1$
by Theorem~\ref{devosseymour} applied to the society $(K\cup G',\Omega_0)$,
and hence $\ell >6k-6$. Let $L$ be the graph obtained from $K\cup G'$
by adding
a new vertex $v$ and joining it to every vertex of $V(\Omega_0)$ and
by adding an edge joining every pair of nonadjacent vertices of $V(\Omega_0)$
that are consecutive in $\Omega_0$. Then $L$
is planar. Let $s$ be the number of vertices of $V(\Omega_0)$ with at least
two neighbors in $G_0$. Then all but $s$ vertices of $K\cup G'$ have
degree in $L$ at least six. Thus the sum of the degrees of vertices of
$L$ is at least $6|V(K\cup G')|-6s+\ell$. On the other hand, the sum of
the degrees is at most $6|V(L)|-12$, because $L$ is planar, and hence
$s\ge k$, as desired.~$\square$\bigskip
\input rural.tex
\input leap.tex
\section{Societies of bounded depth}
\label{sec:transactions}
\showlabel{sec:transactions}
Let $(G,\Omega)$ be a society. A {\em linear decomposition} of $(G,\Omega)$
is an enumeration $\{t_1,\dots,t_n\}$ of $V(\Omega)$ where
$(t_1,\dots, t_n)$ is clockwise, together with a family $(X_i:1\le i\le n)$
of subsets
of $V(G)$, with the following properties:
\par\hangindent0pt\mytextindent{(i)} $\bigcup (G[X_i]:1\le i\le n)=G$,
\par\hangindent0pt\mytextindent{(ii)} for $1\le i\le n$, $t_i\in X_i$, and
\par\hangindent0pt\mytextindent{(iii)} for $1\le i\le i'\le i''\le n$, $X_i\cap X_{i''}\subseteq X_{i'}$.
\noindent The {\em depth} of such a linear decomposition is
$$\max(|X_i\cap X_{i'}|:1\le i <i'\le n),$$
and the depth of $(G,\Omega)$ is the minimum depth of a linear decomposition
of $(G,\Omega)$.
Theorems~(6.1), (7.1) and (8.1) of~\cite{RobSeyGM9} imply the following.
\begin{theorem}\label{gm9}
\showlabel{gm9}
There exists an integer $d$ such that every $4$-connected society
$(G,\Omega)$ either has a separated doublecross,
three crossed paths or a leap of length
five, or some planar truncation of $(G,\Omega)$ has depth at most $d$.
\end{theorem}
In light of Theorems~\ref{thm:leap} and~\ref{gm9},
in the remainder of the paper we concentrate on societies of
bounded depth. We need a few definitions.
Let $(G,\Omega)$ be a society, let
$u_1,u_2,\ldots,u_{4t}$ be clockwise in $\Omega$, and let
$P_1,P_2,\ldots,P_{2t}$ be disjoint bumps in $G$ such that
for $i=1,2,\ldots,2t$
the path $P_{2i-1}$ has ends $u_{4i-3}$ and $u_{4i-1}$, and
the path $P_{2i}$ has ends $u_{4i-2}$ and $u_{4i}$.
In those circumstances we say that $(G,\Omega)$ has
{\em $t$ disjoint consecutive crosses}.
Now let $u_1,v_1,w_1,u_2,v_2,w_2,\ldots,u_t,v_t,w_t$ be clockwise
in $\Omega$, let
$x\in V(G)-\{u_1,v_1,\allowbreak w_1,\ldots,u_t,\allowbreak v_t,w_t\}$,
for $i=1,2,\ldots,t$ let $P_i$ be a path in
$G\backslash x$ with ends $u_i$ and $w_i$
and otherwise disjoint from $V(\Omega)$, let $Q_i$ be a path with
ends $x$ and $v_i$ and otherwise disjoint from $V(\Omega)$,
and assume that the paths $P_i$ and $Q_i$ are
pairwise disjoint, except that the paths $Q_i$ meet at $x$.
Let $W$ be the union of all the paths $P_i$ and $Q_i$.
We say that $W$ is
a {\em windmill with $t$ vanes}, and that the graph $P_i\cup Q_i$ is
a {\em vane} of the windmill.
Finally, let $u_1,u_2,\ldots,u_t$ and $v_1,v_2,\ldots,v_t$ be
vertices of $V(\Omega)$ such that for all $x_i\in\{u_i,v_i\}$
the sequence $x_1,x_2,\ldots,x_t$ is clockwise in $\Omega$.
Let $z_1,z_2\in V(G)-\{u_1,v_1,\ldots,u_t,v_t\}$ be distinct,
for $i=1,2,\ldots,t$ let $P_i$ be a path in $G\backslash z_2$ with ends $z_1$
and $u_i$ and otherwise disjoint from $V(\Omega)$, and let
$Q_i$ be a path in $G\backslash z_1$ with ends $z_2$
and $v_i$ and otherwise disjoint from $V(\Omega)$.
Assume that the paths $P_i$ and $Q_j$ are disjoint, except that the
$P_i$ share $z_1$, the $Q_i$ share $z_2$ and $P_i$ and $Q_i$ are allowed to
intersect.
Let $F$ be the union of all the paths $P_i$ and $Q_i$.
Then we say that $F$ is a {\em fan with $t$ blades}, and we
say that $P_i\cup Q_i$ is a {\em blade} of the fan.
The vertices $z_1$ and $z_2$ will be called the {\em hubs} of the fan.
In Section~\ref{usewars} we prove the following theorem.
\begin{theorem}
\label{thm1}
\showlabel{thm1}
For every two integers $d$ and $t$ there exists an integer $k$ such
that every $6$-connected%
\REMARK{We could lower 6 to 5, but the proof will have to be changed}
$k$-cosmopolitan society $(G,\Omega)$ of depth at most $d$
contains one of the following:
\par\hangindent0pt\mytextindent{(1)}$t$ disjoint consecutive crosses, or
\par\hangindent0pt\mytextindent{(2)}a windmill with $t$ vanes, or
\par\hangindent0pt\mytextindent{(3)}a fan with $t$ blades.
\end{theorem}
Unfortunately, windmills and fans are nearly rural, and so for our
application we need to improve Theorem~\ref{thm1}. We need more definitions.
Let $x,u_i,v_i,w_i,P_i,Q_i$ be as in the definition of a windmill $W$
with $t$ vanes,
let $a,b,c,d\in V(G)$ be such that
$u_1,v_1,w_1,\ldots,u_t,v_t,w_t, a, b, c, d$ is clockwise
in $\Omega$, and let $(P,Q)$ be a cross disjoint from $W$ whose
paths have ends in $\{a,b,c,d\}$. In those circumstances we say that
$W\cup P\cup Q$ is a {\em windmill with $t$ vanes and a cross}.
Now let $u_i,v_i,P_i,Q_i$ be as in the definition of a fan $F$ with
$t$ blades, and let $a,b,c,d\in V(\Omega)$ be such that
all $x_i\in\{u_i,v_i\}$
the sequence $x_1,x_2,\ldots,x_t,a,b,c,d$ is clockwise in $\Omega$.
Let $(P,Q)$ be a cross disjoint from $F$ whose
paths have ends in $\{a,b,c,d\}$. In those circumstances we say that
$W\cup P\cup Q$ is a {\em fan with $t$ blades and a cross}.
Let $z_1,z_2,u_i,v_i,P_i,Q_i$ be as in the definition of a fan $F$ with
$t$ blades, and let $a_1,b_1,c_1,a_2,\allowbreak b_2,c_2\in V(G)$ be such that
all $x_i\in\{u_i,v_i\}$
the sequence $x_1,x_2,\ldots,x_t,a_1,b_1,c_1,\allowbreak a_2,b_2,c_2$
is clockwise in $\Omega$, except that we permit $c_1=a_2$.
For $i=1,2$ let $L_i$ be a path in $G\backslash V(F)$
with ends $a_i$ and $c_i$ and otherwise disjoint from $V(\Omega)$,
and let $S_i$ be a path with ends $z_i$ and $b_i$ and otherwise disjoint
from $V(F)\cup V(\Omega)$.
If the paths $L_1,L_2,S_1,S_2$ are pairwise disjoint, except possibly
for $L_1$ intersecting $L_2$ at $c_1=a_2$, then we say that
$F\cup L_1\cup L_2\cup S_1\cup S_2$ is a {\em fan with $t$ blades and two jumps}.
Now let $u_i,v_i,P_i,Q_i$ be as in the definition of a fan $F$ with
$t+1$ blades, and let $a,b\in V(\Omega)$ be such that
all $x_i\in\{u_i,v_i\}$
the sequence $x_1,x_2,\ldots,x_t,a,x_{t+1},b$ is clockwise in $\Omega$.
Let $P$ be a path in $G\backslash V(F)$ with ends $a$ and $b$, and
otherwise disjoint from $V(F)$. We say that $F\cup P$ is a
{\em fan with $t$ blades and a jump}.
In Section~\ref{sec:lack} we improve Theorem~\ref{thm1} as follows.
\begin{theorem}
\label{thm2}
\showlabel{thm2}
For every two integers $d$ and $t$ there exists an integer $k$ such
that every $6$-connected
$k$-cosmopolitan society $(G,\Omega)$ of depth at most $d$
is either nearly rural, or
contains one of the following:
\par\hangindent0pt\mytextindent{(1)}$t$ disjoint consecutive crosses, or
\par\hangindent0pt\mytextindent{(2)}a windmill with $t$ vanes and a cross, or
\par\hangindent0pt\mytextindent{(3)}a fan with $t$ blades and a cross, or
\par\hangindent0pt\mytextindent{(4)}a fan with $t$ blades and a jump, or
\par\hangindent0pt\mytextindent{(5)}a fan with $t$ blades and two jumps.
\end{theorem}
\noindent
For $t=4$ each of the above outcomes gives a turtle, and hence we
have the following immediate corollary.
\begin{corollary}
\label{thm3}
\showlabel{thm3}
For every integer $d$ there exists an integer $k$ such
that every $6$-connected
$k$-cosmopolitan society $(G,\Omega)$ of depth at most $d$
is either nearly rural, or has a turtle.
\end{corollary}
The next four sections are devoted to proofs of Theorems~\ref{thm1}
and~\ref{thm2}. The proof of Theorem~\ref{thm1} will be completed in
Section~\ref{sec:wars} and the proof of Theorem~\ref{thm2} will be
completed in Section~\ref{sec:lack}.
At that time we will be able to deduce Theorem~\ref{thm:society}.
\section{Crosses and goose bumps}
In this section we prove that a society $(G,\Omega)$ either
satisfies Theorem~\ref{thm1}, or it has
many disjoint bumps.
If $X$ is a set and $\Omega$ is a cyclic permutation,
we define $\Omega\backslash X$ to be $\Omega|(V(\Omega)-X)$.
Let $P_1,P_2,\ldots,P_k$ be a set of pairwise disjoint bumps in
$(G,\Omega)$, where $P_i$ has ends $u_i$ and $v_i$ and
$u_1,v_1,u_2,v_2,\ldots,u_k,v_k$ is clockwise in $\Omega$.
In those circumstances we say that $P_1,P_2,\ldots,P_k$ is a
{\em goose bump in $(G,\Omega)$ of strength $k$.}
\begin{lemma}
\label{EPbumps}
\showlabel{EPbumps}
Let $b,d$ and $t$ be positive integers, and let $(G,\Omega)$ be
a society of depth at most $d$. Then either $(G,\Omega)$ has a goose
bump of strength $b$, or there is a set $X\subseteq V(G)$
of size at most $(b-1)d$
such that the society $(G\backslash X, \Omega\backslash X)$ has no bump.
\end{lemma}
{\noindent\bf Proof. }
Let $(t_1,t_2,\dots,t_n)$ and $(X_1,X_2,\dots, X_n)$ be a linear
decomposition of $(G,\Omega)$ of depth at most $d$, and for $i=1,2,\dots, n-1$
let $Y_i=X_i\cap X_{i+1}$. If $P$ is a bump in $(G,\Omega)$, then the
axioms of a linear decomposition imply that
$$I_P:=\{i\in \{1,2,\dots, n-1\}: Y_i\cap V(P)\ne \emptyset\}$$
is a nonempty subinterval of $\{1,2,\dots, n-1\}$. It follows that
either there exist bumps $P_1,P_2,\dots, P_b$ such that $I_{P_1},
I_{P_2},\dots, I_{P_b}$ are pairwise disjoint, or there exists a set
$I\subseteq \{1,2,\dots, n-1\}$ of size at most $b-1$ such that
$I\cap I_P\ne\emptyset$ for every bump $P$. In the former case
$P_1,P_2,\dots, P_b$ is a desired goose bump, and in the latter
case the set $X:=\bigcup_{i\in I} Y_i$ is as desired.~$\square$\bigskip
The proof of the following lemma is similar and is omitted.
\begin{lemma}
\label{EPcrosses}
\showlabel{EPcrosses}
Let $t$ and $d$ be positive integers, and let $(G,\Omega)$ be
a society of depth at most $d$. Then either $(G,\Omega)$ has
$t$ disjoint consecutive crosses, or there is a set $X\subseteq V(G)$
of size at most $(t-1)d$
such that the society $(G\backslash X, \Omega\backslash X)$ is cross-free.
\end{lemma}
\begin{lemma}
\label{existgoose}
\showlabel{existgoose}
Let $d,b,t$ be positive integers, let $k\ge (b-1)d+(t-1)\binom{(b-1)d}2+1$
and let $(G,\Omega)$ be a $3$-connected
society of depth at most $d$ such that
at least $k$ vertices in $V(\Omega)$ have at least two neighbors in
$V(G)$. Then $(G,\Omega)$
has either a fan with $t$ blades, or
a goose bump of strength $b$.
\end{lemma}
{\noindent\bf Proof. }
By Lemma~\ref{EPbumps} we may assume that there exists a set
$X\subseteq V(G)$ of size at most $(b-1)d$ such that
$(G\backslash X,\Omega\backslash X)$ has no bump. There are at least
$(t-1)\binom{(b-1)d}2 +1$ vertices in $V(\Omega)-X$ with at least two
neighbors in $V(G)$. Let $v$ be one such vertex, and let $H$
be the component of $G\backslash X$ containing $v$. Since
$(G\backslash X,\Omega\backslash X)$ has no bumps it follows that
$V(H)\cap V(\Omega)=\{v\}$. By the fact that $v$ has at least two
neighbors in $G$ (if $V(H)=\{v\}$) or the 3-connectivity of
$(G,\Omega)$ (if $V(H)\ne \{v\}$) it follows that $H$ has at least two
neighbors in $X$. Thus there exist distinct vertices $z_1,z_2$ such that
for at least $t$ vertices of $v\in V(\Omega)-X$ the component of
$G\backslash X$ containing $v$ has $z_1$ and $z_2$ as neighbors.
It follows that $(G,\Omega)$ has a fan with $t$ blades, as desired.~$\square$\bigskip
\section{Intrusions, invasions and wars}
Let $\Omega$ be a cyclic permutation. A {\em base} in $\Omega$ is
a pair $(X,Y)$ of subsets of $V(\Omega)$ such that $|X\cap Y|=2$,
$X\cup Y=V(\Omega)$ and for distinct elements $x_1,x_2\in X$ and $y_1,y_2\in Y$
the sequence $(x_1,y_1,x_2,y_2)$ is not clockwise.
Now let $(G,\Omega)$ be a society. A separation $(A,B)$ of $G$ is
called an {\em intrusion} in $(G,\Omega)$ if there exists a base
$(X,Y)$ in $\Omega$ such that $X\subseteq A$, $Y\subseteq B$ and there
exist disjoint paths $(P_v)_{v\in A\cap B}$, each with one end in
$X$, the other end in $Y$ and with $v\in V(P_v)$.
The intrusion $(A,B)$ is {\em minimal} if there is no intrusion
$(A',B')$ of order $|A\cap B|$ with base $(X,Y)$ such that $A'$ is
a proper subset of $A$.
The paths $P_v$ will be called {\em longitudes}
for the intrusion $(A,B)$. We say that $(A,B)$ is {\em based} at
$(X,Y)$, and that $(X,Y)$ is a base for $(A,B)$.
An intrusion $(A,B)$ in $(G,\Omega)$ is an {\em invasion}
if $|A\cap B\cap V(\Omega)|=2$.
\begin{lemma}
\label{existintrusion}
\showlabel{existintrusion}
Let $d$ be a positive integer, and let $(G,\Omega)$ be a
society of depth at most $d-1$. Then for every base $(X,Y)$ in $\Omega$
there exists an intrusion of order at most $2d$ based at $(X,Y)$.
\end{lemma}
{\noindent\bf Proof. }
Let $(t_1,t_2,\dots, t_n)$ and $(X_1,X_2,\dots, X_n)$ be a linear
decomposition of $(G,\Omega)$ of depth at most $d-1$, and let
$X\cap Y=\{t_i,t_j\}$. Let $i',j'\in \{1,2,\dots, n\}$ be such that
$|i-i'| = |j-j'|=1$, and let $Z:=(X_i\cap X_{i'})\cup
(X_j\cap X_{j'})\cup \{t_i,t_j\}$. It follows from the axioms of a linear
decomposition that $|Z|\le 2d$ and that $Z$ separates $X$ from $Y$ in $G$.
Thus there exists a separation $(A,B)$ of $G$ of order at most $2d$ with
$X\subseteq A$ and $Y\subseteq B$. Any such separation $(A,B)$ with
$|A\cap B|$ minimum is as desired by Menger's theorem.~$\square$\bigskip
An intrusion $(A,B)$ in a society $(G,\Omega)$ is {\em $t$-separating}
if $(G,\Omega)$ has goose bumps $P_1,P_2,\ldots,P_t$ and
$Q_1,Q_2,\ldots,Q_t$ such that $V(P_i)\subseteq A-B$ and
$V(Q_i)\subseteq B-A$ for all $i=1,2,\ldots,t$.
\begin{lemma}
\label{existtsepintrus}
\showlabel{existtsepintrus}
Let $d,s,t$ be positive integers, and let $(G,\Omega)$ be a
society of depth at most $d-1$ with a goose bump of strength $t(s+2d)$.
Then there exist $s$-separating {minimal} intrusions
$(A_1,B_1),(A_2,B_2),\ldots,(A_t,B_t)$ of order at most $2d$ such that
$A_i\cap A_j\subseteq B_i\cap B_j$ for all pairs of distinct indices
$i,j=1,2,\ldots,t$.
\end{lemma}
{\noindent\bf Proof. }
Let ${\cal P}$ be the set of paths comprising a goose bump of strength
$t(s +2d)$. Thus there exist bases $(X_1,Y_1),(X_2,Y_2),\dots
(X_t,Y_t)$ such that the sets $X_i$ are pairwise disjoint and for each
$i=1,2,\dots, t$ exactly $s+2d$ of the paths in ${\cal P}$ have
both ends in $X_i$. By Lemma~\ref{existintrusion} there exists,
for each $i=1,2,\dots, t$, an intrusion $(A_i,B_i)$ of order at
most $2d$ based at $(X_i,Y_i)$.
Let us choose, for each $i=1,2,\dots, t$, an intrusion $(A_i,B_i)$ of order
at most $2d$ based at $(X_i,Y_i)$ in such a way that
\begin{equation}\label{eq:min}
\sum^t_{i=1}|A_i|\mbox{ is minimum.}\end{equation}
We claim that $A_i\cap A_j\subseteq B_i\cap B_j$. To prove the
claim suppose to the contrary that say $x\in A_1\cap A_2-B_1\cap B_2$. Let
\begin{align*}
A'_1&=A_1\cap B_2,\\
B'_1&=A_2\cup B_1,\\
A'_2&=A_2\cap B_1,\\
B'_2&=A_1\cup B_2.\end{align*}
Then $(A'_1,B'_1)$ and $(A'_2,B'_2)$ are separations of $G$ with
$X_1\subseteq A'_1$, $Y_1\subseteq B'_1$, $X_2\subseteq A'_2$ and
$Y_2\subseteq B'_2$. We have
$$|A_1\cap B_1|+|A_2\cap B_2| = |A'_1\cap B'_1| + |A'_2\cap B'_2|.$$
Furthermore, since each longitude for $(A_1,B_1)$ intersects
$A'_1\cap B'_1$ we deduce that $|A'_1\cap B'_1| \ge |A_1\cap B_1|$,
and similarly $|A'_2\cap B'_2|\ge |A_2\cap B_2|$. Thus the last two
inequalities hold with equality, and hence the longitudes for $(A_1,B_1)$
are also longitudes for $(A'_1,B'_1)$, and the longitudes for $(A_2,B_2)$
are longitudes for $(A'_2,B'_2)$. It follows that for $i=1,2$ the
separation $(A'_i,B'_i)$ is an intrusion in $(G,\Omega)$ based at
$(X_i,Y_i)$ of order $|A_i\cap B_i|$. Since $A_1\cap A_2-(B_1\cap B_2)=
(A_1\cap A_2-B_1)\cup (A_1\cap A_2-B_2)$ we may assume that $x\in A_1-B_2$.
But then replacing $(A_1,B_1)$ by $(A'_1,B'_1)$ produces a set of intrusions
that contradict \eqref{eq:min}. This proves our claim that
$A_i\cap A_j\subseteq B_i\cap B_j$ for all distinct integers
$i,j=1,2,\dots, t$.
Since at most $2d$ of the paths in ${\cal P}$ with ends in $X_i$ can intersect
$A_i\cap B_i$, we deduce that each intrusion $(A_i,B_i)$ is $s$-separating.
Moreover, each $(A_i,B_i)$ is clearly minimal by \eqref{eq:min}.~$\square$\bigskip
We need a lemma about subsets of a set.
\begin{lemma}
\label{subsets}
\showlabel{subsets}
Let $d$ and $t$ be positive integers, and let $\cal F$ be a family
of $2^{d+1\choose 2}t^d$ distinct subsets of a set $S$,
where each member of ${\cal F}$ has size at most $d$. Then there exist
a set $X\subset S$ of size at most ${d+1\choose 2}$ and a family
${\cal F}'\subseteq\cal F$ of size at least $t$ such that
$F\cap F'\subseteq X$ for every two distinct sets $F,F'\in{\cal F}'$.
\end{lemma}
{\noindent\bf Proof. }
We proceed by induction on $d+t$. If $d=1$ or $t=1$, then the lemma clearly
holds, and so we may assume that $d,t>1$. Let $F_0\in{\cal F}$ be minimal
with respect to inclusion. If ${\cal F}$ has a subfamily ${\cal F}_1$ of
at least $2^{d+1\choose 2}(t-1)^d$ sets disjoint from $F_0$, then the
result follows from the induction hypothesis applied to ${\cal F}_1$ and
by adding $F_0$ to the family thus obtained. If the family
${\cal F}_2=\{F-F_0: F\in {\cal F}, F\cap F_0\ne\emptyset\}$ includes at
least $2^{\binom{d}2} t^{d-1}$ distinct sets, then the result follows
from the induction hypothesis applied to ${\cal F}_2$ by adding $F_0$ to
the set thus obtained. Thus we may assume neither of the two cases holds.
Thus
$$|{\cal F}|\le 2^{\binom {d+1}2} (t-1)^d-1+2^d2^{\binom{d}2}t^{d-1}-1+1
<2^{\binom {d+1}2}t^d,$$
a contradiction.~$\square$\bigskip
\begin{lemma}
\label{existdisjintrus}
\showlabel{existdisjintrus}
Let $d,s,t$ be positive integers, and let $(G,\Omega)$ be a
society of depth at most $d-1$ with a goose bump of strength
$2^{{2d+1}\choose 2}t^{2d}(s+2d)$.
Then there exist a set $X\subseteq V(G)$ of size at most ${2d+1\choose 2}$
and $s$-separating intrusions
$(A_1,B_1),(A_2,B_2),\ldots,(A_t,B_t)$ in
$(G\backslash X,\Omega\backslash X)$ such that
$A_i\cap A_j=\emptyset$ for all pairs of distinct indices
$i,j=1,2,\ldots,t$.
\end{lemma}
{\noindent\bf Proof. }
Let $T=2^{{2d+1}\choose 2}t^{2d}$. By Lemma~\ref{existtsepintrus} there
exist $s$-separating {minimal} intrusions
$(A_1,B_1),\allowbreak (A_2,B_2),\ldots,(A_T,B_T)$ of order at most $2d$ such that
$A_i\cap A_j\subseteq B_i\cap B_j$ for all pairs of distinct indices
$i,j=1,2,\ldots,t$. By Lemma~\ref{subsets} applied to the sets
$A_i\cap B_i$ there exist a set $X\subseteq \bigcup_{i=1}^T (A_i\cap B_i)$
of size at most
${{2d+1}\choose2}$ and a subset of $t$ of those intrusions, say
$(A_1,B_1),(A_2,B_2),\ldots,(A_t,B_t)$, such that
$A_i\cap B_i\cap A_j\cap B_j\subseteq X$ for all distinct integers
$i,j=1,2,\ldots,t$. It follows that $(A_i-X,B_i-X)$ are as required for
$(G\backslash X,\Omega\backslash X)$.
$\square$\bigskip
Our next objective is to prove, albeit with weaker bounds, that the conclusion of
Lemma~\ref{existdisjintrus} can be strengthened to assert that the
intrusions $(A_i,B_i)$ therein are actually invasions.
Let $(A,B)$ be an intrusion in a society $(G,\Omega)$ based at $(X,Y)$.
A path $P$ in $G[A]$ is a {\em meridian} for $(A,B)$ if its ends are
the two vertices of $X\cap Y$.
If $P$ is a meridian for $(A,B)$ and $(L_v)_{v\in A\cap B}$ are longitudes
for $(A,B)$, then the graph $(P\cup\bigcup_{v\in A\cap B} L_v)\backslash(B-A)$
is called a {\em frame} for $(A,B)$.
\begin{lemma}
\label{existtsepinv}
\showlabel{existtsepinv}
Let $\lambda$ and $s$ be positive integers, let $s'=(s-1)(\lambda-1)+1$,
let $(G,\Omega)$ be a cross-free society,
and let $(A,B)$ be an $s'$-separating {minimal} intrusion in $(G,\Omega)$
of order at most $\lambda$. Then there exists an
$s$-separating {minimal} invasion $(C,D)$
in $(G,\Omega)$ of order at most $\lambda$ with a frame $F$ such that
$V(F)-V(\Omega)\subseteq A$.
\end{lemma}
{\noindent\bf Proof. }
We may assume that
\claim{1}{%
there is no integer $\lambda'\le \lambda$ and an
$((s-1)(\lambda'-1)+1)$-separating minimal intrusion
$(A',B')$ in $(G,\Omega)$ of order
at most $\lambda'$ with $A'$ a proper subset of $A$,
}
\noindent for if $(A',B')$ exists, and it satisfies the conclusion of
the lemma, then so does $(A,B)$. We first show that $(A,B)$ has a meridian.
Indeed, suppose not. Let $(X,Y)$ be a base of $(A,B)$ and let
$X\cap Y=\{u,v\}$; then $G[A]$ has no $u$-$v$ path. Since $(G,\Omega)$ is
cross-free it follows that $G[A]$ has a separation $(A_1,A_2)$ of order
zero such that both $X_1=X\cap A_1$ and $X_2=X\cap A_2$ are
intervals in $\Omega$. It follows that there exist $Y_1,Y_2$ such that
$(X_1,Y_1)$ and $(X_2,Y_2)$ are bases. Thus $(A_1, A_2\cup B\cup
(X_1\cap Y_1))$ and $(A_2, A_1\cup B\cup (X_2\cap Y_2))$ are
minimal
intrusions,
and one of them violates (1). This proves
that $(A,B)$ has a meridian.
Let $M$ be a meridian in $(A,B)$, let $(L_v)_{v\in A\cap B}$ be a
collection of longitudes for $(A,B)$ and let
$F=M\cup \bigcup_{v\in A\cap B} (L_v\backslash(B-A))$.
By the same argument that justifies (1) we may assume that
\myclaim{2} there is no integer $\lambda' <\lambda$ and an
$((s-1)(\lambda'-1)+1)$-separating minimal intrusion $(A',B')$ in $(G,\Omega)$
of order at most $\lambda'$ with frame $F'$ such that $F'\backslash
V(\Omega)$ is a subgraph of $F$.\par
\medskip
We claim that $|A\cap B\cap V(\Omega)|=2$. We first prove that
$A\cap B\cap X=\{u,v\}$. To this end suppose for a contradiction that
$w\in A\cap B\cap X-\{u,v\}$; then $w$ divides $X$ into
two cyclic intervals $X_1$ and $X_2$ with ends $u,w$ and $w,v$,
respectively. Let $Y_1$ and $Y_2$ be the complementary cyclic intervals
so that $(X_1,Y_1)$ and $(X_2,Y_2)$ are bases.
For $i=1,2$ let $A_i$ consist of $w$ and all vertices $a\in A$ such that
there exists a path in $G[A]\backslash w$ with one end $a$ and the
other end in $X_i-\{w\}$, and let $A_3=A-A_1-A_2$. It follows
that $A_1\cap A_2=\{w\}$, for if $P$ is a path in $G[A]\backslash w$ with
one end in $X_1$ and the other end in $X_2$, then $(P,P_w)$ is a cross
in $(G,\Omega)$, a contradiction. Thus $(A_1,A_2\cup A_3\cup B)$ and
$(A_2, A_1\cup A_3\cup B)$ are minimal intrusions based on $(X_1,Y_1)$
and $(X_2,Y_2)$, respectively, with $A_1,A_2\subseteq A$. Thus one of
them violates (2).
Next we show that $|A\cap B\cap Y|=2$, and so we suppose for a
contradiction that there exists $z\in A\cap B\cap Y-\{u,v\}$. We define
$B_1,B_2,B_3, X_1,Y_1,X_2,Y_2$ analogously as in the previous
paragraph, but with the roles of $A$ and $B$ reversed. Similarly
we find that one of $(A\cup B_1\cup B_3,B_2)$ and
$(A\cup B_2\cup B_3,B_1)$ is an
$((s-1)(\lambda'-1)+1)$-separating minimal%
\REMARK{needs justification using longitudes}
intrusion in $(G,\Omega)$ of order at most
$\lambda'$, for some $\lambda'<\lambda$, and so from the symmetry we may assume
that $(A\cup B_1\cup B_3,B_2)$ has this property. Since $(M,P_z)$ is not a
cross in $(G,\Omega)$ it follows that $M$ and $P_z$ intersect. Thus
$M\cup P_z$ includes a meridian for $(A\cup B_1\cup B_3,B_2)$. Finally,
since $Z=B_2\cap (A\cup B_1\cup B_3)\subseteq A\cap B$, the paths
$(L_v)_{v\in Z}$ form longitudes for $(A\cup B_1\cup B_3,B_2)$, contrary
to (2).
Thus we have shown that $A\cap B\cap V(\Omega)=\{u,v\}$. Let $Z$ be the set
of all vertices $z\in A$ such that there is no path in $G[A]$ with
one end $z$ and the other end in $X$, let $C=A-Z$ and $D=B\cup Z$.
Then $(C,D)$ is an intrusion with $C\cap D=A\cap B$ and $F$ is a frame
for $(C,D)$ with $V(F)-V(\Omega)\subseteq C$. Since the order of
$(C,D)$ is at least two, it satisfies the conclusion of the lemma.~$\square$\bigskip
We are ready to deduce the main result of this section.
By a {\em war} in a society $(G,\Omega)$ we mean a set $\cal W$
of minimal invasions such that each invasion in $\cal W$ has a meridian,
and $A\cap A'=\emptyset$ for every two distinct
invasions $(A,B),(A',B')\in\cal W$.
We say that the war $\cal W$ is
{\em $s$-separating} if each invasion in $\cal W$ is $s$-separating,
we say $\cal W$ has {\em order at most $\lambda$} if each member
of $\cal W$ has order at most $\lambda$, and we say that $\cal W$
is a {\em war of intensity $|{\cal W}|$}.
\begin{lemma}
\label{existinvasions}
\showlabel{existinvasions}
Let $s$, $t$ and $d$ be positive integers, and let
$b=2^{{2d+1}\choose 2}(2dt)^{2d}(s(2d-1)+2)$. Then
if a cross-free society $(G,\Omega)$ of depth at most $d-1$ has
a goose bump of strength $b$, then it has a set $X$ of at most
${2d+1}\choose 2$ vertices such that the society
$(G\backslash X,\Omega\backslash X)$
has an $s$-separating war of intensity $t$ and order order at most $2d$.
\end{lemma}
{\noindent\bf Proof. }
Let $s'=(2d-1)(s-1)+1$. By Lemma~\ref{existdisjintrus} there exist a set
$X\subseteq V(G)$ with at most $\binom{2d+1}2$ elements and $s'$-separating
intrusions $(A_1,B_1), (A_2,B_2),\dots, (A_{2dt},B_{2dt})$ in $(G\backslash
X,\Omega\backslash X)$ of order at most $2d$ such that $A_i\cap A_j=
\emptyset$ for every pair $i,j=1,2,\dots, 2dt$ of distinct integers.
By $2dt$ applications of Lemma~\ref{existtsepinv} there exist, for each
$i=1,2,\dots, 2dt$, and $s$-separating minimal invasion $(C_i,D_i)$ in
$(G\backslash X,\Omega\backslash X)$ of order at most $2d$ with a frame
$F_i$ such that $V(F_i)-V(\Omega)\subseteq V(A_i)$. Let $M_i$ be a meridian
for $(C_i,D_i)$, and let $(X_i,Y_i)$ be the base for $(C_i,D_i)$.
Since $(G,\Omega)$ has depth at most $d$ there exists a set
$I\subseteq \{1,2,\dots, 2dt\}$ of size $t$ such that the sets
$\{X_i\}_{i\in I}$ are pairwise disjoint. By symmetry we may assume that
$I=\{1,2,\dots, t\}$. We claim that $(C_1,D_1), (C_2,D_2),\dots ,
(C_t, D_t)$ are as desired. To prove the claim suppose for a
contradiction that say $x\in C_i\cap C_j$. Since $(C_i,D_i)$
is an invasion there exists a path in $G[C_i]$ from $x$ to $X_i\subseteq
Y_j$; therefore this path intersects $C_j\cap D_j$. Thus there exists a
vertex $v\in C_j\cap D_j\cap C_i$; let $L$ be the longitude of $F_j$
that includes $v$. But $L$ connects $v\in C_i$ to a vertex of
$X_j\subseteq Y_i\subseteq D_i$, and hence intersects $C_i\cap D_i
\subseteq V(F_i)$. Thus $F_i$ and $F_j$ intersect. But
$V(F_i)\cap V(F_j)-V(\Omega)\subseteq A_i\cap A_j=\emptyset$ and
$V(F_i) \cap V(F_j)\cap V(\Omega)\subseteq X_i\cap X_j=\emptyset$, a
contradiction. Thus $(C_1,D_1), (C_2,D_2),\dots, (C_t,D_t)$
satisfy the conclusion of the lemma.~$\square$\bigskip
\section{Using wars}\label{usewars}
\label{sec:wars}
\showlabel{sec:wars}
\begin{lemma}
\label{2nbrs}
\showlabel{2nbrs}
Let $l,t,r$ be positive integers such that $r\ge(t-1){l\choose 2}+1$,
let $(G,\Omega)$ be a connected society, and let $Z\subseteq V(G)$ be a set of
size at most $l$ such that the society $(G\backslash Z,\Omega\backslash Z)$
has a war $\cal W$ of intensity $r$ such that for every $(A,B)\in\cal W$
at least two distinct members of $Z$ have at least one neighbor in $A$.
Then $(G,\Omega)$ has a fan with $t$ blades.
\end{lemma}
{\noindent\bf Proof. } There exist distinct vertices $z_1,z_2\in Z$ and a subset
${\cal W}'$ of ${\cal W}$ of size $t$ such that for every $(A,B)\in {\cal W}'$ both
$z_1$ and $z_2$ have a neighbor in $A$. Furthermore, since $(A,B)$
is a minimal intrusion, it follows that
for every vertex $a\in A$ there exists a path in $G[A]$ from $a$ to
$V(\Omega)$. It follows that $(G,\Omega)$ has a fan with $t$ blades,
as desired.~$\square$\bigskip
Let $(A,B)$ be an invasion in a cross-free society $(G,\Omega)$,
based at $(X,Y)$, and let $(L_v)_{v\in A\cap B}$ be longitudes for
$(A,B)$. Let $\Omega'$ be a cyclic permutation in $A$ defined as follows:
for each $u\in Y$, if $u$ is an end of $L_v$, then we replace $u$
by $v$, and otherwise we delete $u$. Then $(G[A],\Omega')$ is a society,
and we will call it the {\em society induced by} $(A,B)$. Since
$(G,\Omega)$ is cross-free the definition does not depend on the choice
of longitudes for $(A,B)$.
Assume now that $(G[A],\Omega')$ is rural. A path $P$ in $G[A]$ is called a
{\em perimeter path} in $(G[A], \Omega')$ if $A\cap B\subseteq V(P)$ and
$G[A]$ has a drawing in a disk with vertices of $\Omega'$ appearing on
the boundary of the disk in the order specified by $\Omega'$ and with
every edge of $P$ drawn in the boundary of the disk.
The next lemma is easy and we omit its proof.
\begin{lemma}
\label{indcrossfree}
\showlabel{indcrossfree}
Let $(A,B)$ be an invasion with longitudes $\{P_v\}_{v\in A\cap B}$
in a cross-free society $(G,\Omega)$.
Then the society induced by $(A,B)$ is cross-free.
\end{lemma}
\begin{lemma}
\label{indplanar}
\showlabel{indplanar}
Let $(G,\Omega)$ be a $5$-connected society, let $Z\subseteq V(G)$ be such that
$(G\backslash Z,\Omega\backslash Z)$ is cross-free, and let
$(A,B)$ be an invasion in $(G\backslash Z,\Omega\backslash Z)$.
If at most one vertex of $Z$ has a neighbor in $A$, then the
society induced in $(G\backslash Z,\Omega\backslash Z)$ by $(A,B)$
is rural and has a perimeter path.
\end{lemma}
{\noindent\bf Proof. } Let $(G[A],\Omega')$ be the society induced in $(G\backslash Z,
\Omega\backslash Z)$ by $(A,B)$. By Lemma~\ref{indcrossfree} it is
cross-free and by Theorem~\ref{2paththm} it is rural. Thus it has a
drawing in a disk $\Delta$ with $V(\Omega')$ drawn on the boundary of $\Delta$
in the order specified by $\Omega'$. When $\Delta$ is regarded as a subset
of the plane, the unbounded face of $G[A]$ is bounded by a walk $W$.
Let $P$ be a subwalk of $W$ containing $A\cap B$. If $P$ is not a path,
then it has a repeated vertex, say $x$, and $G[A]$ has a separation
$(C,D)$ with $C\cap D=\{x\}$ and $A\cap B\cap V(\Omega)\subseteq C$.
Since $(G[A],\Omega')$ is cross-free, the latter inclusion implies
that $D-C$ is disjoint from $V(\Omega)$ or from $A\cap B$. However,
the latter is impossible, which can be seen by considering the
drawing of $G[A]$ in $\Delta$. Thus $(D-C)\cap V(\Omega)=\emptyset$,
and since $(A,B)$ has longitudes we deduce that $|(D-C)\cap A\cap B|\le 1$.
Let $z\in Z$ be such that no vertex of $Z-\{z\}$ has a neighbor in $A$.
Since $(G,\Omega)$ is $4$-connected, the fact that
$((D-C)\cap A\cap B)\cup\{x,z\}$
does not separate $G$ implies that $D-C$ consists of a unique vertex,
say $d$, and $d\in A\cap B$. Furthermore, the only neighbor of $d$ in
$A$ is $x$. But then $(A-\{d\}, B\cup\{x\})$ contradicts the
minimality of $(A,B)$. This proves that $P$ is a path, and it
follows that it is a perimeter path for $(G[A],\Omega')$.~$\square$\bigskip
Let $(G,\Omega)$ be a society. A set $\cal T$ of bumps in $(G,\Omega)$
is called a {\em transaction in $(G,\Omega)$} if there exist
elements $u,v\in V(\Omega)$ such that each member of $\cal T$ has one
end in $u\Omega v$ and the other end in $V(\Omega)-u\Omega v$.
The first part of the next lemma is easy, and the second part
is proved in~\cite[Theorem (8.1)]{RobSeyGM9}.
\begin{lemma}
\mylabel{depthtrans}
Let $(G,\Omega)$ be a society, and let $d\ge1$ be an integer.
If $(G,\Omega)$ has depth $d$, then it has no transaction of cardinality
exceeding $2d$. Conversely, if $(G,\Omega)$ has no transaction of
cardinality exceeding $d$, then it has depth at most $d$.
\end{lemma}
\begin{lemma}
\label{deldepth}
\showlabel{deldepth}
Let $(G,\Omega)$ be a society of depth $d$,
and let $X\subseteq V(G)$. Then the society
$(G\backslash X,\Omega\backslash X)$ has depth at most $2d$.
\end{lemma}
{\noindent\bf Proof. } By Lemma~\ref{depthtrans} the society $(G,\Omega)$ has
no transaction of cardinality exceeding $2d$. Then clearly
$(G\backslash X, \Omega\backslash X)$ has no transaction of
cardinality exceeding $2d$, and hence has depth at most $2d$ by
another application of Lemma~\ref{depthtrans}.~$\square$\bigskip
We need one last lemma before we can prove Theorem~\ref{thm1}.
The lemma we need is concerned with the situation when a society of bounded
depth ``almost" has a windmill with $t$ vanes, except that the
paths $P_i$ are not necessarily disjoint and their ends do not
necessarily appear in the right order.
We begin with a special case when the ends of the paths $P_i$
do appear in the right order.
\begin{lemma}
\label{prewindmill}
\showlabel{prewindmill}
Let $t\ge1$ be an integer, and let $\rho=d(t-1)(t'-1)+1$,
where $t'=d(t-1)^2+t$.
Let $(G,\Omega)$ be a society of depth $d$, let
$(u_1,z_1,v_1,u_2,z_2,v_2,\ldots,u_\rho,z_\rho,v_\rho)$
be clockwise, let $z\in V(G)$, for $i=1,2,\ldots,\rho$
let $P_i$ be a bump with ends $u_i$ and $v_i$, and let
$Q_i$ be a path of length at least one with ends $z$ and $z_i$
disjoint from $V(\Omega)-\{z,z_i\}$.
Assume that the paths $Q_i$ are pairwise disjoint except for $z$,
and that each is disjoint from every $P_j$.
Then $(G,\Omega)$ has either a windmill with $t$ vanes,
or a fan with $t$ blades.
\end{lemma}
{\noindent\bf Proof. }
By the proof of Lemma~\ref{EPbumps} applied to the paths $P_i$
either some $t$ of those paths are vertex-disjoint, in which case
$(G,\Omega)$ has a windmill with $t$ vanes, or there exists
a set $X\subseteq V(G)$ of size at most $(t-1)d$
such that each $P_i$ uses at least one
vertex of $X$.
We may therefore assume the latter.
For $i=1,2,\ldots,\rho$ the path $P_i$ has a subpath $P_i'$
with one end $u_i$, the other end $x_i\in X$ and no internal vertex in $X$.
Thus there exist $x\in X$ and a set $I\subseteq\{1,2,\ldots,\rho\}$
of size $t'$ such that $x=x_i$ for all $i\in I$.
Let $H$ be the union of all $P_i'$ over $i\in I$.
By an application of Lemma~\ref{EPbumps} to the graph $H\backslash x$
we deduce that either $H\backslash x$ has a goose bump of strength $t$,
in which case $(G,\Omega)$ has a windmill with $t$ vanes,
or $H$ has a set $Y$ of size at most $(t-1)d$ such that
$H\backslash Y\backslash x$ has no bumps.
In the latter case for each $i\in I$ there is a path $P''_i$ in $H$ with
one end $u_i$, the other end $y_i\in Y\cup \{x\}$ and otherwise
disjoint from $Y\cup \{x\}$.
Thus there is a vertex $y\in Y\cup \{x\}$ and a set $J\subseteq I$
of size $t$ such that $y_i=y$ for every $i\in J$.
Since $H\backslash Y\backslash x$ has no bumps it follows that
$P''_j$ and $P''_{j'}$ share only $y$ for distinct $j,j'\in J$.
Thus $(G,\Omega)$ has a fan with $t$ blades, as desired.~$\square$\bigskip
Now we are ready to prove the last lemma in full generality.
\begin{lemma}
\label{semiwindmill}
\showlabel{semiwindmill}
Let $t\ge1$ be an integer, and let $\xi=(d+1)\rho$, where $\rho$ is
as in Lemma~\ref{prewindmill}.
Let $(G,\Omega)$ be a society of depth $d$,
let $z\in V(G)$, for $i=1,2,\ldots,\xi$ let
$(u_i,z_i,v_i)$ be clockwise, and let
$(u_1,z_1,u_2,z_2,\ldots,u_\xi,z_\xi)$
be clockwise.
Let $P_i$ be a bump with ends $u_i$ and $v_i$, and let
$Q_i$ be a path of length at least one with ends $z$ and $z_i$
disjoint from $V(\Omega)-\{z,z_i\}$.
Assume that the paths $Q_i$ are pairwise disjoint except for $z$,
and that each is disjoint from every $P_j$.
Then $(G,\Omega)$ has either a windmill with $t$ vanes,
or a fan with $t$ blades.
\end{lemma}
{\noindent\bf Proof. }
Let $(t_1,t_2,\ldots,t_n)$ be
a clockwise enumeration of $V(\Omega)$, and let
$(X_1,X_2,\ldots,X_n)$ be
a corresponding linear decomposition of $(G,\Omega)$ of depth $d$.
Let us fix an integer $i=1,2,\ldots,\rho$, and let
$I=\{(i-1)(d+1)+1,(i-1)(d+1)+2,\ldots,i(d+1)\}$.
For each such $i$ we will construct paths $P^*_i$ and $Q^*_i$
satisfying the hypothesis of Lemma~\ref{prewindmill}.
In the construction we will make use of the paths $P_j$ and $Q_j$
for $j\in I$.
If $(u_j,z_j,v_j,u_{i(d+1)+1})$ is clockwise for some $j\in I$,
then we put $P^*_i=P_j$ and $Q^*_i=Q_j$.
Otherwise, letting $s$ be such that $t_s=u_{i(d+1)}$,
we deduce that $P_j$ intersects $X_{t_s}\cap X_{t_{s+1}}$ for all
$j\in I$. Since $|I|>|X_{t_s}\cap X_{t_{s+1}}|$ it follows that
there exist $j<j'\in I$ such that $P_j$ and $P_{j'}$ intersect.
Let $P_i^*$ be a subpath of $P_j\cup P_{j'}$ with ends $u_j$
and $u_{j'}$, and let $Q^*_i=Q_j$.
This completes the construction. The lemma follows from
Lemma~\ref{prewindmill}.~$\square$\bigskip
\noindent
{\bf Proof of Theorem~\ref{thm1}.}
Let the integers $d$ and $t$ be given,
let $\xi$ be as in Lemma~\ref{semiwindmill},
let $\ell = 2(t-1)d+
\binom{4d+2}2$, let $\tau =(t-1)\binom\ell 2 +\left(2(t-1)d+
\binom{8d+2}2\right) (6\xi-1)+1$, let $b$ be as in
Lemma~\ref{existinvasions} with $s=1$, $t=\tau$ and $d$ replaced
by $4d+1$, and let $k$ be as in Lemma~\ref{existgoose} applied to
$b$, $t$, and $4d$. We will prove that $k$ satisfies the conclusion of
the theorem.
To that end let $(G,\Omega)$ be a $k$-cosmopolitan society of
depth at most $d$, and let $(G_0,\Omega_0)$ be a planar truncation of
$(G,\Omega)$.
Let $S\subseteq V(\Omega_0)$. We say that $S$ is {\em sparse}
if whenever $u_1,u_2\in S$ are such that there does not exist $w\in S$
such that $(u_1,w,u_2)$ is clockwise, then there exist two disjoint
bumps $P_1,P_2$ in $(G_0,\Omega_0)$ such that $u_i$ is an end of $P_i$.
The reader should notice that if $H$ is one of the graphs listed
as outcomes (1)-(3) of Theorem~\ref{thm1}, then $V(H)\cap V(\Omega_0)$
is sparse.
We say that $(G_0,\Omega_0)$ is {\em weakly linked} if for every
sparse set $S\subseteq V(\Omega_0)$ there exist $|S|$ disjoint
paths from $S$ to $V(\Omega)$ with no internal vertex in $V(G_0)$.
Thus if the conclusion of the theorem holds for
some weakly linked truncation of $(G_0,\Omega_0)$, then it holds for
$(G,\Omega)$ as well.
Thus we may assume that $(G_0,\Omega_0)$ is a weakly linked
truncation of $(G,\Omega)$ with $|V(G_0)|$ minimum.
We will prove that $(G_0,\Omega_0)$ satisfies the conclusion of
Theorem~\ref{thm1}.
Since $(G_0,\Omega_0)$ is weakly linked, Lemma~\ref{depthtrans} implies
that $(G_0,\Omega_0)$ has no transaction of cardinality exceeding $2d$,
and hence has depth at most $2d$ by Lemma~\ref{depthtrans}.
By Lemma~\ref{EPcrosses}
there exists a set $Z_1\subseteq V(G_0)$ such that $|Z_1|\le 2(t-1)d$
and the society $(G_0\backslash Z_1,\Omega_1\backslash Z_1)$ is cross-free.
By Lemma~\ref{deldepth} the society $(G_0\backslash Z_1,\Omega_0\backslash Z_1)$
has depth at most $4d$. By Lemma~\ref{existgoose} we may assume that
$(G_0\backslash Z_1,\Omega_0\backslash Z_1)$ has a goose bump of strength
$b$. By Lemma~\ref{existinvasions} there exists a set $Z_2\subseteq
V(G)-Z_1$ such that $|Z_2|\le \binom{4d+2}2$ and in the society
$(G_0\backslash Z,\Omega_0\backslash Z)$ there exists a $1$-separating
war ${\cal W}$ of intensity $\tau$ and order at most $8d+2$, where
$Z=Z_1\cup Z_2$. If there
exist at least $(t-1)\binom\ell 2+1$ invasions $(A,B)\in {\cal W}$
such that at least two distinct vertices of $Z$ have a neighbor in $A$,
then the theorem holds by Lemma~\ref{2nbrs}. We may therefore assume
that this is not the case, and hence ${\cal W}$ has a subset ${\cal W}'$ of
size at least $|Z|(6\xi-1)+1$ such that for every $(A,B)\in{\cal W}' $
at most one vertex of $Z$ has a neighbor in $A$.
Let $(A,B)\in{\cal W}'$ and let $z\in Z$ be such that no vertex in $Z-\{z\}$ has
a neighbor in $A$.
By Lemma~\ref{indplanar} the society $(G_0[A],\Omega')$ induced
in $(G_0\backslash Z,\Omega_0\backslash Z)$ by $(A,B)$ is rural and has
a perimeter path $P$.
It follows that $(A\cup\{z\},B\cup\{z\})$ is a separation of $G_0$.
Let $A\cap B=\{w_0,w_1,\ldots,w_s\}$, and let $L_i$ be the longitude
containing $w_i$. Let the ends of $L_i$ be $u_i\in A$ and $v_i\in B$.
We may assume that $(u_0,u_1,\ldots,u_s)$ is clockwise.
The vertices $w_i$ divide $P$ into paths $P_0,P_1,\ldots,P_s$,
where $P_i$ has ends $w_{i-1}$ and $w_i$.
We claim that no $P_i$ includes all neighbors of $z$.
Suppose for a contradiction that $P_i$ does.
Let $(G,\Omega)$ be
the composition of $(G_0,\Omega_0)$ with a rural neighborhood
$(G_1,\Omega,\Omega_0)$. Let $G'_1=G_1\cup G[A\cup\{z\}]$, let
$G'_0=G_0\backslash (A-B)$ and let $\Omega'_0$
consist of $w_s\Omega w_0$ followed by $w_{s-1},w_{s-2},\ldots,w_i$
followed by $z$
followed by $w_{i-1},w_{i-2},\ldots,w_1$.
Since $(G[A],\Omega')$ is
rural and all neighbors of $z$ belong to $P_i$,
it follows that $(G'_1, \Omega,\Omega'_0)$ is a rural neighborhood
and $(G,\Omega)$ is the composition of $(G'_0,\Omega'_0)$ with this
neighborhood.
Thus $(G_0',\Omega_0')$ is a planar truncation of $(G,\Omega)$.
We claim that $(G_0',\Omega_0')$ is weakly linked.
To prove that let $S'\subseteq V(\Omega_0')$ be sparse.
Since $(A,B)$ is a minimal intrusion there exists a set ${\cal P}'$
of $|S'|$ disjoint paths from $S'$ to $V(\Omega_0)$ with no internal
vertex in $G_0'$; let $S$ be the set of their ends in $V(\Omega_0)$.
Since $S'$ is sparse in $(G_0',\Omega_0')$, it follows that $S$ is sparse
in $(G_0,\Omega_0)$. Since $(G_0,\Omega_0)$ is weakly linked there exists
a set $\cal P$ of $|S|$ disjoint paths in $G$ from $S$ to $V(\Omega)$
with no internal vertex in $G_0$.
By taking unions of members of $\cal P$ and ${\cal P}'$ we obtain a set
of paths proving that $(G_0',\Omega_0')$ is weakly linked, as desired.
Since ${\cal W}$ is 1-separating this contradicts the
minimality of $G_0$, proving our claim that
no $P_i$ includes all neighbors of $z$.
The same argument, but with $G_1'=G_1\cup G[A]$ and $\Omega_0'$ not including
$z$ shows that $z$ has a neighbor in $A-B$.
We have shown, in particular, that exactly one vertex of $Z$
has a neighbor in $A-B$.
Thus there exists a subset ${\cal W}''$ of ${\cal W}'$ of size $6\xi$ and
a vertex $z\in Z$ such that for every $(A,B)\in{\cal W}''$ the vertex $z$
has a neighbor in $A-B$.
Now let $w=(A,B)\in {\cal W}''$, and let the notation be as before.
We will construct paths $P_w$, $Q_w$ such that the
hypotheses of Lemma~\ref{semiwindmill} will be satisfied for at
least half the members $w\in{\cal W}''$.
The facts that $(A,B)$ is a minimal intrusion and that $z$ has a neighbor
in $A-B$ imply that there exists a path $Q_w$ in $G[A\cup\{z\}]$
from $z$ to $z_w\in V(\Omega_0)\cap A$
and a choice of longitudes $(L_v:v\in A\cap B)$
for $(A,B)$ such that $Q_w$ is disjoint from all $L_v$.
Referring to the subpaths $P_i$ of the perimeter path $P$ defined above,
since no $P_i$ includes all neighbors of $z$ it follows that there exists
$v\in A\cap B-V(\Omega_0)$.
We define $P_w$ to be a path obtained from $L_v$ by suitably
modifying $L_v$ inside $B$ such that $P_w$ intersects
$A'$ for at most one $(A',B')\in{\cal W}''-\{(A,B)\}$.
Such modification is easy to make, using the perimeter path of $(A',B')$.
Let $u_w\in A$ and $v_w\in B$ be the ends of $P_w$.
The set ${\cal W}''$ has a subset ${\cal W}'''$ of size $\xi$ such that,
using to the notation of the previous paragraph,
either $(u_w,z_w,v_w)$ is
clockwise for every $w\in {\cal W}'''$ or $(v_w,z_w,u_w)$ is clockwise
for every $w\in {\cal W}'''$, and
for every $w\in {\cal W}'''$ the path $P_w$ is disjoint from
$A'$ for every $(A',B')\in{\cal W}'''-\{w\}$.
The theorem now follows from Lemma~\ref{semiwindmill}.~$\square$\bigskip
\section{Using lack of near-planarity}
\label{sec:lack}
\showlabel{sec:lack}
In this section we prove Theorems~\ref{thm2} and~\ref{thm:society}.
The first
follows immediately from Theorem~\ref{thm1} and the two lemmas below.
\begin{lemma}
\label{planwindmill}
\showlabel{planwindmill}
Let $(G,\Omega)$ be a rurally $5$-connected society that
is not nearly rural, and let $t$ be a positive integer. If
$(G,\Omega)$ has a windmill with $4t+1$ vanes, then it has
a windmill with $t$ vanes and a cross.
\end{lemma}
{\noindent\bf Proof. }
Let $x,u_i,v_i,w_i,P_i,Q_i$ be as in the definition of a windmill $W$
with $4t+1$ vanes. Since $(G\backslash x,\Omega\backslash\{x\})$
is rurally $4$-connected and not rural, it has a cross $(P,Q)$
by Theorem~\ref{2paththm}. We may choose the windmill $W$ and
cross $(P,Q)$ in $(G\backslash x,\Omega\backslash\{x\})$ such that
$W\cup P\cup Q$ is minimal with respect to inclusion.
If the cross does not intersect the windmill, then the lemma
clearly holds, and so we may assume that a vane $P_i\cup Q_i$ intersects
$P\cup Q$. Let $v$ be a vertex that belongs to both $P_i\cup Q_i$
and $P\cup Q$ such that some subpath $R$ of $P_i\cup Q_i$ with one end
$v$ and the other end in $V(\Omega)$ has no vertex in $(P\cup Q)\backslash v$.
If $R$ has at least one edge, then $P\cup Q\cup R$ has a proper
subgraph that is a cross, contrary to the minimality of $W\cup P\cup Q$.
Thus $v$ is an end of $P$ or $Q$. Since $P$ and $Q$ have a total of
four ends, it follows that $P\cup Q$ intersects at most four vanes
of $W$. By ignoring those vanes we obtain a windmill with $4(t-1)+1$
vanes, and a cross $(P,Q)$ disjoint from it. \REMARK{Make clearer.}
The lemma follows.~$\square$\bigskip
\begin{lemma}
\label{planfan}
\showlabel{planfan}
Let $(G,\Omega)$ be a rurally $6$-connected society that
is not nearly rural, and let $t$ be a positive integer. If
$(G,\Omega)$ has a fan with $16t+5$ blades, then it has
a fan with $t$ blades and a cross, or
a fan with $t$ blades and a jump, or
a fan with $t$ blades and two jumps.
\end{lemma}
{\noindent\bf Proof. }
Let $z_1,z_2$ be the hubs of a fan $F_2$ with $16t+5$ blades.
If $(G\backslash \{z_1,z_2\},\Omega\backslash \{z_1,z_2\})$ has a cross,
then the lemma follows in the same way as Lemma~\ref{planwindmill},
and so we may assume not. Since $(G\backslash z_1,\Omega\backslash
\{z_1\})$ has a cross, an argument analogous to the proof of
Lemma~\ref{planwindmill} shows that there exists a subfan $F_1$ of
$F_2$ with $4t+1$ blades (that is, $F_1$ is obtained by ignoring a set of
$12t+4$ blades), and two paths $L_2, S_2$ with ends $a_2,c_2$ and
$b_2,z_2$, respectively, such that $x_1,x_2,\dots, x_{4t+1}$, $a_2,b_2,c_2$
is clockwise in $\Omega$ for every choice of $x_1,x_2,\dots, x_{4t+1}$ as
in the definition of a fan, and the graphs $L_2,S_2\backslash z_2,
F_1$ are pairwise disjoint. By using the same argument and the fact that
$(G\backslash z_2,\Omega\backslash \{z_2\})$ has a cross we arrive at
a subfan $F$ of $F_1$ with $t$ blades and paths $L_1,S_1$ satisfying
the same properties, but with the index 2 replaced by 1. We may
assume that $F, L_1,L_2,S_1,S_2$ are chosen so that $F\cup L_1\cup L_2\cup
S_1\cup S_2$ is minimal with respect to inclusion. This will be referred to
as ``minimality."
If the paths $L_1,L_2,S_1,S_2$ are pairwise disjoint, except possibly
for shared ends and possibly $S_1$ and $S_2$ intersecting,
then it is easy to see that the lemma holds, and
so we may assume that an internal vertex of $L_1$ belongs to $L_2
\cup S_2$. Let $v$ be the first vertex on $L_1$ (in either direction)
that belongs to $L_2\cup S_2$, and suppose for a contradiction that
$v$ is not an end of $L_1$. Let $L'_1$ be a subpath of $L_1$ with
one end $v$, the other end in $V(\Omega)$ and no internal vertex in
$L_2\cup S_2$. Then by replacing a subpath of $L_2$ or $S_2$
by $L'_1$ we obtain either a contradiction to minimality, or a cross
that is a subgraph of $L_1\cup L_2\cup S_1\cup S_2\backslash \{z_1,
z_2\}$, also a contradiction. This proves that $v$ is an end of $L_1$,
and hence both ends of $L_1$ are also ends of $L_2$ or $S_2$.
In particular, $L_1$ and $L_2$ share at least one end.
Suppose first that one end of $L_1$ is an end of $S_2$.
Thus from the symmetry we may assume that $a_1$ is an end of $L_2$ and
$c_1=b_2$; thus $a_2=a_1$, because $a_2,b_2,c_2$ is clockwise.
But now $c_2$ is not an end of $L_1$ or $S_1$, and so the argument of
the previous paragraph implies that no internal vertex of $L_2$
belongs to $S_1\cup L_1$. The paths $S_1,S_2,L_2$ now show that
$(G,\Omega)$ has a fan with $t$ blades and a jump.
We may therefore assume that $a_1=a_2$ and $c_1=c_2$.
Let $H$ be the union of $L_1, L_2, S_1\backslash z_1$, $S_2\backslash z_2$,
and $V(\Omega)$. Then the society
$(H, \Omega)$ is rural, as otherwise $(G\backslash \{z_1, z_2\},\Omega)$
has a cross.
Let $\Gamma$ be a drawing of $(H, \Omega)$ in a disk $\Delta$ such that
the vertices of $V(\Omega)$ are drawn on the boundary of $\Delta$ in the
clockwise order specified by $\Omega$.
Let $\Delta'\subseteq\Delta$ be a disk such that $\Delta'$ includes
every path in $\Gamma$ with ends $a_1$ and $c_1$, and the boundary
of $\Delta'$ includes $a_1\Omega c_1$ and a path $P$ of $\Gamma$
from $a_1$ to $c_1$.
Then $L_1$ and $L_2$ lie in $\Delta'$, and since
$L_i$ is disjoint from $S_i\backslash z_i$ it follows that
$S_1\backslash z_1$ and $S_2\backslash z_2$ are inside $\Delta'$
and, in particular, are disjoint from $P$. By considering $P$, $S_1$ and $S_2$
we obtain a fan with $t$ blades and a jump.~$\square$\bigskip
\noindent
{\bf Proof of Theorem~\ref{thm2}.}
Let $d$ and $t$ be integers, let $k$ be an integer such that
Theorem~\ref{thm1} holds for $d$ and $16t+5$, and let
$(G,\Omega)$ be a $6$-connected $k$-cosmopolitan society of depth at most $d$.
We may assume that $(G,\Omega)$ is not nearly rural, for otherwise the
theorem holds.
By Theorem~\ref{thm1} the society $(G,\Omega)$ has $t$ disjoint
consecutive crosses, or
a windmill with $4t+1$ vanes, or
a fan with $16t+5$ blades.
In the first case the theorem holds, and in the second and third case
the theorem follows from Lemma~\ref{planwindmill} and
Lemma~\ref{planfan}, respectively.~$\square$\bigskip
For the proof of Theorem~\ref{thm:society} we need one more lemma.
Let us recall that presentation of a neighborhood was defined prior to
Theorem~\ref{cosmopolitan}.
\begin{lemma}
\mylabel{nestedtrunk}
Let $d$ and $s$ be integers,
let $(G,\Omega)$ be an $s$-nested society, and let $(G',\Omega')$ be a planar
truncation of $(G,\Omega)$ of depth at most $d$.
Then $(G,\Omega)$ has an $s$-nested planar truncation of depth at most $2(d+2s)$.
\end{lemma}
{\noindent\bf Proof. }
By a vortical decomposition of a society $(G,\Omega)$ we mean a collection
$(Z_v:v\in V(\Omega))$ of sets such that
\par\hangindent0pt\mytextindent{(i)} $\bigcup (Z_v:v\in V(\Omega))=V(G)$ and every edge of $G$ has
both ends in $Z_v$ for some $v\in V(\Omega)$,
\par\hangindent0pt\mytextindent{(ii)} for $v\in V(\Omega)$, $v\in Z_v$, and
\par\hangindent0pt\mytextindent{(iii)} if $(v_1,v_2,v_3,v_4)$ is clockwise in $\Omega$,
then $Z_{v_1}\cap Z_{v_3}\subseteq Z_{v_2}\cup Z_{v_4}$.
\noindent The {\em depth} of such a vortical decomposition is
$\max |Z_u\cap Z_v|$, taken over all pairs of distinct vertices $u,v\in V(\Omega)$
that are consecutive in $\Omega$,
and the depth of $(G,\Omega)$ is the minimum depth of a vortical decomposition
of $(G,\Omega)$.
Thus if $(G,\Omega)$ has depth at most $d$, then the corresponding linear
decomposition also serves as a vortical decomposition of depth at most~$d$.
Let $(G,\Omega)$ be an $s$-nested society, and let it be the composition
of a society $(G_0,\Omega_0)$ with a rural neighborhood $(G_1,\Omega,\Omega_0)$,
where the neighborhood has a presentation $(\Sigma, \Gamma_1, \Delta, \Delta_0)$
with an $s$-nest $C_1,C_2,\ldots,C_s$.
Let $\Delta_0,\Delta_1,\ldots,\Delta_s$ be as in the definition of $s$-nest.
Let $(G',\Omega')$ be a planar truncation of $(G,\Omega)$ of depth at most $d$.
Then $(G,\Omega)$ is the composition of $(G',\Omega')$ with a rural
neighborhood $(G_2,\Omega,\Omega')$, and we may assume that
$(G_2,\Omega,\Omega')$ has a presentation $(\Sigma, \Gamma_2, \Delta, \Delta')$,
where $\Delta_0\subseteq\Delta'$.
We may assume that the $s$-nest $C_1,C_2,\ldots,C_s$ is chosen as follows:
first we select $C_1$ such that $\Delta_0\subseteq \Delta_1$
and the disk $\Delta_1$ is as small as possible,
subject to that
we select $C_2$ such that $\Delta_1\subseteq \Delta_2$
and the disk $\Delta_2$ is as small as possible,
subject to that we select $C_3$, and so on.
Let $\Delta^*$ be a closed disk with $\Delta'\subseteq\Delta^*\subseteq\Delta$.
We say that $\Delta^*$ is {\em normal} if whenever an interior point of
an edge $e\in E(\Gamma_1)$ belongs to the boundary of $\Delta^*$, then
$e$ is a subset of the boundary of $\Delta^*$.
A normal disk $\Delta^*$ defines a planar truncation $(G^*,\Omega^*)$
in a natural way as follows: $G^*$ is consists of all vertices and edges
that of $G$ either belong to $G'$, or their image under $\Gamma_1$ belongs to
$\Delta^*$, and $\Omega^*$ consists of vertices of $G$ whose image under
$\Gamma_1$ belongs to the boundary $\Delta^*$ in the order determined by
the boundary of $\Delta^*$.
Given a normal disk $\Delta^*$ and two vertices $u,v\in V(G)$
we define $\xi_{\Delta^*}(u,v)$, or simply $\xi(u,v)$ as follows.
If $u$ is adjacent to $v$, and the image $e$ under $\Gamma_1$ of the edge $uv$
is a subset of the boundary of $\Delta^*$, and for every internal point
$x$ on $e$ there exists an open neighborhood $U$ of $x$ such that
$U\cap \Delta^*=U\cap\Delta_i$, then we let $\xi(u,v)=i$.
Otherwise we define $\xi(u,v)=0$.
A short explanation may be in order. If the image $e$ of $uv$ is a subset
of the boundary of $\Delta^*$, then this can happen in two ways:
if we think of $e$ as having two sides, either $\Delta^*$ and $\Delta_i$
appear on the same side, or on opposite sides of $e$. In the definition of
$\xi$ it is only edges with $\Delta^*$ and $\Delta_i$ on the same side that
count.
We may assume, by shrinking $\Delta'$ slightly, that the boundary of
$\Delta'$ does not include an interior point of any edge of $\Gamma_2$.
Then $\Delta'$ is normal, and the corresponding planar truncation
is $(G',\Omega')$. Since a linear decomposition
of $(G',\Omega')$ of depth at most $d$ may be regarded as a vortical
decomposition of $(G',\Omega')$ of depth at most $d$,
we may select a normal disk $\Delta^*$ that gives rise to a planar truncation
$(G^*,\Omega^*)$ of $(G,\Omega)$, and we may select a vortical decomposition
$(Z_v:v\in V(\Omega^*))$ of $(G^*,\Omega^*)$ such that
$|Z_u\cap Z_v|\le d+2\xi(u,v)$ for every pair of consecutive vertices of $\Omega^*$.
Furthermore, subject to this, we may choose $\Delta^*$ such that
the number of unordered pairs $u,v$ of distinct vertices of $G$
with $\xi(u,v)=s$ is maximum, subject to that
the number of unordered pairs $u,v$ of distinct vertices of $G$
with $\xi(u,v)=s-1$ is maximum, subject to that
the number of unordered pairs $u,v$ of distinct vertices of $G$
with $\xi(u,v)=s-2$ is maximum, and so on.
We will show that $(G^*,\Omega^*)$ satisfies the conclusion of the theorem.
Let $(t_1,t_2,\ldots,t_n)$ be an arbitrary clockwise enumeration of
$V(\Omega^*)$, and let $X_i:= Z_{t_i}\cup(Z_{t_1}\cap Z_{t_n})$.
Then $(X_1,X_2,\ldots,X_n)$ is a linear decomposition of $(G^*,\Omega^*)$ of
depth at most $2(d+2s)$.
To complete the proof we must show that $(G^*,\Omega^*)$ is $s$-nested, and
we will do that by showing that each $C_i$ is a subgraph of $G^*$.
To this end we suppose for a contradiction that it is not the case,
and let $i_0\in\{1,2,\ldots,s\}$ be the minimum integer such that
$C_{i_0}$ is not a subgraph of $G^*$.
If $C_{i_0}$ has no edge in $G^*$, then we can construct a new society
$(G_3,\Omega_3)$, where $\Omega_3$ consists of the vertices of $C_{i_0}$ in order,
and obtain a contradiction to the choice of $(G^*,\Omega^*)$. Since the
construction is very similar but slightly easier than the one we are
about to exhibit, we omit the details.
Instead, we assume that $C_{i_0}$ includes edges of both $G^*$ and
$G\backslash E(G^*)$.
Thus there exist vertices
$x,y\in V(C_{i_0})\cap V(\Omega^*)$ such that some subpath $P$ of $C_{i_0}$ with
ends $x$ and $y$ has no internal vertex in $V(\Omega^*)$.
Let $B$ denote the boundary of $\Delta^*$.
There are three closed disks with boundaries contained in $B\cup P$.
One of them is $\Delta^*$; let $D$ be the one that is disjoint from $\Delta_0$.
If the interior of $D$ is a subset of $\Delta_{i_0}$ and includes
no edge of $C_{i_0}$,
then we say that $P$ is a {\em good segment}.
It follows by a standard elementary argument that there is a good segment.
Thus we may assume that $P$ is a good segment, and that the notation is
as in the previous paragraph.
There are two cases: either $D$ is a subset of $\Delta^*$, or the interiors
of $D$ and $\Delta^*$ are disjoint.
Since the former case is handled by a similar, but easier construction,
we leave it to the reader and assume the latter case.
Let $(s_0,s_1,\ldots,s_{t+1})$ be clockwise in $\Omega^*$ such that
$s_0,s_1,\ldots,s_{t+1}$ are all the vertices that belong to $D\cap \Delta^*$.
Thus $\{s_0,s_{t+1}\}=\{x,y\}$.
Let $r_0=s_0,r_1,\ldots,r_k,r_{k+1}=s_{t+1}$ be all the vertices of $P$,
in order, let $H$ be the subgraph of $G^*$ consisting of all vertices
and edges whose images under $\Gamma_1$ belong to $D$, and let
$X:=\{s_0,s_1,\ldots,s_{t+1},r_0,r_1,\ldots,r_{k+1}\}$.
We can regard $H$ as drawn in a disk with the vertices
$s_0,s_1,\ldots,s_{t+1},r_k,r_{k-1},\ldots,r_1$ drawn on the boundary of the
disk in order.
We may assume that every component of $H$ intersects $X$.
The way we chose the cycles $C_{i_0}$ implies that every path in
$H\backslash\{s_1,s_2,\ldots,s_k\}$ that joins two vertices of $P$ is
a subpath of $P$.
We will refer to this property as the convexity of $H$.
For $i=0,1,\ldots,k+1$ let $b_i$ be the maximum index $j$ such that
the vertex $s_j$ can be reached from $\{r_0,r_1,\ldots,r_i\}$ by a path in $H$
with no internal vertex in $X$.
We define $b_{-1}:=-1$, and let $R_i$ be the set of all vertices of $H$
that can be reached from $\{r_i,s_{b_{i-1}+1},s_{b_{i-1}+2},\ldots,s_{b_i}\}$
by a path with no internal vertex in $X$.
The convexity of $H$ implies that for $i<j$ the only possible
member of $R_i\cap R_j$ is $s_{b_i}$.
We now define a new society $(G^{**},\Omega^{**})$ as follows.
The graph $G^{**}$ will be the union of $G^*$ and $H$,
and the cyclic permutation is defined by replacing the subsequence
$s_0,s_1,\ldots,s_{t+1}$ of $\Omega^*$ by the sequence
$r_0,r_1,\ldots,r_k,r_{k+1}$.
We define the sets $Z^{**}_v$ as follows.
For $v\in V(\Omega^*)-V(\Omega^{**})$ we let $Z^{**}_v:=Z_v$.
If $v=r_i$ and $b_i>b_{i-1}$ we define $Z^{**}_v$ to be the union of
$R_i\cup\{s_{b_i},r_{i-1}\}$ and all $Z_{s_j}$ for
$j=b_{i-1}+1,b_{i-1}+2,\ldots,b_i$.
If $v=r_i$ and $b_i=b_{i-1}$ we define
$Z^{**}_v:=R_i\cup\{s_{b_i},r_{i-1}\}\cup (Z_{s_{b_i}}\cap Z_{s_{b_i+1}})$.
It is straightforward to verify that $(G^{**},\Omega^{**})$ is a planar
truncation of $(G,\Omega)$ and that
$(Z^{**}_v:v\in V(\Omega^{**}))$
is a vortical decomposition of $(G^{**},\Omega^{**})$.
We claim that $\xi_{\Delta^*}(s_j,s_{j+1})<i_0$ for all $j=0,1,\ldots,t$.
To prove this we may assume that $s_j$ is adjacent to $s_{j+1}$, and
let $e$ be the image under $\Gamma_1$ of the edge $s_js_{j+1}$.
It follows that $e$ is a subset of $\Delta_{i_0}$, and hence if
$s_js_{j+1}\in E(C_k)$ for some $k$, then $k\le i_0$.
Furthermore, if equality holds, then $\Delta_{i_0}$ and $\Delta^*$
lie on opposite sides of $e$, and hence $\xi_{\Delta^*}(s_j,s_{j+1})=0$.
This proves our claim that $\xi_{\Delta^*}(s_j,s_{j+1})<i_0$.
Since for $i=0,1,\ldots,k$ we have
$Z^{**}_{r_i}\cap Z^{**}_{r_{i+1}}\subseteq (Z_{s_{b_i}}\cap Z_{s_{b_i+1}})\cup
\{r_i,s_{b_i}\}$,
and $\xi_{\Delta^{**}}(r_i,r_{i+1})=i_0$,
we deduce that
$$|Z^{**}_{r_i}\cap Z^{**}_{r_{i+1}}|\le
|Z_{s_{b_i}}\cap Z_{s_{b_i+1}}|+2\le
d+\xi_{\Delta^*}(s_{b_i},s_{b_i+1})\le
d+2\xi_{\Delta^{**}}(r_i,r_{i+1}).$$
Thus the existence of $(G^{**},\Omega^{**})$ contradicts
the choice of $(G^*,\Omega^*)$.
This completes our proof that $C_1,C_2,\ldots,C_s$ are subgraphs of $G^*$,
and hence $(G^*,\Omega^*)$ is $s$-nested, as desired.~$\square$\bigskip
\noindent
{\bf Proof of Theorem~\ref{thm:society}.}
Let $d$ be as in Theorem~\ref{gm9}, and let $k$ be
as in Corollary~\ref{thm3} applied to $2(d+2s)$ in place of $d$.
We claim that $k$ satisfies Theorem~\ref{thm:society}.
To prove that let $(G,\Omega)$ be a $6$-connected $s$-nested $k$-cosmopolitan
society that is not nearly rural.
Since $(G,\Omega)$ is an $s$-nested planar truncation of itself,
by Theorem~\ref{gm9} we may assume that $(G,\Omega)$ has
either a leap of length five, in which case it satisfies
Theorem~\ref{thm:society} by Theorem~\ref{thm:leap},
or it has a planar truncation of depth at most $d$.
In the latter case it has an $s$-nested planar truncation $(G',\Omega')$
of depth at most
$2(d+2s)$ by Lemma~\ref{nestedtrunk}, and
the theorem follows from Corollary~\ref{thm3}
applied to the society $(G',\Omega')$.~$\square$\bigskip
\section{Finding a planar nest}
\label{sec:nest}
\showlabel{sec:nest}
In this section we prove a technical result that applies in the
following situation. We will be able to guarantee
that some societies $(G,\Omega)$ contain certain configurations consisting of
disjoint trees connecting specified vertices in $V(\Omega)$.
The main result of this section, Theorem~\ref{legs} below, states that
if the society is sufficiently nested, then we can make sure
that the cycles in some reasonably big nest and the trees of the
configuration intersect nicely.
A {\em target} in a society $(G,\Omega)$ is a subgraph $F$ of $G$ such
that
\par\hangindent0pt\mytextindent{(i)} $F$ is a forest and every leaf of $F$ belongs to
$V(\Omega)$, and
\par\hangindent0pt\mytextindent{(ii)} if $u,v\in V(\Omega)$ belong to a component $T$ of
$F$, then there exists a component $T'\ne T$ of $F$ and $w\in V(T')
\cap V(\Omega)$ such that $(u,w,v)$ is clockwise.
\noindent We say that a vertex $v\in V(G)$ is $F$-{\em special} if
either $v$ has degree at least three in $F$, or $v$ has degree at
least two in $F$ and $v\in V(\Omega)$.
Now let $F$ be a target in $(G,\Omega)$ and let $T$ be a component
of $F$. Let $P$ be a path in $G\backslash V(\Omega)$ with ends $u,v$
such that
$u,v\in V(T)$ and $P$ is otherwise disjoint from $F$. Let $C$ be
the unique cycle in $T\cup P$, and assume that $C$ has at most one
$F$-special vertex. If $C\backslash u\backslash v$ has no $F$-special
vertex, then let $P'$ be the subpath of $C$ that is complementary to
$P$, and if $C\backslash u\backslash v$ has an $F$-special vertex, say
$w$, then let $P'$ be either the subpath of $C\backslash u$ with ends
$v$ and $w$, or the subpath of $C\backslash v$ with ends $u$ and
$w$. Finally, let $F'$ be obtained from $F\cup P$ by deleting all
edges and internal vertices of $P'$. In those circumstances we say that
$F'$ was obtained from $F$ by {\em rerouting}.
A subgraph $F$ of a rural neighborhood $(G,\Omega,\Omega_0)$ is
{\em perpendicular} to an $s$-nest $(C_1,C_2,\dots, C_s)$ if for every
component $P$ of $F$
\par\hangindent0pt\mytextindent{(i)} $P$ is a path with one end in $V(\Omega)$ and the other
in $V(\Omega_0)$, and
\par\hangindent0pt\mytextindent{(ii)} $P\cap C_i$ is a path for all $i=1,2,\dots, s$.
The complexity of a forest $F$ in a society $(G,\Omega)$ is
$$\sum (\mbox{deg}_F(v)-2)^++\sum_{v\in V(\Omega)} (\mbox{deg}_F(v)-1)^+,$$
where the first summation is over all $v\in V(G)-V(\Omega)$ and $x^+$ denotes
$\max (x,0)$.
The following is a preliminary version of the main result of this section.
\begin{theorem}\label{legsbdedtw}
\showlabel{legsbdedtw}
Let $w,s,k$ be positive integers, and let $s'=2w(k+1)+s$. Then for every
$s'$-nested society $(G,\Omega)$ such that $G$ has tree-width
at most $w$ and for every target $F_0$ in $(G,\Omega)$ of complexity at most $k$
there exists a target $F$ in $(G,\Omega)$ obtained from $F_0$ by repeated
rerouting such that $(G,\Omega)$ can be expressed as a composition of
some society with a rural neighborhood $(G',\Omega,\Omega')$ that
has a presentation with an $s$-nest $(C_1,C_2,\dots, C_s)$ such that
$G'\cap F$ is perpendicular to $(C_1,C_2,\dots, C_s)$.
\end{theorem}
{\noindent\bf Proof. } Suppose that the theorem is false for some integers $w,s,k$,
a society $(G,\Omega)$ and target $F_0$, and choose these entities with
$|V(G)|+|E(G)|$ minimum.
Let $(G,\Omega)$ be the composition
of a society $(G_0,\Omega_0)$ with a rural neighborhood $(G_1,\Omega,\Omega_0)$.
Let $\kappa$ be the complexity of $F\cap G_1$ in the society $(G_1,\Omega)$, and
let $s''=2w(\kappa+1)+s$.
Since $(G,\Omega)$ is $s'$-nested and $s''\le s'$ we may choose
a presentation $(\Sigma, \Gamma,\Delta,\Delta_0)$ of $(G_1,\Omega,\Omega_0)$
and an $s''$-nest $(C_1,C_2,\dots, C_{s''})$ for it.
We may assume that $G_0,\Omega_0, G_1,F,\Sigma,
\Gamma, \Delta,\Delta_0,C_1,C_2,\dots, C_{s''}$ are chosen to minimize
$\kappa$. The minimality of $G$ implies that $G=C_1\cup C_2\cup\cdots
\cup C_{s'}\cup F$. Likewise, $C_1\cup C_2\cup\cdots\cup C_{s'}$ is
edge-disjoint from $F$, for otherwise contracting an edge belonging to the
intersection of the two graphs contradicts the minimality of $G$.
By a {\em dive} we mean a subpath of $F\cap G_1$ with both ends in
$V(\Omega_0)$ and otherwise disjoint from $V(\Omega_0)$.
Let $P$ be a dive with ends $u,v$, and let $P'$ be the corresponding
path in $\Gamma$. Then $\Delta_0\cup P'$ separates $\Sigma$;
let $\Delta (P')$ denote the component of $\Sigma-\Delta_0-P'$
that is contained in $\Delta$, and let $H(P)$ denote the subgraph of $G_1$
consisting of all vertices and edges that correspond to vertices or
edges of $\Gamma$ that belong to the closure of $\Delta (P')$. Thus
$P$ is a subgraph of $H(P)$.
We say that a dive $P$ is {\em clean} if $H(P)\backslash V(\Omega_0)$
includes at most one $F$-special vertex, and if it includes one, say
$v$, then $v\in V(P)$, and no edge of $E(F)-E(P)$ incident with
$v$ belongs to $H(P)$.
The {\em depth} of a dive $P$ is the maximum integer $d\in \{1,2,\dots, s'\}$
such that $V(P)\cap V(C_d)\ne\emptyset$, or 0 if no such integer exists.
It follows from planarity that $|V(P)\cap V(C_i)|\ge 2$ for all
$i=1,2,\dots, d-1$.
\myclaim{1}
Every clean dive has depth at most $2w$.\par
\medskip
To prove the claim suppose for a contradiction that $P_1$ is a clean
dive of depth $d\ge 2w+1$. Thus $V(P_1)\cap V(C_d)\ne\emptyset$.
Assume that we have already constructed dives $P_1,P_2,\dots, P_t$ for
some $t\le w$ such that $V(P_i)\cap V(C_{d-i+1})\ne\emptyset$ for all
$i=1,2,\dots, t$ and $H(P_t)\subseteq H(P_{t-1})\subseteq\cdots
\subseteq H(P_1)$. Since $V(P_t)\cap V(C_{d-t+1})\ne\emptyset$, there
exist distinct vertices $x,y\in V(P_t)\cap V(C_{d-t})$. Furthermore, it is
possible to select $x,y$ such that one of subpaths of $C_{d-t}$
with ends $x,y$, say $Q$, is a subgraph of $H(P_t)$ and no internal vertex of
$Q$ belongs to $P_t$.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = 1]{dive.eps}
\end{center}
\caption{Construction of $H(P_{t+1})$.
}
\label{fig:dive}
\end{figure}
\showfiglabel{fig:dive}
We claim that some internal vertex of $Q$ belongs to $F$. Indeed, if not,
then we can reroute $xP_ty$ along $Q$ to produce a target $F'$
and delete an edge of $xP_ty$;
since $P_1$ is clean and $H(P_t)$ is a subgraph of $H(P_1)$ this is
indeed a valid rerouting as defined above. But this contradicts
the minimality of $G$, and hence some internal vertex of $Q$, say
$q$, belongs to $F$. Since $P_1$ is clean and $H(P_t)$ is a subgraph of
$H(P_1)$ it follows that $q$ belongs to a dive $P_{t+1}$ that is
a subgraph of $H(P_t)\backslash V(P_t)$. It follows that
$H(P_{t+1})$ is a subgraph of $H(P_t)$, thus completing the construction. (See Figure~\ref{fig:dive}.)
The dives $P_1,P_2,\dots, P_{w+1}$ just constructed are pairwise
disjoint and all intersect $C_{d-w}$. Since $d\ge 2w+1$ this implies that
$P_1,P_2,\dots, P_{w+1}$ all intersect each of $C_1,C_2,\dots, C_{w+1}$,
and hence $C_1\cup P_1,C_2\cup P_2,\dots, C_{w+1}\cup P_{w+1}$ is a
``screen" in $G$ of ``thickness" at least $w+1$. By
\cite[Theorem~(1.4)]{SeyThoSearch} the graph $G$ has tree-width
at least $w$, a contradiction. This proves (1).
\medskip
Our next objective is to prove that $\kappa=0$. That will take several
steps. To that end let us define a dive $P$ to be {\em special}
if $P\backslash V(\Omega_0)$ contains exactly one $F$-special vertex.
By a {\em bridge} we mean a subgraph $B$ of $G_1\cap F$ consisting of
a component $C$ of $G_1\backslash V(\Omega_0)$ together with all edges from
$V(C)$ to $V(\Omega_0)$ and all ends of these edges.
\myclaim{2}
If a bridge $B$ includes an $F$-special vertex not in $V(\Omega_0)$,
then $B$ includes a special dive.\par
\medskip
To prove Claim (2)
let $B$ be a bridge containing an
$F$-special vertex not in $V(\Omega_0)$.
For an $F$-special vertex $b\in V(B)-V(\Omega_0)$
and an edge $e\in E(B)$ incident with
$b$ let $P_e$ be the maximal subpath of $B$ containing $e$ such that one
end of $P_e$ is $b$ and no internal vertex of $P_e$ is $F$-special
or belongs to $V(\Omega_0)$.
Let $u_e$ be the other end of $P_e$.
The second axiom in the definition of target implies that at most one vertex
of $F$ belongs to $V(\Omega)$.
Since every $F$-special vertex in $V(G_1)-V(\Omega)$ has degree at least three,
it follows that there exists an $F$-special vertex $b\in V(B)-V(\Omega_0)$ such that
$u_{e_1},u_{e_2}\in V(\Omega_0)$ for two distinct edges $e_1,e_2\in E(B)$
incident with $b$.
Then $P_{e_1}\cup P_{e_2}$
is as desired. This proves (2).
\medskip
By (2) we may select a special dive $P$ with
$H(P)$ minimal. We claim that $P$ is clean. For let $v\in V(P)-V(\Omega_0)$
be $F$-special. If some edge $e\in E(F)-E(P)$ incident with $v$ belongs
to $H(P)$, then there exists a subpath $P'$ of $F$ containing $e$
with one end $v$ and the other end in $V(\Omega_0)\cup V(\Omega)$.
But $P'$ is a subgraph of $H(P)$, and hence the other end of $P'$ belongs
to $V(\Omega_0)$ by planarity. It follows that $P\cup P'$ includes a
dive
that contradicts the minimality of $H(P)$.
This proves that the edge $e$ as above does not exist.
It remains to show that no vertex of $H(P)\backslash V(\Omega_0)$ except
$v$ is $F$-special. So suppose for a contradiction that such
vertex, say $v'$, exists. Then $v'\not\in V(P)$, because $P$ is special,
and hence $v'$ belongs to a bridge $B'\ne B$. But $B'$ includes a special
dive by (2),
contrary to the choice of $P$.
This proves our claim that $P$ is clean.
By (1)
$P$ has depth at most $2w$. In particular,
the image under $\Gamma$ of some $F$-special vertex belongs to the
open disk $\Delta_{2w+1}$ bounded by the image under $\Gamma$ of
$C_{2w+1}$. Let $G'_0$ consist of $G_0$ and all vertices and edges of
$G$ whose images under $\Gamma$ belong to the closure of $\Delta_{2w+1}$,
let $G'_1$ consist of all vertices and edges whose images under
$\Gamma$ belong to the complement of $\Delta_{2w+1}$, and let
$\Omega'_0$ be defined by $V(\Omega'_0)=V(C_{2w+1})$ and let the cyclic
order of $\Omega'_0$ be determined by the order of
$V(C_{2w+1})$. Then $(G,\Omega)$ can be regarded as a composition
of $(G'_0,\Omega'_0)$ with the rural neighborhood $(G'_1,\Omega,\Omega'_0)$.
This rural neighborhood has a presentation with a $\sigma$-nest,
where $\sigma=2w\kappa+s$. On the other hand, the complexity
of $F\cap G'_1$ is at most $\kappa-1$, contrary to the minimality of
$\kappa$. This proves our claim that $\kappa=0$.
By repeating the argument of the previous paragraph and sacrificing
$2w$ of the cycles $C_i$ we may assume that $(G_1,\Omega,\Omega_0)$
has a presentation with an $s$-nest $C_1,C_2,\ldots,C_s$ and that
there are no dives.
It follows that every component $P$ of $F\cap G_1$ is a path with
one end in $V(\Omega)$ and the other in $V(\Omega_0)$. To complete the proof
of the theorem we must show that $P\cap C_i$ is a path for all
$ i=1,2,\dots, s$. Suppose for a contradiction that that is not
the case. Thus for some $i\in \{1,2,\dots, s\}$ and some component $P$
of $F\cap G_1$ the intersection $P\cap C_i$ is not a path.
Thus there exist distinct vertices $x,y\in V(P\cap C_i)$ such that
$xPy$ is a path with no edge or internal vertex in $C_i$. Let us choose
$P,i,x,y$ such that, subject to the conditions stated,
$i$ is maximum. If $i<s$ and $xPy$ intersects $C_{i+1}$, then
$P\cap C_{i+1}$ is not a path, contrary to the choice of $i$.
If $i=1$ or $xPy$ does not intersect $C_{i-1}$, then by rerouting one of the
subpaths of $C_i$ with ends $x,y$ along $xPy$ we obtain
contradiction to the minimality of $G$. Thus we may assume
that $i>1$ and that $xPy$ intersects $C_{i-1}$.
Exactly one of the subpaths of $C_i$ with ends $x,y$, say $Q$,
has the property that the image under $\Gamma$ of $xPy\cup Q$ bounds
a disk contained in $\Delta$ and disjoint from $\Delta_0$. If no
component of $F\cap G_1$ other than $P$ intersects $Q$, then by
rerouting $F$ along $Q$ we obtain a contradiction to the minimality of
$G$. Thus there exists a component $P'$ of $F\cap G_1$ other that $P$
that intersects $Q$, say in a vertex $u$. The vertex $u$ divides $P'$
into two subpaths $P_1'$ and $P_2'$. If both $P_1'$ and $P_2'$ intersect
$C_{i+1}$, then $P'$ contradicts the choice of $i$. Thus we may
assume that say $P_1'$ does not intersect $C_{i+1}$. But $P_1'$
includes a subpath $P''$ with both ends on $C_i$ and otherwise
disjoint from $C_1\cup C_2\cup\cdots\cup C_s$, and hence by rerouting $C_i$
along $P''$ we obtain a contradiction to the minimality of $G$. This
completes the proof of the theorem.~$\square$\bigskip
Before we state the main result of this section we need the
following deep result from \cite{RobSeyGM21}.
A {\em linkage} in a graph $G$ is a subgraph of $G$, every component of
which is a path. A linkage $L$ in a graph $G$ is {\em vital} if
$V(L)=V(G)$ and there is no linkage $L'\ne L$ in $G$ such that for
every two vertices $u,v\in V(G)$, the vertices $u,v$ are the ends of a
component of $L$ if and only if they are the ends of a component of $L'$.
\begin{theorem}\label{vitallinkage}
\showlabel{vitallinkage}
For every integer $p\ge 0$ there exists an integer $w$ such that
every graph that has a vital linkage with $p$ components has
tree-width at most $w$.
\end{theorem}
Now we are ready to state and prove the main theorem of this
section. If $F$ is a target in a society $(G,\Omega)$ we say that a
vertex $v\in V(G)$ is {\em critical} for $F$ if $v$ is either
$F$-special or a leaf of $F$. We say that two targets $F,F'$
are {\em hypomorphic} if they have the same set of critical vertices,
say $X$, and $u,v\in X$ are joined by a path in $F$ with no internal
vertices in $X$ if and only if they are so joined in $F'$.
\begin{theorem}\label{legs}
\showlabel{legs}
For every two positive integers $s,k$ there exists an integer
$s'$ such that for every $s'$-nested society $(G,\Omega)$
and for every target $F$ in $(G,\Omega)$ of complexity at most
$k$ there exists a target $F$ in $(G,\Omega)$ obtained from a target
hypomorphic to $F_0$ by repeated rerouting such that $(G,\Omega)$
can be expressed as a composition of some society with a rural
neighborhood $(G', \Omega,\Omega')$ that has a presentation
with an $s$-nest $(C_1,C_2,\dots, C_s)$ such that $G'\cap F$ is
perpendicular to $(C_1,C_2,\dots, C_s)$.
\end{theorem}
{\noindent\bf Proof. } We proceed by induction on $|V(G)| + |E(G)|$. Let $p=k+2$,
and let $w$ be the bound guaranteed by Theorem~\ref{vitallinkage}.
By hypothesis $(G,\Omega)$ is the composition of a society
$(G_0,\Omega_0)$ with a rural neighborhood $(G_1,\Omega,\Omega_0)$,
where $(G_1,\Omega,\Omega_0)$ has a presentation $(\Sigma,\Gamma,\Delta,
\Delta_0)$ and an $s'$-nest $(C_1,C_2,\dots, C_{s'})$. Let
$X$ be the set of all vertices critical for $F$, and
let $L=F\backslash X$. Then $L$ is a linkage in $G\backslash X$.
If it is vital, then $G$ has tree-width at most $|X|+w\le 2k+1+w$,
and hence the theorem follows from Theorem~\ref{legsbdedtw}.
Thus we may assume that $L$ is not vital. Assume first that there
exists a vertex $v\in V(G)-V(L)$. If $v\in V(C_i)$ for some
$i\in \{1,2,\dots, s'\}$, then the theorem follows by induction
applied to the graph obtained from $G$ by contracting one of the
edges of $C_i$ incident with $v$; otherwise, the theorem follows
by induction applied to the graph $G\backslash v$.
Thus we may assume that $V(L)=V(G)$, and hence there exists a linkage
$L'\ne L$ linking the same pairs of terminals. Thus there exists an
edge $e\in E(L)-E(L')$. If $e\in E(C_i)$ for some $i\in \{1,2,\dots, s'\}$,
then the theorem follows by induction by contracting the edge $e$;
otherwise it follows by induction by deleting $e$, because the
linkage $L'$ guarantees that $G\backslash e$ has a target
hypomorphic to $F$.~$\square$\bigskip
\section{Chasing a turtle}
\label{sec:turtle}
\showlabel{sec:turtle}
In this section we prove Theorem~\ref{thm:largejorgensen},
but first we need
the following two theorems.
\begin{theorem}
\label{thm4}
\showlabel{thm4}
There is an integer $s$ such that if an $s$-nested society
$(G,\Omega)$ has a turtle, then $G$ has a $K_6$ minor.
\end{theorem}
{\noindent\bf Proof. }
Let $k$ be the maximum complexity of a turtle, let $s=3$,
and let $s'$ be as in Theorem~\ref{legs}.
We claim that $s'$ satisfies the theorem.
Indeed, let $(G,\Omega)$ be an $s'$-nested society that has a turtle.
Since every turtle is a target, and every target obtained from
a target hypomorphic to a turtle is again a turtle,
we deduce from Theorem~\ref{legs}%
\REMARK{check for problems with rerouting, also for gridlet, etc.}
that $(G,\Omega)$
has a turtle $F$ and can be
expressed as a composition of a society with a rural neighborhood
$(G',\Omega,\Omega')$
that has a presentation with a $3$-nest $(C_1,C_2,C_3)$ such that
$G'\cap F$ is perpendicular to $(C_1,C_2,C_3)$.
It is now fairly straightforward to deduce that $G$ has a $K_6$ minor.
The argument is illustrated in Figure~\ref{fig:turtlek6}.~$\square$\bigskip
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = .7]{turtlek6.eps}
\end{center}
\caption{
A turtle giving rise to a $K_6$ minor.
}
\label{fig:turtlek6}
\end{figure}
\showfiglabel{fig:turtlek6}
\begin{theorem}
\label{thm5}
\showlabel{thm5}
There is an integer $s$ such that if an $s$-nested society
$(G,\Omega)$ has three crossed paths, a separated doublecross or a gridlet,
then $G$ has a $K_6$ minor.
\end{theorem}
{\noindent\bf Proof. }
The argument is analogous to the proof of the previous theorem, using
Figures~\ref{fig:3xpaths}, \ref{fig:gridlet}
and~\ref{fig:dblecross} instead. We omit the details.~$\square$\bigskip
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = .9]{3xpaths.eps}
\end{center}
\caption{
Three crossed paths giving rise to a $K_6$ minor.
}
\label{fig:3xpaths}
\end{figure}
\showfiglabel{fig:3xpaths}
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = .7]{gridletk6.eps}
\end{center}
\caption{
A gridlet giving rise to a $K_6$ minor.
}
\label{fig:gridlet}
\end{figure}
\showfiglabel{fig:gridlet}
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = .7]{dblcrossk6.eps}
\end{center}
\caption{
A separated doublecross giving rise to a $K_6$ minor.
}
\label{fig:dblecross}
\end{figure}
\showfiglabel{fig:dblecross}
\noindent
{\bf Proof of Theorem~\ref{thm:largejorgensen}.}
Let $s$ be an integer large enough that both Theorem~\ref{thm4} and
Theorem~\ref{thm5} hold for $s$.
Let $k$ be an integer such that Theorem~\ref{thm:society} holds for
this integer.
Let $t$ be such that Theorem~\ref{cosmopolitan} holds for $t$ and
the integer $k$ just defined.
Let $h$ be an integer such that Theorem~\ref{thm:plwall} holds with
$t$ replaced by $t+2s$.
Let $w$ be an integer such that Theorem~\ref{thm:grid} holds for the
integer $h$ just defined.
Finally, let $N$ be as in Theorem~\ref{thm:bdedtwjorgensen}.
Suppose for a contradiction that $G$ is a 6-connected graph on at least
$N$ vertices that is not apex. By Theorem~\ref{thm:bdedtwjorgensen}
$G$ has tree-width
exceeding $w$. By Theorem~\ref{thm:grid} $G$ has a wall of height $h$.
By Theorem~\ref{thm:plwall} $G$ has a planar wall $H_0$ of
height $t+2s$. By considering a subwall $H$ of $H_0$
of height $t$ and $s$ cycles of $H_0\backslash V(H)$ we find, by
Theorem~\ref{cosmopolitan}, that the anticompass society $(K,\Omega)$
of $H$ in $G$ is $s$-nested and $k$-cosmopolitan.
By Theorem~\ref{thm:society}
the society $(K,\Omega)$ has a turtle,
three crossed paths, a separated doublecross, or a gridlet.
By Theorems~\ref{thm4} and \ref{thm5} the graph
$G$ has a $K_6$ minor, a contradiction.~$\square$\bigskip
\section*{Acknowledgment}
We would like to acknowledge the contributions of Matthew DeVos and
Rajneesh Hegde, who worked
with us in March 2005 and contributed to this paper,
but did not want to be included as a coauthors.
\section{Leap of length five}
\label{sec:leap}
\showlabel{sec:leap}
A {\em leap of length} $k$ in a society $(G,\Omega)$ is a sequence of $k+1$
pairwise disjoint bumps $P_0,P_1,\dots, P_k$ such that $P_i$ has
ends $u_i$ and $v_i$ and $u_0,u_1,u_2,\dots, u_k, v_0, v_k,v_{k-1},\dots,
v_1$, is clockwise. In this section we prove the following.
\begin{theorem}\label{thm:leap}
\showlabel{thm:leap}
Let $(G,\Omega)$ be a $6$-connected society with a leap of length five.
Then $(G,\Omega)$ is nearly rural, or $G$ has a triangle $C$ such that
$(G\backslash E(C),\Omega)$ is rural, or $(G,\Omega)$ has three crossed paths,
a gridlet, a separated doublecross, or a turtle.
\end{theorem}
The following is a hypothesis that will be common to several lemmas
of this section, and so we state it separately to avoid repetition.
\begin{hypothesis}
\label{hyp:leap}
\showlabel{hyp:leap}
Let $(G,\Omega)$ be a society with no three crossed paths,
a gridlet, a separated doublecross, or a turtle,
let $k\ge1$ be an integer, let
$$(u_0,u_1,u_2,\ldots,u_k,v_0,v_k,v_{k-1},\ldots,v_1)$$
be clockwise, and let $P_0,P_1,\ldots,P_k$ be pairwise disjoint bumps
such that $P_i$ has ends $u_i$ and $v_i$.
Let $\cal T$ be the orderly transaction $(P_1,P_2,\ldots,P_k)$, let $M$ be the frame of $\cal T$ and let
$$Z=u_1\Omega u_k\cup v_k\Omega v_1\cup V(P_2)\cup V(P_3)\cup\cdots\cup
V(P_{k-1})-\{u_1,u_k,v_1,v_k\}.$$
Let $Z_1=v_1\Omega u_1-\{u_0,u_1,v_1\}$ and $Z_2=u_k\Omega v_k-\{v_0,u_k,v_k\}$.
\end{hypothesis}
If $H$ is a subgraph of $G$, then an \emph{$H$-path} is a (possibly trivial) path with both ends in $V(H)$ and otherwise disjoint from $H$.
This is somewhat non-standard, typically an $H$-path is required to have at least one edge, but we use our definition for convenience.
We say that a vertex $v$ of $P_0$ is {\em exposed} if
there exists an $(M\cup P_0)$-path $P$ with one end $v$ and the other in $Z$.
\begin{lemma}
\label{lem:2pathstoZ}
\showlabel{lem:2pathstoZ}
Assume Hypothesis~\ref{hyp:leap} and let $k\ge 3$.
Let $R_1,R_2$ be two disjoint $(M\cup P_0)$-paths in $G$ such that $R_i$
has ends $x_i\in V(P_0)$ and $y_i\in V(M)-\{u_0,v_0\}$, and assume that
$u_0,x_1,x_2,v_0$ occur on $P_0$ in the order listed, where possibly $u_0=x_1$, or $v_0=x_2$, or both.
Then either $y_1\in V(P_1)\cup v_1\Omega u_1$, or
$y_2\in V(P_k)\cup u_k\Omega v_k$, or both.
In particular, there do not exist two disjoint $(M\cup P_0)$-paths from
$V(P_0)$ to $Z$.
\end{lemma}
{\noindent\bf Proof. } The second statement follows immediately from the first,
and so it suffices to prove the first statement.
Suppose for a contradiction that there exist paths $R_1,R_2$
satisfying the hypotheses but not the conclusion of the lemma.
By using the paths $P_2, P_3, \ldots, P_{k-1}$ we conclude that
there exist two disjoint paths
$Q_1,Q_2$ in $G$ such that $Q_i$ has ends
$x_i\in V(P_0)$ and $z_i \in V(\Omega)$,
and is otherwise disjoint from $V(P_0)\cup V(\Omega)$,
and if $Q_i$ intersects some $P_j$ for $j\in\{1,2,\ldots,k\}$,
then $j\in\{2,\ldots,k-1\}$ and $Q_i \cap P_j$ is a path one of whose ends is a common end
of $Q_i$ and $P_j$. Furthermore, $z_1 \in u_1 \Omega v_1-\{u_1,v_1\} $ and $z_2 \in v_k \Omega u_k-\{u_k,v_k\}$.
From the symmetry we may assume that either
$(u_0,v_0,z_2,z_1)$, or $(u_0,z_1,v_0,z_2)$
or $(u_0,v_0,z_1,z_2)$ is clockwise.
In the first two cases $(G,\Omega)$ has a separated doublecross
(the two pairs of crossing bumps are $P_1$ and
$Q_1\cup u_0P_0x_1$, and $P_k$ and $Q_2\cup v_0P_0x_2$, and
the fifth path is a subpath of $P_2$), unless the second case holds and
$z_1 \in u_k \Omega v_0$ or $z_2 \in v_1 \Omega u_0$, or both. By symmetry we may assume that
$z_1 \in u_k \Omega v_0$. Then, if $z_2 \in v_{k-2} \Omega u_0$, $(G,\Omega)$ has a gridlet formed by the paths $P_k,P_{k-1},
u_0P_0x_1 \cup Q_1$ and $v_0P_0x_2 \cup Q_2$. Otherwise, $z_2 \in v_k\Omega v_{k-2} - \{v_k,v_{k-2}\}$ and $(G,\Omega)$ has a turtle
with legs $P_k$ and $v_0P_0x_2 \cup Q_2$, neck $P_1$ and body $u_0P_0x_2 \cup Q_1$.
Finally, in the third case
$(G,\Omega)$ has a turtle or three crossed paths.
More precisely, if $z_2 \in v_0 \Omega v_1 - \{v_1\}$, then $(G,\Omega)$ has a turtle described in the paragraph above.
Otherwise, by symmetry, we may assume that $z_2 \in v_1 \Omega u_0$ and $z_1 \in v_0 \Omega v_k$, in which case $v_0P_0x_2 \cup Q_2$,
$u_0P_0x_1 \cup Q_1$ and $P_2$ are the three crossed paths.~$\square$\bigskip
\begin{lemma}
\mylabel{prejump}
Assume Hypothesis~\ref{hyp:leap} and let $k\ge2$.
Then $(G\backslash V(P_0),\Omega\backslash V(P_0))$ has no $\cal T$-jump.
\end{lemma}
{\noindent\bf Proof. } Suppose for a contradiction that
$(G\backslash V(P_0),\Omega\backslash V(P_0))$
has a $\cal T$-jump.
Thus there is an index $i\in\{1,2,\ldots,k\}$ and a $\cal T$-coterminal
path $P$ in $G\backslash V(P_0\cup P_i)$ with ends
$x\in v_i\Omega u_i$ and $y\in u_i\Omega v_i$.
Let $j\in\{1,2,\ldots,k\}-\{i\}$.
Then using the paths $P_0,P_i,P_j$ and $P$ we deduce that $(G,\Omega)$ has either
three crossed paths or a gridlet,
in either case a contradiction.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:nojump}
Assume Hypothesis~\ref{hyp:leap} and let $k\ge2$.
Let $v\in V(P_0)$ be such that there is no $(M\cup P_0)$-path in $G\backslash v$
from $vP_0v_0$ to $vP_0u_0\cup V(P_1\cup P_2\cup\cdots\cup P_{k-1})
\cup v_k\Omega u_k-\{v_k,u_k\}$
and none from $vP_0u_0$ to
$V(P_2\cup P_3\cup\cdots\cup P_{k})
\cup u_1\Omega v_1-\{u_1,v_1\}$.
Then $(G\backslash v,\Omega\backslash v)$ has no $\cal T$-jump.
\end{lemma}
{\noindent\bf Proof. }
The hypotheses of the lemma imply that every $\cal T$-jump in $(G\backslash v,\Omega\backslash v)$ is disjoint from $P_0$.
Thus the lemma follows from Lemma~\ref{prejump}.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:nocross}
Assume Hypothesis~\ref{hyp:leap}, let $k\ge3$, and let
$v\in V(P_0)$ be such that no vertex in $V(P_0)-\{v\}$ is exposed.
Let $i\in\{0,1,\ldots,k\}$ be such that
$(G\backslash v,\Omega\backslash v)$ has
a $\cal T$-cross $(Q_1,Q_2)$ in region $i$.
Then $i\in\{0,k\}$ and $v$ is not exposed.
Furthermore, assume that $i=0$, and that there exists an $(M\cup P_0)$-path $Q$
with one end $v$ and the other end in $P_1\cup v_1\Omega u_1-\{u_0\}$,
and that $v_0P_0v$ is disjoint from $Q_1\cup Q_2$. Then for some $j\in\{1,2\}$
there exist $p \in V(Q_j\cap u_0P_0v)$ and $q \in V(Q_j\cap Q)$
such that $pP_0v$ and $qQv$ are internally disjoint from $Q_1\cup Q_2$.
\end{lemma}
{\noindent\bf Proof. }
If $i\not\in\{0,k\}$, then the $\cal T$-cross is disjoint from
$P_0$ by the choice of $v$, and hence the $\cal T$-cross and $P_0$ give
rise to three crossed paths.
To complete the proof of the first assertion we may assume that
$i=0$ and that $v$ is exposed. Subject to these assumptions we choose $Q_1$ and $Q_2$ so that
$Q_1 \cup Q_2 \cup P_0$ is minimal. Since $v$ is exposed there exists a $\cal T$-coterminal path $Q'$ from $v$ to
$y \in Z\cap V(\Omega)$ disjoint from $P_0\cup P_1\cup P_k\backslash v$. Let $Q''=Q'\cup vP_0v_0$.
If $Q'' \cap (Q_1 \cup Q_2) = \emptyset$ then $(G,\Omega)$ has a separated doublecross, where
one pair of crossed paths is obtained from the $\cal T$-cross,
the other pair is $P_k$ and $Q''$, and the fifth path is
a subpath of $P_2$. Thus we may assume that there exists $x \in V(Q'') \cap V(Q_j)$ for some $j \in \{1,2\}$ and that
$x$ is chosen so that $xQ''y$ is internally disjoint from $Q_1 \cup Q_2$. For $r=1,2$ let $z_r \in v_1\Omega u_1 - \{v_1,u_1\}$
be an end of $Q_r$ such that $Q_{3-r}$ has one end in $z_r\Omega v_0$ and another in $v_0\Omega z_r$. If $x \in V(Q')$, then $Q_j$ is disjoint from $P_0$, because $v$
is the only exposed vertex and $v \not \in V(Q_1) \cup V(Q_2)$. Thus $z_jQ_jx \cup xQ'y$ is a $\cal T$-jump
disjoint from $P_0$, contrary to Lemma~\ref{prejump}. It follows that $x \in V(v_0P_0v)$, and
$Q'$ is disjoint from $Q_1 \cup Q_2$.
Let $x' \in V(P_0) \cap (V(Q_1) \cup V(Q_2))$ be chosen so that $x'P_0v_0$ is internally disjoint from $Q_1 \cup Q_2$. Without loss of generality, we assume that $x' \in V(Q_1)$. Define $P_0'=v_0P_0x' \cup x'Q_1z_1$. Let $x'' \in vP_0u_0 \cap (V(P'_0) \cup V(Q_2) \cup \{u_0\})$ be chosen so that $vP_0x''$ is internally
disjoint from $P'_0 \cup Q_2$. If $x'' \not \in V(P'_0)$ then the path $Q' \cup vP_0u_0$, if $x'' =u_0$, or the path $Q' \cup vP_0x'' \cup x''Q_2z_2$, if $x'' \in V(Q_2)$ is a $\cal T$-jump, disjoint from $P_0'$, contradicting Lemma~\ref{prejump}. (See Figure~\ref{fig:leaplem46}(a).) If $x'' \in V(P'_0)$ then $x''P_0v \cup Q'$ and $Q_1 \backslash (V(P_0)-\{x'\})$ are paths with one end in $V(P_0')$ and another in $V(\Omega)$, contradicting Lemma~\ref{lem:2pathstoZ}, after we replace $P_0$ by $P_0'$ and $P_1$ by $Q_2$ in $M$. (See Figure~\ref{fig:leaplem46}(b).) This proves the first assertion of the lemma.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = 0.8]{LeapLem46.eps}
\end{center}
\caption{
Configurations considered in the proof of the first assertion of Lemma~\ref{lem:nocross}.
}
\label{fig:leaplem46}
\end{figure}
\showfiglabel{fig:leaplem46}
To prove the second statement of the lemma we assume that $i=0$
and that $Q$ is a path from $v$
to $v'\in v_1\Omega u_1-\{u_0\}$, disjoint from $M\cup P_0\backslash v$,
except that $P_1\cap Q$ may be a path with one end $v'$.
Let
the ends of $Q_1,Q_2$ be labeled as in the definition of $\cal T$-cross.
If $P_0$ is disjoint from $Q_1\cup Q_2$, then $(G,\Omega)$ has
three crossed paths (if $(y_2,u_0,x_1)$ is clockwise) or
a gridlet with paths $Q_1,Q_2,P_0,P_2$ (if $(x_1,u_0,x_2)$
or $(y_1,u_0,y_2)$ is clockwise), or a separated doublecross
with paths $Q_1,Q_2,P_0,P_2,P_k$ (if $(v_1,u_0,y_1)$
or $(x_2,u_0,u_1)$ is clockwise).
Thus we may assume that $P_0$ intersects $Q_1\cup Q_2$.
(Please note that $v_0P_0v$ is disjoint from $Q_1\cup Q_2$ by hypothesis.)
Similarly we may assume that $Q$ intersects $Q_1\cup Q_2$,
for otherwise we apply the previous argument with $P_0$ replaced by
$Q\cup vP_0v_0$.
Let $p\in V(Q_1\cup Q_2)\cap u_0P_0v$ and $q\in V(Q_1\cup Q_2)\cap V(Q)$
be chosen to minimize $pP_0v$ and $qQv$.
If $p$ and $q$ belong to different paths $Q_1,Q_2$, then
$(G,\Omega)$ has a turtle with legs $Q_1,Q_2$, neck
$P_k$ and body $pP_0v_0\cup qQv$.
Thus $p$ and $q$ belong to the same $Q_j$ and the lemma holds.~$\square$\bigskip
In the proof of the following lemma we will be applying Lemma~\ref{orderlyrural1}.
To guarantee that the conditions of Lemma~\ref{orderlyrural1} are satisfied,
we will need a result from~\cite{KawNorThoWolbdtw}.
We need to precede the statement of this result by a few definitions.
Let $M$ be a subgraph of a graph $G$, such that no block of $M$ is a cycle.
Let $P$ be a segment of $M$ of length at least two, and let $Q$ be a path
in $G$ with ends $x,y\in V(P)$ and otherwise disjoint from $M$.
Let $M'$ be obtained from $M$ by replacing the path $xPy$ by $Q$;
then we say that $M'$ was obtained from $M$ by {\em rerouting} $P$ along $Q$, or
simply that $M'$ was obtained from $M$ by {\em rerouting}. Please
note that $P$ is required to have length at least two,
and hence this relation is not symmetric. We say that the
rerouting is {\em proper} if all the attachments
of the $M$-bridge that contains $Q$ belong to $P$.
The following is proved in~\cite[Lemma~2.1]{KawNorThoWolbdtw}.
\begin{lemma
\mylabel{prestable}
Let $G$ be a graph,
and let $M$ be a subgraph of $G$
such that no block of $M$ is a cycle.
Then there exists a subgraph $M'$ of $G$
obtained from $M$ by a sequence of proper reroutings
such that if an $M'$-bridge $B$ of $G$ is unstable, say all its attachments
belong to a segment $P$ of $M'$, then there exist
vertices
$x,y\in V(P)$ such that
some component of $G\backslash\{x,y\}$ includes a vertex of $B$
and is disjoint from $M\backslash V(P)$.
\end{lemma}
\begin{lemma}
\mylabel{lem:1exposed}
Assume Hypothesis~\ref{hyp:leap}, and let $k\ge 4$.
If every leap of length $k-1$ has at most one exposed vertex,
$(G,\Omega)$ is $4$-connected and
$(G\backslash v,\Omega\backslash v)$ is rurally $4$-connected for
every $v\in V(P_0)$,
then $(G,\Omega)$ is nearly rural.
\end{lemma}
{\noindent\bf Proof. }
Since $(G,\Omega)$ has no separated doublecross it follows that
it does not have a $\cal T$-cross both in region $0$ and region $k$.
Thus we may assume that it has no $\cal T$-cross in region $k$.
Similarly, it follows that it does not have a $\cal T$-tunnel under both
$P_1$ and $P_k$, or a $\cal T$-cross in region $0$ and a
$\cal T$-tunnel under $P_k$.
Thus we may also assume that $(G,\Omega)$ has no $\cal T$-tunnel under $P_k$.
If some leap of length $k$ in $(G, \Omega)$ has an exposed vertex, then we
may assume that $v$ is an exposed vertex.
Otherwise, let the leap $(P_0,P_1,\ldots,P_k)$ and $v\in V(P_0)$
be chosen such that either $v=u_0$ or
there exists an $(M\cup P_0)$-path with one end
$v$ and the other end in $P_1\cup v_1\Omega u_1-\{u_0\}$,
and, subject to that, $vP_0v_0$ is as short as possible.
By Lemma~\ref{prestable} we may assume,
by properly rerouting $M$ if necessary, that every
$M$-bridge of $G\backslash v$ is stable.
Since the reroutings are proper the new paths $P_i$ will still be
disjoint from $P_0$, and the property that defines $v$ will continue to hold.
Similarly, the facts that there is no $\cal T$-cross in region $k$
and no $\cal T$-tunnel under $P_k$ remain unaffected.
We claim that $(G\backslash v,\Omega\backslash v)$ is rural.
We apply Lemma~\ref{orderlyrural1} to the society
$(G\backslash v,\Omega\backslash v)$ and orderly transaction $\cal T$.
We may assume that $(G\backslash v,\Omega\backslash v)$
is not rural, and hence
by Lemma~\ref{orderlyrural1} the society $(G\backslash v,\Omega\backslash v)$
has a $\cal T$-jump, a $\cal T$-cross or a $\cal T$-tunnel.
By the choice of $v$ there exists a path $Q$ from $v$ to
$v'\in v_k\Omega u_k-\{v_k,u_k\}$ such that $Q$ does not intersect
$P_k\cup P_0\backslash v$ and intersects at most one of
$P_1,P_2,\ldots,P_{k-1}$. Furthermore, if it intersects $P_i$ for some $i \in \{1,2,\ldots,k-1\}$ then $P_i \cap Q$
is a path with one end a common end of both. (If $v=u_0$ then we can choose $Q$ to be a one vertex path.)
We claim that $v$ satisfies the hypotheses of Lemma~\ref{lem:nojump}.
To prove this claim suppose for a contradiction that $P$ is an
$(M\cup P_0)$-path violating that hypothesis.
Suppose first that $P$ and $Q$ are disjoint.
Then $P$ joins different components of $P_0\backslash v$
by Lemma~\ref{lem:2pathstoZ}. But then
changing $P_0$ to the unique path in $P_0\cup P$ that does
not use $v$ either produces a leap with at least two exposed vertices,
or contradicts the minimality of $vP_0v_0$.
Thus $P$ and $Q$ intersect. Since no leap of length $k$ has two or
more exposed vertices, it follows that $v$ is not exposed.
Thus $P$ has one end in $u_0P_0v$ by the minimality of $vP_0v_0$,
and the other end in $P_k\cup u_k\Omega v_k$, because $v$ is not
exposed.
But then $P\cup Q$ includes a $\cal T$-jump disjoint from $P_0$,
contrary to Lemma~\ref{prejump}.
This proves our claim that $v$ satisfies the hypotheses of
Lemma~\ref{lem:nojump}.
We conclude that $(G\backslash v,\Omega\backslash v)$ has no
$\cal T$-jump.
Assume now that
$(G\backslash v,\Omega\backslash v)$ has a $\cal T$-cross $(Q_1,Q_2)$
in region $i$ for some integer $i\in\{0,1,\ldots,k\}$.
By the first part of Lemma~\ref{lem:nocross} and the fact that there
is no $\cal T$-cross in region $k$ it follows that $i=0$ and $v$ is not exposed.
We have $v \neq u_0$, for otherwise
$V(P_0) \cap (V(Q_1) \cup V(Q_2))=\emptyset$ and
either $Q_1,Q_2,P_0$ are three crossed paths, or
$Q_1,Q_2,P_0,P_3,P_2$ is a separated double cross in $(G, \Omega)$.
Since $v$ is not exposed we deduce that $Q$ satisfies the requirements of
Lemma~\ref{lem:nocross}. By the first part of
Lemma~\ref{lem:nocross} and the assumption made earlier it follows
that $i=0$ and $v$ is not exposed.
But the existence of $Q$ and the second statement of Lemma~\ref{lem:nocross}
imply that some leap of length $k$ has at least two exposed vertices,
a contradiction. (To see that let $j,p,q$ be as in Lemma~\ref{lem:nocross}. Replace $P_1$ by $Q_{3-j}$ and replace
$P_0$ by a suitable subpath of $Q_j \cup pP_0v_0\cup qQv$.)
We may therefore
assume that $(G\backslash v,\Omega\backslash v)$ has a
$\cal T$-tunnel $(Q_0,Q_1,Q_2)$ under $P_i$ for some $i\in\{1,2,\ldots,k\}$.
Then the leap $L'=(P_0,P_1,\ldots,P_{i-1},P_{i+1},\ldots,P_k)$
of length $k-1\ge3$ has a $\cal T'$-cross, where $\cal T'$ is the
corresponding orderly society, and the result follows in the same
way as above.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:cycleC}
Assume Hypothesis~\ref{hyp:leap} and let $k \geq 3$. If there exist at least two
exposed vertices, then there exists a cycle $C$ and three disjoint
$(M\cup C)$-paths $R_1,R_2,R_3$ such that $R_i$ has ends $x_i\in V(C)$
and $y_i\in V(M)$, $C\backslash \{x_1,x_2,x_3\}$ is disjoint from $M$,
$y_1=u_0$, $y_2=v_0$ and $y_3\in Z$.
\end{lemma}
{\noindent\bf Proof. }
Let $x_1$ be the closest exposed vertex to $u_0$ on $P_0$,
and let $x_2$ be the closest exposed vertex to $v_0$.
Let $R_1=x_1P_0u_0$ and let $R_2=x_2P_0v_0$.
For $i=1,2$ let $S_i$ be an $(M\cup P_0)$-path with one end $x_i$
and the other end in $Z$.
By Lemma~\ref{lem:2pathstoZ} $S_1$ and $S_2$ intersect, and so we may
assume that $S_1\cap S_2$ is a path $R_3$ containing an end of both
$S_1$ and $S_2$, say $y_3$. Let $x_3$ be the other end of $R_3$.
Then $P_0\cup S_1\cup S_2$ includes a unique cycle $C$.
The cycle $C$ and paths $R_1,R_2,R_3$ are as desired for the lemma.~$\square$\bigskip
If the cycle $C$ in Lemma~\ref{lem:cycleC} can be chosen to
have at least four vertices,
then we say that the leap $(P_0,P_1,\ldots,P_k)$ is {\em diverse}.
\begin{lemma}
\label{lem:triangle}
\showlabel{lem:triangle}
Assume Hypothesis~\ref{hyp:leap}, let $k\ge4$, and let there be
no diverse leap of length $k$. If $C$ is as in Lemma~{\rm\ref{lem:cycleC}}
and $(G\backslash E(C),\Omega)$ is rurally $4$-connected,
then $(G\backslash E(C),\Omega)$ is rural.
\end{lemma}
{\noindent\bf Proof. } Since the leap $(P_0,P_1,\ldots,P_k)$ is not diverse,
it follows that $C$ is a triangle. Let $R_1,R_2,R_3$ and their ends
be numbered as in Lemma~\ref{lem:cycleC}.
We may assume that $P_0=R_1\cup R_2+x_1x_2$.
Since there is no diverse leap, Lemma~\ref{lem:2pathstoZ} implies that
there is no path in $G\backslash E(C)\backslash V(P_k)$ from $x_2$
to $v_k\Omega u_k$,
and none in $G\backslash E(C)\backslash V(P_1)$ from $x_1$
to $u_1\Omega v_1$. It also implies that no vertex on $P_0$ is exposed in $G \backslash x_1x_3 \backslash x_2x_3$.
As in Lemma~\ref{lem:1exposed}, we can apply Lemma~\ref{prestable} and assume,
by properly rerouting $M$ if necessary, that the conditions of Lemma~\ref{orderlyrural1} are satisfied.
We assume that
the society $(G\backslash E(C),\Omega)$ has a
$\cal T$-jump, a $\cal T$-cross, or a $\cal T$-tunnel, as otherwise by Lemma~\ref{orderlyrural1} $(G\backslash E(C),\Omega)$ is rural.
By the observation at the end of the previous paragraph this
$\cal T$-jump, $\cal T$-cross, or $\cal T$-tunnel
cannot use both $x_1$ and $x_2$;
say it does not use $x_2$.
But that contradicts
Lemma~\ref{lem:nojump} or the first part of Lemma~\ref{lem:nocross},
applied to $v=x_2$ and the graph $G \backslash x_1x_3$, in case of a $\cal T$-jump or a $\cal T$-cross.
Thus we may assume that $(G\backslash E(C)\backslash x_2,\Omega\backslash x_2)$
has a $\cal T$-tunnel $(Q_0,Q_1,Q_2)$
under $P_i$ for some $i\in\{1,2,\ldots,k\}$.
But then the leap $L'=(P_0,P_1,\ldots,P_{i-1},P_{i+1},\ldots,P_k)$
of length $k-1\ge3$ has a $\cal T'$-cross $(Q_1',Q_2')$,
where $\cal T'$ is the
corresponding orderly transaction, $Q_1'$ is obtained from $P_i$
by rerouting along $Q_0$ and $Q_2'$ is the union of $Q_1\cup Q_2$
with the subpath of $P_i$ joining the ends of $Q_1$ and $Q_2$.
By the first half of Lemma~\ref{lem:nocross} applied to the graph $G \backslash x_1x_3$, the leap $L'$,
$v:=x_2$ and the $\cal T'$-cross $(Q_1',Q_2')$ we may assume that $i=1$ and that $y_3 \in v_2\Omega u_2 - \{u_0\}$.
By the second half of Lemma~\ref{lem:nocross} applied to the same entities
and $Q:=R_3+x_3x_2$ there exist $j\in\{1,2\}$,
$p\in V(Q_j'\cap R_1)$ and $q\in V(Q_j'\cap Q)$
such that $pP_0x_2$ and $qQx_2$ are internally disjoint from $Q_1'\cup Q_2'$.
If $j=1$, then $p,q$ belong to the interior of $Q_0$, and the
leap $(P_0,P_1,\ldots,P_k)$ is diverse, as a subpath of $Q_0$ joins a vertex of $R_1$ to a vertex of $Q$ in
$G \backslash x_1x_3$. If $j=2$ then we obtain a diverse leap from $(P_0,P_1,\ldots,P_k)$
by replacing $P_1$ by $Q_1'$ and replacing $P_0$ by
a suitable subpath of $Q\cup v_0P_0p\cup Q_2'$.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:q4}
Assume Hypothesis~\ref{hyp:leap}, let $k\ge3$,
let $(G,\Omega)$ be $4$-connected,
let $C,R_1,R_2,R_3$ be as in Lemma~{\rm\ref{lem:cycleC}},
and assume that $C$ is not a triangle.
Then there exist four disjoint $(M\cup C)$-paths, each with one end in $V(C)$
and the other end respectively in the sets $\{u_0\}$, $\{v_0\}$, $Z$
and $V(P_1\cup P_k)$.
\end{lemma}
{\noindent\bf Proof. } By an application of the proof of the max-flow min-cut theorem
there exist four disjoint $(M\cup C)$-paths, each with one end in $V(C)$
and the other end respectively in the sets $\{u_0\}$, $\{v_0\}$, $Z$
and $V(M)$.
By Lemma~\ref{lem:2pathstoZ} the fourth path does not end in
$V(M)-V(P_1)-V(P_k)$.
The result follows.~$\square$\bigskip
\begin{lemma}
\mylabel{lem:y4}
Assume Hypothesis~\ref{hyp:leap}, let $k\ge3$,
let $C,R_1,R_2,R_3$ be as in Lemma~{\rm\ref{lem:cycleC}},
let $D:=M\cup C\cup R_1\cup R_2\cup R_3$,
and let $R_4$ be a $D$-path with ends
$x_4\in V(C)-\{x_1,x_2,x_3\}$ and $y_4\in V(P_1)$.
Then $x_1,x_2,x_3,x_4$ occur on $C$ in the order listed.
Furthermore, if $R$ is a $D$-path with ends
$x\in V(C)-\{x_1,x_2,x_3\}$ and $y \in V(M)$, then
$x_1,x_2,x_3,x$ occur on $C$ in the order listed and $y\in V(P_1)$.
\end{lemma}
{\noindent\bf Proof. }
The vertices $x_1,x_2,x_3,x_4$ occur on $C$ in the order listed by
Lemma~\ref{lem:2pathstoZ}. Now let $R$ be as stated.
By Lemma~\ref{lem:2pathstoZ} we have $y\in V(P_1\cup P_k)$,
and so by the first part of the lemma we may assume that $y\in V(P_k)$.
By the symmetric statement to the first half of the lemma
it follows that $x_1,x_2,x,x_3$ occur on $C$ in the order listed.
We may assume that $P_0$ is the unique path from $u_0$ to $v_0$ in
$R_1\cup R_2\cup C\backslash x_3$. Then $R_4\cup R\cup C\backslash V(P_0)$
includes a $\cal T$-jump disjoint from $P_0$,
contrary to Lemma~\ref{prejump}.~$\square$\bigskip
We need to further upgrade the assumptions of Hypothesis~\ref{hyp:leap},
as follows.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[scale = 1]{hypothesis13.eps}
\end{center}
\caption{Hypothesis~\ref{hyp:ray}.
}
\label{fig:hypothesis13}
\end{figure}
\showfiglabel{fig:hypothesis13}
\begin{hypothesis}
\mylabel{hyp:ray}
Assume Hypothesis~\ref{hyp:leap}.
Let $C$ be a cycle with distinct vertices $x_1,x_2,x_3$
such that $C\backslash\{x_1,x_2,x_3\}$ is disjoint from $M$.
Let $R_1,R_2,R_3$ be pairwise disjoint $(M\cup C)$-paths such that
$R_i$ has ends $x_i$ and $y_i$, where $y_1=u_0$, $y_2=v_0$, and $y_3\in Z$.
By a {\em ray} we mean an $(M\cup C)$-path from $C$ to $M$,
disjoint from $R_1\cup R_2\cup R_3$.
We say that a vertex $v\in V(P_1)$ is {\em illuminated} if there
is a ray with end $v$.
Let $x_4,x_5\in V(P_1)$ be illuminated vertices such that either $x_4=x_5$, or
$u_1,x_4,x_5,v_1$ occur on $P_1$ in the order listed,
and $x_4P_1x_5$ includes all illuminated vertices.
Let $R_4:=u_1P_1x_4$ and $R_5:=v_1P_1x_5$, and let $y_4:=u_1$
and $y_5:=v_1$.
Let $S_4$ and $S_5$ be rays with ends $x_4$ and $x_5$, respectively,
and let
$A_0:=V(M)-V(P_1)$ and $B_0:=V(C\cup S_4\cup S_5\cup x_4P_1x_5)$. (See Figure~\ref{fig:hypothesis13}.)
\end{hypothesis}
\begin{lemma}
\mylabel{lem:x4x5}
Assume Hypothesis~\ref{hyp:ray}, let $k\ge 3$, and let
$(G,\Omega)$ be $6$-connected.
Then $x_4\ne x_5$, and the path $x_4P_1x_5$ has at least one internal
vertex.
\end{lemma}
{\noindent\bf Proof. } If $x_4=x_5$ or $x_4P_1x_5$ has no internal vertex,
then by Lemma~\ref{lem:y4} the set $\{x_1,x_2,\ldots,x_5\}$ is a cutset
separating $C$ from $M\backslash V(P_1)$, contrary to the
$6$-connectivity of $(G,\Omega)$. Note that
$V(C) - \{x_1,x_2,\ldots,x_5\}$ is non-empty
as it includes an end of a ray.~$\square$\bigskip
Assume Hypothesis~\ref{hyp:ray}. By Lemma~\ref{lem:x4x5} the
paths $R_1,R_2,\ldots,R_5$ are disjoint paths from $A_0$ to $B_0$.
The following lemma follows by a standard ``augmenting path" argument.
\begin{lemma}
\mylabel{lem:augmentation}
Assume Hypothesis~\ref{hyp:ray}, and let $k\ge 2$.
If there is no separation $(A,B)$ of order at most five with
$A_0\subseteq A$ and $B_0\subseteq B$, then there exist
an integer $n$ and internally
disjoint paths $Q_1,Q_2,\ldots,Q_n$ in $G$, where $Q_i$ has distinct
ends $a_i$ and $b_i$ such that
\par\hangindent0pt\mytextindent{(i)}$a_1\in A_0-\{y_1,y_2,\ldots,y_5\}$ and $
b_n\in B_0-\{x_1,x_2,\ldots,x_5\}$,
\par\hangindent0pt\mytextindent{(ii)}for all $i=1,2,\ldots,n-1$, $a_{i+1},b_{i}\in V(R_t)$
for some $t\in\{1,2,\ldots,5\}$, and $y_t,a_{i+1},b_i,x_t$ are
pairwise distinct and occur on $R_t$ in the order listed,
\par\hangindent0pt\mytextindent{(iii)}if $a_i,b_j\in V(R_t)$ for some $t\in\{1,2,\ldots,5\}$
and $i,j\in\{1,2,\ldots,5\}$ with $i>j+1$, then either $a_i=b_j$, or
$y_t,b_j,a_i,x_t$ occur on $R_t$ in the order listed, and
\par\hangindent0pt\mytextindent{(iv)}for $i=1,2,\ldots,n$, if a vertex of $Q_i$
belongs to $A_0\cup B_0\cup V(R_1\cup R_2\cup\cdots\cup R_5)$,
then it is an end of $Q_i$.
\end{lemma}
The sequence of paths $(Q_1,Q_2,\ldots,Q_n)$ as in Lemma~\ref{lem:augmentation}
will be called an {\em augmenting sequence}.
\begin{lemma}
\mylabel{lem:endinP2}
Assume Hypothesis~\ref{hyp:ray}, and let $k\ge 3$.
Then there is no augmenting sequence $(Q_1,Q_2,\ldots,Q_n)$, where
$Q_1$ is disjoint from $P_2$.
\end{lemma}
{\noindent\bf Proof. } Suppose for a contradiction that there is an augmenting
sequence $(Q_1,Q_2,\ldots,Q_n)$, where
$Q_1$ is disjoint from $P_2$, and let us assume that the leap
$(P_0,P_1,\ldots,P_k)$, cycle $C$, paths $R_1,R_2,R_3,S_4,S_5$ and
augmenting sequence $(Q_1,Q_2,\ldots,Q_n)$ are chosen with $n$
minimum. Let the ends of the paths $Q_i$ be labeled as in
Lemma~\ref{lem:augmentation}.
We may assume that
$P_0$ is the unique path from $u_0$ to $v_0$ in
$R_1\cup R_2\cup C\backslash x_3$.
We proceed in a series of claims.
\claim{1}{The vertex $b_n$ belongs to the interior of $x_4P_1x_5$.}
\noindent
To prove (1) suppose for a contradiction that $b_n\in V(C\cup S_4\cup S_5)$.
By Lemma~\ref{lem:y4}, the choice of $x_4,x_5$ and the fact that
$a_n\ne x_4,x_5$ by Lemma~\ref{lem:augmentation}(ii) we deduce
that $a_n\in V(R_i)$ for some $i\in\{1,2,3\}$. Then we can use $Q_n$ to modify
$C$ to include $a_nR_ix_i$ (and modify $R_1,R_2,R_3$ accordingly),
in which case $(Q_1,Q_2,\ldots,Q_{n-1})$ is an augmentation
contradicting the choice of $n$. This proves (1).
\claim{2}{$a_i,b_i\in V(R_j)$ for no $i\in\{1,2,\ldots,n\}$ and no
$j\in\{1,2,\ldots,5\}$.}
\noindent
To prove (2) suppose to the contrary that $a_i,b_i\in V(R_j)$.
Then $1<i<n$ and
by rerouting $R_j$ along $Q_i$ we obtain an augmentation
$(Q_1,Q_2,\ldots,Q_{i-2},Q_{i-1}\cup b_{i-1}R_ja_{i+1}\cup Q_{i+1},Q_{i+2},\ldots,Q_n)$,
contrary to the minimality of $n$. This proves (2).
\claim{3}{$a_i,b_i\in V(R_1\cup R_2\cup R_3)$ for no $i\in\{1,2,\ldots,n\}$.}
\noindent
Using (2) the
proof of (3) is analogous to the argument at the end of the proof
of Claim (1).
\claim{4}{$a_i,b_i\in V(R_4\cup R_5)$ for no $i\in\{1,2,\ldots,n\}$.}
\noindent
By (2) one of $a_i,b_i$ belongs to
$R_4$ and the other to $R_5$. We can reroute $P_1$ along $Q_i$,
and then $(Q_1,Q_2,\ldots,Q_{i-1})$ becomes an augmentation,
contrary to the minimality of $n$.
\claim{5}{For $i=1,2,\ldots,n-1$, the graph $Q_i\cup R_1\cup R_2\cup R_3$
includes no $\cal T$-jump.}
\noindent
This claim follows from (3),
Lemma~\ref{lem:2pathstoZ} and Lemma~\ref{prejump} applied to $P_0$.
\claim{6}{$a_1\not\in v_1\Omega u_1$.}
\noindent
To prove (6) suppose for a contradiction that $a_1\in v_1\Omega u_1$.
Since $a_1\ne y_1$, we may assume from the symmetry that
$a_1\in v_1\Omega y_1-\{y_1\}$.
Then $b_1\in V(P_1\cup R_1)$ by (5). But if $b_1\in V(R_i)$, where $i=1$
or $i=5$,
then by rerouting $R_i$ along $Q_1$
we obtain an augmenting sequence $(Q_2\cup x_1R_ia_2,Q_3,Q_4,\ldots,Q_n)$,
contrary to the choice of $n$.
Thus $b_1\in u_1P_1x_5$. By replacing $P_1$ by the path
$Q_1\cup u_1P_1b_1$ and considering the paths $R_3$ and $S_5\cup R_5$
we obtain contradiction to Lemma~\ref{lem:2pathstoZ}.
This proves (6).
\claim{7}{$a_1\not\in u_k\Omega v_k$.}
Similarly as in the proof of (6), if $a_1\in u_k\Omega v_k$,
then $b_1\in V(R_2)$ by (5),
and we reroute $R_2$ along $Q_1$ to obtain a contradiction to the minimality
of $n$. This proves (7).
\claim{8}{$a_1\in V(P_k)$.}
To prove (8) we may assume by (6) and (7) that $a_1\in Z$.
Then $b_1\in V(R_3\cup P_1)$ by (5).
If $b_1\in V(R_3)$, then
we reroute $R_3$ along $Q_1$ as before.
Thus $b_1\in V(P_1)$. It follows from (5) and the hypothesis $V(P_2) \cap V(Q_1)= \emptyset$ that $a_1\in u_1\Omega u_2-\{u_1,u_2\}$
or $a_1\in v_2\Omega v_1-\{v_1,v_2\}$, and so from the symmetry we
may assume the latter.
Let us assume for a moment that $y_3 \in a_1\Omega v_1$.
We reroute $P_1$ along $Q_1 \cup b_1P_1v_1$.
The union of $R_3$, $R_2$ and a path in $C$ between $x_2$ and $x_3$,
avoiding $x_1,x_4,x_5$, will play the role of $P_0$ after rerouting.
If $b_1 \in x_4P_1v_1 - \{x_4\}$, then
$R_1\cup C\cup S_4 \cup R_4$ includes two disjoint paths that contradict
Lemma~\ref{lem:2pathstoZ} applied to the new frame and new path $P_0$.
Therefore $b_1 \in V(R_4)$, and hence $(u_1P_1a_2 \cup Q_2, Q_3, \ldots, Q_n)$
is an augmenting sequence after the
rerouting, contrary to the choice of $n$.
It follows that $y_3 \not \in a_1\Omega v_1$.
If $b_1\in V(R_5)$, we replace $P_1$ by $Q_1\cup u_1P_1b_1$;
then $(v_1P_1a_2\cup Q_2,Q_3,\ldots,Q_n)$ is an augmenting sequence that
contradicts the choice of $n$.
So it follows that $b_1\in u_1P_1x_5$.
But now $(G,\Omega)$ has a gridlet using the paths $P_0, P_k$,
$Q_1\cup u_1P_1b_1$ and a subpath of $R_5\cup S_5\cup R_3\cup C\backslash V(P_0)$.
This proves (8).
\claim{9}{$n>1$.}
\noindent
To prove (9) suppose for a contradiction that $n=1$.
Thus $b_1$ belongs to the interior of $x_4Px_5$ by (1), and
$a_1\in V(P_k)$ by (8). But then $Q_1$ is a $\cal T$-jump, contrary to (5).
\claim{10}{$b_1\in V(R_3)$.}
\noindent
To prove (10) we first notice that $b_1\in V(R_2\cup R_3)$ by (5), (9) and (1).
Suppose for a contradiction that $b_1\in V(R_2)$.
Then $a_2\in V(R_2)$, but $b_2\not\in V(R_1\cup R_2\cup R_3)$ by (3)
and $b_2\not\in V(P_1)$ by (5), a contradiction. This proves (10).
\medskip
Let $P_{12}$ and $P_{34}$ be two disjoint subpaths of $C$, where the
first has ends $x_1,x_2$, and the second has ends $x_3,x_4$.
By (8) and (10) the path $Q_1 \cup b_1R_3x_3 \cup P_{34} \cup
S_4$ is a $\cal T$-jump disjoint from $R_1 \cup P_{12} \cup R_2$,
contrary to Lemma~\ref{prejump}.~$\square$\bigskip
We are now ready to prove Theorem~\ref{thm:leap}.
\medskip
\noindent {\bf Proof of Theorem~\ref{thm:leap}.}
Let $(G,\Omega)$ be a $6$-connected society with a leap of length five.
Thus we may assume that Hypothesis~\ref{hyp:leap} holds for $k=5$.
By Lemma~\ref{lem:1exposed} either $(G,\Omega)$ is nearly rural,
in which case the theorem holds,
or there exists a leap of length at least four with at least two exposed vertices.
Thus we may assume that there exists a leap of length four with at least two exposed vertices.
Let $C$ be a cycle as in Lemma~\ref{lem:cycleC}. If there is no
diverse leap, then $C$ is a triangle, $(G\backslash E(C),\Omega)$
is rurally $4$-connected and hence rural by Lemma~\ref{lem:triangle}, and the theorem holds.
Thus we may assume that the cycle $C$ is not a triangle,
and so by Lemma~\ref{lem:q4} we may assume that Hypothesis~\ref{hyp:ray} for $k=4$
holds.
By Lemma~\ref{lem:x4x5} and the $6$-connectivity of $G$ there is
no separation $(A,B)$ as described in Lemma~\ref{lem:augmentation},
and hence by that lemma there exists an augmenting sequence
$(Q_1,Q_2,\ldots,Q_n)$.
By Lemma~\ref{lem:endinP2} the path $Q_1$ intersects $P_2$,
and hence $Q_1$ is disjoint from $P_3$, contrary to
Lemma~\ref{lem:endinP2} applied to the leap $(P_0,P_1,P_3,P_4)$
of length three and an augmenting sequence $(Q_1',Q_2,\ldots, Q_n)$, where $Q_1'$ is the union of $Q_1$ and $a_1P_2u_2$ or $a_1P_2v_2$.~$\square$\bigskip
\section{Rural societies}
If $P$ is a path and $x,y\in V(P)$, we denote by $xPy$ the unique
subpath of $P$ with ends $x$ and $y$.
Let $(G,\Omega)$ be a society.
An {\em orderly transaction} in $(G,\Omega)$ is a sequence of $k$
pairwise disjoint bumps ${\cal T}=(P_1,\dots, P_k)$ such that $P_i$ has
ends $u_i$ and $v_i$ and $u_1,u_2,\dots, u_k, v_k,v_{k-1},\dots,
v_1$ is clockwise.
Let $M$ be the graph obtained from
$P_1\cup P_2\cup\cdots\cup P_k$ by adding
the vertices of $V(\Omega)$ as isolated vertices.
We say that $M$ is the {\em frame} of $\cal T$.
We say that a path $Q$ in $G$ is {\em $\cal T$-coterminal} if
$Q$ has both ends in $V(\Omega)$ and is otherwise disjoint from it and
for every $i=1,2,\ldots,k$ the following holds: if $Q$ intersects $P_i$,
then their intersection is a path whose one end is a common end of
$Q$ and $P_i$.
Let $(G,\Omega)$ be a society, and let $M$ and $\cal T$ be as in
the previous paragraph.
Let $i\in\{1,2,\ldots,k\}$ and let $Q$ be
a $\cal T$-coterminal path in $G\backslash V(P_i)$ with one end in
$v_i\Omega u_i$ and the other end in $u_i\Omega v_i$.
In those circumstances we say that $Q$ is a {\em $\cal T$-jump over $P_i$},
or simply a {\em $\cal T$-jump}.
Now let $i\in\{0,1,\ldots,k\}$ and let $Q_1,Q_2$ be two disjoint
$\cal T$-coterminal paths such that $Q_j$ has ends $x_j,y_j$
and $(u_{i},x_1,x_2,u_{i+1},v_{i+1},y_1,y_2,v_{i})$ is clockwise in $\Omega$,
where possibly $u_{i}=x_1$, $x_2=u_{i+1}$, $v_{i+1}=y_1$, or $y_2=v_{i}$,
and $u_0$ means $x_1$, $u_{k+1}$ means $x_2$, $v_{k+1}$ means $y_1$,
and $v_0$ means $y_2$.
In those circumstances we say that $(Q_1,Q_2)$ is a {\em $\cal T$-cross
in region $i$},
or simply a {\em $\cal T$-cross}.
Finally, let $i\in\{1,2,\ldots,k\}$ and let $Q_0$, $Q_{1}$, $Q_2$
be three paths such that $Q_j$ has ends $x_j,y_j$ and is otherwise
disjoint from all members of $\cal T$,
$x_0,y_0\in V(P_i)$,
the vertices $x_1,x_2$ are internal vertices of $x_0P_i y_0$,
$y_1,y_2\not\in V(P_i)$,
$y_1\in u_{i-1}\Omega u_i\cup v_{i}\Omega v_{i-1}$,
$y_2\in u_{i}\Omega u_{i+1}\cup v_{i+1}\Omega v_{i}$,
and the paths $Q_0$, $Q_{1}$, $Q_2$ are pairwise disjoint, except possibly
$x_1=x_2$.
In those circumstances we say that $(Q_0,Q_{1}, Q_2)$ is a
{\em $\cal T$-tunnel
under $P_i$}, or simply a {\em $\cal T$-tunnel}.
Intuitively, if we think of the paths in $\cal T$ as dividing the
society into ``regions", then a $\cal T$-jump arises from a $\cal T$-path
whose ends do not belong to the same region.
A $\cal T$-cross arises from two $\cal T$-paths with ends in the
same region that cross inside that region, and furthermore,
each path in $\cal T$ includes at most two ends of those crossing paths.
Finally, a $\cal T$-tunnel can be converted into a $\cal T$-jump
by rerouting $P_i$ along $Q_0$.
However, in some applications such rerouting will be undesirable,
and therefore we need to list $\cal T$-tunnels as outcomes.
Let $M$ be a subgraph of a graph $G$. An {\em $M$-bridge} in $G$ is a
connected subgraph $B$ of $G$ such that $E(B)\cap E(M)=\emptyset$
and either $E(B)$ consists of a unique edge with both ends in $M$, or for
some component $C$ of $G\backslash V(M)$ the set $E(B)$ consists of all
edges of $G$ with at least one end in $V(C)$. The vertices in
$V(B)\cap V(M)$ are called the {\em attachments} of $B$.
Now let $M$ be such that no block of $M$ is a cycle.
By a {\em segment of $M$} we mean a maximal subpath $P$ of $M$ such that
every internal vertex of $P$ has degree two in $M$.
It follows that the segments
of $M$ are uniquely determined. Now if $B$ is an $M$-bridge of $G$, then we
say that $B$ is {\em unstable} if some segment of $M$ includes all the
attachments of $B$, and otherwise we say that $B$ is
{\em stable}.
A society $(G,\Omega)$ is {\em rurally $4$-connected} if for
every separation $(A,B)$ of order at most three with
$V(\Omega)\subseteq A$ the graph $G[B]$ can be drawn in a disk
with the vertices of $A\cap B$ drawn on the boundary of the disk.
A society is {\em cross-free} if it has no cross.
The following, a close relative of Lemma~\ref{crossinwall},
follows from \cite[Theorem~2.4]{RobSeyGM9}.
\begin{theorem}
\label{2paththm}
\showlabel{2paththm}
Every cross-free rurally $4$-connected society is rural.
\end{theorem}
\begin{lemma}
\mylabel{orderlyrural1}
Let $(G,\Omega)$ be a rurally $4$-connected society,
let ${\cal T}=(P_1,\dots, P_k)$ be an orderly transaction in $(G,\Omega)$,
and let $M$ be the frame of $\cal T$.
If every $M$-bridge of $G$ is stable and $(G,\Omega)$ is not
rural, then $(G,\Omega)$ has a $\cal T$-jump, a $\cal T$-cross, or a $\cal T$-tunnel.
\end{lemma}
{\noindent\bf Proof. }
For $i=1,2,\ldots,k$ let $u_i$ and $v_i$ be the ends of $P_i$ numbered
as in the definition of orderly transaction, and for
convenience let $P_0$ and $P_{k+1}$ be null graphs.
We define $k+1$
cyclic permutations $\Omega_0,\Omega_1,\ldots,\Omega_k$ as follows.
For $i=1,2,\ldots,k-1$ let $V(\Omega_i):= V(P_{i})\cup V(P_{i+1})\cup
u_i\Omega u_{i+1}\cup v_{i+1}\Omega v_i$ with the cyclic order
defined by saying that $u_i\Omega u_{i+1}$ is followed by $V(P_{i+1})$
in order from $u_{i+1}$ to $v_{i+1}$, followed by $v_{i+1}\Omega v_i$
followed by $V(P_{i})$ in order from $v_i$ to $u_i$.
The cyclic permutation $\Omega_0$ is defined by letting $v_1\Omega u_1$
be followed by $V(P_1)$ in order from $u_1$ to $v_1$,
and $\Omega_k$ is defined by letting $u_k\Omega v_k$
be followed by $V(P_k)$ in order from $v_k$ to $u_k$.
Now if for some $M$-bridge $B$ of $G$ there is no index $i\in\{0,1,\ldots,k\}$
such that all attachments of $B$ belong to $V(\Omega_i)$, then
$(G,\Omega)$ has a $\cal T$-jump.
Thus we may assume that such index exists for every $M$-bridge $B$,
and since $B$ is stable that index is unique. Let us denote it by $i(B)$.
For $i=0,1,\ldots,k$
let $G_i$ be the subgraph of $G$ consisting of $P_i\cup P_{i+1}$,
the vertex-set $V(\Omega_i)$ and all $M$-bridges $B$ of $G$ with
$i(B)=i$.
The society $(G_i,\Omega_i)$ is rurally $4$-connected.
If each $(G_i,\Omega_i)$ is cross-free,
then each of them is rural by Theorem~\ref{2paththm}
and it follows that $(G,\Omega)$ is rural.
Thus we may assume that for some $i=0,1,\ldots,k$ the society
$(G_i,\Omega_i)$ has a cross $(Q_1,Q_2)$.
If neither $P_i$ nor $P_{i+1}$ includes three or four ends of the
paths $Q_1$ and $Q_2$, then $(G,\Omega)$ has a $\cal T$-cross.
Thus we may assume that $P_i$ includes both ends of $Q_1$ and
at least one end of $Q_2$.
Let $x_j,y_j$ be the ends of $Q_j$. Since the $M$-bridge of $G$ containing
$Q_2$ is stable, it has an attachment outside $P_i$,
and so if needed, we may replace $Q_2$ by a path with an end outside $P_i$
(or conclude that $(G,\Omega)$ has a $\cal T$-jump).
Thus we may assume that $u_i,x_1,x_2,y_1,v_i$ occur on $P_i$ in
the order listed, and $y_2\not\in V(P_i)$.
The $M$-bridge of $G$ containing $Q_1$ has an attachment outside $P_i$.
If it does not include $Q_2$ and has an attachment outside $V(P_i)\cup\{y_2\}$, then $(G,\Omega)$ has a $\cal T$-jump or $\cal T$-cross,
and so we may assume not. Thus there exists a path $Q_3$ with
one end $x_3$ in the interior of $Q_1$ and the other end $y_3\in V(Q_2)-\{x_2\}$
with no internal vertex in $M\cup Q_1\cup Q_2$.
We call the triple $(Q_1,Q_2,Q_3)$ a {\em tripod}, and the path
$y_3Q_2y_2$ the {\em leg} of the tripod.
If $v$ is an internal vertex of $x_1P_iy_1$, then we say that
$v$ is {\em sheltered} by the tripod $(Q_1,Q_2,Q_3)$.
Let $L$ be a path that is the leg of some tripod, and subject to that
$L$ is minimal.
From now on we fix $L$ and will consider different tripods with leg $L$;
thus the vertices $x_1,y_1,x_2,x_3$ may change, but $y_2$ and $y_3$
will remain fixed as the ends of $L$.
Let $x_1',y_1'\in V(P_i)$ be such that they are sheltered by no tripod
with leg $L$, but every internal vertex of $x_1'P_iy_1'$ is sheltered
by some tripod with leg $L$.
Let $X'$ be the union of $x_1'P_iy_1'$ and all tripods with leg $L$
that shelter some internal vertex of $x_1'P_iy_1'$,
let $X:=X'\backslash V(L)\backslash\{x_1',y_1'\}$
and
let $Y:=V(M\cup L)- x_1'P_iy_1'-\{y_3\}$.
Since $(G,\Omega)$ is rurally $4$-connected we deduce that
the set $\{x_1',y_1',y_3\}$ does not separate $X$ from $Y$ in $G$.
It follows that there exists a path $P$ in $G\backslash \{x_1',y_1',y_3\}$
with ends $x\in X$ and $y\in Y$.
We may assume that $P$ has no internal vertex in $X\cup Y$.
Let $(Q_1,Q_2,Q_3)$ be a tripod with leg L such that either $x$ is sheltered
by it, or $x\in V(Q_1\cup Q_2\cup Q_3)$.
If $y\not\in V(L\cup P_i)$, then by considering the paths $P,Q_1,Q_2,Q_3$
it follows that either $(G,\Omega)$ has a $\cal T$-jump or $\cal T$-tunnel.
If $y\in V(L)$, then there is a tripod whose leg is a proper subpath of
$L$, contrary to the choice of $L$.
Thus we may assume that $y\in V(P_i)$, and that $y\in V(P_i)$ for every
choice of the path $P$ as above.
If $x\in V(Q_1\cup Q_2\cup Q_3)$ then there is a tripod with leg $L$
that shelters $x_1'$ or $y_1'$, a contradiction. Thus $x\in V(P_i)$.
Let $B$ be the $M$-bridge containing $P$.
Since $y\in V(P_i)$ for all choices of $P$ it follows that the attachments
of $B$ are a subset of $V(P_i)\cup\{y_2\}$.
But $B$ is stable, and hence $y_2$ is an attachment of $B$.
The minimality of $L$ implies that $B$ includes a path from $y$ to $y_3$,
internally disjoint from $L$.
Using that path and the paths $P,Q_1,Q_2,Q_3$ it is now easy to
construct a tripod that shelters either $x_1'$ or $y_1'$, a contradiction.~$\square$\bigskip
|
{
"timestamp": "2012-03-13T01:00:17",
"yymm": "1203",
"arxiv_id": "1203.2192",
"language": "en",
"url": "https://arxiv.org/abs/1203.2192"
}
|
\section{Introduction}
The process of electron-positron ($e^+e^-$) pair production by a photon in a strong atomic field has been investigated since many years (see the reviews \cite{HGO1980,Hubbell2000}). The cross section of this process in the leading order in $Z\alpha$ (Born approximation), is known for arbitrary energy $\omega$ of the incoming photon \cite{BH1934,Racah1934}. Here, $Z$ is the atomic charge number and $\alpha\approx 1/137$ is the fine-structure constant (units with $\hbar=c=1$ are employed throughout). The formal expression of the cross section of $e^+e^-$ pair photoproduction, exact in the parameters $\eta=Z\alpha$ and $\omega$, was derived in \cite{Overbo1968}. This expression has a very complicated form which leads to substantial difficulties in numerical computations. The difficulties grow as $\omega$ increases, so that numerical results have been obtained so far only for $\omega<12.5\,$ MeV \cite{SudSharma2006}.
In the high-energy region $\omega\gg m$, with $m$ being the electron mass, a simple expression of the cross section was obtained in \cite{BM1954,DBM1954}, exactly with respect to $\eta$ and in the leading approximation in $m/\omega$. However, this expression provides rather accurate results only at energies $\omega \gtrsim 100\,$MeV. On the other hand, the theoretical description of the total cross section at $\eta\lesssim 1$ and at intermediate photon energies between $5\,$ MeV and $100\,$ MeV has been based for a long time on the extrapolation of the results obtained at $\omega<5\,$MeV, Ref.\cite{Overbo1977}. Finally, results for the spectrum of one of the created particles at intermediate photon energies were practically absent. An important step has been made recently in \cite{LMS2004} in this direction, where the first corrections of the order of $m/\omega$ to the spectrum as well as to the total cross section of $e^+e^-$ photoproduction in a strong atomic field were derived. The correction to the spectrum was obtained in the region where both produced particles are ultrarelativistic. In \cite{AA2010}, the spectrum was obtained in the region where one of the produced particles is ultrarelativistic and the other has an energy of the order of the electron mass. Essentially less is known on the angular distribution of the final particles at intermediate photon energies. Coulomb corrections, i.e., the contributions of higher-order terms of the perturbation theory with respect to $\eta$, are much more important for the angular distribution than for the spectrum. In \cite{BM1954,DBM1954}, the angular distribution of $e^+e^-$ photoproduction was obtained exactly in the parameter $\eta$ in the leading order with respect to $m/\omega$. This result was obtained under the assumption that both created particles are ultrarelativistic and that the angles between their momenta and the momentum of the initial photon are small. Under the same assumptions, the first quasiclassical correction to the angular distribution found in \cite{BM1954,DBM1954}, was derived very recently in \cite{LMS2011}. In this paper a noticeable charge asymmetry in the differential cross section of high-energy $e^+e^-$ photoproduction was predicted. The angular distribution, when one of the particles is not ultrarelativistic, is not known for $\eta\lesssim 1$ although one can expect Coulomb corrections to be important in this case. The investigation of this problem for arbitrary angles between the momenta of the final particles and the photon momentum is in general a complicated task. In the present paper, we consider a particular case of this problem, which admits a relatively simple analytical solution. Namely, for high-energy $e^+e^-$ photoproduction in a strong Coulomb field, we investigate the distribution over the angle $\theta$ between the positron momentum $\bm p_+$ and the photon momentum $\bm k$ at electron energies $\epsilon_-$ much smaller than the positron energy $\epsilon_+$, so that the electron may not be ultrarelativistic. We also assume that
\begin{equation}
\label{condition}
\frac{\epsilon_-}{\epsilon_+}\ll\theta\ll\sqrt{\frac{\epsilon_-}{\epsilon_+}},
\end{equation}
which implies a large transverse positron momentum $Q=\omega\theta\approx\epsilon_+\theta\gg \epsilon_-$. Finally, in the same kinematical region we have also obtained the angular distribution for bound-free pair production, where the final electron is in an arbitrary bound state, as well as for the cross channels, i.e., for bremsstrahlung and for photorecombination.
\section{Calculation of the cross section}
In order to calculate the cross section of $e^+e^-$ photoproduction, differential over the angles of the fast positron and integrated over the angles of the slow electron, we can employ the relation
\begin{equation}
\label{compl}
\sum_{\lambda}\int \frac{d\Omega}{4\pi}\Psi_{\lambda}(\bm{p},\bm{r})\Psi^{\dag}_{\lambda}(\bm{p},\bm{r})=\frac{\pi}{p^2}\sum_{j,\sigma,\mu}U_{j,\sigma,\mu}(p,\bm r)U^{\dag}_{j,\sigma,\mu}(p,\bm r)
\end{equation}
between the positive-energy electron states $\Psi_{\lambda}(\bm{p},\bm{r})$ in a Coulomb field with definite momentum $\bm{p}$ and polarization index $\lambda$, and those $U_{j,\sigma,\mu}(p,\bm r)$ with definite total angular momentum $j$, projection $\mu$ on some quantization axis, and parity $-\sigma$ (see, e.g., \cite{MS1983,AA2010}). In Eq. (\ref{compl}) $d\Omega$ is the solid angle corresponding to the direction of the momentum $\bm{p}$. In this way, the cross section, averaged over the polarization of the incoming photon and summed up over the polarizations of the final electron and positron, has the form (see, e. g., Ref. \cite{AA2010})
\begin{eqnarray}\label{sigmaee0}
\frac{d\sigma}{d\bm p_+}&=&-\frac{\alpha}{8\pi^2}\frac{1}{\omega\beta_-}\sum_{\lambda_+,\,j,\,\sigma,\,\mu}{\cal M}^{\rho}{\cal M}^*_{\rho}
\,,\nonumber\\
{\cal M}^{\rho}&=&
\int d\bm r\,\bar U_{j,\sigma,\mu}(p_-,\bm r)\gamma^\rho V^{(+)}_{\lambda_+}(\bm p_+,\bm r)\mbox{e}^{i\bm k\cdot\bm r}\,.
\end{eqnarray}
In these equations $(\omega,\bm k)$ is the four-momentum of the photon, $\beta_-=p_-/\epsilon_-$ is the modulus of the electron velocity and $\gamma^\rho$ are the Dirac matrices. The wave function $V^{(+)}_{\lambda_+}(\bm p_+,\bm{r})$ is the negative-energy wave function in a strong Coulomb field with $\bm p_{+}$ and $\lambda_+$ being the positron momentum and its polarization index. The asymptotics of this wave functions at large $r$ contains a plane wave and a divergent spherical wave \cite{BLP1982}.
For the negative-energy wave function $V^{(+)}_{\lambda_+}(\bm p_+,\bm r)$ corresponding to an ultrarelativistic particle, one can use the Furry-Sommerfeld-Mauer form \cite{BLP1982}
\begin{equation}\label{FSMV}
V^{(+)}_{\lambda_+}(\bm p_+,\bm r) = \mbox{e}^{\pi\eta/2}\Gamma(1-i\eta)\mbox{e}^{-i\bm p_+\cdot\bm r}\left(1+\frac{i}{2\epsilon_+}\bm{\alpha}\cdot\bm{\nabla}\right)F(-i\eta,1,i(p_+r+\bm p_+\cdot\bm r))v_{\lambda_+},
\end{equation}
where $\bm{\alpha}=\gamma^0\bm{\gamma}$, $F(a,b,z)$ is the confluent hypergeometric function and where
\begin{equation}
\label{v}
v_{\lambda_+}=\begin{pmatrix}
\bm\sigma\cdot\bm \zeta\,\chi_{\lambda_+}\\
\chi_{\lambda_+}
\end{pmatrix}\,,
\end{equation}
with $\bm \zeta=\bm p_+/p_+$, $\bm{\sigma}$ being the Pauli matrices, and $\chi_{\lambda_+}$ being a constant spinor. For the sake of convenience, we choose the system of reference such that the vector $\bm{p}_+$ points along the positive $z$ axis. Thus, $(\bm{k}-\bm{p}_+)\cdot \bm{r}\approx-r(Q\sin\vartheta\,\cos\varphi+\epsilon_-\cos\vartheta)$, where $Q=\omega\theta$ is the modulus of the momentum transfer, $\vartheta$ is the angle between the vectors $\bm p_+$ and $-\bm r$, and $\varphi$ is the azimuth angle of $-\bm{r}$ in the plane perpendicular to $\bm{p}_+$. The main contribution to the quantity ${\cal M}^\rho$ in Eq. ({\ref{sigmaee0}) is given by the region of integration $r\sim 1/\epsilon_-$ and $|\sin\vartheta|\sim \epsilon_-/Q\ll 1$, so that either $\pi-\vartheta\sim \epsilon_-/Q$ or $\vartheta\sim \epsilon_-/Q$. However, for $\pi-\vartheta\sim \epsilon_-/Q$ the argument of the hypergeometric function is very large (of the order of $\epsilon_+/\epsilon_-$), which makes the integrand highly oscillating for such values of $\vartheta$. Therefore, the largest contribution to ${\cal M}^\rho$ comes from the region $\vartheta\sim \epsilon_-/Q\ll 1$. In this region the argument of the confluent hypergeometric function is of the order of $\omega\vartheta^2/\epsilon_-\sim \epsilon_-/(\omega\theta^2)$, which is much larger than unity due to our condition in Eq. (\ref{condition}), but much smaller than $\epsilon_+/\epsilon_-$. As a result, we can use the asymptotics of $V^{(+)}_{\lambda_+}(\bm p_+,\bm r)$ in Eq. (\ref{FSMV}), which is nothing but the eikonal form of this wave function,
\begin{equation}\label{wfV}
V^{(+)}_{\lambda_+}(\bm p_+,\bm r) = (p_+r+\bm p_+\cdot\bm r)^{i\eta} \mbox{e}^{-i\bm p_+\cdot\bm r}v_{\lambda_+}.
\end{equation}
The wave function $U_{j,\sigma,\mu}(p_-,\bm r)$ has the following form \cite{BLP1982}:
\begin{eqnarray}\label{wfU}
&&U_{j,\sigma,\mu}(p_-,\bm r)=
\begin{pmatrix}
f(r)\Omega_{j,l,\mu}(\bm n)\\
-\sigma \,g(r) \Omega_{j,l',\mu}(\bm n)
\end{pmatrix}\, ,\nonumber\\
&&f(r)=\frac{\sqrt{2}}{r}\sqrt{1+\frac{m}{\epsilon_-}}\,\mbox{e}^{(\pi\nu/2)}\frac{|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}
(2p_-r)^\gamma\,\mbox{Im} \left\{e^{i(p_-r+\xi)}F(\gamma-i\nu,2\gamma+1,-2ip_-r)\right\}\,,\nonumber\\
&&g(r)=\frac{\sqrt{2}}{r}\sqrt{1-\frac{m}{\epsilon_-}}\,\mbox{e}^{(\pi\nu/2)}\frac{|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}
(2p_-r)^\gamma\,\mbox{Re} \left\{e^{i(p_-r+\xi)}F(\gamma-i\nu,2\gamma+1,-2ip_-r)\right\}\,,\nonumber\\
&&l=j+\frac{\sigma}{2}\,,\quad l'=j-\frac{\sigma}{2}\,,\quad \nu=\dfrac{\eta}{\beta_-}\,,\quad \kappa=\sigma \left(j+\frac{1}{2}\right)\,,\quad
\gamma=\sqrt{\kappa^2-\eta^2}\,,\nonumber\\
&&\xi=(1-\sigma)\frac{\pi}{2}+\arctan\left[\frac{\nu(\epsilon_--m)}{\epsilon_- (\gamma+\kappa)}\right]\,, \quad\sigma=\pm 1\,,\quad
\mbox{e}^{-2i\xi}=\frac{\kappa+i\nu m/\epsilon_-}{\gamma+i\nu}\,,\, \bm n=\frac{\bm r}{r}
\end{eqnarray}
where $\Omega_{j,l,\mu}(\bm n)$ is a spherical spinor. We direct the quantization axes for the electron spin along the vector $\bm \zeta$. We recall that each component of the spherical spinors $\Omega_{j,j\pm 1/2,\mu}(\vartheta,\varphi)$ is proportional either to $\sin^{\mu-1/2}\vartheta$ or to $\sin^{\mu+1/2}\vartheta$ \cite{BLP1982}. Thus, since $\vartheta\ll 1$, the main contribution to the sum over $\mu$ comes from the terms with $\mu=\pm 1/2$, and the wave function $U_{j,\sigma,\mu}(p_-,\bm r)$ can be written as
\begin{eqnarray}\label{wfU1}
&&U_{j,\sigma,1/2}(p_-,\bm r)=\sqrt{\frac{j+1/2}{4\pi}}(-i)^{j-\sigma/2}\,
\begin{pmatrix}
if(r)\varphi_{1/2}\\
- \,g(r)\varphi_{1/2}
\end{pmatrix}\, ,\nonumber\\
&&U_{j,\sigma,-1/2}(p_-,\bm r)=\sigma\,\sqrt{\frac{j+1/2}{4\pi}}(-i)^{j-\sigma/2}\,
\begin{pmatrix}
-if(r)\varphi_{-1/2}\\
- \,g(r)\varphi_{-1/2}
\end{pmatrix}\, ,\nonumber\\
&&\varphi_{1/2}=\begin{pmatrix}
1\\
0
\end{pmatrix}\, ,\quad \varphi_{-1/2}=\begin{pmatrix}
0\\
1
\end{pmatrix}\, .
\end{eqnarray}
It is convenient to introduce the functions $F$ and $G$ as
\begin{eqnarray}\label{FG}
&&\begin{pmatrix}
F\\
G
\end{pmatrix}\, =\sqrt{\frac{j+1/2}{4\pi}}\int d\bm r\,(p_+r+\bm p_+\cdot\bm r)^{i\eta} \mbox{e}^{i(\bm k-\bm p_+)\cdot\bm r}\,
\begin{pmatrix}
f(r)\\
g(r)
\end{pmatrix}\, .
\end{eqnarray}
It can be easily shown that in terms of these functions, the cross section (\ref{sigmaee0}) has the simple form
\begin{eqnarray}\label{sigmaee1}
\frac{d\sigma}{d\bm p_+}&=&\frac{\alpha}{4\pi^2}\frac{1}{\omega\beta_-}\sum_{j,\,\sigma}\left[|F|^2+|G|^2+2\mbox{Im}(FG^*)\right]
\,.
\end{eqnarray}
Under the condition (\ref{condition}), we can make the replacement in Eq. (\ref{FG}) (see the discussion below Eq. (\ref{v})),
\begin{equation}
p_+r+\bm p_+\cdot\bm r\longrightarrow \frac{1}{2}\omega r\vartheta^2\,,\quad (\bm k-\bm p_+)\cdot\bm r\longrightarrow
-r(Q\vartheta\cos\varphi+\epsilon_-).
\end{equation}
Then we take the integral over $\varphi$, $\vartheta$ and $r$:
\begin{eqnarray}\label{FGphithetar}
&&\begin{pmatrix}
F\\
G
\end{pmatrix}\, =-\frac{4\pi i\eta}{Q^2}\sqrt{\frac{j+1/2}{4\pi}}\left(\frac{2\omega}{Q^2}\right)^{i\eta}\,\frac{\Gamma(1+i\eta)}{\Gamma(1-i\eta)}
\int_0^\infty dr\, r^{-i\eta}\mbox{e}^{-i\epsilon_-r}\,
\begin{pmatrix}
f(r)\\
g(r)
\end{pmatrix}\nonumber\\
&&=-\frac{2^{\gamma+1}\pi i\eta}{Q^2}\sqrt{\frac{j+1/2}{2\pi}}\left(\frac{2\omega p_-}{Q^2}\right)^{i\eta}\,
\frac{|\Gamma(\gamma+1+i\nu)|\Gamma(\gamma-i\eta)\Gamma(1+i\eta)}{\Gamma(2\gamma+1)\Gamma(1-i\eta)}\left(\frac{\beta_-}{1-\beta_-}\right)^{\gamma-i\eta}\,\nonumber\\
&&\times\mbox{e}^{\pi(\nu-\eta-i\gamma)/2}\,
\begin{pmatrix}
-i\sqrt{1+\frac{m}{\epsilon_-}}(\mbox{e}^{i\xi}{\cal F}_1-\mbox{e}^{-i\xi}{\cal F}_2)\\
\sqrt{1-\frac{m}{\epsilon_-}}(\mbox{e}^{i\xi}{\cal F}_1+\mbox{e}^{-i\xi}{\cal F}_2)
\end{pmatrix}\,,
\end{eqnarray}
where
\begin{equation}
{\cal F}_1=F(\gamma-i\eta,\, \gamma-i\nu,\,2\gamma+1,\,-x)\,,\quad
{\cal F}_2=F(\gamma-i\eta,\, \gamma+1-i\nu,\,2\gamma+1,\,-x)\,,\quad x=\frac{2\beta_-}{1-\beta_-}\,,
\end{equation}
with $F(a,b,c,z)$ being the hypergeometric function. By substituting this result in Eq. (\ref{sigmaee1}) and by performing the summation over $\sigma$, we finally obtain,
\begin{eqnarray}\label{sigmaeefinal}
&&\frac{d\sigma}{d\bm p_+}=\frac{2\alpha}{\pi}\frac{\eta^2}{\omega\beta_-Q^4}\mbox{e}^{\pi(\nu-\eta)}
\sum_{j}\left(j+\frac{1}{2}\right)\frac{|\Gamma(\gamma+1+i\nu)|^2 |\Gamma(\gamma-i\eta)|^2}{\Gamma^2(2\gamma+1)}x^{2\gamma}\nonumber\\
&&\times\left[(1-\beta_-)|{\cal F}_1|^2+(1+\beta_-)|{\cal F}_2|^2
+2\nu (1-\beta_-^2)\mbox{Im}\left(\frac{{\cal F}_1^*{\cal F}_2}{\gamma+i\nu}\right)\right]
\,.
\end{eqnarray}
The result for the analogous cross section $d\sigma/d\bm p_-$ of photoproduction at $\epsilon_-\gg\epsilon_+$ is given by Eq. (\ref{sigmaeefinal}) with the replacements $\eta\rightarrow -\eta$, $\beta_-\rightarrow\beta_+$, and $\nu=\eta/\beta_-\rightarrow -\eta/\beta_+$.
\section{Discussion of the results}
At $\epsilon_+\gg \epsilon_-$, $\nu\ll 1$, and $Q=\omega\theta\gg \epsilon_-$, the cross section integrated over the angles of the electron momentum $\bm p_-$ can be easily found from the general expression of the cross section in the Born approximation (see, e.g., \cite{BLP1982}). It has the form,
\begin{eqnarray}\label{sigmaeefinalas2}
&&\frac{d\sigma_B}{d\bm p_+}=\frac{2\alpha}{\pi}\frac{\eta^2}{\omega Q^4}\,\ln\left(\frac{1+\beta_-}{1-\beta_-}\right)\,.
\end{eqnarray}
For $\eta\ll 1$ our result in Eq. (\ref{sigmaeefinal}) is in agreement with this formula. Although the above expression of $d\sigma_B/d\bm p_+$ tends to zero at $\beta_-\to 0$, the cross section (\ref{sigmaeefinal}) in the limit $\beta_-\to 0$ at fixed $\eta$ (when $\nu\to\infty$), is not zero. The most convenient way to obtain this last asymptotics is to substitute the asymptotics
\begin{eqnarray}\label{fgbeta0}
&&f(r)=\frac{2\sigma\sqrt{2\pi\eta\beta_- }}{u}\left[(\kappa-\gamma)J_{2\gamma}(2\sqrt{u})+\sqrt{u}J_{2\gamma+1}(2\sqrt{u})\right]\,,\nonumber\\
&&g(r)=\frac{2\sigma\eta\sqrt{2\pi\eta\beta_-} }{u}J_{2\gamma}(2\sqrt{u})\,,\quad u=2\eta mr\,.
\end{eqnarray}
of the functions $f(r)$ and $g(r)$ at $\beta_-\to 0$ in Eq. (\ref{FGphithetar}) (see Appendix in Ref. \cite{AA2010} and note the different definition of the functions $f(r)$ and $g(r)$ there). As a result we obtain
\begin{eqnarray}\label{FGas1}
&&\begin{pmatrix}
F\\
G
\end{pmatrix}\, =-\frac{2\pi i\sigma\sqrt{\beta_-(j+1/2)}}{Q^2}\left(\frac{2\omega m}{Q^2}\right)^{i\eta}\,
\frac{\Gamma(\gamma-i\nu)\Gamma(1+i\eta)}{\Gamma(1-i\eta)\Gamma(2\gamma+1)}\,(2\eta)^{\gamma+1/2}\nonumber\\
&&\times\mbox{e}^{-\pi(\eta+i\gamma)/2}\,
\begin{pmatrix}
(\kappa-\gamma){\cal G}_1-2i(\gamma-i\eta){\cal G}_2\\
\eta{\cal G}_1
\end{pmatrix}\,,\nonumber\\
&& {\cal G}_1=F(\gamma-i\eta,\,2\gamma+1,\,2i\eta)\,,\quad
{\cal G}_2=\frac{\eta}{2\gamma+1}F(\gamma+1-i\eta,\,2\gamma+2,\,2i\eta)\,.
\end{eqnarray}
By employing these formulas, we arrive at the following asymptotics of the cross section at $\beta_-\to 0$ and fixed $\eta$,
\begin{equation}\label{sigmaeefinalas1}
\frac{d\sigma}{d\bm p_+}=\frac{4\alpha\mbox{e}^{-\pi\eta}}{\omega Q^4}
\sum_{j}\left(j+\frac{1}{2}\right)^3 (2\eta)^{2\gamma+1} \frac{|\Gamma(\gamma-i\eta)|^2}{\Gamma^2(2\gamma+1)}
\left[|{\cal G}_1|^2+2|{\cal G}_2|^2
-2\mbox{Im}\left({\cal G}_1^*{\cal G}_2\right)\right]
\,.
\end{equation}
This expression shows that $d\sigma/d\bm p_+$ has a finite limit at $\beta_-\to 0$. At $\eta\ll 1$, Eq. (\ref{sigmaeefinalas1}) becomes
\begin{eqnarray}\label{sigmaeefinalas11}
&&\frac{d\sigma}{d\bm p_+}=\frac{8\alpha\eta^3}{\omega Q^4}\,.
\end{eqnarray}
The ratio of $d\sigma/d\bm p_+$ at $\beta_-\to 0$, Eq. (\ref{sigmaeefinalas1}) and of its small-$\eta$ limit, Eq. (\ref{sigmaeefinalas11}) is displayed in Fig. \ref{beta0} as a function of $\eta$.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\linewidth]{Figure_1.eps}
\caption{The cross section $d\sigma/d\bm p_+$ at $\beta_-\to 0$ (see Eq. (\ref{sigmaeefinalas1})) in units of $S_0=8\alpha\eta^3/\omega Q^4$ as a function of $\eta$.}\label{beta0}
\end{center}
\end{figure}
It is seen that the contribution of high-order terms in $\eta$ essentially modifies the result obtained in the lowest order in $\eta$. Note that the asymptotics of the cross section $d\sigma/d\bm p_-$ at $\beta_+\to 0$ cannot be obtained from the asymptotic value in Eq. (\ref{sigmaeefinalas1}) via the replacement $\eta\rightarrow -\eta$. In fact, the result, following from the general formula in Eq. (\ref{sigmaeefinal}) after the replacements $\eta\rightarrow -\eta$, $\beta_-\rightarrow\beta_+$, and $\nu=\eta/\beta_-\rightarrow -\eta/\beta_+$, vanishes in the limit $\beta_+\to 0$ because the positron wave functions are exponentially small in this limit.
In part a) of Fig. \ref{beta1} the difference $d\sigma/d\bm p_+-d\sigma_B/d\bm p_+$ as a function of $\beta_-$ is plotted in units of $S_1=\alpha\eta^2/(\omega Q^4)$ in the region $\beta_-$ close to unity, but $\epsilon_+$ still much larger than $\epsilon_-$. The difference $d\sigma/d\bm p_--d\sigma_B/d\bm p_-$, with $d\sigma_B/d\bm p_-$ given by Eq. (\ref{sigmaeefinalas2}) with the replacement $\beta_-\rightarrow\beta_+$, is shown in part b) of Fig. \ref{beta1} as a function of $\beta_+$ for $\beta_+$ close to unity. The figure shows that in both cases the Coulomb corrections tend to zero as $\beta_{\pm}\to 1$. This fact can be explained as follows. The main contribution to the Coulomb corrections is given by the region of integration over distances $r$ of the order of the Compton wavelength $\lambda_C=1/m$, but in our kinematics both $r$ and $\rho=r\vartheta\sim 1/Q$ are much smaller than $\lambda_C$ at $\beta_-\to 1$ but still $\epsilon_+\gg\epsilon_-$, or at $\beta_+\to 1$ but still $\epsilon_-\gg\epsilon_+$, because in both cases $r\sim 1/\min(\epsilon_-,\epsilon_+)$. Also, as expected, the Coulomb corrections tend to increase the cross section with respect to the Born value in the case of fast positron (part a)) and to decrease it in the case of fast electron (part b)). It is interesting to note that at $\eta$ of the order of unity the Coulomb corrections are not symmetric even at $\beta_{\pm}$ close to unity.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{Figure_2.eps}
\caption{The differences $d\sigma/d\bm p_+-d\sigma_B/d\bm p_+$ as a function of $\beta_-$ for $\beta_-$ close to unity (part a)) and $d\sigma/d\bm p_--d\sigma_B/d\bm p_-$ as a function of $\beta_+$ for $\beta_+$ close to unity (part b)) plotted in units of $S_1=\alpha\eta^2/(\omega Q^4)$. In each part the solid curve corresponds to $Z=92$, the dashed curve to $Z=47$ and the dotted curve to $Z=26$.}\label{beta1}
\end{center}
\end{figure}
Finally, in Fig. \ref{betaall} the cross section $d\sigma/d\bm p_+$ as a function of $\beta_-$ (part a)) and the cross section $d\sigma/d\bm p_-$ as a function of $\beta_+$ (part b)) are shown in units of $S_1$ in the whole interval of values of $\beta_-$ and $\beta_+$, and for a few values of the charge number $Z$. In both cases higher-order terms in $\eta$ play an important role in the whole interval of $\beta_{\pm}$ except that in a narrow region close to the point $\beta_{\pm}=1$.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{Figure_3.eps}
\caption{The cross section $d\sigma/d\bm p_+$ as a function of $\beta_-$ (part a)) and the cross section $d\sigma/d\bm p_-$ as a function of $\beta_+$ (part b)). In each part the solid curve corresponds to $Z=92$, the dashed curve to $Z=47$, the dotted curve to $Z=26$ and the dashed-dotted curve to $Z=1$.}\label{betaall}
\end{center}
\end{figure}
\section{Bound-free pair photoproduction and cross channels}
We consider now the high-energy photoproduction cross section with the electron in a bound state having total angular momentum $j$, projection $\mu$ on some quantization axis, parity $-\sigma$, and radial quantum number $n_r$ (see \cite{MS93,AS97}, the review \cite{BS2006} and the references therein). The cross section $d\sigma_{bf}/d\Omega_+$ for $d\Omega_+$ being the solid angle corresponding to the positron momentum $\bm{p}_+$, averaged over the polarization of the incoming photon and summed up over the polarizations of positron and over $\mu$, has the form,
\begin{eqnarray}\label{sigmaeebf}
\frac{d\sigma_{bf}}{d\Omega_+}&=&-\frac{\alpha\omega}{4\pi}\sum_{\lambda_+,\,\mu}{\cal N}^{\rho}{\cal N}^*_{\rho}
\,,\nonumber\\
{\cal N}^{\rho}&=&
\int d\bm r\,\bar U_{j,\sigma,\mu,\,n_r}(\bm r)\gamma^\rho V^{(+)}_{\lambda_+}(\bm p_+,\bm r)\mbox{e}^{i\bm k\cdot\bm r}\,,
\end{eqnarray}
where $V^{(+)}_{\lambda_+}(\bm p_+,\bm{r})$ is given by Eq. (\ref{wfV}) and $U_{j,\sigma,\mu,\,n_r}(\bm r)$ is the positive-energy wave function of the bound state \cite{BLP1982}. Performing the same calculation as above, we obtain at $Q\gg m$,
\begin{equation}\label{sigmaeebffinal}
\begin{split}
\frac{d\sigma_{bf}}{d\Omega_+}=&\frac{2\alpha\eta^2 m\omega(j+1/2)}{Q^4}\left(\frac{2\eta}{N}\right)^{2\gamma+1}
\frac{\Gamma(2\gamma+n_r+1) |\Gamma(\gamma-i\eta)|^2}{\Gamma^2(2\gamma+1)n_r!}\\
&\times\exp\left[-2\eta \arctan\left(\frac{\gamma+n_r}{\eta}\right)\right]\left\{|{\cal F}_{b1}|^2+\frac{n_r}{2\gamma+n_r}|{\cal F}_{b2}|^2
+2n_r\mbox{Im}\left[\frac{{\cal F}_{b1}^*{\cal F}_{b2}}{\eta+i(\gamma+n_r)}\right]\right\}
\,.
\end{split}
\end{equation}
Here we have introduced the notation
\begin{eqnarray}\label{sigmaeebffinalnotation}
&&{\cal F}_{b1}=F(-n_r,\,\gamma-i\eta,\,2\gamma+1,\,y)\,,\quad {\cal F}_{b2}=F(1-n_r,\,\gamma-i\eta,\,2\gamma+1,\,y)\,,\nonumber\\
&&y=\frac{2\eta}{\eta+i(\gamma+n_r)}\,,\quad N=\sqrt{n_r^2+2\gamma n_r+\kappa^2}\,.
\end{eqnarray}
The cross section $d\sigma_{bf}/d\Omega_+$ in units of the quantity $S_{bf}=\alpha m\omega\eta^5/Q^4$ and summed up over $j$ and $n_r$, is shown as a function of $\eta$ in Fig. \ref{figbf} (solid line). The dashed line in this figure indicates the contribution of the ground state ($n_r=0$ and $j=1/2$) to the total cross section. The leading-order value of $d\sigma_{bf}/d\Omega_+$ at $\eta\ll 1$ in the above units gives $8\zeta(3)\approx 9.616$.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\linewidth]{Figure_4.eps}
\caption{The cross section $d\sigma_{bf}/d\Omega_+$ of the bound-free pair photoproduction in units of $S_{bf}=\alpha m\omega\eta^5/Q^4$ as a function of $\eta$. The solid curve corresponds to the sum over all $j$ and $n_r$ and the dashed one to the contribution of the ground state ($n_r=0$ and $j=1/2$).}\label{figbf}
\end{center}
\end{figure}
Fig. \ref{figbf} shows that the Coulomb corrections essentially modify the cross section already at moderate values of $\eta$.
We conclude by briefly discussing the cross sections in the cross channels of $e^+e^-$ photoproduction. The cross section $d\sigma/d\bm k$ of bremsstrahlung of an ultrarelativistic initial electron with energy $\epsilon_1$ in a strong Coulomb field, when the final electron energy $\epsilon_2$ is much smaller than $\epsilon_1$ and $Q\gg\epsilon_2$, is given by the right-hand-side of Eq. (\ref{sigmaeefinal}), with the replacement $\omega\rightarrow\epsilon_1$ and
$\beta_-\rightarrow\beta_2=p_2/\epsilon_2$. Here, we assumed the result to be averaged over the polarization of the initial electron and summed over the polarizations of the two final particles. Similarly, in the same kinematical region, but with the final electron in a bound state, the cross section $d\sigma_{bf}/d\Omega_\gamma$ of bremsstrahlung (radiative recombination), where $d\Omega_\gamma$ is the solid angle corresponding to the photon momentum $\bm{k}$, is given by Eq. (\ref{sigmaeebffinal}). Finally, we mention that similar calculations for the spectrum of photoionization at high photon energies have been performed in Ref. \cite{Pratt_1960} and in Ref. \cite{Pratt_1960b} for the electron being initially in the $K$ and in the $L$ shell, respectively.
\section{Conclusion}
In the present paper we have calculated analytically the cross section $d\sigma/d\bm p_+$ of $e^+e^-$ photoproduction in a Coulomb field exactly in the parameter $\eta=Z\alpha$. The result has been obtained at $\omega\gg m$ and $\epsilon_-\gtrsim m$ (slow electron) and under the assumption $\omega\theta\gg \epsilon_-$. In a wide region of values of $\beta_-=p_-/\epsilon_-$, our results differ essentially from those obtained in the Born approximation. Only in a very narrow region close to the point $\beta_-=1$, the Coulomb corrections vanish. Analogous results concerning the Coulomb corrections are obtained in the complementary case in which the created electron is fast, with the important difference that Coulomb corrections decrease the cross section with respect to the Born value, while they increase it in the case of fast positron.
In the same kinematical region, we have also calculated the cross section $d\sigma_{bf}/d\Omega_+$, when the final electron is in a bound state with arbitrary quantum numbers. The cross section $d\sigma/d\bm k$ of bremsstrahlung in a strong Coulomb field by an ultrarelativistic electron with energy $\epsilon_1$ in the region where the final electron has energy $\epsilon_2\gtrsim m$ coincides with the cross section of $e^+e^-$ photoproduction at $\epsilon _-\gtrsim m$ (slow electron) after the replacement $\beta_-\rightarrow\beta_2$ and $\omega\rightarrow \epsilon_1$.
Our results are obtained for a pure Coulomb field. However, the effects of screening for high-energy photoproduction in our kinematical region are expected to be important only in the very narrow region close to the point $\beta_-=0$ ($\beta_+=0$) in the case of fast positron (electron).
\section*{Acknowledgments}
A. I. M. gratefully acknowledges the Max-Planck-Institute for Nuclear Physics for warm hospitality and financial support during his visit. The work was supported in part by the Ministry of Education and Science of the Russian Federation and the Grant 14.740.11.0082 of Federal special-purpose program “Scientific and scientific-pedagogical personnel of innovative Russia”.
|
{
"timestamp": "2012-03-12T01:02:09",
"yymm": "1203",
"arxiv_id": "1203.2137",
"language": "en",
"url": "https://arxiv.org/abs/1203.2137"
}
|
\section{Introduction}
\label{sec:intro}
Analog network coding (ANC) extends to multihop wireless networks the idea of linear network coding \cite{103liYeungCai}, where an intermediate node sends out a linear combination of its incoming packets. In a wireless network, signals transmitted simultaneously by multiple sources add in the air. Each node receives a \textit{noisy sum} of these signals, \textit{i.e.} a linear combination of the received signals and noise. A communication scheme wherein each relay node merely amplifies and forwards this noisy sum is referred to as analog network coding \cite{107kattiGollakottaKatabi, 110maricGoldsmithMedard}.
The rates achievable with ANC in layered relay networks is analyzed in \cite{110maricGoldsmithMedard, 111liuCai}. In \cite{110maricGoldsmithMedard}, the achievable rate is computed under two assumptions: (A) each relay node scales the received signal to the maximum extent subject to its transmit power constraint, (B) the nodes in all $L$ layers operate in the high-SNR regime, where $\min_{k \in l} P_{R,k} \ge 1/\delta, l = 1, \ldots, L$ for $\delta \ge 0$, where $P_{R,k}$ is the received signal power at the $k^\textrm{th}$ node. It is shown that the rate achieved under these two assumptions approaches network capacity as the source power increases. The authors in \cite{111liuCai} extend this result to the scenarios where the nodes in at most one layer do not satisfy these assumptions and show that achievable rates in such scenarios still approach the network capacity as the source power increases\footnote{However, it is assumed that the noises at the nodes in this particular layer are independent, resulting in the computed ANC rate overestimating the optimal ANC rate in general.}.
However, requiring each relay node to amplify its received signal to the upper bound of its transmit power constraint results in suboptimal end-to-end performance of analog network coding, as we show in \cite{111agnihotriJaggiChen, 112agnihotriJaggiChen}. Further, even in low-SNR regimes amplify-and-forward relaying can be capacity-achieving relay strategy in some scenarios, \cite{107gomadamJafar}.
In this paper we are concerned with analyzing the performance of analog network coding in general layered networks, without the above two assumptions on input signal scaling factors and received SNRs. However, such a characterization of the performance of analog network coding results in a computationally intractable problem in general \cite{111liuCai, 111agnihotriJaggiChen}.
In \cite{112agnihotriJaggiChen}, we establish that a globally optimal set of scaling factors for each node, {\it i.e.} a choice of relaying strategies that optimizes end-to-end throughput over all ANC strategies, can be computed in a layer-by-layer manner. This result allows us to computationally efficiently characterize exactly the optimal ANC rate in a large class of layered networks that cannot be addressed using existing approaches under the assumptions A and B. Further, for general layered relay networks, this result significantly reduces the computational complexity of computing a set of non-trivial achievable rates.
However, even layer-by-layer computation of a network-wide scaling vector that maximizes the end-to-end ANC rate for general layered networks is a computationally hard problem. In this paper, we propose a greedy scheme to bound from below the optimal rate achievable with analog network coding in general layered networks. The proposed scheme allows us to exactly compute the optimal ANC rate in a much wider class of layered networks than those that can be so addressed using existing approaches, including our approach in \cite{112agnihotriJaggiChen}. In particular, for the Gaussian $N$-relay diamond network \cite{111nazarogluOzgurFragouli}, the proposed scheme allows us to exactly compute the optimal rate achievable with analog network coding. To the best of our knowledge, this is the first characterization of the optimal ANC rate for Gaussian diamond network. Further, for general layered networks, our scheme allows for the computation of the optimal rates within a constant gap from the cut-set upper bound asymptotically in the source power.
\textit{Organization:} In Section~\ref{sec:sysModel} we introduce a general wireless layered relay network model and formulate the problem of maximum rate achievable with ANC in such a network. Section~\ref{sec:diamond} addresses the problem of maximum ANC rate achievable in a Gaussian $N$-relay diamond network and shows that a greedy scheme optimally solves this problem. In Section~\ref{sec:genNet} we first generalize the greedy scheme for Gaussian diamond networks to characterize the optimal performance of a specific subnetwork of the general layered network. We then construct a scheme to bound from below the optimal performance of ANC in general layered networks. Section~\ref{sec:exa} illustrates that the proposed scheme leads to the exact computation of the maximum ANC rate in a specific class of layered networks and tight characterization of the optimal rate in the general layered networks asymptotically in the source power. Section~\ref{sec:conclFW} concludes the paper.
\section{System Model}
\label{sec:sysModel}
Consider a $(L+2)$-layer wireless network with directed links. The source $s$ is at the layer `$0$', the destination $t$ is at the layer `$L+1$', and the relay nodes from the set $R$ are arranged in $L$ layers between them. The $l^\textrm{th}$ layer contains $n_l$ relay nodes, $\sum_{l=1}^L n_l = M$. An instance of such a network is given in Figure~\ref{fig:layrdNetExa}. Each node is assumed to have a single antenna and operate in full-duplex mode.
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./layrdNetExa}
\caption{Layered network with 3 relay layers between the source `s' and destination `t'. Each layer contains two relay nodes.}
\label{fig:layrdNetExa}
\end{figure}
At instant $n$, the channel output at node $i, i \in R \cup \{t\}$, is
\begin{equation}
\label{eqn:channelOut}
y_i[n] = \sum_{j \in {\mathcal N}(i)} h_{ji} x_j[n] + z_i[n], \quad - \infty < n < \infty,
\end{equation}
where $x_j[n]$ is the channel input of the node $j$ in the neighbor set ${\mathcal N}(i)$ of node $i$. In \eqref{eqn:channelOut}, $h_{ji}$ is a real number representing the channel gain along the link from node $j$ to node $i$. It is assumed to be fixed (for example, as in a single realization of a fading process) and known throughout the network. The source symbols $x_s[n], - \infty < n < \infty$, are i.i.d. Gaussian random variables with zero mean and variance $P_s$ that satisfy an average source power constraint, $x_s[n] \sim {\cal N}(0, P_s)$. Further, $\{z_i[n]\}$ is a sequence (in $n$) of i.i.d. Gaussian random variables with $z_i[n] \sim {\cal N}(0, \sigma^2)$. We also assume that $z_i$ are independent of the input signal and of each other. We assume that the $i^{\textrm{th}}$ relay's transmit power is constrained as:
\begin{equation}
\label{eqn:pwrConstraint}
E[x_i^2[n]] \le P_i, \quad - \infty < n < \infty
\end{equation}
In analog network coding each relay node amplifies and forwards the noisy signal sum received at its input. More precisely, a relay node $i$ at instant $n+1$ transmits the scaled version of $y_i[n]$, its input at time instant $n$, as follows
\begin{equation}
\label{eqn:AFdef}
x_i[n+1] = \beta_i y_i[n], \quad 0 \le \beta_i^2 \le \beta_{i,max}^2 = P_i/P_{R,i},
\end{equation}
where $P_{R,i}$ is the received power at the node $i$ and choice of the scaling factor $\beta_i$ satisfies the power constraint \eqref{eqn:pwrConstraint}.
One important characteristic of layered networks with unidirectional links is that all paths from the source to destination have same number of hops. Also, each path from the $i^\textrm{th}, i \in R$, relay node to the destination has the same length. Therefore, in a layered network with $L$ layers, all copies of a source signal traveling along different paths arrive at the destination with time delay $L$ and all copies of a noise symbol introduced at a node in $l^\textrm{th}$ layer arrive at the destination with time delay $L-i+1$. Therefore, the outputs of the source-destination channel are free of intersymbol interference. This simplifies the relation between input and output of the source-destination channel and allows us to omit the time-index while denoting the input and output signals.
Using \eqref{eqn:channelOut} and \eqref{eqn:AFdef}, the input-output channel between the source and destination can be written as
\begin{equation}
\label{eqn:sdchnl}
y_t = \bigg[\sum_{(i_1, \ldots, i_{L}) \in K_{L}} \hspace{-0.25in} h_{si_1}\beta_{i_1}h_{i_1 i_2} \ldots \beta_{i_{L}}h_{i_{L} t}\bigg] x_s + \sum_{l=1}^{L} \sum_{j=1}^{n_l}\bigg[\sum_{(i_l, \ldots, i_{L}) \in K_{lj,L}} \hspace{-0.25in} \beta_{lj} h_{lj, i_l} \ldots \beta_{L i_{L}}h_{L i_{L}, t}\bigg] z_{lj} + z_t, \nonumber
\end{equation}
where $K_L$, is the set of $L$-tuples of node indices corresponding to all paths from the source to the destination with path delay $L$. Similarly, $K_{lj,L-l+1}$, is the set of $L-l+1$-tuples of node indices corresponding to all paths from the $j^{\textrm{th}}$ relay of $l^\textrm{th}$ layer to the destination with path delay $L-l+1$.
We introduce \textit{modified} channel gains as follows. For all the paths between the source $s$ and the destination $t$:
\begin{equation}
\label{eqn:modChnlParams}
h_s = \sum_{(i_1, \ldots, i_{L}) \in K_{L}} h_{si_1}\beta_{i_1}h_{i_1 i_2} \ldots \beta_{i_{L}}h_{i_{L} t}
\end{equation}
For all the paths between the $j^{\textrm{th}}$ relay of $l^\textrm{th}$ layer to the destination $t$ with path delay $L-l+1$:
\begin{equation}
\label{eqn:modChnlParams2}
h_{lj} = \sum_{(i_1, \ldots, i_{L-l+1}) \in K_{lj,L-l+1}} \beta_{lj} h_{lj, i_1} \ldots \beta_{L i_{L}}h_{L i_{L}, t}
\end{equation}
In terms of these modified channel gains\footnote{Modified channel gains for even a possibly exponential number of paths as in \eqref{eqn:modChnlParams} and \eqref{eqn:modChnlParams2} can be efficiently computed using line-graphs \cite{103koetterMedard}. Further, the number of such modified channel gains scales polynomially in the size of the graph being considered.}, the source-destination channel in \eqref{eqn:sdchnl} can be written as:
\begin{equation}
\label{eqn:chnlmod}
y_t = h_s x_s + \sum_{l=1}^{L} \sum_{j=1}^{n_l} h_{lj} z_{lj} + z_t
\end{equation}
\textit{Problem Formulation:} For a given network-wide scaling vector $\bm{\beta}=(\beta_{li})_{1 \le l \le L, 1 \le i \le n_l}$, the achievable rate for the channel in \eqref{eqn:chnlmod} with i.i.d. Gaussian input is (\hspace{-0.001cm}\cite{110maricGoldsmithMedard, 111liuCai, 111agnihotriJaggiChen}):
\begin{equation}
\label{eqn:infoRateFin}
I(P_s, \bm{\beta}) = (1/2) \log\big(1 + SNR_t\big),
\end{equation}
where $SNR_t$, the signal-to-noise ratio at the destination $t$ is:
\begin{equation}
\label{eqn:snr}
SNR_t = \frac{P_s}{\sigma^2}\frac{h_s^2}{1 + \sum_{l=1}^{L} \sum_{j=1}^{n_l} h_{lj}^2}
\end{equation}
The maximum information-rate $I_{ANC}(P_s)$ achievable in a given layered network with i.i.d. Gaussian input is defined as the maximum of $I(P_s, \bm{\beta})$ over all feasible $\bm{\beta}$, subject to per relay transmit power constraint \eqref{eqn:AFdef}. In other words:
\begin{equation}
\label{eqn:maxAFrate}
I_{ANC}(P_s) \stackrel{def}{=} \max_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2} I(P_s, \bm{\beta})
\end{equation}
It should be noted that $\beta_{li, max}$ (the maximum value of the scaling factor for $i^\textrm{th}$ node in the $l^\textrm{th}$ layer) depends on the scaling factors for the nodes in the previous $l-1$ layers.
Given the monotonicity of the $\log(\cdot)$ function, we have
\begin{equation}
\label{eqn:eqProb}
\bm{\beta}_{opt} = \argmax_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2} I(P_s, \bm{\beta}) = \argmax_{\bm{\beta}:0 \le \beta_{li}^2 \le \beta_{li, max}^2} SNR_t
\end{equation}
Therefore in the rest of the paper, we concern ourselves mostly with maximizing the received SNRs.
In \cite{112agnihotriJaggiChen}, we discussed the computational complexity of exactly solving the problem \eqref{eqn:maxAFrate} or equivalently the problem \eqref{eqn:eqProb}. Further, we also introduced a key result \cite[Lemma 2]{112agnihotriJaggiChen} that reduces the computational complexity of the problem of computing $\bm{\beta}_{opt}$ by computing it layer-by-layer as a solution of a cascade of subproblems. This result allowed us to characterize the optimal end-to-end rate achievable with analog network coding in communication scenarios that cannot be so addressed using previous approaches. However, each of these subproblems itself is computationally hard for general network scenarios as it involves maximizing the ratio of \textit{posynomials} \cite{107boydkimVandenberghe, 105chiang}, which is known to be computationally intractable in general \cite{105chiang}. Therefore, in this paper, we introduce a greedy scheme that optimally solves these subproblems and consequently the problem \eqref{eqn:eqProb} for a large class of layered networks that cannot be addressed with current schemes. For general layered networks, the proposed scheme allows us to tightly bound from below the optimal ANC performance. However before discussing this scheme, we motivate it by computing the maximum rate of information transfer achievable with analog network coding over the diamond network with $N$ relay nodes.
\section{Diamond Network: The optimal rate achievable with analog network coding}
\label{sec:diamond}
Consider the diamond network of Figure~\ref{fig:diamond}. We can consider diamond network as a layered network with only one layer of relay nodes. Then using \eqref{eqn:modChnlParams}, \eqref{eqn:modChnlParams2}, and \eqref{eqn:snr}, we compute the SNR at the destination $t$ for any scaling vector $\bm{\beta} = (\beta_1, \ldots, \beta_N)$ as
\begin{equation}
\label{eqn:diamondSNR}
SNR_t = \frac{P_s}{\sigma^2} \frac{(\sum_{i=1}^N h_{si} \beta_i h_{it})^2}{1+\sum_{i=1}^N \beta_i^2 h_{it}^2}
\end{equation}
Using \eqref{eqn:maxAFrate}, the problem of computing the maximum ANC rate for this network thus can be formulated as
\begin{equation}
\label{eqn:diamondProb}
\max_{0 \le \bm{\beta}^2 \le \bm{\beta}_{max}^2} SNR_t,
\end{equation}
where $\bm{\beta}_{max} = (\beta_{1,max} \ldots, \beta_{N,max})$ with $\beta_{i,max}^2 = P_i/(h_{si}^2 P_s + \sigma^2), i \in {\mathcal N}, {\mathcal N} = \{1, \ldots, N\}$.
Equating the first-order partial derivatives of the objective function with respect to $\beta_i, i \in \mathcal{N}$, to zero, we get the following $N+1$ conditions for local extrema:
\begin{align}
&\sum_{i \in \mathcal{N}} h_{si} \beta_i h_{it} = 0 \label{eqn:b1bN} \\
&\beta_i = \frac{h_{si}/h_{it}}{\sum_{j \in \mathcal{N}\setminus\{i\}} h_{sj} \beta_j h_{jt}}\bigg(1+\sum_{j \in \mathcal{N}\setminus\{i\}} \beta_j^2 h_{it}^2\bigg) \label{eqn:bi_ito_rest}
\end{align}
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./diamond_N}
\caption{A diamond network with $N$ relay nodes.}
\label{fig:diamond}
\end{figure}
Let $SNR_{\beta_i \beta_j} = \frac{\partial^2 SNR_t}{\partial \beta_i \partial \beta_j}$ denote the second-order partial derivatives of $SNR_t$ with respect to $\beta_i$ and $\beta_j$, $i, j \in \mathcal{N}$ and $H(\bm{\beta})$ denote the determinant of $N \times N$ Hessian matrix.
First, consider the set of stationary points $S_{\bm{\beta}} = \{\bm{\beta}: \bm{\beta} \mbox{ satisfies } \eqref{eqn:b1bN}\}$. For all points in $S_{\bm{\beta}}$ we can prove that
\begin{align*}
SNR_{\beta_1 \beta_1} &> 0\\
H(\bm{\beta}) &= 0
\end{align*}
Therefore, the second partial derivative test to determine if the points in $S_{\bm{\beta}}$ are local minimum, maximum, or saddle points fails. However, we can establish that for every $\bm{\beta} \in S_{\bm{\beta}}$, the following set of conditions holds
\begin{align}
\pd{SNR_t}{\beta_i}\bigg|_{\bm{\beta}+\bm{\delta}} &< 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i < 0, \label{eqn:ineq1}\\
\pd{SNR_t}{\beta_i}\bigg|_{\bm{\beta}+\bm{\delta}} &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i > 0, \label{eqn:ineq2}\\
H(\bm{\beta}) &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i < 0, \label{eqn:ineq3}\\
H(\bm{\beta}) &> 0, \mbox{ if } \sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i > 0, \label{eqn:ineq4}
\end{align}
for all $\bm{\delta}=(\delta_1, \ldots, \delta_N) \rightarrow \bm{0}$. In other words, \eqref{eqn:ineq1} and \eqref{eqn:ineq2} imply that the slope of the function changes sign at $\sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i = 0$, and \eqref{eqn:ineq3} and \eqref{eqn:ineq4} imply that the convexity of the function, however, does not change at $\sum_{i \in \mathcal{N}} h_{si} h_{it} \delta_i = 0$. Therefore, together these imply that \eqref{eqn:b1bN} leads to a local minimum of the objective function.
Next, consider the set of points defined by \eqref{eqn:bi_ito_rest}. For all such points we can prove that
\begin{align*}
SNR_{\beta_1 \beta_1} &< 0\\
H(\bm{\beta}) &> 0
\end{align*}
Therefore, from the second partial derivative test the objective function attains its local maximum at the set of points characterized by \eqref{eqn:bi_ito_rest} above. However, no real solution of the simultaneous system of equations in \eqref{eqn:bi_ito_rest} exists. In other words, no solution of \eqref{eqn:diamondProb} exists where all relay nodes transmit strictly below their respective transmit power constraints. This is illustrated by the following example.
{\textit{\textbf{Example 1} (Three node Diamond Network):}} Consider the Gaussian diamond network of Figure~\ref{fig:diamond} with three relay nodes. For this network, \eqref{eqn:bi_ito_rest} results in:
\begin{align}
\beta_1 &= \frac{h_{s1}/h_{1t}}{h_{s2} \beta_2 h_{2t}+h_{s3} \beta_3 h_{3t}}\big(1+ \beta_2^2 h_{2t}^2 + \beta_3^2 h_{3t}^2\big) \label{eqn:beta1diamond3} \\
\beta_2 &= \frac{h_{s2}/h_{2t}}{h_{s1} \beta_1 h_{1t}+h_{s3} \beta_3 h_{3t}}\big(1+ \beta_1^2 h_{1t}^2 + \beta_3^2 h_{3t}^2\big) \label{eqn:beta2diamond3} \\
\beta_3 &= \frac{h_{s3}/h_{3t}}{h_{s1} \beta_1 h_{1t}+h_{s2} \beta_2 h_{2t}}\big(1+ \beta_1^2 h_{1t}^2 + \beta_2^2 h_{2t}^2\big) \label{eqn:beta3diamond3}
\end{align}
Substituting \eqref{eqn:beta2diamond3} in \eqref{eqn:beta3diamond3}, after a little algebraic manipulation we get:
\begin{equation*}
\label{eqn:beta3condition}
\{\beta_3 h_{3t} h_{s1} \beta_1 h_{1t} - h_{s3}(1+\beta_1^2 h_{1t}^2)\}\{(h_{s1} \beta_1 h_{1t}+h_{s3} \beta_3 h_{3t})^2 + h_{s2}^2(1 + \beta_1^2 h_{1t}^2 + \beta_3^2 h_{3t}^2)\}=0
\end{equation*}
Solving this gives three solutions for $\beta_3$, namely:
\begin{align}
\beta_{3,1} &= \frac{h_{03}/h_{3t}}{h_{s1} \beta_1 h_{1t}} (1+\beta_1^2 h_{1t}^2) \label{eqn:beta3sol1} \\
\beta_{3,2} &= \frac{- h_{s3} h_{s1} \beta_1 h_{1t} + i h_{s2} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{3t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \label{eqn:beta3sol2} \\
\beta_{3,2} &= \frac{- h_{s3} h_{s1} \beta_1 h_{1t} - i h_{s2} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{3t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \label{eqn:beta3sol3}
\end{align}
Substituting each of \eqref{eqn:beta3sol1}, \eqref{eqn:beta3sol2}, and \eqref{eqn:beta3sol3} in \eqref{eqn:beta2diamond3} results in three corresponding solutions for $\beta_2$, namely:
\begin{align*}
\beta_{2,1} &= \frac{h_{02}/h_{2t}}{h_{s1} \beta_1 h_{1t}} (1+\beta_1^2 h_{1t}^2) \\
\beta_{2,2} &= \frac{- h_{s2} h_{s1} \beta_1 h_{1t} + i h_{s3} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{2t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1} \\
\beta_{2,3} &= \frac{- h_{s2} h_{s1} \beta_1 h_{1t} - i h_{s3} \sqrt{(h_{s2}^2 + h_{s3}^2)(1+\beta_1^2 h_{1t}^2) + (h_{s1} \beta_1 h_{1t})^2}}{h_{2t}(h_{s2}^2 + h_{s3}^2)}, i = \sqrt{-1}
\end{align*}
Therefore, we have three possible solutions for optimal $(\beta_2, \beta_3)$, \textit{i.e.} $(\beta_{2,1}, \beta_{3,1})$, $(\beta_{2,2}, \beta_{3,2})$, and $(\beta_{2,3}, \beta_{3,3})$.
Substituting $(\beta_{2,1}, \beta_{3,1})$ in \eqref{eqn:beta1diamond3} results in
\begin{equation*}
\beta_1^2 = -\frac{h_{s2}^2 + h_{s3}^2}{h_{1t}^2 (h_{s1}^2 + h_{s2}^2 + h_{s3}^2)},
\end{equation*}
which leads to complex valued solutions for optimal $\beta_1$.
Similarly, substituting $(\beta_{2,2}, \beta_{3,2})$, and $(\beta_{2,3}, \beta_{3,3})$ in \eqref{eqn:beta1diamond3} results in
\begin{equation*}
\beta_1^2 = -\frac{1 + h_{s2}^2 + h_{s3}^2}{h_{1t}^2 (h_{s2}^2 + h_{s3}^2)}
\end{equation*}
which also leads to complex valued solutions for optimal $\beta_1$.
This allows us to conclude that no real solution of the system of simultaneous equations in \eqref{eqn:beta1diamond3}-\eqref{eqn:beta3diamond3} exists. {\hspace*{\fill}~\IEEEQEDclosed\par}
The above discussion implies that all points satisfying \eqref{eqn:b1bN} lead to the global minimum of the objective function in \eqref{eqn:diamondProb} and the global maximum of the objective function occurs at one of the $N$ hyperplanes (of dimension $N-1$) defined by $\beta_k = \beta_{k,max}, k \in \mathcal{N}$. Next we identify this hyperplane and characterize the corresponding optimal solution.
Consider the system of simultaneous equations in \eqref{eqn:bi_ito_rest} on the $(N-1)$-dimensional hyperplane defined by $\beta_k = \beta_{k,max}$.
\begin{equation}
\label{eqn:bi_ito_rest_on_kth_plane}
\beta_i = \frac{h_{si}}{h_{it}} \frac{1+\beta_{k,max}^2 h_{kt}^2+\sum_{j \in \mathcal{N}\setminus\{i,k\}} \beta_j^2 h_{jt}^2}{h_{sk} \beta_{k,max} h_{kt} + \sum_{j \in \mathcal{N}\setminus\{i,k\}} h_{sj} \beta_j h_{jt}}
\end{equation}
Note that the solution of the above system of equations is the set of scaling-factors for the nodes in the set of relay nodes $\mathcal{N} \setminus \{k\}$ that maximizes $SNR_t$ on hyperplane $\beta_k = \beta_{k,max}$. Solving the system of equations in \eqref{eqn:bi_ito_rest_on_kth_plane} results in the following set of optimal solutions for $\beta_i$ on hyperplane $\beta_k = \beta_{k,max}$:
\begin{equation}
\label{eqn:optBeta_on_kth_plane}
\beta_{i}^k = \frac{h_{si}}{h_{it}} \frac{1+\beta_{k,max}^2 h_{kt}^2}{h_{sk} \beta_{k,max} h_{kt}}, i \in \mathcal{N}\setminus \{k\}
\end{equation}
However, the optimal scaling factors in \eqref{eqn:optBeta_on_kth_plane} for $N-1$ nodes are computed without considering the upper bound $\beta_{i,max}$ on each $\beta_i, i \in \mathcal{N}\setminus\{k\}$. Therefore, taking into consideration the upper bound on the scaling factor for each node, the modified solution is computed as per the following lemma.
\begin{pavikl}
\label{lemma:optBeta}
The optimal scaling vector $\bm{\beta}_{opt}^k=(\beta_{1,opt}^k, \ldots, \beta_{N,opt}^k)$ for $N$ nodes on $\beta_k = \beta_{k,max}$ hyperplane such that each scaling factor satisfies the corresponding upper bound on its maximum value is given as
\begin{equation*}
\beta_{i,opt}^k = \begin{dcases}
\beta_{i,max}, i \in S^k \\
\frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^k,
\end{dcases}
\end{equation*}
where $S^k$ is the set of nodes such that on hyperplane $\beta_k = \beta_{k,max}$, the optimal value of the scaling factor of a node is saturated to its corresponding upper bound, $S^k = \{k\} \cup \{i: \beta_{i}^k \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k\}\}$.
\end{pavikl}
\begin{IEEEproof}
Following the argument similar to the one used to prove the global extrema properties of \eqref{eqn:b1bN} and \eqref{eqn:bi_ito_rest}, we can prove that on the $\beta_k = \beta_{k,max}$ hyperplane, the $SNR_t$ achieves its global minimum at a hyperplane defined by
\begin{equation*}
\sum_{i \in \mathcal{N}\setminus\{k\}} h_{si} \beta_i h_{it} = 0
\end{equation*}
and its global maximum at the points defined by $\beta_i^k$ given in \eqref{eqn:optBeta_on_kth_plane}.
Let $M^k$ denotes the set of nodes for which $\beta_i^k$ computed in \eqref{eqn:optBeta_on_kth_plane} is greater than or equal to the corresponding upper bound $\beta_{i,max}$ on the maximum value of the scaling factor, \textit{i.e.} $M^k = \{i: \beta_{i}^k \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k\}\}$. For all such $\beta_i^k, i \in M^k$, after proving that $\pd{SNR_t}{\beta_i}\big|_{\beta_{i,max}} \ge 0$, we set $\beta_i^k = \beta_{i,max}$ and update $S^k$, the set of nodes such that on hyperplane $\beta_k = \beta_{k,max}$, the optimal value of each node is saturated to its corresponding upper bound; as follows: $S^k = S^k \cup M^k$. As $\beta_i^k$ computed in \eqref{eqn:optBeta_on_kth_plane} for a node $i \not\in S^k$ may no longer be optimal after the above re-assignment of $\beta_i^k, i \in M^k$, we need to solve the following simultaneous system of $N - |S^k| = N-|M^k|-1$ equations with $i \in \mathcal{N} \setminus S^k$:
\begin{equation}
\label{eqn:newSysOeqns}
\beta_i = \frac{h_{si}}{h_{it}} \frac{1+\sum\limits_{j \in S^k} \beta_{j,max}^2 h_{jt}^2+\sum\limits_{j \not\in S^k \cup \{i\}} \beta_{j}^2 h_{jt}^2}{\sum\limits_{j \in S^k} h_{sj} \beta_{j,max} h_{jt} + \sum\limits_{j \not\in S^k \cup \{i\}} h_{sj} \beta_{j} h_{jt}},
\end{equation}
Solving this system of equations results in
\begin{equation}
\label{eqn:recomputedBeta}
\beta_{i,opt}^k = \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^k
\end{equation}
Some of the recomputed scaling factors $\beta_{i,opt}^k, i \not\in S^k$ may violate the corresponding upper bound on their maximum value. All such nodes are added to set $S^k$, thus updating it. Then, the system of equations in \eqref{eqn:newSysOeqns} is solved again for this updated set $S^k$. This iterative process continues until none of the recomputed $\beta_i$ in \eqref{eqn:recomputedBeta} violates its corresponding upper bound. This iterative process is presented formally in terms of an algorithm: \textit{Algorithm 1}, given on the top of the next page.
Note that \textit{Algorithm 1} always halts with either $S^k = \mathcal{N} \setminus \{k\}$ or $S^k \subset \mathcal{N} \setminus \{k\}$ and $\beta_i^k < \beta_{i,max}, i \in \mathcal{N} \setminus S^k$.
\end{IEEEproof}
Using Lemma~\ref{lemma:optBeta}, for each of $N$ hyperplanes, defined as $\beta_k = \beta_{k,max}$, $k \in \mathcal{N}$, we can compute $\bm{\beta}_{opt}^k$, the set of scaling factors for all nodes at which $SNR_t$ attains its maximum on $\beta_k = \beta_{k,max}$ hyperplane. Then the hyperplane at which $SNR_t$ attains its global maximum is identified as follows:
\begin{pavikp}
\label{prop:optHyperplane}
The hyperplane at which $SNR_t$ attains its global maximum is defined as
\begin{equation*}
k^\star = \argmax_{k \in \mathcal{N}} SNR_t(\bm{\beta}_{opt}^k)
\end{equation*}
\end{pavikp}
Combining Proposition~\ref{prop:optHyperplane} and Lemma~\ref{lemma:optBeta}, we can characterize the scaling vector $\bm{\beta}_{opt}$ that solves the problem \eqref{eqn:diamondProb} as follows.
\newpage
\hspace{-1.0em}\hrulefill
\hspace{-1.0em}{\textbf{Algorithm 1}}
\vspace{-0.3cm}\hspace{-1.0em}\hrulefill
\begin{codebox}
\li Initialization:
\zi $S^k = \{k\}$, the set of nodes whose scaling factors are saturated to their respective upper-bounds
\zi on hyperplane $\beta_k = \beta_{k,max}$.
\zi $U^k = \mathcal{N}\setminus\{k\}$ , the set of nodes whose scaling factors are not saturated to their respective upper-bounds
\zi on hyperplane $\beta_k = \beta_{k,max}$.
\li Compute $\beta_{i}^k = {\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}}, i \in U^k$.
\li \While ($\exists \, \beta_{i}^k \ge \beta_{i,max}, i \in U^k$)
\li \hspace{-0.45in} Compute $M^k = \{i: \beta_{i}^k \ge \beta_{i,max}, i \in U^k\}$.
\li \hspace{-0.45in} $\beta_{i}^k = \beta_{i,max}, i \in M^k$.
\li \hspace{-0.45in} $S^k = S^k \cup M^k$.
\li \hspace{-0.45in} $U^k = U^k \setminus M^k$.
\li \hspace{-0.45in} Compute $\beta_{i}^k = {\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum_{j \in S^k} \beta_{j,max}^2 h_{jt}^2}{\sum_{j \in S^k} h_{sj} \beta_{j,max} h_{jt}}}, i \in U^k$.
\End
\end{codebox}
\vspace{-0.2cm}\hrulefill
\begin{pavikt}
\label{thrm:optBeta4diamond}
A network-wide scaling vector $\bm{\beta}_{opt} = (\beta_{1}^{opt}, \ldots, \beta_{N}^{opt})$ that maximizes the $SNR_t$ for a diamond network with the relay nodes performing ANC is given as
\begin{equation*}
\beta_{i}^{opt} = \begin{dcases}
\beta_{i,max}, i = k^\star, k^\star = \argmax\limits_{j \in \mathcal{N}}SNR_t(\bm{\beta}_{opt}^j),\\
\beta_{i,max}, i \in S^{k^\star},\\
{\displaystyle \frac{h_{si}}{h_{it}} \frac{1+\sum\limits_{j \in S^{k^\star}} \beta_{j,max}^2 h_{jt}^2}{\sum\limits_{j \in S^{k^\star}} h_{sj} \beta_{j,max} h_{jt}}, i \not\in S^{k^\star}},
\end{dcases}
\end{equation*}
where $S^{k^\star} = \{k^\star\} \cup \{i: \beta_{i}^{k^\star} \ge \beta_{i,max}, i \in \mathcal{N}\setminus\{k^\star\}\}$.
\end{pavikt}
Based on our approach in this section to compute the optimal ANC rate in the Gaussian diamond networks, in the next section we introduce a greedy scheme to bound from below the maximum end-to-end rate achievable with analog network coding in the general layered networks.
\section{General layered networks: a greedy scheme to lower bound the maximum ANC rate}
\label{sec:genNet}
In a general layered network with $L$ layers of relay nodes, consider layer $l, 1 \le l \le L$, and a node in the next $l+1^\textrm{st}$ layer, denoted as $t_{l+1}$ or with a little abuse of notation as $t$. This scenario is depicted in Figure~\ref{fig:l2lplus1}. For this subnetwork, for any scaling vector $\bm{\beta}$ we have
\begin{equation}
\label{eqn:subnetSNR}
SNR_t = \frac{P_s(\sum_{i=1}^N s_i \beta_i h_{it})^2}{\mathbb{E}(z_t+\sum_{i=1}^N z_i \beta_i h_{it})^2}
\end{equation}
Using \eqref{eqn:maxAFrate} the problem of computing the maximum ANC rate for this subnetwork can be formulated as
\begin{equation}
\label{eqn:l2lplus1prob}
\max_{0 \le \bm{\beta} \le \bm{\beta}_{max}} SNR_t,
\end{equation}
where $\bm{\beta} = (\beta_1, \ldots, \beta_N)$ and $\bm{\beta}_{max} = (\beta_{1,max} \ldots, \beta_{N,max})$ with $\beta_{i,max}^2 = P_i/\mathbb{E}(s_i x_s + z_i)^2, i \in {\mathcal N}, {\mathcal N} = \{1, \ldots, N\}$.
Equating the first-order partial derivatives of the objective function with respect to $\beta_i, i \in \mathcal{N}$, to zero, we get the following $N+1$ conditions for local extrema:
\begin{align}
&\sum_{i \in \mathcal{N}} s_{i} \beta_i h_{it} = 0 \label{eqn:gen_b1bN} \\
&\beta_i = \frac{s_i + s_i \mathbb{E}\big(\sum\limits_{j \in \mathcal{N} \setminus \{i\}} z_j \beta_{j} h_{jt}\big)^2 - \alpha_i \gamma_i}{h_{it}(\alpha_i \mathbb{E}z_i^2 / \sigma^2 - s_i \gamma_i)} \label{eqn:gen_bi_ito_rest}
\end{align}
where
\begin{align*}
\alpha_i &= \sum\limits_{j \in \mathcal{N} \setminus \{i\}} s_j \beta_{j} h_{jt}, \quad\qquad \mbox{(signal component at destination $t$ from all the nodes except node $i$)}\\
\gamma_i &= \sum\limits_{j \in \mathcal{N} \setminus \{i\}} \beta_{j} h_{jt} \mathbb{E}(z_i z_j) / \sigma^2, \quad \mbox{(noise component at destination $t$ from all the nodes except node $i$)}
\end{align*}
As we established the extremal properties of conditions \eqref{eqn:b1bN} and \eqref{eqn:bi_ito_rest} in Section~\ref{sec:diamond}, we can also prove that condition \eqref{eqn:gen_b1bN} leads to the global minimum of the objective function in \eqref{eqn:l2lplus1prob} and the global maximum of the objective function occurs at one of the $N$ hyperplanes defined by $\beta_k = \beta_{k,max}$. Following a sequence of arguments similar to those used to establish Theorem~\ref{thrm:optBeta4diamond} for diamond networks, we can characterize the scaling vector for the nodes in the $l^\textrm{th}$ layer that optimally solve the problem~\eqref{eqn:l2lplus1prob} for the subnetwork under consideration. Note that in this subnetwork, the noises at different nodes in a relay layer are correlated, unlike the noises at relay nodes in the diamond network in Figure~\ref{fig:diamond}. This explains the difference between the $SNR$ expression in \eqref{eqn:subnetSNR} and the one in \eqref{eqn:diamondSNR} for the diamond network, and results in more complex analysis in the present case.
\begin{pavikl}
\label{lemma:optBeta4l2lplus1}
A scaling vector $\bm{\beta}_{opt} = (\beta_{1}^{opt}, \ldots, \beta_{N}^{opt})$ that maximizes the $SNR_t$ for any subnetwork, as in Figure~\ref{fig:l2lplus1}, of the general layered network with the relay nodes in $l^\textrm{th}$ layer performing analog network coding is given as
\begin{equation*}
\beta_{i}^{opt} = \begin{dcases}
\beta_{i,max}, i = k^\star, k^\star = \argmax_{\{\bm{\beta}^j:j \in \mathcal{N}\}} SNR_t(\bm{\beta}^j),\\
\beta_{i,max}, i \in S^{k^\star},\\
\frac{s_i + s_i \mathbb{E}\big(\sum\limits_{j \in S^{k^\star}} z_j \beta_{j,max} h_{jt}\big)^2 - \alpha_j \gamma_j}{h_{it}(\alpha_j\mathbb{E}z_i^2 / \sigma^2 - s_i \gamma_j)}, i \not\in S^{k^\star},
\end{dcases}
\end{equation*}
where $\bm{\beta}^j=(\beta_1^j, \ldots, \beta_N^j)$ with
\begin{align*}
\beta_i^j &= \begin{dcases}
\beta_{j,max}, i = j\\
\frac{s_i + \{s_i \mathbb{E}z_j^2 / \sigma^2 - s_j \mathbb{E}(z_i z_j) / \sigma^2\}\beta_{j,max}^2 h_{jt}^2}{h_{it}\{s_j \mathbb{E}z_i^2 / \sigma^2 - s_i \mathbb{E}(z_i z_j) / \sigma^2\}\beta_{j,max} h_{jt}}, i \neq j,
\end{dcases}\\
\alpha_j &= \sum_{j \in S^{k^\star}} s_j \beta_{j,max} h_{jt}, \qquad\qquad \mbox{(signal component at destination $t$ from the nodes in $S^{k^\star}$)}\\
\gamma_j &= \sum_{j \in S^{k^\star}} \beta_{j,max} h_{jt} \mathbb{E}(z_i z_j) / \sigma^2, \quad \mbox{(noise component at destination $t$ from the nodes in $S^{k^\star}$)}
\end{align*}
and $S^{k^\star} = \{k^\star\} \cup \{i: \beta_{i}^{k^\star} \ge \beta_{i,max}, i \in \mathcal{N}\setminus \{k^\star\}\}$.
\end{pavikl}
Note that Lemma~\ref{lemma:optBeta4l2lplus1} reduces to Theorem~\ref{thrm:optBeta4diamond} when the noise components at the relay nodes are uncorrelated.
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./l2lplus1}
\caption{A subnetwork of general layered network with $L$ relay layers, depicting $l^\textrm{th}$ layer with $N$ relay nodes and a node in the $l+1^\textrm{st}$ layer. The received signal component at node $i, 1 \le i \le N$, in the $l^\textrm{th}$ layer is denoted as $s_i x_s$, where $x_s$ is the source symbol and corresponding noise component is denoted as $z_i$.}
\label{fig:l2lplus1}
\end{figure}
Using Lemma~\ref{lemma:optBeta4l2lplus1}, we can compute $\bm{\beta}_{l,opt}^{l+1, j}$, the scaling vector for the nodes in the $l^\textrm{th}$ layer that maximizes the received SNR for node $j, 1 \le j \le n_{l+1}$, in the $l+1^\textrm{st}$ layer. Among these $n_{l+1}$ scaling vectors for the nodes in the $l^\textrm{th}$ layer, let $\bm{\beta}_{l}^{low}$ denote the one that solves the following problem
\begin{equation}
\label{eqn:bestlbeta}
\bm{\beta}_{l}^{low} = \argmax_{\substack{\bm{\beta}_{l,opt}^{l+1, j} \\ 1 \le j \le n_l}} \prod_{k=1}^{n_{l+1}} (1+SNR_k)
\end{equation}
The following corollary of Lemma~2 in \cite{112agnihotriJaggiChen} establishes that among $n_{l+1}$ such scaling vectors, the scaling vector characterized by $\bm{\beta}_{l}^{low}$ computes the tightest lower bound for the optimal value of the objective function in \eqref{eqn:bestlbeta} as well as \eqref{eqn:eqProb}.
\begin{pavikc}[\hspace{-0.001cm}\cite{112agnihotriJaggiChen}, Lemma 2]
\label{cor:betterBeta}
Consider two scaling vectors $\bm{\beta}_l$ and $\bm{\hat\beta}_l$ for the nodes in $l^\textrm{th}$ layer. If $\prod_{k=1}^{n_{l+1}} (1+SNR_k)\big|_{\bm{\beta}_l} > \prod_{k=1}^{n_{l+1}} (1+SNR_k)\big|_{\bm{\hat\beta}_l}$, then $SNR_t(\bm{\beta}_l) > SNR_t(\bm{\hat\beta}_l)$.
\end{pavikc}
Computing $\bm{\beta}_{l}^{low}$ as above for each layer $l, 1 \le l \le L$, in conjunction with Corollary~\ref{cor:betterBeta}, allows us to construct a network-wide scaling vector $\bm{\beta}_{low} = (\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{L}^{low})$ to compute a lower bound\footnote{Clearly, choosing $\bm{\beta}_{l}^{low}$ as in \eqref{eqn:bestlbeta} for each layer $l$ may lead, in general, to some performance loss at each layer as $\bm{\beta}_{l}^{low}$ may not be the optimal vector of the scaling factors for the nodes in the layer $l$ that solves
\begin{equation*}
\argmax_{0 \le \bm{\beta}_{l} \le \bm{\beta}_{l,max}} \prod_{k=1}^{n_{l+1}} (1+SNR_k)
\end{equation*}
The cumulative effect of this performance loss at each layer is that the end-to-end ANC rate computed at $\bm{\beta}_{low}$ may not lead to the optimal solution of problem~\eqref{eqn:maxAFrate}. However, our results in the next section show that for a large class of layered networks there is no loss in the optimality and for other layered networks, the loss is \textit{small} asymptotically in the network parameters.} to the optimal solution of \eqref{eqn:maxAFrate}. Formally, for a given layered network, $\bm{\beta}_{low}$ is constructed as follows.
\begin{pavikp}
\label{prop:lowBetaGenNet}
Consider a layered relay network of $L+2$ layers, with the source $s$ in layer `$0$', the destination $t$ in layer `$L+1$', and $L$ layers of relay nodes between them. The $l^\textrm{th}$ layer contains $n_l$ nodes, $n_0 = n_{L+1} = 1$. A network-wide scaling vector $\bm{\beta}_{low}=(\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{L}^{low})$ that provides a lower bound to the optimal solution of \eqref{eqn:maxAFrate} for this network, can be computed recursively for $1 \le l \le L$ as
\begin{equation*}
\bm{\beta}_{l}^{low} = \argmax_{\substack{\bm{\beta}_{l,opt}^{l+1, j} \\ 1 \le j \le n_l}} \prod_{k=1}^{n_{l+1}}(1+SNR_{l+1, k}(\bm{\beta}_{1}^{low}, \ldots, \bm{\beta}_{l-1}^{low}, \bm{\beta}_{l,opt}^{l+1, j}))
\end{equation*}
Here $\bm{\beta}_{l}^{low} = (\beta_{l 1}^{low}, \ldots, \beta_{l n_l}^{low})$ is the vector of scaling factors for the nodes in the $l^\textrm{th}$ layer, and $\bm{\beta}_{l,opt}^{l+1, j}$ (computed using Lemma~\ref{lemma:optBeta4l2lplus1}) is the scaling vector for the nodes in the $l^\textrm{th}$ layer that maximizes the received SNR for node $j, 1 \le j \le n_{l+1}$, in layer $l+1$.
\end{pavikp}
In the next section we analyze the performance of the greedy scheme of the Proposition~\ref{prop:lowBetaGenNet} in the context of both a special class of layered networks and the general layered networks.
\section{Illustration}
\label{sec:exa}
We first demonstrate that the greedy scheme of Proposition~\ref{prop:lowBetaGenNet} allows us to exactly compute the optimal ANC rate for a broad class of layered networks. Then, we give an example to show that for the general layered networks, the proposed scheme leads to the optimal rates within a constant gap from the cut-set upper bound asymptotically in the source power.
\textit{\textbf{Example 2} (A class of exactly solvable layered networks):} Let us consider a class of symmetric layered networks where the channel gains along all outgoing links from a node are equal. An instance of such a network is obtained from the network in Figure~\ref{fig:layrdNetSpl_Exa} when $h_{s1}=h_{s2}, h_{13}=h_{14}$, and $h_{23}=h_{24}$. An implication of this property of the channel gains is that the received SNRs at every node in a layer are equal: $SNR_{l, j}=SNR_l, 1 \le j \le n_l, 1 \le l \le L$. In this case, for each layer $l, 1 \le l \le L$, $\bm{\beta}_{l}^{low}$ computed in Proposition~\ref{prop:lowBetaGenNet} is equal to the optimal $\bm{\beta}_l^{opt}$ computed in \cite[Lemma 2]{112agnihotriJaggiChen}. Therefore, $\bm{\beta}_{low}$ is the optimal solution of problem \eqref{eqn:eqProb} for this class of networks.
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./layrdNetSpl_Exa}
\caption{General layered network with 2 relay layers between the source `s' and destination `t'. Each layer contains two relay nodes.}
\label{fig:layrdNetSpl_Exa}
\end{figure}
Consider an instance of the network in Figure~\ref{fig:layrdNetSpl_Exa} when $h_{s1}=h_{s2}, h_{13}=h_{14}$, and $h_{23}=h_{24}$. Such an instance belongs to the class of symmetric networks we are concerned with in this example. Using Proposition~\ref{prop:lowBetaGenNet}, the optimal solution of problem \eqref{eqn:eqProb} for this instance is:
\begin{align*}
&\bm{\beta}_{opt} = \bigg(\beta_{1,max}, \frac{1+\beta_{1,max}^2 h_1^2}{h_2 \beta_{1,max} h_1}, \beta_{3,max}, \beta_{4,max}\bigg), \mbox{ where}\\
&\beta_{1,max}^2 = \frac{P_1}{h_0^2 P_s + \sigma^2} \\
&\beta_{3,max}^2 = \frac{P_3}{S^2 P_s + Z^2 \sigma^2}, \beta_{4,max}^2 = \frac{P_4}{S^2 P_s + Z^2 \sigma^2} \\
&S = h_0(\beta_{1,opt} h_1 + \beta_{2,opt} h_2), Z^2 = 1 + \beta_{1,opt}^2 h_1^2 + \beta_{2,opt}^2 h_2^2
\end{align*}
and we assume that $P_1 h_1^2 > P_2 h_2^2$. {\hspace*{\fill}~\IEEEQEDclosed\par}
{\textit{\textbf{Example 3} (General layered networks):}}
Let us consider the layered network of Figure~\ref{fig:layrdNetSpl_Exa}. We compute a lower bound to the optimal ANC rate for this network using the greedy scheme in the Proposition~\ref{prop:lowBetaGenNet} and compare it with the MAC upper bound in Figure~\ref{fig:rateComparison}. Also, plotted in this figure is the ANC rate achievable when the scaling factors for all relay nodes are set to their respective upper-bounds. We observe that in this case the ANC rate achieved with the greedy scheme of Proposition~\ref{prop:lowBetaGenNet} approaches the capacity within one bit when $P_s > 100$.
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{./rateComparison1t}
\caption{Comparison of the ANC rate achievable with the scheme in Proposition~\ref{prop:lowBetaGenNet} with the MAC upper bound for the layered network in Figure~\ref{fig:layrdNetSpl_Exa} with $P_1=P_2=P_3=P_4=10$, $h_{14}=h_{24}=2$ and all other channel gains are equal to $10$. Also plotted is the ANC rate when the scaling factors for all relay nodes are set to their respective upper-bounds.}
\label{fig:rateComparison}
\end{figure}
\section{Conclusion and Future Work}
\label{sec:conclFW}
We consider the problem of maximum rate achievable with analog network coding in general layered networks. Previously, this problem has been considered under certain assumptions on per node scaling factor and received SNR as without these assumptions the problem was presumed to be intractable. The key contribution of this work is a greedy scheme to exactly compute the optimal rates in a wider class of layered networks than those that can be addressed using prior approaches. In particular, using the proposed scheme for the Gaussian $N$-relay diamond network, to the best of our knowledge, we provide the first exact characterization of the optimal rate achievable with analog network coding. Further, for general layered networks, our scheme allows us to compute optimal rates at most a constant gap away from the cut-set upper bound asymptotically in the source power. In the future, we plan to extend this work to non-layered networks, and to construct the optimal distributed relay schemes.
|
{
"timestamp": "2012-04-24T02:05:52",
"yymm": "1203",
"arxiv_id": "1203.2297",
"language": "en",
"url": "https://arxiv.org/abs/1203.2297"
}
|
\section{\label{thsec1}Introduction}
Quantum coherent oscillations in a quantum two-level system
(qubit) stand for the most basic dynamic manifestation of
quantum coherence between the qubit states.
Motivated by potential applications to quantum
computation \cite{Bra92,All11072138}, as well as
general interest in mesoscopic quantum phenomena,
intensive experimental and theoretical effort has been
devoted to the attempts to study these oscillations
and measurement possibilities of individual qubits.
One of the especially interesting methods of detecting
coherent oscillations is to monitor them continuously with a
mesoscopic electrometer, such as single electron transistor
(SET) \cite{Shn9815400,Dev001039,Mak01357,Cle02176804
Jia07155333,Gil06116806,Gur05073303,Oxt06045328} or
quantum point contact (QPC)
\cite{Buk98871,Sch981238,Nak99786,Spr005820,Aas013376},
whose conductance depends on the charge state of a nearby
qubit.
From the readout of the detector, one is capable of gaining
essential insight into the nontrivial correlation
characteristics between the detector and the measured
system.
In contrast to the projective measurement which takes
place instantaneously, the continuous detection extracts
information of the measured system continually.
However, the detection inevitably acts back on the
system, leading thus to the dephasing of the qubit.
This trade-off between acquisition of quantum state information
and backaction dephasing of the measured system plays the central
role in the process of quantum measurement.
Recently, it was demonstrated that the measurement properties
are intimately associated with the full counting statistics
of the detector \cite{Naz03,Ave05126803}.
Evaluation of the shot noise and higher order cumulants
of quantum measurement have been worked out under Markovian
approximation and in the wide-band limit
(WBL) \cite{Rod05251,Moz04018303,Cle04121303,Wab05165347
Goa01125326,Kor01115403,Sta04136802,Fli04205334,Kie06033312,Wan07125416}.
The Markovian approximation assumes that the correlation time in the reservoirs
is much shorter than the typical response time in the reduced
system, while the WBL neglects the energy-dependent densities of
states in the electrodes.
Yet, these approximations may not be always true in realistic
devices.
Hence, a recent development in noise characteristics has
been devoted to the investigations of the non-Markovian
effects with energy-dependent spectral density in the
environment \cite{Fen08075302,Zhe09124508,Zed09045309}.
Flindt et al \cite{Fli08150601} and Aguado et al \cite{Agu04206601}
studied the noise properties of a charge qubit in a transport
configuration, where the non-Markovian effect of the phonon bath
was effectively accounted for.
The non--Markovian correlations of electrodes were investigated
in the context of transport through quantum dot (QD) systems
\cite{Jin11053704,Bra06026805}, and measurement of a nanomechanical
resonator by QPC \cite{Che11012393}, where radical difference
in dynamics between non-Markovian and Markovian cases was
identified.
The purpose of this paper is to study the non--Markovian
characteristics in continuous measurement of a qubit by a QPC.
Our analysis is based on a generalized time-nonlocal
quantum master equation approach \cite{Yan982721},
which is capable of treating properly the energy exchange
between the qubit and detector.
In comparison with the Markovian case, we find considerable
differences in qubit relaxation and dephasing in the
non-Markovian domain where the information may flow
from the environment back to the reduced system.
Furthermore, the unique non-Markovian dynamics
is reflected in the output noise feature of the detector.
We observe that the non-Markovian memory
effect results in a strong enhancement of the
signal-to-noise ratio (SNR).
It is demonstrated unambiguously that under appropriate
conditions the SNR can even exceed the upper limit of
``4'', leading thus to the violation of the Korotkov-Averin
(K-A) bound.
Our study thus may open new possibilities to improve detector's
measurement efficiency in a direct and transparent way.
This paper is organized as follows. We begin in \Sec{thsec2} with the
model set-up of a charge qubit under the continuous monitoring by a
QPC. In \Sec{thsec3}, we first analyze the reservoir correlation time
under various parameters of the QPC detector, and then study the unique
qubit relaxation and dephasing arising from the non-Markovian processes.
\Sec{thsec4} is devoted to the calculation of noise characteristics
based on the ``$N$''-resolved time-nonlocal quantum master equation
approach.
Numerical results, particularly, the discussions of SNR under various
parameters are presented.
Finally, we summarize the main results and implications of this work
in \Sec{thsec5}.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.8]{Fig_1.eps}
\caption{\label{fig1}
Schematic setup of a solid--state charge qubit
measured continuously by a quantum point contact.
The qubit is represented by an extra electron
tunneling between the coupled quantum dots.
The tunneling amplitude of the QPC is susceptible to
changes in the surrounding electrostatic environment,
and can therefore be used to sense the position
of the extra electron.}
\end{center}
\end{figure}
\section{\label{thsec2}Model description}
The system under investigation is schematically shown
in \Fig{fig1}. The qubit is represented by an extra
electron tunneling between two coupled quantum dots (QDa and QDb).
When the electron occupies the QDa (QDb), the qubit
is said to be in the localized state $|a\ra$ ($|b\ra$).
A nearby QPC serves as the charge detector to continuously
monitor the position of the electron.
Occupation of the electron in different dots
leads to distinct influence on the transport current
through the QPC. It is right this mechanism that makes
it possible to read out the qubit-state information.
The entire system is described by the Hamiltonian
$H_{\rm T} =H_{\rm qu}+H_{\rm D}+H'$, where
\bsube\label{sys-Ham}
\begin{gather}
H_{\rm qu} =\frac{1}{2}\epl\sgm_z+\Omg\sgm_x,\label{Hqubit}
\\
H_{\rm D} =\sum_{k\in {\rm L}} \vpl_k
\hat c_k^\dag \hat c_k
+\sum_{q\in {\rm R}} \vpl_q
\hat c_q^\dag \hat c_q \, ,
\\
H' =\sum_{s=a,b}\sum_{k,q}t^{s}_{kq}
\hat c_k^\dag \hat c_q
\cdot |s\ra\la s|+{\rm h.c.}.\label{Hprim}
\end{gather}
\esube
Here, $H_{\rm qu}$ denotes the qubit Hamiltonian, where
the pseudospin operators are defined as
$\sgm_z\equiv|a\ra\la a|-|b\ra\la b|$ and
$\sgm_x\equiv|a\ra\la b|+|b\ra\la a|$, respectively.
Each dot has only one bound state, i.e., the logic
states $|a\ra$ and $|b\ra$, with level detuning $\epl$
and interdot coupling $\Omg$.
The second component $H_{\rm D}$ depicts the left and right
QPC reservoirs, where $\hat{c}_k$ ($\hat{c}_q$) denotes
the annihilation operator for an electron in the left (right)
QPC reservoir.
The electron reservoirs are characterized by the Fermi
distributions $f_\alf{(\omg)}=\{1+e^{\beta(\omg-\mu_\alf)}\}^{-1}$,
where $\mu_\alf$ is the Fermi energy of the left ($\alf$=L)
or right ($\alf$=R) reservoir, and $\beta=(k_{\rm B}T)^{-1}$ is the
inverse temperature.
Hereafter, the Planck's constant $\hbar$ and the electron charge $e$
are set to unity, i.e. $\hbar=e=1$, unless stated otherwise.
Throughout this work, we define
$\mu^{\rm eq}_{\rm L}=\mu^{\rm eq}_{\rm R}=0$ for the
equilibrium chemical potentials (or Fermi energies)
of the QPC reservoirs.
An applied measurement voltage thus is modeled by the difference
in chemical potentials of the left and right electrodes:
$V=\mu_{\rm L}-\mu_{\rm R}$.
The tunneling Hamiltonian for the QPC detector is
represented by the last component $H'$.
The amplitude $t^s_{kq}$ of electron tunneling
through two reservoirs of the QPC depends explicitly
on the qubit state $|s\ra$ ($s=a$ or $b$).
Thus the quantum operator to be measured is $\sgm_z$.
By denoting $Q_s\equiv |s\ra\la s|$, the qubit--QPC detector
coupling in the $H_{\rm D}$--interaction picture can be rewritten
as $H'(t)=\sum_s[\hat f_s(t)+\hat f^{\dag}_s(t)]\cdot Q_s$,
with
$\hat f_s(t)\equiv e^{\rmi H_{\rm D}t}\big(\sum_{kq}t_{kq}^s
\hat c_k^\dag \hat c_q\big)e^{-\rmi H_{\rm D}t}$.
The effects of the stochastic QPC reservoirs on
measurement are characterized by the reservoir correlation
functions
$C_{ss'}^{(+)}(t-\tau)\equiv\la \hat f_{s}^\dag(t)
\hat f_{s'}(\tau)\ra$ and
$C_{ss'}^{(-)}(t-\tau)\equiv\la \hat f_{s}(t)
\hat f_{s'}^\dag(\tau)\ra$.
By introducing the reservoir spectral density function
\be\label{Jww}
J_{ss'}(\omg,\omg')=\sum_{k,q} t_{kq}^st_{kq}^{s'}
\delta(\omg-\vpl_k)\delta(\omg'-\vpl_q),
\ee
these QPC coupling correlation functions can be recast as
\be\label{CCF}
C_{ss'}^{(\pm)}(t) = \! \int\!\!\!\int\!\! \rmd\omg \rmd\omg'
J_{ss'}(\omg,\omg')f_{\rm L}^{(\pm)}(\omg)
f_{\rm R}^{(\mp)}(\omg') e^{\pm \rmi(\omg-\omg')t},
\ee
where $f_\alf^{(+)}(\omg)$ is the usual Fermi function, and
$f_\alf^{(-)}(\omg)\equiv1-f_\alf^{(+)}(\omg)$.
In order to characterize finite cutoff energy of the QPC reservoirs,
we introduce a single Lorentzian to model the band structure.
For the sake of constructing analytical results, we assume
a simple Lorentzian function of cutoff ``w'' centered at
the Fermi energy for the spectral density \Eq{Jww}.
Moreover, the bias voltage is conventionally described by
a relative shift of the entire energy-bands, thus the
centers of the Lorentzian functions would fix at the
Fermi levels $\mu_{\rm L}$ and $\mu_{\rm R}$. Without loss of
generality, we set \cite{Luo09385801}
\be\label{Jw}
J_{ss'}(\omg,\omg')=\chi_s\chi_{s'}
\frac{\Gam_{\rm L}^0\rmw^2}{(\omg-\mu_{\rm L})^2+\rmw^2}\cdot
\frac{\Gam_{\rm R}^0\rmw^2}{(\omg'-\mu_{\rm R})^2+\rmw^2},
\ee
where $\Gam_{\rm L}^0$ $(\Gam_{\rm R}^0)$ is the maximum
of the spectral in the left (right) electrode;
$\chi_s$ and $\chi_{s'}$ are qubit-QPC coupling
parameters, which are of $\chi_a>\chi_b$, as can be inferred
from \Fig{fig1}.
In the limit of w$\rightarrow\infty$, the QPC spectral density
\Eq{Jw} becomes energy-independent and reduces to the constant
WBL spectral density used in the literature.
\section{\label{thsec3}Non-Markovian Dynamics}
The non--Markovian dynamics of the reduced system is described
by a generalized time-nonlocal quantum master equation.
An equation of this type can be obtained from the partitioning
scheme devised by Nakajima and Zwanzig \cite{Zwa01,Bre02},
or in the real-time diagrammatic technique for the dynamics
of the reduced density matrix on the Keldysh contour \cite{Mak01357},
\be\label{QME}
\dot{\rho}(t)=-\rmi {\cal L}\rho(t)-\!\int_{-\infty}^t\! \rmd \tau
\Pi(t-\tau)\rho(\tau),
\ee
where the first term ${\cal L}(\cdots)\equiv[H_{\rm qu},(\cdots)]$ is the
qubit Liouvillian.
The influence of the QPC detector on the dynamics of the qubit
is described by the memory kernel (the second term), which is
given by \cite{Yan982721},
\begin{align}
\!\!\Pi(t-\tau)\rho(\tau)=
\sum_{ss'}[&Q_s,C_{ss'}(t-\tau)G(t-\tau)Q_{s'}\rho(\tau)
\nonumber \\
&\;\;-C^\ast_{ss'}(t-\tau)G(t-\tau)\rho(\tau)Q_{s'}],
\end{align}
where $C_{ss'}(t-\tau)=C^{(+)}_{ss'}(t-\tau)+C^{(-)}_{ss'}(t-\tau)$,
and $G(t-\tau)\equiv e^{-{\rmi\cal L}(t-\tau)}$ is the
free propagator associated with the qubit Hamiltonian alone.
In deriving \Eq{QME}, the only approximation made is the
second-order perturbation in the system-reservoir coupling.
This equation thus is valid for arbitrary reservoir temperatures,
cutoff energies, and measurement voltages, as long as the
second-order perturbation holds.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.72]{Fig_2.eps}
\caption{\label{fig2}
Real part of reservoir correlation function $C_{ab}(t)$ at
(a) a small cutoff energy w$/\Dlt=1.0$ and (b) a large
cutoff energy w$/\Dlt=10.0$ for different bias voltages:
$V/\Dlt=0.2$ (solid lines), $V/\Dlt=2.0$ (dashed lines),
and $V/\Dlt=5.0$ (dotted lines).
The temperature is $\beta\Dlt=1.0$.}
\end{center}
\end{figure}
The Markovian approximation is valid only when the correlation
time of the reservoir is much shorter than the characteristic
time of the reduced quantum system.
The former one is defined as the time scale at which the
profiles of the QPC reservoir two-time correlation function
decays.
It is closely associated with the following time scales,
the time scale of the QPC spectral density
($\sim$w$^{-1}$), the time scale of the applied bias ($\sim V^{-1})$,
and the time scale of the QPC reservoir temperature ($\sim\beta$).
In what follows, we first study numerically how the reservoir
correlation time is varied as a function of different parameters of the
QPC reservoirs.
\Fig{fig2} shows the real parts of the QPC reservoir correlation
function $C_{ab}(t)$ versus
time for various values of cutoff energy and voltage at a given
temperature $\beta\Dlt=1.0$.
The correlation time decreases as the voltage increases, which is
particularly prominent for a narrow cutoff w$/\Dlt=1.0$
as displayed in \Fig{fig2}(a).
By comparing \Fig{fig2} (a) and (b), it is revealed that
the dependence of reservoir correlation time on the bias
voltage is much weaker than that on the cutoff.
In the limit of w$\rightarrow0$, one finds
$J_{ss'}(\omg,\omg')\propto\dlt(\omg-\mu_{\rm L})\dlt(\omg'-\mu_{\rm R})$, which
leads to QPC correlation
functions proportional to $e^{\pm\rmi Vt}$, i.e. completely
non-local in time.
The opposite limit of WBL (w$\rightarrow\infty$) corresponds to a channel-mixture
regime, where a great number of possible transitions of electron
tunneling between the two reservoirs of the QPC
take place.
It was shown in this regime the QPC reservoir correlation time
and memory effect are remarkably reduced \cite{Lee08224106}.
Thus, the larger the cutoff is, the shorter the
correlation time is. The reservoir correlation time is
mainly restricted by the cutoff.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.72]{Fig_3.eps}
\caption{\label{fig3}
Real part of reservoir correlation function $C_{ab}(t)$ vs time
``$t$'' for (a) w$/\Dlt=10.0$ and (b) w$/\Dlt=1.0$
at different temperatures: $\beta\Dlt=0.2$ (solid lines), $\beta\Dlt=1.0$ (dashed
lines), and $\beta\Dlt=2.0$ (dotted lines).
The measurement voltage is $V/\Dlt=2.0$.}
\end{center}
\end{figure}
To explore the influence of the temperature, we plot in
\Fig{fig3} the reservoir correlation function for various
temperatures.
Analogous to that on the measurement voltage, the correlation
time deceases as the reservoir temperature rises.
Nevertheless, the effect of the temperature on the reservoir
correlation time is less sensitive than that on the voltage,
as can be seen by comparing \Fig{fig2} and \Fig{fig3}.
Therefore, the cutoff energy has the dominant role to play in
determining the QPC reservoir correlation time.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.8]{Fig_4.eps}
\caption{\label{fig4}
Non-Markovian dynamics of the qubit for different values of
cutoff energy: w$/\Dlt=1.0$ (solid lines), w$/\Dlt=5.0$ (dashed
lines), and w$/\Dlt=10.0$ (dotted lines).
(a) The probability of finding the electron in the localized state $|a\ra$:
$\rho_{aa}(t)\equiv\la a|\rho(t)|a\ra$,
(b) in the localized state $|b\ra$: $\rho_{bb}(t)\equiv\la b|\rho(t)|b\ra$,
(c) real part of the off-diagonal matrix element $\rho_{ab}(t)$, and
(d) imaginary part of $\rho_{ab}(t)$.
The Markovian WBL results are also plotted in dash-dotted lines
for comparison.
The qubit is assumed to be symmetric ($\epl=0$), and
initially in the state $\rho_{\rm ini}=|a\ra\la a|$.
Other parameters used are $V/\Dlt=5.0$, $\beta\Dlt=1.0$,
$\eta\Dlt^2=2.0$, and the tunneling amplitudes $\chi_a/\Dlt=1.0$
and $\chi_b/\Dlt=0.8$.}
\end{center}
\end{figure}
With the knowledge of these time scales, we are now
in a position to discuss the non-Markovian dynamics of the
qubit under the continuous measurement of the QPC.
The numerical propagation of the time-nonlocal QME
is facilitated by employing the approach of auxiliary density
operators \cite{Wel06044712,Jin08234703,Cro09073102,Cro10159904}.
The calculation of the time evolution is then reduced
to the propagation of coupled differential equations.
The numerical results of the non-Markovian dynamics
of the qubit are plotted in \Fig{fig4}
for different values of the cutoff energy.
For comparison, we have also plotted the Markovian
result by the dash-dotted lines for the same parameters.
The measurement backaction-induced dephasing leads to
a coherent to incoherent transition of the qubit
electron tunneling.
In the coherent regime, the tunneling leads to
the well-known Rabi oscillations with frequency
given by $\Dlt$, as indicated in \Fig{fig4}(a)
and (b).
For a symmetric qubit ($\epl=0$), the occupation
probability in each dot finally reaches 1/2 for
both Markovian and non-Markovian cases.
However, for a small cutoff, such as w$/\Dlt=1.0$ (solid lines),
the non-Markovian relaxation behavior shows a considerable
difference to the Markovian case (dash-dotted lines).
As the cutoff energy increases, the qubit relaxation gets close
to that of the Markovian result, due to reduced reservoir
correlation time [see the dotted lines in
\Fig{fig4}(a) and (b)].
The backaction-induced dephasing behavior is described
by the off-diagonal density-matrix element, as displayed
in \Fig{fig4}(c) and (d).
The real part of $\rho_{ab}$ approaches a nonzero
constant at long times.
The nonzero stationary result stems from the energy exchange
between the qubit and QPC detector \cite{Li04085315,Li05066803,Luo09385801}.
For both Markovian and non-Markovian cases, the imaginary
part of $\rho_{ab}$ goes to zero in the stationary limit.
However, for a small cutoff energy, the dephasing
rate is much lower than that of the Markovian case, as displayed
by the solid line in \Fig{fig4}(d).
In Markovian processes, information flows continuously
from the qubit to its environment.
Yet, in the presence of non-Markovian behavior, a reversed
flow of information from the environment
back to the reduced system occurs, which
leads to the reduction of the dephasing rate.
To clearly demonstrate this unique feature, we
employ the ``trace distance'' of two quantum states $\rho_1$ and
$\rho_2$, which is defined as \cite{Nie00,Bre09210401,Lai10062115}
\be
D[\rho_1(t),\rho_2(t)]=\frac{1}{2}{\rm tr}|\rho_1(t)-\rho_2(t)|.
\ee
Here the norm is given by $|A|=\sqrt{A^\dag A}$, and
$\rho_{1,2}(t)$ are the dynamical qubit states for a given
pair of initial states $\rho_{1,2}(0)$.
The trace distance describes the probability of distinguishing
those states.
In Markovian processes, the distinguishability between
any two states are continuously reduced, and thus the trace
distance $D(\rho_1,\rho_2)$ decreases monotonically.
The essential property of non-Markovian behavior is
the growth of this distinguishability.
An increase of the trace distance during any time
intervals implies the emergence of non-Markovianity
(inverse flow of the information).
One is therefore inspired to utilize the rate of change of the
trace distance ``$\kappa$'' to exhibit unambiguously
this process
\be
\kappa[t,\rho_{1,2}(0)]=\frac{\rmd}{\rmd t}D[\rho_1(t),\rho_2(t)].
\ee
Apparently, for a Markovian process the monotonically reduction
of the trace-distance implies $\kappa\leq0$.
The existence of $\kappa>0$ during any time intervals
identifies the non-Markovian process.
The numerical results are plotted in \Fig{fig5}.
For a small cutoff energy (w$/\Dlt=1.0$) as shown in \Fig{fig5}(a),
there exist certain times in which $\kappa>0$.
In those regimes, the information flows from the environment
back to the reduced system, i.e. the non-Markovian process. It
explains the suppression of the dephasing rate in \Fig{fig4}(d).
An increase in cutoff energy reduces reservoir correlation time,
and thus leads to the inhibition of the non-Markovianity
[see \Fig{fig5}(b) for w$/\Dlt=10.0$].
While in Markovian processes, measurements tend to wash out
more and more characteristic features of the two states, resulting
thus in an uncovering of these features.
The rate of change of the trace distance ``$\kappa$'' stays
below zero, as shown in \Fig{fig5}(c).
The suppression of the dephasing rate due to non-Markovian
dynamics has an important impact on noise characteristics of
the measurement, which will be discussed
in the next section.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.7]{Fig_5.eps}
\caption{\label{fig5}
The rate of change $\kappa$ of the trace distance as a
function of time for (a) w$/\Dlt=1.0$, (b) w$/\Dlt=10.0$,
and (c) Markovian WBL result.
The initial pair of states used are $\rho_1(0)=|a\ra\la a|$
and $\rho_2(0)=|b\ra\la b|$.
The other parameters are the same as those in \Fig{fig4}.}
\end{center}
\end{figure}
\section{\label{thsec4}Noise Characteristics}
In this section, we first introduce the ``$N$''-resolved
non-Markovian master equation for the calculation of the
output noise characteristics of the QPC detector.
Next, the numerical results for QPC noise are presented,
with special emphasis on the measurement SNR under
various conditions.
\subsection{Particle-Number-Resolved Master Equation}
To achieve the description of the output characteristics, the reduced
density matrix $\rho(t)$ is unraveled into components $\rho^{(N)}(t)$, in
which ``$N$'' is the number of electrons passing though the
QPC during the time span $[0,t]$. The resultant time-nonlocal
``$N$''--resolved quantum master
equation reads \cite{Mak01357,Fli08150601,Agu04206601,Bra06026805,Jin11053704}
\begin{widetext}
\begin{align}\label{CQME}
\dot{\rho}^{(N)}(t)=-\rmi {\cal L}\rho^{(N)}(t)-\!\int_0^t\! \rmd \tau
\bigg\{ \Pi_0(t-\tau)\rho^{(N)}(\tau)
-\sum_{\pm}\Pi_{\pm}(t-\tau)\rho^{(N\pm1)}(\tau)\bigg\}+\varrho^{(N)}(t),
\end{align}
with
\bsube
\begin{gather}
\Pi_0(t-\tau)(\cdots)=\sum_{ss'}\{C_{ss'}(t-\tau)Q_sG(t-\tau)Q_{s'}(\cdots)
+[C_{ss'}(t-\tau)]^\ast G(t-\tau)(\cdots)Q_{s'}Q_{s}\},
\\
\Pi_\pm(t\!-\!\tau)(\cdots)
=\sum_{ss'}\{C^{(\pm)}_{ss'}(t-\tau)G(t-\tau)Q_{s'}(\cdots)Q_s
+[C^{(\pm)}_{ss'}(t-\tau)]^\ast Q_{s}G(t\!-\!\tau)(\cdots)Q_{s'}\}.
\end{gather}
\esube
\end{widetext}
Here, the memory kernel $\Pi_0$ corresponds to ``continuous''
evolution of the system, and $\Pi_\pm$
denotes forward and backward jumps of the transfer of an
electron from the left electrode to the right one.
By summing up \Eq{CQME} over all possible electron numbers
``$N$'', one straightforwardly recovers the unconditional
master equation (\ref{QME}).
Hereafter, we assume that the system evolves from $t_0=-\infty$, such that
the electronic occupation probabilities at $t=0$, where electron counting
begins, have reached the stationary state, i.e.,
$\rho^{(N)}(t=0)=\dlt_{N,0}\rho_{\rm st}$,
with $\rho_{\rm st}=\rho(t\rightarrow\infty)$.
The effects of the memory of its history prior to time
$t=0$ are incorporated in the inhomogeneity $\varrho^{(N)}$ \cite{Ema11085425}.
The unraveling of the density matrix in \Eq{CQME} enables
us to evaluate the probability distribution for the number
of transferred charge $P(N,t)=\tr \{\rho^{(N)}(t)\}$,
where the trace is over degrees of freedom of the reduced system.
In principle, all the cumulants of the current distribution
can be obtained, consisting thus a spectrum of full counting statistics.
For instance, the first cumulant is directly related to the average current
through the QPC, $I(t)=\sum_N N \dot{P}(N,t)$. By using \Eq{CQME},
the current is given by
\bea
I(t)=\int_0^t \rmd \tau\, \tr\{[\Pi_-(t-\tau)-\Pi_+(t-\tau)]\rho(\tau)\}.
\eea
The stationary current thus reads
\be\label{cur}
\bar{I}\equiv
I(t\rightarrow\infty)=\tr\{J_-(z)\rho_{\rm st}\}|_{z\rightarrow0},
\ee
with
\bea
J_\pm(z)=\tilde{\Pi}_-(z)\pm\tilde{\Pi}_+(z).
\eea
Here $\tilde{\Pi}_0(z)$ and $\tilde{\Pi}_\pm(z)$, the resolvents
of the corresponding kernels in \Eq{CQME}, are obtained by performing
the Laplace transform
\bsube
\begin{align}
\!{\ti \Pi}_0(z)(\cdots)=\sum_{ss'}\Big\{&Q_s\,\tilde{{\cal Q}}_{ss'}(z+\rmi{\cal L})Q_{s'}(\cdots)
\nonumber \\
&+\tilde{{\cal Q}}_{ss'}(z^\ast-\rmi{\cal L})(\cdots)\,Q_{s'}Q_s\Big\},
\\
\!{\ti \Pi}_\pm(z)(\cdots)=\sum_{ss'}\Big\{&\tilde{{\cal Q}}^{(\pm)}_{ss'}(z+\rmi{\cal L})Q_{s'}(\cdots)Q_s
\nonumber \\
&+\!Q_s\tilde{{\cal Q}}^{(\pm)}_{ss'}(z^\ast-\rmi{\cal L})(\cdots)Q_{s'}\Big\},
\end{align}
\esube
where ${\cal Q}_{ss'}={\cal Q}_{ss'}^{(+)}+{\cal Q}_{ss'}^{(-)}$, with
\begin{align}\label{Cssw}
\tilde{\mathcal{Q}}_{ss'}^{(\pm)}(z)\equiv
\int_{0}^{\infty}\rmd t C_{ss'}^{(\pm)}(t)e^{-z t}.
\end{align}
In the limit $z\rightarrow\rmi\omg$, it can be further
simplified to
\begin{align}
\tilde{\mathcal{Q}}_{ss'}^{(\pm)}(z)|_{z\rightarrow\rmi\omg}
= \tilde{C}^{(\pm)}_{ss'}(\omg)+\rmi \tilde{D}^{(\pm)}_{ss'}(\omg).
\end{align}
The first term denotes the coupling spectral function
\be\label{Coupl-spec}
\tilde{C}^{(\pm)}_{ss'}(\omg)\equiv \int_{-\infty}^{\infty}
\rmd t C_{ss'}^{(\pm)}(t)e^{-\rmi\omg t},
\ee
which is associated with particle transfer processes, with
interactions between the qubit and QPC being properly
accounted for.
For a Lorentzian band structure [see \Eq{Jw}], it can be evaluated
explicitly as
\begin{align}\label{Cpm}
\tilde{C}_{ss'}^{(\pm)}(\omg)=&\frac{\eta\chi_s\chi_{s'}}{e^{\beta (\omg\pm V)}-1}
\frac{4\rmw^2}{(\omg\pm V)^2+4\rmw^2}\bigg\{\frac{\rmw}{2}\varphi(\omg\pm V)
\nonumber \\
&+\frac{\rmw^2}{\omg\pm V}[\phi(\omg\pm V)-\phi(0)] \bigg\},
\end{align}
where $\eta=2\pi\Gam_{\rm L}^0\Gam_{\rm R}^0$,
$\phi(x)$ and $\varphi(x)$ denote the real and imaginary
parts of the digamma function
$\Psi(\frac{1}{2}+\beta\frac{\rmw+\rmi x}{2\pi})$, respectively.
Note here due to finite cutoff energy of the QPC detector and quasistep
feature in the Fermi functions in \Eq{CCF}, $\tilde{\mathcal{Q}}_{ss'}^{(\pm)}(\omg)$
decays exponentially when $\omg$ goes beyond the cutoff energy. As a result, the
resolvents of the kernels $\tilde{\Pi}_0(z)$ and $\tilde{\Pi}_\pm(z)$ vanish in the
limit $\omg\rightarrow\infty$.
It should also be stressed that the present spectrum functions satisfy
the detailed--balance relation, i.e.
$\tilde{C}_{ss'}^{(+)}(\omg)=e^{-\beta(\omg+V)}\tilde{C}_{ss'}^{(-)}(-\omg)$,
which means that our approach properly accounts for the energy
exchange between the qubit and the detector during
measurement.
This is the reason we get nonzero stationary value for the
real part of the off-diagonal matrix element $\rho_{ab}$,
in contrast to that obtained in Ref. \onlinecite{Gur9715215}.
With the Knowledge of the spectral function, the dispersion function
$\tilde{D}_{ss'}^{(\pm)}(\omg)$
can be obtained via the Kramers-Kronig relation
\be \label{dispersion}
\tilde{D}_{ss'}^{(\pm)}(\omg)=\frac{1}{\pi}\,{\cal P}
\!\!\int_{-\infty}^{\infty} d\omg'
\frac{\tilde{C}_{ss'}^{(\pm)}(\omg')}{\omg-\omg'},
\ee
where ${\cal P}$ stands for Cauchy's principal value.
Physically, the dispersion accounts for the coupling-induced energy
renormalization of the internal
energies \cite{Xu029196,Yan05187,Cal83587,Wei08}.
The second cumulant of the current distribution corresponds to
the shot noise. To study the finite-frequency spectrum, we employ
the MacDonald's formula \cite{Mac62}
\be \label{MacD}
S(\omg)=2\omg\int_0^\infty \rmd t\sin(\omg t)
\frac{\rmd}{\rmd t}[\la N^2(t)\ra-(\bar{I}t)^2],
\ee
with $\la N^2(t)\ra\equiv\sum_NN^2P(N,t)$. By utilizing \Eq{CQME},
it is simplified to
\begin{align}\label{Sw}
S(\omg)=S_0+4\omg {\rm Im}[\tr\{J_-(z)\tilde{N}(z)\}]|_{z\rightarrow\rmi\omg},
\end{align}
where the noise pedestal $S_0=2\tr\{J_+(0){\rho}_{\rm st}\}$ is
the shot noise of the QPC detector alone.
In the WBL and large voltage, it reproduces
the well-known result $S_0=2e\bar{I}$ \cite{Kor995737}.
Rich information about qubit measurement dynamics is contained
in the excess noise (second term) in \Eq{Sw}.
Here, $\tilde{N}(z)$ is the Laplace space counterpart of
$N(t)\equiv\sum_N N\rho^{(N)}(t)$. By employing the ``$N$''-resolved
quantum master equation (\ref{CQME}), it can be solved from the
following algebraic equation
\bea\label{Nz}
z\tilde{N}(z)=-\rmi{\cal L}\tilde{N}(z)-\tilde{\Pi}(z)\tilde{N}(z)
+\frac{J_-(0)\tilde{\rho}_{\rm st}}{z},
\eea
with $\tilde{\Pi}(z)\equiv\tilde{\Pi}_0(z)-\tilde{\Pi}_+(z)-\tilde{\Pi}_-(z)$.
\subsection{Noise characteristics}
The basic physics of the measurement process is the trade--off
between acquisition of information about the state of the qubit
and backaction dephasing of this system.
For a quantum-limited detector, the rates of the two processes
coincide, while for a less efficient detector, the qubit dephasing
is more rapid than information acquisition.
It imposes a fundamental limit on the SNR for
a weakly measured qubit, known as the K-A bound \cite{Kor01165310}.
An interesting feature is that the K-A bound is closely related to
oscillation peak (around the hybridization energy $\Dlt=\sqrt{\epl^2+(2\Omg)^2}$)
in the noise spectrum of the QPC detector, i.e.
the maximum peak height can reach 4 times larger
than the noise pedestal for a quantum-limited detector.
To see how this bound emerges, let us first briefly
derive this inequality for the Markovian case.
We start with the current-correlation function
$K(t)=\la \hat{I}(t+\tau)\hat{I}(t)\ra_{t\rightarrow\infty}$,
where the average is taken over the whole system.
The evolution of the QPC current operator $\hat{I}(t)$ is determined
by the entire system Hamiltonian, which yields \cite{Kor01165310}
\be
K(\tau)=e\bar{I}\delta(\tau)
+\frac{(\delta I)^2}{4}\tr[\sgm_z\sgm_z(\tau)\rho_{\rm st}],
\ee
with $\bar{I}$ the stationary current and $\rho_{\rm st}$
is the steady state of the qubit.
The tr[$\cdots$] denotes the trace over the state of the
reduced system.
The current change $\delta I=I_a-I_b$ reflects the current
response to electron oscillations between the dots, where
$I_{a} (I_b)$ corresponds to the QPC current
when the electron occupies the state $|a\ra(|b\ra)$.
Apparently, $K(\tau)$ reflects the correlation function of
the electron position in the dots given by $\sgm_z$.
The evolution of the $\sgm_z(\tau)$ can be found by
expanding the evolution operator of the entire system to
the second order in the coupling constant, and then averaging
over the reservoir states to obtain the equations of motion,
with dephasing rate given by $\Gam_\rmd$.
In the case of $\epl=0$, the noise spectrum is given
by \cite{Kor01165310}
\be
S(\omg)=S_0+\frac{(\delta I)^2\Gam_\rmd\Dlt^2}{(\omg^2-\Dlt^2)^2
+\Gam_\rmd^2\omg^2},
\ee
where $S_0=2e\bar{I}$ is the output noise of the QPC detector
alone, i.e. the noise pedestal.
At the qubit oscillation frequency $\omg=\Dlt$, the noise
spectrum has a maximum ``signal'' of $(\delta I)^2/\Gam_\rmd$.
The SNR thus is limited:
\be
{\rm SNR}\equiv\frac{S(\Dlt)-S_0}{S_0}\leq4.
\ee
This is the K-A bound. It has been confirmed in
Refs. \onlinecite{Goa01235307,Rus03075303,Shn04840},
generalized in Refs. \onlinecite{Mao04056803,Mao05085320},
and measured in Ref. \onlinecite{Ili03097906}.
However, several schemes have been proposed recently to overcome
the K-A bound, which can be divided into two categories.
The first one concerns with increasing the signal, such
as quantum nondemolition measurements \cite{Ave02207901,Jor05125333}
and quantum feedback control \cite{Wan07155304,Vij1277}.
The second type is to reduce the pedestal noise by employing a strongly
responding SET \cite{Jia09075320} or twin detectors \cite{Jor05220401}.
In this work, we find the non-Markovian processes allow
a violation of the K-A bound on the SNR.
The details together with an interpretation will be provided
later.
\begin{figure}
\begin{center}
\includegraphics*[scale=0.78]{Fig_6.eps}
\caption{\label{fig6}
Noise feature for a symmetric qubit ($\epl=0$) under different measurement
voltages for (a) w$/\Dlt=1.0$ and (b) w$/\Dlt=10.0$.
The inset in (a) and (b) shows the reservoir's
spectral density for corresponding cutoff energies and fixed measurement voltage
$V/\Dlt=5.0$.
The temperature and other parameters are the same as those in \Fig{fig4}.}
\end{center}
\end{figure}
The computed noise is shown in \Fig{fig6} for different
cutoff energies and voltages.
For a small cutoff energy (w$/\Dlt=1.0$), prominent non-Markovian
processes take place, which leads to a strongly suppressed
dephasing rate [cf. \Fig{fig4}(d)].
It is reflected in the noise spectrum as the
narrow width of the oscillation peak.
As the measurement voltage increases, the qubit will be excited,
which leads to the rising peak height (``signal'') and SNR.
However, the SNR cannot exceed the limit of ``4'', even in the
limit of $V/\Dlt\rightarrow\infty$, as we have verified.
In the case of large cutoff energy (w$/\Dlt=10.0$), however,
violation the K-A bound is observed unambiguously [see,
for instance, the dotted curve in \Fig{fig6}(b)].
The violation is due to the presence of non-Markovian processes,
in which reversed flow of information from the environment
back to the reduced system takes place.
This mechanism is analogous to the quantum feedback
scheme \cite{Wan07155304,Vij1277}. Yet, there, one has to
implement an extra procedure in which
the measurement information
in the detector is converted into the evolution of a qubit state.
Our analysis thus serves a direct and transparent way to improve
the efficiency in quantum measurement.
One may ask why the K-A bound is not violated in the case of
a small cutoff (w$/\Dlt=1.0$), where prominent non-Markovian
processes are present.
This is actually associated with the energy that needed to
excite the qubit.
Let us consider the situation of a voltage $V/\Dlt=5.0$.
For a small cutoff (w$/\Dlt=1.0$), the number of channels for electrons
to tunnel though the detector is remarkably suppressed, see the
schematic density spectral in the inset of \Fig{fig6}(a).
It restricts the number of electrons that can provide
energy to excite the qubit, and eventually results in
the SNR below the limit of ``4''.
Unlike the cases of w$/\Dlt=1.0$, the number of channels
are considerably increased for a large cutoff energy (w$/\Dlt=10.0$),
as shown in the inset of \Fig{fig6}(b).
Therefore, sufficient energy is provided to excite the
qubit, which leads eventually to the violation of the K-A bound.
However, even in the case of large cutoff energy, the
number of channels does not necessarily increase with
rising voltage.
For instance, there are less effective channels in the case
of $V/\Dlt=10.0$ than that of $V/\Dlt=5.0$.
One thus observes a suppressed SNR for $V/\Dlt=10.0$
in comparison with that of $V/\Dlt=5.0$, as shown by the
dot-dashed curve in \Fig{fig6}(b).
Furthermore, the qubit-QPC coupling gives rise to a
dynamical renormalization of the qubit energies
[see \Eq{dispersion}], which vanishes at ``w$\rightarrow0$ and
increases with the cutoff \cite{Luo09385801}.
On one hand, it leads to the shift of the oscillation peaks
towards low frequencies, as shown in \Fig{fig6}(b).
One the other hand, the energy renormalization gives rise to
incoherent jumps between the two states.
The detector attempts to localize the electron in one of the
states for a longer time, leading thus to the quantum Zeno effect,
which is manifested as the non-zero noise at zero frequency,
as displayed in \Fig{fig6}(b).
\begin{figure}
\begin{center}
\includegraphics*[scale=0.78]{Fig_7.eps}
\caption{\label{fig7}
Noise spectrum of a symmetric qubit at various temperatures
and cutoff energies (a) w$/\Dlt=1.0$ and (b) w$/\Dlt=10.0$.
The measurement voltage is $V/\Dlt=5.0$.
The other parameters are the same as those in \Fig{fig4}.}
\end{center}
\end{figure}
We are now in a position to discuss the influence of the temperature
on the noise spectrum. The numerical results are plotted in
\Fig{fig7}.
For a small cutoff energy (w$/\Dlt=1.0$), the
presence of a prominent non-Markovian effect inhibits
the dephasing rate, which is insensitive to the temperature, as
shown in \Fig{fig3}(b).
The width of the oscillation peak thus is strongly suppressed
for all the temperatures.
Furthermore, it is found that the height of the oscillation peak
decreases rapidly with rising temperature.
The reason is attributed to the enhanced qubit relaxation rate
as the temperature grows, analogous to the finding in the
the Markovian limit \cite{Li05066803}.
In this regime of small cutoff energy, the SNR cannot exceed
the K-A bound as we have checked.
It is again owing to the limited number of channels that electrons
can transfer through the QPC.
However, in the case of large bandwidth w$/\Dlt=10.0$ strong
violation of the K-A bond is observed at a low temperature
$\beta\Dlt=4.0$ [see \Fig{fig7}(b)].
As the temperature grows, the oscillation peak is again
reduced due to qubit relaxation, similar to the situation
of w$/\Dlt=1.0$.
To complete this section, we discuss the situation under which
the violation of the K-A bound may take place.
In the case of small cutoff energy (w$<\Dlt$), the SNR cannot
exceed the limit of ``4'' under arbitrary voltage and temperature,
even though there is a strong non-Markovian effect.
It is ascribed to the limited number of channels that electrons
can transfer though the detector.
It cannot provide enough energy to fully excite the qubit,
thus restricts the ``efficiency'' of the measurement.
In this sense, a small cutoff energy works as a suppression mechanism
to the ``effectiveness'' of the voltage and temperature.
In the opposite WBL (w$\rightarrow\infty$),
one expects very short reservoir correlation time, approaching
thus to the Markovian case.
Our result reproduces to the previous Markovian ones.
In this case, the information flows purely from the reduced
system to the reservoir and the SNR is limited to 4.
Therefore, the violation of the K-A bound only occurs for
moderate cutoff energies, together with an appropriately
large measurement voltage and a low temperature.
In this regime, enough energy will be provided to excite
the qubit, while relaxation to the ground state takes places
slowly.
Moreover, the presence of finite non-Markovian dynamics
results in the opportunity for the information to flow
from the reservoir back to the system, which eventually
lead to an SNR exceeding the K-A bound.
In comparison with the quantum feedback scheme, the present
work serves as a straightforward and transparent way to
improve the ``efficiency'' in quantum measurement.
\section{\label{thsec5}Conclusions}
In summary, we have investigated the dynamics of a charge qubit
under continuous measurement by a quantum point contact, with
special attention paid to the non-Markovian measurement characteristics.
We identified the regimes where prominent non-Markovian memory effects
are present by analyzing how the reservoir's correlation time is
varied as function of different parameters of the QPC detector.
In comparison with the Markovian case, considerable differences in
qubit relaxation and dephasing behaviors were observed in the
non-Markovian domain.
Furthermore, the non-Markovian dynamics was found to have a vital
role to play in the output noise features of the detector.
In particular, we observed unambiguously that the signal-to-noise
ratio can exceed the limit of ``4'', leading thus to the violation
of the Korotkov-Averin bound.
In comparison with other approaches, such as quantum feedback
scheme, our results might open new possibilities to enhance
measurement efficiency in a straightforward and transparent
way.
\begin{acknowledgments}
Support from the National Natural Science
Foundation of China (11204272, 11147114, and 11004124),
the Zhejiang Provincial Natural Science Foundation
(Y6110467 and LY12A04008) is gratefully acknowledged.
\end{acknowledgments}
|
{
"timestamp": "2013-08-20T02:02:35",
"yymm": "1203",
"arxiv_id": "1203.2233",
"language": "en",
"url": "https://arxiv.org/abs/1203.2233"
}
|
\section{Introduction}
The inequality of Ostrowski \cite{Ostrowski} gives us an estimate for the
deviation of the values of a smooth function from its mean value. More
precisely, if $f:[a,b]\rightarrow \mathbb{R}$ is a differentiable function
with bounded derivative, the
\begin{equation*}
\left\vert f(x)-\frac{1}{b-a}\int\limits_{a}^{b}f(t)dt\right\vert \leq \left[
\frac{1}{4}+\frac{(x-\frac{a+b}{2})^{2}}{(b-a)^{2}}\right] (b-a)\left\Vert
f^{\prime }\right\Vert _{\infty }
\end{equation*}
for every $x\in \lbrack a,b]$. Moreover the constant $1/4$ is the best
possible.
For some generalizations of this classic fact see the book \cite[p.468-484
{mitrovich} by Mitrinovic, Pecaric and Fink. A simple proof of this fact can
be \ done by using the following identity \cite{mitrovich}:
If $f:[a,b]\rightarrow \mathbb{R}$ is differentiable on $[a,b]$ with the
first derivative $f^{\prime }$ integrable on $[a,b],$ then Montgomery
identity holds
\begin{equation}
f(x)=\frac{1}{b-a}\int\limits_{a}^{b}f(t)dt+\int\limits_{a}^{b}P_{1}(x,t)f^
\prime }(t)dt, \label{h}
\end{equation
where $P_{1}(x,t)$ is the Peano kernel defined b
\begin{equation*}
P_{1}(x,t):=\left\{
\begin{array}{ll}
\dfrac{t-a}{b-a}, & a\leq t<x \\
& \\
\dfrac{t-b}{b-a}, & x\leq t\leq b
\end{array
\right.
\end{equation*
Recently, several generalizations of the Ostrowski integral inequality are
considered by many authors; for instance covering the following concepts:
functions of bounded variation, Lipschitzian, monotonic, absolutely
continuous and $n$-times differentiable mappings with error estimates with
some special means together with some numerical quadrature rules. For recent
results and generalizations concerning Ostrowski's inequality, we refer the
reader to the recent papers \cite{Cerone1}, \cite{Duo}, \cite{Dragomir}-\cit
{Liu}, \cite{sarikaya}-\cite{sarikaya2}.
In \cite{Anastassiou} and \cite{sarikaya3}, the authors established some
inequalities for differentiable mappings which are connected with Ostrowski
type inequality by used the Riemann-Liouville fractional integrals, and they
used the following lemma to prove their results:
\begin{lemma}
Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ be differentiable function
on $I^{\circ }$ with $a,b\in I$ ($a<b$) and $f^{\prime }\in L_{1}[a,b]$, the
\begin{equation}
f(x)=\frac{\Gamma (\alpha )}{b-a}(b-x)^{1-\alpha }{\Large J}_{a}^{\alpha
}f(b)-{\Large J}_{a}^{\alpha -1}(P_{2}(x,b)f(b))+{\Large J}_{a}^{\alpha
}(P_{2}(x,b)f^{^{\prime }}(b)),\ \ \ \alpha \geq 1, \label{z}
\end{equation
where $P_{2}(x,t)$ is the fractional Peano kernel defined b
\begin{equation*}
P_{2}(x,t)=\left\{
\begin{array}{ll}
\dfrac{t-a}{b-a}(b-x)^{1-\alpha }\Gamma (\alpha ), & a\leq t<x \\
& \\
\dfrac{t-b}{b-a}(b-x)^{1-\alpha }\Gamma (\alpha ), & x\leq t\leq b
\end{array
\right.
\end{equation*}
\end{lemma}
In this article, we use the Riemann-Liouville fractional integrals to
establish some new weighted integral inequalities of Ostrowski's type. From
our results, the weighted and the classical Ostrowski's inequalities can be
deduced as some special cases.
\section{Fractional Calculus}
Firstly, we give some necessary definitions and mathematical preliminaries
of fractional calculus theory which are used further in this paper. More
details, one can consult \cite{gorenflo}, \cite{samko}.
\begin{definition}
The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$
with $a\geq 0$ is defined a
\begin{eqnarray*}
J_{a}^{\alpha }f(x) &=&\frac{1}{\Gamma (\alpha )}\dint\limits_{a}^{x}(x-t)^
\alpha -1}f(t)dt, \\
J_{a}^{0}f(x) &=&f(x).
\end{eqnarray*}
\end{definition}
Recently, many authors have studied a number of inequalities by used the
Riemann-Liouville fractional integrals, see (\cite{Anastassiou}, \cit
{Belarbi}, \cite{Dahmani}, \cite{Dahmani1}, \cite{sarikaya3}, \cit
{sarikaya4}) and the references cited therein.
\section{Main Results}
Throughout this work, we assume that the weight function $w:\left[ a,b\right]
\rightarrow \lbrack 0,\infty ),$ is integrable, nonnegative and
\begin{equation*}
m(a,b)=\int\limits_{a}^{b}w(t)dt<\infty .
\end{equation*}
In order to prove our main results, we need the following identities:
\begin{lemma}
\label{lm} Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ be a
differentiable function on $I^{\circ }$ with $a,b\in I$ ($a<b$)$,$ $\alpha
\geq 1$ and $f^{\prime }\in L_{1}[a,b]$, then the generalization of the
weighted Montgomery identity for fractional integrals holds:
\end{lemma}
\begin{eqnarray}
m\left( a,b\right) f\left( x\right) &=&\left( b-x\right) ^{1-\alpha }\Gamma
\left( \alpha \right) J_{a}^{\alpha }\left( w\left( b\right) f\left(
b\right) \right) \notag \\
&& \label{1} \\
&&-J_{a}^{\alpha -1}\left( \Omega _{w}\left( x,b\right) f\left( b\right)
\right) +J_{a}^{\alpha }\left( \Omega _{w}\left( x,b\right) f^{^{\prime
}}\left( b\right) \right) \notag
\end{eqnarray
where $\Omega _{w}\left( x,t\right) $ is the weighted fractional Peano
kernel defined by
\begin{equation}
\Omega _{w}\left( x,t\right) :=\left\{
\begin{array}{ll}
\left( b-x\right) ^{1-\alpha }\Gamma \left( \alpha \right)
\dint\limits_{a}^{t}w\left( u\right) du, & t\in \lbrack a,x) \\
\left( b-x\right) ^{1-\alpha }\Gamma \left( \alpha \right)
\dint\limits_{b}^{t}w\left( u\right) du, & t\in \lbrack x,b]
\end{array
\right. \label{2}
\end{equation}
\begin{proof}
By definition of $\Omega _{w}\left( x,t\right) $, we hav
\begin{eqnarray}
&&J_{a}^{\alpha }\left( \Omega _{w}\left( x,b\right) f^{^{\prime }}\left(
b\right) \right) \notag \\
&& \label{3} \\
&=&\frac{1}{\Gamma \left( \alpha \right) }\dint\limits_{a}^{b}\left(
b-t\right) ^{\alpha -1}\Omega _{w}\left( x,t\right) f^{\prime }\left(
t\right) dt \notag \\
&& \notag \\
&=&\left( b-x\right) ^{1-\alpha }\left[ \dint\limits_{a}^{x}\left(
b-t\right) ^{\alpha -1}\left( \dint\limits_{a}^{t}w\left( u\right) du\right)
f^{^{\prime }}\left( t\right) dt\right. \notag \\
&& \notag \\
&=&\left. +\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha -1}\left(
\dint\limits_{b}^{t}w\left( u\right) du\right) f^{^{\prime }}\left( t\right)
dt\right] \notag \\
&& \notag \\
&=&\left( b-x\right) ^{1-\alpha }\left( J_{1}+J_{2}\right) . \notag
\end{eqnarray
Integrating by parts, we can state
\begin{eqnarray}
J_{1} &=&\left( b-x\right) ^{\alpha -1}\left( \dint\limits_{a}^{x}w\left(
u\right) du\right) f\left( x\right) \notag \\
&& \label{41} \\
&&+\left( \alpha -1\right) \dint\limits_{a}^{x}\left( b-t\right) ^{\alpha
-2}\left( \dint\limits_{a}^{t}w\left( u\right) du\right) f\left( t\right)
dt-\dint\limits_{a}^{x}\left( b-t\right) ^{\alpha -1}w\left( t\right)
f\left( t\right) dt \notag
\end{eqnarray
and similary
\begin{eqnarray}
J_{2} &=&\left( b-x\right) ^{\alpha -1}\left( \dint\limits_{x}^{b}w\left(
u\right) du\right) f\left( x\right) \notag \\
&& \label{5} \\
&&+\left( \alpha -1\right) \dint\limits_{x}^{b}\left( b-t\right) ^{\alpha
-2}\left( \dint\limits_{b}^{t}w\left( u\right) du\right) f\left( t\right)
dt-\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha -1}w\left( t\right)
f\left( t\right) dt. \notag
\end{eqnarray
Adding (\ref{41}) and (\ref{5}), we obtain (\ref{1}) which this completes
the proof.
\end{proof}
\begin{remark}
If we choose $\alpha =1$ and $w\left( u\right) =1$, the formula (\ref{1})
reduces to the classical Montgomery Identity given by (\ref{h}).
\end{remark}
\begin{remark}
If we choose $w\left( u\right) =1$, the formula (\ref{1}) reduces to the
fractional Montgomery Identity given by (\ref{z}).
\end{remark}
\begin{theorem}
\label{thm3} Let $f:[a,b]\rightarrow \mathbb{R}$ be differentiable on $(a,b)$
such that $f^{^{\prime }}\in L_{1}[a,b],$ where $a<b.$ If $\left\vert
f^{^{\prime }}(x)\right\vert \leq M$ for every $x\in \lbrack a,b]$ and
\alpha \geq 1$, then the following Ostrowski fractional inequality holds:
\end{theorem}
\begin{eqnarray}
&&\left\vert m\left( a,b\right) f\left( x\right) -\left( b-x\right)
^{1-\alpha }\Gamma \left( \alpha \right) J_{a}^{\alpha }\left( w\left(
b\right) f\left( b\right) \right) -J_{a}^{\alpha -1}\left( \Omega _{w}\left(
x,b\right) f\left( b\right) \right) \right\vert \notag \\
&& \label{22} \\
&\leq &\frac{M\left( b-x\right) ^{1-\alpha }}{\alpha }\left[ A(x)-\left(
b-x\right) ^{\alpha }B(x)\right] \notag
\end{eqnarray
wher
\begin{equation*}
A(x)=\dint\limits_{a}^{x}\left( b-u\right) ^{\alpha -1}w\left( u\right)
du-\dint\limits_{x}^{b}\left( b-u\right) ^{\alpha }w\left( u\right) du
\end{equation*
and
\begin{equation*}
B(x)=\dint\limits_{a}^{x}w\left( u\right) du-\dint\limits_{x}^{b}w\left(
u\right) du.
\end{equation*}
\begin{proof}
From Lemma \ref{lm}, we ge
\begin{eqnarray}
&&\left\vert m\left( a,b\right) f\left( x\right) -\Gamma \left( \alpha
\right) \left( b-x\right) ^{1-\alpha }J_{a}^{\alpha }\left( w\left( b\right)
f\left( b\right) \right) -J_{a}^{\alpha -1}\left( \Omega _{w}\left(
x,b\right) f\left( b\right) \right) \right\vert \notag \\
&& \notag \\
&\leq &\frac{1}{\Gamma \left( \alpha \right) }\left\vert
\dint\limits_{a}^{b}\left( b-t\right) ^{\alpha -1}\Omega _{w}\left(
x,t\right) f^{^{\prime }}\left( t\right) dt\right\vert \notag \\
&& \label{23} \\
&\leq &\frac{M}{\Gamma \left( \alpha \right) }\dint\limits_{a}^{b}\left(
b-t\right) ^{\alpha -1}\left\vert \Omega _{w}\left( x,t\right) \right\vert dt
\notag \\
&& \notag \\
&=&M\left( b-x\right) ^{1-\alpha }\left( \dint\limits_{a}^{x}\left(
b-t\right) ^{\alpha -1}\left( \dint\limits_{a}^{t}w\left( u\right) du\right)
dt+\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha -1}\left(
\dint\limits_{t}^{b}w\left( u\right) du\right) dt\right) \notag \\
&& \notag \\
&=&M(b-x)^{1-\alpha }\left\{ J_{3}+J_{4}\right\} . \notag
\end{eqnarray
Now, using the change of order of integration we ge
\begin{eqnarray*}
J_{3} &=&\dint\limits_{a}^{x}\left( b-t\right) ^{\alpha -1}\left(
\dint\limits_{a}^{t}w\left( u\right) du\right) dt \\
&& \\
&=&\dint\limits_{a}^{x}w(u)\dint\limits_{u}^{x}\left( b-t\right) ^{\alpha
-1}dtdu \\
&& \\
&=&\frac{1}{\alpha }\left[ \dint\limits_{a}^{x}\left( b-u\right) ^{\alpha
-1}w\left( u\right) du-\left( b-x\right) ^{\alpha
}\dint\limits_{a}^{x}w\left( u\right) du\right]
\end{eqnarray*
and similarly
\begin{eqnarray*}
J_{4} &=&\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha -1}\left(
\dint\limits_{t}^{b}w\left( u\right) du\right) dt \\
&& \\
&=&\dint\limits_{x}^{b}w(u)\dint\limits_{x}^{u}\left( b-t\right) ^{\alpha
-1}dtdu \\
&& \\
&=&\frac{1}{\alpha }\left[ \left( b-x\right) ^{\alpha
}\dint\limits_{x}^{b}w\left( u\right) du-\dint\limits_{x}^{b}\left(
b-u\right) ^{\alpha }w\left( u\right) du\right] .
\end{eqnarray*
Using $J_{3}$ and $J_{4}$ in (\ref{23}), we obtain (\ref{22}).
\end{proof}
\begin{remark}
We note that in the special cases, if we take $w\left( u\right) =1$ in
Theorem \ref{thm3}, then it reduces Theorem 4.1 proved by Anastassiou et.
al. \cite{Anastassiou}. So, our results are generalizations of the
corresponding results of Anastassiou et. al. \cite{Anastassiou}.
\end{remark}
|
{
"timestamp": "2012-03-13T01:01:30",
"yymm": "1203",
"arxiv_id": "1203.2280",
"language": "en",
"url": "https://arxiv.org/abs/1203.2280"
}
|
\section{Introduction}
Millisecond oscillations in the X-ray brightness of low mass X-ray
binaries (LMXB) have been observed for a large and growing number
of sources since the launch of the {\it Rossi X-ray Timing Explorer}
({\it RXTE}) in late 1995. A review of those sources which exhibit nearly
coherent oscillations during their Type I X-ray bursts, a class that
includes 4U 1636--53, can be found in Strohmayer (2001). The larger
group that exhibit quasi-periodic oscillations only during their
quiescent non-bursting state, a class that includes 4U 1735--44, have
been reviewed by Van der Klis (2000). Relatively few of the non-transient
X-ray sources exhibiting kHz oscillations have permanent optical
counterparts but the similar systems 4U 1735--44 and 4U 1636--53
are ideal for studies at optical wavelengths with small telescopes.
The many similarities of these two syatems are discussed in Casares
et al (2006) and references therein and will not be repeated here.
A more general summary of the many known X-ray emitting accreting
neutron stars can be found in Watts et al. (2008).
In 1997 we commenced a program of occasional monitoring of 4U 1735--44
and 4U 1636--53. The initial thrust of this work was twofold in nature.
One program was to obtain an updated ephemeris for 4U 1636--53 and to use
this to examine the binary phases of a large sample of X-ray bursts
observed with {\it RXTE}. The intention was to search for possible Doppler
shift modulations in the bursts kHz oscillation frequency induced by the
orbital motion of the neutron star. This work has been reported by Giles et
al. (2002). The initial start on 4U 1735--44 was part of a coordinated
program in 1997 to observe simultaneous X-ray and optical bursts but no
X-ray burst were seen ({\it RXTE} observation 20084). In this paper we
present our data and results for CCD optical photometry of the
4U 1735--44 system obtained since 1997.
\begin{table*}
\centering
\begin{minipage}{175mm}
\begin{center}
\bf Table 1. \rm Optical observations of 4U 1735--44. \\
\vspace*{0.125cm}
\label{symbols}
\begin{tabular}{@{}llrrccc}
\hline
Obs. & & HJD Start & HJD End & & Integration & Number of \\
No. & Date & (-2450000) & (-2450000) & Filter & Time (s) & Exposures \\
\hline
1 & 1997 Aug 1 & 661.92966 & 662.13406 & {\it V\/}, {\it I\/} & 120 & 64 \\
2 & 1997 Aug 29 & 689.89771 & 690.08805 & {\it V\/}, {\it I\/} & 60 & 115 \\
3 & 1999 Sep 16 & 1437.89914 & 1438.11925 & {\it V\/} & 150 & 93 \\
4 & 1999 Sep 20 & 1441.89958 & 1442.11134 & {\it V\/} & 300 & 52 \\
5 & 2000 May 25 & 1689.90368 & 1690.30974 & {\it V\/} & 300 & 92 \\
6 & 2001 Jul 3 & 2093.89874 & 2094.32339 & {\it V\/} & 300 & 110 \\
7 & 2001 Jul 4 & 2094.90880 & 2095.31425 & {\it V\/} & 300 & 111 \\
8 & 2004 May 7 & 3133.09892 & 3133.34262 & {\it V\/} & 300 & 63 \\
9 & 2004 May 14 & 3139.92629 & 3140.35182 & {\it V\/} & 300 & 115 \\
10 & 2004 May 15 & 3141.01611 & 3141.18857 & {\it V\/} & 300 & 46 \\
11 & 2005 May 12 & 3502.95212 & 3503.33362 & {\it V\/} & 300 & 80 \\
12 & 2005 May 13 & 3503.92808 & 3504.31953 & {\it V\/} & 300 & 106 \\
13 & 2007 Apr 20 & 4211.00317 & 4211.19962 & {\it V\/} & 300 & 51 \\
14 & 2007 May 12 & 4231.92835 & 4232.34208 & {\it V\/} & 300 & 58 \\
\hline
\end{tabular}
\medskip
\end{center}
\end{minipage}
\end{table*}
\begin{table*}
\centering
\begin{minipage}{175mm}
\begin{center}
\bf Table 2. \rm Sine curve fits, times of maximun optical light and {\it RXTE} ASM X-ray data for 4U 1735--44. \\
\vspace*{0.125cm}
\label{symbols}
\begin{tabular}{@{}lcrccccc}
\hline
Obs. & Cycle & HJD & Error & Amplitude & Mean $\Delta$ & ASM Daily X-ray & No. ASM \\
No. & Number (N) & (-2450000) & (d) & {\it V\/} mag. & {\it V\/} mag. & Flux Units $cs^{-1}$ & Dwells \\
\hline
1 & 24547 & 662.0435 & 0.0046 & 0.188 $\pm$ 0.028 & 1.190 $\pm$ 0.010 & 8.97 $\pm$ 0.50 & 0 \\
2 & 24691 & 689.9535 & 0.0018 & 0.254 $\pm$ 0.017 & 1.335 $\pm$ 0.006 & 12.89 $\pm$ 0.28 & 8 \\
3 & 28550 & 1437.9459 & 0.0021 & 0.236 $\pm$ 0.018 & 1.345 $\pm$ 0.006 & 10.81 $\pm$ 0.43 & 7 \\
4 & 28571 & 1442.0236 & 0.0032 & 0.180 $\pm$ 0.016 & 1.375 $\pm$ 0.007 & 13.25 $\pm$ 0.23 & 10 \\
5 & 29851 & 1690.1214 & 0.0046 & 0.116 $\pm$ 0.016 & 1.421 $\pm$ 0.006 & 12.97 $\pm$ 0.42 & 9 \\
6 & 31935 & 2094.0848 & 0.0030 & 0.183 $\pm$ 0.018 & 1.499 $\pm$ 0.006 & 10.98 $\pm$ 2.48 & 1 \\
7 & 31940 & 2095.0451 & 0.0030 & 0.145 $\pm$ 0.014 & 1.216 $\pm$ 0.005 & 12.35 $\pm$ 0.50 & 6 \\
8 & 37296 & 3133.2238 & 0.0022 & 0.193 $\pm$ 0.012 & 1.152 $\pm$ 0.005 & 17.85 $\pm$ 0.47 & 0 \\
$9^{ a}$ & & & & & 1.040 $\pm$ 0.007 & 21.34 $\pm$ 0.33 & 18 \\
10 & 37337 & 3141.1581 & 0.0019 & 0.282 $\pm$ 0.015 & 1.202 $\pm$ 0.006 & 18.89 $\pm$ 0.37 & 2 \\
11 & 39204 & 3503.0260 & 0.0058 & 0.114 $\pm$ 0.022 & 1.183 $\pm$ 0.008 & 17.25 $\pm$ 0.43 & 0 \\
12 & 39209 & 3504.0359 & 0.0022 & 0.217 $\pm$ 0.016 & 0.964 $\pm$ 0.005 & 16.76 $\pm$ 0.41 & 8 \\
13 & 42857 & 4211.1238 & 0.0033 & 0.253 $\pm$ 0.027 & 1.299 $\pm$ 0.010 & 14.64 $\pm$ 0.51 & 3 \\
14 & 42965 & 4232.0368 & 0.0045 & 0.179 $\pm$ 0.027 & 1.314 $\pm$ 0.011 & 12.94 $\pm$ 0.39 & 13 \\
\hline
\end{tabular}
\medskip
\end{center}
\hspace*{3.0cm} {a}{ Observation not suitable for a sine curve fit. } \\
\end{minipage}
\end{table*}
The optical counterpart of 4U 1735--44, V926 Scorpii, has been observed
on a number of occasions since its identification in 1977 by McClintock
et al. (1977). A collection of all these photometric data, covering
the interval from 1984 July to 1993 August, was compiled by Augusteijn
et al. (1998) (see references therein) to derive an accurate photometric
ephemeris. Detailed spectroscopic studies have been reported by Smale \&
Corbet (1991), Augusteijn et al. (1998) and Casares et al. (2006).
This latter observation of 4U 1735--44 (and 4U 1636--53) is particularly
relevant for the latter discussion as it established an important
spectroscopic ephemeris, effectively defining a dynamical phase zero,
by observing emission from the optical donor star.
The paper is organised as follows. In Section 2 we report new
photometric light curves of 4U 1735--44 obtained over the period
1997 August to 2007 May. In Section 3 we use our new optical data
to revise the Augusteijn et al. (1998) ephemeris and in Section 4
we explore aspects of the historical X-ray data to look for possible
correlated changes. In Section 5 we briefly discuss the 4U 1735--44 system
based on our new observations.
\section{Optical Observations}
All the optical observations were made between 1997 and 2007 using the
1-m telescope at the Mt. Canopus Observatory, University of Tasmania.
The observations used standard Johnson {\it V\/} and {\it I\/} filters and
the CCD reduction procedure was identical to that described in Giles, Hill
\& Greenhill (1999). All times presented in this paper have been corrected
to Heliocentric Julian Dates (HJD) and a complete journal of the
observations is given in Table 1. Throughout this paper, except
when discussing spectroscopic observations, phase zero
is defined as superior conjunction of the companion star (neutron star
closest to the Earth) when the system optical flux is at a maximum.
\begin{figure*}
\noindent
\begin{minipage}[b]{0.48\linewidth}
\centering\epsfig{file=fig_1a.eps, width=8.6cm }
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering\epsfig{file=fig_1b.eps, width=8.6cm }
\end{minipage}
\caption{The {\it V\/}-band light curves for 4U 1735--44. The solid traces
through the data points mark our best fit sine curves to each nights
observations. The dashed curves show the ephemeris predictions of Augusteijn
et al. (1998) for the same nights with an arbitrary offset and
amplitude. The number to the right in each panel refers to the HJD
starting at zero hours within each light curve. The number to the left
in each panel refers to the observation number listed in Table 2.
Vertical lines mark the cycle count epochs listed in Table 2.
The dashed trace for observation 9 is derived from the new ephemeris
and is not a fit to the data points in that panel. }
\end{figure*}
All the images were reduced with between 3--7 nearby stars being used
as local standards. However, the light curves in Fig. 1 plot the
differential magnitude between 4U 1735--44 and only one of these stars
which can be located on the finder chart in Jernigan et al. (1977).
This brighter secondary standard is star number 6 on their 2S1735--444
chart which we find to have a {\it V\/} magnitude of 16.10 $\pm$ 0.02.
4U 1735--44 is star number 5 on this same chart and is $\sim1.4$ mag fainter.
For the 1997 observations the telescope was equipped with an SBIG CCD
camera having 375 x 242 pixels with an image scale of 0.42 $\times$ 0.49
arcsec pixel$^{-1}$. On the nights of 1997 August 1 and 29 continuous
pairs of {\it V\/} and {\it I\/} integrations were obtained but
the {\it I\/}-band data are not discussed further in this paper.
For the 1999 and later observations the telescope was equipped with
a SITe CCD camera having 512 x 512 pixels with an image scale of
0.42 arcsec pixel$^{-1}$.
The entire data set represented in Tables 1 and 2 are plotted in Fig. 1.
Observation 9 has no clear modulation and has therefore not been fitted
with a sine curve. The data are included since the mean level is used
in section 4 of this paper and it also illustrates this unmodulated state
at the time of highest X-ray flux. Observation 12 is also interesting
as it appears to show two cycles containing a definite dip, midway
in phase between maximum brightness.
\section{Optical Maximum Ephemeris}
The ephemeris for times of maximum optical light given by Augusteijn et al.
(1998) was HJD = 2447288.0143(25) + [ N $\times$ 0.19383351(32) ]
where the errors are indicated in the round brackets ($\pm \sigma$) and
N is the cycle number starting from zero. This ephemeris was based on
observations made between 1984 July 22 and 1993 July 28 and covered a total
of 16,986 binary periods. We have fitted a sine curve to each new night's
observations listed in Table 1 taking the amplitude, phase and mean as free
parameters but fixing the binary period at the value given by Augusteijn
et al. (1998). The appropriate sine curve fits are shown as solid traces
through the respective data points in Fig. 1. From these fits we
derive the times of optical maxima, peak-to-peak amplitudes and mean
intensities listed in Table 2. This table is intended to be complimentary
to the similar table 3 of Augusteijn et al. (1998) and continues
the same cycle number sequence.
The predictions from the previous Augusteijn et al (1998) ephemeris
are also shown on Fig. 1 as the dotted traces in the lower part
of each light curve panel. There is a significant phase shift evident
between the two sets of sine curves. The new observed maxima also fall
increasingly earlier than expected, indicating a different period, over
the additional $\sim$25,979 binary periods to 2007 May. However, a quick
examination shows that the earlier period was close enough to avoid any
cycle count ambiguity through the new data sets and, as stated earler,
the previous cycle sequence is maintained in Table 2.
\begin{figure}
\epsfig{file=fig_2.eps, width=8.6cm }
\caption{The {\it O\/} - {\it C\/} times of maximum optical
light for 4U 1735--44 plotted against time for the best fit linear
model. The open circles denote the earlier points from Augusteijn et al.
(1998) and the filled circles are the new optical data.}
\end{figure}
We then used our new data, together with the data in table 3 of
Augusteijn et al (1998,) to perform the usual {\it O\/} - {\it C\/}
analysis on the entire data set and the results are plotted in Fig. 2.
Our new linear fit for the time of maximum optical light has the ephemeris
HJD = 2445904.0494(90) + [ N $\times$ 0.19383222(29) ]
with a value of $\chi^2 = 253.5$ for 16 dof and a mean phase scatter about
phase zero of 0.061. The relatively poor $\chi^2$ value reflects
the intrinsic scatter of the data points rather than their error bars.
Deriving a highly accurate and reliable
ephemeris for systems like 4U 1735--44 (and 4U 1636--53) is not simple
due to the lack of any sharp eclipse type feature in the optical light
curve and the fact that the profile is quite variable from cycle to cycle.
For these reasoms and the `noise' in the scatter of the {\it O\/} - {\it C\/}
values in Fig. 2. there is still no significant period derivative term.
An upper limit for the $\dot P$ term gives
$\mid P/\dot P\mid$ $\lesssim 6.5 \times$10$^{6}$ yr.
Until relatively recently it had proved impossible to reliably identify
features in the optical spectrum of 4U 1735--44 that originated from the
irradiated donor companion star which could be used to define a dynamical
phase zero. This has been achieved for a number of LMXB systems by following
the Bowen transition feature and Casares et al. (2006) have obtained a
spectroscopic based ephemeris for 4U 1735--44 using observations made in
2003 June. Phase zero, in this spectral definition, corresponds to inferior
conjunction of the companion (donor) star and occurs at
HJD = 2452813.495(3) + [ N $\times$ 0.19383351(32) ] where the period was
taken as the Augusteijn et al. (1998) value. Note that this definition
has a 0.5 phase shift with respect to that used for optical intensity.
Our new photometric ephemeris places the optical maximun at a phase
of 0.47 $\pm$ 0.05 on this spectral ephemeris.
Since the spectral ephemeris date falls well within the sequence of
new optical observations (see Fig. 3) the phase zero error in our new
optical ephemeris is primarily due to the uncertainty in defining when
the optical maximum occurs rather than to period uncertainty accumulations
or the spectral phase zero error.
\begin{figure}
\epsfig{file=fig_3.eps, width=8.6cm }
\caption{The ASM light curve for 4U 1735--44 from 1996 February 21, just
after the launch of {\it RXTE}, to 2011 August 30. The ASM data (1.5--12 keV)
were extracted as one day values and then averaged over a four day interval.
Solid vertical lines mark the dates of the optical observations listed in
Table 1. The dotted vertical line marks the date of the spectral phase zero
observation of Casares et al. (2006).}
\end{figure}
\section{Optical - X-ray correlations}
Since our data set spans an $\sim 10$ year interval and overlaps with much
of the duration of the {\it RXTE} mission we have also examined the All Sky
Monitor (ASM) (Levine et al (1996)) database to look at the X-ray light
curve history for 4U 1735--44. Fig. 3 shows this data together with the
dates on which our optical observations were made.
We have attempted to extract the ASM fluxes corresponding to the exact times of
the observations listed in Table 1 in order to compare them to the corresponding
optical intensities and amplitude modulations listed in Table 2.
ASM X-ray fluxes can be obtained as individual dwell cycle values, effectively 90 s
integrations, or as daily averages which are summations of all the dwell cycles
falling within 24 h intervals. Unfortunately, due to a combination of orbital and
aspect constraints, the dwell cycle values for 4U 1735--44 occur in irregularly
spread groups with gaps containing no observations. Although daily values are virtually
always available they may be calculated from data within 24 h intervals that fall
outside the precise span of any particular stretch of optical observations.
Fig. 1 shows that the durations of the optical observations vary between
4--10 h and Table 2 shows that the number of dwell cycles within each optical
sequence varies from 0--18. Table 2 also shows the nearest, or encompassing,
daily ASM X-ray flux intensities (bands A+B+C, 1.5--12 keV).
For the analysis we have chosen to use daily values as many optical nights
have few or no dwell cycles. A visual inspection of the appropriate ASM
data sections suggests that, in general, the daily values represent reasonable
local estimates even where they do not overlap with the optical data. Any effort
to be more selective rapidy becomes problematic and rather arbitary in nature.
Fig. 4 shows that there is a roughly linear trend between the X-ray flux
and the optical intensity with the optical source brightening, as expected,
when the X-ray flux increases. The linear fit, to all the observations, has a
value of $\chi^2 = 93.9$ for 12 dof and a modest correlation coefficient of 0.64.
The slope of the line gives a $\Delta$ optical increase of 59 percent (0.5 mag.)
for a $\Delta$ X-ray increase of 69 percent. This suggests that the non-X-ray
induced optical flux from the companion is $\lesssim 14$ percent of the
total light from the system as expected from a late type dwarf star.
\begin{figure}
\epsfig{file=fig_4.eps, width=8.6cm }
\caption{The {\it RXTE} ASM one-day X-ray intensity plotted against mean optical
brightness for the data in Table 2. The X axis plots the difference to the
brighter comparison star so lower values (points to the right) are brighter.
The numbers indicate the observation sequence listed in Table 2. }
\end{figure}
In contrast, there is no apparent correlation between the X-ray flux and the
amplitude modulation for 4U 1735--44 in Fig. 5 though the modulation error
bars are relatively large in this case. Perhaps only observation 1 seems
to stand outside the general linear trend in Fig. 4 and this happens
to be both the oldest optical data and also has no contained ASM dwell
cycles. However, the daily X-ray flux value appears to be consistent
with that for adjacent days and dwell cycles and the relative values
of the various calibration stars on this night are also consistent
with the many later CCD sequences.
The entire ASM data set has been folded,
modulo the period, using the new ephemeris and there is no evidence for any
orbital modulation, dips or eclipses. This applies for combined energy bands
(A+B+C) as well as just the low (A, 1.5--3 keV) and high (C, 5--12 keV) bands.
This is also true whether the data is selected for when the source is bright
(above the mean for the whole record) or weak (below the mean).
\begin{figure}
\epsfig{file=fig_5.eps, width=8.6cm }
\caption{The {\it RXTE} ASM one-day X-ray intensity plotted against optical
modulation amplitude (peak-to-peak) for the data in Table 2. No obvious trend
is apparent in this data. Observation 9 is missing from this plot. }
\end{figure}
\section{Discussion}
In the traditional model of an LMXB system there are three regions which
contribute to its optical variability due to X-ray heating. These are the
accretion disc itself, a bright hot spot or bulge on the outer edge of the
accretion disc due to the impact of inflowing material and the hemisphere of
the companion facing the neutron star which is not shadowed by the accretion
disc. For most LMXB systems the reprocessed X-ray optical flux is assumed to
come from the facing hemisphere of the companion star and to dominate the
optical light from the rest of the system (van Paradijs 1983, van Paradijs \&
McClintock 1995). The optical maximum was therefore assumed to occur when
the companion was on the far side of the neutron star but variations about
the mean modulation profile were expected due to gas flows causing
various X-ray shielding effects (Pedersen et al. 1982). Since the
magnetosphere of the neutron star in the 4U 1735--44 system is so small
the Roche lobe overflow through the inner Lagrange point (L1) must form a
standard Keplerian accretion disc (Frank, King \& Raine, 1992).
Smale \& Mukai (1988) investigated the optical flux and modulation created
by the persistent X-ray heating of the companion's facing hemisphere, and
although they assumed this scenario in their modelling they commented that
a thick disc would also likely contribute to the variable optical flux.
We have not attempted any detailed modelling of our observations but a
number of comments can be made. The earlier optical ephemeris due to
Augusteijn et al. (1998) placed maximum optical light for 4U 1735--44
at phase 0.68 $\pm$ 0.06 on the spectral ephemeris of Casares et al.
(2006) which was between the expected maximum visibility of the
irradiated donor and the disc bulge. This was in contrast to
the result for 4U 1636--53 where their spectral ephemeris and the
optical ephemeris of Giles et al. (2002) placed optical maximum
at a more reasonable phase of 0.47 $\pm$ 0.06. Our new photometric
ephemeris for 4U 1735--44 resolves this issue by placing the optical
maximum much closer to phase 0.5 (at 0.47 $\pm$ 0.05).
Indeed, Casares et al. (2006) comment on the uncertainty of the
Augusteijn et al. (1998) ephemeris and Fig. 2 clearly shows how an
ephemeris based on only the 5 earliest data points leads to a
significant phase error of $\sim 0.2$ after the passage of 10-yr.
The correlation between optical flux and X-ray flux, evident
in Fig. 4, is not unexpected, but Fig. 5 shows no apparent correlation
between optical modulation amplitude and X-ray flux.
If the X-ray induced optical flux were sitting on a substantial
and constant base component from the companion star then Fig. 5 would be
expected to show a small increase in modulation percentage (essentially
just $\Delta$ mag. for small values $\lesssim 0.30$) for increasing X-ray
flux. There is no evidence for this. Therefore, the companion contribution
must be minor and may be obscured, on Fig. 5, by the variability in
the relationships between the different components generating, or
obscuring, the optical flux. The X axis in Fig. 5 is still in
magnitudes, rather than on a linear intensity scale, as in Fig. 4.
The X-ray flux and optical intensity were also shown to be positively
correlated in observations of 4U 1636--53 by Shih, Charles \& Cornelisse (2011)
who performed a 100 day, monitoring program of this source in which daily
spot optical measurements were made. Only the lower energy X-ray
intensity (1.5--12 keV {\it RXTE} ASM) showed a positive correlation, the
high energy X-ray flux (15--50 keV {\it Swift} BAT) being
anti-correlated. In the present paper, the optical and X-ray data
have been measured across binary cycles and many hours respectively
and thus hopefully represent more reasonable `steady-state' average values.
There is insufficient data to be certain, but there is a suggestion in the
light curves that at the very highest X-ray flux the optical modulation
has almost gone away and the faster optical variability decreased.
Perhaps this is an indication of an optical state change. The continuing
lack of any orbital modulation signature in the folded X-ray data suggests
$i \le 60^{\circ}$, based on the modelling of Frank, King \& Lasota (1987).
Several key aspects of the standard model for an LMXB system are
supported by the following results: \vspace*{0.2cm} \\
\hspace*{0.2cm} $\bullet$ Optical maximum at spectral phase 0.47 $\pm$ 0.05 \\
\hspace*{0.2cm} $\bullet$ Positive correlation, optical flux -- X-ray flux \\
\hspace*{0.2cm} $\bullet$ No correlation, optical amplitude -- X-ray flux \\
\hspace*{0.2cm} $\bullet$ Non-X-ray induced optical flux is $\lesssim 14$ percent \\
\vspace*{0.1cm}
The dynamical ephemeris for 4U 1735--44 is now obtained by combining the
new period derived in this paper with the phase zero epoch from
Casares et al. (2006). Phase zero, in this spectral definition, again
corresponds to inferior conjunction of the companion star. The ephemeris
is therefore HJD = 2452813.495(3) + [ N $\times$ 0.19383222(29) ].
Clearly, on-going occasional optical photometry and spectroscopy are
desirable for both 4U 1735--44 and 4U 1636--53 to monitor any possible changes,
in order to better understand LMXB systems in general. These two X-ray
binaries continue to be quite similar and are apparently viewed from much
the same perspective. Future simultaneous X-ray and optical studies may be
problematic with the recent decommissioning of the long running {\it RXTE}
mission in 2012 January. A number of alternative, somewhat equivalent,
ASM type systems are presently operating. Perhaps the {\it LOFT} mission,
if selected for flight by ESA, will provide a more continuous X-ray coverage.
\section{Acknowledgements}
We thank John Greenhill for assistance at the Mt. Canopus Observatory
during the early part of this work and Stefan Dieters for helpful
comments. ABG thanks the ACE CRC at the University of Tasmania
for the use of computer facilities. The Mt. Canopus Observatory
received some financial support from David Warren.
|
{
"timestamp": "2012-03-12T01:00:39",
"yymm": "1203",
"arxiv_id": "1203.1988",
"language": "en",
"url": "https://arxiv.org/abs/1203.1988"
}
|
\section{\label{sec.I} Introduction}
The idea of considering a hole or leak in an otherwise closed billiard is very appealing. This was the physical picture used by Pianigiani and
Yorke in their influential work on conditionally invariant measures~\cite{PY:1979}. Nowadays it is an essential element in the theoretical description
of experiments in atomic, acoustic, microwave, and optical cavities~\cite{Friedman,Kuhl2005,Harayama2011}.
Another important ingredient considered in this paper is the effect of random independent and identically distributed perturbations (noise) on trajectories. There are many good reasons to consider it: test the
structural stability of the results, facilitate calculations (e.g., random phase approximations), and mimic the effect of well-defined
physical processes that are too complicated or high-dimensional to be modeled in detail (e.g., molecular diffusion).
Not surprisingly, the effect of noise is a classical problem in dynamical systems. In low-dimensional chaotic Hamiltonian systems
more than three decades of research considered the effect of noise on diffusion and anomalous transport~\cite{rechester,karney,floriani}, on the
trapping of trajectories~\cite{pogorelov,altmann.higherN,rodrigues,kruscha,kruscha2}, on scattering~\cite{seoane,altmann.noise}, etc.
In this paper we consider the effect of white noise on the dynamics of open billiards. We are interested in the generic case of billiards
with mixed phase space in which regions of regular and chaotic motion coexist. We show how the survival probability of trajectories inside
the billiard is modified due to noise, and how the transition times and parameters of this modified survival probability scale with
noise intensity and leak size. The paper has a straightforward organization: we start with the closed billiard (Sec.~\ref{sec.II}),
which we leak (Sec.~\ref{sec.III}), and perturb by noise (Sec.~\ref{sec.IV}).
\section{\label{sec.II} Closed billiard}
\subsection{Definition of the dynamics}\label{ssec.IIdynamics}
The annular billiard is defined by two eccentric circles with ratio $0<r<1$ between the two radii and distance~$0 \le \delta < 1-r$ between
the two centers. Since its introduction by Sait\^o et al. three decades ago~\cite{saito}, the annular billiard has been used to investigate
different physical phenomena~\cite{bohigas,hentschel,egydio}. Here we consider the case~$r=0.3$ and $\delta=0.65$ shown in
Fig.~\ref{fig.1}(a). These parameters were chosen because numerical simulations shown in Fig.~\ref{fig.1}(b)
strongly indicate a mixed phase space with the coexistence of regions of regular and chaotic motion. Below we discuss the main properties of this particular billiard
but we emphasize the generality of our main results to the generic case of mixed-phase-space billiards.
The dynamics in the annular billiard can be easily constructed by noting that after colliding with the outer circle, trajectories can either collide directly with the outer circle or instead first collide once with the inner circle. Trajectories that collide with the inner
circle fulfill the following condition~\cite{saito}
\begin{equation}\label{eq.collision}
|\sin\alpha + \delta\sin(\alpha-2\pi \omega)| \leq r,
\end{equation}
where $\alpha$ is the angle between the velocity and the normal vector at the collision point and $\omega$ is the normalized position of
collision at the outer circle (see Fig.~\ref{fig.1}a).
The significance of the hitting condition in Eq.~(\ref{eq.collision}) is that it can be used to obtain a mapping between two successive collisions at the outer
boundary~\cite{saito}.
We investigate this billiard map, which
corresponds to a Poincar\'e surface of section at the outer boundary. For convenience, time is counted discretely between successive
collisions of the map. The collision time ($t_{col}$, assuming constant velocity~$v=1$) in the {\it chaotic component} of the annular billiard is bounded by
$\min\{1-r-\delta,2\sqrt{1-(r+\delta)^2}\}<t_{\text{coll}}< 2$, so that the scalings and the existence of regimes of decay remain unaffected
by this simplification\footnote{In ergodic closed systems the correspondence between discrete and continuous time is given by the mean
collision time $\left\langle t_{\text{coll}}\right\rangle = \frac{\pi A}{d v}$, where $A$ and $d$ are the area and perimeter of the billiard. In open systems this does not hold~\cite{mortessagne,altmann.rmp}. The specific
quantitative results change differently in different parts of the survival probability, however, the scalings remain unchanged.}.
\subsection{Phase space components}\label{ssec.phasespace}
The phase space in Birkhoff coordinates $(\omega, \sin \alpha)$ of the closed annular billiard is shown in Fig.~\ref{fig.1}(b).
It can be divided in four invariant components (i)-(iv), which are built by the trajectories that:
\begin{itemize}
\item[(i)] do not cross the inner circle of radius $r+\delta$ ($\sin \alpha \ge r+\delta$) and therefore never satisfy
Eq.~(\ref{eq.collision}). Graphically they correspond to orbits close to the outer boundary of the circle and are called {\bf whispering
gallery}. For the billiard used in this paper, the whispering gallery exists for $|\sin \alpha|<0.95$, beyond the dotted line in Fig.~\ref{fig.1}.
\item[(ii)] cross the circle of radius~$r+\delta$ ($\sin \alpha < r+\delta$) but never satisfy Eq.~(\ref{eq.collision}). These
conditions are satisfied by periodic orbits that build one-parameter families of marginally unstable orbits (MUPOs)~\cite{Gaspard}. One trivial
example is the diameter (period $p=2$ and winding number $q=1$ orbit) highlighted in Fig.~\ref{fig.1}. In opposite to the whispering
gallery, these orbits are
usually embedded in a chaotic component
(see below) and affect the dynamics of chaotic trajectories despite having zero measure, see Ref.~\cite{altmann.mupos} for a detailed investigation in the
annular billiard and Refs.~\cite{Fendrik,Gaspard,Akaishi2009,OrestisMushroom,OrestisChaos} in other systems. For the
billiard used in this paper the following MUPOs $(p,q)$ exist: $(2,1), (6, 1), (7, 1), (8, 1), (9, 1), (19, 2), (29, 3), (39, 4)$,
$(49,5), (59, 6), (69, 7), (79, 8), (89, 9),\ldots$.
\item[(iii)] cross the circle of radius~$r+\delta$ ($\sin \alpha < r+\delta$), satisfy Eq.~(\ref{eq.collision}), but remain
close to stable a periodic orbit. These orbits can be periodic, quasi-periodic, or even chaotic (confined inside the last quasi-periodic
circle) and build the so-called Kolmogorov-Arnold-Moser (KAM) islands. For the billiard used in this paper the most prominent examples are the trajectories
around the period $6$ orbit shown as $\blacksquare$ in Fig.~\ref{fig.1}.
\item[(iv)] cross the circle of radius~$r+\delta$ ($\sin \alpha < r+\delta$), satisfy Eq.~(\ref{eq.collision}), are chaotic, and
fill a large component of the phase space.
Despite the
mathematical difficulties to provide rigorous proofs (see Refs.~\cite{Chen,Foltin} for rigorous results in particular cases), it is largely believed that the annular billiard with $\delta\gg0$ contains a large
chaotic component, in which a single trajectory visits a positive area of the phase space \cite{saito,bohigas,hentschel,egydio}. This large component is called {\bf chaotic
sea}. In Fig.~\ref{fig.1}(b) this region corresponds to the large dotted component.
\end{itemize}
\begin{figure*}[!ht]
\includegraphics[width=\linewidth]{figure2.pdf}
\caption{(Color online) The survival probability~$P(t)$ for the open annular billiard. (a) Logarithmic scale
in the $y$ axis; Inset: $x,y$ axes in linear scale, magnification for short times. (b)
Logarithmic scale in both $x,y$ axes. All regimes and transition times in Eq.~(\ref{eq.deterministic}) are depicted: the dashed line
corresponds to the fitting of an exponential with $a=0.72$ and $\gamma=0.016$; and the dotted line corresponds to the fitting of a
power law with~$b=2.492$ and $\beta=1.65$. The transition times are $t_I=14$ (visual inspection) and $t_{II}=1,020$. The leak is introduced in the position $\omega_{c,3}$ with $\mu(I)=0.01$, see Fig.~\ref{fig.1}.}
\label{fig.2}
\end{figure*}
\section{\label{sec.III} Open billiard}
\subsection{Definition of the dynamics}
We open the annular billiard by considering a region~$I$ at the border of the billiard through which the trajectories escape~\cite{PY:1979,paar,Bunimovich,AltmannTel2,beims2010,OrestisStadium,OrestisChaos,altmann.rmp}. Formally, the dynamics of the system with leak~$\tilde{M}$ is defined based on the dynamics of the closed billiard~$M$ as:
\begin{equation}\label{eq.leak}
\tilde{M} = \left\{ \begin{array}{ll}
\text{ escape } & \text{ for } \vec{x} \in I, \\
M & \text{ for } \vec{x} \notin I, \\
\end{array}
\right.
\end{equation}
where $\vec{x}=(\omega,\sin \alpha)$.
Notice that, by convention, escape occurs only one time step after trajectories hit the leak~$I$ so that~$\tilde{M}$ is defined in~$I$.
We are
interested in finite but small~$I$ such that a non-trivial dynamics still exists in the billiard.
Here we
consider leaks placed inside the large chaotic sea, item (iv) of Sec.~\ref{ssec.phasespace}, and we are interested on how the trajectories escape
from this ergodic component of the closed billiard's phase space. The invariant set of the open system is the
so-called {\it chaotic saddle}~\cite{laitamas}, the set of points that never leave the system neither in forward nor in backward times.
In fully chaotic systems the chaotic saddle is a zero measure fractal set. In the case of mixed phase space system discussed here, the
chaotic saddle relevant to the escaping trajectories also contains a similar hyperbolic component~\cite{Jung,AltmannTel2} but additionally it includes a
non-hyperbolic component composed by the borders of the whispering gallery [region (i) of Sec.~\ref{ssec.phasespace}], the MUPOs [region (ii) of
Sec.~\ref{ssec.phasespace}], and of the KAM islands [region (iii) of Sec.~\ref{ssec.phasespace}].
In our simulations of the annular billiard we consider leaks~$I=[\omega_c-\Delta \omega, \omega_c+\Delta \omega] \times [-\Delta
\sin\alpha, +\Delta \sin \alpha]$ centered at three different positions $\omega_c$ (see Fig.~\ref{fig.1}), varying $\Delta \omega$, and a fixed $\Delta
\sin\alpha= 2/3$.
Physically, this configuration corresponds to a dielectric billiard with refraction index $\eta=(\Delta \sin \alpha)^{-1}=1.5$ (glass) with a
perfect mirror boundary everywhere except inside the leak, where trajectories escape for collisions below the critical angle $\alpha_c$
with $\sin \alpha_c = 1/\eta$.
\subsection{Survival probability}\label{ssec.det.surv}
We compute the survival probability~$P(t)$ inside the billiard by starting an ensemble of trajectories distributed according to an
initial density~$\rho_0(\vec{x})$. In our simulations we consider $\rho_0(\vec{x})$ to be uniform inside the leak of the billiard and $0$
elsewhere (for the first iteration the closed billiard $M$ is used).
Physically, these initial conditions correspond to throwing trajectories inside the billiard through the leak. Another
motivation for using this particular $\rho_0(\vec{x})$ comes from the fact that~$P(t)$ in this case corresponds exactly to the distribution of Poincar\'e recurrence times~\cite{AltmannTel2}. The
main decay regimes of $P(t)$ remain unaffected by this choice of $\rho_0(\vec{x})$, but values of the exponents and transition times may
change [see bullet items after Eq.~(\ref{eq.deterministic})].
Figure 2 shows the decay of the survival probability for the particular leak shown in Fig.~\ref{fig.1}. We can identify three different regimes of decay~\cite{Fendrik,AltmannTel2}
\begin{equation}\label{eq.deterministic}
P_{\text{deterministic}}(t) \approx \left\{ \begin{array}{ll}
\text{ irregular } & \text{ for } t < t_I, \\
a e^{-\gamma t} & \text{ for } t_I < t < t_{II}, \\
b t^{-\beta} & \text{ for } t > t_{II}. \\
\end{array}
\right.
\end{equation}
The dynamics of a typical trajectory escaping in each of the regimes in Eq.~(\ref{eq.deterministic}) can be associated to the phase space
structures as:
\begin{itemize}
\item $t<t_I$ irregular: trajectories that collide only a few times in the chaotic sea of the closed billiard [region (iv) of Sec.~\ref{ssec.phasespace}], the spatial density of survival trajectories has
not converged yet. The exact shape in this regime is extremely sensitive to~$\rho_0(\vec{x})$.
\item $t_I<t<t_{II}$ exponential $e^{-\gamma t}$: trajectories explore the hyperbolic component of the chaotic saddle but escape before
coming close to the border of the non-hyperbolic components [(i,ii,iii) of
Sec.~\ref{ssec.phasespace}]. Analogous to the case of fully chaotic systems~\cite{laitamas}, the same exponent $\gamma$ is
observed for different $\rho_0(\vec{x})$~\cite{AltmannTel2}.
\item $t>t_{II}$ power-law $t^{-\beta}$: trajectories get stuck close to the non-hyperbolic components of the saddle (classical references on such
stickiness phenomena are Refs.~\cite{chirikov1,meiss.ott.physicaD,zaslavsky.pr}). The asymptotic exponent changes from $\beta$ to
$\beta'=\beta-1$ for
$\rho_0(\vec{x})$ nonzero at the boundary of the non-hyperbolic components (e.g., if $\rho_0(\vec{x})$ is taken according to the
Liouville measure restricted to the chaotic sea of the closed billiard)~\cite{meiss,laitamas}.
\end{itemize}
The description above applies to typical trajectories escaping in the corresponding regimes. It is instructive to think that for
intermediate times, $t_\beta<t< t_{II}$, both exponential and power-law regimes
coexist~\cite{AltmannTel2,Akaishi2009,OrestisStadium}
\begin{equation}\label{eq.coexist}
P(t) = a e^{-\gamma t} + b t^{-\beta},
\end{equation}
where $t_{\beta}$ is the time needed to approach the non-hyperbolic component of the saddle~\cite{AltmannTel2}.
\begin{figure*}[!ht]
\includegraphics[width=\linewidth]{figure3.pdf}
\caption{(Color online) Scalings of the parameters of $P(t)$ in Eq.~(\ref{eq.deterministic}) with leak
size~$\mu(I)$ for three different leak locations.
(a) Intermediate times escape rate~$\gamma$; The dashed line corresponds to $\gamma^*$
in Eq.~(\ref{eq.mustar}). Inset: dependence of the coefficient $a$ with $\mu(I)$. (b) Transition time $t_{II}$, obtained
using Eq.~(\ref{eq.defnc}); the dashed line corresponds to a scaling $1/\mu(I)$.
The leaks have a fixed height~$\Delta \sin \alpha = 2/3$ and varying width $\Delta \omega \rightarrow 0$, which leads to $\mu(I)\rightarrow0$.
The centers of the leaks are: $\omega_{c,1} = 0.161$ (in a MUPO, black circles),
$\omega_{c,2}=0.5$ (in an unstable periodic orbit, red squares), and
$\omega_{c,3}=0.55$ (in the chaotic sea, green diamonds).
}
\label{fig.3}
\end{figure*}
\subsection{Dependence of the parameters of $P(t)$ on the leak}
The {\bf exponential decay~$\gamma$} in Eq.~(\ref{eq.deterministic}) can be considered a signature of the chaoticity of the map. Following
this reasoning, for small leaks we can approximate the escape at each time step by the area of the leak relative to the area of the
chaotic sea $\mu(I):=\text{Area}(I)/\text{Area(chaotic sea)}$. For the billiard considered here we found numerically that $\text{Area(chaotic sea)} \approx
0.993\times(r+\delta) = 0.943$. Using phase-space areas correspond to using the Liouville measure $d\mu=d\omega d \sin \alpha$ of the closed
system to approximate properties of the open system, and can be shown to be valid for almost all leak positions in
strongly chaotic systems~\cite{altmann.rmp}. This leads to an estimation of the exponential decay as
\begin{equation}\label{eq.mustar}
\gamma^*=-\ln(1-\mu(I)) \approx \mu(I) \text{ for } \mu(I)\rightarrow0.
\end{equation}
Violations of this approximation in the fully chaotic case have been extensively discussed in the recent years and are particularly large
for leaks containing low-period periodic orbits of the closed system~\cite{paar,AltmannTel2,Bunimovich,Demers}. Here we extend these previous
results and verify the
effectiveness of the approximation in Eq.~(\ref{eq.mustar}) for the intermediate-time exponential decay in Eq.~(\ref{eq.deterministic}). In Fig.~\ref{fig.3}(a) we
compare the numerically obtained values to the prediction for different leak sizes centered at three different positions: in the chaotic region, around
an unstable periodic orbit, and around a family of MUPOs. In all cases $\gamma \rightarrow \gamma^*$ is observed in the limit of small
leaks $\mu(I)\rightarrow0$, in agreement with relation~(\ref{eq.mustar}). For large leak sizes Fig.~{fig.3}(a) shows different deviations of this relation, in agreement with the results observed for hyperbolic systems~\cite{paar,AltmannTel2,Bunimovich,Demers}.
The importance of the value of~$\gamma$ is that the same value is obtained for a broad class of smooth initial densities $\rho_0$. In fully chaotic systems the requirement is
that $\rho_0$ intersects the stable manifold of the chaotic saddle. Analogously, the requirement here is that it intersects the stable manifold of the hyperbolic component of the chaotic saddles.
The {\bf power-law exponent~$\beta$} depends on the properties (of the boundary) of the non-hyperbolic sets embedded in the chaotic
component of the phase space (components (i), (ii) and (iii) in the list of Sec.~\ref{sec.II}). If no KAM islands are present, an exponent $\beta=2$ can be obtained for MUPOs~\cite{Fendrik,OrestisStadium,OrestisChaos}. The question of whether the asymptotic regime has a
well-defined and universal power law in the generic KAM case is still under investigation for the case of area-preserving maps (see Refs.~\cite{cristadoro,venegeroles} for the latest results
that indicate $\beta \approx 1.57$). For simplicity we write the asymptotic decay as $t^{-\beta}$, but it is meant to describe the
power-law like behavior usually observed in mixed-phase-space systems~\cite{chirikov1,meiss.ott.physicaD,zaslavsky.pr}.
\subsection{Dependence of the transition times of $P(t)$ on the leak}
Transition time $t_I$ indicates the starting of the exponential decay. It can be interpreted as a convergence time which is proportional
to~$1/|\lambda'|$, where~$\lambda'$ is the negative Lyapunov exponent of the saddle (the time to relax
to the hyperbolic component of the saddle along its stable manifold). Numerical observations usually show an abrupt approach, i.e., the
exponential decay provides a good description of $P(t)$ after a finite (short) time~\cite{AltmannTel2}.
The transition time $t_{II}$ is defined from Eq.~(\ref{eq.coexist}) as the time for which the exponential and power-law contributions are
equal~\cite{AltmannTel2}
\begin{equation}\label{eq.defnc}
a e^{-\gamma t_{II}}= b t_{II}^{-\beta} \Rightarrow p(t_{II})= 2 \gamma a e^{-\gamma t_{II}}.
\end{equation}
The ratio $a/b$ can be interpreted as the proportion between the number of trajectories escaping exponentially to the number of trajectories
escaping algebraically. It depends mainly on the measures of the chaotic and regular components of the phase space and therefore it should not
depend strongly on the measure of the leak $\mu(I)$. Under this assumption we can estimate the scaling of $t_{II}$ on the leak size $\mu(I)$ as~\cite{AltmannTel2}
\begin{equation}\label{eq.tii}
t_{II} \sim \frac{1}{\gamma} \sim \frac{1}{\mu(I)},
\end{equation}
for which additional logarithmic corrections apply~\cite{Akaishi2009}. The scaling in Eq.~(\ref{eq.tii}) has been confirmed in our numerical
simulations for the three different leak positions, see Fig.~\ref{fig.3}(b).
\section{\label{sec.IV} Open noisy billiard}
\subsection{Definition of the dynamics}
Here we consider additive noise perturbations to the trajectories. In the simulations of the annular billiard we have implemented at each
collision a perturbation to the angle~$\alpha$ as
$$ \alpha'=\alpha+\delta,$$
where $\delta$ is an independent normal distributed random variable with zero mean, $\langle \delta \rangle=0$, and standard deviation
$\sigma=\pi \xi$ (noise strength).
In order to prevent the particle from leaving the billiard through the border (non-physical situation), the noise distribution was truncated at $\alpha=\pm \pi/2$. Notice that perturbations in the $\alpha$ direction are perpendicular to the border of the whispering gallery component and to the parameterization of the billiard boundary, having therefore a strong
impact on sliding orbits ($\alpha=\pm \pi/2$)\footnote{In fact, trajectories have a tendency of being repelled
from sliding orbits because the noise perturbation in $\alpha$ is nonzero, truncated in $\alpha = \pm \pi/2$, and added on
each collision. Therefore, we do not expect these orbits to dramatically affect $P(t)$ or the relation between billiard maps and flows in our case.}.
Based on previous observations with different setups~\cite{altmann.higherN,altmann.noise,kruscha,kruscha2,rodrigues}, and in the generality of the
arguments below, we believe our results are valid for additive white noise in general (provided the perturbation in $\alpha$ is
nonzero). It is an interesting open question whether (and which) modifications are needed for multiplicative and colored noise (see Ref.~\cite{pogorelov}).
\begin{figure*}[!ht]
\includegraphics[width=\linewidth]{figure4.pdf}
\caption{(Color online) Survival probability~$P(t)$ for the open annular billiard perturbed by noise with intensity~$\xi=3\times 10^{-4}$. (a) Logarithmic scale in the y axis; (b)
Logarithmic scale in both x and y axis. All regimes and transition times in Eqs.~(\ref{eq.both})-(\ref{eq.noise}) are depicted in (b):
fitting of the asymptotic exponential with $d=9.34\times 10^{-5}$ and $\gamma_\xi=1.06\times 10^{-5}$ (dot-dashed line in brown);
power-law decay with~$c=0.014$ and $\beta_{RW}=0.5$ (dotted line in green). The dashed black line corresponds to the $\xi=0$ case (see
Fig.~\ref{fig.2}). The transition times were estimated as $t_D=887$ and $t_{III}=15,158$. The leak is as in Fig. \ref{fig.2}
($\omega_{c,3}$ with $\mu(I) = 0.01$).}
\label{fig.4}
\end{figure*}
\subsection{Survival probability}
In Sec.~\ref{ssec.det.surv} the trapping of trajectories inside the billiard was connected to invariant structures of the
deterministic phase space. The longer the escape time of trajectories, the closer they approach these invariant structures. This leads to a
connection between temporal scales of the survival probability and spatial scales in the phase space. Noise perturbations affect phase-space scales comparable to~$\xi$. Based on these arguments we expect that for small~$\xi$ the survival probability~$P(t)$ is
modified for long times only:
\begin{equation}\label{eq.both}
P(t) \approx \left\{ \begin{array}{ll}
P_{\text{deterministic}} & \text{ for } t < t_D, \\
P_{\text{noise}} & \text{ for } t > t_D , \\
\end{array}
\right.
\end{equation}
where $P_{\text{deterministic}}$ is given by Eq.~(\ref{eq.deterministic}) and $t_D$ is the transition time. Following Refs.~\cite{altmann.higherN,altmann.noise}, the
noise perturbed survival probability is given by
\begin{equation}\label{eq.noise}
P_{\text{noise}}(t) \approx \left\{ \begin{array}{ll}
c t^{-\beta_{RW}} & \text{ for } t_{D} < t < t_{III}, \\
d e^{-\gamma_\xi t} & \text{ for } t > t_{III}. \\
\end{array}
\right.
\end{equation}
Figure~\ref{fig.4} shows, for $\xi=3\times10^{-4}$, the different decay regimes and transition times of the survival probability given by Eqs.~(\ref{eq.both})-(\ref{eq.noise}).
The dynamics of a typical trajectory escaping in each of these regimes
can be associated to the phase space structures of the deterministic dynamics as:
\begin{itemize}
\item $t<t_D$ deterministic: trajectories escape before the noise perturbation is noticed, $P(t)$ coincides with the $\xi=0$ case.
\item $t_{D} < t < t_{III}$ enhanced trapping: trajectories enter the region corresponding to regular motion of the deterministic dynamics
[components (i) and (iii) of Sec.~\ref{ssec.phasespace}] and perform a one-dimensional random walk inside it~\cite{altmann.higherN,rodrigues,altmann.noise,kruscha,kruscha2}.
\item $t>t_{III}$ asymptotic exponential: trajectories explored all available phase-space.
\end{itemize}
In this description we neglect the effect of noise on~$\gamma$, which has been investigated for a fully chaotic system in
Ref.~\cite{altmann.noise}. The sticky region around MUPOs [region (ii) of Sec.~\ref{ssec.phasespace}] does not contribute to the enhanced
trapping regime ($t_{D} < t < t_{III}$) because MUPOs build a zero measure set. This means that if noise is added to a system in which
MUPOs are the only source of stickiness (e.g., the Stadium~\cite{OrestisStadium} or Drive-belt billiards~\cite{OrestisChaos}), we predict
that the exponential decay will start immediately after $t_{D}$ (i.e., $t_{III}=t_{D}$).
\begin{figure*}[!ht]
\includegraphics[width=\linewidth]{figure5.pdf}
\caption{(Color online) Dependence of the parameters of $P(t)$ in Eq.~(\ref{eq.noise}) with noise intensity~$\xi$. (a)
Fitted $\gamma_\xi$ with scaling $\xi^2$ (red dashed line) and value $\gamma$ (blue dots). Inset: coefficient $d$ vs. $\xi$ with
green dashed line representing $\xi^{-2}$; (b) $t_{III}$
obtained as the time for which $P(t)$ intersects the fitted curve $1.1 d\exp(-\gamma_\xi t_{III})$ (black circles); $t_D$ obtained as the time
for which $P(t)$ intersects $2 P_{\text{deterministic}}(t)$ (red squares); and green dotted lines indicating scalings $\xi^{-0.4}$
(bottom) and $\xi^{-2}$ (top). The leak is the same as in Figs.~\ref{fig.2} and \ref{fig.4}. }
\label{fig.5}
\end{figure*}
\subsection{Dependence of the parameters of $P(t)$ on the leak}
For small noise perturbations the parameter $\beta_{RW}$ in Eq.~(\ref{eq.noise}) can be related to the scaling of the recurrence time
distribution of a one-dimensional random walk (with step size $\sim \xi$) and is therefore given by~\cite{Feller,altmann.higherN}
\begin{equation}\label{eq.betarw}
\beta_{RW}=\frac{1}{2} \text{ for } \xi\rightarrow 0.
\end{equation}
In the derivation of this result in Ref.~\cite{altmann.higherN} (see also~\cite{altmann.noise}) the initial conditions~$\rho_0(\vec{x})$ were
chosen {\it outside} KAM islands of the deterministic closed billiard, in agreement with the case treated here. See
Refs.~\cite{rodrigues,kruscha,kruscha2} for the case of $\rho_0(\vec{x})$ {\it inside} KAM islands. References~\cite{kruscha,kruscha2}
also showed that if $\xi^2$ terms are included, the random walk is biased.
The scaling in Eq.~(\ref{eq.betarw}) is interrupted for long times because of the limited region available for the random walk in the KAM islands and whispering gallery. In the random-walk model this corresponds to adding a reflecting boundary condition~\cite{altmann.higherN,altmann.noise}. The exponent~$\gamma_{\xi}$ of the asymptotic decay can be obtained
considering $P(t)$ to be a continuous and smooth function around $t=t_{III}$. Evaluating
\begin{equation}\label{eq.gammaxi1}
\frac{\partial \log(P(t))}{\partial t} \text{ at } t=t_{III}
\end{equation}
for both terms in Eq.~(\ref{eq.noise}) and equating then leads to $t_{III}=0.5/\gamma_\xi$. Below we show that
$t_{III}\sim1/\xi^2$ [see Eq.~(\ref{eq.tiii}) and Refs.~\cite{altmann.higherN,altmann.noise}], and therefore we obtain
\begin{equation}\label{eq.gamma_xi}
\gamma_\xi \sim \frac{1}{t_{III}} \sim \xi^{2} \text{ for } \xi\rightarrow 0.
\end{equation}
This scaling is confirmed in Fig.~\ref{fig.5}(a). Interestingly, we observe that for larger noise intensities $\gamma_\xi$ experiences a
crossover to $\gamma_\xi\approx \gamma$, i.e., $\gamma_\xi$ is bounded by the escape rate observed for short times which was related to the hyperbolic
component of the chaotic saddle in Sec.~\ref{sec.II}. This observation indicates that once the
trajectories leave the region corresponding to regular motion of the deterministic dynamics, the hyperbolic component of the
saddle controls their escape. For small noise the process of leaving the regular components is slower and therefore the
scaling~(\ref{eq.gamma_xi}) dominates~$\gamma_\xi$. For larger noise, the deterministic
exponential escape is slower than the escape from the islands and therefore $\gamma_\xi\approx\gamma$ is observed. In Ref.~\cite{rodrigues} a
similar but different scaling of $\gamma_\xi$ on $\xi$ was numerically obtained for the case of random maps and for initial conditions
taken inside the KAM islands.
\subsection{Dependence of the transition times of $P(t)$ on the leak}
A theory for $t_D$ can be found in Ref.~\cite{floriani} and predicts that
\begin{equation}\label{eq.tD}
t_D \sim 1/t^\Lambda \text{ for } \xi\rightarrow0,
\end{equation}
with $\Lambda\lessapprox 1$ related to the scaling of {\it Cantori} close to the KAM islands. This scaling is valid for $t_D \gg t_{II}$ because
for $t_D \simeq t_{II}$ the deterministic trapping around the KAM islands has a limited contribution.
The final cut-off time~$t_{III}$ can be estimated from basic properties of diffusion motion. The expected distance~$L$
traveled by a random walker with step-size $\sim \xi$ grows as $L \sim \xi \sqrt{t}$. The time $t_{III}$ corresponds to the
expected time for the $\xi$-perturbed trajectories to travel the (fixed) distance corresponding to the diameter of the (largest) KAM
island. Therefore, we estimate
\begin{equation}\label{eq.tiii}
t_{III} \sim \xi^2, \text{ for } \xi\rightarrow 0.
\end{equation}
Our numerical simulations for the scaling of the transition times are shown in Fig.~\ref{fig.5}(b). We see that for times comparable
to~$t_{II}$ only a weak dependence of $t_D$ on $\xi$ is observed. For times larger than $t_{II}$ our results indicate an increase of this
dependency, consistent with relation~(\ref{eq.tD}). The scaling of $t_{III}$ in Eq.~(\ref{eq.tiii}) is confirmed over a larger interval of $\xi$.
\section{\label{sec.V} Summary of Conclusions}
White noise perturbations have surprising effects on the chaotic dynamics of mixed-phase-space Hamiltonian
systems~\cite{rechester,karney,floriani,pogorelov,altmann.higherN,rodrigues,kruscha,kruscha2,seoane,altmann.noise}. Here we have shown how
these results impact the dynamics of open billiards, in which case the openness can be controlled systematically and the phase space
usually contains MUPOs~\cite{Gaspard,altmann.mupos}. We have combined these results into a complete survival probability which is obtained introducing Eqs.~(\ref{eq.deterministic})
and~(\ref{eq.noise})
into Eq.~(\ref{eq.both}). For small noise intensities, it contains five different regimes with four transition times $(t_I,t_{II}, t_{III},t_D)$
and four physically relevant parameters $(\gamma,\beta,\beta_{RW},\gamma_\xi)$. We have discussed how these quantities depend on the
intensity of the perturbation~$\xi$ and on the position and size~$\mu(I)$ of the leak.
Apart from extending previous results to billiard systems, our paper contains several new findings. First, we have shown that the (intermediate-times) escape rate scales linearly with
leak size and is extremely sensitive to the location of the leak. These results were previously known for fully chaotic systems~\cite{paar,AltmannTel2,Bunimovich,Demers}. Second, we have
shown that the asymptotic exponential decay~$\gamma_\xi$ depends on the noise intensity $\xi$ as $\gamma_\xi \sim \xi^2$, with a transition
towards $\gamma_\xi = \gamma$ for large $\xi$ (Fig.~\ref{fig.5}a). Altogether, our results further emphasize the significance of the intermediate time decay
regimes of the survival probability in weakly chaotic noise-perturbed billiards.
\begin{acknowledgments}
We thank T. T\'el for insightful discussions that led to Fig.~\ref{fig.5}(a), O. Georgiou and M. Matos for the careful reading of the
manuscript. J. C. Leit\~ao acknowledges funding from Erasmus N 29233-IC-1-2007-1-PT-ERASMUS-EUCX-1.
\end{acknowledgments}
|
{
"timestamp": "2012-03-09T02:02:52",
"yymm": "1203",
"arxiv_id": "1203.1791",
"language": "en",
"url": "https://arxiv.org/abs/1203.1791"
}
|
\section{background}
The room-temperature resistivity of a metal (as well as most metallic
electronic materials, e.g., doped semiconductors) is mainly limited by
the electron-phonon interaction \cite{ziman} i.e., by phonon
scattering, with a notable exception being doped graphene with its
very weak electron-phonon coupling \cite{g_phonon1}. The
electron-phonon scattering contribution to the resistivity falls off
strongly at very low temperatures ($T < T_{\rm BG}$) in the so-called
Bloch-Gr\"{u}neissen (BG) regime \cite{ziman}, due to the exponential
suppression of the bosonic thermal occupancy of the phonons. At low
temperatures, therefore, the electronic conductivity of metallic
systems is invariably limited by disorder, i.e., by electron-impurity
scattering, which gives rise to the zero-temperature residual
resistivity of metals at low temperatures where the phonon scattering
contribution has vanished. In general, the impurity scattering
contribution to the metallic resistivity is temperature independent,
at least in three-dimensional (3D) metals, because the impurities are
quenched, and the temperature scale is, therefore, the Fermi
temperature ($T_{\rm F} \sim 10^4$ K in 3D metals), the impurity scattering
contribution to the resistivity is essentially temperature-independent
in the $0-300$ K regime. This, however, is not true if $T_{\rm F}$ is low as
it could be in 2D semiconductor-based systems with their tunable
carrier density where at low densities $T_{\rm F}$ could be just a few
K. In fact, for 2D GaAs hole systems (2DHS), the situation with $T_{\rm F}
<1$ K can easily be reached for a 2D hole density $\sim 10^{10}$
cm$^{-2}$ \cite{holeexp,mills1999}. This situation, which has no known analog in
3D metallic systems, leads to very strong experimentally observed
temperature-dependent 2DHS resistivity in the $T\alt 1$ K temperature
range arising entirely from electron-impurity scattering since the
phonon scattering is completely thermally quenched at such low
temperatures. As an aside, we note that the temperature dependence of
2D electronic resistivity arising from impurity scattering in 2D GaAs
based electron systems is rather weak due to the much lower effective
mass of 2D electrons ($m_{\rm e} \sim 0.07 m_0$, where $m_0$ is the vacuum
electron mass), leading to much higher $T_{\rm F}$, compared with 2D holes
($m_{\rm h} \sim 0.4 m_0$) in GaAs-based 2D heterostructures.
The goal of the current work is to explore the interplay between
impurity scattering and phonon scattering in the temperature-dependent
resistivity of 2DHS in GaAs-based 2D systems. At higher temperature
($\agt 100$ K), the GaAs carrier resistivity is completely dominated by
longitudinal optical (LO) phonon scattering, which has been extensively studied
\cite{kawamura1992} and is not a subject matter of interest here since the
resistivity limited by LO-phonon scattering manifests the strong
exponential temperature dependence $\sim e^{-\hbar\omega_{\rm LO}/k_{\rm B}T}$
with $\hbar\omega_{\rm LO} \sim 36$ meV.
Our interest is in the interplay between acoustic phonon scattering
and impurity scattering in the low to intermediate temperature regime
($T\sim T_{\rm BG} - T_{\rm F}$) where both impurity scattering and acoustic
phonon scattering contributions to resistivity would show nontrivial
temperature dependence. In particular, we are interested in the question
whether an interplay between the two scattering processes could lead
to an approximately constant (i.e. temperature independent)
resistivity over some intermediate temperature range (1 K $<T<$
40 K). One of our motivation comes from a recent experiment \cite{zhou2012}
which discovered such an intermediate-temperature ``resistivity
saturation'' phenomenon in a 2D GaAs-based electron system in the
presence of a parallel magnetic field (which
presumably serves to enhance the electron effective mass due to the
magneto-orbital coupling, thus reducing the Fermi temperature of the
electron system \cite{parallel_B}). A second motivation of our work is estimating the
deformation potential coupling strength for hole-acoustic phonon
scattering in 2DHS in GaAs. It turns out that the electron-phonon
deformation coupling is not known in GaAs, and a quantitative
comparison between our theoretical results and experimental transport
data could lead to an accurate estimation of the deformation potential
coupling in the GaAs-based 2DHS. We mention in this context that the
accurate evaluation of the electron deformation potential coupling in
2D GaAs systems is also based on a quantitative comparison of the
experimental and theoretical transport data \cite{kawamura1992,kawamura1990}.
The basic physics we are interested in (see Fig.~1) is a situation
where the acoustic phonon contribution to the resistivity increases linearly
with increasing temperature ($T>T_{\rm BG}$), but the impurity
contribution decreases with increasing temperature ($T \agt T_{\rm F}$) as
happens in a nondegenerate classical system where increasing
temperature must necessarily increase the conductivity since the
electrons are classically moving ``faster''. (We emphasize that such a
situation is physically impossible in 3D metals since $T>10^{4}$ K
would be required, but is routinely achieved in 2D semiconductor-based
systems where very low carrier density could lead to very low values
of $T_{\rm F}$.) We explicitly consider a situation with $T \ll T_{\rm LO}$, where
$T_{\rm LO} \sim 100$ K in GaAs where LO-phonon start contributing to the
resistivity in a substantial manner. Now we ask the question whether
it is possible for the increasing temperature dependent contribution
to the hole resistivity
arising from enhanced phonon scattering at higher
temperatures could be approximately canceled over a finite temperature
range by the decreasing
temperature dependent contribution arising from impurity scattering
with increasing temperature due to the quantum-classical crossover,
i.e., the nondegeneracy, effects. Some early theoretical work
\cite{hwang2000} indicated that the interplay between phonon
scattering and nondegeneracy may indeed lead to a partial cancellation
of different temperature-dependent contributions, producing an
interesting nonmonotonicity (see Fig.~1) in the resistivity as a
function of temperature in the $1-10$ K range for low-density,
high-mobility 2DHS in GaAs. In the current work, we look into this
issue at great depth in view of the recent experimental work. In
addition, we obtain the appropriate 2DHS deformation potential
coupling constant by comparing our theoretical results to existing
2DHS hole transport data.
\section{Introduction}
\label{sec:motivation}
The temperature dependent transport in 2D systems
has been a subject of intense activity since the observation of an
apparent electronic metal-insulator transition (MIT),
which represents the experimental observation of a transition from
an apparent metallic behavior (i.e., $d\rho/dT>0$, where $\rho$ is the
2D resistivity) to an insulating behaviors (i.e. $d\rho/dT <0$) as the
carrier density is reduced.
The remarkable observation of the anomalous metallic
temperature dependence of the resistivity
is observed mostly in high-mobility low-density 2D semiconductor
systems \cite{dassarma2005,abrahams2001}.
The low temperature anomalous
metallic behavior discovered in 2D semiconductor systems
arises from the physical mechanism of strong temperature dependent
screening of charged impurity scattering \cite{dassarma2005}.
At low temperatures ($T \alt T_{\rm F}$, where $T_{\rm F}=E_{\rm
F}/k_{\rm B}$ is the Fermi
temperature with the Fermi energy $E_{\rm F}$) the main scattering mechanism
in resistivity is due to impurity disorder from unintentional background
charged impurities and/or intentional dopants in the modulation-doping.
The resistivity $\rho_{\rm imp}(T)$ limited by the charged
impurities increases linearly with temperature at lower temperatures
($T<T_{\rm F}$) due to screening effects.
This is a direct manifestation of the weakening of the screened
charged disorder with increasing temperature \cite{dassarma2005,dassarma1999} or
equivalently an electron-electron interaction effect in the so-called
ballistic regime \cite{zna}.
For $T \gg T_{\rm F}$, $\rho(T)$
decreases as $T_{\rm F}/T$ due to nondegeneracy effects and the
quantum-classical crossover occurs
at the intermediate temperature regime around $T\sim T_{\rm F}$. \cite{dassarma1999}
The temperature dependent resistivity \cite{kawamura1992} limited by
acoustic phonons $\rho_{\rm ph}(T)$ undergoes a smooth
transition from a linear-$T$ dependence at high ($T>T_{\rm BG}$)
temperatures to a weak high power $T^{a}$
dependence with $a\geq 3$ as the temperature is reduced below
$k_{\rm B}T_{\rm BG} = 2k_{\rm F}v_{\rm ph}$, where $k_{\rm F}$
is the Fermi wave vector of the 2D hole system and $v_{\rm ph}$ is
the longitudinal or transversal sound velocity.
The characteristic temperature $T_{\rm BG}$ is referred to as the
Bloch-Gr\"{u}neisen (BG) temperature.
Note that $T_{\rm BG}$ is much smaller than the Debye temperature
since the inverse lattice constant greatly exceeds $k_{\rm F}$.
In the BG regime ($T<T_{\rm BG}$) the scattering rate is strongly reduced by the
thermal occupation factors because the phonon population
decreases exponentially and the phonon emission is prohibited by the sharp
Fermi distribution, which gives rise to high power law behavior
($a\geq 5$ with screening effects, but $a\geq 3$ without screening) in
the temperature
dependent resistivity.
Thus, the phonon contribution to the resistivity is negligible
compared to the charged impurities
and the phonon contribution to the resistivity shows very weak
temperature dependence for $T<T_{\rm BG}$.
For temperatures $T>T_{\rm BG}$, since the electron-phonon scattering becomes
proportional to the square of the oscillation amplitude of ions,
the $\rho_{\rm ph}$ depends linearly on the temperature.
We note that at low carrier density where $k_{\rm F}$ is small, $T_{\rm BG}$ can
be very low.
In this paper we investigate the temperature dependent transport
properties of p-type GaAs-based 2DHSs for the temperatures
$T \alt 100$ K by taking into account both hole-phonon and
hole-impurity scatterings. In p-GaAs, holes interact with
acoustic phonons through a short-range deformation potential as well
as through a long range electrostatic potential resulting from the
piezoelectric effect. The precise value of the deformation-potential
coupling constant $D$ is very important to understand 2D p-GaAs
carrier transport properties. For example, the mobility is strongly related
to the deformation potential, $\mu \sim D^{-2}$. However,
the precise value of $D$ for p-GaAs has not been available, so the
value of the n-GaAs deformation potential ($D=12$ eV
\cite{kawamura1992}) is used uncritically.
Thus, the current investigation of the transport properties of p-GaAs
systems is motivated by getting the exact value of the deformation potential
constant $D$ in p-GaAs.
We find that the fitted value of deformation-potential coupling
could change by as much as 60\% (i.e. $D=7.6-12.7$ eV) depending on the
value of the depletion density $n_{\rm depl}$.
When we assume $n_{\rm depl}=0$, we obtain $D=12.7$ eV as the most
suitable value for the p-GaAs acoustic phonon deformation coupling
constant by fitting the available experimental data \cite{noh2003}.
The value of
$D=12.7$ eV is larger than the generally accepted value in bulk GaAs
($D=7$ eV) \cite{wolfe1970} but comparable to the value of the
n-GaAs ($12-14$ eV) \cite{price1985,mendez1984,kawamura1990}.
However, due to the lack of knowledge about $n_{\rm depl}$ we have
the uncertainty in the value of deformation potential coupling.
To precisely determine the value of $D$ for holes it is required a careful
measurement of $\rho(T)$ over the $T=1-50$ K range in a high mobility and
high density ($n \agt 5\times10^{11}$ cm$^{-2}$) sample.
Note that our finding of the apparent dependence of the GaAs 2D hole
deformation potential coupling on the background (and generally
unknown) depletion charge density in the 2DHS is obviously {\it not} a
real effect since the hole-phonon coupling strength cannot depend on
the depletion charge density. The apparent dependence we find arises
from the fact that the calculated resistivity depends strongly on the
depletion density whose precise value is unknown.
Other motivation of our work is to investigate the nontrivial nonmonotonic
behavior of the temperature dependent resistivity observed experimentally.
In the presence of the hole-phonon and hole-impurity scattering
a nonmonotonicity arises from a competition among three
mechanisms \cite{mills1999,hwang2000,lilly2003}: screening which is
particularly important for $T<T_{\rm F}$, nondegeneracy and the associated
quantum-classical crossover for $T \sim T_{\rm F}$, and
phonon scattering effect which becomes increasingly important for $T >5-
10$ K, depending on the carrier density.
In Fig.~\ref{fig:rho_T_schematic} we show the
schematic resistivity behavior of the p-GaAs in the presence of both charged
impurity and phonon scatterings.
At low temperatures ($T<T_1\sim T_{\rm F}$), the scattering arising from charged
impurities dominates and the resistivity increases as temperature increases
due to the screening effects.
For $T > T_2\sim 5-10$ K the scattering by acoustic
phonons plays a major role and limits the carrier mobility of p-GaAs
systems in this temperature range. Note that in general $T_2 \gg
T_{BG}$ for p-GaAs systems. However, at intermediate
temperatures ($T_1<T<T_2$) the
competition between acoustic phonon scattering and impurity scattering
gives rise to a nontrivial transport behavior.
We carefully study the non-trivial transport properties of p-GaAs in the
intermediate temperature range (i.e., $T_1<T<T_2$). Interestingly we
find that the approximate temperature dependent $\rho(T)$ in an intermediate
temperature range can be constant, which arises from the approximate
cancellation between the quantum-classical crossover and phonon
scattering, and is quite general.
When the phonon scattering dominates impurity scattering for $T>T_2$ the
temperature dependence of hole mobility enables one to extract
information on the electron-phonon scattering from mobility
measurements. In this temperature range the valence band deformation
potential can be determined by fitting theoretical calculations to
the existing carrier mobility data. In this paper
we extract the value of the deformation potential by fitting the experimental
mobility data, but uncertainty arises from the unknown values of the
depletion charge density and background impurity charge density.
\begin{figure
\includegraphics[width=\linewidth]{fig1_rho_T_schematic.eps}
\caption{(Color online) (a) Schematic resistivity behavior in the
presence of both hole-impurity
scattering and acoustic phonon scattering as a function of
temperature. Typically $T_1\sim T_{\rm F}$ and $T_2\sim 5-10$ K depending
on the carrier density. }
\label{fig:rho_T_schematic}
\end{figure}
To investigate the temperature dependent transport properties for the
p-GaAs 2DHS we use the finite temperature Boltzmann
transport theory considering scatterings by charged random impurity centers and
by electron-acoustic phonons with finite temperature and finite wave
vector screening through random phase approximation (RPA) \cite{dassarma2005}.
We also include the finite size confinement
effects (i.e., we take into account the extent of the 2D system in the
third dimension and do not assume it to be a zero-width 2D layer).
The effect we neglect in our theory is the inelastic optical
phonon scattering. Polar carrier scattering by LO phonons is
important in GaAs only at relatively high temperatures ($T \agt
100$ K), becoming dominant at room temperatures. Due to the rather
high energy of the GaAs optical phonons ($\sim 36$ meV), LO phonon
scattering is completely suppressed in the temperature regime
($T<100$ K) of interest to us in this work. Resistive scattering by
optical phonons in 2D GaAs system has been considered in the
literature \cite{kawamura1992}. We note that the other scattering
mechanisms (e.g., interface roughness scattering and alloy disorder scattering in
Ga$_x$Al$_{1-x}$As, etc.) are known to be much less quantitatively
important \cite{ando1982} than the mechanisms we are considering (i.e., impurity scattering and acoustic
phonons) in this work.
The rest of the paper is organized as follows. In Sec. III we present
the general theory of the impurity and phonon scatterings, and discuss the power-law behavior
of the resistivity in the low and high temperature limit.
In Sec. IV we show our calculated resistivity taking into account hole-phonon and hole-impurity scatterings,
and demonstrate that there is a temperature-independent region due to the competition between the two scatterings.
Finally we conclude in Sec. V summarizing our results.
\section{Theory}
\label{sec:theory}
To investigate the temperature dependent resistivity $\rho(T)$ (or
equivalently conductivity $\sigma(T) \equiv \rho(T)^{-1}$) of p-GaAs
systems we start with the Drude-Boltzmann semiclassical formula for 2D
transport. Due to the finite extent in the $z$-direction of the real 2D
semiconductor system we have to include a form factor depending on the
details of the 2D structure.
In GaAs heterostructures, the carriers are spatially confined at the
2D interface and there is no longer translational
invariance along the direction normal to the interface, designated as
the $z$ direction. We assume that the confinement profile is described
by the variational wavefunction $\Psi(x,y,z)=\xi_0(z)e^{i (k_x x+k_y
y)}$ \cite{ando1982}, where
\begin{equation}
\label{eq:WF_z}
\xi_0(z)=\sqrt{{1\over 2} b^3 z^2} \exp\left(-{1\over 2} bz\right),
\end{equation}
and $b$ is a variational parameter. For a triangular well, $b$ is
given by \cite{fang1966}
\begin{equation}
b=\left({48\pi m^{\ast} e^2 \over \epsilon_0 \hbar^2}\right)^{1\over
3}\left(n_{\rm depl}+{11\over 32}n\right)^{1\over 3},
\end{equation}
where $m^{\ast}$ is an effective mass, $\epsilon_0$ is the dielectric constant,
$n_{\rm depl}$ is the depletion charge per unit area and $n$ is the 2D carrier density.
The density of states of 2DHS with
parabolic energy dispersion $\varepsilon({\bm k})={\hbar^2 k^2 \over 2
m^{\ast}}$ is given by
\begin{equation}
\label{eq:DOS}
D(\varepsilon)=
\begin{cases}
D_0 & \text{($\varepsilon > 0$)}, \\
0 & \text{($\varepsilon < 0$)},
\end{cases}
\end{equation}
where $D_0={g_{\rm s} m^{\ast} \over 2\pi \hbar^2}$ and $g_{\rm s}=2$ is the spin degeneracy factor.
The carrier density $n$ is given by
\begin{equation}
\label{eq:n}
n=D_0 \int_0^{\infty} d\varepsilon f(\varepsilon)=D_0 k_{\rm B} T
\ln\left[1+\exp\left({\mu(T) \over k_{\rm B} T}\right)\right],
\end{equation}
where $f(\varepsilon)=[e^{\beta (\varepsilon-\mu)}+1]^{-1}$ is the
Fermi distribution function and $\beta=1/(k_{\rm B} T)$.
Alternatively, the chemical potential $\mu(T)$ at a finite temperature
$T$ can be expressed as
\begin{equation}
\label{eq:mu}
\mu(T)=k_{\rm B} T \ln\left[\exp\left({E_{\rm F}/k_{\rm B} T}\right)-1\right],
\end{equation}
where $E_{\rm F}=n/D_0$. Note that $\lim_{T\rightarrow 0}
\mu(T)=E_{\rm F}$.
From Eqs.~(\ref{eq:n}) and (\ref{eq:mu}),
\begin{equation}
{\partial n \over \partial \mu}=D_0
\left[1-\exp\left(-{E_{\rm F}\over k_{\rm B}
T}\right)\right].
\end{equation}
The finite temperature Thomas-Fermi (TF) screening wavevector
$q_{\rm TF}(T)$ is defined by
\begin{equation}
\label{eq:TF_T}
q_{\rm TF}(T)={2\pi e^2 \over \epsilon_0} {\partial n \over \partial
\mu}= q_{\rm TF} \left[1-\exp\left(-{E_{\rm F}\over k_{\rm
B} T}\right)\right],
\end{equation}
where $q_{\rm TF}={2\pi e^2 \over \epsilon_0} D_0={g_{\rm s}
e^2 m^{\ast} \over \epsilon_0 \hbar^2}$.
Note that
\begin{equation}
\label{eq:q_TF_limit}
q_{\rm TF}(T) =
\begin{cases}
q_{\rm TF} & \text{($T\rightarrow 0$)}, \\
q_{\rm TF} \left({E_{\rm F} \over k_{\rm B} T}\right) &
\text{($T \rightarrow \infty$)}.
\end{cases}
\end{equation}
\subsection{Electron-phonon interactions}
The electron-longitudinal acoustic phonon interaction Hamiltonian for
a small phonon wavevector ${\bm Q}$ is
\begin{equation}
H_{\rm DP}({\bm Q})=D {\bm Q}\cdot{\delta{\bm R}},
\end{equation}
where $D$ is the acoustic phonon deformation potential and $\delta {\bm R}$ is the displacement vector.
Note that position operator in a simple harmonic oscillator with mass
$m$ and angular frequency $\omega$ is given by
\begin{equation}
x=\sqrt{\hbar \over 2 m \omega} (a + a^{\dagger}),
\end{equation}
where $a$ and $a^{\dagger}$ is the annihilation and creation operators, respectively.
Similarly, $\delta{\bm R}$ for $\bm Q$ can be expressed in terms of phonon
annihilation and creation operators as
\begin{equation}
\delta{\bm R}({\bm Q})= \sqrt{\hbar \over 2 \rho_{\rm m} \omega_{\bm Q}} \hat{e}_{\bm Q} \left( a_{\bm Q} + a_{-\bm Q}^{\dagger}
\right),
\label{eq:electron_phonon}
\end{equation}
where $\rho_{\rm m}$ is the mass density, $\omega_{\bm Q}=v_{\rm l}
Q$, $v_{\rm l}$ is the longitudinal sound velocity and $\hat{e}_{\bm Q}$ is the phonon polarization unit vector.
Thus the electron-phonon interaction by the deformation potential coupling
can be expressed as
\begin{equation}
H_{\rm ep}=D \sum_{\bm Q} \sqrt{\hbar \over 2 \rho_{\rm m} \omega_{\bm
Q}} {\bm Q} \left( a_{\bm Q} + a_{-\bm Q}^{\dagger} \right) \rho({\bm Q})
\end{equation}
where $\rho({\bm Q})$ is the electron density operator.
For the piezoelectric
scattering in polar semiconductors (i.e. GaAs) the scattering matrix
elements are equivalent to the substitution of the deformation with the
following form \cite{zook1964}
\begin{equation}
D^2 \rightarrow \frac{(eh_{14})^2 A}{Q^2}
\end{equation}
where $h_{14}$ is the basic piezoelectric tensor component and $A$ is
a dimensionless anisotropy factor that depends on the direction of
the phonon wave vector in the crystal lattice. We provide details of
the parameter $A$ in the following subsection.
\subsection{Boltzmann transport theory}
We calculate the temperature dependence of the hole resistivity by
considering screened acoustic-phonon scattering. We include both
deformation potential and piezo-electric coupling of the 2D holes to
3D acoustic phonons of GaAs. Details of the acoustic-phonon scattering
theory are given in Ref.~\cite{kawamura1992}.
The transport relaxation time $\tau(\varepsilon_{\bm k})$ at an energy $\varepsilon_{\bm k}$ and a 2D wavevector $\bm k$ is given by
\begin{equation}
\label{eq:tau_IE}
{1\over \tau(\varepsilon_{\bm k})}=\int {d^2 k' \over (2\pi)^2} W_{{\bm k},{\bm k}'} (1-\cos\phi_{{\bm k}{\bm k}'}) {1-f(\varepsilon_{{\bm k}'}) \over 1-f(\varepsilon_{\bm k})},
\end{equation}
where $W_{{\bm k},{\bm k}'}$ is the scattering probability between ${\bm k}$ and ${\bm k}'$ states, and $\phi_{{\bm k}{\bm k}'}$ is the scattering angle between ${\bm k}$ and ${\bm k}'$ vectors.
First, consider impurity scattering. Assume that impurity charges are
distributed randomly in a 2D plane located at $(-d_{\rm imp})\hat{\bm z}$ with a 2D impurity density $n_{\rm imp}$. Then the effective impurity potential for a 2D wavevector $\bm q$ is given by
\begin{eqnarray}
V_{\rm imp}({\bm q},d)&=&\int_0^{\infty}dz \xi_0^2(z) \left({2\pi e^2 \over \epsilon_0 q} e^{-q(d_{\rm imp}+z)}\right) \nonumber \\
&=&\left({2\pi e^2 \over \epsilon_0 q} e^{-q d_{\rm imp}}\right)\left({b\over b+q}\right)^3.
\end{eqnarray}
From the Fermi's golden rule, $W_{{\bm k},{\bm k}'}$ for the impurity scattering has the following form:
\begin{equation}
W_{{\bm k},{\bm k}'}^{\rm imp}={2\pi \over \hbar} {|V_{\rm imp}({\bm q},d)|^2 \over \epsilon^2({\bm q},T) } n_{\rm imp} \delta (\varepsilon_{\bm k}-\varepsilon_{\bm k}'),
\end{equation}
where ${\bm q}={\bm k}-{\bm k}'$.
The dielectric function $\epsilon({\bm q},T)$ takes into account the screening effects of electron gas at a wavevector ${\bm q}$ and a temperature $T$. We will consider screened scattering within RPA approximation defined by
\begin{equation}
\label{eq:screen_q}
\epsilon({\bm q},T)=1+{q_{\rm s}(q,T)\over q},
\end{equation}
where $q_{\rm s}(q,T)$ is the temperature- and wave vector-dependent
screening wavevector \cite{dassarma2011}.
In the long wavelength limit ($q=0$),
$q_{\rm s}(q,T)$ is given by Eq.~(\ref{eq:TF_T}).
For electron-phonon scattering, $W_{{\bm k},{\bm k}'}$ for the electron-longitudinal acoustic phonon interaction has the following form \cite{ziman}:
\begin{equation}
W_{{\bm k},{\bm k}'}^{\rm ph}={2\pi \over \hbar} \int {dq_z \over 2\pi} {|C({\bm q},q_z)|^2 \over \epsilon^2({\bm q},T) } \Delta (\varepsilon_{\bm k},\varepsilon_{\bm k}') \left|I(q_z)\right|^2,
\end{equation}
where $I_z(q_z)$ is the wavefunction overlap at $q_z$ defined by
\begin{equation}
I_z(q_z)=\int dz \xi_0^2(z) e^{iq_z z}.
\end{equation}
Note that from Eq.~(\ref{eq:WF_z}), $\left|I_z(q_z)\right|^2={b^6 \over (b^2+q_z^2)^2}$.
The factor $C({\bm q},q_z)$ is the matrix element for acoustic phonon scattering.
From Eq.~(\ref{eq:electron_phonon}), for the deformation potential (DP),
\begin{equation}
\label{eq:Cdp}
|C_{\rm DP}({\bm q},q_z)|^2={D^2 \hbar Q \over 2\rho_{\rm m} v_{\rm l}},
\end{equation}
while for the piezoelectric potential (PE) \cite{kawamura1992},
\begin{equation}
\label{eq:Cpe}
|C_{{\rm PE},\lambda}({\bm q},q_z)|^2={(eh_{14})^2 \hbar A_{\lambda}(q,q_z) \over 2\rho_{\rm m} v_{\lambda} Q},
\end{equation}
where ${\bm Q}=({\bm q},q_z)$, $v_{\rm l}$ ($v_{\rm t}$) is the longitudinal (transverse) sound velocity and
\begin{equation}
A_{\rm l}(q,q_z)={9 q_z^2 q^4 \over 2(q_z^2+q^2)^3}, \,
A_{\rm t}(q,q_z)={8 q_z^4 q^2+q^6 \over 4(q_z^2+q^2)^3}.
\end{equation}
The factor $\Delta(\varepsilon,\varepsilon')$ is given by
\begin{equation}
\label{eq:Delta}
\Delta(\varepsilon,\varepsilon')=N_q \delta(\varepsilon-\varepsilon'+\hbar\omega_q)+(N_q+1)\delta(\varepsilon-\varepsilon'-\hbar \omega_q),
\end{equation}
where $N_q=(e^{\beta \hbar \omega_q}-1)^{-1}$ is the phonon occupation number.
Note that the first and second terms in Eq.~(\ref{eq:Delta}) correspond to absorption and emission of phonons, respectively.
Finally, the total transport relaxation time is given by
\begin{equation}
\label{eq:relaxation_time}
{1 \over \tau_{\rm tot}(\varepsilon)}={1 \over \tau_{\rm imp}(\varepsilon)}+{1 \over \tau_{\rm DP}(\varepsilon)}+{1 \over \tau_{\rm PE,l}(\varepsilon)}+{2 \over \tau_{\rm PE,t}(\varepsilon)},
\end{equation}
where in the last term the degeneracy of the transverse modes has been taken into account.
Then electrical conductivity in a 2DHS is given by
\begin{equation}
\label{eq:boltzmann}
\sigma= g_{\rm s} e^2 \int {d^2 k \over (2\pi)^2} {v^2_{\bm k} \over 2} \tau_{\rm tot}(\varepsilon_{\bm k}) \left(-{\partial f \over \partial \varepsilon}\right)_{\varepsilon=\varepsilon_{\bm k}},
\end{equation}
where $v_{\bm k}$ is the mean-velocity at ${\bm k}$. By inverting the
conductivity we have the resistivity, i.e. $\rho(T) = \sigma^{-1}(T)$.
\subsection{Quasi-elastic limit}
For a degenerate system ($k_{\rm B} T \ll E_{\rm F}$), all the scattering events at an energy $\varepsilon$ take place in a thin shell around the energy circle and the scattering can be considered as quasi-elastic. For electron-phonon scatterings, the transport relaxation time in Eq.~(\ref{eq:tau_IE}) is given by
\begin{equation}
\label{eq:tau_QE}
{1\over \tau(\varepsilon)}={2\pi \over \hbar} D_0 \int {d\phi \over 2\pi} \int {dq_z \over 2\pi} {|C(q)|^2 \over \epsilon^2(q)} \left|I(q_z)\right|^2 G(q) (1-\cos\phi),
\end{equation}
where $q=2 k_{\rm F} \sin (\phi/2)$ and $G(q)$ is given by
\begin{eqnarray}
\label{eq:G}
G(q)&=&\beta \int d\varepsilon f(\varepsilon) \left[\left(1-f(\varepsilon+\hbar\omega_q)\right) N_q\right. \\
&+& \left.\left(1-f(\varepsilon-\hbar\omega_q)\right) (N_q+1)\right] \nonumber \\
&=&2\beta \hbar \omega_q N_q (N_q+1). \nonumber
\end{eqnarray}
Thus for DP, Eq.~(\ref{eq:tau_QE}) becomes \cite{kawamura1990,kawamura1992}
\begin{eqnarray}
\label{eq:tau_DP}
{1\over \tau_{\rm DP}(\varepsilon)}&=&{3 D^2 m^{\ast} b k_{\rm B} T\over 16 \hbar^3 \rho_{\rm m} v_l^2} \int_0^{\pi}{d\phi \over \pi} {(1-\cos\phi)\over \epsilon^2(q,T)} \nonumber \\
&\times&(\beta \hbar w_q)^2 N_q (N_q+1),
\end{eqnarray}
while for PE,
\begin{eqnarray}
\label{eq:tau_PE}
{1\over \tau_{{\rm PE},\lambda}(\varepsilon)}&=&{c_{\lambda} (e h_{14})^2 m^{\ast} k_{\rm B} T \over 2 \hbar^3 \rho_{\rm m} v_{\lambda}^2} \int_0^{\pi}{d\phi \over \pi} {(1-\cos\phi)\over q\epsilon^2(q,T)} \nonumber \\
&\times&(\beta\hbar w_q)^2 N_q (N_q+1) f_{\lambda}(q/b),
\end{eqnarray}
where $c_{\rm l}=9/32$, $c_{\rm t}=13/32$ and
\begin{eqnarray}
f_{\rm l}(w)&=&{1+6w+13w^2+2w^3 \over (1+w)^6}, \\
f_{\rm t}(w)&=&{13+78w+72w^2+82w^3+36w^4+6w^5 \over 13(1+w)^6}. \nonumber
\end{eqnarray}
For simplicity, consider $f_{\lambda}(q/b)=1$ case assuming $b\gg 1$ in the extreme 2D limit.
Then, in the high temperature limit,
\begin{eqnarray}
{1\over \tau_{\rm DP}(T)}&\approx& {3 D^2 m^{\ast} b k_{\rm B} T \over 16 \hbar^3 \rho_{\rm m} v_l^2} \propto T, \\
{1\over \tau_{\rm PE,\lambda}(T)}&\approx& {c_{\lambda} (e h_{14})^2 m^{\ast} \over 2 \hbar^3 \rho_{\rm m} v_{\lambda}^2} {4 \over \pi} {T \over T_{\rm BG} } \propto T, \nonumber
\end{eqnarray}
while in the low temperature limit,
\begin{eqnarray}
{1\over \tau_{\rm DP}(T)}&\approx& {3 D^2 m^{\ast} b k_{\rm B} T
\over 16 \hbar^3 \rho_{\rm m} v_l^2} {4\cdot 6! \zeta(6) \over \pi
x_{\rm TF}^2} \left(T\over T_{\rm BG}\right)^5 \propto T^6,
\nonumber \\
{1\over \tau_{\rm PE,\lambda}(T)}&\approx& {c_{\lambda} (e h_{14})^2 m^{\ast} \over 2 \hbar^3 \rho_{\rm m} v_{\lambda}^2} {4\cdot 5! \zeta(5) \over \pi x_{\rm TF}^2} \left(T\over T_{\rm BG}\right)^5 \propto T^5,
\end{eqnarray}
where $x_{\rm TF}=q_{\rm TF}/(2 k_{\rm F})$.
Note that the resistivity is proportional to the inverse relaxation time, thus it follows the same power-law dependence in the high or low temperature limit.
For the charged impurity with temperature-dependent screening wave vector
the asymptotic low- and high- temperature behaviors of 2D resistivity
for a $\delta$-layer system are given
by \cite{dassarma2004}
\begin{equation}
\rho_{\rm imp}(T\ll T_{\rm F}) \sim \rho_0 \left [ 1 +
\frac{2 x_{\rm TF}}{1+x_{\rm TF}}\frac{T}{T_{\rm F}} + C\left ( \frac{T}{T_{\rm F}}
\right )^{3/2} \right ],
\end{equation}
\begin{equation}
\rho_{\rm imp}(T \gg T_{\rm F} ) \sim \rho_1 \frac{T_{\rm F}}{T} \left
[ 1 - \frac{3 \sqrt{\pi} x_{\rm TF}}{4} \left (\frac{T_{\rm F}}{T} \right )^{3/2}
\right ],
\end{equation}
where $\rho_0=\rho(T=0)$, $\rho_1=(h/e^2)(n_{\rm imp}/n \pi x_{\rm TF}^2)$,
and $C= 2.646[x_{\rm TF}/(1+x_{\rm TF})]^2$.
At low temperatures ($T<T_{\rm F}$) the resistivity increases
linearly due to screening (or electron-electron interaction) effects
on the impurity scattering \cite{dassarma2005,dassarma1999}. To get
the linear temperature
dependent resistivity at low temperatures
it is crucial to include the temperature-dependent screening wave
vector \cite{dassarma2005}. At high temperatures ($T>T_{\rm F}$) $\rho(T)$
decreases inverse linearly due to nondegeneracy effects. Thus, it is
expected that
the resistivity has a maximum value and
the quantum-classical crossover occurs at the intermediate
temperature regime around $T_{\rm F}$.
When we consider both hole-phonon and hole-impurity scatterings
the temperature dependent resistivity becomes nontrivial due to the
competition between these independent two scattering mechanisms.
Since the resistivity limited by charged impurities decreases at high
temperatures phonon scattering eventually takes
over and $\rho(T)$ increases with $T$ again, which gives rise to
nonmonotonicity in $\rho(T)$. The nonmonotonicity becomes
pronounced in the systems with strong impurity scattering or at low
carrier density. For weaker impurity scattering the phonon
scattering dominates before the quantum-classical crossover occurs, so
the overall resistivity increases with temperature.
At higher carrier densities, $T_{\rm F}$ is pushed up to the phonon scattering regime,
and the quantum-classical cross-over physics is overshadowed by phonons
so that nonmonotonicity effects are not manifest.
\section{Numerical results}
\label{sec:numerical}
\subsection{Determination of deformation potential}
In this section we provide the numerically calculated temperature
dependence of the hole resistivity by
considering both screened acoustic-phonon scattering and screened
charged impurity scattering.
In the calculation of phonon scattering we use the parameters corresponding to GaAs:
$m^{\ast}=0.38$ $m_{\rm e}$, $v_{\rm l}=5.14\times 10^5$ cm/s,
$v_{\rm t}=3.04\times 10^5$ cm/s, $\rho_{\rm m}=5.3$ g/cm$^3$ and $e h_{14}=1.2\times 10^7$ eV/cm.
For the deformation potential, we fitted several available mobility data set
\cite{stormer1984,noh2003} and the best fitted value we obtained is
$D=12.7$ eV for $n_{\rm depl}= 0$ (see Fig.~\ref{fig:fitting_D}). In the following calculations
we use this value as a deformation potential of p-GaAs.
\begin{figure
\includegraphics[width=\linewidth]{fig2_fitting_D.eps}
\caption{(color online). Mobility as a function of temperature for several values of
$D$ with $n=2\times 10^{11}$ cm$^{-2}$, $n_{\rm imp}=1.22\times
10^{10}$ cm$^{-2}$, $n_{\rm depl}=0$ and $d_{\rm imp}=0$. Black dots
represent the experimental data \cite{noh2003}.}
\label{fig:fitting_D}
\end{figure}
\begin{figure
\includegraphics[width=\linewidth]{fig3_D_ndepl.eps}
\caption{Fitted deformation potential $D$ as a function of depletion density $n_{\rm depl}$ for a sample with $n=2\times 10^{11}$ cm$^{-2}$ and $d_{\rm imp}=0$.}
\label{fig:D_ndepl}
\end{figure}
To obtain the best fitted value of the deformation potential, in
Fig.~\ref{fig:fitting_D}, we calculate the total mobility $\mu =
\sigma/ne$ with Eqs.~(\ref{eq:relaxation_time}) and (\ref{eq:boltzmann}) as a
function temperature for different values of the deformation-potential
constant $D$. Knowing the precise value of the deformation-potential coupling
constant $D$ is very critical because $\mu \sim D^{-2}$, i.e.,
the calculated mobility will be uncertain by a factor of four
with values of $D$ differing by a factor of 2.
In this calculation we consider two different scattering
mechanisms: remote impurity scattering and acoustic phonon
scattering. We first fit the low temperature data ($T \alt 4$ K) to find the
charged impurity density, $n_{\rm imp}$, because the phonon scattering
is severely
suppressed and the charged impurity scattering determines the carrier
mobility in this temperature range.
We set $d_{\rm imp}=0$ for simplicity and carried the effect of impurity by $n_{\rm imp}$.
For $n_{\rm depl}=0$, we find that $n_{\rm imp}=1.22\times
10^{10}$ cm$^{-2}$ gives the best fitted mobility at low temperatures for the data set.
With this impurity density we calculate the mobility data at
high temperatures (20 K$<T<$60 K) by changing deformation potential.
From Fig.~\ref{fig:fitting_D}, we get $D=12.7$ eV as the most suitable value for the p-GaAs
acoustic phonon deformation coupling constant.
In Fig.~\ref{fig:D_ndepl} we show the deformation potential coupling
as a function of the depletion density for a hole density $n=2 \times
10^{11}$ cm$^{-2}$.
The calculated deformation potential $D$ strongly depends on the
depletion density $n_{\rm depl}$ for $n_{\rm depl} <10^{12}$ cm$^{-2}$,
but for higher densities ($n_{\rm depl} > 10^{12}$ cm$^{-2}$ it
decreases slowly, as seen in Fig.~\ref{fig:D_ndepl}.
$n_{\rm depl}$ is a measure of fixed charges in the background and
typically $n_{\rm depl}$ is unknown. Thus, the uncertainty in the
value of deformation potential coupling (both for electrons and for
holes) could be a result of our lack of knowledge about $n_{\rm
depl}$. When $n_{\rm depl}=0$ we get $D=12.7$ eV.
We use $D=12.7$ eV in the remaining calculations with $n_{\rm depl}=0$. However, the
different values of $D$ do not change the results qualitatively.
\begin{figure
\includegraphics[width=\linewidth]{fig4a_relaxation_rate_T10.eps}
\includegraphics[width=\linewidth]{fig4b_relaxation_rate_T1.eps}
\caption{(color online). The relaxation rate (solid line) and
scattering rate (dashed line) as a function of hole energy $E$ for
(a) $T=10$ K and (b) $T=1$ K with $n=10^{11}$ cm$^{-2}$, $n_{\rm
imp}=0$, $n_{\rm depl}=0$ and $D=12.7$ eV. DP and PE represent the
deformation potential and piezoelectric potential contributions,
respectively.}
\label{fig:relaxation_rate}
\end{figure}
\subsection{Acoustic phonon-limited transport}
Using the theoretical model outlined in Sec. III we study hole
transport limited by acoustic phonons in this subsection.
In Fig.~\ref{fig:relaxation_rate} we show the calculated
scattering rates $\tau_{\rm s}^{-1}$ and transport relaxation rates
$\tau_{\rm t}^{-1}$ due to acoustic phonons, as a function of the hole energy.
The relevant transport relaxation rate, $\tau_{\rm t}^{-1}$, has been obtained in Eq.~(\ref{eq:tau_QE}).
The two characteristic times shown in Fig.~\ref{fig:relaxation_rate}
differ by the important $(1-\cos\theta)$ factor
\cite{dassarma_1985}. The scattering rate
$\tau_{\rm s}^{-1}$ is given by making the replacement $(1-\cos\theta)
\rightarrow 1$ in the integrand for the formula for $\tau_{\rm t}^{-1}$
given in Eq.~(\ref{eq:tau_QE}).
$\tau_{\rm t}$ determines the conductivity (or mobility),
$\sigma=ne\mu =ne^2\tau_{\rm t}/m$, where $n$ is the carrier density and
$\mu$ is the mobility, whereas $\tau_{\rm s}$ determines the quantum level
broadening, $\gamma =\hbar/2\tau_{\rm s}$, of the momentum eigenstates.
The scattering time $\tau_{\rm s}$ is related to the imaginary part of the
single-particle self-energy and simply gives the time
between scattering events between a hole and an acoustic phonon.
The difference between $\tau_{\rm t}$ and $\tau_{\rm s}$ arises from the subtle effect
of the wave vector dependent transition rate \cite{dassarma_1985}.
The large angle scattering events (or large momentum transfer)
contribute significantly to the transport scattering events,
but small angle scattering events
where $\cos\theta \approx 1$ makes a negligible
contribution to $\tau_{\rm t}$, while all scattering events contribute
equally to $\tau_{\rm s}$.
Our result for the individual DP and total PE rates are given in
Fig.~\ref{fig:relaxation_rate} for $n=10^{11}$ cm$^{-2}$ at two different
temperatures $T=10$ K and $T=1$ K. In this calculation we take $D=12.7$ eV
which is the best fitted values of the experimental data.
We find that $\tau_{\rm t}/\tau_{\rm s} \approx 1$, since the screened
electron-acoustic mode phonon interactions are of relatively short range.
It has been known that the ratio $\tau_{\rm t}/\tau_{\rm s}$ from remote ionized
impurities is much bigger due to the long-range nature of the electron-impurity
interaction \cite{harrang_1985}.
\begin{figure
\includegraphics[width=\linewidth]{fig5a_rho_T_n.eps}
\includegraphics[width=\linewidth]{fig5b_drho_T_n.eps}
\caption{(color online). (a) Acoustic phonon-limited resistivity of 2DHS as a function
of temperature for $n=10^8$, $10^9$, $10^{10}$, $10^{11}$, $10^{12}$
cm$^{-2}$ and (b) the calculated exponent $a$ in $\rho(T)
\propto T^a$ which is obtained from logarithmic derivatives of (a).}
\label{fig:rho_T_n}
\end{figure}
Figure \ref{fig:rho_T_n}(a) shows acoustic phonon-limited resistivity of
2DHS in the absence of impurity scattering as a function of temperature for
different hole densities $n=10^8$, $10^9$, $10^{10}$, $10^{11}$,
$10^{12}$ cm$^{-2}$. The calculated resistivities clearly demonstrate
the two different regimes: BG region characterized by high power law
behavior for $T < T_{\rm BG}$ and equipartition region with $\rho \sim T$ behavior
for $T > T_{\rm BG}$. The transition temperature $T_{\rm BG}$ increases with
density since $T_{\rm BG}\propto \sqrt{n}$. As the density increases the calculated
resistivity at a fixed temperature decreases.
In Fig.~\ref{fig:rho_T_n}(b) we show the logarithmic derivatives of
the acoustic phonon limited resistivity, which give rise to an
approximate temperature exponent of acoustic phonon limited
resistivity by writing $\rho \sim T^a$, i.e., $a = d \ln \rho/ d \ln T$.
At low temperature BG regime $T < T_{\rm BG}$ the numerically evaluated
exponent $a$ varies from 4 to 6 depending on the carrier density.
But at high temperatures the exponent approaches to 1 as we
expected, i.e., $\rho(T)\propto T$.
\begin{figure
\includegraphics[width=\linewidth]{fig6a_mu_T_n.eps}
\includegraphics[width=\linewidth]{fig6b_alpha_n.eps}
\caption{(color online). (a) Acoustic phonon-limited mobility of 2DHS as a function of
temperature for $n=10^8$, $10^9$, $10^{10}$, $10^{11}$, $10^{12}$
cm$^{-2}$. (b) Density dependence of coefficient $\alpha$ for
$n_{\rm imp}=$0 and 5$\times 10^9$ cm$^{-2}$, where
$1/\mu=1/\mu_0+\alpha T$ in 10 K$<T<$60 K range.}
\label{fig:mu_T}
\end{figure}
Figure \ref{fig:mu_T}(a) shows acoustic phonon-limited mobility of
2DHS in the absence of impurity as a function of temperature for
different hole densities
$n=10^8$, $10^9$, $10^{10}$, $10^{11}$, $10^{12}$ cm$^{-2}$. At high
temperatures ($T>10$ K) the calculated mobilities show very weak
density dependence for the density range $n=10^{10}-10^{12}$ cm$^{-2}$ and
decrease approximately as $\mu \sim T^{-1}$. Thus,
the reciprocal mobility increases linearly with temperature,
i.e., $1/\mu=1/\mu_0+\alpha T$, where $\alpha$ is the slope in the
relation between $\mu^{-1}$ and $T$.
Fig.~\ref{fig:mu_T}(b) shows density dependence of the slope $\alpha$ for
$n_{\rm imp}=$0 and 5$\times 10^9$ cm$^{-2}$ in the temperature range
10 K$<T<$60 K. The slope $\alpha$ first increases with $n$, reaches its
maximum at $n\sim 10^{11}$ cm$^{-2}$, and decreases very slowly for $n \agt 10^{11}$ cm$^{-2}$.
This nonmonotonic behavior is different from
that of the n-type GaAs, in which the slope $\alpha$ has a minimum
value rather than a maximum \cite{kawamura1992,harris1990}.
\begin{figure
\includegraphics[width=\linewidth]{fig7a_rho_T_ni.eps}
\includegraphics[width=\linewidth]{fig7b_rho_T_ni_rescaled.eps}
\caption{(color online). (a) Resistivity of 2DHS as a function of temperature for
$n_{\rm imp}=$0, 1, 2, 3, 5, 7, 10$\times 10^9$ cm$^{-2}$
with $n=10^{10}$ cm$^{-2}$, $n_{\rm depl}=0$ and $d_{\rm imp}=0$. Dotted lines indicate the calculated
resistivity due to the charged impurity scattering alone. (b) Same as (a) but rescaled by
$\rho_0=\rho(T=0.1$ K).}
\label{fig:rho_T_ni}
\end{figure}
\begin{figure
\includegraphics[width=\linewidth]{fig8a_rho_T_di.eps}
\includegraphics[width=\linewidth]{fig8b_rho_T_di_rescaled.eps}
\caption{(color online). (a) Resistivity of 2DHS as a function of temperature for
$d_{\rm imp}=0, 5, 10, 15, 20, 25, 30$ nm with
$n=10^{10}$ cm$^{-2}$, $n_{\rm imp}=5\times 10^9$ cm$^{-2}$ and $n_{\rm
depl}=0$. Dotted lines indicate the calculated resistivity due
to the charged impurity scattering alone. (b) Same as (a) but
rescaled by
$\rho_0=\rho(T=0.1$ K).}
\label{fig:rho_T_di}
\end{figure}
\subsection{Nonmonotonic resistivity in p-GaAs}
In this subsection we study the transport in the presence of both
acoustic phonon and impurity scatterings and the nonmonotonic behavior
in temperature due to the competition between these two scatterings.
In Figs.~\ref{fig:rho_T_ni} and \ref{fig:rho_T_di} we show our
calculated total resistivity $\rho(T)$
arising from screened charged impurity scattering $\rho_{\rm imp}(T)$ and
phonon scattering $\rho_{\rm ph}(T)$ as a function of temperature.
In Fig.~\ref{fig:rho_T_ni}(a) the total resistivity $\rho(T)$ is shown for
different impurity densities $n_{\rm imp}=$0, 1, 2, 3, 5, 7, 10$\times 10^9$ cm$^{-2}$ with a fixed $d_{\rm imp}=0$.
In Fig.~\ref{fig:rho_T_di}(a) the total resistivity $\rho(T)$ is shown for
different impurity location from the interface
$d_{\rm imp}=$0, 5, 10, 15, 20, 25, 30 nm with a fixed impurity density
$n_{\rm imp}=5\times 10^9$ cm$^{-2}$.
Figures \ref{fig:rho_T_ni}(b) and \ref{fig:rho_T_di}(b) are the same as
Figs.~\ref{fig:rho_T_ni}(a) and \ref{fig:rho_T_di}(a), respectively, but
rescaled by $\rho_0=\rho(T=0.1$ K).
In a real system the amount of random disorder depends on
the strength and the spatial distribution of all the
random impurity scattering centers. However,
in these calculations we assume that the charged impurities
are randomly distributed in a 2D plane located at $d_{\rm imp}$ from the interface.
The calculation is carried out with the hole density
$n=10^{10}$ cm$^{-2}$ which corresponds to the Fermi
temperature $T_{\rm F}\approx 0.7$ K. The dotted lines in
Figs.~\ref{fig:rho_T_ni} and \ref{fig:rho_T_di} indicate the
calculated resistivity due to
the charged impurity scattering alone, $\rho_{\rm imp}(T)$. To calculate
the total resistivity $\rho(T)$ we use the total scattering rate of Eq.~(\ref{eq:relaxation_time}) because
the Matthiessen’s rule, which is implicitly assumed
$\rho(T)=\rho_{\rm imp}(T) + \rho_{\rm ph}(T)$, is known to be not strictly
valid at finite temperatures.
As shown in Fig.~\ref{fig:rho_T_ni}, when the charged impurity density $n_{\rm imp}$
increases the impurity scattering effects
become stronger, while the phonon scattering effects are
unaffected. Therefore at high impurity densities the impurity
scatterings are dominant over phonon scatterings.
The calculated $\rho(T)$ increases at lower temperatures ($T<1$ K)
due to screening effects, then the quantum-classical crossover occurs
at the intermediate temperature regime around $T \sim 1.5$ K where nondegeneracy
effects make resistivity decrease as $\rho \sim T^{-1}$.
At higher temperatures ($T \gg 10$ K) phonon
scattering takes over and $\rho(T)$ increases with $T$.
Thus, for large impurity densities
the temperature dependence of the calculated resistivity shows a
nontrivial nonmonotonic behavior, arising
from a competition among three mechanisms discussed above, i.e. screening which is
particularly important for $T<1$ K, nondegeneracy and the associated
quantum-classical crossover for $T \sim T_{\rm F}$, and
phonon scattering effect which becomes increasingly important for $T > 10$ K.
At lower impurity densities the
quantum-classical crossover effects are not particularly shown in
Fig.~\ref{fig:rho_T_ni}
because phonon scattering becomes more important than the classical
behavior $\rho \sim T^{-1}$, and the system makes a transition
from the quantum regime to the phonon scattering dominated regime. The
linear rise in $\rho(T)$ for $T>10$ K in Fig.~\ref{fig:rho_T_ni} is the
phonon scattering effect.
The same results shown in Fig.~\ref{fig:rho_T_ni} are expected by varying the
impurity location because the scattering limited by the remote impurity
becomes weaker as the distance of the impurity from the interface increases.
In Fig.~\ref{fig:rho_T_di} we show the several different kinds of
nonmonotonic behavior by varying the impurity location. When the
impurities are located very close to the interface (top lines in
Fig.~\ref{fig:rho_T_di}) the nonmonotonic
behavior of the resistivity clearly appears in the temperature range
we consider (i.e. $T<100$ K) due to competition among the three
mechanisms discussed above. As the separation increases the
nonmonotonicity becomes weaker because of
the reduction of the charged impurity scattering and the associated
weakening of screening effects. In addition, the increase of
the separation gives rise to
the shift of the local maximum peak to the lower temperature.
For large separations (bottom lines in Fig.~\ref{fig:rho_T_di})
the local maximum peak does not appear in the calculated resistivity
because it shifts to very low temperatures ($T<0.1$ K).
One interesting finding in our calculation is the temperature region
where the calculated resistivity is approximately constant, as indicated
by the dashed box in Figs.~\ref{fig:rho_T_ni} and \ref{fig:rho_T_di}.
The temperature range of the constant resistivity appears when the
increasing resistivity
due to phonon scatterings compensates for the decreasing resistivity
due to the nondegeneracy effects. The flat region depends critically on the
impurity density and the location of the impurities, and can be
observed in experiments by varying the doping density and location. In
Fig.~\ref{fig:rho_T_ni} a flat region spanning around 2 K $< T <$ 10 K
appears at an impurity density $n_{\rm imp} = 3\times 10^9$ cm$^{-2}$
for $d_{\rm imp}=0$. In
Fig.~\ref{fig:rho_T_di} the flat region for 2 K $< T <$ 10 K
appears at $d_{\rm imp}=5$ nm with an impurity density $n_{\rm imp} =
5\times 10^9$ cm$^{-2}$.
It is, therefore, possible in some situations for a complete
accidental cancellation between the increasing temperature dependence
of the phonon-induced resistivity and the decreasing temperature
dependence of the quantum classical crossover effect from impurity
scattering in a narrow intermediate temperature regime. We believe
that this has recently been observed experimentally \cite{zhou2012}, but a
detailed comparison with experiment is not possible due to the
complications of the parallel magnetic field used in the experimental
measurement to induce magneto-orbital coupling.
\section{Conclusion}
\label{sec:discussion}
To conclude,
we have calculated the temperature dependent transport
properties of p-type GaAs-based 2DHSs for temperatures
$T \alt 100$ K by taking into account both hole-phonon and
hole-impurity scatterings. Our theory includes
temperature-dependent screening of both charged impurity scattering and phonon
scattering. We extract the deformation potential $D$ of hole-phonon
coupling constant by fitting the experimentally available mobility data.
We find that the deformation potential coupling
varies (i.e. $D=7.6-12.7$ eV) depending on the
value of the depletion density $n_{\rm depl}$, which is not known.
When we assume $n_{\rm depl}=0$ we obtain $D=12.7$ eV for the p-GaAs
acoustic phonon deformation potential,
which is larger than the generally accepted value in bulk GaAs ($D=7$
eV) \cite{wolfe1970} but comparable to the value of the
n-GaAs ($12-14$ eV) \cite{price1985,mendez1984,kawamura1990} in 2D
electron systems.
We also investigate the nonmonotonicity of $\rho(T)$ arising
from the competition among three mechanisms: screening, nondegeneracy,
and phonon scattering. Both
screening and phonon scattering mechanisms give rise to
monotonically increasing $\rho(T)$ with $T$ (at low temperature for screening, and
at high temperatures for phonons), but nondegeneracy effects produce a
$\rho(T)$ decreasing with increasing $T$ for $T > T_{\rm F}$. Since
phonon scattering is the dominant temperature-dependent scattering
mechanism in GaAs holes for $T \agt 5-10$ K, depending on the density,
the stronger
nonmonotonicity appears when the impurity scattering is dominant
over phonon scattering below $T\sim 5-10$ K.
We carefully study the non-trivial transport properties of p-GaAs at the
intermediate temperature range (i.e., 2 K$<T<$10 K). Interestingly we
find that the approximate temperature independence may appear
in which $\rho(T)$ saturates in an intermediate temperature range,
arising from the approximate
cancellation between the quantum-classical crossover and phonon
scattering. Since this flat region of the temperature dependent
resistivity depends critically on the
impurity density and the location of the impurities, it can be
observed in experiments by varying the doping density and location.
We believe that a recent measurement \cite{zhou2012} has observed this
saturation effect.
\section*{acknowledgments}
The work is supported by the NRI-SWAN and US-ONR. We thank
Dr. H. Noh for sharing unpublished data with us.
|
{
"timestamp": "2012-03-12T01:00:08",
"yymm": "1203",
"arxiv_id": "1203.1929",
"language": "en",
"url": "https://arxiv.org/abs/1203.1929"
}
|
\section{Introduction}
The classical Julia-Wolff-Carath\'eodory theorem (see, {\sl e.g.} \cite{Ab-T, Co-Ma, RS, Shb}) is the most powerful tool for studying properties of bounded holomorphic functions of the unit disc $\mathbb D$ of $\mathbb C$ at a given boundary point. This theorem has been generalized to the unit ball $\mathbb B^n$ of $\mathbb C^n$ by W. Rudin (see \cite{Ru}) and to strongly (pseudo)convex domains and other domains in $\mathbb C^n$ by other authors, notably by M. Abate (see \cite{Abate}, \cite{Ab-T}. See also \cite{Abanew} for the most recent and complete survey on the subject).
In what follows we are mainly interested in the case of mappings fixing a boundary point. Since the group of automorphisms of $\mathbb B^n$ acts bi-transitively on $\partial \mathbb B^n$, without loss of generality we restrict our attention to the point $e_1=(1,0\ldots, 0)\in \partial \mathbb B^n$.
The maps we are working with are not assumed to be
continuous up to the boundary, thus we have to specify the meaning of
the term ``boundary fixed point''. In higher dimensions, in fact,
different approaches to boundary limits are possible. We recall
them here briefly (see \cite{Abate}, \cite{Ru} for more
information).
Let $R\geq 1$ and let $K(e_1,R):=\{z\in \mathbb B^n: |1-z_1|\leq
\frac{R}{2}(1-\|z\|^2)\}$ be a {\sl Kor\'anyi region of vertex $e_1$ and amplitude $R$} (see
\cite[Section 5.4.1]{Ru}, \cite{Co-Ma}). In \cite[Section 2.2.3]{Abate} a slightly different but essentially equivalent definition is given and used. In order not to excessively burden the notation, since we are only working at $e_1$, from now on, when we talk about Kor\'anyi regions, we will always mean Kor\'anyi regions of vertex $e_1$.
Let $f: \mathbb B^n \to \mathbb C^n$
be a holomorphic map. We say that $f$ has {\sl $K$-limit} $L$ at
$e_1$ -- and we write $K\hbox{-}\lim_{z\to e_1}f(z)=L$ -- if for
each sequence $\{z_k\}\subset \mathbb B^n$ converging to $e_1$ such that
$\{z_k\}$ belongs eventually to some Kor\'anyi region, it follows
that $f(z_k)\to L$. We say that $f$ has {\sl restricted $K$-limit}
$L$ at $e_1$ -- and we write $\angle_K\lim_{z\to e_1}f(z)=L$ -- if
for each sequence $\{z_k\}\subset \mathbb B^n$ converging to $e_1$ such
that $\|z_k-\langle z_k,e_1\rangle e_1\|^2/(1-|\langle z_k,e_1\rangle|^2)\to 0$
and $\langle z_k, e_1\rangle\to 1$ non-tangentially in $\mathbb D$ it follows
that $f(z_k)\to L$. Finally, we say that $f$ has {\sl
non-tangential limit} $L$ at $e_1$ and we write $\angle\lim_{z\to
e_1}f(z)=L$, if for each sequence $\{z_k\}\subset \mathbb B^n$ converging
non-tangentially to $e_1$ -- {\sl i.e.}, such that there exists
$C>0$ with $\|z_k-e_1\|\leq C (1-\|z_k\|^2)$ for all $k\geq 1$ --
it follows that $f(z_k)\to L$.
One can show that
\[
K\hbox{-}\lim_{z\to e_1}f(z)=L\Longrightarrow \angle_K\lim_{z\to
e_1}f(z)=L\Longrightarrow\angle\lim_{z\to e_1}f(z)=L,
\]
but the converse to any of these implications is not true in general.
A holomorphic self-map $f:\mathbb B^n\to \mathbb B^n$ has a {\sl boundary regular fixed point} at $e_1$ if $\angle \lim_{z\to e_1}f(z)=e_1$ and \[
\alpha_f(e_1):=\liminf_{z\to e_1}\frac{1-\|f(z)\|}{1-\|z\|}<+\infty.
\]
Now we can formulate the Julia-Wolff-Carath\'eodory Theorem for
$\mathbb B^n$ for boundary regular fixed points in the way we need in
this paper. As is customary, we denote by $\{e_1,\ldots, e_n\}$
the standard orthonormal basis in $\mathbb C^n$ (the symbol $e_1$ denotes
thus both the point and the direction).
\begin{theorem}[Rudin]\label{RudinJWC}
Let $f:\mathbb B^n\to \mathbb B^n$ be holomorphic. Suppose that $e_1$ is a boundary regular fixed point for $f$. Then
$K\hbox{-}\lim_{z\to e_1} f(z)=e_1$. Moreover,
\begin{itemize}
\item[(1$^{'}$)] $\langle df_z(e_1), e_1\rangle$ and $\langle df_z(e_h), e_k\rangle$ are bounded in any Kor\'anyi region for $h,k=2,\ldots, n$.
\item[(1$^{''}$)] $\langle df_z(e_j), e_1\rangle/(1-z_1)^{1/2}$ is bounded in any Kor\'anyi region for $j=2,\ldots, n$.
\item[(1$^{'''}$)] $(1-z_1)^{1/2}\langle df_z(e_1), e_j\rangle$ is bounded in any Kor\'anyi region for $j=2,\ldots, n$.
\item[(2)] $\angle_K\lim_{z\to e_1}\frac{ 1-\langle f(z),e_1\rangle}{1-z_1}=\alpha_f(e_1)$,
\item[(3)] $\angle_K\lim_{z\to e_1}\langle df_z(e_1), e_1\rangle=\alpha_f(e_1)$,
\item[(4)] $\angle_K\lim_{z\to e_1}\langle df_z(e_j), e_1\rangle=0$ for $j=2,\ldots, n$.
\item[(5)] $\angle_K\lim_{z\to e_1}\frac{\langle f(z), e_j\rangle}{(1-z_1)^{1/2}}=0$ for $j=2,\ldots, n$.
\item[(6)] $\angle_K\lim_{z\to e_1}(1-z_1)^{1/2}\langle df_z(e_1), e_j\rangle=0$ for $j=2,\ldots, n$.
\end{itemize}
\end{theorem}
One can interpret Julia-Wolff-Carath\'eodory's theorem as a description of the first jet of a holomorphic self-map of the unit ball at a boundary regular fixed point.
One of the aims of the present paper is to give a corresponding
theorem for infinitesimal generator (that is, {\sl $\mathbb R$-semicomplete holomorphic vector fields}) on
$\mathbb B^n$ having a ``regular singularity'' at $e_1$.
A holomorphic vector field $G:\mathbb B^n \to \mathbb C^n$ is said to be an {\sl infinitesimal generator} if the Cauchy problem
\begin{equation}\label{Cauchy}
\begin{cases}
\stackrel{\bullet}{x}(t)=G(x(t))\\
x(0)=z_0
\end{cases}
\end{equation}
has a solution $x_{z_0}:[0,+\infty)\ni t\mapsto x(t)$ for all $z_0\in \mathbb B^n$. If this is the case, the map $\phi:[0,+\infty)\times \mathbb B^n\mapsto \mathbb B^n$ given by $\phi_t(z):=x_z(t)$ is real analytic and $z\mapsto \phi_t(z)$ is a univalent holomorphic self-map of $\mathbb B^n$ for all fixed $t\in[0,+\infty)$. The family $(\phi_t)$ is a (continuous) semigroup, namely a continuous morphism of semigroups between $(\mathbb R^+,+)$ endowed with the Euclidean topology and $({\sf Hol}(\mathbb B^n,\mathbb B^n),\circ)$ endowed with the topology of uniform convergence on compacta.
Conversely, any (continuous) semigroup of holomorphic self-maps of
$\mathbb B^n$ is associated uniquely to an infinitesimal generator.
Interior fixed points of the semigroups correspond to
singularities of the vector field. At the boundary, the situation is
more complicated (see Sections \ref{due} and \ref{tre}). For the
time being, we say that $e_1$ is a {\sl boundary regular null
point} (or BRNP for short) if it is a boundary regular fixed point
for the associated semigroup of holomorphic self-maps and we say
that $\beta\in \mathbb R$ is the {\sl dilation} of $G$ at $e_1$ if the
flow $\phi_1$ of $G$ at the time $1$ has boundary dilation
coefficient $\alpha_{\phi_1}(e_1)=e^{\beta}$ (see Definition
\ref{defBRNP} for a definition of BRNP which does not involve the
associated semigroup).
Now, a version of the Julia-Wolff-Carath\'eodory Theorem for
infinitesimal generators which we are going to prove is the
following:
\begin{theorem}\label{JWC}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator. Suppose that
\begin{equation}\label{ipo}
\begin{split}
(\ast)\quad&\mathbb B^n\ni z\mapsto\frac{|\langle G(z),e_1\rangle|}{|z_1-1|} \quad \hbox{is bounded in any Kor\'anyi region and} \\ (\ast\ast)\quad& \mathbb B^n\ni z\mapsto\frac{|\langle G(z),e_j\rangle|}{|z_1-1|^{1/2}} \quad \hbox{is bounded in any Kor\'anyi region for $j=2,\ldots, n$.}
\end{split}
\end{equation}
Then $e_1$ is a boundary regular null point for $G$. Moreover, let $\beta\in\mathbb R$ denote the dilation of $G$ at $e_1$. Then
\begin{itemize}
\item[(1$^{'}$)] $\langle dG_z(e_1),e_1\rangle$ and $\langle dG_z(e_h),e_k\rangle$ are bounded in any Kor\'anyi region for $h,k=2,\ldots, n$,
\item[(1$^{''}$)] $\langle dG_z(e_j), e_1\rangle /(1-z_1)^{1/2}$ is bounded in any Kor\'anyi region for $j=2,\ldots, n$,
\item[(1$^{'''}$)] $(1-z_1)^{1/2}\langle dG_z(e_1), e_j\rangle$ is bounded in any Kor\'anyi region for $j=2,\ldots, n$,
\item[(2)] $\angle_K\lim_{z\to e_1}\frac{ \langle G(z),e_1\rangle}{z_1-1}=\beta$,
\item[(3)] $\angle_K\lim_{z\to e_1}\langle dG_z(e_1), e_1\rangle=\beta$,
\item[(4)] $\angle_K\lim_{z\to e_1}\langle dG_z(e_j), e_1\rangle=0$ for $j=2,\ldots, n$.
\end{itemize}
\end{theorem}
In the case where the infinitesimal generator extends smoothly
past $e_1$, Theorem \ref{JWC} is a consequence of Theorem
\ref{RudinJWC} applied to the associated semigroup. However, if no
regularity is assumed, this way of proceeding does not seem to be
possible. Our proof, in fact, does not involve the associated
semigroup, but it is based on the properties of infinitesimal
generators, and it is contained in Section \ref{sectJWC}. In
particular, we shall prove an intermediate version of the
Julia-Wolff-Carath\'eodory theorem assuming only
hypothesis \eqref{ipo}.$(\ast)$ (see Proposition \ref{JWC-1}). In Example \ref{esempio}, we give an example of an infinitesimal generator which satisfies \eqref{ipo}.$(\ast)$ but not \eqref{ipo}.$(\ast\ast)$ and for which some implications of Theorem \ref{JWC} do not hold. In Subsection \ref{discuto} we discuss the (dis)similarities between Theorem \ref{RudinJWC} and Theorem \ref{JWC} and some natural open questions raised up from this work.
Next, in Section \ref{sec-higher}, assuming a $C^3$ regularity at the BRNP $e_1$, we describe the jets space of infinitesimal generators, giving complementary results to the ones obtained in \cite{BZ} for local biholomorphisms of strongly (pseudo)convex domains. In particular, we are interested in finding (minimal, pointwise) necessary and sufficient conditions for an infinitesimal generator to generate a group of automorphisms of $\mathbb B^n$. In case of an interior singularity, the condition is rather simple: an infinitesimal generator $G$ with a singularity at $z_0\in \mathbb B^n$ generates a group of automorphisms of $\mathbb B^n$ if and only if the spectrum of $dG_{z_0}$ is contained in the imaginary axis $i\mathbb R$.
In the case where the singularity is at the boundary, we prove the
following result:
\begin{theorem}\label{rigidity}
Let $G$ be an infinitesimal generator on $\mathbb B^n$ of class $C^3$ at $e_1$. Assume that $e_1$ is a boundary regular null point with dilation $\beta\in \mathbb R$. Then $G$ generates a group of automorphisms if and only if the following conditions are satisfied:
\begin{enumerate}
\item ${\sf Re}\, \langle \frac{\partial G}{\partial z_k}(e_1),e_k\rangle =\frac{\beta}{2}$, for $k=2,\ldots, n$,
\item ${\sf Re}\, \langle \frac{\partial^2 G}{\partial z_1\partial z_k}(e_1),e_1\rangle =\beta$ for $k=1,\ldots, n$,
\item $\langle \frac{\partial^2 G}{\partial z_1\partial z_k}(e_1),e_h\rangle=0$ for $2\leq k<h\leq n$,
\item ${\sf Re}\, \langle \frac{\partial^3 G}{\partial z_1^3}(e_1),e_1\rangle=0$.
\end{enumerate}
Moreover, if the previous conditions are satisfied, then $G\equiv 0$ if and only if $\beta=0$, $\langle \frac{\partial G}{\partial z_k}(e_1),e_k\rangle =0$ for $k=2,\ldots, n$ and $\langle \frac{\partial^2 G}{\partial z_1^2}(e_1),e_1\rangle =0$.
\end{theorem}
The assumption on the $C^3$ regularity of $G$ at $e_1$ can be
lowered by assuming the existence of an expansion of $G$ in any
Kor\'anyi region with vertex $e_1$, but for the sake of clarity, we
will deal only with the $C^3$ case.
The previous result belongs to the family of so-called ``rigidity phenomena'', where some minimal conditions on the maps/infinitesimal generators of $\mathbb B^n$ at one point imply certain specific forms. For instance, the well known Burns-Krantz rigidity theorem \cite{BK} states that a holomorphic self-map of the unit ball which is the identity up to the third order at a boundary point, is the identity {\sl tout court}. Such a result has been extended later to infinitesimal generators (see \cite{ELRS}, and also \cite{Ca}), in the following way: an infinitesimal generator in $\mathbb B^n$ which is $0$ up to the third order at a boundary point of $\mathbb B^n$ is identically zero. In a sense, Theorem \ref{rigidity} is a quantitative version of such rigidity phenomena.
The main idea for the proof is to transfer the information on $G$
to a family of infinitesimal generators on $\mathbb D$ by means of a
method which we call ``slice reduction'' (see Section \ref{tre}),
first introduced in \cite{BCD} and implemented here.
Finally, in Section \ref{quadratic} we show with a couple of examples that, contrarily as one might expect, the slice reductions do not preserve the boundary expansion: while in the one dimensional case the quadratic expansion at a BRNP of an infinitesimal generator is always an infinitesimal generator which generates a semigroup of linear fractional maps, in higher dimension this is no longer the case. Moreover, even if the quadratic expansion is an infinitesimal generator, the generated semigroup might not be linear fractional.
\medskip
This work was carried out while both authors where visiting the
Mittag-Leffler Institute during the program ``Complex Analysis
and Integrable Systems'' in Fall 2011. Both authors thank the
organizers and the Institute for the kind hospitality and the
atmosphere experienced there.
\smallskip
The authors thank Mark Elin, Marina Levenshtein, and Jasmin Raissy for useful comments on a preliminary version of the manuscript.
They also warmly thank the referee for his/her precious comments and remarks which improved a lot the paper. In particular they are in debts for his/her suggestion to use weights in the statement of Theorem \ref{JWC} and for having found a mistake in the original statement of Theorem \ref{rigidity}.
\section{Infinitesimal generators on the unit ball and BRNP's}\label{due}
Infinitesimal generators have been characterized in several ways. In the unit disc $\mathbb D$, the following powerful characterization is due to Berkson-Porta formula \cite{BP}: a holomorphic vector field $g:\mathbb D\to \mathbb C$ is an infinitesimal generator if and only if there exist $\tau\in \overline{\mathbb D}$ and $p:\mathbb D\to \{z\in \mathbb C: {\sf Re}\, z\geq 0\}$ such that
\begin{equation}\label{Berkson-Porta}
g(\zeta)=(\tau-\zeta)(1-\overline{\tau}\zeta)p(\zeta).
\end{equation}
In the multi-dimensional case, several equivalent
characterizations are given both using Euclidean inequalities (see
\cite{RS}), the Kobayashi metric (see \cite{Abate}) and
pluripotential theory (see \cite{BCD}). In what follows, we need a
characterization only for boundary regular fixed points, which we
are going to define.
The function
\[
u_{\mathbb B^n}(z):=-\frac{1-\|z\|^2}{|1-z_1|^2}
\]
is the pluricomplex Poisson kernel with a pole at $e_1$ and its
sublevel sets $\{u_{\mathbb B^n}(z)<-1/R\}$ for $R>0$ are called {\sl
horospheres} with center $e_1$ and radius $R$.
Recall that a function $f:\mathbb B^n\to \mathbb C^m$ is $C^k$ at $e_1$ if $f$ and all its partial derivatives up to order $k$ extend continuously to $e_1$. As the horospheres are smooth ellipsoids, by Whitney's extension theorem, this is equivalent to saying that for each horosphere $E$ with center $e_1$ there exists a function $\tilde{f}$ (depending on $E$) of order $C^k$ defined in a open neighborhood of $\overline{E}$ such that $\tilde{f}|_{E}\equiv f$.
The following characterization of infinitesimal generators in terms of the function $u_{\mathbb B^n}$ has been proved in \cite[Theorem 3.11]{BCD}:
\begin{theorem}\label{char}
Let $G:\mathbb B^n\to \mathbb C^n$ be holomorphic and $C^1$ at $e_1$. If $d(u_{\mathbb B^n})_z\cdot G(z)\leq 0$ for all $z\in \mathbb B^n$ then $G$ is an infinitesimal generator.
\end{theorem}
Now we define BRNPs:
\begin{definition}\label{defBRNP}
Let $G$ be an infinitesimal generator in $\mathbb B^n$. The point $e_1$ is a {\sl boundary regular null point}, or BRNP for short, if there exists $b\in \mathbb R$ such that
\[
(du_{\mathbb B^n})_z\cdot G(z)+ b u_{\mathbb B^n}(z)\leq 0\quad \forall z\in \mathbb B^n.
\]
The number
\[
\beta:=-\inf_{z\in \mathbb B^n} \frac{(du_{\mathbb B^n})_z\cdot G(z)}{u_{\mathbb B^n}(z)}
\]
is called the {\sl dilation} of $G$ at $e_1$.
\end{definition}
According to \cite[Theorem 0.4]{BCD} (see also \cite{ERS}), if $G$
is an infinitesimal generator with the associated semigroup
$(\phi_t)$ then $e_1$ is a BRNP for $G$ with dilation $\beta$ if
and only if for all $t\geq 0$ it follows
\[
u_{\mathbb B^n}(\phi_t(z))\leq e^{-t\beta}u_{\mathbb B^n}(z)\quad \forall z\in \mathbb B^n.
\]
The number $e^{t\beta}$ is the so-called {\sl boundary dilation coefficient} of $\phi_t$ at $e_1$. The previous inequality means that a horosphere of center $e_1$ and radius $R>0$ is mapped into a horosphere with center $e_1$ and radius $e^{t\beta}R$.
\section{Slice reduction of infinitesimal generators and BRNPs}\label{tre}
Let
\[
\mathcal L_{e_1}:=\{v\in\mathbb C^n : \|v\|=1, \langle v, e_1\rangle =\alpha>0\}.
\]
Also, let
\begin{equation}\label{geodesic}
\varphi_v(\zeta):=\alpha(\zeta-1)v+e_1.
\end{equation}
It is easy to see that $\varphi_v:\mathbb D \to \mathbb B^n$ is holomorphic, and it is a complex geodesic, in the sense that it is an isometry
between the Poincar\'e distance in $\mathbb D$ and the Kobayashi distance in $\mathbb B^n$. Furthermore, it is well known (see, {\sl e.g.} \cite{Abate} and \cite[Section 1]{BPT}) that any complex geodesic $\eta:\mathbb D\to \mathbb B^n$ extends holomorphically through the boundary and moreover, if $e_1\in \eta(\partial \mathbb D)$, then there exists an automorphism $\theta$ of the unit disc such that $\eta\circ \theta$ is of the form \eqref{geodesic}.
\begin{remark}\label{plinio}
A direct computation shows that $u_{\mathbb B^n}(\varphi_v(\zeta))=\frac{1}{\alpha^2}u_{\mathbb D}(\zeta)$ for all $\zeta\in \mathbb D$.
\end{remark}
For a vector $w\in \mathbb C^n$ we use the notation $w=(w_1,w'')\in \mathbb C\times \mathbb C^{n-1}$.
The holomorphic map $\rho_v: \mathbb B^n \to \mathbb C^n$ defined by
\begin{equation}\label{retract}
\rho_v(z_1,z''):=\left(\frac{z_1+\frac{1}{\alpha}\langle z'', v''\rangle +\frac{1-\alpha^2}{\alpha^2}(1-z_1), -\frac{(1-z_1)}{\alpha} v''}{1+\frac{1}{\alpha}\langle z'', v''\rangle +\frac{1-\alpha^2}{\alpha^2}(1-z_1)}\right)
\end{equation}
has the property that $\rho_v(\mathbb B^n)=\varphi_v(\mathbb D)$ and, moreover, $\rho_v \circ \varphi_v(\zeta)=\varphi_v(\zeta)$ for all $\zeta\in \mathbb D$.
Finally, we let $\widetilde{\rho}_v:=\varphi_v^{-1}\circ \rho_v : \mathbb B^n\to \mathbb D$, {\sl i.e.}
\begin{equation}\label{projection}
\widetilde{\rho}_v(z_1,z'')=1+\frac{\frac{1}{\alpha^2}(z_1-1)}{1+\frac{1}{\alpha}\langle z'', v''\rangle + \frac{1-\alpha^2}{\alpha^2}(1-z_1)}=\frac{\alpha\langle z,v\rangle}{1-z_1+\alpha\langle z, v\rangle}.
\end{equation}
Note that for every $w=(w_1,w'')\in \mathbb C^n$ it follows that
\[
d(\widetilde{\rho}_v)_{\varphi_v(\zeta)}(w_1,w'')=\frac{1}{\alpha^2}w_1+\frac{1-\alpha^2}{\alpha^2}(\zeta-1)w_1-\frac{1}{\alpha}(\zeta-1)\langle w'',v''\rangle.
\]
\begin{definition}
Let $G:\mathbb B^n\to \mathbb C^n$ be a holomorphic vector field. Let
\[
g_v(\zeta):=d(\widetilde{\rho}_v)_{\varphi_v(\zeta)}(G(\varphi_v(\zeta))).
\]
We call the holomorphic vector field $g_v:\mathbb D\to \mathbb C$ the {\sl slice reduction of $G$ to $v$}.
\end{definition}
More explicitly
\begin{equation}\label{explicit}
\begin{split}
g_v(\zeta)&=\frac{1}{\alpha^2}G_1(\varphi_v(\zeta))+\frac{1-\alpha^2}{\alpha^2}(\zeta-1)G_1(\varphi_v(\zeta))-\frac{1}{\alpha}(\zeta-1)\langle G''(\varphi_v(\zeta)),v''\rangle\\
&=\frac{1}{\alpha^2}\zeta G_1(\varphi_v(\zeta))-\frac{\zeta-1}{\alpha}\langle G(\varphi_v(\zeta)),v\rangle.
\end{split}
\end{equation}
The following version of Julia's lemma for infinitesimal generators was proved in \cite[Theorem 0.4]{BCD} (see also \cite[Theorem p.403]{ERS} and, for the one-dimensional case, see \cite[Theorem 1]{CDP}):
\begin{theorem}\label{one-gv}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator. Then the following are equivalent:
\begin{enumerate}
\item $G$ has BRNP at $e_1$ and dilation $\beta_0\leq \beta\in \mathbb R$,
\item $d(u_{\mathbb B^n})_z\cdot G(z)+\beta u_{\mathbb B^n}(z)\leq 0$ for all $z\in \mathbb B^n$,
\item $\displaystyle{\frac{{\sf Re}\, \langle G(z),z\rangle}{1-\|z\|^2}- {\sf Re}\, \frac{\langle G(z),e_1\rangle }{1-z_1}}\leq\frac{\beta}{2}$, for all $z\in \mathbb B^n$,
\item for each $v\in \mathcal L_{e_1}$ the slice reduction $g_v$ is an infinitesimal generator of the unit disc with BRNP at $1$ and dilation $\leq \beta$.
\item there exists $C>0$ such that for all $v\in \mathcal L_{e_1}$ it follows
\[
\limsup_{(0,1)\ni r\to 1}\frac{|g_v(r)|}{1-r}\leq C.
\]
Moreover, if the previous condition is satisfied, then $1$ is a BRNP for $g_v$ and the following non-tangential limit exist
\[
\angle\lim_{\zeta\to 1}g_v'(\zeta)=\angle\lim_{\zeta\to 1}\frac{g_v(\zeta)}{\zeta-1}=\beta_v\in \mathbb R,
\]
with $\beta_v\leq \beta$ and
\[
\beta_0=\sup_{v\in \mathcal L_{e_1}}\beta_v.
\]
\end{enumerate}
\end{theorem}
A sufficient condition for the existence of BRNP, which we will
use in the sequel, is contained in the following (see \cite{ERS}):
\begin{theorem}\label{semign}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator. Assume that
\begin{enumerate}
\item $\lim_{(0,1)\ni r\to 1}G(re_1)=0$,
\item $\liminf_{r\to 1}{\sf Re}\, \frac{\langle G(re_1),e_1\rangle}{r-1}<+\infty$.
\end{enumerate}
Then
\[
\lim_{(0,1)\ni r\to 1}\frac{\langle G(re_1),e_1\rangle}{r-1}=\beta\in \mathbb R
\]
and $e_1$ is a BRNP for $G$ with dilation $\beta$.
\end{theorem}
Slice reductions of holomorphic vector fields preserve
pluricomplex Green and Poisson functions of strongly convex
domains, as shown in \cite{BCD} (see also \cite{Ca} for somewhat
explicit computations). As a consequence, a holomorphic vector
field is an infinitesimal generator if and only if all its slice
reductions (with respect to all points $\tau\in\partial \mathbb B^n$) are
infinitesimal generators in the unit disc. In what follows, we
need only a boundary version of this fact, which we prove here
explicitly for the unit ball. We start with the following:
\begin{lemma}\label{equal}
Let $G:\mathbb B^n\to \mathbb C^n$ be holomorphic. Then, for all $v=(\alpha,v'')\in \mathcal L_{e_1}$, and for all $\zeta\in \mathbb D$
\begin{enumerate}
\item $\displaystyle{\frac{{\sf Re}\, \langle G(\varphi_v(\zeta)),\varphi_v(\zeta)\rangle}{1-\|\varphi_v(\zeta)\|^2}- {\sf Re}\, \frac{\langle G(\varphi_v(\zeta)),e_1\rangle }{1-\langle\varphi_v(\zeta),e_1\rangle}=
\frac{{\sf Re}\, \langle g_v(\zeta),\zeta\rangle}{1-|\zeta|^2}- {\sf Re}\, \frac{g_v(\zeta) }{1-\zeta}}$
\item for all $\delta\in \mathbb R$ it follows
\[
d(u_{\mathbb D})_{\zeta}\cdot g_v(\zeta)+\delta u_\mathbb D(\zeta)=\alpha^2[d(u_{\mathbb B^n})_{\varphi_v(\zeta)}\cdot G(\varphi_v(\zeta))+\delta u_{\mathbb B^n}(\varphi_v(\zeta))].
\]
\end{enumerate}
\end{lemma}
\begin{proof}
(1) We have $1-\|\varphi_v(\zeta)\|^2=\alpha^2(1-|\zeta|^2)$ and $1-\langle\varphi_v(\zeta),e_1\rangle=\alpha^2(1-\zeta)$. Write $G_1$ for $G_1(\varphi_v(\zeta))$ and $G_2$ for $\langle G(\varphi_v(\zeta)),v\rangle$. Then, taking into account that for all $a\in \mathbb C$ it holds ${\sf Re}\, (a\zeta)+{\sf Re}\, (a\overline{\zeta})=2{\sf Re}\,\zeta {\sf Re}\, a$, and expanding (1), we have
\begin{equation*}
\begin{split}
&\frac{1}{\alpha}\frac{{\sf Re}\, (G_2\overline{\zeta})}{1-|\zeta|^2}-\frac{1}{\alpha}\frac{{\sf Re}\, G_2}{1-|\zeta|^2}+\frac{1}{\alpha^2}\frac{{\sf Re}\, G_1}{1-|\zeta|^2}-\frac{1}{\alpha^2}\frac{{\sf Re}\, G_1}{|1-\zeta|^2}+\frac{1}{\alpha^2}\frac{{\sf Re}\, (G_1\overline{\zeta})}{|1-\zeta|^2}- \frac{|\zeta|^2}{\alpha^2}\frac{{\sf Re}\, G_1}{1-|\zeta|^2}\\+&\frac{|\zeta|^2}{\alpha}\frac{{\sf Re}\, G_2}{1-|\zeta|^2}-\frac{1}{\alpha}\frac{{\sf Re}\, (G_2\overline{\zeta})}{1-|\zeta|^2}+\frac{1}{\alpha^2}\frac{{\sf Re}\,(G_1\zeta)}{|1-\zeta|^2}-\frac{|\zeta|^2}{\alpha^2}\frac{{\sf Re}\, G_1}{|1-\zeta|^2}-\frac{1}{\alpha}\frac{{\sf Re}\,(G_2\zeta)}{|1-\zeta|^2}+\frac{|\zeta|^2}{\alpha}\frac{{\sf Re}\, G_2}{|1-\zeta|^2}\\+&\frac{1}{\alpha}\frac{{\sf Re}\, G_2}{|1-\zeta|^2}-\frac{1}{\alpha}\frac{{\sf Re}\, (G_2\overline{\zeta})}{|1-\zeta|^2}
= \left( \frac{1-|\zeta|^2}{\alpha^2(1-|\zeta|^2)}+\frac{-1+2{\sf Re}\,\zeta-|\zeta|^2}{\alpha^2|1-\zeta|^2} \right) {\sf Re}\, G_1\\+&\left(\frac{|\zeta|^2-1}{\alpha(1-|\zeta|^2)}+\frac{|\zeta|^2-2{\sf Re}\,\zeta+1}{\alpha|1-\zeta|^2}\right){\sf Re}\, G_2=0,
\end{split}
\end{equation*}
as we wanted.
(2) A direct computation shows that
\[
d(u_{\mathbb B^n})_z\cdot G(z)=-2 {\sf Re}\, \left(\frac{\langle G(z),e_1\rangle}{1-z_1}\right) \frac{1-\|z\|^2}{|1-z_1|^2}+2{\sf Re}\, \frac{\langle G(z),z\rangle}{|1-z_1|^2}.
\]
Hence, the result follows from (1) taking into account Remark
\ref{plinio}. Also, see \cite[Eq. (4.7) p.45]{BCD}, where such a
formula has been proved for strongly convex domains.
\end{proof}
Also, we need the following lemma which will be useful to move from BRNP with dilation $>0$ to BRNP with dilation $\leq 0$:
\begin{lemma}\label{H-hyp}
Let $\beta\in \mathbb R$. Define $H_\beta:\mathbb B^n\to \mathbb C^n$ by
\begin{equation}\label{hhh}
H_\beta(z)=\frac{\beta}{2} (e_1-z_1z).
\end{equation}
Then $H_\beta$ generates a group of (hyperbolic) automorphisms of $\mathbb B^n$, with BRNP at $e_1$ with dilation $-\beta$ and
\begin{equation}\label{oroH}
d(u_{\mathbb B^n})_z\cdot H_\beta(z)-\beta u_{\mathbb B^n}(z)\equiv 0\quad \forall z\in \mathbb B^n.
\end{equation}
Moreover, for all $v\in \mathcal L_{e_1}$ the slice reduction is
\[
h_v(\zeta)=\frac{\beta}{2}(1-\zeta^2)=-\beta(\zeta-1)-\frac{\beta}{2}(\zeta-1)^2.
\]
\end{lemma}
\begin{proof}
It is well known that $H_\beta$ is a generator of a group of (hyperbolic) automorphisms (see, {\sl e.g.} \cite{BCDl}) with BRNP at $e_1$ and dilation $-\beta$. Hence $-H_\beta$ is a generator of a group of automorphisms having BRNP at $e_1$ with dilation $\beta$. Applying Theorem \ref{one-gv}.(2) at both $H_\beta$ and $-H_\beta$ we get \eqref{oroH}.
The form of the slice reductions is a direct computation from the very definition.
\end{proof}
In the paper we will use several times the following trick, whose proof is immediate from Theorem \ref{one-gv} and Lemma \ref{H-hyp}, which we state here for the reader convenience:
\begin{corollary}\label{trick}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator and assume $e_1$ is a BRNP for $G$, with dilation $\delta$. Let $\beta\in \mathbb R$ and let $H_\beta$ be given by \eqref{hhh}. Then $G+H_\beta$ is an infinitesimal generator in $\mathbb B^n$ with $e_1$ as BRNP and dilation $\delta-\beta$.
\end{corollary}
Now we can prove a boundary characterization of infinitesimal generators at BRNP:
\begin{proposition}\label{cbDW}
Let $G:\mathbb B^n\to \mathbb C^n$ be holomorphic and $C^1$ at $e_1$. Then the following are equivalent:
\begin{enumerate}
\item $G$ is an infinitesimal generator with BRNP at $e_1$ and dilation $\leq\beta \in \mathbb R$,
\item for each $v\in \mathcal L_{e_1}$ the slice reduction $g_v$ is an infinitesimal generator of the unit disc with BRNP at $1$ and dilation $\leq \beta$.
\item $d(u_{\mathbb B^n})_z\cdot G(z)+\beta u_{\mathbb B^n}(z)\leq 0$ for all $z\in \mathbb B^n$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) implies (2) and (3) by Theorem \ref{one-gv}.
If either (2) or (3) holds, the only aim is to show that $G$ is an
infinitesimal generator, because then (1) follows from Theorem
\ref{one-gv}.
Let $F:=G+H_\beta$, where $H_\beta$ is given by \eqref{hhh}.
Assume (2) holds. By Theorem \ref{one-gv} it follows that $d(u_\mathbb D)_\zeta\cdot g_v(\zeta)+\beta u_\mathbb D(\zeta)\leq 0$ for all $\zeta\in\mathbb D$ and $v\in \mathcal L_{e_1}$. Hence, by Lemma \ref{equal} and \eqref{oroH} it is easy to see that $d(u_{\mathbb B^n})_z\cdot F(z)\leq 0$ for all $z\in \mathbb B^n$. The same conclusion is obtained directly if (3) holds. By Theorem \ref{char} it follows that $F$ is an infinitesimal generator, and so does $G=F-H_\beta$, because infinitesimal generators in the ball form a cone (see \cite[Corollary 2.5.29]{Abate}).
\end{proof}
Finally, we have the following characterization of generators of groups which we will use later.
\begin{proposition}\label{gruppo}
Let $G:\mathbb B^n\to \mathbb C^n$ be holomorphic and $C^1$ at $e_1$. Let $\beta\in \mathbb R$. The following are equivalent:
\begin{enumerate}
\item $G$ generates a group of automorphisms of $\mathbb B^n$ with BRNP $e_1$ and dilation $\beta$,
\item $d(u_{\mathbb B^n})_z\cdot G(z)+\beta u_{\mathbb B^n}(z)\equiv 0$,
\item for each $v\in \mathcal L_{e_1}$ the infinitesimal generator $g_v$ generates a group of automorphisms of $\mathbb D$ with BRNP $1$ and dilation $\beta$,
\item for each $v\in \mathcal L_{e_1}$ it holds $d(u_{\mathbb D})_\zeta\cdot g_v(\zeta)+\beta u_{\mathbb D}(\zeta)\equiv 0$.
\end{enumerate}
Moreover, $G\equiv 0$ (hence the group it generates is the trivial group of automorphisms $\phi_t(z)\equiv z$ for all $t\geq 0$) if and only if for each $v\in \mathcal L_{e_1}$ it follows $g_v\equiv 0$. If this is the case then $\beta=0$.
\end{proposition}
\begin{proof} (2) is equivalent to (4) by Lemma \ref{equal}.
If (1) holds then $-G$ is an infinitesimal generator on $\mathbb B^n$ with BRNP at $e_1$ and dilation $-\beta$. Hence Theorem \ref{one-gv} applied to $G$ and $-G$ implies (2).
If (2) holds, then (1) follows from Proposition \ref{cbDW} applied to $G$ and $-G$.
Similarly, (3) is equivalent to (4).
Finally, by \eqref{explicit} it is easy to see that $G\equiv 0$ if and only if $g_v\equiv 0$ for all $v\in \mathcal L_{e_1}$.
\end{proof}
\section{The Julia-Wolff-Carath\'eodory Theorem for infinitesimal generators}\label{sectJWC}
As a matter of notation, we write $\angle \lim $ for non-tangential limits, $\angle_K\lim$ for restricted $K$-limits and $K-\lim$ for $K$-limits.
\begin{proposition}\label{JWC-1}
Let $G$ be an infinitesimal generator on $\mathbb B^n$. Suppose $\lim_{(0,1)\ni r\to 1}G(re_1)=0$ and
\begin{equation}\label{ipo1}
\mathbb B^n\ni z\mapsto\frac{|\langle G(z),e_1\rangle|}{|z_1-1|}\quad \hbox{is bounded in any Kor\'anyi region.}
\end{equation}
Then $e_1$ is a BRNP for $G$. Moreover, if $\beta\in \mathbb R$ is the dilation of $G$ at $e_1$, then
\begin{itemize}
\item[(1$^{'}$)] $\mathbb B^n\ni z\mapsto \langle dG_z(e_1),e_1\rangle$ is bounded in any Kor\'anyi region,
\item[(1$^{''}$)] $\mathbb B^n\ni z\mapsto \frac{\langle dG_z(e_j),e_1\rangle}{|z_1-1|^{1/2}}$ is bounded in any Kor\'anyi region for $j=2,\ldots, n$,
\item[(2)] $\angle_K\lim_{z\to e_1}\frac{\langle G(z),e_1\rangle}{z_1-1}=\beta$,
\item[(3)] $\angle_K\lim_{z\to e_1}\langle dG_z(e_1), e_1\rangle=\beta$.
\end{itemize}
\end{proposition}
\begin{proof}
By hypotheses of the theorem clearly guarantee that the hypotheses of Theorem \ref{semign} are satisfied so that $e_1$ is a BRNP for $G$.
(1$^{'}$) The proof is based on an
application of the Cauchy formula and it is similar to the one
given by Rudin for the case of holomorphic self-maps of the unit
ball (see \cite[p.180]{Ru}). For the sake of completeness, we
sketch it here.
Let $R\geq 1$ and let $K(e_1,R)=\{z\in \mathbb B^n: |1-z_1|\leq \frac{R}{2}(1-\|z\|^2)\}$ be a Kor\'anyi region. Let $R'>R$ and $\delta:=\frac{1}{3}(\frac{1}{R}-\frac{1}{R'})$. By \cite[Lemma 8.5.5]{Ru} if $z\in K(e_1,R)$ and $\lambda\in \mathbb C$ is such that $|\lambda|\leq \delta|1-z_1|$ and $u''\in \mathbb C^{n-1}$ is such that $\|u''\|\leq \delta |1-z_1|^{1/2}$ then $(z_1+\lambda, z''+u'')\in K(e_1, R')$.
Now, fix $z\in K(e_1,R)$ and let $r=r(z):=\delta |1-z_1|$. By the Cauchy formula
\begin{equation*}
\begin{split}
\langle dG_z(e_1), e_1\rangle &=\frac{1}{2\pi i}\int_{|\zeta|=r} \frac{\langle G(z_1+\zeta, z''), e_1\rangle}{\zeta^2}d\zeta \\&=\frac{1}{2\pi}\int_0^{2\pi} \frac{\langle G(z_1+re^{i\theta}, z''), e_1\rangle}{z_1+re^{i\theta}-1}\left(1-\frac{1-z_1}{re^{i\theta}} \right)d\theta.
\end{split}
\end{equation*}
Now, by the choice of $r$, the points $(z_1+re^{i\theta},z'')\in
K(e_1, R')$, hence by \eqref{ipo1} there exists a constant $C>0$
(which depends only on $R, R'$) such that
\[
\frac{|\langle
G(z_1+re^{i\theta}, z''), e_1\rangle|}{|z_1+re^{i\theta}-1|}\leq C.
\]
Also, $|1-\frac{1-z_1}{re^{i\theta}}|\leq 1+1/\delta$. Hence, the
function $K(e_1,R)\ni z\mapsto \langle dG_z(e_1), e_1\rangle$ is bounded.
(1$^{''}$) We argue as before, but, fixed $z\in K(e_1,R)$, we take $r=r(z):=\delta |1-z_1|^{1/2}$. Hence, for $j=2,\ldots, n$, we have
\begin{equation*}
\begin{split}
\frac{\langle dG_z(e_j), e_1\rangle}{|1-z_1|^{1/2}} &=\frac{1}{2\pi i|1-z_1|^{1/2}}\int_{|\zeta|=r} \frac{\langle G(z+\zeta e_j), e_1\rangle}{\zeta^2}d\zeta \\&=\frac{1}{2\pi\delta}\int_0^{2\pi} \frac{\langle G(z+re^{i\theta} e_j), e_1\rangle}{|1-z_1|}e^{-i\theta} d\theta.
\end{split}
\end{equation*}
By the choice of $r$, the points $z+re_j\in
K(e_1, R')$, $j=2,\ldots, n$, and we can conclude as before.
(2) Let us consider the slice reduction $g_{e_1}(\zeta)=\langle
G(\zeta e_1), e_1\rangle$. By Theorem \ref{semign} it follows that
$\lim_{(0,1)\ni r\to 1}g_{e_1}(r)/(r-1)=\beta$. Since the function $\mathbb B^n
\ni z\mapsto \langle G(z), e_1\rangle /(z_1-1)$ is bounded in any Kor\'anyi
region by \eqref{ipo1}, $\check{\hbox{C}}$irca's theorem
\cite[Theorem 8.4.8]{Ru} implies (2).
(3) By Theorem \ref{one-gv}, we have $\lim_{(0,1)\ni r\to 1}g'_{e_1}(r)=\beta$, that is, $\lim_{(0,1)\ni r\to 1}\langle dG_{re_1}(e_1),e_1\rangle =\beta$. By (1) the map $\mathbb B^n\ni z\mapsto \langle dG_z(e_1),e_1\rangle$ is bounded in any Kor\'anyi region, and once again (3) follows by $\check{\hbox{C}}$irca's theorem \cite[Theorem 8.4.8]{Ru}.
\end{proof}
Assuming slightly more regularity at $e_1$ we can prove the following intermediate result:
\begin{proposition}\label{beta}
Let $G$ be an infinitesimal generator on $\mathbb B^n$. Suppose $\angle\lim_{z\to e_1}G(z)=0$ and
\begin{equation}
\mathbb B^n\ni z\mapsto\frac{|\langle G(z),e_1\rangle|}{|z_1-1|}\quad \hbox{is bounded in any Kor\'anyi region.}
\end{equation}
Then $e_1$ is a BRNP for $G$ and $1$ is a BRNP for $g_v$ for all $v\in \mathcal L_{e_1}$. Moreover, if $\beta\in \mathbb R$ denotes the dilation of $G$ at $e_1$ and $\beta_v$ denotes the dilation of $g_v$ at $1$, then for all $v\in \mathcal L_{e_1}$ it follows $\beta_v=\beta$.
\end{proposition}
\begin{proof}
Let $v\in \mathcal L_{e_1}$. Let $g_v$ be the slice reduction to $v$ of $G$. Write $G=(G_1,G'')$. Taking into account that for all $v\in \mathcal L_{e_1}$ the curve $(0,1)\ni r\mapsto \varphi_v(r)$ tends to $e_1$ non-tangentially it follows that $\lim_{(0,1)\ni r\to1}G(\varphi_v(r))\to 0$. By Theorem \ref{one-gv} and Proposition~\ref{JWC-1}
\begin{equation*}
\begin{split}
\beta_v&=\lim_{(0,1)\ni r\to 1}\frac{g_v(r)}{r-1}\\&=\lim_{(0,1)\ni r\to 1}\frac{\frac{1}{\alpha^2}G_1(\varphi_v(r))+\frac{1-\alpha^2}{\alpha^2}(r-1)G_1(\varphi_v(r))-\frac{1}{\alpha}(r-1)\langle G''(\varphi_v(r)),v''\rangle}{r-1}
\\&=\frac{1}{\alpha^2}\lim_{(0,1)\ni r\to 1}\frac{G_1(\varphi_v(r))}{r-1}
=\frac{1}{\alpha^2}\lim_{(0,1)\ni r\to 1}\frac{G_1(\varphi_v(r))}{\langle \varphi_v(r),e_1\rangle -1}\frac{\langle \varphi_v(r),e_1 \rangle-1}{r-1}\\&=\frac{1}{\alpha^2}\beta \alpha^2=\beta,
\end{split}
\end{equation*}
and we are done.
\end{proof}
\begin{proof}[Proof of Theorem \ref{JWC}] The hypothesis \eqref{ipo} implies that $\angle \lim_{z\to e_1}G(z)=0$ and \eqref{ipo1}. Thus, Proposition \ref{JWC-1} applies and (1$^{''}$), (2) and (3) follow.
(1$^{'}$) The boundness of $\langle dG_z(e_1),e_1\rangle$ in any Kor\'anyi region follows again from Proposition \ref{JWC-1}. The proof that $\mathbb B^n\ni z\mapsto \langle dG_z(e_h),e_k\rangle$ is bounded in any Kor\'anyi region for $h,k=2,\ldots, n$ is similar to the proof of (1$^{''}$) in Proposition \ref{JWC-1}. Thus, we just sketch it here. Let $R,R',\delta$ as in the proof of Proposition \ref{JWC-1}. Fix $z\in K(e_1,R)$ and let $r=r(z):=\delta |1-z_1|^{1/2}$. Then for $h,k=2,\ldots, n$,
\begin{equation*}
\begin{split}
\langle dG_z(e_h), e_k\rangle &=\frac{1}{2\pi i}\int_{|\zeta|=r} \frac{\langle G(z+\zeta e_h), e_k\rangle}{\zeta^2}d\zeta =\frac{1}{2\pi}\int_0^{2\pi} \frac{\langle G(z+re^{i\theta} e_h), e_k\rangle}{r}e^{-i\theta} d\theta\\&=
\frac{1}{2\pi\delta}\int_0^{2\pi} \frac{\langle G(z+re^{i\theta} e_h), e_k\rangle}{|1-z_1|^{1/2}}e^{-i\theta} d\theta.
\end{split}
\end{equation*}
By the choice of $r$, the points $z+re_h\in K(e_1, R')$, $h=2,\ldots, n$. Hence \eqref{ipo}.$(\ast\ast)$ guarantees that $z\mapsto \langle dG_z(e_h), e_k\rangle$ is bounded in $K(e_1,R)$.
(1$^{'''}$) We retain the notations introduced in the proof of Proposition \ref{JWC-1}. Fix $z\in K(e_1,R)$ and let $r=r(z):=\delta |1-z_1|$. Then, for $j=2,\ldots, n$
\begin{equation}\label{boh}
\begin{split}
|1-z_1|^{1/2}\langle dG_z(e_1), e_j\rangle &=\frac{|1-z_1|^{1/2}}{2\pi i}\int_{|\zeta|=r} \frac{\langle G(z_1+\zeta,z''), e_j\rangle}{\zeta^2}d\zeta \\&=\frac{1}{2\pi\delta}\int_0^{2\pi} \frac{\langle G(z_1+re^{i\theta},z''), e_j\rangle}{|1-(z_1+re^{i\theta})|^{1/2}}e^{-i\theta} \left|\frac{1-(z_1+re^{i\theta})}{1-z_1}\right|^{1/2} d\theta.
\end{split}
\end{equation}
Again, by the choice of $r$, the points $(z_1+re^{i\theta},z'')\in K(e_1, R')$. Since $|1-(z_1+re^{i\theta})|/|1-z_1|\leq 1+\delta$, hypothesis \eqref{ipo}.$(\ast\ast)$ guarantees that $z\mapsto |1-z_1|^{1/2}\langle dG_z(e_1), e_j\rangle$ is bounded in $K(e_1,R)$.
(4) Let $v\in \mathcal L_{e_1}$. Let $g_v$ be the slice reduction to $v$ of $G$. Since $\varphi_v'(\zeta)=\alpha v$,
\begin{equation*}
\begin{split}
g'_v(\zeta)&=\frac{1}{\alpha}\langle dG_{\varphi_v(\zeta)}(v),e_1\rangle +(\zeta-1)\frac{1-\alpha^2}{\alpha}\langle dG_{\varphi_v(\zeta)}(v),e_1\rangle
-(\zeta-1)\langle dG''_{\varphi_v(\zeta)}(v),v''\rangle\\&+\frac{1-\alpha^2}{\alpha^2}\langle G(\varphi_v(\zeta)), e_1\rangle -\frac{1}{\alpha}\langle G''(\varphi_v(\zeta)),v''\rangle. \\
\end{split}
\end{equation*}
By Proposition \ref{beta} and Theorem \ref{one-gv} we have $\lim_{(0,1)\ni r \to 1}g'_v(r)=\beta$.
Taking into account that $(0,1)\ni r\mapsto \varphi_v(r)$ tends to $e_1$ non-tangentially, we have $\lim_{r\to 1} G(\varphi_v(r))=0$ and $r \mapsto \langle dG_{\varphi_v(r)}(v),e_1\rangle$ is bounded by (1$^{'}$) and (1$^{''}$). Moreover, by (1$^{'}$) and (1$^{'''}$) it follows that for $v=\alpha e_1+\sum_{j=2}^n v_j e_j\in \mathcal L_{e_1}$
\begin{equation*}
\begin{split}
\lim_{r\to 1}(r-1)\langle dG''_{\varphi_v(r)}(v),v''\rangle&=\lim_{r\to 1}(r-1)\left(\sum_{j=2}^n \alpha \overline{v_j}\langle dG_{\varphi_v(r)}(e_1),e_j\rangle\right.\\&\left.+\sum_{h,k=2}^n v_h\overline{v_k}\langle dG_{\varphi_v(r)}(e_h),e_k\rangle \right)=
\lim_{r\to 1}\sum_{j=2}^n \alpha \overline{v_j}(r-1)\langle dG_{\varphi_v(r)}(e_1),e_j\rangle\\&=
\sum_{j=2}^n\alpha \overline{v_j} \lim_{r\to 1}\frac{(r-1)}{(1-\langle \varphi_v(r),e_1\rangle)^{1/2}}(1-\langle \varphi_v(r),e_1\rangle)^{1/2}\langle dG_{\varphi_v(r)}(e_1),e_j\rangle\\&
=-\sum_{j=2}^n\frac{\overline{v_j}}{\alpha}\lim_{r\to 1}(r-1)^{1/2}(1-\langle \varphi_v(r),e_1\rangle)^{1/2}\langle dG_{\varphi_v(r)}(e_1),e_j\rangle=0.
\end{split}
\end{equation*}
Therefore,
\begin{equation}\label{g-der}
\beta=\lim_{(0,1)\ni r\to 1}g'_v(r)=\frac{1}{\alpha}\lim_{(0,1)\ni r\to 1}\langle dG_{\varphi_v(r)}(v),e_1\rangle.
\end{equation}
Expanding \eqref{g-der}, and taking into account (3), we have
\begin{equation*}
\begin{split}
\beta&=\frac{1}{\alpha}\lim_{(0,1)\ni r\to 1}\left(\langle dG_{\varphi_v(r)}(\alpha e_1),e_1\rangle+\sum_{j=2}^n v_j\langle dG_{\varphi_v(r)} (e_j), e_1\rangle\right)\\&=\beta+\frac{1}{\alpha}\lim_{(0,1)\ni r\to 1}\left(\sum_{j=2}^n v_j\langle dG_{\varphi_v(r)} (e_j), e_1\rangle\right),
\end{split}
\end{equation*}
from which it follows that, for all choices of $v$
\[
\lim_{(0,1)\ni r\to 1}\left(\sum_{j=2}^n v_j\langle dG_{\varphi_v(r)} (e_j), e_1\rangle\right)=0.
\]
For the arbitrariness of $v$ we have
\[
\lim_{(0,1)\ni r\to 1}\langle dG_{\varphi_v(r)} (e_j), e_1\rangle=0 \quad j=2,\ldots, n.
\]
Since the function $\mathbb B^n\ni z\mapsto \langle dG_{z}(e_j),e_1\rangle$ is bounded in every Kor\'anyi region and has limit $0$ along a non-tangential curve, by $\check{\hbox{C}}$irca's theorem \cite[Theorem 8.4.8]{Ru}, it has restricted $K$-limit $0$, and this proves (4).
\end{proof}
\begin{example}\label{esempio}(cfr. \cite[Example 4.2]{BCD}).
Let $G(z_1,z_2):=(0, -z_2/(1-z_1))$. Then $G$ is an infinitesimal generator in $\mathbb B^n$ with BRNP $e_1$ and dilation $\beta=0$. Note that $|\langle G(z), e_1\rangle |\equiv 0$, and Proposition \ref{JWC-1} applies.
However, $\angle\lim_{z\to e_1}G(z)$ does not exist. In fact, $dG_z$ is not bounded in any Kor\'anyi region: a direct computation shows that
\[
dG_z=\left(
\begin{array}{cc}
0 & 0 \\
-\frac{z_2}{(1-z_1)^2} & -\frac{1}{1-z_1} \\
\end{array}
\right).
\]
Moreover, given $v=(\alpha, v_2)\in \mathcal L_{e_1}$ it is easy to see that
\[
g_v(\zeta)=(1-\frac{1}{\alpha^2})(\zeta-1),
\]
hence, the dilation $\beta_v$ of $g_v$ at $1$ is $1-1/\alpha^2$. Thus, $\beta_v<0=\beta$ for all $v\in \mathcal L_{e_1}\setminus \{e_1\}$, and $\beta_{e_1}=0$.
\end{example}
\subsection{(Dis)similarities between the Julia-Wolff-Carath\'eodory theorems for maps and for infinitesimal generators and open questions}\label{discuto}
Hypothesis \eqref{ipo} is stronger than the corresponding starting hypothesis in Rudin's theorem, which involves only finiteness of the liminf defining $\alpha_f(e_1)$. In fact, part of the work in proving Rudin's theorem is devoted to show that such a condition, via Julia's lemma, implies boundness of suitable functions in any Kor\'anyi region. Julia's lemmas for infinitesimal generators (see Theorem \ref{one-gv}) are however -- and, in a certain sense, very naturally -- weaker than those for self-mappings and this forced us to use such a stronger hypothesis. We do not know whether there exists any weaker condition in terms of liminf of some function of $G$ which assures (and it is equivalent to) hypothesis \eqref{ipo}.
It would be also interesting to find an example (if any) of an infinitesimal generator satisfying the hypothesis of Proposition \ref{beta} but not hypothesis \eqref{ipo}.
Moreover, with our techniques, we are not able to prove (or disprove) for infinitesimal generators the statements corresponding to (5) and (6) of Theorem \ref{RudinJWC}. Namely, under the hypothesis of Theorem \ref{JWC}, we do not know whether for $j=2,\ldots, n$ it holds
\begin{equation}\label{nose}
\angle_K\lim_{z\to e_1}\frac{\langle G(z), e_j\rangle}{(1-z_1)^{1/2}}=0, \quad \angle_K\lim_{z\to e_1}(1-z_1)^{1/2}\langle dG_z(e_1), e_j\rangle=0.
\end{equation}
By $\check{\hbox{C}}$irca's theorem \cite[Theorem 8.4.8]{Ru} and Theorem \ref{JWC}.(1$^{'''}$) and using \eqref{boh}, these results hold if one can prove that for $j=2,\ldots, n$
\begin{equation}\label{radile}
\lim_{(0,1)\ni r\to 1}\frac{\langle G(re_1), e_j\rangle}{(1-r)^{1/2}}=0
\end{equation}
In the case of Rudin's theorem, the corresponding radial limit is proven using Julia's lemma and the strong constrain of sending the ball into itself.
Let $f:\mathbb B^n\to \mathbb B^n$ be a holomorphic self-map having a BRFP at $e_1$. Then $G(z):=f(z)-z$ is an infinitesimal generator (see \cite[Corollary 3.3.1]{Shb} and \cite{RS}) and, using Theorem \ref{RudinJWC}, it is not hard to see that $G$ satisfies \eqref{ipo} at $e_1$. Again by Theorem \ref{RudinJWC} it is easy to see that $G$ satisfies \eqref{radile}, and hence \eqref{nose}. Therefore, for the dense subclass of infinitesimal generators of the form $f(z)-z$ with $f:\mathbb B^n\to \mathbb B^n$ holomorphic, the full analogue of Rudin's theorem holds.
Thus, it is reasonable to believe that even in the general case \eqref{radile} holds and should follow from Julia's lemma for infinitesimal generators and the condition of being an infinitesimal generators. However, we are not able to prove the result.
\section{Higher order jets of generators at BRNPs}\label{sec-higher}
Let $G$ be an infinitesimal generator on $\mathbb B^n$ having a boundary regular null point (BRNP) at $e_1$ and assume $G$ is $C^3$ at $e_1$. We can expand $G$ in the form
\begin{equation}\label{expansion}
G(z)=T(z-e_1)+Q_2(z-e_1)+Q_3(z-e_1)+o(|z-e_1|^3),
\end{equation}
where $Q_j$ is a $n$-tuple of homogeneous polynomial of degree $j$ for $j=2,3$.
Then by Theorem \ref{JWC} we can write,
\begin{equation}\label{T}
T=\left(
\begin{array}{cccc}
\beta & 0 &\ldots & 0\\
t_2 & s_{22} &\ldots & s_{2n} \\
\vdots & \vdots & \vdots & \vdots\\
t_n & s_{n2} &\ldots & s_{nn}
\end{array}
\right)
\end{equation}
where $\beta\in \mathbb R$ is the dilation of $G$ at $e_1$ and $t_j, s_{jk}\in \mathbb C$. We set $S=(s_{jk})_{j,k=2,\ldots, n}$.
Also, we write $(x,y)\in \mathbb C\times \mathbb C^{n-1}$ with $y=(y_2,\ldots, y_n)$ and use multi-indices notations. Namely, $y^{J}=y_2^{j_2}\cdots y_n^{j_n}$, if $J=(j_2,\ldots,j_n)$; for a multi-index $I=(i_1,\ldots, i_n)$ we let $|I|=\sum_{j=1}^n i_j$.
We let
\begin{equation}\label{Q2}
Q_2(x,y)=\left( \sum_{I=(i_1,J)\in \mathbb N^n, |I|=2} q^1_{i_1,J}x^{i_1}y^{J},\ldots, \sum_{I=(i_1,J)\in \mathbb N^n, |I|=2} q^n_{i_1,J}x^{i_1}y^{J} \right),
\end{equation}
for some $q^k_{i_1,J}\in \mathbb C$.
Now we characterize boundary jets of infinitesimal generators:
\begin{proposition}\label{basic}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator of class $C^3$ at $e_1$. Assume that $e_1$ is a BRNP with dilation $\beta=0$. Let \eqref{expansion} be the expansion of $G$ at $e_1$, with $T$ given by \eqref{T} and $Q_2$ given by \eqref{Q2}. Then ${\sf Re}\, q^1_{2,0}\geq 0$ and ${\sf Re}\, s_{kk}\leq -|q^1_{0,e_{2k}}|$ for all $k=2,\ldots, n$.
Moreover, ${\sf Re}\, g_v''(1)=0$ for all $v\in \mathcal L_{e_1}$ if and only if
\begin{equation}\label{cond}
{\sf Re}\, q^1_{2,0}={\sf Re}\, s_{kk}=0, \quad k=2,\ldots, n,
\end{equation}
and, $g_v''(1)=0$ for all $v\in \mathcal L_{e_1}$ if and only if $q^1_{2,0}=s_{kk}=0, \quad k=2,\ldots, n$.
If \eqref{cond} holds, then
\begin{itemize}
\item $S$ is anti-Hermitian,
\item $q^1_{0,J}=0$ for all $J\in \mathbb N^n$, $|J|=2$,
\item $q^1_{1,e_k}=q^h_{0,e_k+e_h}$ for $h,k=2,\ldots, n$,
\item $q^h_{0,e_k+e_l}=0$ for all $h,k,l=2,\ldots, n$ with $h\neq k, l$,
\item ${\sf Im}\, q^k_{1,e_k}={\sf Im}\, q^1_{2,0}=0$ and ${\sf Re}\, q^k_{1,e_k}\geq 0$ for $k=2,\ldots, n$,
\item the matrix $\tilde{Q}:=(q^k_{1,e_h})_{h,k=2,\ldots, n}$ is Hermitian and positive semi-definite.
\end{itemize}
Moreover, there exists $\delta\leq 0$ such that
\begin{equation}\label{Q3a0}
Q^1_3(x,y)=\delta x^3+\sum_{j=2}^n (q^1_{1,e_j}+\overline{q}^j_{2,0}) x^2 y_j.
\end{equation}
Finally, if ${\sf Re}\, g_v''(1)=0$ for all $v\in \mathcal L_{e_1}$ then $g_v'''(1)=0$ for all $v\in \mathcal L_{e_1}$ if and only if ${\sf Re}\, q^k_{1,e_k}= 0$, $q_{1,e_k}^h=0$ for $2\leq k<h\leq n$ and $\delta=0$.
\end{proposition}
\begin{proof}
Let $v\in \mathcal L_{e_1}$, and let $g_v:\mathbb D\to \mathbb C$ be the slice reduction of $G$ with respect to $v$. Let $G$ be given by \eqref{expansion}, with $Tv=(T^1v,T''v)\in \mathbb C\times \mathbb C^{n-1}$, $Q_2(v)=(Q_2^1(v), Q_2''(v))\in \mathbb C\times \mathbb C^{n-1}$ and $Q_3(v)=(Q_3^1(v), Q_3''(v))\in \mathbb C\times \mathbb C^{n-1}$. By Theorem \ref{JWC},
\[
T^1v=\langle Tv,e_1\rangle=\langle dG_{e_1}(v), e_1\rangle =\alpha \beta =0.
\]
Thus, a direct computation from \eqref{explicit} shows that
\[
g_v(\zeta)=a_v(\zeta-1)^2+b_v (\zeta-1)^3+o(|\zeta-1|^3),
\]
with
\begin{equation}\begin{split}\label{a_v}
a_v&=Q_2^1(v)-\langle T''v,v''\rangle,\\
b_v&=(1-\alpha^2)Q_2^1(v)+\alpha Q_3^1(v)-\alpha \langle Q_2''(v),v''\rangle.
\end{split}\end{equation}
Now, $g_v(\zeta)=a_v(\zeta-1)^2 +b_v(\zeta-1)^3+o(|\zeta-1|^3)$ is an infinitesimal generator in the unit disc and thus, by Berkson-Porta formula, it has to hold ${\sf Re}\, (a_v+b_v(\zeta-1)+o(|\zeta-1|))\geq 0$.
Therefore, in particular, ${\sf Re}\, a_v\geq 0$ for all $v\in \mathcal L_{e_1}$ (see also, \cite{Sh}).
By writing down explicitly the condition ${\sf Re}\, a_v\geq 0$, we find
that for all $v=(\alpha,v_2,\ldots, v_n)$ with $\alpha\in (0,1]$,
$\sum_{j=2}^n |v_j|^2=1-\alpha^2$,
\begin{equation}\label{jet2}
\begin{split}
&\sum_{2\leq j\leq k\leq n} {\sf Re}\, (q^1_{0,e_j+e_k}v_jv_k)-\sum_{j,k=2}^n {\sf Re}\, (s_{kj}v_j\overline{v}_k)\\&+\alpha \left[\sum_{j=2}^n {\sf Re}\, (q^1_{1,e_j} v_j)-\sum_{k=2}^n {\sf Re}\, (t_k\overline{v}_k)\right]+\alpha^2{\sf Re}\, q_{2,0}^1\geq 0.
\end{split}
\end{equation}
For $\alpha=1, v''=0$, we find ${\sf Re}\, q_{2,0}^1\geq 0$.
When $\alpha\to 0$, the previous inequality implies that the term of degree $0$ in $\alpha$ has to have real part $\geq 0$, namely
\begin{equation}\label{al0}
\sum_{2\leq j\leq k\leq n} {\sf Re}\, (q^1_{0,e_j+e_k}v_jv_k)-\sum_{j,k=2}^n {\sf Re}\, (s_{kj}v_j\overline{v}_k)\geq 0
\end{equation}
for $\|v''\|=1$. Now, fix $k\in\{2,\ldots, n\}$ and substitute $v''$ with $e^{i \theta_k}e_k$ for $\theta_k\in [0,2\pi]$. We obtain ${\sf Re}\, (q^1_{0,2e_k}e^{2i\theta_k})-{\sf Re}\, s_{kk}\geq 0$, which, for the arbitrariness of $\theta_k$, implies ${\sf Re}\, s_{kk}\leq -|q^1_{0,2e_k}|$ for $k=2,\ldots, n$.
Now, ${\sf Re}\, g''_v(1)=0$ if and only if ${\sf Re}\, a_v=0$ for all $v\in \mathcal L_{e_1}$. Therefore, it is clear from the previous considerations that if ${\sf Re}\, g''_v(1)=0$ for all $v\in \mathcal L_{e_1}$, then necessarily ${\sf Re}\, q_{2,0}^1=0$. Moreover, the left hand side of \eqref{al0} is equal to $0$. Therefore, fixing $k\in\{2,\ldots, n\}$ and substituting $v''$ with $e^{i \theta_k}e_k$ for $\theta_k\in [0,2\pi]$ we obtain ${\sf Re}\, (q^1_{0,2e_k}e^{2i\theta_k})-{\sf Re}\, s_{kk}= 0$. Integrating with respect to $\theta_k$ in $[0,2\pi]$ the harmonic term vanishes and we obtain ${\sf Re}\, s_{kk}= 0$ for $k=2,\ldots, n$.
Assume that ${\sf Re}\, s_{kk}=0$ for $k=2,\ldots, n$. Therefore the
non-harmonic part in \eqref{al0} is zero, and we claim that this
implies that \eqref{al0} is, in fact, identically $0$. Indeed, we
rewrite \eqref{al0} as
\[
\sum_{2\leq j\leq k\leq n} {\sf Re}\, (q^1_{0,e_j+e_k}v_jv_k)-\sum_{2\leq j<k\leq n} {\sf Re}\, [(s_{kj}+\overline{s}_{jk})v_j\overline{v}_k]\geq 0.
\]
Taking $|v_k|=1, v_j=0$ for $j\neq k$, we find immediately $q^1_{0,2e_k}=0$, $k=2,\ldots, n$. Next, we take $v_2=\zeta$, $v_3=\pm \zeta$ with $|\zeta|=1/\sqrt{2}$ and $v_4=\ldots=v_n=0$ and we obtain
\[
\pm\left({\sf Re}\, (q^1_{0,e_2+e_3}\zeta^2)-{\sf Re}\, (s_{32}+\overline{s}_{23})|\zeta|^2\right)\geq 0.
\]
Decoupling the harmonic and non-harmonic terms by integrating as before, we obtain
\[
{\sf Re}\, (s_{32}+\overline{s}_{23})=0, \quad q^1_{0,e_2+e_3}=0.
\]
Finally, taking $v_2=\zeta$, $v_3=e^{i\theta} \zeta$ with $|\zeta|=1/\sqrt{2}$, $\theta\in [0,2\pi]$ and $v_4=\ldots=v_n=0$ we obtain
\[
-|\zeta|^2{\sf Re}\, [(s_{32}+\overline{s}_{23})e^{-i\theta}]\geq 0
\]
which implies $s_{32}+\overline{s}_{23}=0$. A similar argument works for the other indices. This proves that $S$ is anti-Hermitian and $q^1_{0,J}=0$ for all $|J|=2$. Moreover, this proves that the terms of degree $0$ in $\alpha$ in \eqref{jet2} are identically zero. Therefore, the condition ${\sf Re}\, s_{kk}=0$ for all $k$ is sufficient for \eqref{jet2} in degree zero in $\alpha$ to be equal to zero for all $v$ when $\alpha\to 0$.
Now, since the terms of degree $0$ in $\alpha$ in \eqref{jet2} are vanishing identically, the terms of degree $1$ in $\alpha$ has to have non negative real part as $\alpha\to 0$, that is
\begin{equation}\label{al1}
\sum_{k=2}^n {\sf Re}\, [(q^1_{1,e_k}-\overline{t}_k)v_k]\geq 0,
\end{equation}
which clearly implies $q^1_{1,e_k}=\overline{t}_k$ for $k=2,\ldots, n$. Hence, if ${\sf Re}\, s_{kk}={\sf Re}\, q^1_{2,0}=0$ then ${\sf Re}\, g''_v(1)=0$ for all $v\in \mathcal L_{e_1}$.
From the previous considerations it follows easily that $g_v''(1)=0$ for all $v\in \mathcal L_{e_1}$ if and only if $q^1_{2,0}=s_{kk}=0$ for $k=2,\ldots, n$.
Now, assume \eqref{cond}. Then for all $v\in \mathcal L_{e_1}$ we have ${\sf Re}\, a_v=0$ namely, $g_v(\zeta)=(\zeta-1)^2[i\tilde{a}_v+b_v(\zeta-1)+o(|\zeta-1|)]$ for all $v\in \mathcal L_{e_1}$, where $\tilde{a}_v\in \mathbb R$. Berkson-Porta's formula (see \cite{Sh}) implies then $b_v\in \mathbb R$ and $b_v\leq 0$.
Taking into account what we have already proved, writing
$Q^1_3(v)=\sum_{|(i_1,J)|=3} p^1_{i_1,J}\alpha^{i_1}v^J$, where we
used the multi-indices notation $v^J=v_2^{j_2}\cdots v_n^{j_n}$,
from \eqref{a_v}, the condition $b_v\leq 0$ becomes
\begin{equation}\label{jet3}
\begin{split}
&(1-\alpha^2)\left(q^1_{2,0}\alpha^2+\alpha \sum_{k=2}^n q^1_{1,e_k}v_k\right) +\alpha \sum_{|(i_1,J)|=3} p^1_{i_1,J}\alpha^{i_1}v^J\\&-\alpha \sum_{k=2}^n\sum_{|(i_1,J)|=2}q^k_{i_1,J}\alpha^{i_1}v^J\overline{v}_k\leq 0.
\end{split}
\end{equation}
For $\alpha=1, v''=0$, we obtain $p^1_{3,0}\leq 0$. And, if $g_v'''(1)=0$ for all $v$, that is $b_v=0$, then $p^1_{3,0}=0$.
Now, as before, we start looking at terms of smallest degree in $\alpha$ when $\alpha\to 0$. Since there are no terms of degree $0$ in $\alpha$, the smallest degree is $1$, and we get
\begin{equation}\label{deg1a}
\sum_{k=2}^n q^1_{1,e_k}v_k+\sum_{|I|=3}p_{0,I}^1v^I-\sum_{k=2}^n\sum_{|J|=2}q^k_{0,J}v^J\overline{v}_k\leq 0,
\end{equation}
for all $v\in \mathbb C^{n-1}$ with $\|v\|=1$. Replacing $v$ by $-v$ the left-hand side of \eqref{deg1a} changes sign. Therefore we deduce that
\begin{equation}\label{deg1}
\sum_{k=2}^n q^1_{1,e_k}v_k+\sum_{|I|=3}p_{0,I}^1v^I-\sum_{k=2}^n\sum_{|J|=2}q^k_{0,J}v^J\overline{v}_k= 0,
\end{equation}
for all $v\in \mathbb C^{n-1}$ with $\|v\|=1$.
Replacing $v$ by $e^{i\theta}v$ for $\theta\in [0,2\pi]$ in \eqref{deg1}, dividing the equation by $e^{i\theta}$ and integrating with respect to $\theta$ in $[0,2\pi]$ the harmonic terms vanish and we obtain
\begin{equation}\label{deg1m}
\sum_{k=2}^n q^1_{1,e_k}v_k-\sum_{k=2}^n\sum_{|J|=2}q^k_{0,J}v^J\overline{v}_k= 0.
\end{equation}
Equation \eqref{deg1} implies then $\sum_{|I|=3}p_{0,I}^1v^I=0$, which is possible only if $p_{0,I}^1=0$ for all $|I|=3$.
Taking $v_k=e_k$ for $k=2,\ldots, n$ in \eqref{deg1m} we obtain $q^1_{1,e_k}=q^k_{0,2e_k}$ for $k=2,\ldots, n$.
Now, let $2\leq k_1< k_2\leq n$ and let $v=\frac{1}{\sqrt{2}}(e^{i\theta_1}e_{k_1}+e^{i\theta_2}e_{k_2})$ for $\theta_1,\theta_2\in [0,2\pi]$. Expanding \eqref{deg1m} with such a choice of $v$, multiplying by $2\sqrt{2}$ and taking into account that $q^1_{1,e_k}=q^k_{0,2e_k}$ for $k=2,\ldots, n$, we obtain
\begin{equation*}
\begin{split}
0=&e^{i\theta_1}(2q^1_{1,e_{k_1}}-q^{k_1}_{0,2e_{k_1}}-q^{k_2}_{0,e_{k_1}+e_{k_2}})+
e^{i\theta_2}(2q^1_{1,e_{k_2}}-q^{k_2}_{0,2e_{k_2}}-q^{k_1}_{0,e_{k_1}+e_{k_2}})
\\&-e^{i(2\theta_2-\theta_1)}q^{k_1}_{0,2e_{k_2}}-e^{i(2\theta_1-\theta_2)}q^{k_2}_{0,2e_{k_1}}
\\ &=e^{i\theta_1}(q^1_{1,e_{k_1}}-q^{k_2}_{0,e_{k_1}+e_{k_2}})+
e^{i\theta_2}(q^1_{1,e_{k_2}}-q^{k_1}_{0,e_{k_1}+e_{k_2}})-e^{i(2\theta_2-\theta_1)}q^{k_1}_{0,2e_{k_2}}-e^{i(2\theta_1-\theta_2)}q^{k_2}_{0,2e_{k_1}},
\end{split}
\end{equation*}
from which we deduce that $q^1_{1,e_{k}}=q^{h}_{0,e_{k}+e_{h}}$ and $q^{k}_{0,2e_{h}}=0$ for $k\neq h\in \{2,\ldots, n\}$.
Finally, let $2\leq k_1<k_2<k_3\leq n$ and consider $v=\frac{1}{\sqrt{3}}(e^{i\theta_1}e_{k_1}+e^{i\theta_2}e_{k_2}+e^{i\theta_3}e_{k_3}$ for $\theta_1,\theta_2,\theta_3\in [0,2\pi]$. Expanding \eqref{deg1m} with such a choice of $v$, multiplying by $3\sqrt{3}$, we obtain
\begin{equation*}
\begin{split}
0&=e^{i\theta_1}(3q^1_{1,e_{k_1}}-q^{k_1}_{0,2e_{k_1}}-q^{k_2}_{0,e_{k_1}+e_{k_2}}-q^{k_3}_{0,e_{k_1}+e_{k_3}})+
e^{i\theta_2}(3q^1_{1,e_{k_2}}-q^{k_2}_{0,2e_{k_2}}-q^{k_1}_{0,e_{k_1}+e_{k_2}}-q^{k_3}_{0,e_{k_2}+e_{k_3}})\\&+
e^{i\theta_3}(3q^1_{1,e_{k_3}}-q^{k_3}_{0,2e_{k_3}}-q^{k_1}_{0,e_{k_1}+e_{k_3}}-q^{k_2}_{0,e_{k_2}+e_{k_3}})-
e^{i(2\theta_2-\theta_1)}q^{k_1}_{0,2e_{k_2}}-e^{i(2\theta_3-\theta_1)}q^{k_1}_{0,2e_{k_3}}\\&-
e^{i(2\theta_1-\theta_2)}q^{k_2}_{0,2e_{k_1}}-e^{i(2\theta_3-\theta_2)}q^{k_2}_{0,2e_{k_3}}-
e^{i(2\theta_1-\theta_3)}q^{k_3}_{0,2e_{k_1}}-e^{i(2\theta_2-\theta_3)}q^{k_3}_{0,2e_{k_2}}\\
&-e^{i(\theta_1+\theta_3-\theta_2)}q^{k_2}_{0,e_{k_1}+e_{k_3}}-e^{i(\theta_2+\theta_3-\theta_1)}q^{k_1}_{0,e_{k_2}+e_{k_3}}-
e^{i(\theta_1+\theta_2-\theta_3)}q^{k_3}_{0,e_{k_1}+e_{k_2}}=0.
\end{split}
\end{equation*}
The first three lines of the previous equation do not give any new information, but the last one implies
that $q^{k}_{0,e_{h}+e_{l}}=0$ for $k,h,l=2,\ldots,n$ and $k\neq h,l$.
Therefore, the term of degree $1$ in $\alpha$, for $\alpha\to 0$ in
\eqref{jet3} identically vanishes. So we look at terms of degree
$2$ in $\alpha$ as $\alpha\to 0$. We have
\begin{equation}\label{deg2}
q^1_{2,0}+\sum_{|I|=2}p^1_{1,I}v^I-\sum_{j,k=2}^n q^k_{1,e_j}v_j\overline{v}_k\leq 0,
\end{equation}
for all $v\in \mathbb C^{n-1}$ with $\|v\|=1$.
Replacing $v_j$ with $e^{i\theta_j}v_j$ for $\theta_j\in [0,2\pi]$ and integrating, we get rid of the harmonic terms and we find
\[
q^1_{2,0}-\sum_{k=2}^n q_{1,e_k}^k|v_k|^2\leq 0.
\]
Taking $v=e_k$ we obtain $q^1_{2,0}-q^k_{1,e_k}\leq 0$. Since ${\sf Re}\, q^1_{2,0}=0$, this implies ${\sf Im}\, q^k_{1,e_k}={\sf Im}\, q^1_{2,0}$ and ${\sf Re}\, q^k_{1,e_k}\geq 0$ for $k=2,\ldots, n$.
Now, the harmonic part in \eqref{deg2} must be real, that is
\[
\sum_{2\leq j\leq l\leq n}p^1_{1,e_j+e_l} v_jv_l-\sum_{k=2}^n\sum_{j=2, j\neq k}^n q^k_{1,e_j}v_j\overline{v}_k\in \mathbb R.
\]
Taking $v=e_k$, this immediately implies $p^1_{1,2e_{k}}=0$. Taking $v=\frac{1}{\sqrt{2}}e^{i\theta}(e_{k_1}+e_{k_2})$ with $2\leq k_1<k_2\leq n$ and $\theta\in [0,2\pi]$, we obtain
\[
e^{2i\theta}p^1_{1,e_{k_1}+e_{k_2}}-(q^{k_1}_{1,e_{k_2}}+q^{k_2}_{1,e_{k_1}})\in \mathbb R,
\]
which implies $p^1_{1,e_j+e_k}=0$ for $j,k=2,\ldots, n$, $j\neq k$. Next, setting $v=\frac{1}{\sqrt{2}}(e_{k_1}+e^{i\theta}e_{k_2})$
with $\theta\in [0,2\pi]$, $2\leq k_1< k_2\leq n$, we obtain $(e^{-i\theta}q^{k_1}_{1,e_{k_2}}+e^{i\theta}q^{k_2}_{1,e_{k_1}})\in \mathbb R$, that is
\[
{\sf Im}\, [e^{-i\theta}(q_{1,e_{k_2}}^{k_1}-\overline{q}^{k_1}_{1,e_{k_2}})]=0,
\]
hence $q_{1,e_{k_2}}^{k_1}-\overline{q}^{k_2}_{1,e_{k_1}}=0$. This, together with \eqref{deg2} implies that the matrix $\tilde{Q}=(q_{1,e_{k}}^h)_{h,k=2,\ldots, n}$ is Hermitian and positive semi-definite.
From the previous considerations, we also note that $b_v=0$ implies ${\sf Re}\, q^k_{1,e_k}= 0$ and $q_{1,e_k}^h=0$ for $2\leq k<h\leq n$, while, this latter condition implies that the term of order $2$ in $\alpha$ as $\alpha\to 0$ in \eqref{jet3} vanishes identically.
Now, we are left to impose the condition that the remaining terms give a non-positive real number. Looking at terms of degree $3$ in $\alpha$ in \eqref{jet3}, we have
\[
0={\sf Im}\,\left[\sum_{j=2}^n (-q^1_{1,e_j}v_j+p^1_{2,e_j}v_j-q^j_{2,0}\overline{v}_j)\right]=\sum_{j=2}^n{\sf Im}\,[(-q^1_{1,e_j}+p^1_{2,e_j}-\overline{q}^j_{2,0})v_j],
\]
for all $v\in \mathbb C^{n-1}$ with $\|v\|=1$, which clearly implies $-q^1_{1,e_j}+p^1_{2,e_j}-\overline{q}^j_{2,0}=0$ for all $j=2,\ldots, n$. In particular, the term of degree $3$ in $\alpha$ always vanishes identically.
Finally, we impose the condition that the terms of degree $4$ in $\alpha$ in \eqref{jet3} are real. This means
\[
{\sf Im}\, \left[ -q^1_{2,0}+p^1_{3,0} \right]=0.
\]
Taking into account that we already proved that $p^1_{3,0}\leq 0$, this implies that ${\sf Im}\, q^1_{2,0}=0$. Note also that if the terms of degree $0,1,2,3$ in \eqref{jet3} vanish, which implies in particular that $q^1_{2,0}=0$, then the terms of order $4$ are vanishing if and only if $p^1_{3,0}=0$.
Summing up, if ${\sf Re}\, g_v''(1)=0$ all $v\in \mathcal L_{e_1}$ then $g_v'''(1)=0$ for all $v\in \mathcal L_{e_1}$ if and only if the terms of degree $2$ and $4$ in $\alpha$ in \eqref{jet3} are identically vanishing (because those of degree $1$ and $3$ always do). This, in turn, is equivalent to ${\sf Re}\, q^k_{1,e_k}= 0$, $q_{1,e_k}^h=0$ for $2\leq k<h\leq n$ and $p^1_{3,0}=0$. And this ends the proof.
\end{proof}
\begin{proposition}\label{almost}
Let $G$ be an infinitesimal generator on $\mathbb B^n$, $C^3$ at $e_1$. Assume that $e_1$ is a BRNP with dilation $0$. Then $G$ generates a group of automorphisms if and only if the following conditions are satisfied:
\begin{enumerate}
\item ${\sf Re}\, \langle \frac{\partial G}{\partial z_k}(e_1),e_k\rangle =0$, for $k=2,\ldots, n$,
\item ${\sf Re}\, \langle \frac{\partial^2 G}{\partial z_1\partial z_k}(e_1),e_1\rangle =0$, for $k=1,\ldots, n$,
\item $\langle \frac{\partial^2 G}{\partial z_1\partial z_k}(e_1),e_h\rangle=0$ for $2\leq k<h\leq n$,
\item ${\sf Re}\, \langle \frac{\partial^3 G}{\partial z_1^3}(e_1),e_1\rangle=0$.
\end{enumerate}
Moreover, if the previous conditions are satisfied, then $G\equiv 0$ if and only if $\langle \frac{\partial G}{\partial z_k}(e_1),e_k\rangle =0$ for $k=2,\ldots, n$ and $\langle \frac{\partial^2 G}{\partial z_1^2}(e_1),e_1\rangle =0$.
\end{proposition}
\begin{proof}
By Proposition \ref{basic}, the hypotheses are equivalent to the fact that for all $v\in \mathcal L_{e_1}$ the slice retraction $g_v$ has the property that $g_v'(1)={\sf Re}\, g_v''(1)=g_v'''(1)=0$. By \cite[Corollary 4]{Sh} this is equivalent to the fact that $g_v$ is a generator of a group of automorphisms of $\mathbb D$ for all $v\in \mathcal L_{e_1}$ (and $g_v\equiv 0$ if and only if $g''_v(1)=0$). Then the statement follows from Proposition \ref{gruppo}.
\end{proof}
\begin{lemma}\label{trickP}
Let $G:\mathbb B^n\to \mathbb C^n$ be an infinitesimal generator, $C^3$ at $e_1$ and with BRNP at $e_1$ and dilation $\beta\in \mathbb R\setminus\{0\}$.
Let $H_\beta$ be given by \eqref{hhh} and let $\tilde{G}:=G+H_\beta$. Then $\tilde{G}$ has a BRNP at $e_1$ with dilation $0$. Moreover, denoting by $\tilde{s}_{jk}$, $\tilde{q}^j_{i_1,J}$ the elements in the expansion of $\tilde{G}$ at $e_1$, we have $\tilde{s}_{kk}=s_{kk}-\frac{\beta}{2}$, $\tilde{q}^1_{2,0}=q^1_{2,0}-\frac{\beta}{2}$, $\tilde{q}^k_{1,e_k}=q^k_{1,e_k}-\frac{\beta}{2}$, $k=2,\ldots, n$ and $\tilde{q}^h_{1,e_k}=q^h_{1,e_k}$ for $2\leq k<h\leq n$.
\end{lemma}
\begin{proof}
By Corollary \ref{trick}, the vector field $G+H_\beta$ is an infinitesimal
generator with BRNP at $1$ and dilation $=0$. Now,
\begin{equation*}
\begin{split}
H_\beta(z)&=\left( -\beta (z_1-1),-\frac{\beta}{2}z_2,\ldots,-\frac{\beta}{2}z_n\right) \\&+ \left(-\frac{\beta}{2}(z_1-1)^2,-\frac{\beta}{2}(z_1-1)z_2,\ldots, -\frac{\beta}{2}(z_1-1)z_n \right).
\end{split}
\end{equation*}
From this the statements follow easily.
\end{proof}
\begin{proof}[Proof of Theorem \ref{rigidity}]
Let $\tilde{G}:=G+H_\beta$. Thanks to Lemma \ref{trickP}, the hypotheses on $G$ implies that $\tilde{G}$ satisfies the hypothesis of Proposition \ref{almost}, and the result follows.
\end{proof}
\section{On the quadratic expansion at BRNPs}\label{quadratic}
In \cite{Sh} it is shown that if $g:\mathbb D \to \mathbb C$ is an infinitesimal generator in $\mathbb D$ which is $C^3(1)$ with expansion $g(z)=z-1+a(z-1)^2+o(|z-1|^2)$ then the quadratic part $z\mapsto z-1+a(z-1)^2$ is always an infinitesimal generator in $\mathbb D$ which generates a semigroup of linear fractional self-maps of the unit disc.
In higher dimension the same result is false, and, even when the quadratic part is an infinitesimal generator, it might not generate a semigroup of linear fractional maps. The underlying reason is that slice reductions at a BRNP of an infinitesimal generator do not preserve the degree of expansion at the boundary (cfr. \eqref{a_v}), so that the quadratic part of the infinitesimal generator in $\mathbb B^n$ might generate a cubic term on some slice reduction. We present the following examples.
\begin{example}
Let $F:\mathbb B^2\to \mathbb C^2$ be given by
\[
F(z_1,z_2)=-\left( z_1-1,\frac{5z_2}{4(2-z_2)}\right).
\]
We claim that $F$ is an infinitesimal generator. Indeed, for each
$z\in\partial\mathbb B^2$ we have
\[
\langle -F(z),z \rangle
=|z_1|^2-\overline{z_1}+\frac{5|z_2|^2}{4(2-z_2)}=(1-\overline{z_1})+|z_2|^2\left(
\frac{5}{4(2-z_2)}-1 \right).
\]
Hence
\begin{equation}\begin{split}
{\sf Re}\,\langle -F(z),z \rangle &\ge 1-\sqrt{1-|z_2|^2} +|z_2|^2{\sf Re}\,\left(
\frac{5}{4(2-z_2)}-1 \right)\\
&\ge|z_2|^2\left(\frac1{1+\sqrt{1-|z_2|^2}} +\frac{5}{4(2+|z_2|)}-1
\right)\\&=r^2\,\frac{5-3\sqrt{1-r^2}-4r\sqrt{1-r^2}}
{4(2+r)(1+\sqrt{1-r^2})}\,,
\end{split}\end{equation}
where $r=|z_2|$.
A standard computation shows that the expression
${5-3\sqrt{1-r^2}-4r\sqrt{1-r^2}}$ is positive on the segment
$[0,1]$. Therefore, one concludes that ${\sf Re}\,\langle F(z),z
\rangle<0$ for all $z\in\partial\mathbb B^2$. Taking into account that $F$ is holomorphic past the boundary of $\mathbb B^2$, one can apply Theorem \ref{one-gv}.(3) with $\beta=0$, and Proposition \ref{cbDW} to see that $F$ is an infinitesimal generator on $\mathbb B^2$ (see also \cite[Corollary 7.1]{RS}).
On the other hand, denote by $\tilde F$ the quadratic expansion of $F$ at $e_1$, namely,
\[
\tilde F(z)=-\left(z_1-1,\frac{5z_2}8\left(1+\frac{z_2}2\right)
\right).
\]
For this mapping
\[
\langle -\tilde F(z),z \rangle
=|z_1|^2-\overline{z_1}+\frac{5|z_2|^2}8\left(1+\frac{z_2}2\right).
\]
In particular, at the point $z_1=\frac1{\sqrt 2},\
z_2=-\frac1{\sqrt 2}$ we have
\[
\langle \tilde F(z),z
\rangle=-\frac{13}{16}+\frac{37}{32\sqrt{2}}\approx 0.005>0.
\]
So, $\tilde F$ is not a semigroup generator on the ball $\mathbb B^2$.
\end{example}
\begin{example}
Let $F:\mathbb B^2\to \mathbb C^2$ be given by
\[
F(z_1,z_2)=-\left( z_1-1,\frac{3z_2}{(2-z_2)}\right).
\]
We claim that $F$ is an infinitesimal generator on $\mathbb B^2$. Indeed, for each
$z\in\partial\mathbb B$ we have
\[
\langle -F(z),z
\rangle=|z_1|^2-\overline{z_1}+\frac{3|z_2|^2}{2-z_2}=(1-\overline{z_1})+|z_2|^2\left(
\frac{3}{2-z_2}-1 \right).
\]
Since the inequality ${\sf Re}\,\frac{1}{2-z_2}>\frac13$ holds for
all $z_2,\ |z_2|<1$, we conclude that ${\sf Re}\,-\langle F(z),z
\rangle>0$ for all $z\in\partial\mathbb B$. As in the previous example taking into account that $F$ is holomorphic past the boundary of $\mathbb B^2$, one can apply Theorem \ref{one-gv}.(3) with $\beta=0$, and Proposition \ref{cbDW} to see that $F$ is an infinitesimal generator on $\mathbb B^2$ (see also \cite[Corollary 7.1]{RS}).
On the other hand, denote by $\tilde F$ the the quadratic expansion of $F$ at $e_1$, namely,
\[
\tilde F(z)=-\left(z_1-1,\frac{3z_2}2\left(1+\frac{z_2}2\right)
\right).
\]
One can easily see that $\tilde F$ generates a semigroup of holomorphic self-maps of $\mathbb B^2$ which does not
consist of linear fractional maps (cfr \cite{BCDl}).
\end{example}
|
{
"timestamp": "2012-09-25T02:02:36",
"yymm": "1203",
"arxiv_id": "1203.1839",
"language": "en",
"url": "https://arxiv.org/abs/1203.1839"
}
|
\section{ Introduction}
\setcounter{equation}{0}
\hspace{5.1 mm}
A great deal of models of applied sciences are described by the parabolic equation:
\vspace{3mm}
\begin {equation} \label{11}
{\cal{L}}_\varepsilon u = \varepsilon u_{xxt}
+ c^2 u_{xx} - u_{tt} - 2 a u_t = -f.
\end{equation}
\vspace{3mm}
The constants
$ a, c^2, \varepsilon $ are all positive and they assume various meanings according to physical problems. As for the source $f$, it can be linear or not.
For instance, the equation (\ref {11}) is involved in the generalized Maxwell- Cattaneo system of equations \cite{jp}-\cite{ps}, in problems of viscoelastic media of Kelvin-Voigt type \cite {r}, or for the study of solids at very low temperatures \cite {jcl}.
Further applications arise in the study of viscoelastic plates with memory, when the relaxation function is given by exponential functions. (\cite {cdf} and references therein).
A typical example of the non linear case
is the {\it
perturbed sine-Gordon equation} which models the flux dynamics in Josephson junctions in
superconductivity \cite{scott}\cite{bp}. In this case, the terms $\varepsilon u_{xxt}$ and $a u_t$ characterize the
dissipative normal electron current flow along and across the junction.
As for the practical applications of superconductors, many areas are involved. In medicine, for instance, Magnetic Resonance Imaging (MRI) has been used since 1977 and is still improving \cite{ebphkba}. Referring to the electric power systems, the high temperature superconductor cables are likely to lead to a lot of benefits as regards the current carrying capacity and for reducing electrical losses.\cite{thjlmoso}\cite{n}.
As for typical boundary value problems related to the equation (\ref{11}), both the Dirichlet conditions and Neumann conditions have interest for practical applications. For instance, in superconductivity, the first case can be referred to periodic conditions according to annular geometry of junction \cite {parm}{\cite {cln}, while in the other case, the phase gradient, proportional to the magnetic field, is specified.(\cite{scr}-\cite{fppcss}). When the source term $f$ is {\it linear}, all these problems can be explicitly solved by means of the Fourier method. The solutions are determined in sect.2-3, together with the related Green functions $ G_\varepsilon , K_\varepsilon $.
When the function $f$ is {\it non linear}, then $ G_\varepsilon , K_\varepsilon $ represent the kernels of the integral equations to which the above mentioned boundary value problems can be reduced. For this, a rigorous analysis of the behavior of these kernels when $\, \varepsilon \rightarrow 0 $ and $\, t \rightarrow \infty\,$ is achieved in sect 4. At last, as first application, the influence of the dissipation on the wave behavior is estimated by an asymptotic approximation uniformly valid also for large $\,t\,$ (sect.5).
\vspace{5.1mm}
\section {Statement of the problem}
\setcounter{equation}{0}
\hspace{5.1mm}
\vspace{3mm}
If $ u _\varepsilon (x,t)$ is a function defined in the strip
\begin {center}
$ \Omega = \{(x,t) : 0 \leq x
\leq
\pi, \ \ t \,\geq \,0 \}$,
\noindent
\end {center}
let ${\cal P}_\varepsilon $ the initial- boundary value
problem related to equation (\ref{11}) with conditions
\vspace{3mm}
\begin{equation} \label{21}
\begin{array}{lll}
u_\varepsilon (x,0)=f_0(x), \ \ \partial_t u_\varepsilon
(x,0)=f_1(x),
& x\in [0,\pi],\vspace{2mm} \\
\end{array}
\end {equation}
\begin {equation} \label{22}
\begin{array}{ll}
u_ \varepsilon(0,t)=\varphi(t), \ \ \ \ \ \ \ u_\varepsilon
(\pi,t)=\psi(t), & t \geq 0,
\end{array}
\end{equation}
\vspace{3mm}
\noindent
where $ f_0, f_1, \psi, \varphi $ are arbitrary date.
\vspace{3mm}
The boundary conditions (\ref{22}) represent only an example of the analysis we are going to apply. Equally, {\em flux-boundary } conditions or {\em mixed-boundary } conditions can be considered too. So, another example is given by the problem ${\cal H}_\varepsilon $ defined in $\Omega $ by (\ref{11})-(\ref{21}) together with the Neumann conditions
\vspace{3mm}
\begin {equation} \label{23}
\begin{array}{ll}
\partial _x \,u_ \varepsilon(0,t)=\varphi_1(t), \ \ \ \ \ \ \ \partial _x \,u_\varepsilon
(\pi,t)=\psi_1(t), & t\geq 0.
\end{array}
\end{equation}
\vspace{3mm}
When $\varepsilon \equiv 0$, the parabolic equation (\ref{11}) turns into the hyperbolic telegraph equation
\vspace{3mm}
\begin{equation} \label{24}
{\cal
L}_0 u_0 \equiv ( c^2 \, \partial_{xx} - \partial_{tt} -2\, a \,\partial_t ) u_0 = - \bar f(x,t,u_0)
\eeq
\vspace{3mm}\noindent
and the problem ${\cal
P}_\varepsilon $ changes into a problem ${\cal
P}_0 $ for $ u_0(x,t)$ which has the same initial-boundary conditions (\ref{21}) - (\ref{22}) of ${\cal
P}_\varepsilon $. When the source term $\bar f$ of (\ref{24}) is {\em linear} $(\bar f= \bar f (x,t))$, ${\cal
P}_0 $ is explicitly solved by means of the well-known Green function:
\vspace{3mm}
\begin{equation} \label {25}
G_0 (x,\xi,t)\,\,= \,\, \frac{2}{\pi } \,\, \,e^{-a \,t } \,\ \sum
_{n=1}^{\infty} \,\, \,\, \frac{ \sin \,(t \,\, \sqrt{c^2 n^2 - a^2}\, \,)}{ \sqrt{c^2 n^2 - a^2}}
\, \, \sin (nx) \,\, \sin
(n\xi)
\end{equation}
\vspace{3mm}
In order to estimate the influence of the dissipative term $\, \varepsilon \, u_{xxt}\,$ on the wave behavior of $\,u_0\,$, the difference
\vspace{3mm}
\begin{equation} \label {26}
v(x,t) = u_\varepsilon - u_0,
\end{equation}
\vspace{3mm}\noindent
is to be evaluated and so the following {\em problem $
\Delta $ } must be analyzed
\vspace{3mm}
\beq \label{27}
\left \{
\begin{array}{lll}
\varepsilon \, v_{xxt}+c^2\, v_{xx} -v_{tt}-2av_t = - F(x,t, u_0 ,v ) & (x,t) \in \Omega \\
\\
v(x,0)=0, \ \ v_t(x,0)=0, \ & x \in [0,\pi], \vspace{2mm} \\
\\
v(0,t)=0, \ \ v(\pi,t)=0, & t\geq 0.
\end{array}
\right.
\eeq
\vspace{3mm}
\noindent
The source term $F$ is given by
\vspace{3mm}
\beq \label{28}
F= f(x,t,u_o+v) -\bar f(x,t,u_0) + \varepsilon \,{u_{0}}_{xxt},
\eeq
\vspace{3mm}\noindent
while, in the linear case, it is $\,f=\bar f\,$ and $ F= \varepsilon \,{u_{0}}_{xxt}$.
\vspace{3mm}
As for the problem ${\cal H}_\varepsilon $, instead of $(\ref{27})_3 $ the following conditions
\vspace{3mm}
\beq \label{29}
v_x(0,t)=0, \ \ v_x(\pi,t)=0, \,\,\,\,\,\,\,t\geq 0,
\eeq
\vspace{3mm}\noindent
must be specified.
\vspace{5.1mm}
\section {Linear case and explicit solutions}
\setcounter{equation}{0}
\hspace{5.1mm}
\vspace{3mm}
\vspace{3mm} Let $ \hat z(s)$ the Laplace-trasform of the function $z(t) $ and let
\vspace{3mm}
\begin{equation} \label {31}
\sigma (s) = \sqrt {\frac{s^2 +2as}{\varepsilon s +c^2}}.
\end{equation}
\vspace{3mm}\noindent
When $F $ is linear and the Laplace trasform is applied to the problem $\Delta$, the transform $\hat v(x,s)$ of the solution $v(x,t)$ is given by
\vspace{3mm}
\begin{equation} \label {32}
\hat v(x,s) = \int_{0}^{\pi}
\hat{G}_\varepsilon
(x,\xi,s) \ \hat F(\xi,s) \, d\xi,
\end{equation}
\vspace{3mm}\noindent where
\vspace{3mm}
\begin{equation} \label{33}
\hat{G}_\varepsilon = \, \frac{1}{2(\varepsilon s +c^2)} \,\,\, [\,\hat{g} (|x-\xi|, \sigma) -
\hat{g} (|x+\xi|, \sigma)\,]
\end{equation}
\vspace{3mm}\noindent and
\vspace{3mm}
\begin{equation} \label {34}
\hat{g}(y,\sigma) = \frac{cosh\,[(\pi-y)\,\sigma]}{
\sigma \, senh (\pi\sigma)}.
\end{equation}
\vspace{3mm}
\noindent
But, for $y \in [0, 2 \pi]$, it results \cite{g}:
\vspace{3mm}
\beq \label{35}
\hat g(y,\sigma) \,= \frac{1}{\pi \sigma^2} \,+ \frac{2}{\pi} \, \, \ \,\sum _{n=1}^{\infty} \,\, \frac{\,\cos\,(ny)}{n^2 + \sigma ^2}
\eeq
\vspace{3mm}
\noindent
and $ \cos\,(n|x-\xi|) \,\, - \,\, \cos\,(n|x+\xi|) \, = 2 \,\sin \,(nx)\,\, \sin\,(n \xi).$ So, by (\ref{33})- (\ref{35}), it follows
\begin{equation} \label{36}
\hat{G}_\varepsilon(x,\xi, s) = \frac{2}{\pi} \, \, \ \,\sum _{n=1}^{\infty} \,\, \, \frac{\,\sin\,(n\xi) \,\, \sin \, (nx)}{s^2 + 2as + (\varepsilon s+c^2 ) n ^2},
\end{equation}
\vspace{3mm}\noindent
which represents the ${\cal L}$ - transform of the Green function related to problem $\Delta$.
By means of elementary formulae, one deduces
\vspace{3mm}
\begin{equation} \label {37}
G_\varepsilon (x,\xi, t) = \frac{2}{\pi } \,\, \sum
_{n=1}^{\infty} \, \,
h_n\,(t, \varepsilon ) \,\, \sin (nx) \,\, \sin
(n\xi),
\end{equation}
\vspace{3mm}\noindent with
\vspace{3mm}
\beq \label{38}
h_n(t,\varepsilon)\,\,=e^{-(\frac{\varepsilon}{2}n^2 +a)\, t } \ \ \frac{\sin \,[\,
t \,\sqrt {c^2n^2- {(\frac {\varepsilon} {2}n^2 +a)^2}}\,]
}{ \sqrt {c^2 n^2- (\frac {\varepsilon }{2}n^2 +a)^2}}.
\eeq
\vspace{3mm}
Then, the esplicit solution $v$ of the problem $\Delta$ is:
\vspace{3mm}
\begin{equation} \label {39}
v (x,t)= \int_{0}^{t} \, \,d \tau \,\,\int_{0}^{\pi} F (\xi,\tau)\,\, \ {G}_\varepsilon (x,\xi, t-\tau) \,d\xi
\eeq
\vspace{3mm}
\noindent
with the Green function $G_\varepsilon$ defined by (\ref{37}),(\ref{38}).
The formal analysis developed so far can be justified as follows. Referring to (\ref{39})- (\ref{37}), the terms
\vspace{3mm}
\begin{equation} \label {310}
F_n\, (t)\, = \, \frac{2}{\pi} \, \, \int_{0}^{\pi} F (\xi,t)\,\, \sin \,(n\xi) \, \, d\xi
\eeq
\vspace{3mm}
\noindent
represent the Fourier coefficients of the sine series of the function $F(x,t)$:
\vspace{3mm}
\beq \label{311}
F(x,t) = \sum
_{n=1}^{\infty} \, \,
F_n\,(t) \,\, \sin (nx) \
\end{equation}
\vspace{3mm}\noindent
and the rapidity of pointwise convergence of this series depends, of course, on the properties of the source $F$. For instance, it can be sufficiently assumed that $\, F, \,\,F_x,\,\, F_{xx}\,$ are continuous in $(0,\pi)$ and more
\vspace {3mm}
\beq \label{312}
F(0,t)=F(\pi,t)=0.
\eeq
\vspace{3mm}
Then, the convergence of (\ref{311}) is uniform everywhere in $[0,\pi]$ and, further, it results:
\vspace{3mm}
\begin{equation} \label {313}
F_n\, (t)\, = \, - \, \frac{1}{n^2} \,\, \frac{2}{\pi} \, \, \int_{0}^{\pi} F_{\xi\xi} (\xi,t)\,\, \sin \,(n\xi) \, \, d\xi.
\eeq
\vspace{3mm}
As consequence, if one puts:
\vspace{3mm}
\begin{equation} \label {314}
v_n(t) \, = \,\, h_n * \, F_n = \,\, \int_{0}^{t} F_n (\tau)\,\,\, h_n(t-\tau) \, \, d\tau,
\eeq
\vspace{3mm}
\noindent
the solution (\ref{39}),(\ref{37}) represents the Fourier sine expansion
of $ v(x,t)$:
\vspace{3mm}
\beq \label{315}
v(x,t) = \sum
_{n=1}^{\infty} \, \,
v_n\,(t ) \,\, \sin (nx).
\end{equation}
\vspace{3mm}
\vspace{3mm}
{\bf Theorem 3.1}-
{\em When }$F(x, \cdot) \in C^2\, ( \Lambda)$ {\em and satisfies }\ref{312}, {\em the solution } $v(x,t) $ {\em of the problem }$\Delta$ {\em can be given the form }:
\vspace{3mm}
\begin{equation} \label {316}
v (x,t) \, = \, - \, \int_{0}^{t} \, \,d \tau \int_{0}^{\pi} F_{\xi\xi} \, (\xi,\tau)\,\, \ {H}_\varepsilon \, (x,\,\xi,\, t-\tau) \,\, d\xi,
\eeq
\vspace{3mm}
\noindent
{\em where} $H_\varepsilon$ {\em is}
\vspace{3mm}
\begin{equation} \label {317}
H_\varepsilon (\xi, x, t) = \frac{2}{\pi } \,\, \sum
_{n=1}^{\infty} \, \, \frac{
h_n\,(t, \varepsilon )}{n^2} \,\, \sin (nx) \,\, \sin
(n\xi)
\end{equation}
\vspace{3mm}\noindent
{\em and the convergence of the series is uniform everywhere in} $[0,\pi]$.
\hbox{}\hspace*{3mm}
\rule{1.85mm}{1.85mm}
\vspace{3mm}
{\bf Remark 3.1}- Theorem 3.1 can be applied also to the problem $ {\cal H}_\varepsilon$, provided that the Green function $G_\varepsilon$ is substituted by the following function:
\vspace{3mm}
\begin{equation} \label {318}
K_\varepsilon (\xi, x, t) =\, \,\frac{h_0(t)}{\pi}\,+\,\frac{2}{\pi } \,\, \sum
_{n=1}^{\infty} \, \,
h_n\,(t, \varepsilon ) \,\, \cos (nx) \,\, \cos
(n\xi),
\end{equation}
\vspace{5.1mm}
\section {Estimates and properties of the series $H_\varepsilon$}
\setcounter{equation}{0}
\hspace{5.1mm}
\vspace{3mm}
The arguments of sine functions in (\ref{38})\@ are real parameters when \@$ K_1 \leq n\, \leq\, K_2$, with
\vspace{3mm}
\beq \label{41}
K_1 = \,\, \frac{c}{\varepsilon}\,\,(\, 1- \sqrt{ 1- \frac{2 \, a \, \varepsilon }{c^2}}\,\,\,\,), \ \ \ K_2 = \,\, \frac{c}{\varepsilon}\,\,(1+ \sqrt{ \,1- \frac{2 \, a \, \varepsilon }{c^2}}\,\,\,\,).
\eeq
\vspace{4mm}
So, if $a\,<\,c$ \@ and $ N \,\equiv \,[K_2]$ , \@ the $h_n$ 's in (\ref{38}) contain trigonometric functions for $1 \, \leq \, n\, \leq \, N $ and hyperbolic functions for $ n \, \geq \, N+1$. Otherwise, if $a>c$, the trigonometric case is related only to $[K_1] \, \leq \, n\, \leq \, N $. This distinction is unimportant to what we are going to demonstrate; however it holds also for the Green function $G_0$ defined by (\ref{25}) and related to the problem ${\cal P}_0 $.
\vspace{3mm}
Let $ \, g_n(x,\xi) = (2/\pi) \,\, \sin(nx) \,\, \sin (n\xi)\,\,$ and consider the series
\vspace{3mm}
\vspace{3mm}
\begin{equation} \label {42}
H_0 \,\,= \,\,\,e^{-a \,t } \,\ \sum
_{n=1}^{\infty} \,\, \,\, \frac{\sin \,(\, t\,\sqrt{c^2 n^2 - a^2}\,)\,}{ \,\,n^2\,\, \sqrt{c^2 n^2 - a^2}}
\, \,g_n(x,\xi)
\end{equation}
\vspace{3mm}\noindent
deduced from $H_\varepsilon$ setting formally $\varepsilon \equiv 0$.
In order to estimate the difference $\,H_\varepsilon -\,H_0$, let
\vspace{3mm}
\begin{equation} \label {43}
A_n \,\,= a+ \frac{\varepsilon}{2} \,\, n^2 ; \,\, \,\, B_n^2 \,\, = {c^2 n^2 - A_n^2};\,\,\,\,\, b_n^2 =\,\,{c^2 n^2 - a^2}
\end{equation}
\vspace{3mm}\noindent
and
\begin{equation} \label {44}
r_n =\, \frac{h_n (t,\varepsilon)}{n^2}\,-\, \frac{h_n(t,0)}{n^2}= e^{-A_n \,\,t} \, \frac{sin (B_n \,t)}{ n^2 \,B_n} \,-\, e^{-at} \, \frac{sin (b_n \,t)}{ n^2 \,b_n}
\end{equation}
\vspace{3mm}\noindent
with $h_n(t,\varepsilon)$ defined by (\ref{38}). It results:
\vspace{3mm}
\beq \label{45}
H_\varepsilon -H_0 \,= R_1 \, + R_2 \, = \sum ^N _{n=1} \,\, \, r_n ( t,\varepsilon ) \,\, g_n \,+\,\sum ^\infty _{n=N+1} \,\, \, r_n ( t,\varepsilon ) \,\, g_n.
\eeq
\vspace{3mm}
If \@ $c_0$ \@ denotes the Euler constant \@$( c_0 \, \simeq \,0,5773)$\@
and $k$ an {\em arbitrary} constant such that \@ $ 0\,<\,k\,<\,1\,$, let $\,b^2_1=c^2-a^2$ and
\vspace{3mm}
\beq \label{46}
\rho (t) \, = \, \frac{t}{b_1} \, (2+at), \ \ \ c_1(\varepsilon) =\frac{\varepsilon + 2 c_0 \, c + (2c)^{2-k} \,\, \varepsilon ^{k-1}}{\pi \, c \, (1-k)}.
\eeq
\vspace{3mm}\noindent
Further, let $c_2 \equiv \, (1/3)\,\,max (1,\pi /cb_1).$ Then one has:
\vspace{3mm}
{\bf Lemma 4.1} - {\em For all } $ t\,\geq 0, \,\,x \in [0,\pi]\,$,{\em \@ when }$\varepsilon$ {\em is vanishing, the following estimates hold:}
\vspace{3mm}
\beq \label{47}
\mid R_1 \mid \,\, \leq \,\, \varepsilon \,\, c_1 (\varepsilon) \,\, \rho ( t) \, e^{-at}
\eeq
\vspace{3mm}
\beq \label{48}
\mid R_2 \mid \,\, \leq \,\, \varepsilon \,\, c_2 \,\, [ e^{-at} \,+ \, \theta \,e^{-\frac{c^2}{2}\,\theta}],
\eeq
\vspace{3mm}\noindent
{\em where} $\theta $ {\em denotes the fast time} $ t/\varepsilon$ {\em and }$ \varepsilon \, c_1 (\varepsilon)$ {\em vanishes with arbitrary order } $ k <1$.
\vspace{3mm}
{\bf Proof}- Referring to the trigonometric terms related to $R_1$, defined in (\ref{45}), by means of the Laplace transform, by (\ref{44}) one deduces that
\vspace{3mm}
\begin{equation} \label {49}
\hat r_n(s, \varepsilon)= \, -\,\frac{\varepsilon }{b_n} \,\, \frac{b_n}{(s+a)^2 \,+\,b_n^2}\,\,\,\frac{s}{(s+A_n)^2\,+\,B_n^2}
\end{equation}
\vspace{3mm}\noindent
hence
\vspace{3mm}
\begin{equation} \label {410}
r_n( t,\varepsilon) = \frac{\varepsilon }{b_n} [ e^{-at} \sin (b_n t)] * [ e^{-A_n t} ( \frac {A_n}{B_n} \sin (B_n \,t)- \cos (B_n t))].
\end{equation}
\vspace{3mm}\noindent
When this convolution is made explicit, by elementary estimates one has
\vspace{3mm}
\begin{equation} \label {411}
\mid r_n \, \,(t,\varepsilon) \mid \,\, \leq \, \, \frac{\varepsilon}{n} \,\, \rho \,\, e^{-a \,t} \ \ \ \ \ \ \ n \, \in [1, \,\, N]
\end{equation}
\vspace{3mm}
\noindent and so
\vspace{3mm}
\begin{equation} \label {412}
\rho^{-1} \, \,e^{at} \mid R_1 \mid \,\,\leq \, \, \frac{2 }{\pi} \,\, \sum ^N_{n=1} \,\, \frac{\varepsilon }{n}\,\, \leq \,\, \frac{2\, \varepsilon}{\pi} \, ( c_0 \, +\ \frac{1}{2N}+ ln\, N).
\end{equation}
\vspace{3mm} \noindent
For each positive
constant \@$\beta \, $, one has $\,\,\ ln \, N \, < \, \beta \,\, N ^{1/\beta}\,\, $ so that for $\, \beta \, = k^{-1} \,\,\, (k<1)\,\, $ the estimate (\ref{47}) follows, with $\rho $ and $c_1$ defined by (\ref{46}). As for $R_2$ one has:
\vspace{3mm}
\beq \label{413}
R_2 = \sum ^\infty _{n=N+1} \frac{1}{n^2} \,\, h_n ( t,\varepsilon ) g_n - e^{-at} \sum ^\infty _{n=N+1} \frac{sen(b_n t)}{n^2\, b_n} g_n = R'_2-R''_2
\eeq
\vspace{3mm}\noindent where the terms $h_n$ defined in (\ref{38}) \@ represent now hyperbolic functions $(B_n^2 <0)$. For this it results:
\vspace{3mm}
\beq \label{414}
h_n(t,\varepsilon)\,\, \leq \,\, t \,\, \,\,\ e^{-\,\,\frac{c^2 n^2 }{\varepsilon n^2 +2a}\,\, t } \ \\ \ \\\,\,\,\,\, \ \ \forall n\geq N+1
\eeq
\vspace{3mm}\noindent
and $ N+1 \geq 2c/\varepsilon $ \@ for \@ $ \varepsilon < 2(c-a). $ \@ As consequence:
\vspace{3mm}
\beq \label{415}
\mid R'_2 \mid \,\, \leq \,\, \frac{2 \,t}{\pi} \,\, e^{-\,\frac{c^2}{2} \, \frac{t}{\varepsilon}}\, \sum ^\infty _{n=1} \,\, \frac{1}{n^2}\,\, =(1/3) \,\, \varepsilon \,\, \theta\,\, e^{-\,\frac{c^2}{2}\, \theta},
\eeq
\vspace{3mm}\noindent
with $\theta = t/\varepsilon$. At last, as $ b_n \,\, > b_1 \,n $, one deduces that
\vspace{3mm}
\beq \label{416}
e^{at} \, \mid R''_2 \mid \,\, \leq \,\, \frac{2 }{\pi\, b_1} \,\, \sum ^\infty _{n=N+1} \,\, \frac{1}{n^3}\,\, \leq \frac{1}{3 b_1} \,\, (N+1)^{-1} \leq \,\, \frac{\pi \, \varepsilon }{3c \, b_1} .
\eeq
\vspace{3mm}\noindent
and (\ref{415})-(\ref{416}) imply the estimate (\ref{48}).
\hbox{}\hspace*{5mm}
\rule{1.85mm}{1.85mm}
\vspace{3mm}
Referring to (\ref{48}), let observe that
\vspace{3mm}
\beq \label{417}
\theta \, e^{-\frac{c^2}{2}\, \theta} \,\, \leq \,\,( 4/c^2 e
)\,\,
e^{-\frac{c^2}{4}\,\theta}
\eeq
\vspace{3mm}\noindent
and let $ b\equiv \, min \, (a, c^2 /4 \varepsilon)$.Thus, by (\ref{46}),(\ref{47}),(\ref{48}) one has
\vspace{3mm}
\beq \label{418}
\mid \, R_1 \, \mid \, +\, \mid \, R_2 \, \mid \,\, \leq \, \varepsilon ^k \, r(t) \, e^ {-b\,t}
\eeq
\vspace{3mm}\noindent
with
\vspace{3mm}
\beq \label {419}
r(t) \, = \varepsilon ^{1-k} \,\, [\, c_1 (\varepsilon) \, \rho (t) \, + \, c_2 \, ( 1+ 4/c^2 e)].
\eeq
\vspace{3mm}\noindent
Then, the following theorem can be stated.
\vspace{3mm}
{\bf Theorem 4.1 }- {\em Whatever the positive constant }$ k<1$ {\em may be, for all}$\,\,t \,\geq \,0\,\,$ {\em and }$x \in [0,\pi]$, {\em it results}:
\vspace{3mm}
\beq \label{420}
\mid \, H_\varepsilon \, - \,H_0\, \mid \, \leq \, \gamma \, \varepsilon ^k \, ( 1+ t+ t^2) \, e^{-b\,t},
\eeq
\vspace{3mm}
\noindent
{\em where the constant } $\, \gamma $ {\em depends only on } $k,\,a,\,c$.
\vspace{3mm}
{\bf Remark 4.1}- The asymptotic analysis of this section and the results of theorem 4.1 can be applied also to the function $ K_\varepsilon $ defined by (\ref{318}) and related to the problem ${\cal H}_\varepsilon$.
\vspace{5.1mm}
\section {Conclusions}
\setcounter{equation}{0}
\hspace{5.1mm}
\vspace{3mm}
To outline a first application and to avoid too many
formulae, let consider only the term depending on the source. Then, referring to the problems ${\,\cal
P}_\varepsilon \,$ and ${\,\cal
P}_0\, $ and putting $\,\Delta H \, = H_\varepsilon \, -\, H_0\,$, it results
\vspace{3mm}
\begin{equation} \label {51}
u_{\varepsilon}\, -\, u_0\,= -\, \int_{0}^{\pi} \, \,d \xi \,\,\int_{0}^{t} f_{\xi\xi} (\xi,\tau)\,\, \ {\Delta H} (x,\xi, t-\tau) \,d\tau.
\eeq
\vspace{3mm}
By assuming that $\, f \in C^2(\Omega)\, $ and that $\, f_{xx}\,$ is bounded also when $ \, t \, \rightarrow \, \infty\, $, let
\begin{equation} \label {52}
||\,u_f \,|| \, = \, \sup_\Omega \,\,|\,f_{xx} \, (x,t)\,|.
\eeq
\vspace{3mm}\noindent
As consequence of theorem 4.1, when $\, \varepsilon \, \rightarrow 0\,$, the following rigorous approximation holds:
\vspace{3mm}
\begin{equation} \label{53}
u_\varepsilon \, =\, u_0\, + \, \varepsilon ^k \,\, r \,\,\,\\\ \ \ \ \\\ \ \ \ \forall (x,t) \in \Omega
\eeq
\vspace{3mm}\noindent
where the error $\, r \, $ is such that
\vspace{3mm}
\begin{equation} \label{54}
|\, r\,| \, \leq \, \gamma _1 \, \,|| u _f|| \,\,\,\\\ \ \ \ \\\ \ \ \ \forall t \geq 0
\eeq
\vspace{3mm}\noindent
and the constant $ \, \gamma _1 \, $ depends only by $ a,\, \, c,\, k.\,$ So, {\em the error of the approximation is neglegible also for large } $ \, t \,\,(t \rightarrow \infty).$
\vspace{3mm}
When $\,f\,$ is non linear, an integral equation like
\vspace{3mm}
\begin{equation} \label {51}
v \,= \int_{0}^{\pi}d \xi \int_{0}^{t} {\Delta H} (x,\xi, t-\tau) F
[\xi,\tau, u(\xi, \tau ), u_\tau(\xi,\tau), u_\xi(\xi,\tau)]d\tau
\eeq
\vspace{3mm}\noindent
must be analyzed. These applications will be dealt successively.
\hspace{5.1mm}
\vspace{3mm}
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\end{document}
|
{
"timestamp": "2012-03-13T01:00:18",
"yymm": "1203",
"arxiv_id": "1203.2193",
"language": "en",
"url": "https://arxiv.org/abs/1203.2193"
}
|
\section{Introduction}
Optical System for Imaging and low Resolution Integrated Spectroscopy (OSIRIS)
is an imager and spectrograph for the optical wavelength range, located
in the Nasmyth-B focus of the 10.4-m {\it Gran Telescopio Canarias} (GTC)
\citep{Cepa05, Cepa10}.
Apart from the standard broad-band imaging and long-slit spectroscopy,
it provides additional capability such as narrow-band tunable
filter (TF) imaging, charge-shuffling and multi-object spectroscopy.
OSIRIS covers the wavelength range from 0.365 to 1.05~$\mu$m with an
unvignetted field of view (FoV) of $7.8\times7.8$\,arcmin$^{2}$, for direct
imaging. Narrow-band imaging is made possible through the use of a TF,
which is in essence a low resolution Fabry-Perot etalon. The filter is tuned
by setting a specific separation between the optical plates of the Fabry-Perot.
OSIRIS has two different TFs: one for the blue range (3750--6750\,\AA;
yet to be commissioned), and another for the red range (6510--9350\,\AA).
In this work, we describe the results for the latter (red) etalon.
The relatively large FoV on a 10-m telescope, combined with the tunable
nature of the filters, opens up a new method of observation, which can be
useful in a variety of astrophysical contexts, both galactic and
extragalactic.
The use of narrow-band filters, adequately defined to map certain
spectral regions, allows the detailed study at seeing-limited resolution, of
the spatial distribution of the ionized gas properties and recent star
formation, and of the associated stellar populations. Flux calibrated images
in the bright lines of hydrogen, oxygen, nitrogen and sulphur in the optical,
allow us to map, among other parameters, the metallicity gradients
in galaxies using imaging techniques rather than the time-consuming
spectroscopic techniques that are in use currently \citep{Rosa10}.
This has motivated recently built large telescopes such as the 11-m South
African Large Telescope \citep{Rang08} or the Magellan Baade 6.5-m telescope
\citep{Veil10} to have TFs for imaging. We are presently engaged in a survey
of galaxies in the local universe (LUS\footnote{http://www.inaoep.mx/$\sim$gtc-lus/}= Local Universe Survey) that includes
all galaxies inside a volume of 3.5~Mpc in radius that are visible from
La Palma. IFU observations of these galaxies present two problems: IFU
observations would represent a huge drop in spatial resolution and, with a
median D$_{25}$ of 4.5$^\prime$, they are too large to be observed even with
the largest available IFU.
\citet{Lara10} used template spectra typical of star-forming galaxies
to estimate the errors in derived flux
for extragalactic point-like sources on the OSIRIS TF images. However,
photometric accuracy of the TF images of extended objects using observed data
is yet to be evaluated. In this study, we explore the capabilities of TF for
obtaining flux calibrated images using observations acquired with OSIRIS TF.
The OSIRIS etalon is placed in a converging beam, which results in the
variation of the effective wavelength over the FoV \citep{Born80}.
For the red etalon, the change is as much as 80\,\AA\, over the 4$^\prime$
radius, thus limiting the monochromatic FoV to less than 2$^\prime$ radius
\citep{Mend11}. In addition, the TF spectral response function
(see Eq.~\ref{eqn_Rlambda}) has a long tail, which difficults the
flux calibration of a
faint line in the neighborhood of a bright one. For example
H$\alpha$ line contaminates even if the TF is tuned to detect the
[\textsc{N\,ii}]$\lambda6583$ line. Given that these characteristics of TF are
well-understood, flux calibrated monochromatic images over the entire FoV can
be obtained using a tailor-made post-observation reconstruction software.
The fact that the filters are tunable, in addition, creates its own problems for
absolute flux calibration due to the non-standard nature of the filters.
Observing spectrophotometric standard stars at each tuned wavelength is
a solution, but it is costly in terms of telescope time.
Alternatively, small telescopes could be used to carry out spectrophotometric
observations of the in-frame field stars. The recent Sloan Digital Sky Survey
(SDSS) spectrophotometric survey of field stars \citep{Beer06, Abaz09} offers
an attractive solution to the problem, that does not require any telescope
time for calibration purposes.
In this paper, we explore the use of this database to spectrophotometrically
calibrate the in-frame SDSS photometric stars.
In \S2, we describe the data used in this work. The method
we followed for reconstructing a monochromatic image and its implementation
are described in \S\S3 and 4, respectively.
The sensitivity of the reconstructed
emission line fluxes to various observational parameters is studied using simulated images in \S5 where we also
suggest guidelines for planning imaging observations with OSIRIS/TF.
The flux calibration technique and accuracy
are described in \S6. H$\alpha$ fluxes of selected \textsc{H\,ii}\ regions in the
reconstructed image are compared to an H$\alpha$ image obtained in a
traditional way in \S7. The results of this study are summarized in \S8.
\section{Description of the Data for Reconstruction}
\subsection{The OSIRIS detector and image formats}
OSIRIS uses two CCDs of $2048\times4102$ pixel format to cover its total
FoV of $8^\prime\times8^\prime$, with a physical gap of
$\sim9^{\prime\prime}$ between the two CCDs.
Each one has 50 additional overscan pixels
at the beginning of the array, thus resulting in a format of $2098\times4102$
for each image. The optical center of OSIRIS lies in the central gap
as described in \citet{Mend11}
and also in the OSIRIS User Manual\footnote{http://www.gtc.iac.es/en/media/documentos/OSIRIS-USER-MANUAL{\_}v1.1.pdf}.
Astrometric calibration is required before being able to mosaic the
images registered in the two CCDs.
By performing astrometry of 10 stars on each CCD that are
distributed over the entire FoV, we found that the image scale is slightly
different for the two CCDs. The left and right CCDs have values of
0.127\arcsec/pixel and 0.129\arcsec/pixel, respectively, with an average
value of 0.128\arcsec/pixel. This average value agrees well with the
theoretically expected value of the plate scale for the GTC/OSIRIS instrument
parameters listed in Table~1 of \citet{Mend11}.
\subsection{OSIRIS TF emission-line imaging of M101}
We illustrate the reconstruction technique using observations of M101 aimed
at obtaining continuum-free images of the H$\alpha$, [\textsc{N\,ii}]$\lambda6583$,
[\textsc{S\,ii}]$\lambda6716$ and [\textsc{S\,ii}]$\lambda6731$ emission lines over a FoV
of $8^\prime\times8^\prime$. The observations were carried out at
the GTC in two observing runs on June 22 and 26 in 2009, the former for the
[\textsc{S\,ii}]$\lambda\lambda 6716, 6731$ lines and the latter for the H$\alpha$,
[\textsc{N\,ii}]$\lambda6583$ lines. Table~1 describes the details of these two
observing runs. The telescope pointing remained the same for all images (wavelengths)
constituting a TF scan. The telescope position was
then dithered by 6\arcsec\ and the entire sequence was repeated. This was done in order
to facilitate removal of any detector artefacts. For a given region in the
galaxy, the two dithered TF images have slightly different wavelength and
hence different response for the detection of a line. Thus it is important
to take into account these response differences before coadding them.
We handled the dithered image sets as independent sets of data
and used them to estimate internal errors in the flux calibration (see \S6).
For the H$\alpha+$[\textsc{N\,ii}] scan, observations were carried out at two dithered
positions (P1 and P2, henceforth), whereas for the [\textsc{S\,ii}] scan, three
dithered positions were used (P1, P2 and P3, henceforth).
\begin{table}
\centering
\caption{\label{Tab:OBs}GTC observing blocks summary}
\begin{tabular}{lcccc}
\hline
Observing Block & $\lambda_{c}$ & FWHM & Order Sorter & Exp. Time \\
\ & (\AA) & (\AA) & Filters & (seconds) \\
\hline
H$\alpha+$[\textsc{N\,ii}] & 6528 & 18 & f648/28 & $2\times180$ \\
& 6548 & 18 & f648/28 & $2\times180$ \\
& 6568 & 18 & f648/28 & $2\times180$ \\
& 6588 & 18 & f657/35 & $2\times180$ \\
& 6608 & 18 & f657/35 & $2\times180$ \\
& 6628 & 18 & f657/35 & $2\times180$ \\
& 6648 & 18 & f657/35 & $2\times180$ \\
& 6668 & 18 & f657/35 & $2\times180$ \\
& 6688 & 18 & f666/36 & $2\times180$ \\
\ [\textsc{S\,ii}] & 6696 & 18 & f666/36 & $3\times180$ \\
& 6716 & 18 & f666/36 & $3\times180$ \\
& 6736 & 18 & f666/36 & $3\times180$ \\
& 6756 & 18 & f666/36 & $3\times180$ \\
& 6776 & 18 & f680/43 & $3\times180$ \\
& 6796 & 18 & f680/43 & $3\times180$ \\
& 6816 & 18 & f680/43 & $3\times180$ \\
\hline
\end{tabular}\\
\end{table}
Initial reduction of the images for correcting for BIAS was done
independently for the two CCDs constituting an image. Reliable flat-field
frames were not available for the set of data we were analyzing, hence
no attempts were made to correct for this. Typical pixel-to-pixel errors
in response are expected to be less than 1\% for the CCD chips used.
Stars in the field of each of the two CCDs
were independently analyzed to obtain the astrometric solution between the
pixel and equatorial coordinates, using the SDSS coordinates of
around 10 stars in the observed field as reference. The two CCDs were then stitched together
to form a single image, free of all geometrical distortions.
The procedure is repeated for the other scans.
\section{Monochromatic Image Reconstruction Technique}
\subsection{Basic formulae for tunable filters}
The effective wavelength $\lambda_r$ at a distance $r$ from the optical
center of the tunable filter (TF) changes following the law
\citep{Mend11, Beck98}:
\begin{equation}
\lambda_r = {\lambda_{\rm c}\over\sqrt{1+6.5247\times10^{-9} r^2}} ,
\label{eqn_lambda}
\end{equation}
where $\lambda_{\rm c}$ is the wavelength at the optical center. The
coefficient of $r^2$ is the inverse square of the effective focal
length of the camera,
with $r$ expressed in physical units of pixels of 15$\mu$m in size.
Due to the geometrical distortions of the images, a given $\lambda_r$
moves to a different radial distance $r^\prime$ in the astrometrically
calibrated images. We used the astrometric solution to map the new
wavelengths $\lambda_{r^\prime}$ at each astrometrically calibrated image
with resampled pixels of 0.125\arcsec\ in size.
This correction, which is not symmetric around the optical center,
increases quadratically with radial distance from the optical center,
reaching values of $\sim2$ and $\sim5$~\AA\ at $4^\prime$ away from the
optical center on the left and right CCDs, respectively.
For a TF of Full Width at Half Maximum FWHM, the filter response for a certain
wavelength $\lambda$ (e.g.~the H$\alpha$ line) in a pixel where the nominal
wavelength is $\lambda_r$, is given by:
\begin{equation}
R_\lambda(r) = \left( 1 + \left[\frac{2(\lambda-\lambda_r)}{FWHM}\right]^2\right)^{-1}.
\label{eqn_Rlambda}
\end{equation}
This equation is an approximation valid when the
free spectral range (i.e.~the wavelength separation
between adjacent orders) is much larger than
the spectral purity (i.e.~the smallest measurable
wavelength difference) --- see, e.g., \citet{Jones02}.
The ratio between free spectral range and spectral purity
is called finesse and is a critical parameter of all TFs.
In the case of OSIRIS, the finesse is of the order of 50
(see OSIRIS User Manual), which allows the use of Eq.~\ref{eqn_Rlambda}
\citep{Jones02}. Spectral purity and FWHM coincide
when the finesse is large. The wavelength at the optical center
($\lambda_c$ in Eq.~\ref{eqn_lambda}) as well as the FWHM are set by tuning
the etalon. In our observations the FWHM
was set to 18\,\AA\ (see Table 1) rendering
a free spectral range of the order of 900\,\AA.
A monochromatic image at a particular wavelength (e.g.~H$\alpha$) over
the entire FoV of the detector can be obtained if we have a sequence of
images where $\lambda_{\rm c}$ between successive images increases by
$\lesssim$FWHM. For the image sequences we have, this criterion was not
strictly met ($\Delta\lambda$ between successive images was 20~\AA\ for a
FWHM=18~\AA). The consequence of this is that there are annular zones with
missing data. However, multiple observations with dithered positions helped
us to fill-in data for these zones.
In what follows we describe the method we have
adopted for reconstructing a monochromatic image from M101 image sequences.
\subsection{Reconstruction Strategy}
The observed count rate $F(x,y)$ at a pixel $x,y$ of an image is related to
the emitted intensity $I_\lambda(x,y)$ by the equation:
\begin{equation}
F(x,y) = \frac{\int{I_\lambda(x,y) R_\lambda(r) d\lambda}}{\kappa} + {\rm Sky}(x,y)
\label{eqn_recon1}
\end{equation}
where $R_\lambda(r)$ (Eq.~\ref{eqn_Rlambda}) is the value of the response
curve at a distance $r=\sqrt{(x-x_0)^2+(y-y_0)^2}$ from the optical
center ($x_0,y_0$), $\kappa$ is the
conversion factor between the intensity and the count rate in units of
erg\,cm$^{-2}$\,s$^{-1}$/(count\,s$^{-1}$), and $Sky(x,y)$ is the count
rate from the sky at a pixel $x,y$. In \S6, we describe a procedure
to determine the value of $\kappa$. Ideally, the sky term can be
estimated if observations of sky frames are taken in each scanned wavelength.
Given that this involves extra telescope time, such observations were not
carried out.
The dependence of sky on $x,y$ is because of two causes (1) an intrinsic
spatial variation of the sky, and (2) a wavelength dependence of sky emission.
The first of these can be assumed to be negligible over the FoV of OSIRIS
and hence the sky variation in the image is principally due to the second
effect. Given the circular symmetry of the variation of wavelength, sky values
are radially symmetric around the optical center,
i.e. ${\rm Sky}(x,y)\equiv {\rm Sky}(r)$.
We determined ${\rm Sky}(r)$ as the average count rate from the sky pixels in
annular zones of width=$FWHM/2$.
In general, $I_\lambda$ includes two contributions: (1) $I_{\rm line}$,
emission line flux integrated over the line profile,
and (2) $I_{\rm cont}$, the continuum flux density.
We first describe the method for reconstructing monochromatic images
in the simplest case of only one emission line
entering the filters in the entire scanned wavelength range.
The emission lines from nebular sources are extremely narrow with
respect to the typical bandwidth of the TF, and hence the count rate from a
line depends only on the value of the response function at the wavelength
of the line, $R_{\rm line}(r)$. Hence the integral can be replaced by a
simple multiplication,
i.e. $\int{I_{\rm line}(x,y) R_\lambda(r) d\lambda} =
I_{\rm line}(x,y) R_{\rm line}(r)$, when line-width $<< FWHM$.
On the other hand, the term due to the continuum from the source is
$\frac{\int{I_{\rm cont}(x,y) R_\lambda(r) d\lambda}}{\kappa}\equiv
{\rm Cont}(x,y)$,
where $Cont(x,y)$ is the observed count rate in the continuum image.
With these approximations Eq.~\ref{eqn_recon1} can be re-written as:
\begin{equation}
F(x,y) = \frac{I_{\rm line}(x,y) R_{\rm line}(r)}{\kappa} + {\rm Cont}(x,y) + {\rm Sky}(r).
\label{eqn_recon2}
\end{equation}
A trivial manipulation of the above equation gives:
\begin{equation}
I_{\rm line}(x,y) = \frac{\kappa \left(F(x,y) - {\rm Cont}(x,y) - {\rm Sky}(r)\right)}{R_{\rm line}(r)}
\label{eqn_recon3}
\end{equation}
By making a simple substitution
\begin{equation}
C(x,y) = F(x,y) - {\rm Cont}(x,y) - {\rm Sky}(r),
\label{eqn_recon5}
\end{equation}
where $C(x,y)$ is the sky and continuum subtracted count rate at a
position $x,y$ of the image, the above equation can be re-written as:
\begin{equation}
I_{\rm line}(x,y) = \frac{\kappa\ C(x,y)}{R_{\rm line}(r)}.
\label{eqn_recon4}
\end{equation}
Due to the long tail of the response curves, an emission line from some
regions of the galaxy is registered by more than one image of the scan.
Hence, in general,
simultaneous equations of the kind of Eq.~\ref{eqn_recon4} can be written
for each image. The line contribution in the two adjacent images
can be combined, with the weights for combining being determined
by the values of the response functions for the line at each pixel.
A generalized equation for recovering the flux from any pixel where the
line is registered by two consecutive images of a scan: $im1$
and $im2$, can be written as:
\begin{equation}
I_{\rm line}(x,y) = {\left[\kappa C(x,y)\right]_{im1} + \left[\kappa C(x,y)\right]_{im2}\over{\left[R_{\rm line}(r)\right]_{im1}+\left[R_{\rm line}(r)\right]_{im2}}}
\label{eqn_recon6}
\end{equation}
For the sake of compactness, from now onwards $R_\lambda(r)$ will be denoted
simply by $R_\lambda$.
\subsection{H$\alpha$ and [\textsc{S\,ii}] scans}
Due to the long tail of the TF response curve, an emission line separated
from the target line by $\sim FWHM$ contributes non-negligibly to the flux
received in the image. This is the case while mapping the H$\alpha$ line
with a tunable filter of FWHM$\sim18$\,\AA, where the contribution of the
flanking [\textsc{N\,ii}] lines to the observed flux cannot be neglected.
The contribution from an unwanted line becomes even more important
while trying to recover the [\textsc{N\,ii}] lines, as the H$\alpha$ line is likely to
dominate the observed flux, rather than the [\textsc{N\,ii}] lines.
Another interesting case is that of the [\textsc{S\,ii}] doublet,
where both lines contribute in most of the images
of a typical TF scan. These cases can be easily treated by adding more
terms --- one term such as $I_{\rm line}(x,y) R_{\rm line}(r)$ for each
emission line --- to the numerator of the first term in Eq.~\ref{eqn_recon2}.
For the scan involving the H$\alpha$ line, the first term in the
{\it numerator} of Eq.~\ref{eqn_recon2} should be replaced by:
\begin{equation}
I_{\rm line}(x,y) R_{\rm line} \rightarrow
I_{{\rm H}\alpha} R_{\lambda{\rm H}\alpha}
+ I_{6583} R_{\lambda 6583}
+ I_{6548} R_{\lambda 6548},
\label{eqn_recon7}
\end{equation}
where $R_{\lambda{\rm H}\alpha}$, $R_{\lambda 6583}$, and
$R_{\lambda 6548}$ are the values of the response function
at a radial distance $r$ from the optical center
for the observed (not the rest-frame) wavelengths of the corresponding lines.
Eq.~\ref{eqn_recon4} for the ${\rm H}\alpha$ line can then be re-written as,
\begin{equation}
I_{{\rm H}\alpha}(x,y) = \frac{\kappa C(x,y)}{R_{\lambda{\rm H}\alpha}+\frac{I_{6583}}{I_{{\rm H}\alpha}}\left(R_{\lambda 6583} + \frac{1}{3} R_{\lambda 6548}\right)},
\label{eqn_recon8}
\end{equation}
where we have substituted the value of the intrinsic flux ratio
of $\frac{I_{6548}}{I_{6583}}=\frac{1}{3}$. The equation has a term
involving $\frac{I_{6583}}{I_{{\rm H}\alpha}}$ in the denominator.
Thus in order to reconstruct the H$\alpha$ line image, we need to know
{\it a priori} the flux ratio images of the contaminating [\textsc{N\,ii}]$\lambda6583$
line with respect to the H$\alpha$ line.
The [\textsc{N\,ii}] line fluxes, on the other hand, require a knowledge of
the H$\alpha$ flux, as can be seen by the equation for the
[\textsc{N\,ii}]$\lambda6583$ line:
\begin{equation}
I_{6583}(x,y) = \frac{\kappa C(x,y) - {I_{{\rm H}\alpha}}(x,y) R_{\lambda{\rm H}\alpha}}{R_{\lambda 6583}}.
\label{eqn_recon9}
\end{equation}
In this equation, we have neglected the [\textsc{N\,ii}]$\lambda6548$ line
contribution given that it contributes less than $2\%$ in image sections
tuned to maximize the [\textsc{N\,ii}]$\lambda6583$ emission.
Thus, in order to obtain [\textsc{N\,ii}]$\lambda6583$ flux map,
we need to know the H$\alpha$ flux. An iterative procedure involving
Eqs.~\ref{eqn_recon8} and ~\ref{eqn_recon9} is necessary for an accurate
recovery of both H$\alpha$ flux and $\frac{I_{6583}}{I_{{\rm H}\alpha}}$ ratio.
Given the exploratory nature of the present study, we have calculated the
H$\alpha$ fluxes by fixing $\frac{I_{6583}}{I_{{\rm H}\alpha}}=0.1$
for all regions. This approximation would result in an overestimation of
the H$\alpha$ fluxes by $>2.5$\%, and underestimation
of the $\frac{I_{6583}}{I_{{\rm H}\alpha}}$ ratios by $\sim0.1$
for regions having intrinsic $\frac{I_{6583}}{I_{{\rm H}\alpha}}\gtrsim0.2$.
The reconstruction equations for the $\lambda6716$ and $\lambda6731$ lines
from the [\textsc{S\,ii}] scan are:
\begin{equation}
I_{6716}(x,y) = \frac{\left[\kappa C(x,y)\right]_{im1} + \left[\kappa C(x,y)\right]_{im2}}
{R^{im1}_{\lambda 6716} +R^{im2}_{\lambda 6716} +\frac{I_{6731}(x,y)}{I_{6716}(x,y)} (R^{im1}_{\lambda 6731}+R^{im2}_{\lambda 6731})},
\label{eqn_recon10}
\end{equation}
and
\begin{equation}
I_{6731}(x,y) = \frac{\left[\kappa C(x,y)\right]_{im1} + \left[\kappa C(x,y)\right]_{im2}}
{R^{im1}_{\lambda 6731} +R^{im2}_{\lambda 6731} +\frac{I_{6716}(x,y)}{I_{6731}(x,y)} (R^{im1}_{\lambda 6716}+R^{im2}_{\lambda 6716})},
\label{eqn_recon11}
\end{equation}
In these equations, $im1$ and $im2$ denote the image sections tuned
to maximize the $\lambda6716$ and $\lambda6731$ lines, respectively.
Note that, in order to recover the $I_{6716}$ image, we need the $I_{6731}$
image and vice versa. Unlike the case of
$\frac{I_{\rm N\,II}}{I_{{\rm H}\alpha}}$, no apriori values for
the intensity ratios of the [\textsc{S\,ii}] lines could be used, given that this ratio
is very sensitive to the electron density of the regions.
We hence resolved Eqs.~\ref{eqn_recon10} and \ref{eqn_recon11}
by assuming a value of
$\frac{I_{6716}}{I_{6731}}$, and iteratively changing that value until the
value at every pixel stabilizes within 10\% in two successive iterations.
We checked that the resulting images
are the same irrespective of the starting value of $\frac{I_{6716}}{I_{6731}}$.
\section{Implementation of the method}
We developed a script under the IRAF\footnote{IRAF is distributed by the National Optical
Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under
cooperative agreement with the National Science Foundation.} environment to implement the method
in a user-friendly way. The first step of the reconstruction process is
obtaining wavelength {\it vs.} pixel look-up images for every image
of the scan. These images are created using Eq.~\ref{eqn_lambda} and
the astrometric solutions
as described in \S3.1. Values of $\lambda_{\rm c}$\footnote{
Wavelength calibration lamps were not supplied in this initial observing run,
and hence we checked/recalibrated the value of $\lambda_{\rm c}$ by comparing
the filter-convolved SDSS spectra of 7 \textsc{H\,ii}\ regions to the profiles of the
corresponding \textsc{H\,ii}\ regions formed using the observed fluxes in successive
images. Error in $\lambda_{\rm c}$ using this method is found to be
$\lesssim2$\AA, which is better than what could be achieved using the sky rings.
}
and FWHM are taken from Table~1.
For each of the emission lines, we then created 2-dimensional response
images using Eq.~\ref{eqn_Rlambda}. The procedure we followed is
illustrated using 1-d cuts on one of these images in Fig.~\ref{fig_lambda}.
Specifically, we chose the image with $\lambda_{\rm c}=6588$~\AA\ for
illustration, as this image is capable of detecting three different
emission lines at different radial bins. The bottom panel
shows the expected value of the response functions for the three
lines as a function of the radial distance from the optical center.
\begin{figure}
\begin{center}
\includegraphics[scale=.5]{f1a.eps}
\includegraphics[scale=.5]{f1b.eps}
\caption{
(Top) An illustration of the variation of the wavelength over the OSIRIS FoV
as expected from Eq.~\ref{eqn_lambda}, with $\lambda_{\rm c}=6588$~\AA\
(observed wavelength of [\textsc{N\,ii}]$\lambda6583$ line in M101) and
FWHM=18~\AA. The positions where [\textsc{N\,ii}]$\lambda6583$, H$\alpha$,
[\textsc{N\,ii}]$\lambda6548$, lines have maximum response are marked.
(Bottom) The response curves, as calculated using Eq.~\ref{eqn_Rlambda},
for each of these lines are shown. The part of the curves above the
horizontal line ($\eta_{\rm line}=0.4$) can be used to reconstruct the
monochromatic images in the corresponding lines.
\label{fig_lambda}}
\end{center}
\end{figure}
In cases such as this, where more than one emission line is registered within
the $8^\prime\times8^\prime$ FoV ($r=4^\prime$), a second line starts
contributing significantly before the line of interest drops to below
$\sim$50\% response, as illustrated in the bottom panel of
Fig.~\ref{fig_lambda}.
It is possible to isolate the contribution of each line
by selecting only those zones where the line of interest has a response
value above a certain level, as described below.
\subsection{Choosing the value for the response cut-off}
The part of a TF image that can be considered monochromatic is decided by the
value for the response cut-off ($\eta_{\rm line}$) parameter.
By carefully choosing $\eta_{\rm line}$, it is possible to reconstruct
a monochromatic image in the line of interest even in the presence of
contaminating lines.
Only those pixels with a response value $R_{\rm line}>\eta_{\rm line}$
will be considered good for reconstructing the monochromatic image in
that $line$. All these pixels belong to an annular zone of particular width
(circle in the case of a line falling close to the optical center).
For example, with $\eta_{\rm line}=0.4$, the image sections for reconstructing
the H$\alpha$ and [\textsc{N\,ii}]$\lambda6583$ lines in the $\lambda_{\rm c}=6588$~\AA\
image correspond to the radial zones where the curve lies
above the horizontal line in Fig.~\ref{fig_lambda}.
Two guidelines are useful for setting the value of $\eta_{\rm line}$:
(1) the response for the contaminating line in the selected image section
has to be less than that for the main line,
(2) there are no annular gaps
in the final reconstructed image. The first condition depends on
the wavelength difference between the contaminating lines, whereas the
difference between central wavelengths of successive images of the scan
($\Delta\lambda_{\rm c}$)
determines the fulfillment of the second condition.
For the case of H$\alpha$ and [\textsc{N\,ii}]$\lambda6583$, the response value
would be the same at a wavelength mid-way between the two lines
(i.e. 6573~\AA\ at rest-frame or 6578~\AA\ for our M101 scan given its
recession velocity of 214 km\,s$^{-1}$). At radial zones where
$\lambda_r<6578$~\AA, the
observed count rates would be predominantly from the H$\alpha$ line,
implying $\eta_{\rm line}\geq0.45$.
On the other hand, the second criterion requires that $\lambda_r$ be separated from
$\lambda_{{\rm H}\alpha}$ by at least $\Delta\lambda_{\rm c}/2$, implying
$\eta_{\rm line} \leq 0.45$ for our M101 scan. Thus $\eta_{\rm line}=0.45$ is the
optimal value for our dataset. The value of $\eta_{\rm line}$ could be
marginally higher than this if dithered images are able to fill in data-less annular
zones, or lower if the contaminating line contribution could be subtracted
to within a few percent accuracy using an iterative procedure.
For example, for our M101 scan, dithering between the images
was sufficient to be able to fill-in the data gaps for $\eta_{\rm line}=0.5$.
On the other hand, the SNR of the [\textsc{N\,ii}]$\lambda6583$ images was not good
enough to iteratively subtract the [\textsc{N\,ii}]$\lambda6583$ contamination in the
predominantly H$\alpha$ pixels. The user would be required to select the
value of $\eta_{\rm line}$ based on the data parameters and the specific
scientific objective.
\subsection{Coadding monochromatic image sections}
In Fig.~\ref{fig_netresp}, we show the response for the
H$\alpha$ line in consecutive images of a TF scan. For a value
of $\eta_{\rm line}=0.4$, in certain ranges of radial zones, the line
is registered in two consecutive images. Thus, for $\eta_{\rm line}=0.4$
there is redundant data for some pixels
for the construction of the monochromatic images. This redundancy can be
used to our advantage to get deeper images, by coadding the pixel values
from both images that contribute to that zone. The response curves
are also coadded to get a net response curve such as shown by the
solid line in Fig.~\ref{fig_netresp}. The coadded line image is
divided by the net response image to get the entire image in the same
flux scale (see Eq.~\ref{eqn_recon6}).
\begin{figure}
\begin{center}
\includegraphics[scale=.50]{f2.eps}
\caption{
The net response curve for the reconstruction of the H$\alpha$ image
for $\eta_{\rm line}=0.4$ is shown by the solid curve.
H$\alpha$ response curves in individual TF images are shown by the
dashed lines. The central wavelength $\lambda_{\rm c}$ (in \AA) for
each image is indicated above the corresponding response function.
\label{fig_netresp}}
\end{center}
\end{figure}
\subsection{Preparation of the continuum and sky images}
Fundamental to the reconstruction process is the determination of pure
count rate in the line $C(x,y)$ from the observed count
rate $F(x,y)$. This is carried out through Eq.~\ref{eqn_recon5},
which involves the estimation of sky and continuum images. The sky count
rates $Sky(r)$ are calculated as the median values in semi-annular zones,
with a width equal to $FWHM/2=9$~\AA.
The bluest wavelength of our scans is $\lambda_c=6528$~\AA, which is the
filter with least contamination from any emission line. We used this image
as a first-guess image of the continuum. Even for this filter, the H$\alpha$
line contributes more than 3\% for pixels closer than 90\arcsec\ from the
optical center. We defined a parameter $\eta_{\rm cont}$ such that only those
pixels for which the response to detect any line does not exceed
$\eta_{\rm cont}$ are considered for continuum (i.e.~only pixels with
$R_{line}<\eta_{\rm cont}$). For example, for the image
with $\lambda_c=6528$\AA, with $\eta_{\rm cont}=0.03$ only pixels that
are more distant than 90\arcsec\ from the optical center satisfy this
condition. Even for these pixels, we estimated the line contribution using
the reconstructed emission line maps, and subtracted it from the continuum
pixels. Individual sections contributing to the continuum are stitched
together (averaged when more than one image contributes to a pixel) to obtain
the final continuum image. This image is input as the new continuum image
and the entire process is repeated. From the simulated data (see \S5.3, {\it
Run 1}), we found that the continuum values converge after 3 iterations.
A small value of $\eta_{\rm cont}$ would result in various annular zones
where there are no pixels satisfying the
condition $R_{\rm line}<\eta_{\rm cont}$.
From simulated data, we found that line-free continuum images could be
obtained even for $\eta_{\rm cont}$ as large as 0.2.
An IRAF package containing the scripts developed as part of this work is
available to users upon request and will be available for downloading
from the LUS webpage http://www.inaoep.mx/$\sim$gtc-lus/.
\section{Flux errors in the reconstructed images}
\subsection{Flux error due to tuning error}
The central wavelength of the OSIRIS TF can be set only to an accuracy
of 1\,\AA\ (See OSIRIS TF User Manual). This uncertainty in tuning the TF
leads to an error in the recovered emission line fluxes.
The error ($\delta F$) in flux ($F$) due to an uncertainty of
$\delta\lambda_{\rm c}$ in setting the central wavelength
depends directly on the first derivative of the response function
(Eq.~\ref{eqn_Rlambda}) with respect to $\lambda$.
\begin{equation}
i.e. \hspace{2cm} {\delta F\over{F}} \equiv {\delta R_\lambda\over{R_\lambda}}
\equiv {1\over{R_\lambda}}\left({\partial R_\lambda\over{\partial\lambda}}\right)\delta\lambda_{\rm c}
\label{eqn_dfbyf1}
\end{equation}
Substituting the value of ${\partial R_\lambda\over{\partial\lambda}}$, we get
\begin{equation}
{\delta F\over{F}} = 4 R_\lambda \left({\delta\lambda_{\rm c} \over{FWHM}}\right) \sqrt{\left({1\over{R_\lambda}} - 1\right)}.
\label{eqn_dfbyf2}
\end{equation}
Maximum error is introduced for image pixels where the TF response $R_\lambda$
for the line of interest is 0.5. Thus, flux errors in the reconstructed image
can be reduced if we use $\eta_{\rm line}>0.5$.
It may be recalled from the analysis in \S4.1 that the lower limit of
$\eta_{\rm line}$ depends on $\Delta\lambda_{\rm c}$. In order to have
$\eta_{\rm line}>0.5$, $\Delta\lambda_{\rm c}$ should be less than the $FWHM$.
We used the Eq.~\ref{eqn_dfbyf2} to calculate the errors in the line
fluxes, and summed the errors in quadrature to calculate the errors in the
flux ratios, for nominal value of $\delta\lambda_{\rm c}=1$~\AA.
Our results are summarized in Fig.~\ref{figures_simul}.
In the three panels, we plot the errors in the calculated H$\alpha$ fluxes
(top), the [\textsc{N\,ii}]$\lambda6583$/H$\alpha$ ratio (middle)
and [\textsc{S\,ii}]$\lambda6717/6731$ ratio (bottom) as a function of the
sampling parameter, for two extreme values of FWHM permitted by the TF.
The value of $\eta_{\rm line}$ is calculated in such a way that there are
neither gaps nor overlapping pixels in the reconstructed image for a given
sampling $\Delta\lambda_{\rm c}$.
The most notable characteristic in these plots is that the
errors in all these three quantities are lower for larger values of FWHM.
This may seem counter-intuitive, and is the result of a sampling error
of 1~\AA\ being a larger fraction of $FWHM=12$~\AA\ than that for $FWHM=18$~\AA.
In other words, the TF response function falls less steeply for larger FWHM,
thus resulting in little errors as compared to that for smaller FWHMs.
As expected, finer sampling results in smaller errors on the derived
quantities, especially when the sampling rate is less than $0.7\times FWHM$.
Line fluxes can be calculated with better accuracy than the flux ratios.
The error in the [\textsc{N\,ii}]$\lambda6583$/H$\alpha$ flux ratio takes into account
the contamination by H$\alpha$ in the [\textsc{N\,ii}]$\lambda6583$ filters. This cross
talk also makes the errors on the [\textsc{N\,ii}]$\lambda6583$ line flux marginally
larger than those for the H$\alpha$ flux.
The calculated errors on the individual [\textsc{S\,ii}] line fluxes are similar to that
for the H$\alpha$ flux.
It may be noted that a recessional velocity of 45\,km\,s$^{-1}$ produces a
shift of 1\AA\ in the wavelength. Thus, in regions where kinematic deviations
of this order or more are likely to exist (e.g. nuclear regions of galaxies),
kinematical data are required for an accurate flux calibration. In the
absence of such data, eq.~15 can be used, where $\delta\lambda_{\rm c}$ is to
be replaced by the uncertainty in the wavelength due to Dopper effect, to
estimate the flux errors due to kinematic deviations.
\begin{figure}
\begin{center}
\includegraphics[scale=.50]{f3.eps}
\caption{
Errors in flux and flux ratios due to an uncertainty of
$\delta\lambda_{\rm c}=1$~\AA\ in setting the central wavelength of TF, as
a function of user-selected TF parameters $\Delta\lambda_{\rm c}$ and FWHM.
The errors are expressed as percentage values of the plotted quantity.
(top) Errors on the H$\alpha$ flux, (middle) errors on
the [\textsc{N\,ii}]$\lambda6583$/H$\alpha$ ratio, and (bottom) errors on
the [\textsc{S\,ii}]$\lambda6717/6731$ ratio. The scale on the top gives the maximum
value of response cut-off ($\eta_{\rm line}$) that the data permit for given
values of $\Delta\lambda_{\rm c}$ and FWHM, without having data gaps in the
reconstructed image (see \S4.1 for details). Errors expected in our dataset
of M101 for an assumed error of $\delta\lambda_{\rm c}=1$~\AA\ are marked
by solid circles.
\label{figures_simul}}
\end{center}
\end{figure}
\subsection{Reconstruction of simulated TF images}
The dataset we have for M101 is reconstructed with $\eta_{\rm line}=0.5$,
implying maximum errors of around 11\% in the recovered H$\alpha$ fluxes
according to Fig.~\ref{figures_simul}. In order to check the implementation
of the equations of \S3 in our reconstruction scripts, we need datasets
with much smaller errors. We hence
created artificial datasets by simulating the observations of an extended
emission line source with a scanning TF, making use of Eqs. 1 and 2.
The simulated dataset also allowed us
to quantify the accuracy of reconstruction
to various instrumental parameters that have non-zero uncertainties.
Two scans were simulated:
(1) TF with a FWHM=18~\AA, $\lambda_c=6528+(i-1)\times20$~\AA,
where $i$ varied from 1 to 9,
(2) TF with a FWHM=12~\AA, $\lambda_c=6528+(i-1)\times10$~\AA,
where $i$ varied from 1 to 18.
The reddest wavelength in both cases is 6688~\AA.
The FWHM in the two settings are among the
extreme values permitted with the red etalon, with the former one simulating
the dataset we have for M101. The intensity of the extended source
is set constant over the FoV of OSIRIS with I(H$\alpha$)=1.
The [\textsc{N\,ii}]$\lambda$6583 line is fixed at 10\% of that of H$\alpha$, which
is among the range of values observed in \textsc{H\,ii}\ regions \citep{Deni02}.
The [\textsc{N\,ii}]$\lambda$6548 line intensity is fixed at one third of that of the
[\textsc{N\,ii}]$\lambda$6583 line. We also added continuum sources at some fixed
positions on the image, in order to test the accuracy of the continuum
subtraction. The intensities of the continuum sources are adjusted such that
the H$\alpha$ emission equivalent widths at the positions of the continuum
sources are 10 for some sources and 100 for others.
Gaussian noise of two different values is added to the images, resulting in
two sets of simulations:
the high and low Signal-to-Noise Ratio (SNR) images correspond to SNRs of
100 and 3 for the
[\textsc{N\,ii}]$\lambda$6583 line image, respectively. A line-free continuum image is
also generated in the simulated set. The lines were redshifted by
214 km\,s$^{-1}$, corresponding to the recession velocity of M101.
\subsection{Parameters that limit the reconstruction accuracy}
{\it Reconstruction with ideal datasets (Run 0):}
The equations described in \S3 should allow us to recover the fluxes of the
H$\alpha$ and [\textsc{N\,ii}]$\lambda$6583 lines, in the absence of
uncertainties in various parameters that control the TF imaging.
We simulated this case by using high
SNR data and subtracting the simulated line-free continuum.
The errors in recovering the H$\alpha$ flux and the flux ratio
[\textsc{N\,ii}]$\lambda$6583/H$\alpha$ are as small as 0.03\% and 0.2\% respectively.
This test establishes that the
equations developed in \S3 are correctly implemented in the script.
However, there are many sources of error in real observational datasets,
especially when trying to maximize the available observing time.
We investigate the accuracy in the recovered flux due to the following 3
sources of error:
(1) the continuum images are not completely free of emission lines,
(2) the sky images are obtained by in-frame object-free pixels, not from an
off-source sky image,
(3) different images within a scan have non-zero ditherings.
We discuss each of these cases in detail below.
{\it Line contamination in continuum images (Run 1):}
Due to the long tail of the TF response, there is non-zero line
contribution in all TF images, including those centered bluer than the
bluest line in a scan (e.g., $\lambda_{\rm c}=6528$ is only $\sim1$ FWHM
blueward of the [\textsc{N\,ii}]$\lambda6548$ line, and $<2$ FWHMs of the bright
H$\alpha$ line).
In order to obtain line-free continuum fluxes from the sequence of images
we have, we followed the iterative procedure described in \S4.3. We obtained
a continuum image from the sequence of simulated images, and compared it to
the simulated pure continuum image. The difference between the two images is
found to be as small as 0.1\% after 3 iterations. Hence, errors
on the fluxes in our reconstructed images of M101 are not dominated by the
limitations of obtaining line-free continuum images.
{\it Non-uniform sky (Run 2):}
Errors in sky value subtraction and flat-field corrections introduce
residuals in the sky value locally. We parametrized this error in terms of the
rms noise value ($\sigma$) of this image. The residual sky value in
different parts of the image is allowed to vary between $-1\sigma$ and
$+1\sigma$, for the simulated set of images.
The [\textsc{N\,ii}]$\lambda$6583/H$\alpha$ ratios are affected by 5\%
in pixels where [\textsc{N\,ii}]$\lambda$6583 is detected with a SNR of $3$.
Thus, for pixels with SNR$>3$ for the line of [\textsc{N\,ii}]$\lambda$6583,
sky subtraction is not a serious problem in our images of M101.
{\it Consequences of Image-dithering (Run 3):}
It is a normal practice to dither images between any two repeat
observations in order to avoid detector blemishes spoiling any interesting
feature. These dithered images are registered to a common coordinate system
before combining them. In a TF observation, the optical centers in the
dithered images correspond to different astrometric coordinates
and hence the corresponding pixels in the registered images do not
have the same wavelength (see Eq.~1). Thus, in the combined image,
a given pixel has contributions from marginally different wavelengths.
We parametrized this effect as an error in the optical center
$\delta r_{\rm c}$. We studied the reconstruction accuracy for
various values of $\delta r_{\rm c}$ between $1\arcsec$ and $5\arcsec$.
A dithering of $1\arcsec$
between different images of a scan can produce errors of the order of
$\sim5$\% in the recovered H$\alpha$ flux for the FWHM=18~\AA\ filters.
The recovered [\textsc{N\,ii}]$\lambda$6583/H$\alpha$ ratio lies between
0.05--0.15 (50\% error over the simulated value of 0.1)
for this case. Combining images that are dithered by more than
$1\arcsec$ would introduce more than 20\% error on the recovered H$\alpha$
flux, making it unusable for most applications.
In the case of our observations of M101, the dithering within a scan
was less than $0.25\arcsec$, and hence the error in the recovered
H$\alpha$ flux due to astrometric registration of all images of a scan
(say P1) is less than a few percent.
\subsection{Recommendations for observing extended sources}
After studying the effect of various parameters on the fluxes in the
reconstructed images, we find that the maximum error on our dataset for M101
arises due to the present uncertainty of $\sim1$~\AA\ in setting the central
wavelength of a TF observation. This uncertainty affects more the images
taken with narrower TF observations, with the accuracy of H$\alpha$ fluxes
being $\sim$10\% and of [\textsc{N\,ii}]$\lambda$6583/H$\alpha$ ratios of $\sim18$\%
for the TF images with $FWHM=18$~\AA, as can be inferred from
Fig.~\ref{figures_simul}. Corresponding errors with $FWHM=12$~\AA\ are 16\%
and 25\%, respectively. Errors may be reduced by carrying out the scan with a
finer sampling. With $FWHM=12$~\AA\, a sampling of around 5~\AA\ would be
required to reduce the error levels to $\sim10$\%, that can be achieved
with 15~\AA\ sampling with $FWHM=18$~\AA. Thus a factor of 3 more exposure
time would be required to map a given emission line over the entire FoV of
OSIRIS with the narrower TF. However, for low SNR pixels or regions for
which sky and/or continuum subtraction errors contribute more than the
plotted errors ($SNR\lesssim 10$), the errors are
expected to increase proportionally with the FWHM, and hence observations with
$FWHM=12$~\AA\ would have an advantage by a factor of 1.5 over those with
$FWHM=18$~\AA. Thus, if the interest is in detecting diffuse faint emissions,
a $FWHM=12$~\AA\ is preferable.
\citet{Lara10} carried out simulations to determine the best combination of
FWHM and sampling ($\Delta\lambda_{\rm c}$) for optimal emission line flux
determinations of emission-line galaxies of redshifts between 0.2 and 0.4 with
GTC/OSIRIS. They found that a FWHM of 12~\AA\ and a sampling of 5~\AA\ are
the optimal combination that allows deblending H$\alpha$ from
the [\textsc{N\,ii}]$\lambda6583$ line with a flux error lower than 20\%. It is
relevant to note that in their simulations, the fluxes were not
corrected for the response curve of the TF, and the quoted flux errors
are due to the unavailability of the redshifts of the detected
galaxies. Hence, it is natural to expect lesser errors for the narrower
filters.
The next source of error comes from image dithering. For an image dithering
of 1\arcsec, the errors could be as large as 5\% for H$\alpha$ and 50\%
for [\textsc{N\,ii}]$\lambda$6583/H$\alpha$. For larger values of dithering, errors
on [\textsc{N\,ii}]$\lambda$6583/H$\alpha$ are unreasonably high.
It is advisable to have the same telescope position (dithering $<1$\arcsec)
for all the images constituting a given TF scan. If more than one scan is
available for a field with dithering of more than 1\arcsec\ between the scans,
it is advisable to reconstruct an emission-line image from each
scan, and then register and combine them. We recommend that a TF scan
intended to obtain a monochromatic image in an emission line should have
one image observed at least one $FWHM$ blueward of the bluest line in the
TF sequence, to facilitate accurate continuum subtraction.
\section{Flux Calibration of Monochromatic Images}
\begin{figure*}
\begin{center}
\centerline{\epsfig{file=f4a.eps, height=0.4\vsize}
\epsfig{file=f4b.eps, height=0.4\vsize}}
\vspace{1cm}
\caption{
An illustration of the calibration procedure adopted, where we fit the
SDSS $griz$ photometry of in-frame field stars with the spectra of
stars in the SDSS spectral catalog. (left) A section of the
OSIRIS continuum image of M101. Four stars with SDSS photometry are identified.
(right) SEDs in $griz$ bands of the 4 identified stars (asterisk), superposed on the
average of 12 best fitting spectra from SDSS spectral catalog (line).
The rms deviation from the mean of 12 best-fit spectra at each wavelength is
shown by the gray band around the mean spectrum. The 12 spectra differ by
less than 1\% ($<rms>_r$ denotes the average rms value within the $r$-band),
which makes this method very attractive for the calibration of OSIRIS TF data.
\label{fig_sdss}}
\end{center}
\end{figure*}
We have explored a new procedure to flux calibrate the TF images using the
in-frame field stars. The procedure, in principle, can be applied to any
optical narrow-band image of a field that contains stars with photometry
in the optical broad-bands. In particular, we used the SDSS photometry
in $griz$ bands of stars in the field of M101, and the recently released SDSS Stellar Spectral
database of stars covering a wide range of spectral types \citep{Abaz09}.
These spectra are of median resolution in the 3500-9000\,\AA\ wavelength range.
In order to have a good spectral library we selected, from the original Stellar
Spectral database, only those stars with $g=$14--18~mag having no gaps,
jumps or emission lines in their spectra.
The general procedure involves obtaining a stellar spectrum that fits
the $griz$ magnitudes of the stars in the field of the TF image.
Basically, we are using the stellar spectrum to interpolate the broad-band
fluxes at the wavelength of the TF. The spectra are integrated in
the $griz$-bands to get their synthetic magnitudes which are then fitted to
the $griz$ magnitudes of a star in the field of interest;
both Spectral Energy Distributions (SEDs) are normalized at the $r$-band.
The best-fit spectrum is chosen by minimizing the $\chi^2$ obtained
in the 4 bands. The spectrum of the best-fit SDSS star is then
de-normalized by multiplying it by the $r$-band flux of the field star.
The resulting spectrum is convolved with the response function of the OSIRIS
TFs (Eq.~\ref{eqn_Rlambda}) to obtain a smoothed spectrum of the field
star. The flux at $\lambda_r$, where $r$ is the radial distance of the star
in the OSIRIS field, is multiplied by the effective bandwidth of the TF, to
estimate the flux intercepted by the TF.
We then carried out aperture photometry of the selected SDSS stars on all the
TF images. This photometry is used to obtain observed count rate of each
star in each TF image.
The estimated flux is divided by the observed count rate of the star in that TF to
obtain the calibration coefficient $\kappa$.
The procedure is repeated for all the good SDSS stars in the observed field
to obtain a set of $\kappa$ values. We note that the
observed count rate is initially corrected for the effects of extinction and
the efficiency of the order sorter filter, and hence the $\kappa$
obtained from different stars in different TFs can be directly compared
with each other.
\subsection{Relative errors in the Calibration coefficients}
The availability of thousands of stellar spectra covering the entire range
of spectral types to fit the photometric data of field stars ensures that
there is at least one spectrum that truly represents the spectrum of the
field star. Generally, there are 10--15 spectra whose $griz$-band $\chi^2$
is within 10\% of the best-fit spectrum.
We obtained a mean and rms of these spectra at every sampled wavelength. The
rms error was found to be less than 1\% for wavelengths between the $g$ and $z$ bands.
Thus the relative flux calibration for different TF settings is better than
1\%. The absolute flux error depends on the error in the SDSS $r$-band magnitude
of the field stars. An illustration of the method followed is
shown in Fig.~\ref{fig_sdss}.
\begin{figure}
\begin{center}
\centerline{\epsfig{file=f5.eps, height=0.65\hsize}}
\caption{
Variation of the calibration coefficient $\kappa$ for 9 images
of the two H$\alpha$ TF scans of M101. We used 8 stars, 4 on each CCD
covering the complete field of view. The coefficients for the right
(dashed line) and left (dotted line) CCDs are separately shown. The solid
line shows the variation without distinguishing on which CCD stars are located.
In the top panel, we show the coefficients for scan 1 (P1) and in the
bottom, for scan 2 (P2).
\label{fig_calha}}
\end{center}
\end{figure}
In Fig.~\ref{fig_calha}, we show the variation of the calibration
coefficient along an H$\alpha$ TF scan containing 9 images.
In this figure, two kinds of systematic variations are seen:
(1) the variation of the coefficient from one TF observation to the next,
and (2) the variation of the coefficient between the two CCDs. \\
{\it Variation of calibration coefficient during a TF scan:} given that
the airmass variation from one observation to the next is taken into account
in obtaining the $\kappa$, this variation was not expected for a photometric
night. The rms dispersion for the 4 stars that are used to calculate $\kappa$\
is around 3\%, which is much smaller than the overall variation (9\% over the
mean value). The trend of the variation is identical for the two scans.\\
{\it Variation of calibration coefficient between the two CCDs:} OSIRIS uses
two CCDs, along with 2 separate electronics to cover the total field of view.
The differences in the background levels between the two CCDs are easily
noticeable. From our analysis, we find that there is a $6\pm3$\% difference
between the efficiencies of the two CCDs for the H$\alpha$ scan.
The most likely reason for this difference is the value of the gain parameter,
which is 0.95 electrons/ADU for the right CCD and 0.91 electrons/ADU for the
left CCD (private communication from GTC technical staff) during the
H$\alpha$ scan.
\begin{table}
\centering
\caption{\label{Tab:CalCoef}Derived values of calibration coefficients}
\begin{tabular}{lcc}
\hline
TF Scan & $\kappa$ & Units \\
\hline
H$\alpha+$[\textsc{N\,ii}]-P1 & $6.54\pm0.27$ & $10^{-18}$~erg\,s$^{-1}$\,cm$^{-2}$/(ADU/s) \\
H$\alpha+$[\textsc{N\,ii}]-P2 & $6.63\pm0.28$ & $10^{-18}$~erg\,s$^{-1}$\,cm$^{-2}$/(ADU/s) \\
\textsc{S\,ii}-P1 & $7.06\pm0.14$ & $10^{-18}$~erg\,s$^{-1}$\,cm$^{-2}$/(ADU/s) \\
\textsc{S\,ii}-P2 & $7.06\pm0.21$ & $10^{-18}$~erg\,s$^{-1}$\,cm$^{-2}$/(ADU/s) \\
\textsc{S\,ii}-P3 & $7.27\pm0.23$ & $10^{-18}$~erg\,s$^{-1}$\,cm$^{-2}$/(ADU/s) \\
\hline
\end{tabular}\\
\end{table}
\begin{figure*}
\begin{center}
\centerline{\epsfig{file=f6.eps, height=0.9\hsize}}
\caption{Reconstructed H$\alpha$ image of the south-west part of M101.
Several \textsc{H\,ii}\ regions, including the massive \textsc{H\,ii}\ complex NGC~5447
($\alpha=14:02:30, \delta=54:16:15$),
as well as several filamentary structures can be seen in this image.
The brightness of the H$\alpha$ structures is shown by the gray bar
at the bottom of the image in units of erg\,s$^{-1}$\,cm$^{-2}$\,arcsec$^{-2}$,
spanning a range from $-3\sigma$ to $512\sigma$ in logarithmic scale.
\label{fig_image}}
\end{center}
\end{figure*}
When these systematic variations in $\kappa$\ are taken into account, the
calibration coefficients obtained using different stars in different CCDs
and in different TF images, agree to within 3\% of each other.
The mean calibration coefficients obtained from the stars on the right CCD
of the first image of the H$\alpha$ and [\textsc{S\,ii}] scans are given in
Table~\ref{Tab:CalCoef}.
The difference in the calibration
coefficients for the two H$\alpha$ scans is of the order of 1\%, whereas
for the [\textsc{S\,ii}] scans this difference is around 3\%.
The mean difference in the calibration coefficients for
the H$\alpha$ and [\textsc{S\,ii}] scans is $\sim7$\%. The $\kappa$\ factors in the
reconstruction equations take into account both these
systematic variations, thus allowing us to combine the monochromatic images
from different images of a scan.
\begin{figure}
\begin{center}
\vspace*{-2cm}
\includegraphics[scale=.47]{f7.eps}
\caption{
Relative errors on the H$\alpha$ fluxes of 23 \textsc{H\,ii}\ regions in M101.
(a) Ratio of the H$\alpha$ fluxes measured on the reconstructed
image for the dithered position P1 to those for position P2; (b) ratio
of the \textsc{H\,ii}\ region fluxes measured on an H$\alpha$+[\textsc{N\,ii}] image taken from NED
to those on our images, both plotted against our H$\alpha$ fluxes. The dotted
horizontal lines denote 1$\sigma$ scatter over the mean ratio.
\label{fig_phot_comp1}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=.47]{f8.eps}
\caption{
Comparison of important diagnostic ratios from our reconstructed images
with those obtained using SDSS spectra for 6 regions in common.
In the left panels the difference between the ratios
(e.g. $\Delta$([\textsc{N\,ii}]$/{\rm H}\alpha$) = ([\textsc{N\,ii}]$\lambda 6583/{\rm H}\alpha{\rm )}_{\rm ours} - $ ([\textsc{N\,ii}]$\lambda 6583/H\alpha {\rm )}_{\rm sdss}$)
is plotted against the SDSS ratios, whereas in the right panels, the difference
between the ratios is plotted against our H$\alpha$ fluxes. In general, the
diagnostic ratios are reproduced within the ranges allowed by the
estimated errors. See text for more details.
\label{fig_sdss_comp}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\vspace*{-1cm}
\includegraphics[scale=.47]{f9.eps}
\caption{
Flux ratios of nebular diagnostic lines plotted against the
H$\alpha$ fluxes for the 23 \textsc{H\,ii}\ regions of Fig.~\ref{fig_phot_comp1}.
In panels (a) and (b), we show [\textsc{N\,ii}]$\lambda$6583/H$\alpha$
and [\textsc{S\,ii}]$\lambda6716+\lambda6731$/H$\alpha$, whereas in panel (c)
we show the density-sensitive ratio [\textsc{S\,ii}]$\lambda6716/\lambda6731$.
The upper limits observed in extragalactic \textsc{H\,ii}\ regions from
\citet{Deni02} are indicated by the horizontal dotted lines in the
top two panels, whereas the theoretically valid range for the [\textsc{S\,ii}]
line ratios from \citet{Oste06} is shown in panel (c). All the regions have
observed ratios in the range expected for \textsc{H\,ii}\ regions, illustrating
the capability of TF imaging for obtaining these diagnostic ratios.
\label{fig_phot_comp2}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=.47]{f10.eps}
\caption{
Nebular diagnostic diagram involving [\textsc{N\,ii}]$\lambda$6583/H$\alpha$ and
[\textsc{S\,ii}]$\lambda6716/\lambda6731$ for \textsc{H\,ii}\ regions in M101. All the observed
points lie within locii of models of \citet{Dopi06} defined by metallicities
between $0.4Z\odot$ and $2Z\odot$, and the logarithmic ionization parameter
$\log R$ between 2 (low pressure regions) and $-6$ (high pressure regions).
\label{fig_phot_comp3}}
\end{center}
\end{figure}
\section{Comparison of reconstructed images}
In Fig.~\ref{fig_image}, we show the reconstructed H$\alpha$ emission-line
image of the south-west part of M101. The H$\alpha$ line being a good
tracer of ionized gas, this image shows the location of \textsc{H\,ii}\ complexes and
ionized filaments. It can be seen from the image that we are able to detect
\textsc{H\,ii}\ regions in the entire field of view.
The $5\sigma$ surface brightness limit
in this image is $8\times10^{-17}$~erg\,cm$^{-2}$\,s$^{-1}$\,arcsec$^{-2}$.
All the regions seen
in the image are recovered in the two independent scans. In the
following paragraphs, we compare the fluxes of recovered emission lines of \textsc{H\,ii}\
regions in these two independent scans. Our H$\alpha$ fluxes are also
compared with the H$\alpha$ fluxes obtained using traditional
narrow-band imaging technique.
We performed photometry of
23 \textsc{H\,ii}\ regions using an aperture radius of 4\arcsec\ on our reconstructed
H$\alpha$, [\textsc{N\,ii}]$\lambda6583$, [\textsc{S\,ii}] images and also on an H$\alpha+$[\textsc{N\,ii}]
image that was obtained in the traditional way using narrow-band
filters (downloaded from NED: Telescope: KPNO Schmidt; Observers:
B. Greenawalt, and R. Walterbos). The selected regions are among the brightest
regions in the FoV.
With a sampling of $\Delta\lambda_{\rm c}=20$~\AA\ for our dataset, $\eta_{\rm line}$ should be
at least 0.45 in order to obtain an image without data gaps.
However, we found systematically larger dispersion in fluxes for regions
with response values less than 0.5. Therefore we carried out the reconstruction
with $\eta_{\rm line}=0.5$. As a result, eight regions
have data in only one of the dithered scans.
In Fig.~\ref{fig_phot_comp1}a, we show a comparison of the fluxes of the
15 \textsc{H\,ii}\ regions
whose fluxes could be measured on the reconstructed images from both
the scans.
The mean ratio of the H$\alpha$ fluxes is unity over 2 orders of
magnitude in flux, confirming the accuracy of the method we adopted for
the flux calibration. The rms dispersion of the H$\alpha$ fluxes of the same
region obtained on images for the two scans is 16\%. This value is within
the flux errors expected for the 1--2~\AA\ error in $\lambda_{\rm c}$.
In order to check whether the reconstruction process rightly reproduces
relative fluxes of different regions over the entire FoV,
we plot the ratio of the H$\alpha+$[\textsc{N\,ii}]$\lambda6583$ fluxes measured from
our reconstructed images (P1 and P2), and
that from traditional narrow-band filters in Fig.~\ref{fig_phot_comp1}b.
Our fluxes are given as the average
when data are available from both scans. As the intention here is
a comparison of relative fluxes, we set the mean value of the flux
ratio to unity. There is $\sim11$\% scatter on this mean value, which is
marginally better than that between P1 and P2, as expected due to the use
of averaged fluxes. There is a marginal trend for the
mean ratio to be $\sim5$\% different between the bright and faint regions,
which is nevertheless smaller than the scatter.
We did not find any trend of these two ratios against the distance of the
region from the optical center.
Three sources of error are included in calculating the sizes of
the error bars plotted in Fig.~\ref{fig_phot_comp1}. They are
(1) photon noise of the object, (2) the error in the subtraction of the sky value and
(3) the error in the recovered flux due to an error in $\lambda_c$.
The last of these errors, which is discussed in detail in \S5,
dominates for the 23 regions for which we performed photometry, contributing
around 10\% for the majority of the regions.
As a second test of the reliability of the flux ratios obtained using the
TF images, we compare the flux ratios of emission
lines relevant for diagnostic diagrams with those values obtained using spectra from the literature for the same regions.
Seventeen \textsc{H\,ii}\ regions in M101 were observed spectroscopically by SDSS
\citep{Shol07}. Six of these lie within the usable FoV of our images (radial
distance from the optical center $\lesssim3.75^\prime$).
Flux ratio of nebular diagnostic lines from our reconstructed images are
compared with those obtained from the SDSS spectra in
Fig.~\ref{fig_sdss_comp}.
Our fluxes were obtained over apertures of $4\arcsec$ radius at the coordinates
associated with the SDSS spectra.
The adopted apertures, though almost 3 times bigger than the fiber sizes of
SDSS spectra (3\arcsec\ diameter), are the minimum area over which reliable fluxes
can be measured in our images. Given that the spectroscopic ratios of giant
\textsc{H\,ii}\ regions are not expected to vary much with aperture size, our relatively
bigger apertures are not expected to introduce additional errors.
The difference in the flux ratios between ours and SDSS values are plotted
both against flux ratios (in the left panel) and H$\alpha$ fluxes
(in the right panel). The plotted error bars take into account all the
errors discussed in the paragraph above. The errors in the spectroscopic
ratios are expected to be almost negligible, and hence we did not include
these errors in our analysis. The majority of the points lie close to the
horizontal line within the plotted errors, indicating that the ratios of
lines relevant for diagnostic purposes can be obtained using TF imaging.
There are a few ratios that deviate from the spectroscopic ratios by more
than the estimated errors. The most important of them is the right-most
point in panel (a). Apart from being the faintest in H$\alpha$,
the spectroscopic [\textsc{N\,ii}]$\lambda6583$/H$\alpha$ value for this region is 0.4,
a value that is too high as compared to the assumed value of 0.1 in
Eqn~\ref{eqn_recon8}.
This is the most likely reason for the large deviation of this region.
Kinematics of individual regions with radial velocities
$>25$~km\,s$^{-1}$ (i.e. 0.5~\AA\ around the H$\alpha$ wavelength) can
also be responsible for the observed deviations of some of
the points. The errors in the measured radial velocities of the regions
from SDSS spectra do not permit us to carry out a more elaborate analysis
of the deviations of individual regions.
In summary, the ratios recovered from the TF images for individual regions
have larger errors than the corresponding spectroscopic ratios.
Nevertheless, the capability of TF images to derive such ratios for
large number of regions, makes TF imaging a scientifically attractive
option, as we illustrate below using the flux ratio of 23 \textsc{H\,ii}\ regions in M101.
In Fig.~\ref{fig_phot_comp2}, we plot the
[\textsc{N\,ii}]$\lambda$6583/H$\alpha$, [\textsc{S\,ii}]$\lambda6716+\lambda6731$/H$\alpha$,
and the [\textsc{S\,ii}]$\lambda6716/\lambda6731$ ratio of \textsc{H\,ii}\ regions against their
H$\alpha$ fluxes in panels (a), (b) and (c), respectively. In each of these
panels, we indicate the observed/theoretical ranges of these ratios in \textsc{H\,ii}\
regions, which cover $\sim10$ times the errors on the ratios. This relatively
large dynamic range, combined with the intrinsic multi-object capability of
TF imaging, makes it competitive with traditional spectroscopic observations.
The [\textsc{N\,ii}] and [\textsc{S\,ii}] lines originate in low ionization zones in \textsc{H\,ii}\ regions.
Their ratio to the H$\alpha$ flux depends not only on the ionization
parameter, but also on the abundance of these ions. The utility of these
ratios for diagnostic purposes has been discussed in \citet{Bald81}.
In Fig.~\ref{fig_phot_comp3}, we show the values for these ratios expected
for a range of ionization parameters ($R$) and metallicities ($Z$), using
the models of \citet{Dopi06}. The observed values of these ratios in our
selected 23 regions lie within the range of model values. The regions having
high values of [\textsc{N\,ii}]/H$\alpha$ have most likely high
nitrogen abundances, as was found by \citet{Deni02} for \textsc{H\,ii}\ regions in a
sample of nearby galaxies. Thus, OSIRIS TF imaging is very promising for
the study of nebular line ratio diagnostics of nearby large galaxies.
\section{Summary}
In this work, we have explored the capability of tunable filter imaging
with the OSIRIS instrument at the 10.4-m GTC, using real data.
The changing wavelength across the field and the non-flat
functional form of the response curves makes it essential to have a
sophisticated analysis package to completely take advantage of the
relatively large field of view of the instrument. With the set-up that we
have used to observe M101, we were able to obtain monochromatic images
in the emission lines of H$\alpha$, [\textsc{N\,ii}]$\lambda6583$ and the [\textsc{S\,ii}] doublet.
We demonstrate that line fluxes and their ratios for
prominent \textsc{H\,ii}\ regions can be obtained to better than $\sim$15\%, which is
basically limited by the current 1~\AA\ uncertainty in setting the central
wavelength of the TF.
Though these errors are much larger than the spectroscopic ones,
the multi-object capability of TF imaging, combined with the relatively
high sensitivity of the flux ratios to variations in density, excitation
and metallicity, makes TF imaging an attractive option for investigating
the point-to-point variation of these physical quantities in galaxies.
We also demonstrate that the emission-line maps
can be flux-calibrated to better than 3\% accuracy, using the $griz$ SDSS
magnitudes of in-frame stars, and the spectral database of SDSS,
without the need to invest extra telescope time to perform the
photometry of spectrophotometric stars.
\acknowledgments
It is a pleasure to thank an anonymous referee, whose thoughtful comments
helped us to improve the original manuscript.
We would also like to thank the GTC/OSIRIS staff members, especially Antonio
Cabrera, for the support provided for this project.
This work is partly supported by CONACyT (Mexico) research grants
CB-2010-01-155142-G3 (PI:YDM), CB-2011-01-167281-F3 (PI: DRG),
CB-2005-01-49847 and CB-2010-01-155046 (PI: RJT), and CB-2008-103365 (PI: ET).
This work has been also partly funded by the Spanish MICINN, Estallidos
(AYA2010-21887, AYA2007-67965) and Consolider-Ingenio (CSD00070-2006) grants.
|
{
"timestamp": "2012-06-27T02:01:33",
"yymm": "1203",
"arxiv_id": "1203.1842",
"language": "en",
"url": "https://arxiv.org/abs/1203.1842"
}
|
\section*{Materials and Methods}
\subsection{Potential drop}
Consider a small region of the channel around position $y$. The line density around this point at equilibrium is $n_l(y)$ and the line density in the presence of current is $n_l(y)+\tilde{n_l}(y)$.
Since there is no interparticle scattering at the length scale of that small region, the energy distribution in the presence of a current is not given by the equilibrium (Fermi-Dirac) distribution. Indeed, the density varies at the short scale of the scattering with impurities or with the external potential, while the distribution relaxes towards equilibrium on the scale of the interparticle interaction mean free path.
Following the standard procedure in mesoscopic physics (see \cite{datta_electronic_1995}, chap.2 part 3), we attribute to the region around $y$ a chemical potential $\mu(y)$, which gives the expected density when inserted in the Fermi distribution, and coincides with the chemical potential in the reservoirs. In this way, the chemical potential variations are local (like the density variations), even if the energy relaxation takes place on a larger length scale, deep in the reservoirs.
We now consider the relation between $\mu$, $\tilde{n_l}$ and the compressibility $\kappa_l = \partial n_l / \partial \mu$. Let $f(\mu,T,E)$ be the Fermi-Dirac distribution at temperature $T$, chemical potential $\mu$ and energy $E$, and $g(E)$ be the density of states,
\begin{eqnarray}
\tilde{n_l} &=& \int f(\mu,T,E) -f(\mu_0,T,E) g(E) dE \\
&=& (\mu-\mu_0) \frac{\partial}{\partial \mu} \int f(\mu_0,T,E) g(E) dE \\
&=& (\mu-\mu_0) \frac{\partial n_l}{\partial \mu} \\
&=& \tilde{\mu} \kappa_l, \label{eq:kappa}
\end{eqnarray}
where we have introduced the local chemical potential at equilibrium $\mu_0$, which follows from the local density approximation at equilibrium, and $\tilde{\mu} = \mu-\mu_0$ the local deviation of the chemical potential away from equilibrium.
Knowing the current $I$ flowing through the channel, we now introduce a local resistivity
\begin{equation}
\rho(y) = \frac{\partial \tilde{\mu}}{\partial y} \frac{1}{I},
\end{equation}
which reflects the local scattering with obstacles. Although the Joule heating happens deep in the reservoirs where excited atoms release their energy via collisions with other atoms, the momentum randomization due to scattering with impurities is local, and leads to a local increase of entropy as information on the direction of motion of particles gets lost.
Experimentally, to get the derivative $\partial \tilde{n}_l/\partial y$ we fit a line to the line density difference at each position $y$ within a window of $\pm 9 \, \mu$m around this point.
To extract $\kappa_l$ at position $y$ we make use of the isotropy
of the compressibility. We consider at fixed position $y$
the variation of $n_{col}$ along the $x$ direction and compare it with
the known trap shape \cite{muller_local_2010}.
Consider the gas around position $y$:
\begin{eqnarray}
\kappa_l(y) &=& \frac{\partial n_l(y)}{\partial \mu}\\
&=& \frac{\partial}{\partial \mu} \iint n(\mu(x,y,z)) dx dz\\
&=& \iint \frac{\partial n}{\partial \mu}(x,y,z) dx dz\\
&=& \iint \frac{\partial n}{\partial x} \left( \frac{\partial \mu}{\partial x} \right)^{-1} dx dz\\
&=& -\iint \frac{\partial n}{\partial x} \left( \frac{\partial V}{\partial x} \right)^{-1} dx dz
\end{eqnarray}
The confinement along the $x$ direction is ensured by the optical dipole trap, and is constant over the channel.
Close to the center of the channel, the confinement along $z$ is mainly ensured by the two repulsive lobes of the laser beam. Therefore, the variations of the potential along the $z$ direction is independent of the variations along the $x$ direction and can therefore be pulled out of the $z$ integration.
\begin{eqnarray}
\kappa_l(y) &=& -\int \left( \frac{\partial V}{\partial x} \right)^{-1} \left( \int \frac{\partial n}{\partial x} dz \right)dx \\
&=& -\int \left( \frac{\partial V}{\partial x} \right)^{-1} \frac{\partial n_{col}}{\partial x} dx \label{kappa} \label{kappa_final} \, ,
\end{eqnarray}
where $n_{col}$ is the column density.
A slice of $\pm 9 \, \mu$m around one position $y$ is taken from
the density picture at equilibrium (figure 4A) and the average along the $y$ direction is calculated.
This results in a one-dimensional density profile along the $x$ direction.
We fit a gaussian to such a profile and use it in equation \eqref{kappa_final},
in order to avoid the noise generated by numerical differentiations and
ratios of those \cite{muller_local_2010}.
This fits the shape of the cloud within the errorbars. In
equation \eqref{kappa_final}, $V$ is taken to be the known gaussian
shaped trapping potential.
\subsection{Mobility}
In electric conduction the mobility relates the drift velocity $v_d$
to a potential gradient (electric field). In our case the gradient in
the chemical potential plays the role of the potential gradient
and the atomic mobility is
\begin{eqnarray}
v_d \cdot \left(\frac{\partial \mu}{\partial y}\right)^{-1} = v_d \cdot \kappa_l \left(\frac{\partial n_l}{\partial y}\right)^{-1},
\end{eqnarray}
where we used equation \eqref{eq:kappa}.
In the data analysis we attribute an infinite atomic mobility
to points where $\frac{\partial \tilde{n}_l}{\partial y}$ is zero
or slightly negative. These points are indicated
with red arrows in figure 4. The upper limit of the red shaded region
is as well set to infinity if the lower edge of the error on
$\frac{\partial \tilde{n}_l}{\partial y}$ extents to negative values.
\subsection{Properties of the confining potential}
The channel is imprinted on the atoms using a 532\,nm wavelength laser beam, with waists $30.2(3)$ and $10.3(3)$\,$\mu$m along the $y$ and $z$ directions, respectively. This beam passes through a holographic plate (Silios) dephasing the upper part of the beam by $\pi$ with respect to the lower part \cite{smith_quasi-2d_2005,meyrath_high_2005}. The beam has been characterized in a test setup and shows a slight asymmetry along the $y$ axis away from the center. The contrast of the central region of the beam compared to the lobes is larger than $0.99$. In the parameter regime explored in the paper, the residual light leads to a repulsive potential along the $y$-axis smaller than $40$\,nK, much smaller than the oscillation frequency along $z$.
The oscillation frequency along the tightly confining direction $z$ has been measured {\it in-situ} using parametric heating on a microscopic cloud \cite{zimmermann_high-resolution_2011}, and the measured frequency agrees with the fitted curvature of the intensity profile observed on the test setup. Along the propagation direction of the beam, the measured curvature of the intensity profile varies by less than 5\% on a length scale of 200\,$\mu$m, and is therefore constant over the transverse radius of the cloud.
\begin{figure*}[htb]
\def S1{S1}
\includegraphics[width=125mm]{figureS1.pdf}
\caption{Properties of the disordered potential. A : central region of the disordered potential as observed with the high-resolution microscope used to image the atoms. The total width of the picture is $30$ $\mathrm{\mu}$m. B : correlation function of the intensity distribution observed in A. C : horizontal cut of B, showing the correlation properties of the disordered potential (in blue). The solid green line shows a Gaussian fit yielding a measured correlation radius of $460$\,nm.}
\label{fig:speckle}
\end{figure*}
\subsection{Properties of the disordered potential}
The disordered potential is created by passing through a light diffuser (Luminit) with a beam at 532 nm, then through a high resolution microscope objective \cite{zimmermann_high-resolution_2011}, corrected for aberrations at that wavelength. The resulting disordered potential is observed directly {\it in-situ} using a second microscope identical to the first. In this way, we directly characterize the potential correlation properties by taking pictures of the potential as seen by the atoms in the glass cell. Figure \ref{fig:speckle} presents a typical observation of the potential characteristics. Figure \ref{fig:speckle}A shows a zoom on the central part of the disordered potential where correlation properties are computed. Figure \ref{fig:speckle}B shows the measured correlation function obtained using the inverse Fourier transform of the power spectrum of A. Figure \ref{fig:speckle}C shows a cut along the horizontal axis, of figure B, together with a Gaussian fit, yielding an observed correlation radius (1/$\sqrt{e}$ radius) in this direction of $460$\,nm. In the other direction, the same fit yields an observed correlation radius of $560$\,nm. To obtain a faithful estimate of the correlation properties of the disorder on the atoms, we deconvolve the correlation properties using the measured, gaussian point spread function of the microscope used to image the pattern \cite{zimmermann_high-resolution_2011}. This yields a correlation radius averaged for the two directions of $370$\,nm. In addition, using the {\it in-situ} images of the light intensity, we also observed the expected exponential probability distribution of intensities.
The envelope of the disorder potential is straightforwardly obtained from a gaussian fit of the observed profile. The total power sent through the microscope, together with the average fitted profile of the envelope, yields the average depth of the disordered potential cited in the text.
\end{document}
|
{
"timestamp": "2012-04-11T02:01:34",
"yymm": "1203",
"arxiv_id": "1203.1927",
"language": "en",
"url": "https://arxiv.org/abs/1203.1927"
}
|
\section{INTRODUCTION}
The Durbin-Watson statistic was originally introduced by the eponymous econometricians Durbin and Watson \cite{DurbinWatson50}, \cite{DurbinWatson51}, \cite{DurbinWatson71} in the middle of last century, in order to detect the presence of a significant first-order autocorrelation in the residuals from a regression analysis. The statistical test worked pretty well in the independent framework of linear regression models, as it was specifically investigated by Tillman \cite{Tillman75}. While the Durbin-Watson statistic started to become well-known in Econometrics by being commonly used in the case of linear regression models containing lagged dependent random variables, Malinvaud \cite{Malinvaud61} and Nerlove and Wallis \cite{NerloveWallis66} observed that its widespread use in inappropriate situations were leading to inadequate conclusions. More precisely, they noticed that the Durbin-Watson statistic was asymptotically biased in the dependent framework. To remedy this misuse, alternative compromises were suggested. In particular, Durbin \cite{Durbin70} proposed a set of revisions of the original test, as the so-called \textit{t-test} and \textit{h-test}, and explained how to use them focusing on the first-order autoregressive process. It inspired a lot of works afterwards. More precisely, Maddala and Rao \cite{MaddalaRao73}, Park \cite{Park75} and then Inder \cite{Inder84}, \cite{Inder86} and Durbin \cite{Durbin86} looked into the approximation of the critical values and distributions under the null hypothesis, and showed by simulations that alternative tests significantly outperformed the inappropriate one, even on small-sized samples. Additional improvements were brought by King and Wu \cite{KingWu91} and lately, Stocker \cite{Stocker06} gave substantial contributions to the study of the asymptotic bias resulting from the presence of lagged dependent random variables. In most cases, the first-order autoregressive process was used as a reference for related research. This is the reason why the recent work of Bercu and Pro\"ia \cite{BercuProia11} was focused on such a process in order to give a new light on the distribution of the Durbin-Watson statistic under the null hypothesis as well as under the alternative hypothesis. They provided a sharp theoretical analysis rather than Monte-Carlo approximations, and they proposed a statistical procedure derived from the Durbin-Watson statistic. They showed how, from a theoretical and a practical point of view, this procedure outperforms the commonly used Box-Pierce \cite{BoxPierce70} and Ljung-Box \cite{LjungBox78} statistical tests, in the restrictive case of the first-order autoregressive process, even on small-sized samples. They also explained that such a procedure is asymptotically equivalent to the \textit{h-test} of Durbin \cite{Durbin70} for testing the significance of the first-order serial correlation. This work \cite{BercuProia11} had the ambition to bring the Durbin-Watson statistic back into light. It also inspired Bitseki Penda, Djellout and Pro\"ia \cite{BitsekiDjelloutProia12} who established moderate deviation principles on the least squares estimators and the Durbin-Watson statistic for the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process.
\medskip
Our goal is to extend of the previous results of Bercu and Pro\"ia \cite{BercuProia11} to $p-$order autoregressive processes, contributing moreover to the investigation on several open questions left unanswered during four decades on the Durbin-Watson statistic \cite{Durbin70}, \cite{Durbin86}, \cite{NerloveWallis66}. One will observe that the multivariate framework is much more difficult to handle than the scalar case of \cite{BercuProia11}. We will focus our attention on the $p-$order autoregressive process given, for all $n\geq 1$, by
\begin{equation}
\label{Int_Mod}
\vspace{1ex}
\left\{
\begin{array}[c]{ccl}
X_{n} & = & \theta_1 X_{n-1} + \hdots + \theta_{\!p} X_{n-p} + \veps_n \vspace{1ex}\\
\veps_n & = & \rho \veps_{n-1} + V_n
\end{array}
\right.
\end{equation}
where the unknown parameter $\theta = \begin{pmatrix} \theta_1 & \theta_2 & \hdots & \theta_{\!p} \end{pmatrix}^{\prime}$ is a nonzero vector such that $\Vert \theta \Vert_1 < 1$, and the unknown parameter $\vert \rho \vert < 1$.
Via an extensive use of the theory of martingales \cite{Duflo97}, \cite{HallHeyde80}, we shall provide a sharp and rigorous analysis on the asymptotic behavior of the least squares estimators of $\theta$ and $\rho$. The previous results of convergence were first established in probability \cite{Malinvaud61}, \cite{NerloveWallis66}, and more recently almost surely \cite{BercuProia11} in the particular case where $p=1$. We shall prove the almost sure convergence as well as the asymptotic normality of the least squares estimators of $\theta$ and $\rho$ in the more general multivariate framework, together with the almost sure rates of convergence of our estimates. We will deduce the almost sure convergence and the asymptotic normality for the Durbin-Watson statistic. Therefore, we shall be in the position to \textcolor{blue}{propose further results on the well-known \textit{h-test} of Durbin \cite{Durbin70} for testing the significance of the first-order serial correlation in the residuals.} We will also explain why, on the basis of the empirical power, this test procedure outperforms Ljung-Box \cite{LjungBox78} and Box-Pierce \cite{BoxPierce70} \textit{portmanteau} tests for stable autoregressive processes. We will finally show by simulation that it is equally powerful than the Breusch-Godfrey \cite{Breusch78}, \cite{Godfrey78} test and the \textit{h-test} \cite{Durbin70} on large samples, and better than all of them on small samples.
\medskip
The paper is organized as follows. Section 2 is devoted to the estimation of the autoregressive parameter. We establish the almost sure convergence of the least squares vector estimator of $\theta$ to the limiting value
\begin{equation}
\label{Int_Tlim}
\theta^{*} = \alpha \left( I_{\! p} - \theta_{\!p} \rho J_{\! p} \right) \beta
\end{equation}
where $I_{\! p}$ is the identity matrix of order $p$, $J_{\! p}$ is the exchange matrix of order $p$, and where $\alpha$ and $\beta$ will be calculated explicitly. The asymptotic normality as well as the quadratic strong law and a set of results derived from the law of iterated logarithm are provided. Section 3 deals with the estimation of the serial correlation parameter. The almost sure convergence of the least squares estimator of $\rho$ to
\begin{equation}
\label{Int_Rlim}
\rho^* = \theta_{\!p} \rho \theta_{\!p}^{*}
\end{equation}
where $\theta_{\!p}^{*}$ stands for the $p-$th component of $\theta^{*}$ is also established along with the quadratic strong law, the law of iterated logarithm and the asymptotic normality. It enables us to establish in Section 4 the almost sure convergence of the Durbin-Watson statistic to
\begin{equation}
\label{Int_Dlim}
D^* = 2(1 - \rho^*)
\end{equation}
together with its asymptotic normality. Our sharp analysis on the asymptotic behavior of the Durbin-Watson statistic remains true whatever the values of the parameters $\theta$ and $\rho$ as soon as $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$, assumptions resulting from the stability of the model. Consequently, we are able in Section 4 to propose a \textcolor{blue}{two-sided statistical test for the presence of a significant first-order residual autocorrelation closely related to the \textit{h-test} of Durbin \cite{Durbin70}. A theoretical comparison as well as a sharp analysis of both approaches are also provided. In Section 5, we give a short conclusion where we briefly summarize our observations on simulated samples. We compare the empirical power of this test procedure with the commonly used \textit{portmanteau} tests of Box-Pierce \cite{BoxPierce70} and Ljung-Box \cite{LjungBox78}, with the Breusch-Godfrey test \cite{Breusch78}, \cite{Godfrey78} and the \textit{h-test} of Durbin \cite{Durbin70}.} Finally, the proofs related to linear algebra calculations are postponed in Appendix A and all the technical proofs of Sections 2 and 3 are postponed in Appendices B and C, respectively. \textcolor{blue}{Moreover, Appendix D is devoted to the asymptotic equivalence between the \textit{h-test} of Durbin and our statistical test procedure.}
\begin{rem}
\label{Int_Rem_DefIJe}
In the whole paper, for any matrix $M$, $M^{\prime}$ is the transpose of $M$. For any square matrix $M$, $\textnormal{tr}(M)$, $\det(M)$, $\VVert M \VVert_1$ and $\rho(M)$ are the trace, the determinant, the 1-norm and the spectral radius of $M$, respectively. In addition, $\lambda_{\text{min}}(M)$ and $\lambda_{\text{max}}(M)$ denote the smallest and the largest eigenvalues of $M$, respectively. For any vector $v$, $\Vert v \Vert$ stands for the euclidean norm of $v$ and $\Vert v \Vert_1$ is the 1-norm of $v$.
\end{rem}
\begin{rem}
Before starting, we denote by $I_{\! p}$ be the identity matrix of order $p$, $J_{\! p}$ the exchange matrix of order $p$ and $e$ the $p-$dimensional vector given by
\begin{equation*}
I_{\! p} = \begin{pmatrix}
1 & 0 & \hdots & 0\\
0 & 1 & \hdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \hdots & 1
\end{pmatrix}, \hspace{1cm}
J_{\! p} = \begin{pmatrix}
0 & \hdots & 0 & 1\\
0 & \hdots & 1 & 0\\
\vdots & \udots & \vdots & \vdots\\
1 & \hdots & 0 & 0
\end{pmatrix}, \hspace{1cm}
e = \begin{pmatrix}
1\\
0\\
\vdots\\
0
\end{pmatrix}.
\end{equation*}
\end{rem}
\bigskip
\section{ON THE AUTOREGRESSIVE PARAMETER}
Consider the $p-$order autoregressive process given by \eqref{Int_Mod} where we shall suppose, to make calculations lighter without loss of generality, that the square-integrable initial values $X_0 = \veps_0$ and $X_{-1}, X_{-2}, \hdots, X_{-p} = 0$. In all the sequel, we assume that $(V_{n})$ is a sequence of square-integrable, independent and identically distributed random variables with zero mean and variance $\sigma^2 > 0$. Let us start by introducing some notations. Let $\Phi^p_{n}$ stand for the lag vector of order $p$, given for all $n \geq 0$, by
\begin{equation}
\label{P1_Phi_Lag}
\Phi^p_{n} = \begin{pmatrix} X_{n} & \hspace{0.1cm} & X_{n-1} & \hspace{0.1cm} & \hdots & \hspace{0.1cm} & X_{n-p+1} \end{pmatrix}^{\prime}.
\end{equation}
Denote by $S_{n}$ the positive definite matrix defined, for all $n \geq 0$, as
\begin{equation}
\label{P1_Sn}
S_{n} = \sum_{k=0}^{n} \Phi^p_{k}\, {\Phi^p_{k}}^{\, \prime} + S
\end{equation}
where the symmetric and positive definite matrix $S$ is added in order to avoid an useless invertibility assumption. For the estimation of the unknown parameter $\theta$, it is natural to make use of the least squares estimator which minimizes
\begin{equation*}
\nabla_{\!n}(\theta) = \sum_{k=1}^n \left( X_k - \theta^{\, \prime}\, \Phi^p_{k-1} \right)^2.
\end{equation*}
A standard calculation leads, for all $n \geq 1$, to
\begin{equation}
\label{P1_Est}
\tn = (S_{n-1})^{\! -1} \sum_{k=1}^{n} \Phi^p_{k-1}\, X_{k}.
\end{equation}
\medskip
\noindent Our first result is related to the almost sure convergence of $\tn$ to the limiting value $\theta^{*} = \alpha \left( I_{\! p} - \theta_{\!p} \rho J_{\! p} \right) \beta$, where
\begin{equation}
\label{P1_Alpha}
\alpha = \frac{1}{(1 - \theta_{\!p} \rho)(1 + \theta_{\!p} \rho)},
\end{equation}
\begin{equation}
\label{P1_Beta}
\beta =
\begin{pmatrix}
\theta_1 + \rho & \hspace{0.1cm} & \theta_2 - \theta_1 \rho & \hspace{0.1cm} & \hdots & \hspace{0.1cm} & \theta_{\!p} - \theta_{\!p-1}\rho \end{pmatrix}^{\prime}.
\end{equation}
\begin{thm}
\label{P1_Thm_CvgTheta}
We have the almost sure convergence
\begin{equation}
\label{P1_CvgTheta}
\lim_{n \rightarrow \infty} \tn = \theta^{*} \textnormal{\cvgps}
\end{equation}
\end{thm}
\begin{rem}
In the particular case where $\rho=0$, we obtain the strong consistency of the least squares estimate in a stable autoregressive model, already proved \textnormal{e.g.} in \cite{LaiWei83}, under the condition of stability $\Vert \theta \Vert_1 < 1$.
\end{rem}
\medskip
\noindent Let us now introduce the square matrix B of order $p+2$, partially made of the elements of $\beta$ given by \eqref{P1_Beta},
\begin{equation}
\label{P1_B}
B =
\begin{pmatrix}
1 & -\beta_1 & -\beta_2 & \hdots & \hdots & \hspace{0.15cm} -\beta_{p-1} \hspace{0.15cm} & \hspace{0.15cm} -\beta_p \hspace{0.15cm} & \hspace{0.15cm} \theta_{\!p} \rho \hspace{0.15cm}\\
-\beta_1 & 1-\beta_2 & -\beta_3 & \hdots & \hdots & -\beta_p & \theta_{\!p} \rho & 0\\
-\beta_2 & -\beta_1-\beta_3 & \hspace{0.15cm} 1-\beta_4 \hspace{0.15cm} & \hdots & \hdots & \theta_{\!p} \rho & 0 & 0\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
\hspace{0.15cm} -\beta_p \hspace{0.15cm} & \hspace{0.15cm} -\beta_{p-1}+\theta_{\!p} \rho \hspace{0.15cm} & -\beta_{p-2} & \hdots & \hdots & -\beta_1 & 1 & 0\\
\theta_{\!p} \rho & -\beta_p & -\beta_{p-1} & \hdots & \hdots & -\beta_2 & -\beta_1 & 1\\
\end{pmatrix}.
\end{equation}
\medskip
\noindent Under our stability conditions, we are able to establish the invertibility of $B$ in Lemma \ref{P1_Lem_InvB}. The corollary that follows will be useful in the next section.
\begin{lem}
\label{P1_Lem_InvB}
Under the stability conditions $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$, the matrix $B$ given by \eqref{P1_B} is invertible.
\end{lem}
\begin{cor}
\label{P1_Cor_InvC}
By virtue of Lemma \ref{P1_Lem_InvB}, the submatrix $C$ obtained by removing from $B$ its first row and first column is invertible.
\end{cor}
\noindent From now on, $\Lambda \in \dR^{p+2}$ is the unique solution of the linear system $B \Lambda = e$, \textit{i.e.}
\begin{equation}
\label{P1_VecLim}
\Lambda = B^{-1} e
\end{equation}
where the vector $e$ has already been defined in Remark \ref{Int_Rem_DefIJe}, but in higher dimension. Denote by $\lambda_0, \hdots, \lambda_{p+1}$ the elements of $\Lambda$ and let $\Delta_{p}$ be the Toeplitz matrix of order $p$ associated with the first $p$ elements of $\Lambda$, that is
\begin{equation}
\label{P1_Lambda}
\Delta_{p} = \begin{pmatrix}
\lambda_0 & \lambda_1 & \lambda_2 & \hdots & \hdots & \lambda_{p-1}\\
\lambda_1 & \lambda_0 & \lambda_1 & \hdots & \hdots & \lambda_{p-2}\\
\vdots & \vdots & \vdots & & & \vdots\\
\vdots & \vdots & \vdots & & & \vdots\\
\lambda_{p-1} & \lambda_{p-2} & \lambda_{p-3} & \hdots & \hdots & \lambda_0\\
\end{pmatrix}.
\end{equation}
\medskip
\noindent Via the same lines, we are able to establish the invertibility of $\Delta_{p}$ in Lemma \ref{P1_Lem_InvL}.
\begin{lem}
\label{P1_Lem_InvL}
Under the stability conditions $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$, for all $p \geq 1$, the matrix $\Delta_{p}$ given by \eqref{P1_Lambda} is positive definite.
\end{lem}
\noindent In light of foregoing, our next result deals with the asymptotic normality of $\tn$.
\begin{thm}
\label{P1_Thm_TlcTheta}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have
the asymptotic normality
\begin{equation}
\label{P1_TlcTheta}
\sqrt{n} \left( \tn - \theta^{*} \right) \liml \cN ( 0, \Sigma_\theta)
\end{equation}
where the asymptotic covariance matrix is given by
\begin{equation}
\label{P1_SigT}
\Sigma_{\theta} = \alpha^2 \left( I_{\! p} - \theta_{\!p} \rho J_{\! p} \right) \Delta_{p}^{-1} \left( I_{\! p} - \theta_{\!p} \rho J_{\! p} \right).
\end{equation}
\end{thm}
\begin{rem}
The covariance matrix $\Sigma_{\theta}$ is invertible under the stability conditions. Furthermore, due to the way it is constructed, $\Sigma_{\theta}$ is bisymmetric.
\end{rem}
\begin{rem}
\textcolor{blue}{In the particular case where $\rho=0$, $\Sigma_{\theta}$ reduces to $\Delta_{p}^{-1}$. This is a well-known result related to the asymptotic normality of the Yule-Walker estimator for the causal autoregressive process that can be found \textnormal{e.g.} in Theorem 8.1.1 of \cite{BrockwellDavis91}.}
\end{rem}
\noindent After establishing the almost sure convergence of the estimator $\tn$ and its asymptotic normality, we focus our attention on the almost sure rates of convergence.
\begin{thm}
\label{P1_Thm_RatTheta}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have
the quadratic strong law
\begin{equation}
\label{P1_LfqTheta}
\lim_{n \rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \tk - \theta^{*} \right) \left( \tk - \theta^{*} \right)^{\prime} = \Sigma_{\theta} \textnormal{\cvgps}
\end{equation}
where $\Sigma_{\theta}$ is given by \eqref{P1_SigT}. In addition, for all $v \in \dR^{p}$, we also have the law of iterated logarithm
\begin{eqnarray}
\label{P1_LliTheta}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} \left( \tn - \theta^{*} \right) & = & -\liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} \left( \tn - \theta^{*} \right), \nonumber\\
& = & \sqrt{v^{\, \prime}\, \Sigma_{\theta}\, v} \textnormal{\cvgps}
\end{eqnarray}
Consequently,
\begin{equation}
\label{P1_LliTheta2}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \left( \tn - \theta^{*} \right) \left( \tn - \theta^{*} \right)^{\prime} = \Sigma_{\theta} \textnormal{\cvgps}
\end{equation}
In particular,
\begin{equation}
\label{P1_LliTheta3}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \big\Vert \tn - \theta^{*} \big\Vert^2 = \textnormal{tr}(\Sigma_{\theta}) \textnormal{\cvgps}
\end{equation}
\end{thm}
\begin{rem}
\label{P1_Rem_RatTheta}
It clearly follows from \eqref{P1_LfqTheta} that
\begin{equation}
\label{P1_LfqTraceTheta}
\lim_{n \rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \big\Vert \tk - \theta^{*} \big\Vert^2 = \textnormal{tr}(\Sigma_{\theta}) \textnormal{\cvgps}
\end{equation}
Furthermore, from \eqref{P1_LliTheta3}, we have the almost sure rate of convergence
\begin{equation}
\label{P1_RatCvgTheta}
\big\Vert \tn - \theta^{*} \big\Vert^2 = O\left( \frac{\log \log n}{n} \right) \textnormal{\cvgps}
\end{equation}
\end{rem}
\begin{proof}
The proofs of Lemma \ref{P1_Lem_InvB} and Lemma \ref{P1_Lem_InvL} are given in Appendix A while those of Theorems \ref{P1_Thm_CvgTheta} to \ref{P1_Thm_RatTheta} may be found in Appendix B.
\end{proof}
\medskip
\noindent To conclude this section, let us draw a parallel between the results of \cite{BercuProia11} and the latter results for $p=1$. In this particular case, $\beta$ and $\alpha$ reduce to $(\theta+\rho)$ and $(1-\theta \rho)^{-1}(1+\theta\rho)^{-1}$ respectively, and it is not hard to see that we obtain the almost sure convergence of our estimate to
\begin{equation*}
\theta^{*} = \frac{\theta + \rho}{1 + \theta \rho}.
\end{equation*}
In addition, a straightforward calculation leads to
\begin{equation*}
\Sigma_{\theta} = \frac{(1 - \theta^2)(1 - \theta \rho)(1 - \rho^2)}{(1 + \theta \rho)^3}.
\end{equation*}
One can verify that these results correspond to Theorem 2.1 and Theorem 2.2 of \cite{BercuProia11}.
\bigskip
\section{ON THE SERIAL CORRELATION PARAMETER}
This section is devoted to the estimation of the serial correlation parameter $\rho$. First of all, it is necessary to evaluate, at stage $n$, the residual set $(\en)$ resulting from the biased estimation of $\theta$. For all $1 \leq k \leq n$, let
\begin{equation}
\label{P2_EstRes}
\ek = X_{k} - \tn^{\: \prime}\, \Phi_{k-1}^p.
\end{equation}
The initial value $\e_0$ may be arbitrarily chosen and we take $\e_0 = X_0$ for a matter of simplification. Then, a natural way to estimate $\rho$ is to make use of the least squares estimator which minimizes
\begin{equation*}
\nabla_{\!n}(\rho) = \sum_{k=1}^n \big( \ek - \rho\, \eek \big)^2.
\end{equation*}
Hence, it clearly follows that, for all $n \geq 1$,
\begin{equation}
\label{P2_Est}
\rn = \left( \sum_{k=1}^{n} \eek^{~ 2} \right)^{\! -1} \sum_{k=1}^{n} \ek\, \eek.
\end{equation}
It is important to note that one deals here with a scalar problem, in contrast to the study of the estimator of $\theta$ in Section 2. Our goal is to obtain the same asymptotic properties for the estimator of $\rho$ as those obtained for each component of the one of $\theta$. However, one shall realize that the results of this section are much more tricky to establish than those of the previous one.
\medskip
\noindent We first state the almost sure convergence of $\rn$ to the limiting value $\rho^{*} = \theta_{\!p} \rho \theta_{\!p}^{*}$.
\begin{thm}
\label{P2_Thm_CvgRho}
We have the almost sure convergence
\begin{equation}
\label{P2_CvgRho}
\lim_{n \rightarrow \infty} \rn = \rho^{*} \textnormal{\cvgps}
\end{equation}
\end{thm}
\noindent Our next result deals with the joint asymptotic normality of $\tn$ and $\rn$. For that purpose, it is necessary to introduce some additional notations. Denote by $P$ the square matrix of order $p+1$ given by
\begin{equation}
\label{P2_P}
P = \begin{pmatrix}
P_{B} & 0\\
P_{L}^{\, \prime} & \varphi
\end{pmatrix}
\end{equation}
where
\begin{eqnarray*}
P_{B} & = & \alpha \big( I_{\! p} - \theta_{\!p} \rho J_{\! p} \big) \Delta_{p}^{-1},\\
P_{L} & = & J_{\! p} \big( I_{\! p} - \theta_{\!p} \rho J_{\! p} \big) \big( \alpha \theta_{\!p} \rho\, \Delta_{p}^{-1} e + \theta_{\!p}^{*}\, \beta \big),\\
\varphi & = & - \alpha^{-1} \theta_{\!p}^{*}.
\end{eqnarray*}
Furthermore, let us introduce the Toeplitz matrix $\Delta_{p+1}$ of order $p+1$ which is the extension of $\Delta_{p}$ given by \eqref{P1_Lambda} to the next dimension,
\begin{equation}
\label{P2_LambdaP}
\Delta_{p+1} = \begin{pmatrix}
\Delta_{p} & J_{\! p}\, \Lambda_{p}^{1}\\
{\Lambda_{p}^{1}}^{\, \prime} J_{\! p} & \lambda_0
\end{pmatrix}
\end{equation}
with $\Lambda_{p}^{1} = \begin{pmatrix} \lambda_1 & \lambda_2 & \hdots & \lambda_{p} \end{pmatrix}^{\prime}$, and the positive semidefinite covariance matrix $\Gamma$ of order $p+1$, given by
\begin{equation}
\label{P2_GamCov}
\Gamma = P \Delta_{p+1} P^{\, \prime}.
\end{equation}
\begin{thm}
\label{P2_Thm_TlcRho}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have
the joint asymptotic normality
\begin{equation}
\label{P2_TlcJoint}
\sqrt{n} \begin{pmatrix}
\tn - \theta^{*}\\
\rn - \rho^{*}
\end{pmatrix} \liml \cN ( 0, \Gamma).
\end{equation}
In particular,
\begin{equation}
\label{P2_TlcRho}
\sqrt{n} \Big( \rn - \rho^{*} \Big) \liml \cN ( 0, \sigma^2_{\rho})
\end{equation}
where $\sigma^2_{\rho} = \Gamma_{p+1,\: p+1}$ is the last diagonal element of $\Gamma$.
\end{thm}
\begin{rem}
The covariance matrix $\Gamma$ has the following explicit expression,
\begin{equation*}
\Gamma = \begin{pmatrix}
\Sigma_{\theta} & \theta_{\!p} \rho\, J_{\! p}\, \Sigma_{\theta}\, e\\
\theta_{\!p} \rho\, e^{\, \prime} \Sigma_{\theta} J_{\! p} & \sigma^2_{\rho}
\end{pmatrix}
\end{equation*}
where
\begin{equation}
\label{P2_SigR}
\sigma_{\rho}^2 = P_{L}^{\, \prime}\, \Delta_{p}\, P_{L} - 2 \alpha^{-1} \theta_{\!p}^{*} {\Lambda_{p}^{1}}^{\, \prime} J_{\! p}\, P_{L} + \left( \alpha^{-1} \theta_{\!p}^{*} \right)^2 \lambda_0.
\end{equation}
\end{rem}
\begin{rem}
\label{P2_Rem_InvGamma}
The covariance matrix $\Gamma$ is invertible under the stability conditions if and only if $\theta_{\!p}^{*} \neq 0$ since, by a straightforward calculation,
\begin{equation*}
\det(\Gamma) = \alpha^{2(p-1)} \left( \theta_{\! p}^{*} \right)^2 \det(\Delta_{p+1}) \left( \frac{\det(I_{\! p} - \theta_{\!p} \rho J_{\! p})}{\det(\Delta_{p})} \right)^2
\end{equation*}
according to Lemma \ref{P1_Lem_InvL} and noticing that $(I_{\! p} - \theta_{\!p} \rho J_{\! p})$ is strictly diagonally dominant, thus invertible. As a result, the joint asymptotic normality given by \eqref{P2_TlcJoint} is degenerate in any situation such that $\theta_{\! p}^{*}=0$, that is
\begin{equation}
\label{JointCLT_Assumption}
\theta_{\!p} - \theta_{\!p-1} \rho = \theta_{\!p} \rho (\theta_{1} + \rho).
\end{equation}
Moreover, \eqref{P2_TlcRho} holds on $\{ \theta_{\!p} - \theta_{\!p-1} \rho \neq \theta_{\!p} \rho (\theta_{1} + \rho)\} \cup \{ \theta_{\!p} \neq 0, \rho \neq 0 \}$, otherwise the asymptotic normality associated with $\rn$ is degenerate. In fact, a more restrictive condition ensuring that \eqref{P2_TlcRho} still holds may be $\{ \theta_{\!p} \neq 0 \}$, \textit{i.e.} that one deals at least with a $p-$order autoregressive process. This restriction seems natural in the context of the study and can be compared to the assumption $\{ \theta \neq 0 \}$ in \cite{BercuProia11}. Theorem 3.2 of \cite{BercuProia11} ensures that the joint asymptotic normality is degenerate under $\{ \theta = -\rho \}$. One can note that such an assumption is equivalent to \eqref{JointCLT_Assumption} in the case of the $p-$order process, since both of them mean that the last component of $\theta^{*}$ has to be nonzero.
\end{rem}
\noindent The almost sure rates of convergence for $\rn$ are as follows.
\begin{thm}
\label{P2_Thm_RatRho}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have
the quadratic strong law
\begin{equation}
\label{P2_LfqRho}
\lim_{n \rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \Big( \rk - \rho^{*} \Big)^2 = \sigma^2_{\rho} \textnormal{\cvgps}
\end{equation}
where $\sigma^2_{\rho}$ is given by \eqref{P2_SigR}. In addition, we also have the law of iterated logarithm
\begin{eqnarray}
\label{P2_LliRho}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \Big( \rn - \rho^{*} \Big) & = & -\liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \Big( \rn - \rho^{*} \Big), \nonumber\\
& = & \sigma_{\rho} \textnormal{\cvgps}
\end{eqnarray}
Consequently,
\begin{equation}
\label{P2_LliRho2}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \Big( \rn - \rho^{*} \Big)^2 = \sigma_{\rho}^2 \textnormal{\cvgps}
\end{equation}
\end{thm}
\begin{rem}
\label{P2_Rem_RatRho}
It clearly follows from \eqref{P2_LliRho2} that we have the almost sure rate of convergence
\begin{equation}
\label{P2_RatCvgRho}
\Big( \rn - \rho^{*} \Big)^2 = O\left( \frac{\log \log n}{n} \right) \textnormal{\cvgps}
\end{equation}
\end{rem}
\medskip
\noindent As before, let us also draw the parallel between the results of \cite{BercuProia11} and the latter results for $p=1$. In this particular case, we immediately obtain $\rho^{*} = \theta \rho \theta^{*}$. Moreover, an additionnal step of calculation shows that
\begin{equation*}
\sigma_{\rho}^2 = \frac{1 - \theta\rho}{(1 + \theta\rho)^3} \left( (\theta + \rho)^2 (1 + \theta\rho)^2 + (\theta \rho)^2 (1 - \theta^2) (1 - \rho^2) \right).
\end{equation*}
One can verify that these results correspond to Theorem 3.1 and Theorem 3.2 of \cite{BercuProia11}. Besides, the estimators of $\theta$ and $\rho$ are self-normalized. Consequently, the asymptotic variances $\Sigma_{\theta}$ and $\sigma_{\rho}^2$ do not depend on the variance $\sigma^2$ associated with the driven noise $(V_{n})$. To be complete and provide an important statistical aspect, it seemed advisable to suggest an estimator of the true variance $\sigma^2$ of the model, based on these previous estimates. Consider, for all $n \geq 1$, the estimator given by
\begin{equation}
\label{P2_EstSig}
\wh{\sigma}_{n}^{\, 2} = \left( 1 - \rn^{~ 2}\, \wh{\theta}_{\!p,\, n}^{\, -2} \right) \frac{1}{n} \sum_{k=0}^{n} \ek^{~ 2}
\end{equation}
where $\wh{\theta}_{\!p,\, n}$ stands for the $p-$th component of $\tn$.
\begin{thm}
\label{P2_Thm_CvgSig}
We have the almost sure convergence
\begin{equation}
\label{P2_CvgRho}
\lim_{n \rightarrow \infty} \wh{\sigma}_{n}^{\, 2} = \sigma^2 \textnormal{\cvgps}
\end{equation}
\end{thm}
\begin{proof}
The proofs of Theorems \ref{P2_Thm_CvgRho} to \ref{P2_Thm_RatRho} are given in Appendix C. The one of Theorem \ref{P2_Thm_CvgSig} is left to the reader as it directly follows from that of Theorem \ref{P2_Thm_CvgRho}.
\end{proof}
\bigskip
\section{ON THE DURBIN-WATSON STATISTIC}
We shall now investigate the asymptotic behavior of the Durbin-Watson statistic for the general autoregressive process \cite{DurbinWatson50}, \cite{DurbinWatson51}, \cite{DurbinWatson71}, given, for all $n \geq 1$, by
\begin{equation}
\label{P3_Dn}
\dn = \left( \sum_{k=0}^{n} \ek^{~ 2} \right)^{\! -1} \sum_{k=1}^{n} \Big( \ek - \eek \Big)^2.
\end{equation}
As mentioned, the almost sure convergence and the asymptotic normality of the Durbin-Watson statistic have previously been investigated in \cite{BercuProia11} in the particular case where $p=1$. It has enabled the authors to propose a two-sided statistical test for the presence of a significant residual autocorrelation. They also explained how this statistical procedure outperformed the commonly used Ljung-Box \cite{LjungBox78} and Box-Pierce \cite{BoxPierce70} \textit{portmanteau} tests for white noise in the case of the first-order autoregressive process, and how it was asymptotically equivalent to the \textit{h-test} of Durbin \cite{Durbin70}, on a theoretical basis and on simulated data. They went even deeper in the study, establishing the distribution of the statistic under the null hypothesis $`` \rho = \rho_0"$, with $\vert \rho_0 \vert < 1$, as well as under the alternative hypothesis $`` \rho \neq \rho_0"$, and noticing the existence of a critical situation in the case where $\theta = -\rho$. This pathological case arises when the covariance matrix $\Gamma$ given by \eqref{P2_GamCov} is singular, and can be compared in the multivariate framework to the content of Remark \ref{P2_Rem_InvGamma}. Our goal is to obtain the same asymptotic results for all $p \geq 1$ so as to build a new statistical procedure for testing serial correlation in the residuals. In this paper, we shall only focus our attention on the test $`` \rho = 0"$ against $`` \rho \neq 0"$, of increased statistical interest. \textcolor{blue}{We shall see below that from a theoretical and a practical point of view, our statistical test procedure clarifies ans outperforms the \textit{h-test} of Durbin. In particular, it avoids the presence of an abstract variance estimation likely to generate perturbations on small-sized samples.} In the next section, \textcolor{blue}{we will observe on simulated data that the procedure proposed in Theorem \ref{P3_Thm_TestD} is more powerful than the \textit{portmanteau} tests \cite{LjungBox78}, \cite{BoxPierce70}, often used} for testing the significance of the first-order serial correlation of the driven noise in a $p-$order autoregressive process.
\medskip
\noindent First, one can observe that $\dn$ and $\rn$ are asymptotically linked together by an affine transformation. Consequently, the asymptotic behavior of the Durbin-Watson statistic directly follows from the previous section. We start with the almost sure convergence to the limiting value $D^{*} = 2(1 - \rho^{*})$.
\begin{thm}
\label{P3_Thm_CvgD}
We have the almost sure convergence
\begin{equation}
\label{P3_CvgD}
\lim_{n \rightarrow \infty} \dn = D^{*} \textnormal{\cvgps}
\end{equation}
\end{thm}
\noindent Our next result deals with the asymptotic normality of $\dn$. It will be the keystone of the statistical procedure deciding whether residuals have a significant first-order correlation or not, for a given significance level. Denote
\begin{equation}
\label{P3_SigD}
\sigma_{D}^2 = 4 \sigma_{\rho}^2
\end{equation}
where the variance $\sigma_{\rho}^2$ is given by \eqref{P2_SigR}.
\begin{thm}
\label{P3_Thm_TlcD}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have the asymptotic normality
\begin{equation}
\label{P3_TlcD}
\sqrt{n} \left( \dn - D^{*} \right) \liml \cN ( 0, \sigma^2_{D}).
\end{equation}
\end{thm}
\begin{rem}
We immediately deduce from \eqref{P3_TlcD} that
\begin{equation}
\label{P2_TlcDChi2}
\frac{n}{\sigma^2_{D}} \left( \dn - D^{*} \right)^2 \liml \chi^2
\end{equation}
where $\chi^2$ has a Chi-square distribution with one degree of freedom.
\end{rem}
\noindent Let us focus now on the almost sure rates of convergence of $\dn$.
\begin{thm}
\label{P3_Thm_RatD}
Assume that $(V_{n})$ has a finite moment of order 4. Then, we have
the quadratic strong law
\begin{equation}
\label{P3_LfqD}
\lim_{n \rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \dk - D^{*} \right)^2 = \sigma^2_{D} \textnormal{\cvgps}
\end{equation}
where $\sigma_{D}^2$ is given by \eqref{P3_SigD}. In addition, we also have the law of iterated logarithm
\begin{eqnarray}
\label{P3_LliD}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \left( \dn - D^{*} \right) & = & -\liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \left( \dn - D^{*} \right), \nonumber\\
& = & \sigma_{D} \textnormal{\cvgps}
\end{eqnarray}
Consequently,
\begin{equation}
\label{P3_LliD2}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \left( \dn - D^{*} \right)^2 = \sigma_{D}^2 \textnormal{\cvgps}
\end{equation}
\end{thm}
\begin{rem}
\label{P3_Rem_RatD}
It clearly follows from \eqref{P3_LliD2} that we have the almost sure rate of convergence
\begin{equation}
\label{P3_RatCvgD}
\left( \dn - D^{*} \right)^2 = O\left( \frac{\log \log n}{n} \right) \textnormal{\cvgps}
\end{equation}
\end{rem}
\noindent We are now in the position to propose the two-sided statistical test built on the Durbin-Watson statistic. First of all, we shall not investigate the particular case where $\theta_{\!p} = 0$ since our procedure is of interest only for autoregressive processes of order $p$. One wishes to test the presence of a significant serial correlation, setting
$$\cH_0\,:\,`` \rho = 0" \hspace{1cm} \text{against}\hspace{1cm} \cH_1\,:\,`` \rho \neq 0".
\vspace{2ex}$$
\begin{thm}
\label{P3_Thm_TestD}
Assume that $(V_{n})$ has a finite moment of order 4, $\theta_{\!p} \neq 0$ and $\theta_{\!p}^{*} \neq 0$. Then, under the null hypothesis $\cH_0\,:\,`` \rho = 0"$,
\begin{equation}
\label{P3_TestH0}
\frac{n}{4 \wh{\theta}_{\!p,\, n}^{\, 2}} \left( \dn - 2 \right)^2 \liml \chi^2
\end{equation}
where $\wh{\theta}_{\!p,\, n}$ stands for the $p-$th component of $\tn$, and where $\chi^2$ has a Chi-square distribution with one degree of freedom. In addition, under the alternative hypothesis $\cH_1\,:\,`` \rho \neq 0"$,
\begin{equation}
\label{P3_TestH1}
\lim_{n \rightarrow \infty} \frac{n}{4 \wh{\theta}_{\!p,\, n}^{\, 2}} \left( \dn - 2 \right)^2 = +\infty \textnormal{\cvgps}
\end{equation}
\end{thm}
\noindent From a practical point of view, for a significance level $a$ where $0 < a < 1$, the acceptance and rejection regions are given by $\cA = [0, z_{a}]$ and $\cR = \hspace{0.15cm} ]z_{a}, +\infty[$ where $z_{a}$ stands for the $(1-a)-$quantile of the Chi-square distribution with one degree of freedom. The null hypothesis $\cH_0$ will not be rejected if the empirical value
\begin{equation*}
\frac{n}{4 \wh{\theta}_{\!p,\, n}^{\, 2}} \left( \dn - 2 \right)^2 \leq z_{a},
\end{equation*}
and will be rejected otherwise.
\begin{rem}
In the particular case where $\theta_{\!p}^{*} = 0$, the test statistic do not respond under $\cH_1$ as described above. To avoid such situation, we suggest to make use of Theorem \ref{P1_Thm_TlcTheta} for testing beforehand whether $\wh{\theta}_{\!p,\, n}$ is significantly far from zero. Besides, testing $\cH_0\,:\,`` \rho = 0"$ with $\theta_{\!p}^{*} = 0$ amounts to testing the significance of the $p-$th coefficient of the model, not rejected under $\{ \theta_{\!p} \neq 0 \}$. Roughly speaking, under $\{ \theta_{\!p} \neq 0 \} \cap \{ \theta_{\!p}^{*} = 0 \}$, we obviously have $\rho \neq 0$ and the use of Theorem \ref{P3_Thm_TestD} would be irrelevant since $\cH_1$ is certainly true.
\end{rem}
\noindent \textcolor{blue}{As previously mentioned, the statistical procedure of Theorem \ref{P3_Thm_TestD} appears to be a substantial clarification of the \textit{h-test} of Durbin \cite{Durbin70}. To be more precise, formula (12) of \cite{Durbin70} suggests to make use of the test statistic
\begin{equation}
\label{StatH}
\wh{H}_{n} = \rn \, \sqrt{\frac{n}{1 - n \wh{\dV}_{n}(\wh{\theta}_{1,\, n}) }}
\end{equation}
where $\wh{\dV}_{n}(\wh{\theta}_{1,\, n})$ is the least squares estimate of the variance of the first element of $\tn$, and to test it as a standard normal deviate. The presence of an abstract variance estimator not only makes the procedure quite tricky to interpret, but also adds some vulnerability on small-sized samples, as will be observed in the next section. The almost sure equivalence between both test statistics is shown in Appendix D.}
\begin{rem}
\textcolor{blue}{The \textnormal{h-test} of Durbin \cite{Durbin70} is based on the normality assumption on the driven noise $(V_{n})$. As a consequence, $(X_{n})$ is a Gaussian process and the maximum likelihood strategy is suitable not only to provide the estimates, but also to determine their conditional distributions. One can observe that all our results hold without any Gaussianity assumption on $(V_{n})$. Hence, Theorem \ref{P3_Thm_TestD} appears to generalize the \textnormal{h-test} of Durbin.}
\end{rem}
\begin{proof}
The proofs of Theorems \ref{P3_Thm_CvgD} to \ref{P3_Thm_RatD} are left to the reader as they follow essentially the same lines as those given in Appendix C of \cite{BercuProia11}. Theorem \ref{P3_Thm_TestD} is an immediate consequence of Theorem \ref{P3_Thm_TlcD}, noticing that $\sigma^2_{\rho}$ reduces to $\theta_{\!p}^{\, 2}$ under $\cH_0$ and using the same methodology as in the proof of Theorem \ref{P2_Thm_CvgRho}.
\end{proof}
\bigskip
\section{\textcolor{blue}{CONCLUSION}}
\textcolor{blue}{We will now briefly summarize our constatations on simulated samples. Following the same methodology as in Section 5 of \cite{BercuProia11} and also being inspired by the empirical work of Park \cite{Park75}, we have compared the empirical power of the statistical procedure of Theorem \ref{P3_Thm_TestD} with the statistical tests commonly used in time series analysis to detect the presence of a significant first-order correlation in the residuals. Assuming that $\theta_{\! p} \neq 0$ was a statistically significant parameter, our observations were essentially the same as those of \cite{BercuProia11} for different sets of parameters. Namely, on large samples $(n=500)$, we have clearly constated the asymptotic equivalence between the \textit{h-test}, the Breusch-Godfrey test and our statistical procedure, as well as the superiority over the commonly used \textit{portmanteau} tests. On small-sized samples $(n=30)$, our procedure has outperformed all tests by always being more sensitive to the presence of correlation in the residuals, except under $\cH_0$ even if the 84\% of non-rejection were quite satisfying. Our expression of the test statistic seems therefore less vulnerable than the one of Durbin for small sizes. To conclude, the extension of this work to the stable $p-$order autoregressive process where the driven noise is also generated by a $q-$order autoregressive process would constitute a substantial progress in time series analysis. The objective would be to propose a statistical procedure to evaluate $\cH_0\,:\,`` \rho_1 = 0,\, \rho_2 = 0,\, \hdots,\, \rho_{q} = 0 "$ against the alternative hypothesis $\cH_1$ that one can find $1 \leq k \leq q$ such that $\rho_k \neq 0$, based on the Durbin-Watson statistic. In \cite{Durbin70}, Durbin gives an outline of such a strategy which seems rather complicated to implement, relying on power series of infinite orders and under a Gaussianity assumption on the driven noise $(V_{n})$. The author strongly believes that it could be possible to obtain the results explicitly and under weaker assumptions, \textit{via} very tedious calculations. A recent approach in \cite{ButlerPaolella08}, based on saddlepoint approximations for ratios of quadratic forms, could form another way to tackle the problem since the Durbin-Watson statistic is precisely a ratio of quadratic forms.}
\bigskip
\section*{Appendix A}
\begin{center}
{\small ON SOME LINEAR ALGEBRA CALCULATIONS}
\end{center}
\renewcommand{\thesection}{\Alph{section}}
\renewcommand{\theequation}
{\thesection.\arabic{equation}} \setcounter{section}{1}
\setcounter{equation}{0}
\subsection*{}
\begin{center}
{\bf A.1. Proof of Lemma \ref{P1_Lem_InvB}.}
\end{center}
We start with the proof of Lemma \ref{P1_Lem_InvB}. Our goal is to show that the matrix $B$ given by \eqref{P1_B} is invertible. Consider the decomposition $B = B_1 + \rho B_2$, where
\begin{equation*}
B_1 =
\begin{pmatrix}
1 & -\theta_1 & -\theta_2 & \hdots & \hdots & \hspace{0.15cm} -\theta_{\!p-1} \hspace{0.15cm} & \hspace{0.15cm} -\theta_{\!p} \hspace{0.15cm} & \hspace{0.15cm} 0 \hspace{0.15cm}\\
\hspace{0.15cm} -\theta_1 \hspace{0.15cm} & 1-\theta_2 & -\theta_3 & \hdots & \hdots & -\theta_{\!p} & 0 & 0\\
-\theta_2 & \hspace{0.15cm} -\theta_1-\theta_3 \hspace{0.15cm} & \hspace{0.15cm} 1-\theta_4 \hspace{0.15cm} & \hdots & \hdots & 0 & 0 & 0\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
-\theta_{\!p} & -\theta_{\!p-1} & -\theta_{\!p-2} & \hdots & \hdots & -\theta_1 & 1 & 0\\
0 & -\theta_{\!p} & -\theta_{\!p-1} & \hdots & \hdots & -\theta_2 & -\theta_1 & 1\\
\end{pmatrix},
\end{equation*}
\begin{equation*}
B_2 =
\begin{pmatrix}
0 & -1 & \theta_1 & \hdots & \hspace{0.15cm} \hdots \hspace{0.15cm} & \hspace{0.15cm} \theta_{\!p-2} \hspace{0.15cm} & \hspace{0.15cm} \theta_{\!p-1} \hspace{0.15cm} & \hspace{0.2cm} \theta_{\!p} \hspace{0.15cm}\\
-1 & \theta_1 & \theta_2 & \hdots & \hdots & \theta_{\!p-1} & \theta_{\!p} & 0\\
\theta_1 & -1+\theta_2 & \theta_3 & \hdots & \hdots & \theta_{\!p} & 0 & 0\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
\vdots & \vdots & \vdots & & & \vdots & \vdots & \vdots\\
\hspace{0.15cm} \theta_{\!p-1} \hspace{0.15cm} & \hspace{0.2cm} \theta_{\!p-2}+ \theta_{\!p} \hspace{0.15cm} & \hspace{0.2cm} \theta_{\!p-3} \hspace{0.2cm} & \hdots & \hdots & -1 & 0 & 0\\
\theta_{\!p} & \theta_{\!p-1} & \theta_{\!p-2} & \hdots & \hdots & \theta_{1} & -1 & 0\\
\end{pmatrix}.
\end{equation*}
\medskip
\noindent It is trivial to see that $\vert \theta_i + \theta_j \vert \leq \vert \theta_i \vert + \vert \theta_j \vert$ for all $1 \leq i,j \leq p$, and the same goes for $1-\vert \theta_i \vert \leq \vert 1 - \theta_i \vert$. These inequalities immediately imply that $B_1$ is strictly diagonally dominant, and thus invertible by virtue of Levy-Desplanques' theorem 6.1.10 of \cite{HornJohnson90}. Hence, $B = (I_{\! p+2} + \rho B_2 B_1^{-1})B_1$ and the invertibility of $B$ only depends on the spectral radius of $\rho B_2 B_1^{-1}$, \textit{i.e.} the supremum modulus of its eigenvalues. One can explicitly obtain, by a straightforward calculation, that
\begin{equation*}
B_2 B_1^{-1} =
\begin{pmatrix}
\hspace{0.15cm} -\theta_1 \hspace{0.15cm} & \hspace{0.15cm} -1-\theta_2 \hspace{0.15cm} & \hspace{0.15cm} \theta_1-\theta_3 \hspace{0.15cm} & \hdots & \hspace{0.15cm} \theta_{\!p-2}-\theta_{\!p} \hspace{0.15cm} & \hspace{0.15cm} \theta_{\!p-1} \hspace{0.15cm} & \hspace{0.15cm} \theta_{\!p} \hspace{0.15cm}\\
-1 & 0 & \hdots & \hdots & \hdots & \hdots & 0\\
0 & -1 & 0 & \hdots & \hdots & \hdots & 0\\
\vdots & \ddots & \ddots & \ddots & & & \vdots \\
\vdots & & \ddots & \ddots & \ddots & & \vdots \\
0 & \hdots & \hdots & 0 & -1 & 0 & 0 \\
0 & \hdots & \hdots & \hdots & 0 & -1 & 0\\
\end{pmatrix}.
\end{equation*}
\medskip
\noindent The sum of the first row of $B_2 B_1^{-1}$ is $-1$, involving \textit{de facto} that $-1$ is an eigenvalue of $B_2 B_1^{-1}$ associated with the $(p+2)-$dimensional eigenvector $\begin{pmatrix} 1 & 1 & \hdots & 1 \end{pmatrix}^{\prime}$. By the same way, it is clear that $1$ is an eigenvalue of $B_2 B_1^{-1}$ associated with the eigenvector $\begin{pmatrix} 1 & -1 & \hdots & (-1)^{p+1} \end{pmatrix}^{\prime}$. Let $P(\lambda) = \det(B_2 B_1^{-1} - \lambda I_{\!p+2})$ be the characteristic polynomial of $B_2 B_1^{-1}$. Then, $P(\lambda)$ is recursively computable and explicitly given by
\begin{equation}
\label{A1_InvB_PolyP}
P(\lambda) = (-\lambda)^{p+2} + \sum_{k=1}^{p+2} b_{k}\, (-\lambda)^{p+2-k}
\end{equation}
where $(b_{k})$ designates, for $k \in \{1, \hdots, p+2\}$, the elements of the first line of $B_2 B_1^{-1}$. Since $-1$ and 1 are zeroes of $P(\lambda)$, there exists a polynomial $Q(\lambda)$ of degree $p$ such that $P(\lambda) = (\lambda^2-1)Q(\lambda)$, and a direct calculation shows that $Q$ is given by
\begin{equation}
\label{A1_InvB_PolyQ}
Q(\lambda) = (-\lambda)^{p} - \sum_{k=1}^{p} \theta_{k}\, (-\lambda)^{p-k}.
\end{equation}
Furthermore, let $R(\lambda)$ be the polynomial of degree $p$ defined as
\begin{equation}
\label{A1_InvB_PolyR}
R(\lambda) = \lambda^{p} - \sum_{k=1}^{p} \vert\, \theta_{k} \vert\, \lambda^{p-k},
\end{equation}
and note that we clearly have $R(\vert \lambda \vert) \leq \vert Q(\lambda) \vert$, for all $\lambda \in \dC$. Assume that $\lambda_0 \in \dC$ is an eigenvalue of $B_2 B_1^{-1}$ such that $\vert \lambda_0 \vert > 1$. Then,
\begin{eqnarray*}
R(\vert \lambda_0 \vert) & = & \vert \lambda_0 \vert^{p} - \sum_{k=1}^{p} \vert\, \theta_{k} \vert \vert \lambda_0 \vert^{p-k} = \vert \lambda_0 \vert^{p} \left( 1 - \sum_{k=1}^{p} \vert\, \theta_{k} \vert \vert \lambda_0 \vert^{-k} \right), \\
& \geq & \vert \lambda_0 \vert^{p} \left( 1 - \sum_{k=1}^{p} \vert\, \theta_{k} \vert \right) > 0
\end{eqnarray*}
as soon as $\Vert \theta \Vert_1 < 1$. Consequently, $\vert Q(\lambda_0) \vert > 0$. This obviously contradicts the hypothesis that $\lambda_0$ is an eigenvalue of $B_2 B_1^{-1}$. \textcolor{blue}{This strategy is closely related to the classical result of Cauchy on the location of zeroes of algebraic polynomials, see \textit{e.g.} Theorem 2.1 of \cite{MilovanovicRassias00}.} In conclusion, all the zeroes of $Q(\lambda)$ lie in the unit circle, implying $\rho(B_2 B_1^{-1}) \leq 1$. Since 1 and $-1$ are eigenvalues of $B_2 B_1^{-1}$, we have precisely $\rho(B_2 B_1^{-1}) = 1$, and therefore $\rho(\rho B_2 B_1^{-1}) = \vert \rho \vert < 1$. This guarantees the invertibility of $B$ under the stability conditions, achieving the proof of Lemma \ref{P1_Lem_InvB}. Finally, Corollary \ref{P1_Cor_InvC} immediately follows from Lemma \ref{P1_Lem_InvB}. As a matter of fact, since $B$ is invertible, we have $\det(B) \neq 0$. Denote by $b$ the first diagonal element of $B^{-1}$. Since $\det(C)$ is the cofactor of the first diagonal element of $B$, we have
\begin{equation}
\label{A1_InvC_Cof}
b = \frac{\det(C)}{\det(B)}.
\end{equation}
However, it follows from \eqref{P1_VecLim} that $b = \lambda_0$. We shall prove in the next subsection that the matrix $\Delta_{p}$ given by \eqref{P1_Lambda} is positive definite. It clearly implies that $\lambda_0 > 0$ which means that $b > 0$, so $\det(C) \neq 0$, and the matrix $C$ is invertible.
\hfill
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\bigskip
\subsection*{}
\begin{center}
{\bf A.2. Proof of Lemma \ref{P1_Lem_InvL}.}
\end{center}
Let us start by proving that the spectral radius of the companion matrix associated with model \eqref{Int_Mod} is strictly less than 1. By virtue of the fundamental autoregressive equation \eqref{A11_NewAR} detailed in the next section, the system \eqref{Int_Mod} can be rewritten in the vectorial form, for all $n \geq p+1$,
\begin{equation}
\label{A1_InvL_X}
\Phi_{n}^{p+1} = C_{\! A} \Phi_{n-1}^{p+1} + W_{n}
\end{equation}
where $\Phi_{n}^{p+1}$ stands for the extension of $\Phi_{n}^{p}$ given by \eqref{P1_Phi_Lag} to the next dimension, $W_{n} = \begin{pmatrix} V_{n} & 0 & \hdots & 0 \end{pmatrix}^{\prime}$ and where the companion matrix of order $p+1$
\begin{equation}
\label{A1_InvL_CompMat}
C_{\! A} = \begin{pmatrix}
\theta_1 + \rho ~ & \theta_2 - \theta_1 \rho ~ & \hdots & \theta_{\!p} - \theta_{\!p-1} \rho ~ & -\theta_{\!p} \rho \\
1 & 0 & \hdots & 0 & 0 \\
0 & 1 & \hdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \hdots & 1 & 0
\end{pmatrix}.
\end{equation}
\medskip
\noindent Let $P_{\! A}(\mu) = \det(C_{\! A} - \mu I_{\!p+1})$ be the characteristic polynomial of $C_{\! A}$. Then, it follows from Lemma 4.1.1 of \cite{Duflo97} that
\begin{eqnarray}
P_{\! A}(\mu) & = & (-1)^{p} \left( \mu^{p+1} - (\theta_1 + \rho) \mu^{p} - \sum_{k=2}^{p} \left( \theta_{\!k} - \theta_{k-1} \rho \right) \mu^{p+1-k} + \theta_{\!p} \rho \right), \nonumber \\
\label{A1_InvL_PolyPA}
& = & (-1)^{p}\,\, ( \mu - \rho ) \left( \mu^{p} - \sum_{k=1}^{p} \theta_{\!k} \mu^{p-k} \right) = (-1)^{p} (\mu - \rho) P(\mu)
\end{eqnarray}
where the polynomial
\begin{equation*}
P(\mu) = \mu^{p} - \sum_{k=1}^{p} \theta_{\!k}\, \mu^{p-k}.
\end{equation*}
Assume that $\mu_0 \in \dC$ is an eigenvalue of $C_{\! A}$ such that $\vert \mu_0 \vert \geq 1$. Then, under the stability condition $\vert \rho \vert < 1$, we obviously have $\mu_0 \neq \rho$. Consequently, we obtain that $P(\mu_0) = 0$ which implies, since $\mu_0 \neq 0$, that
\begin{equation}
\label{A1_InvL_CondVP}
1 - \sum_{k=1}^{p} \theta_{\!k}\, \mu_0^{-k} = 0.
\end{equation}
Nevertheless,
\begin{equation*}
\left\vert \sum_{k=1}^{p} \theta_{\!k}\, \mu_0^{-k} \right\vert \leq \sum_{k=1}^{p} \vert\, \theta_{\!k} \vert \vert \mu_0^{-k} \vert \leq \sum_{k=1}^{p} \vert\, \theta_{\!k} \vert < 1
\end{equation*}
as soon as $\Vert \theta \Vert_1 < 1$ which contradicts \eqref{A1_InvL_CondVP}. Hence, $\rho(C_{\! A}) < 1$ under the stability conditions $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$. Hereafter, let $(Y_{n})$ be the stationary autoregressive process satisfying, for all $n \geq p+1$,
\begin{equation}
\label{A1_InvL_Y}
\Psi_{n}^{p+1} = C_{\! A} \Psi_{n-1}^{p+1} + W_{n}
\end{equation}
where
\begin{equation*}
\Psi^{p+1}_{n} = \begin{pmatrix} Y_{n} & \hspace{0.1cm} & Y_{n-1} & \hspace{0.1cm} & \hdots & \hspace{0.1cm} & Y_{n-p} \end{pmatrix}^{\prime}.
\end{equation*}
It follows from \eqref{A1_InvL_Y} that, for all $n \geq p+1$,
\begin{equation*}
Y_{n} = (\theta_1 + \rho) Y_{n-1} + \sum_{k=2}^{p} (\theta_{\!k} - \theta_{\!k-1} \rho) Y_{n-k} - \theta_{\!p} \rho Y_{n-p-1} + V_{n}.
\end{equation*}
By virtue of Theorem 4.4.2 of \cite{BrockwellDavis91}, the spectral density of the process $(Y_{n})$ is given, for all $x$ in the torus $ \dT = [-\pi, \pi]$, by
\begin{equation}
\label{A1_InvL_SpecDens}
f_{Y}(x) = \frac{\sigma^2}{2 \pi \vert A(e^{- \mathrm{i} x}) \vert^{\, 2}}
\end{equation}
where the polynomial $A$ is defined, for all $\mu \neq 0$, as
\begin{equation}
\label{A1_InvL_PolyA}
A(\mu) = (-1)^{p}\, \mu^{p+1} P_{\! A}(\mu^{-1}),
\end{equation}
in which $P_{A}$ is the polynomial given in \eqref{A1_InvL_PolyPA}, and $A(0) = 1$. In light of foregoing, $A$ has no zero on the unit circle. In addition, for all $k \in \dZ$, denote by
\begin{equation*}
\wh{f}_{k} = \int_{\dT} f_{Y}(x) e^{- \mathrm{i} k x} \, \mathrm{d}x
\end{equation*}
the Fourier coefficient of order $k$ associated with $f_{Y}$. It is well-known that, for all $p \geq 1$, the covariance matrix of the vector $\Psi_{n}^{p}$ coincides with the Toeplitz matrix of order $p$ of the spectral density $f_{Y}$ in \eqref{A1_InvL_SpecDens}. More precisely, for all $p \geq 1$, we have
\begin{equation}
\label{A1_InvL_ToepCov}
T_{\!p}(f_{Y}) = \left( \wh{f}_{i-j} \right)_{1\, \leq\,\, i,\, j\, \leq\, p} = \sigma^2 \Delta_{p}
\end{equation}
where $\Delta_{p}$ is given by \eqref{P1_Lambda} and $T$ stands for the Toeplitz operator. As a matter of fact, since $\rho(C_{\! A}) < 1$, we have
\begin{equation*}
\lim_{n\rightarrow \infty} \dE\Big[ \Phi_{n}^{p} {\Phi_{n}^{p}}^{\: \prime} \Big] = \dE\Big[ \Psi_{p}^{p} \Psi_{p}^{p \:\, \prime} \Big] = \sigma^2 \Delta_{p}.
\end{equation*}
Finally, we deduce from Proposition 4.5.3 of \cite{BrockwellDavis91}, \textcolor{blue}{or from the properties of Toeplitz operators deeply studied in \cite{GrenanderSzego58},} that
\begin{equation}
\label{A1_InvL_MinVP}
2\pi m_{f} \leq \lambda_{\min}(T_{\!p}(f_{Y})) \leq \lambda_{\max}(T_{\!p}(f_{Y})) \leq 2 \pi M_{f}
\end{equation}
where
\begin{equation*}
m_{f} = \min_{x \, \in \, \dT} f_{Y}(x) \hspace{0.5cm} \text{and} \hspace{0.5cm} M_{f} = \max_{x \,\in \, \dT} f_{Y}(x).
\end{equation*}
Therefore, as $m_{f} > 0$, $T_{\!p}(f_{Y})$ is positive definite, which clearly ensures that for all $p \geq 1$, $\Delta_{p}$ is also positive definite. This achieves the proof of Lemma \ref{P1_Lem_InvL}.
\hfill
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\bigskip
\section*{Appendix B}
\begin{center}
{\small PROOFS OF THE AUTOREGRESSIVE PARAMETER RESULTS}
\end{center}
\renewcommand{\thesection}{\Alph{section}}
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\setcounter{equation}{0}
\subsection*{}
\begin{center}
{\bf B.1. Preliminary Lemmas.}
\end{center}
We start with some useful technical lemmas we shall make repeatedly use of. The proof of Lemma \ref{A1_Lem_StabV} may be found in the one of Corollary 1.3.21 in \cite{Duflo97}.
\begin{lem}
\label{A1_Lem_StabV}
Assume that $(V_{n})$ is a sequence of independent and identically distributed random variables such that, for some $a \geq 1$, $\dE[|V_1|^a]$ is finite. Then,
\begin{equation}
\label{A1_Lem_StabV_Lgn}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \vert V_k \vert^a = \dE[|V_1|^a] \textnormal{\cvgps}
\end{equation}
and
\begin{equation}
\label{A1_Lem_StabV_Sup}
\sup_{1 \leq k \leq n} \vert V_k \vert = o(n^{1/a}) \textnormal{\cvgps}
\end{equation}
\end{lem}
\begin{lem}
\label{A1_Lem_StabX}
Assume that $(V_{n})$ is a sequence of independent and identically distributed random variables such that, for some $a \geq 1$, $\dE[|V_1|^a]$ is finite. If $(X_n)$ satisfies \eqref{Int_Mod} with $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$, then
\begin{equation}
\label{A1_Lem_StabX_Lgn}
\sum_{k=0}^{n} \vert X_k \vert^a = O(n) \textnormal{\cvgps}
\end{equation}
and
\begin{equation}
\label{A1_Lem_StabX_Sup}
\sup_{0 \leq k \leq n} \vert X_k \vert = o(n^{1/a}) \textnormal{\cvgps}
\end{equation}
\end{lem}
\begin{rem}
In the particular case where $a=4$, we obtain that
\begin{equation*}
\sum_{k=0}^{n} X_k^4=O(n) \textnormal{\cvgps} \hspace{1cm} \text{and} \hspace{1cm} \sup_{0 \leq k \leq n} X_k^2 = o(\sqrt{n}) \textnormal{\cvgps}
\end{equation*}
\end{rem}
\begin{proof}
The reader may find an approach following essentially the same lines in the proof of Lemma A.2 in \cite{BercuProia11}, merely considering the stability condition $\Vert \theta \Vert_1 < 1$ in lieu of $\vert \theta \vert < 1$.
\end{proof}
\begin{lem}
\label{A1_Lem_LimSn}
Assume that the initial values $X_0, X_1, \hdots, X_{p-1}$ with $\veps_0=X_0$ are square-integrable and that $(V_{n})$ is a sequence of independent and identically distributed random variables with zero mean and variance $\sigma^2 > 0$. Then, under the stability conditions $\Vert \theta \Vert_1 < 1$ and $\vert \rho \vert < 1$, we have the almost sure convergence
\begin{equation}
\label{A12_LimSn}
\lim_{n\rightarrow \infty} \frac{S_{n}}{n} = \sigma^2 \Delta_{p} \textnormal{\cvgps}
\end{equation}
where the matrix $\Delta_{p}$ is given by \eqref{P1_Lambda}.
\end{lem}
\begin{proof}
By adopting the same approach as the one used to prove Theorem 2.2 in \cite{BercuProia11}, it follows from the fundamental autoregressive equation \eqref{A11_NewAR}, that will be detailed in the next section, that for all $0 \leq d \leq p+1$,
\begin{equation*}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} X_{k-d} V_{k} = \sigma^2 \delta_{d} \textnormal{\cvgps}
\end{equation*}
where $\delta_{d}$ stands for the Kronecker delta function equal to 1 when $d=0$, and 0 otherwise. Denote by $\ell_{d}$ the limiting value which verifies, by virtue of Lemma \ref{A1_Lem_StabX} together with Corollary 1.3.25 of \cite{Duflo97},
\begin{equation*}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} X_{k-d} X_{k} = \ell_{d} \textnormal{\cvgps}
\end{equation*}
Finally, let also $L \in \dR^{p+2}$ and, for $0 \leq d \leq p+1$, $L_{p}^{d} \in \dR^{p}$ be vectors of limiting values such that,
\begin{equation*}
L = \begin{pmatrix}
\ell_{0} & \ell_{1} & \hdots & \ell_{p+1}
\end{pmatrix}^{\prime} \hspace{0.5cm} \text{and} \hspace{0.5cm}
L_{p}^{d} = \begin{pmatrix}
\ell_{d} & \ell_{d-1} & \hdots & \ell_{d-p+1}
\end{pmatrix}^{\prime}.
\end{equation*}
From \eqref{A11_NewAR}, an immediate development leads to
\begin{equation*}
\sum_{k=1}^{n} X_{k-d} X_{k} = \beta^{\prime} \sum_{k=1}^{n} \Phi_{k-1}^p X_{k-d} - \theta_{\!p} \rho \sum_{k=1}^{n} X_{k-p-1} X_{k-d} + \sum_{k=1}^{n} X_{k-d} V_{k},
\end{equation*}
considering that $X_{-1}, X_{-2}, \hdots, X_{-p} = 0$. Consequently, we obtain a set of relations between almost sure limits, for all $0 \leq d \leq p+1$,
\begin{equation}
\label{A12_SysLinLim}
\ell_{d} = \beta^{\, \prime} L_{p}^{d-1} - \theta_{\!p} \rho \ell_{d-p-1} + \sigma^2 \delta_{d}
\end{equation}
where $\ell_{-d} = \ell_{d}$. Hereafter, if $d$ varies from 0 to $p+1$, one can build a $(p+2) \times (p+2)$ linear system of equations verifying
\begin{equation}
\label{A12_SysLinMat}
B L = \sigma^2 e
\end{equation}
where $B$ is precisely given by \eqref{P1_B}. We know from Lemma \ref{P1_Lem_InvB} that under the stability conditions, the matrix $B$ is invertible. Therefore, it follows that
\begin{equation*}
L = \sigma^2 B^{-1} e,
\end{equation*}
meaning \textit{via} \eqref{P1_VecLim} that $L = \sigma^2 \Lambda$, or else, for all $0 \leq d \leq p+1$, $\ell_{d} = \sigma^2 \lambda_{d}$, which completes the proof of Lemma \ref{A1_Lem_LimSn}.
\end{proof}
\subsection*{}
\begin{center}
{\bf B.2. Proof of Theorem \ref{P1_Thm_CvgTheta}.}
\end{center}
We easily deduce from \eqref{Int_Mod} that the process $(X_n)$ satisfies the fundamental autoregressive equation given, for all $n\geq p+1$, by
\begin{equation}
\label{A11_NewAR}
X_{n} = \beta^{\, \prime} \Phi_{n-1}^p - \theta_{\!p} \rho X_{n-p-1} + V_{n}
\end{equation}
where $\beta$ is given by \eqref{P1_Beta}. On the basis of \eqref{A11_NewAR}, consider the summation
\begin{equation}
\label{A11_InitSum}
\sum_{k=1}^{n} \Phi_{k-1}^p X_{k} = \sum_{k=1}^{n} \Phi_{k-1}^p \beta^{\, \prime} \Phi_{k-1}^p - \theta_{\!p} \rho \sum_{k=1}^{n} \Phi_{k-1}^p X_{k-p-1} + \sum_{k=1}^{n} \Phi_{k-1}^p V_{k}.
\end{equation}
\medskip
\noindent First of all, an immediate calculation leads to
\begin{equation}
\label{A11_T1}
\sum_{k=1}^{n} \Phi_{k-1}^p \beta^{\, \prime} \Phi_{k-1}^p = (S_{n-1} - S) \beta
\end{equation}
where $S_{n-1}$ and $S$ are given in \eqref{P1_Sn}. Let us focus now on the more intricate term
\begin{equation*}
\sum_{k=1}^{n} \Phi_{k-1}^p X_{k-p-1}
\end{equation*}
in which we shall expand each element of $\Phi_{k-1}^p$ according to \eqref{A11_NewAR}. A direct calculation infers the equality, for all $n \geq p+1$,
\begin{equation}
\label{A11_T2}
\sum_{k=1}^{n} \Phi_{k-1}^p X_{k-p-1} = S_{n-1}\, J_{\! p}\, \beta - \theta_{\!p} \rho \sum_{k=1}^{n} \Phi_{k-1}^p X_{k} + J_{\! p} \sum_{k=1}^{n} \Phi_{k-1}^p V_{k} + \xi_{n}
\end{equation}
where Lemma \ref{A1_Lem_StabX} ensures that the remainder term $\xi_{n}$ is made of isolated terms such that $\Vert \xi_{n} \Vert = o(n)$ a.s. Let also $M_{n}$ be the $p-$dimensional martingale
\begin{equation}
\label{A11_Mn}
M_{n} = \sum_{k=1}^{n} \Phi_{k-1}^p V_{k}.
\end{equation}
We deduce from \eqref{A11_InitSum} together with \eqref{A11_T1} and \eqref{A11_T2} that
\begin{equation*}
\sum_{k=1}^{n} \Phi_{k-1}^p X_{k} = \alpha S_{n-1} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \beta + \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} + \alpha \xi_{n}
\end{equation*}
where $\alpha$ is given by \eqref{P1_Alpha}. Thus, taking into account the expression of the estimator \eqref{P1_Est}, we get the main decomposition, for all $n \geq p+1$,
\begin{equation}
\label{A11_DecompEst}
\tn = \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \beta + \alpha (S_{n-1})^{\! -1} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} + \alpha (S_{n-1})^{\! -1} \xi_{n}.
\end{equation}
\medskip
\noindent For all $n \geq 1$, denote by $\cF_{n}$ the $\sigma-$algebra of the events occurring up to stage $n$, $\cF_{n} = \sigma(X_0, \hdots, X_{p}, V_1, \hdots, V_n)$. The random sequence $(M_n)$ given by \eqref{A11_Mn} is a locally square-integrable real vector martingale \cite{Duflo97}, \cite{HallHeyde80}, adapted to $\cF_{n}$, with predictable quadratic variation given, for all $n \geq 1$, by
\begin{eqnarray}
\langle M \rangle_n & = & \sum_{k=1}^n \dE[(\Delta M_k) (\Delta M_k)^{\prime}|\cF_{k-1}], \nonumber\\
\label{A11_Proc}
& = & \sigma^2 \sum_{k=1}^{n} \Phi^p_{k-1} \Phi^{p \,\, {\prime}}_{k-1} = \sigma^2 (S_{n-1} - S)
\end{eqnarray}
where $\Delta M_{k}$ stands for the difference $M_{k} - M_{k-1}$. We know from Lemma \ref{A1_Lem_LimSn} that
\begin{equation}
\label{A11_LimProc}
\lim_{n\rightarrow \infty} \frac{S_{n}}{n} = \sigma^2 \Delta_{p} \textnormal{\cvgps}
\end{equation}
and $\Delta_{p}$ is positive definite as a result of Lemma \ref{P1_Lem_InvL}. Then, \eqref{A11_LimProc} implies that
\begin{equation}
\label{A11_CondVP1}
\lim_{n\rightarrow \infty} \frac{\text{tr}(S_{n})}{n} = \sigma^2 p\, \lambda_0 \textnormal{\cvgps}
\end{equation}
where $\lambda_0 > 0$. Moreover, since $\Delta_{p}$ is positive definite, we also have that
\begin{equation}
\label{A11_CondVP2}
\lambda_{\text{max}}(S_{n}) = O\left(\lambda_{\text{min}}(S_{n}) \right) \textnormal{\cvgps}
\end{equation}
Consequently, we deduce from \eqref{A11_Proc}, \eqref{A11_CondVP1}, \eqref{A11_CondVP2} and the strong law of large numbers for vector martingales given \textit{e.g.} in Theorem 4.3.15 of \cite{Duflo97}, or \cite{DufloSenoussiTouati90} that,
\begin{equation}
\label{A11_LGN}
\lim_{n\rightarrow \infty} \langle M \rangle_{n}^{-1} M_{n} = 0 \textnormal{\cvgps}
\end{equation}
and obviously,
\begin{equation}
\label{A11_LGNComb}
\lim_{n\rightarrow \infty} (S_{n-1})^{\! -1} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} = 0 \textnormal{\cvgps}
\end{equation}
As mentioned above, $(V_{n})$ having a finite moment of order 2 implies, \textit{via} Lemma \ref{A1_Lem_StabX} and \eqref{A11_LimProc}, that
\begin{equation}
\label{A11_LGNReste}
\lim_{n\rightarrow \infty} (S_{n-1})^{\! -1} \xi_{n} = 0 \textnormal{\cvgps}
\end{equation}
Finally, \eqref{A11_DecompEst} together with \eqref{A11_LGNComb} and \eqref{A11_LGNReste} achieve the proof of Theorem \ref{P1_Thm_CvgTheta},
\begin{equation*}
\lim_{n\rightarrow \infty} \tn = \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \beta \textnormal{\cvgps}
\end{equation*}
\hfill
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\subsection*{}
\begin{center}
{\bf B.3. Proof of Theorem \ref{P1_Thm_TlcTheta}.}
\end{center}
The main decomposition \eqref{A11_DecompEst} enables us to write, for all $n \geq p+1$,
\begin{equation}
\label{A12_DecompDiff}
\sqrt{n} \left( \tn - \theta^{*} \right) = \alpha \sqrt{n}\, (S_{n-1})^{\! -1} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} + \alpha \sqrt{n}\, (S_{n-1})^{\! -1}\, \xi_{n}.
\end{equation}
\medskip
\noindent On the one hand, we have from Lemma \ref{A1_Lem_StabX} with $a=4$ that $\Vert \xi_{n} \Vert = o(\sqrt{n})$ a.s. assuming the existence of a finite moment of order 4 for $(V_{n})$. Hence, \textit{via} \eqref{A11_LimProc},
\begin{equation}
\label{A12_TLCReste}
\lim_{n\rightarrow \infty} \sqrt{n}\, (S_{n-1})^{\! -1}\, \xi_{n} = 0 \textnormal{\cvgps}
\end{equation}
On the other hand, we shall make use of the central limit theorem for vector martingales given \textit{e.g.} by Corollary 2.1.10 of \cite{Duflo97}, to establish the asymptotic normality of the first term in the right-hand side of \eqref{A12_DecompDiff}. Foremost, it is necessary to prove that the Lindeberg's condition is satisfied. We have to prove that, for all $\veps > 0$,
\begin{equation}
\label{A12_LindCond}
\frac{1}{n} \sum_{k=1}^{n} \dE \left[ \Vert \Delta M_{k} \Vert^2 ~ \rI_{\left\{ \Vert \Delta M_{k} \Vert\, \geq\, \veps \sqrt{n} \right\}} \vert \cF_{k-1} \right] \limp 0
\end{equation}
where $\Delta M_{k} = M_{k} - M_{k-1} = \Phi_{k-1}^{p} V_{k}$. We have from Lemma \ref{A1_Lem_StabX} with $a=4$ that
\begin{equation}
\label{A12_SommePhi4}
\sum_{k=1}^{n} \Vert \Phi_{k-1}^p \Vert^4 = O(n) \textnormal{\cvgps}
\end{equation}
Moreover, for all $\veps > 0$,
\begin{eqnarray*}
\frac{1}{n} \sum_{k=1}^{n} \dE \left[ \Vert \Delta M_{k} \Vert^2 ~ \rI_{\left\{ \Vert \Delta M_{k} \Vert\, \geq\, \veps \sqrt{n} \right\} } \vert \cF_{k-1} \right] & \leq & \frac{1}{\veps^2 n^2} \sum_{k=1}^{n} \dE \left[ \Vert \Delta M_{k} \Vert^4 \vert \cF_{k-1} \right], \\
& \leq & \frac{\tau^4}{\veps^2\, n^2} \sum_{k=1}^{n} \Vert \Phi_{k-1}^p \Vert^4
\end{eqnarray*}
where $\tau^4$ stands for the moment of order 4 associated with $(V_{n})$. Consequently, \eqref{A12_SommePhi4} ensures that
\begin{equation*}
\frac{1}{n} \sum_{k=1}^{n} \dE \left[ \Vert \Delta M_{k} \Vert^2 ~ \rI_{\left\{ \Vert \Delta M_{k} \Vert\, \geq\, \veps \sqrt{n} \right\} } \vert \cF_{k-1} \right] = O\left( n^{-1} \right) \textnormal{\cvgps}
\end{equation*}
and the Lindeberg's condition \eqref{A12_LindCond} is satisfied. We conclude from the central limit theorem for vector martingales together with Lemma \ref{P1_Lem_InvL} and Lemma \ref{A1_Lem_LimSn} that
\begin{equation}
\label{A12_TLC}
\sqrt{n}\, \langle M \rangle_{n}^{-1} M_{n} \liml \cN\left( 0,\sigma^{-4} \Delta_{p}^{-1} \right)
\end{equation}
where $\Delta_{p}$ is given by \eqref{P1_Lambda}, which leads to
\begin{equation}
\label{A12_TLCComb}
\alpha \sqrt{n}\, (S_{n-1})^{-1} \left(I_{\! p} - \theta_{\!p} \rho J_{\! p} \right) M_{n} \liml \cN\left( 0, \Sigma_{\theta} \right).
\end{equation}
Finally, \eqref{A12_DecompDiff}, \eqref{A12_TLCReste} and \eqref{A12_TLCComb} complete the proof of Theorem \ref{P1_Thm_TlcTheta}.
\hfill
$\mathbin{\vbox{\hrule\hbox{\vrule height1ex \kern.5em\vrule height1ex}\hrule}}$
\subsection*{}
\begin{center}
{\bf B.4. Proof of Theorem \ref{P1_Thm_RatTheta}.}
\end{center}
Let $(W_{n})$ be the sequence of standardization matrices defined as $W_{n} = \sqrt{n}\, I_{\! p}$. Consider the locally square-integrable real vector martingale $(M_{n})$ with predictable quadratic variation $\langle M \rangle_{n}$ given by \eqref{A11_Proc}. Via Lemma \ref{A1_Lem_LimSn}, we have the almost sure convergence
\begin{equation}
\label{A13_H1}
\lim_{n\rightarrow \infty} W_{n}^{-1}\, \langle M \rangle_{n}\, W_{n}^{-1} = \sigma^4 \Delta_{p} \textnormal{\cvgps}
\end{equation}
where $\Delta_{p}$ is given by \eqref{P1_Lambda}. For all $n \geq 0$, denote
\begin{equation}
\label{A13_Tn}
T_{n} = \sum_{k=1}^{n} X_{k}^4
\end{equation}
with $T_0=0$. From Lemma \ref{A1_Lem_StabX} with $a=4$, we have that $T_{n} = O(n)$ a.s. Thus,
\begin{eqnarray*}
\sum_{n=1}^{\infty} \frac{X_{n}^4}{n^2} & = & \sum_{n=1}^{\infty} \frac{T_{n} - T_{n-1}}{n^2} = \sum_{n=1}^{\infty} \left( \frac{2n + 1}{n^2\, (n+1)^2} \right) T_{n}, \\
& = & O\left( \sum_{n=1}^{\infty} \frac{T_{n}}{n^3} \right) = O\left( \sum_{n=1}^{\infty} \frac{1}{n^2} \right) < +\infty \textnormal{\cvgps}
\end{eqnarray*}
which immediately implies that
\begin{equation}
\label{A13_H3}
\sum_{n=1}^{\infty} \frac{\Vert \Phi_{n-1}^p \Vert^4}{n^2} < +\infty \textnormal{\cvgps}
\end{equation}
From \eqref{A13_H1} and \eqref{A13_H3}, we can deduce that $(M_{n})$ satisfies the quadratic strong law for vector martingales given \textit{e.g.} by Theorem 2.1 of \cite{ChaabaneMaaouia00},
\begin{equation}
\label{A13_LFQ}
\lim_{n\rightarrow \infty} \frac{1}{\log n^p} \sum_{k=1}^{n} \left[1 - \frac{k^p}{(k+1)^p} \right] W_{k}^{-1} M_{k} M_{k}^{\, \prime}\, W_{k}^{-1} = \sigma^4 \Delta_{p} \textnormal{\cvgps}
\end{equation}
Hereafter, it follows from \eqref{A11_DecompEst} that, for all $n \geq p+1$,
\begin{eqnarray}
\label{A13_QuadForm}
\left( \tn - \theta^{*} \right) \left( \tn - \theta^{*} \right)^{\prime} & = & \alpha^2 (S_{n-1})^{\! -1} \Big[ K_{\! p}\, M_{n} + \xi_{n} \Big] \Big[ M_{n}^{\, \prime}\, K_{\! p} + \xi_{n}^{\prime} \Big] (S_{n-1})^{\! -1}, \nonumber\\
& = & \alpha^2 (S_{n-1})^{\! -1}\, K_{\! p}\, M_{n}\, M_{n}^{\, \prime}\, K_{\! p}\, (S_{n-1})^{\! -1} + \zeta_{n}
\end{eqnarray}
where $K_{\! p} = (I_{\! p} - \theta_{\!p} \rho J_{\! p})$ and the remainder term
\begin{equation*}
\zeta_{n} = \alpha^2 (S_{n-1})^{\! -1} (\xi_{n}\, M_{n}^{\, \prime}\, K_{\! p} + K_{\! p}\, M_{n}\, \xi_{n}^{\, \prime} + \xi_{n}\, \xi_{n}^{\, \prime}) (S_{n-1})^{\! -1}.
\end{equation*}
However, we have from Lemma \ref{P1_Lem_InvL} and Lemma \ref{A1_Lem_LimSn} that
\begin{equation}
\label{A13_LimInvSn}
\lim_{n\rightarrow \infty} n(S_{n-1})^{\! -1} = \sigma^{-2} \Delta_{p}^{-1} \textnormal{\cvgps}
\end{equation}
As a result, \eqref{A13_LFQ}, \eqref{A13_LimInvSn} and a set of additional steps of calculation lead to the almost sure convergence
\begin{equation}
\label{A13_LFQComb}
\lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} (S_{k-1})^{\! -1}\, K_{\! p}\, M_{k}\, M_{k}^{\, \prime}\, K_{\! p}\, (S_{k-1})^{\! -1} = K_{\! p}\, \Delta_{p}^{-1} K_{\! p} \textnormal{\cvgps}
\end{equation}
since $K_{\! p}\, \Delta_{p}^{-1} = \Delta_{p}^{-1} K_{\! p}$ due to the bisymmetry of $\Delta_{p}^{-1}$. Assuming a finite moment of order 4 for $(V_{n})$, one can easily be convinced that $\zeta_{n}$ is going to play a negligible role compared to the first one in the right-hand side of \eqref{A13_QuadForm}. Indeed, we clearly have that $\Vert M_{n} \Vert \Vert \xi_{n} \Vert = o(n^{3/4} \sqrt{\log{n}})$ a.s. It follows that
\begin{equation}
\label{A13_LFQReste}
\sum_{k=1}^{n} \zeta_{k} = O(1) \textnormal{\cvgps}
\end{equation}
Finally, \eqref{A13_LFQComb} and \eqref{A13_LFQReste} complete the proof of the first part of Theorem \ref{P1_Thm_RatTheta},
\begin{equation*}
\lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \tk - \theta^{*} \right) \left( \tk - \theta^{*} \right)^{\prime} = \Sigma_{\theta} \textnormal{\cvgps}
\end{equation*}
since $\Sigma_{\theta} = \alpha^2 K_{\! p}\, \Delta_{p}^{-1} K_{\! p}$.
\medskip
\noindent The law of iterated logarithm \eqref{P1_LliTheta} is much more easy to handle. It is based on the law of iterated logarithm for vector martingales given \textit{e.g.} by Lemma C.2 in \cite{Bercu98}. Under the assumption \eqref{A13_H3} already verified, for any vector $v \in \dR^p$, we have
\begin{eqnarray}
\label{A13_LLI}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} (S_{n-1})^{\! -1} M_{n} & = & -\liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} (S_{n-1})^{\! -1} M_{n}, \nonumber\\
& = & \sqrt{ v^{\, \prime} \Delta_{p}^{-1} v } \textnormal{\cvgps}
\end{eqnarray}
Via \eqref{A13_LLI} and the negligibility of $\zeta_{n}$, we immediately get
\begin{eqnarray}
\label{A13_LLIComb}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} \left( \tn - \theta^{*} \right) & = & -\liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} v^{\, \prime} \left( \tn - \theta^{*} \right), \nonumber\\
& = & \alpha \sqrt{ v^{\, \prime} K_{\! p}\, \Delta_{p}^{-1} K_{\! p}\, v } \textnormal{\cvgps}
\end{eqnarray}
Since \eqref{A13_LLIComb} is true whatever the value of $v \in \dR^{p}$, we obtain a matrix formulation of the law of iterated logarithm,
\begin{equation}
\label{A13_LLIQuadForm}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \left( \tn - \theta^{*} \right) \left( \tn - \theta^{*} \right)^{\prime} = \Sigma_{\theta} \textnormal{\cvgps}
\end{equation}
Passing through the trace in \eqref{A13_LLIQuadForm}, we find that
\begin{equation}
\label{A13_LLIEncadrement}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right) \big\Vert \tn - \theta^{*} \big\Vert^2 = \text{tr}(\Sigma_{\theta}) \textnormal{\cvgps}
\end{equation}
which completes the proof of Theorem \ref{P1_Thm_RatTheta}.
\hfill
$\mathbin{\vbox{\hrule\hbox{\vrule height1ex \kern.5em\vrule height1ex}\hrule}}$
\bigskip
\section*{Appendix C}
\begin{center}
{\small PROOFS OF THE SERIAL CORRELATION PARAMETER RESULTS}
\end{center}
\renewcommand{\thesection}{\Alph{section}}
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\subsection*{}
\begin{center}
{\bf C.1. Proof of Theorem \ref{P2_Thm_CvgRho}.}
\end{center}
Let us introduce some additional notations to make this technical proof more understandable. Recall that, for all $d \in \{0, \hdots, p+1\}$, we have the almost sure convergence
\begin{equation}
\label{A21_Lim}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} X_{k-d} X_{k} = \sigma^2 \lambda_{d} \textnormal{\cvgps}
\end{equation}
Let $\Lambda_{p}^{0}$\,, $\Lambda_{p}^{1}$ and $\Lambda_{p}^{2}$ be a set of $p-$dimensional vectors of limiting values such that, for $d = \{0, 1, 2\}$,
\begin{equation}
\label{A21_L012}
\Lambda_{p}^{d} = \begin{pmatrix}
\lambda_{d} & \lambda_{d+1} & \hdots & \lambda_{d+p-1}
\end{pmatrix}^{\prime},
\end{equation}
and note that the almost sure convergence follows,
\begin{equation}
\label{A21_Lim012}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \Phi_{k-d}^p X_{k} = \sigma^2 \Lambda_{p}^{d} \textnormal{\cvgps}
\end{equation}
For all $n \geq 1$, denote by $A_{n}$ the square matrix of order $p$ defined as
\begin{equation}
\label{A21_Pn}
A_{n} = \sum_{k=1}^{n} \Phi^p_{k}\, \Phi^{p \: \prime}_{k-1}.
\end{equation}
Following a reasoning very similar to the proof of Theorem \ref{P1_Thm_CvgTheta}, it is possible to obtain the decomposition, for all $n \geq p+1$,
\begin{equation}
\label{A21_DevL0}
\sum_{k=1}^{n} \Phi_{k}^p X_{k} = A_{n}\, \theta^{*} + \alpha \sum_{k=1}^{n} \Phi_{k}^{p}\, V_{k} - \alpha\, \theta_{\!p} \rho\, J_{\! p} \sum_{k=1}^{n} \Phi_{k-2}^p V_{k} + \eta_{n}
\end{equation}
where the residual $\eta_{n}$ is made of isolated terms such that $\Vert \eta_{n} \Vert = o(n)$ a.s. As an immediate consequence, we have the relation between the limiting values
\begin{equation}
\label{A21_Rel0}
\Lambda_{p}^{0} = A_{p}\, \theta^{*} + \alpha e
\end{equation}
where the almost sure limiting matrix of $\sigma^{-2} A_{n}/n$ is given by
\begin{equation}
\label{A21_P}
A_{p} = \begin{pmatrix}
\lambda_1 & \lambda_2 & \lambda_3 & \hdots & \hdots & \lambda_{p}\\
\lambda_0 & \lambda_1 & \lambda_2 & \hdots & \hdots & \lambda_{p-1}\\
\vdots & \vdots & \vdots & & & \vdots\\
\vdots & \vdots & \vdots & & & \vdots\\
\lambda_{p-2} & \lambda_{p-3} & \lambda_{p-4} & \hdots & \hdots & \lambda_1\\
\end{pmatrix}.
\end{equation}
The reader may find more details about the way to establish these almost sure convergences \textit{e.g.} in the proof of Lemma \ref{A1_Lem_LimSn}. Likewise, one proves that
\begin{equation}
\label{A21_Rel2}
\Lambda_{p}^{2} = A_{p}^{\, \prime}\, \theta^{*} - \alpha\, \theta_{\!p} \rho J_{\! p}\, e.
\end{equation}
Finally, the very definition of the estimator $\tn$ directly implies another relation, involving the matrix $\Delta_{p}$ given by \eqref{P1_Lambda},
\begin{equation}
\label{A21_Rel1}
\Lambda_{p}^{1} = \Delta_{p}\, \theta^{*}.
\end{equation}
Relations \eqref{A21_Rel0}, \eqref{A21_Rel2} and \eqref{A21_Rel1} will be useful thereafter. Let us now consider the expression of $\rn$ given by \eqref{P2_Est}. On the one hand, in light of foregoing,
\begin{eqnarray}
\label{A21_LimNum}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \ek\, \eek & = & \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \Big( X_{k} - \tn^{~ \prime}\, \Phi_{k-1}^p \Big) \Big( X_{k-1} - \tn^{~ \prime}\, \Phi_{k-2}^p \Big), \nonumber\\
& = & \sigma^2 \left( \lambda_1 - \left( {\Lambda_{p}^{0}}^{\, \prime} + {\Lambda_{p}^{2}}^{\, \prime} \right) \theta^{*} + {\theta^{*}}^{\, \prime}\! A_{p}\, \theta^{*} \right), \nonumber \\
& = & \sigma^2 \left( \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \alpha \theta_1^{*} \right) \textnormal{\cvgps}
\end{eqnarray}
On the other hand, similarly,
\begin{eqnarray}
\label{A21_LimDen}
\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \eek^{~ 2} & = & \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \Big( X_{k-1} - \tn^{~ \prime}\, \Phi_{k-2}^p \Big)^2, \nonumber \\
& = & \sigma^2 \left( \lambda_0 - 2 {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} + {\theta^{*}}^{\, \prime}\! \Delta_{p}\, \theta^{*} \right), \nonumber\\
& = & \sigma^2 \left( \lambda_0 - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} \right) \textnormal{\cvgps}
\end{eqnarray}
\textit{Via} the set of relations \eqref{A12_SysLinLim}, we find that $\lambda_0 = \beta^{\, \prime} \Lambda_{p}^{1} - \theta_{\!p} \rho \lambda_{p+1} + 1$ for $d = 0$, and $\lambda_{p+1} = \beta^{\, \prime} J_{\! p}\, \Lambda_{p}^{1} - \theta_{\!p} \rho \lambda_{0}$ for $d = p+1$, in particular. Hence, with $\theta^{*} = \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \beta$,
\begin{eqnarray}
\label{A21_Simpl1}
\lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \alpha \theta_1^{*} & = & \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \alpha \theta_1^{*} ( \lambda_0 - \beta^{\, \prime} \Lambda_{p}^{1} + \theta_{\!p} \rho \lambda_{p+1} ), \nonumber\\
& = & \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \alpha \theta_1^{*} ( \lambda_0 - \beta^{\, \prime} \Lambda_{p}^{1} + \theta_{\!p} \rho ( \beta^{\, \prime} J_{\! p} \Lambda_{p}^{1} - \theta_{\!p} \rho \lambda_{0} ) ), \nonumber\\
& = & \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \theta_1^{*} ( \lambda_0 - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} ), \nonumber\\
& = & \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - (\theta_1 + \rho) ( \lambda_{0} - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} ) + \theta_{\!p} \rho \theta_{\!p}^{*} ( \lambda_{0} - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} )
\end{eqnarray}
since one has to note that $\theta_1^{*} = \theta_1 + \rho - \theta_{\!p} \rho \theta_{\!p}^{*}$. \textit{Via} \eqref{A21_Rel1}, $\lambda_1 = {\Lambda_{p}^{0}}^{\, \prime} \theta^{*}$. Thus,
\begin{eqnarray}
\label{A21_Simpl2}
\lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} & = & {\theta^{*}}^{\, \prime} ( \Lambda_{p}^{0} - \Lambda_{p}^{2}\, ), \nonumber\\
& = & {\theta^{*}}^{\, \prime}\! A_{p}^{\, \prime}\, \theta^{*} - {\theta^{*}}^{\, \prime}\! A_{p}\, \theta^{*} + \alpha (\theta_1 + \rho), \nonumber\\
& = & \alpha (\theta_1 + \rho) ( \lambda_0 - \beta^{\, \prime} \Lambda_{p}^{1} + \theta_{\!p} \rho \lambda_{p+1} ), \nonumber\\
& = & (\theta_1 + \rho) ( \lambda_{0} - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} ).
\end{eqnarray}
To conclude, \eqref{A21_Simpl1} together with \eqref{A21_Simpl2} lead to
\begin{equation*}
\lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} - \alpha \theta_1^{*} = \theta_{\!p} \rho \theta_{\!p}^{*} ( \lambda_{0} - {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} )
\end{equation*}
which, \textit{via} \eqref{A21_LimNum} and \eqref{A21_LimDen}, achieves the proof of Theorem \ref{P2_Thm_CvgRho},
\begin{equation*}
\lim_{n\rightarrow \infty} \rn = \theta_{\!p} \rho \theta_{\!p}^{*} \textnormal{\cvgps}
\end{equation*}
\hfill
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\subsection*{}
\begin{center}
{\bf C.2. Proof of Theorem \ref{P2_Thm_TlcRho}.}
\end{center}
First of all, we have already seen from \eqref{A11_DecompEst} that, for all $n \geq p+1$,
\begin{equation}
\label{A22_DecompTheta}
S_{n-1} \left( \tn - \theta^{*} \right) = \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} + \alpha \xi_{n}
\end{equation}
where Lemma \ref{A1_Lem_StabX} involves $\Vert \xi_{n} \Vert = o(\sqrt{n})$ a.s., assuming a finite moment of order 4 for $(V_{n})$. Our goal is to find a similar decomposition for $\rn - \rho^{*}$. For a better readability, let us introduce two specific notations $Y_{n}$ and $Z_{n}$ given by
\begin{equation*}
Y_{n} = X_{n} - \rho^{*} X_{n-1} \hspace{1cm} \text{and} \hspace{1cm} Z_{n} = X_{n-1} - \rho^{*} X_{n}.
\end{equation*}
We also note $Y_{n}^{p} = \begin{pmatrix} Y_{n} & Y_{n-1} & \hdots & Y_{n-p+1} \end{pmatrix}^{\prime}$ and $Z_{n}^{p} = \begin{pmatrix} Z_{n} & Z_{n-1} & \hdots & Z_{n-p+1} \end{pmatrix}^{\prime}$. Denote by $F_{n}$ the recurrent $p-$dimensional expression that appears repeatedly in the decomposition, given, for all $n \geq 1$, by
\begin{equation}
\label{A22_Fn}
F_{n} = \Phi_{n}^p\, {\theta^{*}}^{\, \prime} Z_{n}^{p} - \left( Z_{n-1}^p + Y_{n}^p \right) X_{n}.
\end{equation}
From the residual estimation \eqref{P2_EstRes}, the development of $\rn - \rho^{*}$ reduces to
\begin{equation}
\label{A22_DecompRho}
J_{n-1} \Big( \rn - \rho^{*} \Big) = W_{n} + \left( \tn - \theta^{*} \right)^{\prime} H_{n}
\end{equation}
where $H_{n}$ is a $p-$dimensional vector and, for all $n \geq p+1$,
\begin{eqnarray}
\label{A22_Jn}
J_{n} & = & \sum_{k=0}^{n} \ek^{~ 2},\\
\label{A22_Wn}
W_{n} & = & \sum_{k=1}^{n} Z_{k} X_{k} + {\theta^{*}}^{\, \prime} \sum_{k=1}^{n} F_{k} + \nu_{n},\\
\label{A22_Hn}
H_{n} & = & \sum_{k=1}^{n} \left( Z_{k}^{p}\, {\theta^{*}}^{\, \prime}\, \Phi_{k}^p + F_{k} \right) + \sum_{k=1}^{n} \Phi_{k}^p \left( \tn - \theta^{*} \right)^{\prime} Z_{k}^{p} + \mu_{n},
\end{eqnarray}
with $\Vert \mu_{n} \Vert = o(\sqrt{n})$ a.s. and $\nu_{n} = o(\sqrt{n})$ a.s. The reasoning develops in two stages. At first, we shall prove that $W_{n}$ reduces to a martingale, except for a residual term. Then, using Theorem \ref{P1_Thm_TlcTheta} and the central limit theorem for vector martingales, we will be in the position to prove the joint asymptotic normality of our estimates.
\medskip
\noindent Let $C$ be the square submatrix of order $p+1$ obtained by removing from $B$ given by \eqref{P1_B} its first row and first column,
\begin{equation}
\label{A22_C}
C =
\begin{pmatrix}
1-\beta_2 & -\beta_3 & \hdots & \hdots & -\beta_p & \theta_{\!p} \rho & 0\\
-\beta_1-\beta_3 & \hspace{0.15cm} 1-\beta_4 \hspace{0.15cm} & \hdots & \hdots & \theta_{\!p} \rho & 0 & 0\\
\vdots & \vdots & & & \vdots & \vdots & \vdots\\
\vdots & \vdots & & & \vdots & \vdots & \vdots\\
-\beta_{p-1}+\theta_{\!p} \rho \hspace{0.15cm} & -\beta_{p-2} & \hdots & \hdots & -\beta_1 & 1 & 0\\
-\beta_p & -\beta_{p-1} & \hdots & \hdots & -\beta_2 & -\beta_1 & \hspace{0.15cm} 1 \hspace{0.15cm}\\
\end{pmatrix}.
\end{equation}
\medskip
\noindent By Corollary \ref{P1_Cor_InvC}, we have already seen that the matrix $C$ is invertible under the stability conditions. Denote by $N_{n}$ be the $(p+1)-$dimensional martingale
\begin{equation}
\label{A22_Nn}
N_{n} = \sum_{k=1}^{n} \Phi_{k-1}^{p+1} V_{k}
\end{equation}
where $\Phi_{n}^{p+1}$ stands for the extension of $\Phi_{n}^{p}$ to the next dimension. A straightforward calculation based on \eqref{A11_NewAR} shows that the following linear system is satisfied,
\begin{equation*}
C \sum_{k=1}^{n} \Phi_{k-1}^{p+1} X_{k} = T \sum_{k=1}^{n} X_{k}^2 + N_{n}
\end{equation*}
in which $T$ is defined as
\begin{equation}
\label{A22_T}
T = \begin{pmatrix} \beta_1 & \hspace{0.1cm} & \beta_2 & \hspace{0.1cm} & \hdots & \hspace{0.1cm} & \beta_{p} & \hspace{0.1cm} & -\theta_{\!p} \rho \end{pmatrix}^{\prime}.
\end{equation}
As a result of the invertibility of $C$, we get the substantial equality, for all $n \geq p+1$,
\begin{equation}
\label{A22_SommeXn}
\sum_{k=1}^{n} \Phi_{k-1}^{p+1} X_{k} = C^{-1} T \sum_{k=1}^{n} X_{k}^2 + C^{-1} N_{n}.
\end{equation}
A large manipulation of $W_{n}$ given in \eqref{A22_Wn} still based on the fundamental autoregressive form \eqref{A11_NewAR} shows, after further calculations, that there exists an isolated term $\nu_{n}$ such that $\nu_{n} = o(\sqrt{n})$ a.s., and, for all $n \geq p+1$,
\begin{eqnarray*}
W_{n} & = & \sum_{k=1}^{n} Z_{k} X_{k} - {\theta^{*}}^{\, \prime} \sum_{k=1}^{n} Z_{k-1}^p X_{k} - \alpha\, {\theta^{*}}^{\, \prime} \sum_{k=1}^{n} \left( \Phi_{k}^p - \theta_{\!p} \rho\, J_{\! p}\, \Phi_{k-2}^p \right) V_{k}\\
& & \hspace{1cm} + \hspace{0.15cm} \alpha\, \rho^{*} {\theta^{*}}^{\, \prime} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \sum_{k=1}^{n} \Phi_{k-1}^p V_{k} + \nu_{n},
\end{eqnarray*}
leading, together with \eqref{A22_SommeXn}, to
\begin{equation}
\label{A22_SimplWn}
W_{n} = \left( G^{\, \prime} C^{-1}\, T - \rho^{*} - \alpha\, \theta_1^{*} \right) \sum_{k=1}^{n} X_{k}^2 + G^{\, \prime} C^{-1} N_{n} + L_{n} + \nu_{n}
\end{equation}
where, for all $n \geq p+1$,
\begin{equation}
\label{A22_Ln}
L_{n} = \alpha\, {\theta^{*}}^{\, \prime} \left( \rho^{*} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) M_{n} - \sum_{k=1}^{n} \left( \Phi_{k}^p - \theta_{\!p} \rho\, J_{\! p}\, \Phi_{k-2}^p \right) V_{k} \right) + \alpha\, \theta_1^{*} \sum_{k=1}^{n} X_{k} V_{k},
\end{equation}
and where the $(p+1)-$dimensional vector $G$ is given by
\begin{equation}
\label{A22_G}
G = \rho^{*} \vartheta^{*} + \alpha\, \theta_1^{*}\, T - \delta^{*}
\end{equation}
with $\vartheta^{*} = \begin{pmatrix} \theta_1^{*} & \theta_2^{*} & \hdots & \theta_{\!p}^{*} & 0 \end{pmatrix}^{\prime}$ and $\delta^{*} = \begin{pmatrix} -1 & \theta_1^{*} & \hdots & \theta_{\!p-1}^{*} & \theta_{\!p}^{*} \end{pmatrix}^{\prime}$.
In terms of almost sure limits, by using the same methodology as \textit{e.g.} in the proof of Lemma \ref{A1_Lem_LimSn}, \eqref{A22_SommeXn} directly implies
\begin{equation}
\label{A22_LimGCT}
\lambda_0\, C^{-1}\, T = \Lambda_{p+1}^{1}
\end{equation}
where $\Lambda_{p+1}^{1} = \begin{pmatrix} \lambda_1 & \lambda_2 & \hdots & \lambda_{p+1} \end{pmatrix}^{\prime}$ is the extension of $\Lambda_{p}^{1}$ in \eqref{A21_L012} to the next dimension. Hence, following the same lines as in the proof of Theorem \ref{P2_Thm_CvgRho},
\begin{eqnarray*}
\lambda_0 \left( G^{\, \prime} C^{-1}\, T - \rho^{*} - \alpha\, \theta_1^{*} \right) & = & G^{\, \prime} \Lambda_{p+1}^{1} - \lambda_0 \left( \rho^{*} + \alpha\, \theta_1^{*} \right), \\
& = & \rho^{*} ( {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} - \lambda_0 ) + \alpha\, \theta_1^{*} ( T^{\, \prime} \Lambda_{p+1}^{1} - \lambda_0) + ( \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} ), \\
& = & \theta_1^{*} ( \alpha {\Lambda_{p}^{1}}^{\, \prime} (I_{\! p} - \theta_{\!p} \rho J_{\! p} ) \beta - \alpha ( 1 - \theta_{\!p} \rho) ( 1 + \theta_{\!p} \rho) \lambda_0 ) \\
& & \hspace{1cm} + \rho^{*} ( {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} - \lambda_0 ) + ( \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*} ), \\
& = & \theta_1^{*} ( {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} - \lambda_0 ) + \rho^{*} ( {\Lambda_{p}^{1}}^{\, \prime} \theta^{*} - \lambda_0 ) + ( \lambda_1 - {\Lambda_{p}^{2}}^{\, \prime} \theta^{*}), \\
& = & -\alpha (\rho^{*} + \theta_1^{*}) + \alpha (\rho^{*} + \theta_1^{*}) = 0.
\end{eqnarray*}
One can see from Lemma \ref{P1_Lem_InvL} that $\lambda_0 > 0$. The latter development ensures that the pathological term of \eqref{A22_SimplWn} vanishes, as it should. Finally, $W_{n}$ reduces to
\begin{equation}
\label{A22_SimplWn2}
W_{n} = G^{\, \prime} C^{-1} N_{n} + L_{n} + \nu_{n},
\end{equation}
and one shall observe that $G^{\, \prime} C^{-1} N_{n} + L_{n}$ is a locally square-integrable real martingale \cite{Duflo97}, \cite{HallHeyde80}. One is now able to combine \eqref{A22_DecompTheta} and \eqref{A22_DecompRho}, \textit{via} \eqref{A22_SimplWn2}, to establish the decomposition, for all $n \geq p+1$,
\begin{equation}
\label{A22_DecompRho2}
J_{n-1} \Big( \rn - \rho^{*} \Big) = G^{\, \prime} C^{-1} N_{n} + L_{n} + \alpha M_{n}^{\prime} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) (S_{n-1})^{\! -1} H_{n} + r_{n}
\end{equation}
where the remainder term $r_{n} = \alpha\, \xi_{n}^{\prime} (S_{n-1})^{\! -1} H_{n} + \nu_{n}$ is such that $r_{n} = o(\sqrt{n})$ a.s. Taking tediously advantage of the $(p+2) \times (p+2)$ linear system of equations \eqref{A12_SysLinLim}, one shall observe that $G^{\, \prime} C^{-1} = \alpha \begin{pmatrix} U_{p}^{\, \prime} & u_{p+1} \end{pmatrix}$ with
\begin{equation*}
U_{p} =
\begin{pmatrix}
1+\beta_2 & \hspace{0.25cm} \beta_3 - \beta_1 & \hspace{0.25cm} \hdots & \hspace{0.25cm} \beta_{p} - \beta_{p-2} & \hspace{0.25cm} -\beta_{p-1} - \theta_{\!p} \rho
\end{pmatrix}^{\prime},
\end{equation*}
and $u_{p+1} = -\alpha^{-1} \theta_{\!p}^{*} - \theta_{\!p} \rho\, \theta_{1}^{*}$. The combination of \eqref{A22_Ln} and \eqref{A22_SimplWn2} results in
\begin{equation}
\label{A22_SimplWn3}
W_{n} = \alpha \left( U_{p} + (I_{\! p} - \theta_{\!p} \rho J_{\! p}) ( \rho^{*} \theta^{*} - \tau^{*} ) \right)^{\prime} M_{n} - \theta_{\!p}^{*} \sum_{k=1}^{n} X_{k-p-1} V_{k} + \nu_{n}
\end{equation}
where $\tau^{*} = \begin{pmatrix} \theta_{2}^{*} & \theta_{3}^{*} & \hdots & \theta_{\!p}^{*} & 0 \end{pmatrix}^{\prime}$. Consequently, it follows from \eqref{A22_DecompTheta} together with \eqref{A22_DecompRho2} and \eqref{A22_SimplWn3} that
\begin{equation}
\label{A22_SystMatn}
\sqrt{n}
\begin{pmatrix}
\tn - \theta^{*}\\
\rn - \rho^{*}
\end{pmatrix} =
\frac{1}{\sqrt{n}} P_{n} N_{n} + R_{n}
\end{equation}
where the square matrix $P_{n}$ of order $p+1$ is given by
\begin{equation}
\label{A22_Pn}
P_{n} = \begin{pmatrix}
P_{n}^{(1,1)} & 0\\
P_{n}^{(2,1)} & P_{n}^{(2,2)}
\end{pmatrix}
\end{equation}
with
\begin{eqnarray*}
P_{n}^{(1,1)} & = & n (S_{n-1})^{\! -1} \alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}),\\
P_{n}^{(2,1)} & = & n (J_{n-1})^{\! -1} \left( \alpha \left( U_{p} + (I_{\! p} - \theta_{\!p} \rho J_{\! p}) ( \rho^{*} \theta^{*} - \tau^{*} ) \right)^{\prime} + \alpha H_{n}^{\, \prime} (S_{n-1})^{\! -1} (I_{\! p} - \theta_{\!p} \rho J_{\! p}) \right),\\
P_{n}^{(2,2)} & = & -n (J_{n-1})^{\! -1} \theta_{\!p}^{*},
\end{eqnarray*}
and where the $(p+1)-$dimensional remainder term
\begin{equation}
\label{A22_Rn}
R_{n} = \sqrt{n} \begin{pmatrix}
\alpha (S_{n-1})^{\! -1} \xi_{n}\\
(J_{n-1})^{\! -1} r_{n}
\end{pmatrix}
\end{equation}
is such that $\Vert R_{n} \Vert = o(1)$ a.s. Via some simplifications on $H_{n}$, \eqref{A21_Rel0}, \eqref{A21_Rel2} and \eqref{A21_Rel1}, we obtain that
\begin{equation}
\label{A22_ConvHn}
\lim_{n \rightarrow \infty} \frac{H_{n}}{n} = -\alpha (I_{\! p} - \theta_{\!p} \rho J_{\! p}) e \textnormal{\cvgps}
\end{equation}
Furthermore, it is not hard to see, \textit{via} Lemma \ref{A1_Lem_LimSn}, \eqref{A21_LimDen}, \eqref{A22_ConvHn} and some simplifications on $P_{n}^{(2,1)}$, that
\begin{equation}
\label{A22_CvgPn}
\lim_{n \rightarrow \infty} P_{n} = \sigma^{-2} P \textnormal{\cvgps}
\end{equation}
where $P$ is the limiting matrix precisely given by \eqref{P2_P}. The locally square-integrable real vector martingale $(N_{n})$ introduced in \eqref{A22_Nn} and adapted to $\cF_{n}$ has a predictable quadratic variation $\langle N \rangle_{n}$ such that
\begin{equation}
\label{A22_LimProc}
\lim_{n\rightarrow \infty} \frac{\langle N \rangle_{n}}{n} = \sigma^4 \Delta_{p+1} \textnormal{\cvgps}
\end{equation}
where $\Delta_{p+1}$ is given by \eqref{P2_LambdaP}. This convergence can be achieved following \textit{e.g.} the same lines as in the proof of Lemma \ref{A1_Lem_LimSn}. On top of that, we also immediately deduce from \eqref{A12_SommePhi4} that $(N_{n})$ satisfies the Lindeberg's condition. We conclude from the central limit theorem for martingales, given \textit{e.g.} in Corollary 2.1.10 of \cite{Duflo97}, that
\begin{equation}
\label{A22_TLC}
\frac{1}{\sqrt{n}} N_{n} \liml \cN\left( 0,\sigma^{4} \Delta_{p+1} \right).
\end{equation}
Whence, from \eqref{A22_SystMatn}, \eqref{A22_Rn}, \eqref{A22_CvgPn}, \eqref{A22_TLC} and Slutsky's lemma,
\begin{equation}
\label{A22_TLCComb}
\sqrt{n}
\begin{pmatrix}
\tn - \theta^{*}\\
\rn - \rho^{*}
\end{pmatrix} \liml \cN\left( 0, P \Delta_{p+1} P^{\, \prime} \right).
\end{equation}
This concludes the proof of Theorem \ref{P2_Thm_TlcRho} where, for readability purposes, we omitted most of the calculations which the attentive reader might easily deduce.
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\subsection*{}
\begin{center}
{\bf C.3. Proof of Theorem \ref{P2_Thm_RatRho}.}
\end{center}
In the proof of Theorem \ref{P2_Thm_TlcRho}, we have established a particular relation that we shall develop from now on, to achieve the proof of Theorem \ref{P2_Thm_RatRho}. Indeed, from \eqref{A22_SystMatn}, for all $n \geq p+1$,
\begin{equation}
\label{A23_DecompRho}
\rn - \rho^{*} = n^{-1} \pi_{n}^{\prime} N_{n} + (J_{n-1})^{\! -1}\, r_{n}
\end{equation}
where $N_{n}$ and $J_{n-1}$ are given by \eqref{A22_Nn} and \eqref{A22_Jn}, respectively, where $r_{n}$ is such that $r_{n} = o(\sqrt{n})$ a.s. and where $\pi_{n}$ of order $p+1$ is given from \eqref{A22_Pn} by
\begin{equation}
\label{A23_Pin}
\pi_{n} = \begin{pmatrix} P_{n}^{(2,1)} & P_{n}^{(2,2)} \end{pmatrix}^{\prime}.
\end{equation}
Denote by $\pi$ the almost sure limit of $\pi_{n}$, accordingly given by
\begin{equation}
\label{A23_Pi}
\pi = \sigma^{-2} \begin{pmatrix} P_{L}^{\prime} & \varphi \end{pmatrix}^{\prime}
\end{equation}
where $P_{L}$ and $\varphi$ are defined in \eqref{P2_P}. Hence, \eqref{A23_DecompRho} can be rewritten as
\begin{equation}
\label{A23_DecompRho2}
\rn - \rho^{*} = n^{-1} \pi^{\prime} N_{n} + n^{-1} \left( \pi_{n} - \pi \right)^{\prime} N_{n} + (J_{n-1})^{\! -1}\, r_{n}.
\end{equation}
One can note that $(\pi^{\prime} N_{n})$ is a locally square-integrable real martingale with predictable quadratic variation given, for all $n \geq 1$, by
\begin{equation}
\label{A23_Proc}
\langle \pi^{\prime} N \rangle_{n} = \sigma^2 \pi^{\prime} (T_{n-1} - T)\, \pi
\end{equation}
where the square matrix $T_{n}$ of order $p+1$ is the extension of $S_{n}$ given by \eqref{P1_Sn} to the next dimension defined, for all $n \geq 1$, as
\begin{equation}
\label{A23_Omegan}
T_{n} = \sum_{k=1}^{n} \Phi^{p+1}_{k} {\Phi^{p+1}_{k}}^{\: \prime} + T,
\end{equation}
and $T$ is a symmetric positive definite matrix. In addition, $(\pi^{\prime} N_{n})$ satisfies a nonexplosion condition summarized by
\begin{equation*}
\lim_{n\rightarrow \infty} \frac{\pi^{\prime}\, \Phi^{p+1}_{n}\, {\Phi^{p+1}_{n}}^{\: \prime} \pi}{\pi^{\prime}\, T_{n}\, \pi} = 0 \textnormal{\cvgps}
\end{equation*}
by application of Lemma \ref{A1_Lem_StabX} with $a=4$. By virtue of the quadratic strong law for martingales given \textit{e.g.} by Theorem 3 of \cite{Bercu04} or \cite{BercuCenacFayolle09},
\begin{equation}
\label{A23_LFQ}
\lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \frac{\pi^{\prime}\, N_{k}}{\pi^{\prime}\, T_{k-1}\, \pi} \right)^2 = \frac{1}{\pi^{\prime}\, \Delta_{p+1}\, \pi} \textnormal{\cvgps}
\end{equation}
where $\Delta_{p+1}$ given by \eqref{P2_LambdaP} is the almost sure limit of $\sigma^{-2} T_{n}/n$. We refer the reader to Lemma \ref{A1_Lem_LimSn} to have more details on the latter remark. Note that $\pi^{\prime}\, \Delta_{p+1}\, \pi > 0$ since $\Delta_{p+1}$ is a positive definite matrix, as a result of Lemma \ref{P1_Lem_InvL}. The same goes for $\pi^{\prime}\, T_{n}\, \pi$, for all $n \geq 1$, assuming a suitable choice of $T$. Besides, the almost sure convergence of $\pi_{n}$ to $\pi$, the finite moment of order 4 for $(V_{n})$ together with \eqref{A23_LFQ} ensure that
\begin{eqnarray}
\label{A23_Reste}
\sum_{k=1}^{n} \left( \frac{ \left( \pi_{k} - \pi \right)^{\prime} N_{k}}{k} + \frac{r_{k}}{J_{k-1}} \right)^2 & = & O\left( \sum_{k=1}^{n} \frac{ \left( \left( \pi_{k} - \pi \right)^{\prime} N_{k} \right)^2}{k^2} + \sum_{k=1}^{n} \frac{r_{k}^{\, 2}}{J_{k-1}^2} \right), \nonumber \\
& = & O(1) + o\left( \sum_{k=1}^{n} \frac{\left( \pi^{\prime} N_{k} \right)^2}{k^2} \right), \nonumber \\
& = & o(\log n) \textnormal{\cvgps}
\end{eqnarray}
since $r_{n}$ is made of isolated terms of order 2 and $J_{n} = O(n)$ a.s. It follows that
\begin{eqnarray*}
\lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \Big( \rk - \rho^{*} \Big)^2 & = & \lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \frac{\pi^{\prime} N_{k}}{k} \right)^2, \\
& = & \lim_{n\rightarrow \infty} \frac{1}{\log n} \sum_{k=1}^{n} \left( \frac{\pi^{\prime} N_{k}}{\pi^{\prime}\, T_{k-1}\, \pi} \right)^2 \left( \frac{\pi^{\prime}\, T_{k-1}\, \pi}{k} \right)^2, \\
& = & \frac{\sigma^4 (\pi^{\prime}\, \Delta_{p+1}\, \pi)^2}{\pi^{\prime}\, \Delta_{p+1}\, \pi} = \sigma^4 \pi^{\prime}\, \Delta_{p+1}\, \pi \textnormal{\cvgps}
\end{eqnarray*}
\textit{via} \eqref{A23_LFQ} and \eqref{A23_Reste}, since the cross-term also plays a negligible role compared to the leading one. The definition of $\pi$ in \eqref{A23_Pi} combined with the one of $\Gamma$ in \eqref{P2_GamCov} achieves the proof of the first part of Theorem \ref{P2_Thm_RatRho}.
\medskip
\noindent Furthermore, it follows from the law of iterated logarithm for martingales \cite{Stout70}, \cite{Stout74}, see also Corollary 6.4.25 of \cite{Duflo97}, that
\begin{eqnarray*}
\limsup_{n \rightarrow \infty} \left( \frac{\langle \pi^{\prime} N \rangle_{n}}{2 \log \log \langle \pi^{\prime} N \rangle_{n}} \right)^{\! 1/2} \frac{\pi^{\prime} N_{n}}{\langle \pi^{\prime} N \rangle_{n}} & = & - \liminf_{n \rightarrow \infty} \left( \frac{\langle \pi^{\prime} N \rangle_{n}}{2 \log \log \langle \pi^{\prime} N \rangle_{n}} \right)^{\! 1/2} \frac{\pi^{\prime} N_{n}}{\langle \pi^{\prime} N \rangle_{n}}, \\
& = & 1 \textnormal{\cvgps}
\end{eqnarray*}
since we have \textit{via} \eqref{A13_H3} that
\begin{equation}
\label{A23_CondLIL}
\sum_{k=1}^{\infty} \frac{(\pi^{\prime} \Phi_{k-1}^{p})^4}{k^2} < +\infty \textnormal{\cvgps}
\end{equation}
Recall that we have the almost sure convergence
\begin{equation}
\label{A23_LIL1}
\lim_{n\rightarrow \infty} \frac{\langle \pi^{\prime} N \rangle_{n}}{n} = \sigma^4 \pi^{\prime} \Delta_{p+1} \pi \textnormal{\cvgps}
\end{equation}
Therefore, we immediately obtain that
\begin{eqnarray}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \frac{\pi^{\prime} N_{n}}{\langle \pi^{\prime} N \rangle_{n}} & = & - \liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \frac{\pi^{\prime} N_{n}}{\langle \pi^{\prime} N \rangle_{n}}, \nonumber \\
\label{A23_LIL2}
& = & \sigma^{-2} (\pi^{\prime} \Delta_{p+1} \pi)^{\! -1/2} \textnormal{\cvgps}
\end{eqnarray}
As in the previous proof and by virtue of the same arguments, one can easily be convinced that the remainder term in the right-hand side of \eqref{A23_DecompRho2} is negligible. It follows from \eqref{A23_DecompRho2} together with \eqref{A23_LIL1} and \eqref{A23_LIL2} that,
\begin{eqnarray*}
\limsup_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \Big( \rn - \rho^{*} \Big) & = & - \liminf_{n \rightarrow \infty} \left( \frac{n}{2 \log \log n} \right)^{\! 1/2} \Big( \rn - \rho^{*} \Big), \\
& = & \sigma^{2} \sqrt{ \pi^{\prime} \Delta_{p+1} \pi } \textnormal{\cvgps}
\end{eqnarray*}
which achieves the proof of Theorem \ref{P2_Thm_RatRho}.
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\bigskip
\section*{\textcolor{blue}{Appendix D}}
\begin{center}
{\small \textcolor{blue}{COMPARISON WITH THE H-TEST OF DURBIN}}
\end{center}
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\medskip
\textcolor{blue}{We shall now compare our statistical procedure with the well-known \textit{h-test} of Durbin \cite{Durbin70}. Let us assume that $\cH_0$ is true, that is $\rho=0$. Then, the least squares estimate of the variance of $\tn$ is given by
\begin{equation}
\label{HT_EstVar}
\wh{\dV}_{n}(\tn) = \wh{\sigma}_{n}^{\, 2}\, S_{n-1}^{-1}
\end{equation}
where $S_{n}$ is given in \eqref{P1_Sn} and $\wh{\sigma}_{n}^{\, 2}$ is the strongly consistent least squares estimate of $\sigma^2$ under $\cH_0$, defined as
\begin{equation}
\label{HT_EstRes}
\wh{\sigma}_{n}^{\, 2} = \frac{1}{n} \sum_{k=0}^{n} \ek^{~ 2}.
\end{equation}
For this proof, we use a Toeplitz version of $S_{n}$ given by
\begin{equation*}
S_{n}^{\, p} = \begin{pmatrix}
s_{n}^{\, 0} & s_{n}^{\, 1} & s_{n}^{\, 2} & \hdots & s_{n}^{\, p-1} \\
s_{n}^{\, 1} & s_{n}^{\, 0} & s_{n}^{\, 1} & \hdots & s_{n}^{\, p-2} \\
s_{n}^{\, 2} & s_{n}^{\, 1} & s_{n}^{\, 0} & \hdots & s_{n}^{\, p-3} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
s_{n}^{\, p-1} & s_{n}^{\, p-2} & s_{n}^{\, p-3} & \hdots & s_{n}^{\, 0}
\end{pmatrix}
\end{equation*}
where, for all $0 \leq h \leq p$, $s_{n}^{\, h} = \sum_{k=0}^{n} X_{k} X_{k-h}$, and we easily note that $S_{n}^{\, p} = S_{n} + o(n)$ a.s. We assume for the sake of simplicity that $S_{n}^{\, p}$ is invertible, saving us from adding a positive definite matrix $S$ without loss of generality. We also define
\begin{equation*}
\Pi_{n}^{h} = \Big( s_{n}^{\, 1} \hspace{0.3cm} s_{n}^{\, 2} \hspace{0.3cm} \hdots \hspace{0.3cm} s_{n}^{\, h} \Big)^{\prime} \hspace{0.5cm} \text{and} \hspace{0.5cm} \wh{\vartheta}_{n}^{\, p-1} = \begin{pmatrix} \wh{\vartheta}_{1,\, n} & \wh{\vartheta}_{2,\, n} & \hdots & \wh{\vartheta}_{p-1,\, n} \end{pmatrix}^{\prime}
\end{equation*}
with $\Pi_{n} = \Pi_{n}^{p}$, $\pi_{n} = \Pi_{n}^{p-1}$ and $\wh{\vartheta}_{n} = (S_{n}^{\, p})^{-1}\, \Pi_{n}$ is the Yule-Walker estimator. First, a simple calculation from \eqref{HT_EstRes} shows that
\begin{equation}
\label{HT_Sig2}
n\, \wh{\sigma}_{n}^{\, 2} = s_{n}^{\, 0} - \Pi_{n}^{\, \prime}\, \wh{\vartheta}_{n} \vspace{0.2cm}
\end{equation}
where $\wh{\sigma}_{n}^{\, 2}$ is built from $\wh{\vartheta}_{n}$. In addition, the first diagonal element of $(S_{n}^{\, p})^{-1}$ is the inverse of the Schur complement of $S_{n}^{\, p-1}$ in $S_{n}^{\, p}$, given by
\begin{equation}
\label{HT_InvSn11}
s_{n}^{\, 0} - \pi_{n}^{\, \prime}\, (\, S_{n}^{\, p-1} \,)^{-1}\, \pi_{n}. \vspace{0.2cm}
\end{equation}
The conjunction of \eqref{HT_Sig2} and \eqref{HT_InvSn11} leads to
\begin{equation}
\label{HT_Part1}
1 - n \wh{\dV}_{n}(\wh{\vartheta}_{1,\, n}) = \frac{\alpha_{n} - \beta_{n}}{\alpha_{n}}
\end{equation}
with
\begin{equation*}
\alpha_{n} = s_{n}^{\, 0} - \pi_{n}^{\, \prime}\, (\, S_{n}^{\, p-1} \,)^{-1}\, \pi_{n} \hspace{0.5cm} \text{and} \hspace{0.5cm} \beta_{n} = s_{n}^{\, 0} - \Pi_{n}^{\, \prime}\, (\, S_{n}^{\, p} \,)^{-1}\, \Pi_{n}. \vspace{0.2cm}
\end{equation*}
We also easily establish, \textit{via} some straightforward calculations, that
\begin{equation*}
\pi_{n} = k_{n} \left( I_{\! p-1} + \wh{\vartheta}_{p,\, n}\, J_{\! p-1} \right) S_{n}^{\, p-1}\, \wh{\vartheta}^{\, p-1}_{n} \hspace{0.5cm} \text{with} \hspace{0.5cm} k_{n} = \left( 1 - \wh{\vartheta}_{p,\, n}^{~ 2} \right)^{\!-1}
\end{equation*}
leading, since $S_{n}^{\, p-1}$ is bissymetric and commutes with $J_{\! p-1}$, to
\begin{equation*}
\alpha_{n} = s_{n}^{\, 0} - k_{n}\, \pi_{n}^{\, \prime}\, \wh{\vartheta}^{\, p-1}_{n} - k_{n}\, \wh{\vartheta}_{p,\, n}\, \pi_{n}^{\, \prime}\, J_{\! p-1}\, \wh{\vartheta}^{\, p-1}_{n} \hspace{0.5cm} \text{and} \hspace{0.5cm} \pi_{n}^{\, \prime}\, J_{\! p-1}\, \wh{\vartheta}^{\, p-1}_{n} = s_{n}^{\, p} - \wh{\vartheta}_{p,\, n}\, s_{n}^{\, 0}. \vspace{0.2cm}
\end{equation*}
Hence, from the previous results,
\begin{eqnarray}
\label{HT_LienAlphaBeta}
k_{n}^{-1}\, \alpha_{n} & = & k_{n}^{-1} \left( s_{n}^{\, 0} - k_{n}\, \pi_{n}^{\, \prime}\, \wh{\vartheta}^{\, p-1}_{n} - k_{n}\, \wh{\vartheta}_{p,\, n}\, s_{n}^{\, p} + k_{n}\, \wh{\vartheta}_{p,\, n}^{~ 2}\, s_{n}^{\, 0} \right),\nonumber \\
& = & s_{n}^{\, 0} - \pi_{n}^{\, \prime}\, \wh{\vartheta}^{\, p-1}_{n} - \wh{\vartheta}_{p,\, n}\, s_{n}^{\, p},\nonumber \\
& = & s_{n}^{\, 0} - \Pi_{n}^{\, \prime}\, \wh{\vartheta}_{n} = \beta_{n}. \vspace{0.2cm}
\end{eqnarray}
We now easily conclude from \eqref{HT_Part1} and \eqref{HT_LienAlphaBeta} that
\begin{equation*}
1 - n \wh{\dV}_{n}(\wh{\vartheta}_{1,\, n}) = \wh{\vartheta}_{p,\, n}^{~ 2}.
\end{equation*}
Considering now that $(\, S_{n}^{\, p} \,)^{-1} = S_{n}^{-1} + o(n^{-1})$ a.s. and making use of $\tn$ given by \eqref{P1_Est}, it is straightforward to obtain that $\tn = \wh{\vartheta}_{n} + o(1)$ a.s. and that
\begin{equation*}
1 - n \wh{\dV}_{n}(\wh{\theta}_{1,\, n}) = \wh{\theta}_{p,\, n}^{~ 2} + o(1) \cvgps
\end{equation*}
which ends the proof.} \hfill
$\mathbin{\vbox{\hrule\hbox{\vrule height1ex \kern.5em\vrule height1ex}\hrule}}$
\bigskip
\noindent{\bf Acknowledgments.} \textit{The author thanks Bernard Bercu for all his advices and suggestions during the preparation of this work. \textcolor{blue}{The author also thanks the Associate Editor and the two anonymous Reviewers for their suggestions and constructive comments which helped to improve the paper substantially.}}
\nocite{*}
\bibliographystyle{acm}
|
{
"timestamp": "2013-01-03T02:02:38",
"yymm": "1203",
"arxiv_id": "1203.1871",
"language": "en",
"url": "https://arxiv.org/abs/1203.1871"
}
|
\section{Introduction}
Most galactic nuclei are thought to host massive black holes and dense
clusters of stars whose structure and kinematics are correlated with
global galaxy properties \citep{geb96, fm00,geb00}. Such correlations
raise questions of great interest related to galaxy formation and the
growth of nuclear black holes. The nearby large spiral galaxy M31 has
an off--centered peak in a double--peaked brightness distribution
around its nuclear black hole \citep{lds74, lau93, lau98, kb99}. This
lopsided brightness distribution could arise naturally if the apoapses
of many stellar orbits, orbiting the black hole, happened to be
clustered together \citep{tre95}. Since then, kinematic and dynamical
models of such an eccentric disc have been constructed by several
authors \citep{bac01, ss01, ss02, pt03}. Of particular interest to
this work is the model of \citet{ss02}, which included a few per cent
of stars on retrograde (i.e. counter--rotating) orbits. They proposed
that these stars could have been accreted to the centre of M31 in the
form of a globular cluster that spiraled in due to dynamical
friction. This proposal was motivated by the work of \citet{tou02},
who demonstrated that a Keplerian axisymmetric disc is susceptible to
a linear lopsided instability in the $m = 1$ mode, even when a small
fraction of the disc mass is in retrograde motion.
\citet{tou02} considered the linearized secular dynamics of particles
orbiting a point mass, wherein particle orbits may be thought of as
slowly deforming elliptical rings of small eccentricities. The $m=1$
counter--rotating instability was studied analytically for a two--ring
system, and numerically for a many--ring system. The corresponding
problem for continuous discs was then studied by \citet{st10}, who
proposed a simple model with dynamics that could be studied largely
analytically in the Wentzel--Kramers--Brillouin (WKB)
approximation. Their model consisted of a two--component
\emph{softened} gravity disc, orbiting a massive central black
hole. Softened gravity was introduced by \citet{mil71} to simplify the
analysis of the dynamics of stellar systems. In this form of
interaction, the Newtonian $1/d$ gravitational potential is replaced
by $1/\sqrt{d^2 + b^2}$, where $b>0$ is called the {\it softening
length}. In the context of waves in discs, it is well known that the
softening length mimics the epicyclic radius of stars on nearly
circular orbits \citep{bt08}. Therefore, a disc composed of cold
collisionless matter, interacting via softened gravity, provides a
surrogate for a \emph{hot} collisionless disc.
\citet{st10} used a short--wavelength (WKB) approximation, derived
analytical expressions for the dispersion relation and showed that the
frequency $\omega$ is smaller than the Keplerian orbital frequency by
a factor proportional to the small quantity $\varepsilon = M_d/M\,$
(which is the ratio of the disc mass to mass of the central object);
in other words, the modes are \emph{slow}. The WKB dispersion
relation was used to argue that equal mass counter--rotating discs
with the same surface density profiles (i.e. when there is not net
rotation) could have unstable modes. They also argued that, for an
arbitrary mass ratio, the discs must be unrealistically hot to avoid
an instability. \citet{st10} then used Bohr--Sommerfeld quantization
to construct global modes, within the WKB approximation. A matter of
concern is that the wavelengths of the modes could be of order the
scale length of the discs; the modes being large--scale it is possible
that the WKB approximation itself is invalid. Another limitation is
that \citet{st10} could construct (WKB) global modes only for the case
of equal mass discs. Therefore it is necessary to address the full
eigenvalue problem to understand the systematic behaviour of
eigenvalues and eigenfunctions. To this end, we formulate the
eigenvalue problem for the linear, slow, $m=1$ modes in a
two--component, softened gravity, counter--rotating disc. Due to the
long--range nature of gravitational interactions, we have to deal with
a pair of coupled integral equations defining the eigenvalue
problem. We draw some general conclusions and then proceed to solve
the equations numerically for eigenvalues and eigenfunctions.
In \S~2 we introduce the unperturbed two--component nearly Keplerian
disc, define the apse precession rates, discuss the potential theory
for softened gravity, and derive the coupled, linear integral
equations that determine the eigenvalue problems for slow $m=1$
modes. A derivation of the relationship between the softened Laplace
coefficients (used in the potential theory of \S~2 and \S~3) and the
usual (unsoftened) Laplace coefficients is given in the Appendix. We
specialize to discs with similar surface density profiles in \S~3,
when the two coupled equations can be cast as a single integral
equation in a new mode variable; this allows us to draw some general
conclusions about the eigenvalue problem. We also discuss in detail
the numerical method to be employed. Our results are presented in
\S~4, where the properties of the stable, unstable and overstable
modes are discussed. Conclusions are offered in \S~5, where we seek to
provide a global perspective on the correlations that occur between
the pattern speeds, growth rates and eigenfunctions, as well as the
variations of these quantities on the mass fraction in retrograde
orbits.
\section{Formulation of the linear eigenvalue problem}
We consider linear non--axisymmetric perturbations in two
counter--rotating discs orbiting a central point mass $M$. The discs
are assumed to be coplanar and consist of cold collisionless particles
which attract each other through softened gravity. However, the
central mass and the disc particles interact via the usual
(unsoftened) Newtonian gravity. Softened gravity is known to mimic the
effects of velocity dispersion, so our discs are surrogates for hot
stellar discs. We assume that the total mass in the discs, $M_d$, is
small in comparison to the central mass. Since $\varepsilon \equiv
M_d/M \ll 1$, the dynamics is dominated by the Keplerian attraction of
the central mass, and the self--gravity of the discs is a small
perturbation which enables slow modes. Below we formulate the linear
eigenvalue problem of slow modes.
\subsection{Unperturbed discs}
We use polar-coordinates $\mathbf r\equiv (R\,,\phi)$ in the plane of
the discs, with the origin at the location of the central
mass. Throughout this paper the superscripts `$+$' and `$-$' refer to
the prograde and retrograde components, respectively. The unperturbed
discs are assumed to be axisymmetric with surface densities
$\sigd{\pm}(R)$. The disc particles orbit in circles with velocities,
${\mathbf v}_{d}^{\pm} = \pm\,R\Omega(R) {\mathbf e}_{\phi}$, where
$\Omega(R) > 0$ is the angular speed determined by the unperturbed
gravitational potential,
\begin{equation}
\Phi(R) \;=\; -\,\frac{GM}{R} \;+\; \Phi_{d}(R)\,.
\end{equation}
\noindent
The first term on the right side is the Keplerian potential due
to the central mass, and $\Phi_d(R)$ is the softened gravitational
potential due to the combined self-gravities of both the discs:
\begin{equation}
\Phi_{d}(\mathbf r) \;=\; -\,G\int\,\frac{\sigd{+}({\mathbf
r}') + \sigd{-}({\mathbf r}')}{\sqrt{\left|{\mathbf r}\,-\,{\mathbf r}'\right|^2+b^2}}\;{\mathrm{d}} ^2r' \,,
\label{eq:miller}
\end{equation}
\noindent
where $b$ is the Miller softening length; the potential
$\Phi_d(R)$ is $O(\epsilon)$ compared to $GM/R$. Test particles for
nearly circular prograde orbits have azimuthal and radial frequencies,
$\Omega$ and $\kappa$, given by
\begin{align}
\Omega^{2}(R) &\;=\; \frac{GM}{R^3} \;+\; \frac{1}{R}\der{\Phi_d}{R},\\[1em]
\kappa^{2}(R) &\;=\; \frac{GM}{R^3} \;+\; \frac{3}{R}\der{\Phi_d}{R} \;+\;
\dder{\Phi_d}{R}\,.
\end{align}
\noindent
The line of apsides of a nearly circular eccentric particle orbit
of angular frequency $\pm\Omega(R)$, subjected only to gravity,
precesses at a rate given by $\pm\dot{\varpi}(R)$, where
\begin{align}
\dot{\varpi}(R) &\;=\; \Omega(R) \;-\; \kappa(R) \nonumber\\[1em]
&\;=\; -\frac{1}{2\Omega(R)}\left(\frac{2}{R}\frac{\dmath}{\dmath R} + \dder{}{R}\right)\Phi_d(R)
\;+\; O(\varepsilon^2)\,.
\label{eq:pom}
\end{align}
\noindent
The cancellation of the $O(1)$ term, $(GM/R^3)$, which is common
to both $\Omega^2$ and $\kappa^2$ makes $\dot{\varpi}\sim
O(\varepsilon)$. This is the special feature of nearly Keplerian discs
which is responsible for the existence of (slow) modes whose
eigenfrequencies are $\sim O(\varepsilon)$ when compared with orbital
frequencies.
\subsection{Perturbed discs}
Let $\bfpar{v}{a}{\pm}({\mathbf r}, t) = u_{a}^{\pm}(\mathbf{r}{}{},t)
{\mathbf e_R} + v_{a}^{\pm}(\mathbf{r}{}{},t) {\mathbf e_{\phi}}$ and
$\siga{a}{\pm}({\mathbf{r}},t)$ be infinitesimal perturbations to the
velocity fields and surface densities of the $\pm$ discs,
respectively. These satisfy the following linearized Euler and
continuity equations:
\begin{align}
\DER{\bfpar{v}{a}{\pm}}{t} \;+\; & (\bfpar{v}{d}{\pm}\cdot\nabla)\bfpar{v}{a}{\pm} \;+\; (\bfpar{v}{a}{\pm}\cdot \nabla)\bfpar{v}{d}{\pm} \;=\; -\nabla\Phi_a\,,\label{eq:per1}\\[1em]
\DER{\siga{a}{\pm}}{t} \;+\; & \nabla \cdot (\sigd{\pm}\bfpar{v}{a}{\pm} \;+\; \siga{a}{\pm}\bfpar{v}{d}{\pm}) \;=\; 0,
\label{eq:per2}
\end{align}
\noindent
where $\Phi_a({\bf{r}},t)$ is the perturbing potential. Fourier
analyzing the perturbations in $t$ and $\phi$, we seek solutions of
the form, $X_{a}({\mathbf r}, t) = \sum_m X_{a}^{m}(R)\exp[{\rm
i}(m\phi - \omega t)]\,$. Then
\begin{align}
u_{a}^{m\pm} \;=\;& -\frac{\rm i}{D_{m}^{\pm}}\left[(\pm m\Omega - \omega)\frac{\dmath}{\dmath R}
\;\pm\;
\frac{2m\Omega}{R}\right]\Phi_{a}^{m}\,,\label{eq:per3}\\[1em]
v_{a}^{m\pm} \;=\;& \frac{1}{D_{m}^{\pm}}\left[\pm\frac{\kappa^2}{2\Omega}\frac{\dmath}{\dmath R} \;+\; \frac{m}{R}(\pm m\Omega - \omega)\right]\Phi_{a}^{m}\,,\label{eq:per4}
\\[1em]
{\rm i}(\pm m\Omega -& \omega)\siga{a}{m\pm} \;+\;
\frac{1}{R}\frac{\dmath}{\dmath R}(R\sigd{\pm}u_{a}^{m\pm})
\;+\; \frac{{\rm i}m}{R}\sigd{\pm}v_{a}^{m\pm} = 0\,,\label{eq:per5}
\end{align}
\noindent
where
\begin{align}
D_{m}^{\pm} \;=\;& \kappa^2 \;-\; (\pm m\Omega - \omega)^2\,.\label{eq:per6}
\end{align}
\noindent
The above equations determine $u_{a}^{m\pm}$, $v_{a}^{m\pm}$ and
$\siga{a}{m\pm}$ in terms of the perturbing potential
$\Phi_{a}^{m}\,$; this would be the solution were $\Phi_{a}^{m}$ due
to an external source.
We are interested in modes for which $\Phi_{a}^{m}$ arises from self
gravity. In this case it depends on the total perturbed surface
density, $\left[\siga{a}
{m+}(R)+\siga{a}{m-}(R)\right]\,$. Manipulating the Poisson integral
given in Eq.~\ref{eq:miller}, we obtain
\begin{align}
\Phi_{a}^{m}(R) \;=\;& \int_{0}^{\infty}R'{\mathrm{dR'}}\,P_{m}(R,R')\,[\Sigma_{a}^{m+}(R')+\Sigma_{a}^{m-}(R')]\,,
\label{eq:psn}
\end{align}
\noindent
where the kernel
\begin{align}
P_{m}(R,R') \;=\;& -\frac{\pi G}{R_{>}}B_{1}^{(m)}(\alpha,\beta) \;+\;
\frac{\pi GR}{R'^{2}}(\delta_{m,1} + \delta_{m,-1})\,.
\label{eq:pm}
\end{align}
\noindent
The second term on the right side is the indirect term arising from
the fact that our coordinate system can be a non--inertial frame,
because its origin is located on the central mass. The first term is
the direct term coming from the perturbed self--gravity. Here $R_{<} =
\min(R,R')\,$, $R_{>} = \max(R,R')\,$, $\alpha = R_{<}/R_{>}$ and
$\beta = b/R_{>}\,$. The functions,
\begin{align}
B_{s}^{(m)}(\alpha,\beta) \;=\;& \frac{2}{\pi}\int_{0}^{\pi}{\mathrm{d\theta}}\,\frac{\cos m\theta}{(1-2\alpha \cos\theta + \alpha^2 + \beta^2)^{s/2}}\,,
\label{eq:BSM}
\end{align}
\noindent
are ``softened Laplace coefficients'', introduced in
\cite{tou02}. They can be expressed in terms of the usual (unsoftened)
Laplace coefficients, as shown in Appendix~A. We note that the
unperturbed disc potential $\Phi_d$ can be obtained from the
unperturbed disc density, $\sigd{+}(R) + \sigd{-}(R)\,$, by using
Eq.~(\ref{eq:psn}) with $m=0\,$.
\subsection{Slow $m = 1$ modes}
Modes with azimuthal wavenumber $m = \pm 1$ are slow in the sense that
their eigenfrequencies, $\omega$, are smaller than the orbital
frequency, $\Omega$, by a factor $\sim O(\varepsilon)\,$. Without loss
of generality we may choose $m=1$. In the slow mode approximation
\citep{tre01}, we use the fact that $\Omega\gg\omega$ in
Eqs.~(\ref{eq:per3})---(\ref{eq:per6}), and write
\begin{align}
u_{a}^{1\pm} \;=\;& \mp\,\frac{\rm i}{D_{1}^{\pm}}\left[\Omega\frac{\dmath}{\dmath R} \;+\;
\frac{2\Omega}{R}\right]\Phi_{a}^{1}\,,\label{eq:per7}\\[1em]
v_{a}^{1\pm} \;=\;& \pm\,\frac{1}{D_{1}^{\pm}}\left[\frac{\Omega}{2}\frac{\dmath}{\dmath R}
\;+\; \frac{\Omega}{R}\right]\Phi_{a}^{1}\,,\label{eq:per8}\\[1em]
\pm\,{\rm i}\,\Omega \Sigma_{a}^{1\pm} \;+\;& \frac{1}{R}\frac{\dmath}{\dmath R}(R\Sigma_{d}^{\pm}u_{a}^{1\pm}) \;+\;\frac{\rm i}{R}\Sigma_{d}^{\pm}v_{a}^{1\pm} \;=\; 0\,,
\label{eq:per9}
\end{align}
\noindent
where
\begin{align}
D_{1}^{\pm} \;=\;& \pm 2\Omega(\omega \;\mp\; \dot{\varpi}).\label{eq:per10}
\end{align}
\noindent
Eqs.~(\ref{eq:per7}) and (\ref{eq:per8}) imply the following relations between the perturbed velocity amplitudes:
\begin{align}
u_{a}^{1\pm} \;=\;& -2 {\rm i}v_{a}^{1\pm}\,,\nonumber\\[1ex]
D_{1}^{-}u_{a}^{1-} \;=\;& - D_{1}^{+}u_{a}^{1+}\,,\nonumber\\[1ex]
D_{1}^{-}v_{a}^{1-} \;=\;& - D_{1}^{+}v_{a}^{1+}\,.
\label{eq:per11}
\end{align}
\noindent
We use Eqs.~(\ref{eq:per11}) in the continuity equation
(\ref{eq:per9}) to eliminate $u_{a}^{1\pm}$ and write,
\begin{equation}
\pm\,\Omega\Sigma_{a}^{1\pm} \;=\; \frac{2}{R^{1/2}}\frac{\dmath}{\dmath R}(R^{1/2}
\sigd{\pm}\per{v}{\pm})\,.
\label{eq:per12}
\end{equation}
\noindent
Combining Eqs.~(\ref{eq:psn}), (\ref{eq:per8}) and (\ref{eq:per10})---(\ref{eq:per12}) we obtain
\begin{align}
\left[\omega \;\mp\; \dot{\varpi}(R)\right]\per{v}{\pm}(R) \;=\;&
\int_{0}^{\infty}\frac{{\mathrm{d}R'}\,R'^{1/2}}{2R^{2}\Omega(R')}\,\left\{\DER{}{R}\left[R^2 P_{1}(R,R')\right]\right\}\times\nonumber\\[1em]
&\quad\left\{\der{}{{R'}} \left[R'^{1/2}\sigd{+}(R')\per{v}{+}(R') \;-\;
R'^{1/2}\sigd{-}(R')\per{v}{-}(R')\right] \right\}\,.
\label{eq:per13}
\end{align}
\noindent
We rewrite this by defining
\begin{equation}
z^{\pm}(R) \;=\; \left[\frac{R^2\sigd{\pm}(R)}{\Omega(R)}\right]^{1/2}\,
\per{v}{\pm}\,,
\label{zpmdef}
\end{equation}
\noindent
use the fact that $\Omega(R) \propto R^{-3/2}$ for a Keplerian flow, and integrate by parts to obtain,
\begin{align}
\left[\omega \;\mp\; \dot{\varpi}(R)\right]z^{\pm}(R) \;=\;& -\int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}\,
2\mathcal{F}(R,R')\left[\frac{\sigd{+}(R')\sigd{\pm}(R)}{\Omega(R')\Omega(R)}\right]^{1/2}z^{+}(R')\nonumber\\[1em]
& + \,\, \int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}\,2\mathcal{F}(R,R')
\left[\frac{\sigd{-}(R')\sigd{\pm}(R)}{\Omega(R')\Omega(R)}\right]^{1/2}z^{-}(R')\,,\label{eq:per15}
\end{align}
\noindent
where
\begin{equation}
\mathcal{F}(R,R') \;=\; \left(1+\frac{1}{2}\DER{}{\,\ln R'}\right)\left(1+\frac{1}{2}\DER{}{\,\ln R}\right)P_{1}(R,R')\,.
\label{eq:F}
\end{equation}
\noindent
It is convenient to write $\sigd{-}(R) = \eta(R)\sigd{}(R)$ and
$\sigd{+}(R)= (1-\eta(R))\sigd{}(R)\,$, where $\eta(R)$ is the local
mass fraction in the unperturbed counter-rotating component; by
definition, $ 0 \le \eta(R) \le 1$. Then, Eq.~$(\ref{eq:per15})$ can
be recast as
\begin{align}
\nonumber \omega z^{+}(R) \;=\; +\dot{\varpi} z^{+}(R) \;+\;& \int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}[(1-\eta(R'))(1-\eta(R))]^{1/2}\mathcal{K}(R,R')z^{+}(R')\\[1em]
\nonumber -\;& \int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}[\eta(R')(1-\eta(R))]^{1/2}\mathcal{K}(R,R')z^{-}(R')\,,\\[2em]
\nonumber \omega z^{-}(R) \;=\; -\dot{\varpi} z^{-}(R) \;+\;& \int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}[(1-\eta(R'))\eta(R)]^{1/2}
\mathcal{K}(R,R')z^{+}(R')\\[1em]
-\;& \int_{0}^{\infty}\frac{{\mathrm{d}}R'}{R'}[\eta(R')\eta(R)]^{1/2}\mathcal{K}(R,R')z^{-}(R'),
\label{eq:coupled}
\end{align}
\noindent
where the kernel
\begin{align}
\mathcal{K}(R,R') \;=\;& -\,2\left[\frac{\sigd{}(R')\sigd{}(R)}{\Omega(R')\Omega(R)}\right]^{1/2}\,\mathcal{F}(R,R')\nonumber\\[1em]
\;=\;& -\,2\left[\frac{\sigd{}(R')\sigd{}(R)}{\Omega(R')\Omega(R)}\right]^{1/2}\,\left(1+\frac{1}{2}\DER{}{\,\ln R'}\right)\left(1+\frac{1}{2}\DER{}{\,\ln R}\right)P_{1}(R,R')\nonumber\\[1em]
\;=\;& 2\pi G\left[\frac{\sigd{}(R')\sigd{}(R)}{\Omega(R')\Omega(R)}\right]^{1/2}\,\left(1+\frac{1}{2}\DER{}{\,\ln R'}\right)\left(1+\frac{1}{2}\DER{}{\,\ln R}\right)\frac{B_{1}^{(1)}(\alpha,\beta)}{R_{>}}\,.
\label{kerdef}
\end{align}
\noindent
Therefore the kernel $\mathcal{K}(R,R')$ is a real symmetric
function of $R$ and $R'$.\footnote{The contribution from the indirect
term in $P_1(R,R')$ vanishes, because $\left(2 +
\partial/\partial\,\ln R'\right)R'^{-2} = 0\,$.}
Using Eqs.~(\ref{eq:per11}) and (\ref{zpmdef}), we can relate the eigenfunctions
$z^+(R)$ and $z^-(R)$ to each other:
\begin{equation}
\sqrt{(1-\eta(R))}\left[\omega - \dot{\varpi}(R)\right]z^+(R) \;=\;
\sqrt{\eta(R)}\left[\omega + \dot{\varpi}(R)\right]z^-(R)\,.
\label{zpmrel}
\end{equation}
\noindent
This relation can, in principle, be used to eliminate one of
$z^+(R)$ or $z^-(R)$ from the coupled Eqs.~(\ref{eq:coupled}), in
which case the eigenvalue problem can be formulated in terms of a
single function (which can be either $z^+(R)$ or $z^-(R)$). However,
such a procedure results in a further complication: the eigenvalue,
$\omega$, will then occur inside the $R'$ integral in the combination,
$(\omega \pm \dot{\varpi})/(\omega \mp \dot{\varpi})$, and this makes further analysis
difficult. Eqs.~(\ref{eq:coupled}) are symmetric under the
(simultaneous) transformations, $\left\{`+'\,, \eta(R)\,,
\omega\,\right\} \;\to\; \left\{`-'\,, \left[1-\eta(R)\right]\,,
-\omega\,\right\}$, which interchange the meanings of the terms
prograde and retrograde. It seems difficult to obtain general results
when $\sigd{+}(R)$ and $\sigd{-} (R)$ have different functional
forms. Below we consider the case when the mass fraction, $\eta$, is a
constant; i.e. when both $\sigd{+}(R)$ and $\sigd{-}(R)$ have the same
radial profile.
\section{The eigenvalue problem for constant $\eta$ discs}
When the counter--rotating discs have the same unperturbed surface
density profiles, i.e. $\sigd{+}(R) \propto \sigd{-}(R)\,$, some
general results can be obtained. This case corresponds to the choice
$\eta = \mbox{constant}$, so that $\sigd{-}(R) = \eta\sigd{}(R)$ and
$\sigd{+}(R)= (1-\eta)\sigd{}(R)\,$. Then the eigenfunctions $z^+(R)$
and $z^-(R)$ are related to each other by,
\begin{equation}
\left[\omega - \dot{\varpi}(R)\right]\sqrt{\eta}\,z^+(R) \;=\;
\left[\omega + \dot{\varpi}(R)\right]\sqrt{(1-\eta)}\,z^-(R)\,.
\label{zpmnewrel}
\end{equation}
\noindent
Let us define a new function, $Z(R)$, which is a linear combination
of $z^+(R)$ and $z^-(R)\,$:
\begin{equation}
Z(R) \;=\; \sqrt{1-\eta}\,z^+(R) \;-\; \sqrt{\eta}\,z^-(R)\,.
\label{Zdef}
\end{equation}
\noindent
Then equations (\ref{eq:coupled}) can be manipulated to derive a closed
equation for $Z(R)\,$:
\begin{equation}
\left[\frac{\omega^2 - \dot{\varpi}^2}{(1-2\eta)\omega + \dot{\varpi}}\right]\,Z(R) \;=\;
\int_{0}^{\infty}\frac{\mathrm{d}R'}{R'}\,\mathcal{K}(R,R')\,Z(R')\,,
\label{Zeqn}
\end{equation}
\noindent
We note that, in this integral eigenvalue problem for the single
unknown function $Z(R)\,$, the (as yet undetermined) eigenvalue
$\omega$ occurs outside the integral. Once the problem is solved and
$Z(R)$ has been determined, we can use Eq.~(\ref{zpmnewrel}) and
(\ref{Zdef}) to recover $z^{\pm}(R)\,$:
\begin{equation}
z^+(R) \;=\; \sqrt{1-\eta}\,\frac{\omega + \dot{\varpi}(R)}{(1-2\eta)\omega + \dot{\varpi}}\,Z(R)\,,\qquad
z^-(R) \;=\; \sqrt{\eta}\,\frac{\omega - \dot{\varpi}(R)}{(1-2\eta)\omega + \dot{\varpi}}\,Z(R)\,.
\label{zpmZexp}
\end{equation}
\noindent
Some general conclusions can be drawn:
\begin{enumerate}
\item In Eq.~(\ref{Zeqn}), the kernel $\mathcal{K}(R,R')$ is real
symmetric. Therefore, the eigenvalues, $\omega$, are either real or
come in complex conjugate pairs.
\item When $\eta = 0$, the counter--rotating component is absent,
which is the case studied by \citet{tre01}; then the left side of
Eq.~(\ref{Zeqn}) becomes $\left(\omega - \dot{\varpi}\right)Z\,$. Since the
kernel $\mathcal{K}(R,R')$ is real symmetric, the eigenvalues
$\omega$ are real, so the slow modes are stable and oscillatory in
time. Then the eigenfunctions, $Z(R)$ may be taken to be
real. Therefore $z^+(R) = Z(R)$ is a real function, and $z^-(R) =
0\,$.
\footnote{When $\eta=1$, the eigenvalues, $\omega$, are again real,
with $z^-(R) = Z(R)$ a real function, and $z^+(R) = 0\,$.}
\item When $\eta = 1/2\,$, there is equal mass in the
counter--rotating component, and the surface densities of the $\pm$
discs are identical to each other. This case may also be thought of
as one in which there is no net rotation at any radius. Then
Eq.~(\ref{Zeqn}) becomes
\begin{equation}
\dot{\varpi}^{-1}(R)\left(\omega^2 - \dot{\varpi}^2(R)\right)Z(R) \;=\;
\int_{0}^{\infty}\frac{\mathrm{d}R'}{R'}\,\mathcal{K}(R,R')\,Z(R')\,,
\label{Zhalfeqn}
\end{equation}
\noindent
Since the kernel $\mathcal{K}(R,R')$ is real symmetric,
$\omega^2$ must be real. There are two cases to consider, when the
eigenvalues, $\omega$, are either real or purely imaginary.
$\bullet$ When $\omega$ is real, the slow modes are stable and
oscillatory in time. The eigenfunctions $z^{\pm}(R)$ can be taken to
be real functions.
$\bullet$ When $\omega$ is imaginary, the eigenvalues come in pairs
that are complex conjugates of each other, corresponding to
non--oscillatory growing/damped modes. Let us set $\eta = 1/2$ and
$\omega = \rm i \gamma\,$ (where $\gamma$ is real) in
Eq.~(\ref{zpmZexp}):
\begin{equation}
z^+(R) \;=\; \frac{\rm i \gamma + \dot{\varpi}(R)}{2^{1/2}\dot{\varpi}(R)}\,Z(R)\,,
\qquad\qquad
z^-(R) \;=\; \frac{\rm i \gamma - \dot{\varpi}(R)}{2^{1/2}\dot{\varpi}(R)}\,Z(R)
\end{equation}
\noindent
The function $Z(R)$, which is a solution of Eq.~(\ref{Zhalfeqn}),
can be taken to be a real function multiplied by an arbitrary complex
constant. It is useful to note two special cases: (i) when $Z(R)$ is
purely imaginary, then $z^+(R)$ and $z^-(R)$ are complex conjugates of
each other; (ii) when $Z(R)$ is real, $z^+(R)$ is equal to minus one
times the complex conjugate of $z^-(R)$.
\end{enumerate}
To make progress for other values of $\eta$, it seems necessary to
address the eigenvalue problem numerically; the rest of this paper is
devoted to this.
\subsection{Application to Kuzmin discs}
For numerical explorations of the eigenvalue problem, we consider the
case when both the unperturbed $\pm$ discs are Kuzmin discs, with
similar surface density profiles: $\sigd{-}(R) = \eta\sigd{}(R)$ and
$\sigd{+}(R) = (1-\eta) \sigd{}(R)\,$, where
\begin{equation}
\sigd{}(R) \;=\; \frac{aM_d}{2\pi(R^2 + a^2)^{3/2}}
\end{equation}
\noindent
is the total surface density, $M_d$ is the total disc mass and
$a$ is the disc scale length. Kuzmin discs, being centrally
concentrated, are reasonable candidates for unperturbed
discs. Moreover, earlier investigations of slow modes
\citep{tre01,st10} have explored modes in Kuzmin discs, so we find
this choice useful for comparisons with earlier work. The
characteristic values of orbital frequency and surface density are
given by $\Omega^{\star} = \sqrt{{GM}/{a^3}}$, and $\sigd{*} =
{M_d}/{a^2}$, respectively. The coupled Eqs. ~(\ref{eq:coupled}) can
be cast in a dimensionless form in terms of these physical scales. The
net effect is to rescale the eigenvalue $\omega$ to $\sigma$, where
\begin{equation}
\sigma = \left( \frac{\Omega^{\star} a}{G\Sigma^{\star}}\right) \omega\,.
\label{sigma}
\end{equation}
\noindent
In the following section, all quantities are to be taken as
dimensionless; however, with some abuse of notation, we shall continue
to use the same symbols for them.
\subsection{Numerical method}
Our method is broadly similar to \citet{tre01}. In order to calculate
eigenvalues and eigenfunctions numerically, we approximate the
integrals by a discrete sum using an $N$--point quadrature rule. The
presence of the term ${\mathrm{dR}}/R$ in the integrals suggests that
a natural choice of variables is $u = \log(R)$ and $v = \log(R')$,
where, as mentioned above, $R$ stands for the dimensionless length
$R/a$. The use of a logarithmic scale is numerically more efficient,
because it induces spacing in the coordinate space that increases with
the radius. This handles naturally a certain expected behaviour of the
eigenfunctions: since the surface density in a Kuzmin disc is a
rapidly decreasing function of the radial distance, we expect the
eigenfunctions to also decrease rapidly with increasing
radius. Therefore, discretization of the coupled equations
(\ref{eq:coupled}) follows the schema:
\begin{equation}
\int_{0}^{\infty} \frac{\mathrm{d}R'}{R'}\,
\mathcal{K}(R,R')\,z^{+}(R') \quad\longrightarrow\quad \sum_{i=1}^{N}w_{i}\,\mathcal{K}(e^{u},e^{v_i})\,
z^{+}(e^{v_i})\,,
\end{equation}
\noindent
where $w_i$'s are suitably chosen weights. Then the discretized
equations can be written as a matrix eigenvalue problem:
\begin{equation}
\bf{A}\,\zeta \;=\; \sigma\,\zeta\,,\label{eigeqn}
\end{equation}
\noindent
where
\begin{equation}
\bf{A} \;=\; \left[\begin{array}{cc}
(1-\eta)w_j {\mathcal K}_{ij} + \dot{\varpi}_j\delta_{ij} \quad&\quad -\sqrt{\eta(1-\eta)} w_j{\mathcal K}_{ij}\\[1em]
\sqrt{\eta(1-\eta)} w_j{\mathcal K}_{ij} \quad&\quad -\eta w_j {\mathcal K}_{ij} - \dot{\varpi}_j\delta_{ij}
\end{array}\right]\,,\qquad \mbox{and}\qquad
\bf{\zeta} \;=\; \left(\begin{array}{c}
z^{+}_{i}\\[1em] z^{-}_{i}
\end{array}\right) \,.
\end{equation}
\noindent
The $2N\,\times\,2N$ matrix $\bf{A}$ has been represented above
in a $2\,\times\,2$ block form, where each of the 4 blocks is a
$N\,\times\,N$ matrix, with row and column indices $i$ and $j$. Note
that $\delta_{ij}$ is the Kronecker delta symbol, and no summation is
implied over the repeated $j$ indices. Thus we have an eigenvalue
problem for eigenvalues $\sigma$, and eigenvectors given by the $2N$
dimensional column vector $\zeta\,$. The use of unequal weights
destroys the natural symmetry of the kernel, but this is readily
restored through a simple transformation given in \S~8.1 of
\citet{prs92}. The grid for our numerical calculations covers the
range $-7 \le \log R \le 5$, which is divided into $N = 4000$ points;
larger values of $N$ give similar results.
We note some differences with \citet{tre01} concerning details of the
numerical method and assumptions. The major difference is in the
treatment of softening: In \citet{tre01}, a dimensionless softening
parameter $\beta = b/R\,$ was introduced, and the eigenvalue problem
for slow modes was solved by holding the parameter $\beta$ constant,
This renders the physical softening length, $b$, effectively dependent
on radius, making it larger at larger radii, thereby not corresponding
to any simple force law between two disc particles located at
different radii. We have preferred to keep $b$ constant, so that the
force law between two disc particles is through the usual Miller
prescription. Other minor differences in treatment are: (i) in
\citet{tre01}, the disc interior to an inner cut-off radius was
assumed to be frozen. In contrast we use a straightforward inner
cut-off radius of $10^{-5}\,$, as mentioned above; (ii) \citet{tre01}
uses a uniform grid in $\log R$, with four--point quadrature in the
intervals between consecutive grid points; we also use a uniform grid
in $\log R$ but instead employ a single $N$--point quadrature for
integration.
Were we dealing with unsoftened gravity (i.e. the case when $b=0\,$),
the diagonal elements of the kernel would be singular. Hence, when the
softening parameter $b$ is much smaller than the grid size, accuracy
is seriously compromised by round-off errors. Typically, the usable
lower limit for $b$ is $\sim 10^{-2}$.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_1.ps}
\caption{Slow g--modes in a single, prograde ($\eta=0$) Kuzmin disc
with $\lambda = 0.1$ and $b = 10^{-2}$. The panels are labeled by
the scaled eigenvalue $\sigma$.}
\label{fig1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_2.ps}
\caption{Slow p--modes in single, prograde ($\eta=0$) Kuzmin disc
with no external source (i.e $\lambda = 1$). The panels are labeled
by the scaled eigenvalue $\sigma$, and the softening parameter, $b$
(which has been scaled with respect to $a$, the disc scale
length).}
\label{fig2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_3.ps}
\caption{Growth rate versus number of nodes for $\eta = 0.5$, for two
values of softening, $b=0.1$ and $b=0.05$.}
\label{fig3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_4.ps}
\caption{Eigenfunctions $Z(R)$ are plotted as a function of the
radial coordinate $R$, for $\eta=0.5\,$. The panels are labeled by
the values of the growth rate, $\Gamma$, and softening, $b$.}
\label{fig4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_5.ps}
\caption{Gray--scale plots of surface density perturbations,
$\siga{a}{\pm}(R, \phi, t)$ at time $t=0$ for the parameter values,
$\eta=0.5\,$ and $b=0.1\,$, and $\Gamma = 0.0230$. White/black
correspond to the maximum positive/negative values of the
perturbations.}
\label{fig5}
\end{figure}
\section{Numerical results}
We obtained the eigenvalues and eigenfunctions of equation
(\ref{eigeqn}) using the linear algebra package LAPACK
\citep{lapack}. We now present the results of our calculations for
specific values of $\eta$. As noted earlier, interchanging the meaning
of prograde and retrograde orbits leave the results invariant under
the transformation $(\eta,\omega) \to (1-\eta\, ,-\omega)$; therefore,
we present results below only for $0 \le \eta \le 1/2\,$.
\subsection{No counter-rotation: $\eta = 0$}
We are dealing with a single disc whose particles rotate in the
prograde sense. The eigenvalue problem for this case was studied
first by \citet{tre01}, who also showed that the eigenvalues are real;
in other words, the disc supports stable slow modes. We consider this
case first to benchmark our numerical method as well as assess the
differences in results that may arise due to the manner in which
softening is treated. To facilitate comparison we use the same
nomenclature as \citet{tre01}. Briefly, modes corresponding to
positive and negative eigenvalues are referred to as ``p--modes'' and
``g-modes'', respectively; we also introduce a parameter $\lambda
=(1+f)^{-1}$, where $f$ is a constant that mimics additional
precession due to an external source of the form $\dot{\varpi}_{e}(R) =
f\dot{\varpi}_{d}(R)$; we define eccentricity ${\rm e}_K$ through
\begin{equation}
{\rm e}_K\;=\; 2\,\left(\frac{GM}{R}\right)^{-1/2}\per{v}{}\,,
\label{eq:ecc}
\end{equation}
\noindent
and use the normalization,
\begin{equation}
\int \frac{\mathrm{d}\mathrm{R}}{R}\,{\rm e}_K^2(R) \;=\; 1\,.
\end{equation}
Our results corresponding to g--modes, for $\lambda = 0.1$ and $\beta
= 10^{-2}$, are presented in Fig.~\ref{fig1}, where we plot modes with
three or fewer nodes and give their eigenvalues. Results for p--modes
for $\lambda = 1$ (no external source) and various values of softening
parameter $b$, are displayed in Fig.~\ref{fig2}. These figures are to
be compared with Fig.~3 and Fig.~6 of \cite{tre01}: the eigenfunctions
are of broadly similar form, but the eigenvalues differ from those in
\cite{tre01} by upto $\sim 30\%\,$.
\subsection{Equal counter-rotation (or no net rotation): $\eta = 1/2$}
This case was studied by \citet{st10}, who derived the following local
or ``WKB'' dispersion relation:
\begin{equation}
\omega^2 = \dot{\varpi}\left(\dot{\varpi} + \frac{\pi G\sigd{}(R)}{\Omega(R)}|k|\exp(-|k|b)\right)\,.
\label{dis:rel}
\end{equation}
\noindent
From this expression they concluded: if $\dot{\varpi}$ happens to be
positive then $\omega$ is real (and the disc is stable), but $\dot{\varpi}$ is
negative for most continuous discs which implies that $\omega$ can be
either real or purely imaginary. \citet{st10} also studied global
modes using Bohr--Sommerfeld quantization, which will be discussed
later in this section.
We have proved in the last section that the eigenvalues are either
real (stable oscillatory modes), or purely imaginary (non--oscillatory
growing and damped modes). Here we focus on the growing (unstable)
modes; we define the \emph{growth rate} of perturbations as,
\begin{equation}
\Gamma = \sqrt{\frac{4}{3}} b |\sigma|\,.\label{grthrt}
\end{equation}
\noindent
in order to facilitate comparisons with \citet{st10}. In
Fig.~{\ref{fig3}}, we plot $\Gamma$ for $b$ equal to $0.1$ and $0.05$,
versus the number of nodes of $Z(R)$. \citet{st10} found two separate
branches in the spectrum, corresponding to long and short wavelengths
(for each value of $b$). Comparing our Fig.~ {\ref{fig3}} with their
Figs.~3 \& 4, we see that our results are more consistent with their
short--wavelength branch than with their long--wavelength branch.
This disagreement is probably because the long--wavelength branch
corresponds to $kR\sim 1$, where WKB approximation breaks
down. Moreover, the agreement between our results and their
short--wavelength branch holds only in a broad sense, because there
are differences in the numerical values of the eigenvalues. We trace
this difference to the fact that \citet{st10} used an analytical
result for the precession frequency corresponding to unsoftened
gravity, whereas we have consistently used softened gravity for all
gravitational interactions between disc particles. This probably also
results in another difference between our results: according to
\citet{st10}, $10^{-3} < \Gamma < 10^{-2}$; however, as can be seen
from our Fig.~{\ref{fig3}}, we obtain values of $\Gamma$ both inside
and outside this range. Changing the value of $b$ causes a horizontal
shift in the spectrum, which is consistent with their results.
In Fig~{\ref{fig4}}, we plot a few of these eigenfunctions as a
function of $R$ for both values of $b$ equal to $0.1$ and $0.05$. Note
that, from the discussion in the previous section, $Z(R)$ can always
be chosen to be a real function of $R$. The smallest number of nodes
corresponds to the largest value of the growth factor. The
eigenfunctions with the fewest nodes have significant amplitudes in a
small range of radii around $R\sim 1$, and this range increases with
the number of nodes (and correspondingly, the growth rate
decreases). Fig~{\ref{fig5}} is a gray--scale plot of the surface
density perturbations in the $\pm$ discs, $\siga{a}{\pm}(R,\phi,
t=0)$. Note the relative phase shift between the $\pm$
perturbations. For other value of $b$ shown in Fig~{\ref{fig4}} we get
similar patterns.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_6.ps}
\caption{Distribution of eigenvalues in the complex $\sigma$ plane,
for $\eta = 0.1, 0.25$ and $0.4$. Panels labeled $\mathnormal{a}$
give an overview, whereas the panels labeled $\mathnormal{b}$
provide an close--up view near the origin.}
\label{fig6}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_7.ps}
\caption{Real parts of the ``most unstable'' eigenfunctions $Z(R)$,
plotted as a function of the radial coordinate, $R$ for $b=0.1$ and
for $\eta = 0.1, 0.25$ and $0.4$. Panels are labeled by the real
and imaginary parts of the eigenvalues.}
\label{fig7}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_8.ps}
\caption{Gray--scale plots of surface density perturbations,
$\siga{a}{\pm}(R, \phi, t)$ at time $t=0$ for the parameter values,
$\eta=0.25\,$ and $b=0.1\,$, and $\sigma = 0.1408 + \rm i 0.2332$.
White/black correspond to the maximum positive/negative values of
the perturbations.}
\label{fig8}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fig_9.ps}
\caption{Real parts of two pairs of eigenfunctions $Z(R)$ (from two
arms of a branch), plotted as a function of the radial coordinate
$R$, for $b=0.1$ and $\eta = 0.25$. Panels are labeled by the real
and imaginary parts of the eigenvalues.}
\label{fig9}
\end{figure}
\subsection{Other values of $\eta$}
We present results for values of $\eta$ other than $0$ and
$1/2$. These are particularly interesting, not only because they were
not explored by \citet{st10}, but because the eigenvalues can be truly
complex, corresponding to growing and damped modes which precess with
steady pattern speeds. We write the eigenvalues as $\sigma =
\sigma_{\mathnormal{R}} + \rm i \sigma_{\mathnormal{I}}$. In
Fig~{\ref{fig6}}, we display the eigenvalues in the complex
$\sigma$--plane, for softening parameter $b=0.1$ and for $\eta$ equal
to $0.1$, $0.25$ and $0.4$. Panels on the left, labeled
$(\mathnormal{a})$, provide an overview, whereas the panels on the
right, labeled $(\mathnormal{b})$, provide a close--up view of the
distribution of eigenvalues near the origin of the complex
$\sigma$--plane; this distribution is similar to Fig.~3 of
\citet{tou02}. We are able to provide much more detail, essentially
because we are dealing with continuous discs rather than a finite
number of rings.
As $\eta$ increases from $0$, the eigenvalues go from real to complex,
a bifurcation that has been traced in \citet{tou02} to a phenomenon
identified by M.~J.~Krein due to the resonant crossing of stable
modes. The complex eigenvalues come in complex conjugate pairs, so
there are two branches to the distribution. As $\eta$ increases, the
branches progressively separate and, for $\eta = 1/2$ must lie along
the positive and negative imaginary axes. It is intriguing that each
of these two branches consists of more than one arm. In the close--up
views provided by the $(\mathnormal{b})$ panels, it appears as if each
of the branches has two arms; however, more detailed investigations
are required to determine if there are more arms. The arms of each of
the branches are most widely separated when $\eta=0.25$, which is the
value of $\eta$ exactly midway in its range $0 \leq \eta \leq
0.5\,$. The separations decrease as $\eta$ approaches either $0$ or
$1/2$; this is natural because, for $\eta=0$ both branches must lie on
the real axis and, for $\eta=1/2$ both branches must lie on the
imaginary axis.
The eigenfunctions are in general complex, and have a rich structure
as functions of their eigenvalues. Since our interest is in the
unstable modes, we now display in Fig.~{\ref{fig7}} plots of the $Z_R
= \Re[Z(R)]$ in Eq.~(\ref{Zdef}), corresponding to the ``most
unstable'' modes (for softening parameter $b=0.1$ and for $\eta$ equal
to $0.1$, $0.25$ and $0.4$). In other words, for some chosen value of
$\sigma_R$, we display the real part of the eigenfunction
corresponding to the largest value of $\sigma_{\mathnormal{I}}$. For a
fixed value of $\eta$, the number of nodes of the eigenfunctions
decreases with increasing pattern speed and growth
rate. Fig~{\ref{fig8}} is a gray--scale plot of the surface density
perturbations in the $\pm$ discs, $\siga{a}{\pm}(R,\phi, t=0)$, for
the parameter values $\eta=0.25$ and $b=0.1$, and $\sigma = 0.1408 +
\rm i 0.2332$ .
It is also of interest to ask how eigenfunctions from two different
arms of the same branch behave. To do this, we picked two
eigenfunctions with nearly the same value of $\sigma_R$, but with
values of $\sigma_I$ corresponding to two different arms of one
branch; Fig.~{\ref{fig9}} shows two such pairs of eigenfunctions for
$\eta = 0.25$. We have looked at pairs of such eigenfunctions for
other values of $\eta$, but do not display them, here we note what
seems to be a general trend: (i) the two members of a pair are more
similar to each other when the values of their $\sigma_R$ are closer
to each other; (ii) the member of a pair with the smaller value of
$\sigma_I$ is more displaced toward larger radii.
\section{Conclusions}
We study linear, \emph{slow} $m=1$ modes in softened gravity,
counter--rotating Keplerian discs. The eigenvalue problem is
formulated as a pair of coupled integral equations for the $\pm$
modes. We then specialize to the case when the two discs have similar
surface density profiles but different $\pm$ disc masses. It is of
great interest to study the properties of the modes as a function of
$\eta$, which is the fraction of the total disc mass in the retrograde
population. Recasting the coupled equations as a single equation in a
new modal variable, we are able to demonstrate some general
properties: for instance, when $\eta=1/2$, the eigenvalues must be
purely imaginary or, equivalently, the modes are purely unstable. In
other words, when the $\pm$ discs have identical surface density
profiles then there are growing $m=1$ modes with zero pattern speed, a
conclusion which is consistent with \citet{ara87, pp90, sm94, ljh97,
tou02, tre05}. To study modes for general values of $\eta$, the
eigenvalue problem needs to be solved numerically. Our method is
broadly based on \citet{tre01}, but there are some differences whose
details have been discussed in the text. The main point of departure
is in the way that softening has been treated. In \citet{tre01}, a
dimensionless softening parameter $\beta = b/R\,$ was introduced, and
the eigenvalue problem was solved by holding the parameter $\beta$
constant. This procedure renders the physical softening length, $b$,
effectively dependent on radius (making it larger at larger radii),
thereby not corresponding to any simple force law between two disc
particles located at different radii. We have preferred to keep $b$
constant, so that the force law between two disc particles is through
the usual Miller prescription.
We calculate eigenvalues and eigenfunctions numerically for discs with
surface density profiles of Kuzmin form. Kuzmin discs, being centrally
concentrated, are reasonable candidates for unperturbed
discs. Moreover, earlier investigations of slow modes
\citep{tre01,st10} have explored modes in Kuzmin discs, so this choice
is particularly useful for comparisons with earlier work. Comparing
our results with those of \citet{tre01} for $\eta=0$ (when the slow
modes are stable), we find that the eigenfunctions are of broadly
similar form, but the eigenvalues differ by up to $\sim 30\%\,$; this
is a result of the different ways in which we have treated
softening. For the case of no net rotation ($\eta=1/2$), we find that
the growth rates (of the unstable modes) we calculate are broadly
consistent with the short--wavelength branch of the global WKB modes
determined earlier by \citet{st10}, but not their long--wavelength
branch. This disagreement probably arises because the long--wavelength
branch corresponds to wavelengths of order the disc scale length where
WKB approximation breaks down. Moreover, the agreement between our
results and their short--wavelength branch holds only in a broad
sense, because there are differences in the numerical values of the
eigenvalues. We trace this difference to the fact that \citet{st10}
used an analytical result for the precession frequency corresponding
to unsoftened gravity, whereas we have consistently used softened
gravity for all gravitational interactions between disc particles.
We have also investigate eigenmodes for values of $\eta$ other than
$0$ and $1/2$. These cases are particularly interesting, not only
because they were not explored by \citet{st10}, but because the
eigenvalues can be truly complex, corresponding to growing (and
damped) modes with non zero pattern speeds. We have presented results
for $\eta = 0.1, 0.25$ and $0.4$ in the previous sections. Based on
these, we interpolate and offer the following conclusions about the
properties of the eigenmodes and their physical implications, for all
values of $\eta$ (which is the mass fraction in the retrograde
population):
\begin{enumerate}
\item For a general value of $\eta$ (between $0$ and $1/2$), the
distribution of eigenvalues in the complex plane has two
branches. These branches are symmetrically placed about the real
axis, because the eigenvalues come in complex conjugate pairs.
\item The pattern speed appears to be non negative for all values of
$\eta$, with the growth (or damping) rate being larger for larger
values of the pattern speed.
\item For a fixed value of $\eta$, the number of nodes of the
eigenfunctions decreases with increasing pattern speed and growth
(or damping) rate.
\item For a value of pattern speed in a chosen narrow interval, the
growth (or damping) rate increases as $\eta$ increases from $0$ to
$1/2$.
\item Each of the two branches in the complex (eigenvalue) plane has
at least two arms. When $\eta=0$, the eigenvalues are all real, so
both branches lie on the real axis, with zero spacing between the
arms. As $\eta$ increases, the branches lift out of the real axis,
and the arms separate. It appears as if the maximum separation
between the arms happens when $\eta=1/4$. As $\eta$ increases
further, the branches continue to rise with greater slope, while the
arm separation begins decreasing. Finally, when $\eta=1/2$, the arm
separation decreases to zero as the branches lie on the imaginary
axis.
\end{enumerate}
Observations of lopsided brightness distributions around massive black
holes are somewhat more likely to favour the detection of modes with
fewer nodes than modes with a large number of nodes, because the
former suffer less cancellation due to finite angular resolution. From
items (ii) and (iii) above, we note that the modes with a small number
of nodes also happen to be those with larger values of the pattern
speed and growth rate, both qualities that enable detection to a
greater degree. Having said this, it would be appropriate to note some
limitations of our work. Softened gravity discs are, after all,
surrogates for discs composed of collisionless particles (such as
stars) with non zero thickness and velocity dispersions. It is
necessary to formulate the eigenvalue problem for truly collisionless
discs, in order to really deal with stellar discs around massive black
holes. Meanwhile, our results will serve as a benchmark for future
investigations of modes in these more realistic models.
\def{\it et~al.}{{\it et~al.}}
\def{Astroph.\@ J. }{{Astroph.\@ J. }}
\def{Mon.\@ Not.\@ Roy.\@ Ast.\@ Soc.}{{Mon.\@ Not.\@ Roy.\@ Ast.\@ Soc.}}
\def{Astron.\@ Astrophys.}{{Astron.\@ Astrophys.}}
\def{Astron.\@ J.}{{Astron.\@ J.}}
\def{Astroph.\@ Space \@ Science}{{Astroph.\@ Space \@ Science}}
\def{Publ.\@ Astron.\@ Soc.\@ Japan}{{Publ.\@ Astron.\@ Soc.\@ Japan}}
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|
{
"timestamp": "2012-03-13T01:00:58",
"yymm": "1203",
"arxiv_id": "1203.2239",
"language": "en",
"url": "https://arxiv.org/abs/1203.2239"
}
|
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
\thispagestyle{plain}
\indent Paley graphs are named after Raymond Paley (7 January 1907 -- 7 April 1933). He was born in Bournemouth, England. He won a Smith's Prize in 1930 and was elected a fellow of Trinity College, Cambridge, where he showed himself as one of the most brilliant students among a remarkable collection of fellow undergraduates.
\\[10pt]
\indent In Paley graphs, finite fields form their sets of vertices. So to understand Paley graphs we will start our work with classification of finite fields and study their properties. We will show that any finite field $\F$ has $p^n$ elements, where $p$ is prime and $n \in \N$. Moreover, for every prime power $p^n$ there exists a field with $p^n$ elements, and this field is unique up to isomorphism.
In the last section of Chapter \ref{Chap1} we will see how one can construct the finite field for any prime power $p^n$, and we will give the explicit construction of the fields of 9, 16, and 25 elements.
\\[10pt]
\indent To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. In the first section of Chapter \ref{Chap2}
we will give some basic definitions and properties from graph theory which we will use in te study of the Paley graphs.
In the second section we will give the definition of the Paley graph and we will give examples of the Paley graphs of order 5, 9, and 13. Finally, we will study some important properties of the Paley graphs.
In particular, we will show that the Paley graphs are connected, symmetric, and self-complementary.In \cite{W.P1}, Peisert proved that the Paley graphs of prime order are the only self-complementary symmetric graphs of prime order; furthermore, in \cite{W.P2}, he proved that any self-complementary and symmetric graph is isomorphic to a Paley graph, a $\mathcal{P}^*$-graph, or the exceptional graph $G(23^2)$ with $23^2$ vertices.
Also we will show that the Paley graph of order $q$ is $\frac{q-1}{2}$-regular, and every two adjacent vertices have $\frac{q-5}{4}$ common neighbors, and every two non-adjacent vertices have $\frac{q-1}{4}$ common neighbors, which means that the Paley graphs are strongly regular with parameters ($q,\ \frac{q-1}{2},\ \frac{q-5}{4},\ \frac{q-1}{4}$), see also \cite{Bollobas}.
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\begin{center}{\bf Fig. 0.1. }The Paley graph of order $17$
\end{center}
In \cite{ontheAdjpaley}, Ananchuen and Caccetta proved that for every 3-element subset $S$ of the vertices of the Paley graph with at least 29 vertices, and for every subset $T$ of $S$, there is a vertex $x \notin S$ which is joined to every vertex in $T$ and to no vertex in $S \setminus T$; that is, the Paley graphs are 3-existentially closed.
\\[10pt]
\indent Paley graphs are generalized by many mathematicians. In the first section of Chapter \ref{Chap3} we will see three examples of these generalizations and some of their basic properties.
In \cite{ontheAdjGpaley}, Ananchuen introduced two of these generalizations. The cubic Paley graphs, in which pairs of elements of a finite field are connected by an edge if and only if they differ in a cubic residue, and the quadruple Paley graphs, in which pairs of elements of a finite field are connected by an edge if and only if they differ in a quadruple residue. The third generalization is called the generalized Paley graphs, in this family of graphs, pairs of elements of a finite field are connected by an edge if and only if their difference belongs to a subgroup $S$ of the multiplicative group of the field. This generalization is given by Lim and Praeger in \cite{onGenPal}.
In the second section of Chapter \ref{Chap3} we will define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge if and only if there difference belongs to the $m$-th power of the multiplicative group of the field, for any odd integer $m > 1$, and we call them the $m$-Paley graphs.
Since the cubic Paley graphs are 3-Paley graphs, we can say that the cubic Paley graphs are a special case of the family of $m$-Paley graphs. Also, we will give some examples of this family.
In the third section we will show that the $m$-Paley graph of order $q$ is complete if and only if $\gcd (m, q-1)= 1$ and when $d= \gcd (m, q-1) > 1$, the $m$-Paley graph is $\frac{q-1}{d}$-regular.
Also we will prove that the $m$-Paley graphs are symmetric but not self-complementary. In particular, $m$-Paley graphs are not in the Peisert's list. Since strongly regular graphs must be self-complementary, we see that the $m$-Paley graphs are not strongly regular.
We will show also that the $m$-Paley graphs of prime order are connected but the $m$-Paley graphs of order $p^n, \ n > 1 $ are not necessary connected, for example they are disconnected if $\gcd (m,p^n-1)= \frac{p^n-1}{2}$.
\chapter{Finite Fields}
\label{Chap1}
This chapter provides an introduction to some basic properties of finite fields and their structure. This introduction will be useful for understanding the properties of Paley graphs in which the elements of a finite field represent the set of vertices.
\begin{section}{Basic definitions and properties}
\label{Sec1}
\noindent {\bf Definition:}
A {\it field} \ $\F$ is a set of at least two elements, with two operations $\oplus$ and $\ast$, for which the following axioms are satisfied:
\begin{enumerate}
\item The set \ $\F$ under the operation $\oplus$ forms an abelian group (whose identity is denoted by $0$).
\item The set \ $\F^{*} = \F \setminus \lbrace0\rbrace$ under the operation $\ast$ forms an abelian group (whose identity is denoted by 1).
\item Distributive law: For all $a$, $b$, $c \in \F$, we have $(a \oplus b)\ast c = (a \ast c) \oplus (b \ast c)$.
\end{enumerate}
\noindent Note that it is important to allow $0$ to be an exceptional element with no inverse, because if $0$ had an inverse then $1 = 0^{-1} \ast 0 = 0$, it would follow that $x = x\ast 1 = 0 $ for all $ x \in \F $; therefore, $\F$ would consist of only one element 0.
\\[10pt]
{\bf\large{Examples}}
\begin{enumerate}
\item $\Q$, $\R$, and $\C$ are fields with respect to the usual addition and multiplication.
\item The subring $\Q[i] := \lbrace a + bi \in \C$ : $a$, $b \in \Q \rbrace$ of $\C$ is a field, called the field of Gaussian rationals.
\item The ring of integers modulo $p$, $\Z_{p}$, is a field if $p$ is a prime number.
\item A commutative division ring (a ring in which every nonzero element has a multiplicative inverse) is a field.
\item A finite integral domain (a commutative ring with no zero divisors) is a field.
Indeed, let $R$ be an integral domain and $0 \neq a \in R$. The map $x \rightarrow ax$, $x \in R$, is injective because $R$ is an integral domain $(a(x_{1} - x_{2}) = 0 \Leftrightarrow (x_{1} - x_{2}) = 0)$. If $R$ is finite, the map is surjective as well, so that $ax = 1$ for some $x$, i.e.,
every nonzero element $a$ has an inverse.
\item The quotient ring $R/M$ such that $R$ is a commutative ring and $M$ is a maximal ideal, is a field.
Indeed, since $R$ is commutative, the ring $R/M$ is commutative, where $(x +M)(y +M) = xy +M = yx +M = (y+M)(x +M)$. It also has an identity $1_{R/M} = 1_{R} + M$. Moreover, $1_{R} \notin M$ because if $1_{R} \in M$, then $M = R$, which contradicts the definition of maximal ideal. In order to show that $R/M$ is a field, it remains to prove that every nonzero element $x+M \in R/M$ has an inverse. So we fix $x \notin M$ and consider the set
$I = M +xR =\{ m+xr : m \in M, \ r \in R \} $. First we need to check that $I$ is an ideal. Let $m_{1},m_{2} \in M$ and $r_{1}, r_{2}, s \in R$,
then $0 = 0 + x0 \in I$, $s((m_1 + xr_1) - (m_2 + xr_2)) = s(m_1-m_2) + s(x(r_1-r_2)) = s(m_1-m_2) + xs(r_1-r_2) \in I$.
Hence $I$ is an ideal. Now $M \subsetneq I$ because $x \in I$ but $x \notin M$. Since $M$ is maximal, it follows that $I = R$,
and in particular $1_R \in I$. So there exist $m \in M$ and $y \in R$ such that $1_R = m+xy$.
Then $(x+M)(y+M) = xy+M =(1_R - m) +M = 1_R +M$, so $x+M$ has an inverse in $R/M$.
\end{enumerate}
\noindent {\bf Definition:} The {\it characteristic} of a field $\F$ (denoted by Char $\F$) is the smallest positive integer $n$ such that $n1=0$, where $n1$ is an abbreviation for $1 + 1 + \cdots +1 (n \textrm{ ones})$. If $n1$ is never $0$, we say that \ $\F$ has characteristic $0$.
\\[10pt]
\noindent Note that the characteristic of a field can never be equal to $1$, since $1= 1 \ast 1 \neq 0$. If Char $\F \neq 0$, then Char $\F$ must be a prime number. For if Char $\F = n = rs$ where $r$ and $s$ are positive integers greater than $1$, then $(r1)(s1) = n1 = 0$, so either $r1$ or $s1$ is $0$, which contradicts the minimality of $n$.
\\[10pt]
\noindent {\bf Definition:}
If \ $\F$ and $\E$ are fields and \ $\F \subseteq \E$, we say that $\E$ is an {\it extension} of \ $\F$ and \ $\F$ is a {\it subfield} of \ $\E$, and we write \ $\F \leq \E$.
\\[10pt]
\noindent Note that if \ $\F \leq \E$ then we can consider $\E$ as a vector space over $\F$, because if we consider the elements of $\E$ as vectors and the elements of $\F$ as scalars, then the axioms of a vector space are satisfied. The dimension of this vector space is called the degree of the extension and is denoted by $[\E:\F]$. If $[\E:\F]= n$ with $n < \infty$, we say that $\E$ is a {\it finite extension} of $\F$.
\\[10pt]
\noindent {\bf Definition:}
A minimal subfield $\F_{p}$ of a field $\F$ with Char $\F = p$ is called a {\it prime field}.
\\[10pt]
\noindent Note that the only subfield of the prime field $\F_{p}$ is $\F_{p}$ itself. Let Char $\F= p$ then $\lbrace1,1+1,1+1+1,\dots,0=1+1+\cdots+1(p \textrm{ ones} )\rbrace$ form a subfield of $\F$ which is a prime field isomorphic to $\Z_{p}$, i.e., every finite field with characteristic $p$ has a prime subfield which is isomorphic to $\Z_{p}$.
\\[10pt]
\noindent {\bf Definition:}
A {\it nonzero polynomial }$f(x)$ of degree $m$ over a field $\F$ is an expression of the form
\begin{center}
$f(x) = f_{0} + f_{1} x + f_{2} x^{2} + \cdots + f_{m} x^{m}$
\end{center}
where $f_{i} \in \F \textrm{, }0 \leq i \leq m \textrm{, and }f_{m} \neq 0$. The degree of $f(x)$ is denoted by $\textrm{deg}{\hskip 0.7mm} f(x)$. The polynomial $f(x) = 0 $ is called the {\it zero polynomial} and the set of all polynomials over $\F$ is denoted by $\F[x]$.
\\[10pt]
\noindent {\bf Definition:}
A nonzero polynomial \ $f(x)= f_{0} + f_{1} x + f_{2} x^{2} + \cdots + f_{m} x^{m}$ over \ $\F$ with $f_{m}=1$ is called {\it monic}.
\\[10pt]
\noindent Note that the set $\F[x]$ is a ring, its additive identity is the zero polynomial $f(x) = 0$ and its multiplicative identity is $f(x) = 1$. However $\F[x]$ is not a field, because the polynomials of degree greater than 0 have no inverse.
\\[10pt]
\noindent {\bf Definition:}
A nonzero polynomial $f(x) \in \F[x]$ is called {\it irreducible} over $\F$ if $\textrm{deg}{\hskip 0.7mm}f(x) \geq 1$ and $f(x) = g(x)h(x)$ with $g(x)$, $h(x) \in \F[x]$ gives either $\textrm{deg}{\hskip 0.7mm} g(x) = 0$ or $\textrm{deg}{\hskip 0.7mm}h(x) = 0$.
\\[10pt]
\noindent In other words, up to a nonzero constant factor the only divisors of $f(x)$ are $f(x)$ itself and $1$.
\\[10pt]
\noindent {\bf Definition:}
Let $f(x)$ be a nonzero polynomial of $\F[x]$. A finite extension $\E$ of $\F$ is called the {\it splitting field} of $f(x)$ if $\E$ is the smallest extension field of $\F$ in which $f(x)$ can be written as
$$\lambda (x - \alpha_{1}) \cdots (x - \alpha_{k})$$
for some $\alpha_{1} , \dots , \alpha_{k} \in \E$ and $\lambda$ in $\F$.
\\[10pt]
Note that the splitting field $\E$ can be written as $\E = \F[\alpha_{1} , \dots , \alpha_{k}]$ which denotes the field generated by $\alpha_{1} , \dots , \alpha_{k}$ over $\F$, because $\F[\alpha_{1} , \dots , \alpha_{k}]$ is the smallest field containing $\F$ and $(\alpha_{1} , \dots , \alpha_{k})$.
\\[10pt]
\noindent {\it Example: }The field of complex numbers is the splitting field of $x^{2} + 1$ over the field of real numbers.
\end{section}
\begin{section}{Classification of the finite fields}
\label{sec2}
\begin{theo}
The number of elements of a finite field $\F$ is equal to $p^{n}$, where $p$ is prime and $n \in \N$.
\end{theo}
\proof
Let $\textrm{Char}{\hskip 0.5mm}\F = p$ where $p$ is a prime number, then $\F_{p}$ is a subfield of $\F$, so we can consider $\F$ as a vector space over $\F_{p}$. Since $\F$ is finite, the dimension of $\F$ over $\F_{p}$ is finite.
\\[10pt]
\indent Let $[\F:\F_{p}] =n$ then there exists a basis {$v_{1}, v_{2}, \dots , v_{n}$} in $\F$. With respect to this basis every element $x \in \F$ can be written uniquely as
$$x = a_{1} v_{1} + a_{2} v_{2} + \cdots + a_{n} v_{n}, \textrm{ where } a_{1}, a_{2},\dots ,a_{n} \in \F_{p}.$$
\indent Since $\vert \F_{p} \vert= p$, every $a_{i}$, $i = 1,2, \dots ,n$, can be equal to one of $p$ values. Therefore, $\F$ has $p^{n}$ elements. \hspace{\stretch{1}} $\Box$
\\[10pt]
\noindent Now we will consider the following, more difficult, question: Does a field with $p^{n}$ elements exist, for each prime $p$ and each $n \in \N$? This question will be answered affirmatively in Theorem \ref{p^n}. In order to prove this theorem, we need Theorems \ref{theo EX&UN} and \ref{theo f&f'}.
\begin{theo}
\label{theo EX&UN}
Let $f(x)$ be an irreducible polynomial with degree $\geqslant 1$ in $\F[x]$. Then a splitting field for $f(x)$ over $\F$ exists and any two such splitting fields are isomorphic.
\end{theo}
\proof
First we will prove that there exists an extension field of $\F$ in which $f(x)$ has a root.
\\[10pt]
\indent Since $f(x)$ is irreducible, the principal ideal $(f(x))$ is a maximal ideal in the ring $\F[x]$.
To prove this, let $I$ be an ideal in $\F[x]$ with $(f(x)) \subsetneq I \subseteq \F[x]$ and let $g(x) \in I \setminus (f(x))$. Since $f(x)$ is irreducible and $f(x) \nmid g(x)$, we have $(f(x),g(x))= 1$. Therefore, there exist two polynomials $h(x)$, $k(x) \in \F[x]$ with $f(x)h(x)+g(x)k(x) = 1$. We see that $ 1 \in I$ and thus $I = \F[x]$, so $(f(x))$ is a maximal ideal in the commutative ring $\F[x]$. We conclude that $\F[x]/(f(x))$ is a field.
\\[10pt]
\indent Consider the homomorphism
$$\sigma: \F \rightarrow \F[x]/(f(x)) \textrm{ defined by } \sigma(a) = \overline{a} = a + (f(x)),\textrm{ }a\in \F,$$
\noindent which is injective. Indeed, $\F$ is a field, which means that $\textrm{Ker} \sigma$ equals $\F$ or $(0)$ but $\textrm{Ker} \sigma \neq \F$ because $\overline{1} \neq \overline{0}$. So $\textrm{Ker} \sigma = (0) $ and $\F$ is isomorphic to $\sigma (\F)$.
\\[10pt]
\indent Now we can consider $\F[x]/(f(x))$ as a finite extension of $\F$, which has a root $\alpha = x+(f(x))$ of $f(x)$. Thus for a field $\F$ and an irreducible polynomial $f(x)$ over $\F$ there exists an extension in which $f(x)$ has a root.
\\[10pt]
\indent Now we will prove by induction on $\textrm{deg} {\hskip 0.7mm} f(x)$ the existence of the splitting field of $f(x)$.
If $\textrm{deg} {\hskip 0.7mm} f(x) = 1$ then $f(x)$ has one root. Thus, as we have shown, $\F$ has an extension field $\E$ which contains a root $\alpha$ of $f(x)$, so the field $\F[\alpha]$ is the splitting field for $f(x)$.
Now assume that for each irreducible polynomial $g(x)$ of degree $m < n$ there exists a splitting field of $g(x)$.
Let $\textrm{deg} {\hskip 0.7mm} f(x) = n$, we have shown that there exists an extension field $\E_{1}$ of $\F$ with root $\alpha_{1}$ of $f(x)$. So in $\E_{1}$, $f(x)$ can be written as
\begin{center}
$f(x)=(x-\alpha_{1})g(x)$, with $\textrm{deg} {\hskip 0.7mm} g(x) = n-1$.
\end{center}
By induction hypothesis, there exists a splitting field in which $g(x)$ can be written as
\begin{center}
$g(x)=c(x-\alpha_{2}) \cdots (x-\alpha_{n})$, with $c \in \F$.
\end{center}
Thus the field $\E = \F[\alpha_{1},\alpha_{2},\dots,\alpha_{n}]$ is the splitting field of $f(x)$ over $\F$ in which all the roots of $f(x)$ are contained.
\\[10pt]
\indent We have proved the existence of the splitting field of $f(x)$, let us now prove the uniqueness up to isomorphism.
Let $\E$ and $\K$ be two splitting fields of $f(x)$, then there exists a nontrivial isomorphism $\theta: \F \longrightarrow \F$ such that $\E = \F[\alpha_{1},\dots,\alpha_{n}]$ and $\K = \theta\F[\beta_{1},\dots,\beta_{n}]$ with $f(x) = c_{1}(x-\alpha_{1}) \cdots (x-\alpha_{n})$ over $\E$ and $\theta f(x) = c_{2}(x-\beta_{1}) \cdots (x-\beta_{n})$ over $\K$ and $c_{1} , c_{2} \in \F$.
We will prove by induction on $\textrm{deg} {\hskip 0.7mm} f(x)$, that $\theta$ can be extended to an isomorphism $\overline{\theta}: \E \longrightarrow \K$.
If $\textrm{deg} {\hskip 0.7mm} f(x)= 1$ then $\E = \F[\alpha]$ and $\K = \theta\F[\beta]$. Thus the mapping $\theta_{1}: \E \longrightarrow \K$ defined by $\theta_{1}(\alpha)= \beta$ and $\theta_{1}(c)= \theta(c)$ for all $c \in \F$, is an isomorphism.
If $\textrm{deg} {\hskip 0.7mm} f(x)= n$, then for a root $\alpha_{1}$ of $f(x)$ and a root $\beta_{1}$ of $\theta f(x)$, there exists an isomorphism $\theta_{1}:\F[\alpha_{1}] \longrightarrow \theta\F[\beta_{1}]$. So we can write $f(x)$ as
\noindent $f(x)= c_{1}(x-\alpha_{1})g(x)$ over $\F[\alpha_{1}]$ and $\theta f(x)= c_{2}(x-\beta_{1})\theta_{1}g(x)$ over $\theta\F[\beta_{1}]$.
Since $\textrm{deg} {\hskip 0.7mm} g(x)= n-1 =$ $\textrm{deg} {\hskip 0.7mm} \theta_{1}g(x)$, by induction hypothesis, there exists an isomorphism $\theta_{2} : \F[\alpha_{2},\dots,\alpha_{n}] \longrightarrow \theta\F[\beta_{2},\dots,\beta_{n}]$ such that, after a permutation of $\beta_{2},\dots,\beta_{n}$, $\theta_{2}(\alpha_{i})=\beta_{i}$ for $i= 2,\dots,n$.
\\[10pt]
Thus we can write $g(x)$ and $\theta g(x)$ as
$g(x)= c_{3}(x-\alpha_{2})\cdots(x-\alpha_{n})$ over $\F[\alpha_{2},\dots,\alpha_{n}]$,
$\theta g(x)= c_{4}(x-\beta_{2})\cdots(x-\beta_{n})$ over $\theta\F[\beta_{2},\dots,\beta_{n}]$.
\\[10pt]
So $f(x)= c_{1}(x-\alpha_{1})g(x)= c(x-\alpha_{1})\cdots(x-\alpha_{n})$ over $\F[\alpha_{1},\dots,\alpha_{n}]$ and $\theta f(x)= c_{2}(x-\beta_{1})\theta_{1}g(x)=\theta(c)(x-\beta_{1})\cdots(x-\beta_{n})$ over $\theta\F[\beta_{1}, \dots ,\beta_{n}]$ such that $c=c_{1}c_{3}$ and $\theta(c)= c_{2}c_{4}$. Then $\theta$ can be extended to $\overline{\theta}:\E \longrightarrow \K$ such that $\overline{\theta}(\alpha_{i})=\beta_{i}$ for $i= 1,\dots,n$ and $\overline{\theta}(c) =\theta(c)$ for all $c \in \F$. \hspace{\stretch{1}} $\Box$
\begin{theo}
\label{theo f&f'}
Let $\F$ be a field and $f(x)$ be a polynomial over $\F$. Then $f(x)$ has no roots with multiplicity $\geq 2$ if and only if the greatest common divisor of a polynomial $f(x)$ and $f^{'}(x)$ has degree $0$. i.e., $f(x)$ and $f^{'}(x)$ has no common root.
\end{theo}
\proof
Assume that the greatest common divisor of $f(x)$ and $f^{'}(x)$ has degree $0$, we will prove that $f(x)$ has no roots with multiplicity $\geq 2$ by contradiction. Let $f(x)$ have at least one root with multiplicity $r \geq 2$, then we can write $f(x)$ as
$$f(x) = c(x-a_{1})^{r}(x-a_{2})\cdots(x-a_{n-r})$$
where $n = \textrm{deg} {\hskip 0.7mm} f(x) , c \in \F$ and $a_{1},a_{2},\dots,a_{n-r}$ are all roots of $f(x)$ such that $a_{1}$ is repeated $r$ times. Then
\begin{center}
$f^{'}(x) = cr(x-a_{1})^{r-1}(x-a_{2}) \cdots (x-a_{n-r})+c(x-a_{1})^{r}(x-a_{3}) \cdots (x-a_{n-r})+ \cdots +c(x-a_{1})^{r}(x-a_{2}) \cdots (x-a_{n-r-1}) = c(x-a_{1})^{r-1}[r(x-a_{2}) \cdots (x-a_{n-r})+(x-a_{1})(x-a_{3}) \cdots (x-a_{n-r})+ \cdots +(x-a_{1})(x-a_{2}) \cdots (x-a_{n-r-1})]$.
\end{center}
It follows that $(x-a_{1})^{r-1}\mid f(x)$ and $(x-a_{1})^{r-1}\mid f^{'}(x)$, so the greatest common divisor of $f(x)$ and $f^{'}(x)$ has degree $\geq r-1 \neq 0$ because $r \geq 2 $.
\\[10pt]
\indent Now assume that $f(x)$ has no roots with multiplicity $\geq 2$, we prove that the greatest common divisor of a polynomial $f(x)$ and $f^{'}(x)$ has degree $0$. We can write $f(x)$ as
$$f(x) = c(x-a_{1})^{r}(x-a_{2})\cdots(x-a_{n})$$
where $n = \textrm{deg} {\hskip 0.7mm} f(x), c \in \F$ and $a_{1},a_{2},\dots,a_{n}$ are all roots of $f(x)$.
Then
$f^{'}(x)= c(x-a_{2})\cdots(x-a_{n})+c(x-a_{1})(x-a_{3})\cdots(x-a_{n})+\cdots+c(x-a_{1})(x-a_{2})\cdots(x-a_{n-1})=c[(x-a_{2})\cdots(x-a_{n})+(x-a_{1})(x-a_{3})\cdots(x-a_{n})+\cdots+(x-a_{1})(x-a_{2})\cdots(x-a_{n-1})]$.
It follows that $c$ is the only common divisor of $f(x)$ and $f^{'}(x)$. \hspace{\stretch{1}} $\Box$
\begin{theo}
\label{p^n}
For every prime $p$ and $n \in \N$ there is a field with $p^{n}$ elements.
\end{theo}
\proof
Consider the polynomial $f(x)= x^{p^{n}}-x$ over a field $\F$. If $f(x)$ is irreducible, then by Theorem \ref{theo EX&UN} there exists a splitting field $\E$ of $f(x)$ which is unique up to isomorphism. If $f(x)$ is not irreducible, set $\E = \F$. Let $K$ be the set of all zeros of $f(x)$ in $\E$, then $K = \{ a \in \E : a^{p^{n}}=a \}$.
\\[10pt]
\indent Since $f^{'}(x) = p^{n}x^{(p^{n}-1)}-1$, $f(x)$ and $f^{'}(x)$ has no common zero, where $f(a)=0 \Rightarrow a^{p^{n}}=a \Rightarrow a^{p^{n}-1}=1 \Rightarrow f^{'}(a)=p^{n}a^{p^{n}-1}-1=p^{n}-1 \neq 0$.
It follows from Theorem \ref{theo f&f'} that all zeros of $f(x)$ are distinct. Thus the set $K$ has $p^{n}$ elements.
\\[10pt]
\indent We will prove that $K = \E$. In order to show that, we need to prove that $K$ is a field, in which each element is a root of $f(x)$, then $K$ is a splitting field of $f(x)$ with $p^{n}$ elements.
It is clear that $K \subseteq \E$ and by using Theorem \ref{theo EX&UN} $K$ isomorphic to $\E$. Thus $K = \E$.
\\[10pt]
\indent Clearly $0,1 \in K$. Let $a , b \in K$, to prove that $K$ is closed under addition, we need to prove that $(a+b)^{p^{n}}= a^{p^{n}}+b^{p^{n}}= a+b$. In the polynomial expansion
\begin{displaymath}
(a+b)^{p^{n}} = \sum_{i=0}^{p^{n}} \binom{p^{n}}{i} a^{i} b^{p^{n}-i} : \binom {p^{n}}{i}= \dfrac{p^{n}!}{i! (p^{n}-i)!} \end{displaymath}
we can see that all binomial coefficients are divisible by $p^{n}$ except the first and the last, and since the finite field $\E$ has $\textrm{Char} {\hskip 0.7mm} \E = p$, all binomial coefficients are $0$ except the first and the last. That is, $$(a+b)^{p^{n}}= a^{p^{n}}+b^{p^{n}}=a+b.$$
Clearly, $K$ is closed under multiplication or $(ab)^{p^{n}}= a^{p^{n}}b^{p^{n}}$, because $\E$ is commutative. Since $\forall \ a \neq 0 ,\ a^{p^{n}} = a$, we have $(a^{-1})^{p^{n}} = (a^{p^{n})^{-1}} = a^{-1} $. So the inverse of any element in $K$ belongs to $K$.
Thus $K$ is subfield of $\E$ or $K = \E$ is a field with $p^{n}$ element. \hspace{\stretch{1}} $\Box$
\\[10pt]
\noindent {\bf Notation} : We will denote by $\F_{p^n}$ the field with $p^n$ elements.
\\[10pt]
Clearly $(\F_{p^n})^*=\F_{p^n} \setminus \{0\}$ and $(\F^{*}_{p^n})^2=\{\ a^2 : a \in \F^{*}_{p^n} \}$ form a group under multiplication. In general, $(\F^{*}_{p^n})^m=\{\ a^m : a \in \F^{*}_{p^n}$, $m \in \N\}$, form a group under multiplication.
\\[10pt]
\begin{theo}
\label{TH:cyclic}
The multiplicative group $\ \F^*_{p^n}$ is cyclic.
\end{theo}
\proof
Let $p^n \geqslant 3$ and $h=p^n-1=p^{r_1}_1p^{r_2}_2\cdots p^{r_m}_m$ be the prime factorization of $|\F^*_{p^n}|$. Let $a_i$, for every $1 \leqslant i \leqslant m$, be an element in $\F_{p^n}$ with $a^{\frac{h}{p_i}}_i\neq 1$. To prove the existence of $a_i$, consider the polynomial $x^{\frac{h}{p_i}}-1$ which has at most $\frac{h}{p_i}$ roots in $\F_{p^n}$. Since $\frac{h}{p_i} < h$, it follows that there exists $a_i \in \F_{p^n}$ with $a^{\frac{h}{p_i}}_i-1 \neq 0$ or $a^{\frac{h}{p_i}}_i \neq 1$.
Now set $b_i=a^{{h}/{p^{r_i}_i}}_i$, it follows that $b_i \neq 1$ and $b^{p^{r_i}}_i = 1$, then the order of $b_i$ must be in the form $p^{s_i}_i$ with $1 \leq s_i \leq r_i$. Since $$b^{p^{r_i-1}_i}_i = (a^{{h}/{p^{r_i}_i}}_i)^{p^{r_i-1}_i}= a^{\frac{h}{p_i}}_i \neq 1,$$ the order of $b_i$ is $p^{r_i}_i$.
Now we will prove that $b= b_1b_2\cdots b_m$ is a generator of the group $\ \F^*_{p^n}$, that is the order of $b$ is $h$. We know that $b^h=b^h_1b^h_2\cdots b^h_m=1$.
Assume that the order of $b$ is not $h$, then the order of $b$ is a proper divisor of $h$. So the order of $b$ is a divisor of at least one of the $m$ integers $\frac{h}{p_i}$ and $1 \leq i \leq m$, say $\frac{h}{p_1}$. Thus
$$b^{h/p_1}=b^{h/p_1}_1b^{h/p_1}_2\cdots b^{h/p_1}_m=1.$$ Since $p^{r_i}_i | \frac{h}{p_1}$ for every $2 \leq i \leq m$, it follows that $b^{h/p_1}_i=1$ for every $2 \leq i \leq m$. Thus $b^{h/p_1}_1=1$, which is impossible, because the order of $b_1$ is $p^{r_1}_1 \nmid \frac{h}{p_1}$.
So the order of $b$ is $h$, in other words, the multiplication group $\ \F^*_{p^n}$ is cyclic.
\hspace{\stretch{1}} $\Box$
\\[10pt]
\noindent We remark here that $\F_{p^n}$ with $n > 1$ is never $\Z_{p^n}$, in the following section we construct some of such fields.
\end{section}
\begin{section}{Construction of the finite fields}
\label{sec3}
To construct a field with $p^{n}$ elements, we use an irreducible monic polynomial $f(x) \in \Z_{p}[x]$ with $\textrm{deg} {\hskip 0.7mm} f(x) = n$. The elements of the field $\Z_{p}[x]/(f(x))$ can be written in the form
$$ a_{0} + a_{1} x + a_{2} x^{2} + \cdots + a_{n-1} x^{n-1}, \textrm { where } a_{i} \in \Z_{p} \textrm { for all } \ i = 0,1,\dots,n-1.$$
Since there are $p$ possible values for each $a_{i}$, the field $\Z_{p}[x]/(f(x))$ has $p^{n}$ elements.
To find an irreducible polynomial, we list all possible monic polynomials of degree $n$ ($p^{n}$ possible monic polynomials), which is not always an easy process especially for large $n ,p$.
Clearly, any polynomial without a constant term is not irreducible ($x$ is a factor), so these $p^{n-1}$ polynomial will not be considered.
For each of the remaining $p^{n} - p^{n-1}$ polynomials, we could substitute one by one all the field elements for $x$. If none of these substitutions is equal to zero, the polynomial is irreducible (i.e., it has no root in the field).
\\[10pt]
\indent Let $a$ be a zero of the chosen polynomial, then the elements of $\Z_{p}[x]/(f(x))$ can be written in its vector form representation using the basis $$\{1, a, a^{2},\dots, a^{n-1}\}.$$
\indent We can also generate a multiplicative representation of the field by using the fact that the multiplicative group of the field is cyclic. So if we can find a primitive element (i.e., a generator of the cyclic group), we will have a representation of the elements.
\\[10pt]
{\it Example 1}: We will construct a field of \ $16 = 2^{4}$ \ elements, here $p=2 , n=4$. We start with a field of order $2$ which is \ $\Z_{2}= \{0,1\}$ and an irreducible polynomial over $\Z_{2}$ of degree $4$. We can easily list all possible polynomials of degree $4$ over $\Z_{2}$. There are $16$ of them :
\begin{flushleft}
$x^4 ,\quad x^4 + 1,\quad x^4 + x,\quad x^4 + x^2,\quad x^4 + x^3,\quad x^4 + x + 1,\quad x^4 + x^2 + 1$,\\
$x^4 + x^3 + 1,\quad x^4 + x^2 + x,\quad x^4 + x^3 + x,\quad x^4 + x^3 + x^2,\quad x^4 + x^2 + x + 1$,\\
$x^4 + x^3 + x + 1,\quad x^4 + x^3 + x^2 + 1,\quad x^4 + x^3 + x^2 + x,\quad x^4 + x^3 + x^2 + x +1$.
\end{flushleft}
Every polynomial without a constant term has root $0$. So we will consider just the following polynomials
\begin{flushleft}
$x^4 + 1,\quad x^4 + x + 1,\quad x^4 + x^2 + 1,\quad x^4 + x^3 + 1,\quad x^4 + x^2 + x + 1$,\\
$x^4 + x^3 + x + 1,\quad x^4 + x^3 + x^2 + 1,\quad x^4 + x^3 + x^2 + x +1$.
\end{flushleft}
In this set, every polynomial with even number of terms has root $1$. So we will consider just the following polynomials
\begin{flushleft}
$x^4 + x + 1,\quad x^4 + x^2 + 1,\quad x^4 + x^3 + 1,\quad x^4 + x^3 + x^2 + x +1$.
\end{flushleft}
All of them have no roots in $\Z_{2}$, then all of them are irreducible polynomials over $\Z_{2}$.
Consider one of them, say $x^4 + x + 1$. Let $a$ be a root of $x^4 + x + 1$, then the elements of the field $\Z_{2}[x]/(x^4 + x + 1)$ can be obtained by two methods:
The first method is additive, in which we construct all linear combinations of $1, a, a^2$ and $a^3$. They are :
\begin{flushleft}
$0,\quad 1,\quad a,\quad a^2,\quad a^3,\quad a + 1,\quad a^2 + 1,\quad a^3 + 1,\quad a^2 + a,\quad a^3 + a$,\\
$a^3 + a^2,\quad a^2 + a + 1,\quad a^3 + a + 1,\quad a^3 + a^2 + 1,\quad a^3 + a^2 + a$,\\
$a^3 + a^2 + a + 1$.
\end{flushleft}
The second method is multiplicative. Since $a^4 = -a - 1 = a + 1$, we can write down the powers of $a$ as the following:
\begin{displaymath}
\begin{array}{lllllllll}
a^1 &=& a & a^2 &=& a^2 & a^3 &=& a^3 \\
a^4 &=& a + 1&a^5 &=& a^2 + a&a^6 &=& a^3 + a^2\\
a^7 &=& a^3 + a + 1&a^8 &=& a^2 + 1&a^9 &=& a^3 + a\\
a^{10} &=& a^2 + a + 1& a^{11} &=& a^3 + a^2 + a&a^{12} &=& a^3 + a^2 + a + 1\\
a^{13} &=& a^3 + a^2 + 1& a^{14} &=& a^3 + 1& a^{15} &=& 1,
\end{array}
\end{displaymath}
which means that $a$ is a generator of the cyclic group
$$(\Z_{2}[x]/(x^4 + x + 1))^{*} = \Z_{2}[x]/(x^4 + x + 1) \setminus \{0\}.$$
Notice also that the terms on the right are all the possible terms that can be written as linear combinations of the basis $\{1, a, a^2, a^3\}$ over $\Z_{2}$. When working with finite fields it is convenient to have both of the above representations, since the terms on the left are easy to multiply and the terms on the right are easy to add.
\\[10pt]
\indent Now suppose we had chosen a root of the second irreducible polynomial $x^4 + x^2 + 1$, say, $b$. We would then have $b^4= b^2 + 1$ and the powers of $b$ will be
\begin{displaymath}
\begin{array}{lllllllll}
b^1 &=& b&b^2&=& b^2& b^3 &=& b^3\\
b^4 &=& b^2 + 1& b^5 &=& b^3 + b& b^6 &=& b^4 + b^2 = b^2 + 1 + b^2 = 1,
\end{array}
\end{displaymath}
which means that $b$ cannot be a generator of the group $(\Z_{2}[x]/(x^4 + x^2 + 1))^{*}$.
\\[10pt]
{\it Example 2}: Now we will construct a field of \ $9 = 3^{2}$ \ elements, that is, $p=3 , n=2$. We start with a field of order $3$ which is \ $\Z_{3}= \{0, 1, 2\}$ and an irreducible polynomial over $\Z_{3}$ of degree $2$. We can easily list all possible monic polynomials over $\Z_{3}$. They are :
\begin{flushleft}
$x^2,\quad x^2 + 1,\quad x^2 + 2,\quad x^2 + x,\quad x^2 + 2x,\quad x^2 + x + 1,\quad x^2 + x + 2,\quad x^2 + 2x + 1,$\\
$x^2 + 2x + 2$.
\end{flushleft}
Every polynomial without a constant term has root $0$. So we will consider just the following polynomials
\begin{flushleft}
$x^2 + 1
,\quad x^2 + 2
,\quad x^2 + x + 1,\quad
x^2 + x + 2
,\quad x^2 + 2x + 1,\quad
x^2 + 2x + 2.$
\end{flushleft}
In this set, $x^2 + 2$ and $x^2 + x + 1$ have root $1$ and $x^2 + 2x + 1$ has root $2$. So we will consider just the following polynomials
\begin{center}
$x^2 + 1,\quad x^2 + x + 2,\quad x^2 + 2x + 2$.
\end{center}
All of them have no roots in $\Z_{3}$, then all of them are irreducible polynomials over $\Z_{3}$.
Consider one of them, say $x^2 + x + 2$. Let $a$ be a root of $x^2 + x + 2$, then the elements of the field $\Z_{2}[x]/(x^2 + x + 2)$ can be obtained by two methods:
The first method is additive, in which we construct all linear combinations of $1$ and $a$. They are :
\begin{center}
$0,\quad 1,\quad 2,\quad a,\quad 2a,\quad a + 1,\quad a + 2,\quad 2a + 1,\quad 2a + 2$.
\end{center}
The second method is multiplicative. Since $a^2 = -a - 2 = 2a + 1$, we can write out the powers of $a$ as follows:
\begin{displaymath}
\begin{array}{llllllllllll}
a^1 &=& a
&a^2 &=& 2a + 1
&a^3 &=& 2a + 2
&a^4 &=& 2\\
a^5 &=& 2a&
a^6 &=& a + 2
&a^7 &=& a + 1
&a^8 &=& 1.
\end{array}
\end{displaymath}
In other words, $a$ is a generator of the cyclic group
$$(\Z_{3}[x]/(x^2 + x + 2))^{*} = (\Z_{2}[x]/(x^2 + x + 2) )\setminus \{0\}.$$
\noindent {\it Example 3}: Finally, we will construct a field of \ $25 = 5^{2}$ \ elements. Here, $p=5 , n=2$, a field of order $5$ is \ $\Z_{5}= \{0, 1, 2, 3, 4\}$. To find an irreducible polynomial over $\Z_{5}$ of degree $2$, we list all possible monic polynomials of degree $2$ over $\Z_{5}$:
\begin{flushleft}
$x^2,\quad x^2 + 1,\quad x^2 + 2,\quad x^2 + 3,\quad x^2 + 4,\quad x^2 + x,\quad x^2 + 2x,\quad x^2 + 3x$,\\
$x^2 + 4x,\quad x^2 + x + 1,\quad x^2 + x + 2,\quad x^2 + x + 3 ,\quad x^2 + x + 4,\quad x^2 + 2x + 1$,\\
$x^2 + 2x + 2,\quad x^2 + 2x + 3,\quad x^2 + 2x + 4,\quad x^2 + 3x + 1,\quad x^2 + 3x + 2,\quad x^2 + 3x + 3$,\\
$x^2 + 3x + 4,\quad x^2 + 4x + 1,\quad x^2 + 4x + 2,\quad x^2 + 4x + 3,\quad x^2 + 4x + 4$.
\end{flushleft}
Every polynomial without a constant term has root $0$,\\
$x^2 + 4$, $x^2 + x + 3$, $x^2 + 2x + 2$ and $x^2 + 3x + 1$ have root $1$,\\
$x^2 + 1$ ,$x^2 + 3x$, $x^2 + x + 4$, $x^2 + 2x + 2$ and $x^2 + 4x + 3$ have root $2$,\\
$x^2 + 1$, $x^2 + 2x$, $x^2 + x + 3$, $x^2 + 3x + 2$ and $x^2 + 4x + 4$ have root $3$,\\
and $x^2 + 4$, $x^2 + x$, $x^2 + 2x + 1$, $x^2 + 3x + 2$ and $x^2 + 4x + 3$ have root $4$.\\
So we will consider the remaining polynomials:
\begin{flushleft}
$x^2 + 2,\quad x^2 + 3,\quad x^2 + x + 1,\quad x^2 + x + 2,\quad x^2 + 2x + 3,\quad x^2 + 2x + 4$,\\
$x^2 + 3x + 3,\quad x^2 + 3x + 4,\quad x^2 + 4x + 1,\quad x^2 + 4x + 2$.
\end{flushleft}
All of them have no roots in $\Z_{5}$, then all of them are irreducible polynomials over $\Z_{5}$.
Consider one of them, say $x^2 + 2$. Let $a$ be a root of $x^2 + 2$, then the elements of the field $\Z_{5}[x]/(x^2 + 2)$ are all linear combinations of $1$ and $a$. They are :
\begin{flushleft}
$0,\quad 1,\quad2,\quad3,\quad 4,\quad a,\quad2a,\quad3a,\quad4a,\quad a + 1,\quad a + 2,\quad a + 3,\quad a + 4$, \\
$ 2a + 1,\quad 2a + 2,\quad 2a + 3,\quad 2a + 4,\quad3a + 1,\quad 3a + 2, \quad3a + 3,\quad 3a + 4$,\\
$4a + 1,\quad 4a + 2,\quad4a + 3,\quad4a + 4$.
\end{flushleft}
\end{section}
\chapter{Paley Graph}
\label{Chap2}
In this chapter we will give some basic definitions and properties of graph theory and we will study in details Paley graphs and some of its properties.
\begin{section}{Basic definitions and properties}
\noindent {\bf Definition:}
A {\it graph} $G$ is a pair $( V, E )$ of sets satisfying $E \subseteq P_2(V)$, where $P_2(V)$ is the set of all subsets of $V$ with two elements. The elements of\ $V$ are called {\it vertices} and the elements of $E$ are called {\it edges}.
\\[10pt]
\noindent Note that the set of vertices of a graph $H = ( W, F)$ is denoted by $V(H)$ and the set of edges is denoted by $E(H)$. An edge $e = \{x, y\}$ is sometimes written as $xy$.
\\[10pt]
\noindent {\bf Definition:}
The {\it order} of a graph $G$ is the number of its vertices and is denoted by $|G|$. A graph $G$ is called a {\it finite graph} or an {\it infinite graph} depending on the order of $G$. If $|G| = 0$ then $G$ is called the {\it empty graph}.
\\[10pt]
\noindent In order to draw a graph $G$, we can represent its vertex set $V(G)$ by dots, we join two of these dots by a line if and only if the two corresponding vertices form an edge in $E(G)$.
\begin{displaymath}
\def\objectstyle{\scriptscriptstyle}
\xy /r2.8pc/:, {\xypolygon6"A"{~>{}{\bullet}}}, "A1"!{+U*+!L{\textrm{\normalsize1}}}
,"A2"!{+RD*++!D{\textrm{\normalsize2}}},"A6"!{+RD*+!UL{\textrm{\normalsize6}}}
,"A3"!{+RD*++!DR{\textrm{\normalsize3}}}, "A4"!{+LDD*+!R{\textrm{\normalsize4}}},
"A5"!{+LD*+!UR{\textrm{\normalsize5}}}, "A4";"A2"**@{-}, "A1";"A3"**@{-}, "A5";"A2"**@{-}
, "A1";"A2"**@{-}, "A1";"A5"**@{-}, "A2";"A3"**@{-}, "A1";"A4"**@{-}
\endxy
\end{displaymath}
\begin{fig}
\begin{center}
A graph $G$ with $V(G)=\{1,2,3,4,5,6\}$ and $E(G)=\{\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,3\},\{2,4\},\{2,5\}\}$.
\end{center}
\end{fig}
\noindent {\bf Definition:}
Two vertices $x$, $y$ of $G$ are {\it adjacent}, or {\it neighbors}, if $xy$ is an edge of $G$, and the set of all neighbors of $x$ is denoted by $N(x)$.
The order of the set $N(x)$ is called the {\it degree} of $x$ and is denoted by $d(x)$. We say that a vertex $x$ is {\it isolated} if $d(x) = 0$.
\\[10pt]
\noindent Note that for each graph $G = ( V, E )$ we have \begin{displaymath}
2|E(G)| = \sum_{x\in V(G)} d(x).\end{displaymath}
\noindent {\bf Definition:}
A graph $G$ is called {\it k-regular} if all the vertices of $G$ have the same degree $k$.
If $|V(G)|= n$ and $G$ is an ($n$-1)-regular graph then $G$ is called a {\it complete} graph. We will denote by $K_n$ the complete graph on $n$ vertices.
\\[10pt]
\noindent Note that in a $k$-regular graph $G$ we have $2|E(G)| = k |V(G)|$, it follows that $k$ or $|V(G)|$ is even.
\\[10pt]
\noindent {\bf Definition:}
A {\it path} of length $n$ in a graph $G$ is the sequence
$$x_{1}e_{1}x_{2} \cdots e_{n-1}x_{n} \textrm{ with } x_{i} \in V(G),\ e_{i}=\{x_{i},x_{i+1}\}\in E(G)$$ for all $i \in \{ 1, 2, \dots, n-1\}$, and $x_{i}\neq x_{j}$ for all $i\neq j$.
Let $x_{1}e_{1}x_{2}e_{2} \cdots x_{n-1}e_{n-1}x_{n}$ be a path then the sequence
$$x_{1}e_{1}x_{2}e_{2} \cdots x_{n-1}e_{n-1}x_{n}e x_{1} \textrm{ with } e=\{x_{n},x_{1}\}\in E(G)$$
\noindent is called a {\it cycle} of length $n$ which will be denoted by $C_n$.
\\[10pt]
\noindent As an example, in figure \ref{fig2} we see the cycle and the complete graph of $5$ vertices.
\begin{displaymath}
\def\objectstyle{\scriptscriptstyle}
\xy /r3pc/:, {\xypolygon5"B"{\bullet}}, "B1"!{+U*+!L{\textrm{\normalsize1}}}
,"B2"!{+RD*++!D{\textrm{\normalsize2}}}
,"B3"!{+RD*+!R{\textrm{\normalsize3}}}, "B4"!{+LDD*+!UR{\textrm{\normalsize4}}},
"B5"!{+LD*+!UL{\textrm{\normalsize5}}}
\endxy
\quad\quad\quad
\xy /r3pc/:, {\xypolygon5"A"{\bullet}}, "A1"!{+U*+!L{\textrm{\normalsize1}}}
,"A2"!{+RD*++!D{\textrm{\normalsize2}}} ,"A3"!{+RD*+!R{\textrm{\normalsize3}}},
"A4"!{+LDD*+!UR{\textrm{\normalsize4}}}, "A5"!{+LD*+!UL{\textrm{\normalsize5}}},
"A4";"A1"**@{-}, "A1";"A3"**@{-}, "A5";"A2"**@{-}, "A2";"A4"**@{-}, "A3";"A5"**@{-}
\endxy
\end{displaymath}
\begin{center}
$C_5$\quad\quad\quad\quad\quad
\quad\quad\quad\quad $K_5$\end{center}
\begin{fig}\label{fig2}
\begin{center}
\end{center}
\end{fig}
\noindent {\bf Definition:}
Let $G$ be a $k$-regular graph with $|G|=n$. If there are two integers $\lambda, \mu$ such that
every two adjacent vertices have $\lambda$ common neighbors and
every two non-adjacent vertices have $\mu$ common neighbors,
\noindent then $G$ is called a {\it {\bf s}trongly {\bf r}egular {\bf g}raph} with parameters $(n, k, \lambda, \mu)$ and is denoted by {\bf srg}$(n, k, \lambda, \mu)$.
\\[10pt]
\indent Clearly, every strongly regular graph is regular, but not vice versa. For example $C_6$ is 2-regular but not a strongly regular graph.
\\[10pt]
\indent Note that in any strongly regular graph srg($n,k,\lambda,\mu$), its parameters are related by
\begin{equation}\label{eq1} \mu (n-k-1)=k(k-\lambda-1). \tag{*} \end{equation}
\indent In order to show that, consider a vertex $x \in V(G)$. We remind that $N(x)$ is the set of all neighbors of $x$. Let $N^{'}(x)$ be the set of all non-adjacent vertices of $x$, then $n = |N(x)|+1+|N^{'}(x)|$. We will prove that both sides of (\ref{eq1}) are equal to the number of edges between $N(x)$ and $N^{'}(x)$.
\\[10pt]
\indent Since $G$ is $k$-regular, $|N^{'}(x)|= n -|N(x)|-1 = n-k-1$. Let $y \in N^{'}(x)$, then $y$ and $x$ have $\mu$ common neighbors, that is $\mu$ equals the number of edges between $y$ and $N(x)$. It follows that $\mu (n-k-1)$ is the number of edges between $N(x)$ and $N^{'}(x)$.
\\[10pt]
\indent Let $z \in N(x)$ then $z$ and $x$ have $\lambda$ common neighbors. Since $|N(x)|= k$, it follows that the number of neighbors of $z$ which are not adjacent to $x$ is equal to $k-\lambda -1$. Thus $k-\lambda -1$ is the number of edges between $z$ and $N^{'}(x)$, so $k(k-\lambda -1)$ equals the number of edges between $N(x)$ and $N^{'}(x)$. Therefore, (\ref{eq1}) is proved.
\\[10pt]
\noindent {\bf Definition:}
A graph $G$ is called {\it connected } if every two vertices are connected by a path.
\\[10pt]
\indent Note that every complete graph is connected, regular, and strongly regular. Both strongly regular and regular graphs are not necessary connected and also connected graphs are not necessary complete, strongly regular, or regular.
For example, a cycle is a connected graph which is not complete, a path is a connected graph which is not regular, and the graph $G$ with $V(G)=\{1,2,3,4\}$ and $E(G)=\{\{1,2\},\{3,4\}\}$ is strongly regular which is not connected.
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\begin{fig}
\begin{center}
\end{center}
\end{fig}
\noindent {\bf Definition:}
Let $G=(V, E) \textrm{ and } G^{'}=(V^{'}, E^{'})$ be two graphs. $G$ is {\it isomorphic} to $G^{'}$ if there is a bijection $f : V\longrightarrow V^{'}$ such that $xy \in E$ if and only if $f(x)f(y)\in E^{'}$; we denote this by $G \cong G^{'}$.
An isomorphism from a graph $G$ to itself is called an {\it automorphism}. The set of all automorphismus of a graph $G$ form a group under composition, and it is denoted by $Aut(G)$.
\\[10pt]
\noindent {\bf Definition:}
Let $G = (V, E)$ be a finite graph. The {\it complementary} graph of $G$ is a graph $\overline{G}$ with $V(\overline{G})=V(G)$ and $E(\overline{G})= P_2(V) \setminus E(G)$. That is, $xy \in E(\overline{G})$ if and only if $xy \notin E(G)$.
A graph $G$ is called {\it self-complementary} if it is isomorphic to its complement.
\\[10pt]
For example $C_5$ is self-complementary, see figure 2.1.4.
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\begin{center}
$C_5\quad\quad\quad\quad\cong\quad\quad\quad\quad\overline{C_5}$
\end{center}
\begin{fig}
\begin{center}
\end{center}
\end{fig}
\noindent {\bf Definition:}
Let $G$ be a group and let $X$ be a non-empty set. We say that $G$ acts on $X$ if there is a map $ \phi:G\times X \rightarrow X$ such that the following conditions hold for all $x \in X$:
\begin{enumerate}
\item $\phi(e,x)=x$ where $e$ is the identity element of $G$.
\item $\phi(g,\phi(h,x))=\phi(gh,x) \ \forall g,h \in G$.
\end{enumerate}
In this case, $G$ is called a transformation group, $X$ is a called a $G$-set, and $\phi$ is called the group action.
The group action is called {\it transitive} (we also say that $G$ acts {\it transitively} on $X$)
if for every $x,y \in X$, there exists $g \in G$ such that $\phi(g,x)= y$.
\\[10pt]
\noindent {\bf Definition:}
A graph $G$ is called {\it symmetric} if its automorphism group acts transitively on the vertices and edges.
\\[10pt]
\indent For example every cycle graph or complete graph is symmetric graph and every symmetric graph is a regular graph, but not vice versa.
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\begin{fig}
\begin{center}
Symmetric graph which is regular and regular graph which is not symmetric.
\end{center}
\end{fig}
\end{section}
\begin{section}{Paley graphs}
\begin{subsection}{Definition and examples}
Before we define the Paley graph, we need the following definition.
\\[10pt]
{\bf Definition:} Let $q$ and $r$ be two positive integers with gcd $(q,r)=1$, then $r$ is a {\it quadratic residue} of $q$ if and only if $x^2 \equiv r \ (\textrm{mod}\ q$) has a solution, and $r$ is a {\it quadratic nonresidue} of $q$ if and only if $x^2 \equiv r \ (\textrm{mod}\ q$) has no solution.
\\[10pt]
{\bf Definition:} Let $p$ be a prime number and $n$ be a positive integer such that $p^n \equiv 1\ (\textrm{mod}\ 4)$. The graph $P =(V, E)$ with $$V(P)=\F_{p^n} {\textrm \ and \ } E(P)=\{\{x,y\} : x,y \in \F_{p^n},\ x-y\in (\F^{*}_{p^n})^2\}$$ is called the {\it Paley graph} of order $p^n$.
\\[10pt]
\indent Note that the set $E(P)$ in the definition of Paley graph is well defined because $ x-y\in (\F^*_{p^n})^2$ if and only if $y-x\in (\F^*_{p^n})^2$. Since $x-y= -1(y-x)$, we need only to show that $-1 \in (\F^*_{p^n})^2$.
We have $p^n \equiv 1\ (\textrm{mod}\ 4)$, so $4 \mid (p^n -1)$. Let $g$ be a generator of the group $\F^*_{p^n}$ then $p^n-1$ is the least positive integer such that $g^{p^n-1} = 1$. We can rewrite this as $g^{p^n-1}-1=(g^{\frac{p^n-1}{2}}-1)(g^{\frac{p^n-1}{2}}+1)=0$. Since $g^{\frac{p^n-1}{2}}$ cannot be equal to $1$, it follows that $g^{\frac{p^n-1}{2}}=(g^{\frac{p^n-1}{4}})^2 =-1$ which means that $g^{\frac{p^n-1}{4}}$ is a square root of $-1$.
\\[10pt]
\indent Note that if the Paley graph has prime order $p$, then we can consider the field of integers modulo $p$, $\Z_p$, as its vertex set.
\\[10pt]
\indent However, we cannot consider $\Z_{p^n}$ with $n > 1$ as a vertex set of the Paley graph of order $p^n$, because as we have seen in the previous chapter, there exists a unique field $\F_{p^n}$ of order $p^n$ which is not $\Z_{p^n}$, such a field will represent the set of vertices of the Paley graph. To get this field $\F_{p^n}$ we need the construction that was developed in the previous chapter.
\\[10pt]
\indent The list of integers which can be considered as an order of the Paley graph starts with $5$, $9$, $13$, $17$, $25$, $29$, $37$, $41$. In the following examples, we show the Paley graphs explicitly for the first three cases.
\\[10pt]
{\it Example 1:}
The Paley graph of order $5$ is the cycle $C_5$.
\\[10pt]
\indent In order to see that, let $P=(V, E)$ be the Paley graph of order $5$ then $V(P)=\Z_5=\{0,1,2,3,4\}$ and $(\Z^*_5)^2=\{1,4\}$, it follows that $$E(P)=\{\{0,1\},\{1,2\},\{2,3\},\{3,4\},\{4,0\}\}.$$
{\it Example 2:}
Let $P=(V, E)$ be the Paley graph of order $9=3^2$. Here we have $p=3$, $n=2$, then $V(P)=\F_{3^2}$, the field of order $9$, can be written as $$\F_{3^2}=\{0,1,2,a,2a,1+a,1+2a,2+a,2+2a\}\cong \Z_3[x]/(x^2+1)$$ where $a$ is a root of $x^2+1$. Since
$$
\begin{array}{cllcllcll}
1^2&=&1,
&2^2&=&1,
&a^2&=&-1=2,\\
(2a)^2&=&2,&
(1+a)^2&=&2a,&
(1+2a)^2&=&a,\\
(2+a)^2&=&a,
&(2+2a)^2&=&2a,
\end{array}
$$
we have $(\F^*_{3^2})^2=\{1,2,a,2a\}$. Thus $E(G)=\{\{0,1\},\{0,2\},\{0,a\},\{0,2a\},$
$\{1,2\},\{1,1+a\},\{1,1+2a\},\{2,2+a\},\{2,2+2a\},\{a,1+a\},\{a,2+a\},\{a,2a\}$,
$\{2a,1+2a\},\{2a,2+2a\},\{1+a,2+a\},\{1+a,1+2a\},\{1+2a,2+2a\}$,
$\{2+a,2+2a\}\}$.
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\begin{fig}
\begin{center}The Paley graph of order $9$
\end{center}
\end{fig}
\noindent{\it Example 3:}
Let $P=(V, E)$ be the Paley graph of order $13$. Here we have $p=13$, $n=1$ then $V(P)=\Z_{13}=\{0,1,2,3,4,5,6,7,8,9,10,11,12\}$ and $(\Z^*_{13})^2=\{1,3,4,9,10,12\}$. It follows that each vertex $x$ in $V(P)$ is adjacent exactly to $6$ vertices $x+1$, $x+3$, $x+4$, $x+9$, $x+10$, and $x+12$. So $E(P)=\{\{x,x+1\},\{x,x+3\},\{x,x+4\},\{x,x+9\},\{x,x+10\},\{x,x+12\}\ \forall x\in \Z_{13}\}$.
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\begin{fig}
\begin{center}The Paley graph of order $13$
\end{center}
\end{fig}
\end{subsection}
\begin{subsection}{Properties}
In the previous examples we can see that the Paley graphs of order 5, 9, and 13 are connected, symmetric, self-complementary, and strongly regular.
The following Propositions prove that these properties are true for every order $q$.
\begin{prop}
\label{Paley Symm}
The Paley graphs are symmetric.
\end{prop}
\proof
Let $P$ be the Paley graph of order $q = p^n$. To prove that $P$ is symmetric we need to prove that the automorphism group $Aut(P)$ acts transitively on $V(P)$ and $E(P)$. In other words, we need to prove that
for every two vertices $x,y \in V(P)$ there exists $\phi \in Aut(P)$ such that $\phi(x)=y$, and
for every two edges $\{x_1,y_1\},\{x_2,y_2\} \in E(P)$ there exists $\theta \in Aut(P)$ such that $\theta(x_1)=x_2,\theta(y_1)=y_2$.
\\[10pt]
\indent Fix $a,b \in V(P)$ with $a \in (\F^{*}_{p^n})^2$ and define the nontrivial function $$\phi : V(P) \rightarrow V(P) {\textrm \ with \ } \phi(x)=ax+b \ \forall x\in V(P).$$
\indent We show that $\phi$ is an automorphism. Easily, we can see that $\phi$ is one to one, because $$\phi(x_1)-\phi(x_2)=0\Leftrightarrow (ax_1+b)-(ax_2+b)=0\Leftrightarrow a(x_1-x_2)+b-b=0$$
$$\Leftrightarrow x_1-x_2=0.$$
Since for every $y \in V(P)$, we have $$a^{-1}y-a^{-1}b=x \in V(P) \textrm{\ with \ }\phi(x)=a(a^{-1}y-a^{-1}b)+b=y.$$
Thus $\phi$ is onto.
\\[10pt]
\indent Since $\{x,y\} \in E(P) \Leftrightarrow x-y \in (\F^{*}_{p^n})^2 \Leftrightarrow a(x-y)+b-b \in (\F^{*}_{p^n})^2 \Leftrightarrow (ax+b)-(ay+b) \in (\F^{*}_{p^n})^2 \Leftrightarrow \phi(x)-\phi(y) \in (\F^{*}_{p^n})^2 \Leftrightarrow \{\phi(x),\phi(y)\} \in E(P)$, this proves that $\phi \in Aut(P)$.
\\[10pt]
\indent Moreover, for every two vertices $x,y \in V(P)$, take $a=1 \in (\F^{*}_{p^n})^2$ and $b=y-x \in V(P)$, the mapping $\phi : V(P) \rightarrow V(P)$ defined by $\phi(x)=ax+b$ is an automorphism with $\phi(x)=y$. Thus $Aut(P)$ acts transitively on $V(P)$.
\\[10pt]
\indent Finally, for every two edges $\{x_1,y_1\},\{x_2,y_2\} \in E(P)$ we can find $$a=(x_2-y_2)(x_1-y_1)^{-1} \in (\F^{*}_{p^n})^2 {\textrm \ and \ } b=x_2-ax_1 \in V(P)$$ so that $\theta : V(P) \rightarrow V(P)$ with $\theta(x)=ax+b$ is an automorphism with $\theta(x_1)=x_2,\theta(y_1)=y_2$. Thus $Aut(P)$ acts transitively on $E(P)$.
\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\begin{prop}
Let $P$ be the Paley graph of order $q = p^n$, then $P$ is a self-complementary graph.
\end{prop}
\proof
Let $r$ be a quadratic nonresidue modulo $q$, consider the function $$f: V(P) \longrightarrow V(\overline{P})\textrm{ defined by }f(x)=rx.$$
The function $f$ is well defined, because
$$\{x,y\} \in E(P) \Leftrightarrow (x-y) \in (\F^{*}_{p^n})^2 \Leftrightarrow f(x)-f(y)=rx-ry=r(x-y) \notin (\F^{*}_{p^n})^2$$ $$\Leftrightarrow \{f(x), f(y)\} \in E(\overline{P}).$$
Now we prove that $f$ is a bijection. Clearly, $f$ is injective, since $$(x-y)=0 \Leftrightarrow 0=r(x-y)=rx-ry=f(x)-f(y).$$
\indent Since gcd $(r,q)=1$, there exist $$a , b \in \Z \textrm{ with }1=qa+rb \Leftrightarrow rb\equiv 1 \ (\textrm {mod } q).$$
Thus $f(bx)= rbx = x$, so $f$ is surjective.
\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\begin{prop}
Let $P$ be the Paley graph of order $q = p^n$, then $P$ is a strongly regular graph with parameters
\begin{displaymath}(q, \frac{q-1}{2}, \frac{q-5}{4}, \frac{q-1}{4}). \end{displaymath}
\end{prop}
\proof First, we prove that each vertex has degree $\frac{q-1}{2}$.
\\[10pt]
\indent Fix $x \in V(P)$, we have $N(x)=\{z\in V(P) : x-z = s \in (\F^{*}_{p^n})^2\}$. If $x-z_1 = s, x-z_2 = s$ then $z_1= x-s = z_2$, so for all $s\in (\F^{*}_{p^n})^2$ there exists a unique $z\in V(P)$ such that $x-z = s$.
\\[10pt]
\indent Thus there exists a one to one correspondence between the number of elements of $N(x)$ and the number of elements of $(\F^{*}_{p^n})^2$, so all vertices have the same degree $d(x)=|N(x)|=|(\F^{*}_{p^n})^2|$.
\\[10pt]
\indent Now we calculate $|(\F^{*}_{p^n})^2|$, we have $|\F^{*}_{p^n}|=q-1$ and if $x\neq y \in \F^{*}_{p^n}$ then $x^2=y^2 \Leftrightarrow 0=x^2-y^2=(x-y)(x+y)\Leftrightarrow x=-y$. Thus $|(\F^{*}_{p^n})^2|=\frac{q-1}{2}$.
\\[10pt]
\indent Second, we prove that every two adjacent vertices have $\frac{q-5}{4}$ common neighbors and every two non-adjacent vertices have $\frac{q-1}{4}$ common neighbors.
\\[10pt]
\indent Let $x \in V(P)=(\F^{*}_{p^n}), A= N(x), B= V(P)\setminus(A\cup\{x\})$. If $y \in A$ and $z \in B$, we want to prove that $|A\cap N(y)|=\frac{q-5}{4}$ and $|A\cap N(z)|=\frac{q-1}{4}$.
\\[10pt]
\indent Because $P$ is symmetric, we can assume that there is an integer $l$ with every vertex $y \in A$ is joined to $l$ vertices in $B$ ($|N(y)\cap B|=l$). Moreover, because $P$ is self-complementary, every vertex $z \in B$ is not joined to $l$ vertices in $A$ ($|(V(P)\setminus N(z))\cap A|=l$).
\\[10pt]
\indent To find $l$ we calculate $|A||B|$ from two sides. First $|A|= |N(x)|=\frac{q-1}{2}$, $|B|=|V(P)|-|A\cup\{x\}|=q-(\frac{q-1}{2}+1)=q-\frac{q+1}{2}=\frac{2q-q-1}{2}=\frac{q-1}{2}$, which means that $|A||B|=(\frac{q-1}{2})^2$.
\\[10pt]
\indent Second $|A||B|=|A\times B|=|\{(a,b): a\in A,b\in B\textrm { and }\{a,b\} \in E(P)\}|+$
$|\{(a,b): a\in A,b\in B\textrm { and }\{a,b\} \notin E(P)\}|=\frac{q-1}{2}l+l\frac{q-1}{2}=2l\frac{q-1}{2}$.
\\[10pt]
\indent So $|A||B|=(\frac{q-1}{2})^2=2l\frac{q-1}{2}$, which gives $2l=\frac{q-1}{2}$ or $l=\frac{q-1}{4}$.
\\[10pt]
\indent Now we can calculate $|A\cap N(y)|$ and $|A\cap N(z)|$. Since
\\[10pt]
$\frac{q-1}{2}=|N(y)|=|A\cap N(y)|+|B\cap N(y)|+|\{x\}\cap N(y)|=|A\cap N(y)|+l+1=
|A\cap N(y)|+\frac{q-1}{4}+1=|A\cap N(y)|+\frac{q+3}{4},$
\\[10pt]
we have $|A\cap N(y)|=\frac{q-1}{2}-\frac{q+3}{4}=\frac{2q-2-q-3}{4}=\frac{q-5}{4}$.
\\[10pt]
\indent Since $\frac{q-1}{2}=|A|=|(V(P)\setminus N(z))\cap A|+|N(z)\cap A|=l+|N(z)\cap A|$ $=\frac{q-1}{4}+|N(z)\cap A|,$
\\[10pt]
hence $|N(z)\cap A|=\frac{q-1}{2}-\frac{q-1}{4}=\frac{q-1}{4}.$
\\[10pt]
Then $P$ is a strongly regular graph with parameters
$$(q, \frac{q-1}{2}, \frac{q-5}{4}, \frac{q-1}{4}).$$\hspace{\stretch{1}} $\Box$
\begin{cor}
The Paley graphs are connected.
\end{cor}
\proof
Let $P$ be the Paley graph of order $q = p^n$. Let $x,y$ be two vertices in $V(P)$, then $x,y$ are adjacent or non-adjacent. If $x,y$ are adjacent, then there exists a path of length $1$ connected $x$ and $y$.
\\[10pt]
\indent If $x,y$ are non-adjacent, then $x$ and $y$ have at least one common neighbor $z$ because $q\geq 5$ means that $\frac{q-1}{4} \geq 1$. So there exists a path $xe_1ze_2y$ with $e_1=\{x,z\},e_2=\{z,y\}$ of length $2$ connected $x$ and $y$.
\\[10pt]
\indent Thus in all cases every two vertices in $V(P)$ are connected by a path.\hspace{\stretch{1}} $\Box$
\\[10pt]
\indent Now we know that the Paley graphs are self-complementary symmetric graphs. So the question now is: Are there any self-complementary symmetric graphs other than Paley graphs?
\\[10pt]
\indent Peisert proved in \cite{W.P1} that the Paley graphs of prime order are the only self-complementary symmetric graphs of prime order and he proved in \cite{W.P2} that a graph $G$ is self-complementary and symmetric if and only if $ |G| = p^n$ for some prime $p,\ p^n \equiv 1$ (mod $4$), and $G$ is a Paley graph or a $\mathcal{P}^*$-graph or is isomorphic to the exceptional graph $G(23^2)$.
\\[10pt]
\indent The $\mathcal{P}^*$-graph is a graph with $V(\mathcal{P}^*)=\F_{p^n}$ and two vertices are adjacent if their difference belongs to the set $M =\{ g^j : j \equiv 0 ,1$ (mod $4$)$\}$ , where $g$ is a primitive root of the field. The graph $G(23^2)$ has $23^2$ vertices and is described in Section 3 in \cite{W.P2}.
\\[10pt]
\indent One of the properties of the Paley graphs is the 3-existentially closed property. As in \cite{Hadamard M.}, for a fixed integer $n \geq 1$, a graph $G$ is $n$-existentially closed, if for every $n$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex $x \notin S$ which is joined to every vertex in $T$ and to no vertex in $S \setminus T$.
\\[10pt]
\indent The $n$-existentially closed graphs were first studied in \cite{Erdos}, where they were called graphs with property $P (n)$. Ananchuen and Caccetta, in \cite{ontheAdjpaley}, proved that all Paley graphs with at least 29 vertices are 3-existentially closed, and before \cite{Hadamard M.} they were the only known examples of strongly regular 3-existentially closed graphs. Now in \cite{Hadamard M.} we can find a new infinite family of 3-existentially closed graphs, that are strongly regular but not Paley graphs. For further background on $n$-existentially closed graphs the reader is directed to \cite{on the Adjacency}.
\\[10pt]
\indent Another property of the Paley graphs is also interesting. To understand it we need the following definition.
\\[10pt]
\noindent {\bf Definition:} If $m,\ n \in \N \cup \{0\}$ and $k \in \N$, a graph $G$ is said to have the property $P(m,n,k)$, if for any disjoint subsets $A$ and $B$ of $V(G)$ with $|A|=m$ and $|B|=n$ there exist at least $k$ other vertices, each of which is adjacent to every vertex in $A$ but not adjacent to any vertex in $B$.
The set of graphs which have the property $P(m,n,k)$ is denoted by $\mathcal{G}(m,n,k)$.
\\[10pt]
In \cite{ontheAdjGpaley} and \cite{ongraphs}, it has been proved that the Paley graph
$$P_q \in \mathcal{G}(1,n,k) \textrm{ for every } q >{\big (}(n-2)2^n+2{\big )}\sqrt{q}+(n+2k-1)2^n-2n-1;$$
$$P_q \in \mathcal{G}(n,n,k) \textrm{ for every } q >{\big (}(2n-3)2^{2n-1}+2{\big )}\sqrt{q}+(n+2k-1)2^{2n-1}-2n^2-1;$$
$$\textrm{and }P_q \in \mathcal{G}(m,n,k) \textrm{ for every } q >{\big (}(t-3)2^{t-1}+2{\big )}\sqrt{q}+(t+2k-1)2^{t-1}-1,$$
where $t \geq m+n.$
\end{subsection}
\end{section}
\chapter{Generalizations of The Paley Graphs}
\label{Chap3}
There are many generalizations of the Paley graphs. We will see some examples of these generalizations
and some of its properties in the following section. In the second section we will define a new generalization, and we will study some of its properties in the third section.
\begin{section}{Examples of some generalizations}
Since two vertices in the Paley graphs are adjacent if and only if their difference is a quadratic residue, we can generate other classes of graphs by using higher order residues. For example, in \cite{ontheAdjGpaley}, by using the cubic and quadruple residues Ananchuen has defined the cubic Paley graphs and the quadruple Paley graphs.
\begin{subsection}{The cubic and the quadruple Paley graphs}
\noindent {\bf Definition:} Let $q=p^n$ with odd prime $p$, $n \in \N$, and $q \equiv 1\ (\textrm{mod}\ 3)$. The graph $G^{(3)}_q$ with $$V(G^{(3)}_q)=\F_q \textrm{ and } E(G^{(3)}_q)=\{\{x,y\} : x,y \in \F_q, x-y \in (\F^*_q)^3\}$$ is called {\it the cubic Paley graph}.
\\[10pt]
\indent Note that the set $E(G^{(3)}_q)$ in the definition is well defined because: $-1 = {-1}^3 \in (\F^*_q)^3$ implies that,
$\{x,y\}$ is defined to be an edge if and only if $\{y,x\}$ is defined to be an edge.
\\[10pt]
\noindent {\bf Definition:} Let $q=p^n$ with odd prime $p$, $n \in \N$, and $q \equiv 1\ (\textrm{mod}\ 8)$. The graph $G^{(4)}_q$ with $$V(G^{(4)}_q)=\F_q \textrm{ and } E(G^{(4)}_q)=\{\{x,y\} : x,y \in \F_q, x-y \in (\F^*_q)^4\}$$ is called {\it the quadruple Paley graph}.
\\[10pt]
\indent Note that the set $E(G^{(4)}_q)$ in the definition is well defined because :
We have $q \equiv 1\ (\textrm{mod}\ 8)$, so $8 \mid (q -1)$. If $g$ is a generator of the group $\F^*_{q}$ then $g^{\frac{q-1}{2}}=(g^{\frac{q-1}{8}})^4 =-1$ which means that $-1 \in ({\F_q}^*)^4$. Thus
$\{x,y\}$ is defined to be an edge if and only if $\{y,x\}$ is defined to be an edge.
\\[10pt]
\indent The following figure gives an example:
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\begin{fig}
\begin{center}The cubic Paley graph $G^{(3)}_{13}$ and the quadruple Paley graph $G^{(4)}_{17}$
\end{center}
\end{fig}
\vspace{10pt}
\noindent In \cite{ontheAdjGpaley}, Ananchuen has proved that the cubic Paley graphs
$$G^{(3)}_q \in \mathcal{G}(2,2,k) \textrm{ for every } q > [\frac{1}{4}(79+3\sqrt{36k+701})]^2;$$
$$G^{(3)}_q \in \mathcal{G}(m,n,k) \textrm{ for every } q > (t2^{t-1}-2^t+1)2^m\sqrt{q}+(m+2n+3k-3)2^{-n}3^{t-1},$$
where $t \geq m+n$; and the quadruple Paley graphs
$$G^{(4)}_q \in \mathcal{G}(m,n,k) \textrm{ for every } q > (t2^{t-1}-2^t+1)3^m\sqrt{q}+(m+3n+4k-4)3^{-n}4^{t-1},$$
where $t \geq m+n$.
\\[10pt]
\noindent Ananchuen and Caccetta have proved in \cite{CubQuadnec} that the cubic Paley graphs are $n$-existentially closed whenever
$q \geq n^2 2^{4n-2}$ and the quadruple Paley graphs are $n$-existentially closed whenever $q \geq 9n^2 6^{2n-2}$.
\end{subsection}
\begin{subsection}{The generalized Paley graphs}
In \cite{onGenPal}, Lim and Praeger have defined the following generalization of the Paley graphs.
\\[10pt]
\noindent {\bf Definition:} Let $\F_q$ be a finite field of order $q$, and let $k$ be a divisor of $q - 1$ such that $k \geq 2$, and if $q$ is odd then $\frac{q-1}{k}$ is even. Let $S$ be the subgroup of order $\frac{q-1}{k}$ of the multiplicative group $\F^*_q$. Then the generalized Paley graph GPaley$(q,\frac{q-1}{k})$ is the graph with vertex set $\F_q$ and edges all pairs $\{x, y\}$ such that $x - y \in S$.
\\[10pt]
\indent Note that in the definition they require $\frac{q-1}{k}$ to be even when $q$ is odd, and hence in
all cases $S = -S$, so that the adjacency relation is symmetric ($x - y \in S$ if and only if $y - x \in S$).
Also we can see that if $q \equiv 1\mod 4$ and $k=2$, then GPaley$(q,\frac{q-1}{k})$ is the Paley graph $P_q$.
\\[10pt]
\indent As an example consider $q=11$ then $q-1=10$, and since $\frac{q-1}{k}$ should be even and $k \geq 2$, we have only one choice $k=5$. Then $|S|=\frac{10}{5}=2$ and $S=\{1,10\}$, it follows that GPaley$(11,2)$ is the cycle $C_{11}$.
\\[10pt]
\indent Moreover, they have studied in \cite{onGenPal} the automorphism groups of this generalized Paley graphs, and in some cases, compute their full automorphism groups. Moreover they have determined precisely when these graphs are connected.
\end{subsection}
\end{section}
\begin{section}{Definition and examples}
Now we will give a new generalization of the Paley graphs.
\\[10pt]
\noindent {\bf Definition:} Let $q=p^n$ with odd prime $p$, $n \in \N$, and $m \geq 3$ be an odd integer. We will denote by $m\textrm{-}P_q$, the graph with $V(m\textrm{-}P_q)=\F_q$ and $E(m\textrm{-}P_q)=\{\{x,y\} : x,y \in \F_q, x-y \in ({\F_q}^*)^m\}$. Such a graph will be called $m$-Paley graph.
\\[10pt]
\indent Note that the set $E(m\textrm{-}P_q)$ in the definition is well defined because : $-1 = {-1}^m \in ({\F_q}^*)^m$ implies that
\begin{center}
$x-y\in (\F^*_{p^n})^m$ if and only if $y-x\in (\F^*_{p^n})^m$.
\end{center}
\indent The list of integers which can be considered as the order of $m$-Paley graph starts with $3$,$5$,$7$,$9$,$11$,$13$,$17$,$19$,$23$,$25$,$27$,$29$,$31$.
In the following examples, we will show the $m$-Paley graphs explicitly for the first three cases.
\\[10pt]
{\it Example 1:}
The $m$-Paley graph of order $3$, for every odd integer $m \geq 3$, is the cycle $C_3$.
In order to see that, let $m\textrm{-}P_3=(V, E)$ be the $m$-Paley graph of order $3$ then $V(m\textrm{-}P_3)=\Z_3=\{0,1,2\}$ and $(\Z^*_3)^m=\{1,2\}$.
It follows that $E(m\textrm{-}P_3)=\{\{0,1\},\{1,2\},\{2,0\}\}.$
\\[10pt]
{\it Example 2:}
The $m$-Paley graph of order $5$, for every odd integer $m \geq 3$, is the complete graph $K_5$.
In order to see that, let $m\textrm{-}P_5=(V, E)$ be the $m$-Paley graph of order $5$ and $m=2k+1$ with positive integer $k$, then $V(m\textrm{-}P_5)=\Z_5=\{0,1,2,3,4\}$ and $1^m=1$, $4^m={-1}^m=-1$. To calculate $2^m$, we have two cases:
If $k$ is odd then $2^m=2^{2k+1}=(2^2)^k 2=(-1)^k 2=3$ and if $k$ is even then $2^m=2^{2k+1}=(-1)^k 2=2$.
To calculate $3^m$, we have also two cases:
If $k$ is odd then $3^m=3^{2k+1}=(3^2)^k 3=(-1)^k 3=2$ and if $k$ is even then $3^m=3^{2k+1}=(-1)^k 3=3$.
Which gives $(\Z^*_5)^m=\{1,2,3,4\}$ for every odd integer $m$, it follows that $E(m\textrm{-}P_5)=\{\{0,1\},\{0,2\},\{0,3\},\{0,4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},$ \\
$\{3,4\}\}$.
\\[10pt]
{\it Example 3:}
Let $m\textrm{-}P_7=(V, E)$ be the $m$-Paley graph of order $7$, then $V(m\textrm{-}P_7)=\Z_7=\{0,1,2,3,4,5,6\}$.
Consider $m=3$, then $1^3=1$, $2^3=1$, $3^3=6$, $4^3=1$, $5^3=6$, $6^3=6$. It follows that
$(\Z^*_7)^3=\{1,6\}$ and
\noindent $E(3\textrm{-}P_7)=\{\{0,1\},\{1,2\},\{2,3\},\{3,4\},\{4,5\},\{5,6\},\{6,0\}\}$.
Consider $m=5$, then $1^5=1$, $2^5=4$, $3^5=5$, $4^5=2$, $5^5=3$, $6^5=6$. It follows that
\noindent $(\Z^*_7)^5=\{1,2,3,4,5,6\}$ and $5\textrm{-}P_7$ is the complete graph $K_7$.
Consider $m=7$, then $1^7=1$, $2^7=2$, $3^7=3$, $4^7=4$, $5^7=5$, $6^7=6$. It follows that
\noindent $(\Z^*_7)^7=\{1,2,3,4,5,6\}$ and $7\textrm{-}P_7$ is the complete graph $K_7$.
Consider $m=9$, then $1^9=1$, $2^9=1$, $3^9=6$, $4^9=1$, $5^9=6$, $6^9=6$. It follows that
$(\Z^*_7)^9=\{1,6\}$ and
\noindent $E(9\textrm{-}P_7)=\{\{0,1\},\{1,2\},\{2,3\},\{3,4\},\{4,5\},\{5,6\},\{6,0\}\}$.
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\begin{center}
$3\textrm{-}P_7$, $9\textrm{-}P_7$ \quad\quad\quad\quad\quad
\quad\quad\quad\quad\quad $5\textrm{-}P_7$, $7\textrm{-}P_7$\end{center}
\begin{fig}
\begin{center}
\end{center}
\end{fig}
\noindent We can see that if gcd $(m,6=7-1)= 1$ then $m\textrm{-}P_7$ is the complete graph $K_7$, and if gcd $(m,6=7-1)= 3$ then $m\textrm{-}P_7$ is the cycle $C_7$.
\end{section}
\begin{section}{Properties}
We have seen in the previous examples that the $m$-Paley graphs $m\textrm{-}P_q$ are the complete graph $K_q$ in some cases and in other cases are not. The question, which we consider now, is : When are the $m$-Paley graphs complete? The answer to this question can be found in the following theorem.
\begin{theo}
\label{mPaleycom}
Let $m\textrm{-}P_q=(V,E)$ be the $m$-Paley graph of order $q$ and $d=$ gcd $(m,q-1)$, then
\begin{center}
$m\textrm{-}P_q$ is complete if and only if $d=1$.
\end{center}
\end{theo}
\proof
$(\Rightarrow)$ If $m\textrm{-}P_q$ is complete then $d=1$: Assume that $m\textrm{-}P_q$ is complete, then for all $x\neq y \in \F_q$ we have $\{x,y\} \in E(m\textrm{-}P_q)$, it follows $x-y \in ({\F^*_q})^m$.
Clearly $(\F^*_q)^m \subseteq \F^*_q$. If $a \in \F^*_q $ then we can find $x,\ y \in \F_q$ with $a=x-y \in ({\F^*_q})^m$. It follows that $(\F^*_q)^m=\F^*_q$, which means that for all $a \in {\F^*_q}$ there exists $b \in {\F^*_q}$ with $a=b^m$.
\\[10pt]
\indent By Theorem \ref{TH:cyclic} we know that ${\F^*_q}$ is cyclic. Let $g$ be a generator of ${\F^*_q}$, then for all $i \in \{1,2,\dots,q-1\}$ there exists $j \in \{1,2,\dots,q-1\}$ with $g^i=(g^j)^m$ or $g^{i-jm}=1$. Since order $g$ is $q-1$, we have $q-1\mid i-jm$, so $i-mj=(q-1)k$ for some $k$. Thus $i=mj+(q-1)k$.
\\[10pt]
\indent Now to prove that $d=1$, assume the contrary. Let $d > 1$, then $$m=dm_1, \ (q-1)=dk_1{\textrm \ for\ some\ } m_1,k_1.$$
\indent So we can write $i$ in the form $i=dm_1j+dk_1k=d(m_1j+k_1k)$, which means that $i$ must be a multiple of $d$, but $i$ is an arbitrary element of the set $\{1,2,\dots,q-1\}$. Thus we have a contradiction, then $d=1$.
\\[10pt]
\noindent$(\Leftarrow)$ If $d=1$ then $m\textrm{-}P_q$ is complete: Assume that $d=1$ and $m\textrm{-}P_q$ is not complete, then in the set $\{g^m,(g^2)^m,\dots,(g^{q-1})^m=1\}$ there exists two equal elements. Let $(g^i)^m=(g^j)^m$ with $i,j \in \{1,2,\dots,q-1\}, i>j$, then $g^{(i-j)m}=1$, so $q-1\mid(i-j)m$.
\\[10pt]
\indent From the assumption $\gcd (m,q-1)=d=1 \Rightarrow q-1\mid i-j$, which is impossible because $i-j < q-1$. Then $m\textrm{-}P_q$ is complete.\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\noindent Note that the completeness of the $m$-Paley graph means that $(\F^*_q)^m=\F^*_q$. So as an application of Theorem \ref{mPaleycom} we have the following corollary.
\vspace{10pt}
\begin{cor}
In the field $\F_q$, the equation $x^m=a$, $a \in \F_q$ has exactly one solution if and only if $d=$ gcd $(m,q-1)=1$.
\end{cor}
\vspace{10pt}
\noindent Note that $d$ is an odd integer because $m$ is an odd integer. Now the question is: How does the $m$-Paley graphs, if gcd $(m,q-1)=d > 1$, look like? The following proposition has the answer of this question.
\vspace{10pt}
\begin{prop}
\label{mPaleyreg}
Let $m\textrm{-}P_q=(V,E)$ be the $m$-Paley graph of order $q$,
\begin{center}
if $d=$ gcd $(m,q-1) > 1$, then $m\textrm{-}P_q$ is $\frac{q-1}{d}$-regular.
\end{center}
\end{prop}
\proof
Let $x$ be any vertex in $V(m\textrm{-}P_q)$, then $y \in V(m\textrm{-}P_q)$ is adjacent to $x$ if and only if there exists $z \in (\F^*_q)^m$ with $x-y=z$. Which means that $|N(x)|=|(\F^*_q)^m|$ for all $x \in V(m\textrm{-}P_q)$, then $m\textrm{-}P_q$ is $|(\F^*_q)^m|$-regular.
\\[10pt]
\indent Let us now prove that $|(\F^*_q)^m|=\frac{q-1}{d}$. Let $g$ be a generator of the group $\F^*_q$ and $(g^i)^m=(g^j)^m$ for some $i,j \in \{1,2,\dots,q-1\}$, then $g^{(i-j)m}=1$, which implies that $q-1\mid(i-j)m$.
\\[10pt]
\indent Since $d\mid q-1,\ d\mid m$, it follows that $\frac{q-1}{d} \mid \frac{m}{d}(i-j)$. Since gcd $(\frac{q-1}{d},\frac{m}{d})=1$, we have $\frac{q-1}{d} \mid (i-j) \Leftrightarrow i\equiv j$ (mod $\frac{q-1}{d}$). So the following $d$ elements $$g^i,g^{i+\frac{q-1}{d}},g^{i+2\frac{q-1}{d}},\dots,g^{i+(d-1)\frac{q-1}{d}}$$ are the same. Thus $|(\F^*_q)^m|=\frac{q-1}{d}$ and $m\textrm{-}P_q$ is $\frac{q-1}{d}$-regular.\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\begin{cor}
In the field $\F_q$, if $d=$ gcd $(m,q-1) > 1$, then the equation $x^m=a$, $a \in \F^*_q$ has exactly $d$ solutions.
\end{cor}
\noindent As a special case, if $d=m$ and $a=1$ we get Lagrange's lemma.
\vspace{10pt}
We can see that the $m$-Paley graphs are not strongly regular in general. In example $3$ we have seen that $3\textrm{-}P_7$, $9\textrm{-}P_7$ are $C_7$ which is not strongly regular. So the question now is: Are the $m$-Paley graphs symmetric or self-complementary?
\vspace{10pt}
\begin{prop}
The $m$-Paley graphs are symmetric.
\end{prop}
\proof
By the same proof of Proposition \ref{Paley Symm} with replacing $(\F^*_q)^2$ by $(\F^*_q)^m$ we get the result.\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\begin{prop}
The $m$-Paley graphs are not self-complementary.
\end{prop}
\proof
Clearly, a self-complementary graph of order $q$ should have $\frac{q(q-1)}{4}$ edges. Let $m\textrm{-}P_q=(V,E)$ be the $m$-Paley graph of order $q$.
If $d=$gcd$(m,q-1) > 1$, then by Proposition \ref{mPaleyreg}
\begin{displaymath}
|E(m\textrm{-}P_q)|=\frac{1}{2} \sum_{x\in V(m\textrm{-}P_q)} d(x) =\frac{1}{2}q\frac{q-1}{d}.
\end{displaymath}
Since $d\geqslant 3$, it follows that $|E(m\textrm{-}P_q)|<\frac{q(q-1)}{4}$.
If $d=$ gcd $(m,q-1)=1$, then by Proposition \ref{mPaleycom}
$$|E(m\textrm{-}P_q)|=q\frac{q-1}{2}>\frac{q(q-1)}{4}.$$
Thus the $m$-Paley graphs are not self-complementary.\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\noindent Now let us ask the following question: Are the $m$-Paley graphs connected?
\noindent Clearly, the $m$-Paley graph of order $q$ with $d= \gcd (m,q-1) = 1$, which is the complete graph $K_q$, is connected. So the case which we will study is the $m$-Paley graph of order $q$ with $d=\gcd (m,q-1) > 1$.
\\[10pt]
\noindent Note that $d \neq q-1$ because $q-1$ is even and $d$ must be odd. So
\hspace{100pt} $\frac{q-1}{2} \geq d=$gcd$(m,q-1) \geq 1$.
\vspace{10pt}
\begin{prop}
\label{Prop:prim}
Let $m\textrm{-}P_q=(V,E)$ be the $m$-Paley graph of order $q$,
if $d=$ \normalfont{gcd} $(m,q-1) > 1$ and $q$ is prime, then $m\textrm{-}P_q$ is connected.
\end{prop}
\proof
Since $q$ is prime, $\F_q = \Z_q$. For every $x < y \in \F_q$, we have the sequence $x,x+1,x+2,\dots,x+(y-x-1)=y-1,x+(y-x)=y \in \F_q$.
So $\{x,x+1\},\{x+1,x+2\},\dots,\{y-1,y\} \in E(m\textrm{-}P_q)$, because $1\in (\F^*_q)^m$ for each odd integer $m$. Thus $x\{x,x+1\}x+1\{x+1,x+2\}x+2\cdots y-1\{y-1,y\}y$ is a path in $m\textrm{-}P_q$ between $x$ and $y$. So $m\textrm{-}P_q$ is connected.
\hspace{\stretch{1}} $\Box$
\vspace{10pt}
\noindent Note that as a special case of Proposition \ref{Prop:prim}, if $d=\gcd(m,q-1)= \frac{q-1}{2}$ and $q$ is prime, then $m\textrm{-}P_q$ is the cycle $C_q$, because $m\textrm{-}P_q$ is connected and $2$-regular.
\begin{prop}
Let $m\textrm{-}P_q=(V,E)$ be the $m$-Paley graph of order $q$,
if $d=\gcd(m,q-1)= \frac{q-1}{2}$ and $q$ is not prime, then $m\textrm{-}P_q$ is disconnected.
\end{prop}
\proof
Let $q=p^n, n > 1$, then $\F_q = \Z_p[x] /(f(x))$ where $f(x)$ is an irreducible polynomial of degree $n$ over $\Z_p$.
\\[10pt]
\indent Since $1 \in (\F^*_q)^m$, the path $0\{0,1\}1\{1,2\}2\cdots p-1\{p-1,0\}0\ $ in $m\textrm{-}P_q$ form a cycle, say $C_p$. By using Proposition \ref{mPaleyreg} with $d=$ gcd $(m,q-1)= \frac{q-1}{2}$, it follows that $m\textrm{-}P_q$ is $2$-regular.
\\[10pt]
\indent Since $n > 1$, it follows that the set $\F_q \setminus\Z_p$ is not empty. Let $a \in \F_q \setminus\Z_p$, then $a$ is not adjacent to any vertex in $C_p$. So we cannot find a path in $m\textrm{-}P_q$ between any vertex in $\F_q \setminus\Z_p$ and any vertex in $C_p$.
Thus $m\textrm{-}P_q$ is disconnected.\hspace{\stretch{1}} $\Box$
\\[20pt]
{\it Example:} Take $q=27, \ m=13$, then $d=\gcd(13,27-1)=13=\frac{27-1}{2}$. The following figure shows that the $13\textrm{-}P_{27}$ is disconnected
\begin{displaymath}
\def\objectstyle{\scriptscriptstyle}
\xy /r6pc/:, {\xypolygon27"B"{~>{}\bullet}}, "B1"!{+U*+!L{\scriptstyle 0}}
,"B2"!{+RD*+!L{\scriptstyle 1}}
,"B3"!{+RD*+!L{\scriptstyle 2}}, "B4"!{+LDD*+!L{\scriptstyle a}}
,"B5"!{+RD*+!LD{\scriptstyle a+1}}, "B6"!{+LDD*+!LD{\scriptstyle a+2}}
,"B7"!{+RD*++!LD{\scriptstyle a^2}},
"B8"!{+RD*++!D{\scriptstyle a^2+1}}, "B9"!{+LDD*+!RD{\scriptstyle a^2+2}},
"B10"!{+RD*++!R{\scriptstyle 2a}}, "B11"!{+LDD*+!R{\scriptstyle 2a+1}},
"B12"!{+RD*+!R{\scriptstyle 2a+2}}, "B13"!{+LDD*+!R{\scriptstyle 2a^2}},
"B14"!{+RD*+!R{\scriptstyle 2a^2+1}}, "B15"!{+LDD*+!R{\scriptstyle 2a^2+2}},
"B16"!{+RD*+!R{\scriptstyle a^2+a}}, "B17"!{+LDD*+!R{\scriptstyle a^2+a+1}},
"B18"!{+RD*+!R{\scriptstyle a^2+a+2}}, "B19"!{+LDD*+!R{\scriptstyle a^2+2a}},
"B20"!{+RD*!RU{\scriptstyle a^2+2a+1}}, "B21"!{+LDD*++!U{\scriptstyle a^2+2a+2}},
"B22"!{+RD*!LU{\scriptstyle 2a^2+a}}, "B23"!{+LDD*++!L{\scriptstyle 2a^2+a+1}},
"B24"!{+RD*+!L{\scriptstyle 2a^2+a+2}}, "B25"!{+LDD*+!L{\scriptstyle 2a^2+2a}},
"B26"!{+RD*+!L{\scriptstyle 2a^2+2a+1}}, "B27"!{+LDD*+!L{\scriptstyle 2a^2+2a+2}},
"B3";"B1"**@{-},"B3";"B2"**@{-},"B1";"B2"**@{-},"B5";"B4"**@{-},"B6";"B4"**@{-},"B5";"B6"**@{-},
"B7";"B8"**@{-},"B7";"B9"**@{-},"B8";"B9"**@{-},"B10";"B11"**@{-},"B10";"B12"**@{-},"B11";"B12"**@{-},
"B13";"B14"**@{-},"B13";"B15"**@{-},"B15";"B14"**@{-},"B16";"B17"**@{-},"B16";"B18"**@{-},"B17";"B18"**@{-},
"B19";"B20"**@{-},"B19";"B21"**@{-},"B20";"B21"**@{-},"B22";"B23"**@{-},"B23";"B24"**@{-},"B22";"B24"**@{-},
"B25";"B26"**@{-},"B25";"B27"**@{-},"B26";"B27"**@{-},
\endxy
\end{displaymath}
\begin{fig}
\begin{center}
The $13\textrm{-}P_{27}$ graph, where $a$ is a root of an irreducible polynomial $f(x)$ of degree $3$ over $\Z_3$
\end{center}
\end{fig}
\noindent So now we know that the $m$-Paley graphs are not always connected and the case : $q= p^n,\ n > 1 $ and $ 1 < d=\gcd (m,q-1) < \frac{q-1}{2}$ is still open.
\end{section}
\newpage
|
{
"timestamp": "2012-03-09T02:03:12",
"yymm": "1203",
"arxiv_id": "1203.1818",
"language": "en",
"url": "https://arxiv.org/abs/1203.1818"
}
|
\section{Supplementary Material}
A detailed account of the extended Brandt-Pesch-Tewordt (BPT) quasiclassical approximation in the vortex phase has been given in a series of papers \cite{AHoughton1998,Anton,AntonI,AntonII,Anton2010}. In this supplementary material we summarize the salient ingredients of this approximation and the numerical details of our quantitative studies.
\begin{figure*}[top]
\rotatebox[origin=c]{0}{\includegraphics[width=2.\columnwidth]{supplementary_fig1.pdf}}
\caption{FIG.~S1. (Color online) Order parameters and phase diagrams for the model pairing states in $A_y$Fe$_{2-x}$Se$_2$. Panels (a)-(e): FS maps of the order parameter values for various pairing symmetries as indicated in each Panel. As shown in the color scale on the left, red means negative order parameter (OP), white denotes nodal regions, and blue indicates positive OP. The dotted lines depict the nodal line. Panels (f)-(g): Phase diagram of the fourfold oscillations for $s^{\pm}$-pairing extracted from heat-capacity, $C_{4\alpha}$, and thermal conductivity, $\kappa_{4\alpha}$, respectively.} \label{fig4}
\end{figure*}
{\bf Fermi surface parameterization.$-$}
In order to calculate thermal properties in rotating fields, we use as input Fermi surfaces (FSs) and dispersions of $A_y$Fe$_{2-x}$Se$_2$ derived from first-principles electronic structure calculations to obtain an effective tight-binding model in the 2-Fe unit-cell notation given in Refs.~\cite{Das_FeSe,Dasmodulated}. In accord with experimental observations, and essential for self-consistently calculated Green's functions for in-plane field rotation, a weak $k_z$ dispersion is added with hopping parameter $t_z\sim0.3t$,
where $t$ is the in-plane nearest neighbor hopping. Finally, a FS parameterization is obtained from the effect tight-binding model.
For each FS integration, we used 800 points along the Fermi line in the $(k_x,k_y)$ plane and 9 $k_z$ slices in the Brillouin zone. We confirmed numerical convergence by checking that the results do not change when we increase the number of $k$ points; this test was done for several points in the $T$-$H$ plane.
{\bf Multiband gap symmetries.$-$}
The superconducting (SC) pairing functions of all order parameters (OPs) considered for $A_y$Fe$_{2-x}$Se$_2$ are given in the main text. Fig.~S1 shows the two-dimensional view of of the corresponding OP on each Fermi surface sheet. The conventional $s$-wave gap is isotropic at all momenta as shown in Fig.~S1(a). The extended $\tilde{d}_{xy}$ order parameter is nearly isotropic on each of the FSs, but changes sign between FS sheets located at different M and equivalent points in the Brillouin zone (BZ), see Fig.~S1(b). This is one of the candidate order parameters for this system, which gives rise to a spin resonance \cite{Das_FeSe,Dasmodulated}. The $s^{\pm}$ gap has a nodal line at the mid-point between the $\Gamma$ to M direction, but that nodal line does not touch either FS. Importantly, due to the absence of a hole pocket at the $\Gamma$ point, there is no sign change in the SC gap in $A_y$Fe$_{2-x}$Se$_2$ for this pairing. The nodal line passes through the zone boundary and zone diagonal for $d_{xy}$ and $d_{x^2-y^2}$-pairings as presented by dashed lines in Figs.~S1(d) and S1(e).
{\bf Computational details.$-$}
For each pairing symmetry, the coupled order parameters are computed self-consistently at each given magnetic field (${\bm H}$ applied at angle $\alpha$ to the (100) direction, and temperature ($T$). To plot the phase diagram in Fig.~3 in the main text, we took a mesh of 35 field points between zero and $H_{c2}$, 100 temperature points from zero to $T_{c0}$, and used 31 $\alpha$ points from zero to $90^o$ to extract the anisotropic terms in the heat capacity and the thermal conductivity. The phase diagram for $s^{\pm}$ pairing is very similar to that of the nodeless $s$ and $\tilde{d}_{xy}$ gaps, see Fig.~S1(f)-S1(g), and thus not included in the main text. For all calculations, we considered purely intraband impurity scattering in the clean limit, $1/\tau_{imp}=0.01 \times 2\pi T_{c0}$, where $T_{c0}$ is the bare transition temperature and the scattering phase shift is chosen to be $\delta=\pi/2$ (unitarity limit).
{\bf Methods and formulas.$-$}
If the Fermi velocity in band $n$ is denoted by $\bm{v}_n({\bm k_f})$, the corresponding normal-state DOS at the Fermi level is $N_{fn}({\bm k_f})\sim1/|\bm{v}_n({\bm k_f})|$. The wavevector ${\bm k_f}$ lies on the respective FS.
When a magnetic field ${\bm H}$ is applied along an angle $\alpha$ with respect to the $x$-axis, the relevant parameter in the SC state is the component of the Fermi velocity normal to the applied field, which, in energy units, becomes $\bm{\bar{v}}_{fn}(\phi,{\bm H})\equiv {\bm v}_n^\perp({\bm k_f})/2\Lambda$,
where $\Lambda=(\hbar c/2|e|H)^{1/2}$ is the magnetic length of order of the intervortex distance, $\phi$ is the FS angle with respect to the $k_x$ axis,
and $\bm{v}_n^\perp({\bm k_f})$ is the rescaled component of the Fermi velocity perpendicular to ${\bm H}$. The unit-cell averaged Green's function at the fermionic Matsubara frequency, $\omega_\nu$,
follows the notation of Eqs.~(46)-(48) in Ref.~\onlinecite{AntonI}:
\begin{equation}
g_n(i\widetilde\omega_{\nu}, {\bm k_f}; {\bm H}) =
\frac{-i\pi}{
\sqrt{ 1-\left(\frac{\widetilde\Delta_n({\bm k_f}; {\bm H})}{i\widetilde\omega_\nu-\Sigma_n(i\widetilde\omega_\nu,{\bm k_f}; {\bm H})}\right)^2}}.
\end{equation}
Here $\widetilde\Delta_n$ and $\widetilde\omega_{\nu}$ are the order parameter and Matsubara frequency renormalized by the impurity self-energy, which is evaluated in the $T$-matrix approximation.
The self-energy $\Sigma_n$ is
given by $(i\widetilde\omega_\nu-\Sigma_n)^{-2}=i\sqrt{\pi} W_n^{\prime}(i\widetilde\omega/\bar{v}_{fn})$.
$W^{\prime}(z)$ is the first derivative of the complex-valued function $W(z)=\exp{(-z^2)}{\rm erfc}(-iz)$. In contrast to the Doppler shift approximation, both the real and the imaginary parts of $\Sigma_n$ contribute to the SC DOS, and their interplay as a function of energy, $H$ and $T$,
determine the sign reversal in the fourfold oscillation of the SC DOS.
These effects have been extensively studied earlier using a single quasi-cylindrical FS and nodal gap, and a minimal 2D model for two-band systems, see for example Refs.~\onlinecite{Anton,AntonI,AntonII,Anton2010}.
The order parameters are calculated self-consistently from the coupled gap equations of the two-band model
\begin{eqnarray}
&&\Delta_n({\bm k_f};{\bm H}) = \nonumber\\
&&
T \sum_{\omega_\nu} \sum_{n'}
\Big\langle V_{n n'}({\bm k_f}, {\bm k_f}')
N_{fn'}({\bm k_f}') f_{n'}(i\omega_\nu, {\bm k_f}'; {\bm H}) \Big\rangle_{FS} .
\end{eqnarray}
Here $f_{n'}$ is the anomalous Gorkov function (off-diagonal Green's function).
We used a factorized pairing potential at the Fermi surface as $V_{n n'}({\bm k_f}, {\bm k_f}') = V_{n n'} {\cal Y}_n(\phi) {\cal Y}_{n'}(\phi')$, with ${\cal Y}_n(\phi)$ the azimuthal angle dependence of the SC gap, see Fig.~1 in main text. For simplicity, we consider purely interband pairing $V_{12}=V_{21}=-V$ and eliminate $V$ in favor of the bare transition temperature $T_{c0}$ using weak-coupling theory \cite{Anton2010}.
The specific heat, $C=C_1+C_2$, and thermal conductivity, $\kappa=\kappa_1+\kappa_2$, are calculated
from the approximate expressions
\begin{eqnarray}
&&C_n(\alpha) \! \approx \!\!
\int_{-\infty}^\infty \!
d\omega \frac{\omega^2 \langle N_n(\omega, {\bm k_f}; {\bm H}) \rangle_{FS}}{4 T^2 {\rm cosh}(\omega/2T)^2} ,
\label{eq:C}
\\
&&\kappa_{n}^{xx}(\alpha) \! \approx \!\!
\int_{-\infty}^\infty \!\!
d\omega \frac{ \omega^2 \langle v_{n}^x({\bm k_f})^2 N_n(\omega, {\bm k_f}; {\bm H})\tau_{n}(\omega,{\bm k_f}; {\bm H}) \rangle_{FS} }{2 T^2 {\rm cosh}(\omega/2T)^2},
\nonumber\\
\end{eqnarray}
We note that close to the transition the full expression for the heat capacity computed from the entropy includes the temperature derivative of the gap functions $\Delta_n$. However, inclusion of these terms only minimally affects quantitative aspects of the results away from the transition.
The field-induced SC DOS in each band, $N_n/N_{nf}=-{\rm Im}\ g_n/\pi$, is calculated using analytical continuation $i\omega_\nu\rightarrow\omega+i\delta$ to obtain the retarded Green's functions, and
the {\it transport} lifetime due to both impurity and vortex scattering \cite{Vekhter99,AntonII,Anton2010}:
\begin{widetext}
\begin{eqnarray}
\frac{1}{2\tau_{{n}} (\omega,{\bm k_f}; {\bm H})} =
- {\rm Im}\, \Sigma_n(\omega,{\bm k_f};{\bm H}) +
\sqrt{\pi}{1 \over |{\bar{\bm v}}_{fn}({\bm k_f}; {\bm H})|}
\frac{{\rm Im}\, [g_n(\omega,{\bm k_f};{\bm H}) \, W(\tilde{\omega}/|{\bar{\bm v}}_{fn}({\bm k_f}; {\bm H})|)]}
{{\rm Im} \, g_n(\omega,{\bm k_f};{\bm H})} |\Delta_n({\bm k_f}; {\bm H})|^2
\,.
\label{eq:tauH}
\end{eqnarray}
\end{widetext}
Since the function $x^2/{\rm cosh}(x/2)^2$ has a peak at $x\sim 2.5T$, the heat capacity at low temperatures predominantly probes the anisotropy in the total SC DOS, $N(\omega=2.5T,{\bm k_f}; {\bm H})$. Using the expansion of the error function, we obtain two limiting values for $W^{\prime}(z)$: $W^{\prime}(0)=2i/\sqrt{\pi}$ and $W^{\prime}(z\gg 1)\approx-i/\sqrt{\pi}z^2$. Thus the SC DOS for each band $n$ becomes
\begin{widetext}
\begin{eqnarray}
N_n(\omega; {\bm H}) = \langle N_n(\omega,{\bm k_f};{\bm H})\rangle_{FS}
\approx
\begin{cases}
\left\langle N_{fn}({\bm k_f})\left[1+\left(\frac{\widetilde\Delta_n({\bm k_f}; {\bm H})}{|\bar{\bm v}_{fn}({\bm k_f}; {\bm H})|}\right)^2\right]^{-1/2}\right\rangle_{FS} ,
&\omega\ll \bar{\bm v}_{fn} , \cr\nonumber
\left\langle N_{fn}({\bm k_f})\left[1- \left(\frac{\widetilde\Delta_n({\bm k_f}; {\bm H})}{\widetilde\omega}\right)^2\right]^{-1/2}\right\rangle_{FS} ,
&\omega\gg \bar{ \bm v}_{fn} ,\cr
\end{cases}\\
\label{Eq:N}
\end{eqnarray}
\end{widetext}
where $\widetilde\Delta_n$ and $\widetilde\omega$ are the impurity renormalized order parameter and quasiparticle energy, respectively.
The first line above only makes physical sense when the BPT approximation is valid at low energies, i.e., for nodal gaps. In that case at low $T$ (low energy) and at low fields, where $\Delta_n({\bm k_f}; {\bm H})$ only weakly depends on the direction of the field, the SC DOS depends predominantly on the orientation of
$|{\bar{ \bm v }}_{fn}({\bm k_f};{\bm H})|$ relative to the minima of $\Delta_n({\bm k_f}; {\bm H})$. At $\omega=0$ the inversion of the SC DOS as a function of the field for nodal gaps can be obtained in analogy to Refs.~\cite{UdagawaI,AntonI}.
At higher energies, the second line of Eq.~\eqref{Eq:N} gives the BCS result for the SC DOS and therefore field-angle variation enters via the anisotropy of the upper critical field that influences $\Delta_n({\bm k_f}; {\bm H})$ in the vicinity of the transition. This result is valid for both nodal and nodeless gaps, including the fully isotropic situation. Crucially, for anisotropic Fermi surfaces the $H_{c2}$ anisotropy in the order parameter is weighted by the normal-state DOS, $N_{fn}({\bm k_f})$, leading to a complex behavior including the switching of the minima and maxima found in our Letter. In this regime, however, the energy width of the Fermi weighting factor in the integral exceeds the gap amplitude and a full numerical evaluation is required. The results of such a self-consistent analysis are presented in the main text. All our results are consistent with the general observations based on this expansion.
|
{
"timestamp": "2012-10-04T02:06:48",
"yymm": "1203",
"arxiv_id": "1203.2211",
"language": "en",
"url": "https://arxiv.org/abs/1203.2211"
}
|
\section{Introduction}
The standard classification scheme for the explosions of massive stars
consists of two different branches: stars retaining their hydrogen
envelope at the time of the explosion produce Type II supernovae
(SNe~II), whilst those losing it before the explosion produce Type Ib
or Ic SNe, depending on the strength of He lines in the spectra. In
addition, some SNe~Ic, labelled BL-Ic, show broad lines in the
spectra, signature of a very high kinetic energy/ejected mass
ratio. Over the past 5 years, the above scenario has changed
dramatically with the discovery of new classes of transients that may
originate from different explosion channels. Pair-instability,
pulsational pair-instability and magnetar-powered explosions have been
suggested to explain the properties of a group of hyper-luminous and
slowly evolving transients (SN~2007bi, \citealt{2009Natur.462..624G};
SN~2006gy, \citealt{2007ApJ...666.1116S}; SN~2010gx,
\citealt{2010ApJ...724L..16P}, \citealt{2011Natur.474..487Q}), while
fall-back on the collapsed remnant, electron-capture and
failed-deflagration scenarios have been proposed to explain faint and
fast-evolving transients (e.g. SN~2008ha,
\citealt{2009Natur.459..674V}, \citealt{2009AJ....138..376F};
SN~2005E, \citealt{2010Natur.465..322P}). Alternatively, some of these
events can still be explained if a more \emph{canonical} core-collapse
scenario but an extended range of physical conditions of the
progenitor at the moment of its explosion is invoked (e.g. SN~2006gy,
\citealt{2009ApJ...691.1348A}; SN~2007bi,
\citealt{2010A&A...512A..70Y,2010ApJ...717L..83M}; SN~2005cz,
\citealt{2010Natur.465..326K}). Here we present the observations of a
stripped-envelope supernova that evolved in an extremely slow fashion,
much more slowly than any other \emph{spectroscopically normal}
core-collapse SN studied in the literature.
\section{Observations}
SN~2011bm was discovered by the ``La Sagra Sky Survey'' on 2011 April
5 in the galaxy IC 3918\footnote{The following parameters have been
used for IC 3918 in this paper: $\mu$=34.90 mag (radial velocity
corrected for in-fall onto Virgo of 2799 $km$ $s^{− 1}$ and a Hubble
constant of 72 $km$ $s^{- 1}$ $Mpc^{- 1}$.), $z$=0.0221 (from narrow
emission lines of the host galaxy), $E(B-V)_\mathrm{IC~3918}$=0.032
mag (assuming a similar dust-to-gas ratio in IC~3918 and in the
Milky Way and comparing the equivalent width of the narrow Na\,{\sc i}~D\/
absorption lines from IC~3918 with that from the Milky Way),
$E(B-V)_\mathrm{Gal}$=0.032 mag
\protect\citep{1998ApJ...500..525S}.} \citep{2011CBET.2695....1R}
and classified on April 11 at the 1.82-m Mt.\,Ekar Copernico Telescope
as a type Ic SN close to maximum light
\citep{2011CBET.2695....2R}. The Palomar Transient Factory (PTF)
claimed an independent discovery of SN~2011bm on 2011 March 29 and a
stringent pre-discovery limit on March 23 down to 20.8 mag in the R
band \citep{2011ATel.3288....1G}. Since the Panoramic Survey Telescope
and Rapid Response System-1 (PS1) 3$\pi$ survey has already been very
useful to constrain the explosion epoch of other nearby SNe
\citep[e.g. SN~2010ay, ][]{2011arXiv1110.2363S}, we retrospectively
inspected PS1 data and detected SN 2011bm in $r_{P1}$- and
$i_{P1}$-band images on March 26.5 UT at magnitudes $r$=18.71 mag and
$i$=18.82 mag, and in a $g_{P1}$-band image on March 29.6 UT at a
magnitude $g$=18.78 mag. This is the earliest detection of the
supernova, and strongly constrains the explosion to have occurred
between 2011 March 23 and March 26. After announcement, we
immediately started a follow-up campaign in the framework of the NTT
European Large Programme (ELP) collaboration and we extensively
monitored SN~2011bm using the telescopes available to us. We collected
a large amount of data in the optical domain, complemented by
near-infrared data, especially useful to investigate the presence of
He in the SN ejecta (cf. Tables~\ref{tab1} and \ref{tab2}). A sub-set
of the spectroscopic and photometric data collected by the NTT ELP
collaboration are shown in Figs.~\ref{fig1}a and \ref{fig1}b,
respectively. In Fig.~\ref{fig1}c the expansion velocity as derived
from the position of the minima of Fe\,{\sc ii}{} lines\footnote{Average of
the velocities obtained for the lines at $\lambda$4924,
$\lambda$5018 and $\lambda$5169.} is shown, and compared with those
of normal SNe~Ib/c. SN~2011bm shows typical velocities, but a slow
velocity evolution similar to those of SNe 2009jf or 2007gr.
\begin{figure*}
\plotone{f1.eps}
\caption{Spectroscopic and photometric follow-up of SN~2011bm. All
data have been reduced with standard IRAF routines, using the QUBA
pipeline (see \citealt{2011MNRAS.416.3138V} for information). For
the NOT spectra, a second order correction was performed using the
method described in \protect\citet{2007AN....328..948S}.
\textbf{a}: Sub-sample of our spectra in the host-galaxy frame. The
dashed line marks the position of the narrow absorption line at
$\sim$\,6390\,\AA{} visible in our earliest four spectra.
\textbf{b}: Multi-band light curves of SN~2011bm. \textbf{c}:
Photospheric velocities (measured from Fe\,{\sc ii}\ $\lambda$4924,
$\lambda$5018 and $\lambda$5169) for a sample of stripped-envelope
core-collapse SNe. Data sources: SN~2007gr,
\citet{2009A&A...508..371H}; SN~2008D, \citet{2008Sci...321.1185M};
SN~2007Y, \citet{2009ApJ...696..713S}; SN~2008ax,
\citet{2011MNRAS.413.2140T}; SN~2009jf, \citet{2011MNRAS.416.3138V};
SN~1999ex, \citet{2002AJ....124.2100S}.}
\label{fig1}
\end{figure*}
The classification spectrum taken on April 11 (18 days after the
explosion) is rather blue ($B-V$ $\sim$ 0.35 mag). Such a blue color
is unusual for SN~Ic spectra at this epoch. Early spectra of SN~2011bm
show the classical type Ib/c SN features: Fe\,{\sc ii}, w, Ca\,{\sc ii}{} and a
faint P-Cygni absorption at $\sim$\,6100 \AA, usually identified as
Si\,{\sc ii}. A narrow line is visible at $\sim$\,6530\,\AA{}
($\sim$\,6390\,\AA{} in the host galaxy rest frame) in the first four
spectra (see Fig.~\ref{fig1}a; marked with a vertical dotted line). A
similar feature was also observed in very early spectra of SN~2007gr
up to few days before maximum, and was identified as \CII{}
$\lambda$6580, with H$\alpha${} or He\,{\sc i}{} $\lambda$6678 as possible
alternatives. Using the spectrum synthesis code
SYNOW\footnote{http:/$\!$/www.nhn.ou.edu/\~{}parrent/synow.html}, we
confirm that \CII{} is a consistent identification for this
feature. This supports the idea that our 18-day spectrum of SN~2011bm
is similar to those of younger, canonical SNe Ic. Prominent features
of He\,{\sc i}{} or H are not detected in the optical domain, and there is no
evidence for the He\,{\sc i}{} line at $\sim$ 2 micron in our near-infrared
SOFI spectrum obtained on 2011 May 9, consistent with the
classification as a SN Ic.
The best match for the spectra of SN 2011bm is found with those of
SN~2007gr \citep{2009A&A...508..371H}, persisting along the whole
evolution (Fig.~\ref{fig2}a and \ref{fig2}b). What is different,
however, is the time scale of the transition from optically thick to
optically thin ejecta. At 18 days past core-collapse the spectrum of
SN~2011bm is still blue (like those of normal type Ic SNe a few days
after the explosion). Three months after the explosion SN~2011bm is
still optically thick, and the best match is found with spectra of
SN~2007gr at a phase of only 1 month. Later on, at 305 days the
spectrum of SN~2011bm shows nebular lines, though the detection of
[w] $\lambda$5577 is unexpected, as this line usually disappears in
the spectra of SNe~Ic by $\sim$\,150-–200 days after core-collapse.
The spectrum of SN 2007gr at 172 days shows a stronger Mg\,{\sc i}]
$\lambda$4571 line than that of SN 2011bm. This may be due to a
lower abundance of magnesium in SN 2011bm than in SN~2007gr or to a
different ionisation/excitation state of the ejecta in the two SNe
or simply to the fact that the O-Ne-Mg layer is still not completely
exposed, while we are still seeing the C/O-rich layer. In the latter
case, the Mg\,{\sc i}] $\lambda$4571 line is expected to become more
prominent with time.
\begin{figure*}
\hspace{0.5in}
\epsscale{1}
\plotone{f2.eps}
\caption{\textbf{a}: Spectral evolution of SN~2011bm, and comparison
with spectra of SN~2007gr. The spectra of SN~2007gr have been scaled
to match those of SN~2011bm. \textbf{b}: Infrared spectra of
SN~2011bm compared with spectra of SN~2007gr.}
\label{fig2}
\end{figure*}
Even more intriguing is the luminosity evolution of SN~2011bm. After
the earliest detection by PS1, the light curve rises for several
weeks, reaching its maximum in the R-band on 2011 May 2 (JD =
2455684.3), 40 days after the explosion (see Fig. \ref{fig1}b). No
other \emph{normal} SN Ib/c has shown such a slowly rising light
curve. The post-maximum evolution is qualitatively similar to those of
other SNe~Ic, with a more rapid initial luminosity drop followed by a
slower decline. However, for SN~2011bm the inflection points occurs
much later than in other SNe Ic. Comparing the light curves of
SN~2011bm with those of SNe~1998bw \citep[R band,
][]{2001ApJ...555..900P} and 1997ef \citep[V band,
][]{2000ApJ...534..660I}, which are among the most slowly evolving
BL-Ic SNe available in the literature, we find that SN~2011bm is
evolving $\sim$\,1.5 times more slowly than SN~1997ef and $\sim$\,2.2
times more slowly than SN~1998bw\footnote{The time evolution is
computed stretching the light curves around maximum until the light
curves shapes match.}. SN 2011bm also evolves 2.7 times more slowly
that SN~2007gr \citep{2009A&A...508..371H} in the R band, in good
agreement with the different time scales of the spectral evolution
highlighted in Fig.~\ref{fig2}.
\section{Physical Parameters}
A direct estimate of the host-galaxy metallicity can be obtained from
the fluxes of nebular emission lines in the vicinity of the SN
location. We measured the following narrow lines of the host-galaxy
detected in our WHT spectrum of 2011 December 19: [O\,{\sc ii}]
$\lambda\lambda$3726,3729, H$\beta${}, [O\,{\sc iii}] $\lambda$4959,
[O\,{\sc iii}] $\lambda$5007, H$\alpha${}, [N\,{\sc ii}] $\lambda$6583.
Using different metallicity indicators, we obtain the following
values: the N2 index calibration of \citet{2004MNRAS.348L..59P} gives
12+log(O/H) = 8.41 dex, while the O3N2 index provided by the same
authors gives a value of 8.31 dex. The method adopted by
\citet{2004ApJ...617..240K} (see their eq.~18) yields 12+log(O/H) =
8.58 dex\footnote{With the caveat that this value may be slightly
dependent on the used value for the galactic extinction.}. These
values are only marginally sub-solar. From the H$\alpha${} emission we also
estimate a star formation rate of $\sim$ 0.06 M$_{\odot}${} $yr^{-1}$ that is
typical of H\,{\sc ii}{} regions and similar to that of other stripped
envelope SNe \citep{2008MNRAS.383.1485V}. Our observations of
SN~2011bm can be used to understand the physical conditions of the
progenitor at the moment of the explosion. Important insights come
from the nebular spectra, the shape of the light curve and the
bolometric luminosity. The spectrum obtained on 2011 December 18
($\sim$\,270 days after the explosion) provides additional information
on the progenitor's properties. The presence of the [w]
$\lambda$5577 feature is remarkable: this line is usually visible only
up to $\sim$\,150--200 days after the explosion, because it requires a
relatively high electron density. Using eq.~2 of
\citet{1996ApJ...456..811H} and a temperature of 4000--4500
K\footnote{This excitation temperature for SNe Ib/c at late phases is
obtained from specific models of nebular spectra
\protect\citep{2001ApJ...559.1047M,2007ApJ...670..592M,2010MNRAS.408...87M}.},
the flux ratio [w] $\lambda$6300\,/$\lambda$5577 of about 20--40 ,
as measured from our TNG spectrum, indicates an electron density of
$\geq$ $10^8$ cm$^{-3}$. The flux of the [w]
$\lambda\lambda6300,6364$ feature measured at 270 days ($3.7 \times
10^{-14}$ erg cm$^{-2}$ s$^{-1}$) can also be used to roughly estimate
the ejected oxygen mass. Using the equation from
\citet{1986ApJ...310L..35U} that is appropriate for our
electron-density and temperature regimes, we obtain an oxygen mass of
$M_\mathrm{O}$\,$\sim$\,5--10 M$_{\odot}$. As a comparison, the oxygen masses
obtained with the same method for SNe 1990I \citep[0.7--1.35
M$_{\odot}$,][]{2004A&A...426..963E} and 2009jf \citep[1.34
M$_{\odot}$,][]{2011MNRAS.413.2583S} are significantly smaller. Also the
oxygen mass of SN~1998bw \citep[$\sim$\,3
M$_{\odot}$,][]{2001ApJ...559.1047M}, computed via spectral modelling, is a
factor of 2 smaller than that of SN~2011bm. Such a large oxygen mass
in the ejecta is expected when the progenitor is either a very massive
star ($\geq$ 30 M$_{\odot}${}) or a lower-mass star formed in a very low
metallicity environment
\citep{1996ApJ...460..408T,2003ApJ...592..404L,2005A&A...433.1013H,2006NuPhA.777..424N}.
Given the metallicity measurement reported above, it is likely that
the progenitor of SN~2011bm must have been very massive. Since most
explosion models of massive stars predict that the oxygen mass
represents 60 to 70$\%$ of the total ejected mass \citep[see
e.g.][]{2003ApJ...592..404L,2006NuPhA.777..424N}, we can estimate
the total ejected mass to be in the range 7--17 M$_{\odot}${}.
In Fig.~\ref{fig3}a we show the quasi-bolometric light curve of
SN~2011bm obtained by integrating our multi-band (UBVRI)
observations. We also show the {\it uvoir} bolometric luminosity
obtained for the few epochs in which JHK photometry of SN~2011bm is
available (labelled as $uvoir$ in Fig. \ref{fig3}a) and a computed
{\it uvoir} bolometric light curve using the fractional NIR
contribution of SN~2009jf \citep{2011MNRAS.416.3138V}. The late-time
tail of this light curve is flatter than those of other
stripped-envelope SNe, but still steeper than of H-rich core-collapse
SNe powered by $^{56}$Co decay. This is a clear indication that the
$\gamma$-rays produced in the $^{56}$Co decay are not fully
trapped. However, a larger trapping fraction than in other type Ic SNe
confirms that the ejecta of SN~2011bm are quite massive. We used the
toy model described in \cite{2008MNRAS.383.1485V}\footnote{The toy
model is based on the Arnett's rule
\protect\citep{1982ApJ...253..785A} for the photospheric phase,
equations 1 and 2 from \protect\cite{1997A&A...328..203C} and a
simple $\gamma$-ray deposition function
\protect\citep{1997ApJ...491..375C}.} to independently estimate the
physical conditions at the moment of the explosion. The $uvoir$
bolometric light curve can be reproduced with an explosion of a
massive star releasing 7-16 foe of kinetic energy in ejecta of 7-10
M$_{\odot}${}, 0.6-0.7 M$_{\odot}${} of which is synthesized $^{56}$Ni. These values
are in agreement with those given above. The Nickel mass is
surprisingly high, if we consider that SN 2011bm was less luminous
than SN 1998bw at maximum.
\begin{figure*}
\hspace{0.5in}
\epsscale{1}
\plotone{f3.eps}
\caption{\textbf{a}: Pseudo-bolometric (UBVRI) light curve of
SN~2011bm, compared with a set of pseudo-bolometric light curves of
other stripped-envelope SNe. The blue solid line is the {\it uvoir}
bolometric luminosity of SN~2011bm, computed with the UBVRI-light
curve mentioned before corrected by the fractional NIR contribution
of SN~2009jf \citep{2011MNRAS.416.3138V}. The resulting {\it uvoir}
bolometric light curve is consistent with the {\it uvoir} bolometric
luminosity obtained for the few epochs in which JHK photometry of
SN~2011bm is available (labelled as $uvoir$). The dashed line
corresponds to the fit to the computed {\it uvoir} bolometric
luminosity with the toy model described in
\protect\citet{2008MNRAS.383.1485V}. \textbf{b}: Synthesized
$^{56}$Ni mass versus ejecta mass for a heterogeneous sample of
core-collapse SNe. Data are from
\protect\citet{2009ApJ...692.1131T,2009A&A...508..371H,2011MNRAS.416.3138V,2009ApJ...696..713S}.
}
\label{fig3}
\end{figure*}
\section{Discussion and conclusion}
The extremely slow evolution of the light curve and spectra of
SN~2011bm have never been observed in other \emph{normal}
stripped-envelope SNe. Despite this, its spectral features are very
similar to those observed in typical SNe~Ic, and much narrower than in
broad-lined events. These observables are consistent with the
explosion of a very massive star (with a main-sequence mass of $\geq
30$ M$_{\odot}${}) that ejected 7--17 M$_{\odot}${} of material, 5--10 M$_{\odot}${} of which
is oxygen and 0.6--0.7 M$_{\odot}${} is $^{56}$Ni). The kinetic energy lies
between 7 and 17 foe. SN 2011bm is the first stripped-envelope SN
with such a slow evolution and such peculiar physical parameters. The
intrinsic rareness of these events supports the idea that they arise
from very massive stars. To our knowledge, only one other
\emph{normal} stripped envelope SN has been proposed to come from a
similarly massive star \citep[SN~1984L, ][]{1991ApJ...379L..13S}.
An alternative, more exotic scenario to explain the luminous and broad
light curve of SN~2011bm without invoking exceptionally high $^{56}$Ni
and total ejected masses is that of a magnetar-powered core-collapse
SN. In this case the light curve of SN~2011bm would be dominated by
the energy released in the spin-down of a newly formed magnetar.
However, the slope of the late-time light curve of SN~2011bm (until a
phase of 300 days) is consistent with that expected in a normal
radioactively-powered SN, without the need of an additional source as
proposed in models of magnetar-driven events
\citep{2010ApJ...717..245K,2010ApJ...719L.204W}\footnote {In addition,
the handful of candidate magnetar-powered SNe recently discovered
have never shown a late-time light-curve flattening onto the
$^{56}$Co tail \citep[see
e.g.][]{2010ApJ...724L..16P,2011Natur.474..487Q}.}.
Another characteristic of a magnetar-powered SN would be a bi-polar
geometry of its ejecta \citep{2010ApJ...717..245K}. Evidence of a
bi-polar explosion may be found in the profiles of the oxygen lines in
nebular spectra. Double-peaked profiles would support the idea of a
bi-polar explosion, whereas the absence of a double peak is not
sufficient to rule out such an explosion geometry. Following the
approach of \citet{2009MNRAS.397..677T}, we inspected the oxygen
feature and, though a weak double peak is marginally visible, it
appears consistent with the doublet nature of [w]
$\lambda\lambda6300,6364$. In conclusion, our observations do neither
require nor support a magnetar-SN scenario. Instead, the traditional
framework of a $^{56}$Ni-powered core-collapse SN from a very massive
star seems to provide a coherent explanation. This makes SN~2011bm the
most massive and $^{56}$Ni-rich \emph{normal} (i.e. not broad-lined)
SN~Ic ever observed. Further analysis and modelling will certainly
help in better understanding this exceptional event.
\acknowledgments S.V. is grateful to H. Wang for hospitality at the
UCLA. The authors thank E.~Gall, R.~Pakmor and I.~Maurer for
assistance with the observations. S.B., E.C., M.T.B. and M.T. are
partially supported by the PRIN-INAF 2009 with the project
\emph{Supernovae Variety and Nucleosynthesis
Yields}. S.T. acknowledges support by the TRR 33 \emph{The Dark
Universe} of the German Research Foundation. The PS1 Surveys have
been made possible through contributions of the Institute for
Astronomy, the University of Hawaii, the Pan-STARRS Project Office,
the Max-Planck Society and its participating institutes, the Max
Planck Institute for Astronomy, Heidelberg and the Max Planck
Institute for Extraterrestrial Physics, Garching, The Johns Hopkins
University, Durham University, the University of Edinburgh, Queen's
University Belfast, the Harvard-Smithsonian Center for Astrophysics,
and the Las Cumbres Observatory Global Telescope Network,
Incorporated, the National Central University of Taiwan, and the
National Aeronautics and Space Administration under Grant
No. NNX08AR22G issued through the Planetary Science Division of the
NASA Science Mission Directorate.
{\it Facilities:} \facility{ESO-NTT (EFOSC2 and SOFI)},
\facility{Liverpool Telescope (RATCAM)}, \facility{1.82m Asiago
Mt. Ekar Telescope (AFOSC)}, \facility{Calar Alto 2.2m Telescope
(CAFOS)}, \facility{LBT (LUCIFER)}, \facility{NOT (ALFOSC)},
\facility{PS1}, \facility{TNG (LRS and NICS)}, \facility{Faulkes
Telescope North (FS02)}, \facility{William Herschel Telescope (ISIS
and ACAM)}.\\
|
{
"timestamp": "2012-03-12T01:00:14",
"yymm": "1203",
"arxiv_id": "1203.1933",
"language": "en",
"url": "https://arxiv.org/abs/1203.1933"
}
|
\section{Introduction}
Spontaneous synchronization appears in a large variety of systems in nature. Well-known examples include biological systems such as fireflies flashing in unison or crickets chirping together \cite{Sismondo1990}, rhythmic applause \cite{Neda2000,Neda2000a}, pacemaker cells in the heart \cite{Peskin1975}, the menstrual cycles of women \cite{Stern1998}, oscillating chemical reactions, mechanically coupled metronomes, pendulum clocks hung on the same wall, and many other systems.
Several mathematical models have been proposed to explain and describe the spontaneous synchronization phenomena in large interacting ensembles. Most of these models can be grouped in one of two broad categories that are distinguished by the nature of coupling between the oscillators: those that are based on phase coupling and those that are based on a pulse-like coupling.
The prototypical model for phase coupled oscillators is the Kuramoto model \cite{Kuramoto1987}. The Kuramoto model consists of an ensemble of globally coupled rotators where the state of each unit is described by a $\theta \in [0, 2\pi)$ periodic phase variable. When not coupled to the others, a single oscillator rotates with its natural angular frequency, $\omega$. The angular frequencies of the oscillators are distributed according to a unimodal distribution $g(\omega)$. In the presence of coupling the equation of motion of an oscillator is
\[
\frac{d \theta_i}{dt} = \omega_i + K \sum_{j=1}^{N} \sin (\theta_j - \theta_i),
\quad i = 1, \cdots, N,
\]
where $K$ is a coupling constant. This interaction naturally leads to synchronization because it tries to minimize the phase difference between oscillators. The synchronization level of oscillators can be characterized by the order parameter $r \in [0,1]$ defined by the equation $r e^{i \varphi} = \frac{1}{N} \sum_{j=1}^N e^{i \theta_j}$.
The specially chosen trigonometric form of the coupling makes it possible to study this model using analytic methods. Kuramoto showed that in the limit of a very large number of oscillators, there exists a critical value of the coupling constant, $K_c$, so that if $K < K_c$ then the phases of oscillators are distributed randomly ($r=0$),
while if $K > K_c$, then the oscillators become partially synchronized ($r > 0$) \cite{Kuramoto1987}.
Many systems in nature however can't be assigned an associated periodic phase variable, thus the Kuramoto model is not a suitable description for them. In the case of systems where the interaction between oscillators is pulse-driven (such as in the case of fireflies, firing of neurons, rhythmic clapping), \emph{integrate and fire} type synchronization models are used \cite{FitzHugh1961,Nagumo1962,Bottani1996,Pikovsky1997}. The integrate and fire model is based on the assumption that each oscillator has a monotonically increasing state variable. When the state variable reaches a threshold value, the oscillator ``fires'': it emits a pulse, and its state variable is re-set to zero. When an oscillator detects a pulse in the system, its state variable suddenly increases by a constant value $K$, which may cause it to reach its own threshold. It is easy to see that in this system the firing of an oscillator may trigger an avalanche of pulses, causing many oscillators to fire within a short period of time. As a result of this, synchronization will emerge. The order parameter used to characterize the synchronization level of these systems is usually defined as the size of the largest avalanche compared to the total number of units present. Similarly to the case of the Kuramoto model, it can be shown that in the case of global coupling there is a critical value of the constant $K$ above which the oscillators will fire in a partially synchronized manner.
Both of these basic model categories have many variations, where other interactions are also considered or in which the coupling is not global. Their statistical behaviour and the appearance of synchronization depends on the strength as well as the topology of the coupling. There are a number of review works available on the topic of the statistical mechanics of spontaneous synchronization \cite{Strogatz2000,Strogatz2003,Pikovsky2001}.
A new, simple model that leads to synchronization in a non-trivial manner was recently introduced by Nikitin et al. \cite{Nikitin2001,Neda2003}. It was inspired by the study of rhythmic applause \cite{Neda2000,Neda2000a}. Modifications of this basic model were studied in several subsequent papers \cite{Neda2003,Sumi2009,Horvat2009}, by using numerical modelling or an experimental realization of the system \cite{Sumi2009}.
In this model, the oscillators are coupled through emitted pulses, similarly to integrate and fire type models. The interaction between the oscillators does not however favour synchronization in a direct way. The considered coupling intends to keep the average output in the system close to a threshold level. Unexpectedly, synchronization appears as a side effect of this optimization interaction. In this model the oscillators are bimodal, being able to operate in either a slow or fast (long or short period) oscillation mode, and therefore contribute to the total output of the ensemble with a higher or lower average output. At the beginning of each oscillation period, an oscillator unit will decide whether to choose the slow or fast mode depending on the detected total output level in the system. The periods of the oscillation modes are also randomly fluctuating.
Previous numerical studies performed on different variations of this model have shown that for certain parameter values the ensemble of bimodal oscillators will partially synchronize and produce a periodic output signal \cite{Nikitin2001,Neda2003,Sumi2009,Horvat2009}. The studies were performed as a function of the randomness of the periods of the modes and the threshold value. The effect of changing the ratio of the periods of the two modes was however not investigated in detail, and the phase space of the model was not mapped with high accuracy before. In the present work we focus on exploring the behaviour of the model as a function of the threshold level and the ratio of mode periods, and explore the phase space with a much higher accuracy than before. We have found that this simple model of bimodal oscillators has a phase space with a complex and surprisingly non-trivial structure.
\section{The two-mode stochastic oscillator model}
\subsection{Description of the model}
The basic version of the model considers an ensemble of $N$ identical bimodal, globally coupled, stochastic oscillators \cite{Nikitin2001}. At any time, an oscillator can either be active, emitting a signal of strength $1/N$, or inactive, emitting no signal. Therefore the total output level of the system can vary between 0 and 1. These oscillators can be intuitively thought of as flashing units. For simplicity, from now on we shall refer to active ones as \emph{lit} and inactive ones as being in an \emph{unlit} or \emph{dark} state. In accordance with this intuitive picture, the sum of the units' output levels can be thought of as the total light intensity in the system.
The units are stochastic bimodal oscillators. They can operate in two oscillation modes, one with a shorter and one with a longer period. These will be referred to as mode 1 and mode 2, respectively. The periods of the modes are random, and their mean values are denoted by $\tau_1$ and $\tau_2$.
An oscillation period consists of three phases, $A$, $B$ and $C$. During phase $A$ and $B$ the units are dark, while during phase $C$ they are lit. The duration of phase $A$, $\tau_A$, is a random variable drawn from the interval $[0, 2\tau^*]$ with a uniform distribution. The mean value of $\tau_A$ is $\langle \tau_A \rangle = \tau^*$. In this paper we shall assume that $\tau^* \ll \tau_1$. The duration of phase $B$, $\tau_B$, can have two values, $\tau_{B1}$ and $\tau_{B2}$, corresponding to the two oscillation modes. The duration of the lit phase, $\tau_C$, is fixed. The average lengths of the periods of the modes is the sum of the mean durations of these three phases: $\tau_1 = \langle \tau_A \rangle + \langle \tau_{B1} \rangle + \langle \tau_C \rangle = \tau^* + \tau_{B1} + \tau_C$ and similarly $\tau_2 = \tau^* + \tau_{B2} + \tau_C$. Since the units stay lit for a greater fraction of the short period mode than the long one, the average light intensity will be larger when the units are oscillating in the short period mode.
The coupling between the oscillators is realized through an interaction that strives to optimize the total light intensity in the system, denoted $f$. At the beginning of each period, a unit decides which mode to follow based on whether the total light intensity, $f$, is greater or smaller than a threshold level $f^*$:
\begin{itemize}
\item If $f \le f^*$, the shorter period mode will be chosen. Since an oscillating unit stays lit for a greater fraction of a full period when it is operating in the short mode, this will help in increasing the average light intensity in the system.
\item If $f > f^*$, the longer period mode will be chosen, reducing the average total light intensity in the system.
\end{itemize}
By this dynamic, each oscillating unit individually aims to achieve a total output intensity as close to $f^*$ as possible, based on their instantaneous measurements of the output level. As a side effect of this optimization procedure, synchronization can emerge: the total output intensity of the system becomes a periodic function and the units will flash in unison \cite{Nikitin2001,Neda2003,Sumi2009,Horvat2009}.
The simple model presented in the previous paragraphs differs from the original one described in \cite{Nikitin2001} and \cite{Neda2003} only in the distribution of the duration of the stochastic phase, $\tau_A$. In the original model, $\tau_A$ was exponentially distributed, and the behaviour of the system was studied as a function of the variables $\tau^* = \langle \tau_A \rangle$ and $f^*$. The present paper focuses on the case when $\tau^* \ll \tau_1$, therefore the precise statistical distribution of $\tau_A$ does not influence the results significantly. The reason for choosing a uniform distribution for this study is that using a distribution defined on a bounded interval simplifies numerical modelling of the system. In this paper the system is studied as a function of the parameter $f^*$ and the ratio of the average periods of the two oscillation modes, $\tau_2 / \tau_1$.
There are several variations possible on the basic version of the model. Some of these variations have been previously shown to also lead to synchronization. In \cite{Horvat2009} a version of the model with a duration dark phase and a variable duration lit phase was studied, while in \cite{Sumi2009} it was shown that synchronization emerges also when using multimodal oscillators. The lit phase can occur at the beginning or at the end of the oscillation period, leading to different behaviours. Finally, in this paper we will show that synchronization will occur even when a reversed optimization is used, that aims to achieve an output as different from the threshold $f^*$ as possible.
Three versions of the bimodal oscillator model will be considered: {\bf model 1.}\ the basic model described above with a fixed-duration lit phase, and a variable duration dark phase; {\bf model 2.}\ a model with variable duration lit phase and a fixed-duration dark phase; and {\bf model 3.}\ fixed-duration lit phase and variable duration dark phase with a reversed choice of the long or short modes depending on the $f^*$ value. This last case will be referred to as ``anti-optimization'' because the oscillators strive to achieve an output as different from $f^*$ as possible. Partial synchronization will emerge in all three cases.
\subsection{The order parameter}
We need a quantitative measure to characterize the synchronization level of the system. The order parameter used in previous studies \cite{Nikitin2001,Neda2003,Sumi2009} measures the periodicity level of the output signal. Let the output signal be denoted by $f(t)$, and define the function
\[
\Delta(T) = \frac{1}{2M} \lim_{x \rightarrow \infty} \frac{1}{x},
\int_0^x | f(t) - f(t + T) | \, dt
\]
where:
\[
M = \lim_{x \rightarrow \infty} \frac{1}{x}
\int_0^x | f(t) - \langle f(t) \rangle | \, dt
\]
Here $\langle f(t) \rangle$ denotes the mean value of the function $f(t)$ over the interval $[0, \infty]$. Normalization by $2M$ ensures that the value of $\Delta(T)$ will be between 0 and 1. It is clear that if $f(t)$ is a perfectly periodic signal, then $\Delta(T) = 0$ for all integer multiples of the period, as well as $T = 0$. If the signal is approximately periodic, then the function $\Delta(T)$ will have minima at integer multiples of the period. The location of the first deep minimum, $T_m$, will correspond to the period of the signal. The level of periodicity can then be characterized by $p = 1/\Delta(T_m)$. Note that the value of $p$ can vary between $1$ and $\infty$.
When using numerical modelling to simulate the system, the output level is computed at discrete points in time. Unfortunately the periodicity measure $p$ turned out not to be practical when the output signal is highly periodic and is known at discrete points only. The finite time resolution limits the precision of finding $T_m$, which in turn might have a significant effect on the computed value of the periodicity level $p$. The behaviour of $p$ as a function of the model parameters will no longer be characteristic of the dynamics, but will reflect the discretization of time variables. This becomes apparent only when modelling a larger number of oscillators than has been done previously and sampling the parameter space with a higher resolution.
Therefore, here we chose a different order parameter to characterize the synchronization level in the system. It has been observed in previous numerical studies that when synchronization emerges, the amplitude of the total light intensity function, $f(t)$, will be high as well. Therefore it is possible to use the amplitude of the signal to detect partial synchronization. It is practical to use the standard deviation of the signal to characterize its amplitude. The standard deviation is defined as follows:
\[
\sigma = \lim_{x \rightarrow \infty} \sqrt{ \frac{1}{x}
\int_0^x (f(x) - \langle f(x) \rangle)^2 \, dx }
\]
where
\[
\langle f(x) \rangle = \lim_{x \rightarrow \infty} \frac{1}{x} \int_0^x f(x) \, dx.
\]
This proved to be a robust measure that is not sensitive to outliers in the signal and characterizes intuitively well the ``flashing'' behaviour of the system. A nonzero $\sigma$ value corresponds to a partially synchronized flashing dynamics.
\section{Details of the computer simulation method}
The oscillator system was studied by extensive computer simulations. The value of the $\sigma$ order parameter was investigated as a function of the threshold parameter, $f^*$, and the ratio of the average periods of the two oscillation modes, $\tau_2/\tau_1$. The period of the output signal was estimated using a simple auto-correlation method, and the order parameter, $\sigma$, was calculated from a time average on an interval given by a large \emph{integer} number of periods (when the signal was periodic), for increased accuracy.
\subsection{Efficient simulation methods}
Previous studies \cite{Nikitin2001,Neda2003,Sumi2009,Horvat2009} used a direct method of simulating the ensemble of bimodal oscillators.
The easiest way to simulate the ensemble on a computer is by updating the state of each oscillator in discrete time steps.
Obtaining the results presented in this paper required a huge computational effort and it was made possible by choosing an efficient way to model the
system and map the parameter space. The most direct way to simulate the oscillators is the following: let the model be discretized in time and let $\Delta t$ be the time step. Then, let the state of each oscillator be stored separately, making the oscillator the basic unit of the model, and in each time step update the state of each oscillator. This method requires computer time proportional to $N \ensuremath{t_{\mathrm{max}}} / \Delta t$ to simulate the dynamics of $N$ oscillators for a time-interval \ensuremath{t_{\mathrm{max}}}, i.e.\ the simulation will slow down proportionally to the time resolution $\Delta t$.
A significant speedup can be achieved if the basic unit of the simulation is chosen to be the events that happen to oscillators instead of oscillator states. Events can be an oscillator turning on (lighting up), an oscillator turning off (darkening), or the start of a new oscillation period. Events are processed sequentially, in chronological order. Processing an ``on'' event causes the total output intensity to increase by $1/N$, while an ``off'' event causes it to decrease by $1/N$. A ``period start'' event causes a new event to be created in the future. Since there is an upper bound on the time length of a period, events need to be stored only up to a fixed time ahead in the future. A fixed-length array, containing only the number of each type of event for successive periods of time of length $\Delta t$ can be used for this. This method requires a time proportional to $\alpha N \ensuremath{t_{\mathrm{max}}} + \beta \ensuremath{t_{\mathrm{max}}}/\Delta t$, where the constants $\alpha$ and $\beta$ depend on implementation details. Since in practical implementations $\beta \ll \alpha$, increasing the time resolution does not significantly increase the simulation time, making accurate and fast simulation possible.
\subsection{Mapping the phase space}
Another essential optimization technique used in the present study was sampling the phase space adaptively. The most common way of mapping the phase space of a system is by simulating the dynamics of the system and calculating the value of the order parameter for each point in a rectangular lattice of points in the phase space. Increasing the resolution of the lattice twofold causes a $2^n$-fold increase in the number of sample points and consequently a similar increase in the required computation time.
\begin{figure}[bt!]
\begin{center}
\includegraphics{figure1}
\end{center}
\caption{Phase-space of the model 1 system. The simulation parameters were $N = 10000$ and $\tau_C = 0.15$. (a) The order parameter $\sigma$ as a function of the threshold level $f^*$ and the ratio of the average periods of oscillation modes $\tau_2 / \tau_1$. The colour indicates the value of $\sigma$: purple regions correspond to no synchronization, for more details see the attached colour-code. (b) Illustration of the adaptively subdivided mesh.}
\label{fig:mesh}
\end{figure}
A better approach is using adaptive sampling, i.e.\ increasing the number of sample points only in the regions where the behaviour of the order parameter is ``interesting''. An adaptive sampling method is defined by two choices: 1.\ a quantitative definition of ``interesting'' regions, i.e.\ the criterion for adding more sampling points 2.\ choosing the exact location of new sample points. Appropriate choices for these should depend on the behaviour of the order parameter as a function of system parameters in the given model. In our system, the parameter space is two-dimensional and the value of the order parameter is constrained to be in the interval $[0,1]$. The order parameter varies smoothly inside regions that are separated by abrupt and discontinuous transitions, as seen in figure~\ref{fig:mesh}a for the case of model 1. Due to the Monte Carlo simulation technique used, the results might be noisy. Based on these considerations, a simple adaptive sampling scheme was chosen. We start with an arbitrary set of roughly equally spaced sample points and compute the function value (order parameter) in them. Then in each refinement step, compute the Delaunay triangulation of the point set, and insert a new sample point in the midpoint of each triangle edge if the edge is longer than a threshold and the function values in the two ends differ by more than another threshold.
This method will trace the shape of the discontinuities very well. Since most sample points get inserted close to the discontinuities, the increase in the number of points is close to linear than quadratic in the resolution. Adaptive sampling makes it possible to map the parameter space of the system with high precision with relatively few sample points. The adaptively subdivided mesh for the case of model 1 is illustrated in figure~\ref{fig:mesh}b. The disadvantage of using such a method is that only those features will be discovered with certainty that have a size comparable to the resolution of the initial mesh. The features that \emph{are} discovered are mapped with high precision, and the method can produce a detailed looking output, as in figure~\ref{fig:mesh}a. This may be misleading and one must be aware that the precision of the mapping differs from region to region.
\section{Results}
\begin{figure}[bt]
\begin{center}
\includegraphics{figure2a}
\includegraphics{figure2b}
\includegraphics{figure2c}
\end{center}
\caption{The phase space of model 1, 2 and 3. The order parameter $\sigma$ is shown as a function of the threshold level $f^*$ and the ratio of the average periods of oscillation modes $\tau_2 / \tau_1$. The colour indicates the value of the $\sigma$ order parameter: purple regions correspond to $\sigma = 0$, i.e.\ no synchronization, see also the attached legend. (a) The phase space of model 1. The simulation parameters were $N = 10000$ oscillators and $\tau_C = 0.15$. (b) The phase space of model 2. $N = 10000$, $\tau_B = 0.8$. (c) The phase space of model 3. $N = 10000$, $\tau_C = 0.15$. }
\label{fig:phase-space-wide}
\end{figure}
Previous numerical studies performed on model 1 and model 2 have found that there is an island-like region of the $f^*$--$\tau^*$ phase-space where synchronization emerges as a side effect of the oscillating units striving to achieve a total output that is close to $f^*$. Synchronization would occur for the largest interval of $f^*$ when $\tau^*$ (the parameter characterizing the randomness of the periods) was small \cite{Nikitin2001,Neda2003,Sumi2009,Horvat2009}. The behaviour of the system has not previously been studied as a function of the periods of the oscillation modes, $\tau_1$ and $\tau_2$. In this paper we focus on mapping the $f^*$ -- $\tau_2/\tau_1$ phase space when $\tau^*$ is much smaller than $\tau_1$ and $\tau_2$. In order for the system to reach equilibrium, and in order that the stationary state to be independent of the initial state, it is necessary that $\tau^* > 0$. For all simulations we have fixed the values of $\tau_1$ and $\tau^*$ to be $\tau_1 = 1.0$ and $\tau^* = 0.03$. It is important to note that this value of $\tau^*$ does not approximate the $\tau^* \rightarrow 0$ limit well. Reducing its value further will result in some noticeable changes in the structure of the phase space. However, since the time needed to reach equilibrium was found to grow proportionally with $1/(\tau^*)^2$, the available computation resources imposed a limit on reducing the value of $\tau^*$.
As described already in section 2, three versions of the model were studied:
\begin{description}
\item[model 1] is the basic model described in \cite{Nikitin2001,Neda2003}. The duration of the lit phase, $\tau_C=0.15$, is fixed while the duration of the dark phase, $\tau_B$, can have a greater and a smaller value ($\tau_{B2}$ and $\tau_{B1}$). The oscillation modes are chosen so as to \emph{optimize} the output intensity $f$ towards $f^*$, i.e.\ minimize the difference between $f$ and $f^*$: if $f \le f^*$ then the short mode ($\tau_1$) is chosen while if $f > f^*$, the long mode is chosen ($\tau_2$). The mapped phase space is shown in figure~\ref{fig:phase-space-wide}a.
\item[model 2] has a fixed duration dark phase. This means that $\tau_B$ is fixed while $\tau_C$ can have a greater and a smaller value, $\tau_{C2}$ and $\tau_{C1}$. The oscillation modes are chosen to minimize the difference between $f$ and $f^*$ (\emph{optimization}): if $f \le f^*$ then mode 2 (the longer mode) is chosen, increasing the average light intensity; if $f > f^*$ then mode 1 (the shorter mode) is chosen. The phase space is shown in figure~\ref{fig:phase-space-wide}b.
\item[model 3] is similar to model 1, except that the oscillation modes are chosen so as to make $f$ as different from $f^*$ as possible: if $f \le f^*$ then $\tau_2$ is chosen, while if $f > f^*$ then $\tau_1$ is chosen (\emph{anti-optimization}). The phase space of this model is shown in figure~\ref{fig:phase-space-wide}c.
\end{description}
The time resolution chosen for all simulations presented in the above figures was $\Delta t = 1/1000$. The simulations were performed up to time $\ensuremath{t_{\mathrm{max}}} = 10000$ to ensure that the system reaches a steady state and the order parameter does not change any more. Then, the period of the signal was estimated using a simple auto-correlation method and the order parameter $\sigma$ was calculated based on an integer number of periods covering approximately the last 500 time units in the data. The simulations were run for an ensemble containing a number of oscillators ranging from $N=10$ to $N=100{,}000$. We found that the order-parameter curves do not change significantly when increasing $N$ above 3000. This is nicely visible if one studies finite-size effects for horizontal and vertical sections in figure~\ref{fig:phase-space-wide}a. Characteristic results are shown in figures~\ref{fig:section}a and \ref{fig:section}b, respectively.
\begin{figure}[hbt]
\begin{center}
\includegraphics{figure3a}
\includegraphics{figure3b}
\end{center}
\caption{Finite-size effects for the system. The order parameter $\sigma$ in model 1 as a function of: (a) $\tau_2/\tau_1$ for $f^* = 0.5$; and (b) $f^*$ for $\tau_2/\tau_1 = 1.1$, for different numbers of oscillators. The curves do not change visibly when the number of oscillators is increased above $N = 3000$. The $\tau_C = 0.15$ value was used.}
\label{fig:section}
\end{figure}
\begin{figure}[p]
\begin{center}
\includegraphics[width=0.9\textwidth]{figure4}
\end{center}
\caption{The shape of the output signal for model 1 is shown for various points in the phase space.
($N = 10000$ oscillators and $\tau_C = 0.15$). The value of the $\sigma$ order parameter is illustrated with the same colour-code as in figure~\ref{fig:phase-space-wide}a. }
\label{fig:shape1}
\end{figure}
\begin{figure}[p]
\begin{center}
\includegraphics[width=0.9\textwidth]{figure5}
\end{center}
\caption{The shape of the output signal for model 2 is illustrated for various points in the phase space.
($N = 10000$ oscillators, $\tau_B = 0.8$). The value of the $\sigma$ order parameter is illustrated with the same
colour-code as in figure~\ref{fig:phase-space-wide}b. }
\label{fig:shape2}
\end{figure}
The $f^*$ -- $\tau_2 / \tau_1$ phase space of the system has a complex structure in the case of all three models, and consists of several partially synchronized regions. The regions are separated by discontinuities in the value of the order parameter $\sigma$. In each of these regions, the output intensity function of the system, $f(t)$, is periodic but has a different shape. Some of the shapes of $f(t)$ that occur for different parameter values in model 1 and model 2 are shown in figure~\ref{fig:shape1} and \ref{fig:shape2}. These widely different global signals suggest that the dynamics of this simple system is extremely rich and many different phases are possible. The abrupt appearance of synchronization on the region boundaries and the sudden changes in the shape of the output function resemble phase transitions.
\section{Summary}
Recently, a new type of synchronization model was introduced for pulse-coupled bimodal stochastic oscillators. This model does not contain an explicit phase difference minimizing force. Instead, each oscillator chooses a faster or a slower oscillation mode so as to minimize the difference between the system's total output level and a threshold value $f^*$. Previously this model was studied as a function of the threshold level and the randomness of the oscillation modes.
In this work we have studied three variations of this model, and mapped their behaviour as a function of the threshold level, $f^*$, and the ratio of the average periods of the oscillation modes, $\tau_2 / \tau_1$. It was found that the ratio of the oscillation modes, which was not considered as a parameter of this model before, has a significant influence on the behaviour of the system. The $f^*$ -- $\tau_2 / \tau_1$ phase space has a complex structure with several regions, separated by sharp discontinuities of the chosen order parameter. The shape of the total output intensity function differs between these regions.
An interesting finding of the present model is that synchronization in such models appears under unexpectedly wide range of conditions. All previous studies have considered a dynamic which intends to minimize the difference between the total output intensity of the system and a threshold level $f^*$. Here, we have found that synchronization will emerge even when an ``anti-optimizing'' dynamics is used that will maximize the difference between the output level and $f^*$.
The model described in this work is interesting because synchronization emerges not as a result of an explicitly phase difference minimizing interaction, but as a side effect of a simple optimizing (or anti-optimizing) dynamics. We have found that despite its simplicity, the behaviour of the model changes in an unexpectedly complex manner as a function of the studied parameters, and as a result of this the phase space of the model is also rather complex.
\ack
Work supported from a Romanian IDEI research grant: PN-II-ID-PCE-2011-3-0348. The work of Sz.~Horv\'at is supported by the European Social Fund through a POSDRU 6/1.5/S/3 2007--2013 PhD grant. The Triangle fast Delaunay triangulator software was used in the implementation of the adaptive sampling method.
\section*{References}
\bibliographystyle{iopart-num}
|
{
"timestamp": "2012-03-09T02:01:21",
"yymm": "1203",
"arxiv_id": "1203.1699",
"language": "en",
"url": "https://arxiv.org/abs/1203.1699"
}
|
\section{Introduction}
\label{sec:intro}
Five--dimensional Yang--Mills theories compactified on a circle have a
light scalar mode, whose mass renormalization in perturbation theory
is protected by the remnant of the higher--dimensional gauge
symmetry. This light scalar is the static Kaluza--Klein mode coming
from the compactification of the fifth component of the original gauge
field, when periodic boundary conditions are
imposed~\cite{Kaluza:1921tu,Klein:1926tv}. At tree level, this mode is
massless. At low energies, the physics of compactified
extra--dimensional theories can be described by an effective
lagrangian for a four--dimensional gauge theory coupled to an
elementary scalar particle in the adjoint representation of the gauge
group. This effective description is valid only up to the
compactification energy scale $\Lambda_{\rm R} \sim R^{-1}$, where $R$ is the
radius of the extra dimension. At the compactification scale, other
massive vector modes become relevant for the dynamics, and their
coupling to the low--energy spectrum described above can no
longer be neglected.\\
Quantum corrections usually yield divergences in the
mass of scalar particles: in a generic renormalizable quantum field
theory, the scalar mass receives contributions proportional to the
square of the ultra--violet (UV) cut--off. However, the mass of the
scalar field coming from the compactification of a higher--dimensional
gauge field remains finite, as suggested by one--loop and two--loop
calculations
~\cite{Hosotani:1983xw,Hatanaka:1998yp,Cheng:2002iz,vonGersdorff:2002as,Hosotani:2005fk,Hosotani:2007kn}.\\
These perturbative calculations are performed using the explicit
four--dimensional effective field theory that describes the
five--dimensional system at low energies. The same result has been
obtained using the full five--dimensional gauge theory which
explicitly includes all the higher energy
contributions~\cite{DelDebbio:2008hb}. Since the extra--dimensional
gauge theory is believed to be non--renormalizable, it can only be
defined as a regulated theory with an ultra--violet cut--off
$\Lambda_{\rm UV}$ always in place. Interestingly the quantum
corrections to the scalar mass are independent of $\Lambda_{\rm UV}$:
\begin{equation}
\label{eq:one-loop-mass}
\delta m^2 \; = \; \frac{9 g_5^2 N_c}{32 \pi^3 R^3} \zeta (3)
\; ,
\end{equation}
where $\zeta$ is the Riemann Zeta--function, $N_c$ is the number of
colours and $g_5^2$ is the dimensionful coupling constant of the
five--dimensional Yang--Mills theory.\\
It must be stressed, that Eq.~\eqref{eq:one-loop-mass} is valid
only in the regime where there is a scale separation $\Lambda_{\rm R} \ll
\Lambda_{\rm UV}$ between the
compactification scale $\Lambda_{\rm R}$ and the cut--off $\Lambda_{\rm UV}$,
because in this case the details of the regularization can be
neglected. In this energy region, the highly energetic modes at the
cut--off scale see the extra dimension as non--compact and therefore
do not contribute to the scalar mass corrections, due to the
higher--dimensional gauge symmetry.\\
All the aforementioned results make the compactification mechanism a
very interesting and promising scenario to protect the mass of scalar
particles from cut--off effects. Moreover, since the early work of
Ref.~\cite{Antoniadis:1990ew}, higher--dimensional theories have
gained significant phenomenological interest.\\
In this work, we study this mechanism
in the simple case where the extra dimension is compactified on a
circle $S^1$. In particular, we would like to explore the validity of
the perturbative prediction Eq.~\eqref{eq:one-loop-mass} in the
strongly--coupled regime of the theory. While we do not expect the
proportionality constant to remain unchanged, we want to check whether
the non--perturbative dynamics preserves the independence of the UV
cut--off, and the functional dependence on the compactification
radius. This is a non-trivial task, since in the non--perturbative
regime the states in the spectrum are not the excitations of the
elementary fields in the action.\\
To be able to access the non--perturbative regime, we use Monte Carlo
simulations of a lattice gauge theory in five dimensions. We then
search for a region in the parameter space of the lattice theory where
the hierarchy of scales is such that the low--energy physics is
described by a four--dimensional effective theory with a light
scalar particle.\\
In recent years, there have been several numerical studies of the simplest
of these extra--dimensional theories on the lattice, namely the $\SU{2}$
pure gauge theory on a five--dimensional torus with anisotropic
lattice spacings, $a_4$ in the four--dimensional space and $a_5$ in
the extra fifth
dimension~\cite{Ejiri:2000fc,Ejiri:2002ww,deForcrand:2010be,Knechtli:2010sg,Farakos:2010ie}.
A pioneering study of the same model on isotropic lattices was done in
the late seventies~\cite{Creutz:1979dw}.\\
The aim of this work is to explore the parameter space of the lattice
model and to define the scales separation by studying the behaviour of
observables such as the string tension and the mass of scalar states;
this goes in the direction of improving previous recent
results~\cite{deForcrand:2010be} and trying to clarify the status of
Eq.~\eqref{eq:one-loop-mass} in the non--perturbative regime. Using
numerical simulations in the region of phase space where there is a
hierarchy of scales $\Lambda_{\rm R} \ll \Lambda_{\rm UV}$, we are able to study the
parametric dependence of the non--perturbative scalar mass on the
cut--off $\Lambda_{\rm UV}$ and on the compactification scale
$\Lambda_{\rm R}$. However, in order to fully understand the nature of the
effective theory and of the scalar particle, more studies are needed
that are beyond the scope of this work. In particular, matching
simulations between this five--dimensional gauge model and the
four--dimensional gauge theory with an adjoint scalar field in the
action could be performed, following what was done to test dimensional
reduction in lower dimensions~\cite{Hart:1999dj}.\\
In Sec.~\ref{sec:lattice-setup} we describe the lattice setup used in
our simulations of the $\SU{2}$ Yang--Mills theory in five
dimensions. In Sec.~\ref{sec:scale-separation} we explain the
separation of scales that we expect to find in the lattice model, and
we analyse the perturbative predictions for the behaviour of the
desired hierarchy of scales as we scan the bare parameter space.
Next, we provide a description of the phase diagram of the model in
Sec.~\ref{sec:phase-diagram} and compare our findings with previous
studies. Once the phase diagram has been mapped out and the
interesting region has been identified, we present the details of our
measurements and compare them with the perturbative expectations in
Sec.~\ref{sec:results}. Finally we present a critical discussion of
the results and future developments of these ideas.
\section{The lattice model}
\label{sec:lattice-setup}
The continuum, five--dimensional pure gauge theory is defined by the
Euclidean action
\begin{equation}
\label{eq:continuum-action}
\mathcal S \; = \; \int \mbox{d}^4 x \: \int_0^{2\pi R} \mbox{d} x_5 \:
\frac{1}{2g_5^2} {\rm Tr}~ F_{MN}^2
\; ,
\end{equation}
where periodic boundary conditions are imposed along the fifth
direction (whose coordinate is $x_5$) in order to make it compact.
The field--strength tensor is the extra--dimensional
generalization of the four--dimensional one
\begin{equation}
\label{eq:field-tensor}
F_{MN} \; = \; \partial_M A_N - \partial_N A_M + i [A_M,A_N]
\quad M,N=1,\dotsc,5 \; .
\end{equation}
This continuum theory has an infinite four--dimensional volume, but it
is defined only on a finite and compact fifth dimension of
length $L_5 = 2 \pi R$, where $R$ is the compactification radius.\\
Since this theory is perturbatively non--renormalizable, the
ultra--violet cut--off $\Lambda_{\rm UV}$ cannot be removed. For the same
reason we consider the action in Eq.~\eqref{eq:continuum-action} only
as the simplest non--trivial example of effective theory in five
dimensions: an arbitrary number of operators and couplings could be
added in principle. Cut--off effects
are expected to be irrelevant in the low--energy regime of the
theory defined by the action
in Eq.~\eqref{eq:continuum-action}, i.e. at scales $E \ll \Lambda_{\rm UV}$.\\
The continuum action is regularized on a five--dimensional lattice,
where the finite lattice spacing determines the shortest propagating
wavelength. Two independent lattice spacings $a_4$ and $a_5$ can be defined on the
lattice, which correspond respectively to the lattice spacing in the
four--dimensional subspace, and in the extra fifth direction; the
bigger of the two defines the inverse of the cut--off $\Lambda_{\rm UV}$. The
gauge potential $A_M(x_M)$ is replaced on the lattice by gauge link
variables $\mathcal U_M(x)$ joining the site $x$ and the site $x + a\hat{M}$,
where $a = a_4$ if $M=1,2,3,4$ and $a = a_5$ if $M=5$. Periodic
boundary conditions for the gauge
links are imposed in all five directions.\\
We choose the \emph{anisotropic} lattice Wilson action for $\SU{N_c}$
gauge theories:
\begin{equation}
\label{eq:aniso-lattice-action-E}
\mathcal S_W \; = \; \beta_4 \sum_{x;1\leq \mu <
\nu \leq 4} \left[ 1 - \frac{1}{N_c} {\rm Re}\hskip 1pt {\rm Tr}~ P_{\mu\nu}(x)
\right] +
\beta_5 \sum_{x;1\leq \mu \leq 4}
\left[ 1 - \frac{1}{N_c} {\rm Re}\hskip 1pt {\rm Tr}~ P_{\mu 5}(x)
\right]
\; ,
\end{equation}
where $P_{\mu \nu}$ is the four--dimensional plaquette ($\mu$
and $\nu$ run from $1$ to $4$)
\begin{equation}
\label{eq:4d-plaq}
P_{\mu\nu}(x) \; = \; \mathcal U_{\mu}(x)\mathcal U_{\nu}(x+\hat{\mu}a_4)\mathcal U_{\mu}^\dag(x+\hat{\nu}a_4)
\mathcal U_{\nu}^\dag(x)
\; ,
\end{equation}
and $P_{\mu 5}$ is the
plaquette abutting on an extra--dimensional slice
\begin{equation}
\label{eq:5-plaq}
P_{\mu5}(x) \; = \; \mathcal U_{\mu}(x)\mathcal U_{5}(x+\hat{\mu}a_4)\mathcal U_{\mu}^\dag(x+\hat{5}a_5)
\mathcal U_{5}^\dag(x)
\; .
\end{equation}
The sum is intended to be on all the lattice sites $x$ of the full five
dimensional lattice volume.\\
This lattice setup is the same used
in Ref.~\cite{Ejiri:2000fc}. However, a
different parametrization for the Wilson action can be
used~\cite{deForcrand:2010be}:
\begin{equation}
\label{eq:aniso-lattice-action-P}
\mathcal S_W \; = \; \frac{\beta}{\gamma} \sum_{x;1\leq \mu \leq
\nu \leq 4} \left[ 1 - \frac{1}{N_c} {\rm Re}\hskip 1pt {\rm Tr}~ P_{\mu\nu}(x)
\right] +
\beta \gamma \sum_{x;1\leq \mu \leq 4}
\left[ 1 - \frac{1}{N_c} {\rm Re}\hskip 1pt {\rm Tr}~ P_{\mu 5}(x)
\right]
\; ,
\end{equation}
where the lattice coupling constant is
\begin{equation}
\label{eq:beta}
\beta \; = \; \sqrt{\beta_4 \beta_5}
\; ,
\end{equation}
and the second parameter is the bare anisotropy
\begin{equation}
\label{eq:gamma}
\gamma \; = \; \sqrt{\frac{\beta_5}{\beta_4}}
\; .
\end{equation}
The bare anisotropy is related to the ratio of the lattice spacings
$\xi = a_4/a_5$. At tree level $\gamma=\xi$, but quantum corrections
make $\xi$ deviate from this value. The relation between $\xi$ and
$\gamma$ for this action has already been studied in bare parameter
space and a useful map relating this two
quantities can be found in Ref.~\cite{Ejiri:2000fc}.\\
In order to obtain Eq.~\eqref{eq:continuum-action} in the classical
continuum limit of the action in Eq.~\eqref{eq:aniso-lattice-action-E},
we must require the following relations for the
lattice parameters (coupling constants) $\beta_4$ and $\beta_5$:
\begin{eqnarray}
\label{eq:beta4-beta5}
\beta_4 & = & \frac{2N_ca_5}{g_5^2} \\
\beta_5 & = & \frac{2N_ca_4^2}{g_5^2a_5} \; .
\end{eqnarray}
Similarly, for the action in Eq.~\eqref{eq:aniso-lattice-action-P} we
have
\begin{equation}
\label{eq:lattice-coupling}
\beta \; =\; \frac{2N_c}{g_5^2} a_4
\; ,
\end{equation}
and
\begin{equation}
\label{eq:gamma-xi}
\gamma \; = \; \xi \; = \; \frac{a_4}{a_5}
\; .
\end{equation}
In this work we use the Wilson action
Eq.~\eqref{eq:aniso-lattice-action-E} with $\beta_4$ and $\beta_5$ as
bare parameters, and therefore our results will be presented as
functions of these two quantities. However some of the features of the
phase diagram are better explained in terms of $\beta$ and $\gamma$,
in particular when comparing our findings
to existing results~\cite{deForcrand:2010be}.\\
Finally there are two more parameters in the lattice model that can be
adjusted in order to realize the desired separation of scales; they
are $N_4$, the number of lattice sites in each of the usual four
directions, and $N_5$, the number of lattice sites in the extra
dimension. Together with the corresponding lattice spacings, they
determine the \emph{physical} size of the system: $L_4 = a_4 N_4$ in
four dimensions and $L_5 = 2 \pi R = a_5 N_5$ in the fifth dimension.\\
In the following we restrict ourself to the non--Abelian gauge
group $\SU{2}$, thus setting $N_c=2$ in the above
definitions.
\section{Dimensional reduction and scale separations}
\label{sec:scale-separation}
We have already mentioned that the theory described by the action in
Eq.~\eqref{eq:continuum-action} is perturbatively non--renormalizable
because the five--dimensional coupling constant $g_5^2$ has negative
mass dimension $M^{-1}$. Moreover, the theory possesses another
intrinsic scale when the extra dimension is compactified on the
circle: the compactification scale $\Lambda_{\rm R} \sim R^{-1}$. This scale
is the analogue of the temperature scale in the formulation of
finite--temperature field theories compactified on a circle.\\
Upon compactification, the gauge fields are decomposed into Fourier
modes (called Kaluza--Klein modes in this context, or Matsubara modes
in finite--temperature field theories). At the classical level the
spectrum of the theory contains massless vectors, coming from the
gauge field components in the four--dimensional subspace, and a
massless scalar, coming from the gauge component in the extra compact
direction. All the higher modes acquire masses proportional to
$\Lambda_{\rm R}$. In the quest for an effective description of the
low--energy physics of the theory, one can integrate out the states at
energies greater than the compactification scale, leaving a
four--dimensional gauge field coupled to an adjoint massless
scalar. However, this dimensional reduction is a sensible
description only if there is a scale separation $\Lambda_{\rm R} \ll
\Lambda_{\rm UV}$: the physics of the compactified theory is not affected by
the details of the regularization. As discussed below, this condition
is only satisfied in a specific region of the lattice bare parameter
space.\\
If we focus on the low--energy $E \ll \Lambda_{\rm R}$ and weakly--coupled
regime, we expect a perturbative spectrum, where the elementary scalar
particle acquires a mass through radiative corrections, while the
gauge vectors remain massless. As we explore more strongly--coupled
regimes, the theory develops a non--perturbative mass gap related to
confinement. Our aim is to study what happens to the low--lying
spectrum of scalar particles in this non--perturbative regime. In
particular we would like to understand if there exists a region in the
bare parameter space of the five--dimensional theory where the
non--perturbative dynamics can be described by a four--dimensional
effective gauge theory coupled to a light adjoint scalar, whose mass
is decoupled from the cut--off scale as suggested by the one--loop
equation Eq.~\eqref{eq:one-loop-mass}. Moreover, it would be
interesting to find a region where the scalar mass is of the order of
the mass gap in the gauge sector.\\
A previous study has shown that there is a region of the phase diagram
of the lattice model where a scale separation between the static modes
of the four--dimensional gauge fields and their higher Kaluza--Klein
modes is observed~\cite{deForcrand:2010be,Kurkela:2009dg}, indicating
that the theory undergoes dimensional reduction similar to the case of
four--dimensional hot gauge theories~\cite
Datta:1998eb}. However, in that same region, the static mode of the
fifth component of the gauge field appears to be completely decoupled,
with a mass at the scale of the cut--off, and hence outside the regime
of validity of Eq.~\eqref{eq:one-loop-mass}.\\
Let us summarize now the hierarchy of scales that we would like to
find non--perturbatively in the lattice theory described in
Sec.~\ref{sec:lattice-setup}.
\begin{itemize}
\item A separation between the compactification scale and the cut--off
\begin{equation}
\label{eq:separation-cutoff-radius}
\frac{\Lambda_{\rm UV}}{\Lambda_{\rm R}} \; \gg \; 1
\; .
\end{equation}
\item The mass gap identified by the string tension $\sqrt{\sigma}$ in
four dimensions must be separated both from the cut--off and from
the compactification scales:
\begin{equation}
\label{eq:separation-sigma-cutoff}
\sqrt{\sigma} \; \ll \; \Lambda_{\rm UV} \; {\rm ;} \qquad \sqrt{\sigma} \; \ll \; \Lambda_{\rm R}
\; .
\end{equation}
In fact, only if the above relations are true, we expect the long
distance physics to be independent of the actual regularization of
the theory, and not to be sensitive to contributions from higher
modes. In other words, since the string tension gives the inverse of
the four--dimensional correlation length, when $\sqrt{\sigma}$ is
small compared to the cut--off, then the characteristic length of
the system is much larger than the lattice spacing, and the details
of the discretization of the theory should become insignificant.
\item Light scalar states should be at the energy scale defined by the
mass gap, and hence their mass, generically
referred to as $m_5$, should be separated from the cut--off and from
the energy scale of the other Kaluza--Klein massive modes:
\begin{equation}
\label{eq:separation-mass-cutoff}
m_5 \; \simeq \; \sqrt{\sigma}\; {\rm ;} \qquad m_5 \; \ll \;
\Lambda_{\rm UV} \; {\rm ;} \qquad m_5 \; \ll \; \Lambda_{\rm R} \; .
\end{equation}
\item Finally, we need to check the dependence of the scalar mass from
the cut-off and the compactification radius. We would like to find a
region in the space of bare parameters, where we have a scaling
similar to the one Eq.~\eqref{eq:one-loop-mass} obtained for an
elementary scalar particle in the weakly-coupled regime:
\begin{equation}
\label{eq:light-scalar}
m_5^2 \; \propto \; \frac{g_5^2}{R^3}
\; .
\end{equation}
\end{itemize}
In a strongly--coupled theory the different energy scales described
above are dynamically generated, and need to be measured by numerical
simulations. In the following discussions, we choose to express every
scale in units of the four--dimensional string tension
$\sqrt{\sigma}$; hence the other three scales in the theory are
characterized by three dimensionless ratios. The ultra--violet
cut--off $\Lambda_{\rm UV}$, given by the inverse of the largest lattice
spacing of the model, is
\begin{equation}
\label{eq:cutoff-lattice}
\frac{\Lambda_{\rm UV}}{\sqrt{\sigma}} \; \equiv \; \frac{1}{a_4\sqrt{\sigma}}
\; ,
\end{equation}
because we will be dealing with anisotropies $\xi = \frac{a_4}{a_5}
\geq 1$. Similarly, the compactification scale $\Lambda_{\rm R}$ is
\begin{equation}
\label{eq:radius-lattice}
\frac{\Lambda_{\rm R}}{\sqrt{\sigma}} \; \equiv \; \frac{1}{2 \pi R\sqrt{\sigma}} \; = \; \frac{1}{a_5 N_5 \sqrt{\sigma}}
\; .
\end{equation}
Finally, the scalar mass $m_5$ can be expressed as the ratio of the
scalar mass and the string tension both measured in units of the
lattice spacing $a_4$ in our simulations:
\begin{equation}
\label{eq:scalar-lattice}
\frac{m_5}{\sqrt{\sigma}} \; = \; \frac{a_4m_5}{a_4\sqrt{\sigma}}
\; .
\end{equation}
In Fig.~\ref{fig:scales} we summarize pictorially the scale separations in the theory.\\
\FIGURE[ht]{
\begin{picture}(300,40)
\thinlines
\multiput(80,20)(5,0){7}{\line(1,1){10}}
\thicklines
\put(50,25){\vector(1,0){200}}
\put(100,18){\line(0,1){15}}
\put(95,0){$1$}
\put(110,40){$\displaystyle \frac{m_5}{\sqrt{\sigma}}$}
\put(150,20){\line(0,1){10}}
\put(145,5){$\displaystyle \frac{1}{2\pi R \sqrt{\sigma}}$}
\put(220,20){\line(0,1){10}}
\put(215,5){$\displaystyle \frac{1}{a_4 \sqrt{\sigma}}$}
\put(260,20){$\displaystyle \frac{E}{\sqrt{\sigma}}$}
\end{picture}
\label{fig:scales}
\caption{The figure shows the desired separation of energy scales.
The scales are all expressed in terms of the four--dimensional
string tension that characterizes the low--energy physics of the
theory. The region of energies where we expect the scalar mass to
lie is highlighted by the shaded band.} }
Let us note that the separation of the UV and the compactification
scales can be entirely expressed in terms of bare parameters of the
lattice model at tree level:
\begin{equation}
\label{eq:separation-lattice}
\frac{\Lambda_{\rm UV}}{\Lambda_{\rm R}} \; \equiv \; \frac{a_5 N_5}{a_4} \; = \; \frac{N_5}{\xi} \; \sim \; \frac{N_5}{\gamma}
\; ,
\end{equation}
where the last step follows from Eq.~\eqref{eq:gamma-xi} and is a
valid approximation only in the weak--coupling limit. However, the
scalar mass in Eq.~\eqref{eq:scalar-lattice} must be measured
non--perturbatively, because it would be divergent in the perturbative
regime ($\sqrt{\sigma}=0$).\\
The three energy scales of the system, $\Lambda_{\rm UV}$, $\Lambda_{\rm R}$ and
$m_5$ can be studied by adjusting the three bare parameters of the
lattice model $\beta_4$, $\beta_5$ and $N_5$ (here $N_4$ must be large
enough that the four--dimensional subspace can be considered in the
infinite volume limit). Fixing a point in the space
($\beta_4$,$\beta_5$,$N_5$), or equivalently ($\beta$,$\gamma$,$N_5$),
will dynamically determine the two lattice spacings $a_4$ and $a_5$,
together with the extent of the extra dimension $a_5N_5$. Therefore,
measuring the three scales with lattice simulations at different points
of this bare parameter space is a powerful tool to explore the
dependence of the scalar mass on $a_4$ and $R$. In particular, the
ability to investigate separately these functional dependences is the
major breakthrough of this work: contrary to what was done in
Ref.~\cite{deForcrand:2010be}, where the separation $\frac{2\pi
R}{a_4}$ was kept fixed in the numerical simulations, we explore a
region of the phase space where $R$ and $a_4$ vary independently and
we are able to follow, non--perturbatively, lines of constant scalar
mass.\\
In order to gain some insight in the behaviour of the lines of
constant physics for this model, we can use perturbative results as a
guide, with the caveat that they are expected to provide a sensible
description of the data only in the weak--coupling regime. From the
one--loop renormalization group equation of a four--dimensional
Yang--Mills theory, we expect the asymptotic scaling relation
\begin{equation}
\label{eq:string-4d}
\sigma \; \sim \; \frac{1}{(2 \pi R)^2}
\exp\left\{-\frac{1}{b_0 g_4^2(\Lambda_{\rm R})}\right\}
\; ,
\end{equation}
where $b_0$ is the first term in the perturbative $\beta$--function of
the four--dimensional theory ($b_0=11/24\pi^2$ for $\SU{2}$) and
$g_4^2(\Lambda_{\rm R})$ is the effective dimensionless coupling constant at
the compactification scale. In terms of the lattice parameters of the
model, we can rewrite Eq.~\eqref{eq:string-4d} as
\begin{equation}
\label{eq:string-4d-latt}
a_4^2 \sigma \; \sim \; \frac{\gamma^2}{N_5^2}
\exp\left\{-\frac{\beta N_5}{2 N_c b_0 \gamma}\right\}
\; .
\end{equation}
Notice that Eq.~\eqref{eq:string-4d-latt} is obtained by trading
$g_4^2(\Lambda_{\rm R})$ for the lattice tree--level relation in
Eq.~\eqref{eq:lattice-coupling}, evaluated at the compactification
energy scale $\Lambda_{\rm R}$ (at tree level $g_4^2(\Lambda_{\rm R})=
\frac{g_5^2(\Lambda_{\rm R})}{2 \pi R}$). This asymptotic behaviour has been
checked numerically on the lattice in a particular region of the
parameter space of the model and in the limit $a_5 \rightarrow
0$~\cite{deForcrand:2010be}. Furthermore, if we assume the scalar mass
to behave perturbatively according to Eq.~\eqref{eq:one-loop-mass} in
the dimensionally reduced theory, we have the following expression for
the mass $m_5$ in units of the lattice spacing $a_4$
\begin{equation}
\label{eq:m2a2}
(m_5 a_4)^2 \; = \; \frac{2 N_c \gamma^3}{\beta N_5^3}
\; .
\end{equation}
The latter equation can be divided by Eq.~\eqref{eq:string-4d-latt} to
express the mass in units of the string tension:
\begin{equation}
\label{eq:m2-sigma}
\frac{m_5}{\sqrt{\sigma}} \; \sim \; \sqrt{\frac{2 N_c \gamma}{\beta
N_5}} \exp\left\{\frac{\beta N_5}{4 N_c b_0 \gamma}\right\}
\; .
\end{equation}
We can therefore plot the perturbative predictions from
Eq.~\eqref{eq:string-4d-latt} and Eq.~\eqref{eq:m2-sigma} in the plane
($\beta, N_5/\gamma$). This is shown in Fig.~\ref{fig:perturbative},
where some isosurfaces are labelled in order to understand the
functional behaviour. When these perturbative formulae are used,
the scalar mass in units of the string tension
has a minimum value in the bare parameter space. Moreover, in this
weak--coupling limit, the scalar mass is always above the scale set by
the string tension therefore decoupling from the low--energy theory
~\cite{deForcrand:2010be}.
\FIGURE[ht]{
\epsfig{figure=FIGS/perturbative_new.eps,width=0.65\textwidth,clip}
\label{fig:perturbative}
\caption{The lines of constant lattice spacing in units of the
string tension $a_4\sqrt{\sigma}$ are shown as dashed black lines
in the plane ($\beta,N_5/\gamma$). The lines of constant scalar
mass in units of the string tension $m_5/\sqrt{\sigma}$ are shown
with solid red lines. This plot is similar to the drawing in Fig.7
of Ref.~\cite{deForcrand:2010be}, where the authors also took into
account the features of the phase diagram which we will describe
in Sec.~\ref{sec:phase-diagram} (note that the coordinate
$\beta$ is called $\beta_5$ in Ref.~\cite{deForcrand:2010be}, and
$N_5/\gamma$ is called $\tilde{N}_5$).}
}
Keeping this in mind and assuming that Fig.~\ref{fig:perturbative}
represents the actual lines of constant physics, we
can speculate about how to reach a continuum limit for this
lattice model. As it was firstly noted in Ref.~\cite{deForcrand:2010be}, two
different four--dimensional continuum theories can be described as the
lattice spacing $a_4$ vanishes. The one we are interested in for this
study is a $\SU{2}$ Yang--Mills theory coupled to an adjoint scalar
field: this theory is described by the lattice model following a line
of constant $m_5/\sqrt{\sigma}$ (one of the solid red lines in
Fig.~\ref{fig:perturbative}) towards smaller values of $\beta$ and
bigger values of $N_5/\gamma$. In this direction, $a_4\sqrt{\sigma}$
decreases, while the scalar mass is kept fixed, and the effects of the
regularization are suppressed by powers of $a_4\sqrt{\sigma}$. A
remarkable feature of this approach to the continuum limit is that the
direction to be taken in the bare parameter space goes towards higher
values of $N_5/\gamma$. This means that the size of the extra
dimension $2\pi R$ increases in units of the lattice spacing $a_4$,
while the theory dimensionally reduces to four dimensions as already
suggested in the D--theory non--perturbative approach to quantum field
theories~\cite{Chandrasekharan:1996ih,Brower:1997ha}.\\
Let us stress again that Eq.~\eqref{eq:string-4d-latt} to
Eq.~\eqref{eq:m2-sigma} are found using one--loop continuum
perturbative results and tree--level relations between the lattice
parameters and the continuum ones. The lines of constant values for
the cut--off $1/a_4 \sqrt{\sigma}$ and for the scalar mass
$m_5/\sqrt{\sigma}$ must be determined non--perturbatively using
numerical simulations, and we shall see if and how they deviate from
the perturbative expectations. In particular, it would be interesting
to see if the hierarchy between the scalar mass and the string tension
still holds at stronger couplings.
\section{The phase diagram}
\label{sec:phase-diagram}
In this section, we briefly describe a further issue arising in the
study of the lattice model. Indeed, the perturbative predictions we
referred to in Sec.~\ref{sec:scale-separation} do not take into
account the rich phase structure of the lattice theory. Since it is
crucial for our purposes to simulate the theory in the correct phase,
let us first discuss the current understanding of the phase diagram of
the $\SU{2}$ pure gauge theory in five dimensions described by the
action in Eq.~\eqref{eq:aniso-lattice-action-E}.\\
The first feature, which was already investigated in the early studies
in Ref.~\cite{Creutz:1979dw}, concerns the lattice model on the line
$\beta_4 = \beta_5$, or equivalently $\gamma = 1$. This isotropic
model, where the lattice spacings are the same, $a_4 = a_5$, has a
bulk phase transition when all the dimensions are equal. This phase
transition is independent of the physical volume of the system; it is
signalled by a sudden jump of the plaquette expectation value and by a
hysteresis cycle. The bulk line separates a confined phase that is
connected to the strong coupling regime from a Coulomb--like phase
connected to the weak coupling regime. An interesting feature of the
isotropic model is that the bulk transition disappears when the
lattice size in any one dimension is decreased below a critical size,
$L_{c}$, which is the critical length of the Polyakov loop in that
direction. Below $L_c$ centre symmetry is broken. In this case the
phase transition becomes a second order one in the same universality
class of the four--dimensional Ising model: the position of the
critical point scales with the four--dimensional volume and with the
number of sites $N_5$ in the extra dimension. This has been verified
numerically for $N_5=2$ with a Binder cumulant finite--size scaling in
Ref.~\cite{deForcrand:2010be}. We have performed a scaling analysis of
the susceptibility of the Polyakov loop in the compact direction
$L_5$, and obtained compatible results. However, when the number of
points in the compact fifth dimension is increased to $N_5=4$, we
could not locate the second order phase transition before hitting
again the bulk transition; this is true up to $N_4=14$, which is the
biggest lattice we explored at $\gamma = 1$. A bigger aspect ratio
$N_4/N_5$ is probably needed at $\gamma = 1$ in order to see the
effects of the compactification (namely a thermal--like second order
phase transition), our computational resources did not allow us to
further explore this issue. A very recent study of the phase diagram
at very small anisotropies $\gamma < 1$ was presented in
Ref.~\cite{Knechtli:2010sg} and the authors claim that if any one of
the dimensions becomes smaller than a minimal lattice size $L^{\rm
min}(\gamma)$, no sign of the bulk phase transition can be
detected. At $\gamma=1$, their simulations hint at $2 < L^{\rm min} <
6$, and the results are also supported by
Ref.~\cite{Farakos:2010ie}.\\
In the following, we are interested in the region of the parameter
space where $\gamma > 1$. Clearly, in this case
$a_4 > a_5$, and hence the extra dimension can be easily
made small enough to obtain dimensional reduction as described
above. The phase diagram in this region is known at $N_5 = 4$ and $N_5
= 6$~\cite{Ejiri:2000fc}. We performed a similar study on bigger
four--dimensional lattices and obtained compatible results. We want to
stress that our aim is not to study the details of these phase
transitions; therefore we simply determined the critical lines in the
parameter space searching for the location of the peak in the
susceptibility of the compact Polyakov loop. As shown in
Fig.~\ref{fig:compare-phase-diagram} our results compare favourably to
Ref.~\cite{Ejiri:2000fc}, which also provides a cross--check of the
validity of our simulation code. \\
\FIGURE[ht]{
\epsfig{figure=FIGS/phase_diagram_cfr_ejiri_knechtli.eps,width=0.75\textwidth,clip}
\label{fig:compare-phase-diagram}
\caption{Phase diagram of the five--dimensional $\SU{2}$ pure gauge
lattice model. In red we report the position of the bulk phase
transition characteristic of the system with no compact
dimensions. Both results from Ref.~\cite{Ejiri:2000fc}, and from
Ref.~\cite{Knechtli:2010sg}, are shown. Moreover, we report the
lines of second order phase transition at $N_5=2$ (blue), $N_5=4$
(black) and $N_5=6$ (green). Our results are plotted with filled
circles and come from simulations on the lattices shown in the
legend of the plot. They are compatible with other results in the
literature.} }
The main feature of the phase diagram in this region is that, for
fixed $N_5$, there is a line of second order phase transition that
merges into the bulk one as the anisotropy is decreased below a
critical value $\gamma_c$. Above this $\gamma_c$, which depends on
$N_5$, the transition line separates a phase where the centre symmetry
in the extra compact direction is not broken (at smaller $\beta_5$)
from the phase where the symmetry is broken and the compact Polyakov
loop acquires a non--zero expectation value. We refer to this phase as
the dimensionally reduced one, following the terminology in
Ref.~\cite{deForcrand:2010be}. However, for $\gamma < \gamma_c$ the
bulk phase transition line separates a confined phase (at smaller
$\beta_4$) from a Coulomb--like phase extending to the weak--coupling
regime, exactly as we described in the isotropic case. This pattern of
phase transitions is shown in Fig.~\ref{fig:split-phase-diagram} using
the data already shown in Fig.~\ref{fig:compare-phase-diagram}, but
now separating the phase diagram at $N_5=4$ from the one at
$N_5=6$. Since the second order phase transition is physical, its
location changes as we change $N_5$. Note also that, at fixed $N_5$,
there is no sign of a bulk phase transition for $\gamma >
\gamma_c$. The emerging physical picture tells us that the
disappearance of the bulk phase transition happens when the
five--dimensional system compactifies; in other words, $\gamma_c$
defines a critical lattice spacing in the extra dimension $a_{5c}$
such that $L_{5c} =a_{5c} N_5$.
\FIGURE[ht]{
\epsfig{figure=FIGS/phase_diagram_splitview.eps,width=0.85\textwidth,clip}
\label{fig:split-phase-diagram}
\caption{Phase diagram of the five--dimensional $\SU{2}$ pure gauge
lattice model in the ($\beta_4,\beta_5$) plane for different values
of $N_5$. The data are the same as in
Fig.~\ref{fig:compare-phase-diagram}. The bulk phase transition
separating a confined from a Coulomb--like phase disappears for
$\gamma > \gamma_c$ into a physical thermal--like phase
transition. The location of this transition changes in the parameter
space as we change $N_5$. The region we are interested in studying
is the one labelled by $L_5 < L_{5c}$.} }
Ref.~\cite{Ejiri:2000fc} presents estimates for $\gamma_c$, for both
$N_5=4$ and $N_5=6$, and for the renormalized anisotropy $\xi$ in
those points, so that the critical radius $R_c$ of
the extra dimension in units of the four--dimensional lattice
spacing can be computed. The data in Ref.~\cite{Ejiri:2000fc} suggest that $L_{5c}
\sim 2.25 a_4$ for $N_5=4$, and $L_{5c} \sim 2.16 a_4$ for $N_5=6$:
for extra dimensions bigger than these approximate values, the system
shows a bulk phase transition characteristic of the five--dimensional
model. The interesting region for our purposes, is at $\gamma >
\gamma_c$ and above the line of second order phase transition, where
the extra dimension is smaller than its critical value $L_5 < L_{5c}$.
\section{Lines of constant physics}
\label{sec:results}
\subsection*{Strategy of lattice simulations}
\label{sec:strategy}
Our main goal is to study whether a light scalar particle does exist
in the low--energy spectrum of the five--dimensional theory. The
strategy of the simulations is very straightforward in principle. The
lattice model we described in Sec.~\ref{sec:lattice-setup} has four
tunable parameters: the two coupling constants $\beta_4$ and
$\beta_5$, and the number of sites $N_5$ and $N_4$. If we assume, for
the moment, that the spectrum does not depend on $N_4$ (e.g. we are in
the infinite volume limit of the lattice theory), we are left with
three parameters. Fixing the bare coupling constants dynamically
determines the two lattice spacings, whereas fixing $N_5$ determines
the length of the extra dimension. In other words, by fixing a point
in this three--dimensional bare parameter space, we are choosing a
system with a given separation of scales between the ultra--violet
cut--off $\Lambda_{\rm UV}$, the compactification scale $\Lambda_{\rm R}$ and the
scalar mass $m_5$ (all the energy scales are again expressed in units
of the string tension).\\
The three energy scales of the system can only be determined a
posteriori by measuring physical observables with numerical Monte
Carlo simulations. \\
Let us focus first on the determination of the cut--off scale. From
Eq.~\eqref{eq:cutoff-lattice} it is clear that a measure of the
four--dimensional string tension in units of the lattice spacing
yields the separation between the low-energy scale
and the cut--off. The string tension $\sqrt{\sigma}$ in units of the
four dimensional lattice spacing $a_4$ can be extracted using
different observables. We choose to measure correlation functions of
Polyakov loops winding around the three spatial directions: the string
tension can then be extracted from the mass of the lightest state that
propagates.\\
The non--perturbative scalar mass instead can be obtained from the
ratio of two lattice observables as expressed in
Eq.~\eqref{eq:scalar-lattice}. Having obtained the string tension, we
only need to measure $m_5$ in units of the lattice spacing $a_4$.
Since it is the mass of a scalar particle, we use correlation
functions of operators that only project on the $0^{++}$
representation of the symmetry group of the cube (with positive parity
and charge) following standard spectroscopic notation. Due to the
presence of the extra dimension, different types of basis operators
can be used in the correlation functions; in particular we distinguish
those created using Polyakov loops wrapping around the compact fifth
dimension from those created using Wilson loops embedded in the three
large spatial directions. We generically refer to the first kind of
operators as the scalar ones, while the second set is referred to as
glueballs. In the following, we will focus mostly on masses extracted
from correlators of the scalar operators, but part of our analysis
will be dedicated to glueballs as well. In this respect, we greatly
improve the exploration of the scalar spectrum as first presented in
Ref.~\cite{deForcrand:2010be}. More details on the operators and on
the noise--reduction techniques we used are given in the Appendix.\\
The last scale we need to set is the compactification scale
$\Lambda_{\rm R}$; unfortunately, we were not able to measure a third
independent observable that could be used for this purpose. In
particular, we would need a measure of the extra dimensional lattice
spacing $a_5$ that needs to be done in the confined phase.
However this problem can be easily overcome. As we
noted in Eq.~\eqref{eq:separation-lattice}, the separation between the
cut--off and the compactification scale is determined, at leading
order, by the bare parameters of the lattice model. Therefore, knowing
$\Lambda_{\rm UV}$ from a measure of $a_4\sqrt{\sigma}$ at one point
$(\beta_4,\beta_5,N_5)$ is sufficient to approximately estimate
$\Lambda_{\rm R}$. As we already mentioned, the last step of
Eq.~\eqref{eq:separation-lattice} is only valid in the weak--coupling
limit that is reached, at fixed $\gamma$, when $\beta \rightarrow
\infty$. Although this seems like a reasonable approximation in
Ref.~\cite{deForcrand:2010be} where the values of $\beta$ are large,
we try to estimate the systematic deviation of
$\frac{\Lambda_{\rm UV}}{\Lambda_{\rm R}}$ from its tree--level value
$\frac{N_5}{\gamma}$. As we will show in the following, our
simulations are performed in a different region of the phase diagram
with respect to Ref.~\cite{deForcrand:2010be} and our $\beta$ values
are smaller.
\TABLE[htb]{
\begin{tabular}[h]{|c|c|c|c|}
\hline
\multicolumn{4}{|c|}{$\xi = a\gamma^2 + b\gamma + c$} \\
\hline \hline
$a$ & $b$ & $c$ & $\tilde{\chi}^2$ \\
\hline
-- & 1.600(15) & -0.446(37) & 0.61 \\
\hline
-0.03(1) & 1.767(62) & -0.641(76) & 0.32 \\
\hline
-0.06(1) & 1.950(45) & $1-a-b$ & 0.77 \\
\hline
\end{tabular}
\label{tab:fittedxi}
\caption{Parameters of the fitted function $\xi = \xi(\gamma)$. The
all range of data was used, from $\gamma \sim 1.224$ to $\gamma =
4$.}
}
We expect corrections to Eq.~\eqref{eq:gamma-xi} due to
quantum fluctuations. Since the non--perturbative relation between the
bare anisotropy $\gamma$ and the renormalized one $\xi$ had already
been studied for this system, we interpolated the data available in
Ref.~\cite{Ejiri:2000fc}, in order to estimate the ratio
$\frac{\Lambda_{\rm UV}}{\Lambda_{\rm R}}$ for the points we simulated. The relation
$\xi = \xi(\gamma)$ is shown in Fig.~\ref{fig:plottedxi}.
\FIGURE[ht]{
\epsfig{figure=FIGS/plottedxi.eps,width=0.85\textwidth,clip}
\label{fig:plottedxi}
\caption{The relation $\gamma \sim \xi$ is corrected by quantum
fluctuations. We interpolated data from Ref.~\cite{Ejiri:2000fc} that
were obtained by measuring $\xi$ non--perturbatively using ratios of
suitable correlation functions. All the data are in the correct phase
where $\sqrt{\sigma} \neq 0$. In the plot we show three different fits
(with $\tilde{\chi}^2 \sim 0.5 - 0.7$) and the interpolation used to
obtain $\xi$ in the regions where we performed the simulations (both
for $N_5=4$ and for $N_5=6$). They all compare well and are hardly
distinguishable.}
}
We performed three different fits of the data: a linear fit, a
quadratic one, and a quadratic fit imposing $\xi(1)=1$. The details of
the fits are summarised in Tab.~\ref{tab:fittedxi}. In practice, to
obtain $\xi$ for the points in our simulations, we used a cubic
interpolation nested inside a bootstrap procedure for the errors. The
result is again shown in Fig.~\ref{fig:plottedxi} together with the
$1$--$\sigma$ contour. The errors are such that all the lower order
fits are compatible with this interpolation. Although we only use
interpolated values of $\xi$, it must be noted that $\xi =
\xi(\gamma)$ could in principle also depend on the other bare
parameter $\beta$.
However, in the region where $\xi$ was initially measured
non--perturbatively, the value of $\xi$ is shown to be fairly
insensitive to the values of $\beta$
(cfr. Fig.~1 in Ref.~\cite{Ejiri:2000fc}). This is true in particular
for the values of $\xi$ that we are going to use, namely $1.7 \leq \xi
\leq 3.0$. We performed our simulations at $\beta$ values
that are inside (or just slightly off) the region where $\xi$ was
observed to be independent of it. Hence we expect the systematic
errors of this interpolation procedure to be under control.\\
Before showing the details of the simulations and the results, let us
summarize the main steps of this study:
\begin{enumerate}
\item we fix a point in the three--dimensional parameter space
$(\beta_4,\beta_5,N_5)$ that is in the dimensionally reduced phase;
\item on this point we measure $a_4\sqrt{\sigma}$ and $a_4m_5$ from
correlation functions of suitable operators;
\item the measured observables determine the cut--off scale and the
scalar mass from Eq.~\eqref{eq:cutoff-lattice} and
Eq.~\eqref{eq:scalar-lattice};
\item we use the available data for $\xi$ to estimate the
compactification scale using Eq.~\eqref{eq:separation-lattice} and
the measured cut--off scale (this yields a better determination of
the anisotropy than the one coming from
tree--level relation $\gamma = \xi$, and allows us to estimate the
errors due to $\beta \neq \infty$);
\item we then move to a different point $(\beta_4,\beta_5,N_5)$ and
repeat the procedure;
\item having done this for a certain number of points allows us to
study the dependence of the energy scales on the bare parameters and
to determine lines of constant physics;
\item more importantly, this allows us to study the behaviour of $m_5$ as a
function of $\Lambda_{\rm UV}$ or $\Lambda_{\rm R}$ and to disentangle cut--off
effects from compactification effects.
\end{enumerate}
\subsection*{Results from lattice simulations}
\label{sec:results-lattice}
\FIGURE[hb]{
\begin{tabular}{cc}
\epsfig{figure=FIGS/region_b4b5_n5-4.eps,width=0.47\textwidth,clip}
&
\epsfig{figure=FIGS/region_b4b5_n5-6.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:points}
\caption{The plots show the region of the phase diagram that we explored with
numerical simulations, both for $N_5=4$ (a) and $N_5=6$ (b). The
location of the second order phase transition is also shown.
The blue squares are points where the scalar mass $a_4m_5$ was reliably extracted, whereas
the green circles represent points where we were able to measure the string tension $a_4\sqrt{\sigma}$.}
}
We performed simulations at two different values of $N_5$ and several
different four--dimensional volumes. The smaller lattice has $N_4=10$
and $N_5=4$. This volume is also the one we used to locate the
position of the second order phase transition in the left panel of
Fig.~\ref{fig:split-phase-diagram}. On this lattice, we generated
$\mathcal{O}(800000)$ configurations and the correlators of the
interesting observables were binned over $20$ configurations after
thermalization. We chose a wide range of values for $\beta_4$ and for
$\beta_5$, starting very close to the line of second order phase
transition. In this region we expect a light scalar in units of
the lattice spacing because the scalar mass is the inverse of the correlation
length, and the latter diverges at the critical point. From the phase
structure discussed above, we also expect to find a finite string
tension. The details of the simulated points on this
volume are reported in Tab.~\ref{tab:points-n5-4}.\\
Similarly, for $N_5=6$ we simulated on lattices with $N_4=12$,
generating $\mathcal{O}(600000)$ configurations, binning the
observables over $20$ configurations. The parameters of the
simulations on this second volume are summarised in
Tab.~\ref{tab:points-n5-6}. In the tables we show both the bare
parameters that fix the location of the point in the phase diagram,
and the values of $\gamma$ and the corresponding interpolated value of
$\xi$. The size of the non--perturbative effects on the anisotropy can
be extracted from these numbers. Moreover, from the same tables, one
can compare the separation of scales in
Eq.~\eqref{eq:separation-lattice} to the naive estimate using the bare
parameters, i.e. $\frac{N_5}{\gamma}$. What we notice is that the
naive expectation is systematically larger than what is obtained by
measuring the anisotropy non--perturbatively. As a result, we were
only able to explore the following range
\begin{equation}
\label{eq:separation-explored}
1.7 \; \lesssim \; \frac{N_5}{\xi} = \frac{\Lambda_{\rm UV}}{\Lambda_{\rm R}} = \frac{2
\pi R}{a_4} \; \lesssim \; 2.3
\; ,
\end{equation}
where the upper limit is close to the critical value $L_{5c}/a_4 \sim
2.16
- 2.25$ that we identified in Sec.~\ref{sec:phase-diagram}.\\
Since this is the first time that this particular region of the phase
space is explored with lattice simulations, we performed a broad scan,
aiming primarily at identifying the interesting region. As a
consequence, there are lattices for which we were unable to measure
precisely both the string tension and the scalar mass. In
Fig.~\ref{fig:points}, we show all the points reported in
Tab.~\ref{tab:points-n5-4} and in Tab.~\ref{tab:points-n5-6}, but at
the same time we identify the ones where either $a_4m_5$ or
$a_4\sqrt{\sigma}$ could not be extracted satisfactorily.\\
Our lattice data suggest that the lattice spacing $a_4$
changes dramatically in these regions of the phase space. As shown in
Fig.~\ref{fig:points}, the string
tension $a_4\sqrt{\sigma}$ can only be measured in a small subset of
points; the points closer to the line of second order phase transition
are characterized by spatial Polyakov loops whose mass is too high for
a signal to be extracted reliably. Since the mass of the loops is
given by $N_4 a_4 \sigma$, we see that in this region the lattice
spacing $a_4$ is getting larger in units of the string tension.
Following our discussion in Sec.~\ref{sec:scale-separation}, we
regard the region close to the phase transition line as the one
characterized by a small cut--off $\Lambda_{\rm UV}$. In this region, there
is not a clear separation between the low--energy physics and the
cut--off, and we expect to observe large discretization errors. To
make things even more interesting, we find the scalar mass $a_4m_5$ to
be small in this same region, where $a_4$ is large. In fact, it
turns out to be very difficult to find points in the phase diagram
where both $\sqrt{\sigma}$ and $m_5$ are separated from the cut--off
scale at the same time. This results in a scalar mass $m_5 \gtrsim
\sqrt{\sigma}$ for all the points on which we were able to reliably
measure the string tension, indicating the same hierarchy expected
from perturbation theory (cfr. Fig.~\ref{fig:perturbative}). On the
other hand, a light non--perturbative scalar
does exist very close to the second order transition line,
where $a_4m_5$ is small and $a_4\sqrt{\sigma}$ is large.\\
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{figure=FIGS/b4fixed_scalar_n5-4.eps,width=0.47\textwidth,clip}
&
\epsfig{figure=FIGS/b5fixed_scalar_n5-4.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:scalar-n5-4}
\caption{(a) The scalar mass in units of the lattice spacing
$a_4m_5$ as a function of $\beta_5$ and for four different values
of $\beta_4$ at $N_5=4$. The approximate location of the critical
region is shown by the shaded regions for the different values of
$\beta$. (b) At fixed $\beta_5$, we show the behaviour of
$a_4m_5$. Smaller values of $\beta_4$ are closer to the phase
transition line, but we do not have an estimate of its location in
this direction. Open symbols correspond to alternative fitting
ranges in the effective mass plateaux for the scalar state (see
Appendix). When not quoted, errors are smaller than symbols.}
}
A more quantitative statement can be made by looking at the measured
observables as functions of the bare parameters. For example, our data
allow us to study the behaviour of $a_4\sqrt{\sigma}$ at fixed value
of $\beta_4$ as we change $\beta_5$, and vice versa. The same can be
done with $a_4m_5$ and therefore with the ratio
$m_5/\sqrt{\sigma}$. In Fig.~\ref{fig:scalar-n5-4}(a) we select four
different values of $\beta_4$ and we plot the mass $a_4m_5$ obtained
from scalar operators as a function of
$\beta_5$. Fig.~\ref{fig:scalar-n5-4}(b) shows the dependence of the
scalar mass as a function of $\beta_4$ for fixed $\beta_5$. The values
of $a_4m_5$ are taken from Tab.~5
where we
summarise our results for $N_5=4$, whereas we report the results for
$N_5=6$ in Tab.~
. As we have already mentioned,
we notice that the scalar mass approaches the cut--off scale $a_4m_5
\gtrsim 1$ as we move away from the line of second order phase
transition. This happens both in the $\beta_4$ and $\beta_5$
directions. Similarly, following Ref.~\cite{deForcrand:2010be}, we can
move in the parameter space along a line of fixed $\gamma$, while
changing $\beta$.
We choose $\gamma \approx 1.54$ in order to obtain a separation of
scales $\Lambda_{\rm UV}/\Lambda_{\rm R} \approx 2$ after taking into account the
renormalized anisotropy. In the interval $\beta \in [ 1.71,\; 1.77 ]$,
we accurately study the low--lying spectrum of scalar particles
employing our larger set of operators with the inclusion of
glueballs. Using a variational method, detailed in the Appendix, we
studied the operator content of the different mass eigenstates in the
scalar channel. We extracted the mass of the scalar ground state and
its first excitation. The resulting masses are shown in
Fig.~\ref{fig:scalar-gamma}, where we compare the
non--perturbative scalar masses calculated via the variational ansatz
with the masses obtained solely from effective mass plateaux of scalar
operators. It is clear from the results in the plot that a variational
analysis is crucial to identify the lightest scalar state as $\beta$
is increased.\\
Further information can be obtained by studying the change in the
operator content of the scalar eigenstates as $\beta$ increases. We
measure the normalized projection of the mass eigenstates onto each
operator used in the correlation matrix. The projection of the
extracted ground state is shown in Fig.~\ref{fig:mixing-ground}. The
plot clearly shows how the contribution of the scalar operators to the
ground state decreases as $\beta$ increases. At higher values of
$\beta$, glueball operators have a larger overlap onto the ground
state. On the other hand, we clearly see that at lower values of
$\beta$, closer to the line of second order phase transition, the
scalar state has a dominant contribution from the
extra--dimensional operators.\\
The relative mixing of the first excited state onto the operators in
the variational set is shown in Fig.~\ref{fig:mixing-exc}. The points
where the mixing is calculated are the same as in
Fig.~\ref{fig:mixing-ground}. Up to $\beta = 1.75$, the first excited
state is dominated by a projection onto the scalar operators,
suggesting an extra--dimensional nature for this particle.\\
Again, we conclude that the scalar mass becomes heavy in units of the
cut--off scale while moving away from the critical line, as shown in
Fig.~\ref{fig:scalar-gamma}. This suggests that at $\beta \gtrsim
1.77$ for $N_5=4$ the scalar particle becomes heavy; from data in
Ref.~\cite{deForcrand:2010be} taken at $1.83 \leq \beta \leq 1.91$ at
the same $N_5$ (but at $\gamma=2$) it can be shown that $m_5 \gtrsim 2
\Lambda_{\rm UV}$ and therefore the scalar particle cannot be considered a
low--energy degree of freedom of the theory.\\
\FIGURE[hbt]{
\epsfig{figure=FIGS/variational_masses_cfr_scalar.eps,width=0.77\textwidth,clip}
\label{fig:scalar-gamma}
\caption{The scalar mass in units of the lattice spacing
$a_4m_5$ as a function of $\beta$ for a fixed value of
$\gamma=1.5433$. The shaded area is the approximate location of the
second order phase transition. Open symbols refer to masses
obtained from a variational procedure. Filled symbols are masses
extracted from diagonal correlators of scalar operators.}
}
\FIGURE[ht]{
\begin{tabular}{ccc}
\epsfig{figure=FIGS/proj_ground_1.72.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_ground_1.73.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_ground_1.74.eps,width=0.31\textwidth,clip}
\\
\epsfig{figure=FIGS/proj_ground_1.75.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_ground_1.76.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_ground_1.77.eps,width=0.31\textwidth,clip}
\\
\end{tabular}
\label{fig:mixing-ground}
\caption{Relative projection of the ground state onto each
of the operators in the variational set. Filled symbols
correspond to the set of smeared scalar operators, whereas the
open symbols refer to the smeared versions of glueball operators.}
}
\FIGURE[ht]{
\begin{tabular}{ccc}
\epsfig{figure=FIGS/proj_exc_1.72.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_exc_1.73.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_exc_1.74.eps,width=0.31\textwidth,clip}
\\
\epsfig{figure=FIGS/proj_exc_1.75.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_exc_1.76.eps,width=0.31\textwidth,clip}
&
\epsfig{figure=FIGS/proj_exc_1.77.eps,width=0.31\textwidth,clip}
\\
\end{tabular}
\label{fig:mixing-exc}
\caption{Relative projection of the first excited state onto each
of the operators in the variational set. Filled symbols
correspond to the set of smeared scalar operators, whereas the
open symbols refer to the smeared versions of glueball operators.}
}
While the scalar mass becomes smaller as we approach the critical
line, the opposite happens to the string tension. Its behaviour in
bare parameter space is best illustrated by the data at $N_5=6$. All
the points where we were able to extract the string tension
$a_4\sqrt{\sigma}$ are summarized in Tab.~\ref{tab:torelons-n5-4} for
$N_5=4$, and Tab.~\ref{tab:torelons-n5-6} for $N_5=6$. In
Fig.~\ref{fig:string-n5-6}(a) the string tension is shown at three
different values of $\beta_4$: the common feature of the data is that
the string tension increases as the critical line is
approached. As discussed above, this behaviour can be interpreted as
an increase of the lattice spacing $a_4$ in units of the physical
string tension $\sqrt{\sigma}$. A similar functional dependence of
$a_4\sqrt{\sigma}$ is shown in Fig.~\ref{fig:string-n5-6}(b), where
$\beta_5$ is fixed. At lower values of $\beta_4$, closer to the line
of phase transition, the string tension grows and it becomes very
difficult to extract a signal from our numerical simulations. We can
easily infer from the data that the string tension will decrease with
increasing $\beta$ at fixed $\gamma$, as already reported in
Ref.~\cite{deForcrand:2010be}. This behaviour is expected since $\beta
\rightarrow \infty$ is the weak--coupling limit
of the theory, and accordingly the string tension should vanish.\\
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{figure=FIGS/b4fixed_string_n5-6.eps,width=0.47\textwidth,clip}
&
\epsfig{figure=FIGS/b5fixed_string_n5-6.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:string-n5-6}
\caption{(a) The string tension in units of the lattice spacing
$a_4\sqrt{\sigma}$ as a function of $\beta_5$ and for three different values
of $\beta_4$ at $N_5=6$. The approximate location of the critical
region is shown for the different $\beta_4$. (b) At fixed
$\beta_5$, we show the behaviour of $a_4\sqrt{\sigma}$. Smaller values of
$\beta_4$ are closer to the phase transition line, but we do not
have an estimate of its location in this direction. Open symbols
correspond to alternative fitting ranges in the effective mass
plateaux for the torelon state (see Appendix). When not quoted,
errors are smaller than symbols.}
}
From the previous discussion we have identified the lines of constant
physics in the phase diagram at fixed $N_5$. Moreover, we find similar
features by going from $N_5=4$ to $N_5=6$. The lines of constant
cut--off $\Lambda_{\rm UV}$ are represented by contour lines of
$a_4\sqrt{\sigma}$. These lines start close to the line of second
order phase transition for $\gamma \sim \gamma_c$, but then move away
from it as $\gamma$ is increased. To summarize, at fixed $\gamma$, the
lowest $\beta$ corresponds to the lowest $\Lambda_{\rm UV}$; a larger
separation between the low--energy physics and the cut--off is found
at bigger values of $\beta$, and this is the region where the lattice
discretization starts to become irrelevant and we can safely extract
the low--energy physics from numerical simulations
(cfr. Eq.~\eqref{eq:separation-sigma-cutoff}). What we are really
interested in is the behaviour of the scalar mass $m_5$ in units of
the string tension $\sqrt{\sigma}$. By looking at the ratio of
$a_4m_5$ over $a_4\sqrt{\sigma}$, we can deduce the lines of constant
scalar mass. Unfortunately, we cannot use all the measured values of
$a_4m_5$, because we also need a measure of $a_4\sqrt{\sigma}$ on the
same point. The general pattern of these lines is again quite clear:
the lightest scalar is found closer to the second order critical line,
but it soon starts decoupling from the low--energy physics as we move
away from it. There is only a small patch of the phase space we
explored where Eq.~\eqref{eq:separation-cutoff-radius},
Eq.~\eqref{eq:separation-sigma-cutoff} and
Eq.~\eqref{eq:separation-mass-cutoff} hold simultaneously. The
lightest mass $m_5$ we measured is of order
$2\sqrt{\sigma}$.\\
Using the non--perturbative lines of constant physics, we can try to
discuss the different types of continuum limit. Our findings
can be compared with the perturbative picture reported
in Ref.~\cite{deForcrand:2010be}, bearing in mind that our results are
obtained for fixed values of $N_5$ and $N_4$ and therefore could be
affected by finite--volume effects.
For this comparison, we shall refer in particular to Fig.~7 of
Ref.~\cite{deForcrand:2010be}. First let us relate our choice of
parameters with the definitions in Ref.~\cite{deForcrand:2010be}: the
horizontal axis in Fig.~7, is labelled by $\beta_5$, which corresponds
to what we call $\beta$ (cfr. Eq.~\eqref{eq:beta}) in this work; the
vertical axis is labelled by $\tilde{N}_5$, which is defined as
$N_5/\gamma$. In the following we use only our parametrization,
and the reader should refer to the discussion above for any
comparison.
At any fixed value for $\beta_4$ in the dimensionally reduced phase,
there is a lower bound for $\beta_5$, given by the location of the
critical point. By increasing $\beta_5$, we cross lines of decreasing
lattice spacing $a_4$, therefore moving towards a continuum limit,
meaning that the lattice discretization effects vanish. At the same
time we cross lines of increasing scalar mass $m_5$, which inevitably
decouples from the low--energy spectrum: the low--energy effective
theory described in this region is four--dimensional, and contains
only gauge degrees of freedom. A similar limit occurs at fixed
$\beta_5$ and increasing $\beta_4$. However, by following a line of
constant scalar mass in the phase diagram, we cross lines of different
fixed lattice spacing. In particular, moving towards smaller $\beta_4$
and bigger $\beta_5$ the lattice spacing decreases, allowing us to
reach the desired separation between the cut--off and the low--energy
physics with a constant value of the scalar mass. The low--energy
dynamics is then described by an effective four--dimensional theory
with a light adjoint scalar in the low--energy spectrum, having
started with a five--dimensional theory with only gauge degrees of
freedom. This being an effective description, it is expected to hold
only up to the energy scales given by the compactification radius, as
we already mentioned in Sec.~\ref{sec:scale-separation}. What we have
learned from our non--perturbative map of the energy scales in the
phase diagram of the lattice model is that it requires a certain
amount of fine tuning to pin down the location of a line of constant
mass and to follow it. Moreover, the behaviour of the cut--off scale
near to the line of second order phase transition
(cfr. Fig.~\ref{fig:string-n5-6}) makes it very difficult to determine
$\frac{m_5}{\sqrt{\sigma}}$ non--perturbatively, thereby limiting our
ability to reach values of the scalar mass that are smaller than the
square root of the string tension. This is an important result for
future studies in this context, and it was not anticipated before
using perturbative arguments. For example, looking at the perturbative
results in Fig.~\ref{fig:perturbative}, or equivalently at Fig.~7 in
Ref.~\cite{deForcrand:2010be}, where the line of phase transitions in
the $a_5 \rightarrow 0$ is taken into account, we note that the lines
of fixed $a_4\sqrt{\sigma}$ go straight into the critical line. This
behaviour is not supported by our non--perturbative results: those
lines cannot cross the point where the phase transition occurs,
because $a_4\sqrt{\sigma}$ increases as we approach that point. Any
attempt to follow a line of constant scalar
mass would have to deal with this problem.\\
\subsection*{Compactification effects on the scalar mass}
\label{sec:scalar-mass}
So far we have only explored the behaviour of energy scales in the
bare parameter space. However, each point we have simulated on the phase
diagram corresponds to a precise location in the space given by the
three energy scales we are interested in, that are $\Lambda_{\rm UV}$,
$\Lambda_{\rm R}$ and $m_5$. We can therefore translate our results at
$N_5=4$ and $N_5=6$ into a common set of points
$(\Lambda_{\rm UV},\Lambda_{\rm R},m_5)$. This approach allows us to study $m_5$ as
a function of the other two energy scales, instead of the bare
parameters. From now on we express the energies $\Lambda_{\rm UV}$ and
$\Lambda_{\rm R}$ using their length counterpart, $a_4\sqrt{\sigma}$ and
$R\sqrt{\sigma}$ respectively. These two length scales are related to
each other by Eq.~\eqref{eq:separation-lattice} and they are both
measured non--perturbatively: the first is directly measured, whereas
the second relies on the interpolated data of $\xi$
from Ref.~\cite{Ejiri:2000fc}.\\
The range of values of $a_4\sqrt{\sigma}$ and $R\sqrt{\sigma}$ spanned
in our simulations is shown in Fig.~\ref{fig:radius-vs-string}, and the
data we used are summarized in Tab.~\ref{tab:results-n5-4} and
Tab.~\ref{tab:results-n5-6}. In the following plots, we report results
from $N_5=4$ together with the ones from $N_5=6$. When more than one
value for $a_4\sqrt{\sigma}$ or $a_4m_5$ is extracted for the same
$(\beta_4,\beta_5)$ point, we apply the following procedure: if the
values are compatible within one standard deviation, we plot the
weighted average as central value, and the weighted error as the
statistical error; we also use the spread of the results to estimate
the systematic error due to the choice of the effective mass
plateaux. If the values are not compatible, we use the average for the
central value, whereas the systematic error is chosen to comprise both
the lowest and the highest values.\\
With our available data, we can explore the behaviour of the scalar
mass $m_5$ in the following region of lattice spacing $a_4$
\begin{equation}
\label{eq:scale-interval-a4}
0.15 \; < \; a_4\sqrt{\sigma} \; < \; 0.40
\; ,
\end{equation}
and compactification radius $R$
\begin{equation}
\label{eq:scale-interval-R}
0.05 \; < \; R\sqrt{\sigma} \; < \; 0.12
\; .
\end{equation}
\FIGURE[ht]{
\epsfig{figure=FIGS/radius_vs_string_sys.eps,width=0.77\textwidth,clip}
\label{fig:radius-vs-string}
\caption{The points in the phase space are mapped into the physical
space of the energy scales of the system. In the plot we report the
length scale corresponding to the ultra--violet cut--off, and the
one corresponding to the compactification energy. Using both data
from $N_5=4$ and $N_5=6$, we can span a larger region of this
space. Some points have two different type of error bars described
in the text: the thicker one is statistical, whereas the thinner
is systematic.}
}
The major advantage of interpreting the data in terms of these
physical quantities
is that we can disentangle compactification effects from
cut--off effects. It is clear from Fig.~\ref{fig:radius-vs-string}
that we have points at different values of the lattice spacing, but at
the same value of the compactification radius. The scalar mass on
those particular points can therefore be studied at fixed
compactification scale and different cut--off scale. On the other
hand, we also have points at the same value of the lattice spacing,
but at different radii, which can be used to study the behaviour of
the scalar mass at fixed cut--off scale. From
Fig.~\ref{fig:radius-vs-string} we can also infer that increasing
$N_5$ would allow us to explore a wider range of cut--off values for
fixed
compactification scale.\\
Our main goal is to clarify the validity of the result in
Eq.~\eqref{eq:one-loop-mass} where the perturbative scalar mass is
expected to depend strongly on the compactification scale. In our
lattice model we would like to see if there are leading cut--off
corrections to this expected behaviour when we look at the
non--perturbatively measured scalar mass. The simplest way of looking
for these corrections is to study the dependence of the scalar mass on
the lattice spacing. However, our values for the lattice spacing
usually correspond to different values of the compactification
radius. It is clear from this discussion that the study in
Ref.~\cite{deForcrand:2010be} cannot give any hints about
Eq.~\eqref{eq:one-loop-mass}: the lattice spacing always changes
together with the compactification radius, because their ratio is
forced to be constant. Nothing can be said about the dependence of
$m_5$ at fixed compactification scale nor at fixed cut--off scale from
the results of these earlier studies. Using our data, we can plot
$m_5$ as a function of $a_4$ and separately as a function of $R$. The
plots are shown in Fig.~\ref{fig:scalar}: Fig.~\ref{fig:scalar}(a)
shows the scalar mass dependence on the lattice spacing in the range
defined in Eq.~\eqref{eq:scale-interval-a4}, whereas
Fig.~\ref{fig:scalar}(b) shows its behaviour as a function of the
compactification radius in the range of
Eq.~\eqref{eq:scale-interval-R}. The observed range for the scalar
mass in units of the string tension is
\begin{equation}
\label{eq:m5sigma-interval}
2 \; < \; \frac{m_5}{\sqrt{\sigma}} \; < \; 10
\; .
\end{equation}
The important thing to notice in this analysis is how the scalar mass
changes between the two plots. While some of the points are
insensitive to the two different choices of variables, it is striking
to see points with the same mass but far away in
Fig.~\ref{fig:scalar}(a) fall on top of each other once expressed in
terms of the compactification radius in Fig.~\ref{fig:scalar}(b). If
Eq.~\eqref{eq:one-loop-mass} holds, then the combination $m_5R$ should
be independent of $R$ at leading order, while retaining any dependence
on the cut--off $a_4$. In order to separate the scalar particle from
the Kaluza--Klein modes, this variable should be less than one as we
stated in Eq.~\eqref{eq:separation-mass-cutoff}. In
Fig.~\ref{fig:m5R-vs-string} we plot $m_5R$ as a function of
$a_4\sqrt{\sigma}$. The data show a scalar mass in units of the
compactification radius in the range
\begin{equation}
\label{eq:m5R-interval}
0.2 \; < \; m_5R \; < \; 0.5
\; .
\end{equation}
Such range is smaller than the one spanned by $m_5/\sqrt{\sigma}$ by
a factor of $2$ for the same interval of lattice
spacings. This evidence support the observation that the dependence on
$a_4\sqrt{\sigma}$ is milder than the one shown in
Fig.~\ref{fig:scalar}(a) and it is compatible with the perturbative
expectation in Eq.~\eqref{eq:one-loop-mass}. The product $m_5R$ does
not show any sign of quadratic divergences as the lattice spacing is
reduced. However, we must recall that all the simulations were
performed on a fixed value of $N_4$, therefore the points at the
smallest values of $a_4\sqrt{\sigma}$ are the ones on the smallest
physical volumes and finite--size effects could be present. On the
other hand, large values of $a_4\sqrt{\sigma}$ point in the direction
of larger discretization effects.
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{figure=FIGS/scalar_vs_string_sys.eps,width=0.47\textwidth,clip}
&
\epsfig{figure=FIGS/scalar_vs_radius_sys.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:scalar}
\caption{(a) The scalar mass as a function of the lattice spacing
for $N_5=4$ and $N_5=6$; (b) The same scalar mass is also shown as
a function of the compactification radius. Points at the same
value of the radius have the same mass within the statistical
error. Each of the point in (a) corresponds to a point in
(b). Systematic errors due to the choice of the effective mass
plateaux are reported for some points using thinner error bars.}
}
\FIGURE[hbt]{
\epsfig{figure=FIGS/scalarR_vs_string_sys.eps,width=0.77\textwidth,clip}
\label{fig:m5R-vs-string}
\caption{The scalar mass in units of the compactification radius
$m_5R$ is shown to be mildly dependent on the lattice spacing
$a_4\sqrt{\sigma}$. The grey band $0.3 \leq m_5R \leq 0.4$
includes all points within two standard deviations and it is drawn
to guide the eye. Also in this case, systematic errors are shown
with thinner error bars, whereas the thicker ones represent
statistical standard deviations.}
}
\section{Conclusions}
\label{sec:conclusions}
Lattice theories in more than four dimensions prove to be very
interesting. They provide a sensible and well--defined regularization
of non--renormalizable gauge theories that can be used as UV
completions to calculate phenomenologically interesting
quantities.\\
In this work we presented a non--perturbative study of pure SU($2$)
gauge theory in five dimensions. The system was discretized on
anisotropic lattices and we investigated the so--called dimensionally
reduced phase, where a light scalar particle is expected
in perturbation theory due to
compactification of the extra dimension.\\
If the scales of the theory are properly separated, we expect the
low--energy dynamics of this theory to be described by a four--dimensional
gauge theory coupled to a scalar field. We have measured the mass of
the non--perturbative scalar states in a specific region of the bare parameter space,
where we expect to find the desired separation between physical
scales. We have also determined numerically the four--dimensional
lattice spacing in units of the string tension. This allowed us to
describe the lines of constant scalar mass and constant ultra--violet
cut--off as they arise non--perturbatively.
The scale separation of Eq.~\eqref{eq:separation-mass-cutoff} is
obtained from simulations in a given region of the phase diagram, and
is shown in Fig.~\ref{fig:scalar} and Fig.~\ref{fig:m5R-vs-string}.
The final picture seems to
confirm the observation in Ref.~\cite{deForcrand:2010be} about the
possibility of effectively describing a four--dimensional Yang--Mills
theory with a scalar adjoint particle in the continuum limit. While
that observation was based entirely on perturbative results, our
numerical simulations show how this continuum limit could be actually
reached by following lines of constant scalar mass in the parameter
space of the model. As we described in Sec.~\ref{sec:results}, this is
not a straightforward procedure and it definitely requires some sort
of fine tuning.
Even though the search for a light scalar requires fine tuning in this
simple model, we have shown that its mass is only very mildly affected
by the ultra--violet cut--off, whereas it strongly depends on the
radius of the compactified extra dimension. This is entirely
compatible with the perturbative result of
Eq.~\eqref{eq:one-loop-mass} and it is the first non--perturbative
evidence that the mass of scalar particles coming from a
compactification mechanism does not have a quadratic dependence on the
cut--off. We need to bear in mind that our results are obtained at
finite lattice spacings, both $a_4$ and $a_5$, and at finite
volume. Therefore, it would be ideal to extend our study on larger
lattices, with $N_4 > 12$ and $N_5 > 6$, in order to reduce the
systematics of finite--size effects and discretization effects.\\
We can consider our study as a starting point for exploring in more
details the realization of gauge theories with light scalar particles
in the framework of dimensional reduction. In particular, it would be
interesting to compare the non--perturbative spectrum of the
five--dimensional model obtained in this study with the
non--perturbative spectrum of a four--dimensional gauge theory with a
scalar degree of freedom. A similar comparison has been carried on
between four-- and three--dimensional gauge
theories~\cite{Hart:1999dj}, where the
super--renormalizability of the latter helped in the definition of
physical observables. Another interesting future
extension of this work could be the inclusion of
fermionic degrees of freedom, which are expected to
further reduce the scalar mass~\cite{DelDebbio:2008hb}, at least at a
one--loop level. As a consequence, they might allow the description of
theories with extended supersymmetry on the lattice without fine
tuning the scalar mass.
\acknowledgments It is a pleasure to thank Rodolfo Russo and Richard
Kenway for discussions and comments on this manuscript. This work has
made use of the resources provided by the Edinburgh Compute and Data
Facility (ECDF). ( http://www.ecdf.ed.ac.uk/). The ECDF is partially
supported by the eDIKT initiative (http://www.edikt.org.uk).\\
ER is supported by a SUPA prize studentship. ER also aknowledges
hospitality and support from the INFN, Laboratori Nazionali di
Frascati, during the final stage of this work.
\section*{Appendix}
\label{sec:appendix}
\subsection*{Extracting the string tension and the scalar mass}
\label{sec:operators}
The string tension and the mass of the ground state in the scalar
channel have been
measured at different points $(\beta_4,\beta_5,N_4,N_5)$ in the bare
parameter space. We use standard lattice spectroscopic techniques and
we extract
the masses from $2$--point functions of suitable lattice operators
coupling to the states of interest and correlated in the time
direction (which is taken to be one of the $4$ directions with $N_4$
lattice sites). The correlators are then averaged over the $N_5$ slices in
the extra dimension.\\
To extract the string tension we use Polyakov loop operators $L_i$
winding around the $3$ spatial dimensions ($i\in \{1,2,3\}$). These
operators are non--local and have a non--zero charge under the centre
symmetry group. They couple to torelon states whose mass grows
linearly with the size of the lattice. The string tension is the
coefficent of this linear dependence; this procedure yields the right
string tension if the open--close duality between Wilson loops and
Polyakov loops holds.
More specifically, the mass of the torelon states are related to the
string tension as follows:
\begin{equation}
\label{eq:torellon-mass}
a_4m_{\rm tor}(N_4) \; = \; a_4^2\sigma N_4 - \frac{\pi (D-2)}{6 N_4}
\; ,
\end{equation}
where $D$ is the number of spacetime dimensions.
From the above relation Eq.~\eqref{eq:torellon-mass} we can extract the string tension as
\begin{equation}
\label{eq:string-tension}
a_4^2 \sigma \; = \; \frac{a_4m_{\rm tor}(N_4)}{N_4} + \frac{\pi (D-2)}{6 N_4^2}
\; ,
\end{equation}
and we set $D=4$ in our analysis.\\
The systematic error in extracting the string tension using
Eq.~\eqref{eq:string-tension} is known to be small for long
Polyakov loops, i.e. loops such that $N_4 a_4 \sqrt{\sigma}
>3$. Unfortunately, measures of Polyakov loop operators are
difficult because of the poor signal--to--noise ratio; specific
techniques are usually needed in order to enhance the signal, and
obtain statistically accurate results. In this work, we use an improved diagonal
spatial smearing with a further step of blocking as first described
in Ref.~\cite{Lucini:2004my}. The set of parameters used here is the
same as in Ref.~\cite{Lucini:2004my}, namely
$(p_a,p_d)=(0.40,0.16)$ (cfr. Fig.~\ref{fig:smear-path}).
\FIGURE[b]{
\epsfig{file=FIGS/smearing.eps,width=0.61\textwidth,clip}
\label{fig:smear-path}
\caption{Example of smearing of a single link. Sum of orthogonal
staples and diagonal staples are weighted with two independent
parameters: $p_a$ and $p_d$. The figure is taken
from~\cite{Lucini:2010nv}.}
}
The diagonal correlators of spatial Polyakov loops at different
blocking levels are analysed using a single--state hyperbolic cosine
fit to extract the effective mass, and jackknife bins are used to
estimate the statistical errors. For all the points where we measure
the string tension, the best projection onto the ground state is
obtained at the maximum blocking level. This was confirmed using a
variational procedure on the set of operators including all the
different blocking levels. For example, in
Fig.~\ref{fig:variational-comparison} we show the comparison between
the mass extracted from diagonal correlators of the operator at the
highest level of blocking and the one coming from the variational
procedure. The comparison was done on a lattice with a longer temporal
direction $L_t = 2 L_4$ and using the same fitting window for the
effective mass plateaux in both cases.\\
On the points $(\beta_4,\beta_5,N_4,N_5)$ used for the measurements,
we extracted the effective mass plateaux for the spatial Polyakov
loops only at large temporal distances. The smearing and blocking
algorithm allows for the extraction of a better signal, even though
the parameters $(p_a,p_d)$ are not optimized for the broad range of
lattice spacings $a_4$ explored in this work. In many cases, we still
have small overlaps with the ground state, and consequently the
single--state behaviour of the correlator can only be extracted at
large temporal distances. An example of such cases is shown in
Fig.~\ref{fig:tor-plateaux}(a), whereas in
Fig.~\ref{fig:tor-plateaux}(b) we show one
of the points where the plateaux is reached already at $t/a_4=2$.\\
A summary of all the torelon masses and their corresponding string
tensions is reported in Tab.~\ref{tab:torelons-n5-4}, and
Tab.~\ref{tab:torelons-n5-6}. The fitting range for the effective mass
plateaux is also shown in the tables. Moreover, since the length of
the Polyakov loops in lattice units is different between the $N_5=6$
lattices and the $N_5=4$ ones, we also report the physical size
$L_4\sqrt{\sigma}$. As mentioned above, finite--size effects can be
kept under control when $L_4\sqrt{\sigma}$ is large; in other words we
would like our physical lattice volume to be much larger than the
typical correlation length of the system, given by the
inverse of the string tension.\\
\FIGURE[ht]{
\epsfig{file=FIGS/plateaux_0.88_3.60_cfrvar.eps,width=0.77\textwidth,clip}
\label{fig:variational-comparison}
\caption{Comparison between the ground state effective mass
extracted from the diagonal correlator of the highest blocking
level (black points) and the one extracted from the variational
procedure (red circles). The operators were
$L_4=12a_4$ spatial Polyakov loops at $4$ different blocking levels,
and their correlator was measured along a $L_t=2L_4=24a_4$
temporal distance. The correlator was averaged over the $N_5=6$
extra dimension slices.}
}
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{file=FIGS/plateaux_0.90_3.45.eps,width=0.47\textwidth,clip}
&
\epsfig{file=FIGS/plateaux_0.86_3.70.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:tor-plateaux}
\caption{Example of plateaux of torelon effective masses. (a) The
final mass comes from a weighted fit of the points in the plateaux
that is reached only at $t/a_4=3$ due to the small overlap of the
operator onto the ground state. (b) At lower masses the plateaux
is longer and the signal is extracted more reliably.}
}
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{file=FIGS/plateaux_0.89_3.55_scalar.eps,width=0.47\textwidth,clip}
&
\epsfig{file=FIGS/plateaux_0.90_3.45_scalar.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:scal-plateaux}
\caption{Example of plateaux for one of the highest scalar
masses (a) and one of the lowest (b). (a) The operator overlaps
poorly on the ground state and the
plateaux is reached at large temporal distances. In this case we
tried to estimate the systematic error on the fitting range, by
choosing two different fitting windows. (b) The plateaux is
reached already at $t/a_4=2$.}
}
For the mass of the static scalar mode, we use compact
Polyakov loop operators, that is gauge--invariant combinations of
Polyakov loops winding around the extra fifth dimension. Such
operators transform as scalars under the cubic symmetry group and they
only carry a site index in the four--dimensional subspace.
In particular, we choose two different combinations
\begin{equation}
\label{eq:scalar-1}
\mathcal O_1(t) \; = \; \sum_x {\rm Tr}~[L_5(x,t)] \; ; \qquad
L_5(x,t)=\prod_{j=1}^{N_5} \mathcal U_5(x+ja_5\hat{5},t)
\; ,
\end{equation}
and
\begin{equation}
\label{eq:scalar-2}
\mathcal O_2(t) \; = \; \sum_x {\rm Tr}~[\phi(x,t) \phi^\dag(x,t)] \; ; \qquad
\phi(x,t)=\frac{L_5-L_5^\dag}{2}
\; .
\end{equation}
The sum $\sum_x$ is an average over the spatial volume in order to
obtain zero--momentum operators on a fixed timeslice $t$. We average
the correlators over the
extra--dimensional coordinate, as in the previous case.\\
The first operator $\mathcal O_1$ is the same one used in
Ref.~\cite{deForcrand:2010be}. For the operator in
Eq.~\eqref{eq:scalar-2} it is possible to apply a smearing procedure
following the one introduced in Ref.~\cite{Irges:2006hg} for a scalar
Higgs field. The operator $\phi$ is replaced by a smeared version that
consists of a gauge--invariant combination of parallel transporters in
the
three--dimensional spatial subspace.\\
For this observable, we found the lowest smearing level of $\mathcal O_2$ to
have the largest projection onto the ground state. The effective
masses extracted from $\mathcal O_1$ and the lowest smearing level of $\mathcal O_2$
are always compatible. In some cases, usually at very low masses
$a_4m_5$, the smeared operators show better plateaux, but we have not
yet studied their projection onto the ground state with a more
systematic variational procedure. At this stage we have not
implemented more efficient noise--reduction techniques, such as a
better choice for the smearing parameters, a multi--level
algorithm~\cite{Luscher:2001up}, or a multi--hit procedure. As
a consequence, we obtain plateaux like the ones shown in
Fig.~\ref{fig:scal-plateaux}(a). The mass is extracted from a weighted
fit of three, or sometimes even two, points at very large temporal
distance, where the signal--to--noise ratio is quite small. Clearly
there also points where the scalar mass is small, and the effective
mass plateaux is well behaved. An example can be found in
Fig.~\ref{fig:scal-plateaux}(b). We summarize the scalar masses
$a_4m_5$, and the fitting windows for the plateaux in
Tab.~
, and Tab.~
.\\
\FIGURE[hb]{
\epsfig{file=FIGS/volume_effects_V12_V16.eps,width=0.77\textwidth,clip}
\label{fig:volume-effects}
\caption{For three different points reported on the $x$ axis, we
show the string tension and the scalar mass in units of the
cut--off length. Two volumes are compared and sizable finite--size
effects can be ruled out.}
}
An attempt to estimate the finite--volume effects on the observables
$a_4\sqrt{\sigma}$ and $a_4m_5$ has been performed at $N_5=6$. For
three different points in the phase space $(\beta_4,\beta_5)$, we
simulate two different four--dimensional lattice sizes, $N_4=12$ and
$N_4=16$. The three points have a very similar string tension at
$N_4=12$, but on that volume $L_4\sqrt{\sigma}$ turns out to be
smaller than $3$. The results are summarized in
Tab.~\ref{tab:volume-effects}. The string tension and the scalar mass
are not affected by the change of four--dimensional volume. In
Fig.~\ref{fig:volume-effects} the volume dependence is shown for both
the observables. The larger statistical error for $a_4\sqrt{\sigma}$
on the largest volume is due to the larger torelon mass at $N_4=16$
(the number of configurations is the same for both volumes).\\
\TABLE[t]{
\begin{tabular}[h]{c|c|c||c|c|}
\cline{2-5}
&\multicolumn{2}{|c||}{$a_4\sqrt{\sigma}$}&\multicolumn{2}{|c|}{$a_4m_5$}\\
\hline
$N_4$ & 12 & 16 & 12 & 16 \\
\hline
(0.845,3.80) & 0.2358(11) & 0.2364(25) & 1.05(31) & 1.44(9)\\
(0.870,3.65) & 0.2358(11) & 0.2359(27) & 0.998(93) & 1.08(26)\\
(0.870,3.65) & 0.2383(12) & 0.2392(25) & 0.777(78) & 0.956(99)\\
\hline
\end{tabular}
\label{tab:volume-effects}
\caption{Comparison between the observables on two different
four--dimensional volumes and fixed $N_5=6$. The string tensions
are independent of $N_4$ and the scalar masses are compatible
within one standard deviation.}
}
\subsection*{Mixing with scalar glueball states}
\label{sec:glueballs}
Since we are studying a strongly coupled Yang--Mills theory, the
low--energy dynamics could be affected by the presence of
non--perturbative states, such as glueballs.
It is well known that the lightest glueball state appears in
the scalar channel. However, this is the same symmetry channel where
we perturbatively expect to see a light particle due to the
compactification mechanism. It is therefore mandatory to check whether
these two states mix in order to shed light on the non--perturbative
fate of Eq.~\eqref{eq:one-loop-mass}.\\
Lattice calculations of glueball masses suffer from the aforementioned problems
in relation to torelon masses: to obtain an accurate
estimate of the mass from correlation functions, one needs to adopt
noise--reduction techniques. We used a combination of the improved
diagonal smearing described in Fig.~\ref{fig:smear-path} and a
variational ansatz. We used three different spatially shaped Wilson loops
in order to construct glueball operators. This procedure has been very
succesfull in extracting highly accurate glueball
masses in three and four--dimensional $\SU{N}$ gauge
theories~\cite{Lucini:2004my,Lucini:2010nv,Morningstar:1999rf}.\\
To create operators coupling to glueball states in four dimensions, we
use the four--links plaquette, the six--links rectangular
plaquette and the six--links chair shown in
Fig.~\ref{fig:glue-path}. Symmetrized combinations of these operators
projecting only onto the scalar representation of the three--dimensional
cubic symmetry group are then correlated together with operators in
Eq.~\eqref{eq:scalar-1} and Eq.~\eqref{eq:scalar-2}; we
refer to these scalar glueball operators as $\mathcal O_a$, $\mathcal O_b$ and
$\mathcal O_c$ built starting respectively from the path a), b) and c) in
Fig.~\ref{fig:glue-path} (we always use zero--momentum
projections).\\
\FIGURE[b]{
\epsfig{file=FIGS/glue_path.eps,width=0.61\textwidth,clip}
\label{fig:glue-path}
\caption{Wilson loops used in the construction of glueball operators
in the scalar channel. Each of these three operators is smeared
according to Fig.~\ref{fig:smear-path} in order to construct a larger
variational ansatz.}
}
We expect that the operators in Fig.~\ref{fig:glue-path} will couple
mainly onto glueball states as they are built entirely from links in
the three--dimensional spatial subspace of the lattice. On the other
hand, we suggest that the operators in Eq.~\eqref{eq:scalar-1} and
Eq.~\eqref{eq:scalar-2} will couple mainly with states of
extra--dimensional nature because they are
built from links in the extra direction. Inevitably, due to the
non--perturbative nature of the theory, the masses of the scalar
state extracted from correlators of the latter type of operators could
be affected by non--negligible mixing with glueball states. We studied
the contribution of this mixing, and the results are reported and explained in
Sec.~\ref{sec:results}.\\
To estimate the mixing in the scalar spectrum of the lattice theory, we used
the following procedure:
\begin{itemize}
\item compute the full correlation matrix $C_{\alpha \beta}(t)$, where the
lower indices run over the scalar operators of the following type:
$\mathcal O_2$, $\mathcal O_a$, $\mathcal O_b$, $\mathcal O_c$
\item employ a variational procedure to find a linear combination of
the correlated operators such that the propagating state is the
lightest (or apply the procedure on the orthogonal space to get the
excited ones)
\item decompose the approximate mass eigenstates obtained from the
previous step into their
projections onto the basis operators $\mathcal O_2$, $\mathcal O_a$, $\mathcal O_b$,
$\mathcal O_c$.
\end{itemize}
The last step of this variational analysis gives us informations about
the nature of the propagating state. If the main projection is onto
glueball operators $\mathcal O_a$, $\mathcal O_b$ and $\mathcal O_c$, the mass extracted is
likely to be associated to a glueball state rather than a scalar of
extra--dimensional origin. At the same time, a projection onto $\mathcal O_2$
of more than $50\%$ indicates that the state investigated is probably
a scalar coming from the compactification mechanism.\\
Due to the large computational cost, we measured the full correlation
matrix $C_{\alpha \beta}(t)$ only on a subset of the
points reported in Tab.~\ref{tab:points-n5-4}: we choose points at
fixed $\gamma \approx 1.54$ and we investigate how the mixing of the
extracted states changes as we increase $\beta$, moving away from the
line of second order phase transition. On these points, the masses
extracted using only correlators of $\mathcal O_1$ and $\mathcal O_2$, as described
in the previous section, has been shown in
Fig.~\ref{fig:scalar-gamma}. In order to extract a more reliable
plateau, we increased the number of lattice points in the temporal
direction $L_t=2 L_4$; this allowed us to follow the plateaux of the
effective mass for a wider range of temporal distances, usually
corresponding to a fitting range $t_{\rm min}-t_{\rm max} = [3-7]$
in units of the lattice spacing $a_4$ (cfr. for example the fitting
ranges of Tab.~5). The results for the spectrum of the theory in the
scalar channel at $\gamma \approx 1.54$ is summarized in
Fig.~\ref{fig:scalar-gamma}. An example of the effective mass plateax
for the ground state and its low--energy excitations is also shown in
Fig.~\ref{fig:plateaux-scalar} for two values of $\beta$; we compare
the results of the variational procedure, with the results obtained
from diagonal correlators of pure scalar operators.\\
\FIGURE[ht]{
\begin{tabular}{cc}
\epsfig{file=FIGS/eff_mass_1.72.eps,width=0.47\textwidth,clip}
&
\epsfig{file=FIGS/eff_mass_1.76.eps,width=0.47\textwidth,clip}
\\
{\scriptsize (a)} & {\scriptsize (b)} \\
\end{tabular}
\label{fig:plateaux-scalar}
\caption{Example of effective mass plateaux for two different values
of $\beta$. (a) At $\beta = 1.72$ the mass of the low--lying
scalar state obtained from the variational ansatz is compatible
with the one we measured using only scalar operators. (b) At
$\beta = 1.76$ the scalar operator yelds a mass which is
compatible with the second excitations of the scalar spectrum.}
}
In the range of parameters explored with the full variational ansatz,
we notice that the relative mixing of the scalar states
in the spectrum with the different operators in the correlator matrix
changes with $\beta$. The mixing of the extracted ground state
is shown in Fig.~\ref{fig:mixing-ground}. The relative projections for
both sets of operators are shown for different values of $\beta$. The
plot clearly shows the contribution of the operator $\mathcal O_2$ to the ground
state of the scalar channel decreasing as $\beta$ increases. We
recall here that increasing $\beta$ at fixed anisotropy corresponds to
going towards the weak--coupling limit. This is the same limit taken
in the simulations of Ref.~\cite{deForcrand:2010be}, where it has been
shown how the mass extracted from correlators of our $\mathcal O_1$ diverges
and decouples from the low--energy spectrum. It is therefore not
surprising that the lightest glueballs become relevant to the dynamics
of the
theory in this region of the parameter space. On the other hand, we
clearly see that at lower values of $\beta$, closer to the line of
second order phase transition, the scalar state has a dominant
contribution from the extra--dimensional operator $\mathcal O_2$ and
it increases as we lower the values of $\beta$.\\
Another interesting mixing we looked at is shown in
Fig.~\ref{fig:mixing-exc}. The plots show the relative mixing of the
first excited state onto the operators in the variational set. The
points where the mixing is calculated are the same as in
Fig.~\ref{fig:mixing-ground}. Up to $\beta = 1.75$, the first excited
state is dominated by a projection onto the scalar operator $\mathcal O_2$,
suggesting an extra--dimensional nature for this particle. What it is
not shown is that at $\beta = 1.76$, we find the second excited state
to project mostly onto $\mathcal O_2$ (cfr. Fig.\ref{fig:plateaux-scalar}(b)).\\
\bibliographystyle{JHEP}
|
{
"timestamp": "2012-07-04T02:03:47",
"yymm": "1203",
"arxiv_id": "1203.2116",
"language": "en",
"url": "https://arxiv.org/abs/1203.2116"
}
|
\section{Introduction}
Superheavy nuclei of $Z \ge 104$ were synthesized either in
``cold'' fusion reactions on closed-shell $^{208}$Pb and
$^{209}$Bi target nuclei bombarded by projectiles ranging from Ti
to Zn or in "hot" fusion reactions, in which the heaviest
available actinide targets were bombarded with the neutron rich
$^{48}$Ca projectiles. See review articles
\cite{Hofmann-review,Hof-Munz-review} and \cite{Oganessian},
respectively. In the cold fusion reactions only one neutron is
emitted from the compound nucleus to form the final
compound-residue nucleus in its ground state. In hot fusion
reactions more neutrons are emitted. At each step of the
deexcitation cascade the neutron evaporation competes with the
dominating process of fission. Therefore the synthesis cross
section represents only a small part of the fusion cross section.
A characteristic feature of the fusion-evaporation reactions
leading to the synthesis of superheavy nuclei is enormous
hindrance of the fusion process itself. Consequently, the cross
sections for the synthesis of heaviest elements are measured in
picobarns or even femtobarns. It is believed that the hindrance is
caused by the highly dissipative dynamics of the fusing system in
its passage over the saddle point on the way through the
multidimensional potential energy surface from the initial
configuration of two touching nuclei into the configuration of the
compound nucleus. Zagrebaev and Greiner developed a method of
solving Langevin equations of motion to describe this stochastic
stage of the fusion process \cite{Zagreb-Langevin}. In spite of
very time consuming Langevin trajectory calculations, in which
only one of say a million trajectories leads to formation of the
compound nucleus, the model is used effectively to calculate
synthesis cross sections for various reactions \cite{Zagreb-08}.
Another approach to the process of fusion of a ``dinuclear
system'' (DNS) was proposed in Ref. \cite{DNS}. It was assumed in
this model that the dinuclear system stays in contact
configuration and undergoes successive transfer of all nucleons
from the lighter nucleus to the heavier partner (in competition
with the quasi-fission processes). Applications of this concept
have been used in recent years by several groups. In still another
approach, the ``fusion-by-diffusion'' model \cite{FBD-05}, the
stochastic process of shape fluctuations that lead to the
overcoming the saddle point was described as the solution of the
Smoluchowski diffusion equation in the deformation space along the
fission valley.
The cold fusion reactions leading to the synthesis of nuclei of $Z
\le 113$ were studied systematically with the DNS model in Ref.
\cite{Cherepanov}, with the fusion-by-diffusion (FBD) model
\cite{FBD-05}, \cite{FBD-11} and with the Langevin dynamics model
\cite{Zagreb-08}. The hot fusion reactions leading to the
synthesis of the heaviest nuclei of $Z \ge 114$ have not been
studied so systematically. In Ref. \cite{Zagreb-08} excitation
functions for some selected reactions were calculated although
they were not confronted with experimental cross sections. Most of
the publications on this topic concentrated on the predictions
concerning possible ways of synthesis of the heaviest elements of
$Z=119$ and 120 \cite{Zagreb-08,Liu-120,Adamian-2009,Siwek-120,
Nasirov-120,Nasirov-120a,Zagreb-12,Wang-12}. Only very recently,
an extensive study of cold and hot fusion reactions in terms of a
phenomenological approach based on the DNS model was reported
\cite{Scheid-syst}.
There is one important aspect of all the models of the synthesis
of superheavy nuclei that was not treated with proper attention so
far. This is the question of the choice of theoretical fission
barriers and ground-state masses which have to be adopted for the
description of the deexcitation of the compound nucleus. It is
well known that calculations of the cross sections for synthesis
of superheavy nuclei are extremely sensitive to the height of the
fission barrier, especially in case of ``hot'' fusion reactions,
in which three or four neutrons are emitted from the compound
nucleus. When the barrier heights are not known precisely, an
error in evaluation of the $\Gamma_n/\Gamma_f$ ratio in each step
of the ($xn$) deexcitation cascade accumulates $x$ times leading
to enormous errors in the calculation of the synthesis cross
sections. (Here, $\Gamma_n$ and $\Gamma_f$ denote the neutron
decay width and fission width, respectively.) Thus, precise
knowledge of theoretical fission barriers and neutron binding
energies (ground-state masses) is crucial for reasonable
predictions of the synthesis cross sections.
In the last decade the mass tables of M\"oller et al.
\cite{Moller} have most frequently been used in the field of
superheavy nuclei. Unfortunately, fission barriers heights are not
given in these tables. Therefore, in most of the mentioned above
calculations of the synthesis cross sections the ground-state
shell effect of the compound nucleus (that is listed in these
tables) was used as the barrier height. In this simplification,
both the macroscopic deformation energy and the shell effect at
the saddle configuration are neglected. It seems, therefore, that
these approximate values of the fission barrier are not
sufficiently accurate to guarantee reliable predictions of the
synthesis cross sections. (Absolute value of both these neglected
effects may be of about 1--2 MeV each, while a 1 MeV-shift of the
barrier height may result in a change of the calculated cross
section of $3n$ or $4n$ reaction by 2--3 orders of magnitude.)
Only in recent years systematic compilations of theoretical
fission barriers of superheavy nuclei (combined with the necessary
information on the ground-state masses) have became available in
literature. Calculations in framework of the macro-microscopic
approach were reported by Muntian et al. \cite{Muntian} and later
by M\"oller at al. \cite{Moller-barriers}. The model
\cite{Muntian} has been extended recently by Kowal et al.
\cite{Kowal-10,Kow-Sob} by the inclusion of nonaxiality as an
important new degree of freedom. Fission barriers of superheavy
nuclei have been calculated also in a number of other papers
within various models (see Ref. \cite{Ring}, Table IV for a
review), however no sufficiently systematic information on the
fission barriers and, simultaneously, ground-state masses has been
provided.
In the present study we adopt the ``fusion-by-diffusion'' (FBD)
model \cite{FBD-05,FBD-11} for calculating the synthesis cross
sections of the heaviest nuclei in hot fusion ($xn$) reactions by
using the information on the fission-barrier heights
\cite{Kowal-10,Kow-Sob} and other properties of the superheavy
nuclei obtained within the Warsaw macroscopic-microscopic model
\cite{Muntian}.
The whole set of experimental data
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2}
on the synthesis of new superheavy elements of $Z$ = 114--118
(obtained in Dubna by Oganessian and coworkers and later in a
series of confirming experiments at GSI Darmstadt and LBNL
Berkeley) was analyzed. Based on this test of the model
predictions, the calculations were then performed for
experimentally unexplored yet reactions aimed at the synthesis of
new elements of $Z$ = 119 and 120.
\section{Review of the FBD model}
The fusion-by-diffusion (FBD) model \cite{FBD-05,FBD-11} serves
for calculating cross sections for the synthesis of superheavy
nuclei. Recently the model was modified in order to describe both
cold fusion ($1n$) and hot fusion ($xn$) reactions. In this
extended version \cite{FBD-11}, for each angular momentum $l$ the
partial evaporation-residue cross section $\sigma_{ER}(l)$ for
production of a given final nucleus in its ground state is
factorized as the product of the partial capture cross section
$\sigma_{cap}(l)=\pi \lambdabar^2(2l+1)T(l)$, the fusion
probability $P_{fus}(l)$ and the survival probability
$P_{surv}(l)$:
\begin{equation}
\label{factorize}
\sigma_{ER} = \pi \lambdabar^2 \sum_{l = 0}^{\infty}(2l+1)
T(l)\cdot P_{fus}(l)\cdot P_{surv}(l).
\end{equation}
The capture transmission coefficients $T(l)$ are calculated in a
simple sharp cut-off approximation, where the upper limit
$l_{max}$ of full transmission, $T(l)=1$, is determined by the
capture cross section known from the systematics described in
Refs. \cite{FBD-11,KSW04}. Here $\lambdabar$ is the wave length,
$\lambdabar^2=\hbar^2/2\mu E_{c.m.}$, and $\mu$ is the reduced
mass of the colliding system. The fusion probability $P_{fus}(l)$
is the probability that the colliding system, after reaching the
capture configuration (sticking), will eventually overcome the
saddle point and fuse, thus avoiding reseparation. The other
factor in Eq. (1), the survival probability $P_{surv}(l)$, is the
probability for the compound nucleus to decay to the ground state
of the final residual nucleus via evaporation of light particles
and $\gamma$ rays, thus avoiding fission.
The cross sections for the synthesis of superheavy nuclei are
dramatically small because the fusion probability $P_{fus}(l)$ is
hindered (in some reactions even by several orders of magnitude)
due to the fact that the saddle configuration of the heaviest
compound nuclei is much more compact than the configuration of two
colliding nuclei at sticking. It is assumed in the FBD model that
after the contact of the two nuclei, a neck between them grows
rapidly at an approximately fixed mass asymmetry and constant
length of the system. This ``neck zip'' is expected to carry the
system towards the bottom of the asymmetric fission valley. This
is the ``injection point'', from where the system starts its climb
uphill over the saddle in the process of thermal fluctuations in
the shape degrees of freedom. Theoretical justification of the
above picture of fast zipping the neck was given in Ref.
\cite{Boilley}, where the later stage of the stochastic climb
uphill was described by solving the two-dimensional Langevin
equation. Theoretical location of an effective injection point can
be deduced from this model \cite{Boilley}. Also in a modified
fusion-by-diffusion model \cite{Xe+Xe} the location of the
injection point was estimated theoretically. In our model we rely,
however, on empirical determination of the injection point. Its
location in the asymmetric fission valley, $s_{inj}$, is the only
adjustable parameter of the FBD model.
By solving the Smoluchowski diffusion equation, it was shown in
Ref. \cite{Acta} that the probability of overcoming a parabolic
barrier for the system injected on the outside of the saddle point
at an energy $H$ below the saddle is:
\begin{equation}
\label{hindrance} P_{fus} = \frac{1}{2}(1-{\rm erf}\sqrt{H/T}\,),
\end{equation}
where $T$ is the temperature of the fusing system. The energy
threshold $H$ opposing fusion in the diffusion process is thus the
difference between the energy of the saddle point $E_{saddle}$ and
the energy of the combined system at the injection point
$E_{inj}$, where $E_{inj}$ is calculated using algebraic
expressions given in Ref. \cite{FBD-11} which approximate the
potential energy surface along the fission valley. The energy of
the saddle point is given by the adopted theoretical value of the
fission barrier $B_f$ and the ground-state energy of the compound
nucleus. The corresponding values of the rotational energy at the
injection point and at the symmetric saddle point are calculated
assuming the rigid-body moments of inertia at these configurations
\cite{FBD-11}.
As regards the survival probability $P_{surv}$, the standard
statistical-model calculation were done by applying the Weisskopf
formula for the particle (neutron) emission width $\Gamma_n$, and
the conventional expression of the transition-state theory for the
fission width $\Gamma_f$. The level density parameters $a_n$ and
$a_f$ for neutron evaporation and fission channels were calculated
as proposed by Reisdorf \cite{Reisdorf}, with shell effects
accounted for by the Ignatyuk formula \cite{Ignatyuk}. All details
regarding the calculations of the survival probability $P_{surv}$
can be found in our recent paper \cite{FBD-11}. In case of
calculating multiple evaporation ($xn$) channels a simplified
algorithm avoiding the necessity of using the Monte Carlo method
was applied \cite{Cap-xn}.
\section{Calculations for Z=114-120 elements with the
macro\-scopic-microscopic barriers}
As pointed out in the Introduction, calculations of the cross
sections for synthesis of superheavy nuclei are extremely
sensitive to the height of the fission barrier, especially in case
of ``hot fusion'' reactions because at each step of deexcitation
cascade the competition between neutron emission and fission
strongly depends on the difference of energy thresholds for these
two decay modes. Therefore, in attempts to reasonably calculate
the synthesis cross sections, the choice of realistic and
consistent theoretical information on the fission barrier heights
and the ground-state masses is essential. In our previous
applications of the FBD model, devoted mostly to analysis of cold
fusion reactions (of $Z$ of the compound nucleus $Z_{CN}\le 113$),
fission barriers based on the Thomas-Fermi model
\cite{Thomas-Fermi} were used. In Ref. \cite{KSW-09} it was
observed, however, that for heavier nuclei of $Z_{CN} \ge 114$
produced in hot fusion reactions the fission barriers based on the
Thomas-Fermi model are evidently too high, while barriers based on
the Warsaw macroscopic-microscopic model \cite{Muntian} lead to
better agreement with experimental observations. Therefore results
of the new macroscopic-microscopic calculations of the Warsaw
group \cite{Kowal-10}, involving an extended multi-dimensional
deformation space, have been chosen as the saddle-point and
ground-state input to the FBD model. The published \cite{Kowal-10}
results for even-even nuclei have been supplemented with
unpublished yet results for odd-$Z$ and/or odd-$N$ nuclei
\cite{Kow-Sob}.
In the first stage of calculations a complete set of experimental
data
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2}
on the synthesis of $Z$=114-118 elements in
reactions induced by $^{48}$Ca projectiles on $^{242,244}$Pu,
$^{243}$Am, $^{245,248}$Cm, $^{249}$Bk and $^{249}$Cf targets was
analyzed with the aim to determine location of the injection point
$s_{inj}$. Here $s_{inj}$ is defined as the excess of the total
length of the combined system over the length of the initial
system (at the touching configuration) when the neck-zip process
brings the system to the asymmetric fission valley.
In order to determine systematics of $s_{inj}$ for the set of hot
fusion reactions
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2},
the individual values of $s_{inj}$ were deduced for each reaction
and each particular xn channel by adjusting the assumed
$s_{inj}$-value to the experimental synthesis cross section at the
maximum of a given xn excitation function. The compilation of so
deduced $s_{inj}$-values is displayed in Fig. 1 as a function of
the kinetic energy excess $E_{c.m.}-B_0$ above the Coulomb barrier
$B_0$. (For the definition of $B_0$ see Ref. \cite{FBD-11}.)
It should be commented here that values of $s_{inj}$ are inferred
from the synthesis cross sections in a model-dependent way,
assuming particular ground-state masses and fission barriers.
Therefore the result of this procedure obviously depends to some
extent on these theoretical input data used in the calculations.
Consequently, the systematics of $s_{inj}$ obtained in
calculations employing different sources of the theoretical input
data may appear different (cf. the $s_{inj}$ systematics obtained
in recent calculations of cold fusion reactions \cite{FBD-11}
analyzed assuming masses and fission barriers based on the
Thomas-Fermi model \cite{Thomas-Fermi}).
It is clearly seen from Fig. 1 that the injection distance
$s_{inj}$ increases with the decreasing energy $E_{c.m.}-B_0$, in
agreement with expectations based on the dynamics of
nucleus-nucleus collisions, for example the classical trajectory
calculations \cite{Feldmeier}. Very good correlation between the
$s_{inj}$-values and the corresponding energies $E_{c.m.}-B_0$ can
be viewed as an argument in favor of the fission barriers of Kowal
et al. \cite{Kowal-10,Kow-Sob} because such a striking correlation
would be very unlikely if the theoretical barrier heights were
inconsistent with experimental values.
A linear fit to the dependence of $s_{inj}$ on $E_{c.m.}-B_0$ in
Fig. 1,
\begin{equation}
s_{inj}\approx 4.09 \ {\rm fm}- 0.192(E_{c.m.}-B_0) \ {\rm
fm/MeV}, \label{sinj}
\end{equation}
represents the only empirical input to our model and once this
systematics of the injection-point distance is determined in form
of Eq. (\ref{sinj}), one can use the FBD model to calculate
excitation functions of fusion-evaporation reactions without any
adjustable parameters.
In Fig. 2 we present a comparison of our FBD model predictions of
excitation functions for different $xn$ channels with experimental
synthesis cross sections (assigned to the corresponding xn
channels) in the following hot fusion reactions:
$^{244}$Pu($^{48}$Ca,xn)$^{292-x}$114
\cite{Oganessian,Oga-244Pu,Oga-242Pu,GSI-244Pu,GSI-244Pu2},
$^{243}$Am($^{48}$Ca,xn)$^{291-x}$115
\cite{Oganessian,Oga-243Am,Oga-243Am2},
$^{245}$Cm($^{48}$Ca,xn)$^{293-x}$116 \cite{Oganessian,Oga-244Pu},
$^{248}$Cm($^{48}$Ca,xn)$^{296-x}$116
\cite{Oganessian,Oga-242Pu,GSI-248Cm},
$^{249}$Bk($^{48}$Ca,xn)$^{297-x}$117 \cite{Oga-249Bk} and
$^{249}$Cf($^{48}$Ca,xn)$^{297-x}$118 \cite{Oganessian,Oga-249Cf}.
The largest deviations of our general fit to the data approach a
factor of 10 that corresponds effectively to a difference of about
0.5 MeV in the assumed height of the theoretical fission barrier.
Given this high sensitivity of the model predictions to the
assumed fission barrier heights, the overall agreement between the
FBD predictions and measured cross sections is quite satisfactory.
(It is rather unlikely that the accuracy of the theoretical
predictions of individual fission barriers might be much better
than $\pm 0.5$ MeV.)
It is instructive to compare results of calculations presented in
Figs. 1 and 2 with predictions for an alternative set of
theoretical fission barriers. In Fig. 3 we present individual
values of the injection distance $s_{inj}$ deduced for the same
set of data on hot fusion reactions
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2},
but obtained assuming fission barriers of M\"oller et al.
\cite{Moller-barriers}, the only alternative, complete set of
necessary information available in literature. The barriers of
M\"oller et al. are considerably higher than barriers of Kowal et
al. \cite{Kowal-10,Kow-Sob}, thus resulting in larger values of
the calculated survival probability $P_{surv}$. Consequently, the
procedure of ``calibrating'' the individual $s_{inj}$ values by
fitting the predictions to experimental cross sections resulted in
larger values of the determined injection distance $s_{inj}$.
Contrary to the consistent systematics of $s_{inj}$ values shown
in Fig. 1, Fig. 3 demonstrates the evident inconsistency of the
set of $s_{inj}$ values obtained for the barriers of Ref.
\cite{Moller-barriers}. It is seen from Fig. 3 that the $s_{inj}$
values range from 5.5 fm to 8.5 fm and are too large to have a
reasonable physical meaning. (In most cases, they correspond to
the injection distance that exceeds the distance of the scission
configuration.) Most importantly, the individual points in Fig. 3
seam to be almost randomly scattered and do not show any
correlation with energy.
There is one more inconsistency that can be noticed when the
fission barriers of Ref. \cite{Moller-barriers} and the
ground-state masses \cite{Moller} are used. Namely, for these high
fission barriers and corresponding $Q$-values, the predicted
positions of the maxima of the $xn$ excitation functions are
shifted by some 5--7 MeV toward lower energies as compared with
the data (and also with respect to the predictions for barriers of
Ref. \cite{Kowal-10,Kow-Sob}). This effect is illustrated in Fig.
4, where the data for $3n$ and $4n$ channels in the
$^{243}$Am($^{48}$Ca,xn)$^{291-x}$115 reaction are compared with
the excitation functions calculated for these two reaction
channels. This considerable energy shift, seen also for other
reactions, stems from the fact that for the M\"oller's barriers
\cite{Moller-barriers} and the corresponding ground-state masses
\cite{Moller}, the fission barrier $B_f$ is larger than the
neutron binding energy $B_n$ for all the compound nuclei formed in
the studied reactions. Consequently, the $\Gamma_n/\Gamma_f$ ratio
rises very fast at low excitation energies thus influencing the
position and shape of the $xn$ excitation functions.
From Figs. 1 and 2 it is seen that contrary to the generally
higher fission barriers of Ref. \cite{Moller-barriers}, the input
data of Kowal et al. \cite{Kowal-10,Kow-Sob} give a reasonable
agreement of the calculated and measured cross sections as well as
the very clear correlation between $s_{inj}$ and $E_{c.m.}-B_0$
that ``calibrates'' the injection distance $s_{inj}$. This
entitles us to believe that the set of theoretical fission-barrier
heights and ground-state masses \cite{Kowal-10,Kow-Sob} is quite
adequate for a wide range of the heaviest nuclei considered in
this study. Therefore we are going to use them for predictions of
cross sections of yet unexplored reactions aimed at the synthesis
of new elements $Z$ = 119 and 120.
Regarding possibilities to produce the element $Z$ = 119 we
consider, first of all, the most preferred reactions induced by
the favorable beam of $^{48}$Ca on two isotopes of einsteinium,
$^{252}$Es and $^{254}$Es. These extremely difficult-to-produce
targets might possibly be available in the near future. Therefore
we present in Figs. 5(a) and 5(b) the predicted energy dependence
of the $xn$ cross sections in reactions on these two isotopes. The
largest cross section, which turns out to be at the edge of
experimental possibilities (about 0.2 pb in 4n channel at
$E_{c.m.}\approx 220$ MeV), is predicted for the
$^{252}$Es($^{48}$Ca,xn)$^{300-x}$119 reaction. Surprisingly, the
cross section in the reaction on a more neutron-rich target,
$^{254}$Es($^{48}$Ca,xn)$^{302-x}$119, is by one order of
magnitude lower (only about 15 fb). This is a consequence of lower
fission barriers \cite{Kowal-10,Kow-Sob} in the chain of
subsequent neutron-emitting nuclei, $B_f$ = 4.87 MeV, 4.98 MeV,
5.77 MeV in $^{302}$119, $^{301}$119 and $^{300}$119, while in a
chain of neutron decays starting from the $^{300}$119 nucleus, the
predicted fission barriers are 5.77 MeV, 5.55 MeV and 6.03 MeV,
respectively. Very recently Zagrebaev et al. \cite{Zagreb-12} have
reported a prediction for the same reaction,
$^{254}$Es($^{48}$Ca,xn)$^{302-x}$119 (about 0.3 pb for $3n$
channel). No prediction for the
$^{252}$Es($^{48}$Ca,xn)$^{300-x}$119 reaction was given.
In case of inaccessibility of Es targets, the most promising
target-projectile combination to synthesize the element $Z=119$ is
the $^{249}$Bk($^{50}$Ti,xn)$^{299-x}$119 reaction. Predictions
for this reaction are shown in Fig. 5(c). Both $3n$ and $4n$
channels are expected to have comparable cross sections of about
30 fb (at maximum) at $E_{c.m.} \approx 225$ and 232 MeV,
respectively. Almost an equally small cross section for the
$^{249}$Bk($^{50}$Ti,xn)$^{299-x}$119 reaction (about 60 fb) was
predicted in Ref. \cite{Zagreb-08}, and somewhat larger value
(about 110 fb) in Ref. \cite{Wang-12}. Unfortunately, such small
cross sections seem to be beyond the reach of present-state
experiments. More optimistic predictions for the same reaction
appeared recently in Ref. \cite{Liu-119}, however a relatively
large cross section (about 0.6 pb) was obtained for probably
overestimated values of the fission barrier taken as the pure
ground-state shell effect from tables of Ref. \cite{Moller}.
Prospects for the synthesis of element $Z$ = 120 are considerably
worse than those for $Z$ = 119. First of all, there is no chance
to use the favorable beam of $^{48}$Ca because the complementary
$^{257}$Fm target cannot be produced. We consider therefore
reactions with $^{50}$Ti beam on two available isotopes of
californium, $^{249}$Cf($^{50}$Ti,xn)$^{299-x}$120 and
$^{251}$Cf($^{50}$Ti,xn)$^{301-x}$120, which seem to be best
choice. Excitation functions for these two reactions are shown in
Figs. 5(d) and 5(e). The largest cross section is expected in the
former reaction (about 6 fb at maximum in both $3n$ and $4n$
channels), in the latter reaction the maximum cross section is
about 3 fb for $4n$ channel. Again, similarly as in case of
reactions on two isotopes of einsteinium discussed above, a
smaller cross section for more neutron rich compound nucleus is
associated with respectively lower fission barriers predicted in
Refs. \cite{Kowal-10,Kow-Sob}.
In Fig. 5(f) we present results of calculations for the
$^{248}$Cm($^{54}$Cr,xn)$^{302-x}$120 reaction that is a more
symmetric combination of even-$Z$ target and projectile, next to
Ti + Cf. The obtained cross sections of the order of 1 fb for $3n$
and $4n$ reaction channels clearly demonstrate that fusion
processes are too strongly hindered in more symmetric systems. For
completeness, we calculated also cross sections in two reactions
of much more symmetric systems,
$^{238}$U($^{64}$Ni,xn)$^{302-x}$120 and
$^{244}$Pu($^{58}$Fe,xn)$^{302-x}$120 (not shown in figures), for
which attempts to produce the element $Z$ = 120 were done
\cite{GSI-120}, \cite{Dubna-120}. The calculated $3n$ and $4n$
cross sections in these two reactions are dramatically small,
about 0.3 fb and 0.1 fb, respectively. Note that experimental
upper limits for these two reactions had been established at 90 fb
\cite{GSI-120} and 400 fb \cite{Dubna-120}, respectively.
Our calculations show that if the fission barriers of Refs.
\cite{Kowal-10,Kow-Sob} were correct, there is no chance to
synthesize the element $Z$ = 120, even in the most favorable
reaction $^{249}$Cf($^{50}$Ti,xn)$^{299-x}$120, for which the
predicted cross section is only 6 fb. Note that other model
calculations for the $^{249}$Cf($^{50}$Ti,xn)$^{299-x}$120
reaction, published previously \cite{Zagreb-08,Liu-120,Siwek-120,
Nasirov-120,Nasirov-120a,Wang-12}, predicted considerably larger
cross sections though also too small to be measurable (typically
of the order of 50 fb). The dispersion of these different
theoretical results has to be linked, first of all, to different
fission barriers and ground-state masses used in these
calculations.
We would like to emphasize that our predictions concerning the
synthesis of $Z=119$ and 120 nuclei are based on the {\em
consistency} of the FBD model calculations with the adopted
ground-state {\em masses} and {\em fission barriers} of Refs.
\cite{Kowal-10,Kow-Sob} and with {\em all} the existing {\em
experimental data} on the synthesis of superheavy nuclei in hot
fusion reactions
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2}.
Therefore the accuracy of these predictions is expected to be
comparable with the accuracy of our overall fit to the data for
the synthesis of $Z$ = 114--118 nuclei, shown in Fig. 2.
In summary, we analyzed a complete set of existing data on hot
fusion reactions leading to the synthesis of superheavy nuclei of
$Z$ =114-118
\cite{Oganessian,Oga-243Am,Oga-244Pu,Oga-242Pu,Oga-249Cf,LBL-242Pu1,Oga-249Bk,GSI-244Pu,LBL-242Pu2,GSI-248Cm,GSI-244Pu2,Oga-243Am2}
in terms of a new $l$-dependent version of the FBD
model with fission barriers and ground-state masses taken from
the macroscopic-microscopic model of Kowal et al.
\cite{Kowal-10,Kow-Sob}. By ``calibrating'' the assumed
injection-point distances ($s_{inj}$) to the measured cross
sections, perfect systematics of $s_{inj}$-values have been
established for a wide range of hot fusion reactions enabling,
hopefully, reliable predictions of the synthesis cross sections
for yet unexplored reactions. Regarding prospects to produce the
new element $Z$ = 119, our calculations prefer the
$^{252}$Es($^{48}$Ca,xn)$^{300-x}$119 reaction, for which the
synthesis cross section of about 0.2 pb in $4n$ channel at
$E_{c.m.}\approx 220$ MeV is expected. According to the
microscopic-macroscopic model predictions \cite{Kowal-10,Kow-Sob},
fission barriers for heavier isotopes of the element $Z$ = 119 are
significantly lower leading to a considerably smaller cross
section in the alternative $^{254}$Es($^{48}$Ca,xn)$^{302-x}$119
reaction. Also the reaction $^{249}$Bk($^{50}$Ti,xn)$^{299-x}$119
gives little chances for a measurable cross section (the predicted
cross section is about 30 fb for both $3n$ and $4n$ channels). The
most favorable reaction to synthesize the element $Z$ = 120 is the
$^{249}$Cf($^{50}$Ti,xn)$^{299-x}$120 reaction, but the predicted
cross section is only 6 fb (for $3n$ and $4n$ channels).
|
{
"timestamp": "2012-07-27T02:00:36",
"yymm": "1203",
"arxiv_id": "1203.2252",
"language": "en",
"url": "https://arxiv.org/abs/1203.2252"
}
|
\section{Analysis}
\label{section:analysis}
\subsection{The NFW halo}
\label{section_nfw_stable}
The DM halo is constructed using a Monte Carlo sampling
technique. First, for each particle, the three position coordinates in
spherical coordinates ($r,\ \theta,\ \phi$) are generated. $r$ is
drawn from the density profile $\rho_{\rm NFW}$ using a standard
acceptance-rejectance technique, $\phi$ and $\cos(\theta)$ are drawn
from uniform distributions over the intervals $[0,2\pi]$ and $[-1,1]$,
respectively. Next, $v_{\rm r}$, $v_{\rm \theta}$ and $v_{\rm \phi}$
are drawn from the isotropic distribution function for the NFW model,
again with an acceptance-rejectance technique. This isotropic
distribution function was constructed from the NFW density profile
using the standard Eddington formula \citep{buyle07}. For each
particle, a symmetric partner was constructed with position
coordinates $(r,\ -\theta,\ -\phi)$ and velocity coordinates $(-v_{\rm
r}, -v_{\rm \theta}, -v_{\rm \phi})$. This drastically improved the
stability of the central parts of the halos. The very inner part of
the steep cusp of the NFW model is populated by relatively few
particles, destroying its spherical symmetry and introducing
unbalanced angular momenta. This initial deviation leads to the
ejection of particles from the cusp and triggers a more widespread
dynamical response of the DM halo, over time erasing the inner
cusp. Introducing the partner particles, cancelling out the angular
momenta and increasing the symmetry of the particles' spatial
distribution, greatly alleviates these problems. Such techniques for
constructing ``quiet'' initial conditions have been applied before
with great success, see e.g. \citep{selwood86}. The
improvement of the stability of the DM halo in simulations with a
``quiet'' start over simulations without a ``quiet'' start is
illustrated in the top panel of Fig. \ref{fig:NFWstability} where
the density distribution of both haloes at $z=0$ is plotted as red
and green dots, respectively.
\begin{figure}
\centering \includegraphics[width=0.48\textwidth]{figures/compDensity.eps}
\caption{The density profile of the N03 NFW halo for different
simulations: in the upper panel only DM was included, in the central panel
DM, gas is included but star formation was turned off. The bottom
panel shows the results of a simulation with DM, gas and star
formation. \label{fig:NFWstability}}
\end{figure}
First, to test the stability of the NFW halos, we ran several simulations for the N03 and N05 mass models:
\begin{description}
\item[\textbf{Run 1:}] only DM
\item[\textbf{Run 2:}] DM and gas but no star formation
\item[\textbf{Run 3:}] DM and gas and star formation
\end{description}
For these test simulations an $n_{\rm SF}$ of $0.1\ \rm cm^{-3}$
\citep{katz96} and $\epsilon_{\rm FB}$ of $0.1$ \citep{thornton98} was
used.
Fig. \ref{fig:NFWstability} shows the density profile of the test
simulations for the N03 mass model. From the upper panel, it is evident that the DM density of the
DM-only simulation remains stable and cusped until
the end of the simulation. The simulations presented in the middle and bottom panels,
show a clear conversion of the cusp into a core over time. Moreover, the width of the core depends on the mass of
the system, with more massive halos having larger cores.
Our simulations largely confirm the results from \cite{read05}, where a rapid removal of gas results in a conversion from cusp to core as stated first by \cite{navarro96}.
As gas cools and flows into the halo, the center of the dark matter
halo is adiabatically compressed. Without star formation, the central gas pressure builds up,
eventually stops further inflow, and even makes the gas re-expand somewhat. This re-expansion
happens rapidly enough for the DM halo to respond non-adiabatically:~the central DM density
experiences a net lowering and the cusp is transformed into a core. With star formation turned on,
feedback is responsible for a fast removal of gas from the central parts of the
DM halo, with the same effect:~a conversion from a cusp to a core.
Unlike us, \cite{governato10} found that the density threshold
for star formation needed to be high enough for a cusp-to-core
conversion to occur. Only for $n_{\rm SF} \gtrsim 10$~cm$^{-3}$ does
supernova feedback lead to sufficient gas motions to flatten the
cusp in their simulated dwarfs, which are taken from a larger
cosmological simulation. In contrast, in our more idealized,
initially spherically symmetric setup, even a low density threshold
leads to sufficient gas outflow for the cusp to flatten.
\subsection{Star formation histories}
\label{section_SFH}
\begin{figure}
\centering \includegraphics[width=0.5\textwidth]{figures/SFR.eps}
\caption{Top panel:~the SFR of several N07 models as a function of time. Bottom panel:~the stellar
mass as a function of time.\label{fig:SFR}}
\end{figure}
In Fig. \ref{fig:SFR}, we show the star-formation histories (SFHs) of
different realizations of the N07 mass model. Also, in table \ref{table_finalprop}, the starting time of star formation is tabulated along with the final total stellar mass. Several conclusions can be drawn:
\begin{itemize}
\item The delay between the start of the simulation and the start of the
first star-formation event is an increasing function of $n_{\rm SF}$. This appears
logical:~it takes longer for the gas to collapse to higher densities and ignite star formation.
Comparing different mass models, star formation starts earlier in more massive models for a given
$n_{\rm SF}$. This is most likely due to the more massive models having steeper gravitational
potential wells, increasing their ability to compress the inflowing gas.
\item If $n_{\rm SF}$ is increased while $\epsilon_{\rm FB}$ is kept fixed, more stars are formed
(e.g. going from the green to the blue curve or similarly from the cyan to the magenta curve in Fig. \ref{fig:SFR}). This is because gas collapses
to higher densities and the feedback is no longer able to sufficiently heat and expel this gas and to interrupt star formation.
\item Related to the previous point, the SFR also becomes more rapidly varying if $n_{\rm SF}$ is increased while
$\epsilon_{\rm FB}$ is kept fixed. The reason is that in the small high-density star-forming regions, feedback can only
locally interrupt star formation during short timespans. At lower $n_{\rm SF}$, star formation is more widespread, leading
to more global behavior:~as supernovae go off, star formation can be completely halted.
\item Increasing $\epsilon_{\rm FB}$ while $n_{\rm SF}$ is kept fixed leads to a decrease in star formation (e.g. going from the blue
to the cyan curve in Fig. \ref{fig:SFR}). This is because once feedback is strong enough, it is able to extinguish
star formation, even at high gas densities.
\item The most low-mass models fail to form stars for high $n_{\rm SF}$ values. E.g.
no stars form in the N03 models for $n_{\rm SF}>0.1$~cm$^{-3}$. This is due to the masses of these models being too small for gas to collapse to densities where stars can be formed. This point is further elaborated in the next paragraph.
\end{itemize}
\subsection{Density distribution of the ISM}
\label{section_ISM}
\begin{figure}
\centering \includegraphics[width=0.5\textwidth]{figures/N03densityComp.eps}
\caption{The density distribution of the ISM at different times for the least massive galaxy, N03, with different density threshold and a fixed feedback efficiency of 0.1.\label{fig:N03}}
\end{figure}
In Fig. \ref{fig:N03}, the density of the ISM is plotted as a function of radius. For the N03 model in the left panel a density threshold of $0.1$~cm$^{-3}$ was used while for the model in the right panel, the density threshold was set to a value of $6$~cm$^{-3}$. The red points show the gas distribution at the moment just before the start of star formation in the case of $n_{\rm SF}$ = $0.1$~cm$^{-3}$. Since up to that moment, all models have experienced the same evolution, there is no difference between the red points in both panels.
As can be seen in the left panel, the gas density in this N03 model reaches the star-formation threshold and star formation occurs. Moreover, the influence of supernova feedback can be seen in the green and blue points, where gas expands to larger radii and lower densities after having been heated. As is clear from the right panel, for $n_{\rm SF}$ = $6$~cm$^{-3}$ the gas simply keeps falling in. It will continue to do so during the first 4 Gyr until the built-up central pressure causes the gas to re-expand again. No stars are formed during the course of this simulation.
As the density threshold is increased to higher values, star formation tends to occur more and more in small collapsed clumps. This becomes clear when comparing the panels from Figs. \ref{fig:N03} and \ref{fig:N07}. The latter shows the gas density distributions of two N07 models with $n_{\rm SF}$ = $6$~cm$^{-3}$ and $n_{\rm SF}$ = $50$~cm$^{-3}$. While the $n_{\rm SF}$ = $50$~cm$^{-3}$ model only exhibits star formation in a small number of discrete high-density clumps, the $n_{\rm SF}$ = $6$~cm$^{-3}$ model lacks such well-defined clumps and star formation occurs more widespread.
\begin{figure}
\centering \includegraphics[width=0.5\textwidth]{figures/N07densityComp.eps}
\caption{The density distribution of the ISM at different times for the N07 model, with different density thresholds and a fixed feedback efficiency of 0.7. \label{fig:N07}}
\end{figure}
\subsection{Scaling relations}
\label{section_scaling_relations}
In this section we discuss the properties of each of our models and draw some conclusions regarding the influence of the $n_{\rm SF}$ and $\epsilon_{\rm FB}$ parameters on the models. An overview of some basic properties can be found in Table \ref{table_finalprop}.
\begin{table*}
\begin{minipage}[ht]{\linewidth}
\begin{center}
\caption{Final properties of our large set of simulations. Columns: (1) model number (see Table \ref{Models_table}), (2) density treshold for star formation, (3) feedback efficiency, (4) final stellar mass, (5) starting time of star formation, (6) half-light radius, (7) mean surface \textbf{brightness} within the half-light radius, (8) central one dimensional velocity dispersion, (9) mass-weighted metallicity, (10) central surface brightness, (11) S\'ersic parameter, (12) circular velocity.} \label{table_finalprop} \input{table_finalprop}
\end{center}
\end{minipage}
\end{table*}
\subsubsection{Half-light radius $R_{e}$}
The half-light radius, or effective radius, denoted by $R_{e}$, encloses half of
a galaxy's luminosity. In panel a.) of Fig. \ref{fig:bigFig},
$R_{e}$ is plotted as a function of the $V$-band magnitude.
The following trends can be observed in this figure:
\begin{itemize}
\item For a fixed $n_{\rm SF}$, the effective radius varies only very slightly throughout the $\epsilon_{\rm FB}$-range and this without a clear trend between $R_{e}$ and $\epsilon_{\rm FB}$. However, for a fixed $n_{\rm SF}$ and dark-matter mass the stellar mass and consequently the luminosity decrease with increasing $\epsilon_{\rm FB}$. This is due to star formation being shut down more rapidly when feedback is more effective. As a result, galaxies tend to have higher stellar densities for smaller $\epsilon_{\rm FB}$.
\item For a fixed $\epsilon_{\rm FB}$ of 0.1, an increase of $n_{\rm SF}$ from $0.1$ to $6$~cm$^{-3}$ results in a decrease of the effective radius. This is due to the size of the region where the SFC are fulfilled, which is much smaller for $n_{\rm SF}$ = $6$~cm$^{-3}$ than for $n_{\rm SF}$ = $0.1$~cm$^{-3}$, and the feedback is too weak to overcome this. In the case of an increase of $n_{\rm SF}$ from $6$ to $50$~cm$^{-3}$, the effective radius increases which is caused by the higher star formation peaks resulting in more supernovae explosions which redistribute the gas more efficiently.
\item The simulations with high density threshold, $n_{\rm SF} > 0.1\ \rm cm^{-3}$, and high feedback efficiency, $\epsilon_{\rm FB} > 0.1$, have effective radii which are in agreement with the observations.
\end{itemize}
\textit{From this scaling relation we can constrain the $\epsilon_{\rm FB}$-parameter to be higher then 0.1 to produce galaxies with effective radii in agreement with observations of dwarf galaxies.}
\subsubsection{The fundamental plane}
The fundamental plane (FP) is an observed relation between the effective radius, $R_{\rm e}$, the mean surface brightness within the effective radius, $I_{\rm e}$, and the central velocity dispersion, $\sigma_{\rm c}$ of giant elliptical galaxies. It is a linear relation, given by
\begin{equation}
\log( R_{\rm e} ) = -0.629 - 0.845 \log( I_{\rm e} ) + 1.38 \log( \sigma_{\rm c} ),
\end{equation}
between the logarithms of these quantities \citep{burstein97}. In panel b.) of Fig. \ref{fig:bigFig}, we plot the ''vertical'' deviation of the simulated galaxies from the giant galaxies' FP.
Dwarf galaxies generally lie above the FP in this projection. This is thought to be a consequence of their having shallower gravitational potential wells than giant galaxies. This, together with
the feedback, results in more diffuse systems. Models with a high star-formation threshold in combination with a low supernova feedback turn out to be very compact. They actually populate
the FP at low luminosities. However, this region of the three-dimensional space spanned by $\log(R_{\rm e})$, $\log( I_{\rm e} )$, and $\log( \sigma_{\rm c} )$ is observed to be devoid of galaxies.
Hence, models with low stellar feedback, $\epsilon_{\rm FB}$ up to $0.3 $, and high density thresholds, $n_{\rm SF} > 0.1$~cm$^{-3}$, can be rejected.
\begin{figure*}
\begin{minipage}[ht]{\linewidth}
\begin{center}
\centering
\includegraphics[width=0.95\textwidth,clip]{figures/bigFig.eps}
\caption{Some scaling relations and the surface brightness parameters as a function of the magnitude. In a.), the half-light radius $R_{\rm e}$ is plotted, b.) shows the vertical deviation of the simulated dwarf galaxies from the giant galaxies' FP, in c.) the $V-I$ color is plotted, d.) shows the iron content $[Fe/H]$. In panel e.) and f.), the S\'ersic index $n$ and central surface brightness $\mu_{0}$ are plotted. All these quantities are plotted against the $V$-band magnitude, except the FP which are plotted as a function of the $B$-band luminosity. The models with a density threshold of $6 \rm ~cm^{-3}$ and $50 \rm ~cm^{-3}$ are represented by blue-green diamonds and yellow-red triangles, respectively, where the colorscales represent a varying feedback efficiency.
For each color, the datapoints are connected by a line showing the mass evolution of the models. In the case of $n_{\rm SF}=0.1\ cm^{-3}$, represented by the black line, the models from N03 until N09 are plotted. In the cases of higher densities, represented by the colored lines, the datapoints are from models N05-N09.
Our models are compared with observational data
obtained from \citet{derijcke05}, \citet{graham03}
, LG data come from \citet{peletier93},
\citet{peletier93}, \citet{irwin95}, \citet{saviane96},
\citet{grebel03}, \citet{mcconnachie06},
\citet{mcconnachie07}, \citet{zucker07}, Perseus data from
\citet{derijcke09}, Antlia data from
\citet{castelli08}. For the $[Fe/H]-M_{V}$ plot, data from \citet{grebel03}, \citet{sharina08} and \citet{lianou10} was used, the yellow and magenta dots represent data from dSph and dIrr galaxies, respectively. \label{fig:bigFig}}
\end{center}
\end{minipage}
\end{figure*}
\subsubsection{Color $V-I$}
Fig. \ref{fig:bigFig}, panel c.) shows the $V-I$ color in function of the $V$-band magnitude.
The color scatter between the different models is rather small. The observed galaxies follow a mass-metallicity relation so the metallicity generally increases with the galaxy (stellar) mass, resulting in increasing $V-I$ values for increased galaxy mass. Within the relatively small mass range covered by the models, color
is only a very weak function of stellar mass. For a fixed feedback efficiency, when increasing the density threshold the $V-I$ also increases slightly resulting in bluer galaxies for the models with low density threshold. This is due to the effect that stars are formed in more metal enriched regions in the models with high density threshold.
When the density threshold is kept constant and only the feedback efficiency is increased the $V-I$ slightly decreases, so the models get slightly bluer due to a dilution of the gas when it is more spread out by supernovae explosions.
\subsubsection{Metallicity}
In panel d.) of Fig. \ref{fig:bigFig} a plot of iron content [Fe/H] as a function of the $V$-band magnitude is shown. The mass-weighted value of [Fe/H] is a measure of the metallicity of a galaxy. The yellow and magenta dots represent observational data from dwarf spheroidal and dwarf elliptical galaxies and irregular dwarf galaxies, respectively.
Some general conclusions we can take away from this figure are:
\begin{itemize}
\item Low-mass models with low density threshold, $n_{\rm SF} \approx 0.1$~cm$^{-3}$, and low feedback, $\epsilon_{\rm FB} \approx 0.1$, keep forming stars throughout cosmic history
and do not expel enriched gas. As a consequence, they turn out to be too metal rich, compared with observed dwarf galaxies. Models with higher $n_{\rm SF}$ compare much
more favorably with the data in this respect.
\item For a fixed $n_{\rm SF}$, increasing $\epsilon_{\rm FB}$, produces more metal poor galaxies. This is likely due to the fact that the increased feedback
extinguishes star formation more rapidly and disperses the metal enriched gas more widely.
\item Increasing $n_{\rm SF}$ at fixed $\epsilon_{\rm FB}$ and fixed mass, results in an increase of the metallicity and of the stellar mass when going from $n_{\rm SF} = 0.1$~cm$^{-3}$ to $n_{\rm SF} = 6$~cm$^{-3}$. A further increase of $n_{\rm SF}$ at fixed $\epsilon_{\rm FB}$, up to $n_{\rm SF} = 50$~cm$^{-3}$, has a much smaller impact on metallicity and stellar mass. The former is likely due to more vigorous star formation in less easily dispersable high density regions.
\end{itemize}
\subsubsection{Surface brightness profiles}
We fitted a S\'ersic profile, of the form
\begin{equation}
I(R)=I_{0}e^{-\left(\frac{R}{R_{0}}\right)^{1/n}},
\end{equation}
to the surface brightness profiles of the simulated galaxies.
The S\'ersic parameter $n$ and the central surface brightness $\mu_{0}$ are plotted respectively in the panels e.) and f.) of Fig. \ref{fig:bigFig} as a function of the $V$-band magnitude.
\begin{itemize}
\item For a fixed $n_{\rm SF}$, when increasing the $\epsilon_{\rm FB}$, there is a weak trend for the S\'ersic parameter $n$ and the central surface brightness to decrease. More vigorous
feedback appears to result in more diffuse dwarf galaxies, as one would expect.
\item As an echo of the $R_{\rm e}-M_{\rm V}$ relation, simulations with high density threshold, $n_{\rm SF} > 0.1\ \rm cm^{-3}$, and low feedback efficiency, $\epsilon_{\rm FB} = 0.1-0.3$, are systematically too compact, with $\mu_0 \sim 20$~mag~arcsec$^{-2}$, compared with the observations.
\item The models with high density thresholds and strong feedback are in general agreement with the observations.
\end{itemize}
\subsubsection{The Tully-Fisher relation}
\begin{figure}
\centering \includegraphics[width=\columnwidth]{figures/TF_FJ.eps}
\caption{The top panel shows the Tully-Fisher relation between the circular
velocity and the luminosity in the B-band. The full gray line shows the TF relation for early type galaxies, the dashed gray line is the TF relation of spiral galaxies as determined by \citet{derijcke07}. The lower panel shows the Faber-Jackson relation between the velocity
dispersion and the luminosity in the B-band.}
\label{fig:TF_FJ}
\end{figure}
Panel a.) of Fig. \ref{fig:TF_FJ} shows the B-band Tully-Fisher
relation (TFR) between the circular velocity, denoted by $V_{c}$, and the luminosity in the B-band, $L_{\rm B}$. The simulations are compared with observational data and with the Tully-Fisher relation for early-type (full gray line) and for spiral galaxies (dotted gray line) that was determined by \cite{derijcke07}.
All simulations predict that the TFR becomes substantially shallower in the dwarf regime, below luminosities of the order of $L_{\rm B} \sim 10^7\,L_{\odot, \rm B}$. This can be seen as a consequence of the
very steep $M_{\rm star}-M_{\rm halo}$ relation in the dwarf galaxy regime (see paragraph \ref{subsubMstMh}). For a fixed $n_{\rm SF}$, an increase in feedback efficiency does not influence $V_{c}$ very much since there are so few stars that $V_c$ is set by the dark-matter halo. The effect on the stellar mass, and consequently on $L_{\rm B}$, is, however,
quite large. Therefore, increasing $\epsilon_{\rm FB}$ at fixed $n_{\rm SF}$ and dark-matter mass causes galaxies to shift leftwards in panel a.) of Fig. \ref{fig:TF_FJ}. Except for this effect, once $n_{\rm SF}$ and $\epsilon_{\rm FB}$ are raised above their minimum values of $0.1$~cm$^{-3}$ and $0.1$, respectively, there is no significant differences between the TFRs traced by the different series of models.
\subsubsection{The Faber-Jackson relation}
The Faber-Jackson (FJR) relation, plotted in panel b.) of Fig. \ref{fig:TF_FJ} is the relation between the stellar central velocity dispersion and
the luminosity in the $B$-band. The stellar central velocity dispersion is a projection of the velocity dispersion along the line of
sight. This is measured by fitting an exponential function to the dispersion profile and retaining the maximum of the function as the
central value.
From this figure we see:
\begin{itemize}
\item For a fixed $n_{\rm SF}$, when increasing the $\epsilon_{\rm FB}$, the velocity dispersion decreases first after which it settles around a value which depends on the dark-matter mass of the model.
\item For a fixed $\epsilon_{\rm FB}$, when increasing $n_{\rm SF}$, only a minor influence on the velocity dispersion is observed.
\end{itemize}
\subsubsection{The $M_{\rm star}$-$M_{\rm halo}$ relation} \label{subsubMstMh}
In Fig. \ref{fig:MstarMhalo}, the $M_{\rm star}$-$M_{\rm halo}$ relation of the simulations at $z=0$ is plotted.
We can make similar conclusions here as were made in the SFH section:
\begin{itemize}
\item If $n_{\rm SF}$ is fixed, the stellar mass will decrease if the $\epsilon_{\rm FB}$ is increased. This is what was expected because with more
feedback the gas is distributed over a larger area and
the infall of the gas to the appropriate density threshold will take
longer.
\item If $\epsilon_{\rm FB}$ is fixed, for increasing $n_{\rm SF}$, the stellar mass increases too. When feedback is very small, the gas density will stay high and the star formation will not be interrupted, resulting in a high stellar mass. The effect is smaller for higher feedback.
\end{itemize}
In Fig. \ref{fig:MstarMhalo}, our different sets of
models are found to be in agreement with the results from
the Aquila simulation where a density threshold of 10 $\rm cm^{-3}$ and a feedback efficiency of 0.7 was used.
While the initial conditions of our dwarf galaxy simulations are admittedly quite simplified, they do have high spatial resolution
and realistic implemented physics. It is therefore encouraging that they compare favorably with cosmological simulations
like the Aquila simulation, which have cosmologically well motivated initial conditions but in which dwarf galaxies are very close to the resolution limit \citep{sawala11b}.
However it is impossible by further tuning of the feedback efficiency and/or the density threshold to reproduce the trend that was derived by \cite{guo10}.
By increasing the density threshold and feedback efficiency, the stellar mass is reduced by almost two orders of magnitude, but there still remains a difference of many orders of magnitude between our simulations and the $M_{\star}$-$M_{halo}$ relation from \cite{guo10}. It is also interesting to notice that although our models do not reproduce the relation, they do have a very similar slope.
\section{Numerical details}
\label{section:numerical_details}
We use a modified version of the Nbody-SPH code {\sc Gadget-2}
\citep{springel05}. The original {\sc Gadget-2} code was extended with
star formation, feedback and radiative cooling by \cite{valcke08}.
While the initial conditions of the simulations are cosmologically
motivated (see below), we do not perform full cosmological
simulations. Our approach yields a high mass resolution at
comparatively low computational cost. Still, previous work by
\cite{valcke08}, \cite{valcke10} and \cite{schroyen11} has shown that
with this code realistic dwarf galaxies, following the known
photometric and kinematic scaling relations, can be produced.
We set up the simulations using 200,000 gas particles and 200,000 DM particles.
Depending on the model's total mass, this results in gas particle masses in the range of
$350-2,620 M_{\odot}$ and DM particle masses in the range of $1,650-12,380 M_{\odot}$.
We use a gravitational softening length of 0.03~kpc.
Our results are visualized with our own software package HYPLOT. This
is freely available from SourceForge\footnote{http://sourceforge.net/projects/hyplot/}
and is used for all the figures
in this paper.
\subsection{Initial conditions}
Our models are set up, as in \cite{valcke08,valcke10,schroyen11}, with a spherically symmetric dark matter halo
and a homogeneous gas cloud. This gas cloud has a density of $5.55~\rho_{\rm crit}$, with $\rho_{\rm crit}$ the critical density of the universe at the halo's formation redshift, here taken to be $z_{\rm c}=4.3$. This is equivalent with a number density for the gas of 0.0011 hydrogen atoms per cubic centimeter.
We use a flat $\Lambda$-dominated cold dark matter cosmology with the following cosmological parameters:
$h = 0.71, \Omega_{\rm tot} = 1, \Omega_{\rm m} = 0.2383, \Omega_{\rm DM} = 0.1967$. The baryonic mass fraction will be the difference between $\Omega_{\rm m}$ and $\Omega_{\rm DM}$, in practice it will have a value that is 0.2115 times that of the dark-matter.
At the start of the simulations the gas particles are initially at rest, their initial metallicities are set to $10^{-4}\ Z_{\odot}$ and their initial temperature is $10^4$ K.
The dark matter halo has a NFW density profile \citep{NFW}:
\begin{equation}
\rho_{\rm NFW}(r) = \frac{\rho_{\rm s}}{(r/r_{\rm s})(1+r/r_{\rm s})^{2}}
\end{equation}
where $\rho_{\rm s}$ and $r_{\rm s}$ are, respectively, the
characteristic density and the scale radius. In order to fix the values
of these parameters, we use the correlation between them found by
\cite{wechsler02} and \cite{gentile04}, which makes the NFW density
distribution essentially a one-parameter family of the dark matter
virial mass, $M_{\rm DM}$. The relations we use for $\rho_{\rm s}$,
$r_{\rm s}$ and the concentration parameter $c$ (=$r_{\rm max}/r_{\rm
s}$) are~:
\begin{eqnarray}
c &\simeq& 20 \left(\frac{M_{\rm DM}}{10^{11}M_{\sun}}\right)^{-0.13}
\\ r_{\rm s} &\simeq& 5.7 \left(\frac{M_{\rm
DM}}{10^{11}M_{\sun}}\right)^{0.46}~{\rm kpc} \\ \rho_{\rm s}
&\simeq& \frac{101}{3}\frac{c^{3}}{\ln(1+c)-c/(1+c)}\rho_{\rm crit}.
\end{eqnarray}
Here, $r_{\rm max}$ is the halo's
virial radius. At $r_{\rm max}$, the DM halo is truncated and the
density drops to zero, so the entire mass $M_{\rm DM}$ is situated
inside the radius $r_{\rm max}$.
\subsection{Criteria for Star formation}
\label{subsection:SFC}
Star formation is assumed to take place in cold, dense, converging and
gravitationally unstable molecular clouds \citep{katz96}. Gas
particles that fulfill the star formation criteria (SFC) are eligible
to be turned into stars. These SFC are:
\begin{eqnarray}
\rho_{\rm g} & \geq & \rho_{\rm SF} \\ T & \leq & T_{\rm c} = 15000 \mathrm{K}
\\ \vec{\nabla}.\vec{v} & \leq & 0,
\end{eqnarray}
with $\rho_{\rm g}$ the gas density, $T$ its temperature and $\vec{v}$
its velocity field. $\rho_{\rm SF}$ is the density threshold for star
formation. We employ a Schmidt law \citep{schmidt59} to convert gas
particles that fulfill the SFC into stars:
\begin{equation}
\frac{\mathrm{d}\rho_{\rm s}}{\mathrm{d}t} =
-\frac{\mathrm{d}\rho_{\rm g}}{\mathrm{d}t} =
c_{\star}\frac{\rho_{\rm g}}{t_{\rm g}}, \label{cstar}
\end{equation}
with $\rho_{\rm s}$ the stellar density and $c_\star$ the
dimensionless star formation efficiency. The timescale $t_{\rm g}$ is
taken to be the dynamical time for the gas $1/\sqrt{4 \pi G \rho_{\rm
g}}$. Here, we choose $c_{\star}=0.25$. \cite{stinson06} showed that
the influence on the mean SFR of the value of $c_{\star}$ with values in
the range of 0.05 to 1 is negligible. Lowering $c_{\star}$ reduces the
star formation efficiency as well as the amount of supernova feedback, causing
more particles to fulfill the density and temperature criteria. This
compensates for the lower value of $c_{\star}$, producing a SFR which is
roughly independent of $c_\star$.
\cite{revaz09} also investigated the influence of $c_\star$ by varying it between the values of 0.01 and 0.3. They concluded that the star formation history is mainly determined by the initial total mass with a minor influence of $c_\star$. Self-regulating models, in which star formation occurs in recurrent bursts due to the interplay between cooling and supernova feedback, were achieved for $c_{\star}\sim0.2$.
Such models best resemble real dwarf galaxies.
\subsection{Feedback}
We consider feedback from star particles by supernova Ia (SNIa),
supernova II (SNII) and stellar winds (SW). They deliver energy and
mass to the ISM and enrich the gas. Feedback is
distributed over the gas particles in the
neighborhood of the star particle according to the SPH smoothing kernel. Each star particle represents a
single-age, single-metallicity stellar population (SSP). The stars within each SSP
are distributed according to a Salpeter initial mass function~:
\begin{equation}
\Phi(m)\mathrm{d}m = Am^{-(1+x)}\mathrm{d}m,
\end{equation}
with $x=1.35$ and $A=0.06$. The limits for the stellar masses are
$m_{\mathrm{l}} = 0.01~\mathrm{M_{\odot}}$ and $ m_{\mathrm{u}} =
60~\mathrm{M_{\odot}}$. The energy release of a SN is set to $E_{\rm
tot} = 10^{51}$~erg and that by a SW to $E_{\rm tot} = 10^{50}$~erg
\citep{thornton98}. The actual energy injected into the ISM is
implemented as $\epsilon_{\rm FB} \times E_{\rm tot}$, where
$\epsilon_{\rm FB}$ is a free parameter.
\subsection{Cooling}
Metallicity-dependent radiative cooling is implemented using the cooling curves
from \cite{sutherland93}. With this recipe it is possible
to cool gas to a minimum temperature of $10^{4}$~K.
We also implemented the \cite{maio07} cooling curves, making it possible for particles to cool below $10^{4}$~K.
\subsection{Production runs}
In Table \ref{Models_table}, we give an overview of the parameters
that were used to set up the models. A benefit of our code is that we can retain the same initial conditions
and easily adapt our parameters to perform a detailed parameter
survey. In the remainder, we will quantify the density threshold by $n_{\rm SF}$ expressed in hydrogen ions
per cubic centimeter (so $\rho_{\rm SF} = 1~{\rm amu} \times n_{\rm SF}$). At the start of the simulations,
the models only contain
gas and dark matter. During the first few $10^8$~years, the gas collapses
in the gravitational potential well of the DM. The simulations run for
12.22 Gyr, till $z=0$.
\begin{table}
\caption{Details of the basic spherical dwarf galaxy models that were
used in the simulations. Initial masses for the DM halo and gas
are given in units of $10^{6} \mathrm{M_{\odot}}$, radii in
kpc.\label{Models_table}} \input{Models_table}
\end{table}
In the literature, a large variety of values for the density threshold can be found. \cite{stinson06} use a low density
threshold of $0.1$~cm$^{-3}$ while \cite{governato10} use a high density threshold of $100\ \rm cm^{-3}$ which, these authors
argue, is a better representation of the conditions in star-forming regions in real galaxies. The simulations of \cite{sawala11a}
have been performed with a density threshold of $10\ \rm cm^{-3}$. In this paper, we increase the density threshold from $n_{\rm SF}=0.1$~cm$^{-3}$, over
$n_{\rm SF}=6$~cm$^{-3}$ to $n_{\rm SF}=50$~cm$^{-3}$. For the fiducial series of low-density threshold simulations, we matched the $n_{\rm SF}=0.1$~cm$^{-3}$
with a feedback efficiency of $\epsilon_{\rm FB}=0.1$. For the intermediate-density threshold simulations, with $n_{\rm SF}=6$~cm$^{-3}$, we varied the
feedback efficiency between $\epsilon_{\rm FB}=0.1$ and $0.9$. Finally, for the high-density threshold simulations, with $n_{\rm SF}=50$~cm$^{-3}$, we varied the
feedback efficiency between $\epsilon_{\rm FB}=0.3$ and $0.9$.
\section{Introduction}
Dwarf galaxies are the most common type of galaxy in the local
universe but also the faintest and least easy to observe. In the
$\Lambda$CDM cosmology, our universe consists of matter, both
luminous and dark, and dark energy, which is responsible for the
accelerating expansion of the universe. Galaxies form when gas
collapses in dark matter halos. Baryons, be it in the form of gas, dust
or stars, are the most accessible form of matter, emitting radiation
over the whole electromagnetic spectrum. Dark matter, on the other
hand, as it only interacts gravitationally, is much more difficult to
``observe''.
There have been many attempts to estimate dark halo masses and
mass-to-light ratios for galaxies and clusters of galaxies from direct
observations. These include methods that make use of gravitational
lensing \citep{mandelbaum06, liesenborgs09},
dynamical modeling of the observed properties of a kinematical tracer
such as stars or planetary nebulae \citep{kronawitter00, derijcke06, napolitano11, barnabe09}.
One thing virtually all these works have in
common is the relatively limited size of the data set they are based
on. \cite{guo10} determined the halo mass as a function of stellar
mass for a large sample of galaxies using a statistical analysis of
the Sloan Digital Sky Survey, which yields the stellar masses, and the
Millennium Simulations, which yield the dark-matter masses. In the
range of the most massive halos and bright galaxies, the derived
$M_{\rm star}$-$M_{\rm halo}$ relation, which is of the form $M_{\rm
star} \propto M_{\rm halo}^{0.36}$, is found to be in good
agreement with gravitational lensing data \citep{mandelbaum06}. Below
a halo mass of $M_{\rm halo} \sim 10^{11.4}$~$M_\odot$, this relation
becomes much steeper:~$M_{\rm star} \propto M_{\rm
halo}^{3.26}$. \cite{guo10} extrapolate the latter relation into the
dwarf regime, where $M_{\rm halo} \lesssim 10^{10}$~$M_\odot$. This
leads then to the prediction that faint dwarf galaxies with stellar
masses of the order of $M_{\rm star} \sim 10^6$~$M_\odot$ should live
in comparatively massive $M_{\rm halo} \sim 10^{10}$~$M_\odot$
dark-matter halos.
The \cite{guo10} $M_{\rm star}$-$M_{\rm halo}$ relation was compared
with that found in simulations of dwarf galaxies
\citep{valcke08,stinson07,stinson09,governato10,pelupessy04,mashchenko08}
by \cite{sawala11a} and \cite{sawala11b}. They found that simulated
dwarf galaxies had stellar masses that were at least an order of
magnitude higher at a given halo mass than predicted by
\cite{guo10}. There could be several causes for numerical dwarf
galaxies to be overly prolific star formers:
\begin{itemize}
\item The star formation efficiency could be too high because of an
underestimation of the feedback efficiency. \cite{stinson06}
investigated the influence of the feedback efficiency on the mean
star formation rate (SFR). The general trend they have observed was a
decrease of the mean SFR when increasing the feedback efficiency.
\item \cite{stinson06} also reported finding a decreasing mean SFR with
increasing density threshold for star formation. Recently, high
density thresholds for star formation have come in vogue, see
e.g. \cite{governato10}.
\item Dwarf galaxies, due to their low masses, are expected to be
particularly sensitive to reionisation. Not properly taking into
account the effects of reionisation may lead to an overestimation of
the gas content of dwarfs and an underestimation of the gas cooling
time.
\item Dwarf galaxies are metal poor and hence also dust poor. This
lowers the production of H$_2$ molecules and causes poor
self-shielding of molecular clouds \citep{buyle06} which
could be expected to inhibit star formation. Not taking these
effects into account will lead to an overestimation of the
SFR \citep{gnedin09}.
\end{itemize}
Using the high values for the density threshold above which gas
particles become eligible for star formation, denoted by $\rho_{\rm SF}$, as promoted by \cite{governato10}, in combination with
radiative cooling curves that allow the gas to cool below $10^4$~K
\citep{maio07}, makes the gas collapse into small, very dense and cool clouds
before star formation ignites. If the supernova feedback
$\epsilon_{\rm FB}$, defined as the fraction of the average energy
output of a supernova that is actually absorbed by the interstellar
medium (ISM), is too weak to sufficiently heat and/or disrupt such a
star-forming cloud, one can consequently expect the mean SFR to be
very high, leading to overly massive (in terms of $M_{\rm star}$)
dwarfs. Therefore, one could hope to remedy this situation by
increasing $\epsilon_{\rm FB}$ accordingly. In that case, a
correlation between $\epsilon_{\rm FB}$ and $\rho_{\rm SF}$ would be
expected to exist.
In the present paper, we analyze a large suite of numerical
simulations of isolated, spherically symmetric dwarf galaxies in which
we varied both the feedback efficiency $\epsilon_{\rm FB}$ and the
density threshold $\rho_{\rm SF}$. Our goal is to investigate {\em (i)}
if such a correlation between $\epsilon_{\rm FB}$ and $\rho_{\rm SF}$
exists and, if it exists, how to break it, {\em (ii)} which
$\epsilon_{\rm FB}$/$\rho_{\rm SF}$-combinations lead to viable dwarf
galaxy models in terms of the observed photometric and kinematic
scaling relations, and {\em (iii)} how well these models approximate
the aforementioned $M_{\rm star}$ -$M_{\rm halo}$ relation.
In section \ref{section:numerical_details}, we give more details about the numerical methods that
are used in our code. An analysis of the simulations is given in
section \ref{section:analysis}, where some details are given of the NFW halo that is used
for the simulations and a large set of scaling relations are
plotted comparing our models to observations. In section \ref{section:results} we discuss the obtained results and
conclude.
\section*{Acknowledgements}
We thank Volker Springel for
making publicly available the {\sc Gadget-2} simulation code and Till Sawala for making available to us his
data from the Aquila simulation. We also would like to thank the anonymous referee for his/her stimulating remarks
that have greatly improved the manuscript.
\input{article_bibliography}
\label{lastpage}
\end{document}
\section{Discussion and conclusions}
\label{section:results}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figures/Mstar_Mhalo_2C.eps}
\caption{ The stellar mass versus the DM halo mass, plotted in
comparison with the models by \citet{sawala11b}. The gray dots show data from gravitational lensing from \citet{mandelbaum06}. The black
line is the trend for this relation that was determined by
\citet{guo10}. \label{fig:MstarMhalo}}
\end{figure}
\subsection{Cusp to core}
Whether the halo density profile is cusped or cored has been a point of discussion for quite some time. Observationally, evidence for cored DM profiles is found \citep{gentile04}, but from cosmological DM simulations a cusped density profile is deduced \citep{NFW, moore96}. The inherent limitation due to the angular resolution of the observations is ruled as a cause of the observed flat density profiles by \cite{deblok02}. \cite{gentile05} also excluded the possibility of non-circular gas motions which might result in a rotation curve that is best fitted by a cored halo, while the dark matter halo actually has a cuspy profile. However, from the simulation point of view, \cite{mashchenko06} mentioned a natural transition of a cusp to a flattened core when the dark matter halo is gravitationally heated by bulk gas motions.
Our simulations are set up with a cusped NFW halo in agreement with cosmological simulations. The infall of gas causes an adiabatic compression of the dark halo. When gas is evacuated from the central
regions, be it by a fast re-expansion as the gas pressure builds up or by supernova feedback, the dark-matter halo reacts non-adiabatically and kinetic energy of the gas is transferred to the dark matter. This results in a flattening of the central density and so the cusp is converted into a core. We can conclude that the conversion of the cusped halo density profile to a cored profile is realized by the removal of baryons
from the galaxy center \citep{read05}, whether this is due to a re-expansion of the gas or by feedback effects or by another process.
\subsection{Degeneracy}
By increasing both the density threshold and the feedback efficiency, the simulated galaxies move along the observed kinematic and photometric scaling relations.
These two parameters, the feedback efficiency $\epsilon_{\rm FB}$ and the density threshold $n_{\rm SF}$, correlate with each other and an increase of the one can be counteracted by an increase of the other, resulting in galaxies with similar properties. To be more specific:~the individual galaxies are drastically different for different parameter values but they all line up along the same scaling relations and
can therefore be seen as good analogs of observed dwarf galaxies.
The feedback efficiency quantifies the fraction of the $10^{51}$~ergs of energy that are released during a supernova explosion and thermally injected into the ISM. For each value of the density threshold we can determine the feedback efficiency range for which the models are in agreement with the observations, although we are not able to deduce a unique $n_{\rm SF}$/$\epsilon_{\rm FB}$-combination which would be the ``correct`` representation of the physical processes that happen in galaxies.
For a certain density threshold, a lower limit of the corresponding $\epsilon_{\rm FB}$-parameter can be determined from the effective radius: the galaxies become too centrally concentrated when the feedback is too low.
From the scaling relations we cannot deduce an upper limit for the $\epsilon_{\rm FB}$-parameter, but one could argue that the ISM cannot receive more energy than there is released by the supernova explosion, resulting in a maximal value for the feedback efficiency of 1.
In the case of a density threshold of $n_{\rm SF}=0.1$~cm$^{-3}$, the models are generally in good agreement with the observations besides the somewhat high metallicities. This is also the reason why the feedback efficiency was not varied in this case. If we compare the high density threshold models, $n_{\rm SF}>0.1$~cm$^{-3}$, with the observations we can conclude that the feedback efficiency should be larger then $\sim 0.3$. For a density threshold of $n_{\rm SF}=6$~cm$^{-3}$, we prefer a value of 0.7 for the feedback. Similarly we prefer a feedback efficiency of 0.9 in the case of a density threshold of $n_{\rm SF}=50$~cm$^{-3}$
The fact that different $n_{\rm SF}$/$\epsilon_{\rm FB}$-combinations result in simulated galaxies with properties that are in agreement with the observations invokes a warning for future simulations and indicates that there is still some work left to determine the density of the star forming regions and the fraction of supernova energy that is absorbed by the ISM, quantities which are hard to determine observationally.
There are however other parameters that might influence the starformation rate and our degeneracy, which are not investigated here:
\itemize{
\item{Given the fact that the star-formation efficiency $c_\star$ was found by other authors not to have a significant impact on stellar mass, we did not investigate it in detail in this paper.}
\item{The choice of the IMF, for which in our simulations a Salpeter IMF is used, determines the mass distribution of stars. The fraction of high-mass stars influences the number of SNIa and SNII explosions and as a consequence it will influence the amount of feedback and the chemical evolution. However, given
the large number of IMF parameterizations available in the literature, testing them is a very daunting task which falls outside the scope of this paper. Moreover,
part of the IMF-variation is quantified approximately by the variation in $\epsilon_{\rm FB}$ which we do investigate. }
\item{There are other possible feedback implementations, next to the release of feedback energy as thermal energy to the gas. It also could be released as kinetic energy by kicking the gas particles or by blast-wave feedback \citep{mayer08}.}
\item Other implementations of star formation, e.g. based on a subgrid model of H$_2$-formation \citep{pelupessy04}, are possible.
}
\subsection{The dwarf galaxy dark-matter halo occupancy}
To conclude, Fig. \ref{fig:Mst_Mh_f} shows the models which best agree with the observations for each density threshold that was used in our analysis.
Increasing $n_{\rm SF}$ together with $\epsilon_{\rm FB}$ leads to a strong reduction, of almost two orders of magnitude, of the stellar mass, especially in the most massive models.
However, with the physics included in our simulations, we are unable to reproduce the $M_{\rm star}-M_{\rm halo}$ relation of \cite{guo10}. Surprisingly, the best models trace a $M_{\rm star}-M_{\rm halo}$ relation
with a slope that is similar to that of the relation of \cite{guo10}. Our simulations are in agreement with results from cosmological simulations, which have, however, much lower spatial resolution in the dwarf regime \cite{sawala11b}. We did not explore yet higher values for $n_{\rm SF}$ and $\epsilon_{\rm FB}$ because it is clear from Fig. \ref{fig:Mst_Mh_f} that the reduction of $M_{\rm star}$ stagnates for
high $n_{\rm SF}$-values. Moreover, to compensate for the high density threshold, an unphysical large value for $\epsilon_{\rm FB}$, higher than 1, would be required. Thus, we arrive at $(n_{\rm SF}=6\,{\rm cm}^{-3},\, \epsilon_{\rm FB}\sim 0.7)$ and $(n_{\rm SF}=50\,{\rm cm}^{-3},\, \epsilon_{\rm FB}\sim 0.9)$
as the models which are in best agreement with the observed photometric and kinematical scaling relations and with the $M_{\rm star}-M_{\rm halo}$ relation derived directly from cosmological simulations.
\begin{figure}
\centering \includegraphics[width=\columnwidth]{figures/Mstar_Mhalo_2C_final.eps}
\caption{The $M_{\rm star}-M_{\rm halo}$ of our best models for different density threshold compared to the relation of \citealt{guo10}, other simulations from \citealt{sawala11b} and observations from \citealt{mandelbaum06}.
\label{fig:Mst_Mh_f}}
\end{figure}
While it appears impossible to place isolated dwarf galaxies on the $M_{\rm star}-M_{\rm halo}$ relation of \citet{guo10}, it is possible to envisage external influences that may further reduce
$M_{\rm star}$, as already mentioned in the Introduction:
\begin{itemize}
\item Not properly taking into
account the effects of reionisation may lead to an overestimation of
the gas content of dwarfs and an underestimation of the gas cooling
time. However, even taking into account reionisation, the dwarf galaxies simulated by \citet{sawala11a}
had much too high stellar masses.
\item At a given gas density, the star-formation efficiency of dwarf galaxies could be lower than that of more massive stellar systems because of their lower metallicity and hence lower dust content.
This could be mimicked by reducing the star-formation efficiency parameter $c_\star$ (see eq. (\ref{cstar})) in the dwarf regime. However, \citet{stinson06} have shown that, because of self-regulation, the star-formation rate is very insensitive to this parameter:~varying $c_\star$ between 0.05 and 1 left the mean star-formation rate virtually unchanged.
\item External processes such as ram-pressure stripping and tidal stirring may lead to a premature cessation of star formation and hence lower $M_{\rm star}$ \citep{mayer06}. However, these processes are
only effective if the gravitational potential wells of dwarf galaxies are sufficiently shallow and if they are stripped early enough in cosmic history, before they converted their gas into stars. It is
unclear whether these constraints are met. In \citet{derijcke10}, and references therein, it was argued that the number of red-sequence, quenched dwarf galaxies increased significantly over the last half of the Hubble time and that the dwarf galaxies now residing in the Fornax cluster were accreted less than a few crossing times age (i.e. less than a few Gyr). This timescale would have left dwarf galaxies ample time to form stars before entering the cluster.
\end{itemize}
|
{
"timestamp": "2012-03-09T02:04:08",
"yymm": "1203",
"arxiv_id": "1203.1863",
"language": "en",
"url": "https://arxiv.org/abs/1203.1863"
}
|
\section{Introduction}
One of the most important problems in theoretical physics is the quantization
of gravity. There are several approaches in this direction, such as
string theories and loop quantum gravity. Although we are still far
from a complete theory of quantum gravity, some common notions are
obtained. For example, at small scale or Planck scale, the smooth
Riemann manifold structure of the large spacetime will disappear and
the spacetime will manifest some quantum or discrete properties. The
first discrete spacetime model was suggested by Snyder \cite{snyder1947}
at 1947 and physical models on noncommutative spacetime are studied
extensively during last decades. Here we refer two reviews \cite{Douglas2001,Szabo2003}.
The most common noncommutative spacetime model is the canonical model,
where the spacetime operators satisfying the following relations
\begin{equation}
[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu},\label{eq:canonc}
\end{equation}
where $\theta^{\mu\nu}$ is constant noncommutative parameters. This
model is not only a simple one but can be derived from string theory
when there are some backgroud gauge fields on branes \cite{dou,cheung1998,chu1999,seiberg1999}.
Using Moyal-Weyl correspondence, noncommutative field $\phi(\hat{x})$
defined on eq.(\ref{eq:canonc}) can be described by common field
$\phi(x)$ with Moyal product \cite{moyal},
\begin{equation}
\phi_{1}*\phi_{2}(x)=\exp(\frac{i}{2}\theta^{\mu\nu}\partial_{\mu}^{x}\partial_{\nu}^{y})\phi_{1}(x)\phi_{2}(y)|_{y\rightarrow x}.\label{eq:moyalproduct}
\end{equation}
We encode the noncommutativity or nonlocality into the Moyal product.
The above noncommutative models have one problem that it breaks Lorentz
invariance explicitly. In \cite{Chaichian:2004yh,Wess:2003da}, the
authors suggested a twisted Lorentz symmetry to resolve this problem.
Recently, an interesting Lorentz covariant noncommutative model is
suggested \cite{Falomir:2009cq,Gomes:2010xk}. Some phenomenological
applications of this model are also been studied\cite{Das:2011tj}.In
this model, noncommutativity is related with the spin of the fields.
For a scalar particle, no spacetime nocommutativity can be felt. But
a Dirac particle can feel it. The commutation relations of this model
for a Dirac field are
\begin{eqnarray}
[\hat{x}^{\mu},\hat{x}^{\nu}] & = & -i\theta\epsilon^{\mu\nu\rho\sigma}\hat{S}_{\rho\sigma}+\frac{i}{2}\theta^{2}\epsilon^{\mu\nu\rho\sigma}\hat{W}_{\rho}\hat{p}_{\sigma},\label{eq:spinnc1}\\
{}[\hat{x}^{\mu},\hat{p}^{\nu}] & = & i\eta^{\mu\nu}.\label{eq:spinnc2}
\end{eqnarray}
$W_{\rho}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\hat{S}_{\rho\sigma}\hat{p}_{\nu}$
is the Pauli-Lubanski vector operator. And $\hat{S}_{\rho\sigma}$
are commutators between Dirac $\gamma$ matrices and defined as
\begin{equation}
\hat{S}_{\rho\sigma}=-\frac{i}{4}(\gamma_{\rho}\gamma_{\sigma}-\gamma_{\sigma}\gamma_{\rho})=\frac{i}{2}\gamma_{\rho\sigma}.
\end{equation}
In this paper, we discuss the quantum microcausality of spin-noncommutative
quantum field theories. The microcausality of noncommutative quantum
field theories based on eq.(\ref{eq:canonc}) has been discussed by
several authors. In\cite{Chaichian:2002vw,Ma:2006qx}, the authors
have studied microcausality by computing the commutator of operators
of observables $\mathcal{O}=:\phi*\phi:$ at spacelike seperation
and found that microcausality is obeyed provided that $\theta^{0i}=0$.
But it is showed that the commutators for observables with partial
derivatives of fields do not vanish at spacelike seperation\cite{Greenberg:2005jq}.
Similar results are obtained by other authors \cite{Haque:2007rb}
in a different approach in noncommutative scalar and Yukawa theories.
The microcausility of noncommutative scalar field theory is revisited
in a more mathematical method-distributions theory in\cite{Soloviev:2008yb}
and the author proved that microcausality is violated for observables
$\mathcal{O}$ even for space-space noncommutativity. All the above
negative results are obtained for noncommutative field theories based
on eq.(\ref{eq:canonc}). Maybe we can see these results from a simple
viewpoint-they violate common Lorentz covariance. How about a Lorentz
covariant noncommutative field theory? Spin-noncommutative field theories
are suitable models for consideration. In section 2, we study two
examples in spin-noncommutative Dirac field theory.
\section{CALCULATION OF CAUSALITY}
For spin noncommutative algebras eq.(\ref{eq:spinnc1}) and eq.(\ref{eq:spinnc2})
of a massive Dirac spinor field with mass $m$, we have the following
representation \cite{Gomes:2010xk},
\begin{equation}
\hat{x}^{\mu}=x^{\mu}-\frac{i\theta}{2}\gamma^{5}\gamma^{\mu\nu}\partial_{\nu},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{p}^{\mu}=-i\partial^{\mu}.\label{eq:rep}
\end{equation}
Similar to Moyal product, one can define a new star product into which
we can encode the spin noncommutativiy
\begin{equation}
(f*g)(x)=f(x)\exp(\frac{i\theta}{2}\overset{\leftharpoonup}{\partial}_{\mu}\gamma^{5}\gamma^{\mu\nu}\overset{\rightharpoonup}{\partial}_{\nu})g(x).\label{eq:star}
\end{equation}
In the expansion of Dirac fields, we use the normalization and convention
as
\begin{eqnarray*}
\psi(x) & = & \int\frac{d^{3}p}{(2\pi)^{3/2}}\frac{1}{\sqrt{2\omega_{\mathrm{p}}}}\sum_{s}(\hat{b}_{\mathrm{p},s}u(\mathrm{p},s)e^{-ip\cdot x}+\hat{d}_{\mathrm{p},s}^{+}v(\mathrm{p},s)e^{ip\cdot x}),\\
\bar{\psi}(x) & = & \int\frac{d^{3}p}{(2\pi)^{3/2}}\frac{1}{\sqrt{2\omega_{\mathrm{p}}}}\sum_{s}(\hat{b}_{\mathrm{p},s}^{+}\bar{u}(\mathrm{p},s)e^{ip\cdot x}+\hat{d}_{\mathrm{p},s}\bar{v}(\mathrm{p},s)e^{-ip\cdot x}).
\end{eqnarray*}
where $\omega_{\mathrm{p}}=\sqrt{\mathrm{p}^{2}+m^{2}}$ and $u,v$
are the positive and negative energy solutions of Dirac equation.
The creation and annihilation operators obey anticommutation relations
\[
\{\hat{b}_{\mathrm{p},s},\hat{b}_{\mathrm{q},r}^{+}\}=\delta^{3}(\mathrm{p}-\mathrm{q})\delta_{sr},\,\,\,\,\,\,\,\,\,\,\,\{\hat{d}_{\mathrm{p},s},\hat{d}_{\mathrm{q},r}^{+}\}=\delta^{3}(\mathrm{p}-\mathrm{q})\delta_{sr}.
\]
To study microcausality, we choose Hermitian operator $\mathcal{O}(x)=:\bar{\psi}(x)*\psi(x):$
as a sample observable. First, we consider the vacuum expectation
value $\langle0|[\mathcal{O}(x),\mathcal{O}(y)]|0\rangle$.
Using the definition of the star product, the observable $\mathcal{O}(x)$
can be written in explicit form
\begin{eqnarray}
\mathcal{O}(x) & = & \int\frac{d^{3}p_{1}}{(2\pi)^{3/2}}\frac{1}{\sqrt{2\omega_{\mathrm{p_{1}}}}}\int\frac{d^{3}p_{2}}{(2\pi)^{3/2}}\frac{1}{\sqrt{2\omega_{\mathrm{p_{2}}}}}\sum_{s_{1}}\sum_{s_{2}}\nonumber \\
& & (\hat{b}_{\mathrm{p_{1}}s_{1}}^{+}\hat{b}_{\mathrm{p}_{2}s_{2}}\bar{u}(\mathrm{p}_{1},s_{1})M(p_{1},p_{2})u(\mathrm{p}_{2},s_{2})e^{i(p_{1}-p_{2})\cdot x}\nonumber \\
& & +\hat{b}_{\mathrm{p_{1}}s_{1}}^{+}\hat{d}_{\mathrm{p}_{2}s_{2}}^{+}\bar{u}(\mathrm{p}_{1},s_{1})M(p_{1},-p_{2})v(\mathrm{p}_{2},s_{2})e^{i(p_{1}+p_{2})\cdot x}\nonumber \\
& & +\hat{d}_{\mathrm{p_{1}}s_{1}}\hat{b}_{\mathrm{p}_{2}s_{2}}\bar{v}(\mathrm{p}_{1},s_{1})M(-p_{1},p_{2})u(\mathrm{p}_{2},s_{2})e^{-i(p_{1}+p_{2})\cdot x}\nonumber \\
& & -\hat{d}_{\mathrm{p_{2}}s_{2}}^{+}\hat{d}_{\mathrm{p}_{1}s_{1}}\bar{v}(\mathrm{p}_{1},s_{1})M(-p_{1},-p_{2})v(\mathrm{p}_{2},s_{2})e^{-i(p_{1}-p_{2})\cdot x}),\label{eq:O(x)}
\end{eqnarray}
where the momentum-dependent matrix $M$ is defined as
\begin{equation}
M(p,q)=\exp(\frac{i}{2}\theta\gamma^{5}\gamma^{\mu\nu}p_{\mu}q_{\nu}).\label{eq:M}
\end{equation}
From the definition, we can see that $M(p,q)=M(-p,-q),\,\,\,\,\, M(p,-q)=M(-p,q)$.
Then the vacuum expectation of the commutator of two observables are
\begin{eqnarray}
& & \langle0|[\mathcal{O}(x),\mathcal{O}(y)]|0\rangle=\int\frac{d^{3}p_{1}}{(2\pi)^{3/2}}\frac{1}{2\omega_{\mathrm{p_{1}}}}\int\frac{d^{3}p_{2}}{(2\pi)^{3/2}}\frac{1}{2\omega_{\mathrm{p_{2}}}}\nonumber \\
& & \lbrace\mathrm{tr}[(\slashed{p}{}_{1}+m)M(p_{1},-p_{2})(\slashed{p}{}_{2}-m)M(-p_{2},p_{1})](e^{-i(p_{1}+p_{2})(x-y)}-e^{i(p_{1}+p_{2})(x-y)})\rbrace.\label{eq:vv}
\end{eqnarray}
For the commutative limit, the matrix $M=1$, the above equation becomes
\begin{equation}
\int\frac{d^{3}p_{1}}{(2\pi)^{3/2}}\frac{1}{2\omega_{\mathrm{p_{1}}}}\int\frac{d^{3}p_{2}}{(2\pi)^{3/2}}\frac{1}{2\omega_{\mathrm{p_{2}}}}(4p_{1}\cdot p_{2}-4m^{2})(e^{-i(p_{1}+p_{2})(x-y)}-e^{i(p_{1}+p_{2})(x-y)}).\label{eq:comlim1}
\end{equation}
When $x-y$ is spacelike, we can take $x_{0}=y_{0}$ due to the Lorentz
invariance of the above equation. Then it is easy to see that eq.(\ref{eq:comlim1})
equals to zero. For noncommutative case, the difficulty in the integral
is the trace term in eq.(\ref{eq:vv}) , which can be reduced to
\begin{equation}
\mathrm{tr}[\slashed{p}_{2}M(-p_{2},p_{1})\slashed{p}_{1}M(p_{1},-p_{2})-m^{2}].\label{eq:tr1}
\end{equation}
It is hard to get an explicit expression for the above trace, but
we know that the above trace is a Lorentz scalar. So it must be a
function of the form $F(p_{1}\cdot p_{2},m^{2})$ . Using the same
analysis of the commutative case, the eq.(\ref{eq:vv}) equals to
zero. We conclude that the microcausality is obeyed by vacuum expectation
of the commutator.
Then let us check the matrix element between vacuum and a two-particle
state. We choose the two particle state as $|q_{1},r_{1};q_{2},r_{2}\rangle=\hat{b}_{\mathrm{q}_{1}r_{1}}^{+}\hat{d}_{\mathrm{q}_{2}r_{2}}^{+}|0\rangle$
,
\begin{eqnarray}
& & \langle0|[\mathcal{O}(x),\mathcal{O}(y)]|q_{1},r_{1};q_{2},r_{2}\rangle=\frac{1}{2\sqrt{\omega_{q_{1}}\omega_{q_{2}}}}e^{-iq_{1}\cdot x-iq_{2}\cdot y}\int\frac{d^{3}p}{(2\pi)^{3/2}}\frac{1}{2\omega_{\mathrm{p}}}\lbrace(e^{-ip\cdot(x-y)}-e^{ip\cdot(x-y)})\nonumber \\
& & \bar{v}(q_{2},r_{2})[M(-q_{2},p)(\slashed{p}+m)M(p,q_{1})-M(-q_{2},-p)(\slashed{p}-m)M(-p,q_{1})]u(q_{1},r_{1})\rbrace.\label{eq:vt}
\end{eqnarray}
At this case, we choose a definite two-particle state, so the above
matrix element is not Lorentz invariant. For simplicity, the commutator
is taken to be equal time. It is conceivable that the above integral
is not zero and thus the microcausility is violated. But it is not
easy to obtain a explicitly analytic results from it. In order to
do this, we should take some limit and choose special spinor states
for the two particles. In chiral representation, the matrix in the
second line of the above equation
\[
[M(-q_{2},p)(\slashed{p}+m)M(p,q_{1})-M(-q_{2},-p)(\slashed{p}-m)M(-p,q_{1})]
\]
can be written explicitly as
\[
\left(\begin{array}{cc}
m(e^{\vec{A}\cdot\vec{\sigma}}e^{\vec{C}\cdot\vec{\sigma}}+e^{-\vec{A}\cdot\vec{\sigma}}e^{-\vec{C}\cdot\vec{\sigma}}) & e^{\vec{A}\cdot\vec{\sigma}}(p_{0}-\vec{\mathrm{p}}\cdot\vec{\sigma})e^{\vec{D}\cdot\vec{\sigma}}-e^{-\vec{A}\cdot\vec{\sigma}}(p_{0}-\mathrm{p}\cdot\vec{\sigma})e^{-\vec{D}\cdot\vec{\sigma}}\\
e^{\vec{B}\cdot\vec{\sigma}}(p_{0}+\vec{\mathrm{p}}\cdot\vec{\sigma})e^{\vec{C}\cdot\vec{\sigma}}-e^{-\vec{B}\cdot\vec{\sigma}}(p_{0}+\vec{\mathrm{p}}\cdot\vec{\sigma})e^{-\vec{C}\cdot\vec{\sigma}} & m(e^{\vec{B}\cdot\vec{\sigma}}e^{\vec{D}\cdot\vec{\sigma}}+e^{-\vec{B}\cdot\vec{\sigma}}e^{-\vec{D}\cdot\vec{\sigma}})
\end{array}\right)
\]
where
\begin{eqnarray}
A & = & \frac{i\theta}{2}(p^{0}\vec{\mathrm{q}}_{2}-q_{2}^{0}\vec{\mathrm{p}}+i\vec{\mathrm{q}}_{2}\times\vec{\mathrm{p}})\\
B & = & \frac{i\theta}{2}(p^{0}\vec{\mathrm{q}}_{2}-q_{2}^{0}\vec{\mathrm{p}}-i\vec{\mathrm{q}}_{2}\times\vec{\mathrm{p}})\\
C & = & \frac{i\theta}{2}(p^{0}\vec{\mathrm{q}}_{1}-q_{1}^{0}\vec{\mathrm{p}}+i\vec{\mathrm{q}}_{1}\times\vec{\mathrm{p}})\\
D & = & \frac{i\theta}{2}(p^{0}\vec{\mathrm{q}}_{1}-q_{1}^{0}\vec{\mathrm{p}}-i\vec{\mathrm{q}}_{1}\times\vec{\mathrm{p}}).
\end{eqnarray}
If we take the momenta of the particles to be zero, i.e. $q_{1}=q_{2}=(m,0,0,0)$
and the seperation to be $\vec{x}-\vec{y}=(0,0,z)$, it turns out
that the integral in eq.(\ref{eq:vt}) is zero. Microcausality is
preserved by this special case. Then we should consider other cases.
For the sake of simplicity, we take massless limit and the momenta
are chosen to be $q_{1}=q_{2}=(q,0,0,q)$ and the seperation to be
$\vec{x}-\vec{y}=(0,0,z)$. The spinor states of the two particles
are $u=v\propto\left(\begin{array}{c}
\eta\\
0
\end{array}\right),\eta=\left(\begin{array}{c}
1\\
0
\end{array}\right)$. Then the integral (up to an irrelevant coefficient) in eq.(\ref{eq:vt})
is
\begin{eqnarray}
\frac{1}{4(z^{2}-\theta^{2}q^{2})^{2}}\int_{0}^{\infty}\mathrm{dp}[-2i\theta qz\cos(z\mathrm{p})(e^{2i\theta q\mathrm{p}}+e^{-2i\theta q\mathrm{p}})+4i\theta qz\cos(z\mathrm{p})\nonumber \\
-(\theta^{2}q^{2}+z^{2})\sin(z\mathrm{p})(e^{2i\theta q\mathrm{p}}-e^{-2i\theta q\mathrm{p}})+4i\theta q(z^{2}-\theta^{2}q^{2})\mathrm{p}\sin(z\mathrm{p})]\nonumber \\
=\frac{\pi}{4(z^{2}-\theta^{2}q^{2})^{2}}[i(z-\theta q)^{2}\delta(z+2\theta q)+i(z+\theta q)^{2}\delta(z-2\theta q)+4i\theta qz\delta(z)\nonumber \\
-4i\theta q(z^{2}-\theta^{2}q^{2})\delta^{'}(z)].\label{eq:integral}
\end{eqnarray}
This matrix element is not zero. So microcausality is violated by
this case. This result is similar to the one in canonical noncommutative
models where the violation is also proportional to sum of $\delta$-functions
\cite{Greenberg:2005jq}. We can also see that for commutative limit,
$\theta\rightarrow0$, the integral eq.(\ref{eq:integral}) is zero
and microcausality is preserved as is expected. If $z\rightarrow0$,
the integral is zero too as it should be for a commutator of the same
operator.
\section{conclusion and discussion}
In this report, we discuss the problem of microcausality of an interesting
Lorentz invariant noncommutative field theory where the noncommutativity
is induced by the spin of the field. We take $\mathcal{O}(x)=:\bar{\psi}(x)*\psi(x):$
as a sample observable and show explicitly by two examples that microcausality
is violated for the theory in general. In the canonical noncommutative
theories with constant noncommutative parameters $\theta^{\mu\nu}$,
the noncommutativity appears as extra phase factors depending on external
or internal momenta. If there is only space-space noncommutativity\cite{Gomis:2000zz},
the microcausality is preserved by observables without time derivatives\cite{Greenberg:2005jq},
and for general observables, microcausality is violated. In this spin-induced
noncommutative model, the noncommutativity also depends on the momenta
of external or internal momenta. From eq.(\ref{eq:integral}), one
can see that the violation of the microcausality consists of sum of
$\delta$-functions and nonlocality is proportional to the momenta
of external particles. These features are similar to the ones found
in canonical Moyal-type noncommutative theories \cite{Greenberg:2005jq}.
In Moyal-type noncommutative theories, there are UV/IR mixing problems.
One interesting thing is to explore whether there are similar problems
in quantum field theories on this spin-induced noncommutative spacetime.
In classical special relativity and common quantum field theories,
Lorentz covariance preserves causality and guarantees that one can
not travel back through time. But on noncommutative spacetime, even
Lorentz covariance can not preserve causality. It is also noted that
another Lorentz-covariant noncommutative model is proposed by\cite{Carlson:2002wj}
and the Lorentz violation appears when noncommutative parameters integrated
over all possible values and directions\cite{Saxell:2008zj}. We emphasize
that this nonlocality spreading procedure don't appear in spin-induced
noncommutative theory. In both cases, Lorentz covariance of the action
doesn't mean microcausality or local commutativity. This fact supports
a viewpoint, proposed by Greenberg\cite{Greenberg:2004vt}, that the
Lorentz covariance of time-ordered product of fields leads to microcausality.
Anyway, it is a really nontrival work to construct a quantum theory
of spacetime to be consistent with causality.
\begin{acknowledgments}
The authors thank V. Kupriyanov and A.J. da Silva for the communication
with them and help from them. This work is supported by the Fundamental
Research Funds for the Central Universities with the contract number
2009QNA3015.\end{acknowledgments}
|
{
"timestamp": "2012-03-12T01:00:41",
"yymm": "1203",
"arxiv_id": "1203.1992",
"language": "en",
"url": "https://arxiv.org/abs/1203.1992"
}
|
\section{Introduction}
Debris disks represent an important aspect of the evolution of extra-solar planetary systems. By studying the formation and evolution of debris disks, we can learn a great deal about the formation and evolution of our solar system (Wyatt 2008, Zuckerman 2001). Debris disks consist of small grains of dust that have been ground down by eons of collisions between rocky bodies in orbit around the central star. It is thought that our own solar system went through an especially prominent debris disk phase during the Late Heavy Bombardment (Booth et al. 2009). The dust grains produced by violent collisions of rocky bodies can be observed via thermal emission from infrared to centimeter wavelengths. For a more detailed analysis of circumstellar disks, see Heng $\&$ Tremaine (2010).
The DEBRIS (Disk Emission via a Bias-free Reconnaissance in the Infrared/Submillimeter) project aims to characterize the types of stars which host circumstellar disks using the {\it Herschel Space Observatory}, hereafter {\it Herschel}. The survey includes a roughly equal number of stars from each spectral type A-M, with 446 stars in total (Phillips et al. 2010, Matthews et al. 2010). DEBRIS is an unbiased survey, meaning that it did not specifically target stars with observed IR excesses. We want to know how debris disk incidence trends with various stellar parameters, and how those parameters evolve with time. Our motivation for determining the ages of stars is to better characterize the evolution of these disks.
A concerted effort has been made in the past five years to derive age determination equations based on a variety of parameters. In Section 2, we discuss the age determination methods available for solar type (F, G, and K-type) stars. In Section 2.1, we explore the possibility of gyrochronology, the calculation of stellar age based on the spin-down of a star over its lifetime. In Section 2.2, we discuss chromospheric emission, the use of the evolution of a star's magnetic field as an age tracer. In Section 2.3, we discuss the derivation of stellar age using X-ray emission. In 2.4, we briefly discuss the use of lithium depletion as an age tracer. In Section 2.5, we compare four age determination methods (gyrochronology, chromospheric, X-ray, vsin$i$) against each other. In 2.6, we discuss which ages represent the closest estimate of the ``true" age. In Section 3.1, we discuss the use of isochrone dating in A-type stars. Section 3.2 describes other possible methods for A star age determination. M-type stars are not treated in this paper, as their evolutionary properties are currently not well understood, and few of them have been found to host debris disks.
\section{\large F, G, and K-type stars}
\subsection{Method 1: Gyrochronology}
Developed by Barnes (2003), this method has been used by many researchers in recent years to determine relatively precise ages for field stars. The basic idea is that magnetic winds carry angular momentum away from the star, causing the outer convective envelope to spin slower than the radiative zone beneath. This creates a shear, which slows down the overall rotation of the star with time. This period change is predictable, and can be calibrated using the known age of the Sun. Barnes (2007) suggested the following relation between age (t), rotation period (P$_{rot}$), and B-V color:
\begin{equation}
P_{rot}(B-V,t)=a[(B-V)_{0}-c]^{b}t^{n}
\end{equation}
Mamajek $\&$ Hillenbrand (2008, hereafter MH08) calibrated this relation using four clusters with known ages ($\alpha$ Per - 85 Myr, Pleiades - 130 Myr, M34 - 200 Myr, and Hyades - 625 Myr). They found that a=0.407, b=0.325, c=0.495, and n=0.566 for t in Myrs and P$_{rot}$ in days. This relation could only be calibrated for stars with 0.495$<$(B-V)$_{0}$$<$1.4 and was not fit to any clusters older than the Hyades due to lack of rotational data in the literature. Therefore, their low quoted error of $\sim$15 $\%$ is only valid for ages less than 625 Myr. While no cluster data were available beyond this age limit, MH08 showed that the relation recovered similar ages for both components of binary systems out to 3 Gyr (with 20-25 $\%$ error). The error in age for stars in our sample older than $\sim$1 Gyr is taken to be $\sim$20$\%$.
Several studies use short term (on the order of weeks) variations in the strength of Ca II H \& K emission cores to measure rotation period (Donahue et al. 2006). Ca II emission variations trace the rotation of areas of increased magnetic flux (starspots) across the surface of the star. Thus, these fluxuations likely represent a true rotational period. These variations differ from the longer term (on the order of Myrs) magnetic variations discussed in Section 2.2.
Gyrochronology using true rotational periods was possible for 35 F, G, and K type DEBRIS stars. We gathered rotation period data from various sources (Baliunas et al. 1996 - 5 stars, Baliunas et al. 1983 - 2 stars, Donahue et al. 1996 - 16 stars, Wright et al., 2011 - 12 stars) and used B-V color data from the Hipparcos catalog (Perryman et al. 1997). Table 1 contains our age results and the relevant parameters. Figure 1 plots B-V vs. P$_{rot}$ for these data with so-called $gyrochrones$ (using Equation 1 with coefficient values from MH08) plotted for 0.3, 2, 5, and 7 Gyr.
Gyrochronology is less efficient for dating younger stars, since initial conditions (such as initial rotational period) become less important over time. There is also a recognized rotation period saturation limit above a mass-dependent maximum rotational period (Barnes 2010). Beyond this limit (t$_{sat}$), rotation and age cease to be related by the gyrochronology equation of MH08. For a solar twin (B-V$\sim$0.65), the rotation saturation occurs $\sim$7 Gyr.
\subsubsection{Using vsin$i$ as a Proxy for Measured Rotation Periods}
To use the gyrochronology equation to calculate age, one must know the rotation period of a star. However, rotation is very difficult to measure directly, especially for slowly rotating stars. Most ``rotation rates" in the literature are actually vsin$i$ measurements taken from spectroscopic observations (see Rutten et al. 1987, Valenti et al. 2005). One might estimate a rotation period from a vsin$i$ measurement by assuming that the average inclination ($i$) over a sample should be $\pi$/2. We tested the validity of using vsin$i$ to estimate ages by calculating ages of stars using rotational period (P$_{rot}$) and vsin$i$. We found that vsin$i$-based age calculations are not well-correlated with P$_{rot}$-based age calculations (see Figure 2). On average, the ages derived using vsin$i$ differed from the ages derived from P$_{rot}$ by $\sim$98$\%$ (see Figure 2). We therefore prefer gyrochronology ages based on P$_{rot}$ over those based on vsin$i$. We were able to find vsin$i$ ages for 113 stars. Those data are presented in the right-hand column of Table 2.
Direct comparison of the measured rotational period to the rotational period predicted by vsin$i$ showed reasonably good agreement between the two at short rotational period (<15 days); the two periods (measured and calculated) disagree on average by $\sim$45$\%$. The agreement at short rotational period is likely a result of the fact that both vsin$i$ and rotational period are easier to measure for short rotational period (i.e. high velocity). That is, stars with short rotation periods are easy to monitor photometrically. In addition, stars with high rotational velocities are easy to measure spectroscopically.
\subsection{Method 2: Chromospheric Activity}
This method was first introduced by Baliunas et al. (1983) and used by MH08 to calculate the age of field stars. The physics behind chromospheric emission is an ongoing topic of research in solar and extra-solar astronomy. Angular momentum transport through convection is more efficient than through radiation. Thus, for stars with an outer convective envelope, the outer layer spins down faster than the inner layer, creating a shear at the interface between the layers. This shear induces a magnetic field that is then perturbed and twisted by convective motions. The magnetic field lines carry this stress up into the chromosphere where magnetic heating causes emission lines to form in the cores of Ca II H and K absorption lines. Since magnetic activity decreases with stellar age, this emission will also decline as a function of stellar age. This method has been used successfully to recover the age of stars in clusters with known ages (MH08, Wright et al. 2004). Chromospheric heating is characterized by the parameter $R^\prime_{HK}$; the larger is $R^\prime_{HK}$ the more active - and therefore younger - is a given star.
Many researchers have taken optical spectra, measured the strength of the Ca II H $\&$ K lines and published $R^\prime_{HK}$ values (Soderblom et al. 1985, Henry et al. 1996, Gray et al. 2003, Maldonado et al. 2010, Wright et al. 2004, and Gray et al. 2006). We used these $R^\prime_{HK}$ values to calculate chromospheric ages for stars in the DEBRIS sample with the following formula from MH08 (for t in years):
\begin{equation}
log(t)=-38.05-17.91\ log(R'_{HK})-1.67\ log(R'_{HK})^{2}
\end{equation}
MH08 derived this empirical relation by fitting a curve to stars in clusters with known ages out to 4 Gyr. The applicability of this equation is limited by the sample of clusters used for the fit, and is therefore only accurate between
\begin{displaymath}
-5.1<log(R'_{HK})<-4.0
\end{displaymath}
This corresponds to an age range of $\sim$0.05 Gyr to 6 Gyr for a solar twin (B-V$\sim$0.65).
Wright et al. 2004 (hereafter W04) provide an equation for calculating ages from chromospheric activity which originally came from Donahue (1993). For stars in common, we derive similar ages to MH08 (see Figure 3), using the more recent equation from MH08. While there is a close correlation between the ages, it would seem that for stars older than 2 Gyr, the MH08 relation (Equation 2) produces older ages than W04, whereas for young stars ($<$2 Gyr), the opposite may be the case. This discrepancy was discussed at length in Song et al. (2004) (hereafter S04). S04 derived ages for stars less than a few hundred Myrs old using lithium abundance (discused in Section 2.4), X-ray activity (discussed in Section 2.3), and Galactic UVW space motions. Comparing those ages to ages derived using the relation from W04, S04 found that chromospheric ages from W04 were systematically older than ages derived using other methods. Discrepancies such as this are a strong motivator for revised activity-age relations, and a reason that we choose to use the more modern relation from MH08 to calculate ages for our sample; in comparison with the W04 chromospheric ages, young star chromospheric ages from MH08 are in better agreement with the S04 ages.
A total of 255 chromospheric ages were determined for DEBRIS stars with 35 stars overlapping with our calculated gyrochronology ages. A plot of gyrochronology age vs. chromospheric age can be found in Figure 4 and shows moderately good agreement between gyrochronology ages and chromospheric ages. MH08 quotes a typical error for chromospheric ages of 60$\%$.
There is an intrinsic limitation in using magnetic activity as an age tracer. We know that our Sun undergoes an 11 year activity cycle during which its log($R^\prime_{HK}$) changes from -4.83 during minimum to -4.96 during maximum (MH08). It is thought that solar-type stars undergo similar activity cycles. Using snapshot spectra of field stars, it is not possible to tell whether the star is in an activity minumum or maximum. In the case of the Sun, its calculated age at minimum and maximum is 3.64 Gyr and 5.85 Gyr respectively. This still gives a reasonable estimate of the solar age (4.57 Gyr). Thus, while multiple epoch surveys are preferable, we can still use single epoch surveys to estimate stellar age.
The only way to determine the $average$ magnetic activity over a stellar cycle (which is the parameter that actually decreases with age) is to conduct a long-term observation campaign. Such observations were made for 1296 stars at Mount Wilson observatory over a 17 year period from 1966-1983 (Duncan et al. 1991). In this paper, mean ``S-values" are listed for each star. These S values represent emission flux densities in Ca II H $\&$ K lines, and can be converted to $R^\prime_{HK}$.
When the average activity level of a star is used in the calculation of $R^\prime_{HK}$, the error due to stellar magnetic variation on short timescales should be significantly reduced. Two of our sources of $R^\prime_{HK}$ made efforts to take this variability into account in their measurements. W04 used median activity levels from 6 years of observation at Keck Observatory and 17 years of observation at Lick Observatory to calibrate their published $R^\prime_{HK}$ values. Duncan et al. (1991) contains Mount Wilson data as described above. When possible, we took our $R^\prime_{HK}$ values from one of these two sources. Otherwise, we used $R^\prime_{HK}$ values from single-epoch observing papers such as Gray et al. (2003 $\&$ 2006), Maldonado et al. (2010), and Henry et al. (1996). For stars in Table 2 observed with multi-epoch surveys, the chromospheric age should be trusted before the X-ray ages, since the X-ray age relation was derived from the chromospheric age relation (X-ray ages are described in detail in Section 2.3).
With the advent of long-term monitoring of solar-type stars (mostly in search of exoplanet transits), it will soon be possible to quantify this magnetic variation for a larger fraction of solar-type stars.
\subsection{Method 3: X-Ray Emission}
X-rays trace magnetic heating of the stellar corona. Although it is not well known exactly how the corona is heated, MH08 suggests that it is closely related to the strength of the magnetic field, and also to chromospheric heating.
X-ray count rates and hardness ratios (HR1) for the 0.1-2.4 keV band are readily available from the ROSAT survey (Voges et al. 1999). We first calculated the X-ray luminosity for 100 DEBRIS stars by the following equation from MH08:
\begin{equation}
L_x = 4\pi D^2 C_x f_x
\end{equation}
\noindent where D is distance in cm, f$_x$ is the ROSAT count rate (counts s$^{-1}$), and C$_x$ is a conversion factor defined by the following equation from MH08:
\begin{equation}
C_x=(8.31+5.3HR1) \times 10^{-12}
\end{equation}
Using stellar radii and T$_{eff}$ values from several sources (Allende Prieto et al. 1999, Valenti et al. 2005, Takeda et al. 2007), we were able to calculate L$_{bol}$ from L$_{bol}$=4$\pi$R$^{2}$$\sigma$T$_{eff}^{4}$. MH08 fit a relation between log($R^\prime_{HK}$) and log(L$_{x}$/L$_{bol}$) to derive an age relation for X-ray activity:
\begin{equation}
log(t)=1.20-2.307\ log(R_{x}) - 0.1512\ log(R_{x})^{2}
\end{equation}
\noindent where t is in years and R$_{x}$=L$_{x}$/L$_{bol}$. In MH08, this relation was never fit to cluster data. Rather, R$_{x}$ was used to predict $R^\prime_{HK}$ by:
\begin{equation}
logR'_{HK}=-4.54+0.289\ (log(R_x)+4.92)
\end{equation}
This relation was combined with Equation 2 and used to derive Equation 5. Therefore we expect scatter in the R$_x$ ages (when compared to gyrochronology ages) both from uncertainty of the actual X-ray count measurements, and from the scatter in the R$_x$ - $R^\prime_{HK}$ relation (which MH08 estimates at 55$\%$). X-ray ages and relevant parameters can be found in Table 2. We compared ages determined using X-ray flux to ages determined by gyrochronology. The results are shown in Figure 5. A clear correlation is apparent, with an average scatter of $\sim$84$\%$. Since gyrochronology produces ages with the smallest error, we can conclude that X-rays are legitimate tracers of stellar age.
While X-rays are useful for determining the ages of young, active stars, there is a saturation limit beyond which age and magnetic activity cease to be correlated by Equation 5. This ``saturation" limit defines a minimum age (and therefore maximum magnetic activity level) which can be reliably determined by the methods in this paper. The actual cutoff for this limit varies between studies. According to MH08, the relation between R$_{x}$ and $R^\prime_{HK}$ ceases to correlate above logR$_{x}$=-4. However Zuckerman $\&$ Song (2004) find moderately tight isochrones for young nearby stars and the Pleiades cluster in logR$_{x}$-color space above logR$_{x}$=-4. They do place a lower limit on the age which can reliably determined using X-rays ($\sim$100 Myr).
\subsection{Lithium Depletion}
This method was used by Baumann et al. (2010) to determine the age of planet-hosting stars. Lithium is depleted from the photosphere by way of convective mixing into interior regimes where the Li is burned. This process is dependent on stellar rotation; The faster a star rotates, the slower is this process, since there is less differential rotation and thus less mixing. There exist isochrones in lithium abundance - rotation velocity space which can be used to determine the age of a star in isolation (e.g., Chen et al. 2001, Zuckerman $\&$ Song 2004). Since rotation period is also necessary for age-determination using lithium depletion, we prefer to use gyrochronology - which has a direct equation for determining age - rather than the less precise method of matching isochrones.
In addition, the accuracy of the derived age using lithium depletion varies depending on spectral type. It is most accurate for late-K to early M-type stars. Even so, the isochrones are not well-defined in EW(Li) - age space (see Figure 3 of Zuckerman $\&$ Song 2004). Ages determined using litium abundances are certainly not as accurate as our other methods. We therefore choose not to use lithium depletion to date stars in the DEBRIS survey.
\subsection{Comparing Four Age Determination Methods}
We began our analysis by assuming that gyrochronology provides the most accurate estimation of stellar age. We then compared the other three methods (chromospheric activity, X-ray emission, and vsin$i$ measurements) to gyrochronology in a quantitative way. Two stars were left out of the analysis; HIP 19849 and HIP 25647. HIP 19849 is an extremely slow rotator - thus, its gyrochronology age may not be accurate. Also, HIP 25647 has a very low gyrochronology age (2 Myr). Its gyrochronology age is unreliable as well.
\bf Chromospheric Ages vs. Gyrochronology Ages\rm: There are 35 stars in the DEBRIS sample for which we have both chromospheric ages and gyrochronology ages. We disregard the two ourliers mentioned above and restrict our analysis to the remaining 33 stars. We define a ``discrepancy" factor, f$_{c}$, by:
\begin{displaymath}
f_{c}=\frac{t_{chromo}}{t_{gyro}}
\end{displaymath}
For the sample of 34 stars, the standard deviation ($\sigma_{c}$) of f$_{c}$ was 0.872 around a median discrepancy of f$_{c,m}$=1.15 (and a mean of 1.32). This includes ages derived using $R^\prime_{HK}$ values from both single and multiple epoch observing campaigns.\footnotemark{}
\footnotetext{This analysis should only be used as a comparative tool, and cannot be used to reliably determine the errors associated with these age determination methods (since we do not know the true age of the star). When this method was used to predict ages of stars in clusters (MH08) it yielded a 60$\%$ error. We take this to be the typical error associated with chromospheric age dating.}
\bf X-Ray Ages vs. Gyrochronology Ages\rm: There are 17 stars in the DEBRIS sample for which we have both X-ray ages and gyrochronology ages. Again, we disregard HIP 19849 and HIP 25647. For the remaining 16 stars, we again defined a discrepancy factor, f$_{x}$, by:
\begin{displaymath}
f_{x}=\frac{t_{xray}}{t_{gyro}}
\end{displaymath}
We found that $\sigma_{x}$=0.518 and f$_{x,m}$=0.89 (the mean of the discrepancies was found to be 1.32). This seems to suggest that X-ray ages are more precise than chromospheric ages.
\bf Chromospheric Ages vs. X-Ray Ages\rm: We compared ages calculated using log(R$_{x}$) to ages calculated using log$R^\prime_{HK}$ to produce Figure 6. Chromospheric ages are, on average, higher than X-ray ages. Otherwise, there is little to no correlation between the two age determination methods.
\bf vsin$i$ Ages vs. P$_{rot}$ Ages\rm: We also compared ages calculated using P$_{rot}$ to ages calculated using vsin$i$. We define a discrepancy factor f$_{v}$ by:
\begin{displaymath}
f_{v}=\frac{t_{vsini}}{t_{gyro}}
\end{displaymath}
We found that $\sigma_{v}$=1.6 around a median f$_{v,m}$ of 0.94.
Since the same gyrochronology equation is used to calculate age in both cases, what we really wanted to compare was the true P$_{rot}$ of a star to the P$_{rot}$ calculated from vsin$i$. For this purpose, we define a discrepancy factor f$_{p}$ by:
\begin{displaymath}
f_{p}=\frac{P_{vsini}}{P_{rot}}
\end{displaymath}
We find that $\sigma_{p}$=0.66 around a median f$_{p,m}$ of 0.97. The scatter in the relationship between ages calculated using vsin$i$ and ages calculated using P$_{rot}$ is largely due to the scatter in the relationship between the periods measured using vsin$i$ and the measured rotational period. From this analysis, we conclude that while vsin$i$ may be used to predict a rotational period with an error of $\sim$60$\%$, the scatter that is introduced into the gyrochronology equation from this prediction is quite high. We suggest that chromospheric and X-ray ages be taken to be more precise than vsin$i$ ages.
\subsection{Which Age Should be Used?}
In general, the gyrochronology age should be trusted above any other available age (except in the two cases mentioned above - HIP 19849 and HIP 25647). When the gyrochronology age is unavailable or unreliable, one needs to be able to make a choice between the remaining two methods of age determination. We first define a discrepancy factor f$_{cx}$:
\begin{displaymath}
f_{cx}=\frac{t_{chromo}}{t_{xray}}
\end{displaymath}
There are 62 stars in Table 2 with f$_{cx}$ $<$2. Of the remaining 30 stars, 13 stars have $R^\prime_{HK}$ values taken from multi-epoch surveys. Although our analysis in Section 2.4 suggested that X-ray ages agree better with gyrochronology ages, we know that X-ray ages depend on snapshot observations of X-ray emission. This means that ages derived from X-ray observations will be affected by the short-term variability of the stellar magnetic cycle. Thus, for these 13 stars, we believe the chromospheric ages to be the most accurate.
Finally, we were left with 17 stars which were derived using $R^\prime_{HK}$ values taken from single-epoch surveys. For these stars, we were able to use other age determination methods such as galactic space motion (Zuckerman $\&$ Song 2004), gyrochronology ages, or isochrone ages from the literature to make an educated choice between the X-ray age and the chromospheric age.
\bf HIP 13402\rm: This star has a gyrochronology age of 0.256 Gyr. This agrees best with its X-ray age 0.14 Gyr.
\bf HIP 17420, 86036, 102485\rm: These stars have isochrone ages in the literature which agree best with the derived chromospheric ages.
\bf HIP 80686, 61174\rm: These stars have isochrone ages in the literature which agree best with the derived X-ray ages.
\bf HIP 104440, 2762, 14879, 85295, 67153\rm: For these stars, we recommend taking the chromospheric age as the true age, because their UVWs (Anderson $\&$ Francis 2011) are most compatible with the chromospheric age.
\bf HIP 73695, 89805, 114948\rm: For these stars, we recommend taking the X-ray age as the true age, because their UVWs (Anderson $\&$ Francis 2011) are most compatible with the X-ray age.
\bf HIP 98470, 67422, 5896\rm: For these stars, we are unable to choose between their X-ray ages and chromospheric ages. Further long-term observations are needed to constrain their ages.
\subsection{FGK Type Star Summary}
(1) Gyrochronology produces the lowest errors of any of our age determination methods (typical errors of 15-20$\%$). It is most precise for intermediate age stars (100 Myr $<$ t $<$ t$_{sat}$ where t$_{sat}$ is B-V dependent). However, rotation periods are difficult to measure and vsin$i$ values are an inadequate proxy.
(2) Ca II chromospheric emission is a legitimate tracer for stellar age; $R^\prime_{HK}$ is an easily measured parameter, and is readily available in the literature for many field stars. Chromospheric emission is most precise in the same age regime as gyrochronology (described above). MH08 quotes a typical error of 60$\%$, although our analysis suggests an error of 87$\%$.
(3) X-ray flux is a valid tracer of stellar age. Typical errors in the age are estimated at $\sim$70$\%$ (MH08).
(4) Lithium depletion has been shown to correlate with stellar age in past studies (Chen et al. 2001); however it is most useful for young stars ($<$100 Myr) and requires knowledge of the stellar rotation period (Zuckerman $\&$ Song 2004).
\section{\large A Type Stars}
\subsection{Method: Isochrones in Log(g) - log(T$_{eff}$) Space}
Since A stars evolve quickly, their ages can be reliably determined (errors of about 100-300 Myr) from their position on an HR diagram. We began with three separate sets of isochrones in log(g) - log(T$_{eff}$) space. The first is from Li $\&$ Han (2008), the second is the Padova set of isochrones (Bertelli et al. 2009) and the third is the YREC set of isochrones (Pinsonneault et al. 2004). Values of log(g) come from a variety of sources (Gray et al. 2003, Gray et al. 2006, Lafrasse et al. 2010, Soubiran et al. 2010, Allende Prieto et al. 1999, King et al. 2003, Gerbaldi et al. 1999, and Song et al. 2001). Values of T$_{eff}$ came from Phillips et al. (2010). We estimated the age of these stars using all three sets of isochrones, then compared those resulting ages to each other and to ages published in the literature. We concluded that the YREC isochrones provide a better match to literature ages. Not all of the A stars in the DEBRIS sample fall in the area of log(g)-T$_{eff}$ space that is covered by the YREC isochrones. In some cases, we were able to use Li $\&$ Han (2008) isochrone ages. We filled in any missing ages using ages from Rieke et al. (2005), which uses Y2 isochrones from Yi et al. (2003).
In total, we were able to determine ages for all 83 A stars in the DEBRIS survey - 65 from YREC isochrones, 15 from Li $\&$ Han (2008) isochrones, 2 from Rieke et al. (2005), and 1 from cluster membership. HIP 88726 was found to be a member of the Beta Pictoris moving group, and was assigned an age of 0.012 Gyr according to Zuckerman $\&$ Song (2004). Figure 7 shows a histogram of our derived A star ages (83 stars).
\subsection{Other Methods for A Stars}
Gyrochronology will not work as well for A-type stars as it does for F, G, and K-type stars. Since A stars evolve quickly, they do not spend as much time on the main sequence, and thus their rotation does not brake in the same way as do solar-types.
Furthermore, according to MH08, the correlation between magnetic activity and age breaks down for (B-V)$_0$ $<$ 0.5. Below this limit (which includes A stars) convective envelopes are thin or nonexistent and therefore magnetic field strength caused by rotational shear between the convective and radiative layers diminishes. We therefore prefer isochrone dating to measure the ages of A stars.
\section{\large Conclusions and Future Work}
In total, we were able to reliably determine ages for $\sim$96$\%$ of A-K type stars in the Herschel DEBRIS project. Our motivation for this work was to provide the ages of DEBRIS target stars as a diagnostic for debris disk evolution. Forthcoming papers (i.e. Thureau et al. 2012, in prep, Sibthorpe et al. 2012, in prep) will elaborate on specific disk characteristics as a function of stellar age.
While DEBRIS was an unbiased survey, we hope to expand our age determination project to include any stars observed by Herschel with observed infrared excesses.
Partial support for this work, part of the NASA Herschel Science Center Key Program Data Analysis Program, was provided by NASA through a contract (No. 1353184, PI: H. M. Butner) issued by the Jet Propulsion Laboratory, California Institute of Technology under contract with NASA. The program, DEBRIS, or Disc Emission via a Bias-free Reconnaissance in the Infrared/Submillimetre, is a Herschel Key Program (P. I. Matthews). Herschel is the 4th cornerstone mission of the European Space Agency (ESA) science program.
We thank Ben Zuckerman for his valuable insight, David Rodriguez and Erik Mamajek for their helpful input, and Greg Henry for allowing us access to his rotation period data (Wright et al., in press) prior to publication.
|
{
"timestamp": "2012-03-30T02:03:52",
"yymm": "1203",
"arxiv_id": "1203.1966",
"language": "en",
"url": "https://arxiv.org/abs/1203.1966"
}
|
\section{Introduction}
\label{intro}
Type I X-ray bursts are thermonuclear explosions triggered by unstable
burning in the oceans of accreting neutron stars. The ocean is the low density fluid layer of
light elements that builds up via accretion on top of the solid
crust (see \citealt{Chamel08} for a discussion of the
conditions under which a Coulomb plasma of ions will crystallize to
form a solid). The basic cause of bursts, an imbalance
between nuclear heating and radiative cooling in settling material, can be explained using
simple one-dimensional calculations (for reviews see \citealt{Lewin93}
and \citealt{Bildsten98b}). The past decade, however, has revealed major
complexities that can no longer be understood within the context of
such simple models \citep{Strohmayer06}.
One such complexity is the development of asymmetric brightness patches, known as burst oscillations, in about 10\% of the bursts observed with high time resolution instrumentation \citep{Galloway08}. What drives this process remains unexplained, and requires us to consider flame spreading and other multidimensional effects. The highly atypical properties of burst oscillations from the accretion-powered pulsars also suggest an important dynamical role for the magnetic field. This review article provides an overview of our current understanding
of burst oscillations: the techniques employed to detect and analyse
them, their key observational properties, the status of theoretical
models, and the ways in which they can be used to constrain stellar spin rates and the dense
matter equation of state.
\subsection{Burst oscillations - a brief history}
Although a number of searches for periodic phenomena in X-ray bursts were carried out prior to the mid 1990s, and some tentative detections were claimed, none have stood the test of time (\citealt{Lewin93}, \citealt{Jongert96}). Conclusive
detection of strong periodic signals in thermonuclear bursts came only with
the launch of the {\it Rossi X-ray Timing Explorer} (RXTE) on December
30th, 1995.
Observations of the burster 4U 1728-34 in February 1996, only weeks
after the start of science operations, led to the discovery of a
strong 363 Hz signal in six X-ray bursts
\citep{Strohmayer96b}. The
authors dubbed this phenomenon `burst oscillations'. Many
of the properties that we now consider hallmarks of burst
oscillations were apparent even in these first observations: high
coherence, an upwards
drift of $\sim 1$ Hz towards an asymptotic maximum frequency as the burst progressed, amplitudes of up to $\sim 10$
\% root mean square (rms), and the
apparent disappearance of the signal during the burst peak.
\citet{Strohmayer96b} concluded that rotational modulation of a bright
spot on the burning surface was the most likely explanation for the
burst oscillations. This was based on three arguments: the
expected evolutionary link between accreting neutron stars in Low Mass X-ray Binaries and the
millisecond radio pulsars (with the former being progenitors of the latter
via the spin recycling scenario, see \citealt{Bhattacharya91}); the
ability of a localised hotspot to explain the observed amplitudes; and
the high coherence of the oscillations seen in the burst tails.
Between 1996 and 2002, burst oscillations were discovered in
bursts from eight more sources (see Table \ref{sources}). Although there
was still no independent confirmation of spin rate, the link between
burst oscillation frequency and
stellar rotation was nonetheless strengthened. The fact that
the same frequency was seen in multiple bursts from any given source,
and the high stability of the asymptotic frequency of the
drifting burst oscillations, both pointed to
rotational modulation of a bright spot that was near stationary in the
rotating frame of the star (\citealt{Strohmayer98b}, \citealt{Muno02a}). The
detection of burst oscillations during a superburst
(a longer, more energetic burst thought to be due to unstable
carbon burning) from 4U 1636-536 also supported this interpretation. The frequency matched that seen in the normal Type I bursts despite
the different burst physics, suggesting an external clock
\citep{Strohmayer02a}.
In October 2002, the 401 Hz accretion-powered millsecond X-ray pulsar SAX
J1808.4-3658 went into outburst, and \citet{Chakrabarty03} reported the first robust detection of burst oscillations from a source with an independent measure of the
spin rate (analysis of a burst during a previous accretion episode had yielded a marginal detection, \citealt{intzand01}). Although the frequency of the burst
oscillations drifted by several Hz in the burst rise, the frequency in
the tail was within $\approx 6 \times 10^{-3} $ Hz of the spin
frequency. A second accretion-powered pulsar with burst
oscillations at the spin frequency, XTE J1814-338, followed shortly thereafter
\citep{Strohmayer03}, confirming the status of burst oscillation
sources as nuclear-powered pulsars.
Since this time, burst oscillations have been found in several
additional sources (Table \ref{sources}), including three more
accretion-powered pulsars and two intermittent pulsars (sources
that show accretion-powered pulsations only sporadically). While
rotational modulation of a brightness asymmetry on the burning surface remains an integral
part of all models, the root cause of such an asymmetry remains
unresolved.
\section{Burst oscillation observations}
\subsection{Principles of detection}
\label{detection}
The standard analysis technique, when searching for periodic or
quasi-periodic signals, is to use Fast Fourier Transforms to produce a
power spectrum. For a comprehensive review of this topic, the reader
is referred to \citet{vanderKlis89}. Here I summarize only the key
points that are essential to an understanding of burst oscillation
detection (this is of particular relevance to the discussion in
Table \ref{tentsources} about tentative detections) and subsequent analysis of
properties like frequency and amplitude.
A series of X-ray photon arrival times with total duration $T$ is binned
to form a time series $x_k(t)$, the number of photons (`counts') in time bin
$t_k$ ($k = 1...N$). The time resolution $\Delta t = T/N$ is limited
by the native time resolution of the instrument
(typically 1 - 125 $\mu$s for RXTE, depending on data mode), but is
also set by the desired Nyquist frequency $f_\mathrm{Ny}$. This is the maximum frequency that can be studied for a given time resolution, and is given by $f_\mathrm{Ny} = 1/(2\Delta t)$. The
power spectrum $P_j$ at the Fourier frequencies $\nu_j = j/T$ ($j = 0,
2, ..., N/2$ where $\nu_{N/2} = f_\mathrm{Ny}$), using the standard Leahy
normalisation \citep{Leahy83}, is then given by
\begin{equation}
P_j = \frac{2}{N_\gamma} \left[\left(\sum^N_{k=1} x_k \cos
2\pi \nu_j t_k\right)^2 + \left(\sum^N_{k=1} x_k \sin 2\pi
\nu_jt_k\right)^2\right]
\end{equation}
where $N_\gamma$ is the total number of photons.
In the absence of any deterministic signal, the Poisson statistics of
photon counting yield powers that are distributed as $\chi^2$
with two degrees of freedom (d.o.f.). The presence of a periodic signal at a given frequency will boost the power in the appropriate frequency bin. If we detect a large power, however, we must first evaluate its significance by computing the probability of obtaining such a power through noise alone. By this we mean the
chances of getting such a high value of the power,
in the absence of a periodic signal, due to the natural fluctuations
in powers that are a by-product of photon counting statistics. We
compute this using the known properties of the $\chi^2$ distribution,
taking into account the numbers of trials (see below).
Many burst oscillation papers use the $Z_n^2$ statistic, instead of the
power spectrum. Although very similar to the standard power spectrum computed from a Fourier
transform, it does not require that the photon arrival times be binned. It is
defined as
\begin{equation}
Z^2_n = \frac{2}{N_\gamma}\sum^n_{k=1}\left[\left(\sum^{N_\gamma}_{j=1} \cos
k\phi_j\right)^2 + \left(\sum^{N_\gamma}_{j=1} \sin
k\phi_j\right)^2\right]
\end{equation}
where here $n$ is the number of harmonics (most commonly taken to be 1), $k = 1, 2, ... n$ is the index used to sum over harmonics, $N_\gamma$ is the total number of photons, and $j$ is an index applied to
each photon. The phase $\phi_j$ for each photon is defined as
\begin{equation}
\phi_j = 2\pi \int_{t_0}^{t_j} \nu(t) dt
\label{phasemodel}
\end{equation}
where $\nu(t)$ is the frequency under consideration and
$t_j$ is the arrival time of each photon relative to some reference
time $t_0$. Frequency $\nu(t)$ is most commonly taken to be constant (so that it can be taken outside the integral) but on occasion may be a function of time, for example if one wants to correct for smearing due to orbital Doppler shifts as the neutron star orbits the center of mass of the binary. In the absence of a deterministic (e.g. periodic) signal, $Z_n^2$ is distributed as
$\chi^2$ with $2n$ d.o.f.. An advantage of $Z_n^2$ is that it offers an efficient way of summing harmonics.
When searching for drifting, or quasi-periodic signals, it is
common to average powers from neighbouring frequency bins. In searches for weak
signals one can also stack, or take an average of, power
spectra from many independent data segments (time windows) or bursts. This affects
the number of degrees of freedom in
the theoretical $\chi^2$ distibution of noise powers, but the modified
theoretical distribution is known (see
\citealt{vanderKlis89} for more details).
To assess whether a periodic signal is indeed present, we first take the power spectrum and identify the frequency bins with the highest powers. One then
computes the probability of obtaining such high powers
through Poisson noise alone, using the properties of the $\chi^2$ distribution
with the appropriate number of degrees of freedom. One must then take
into account the number of trials that have been made - the more trials, the more likely one is to obtain a high power via statistical fluctuations alone. The number of trials is the product of the number of
independent frequency bins, time windows, energy bands, bursts and (if appropriate) sources searched. Assessing the number of trials
properly is complicated by the fact that many burst oscillation
searches use overlapping time bins or oversampled frequency bins to
maximise candidate signal power. Failure to account properly for this
can result in the significance of a candidate signal being
overestimated (or underestimated, if overlapping bins are treated as being independent). If the probability of such a high value of the power arising through
Poisson noise alone is below a certain threshold after taking into account
numbers of trials, then it is deemed significant, with the quoted
significance referring to the chances of having obtained such a signal
through Poisson noise alone.
A complication comes from the fact that noise powers (the power distribution in the absence of a periodic signal) are
not always distributed in accordance with the $\chi^2$ that would be expected from constant Poisson noise. What we are
searching for in an X-ray burst is a periodic signal superimposed on
the overall deterministic rise and decay of a burst lightcurve. These variations give low frequency power and add sidebands to noise powers, boosting their level (for a nice discussion of these issues, see
\citealt{Fox01}). This problem is particularly acute in the rising
phase of the burst when the lightcurve is changing rapidly. There may also be other noise components from astrophysical or detector processes, such as red noise associated with accretion, which may continue during the burst and contribute to the overall emission.
Assessing the true distribution that the powers should take, in the absence of a periodic signal, is
most easily done using Monte Carlo simulations. One can for example make a model
of the burst lightcurve, by fitting the rise
and decay with exponential functions, and then use this as a basis to generate a large sample of
fake lightcurves with Poisson counting statistics but without any periodic signals.
Taking power spectra from these fake lightcurves then yields the true, most likely
frequency-dependent, distribution of noise powers. One other
advantage of the Monte Carlo method is
that any pecularities of the data analysis process (such as the use of
overlapping time windows) can be replicated precisely. As a rule of
thumb, unless a
candidate signal is
particularly strong, or is repeated in multiple independent
bursts or time windows (thereby boosting
significance), Monte
Carlo simulations should really be done to obtain an accurate
assessment of significance. Alternatively, one can also try to fit the distribution of measured powers (excluding the frequency bins containing the candidate signals), however this is somewhat risky since it by definition excludes extreme values from the fit.
One final factor that must be considered, for the accretion-powered pulsars
with burst oscillations, is the likely continuation of the channeled
accretion process during the burst. One must therefore ask
whether the signal detected during the burst could be due to the
accretion-powered pulsations. If the accretion process (accretion rate, and fractional amplitude of accretion-powered pulsations) remains unchanged during the burst, the addition of a large number of unpulsed photons during the burst would cause the measured fractional amplitude to fall (by a factor $N_\mathrm{bur}/N_\mathrm{acc}$ where $N_\mathrm{bur}$ is the number of photons due to burst emission and $N_\mathrm{acc}$ the number due to accretion in the data segment being considered). In all cases so far, fractional amplitudes remain similar despite the fact that burst
flux far exceeds accretion flux, indicating that the burst emission itself
must also be pulsed (provided that the assumption of unchanged
accretion holds). Whether this assumption is reasonable remains a matter of debate. Naively one might expect radiation
pressure from a burst to hinder the accretion process. However the radiation may also
remove angular momentum, increasing the accretion rate (\citealt{Walker89}, \citealt{Miller93}, \citealt{Ballantyne05}).
\subsection{Measurable properties}
Once a burst oscillation signal has been identified and deemed statistically
significant, one can measure several observational properties. Some
properties can be calculated directly from the power spectrum or $Z_n^2$ statistic (for the rest of this subsection, these two
terms can be used interchangeably), whereas
others require pulse profile modelling. The latter technique involves selecting a
frequency model for the signal, $\nu(t)$, and calculating a phase for each photon using Equation (\ref{phasemodel}), using this to assign
each photon to a phase bin, and
thereby building up a pulse profile (number of photons per phase bin). The main intrinsic properties of interest are frequency,
amplitude, and phase lags. The first two can be measured from the
power spectrum or using pulse profile modelling, but to obtain phase lags one must use pulse
profile modelling since the power spectrum discards phase information.
In terms of frequency, we are interested in both the absolute value
and the properties of any drift. The latter is most commonly
visualised using dynamical power spectra. These are computed by taking
power spectra of short, usually overlapping, segments of data from the
burst lightcurve, as illustrated in Figure \ref{dps}. The power spectra
are then commonly plotted as contours, overlaid on the
burst lightcurve. Several examples of bursts with oscillations, from different sources, are shown Figure \ref{bplots}. One can then see the
frequency drift that occurs as the burst progresses. Another
property, related to frequency, that is often given is the coherence
$Q$
of the burst oscillation train. This is given by
\begin{equation}
Q = \frac{\nu}{\Delta \nu_\mathrm{FWHM}},
\end{equation}
where $\nu$ is the frequency of the peak in the power spectrum and
$\Delta \nu_\mathrm{FWHM}$ its full width at half-maximum, obtained by
fitting a Gaussian or Lorentzian function to the broad peak in the
power spectrum. Note that $Q$ values are often quoted for peaks where the frequency drift has
been incorporated into the model (as in Equation \ref{phasemodel}), thereby rendering the peak in the power
spectrum narrower, and hence more coherent (see for example
\citealt{Strohmayer99}). In this case the frequency $\nu$ that is
used in computing $Q$ is normally the
asymptotic maximum frequency (Sections \ref{intro} and \ref{frequencies}).
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, clip]{WattsFig1.pdf}
\caption{Generating a dynamical power spectrum. The burst lightcurve (solid black line) is first split into segments (top left). In this case, there are three overlapping time windows, $\Delta t_0$, $\Delta t_1$ and $\Delta t_2$. The time stamps $T_0$, $T_1$ and $T_2$ refer to the midpoints of each window. Computing a power spectrum for each segment (center left), one can see that both power and frequency vary with time (blue = low power, purple = medium power, red = high power). This can be plotted in power-frequency-time space (bottom left). The resulting contours of power (purple = medium, red = high) are projected onto the frequency-time plane and the resulting dynamical power spectrum overplotted on the lightcurve (right). }
\label{dps}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, clip]{WattsFig2.pdf}
\caption{Light curves (grey) and dynamical power spectra (black) showing burst oscillations for bursts from several sources: two persistent accretion-powered pulsars (SAX J1808.4-3658 and XTE J1814-338), two intermittent pulsars (HETE J1900.1-2455 and Aql X-1) and two non-pulsars (4U 1636-536 and 4U 1728-34), using data from RXTE. The dynamical power spectra use overlapping 2 s windows, with new windows starting at 0.25 s intervals. We use a Nyquist frequency of 2048 Hz and an interbin response function to reduce artificial drops in amplitude as the frequency drifts between Fourier bins. The contours show Leahy normalized powers of 10-100, increasing in steps of 10. The dashed lines on the pulsar plots indicate the spin frequency determined from accretion-powered pulsations. }
\label{bplots}
\end{center}
\end{figure}
The amplitude of the pulsations can be computed directly from the power
spectrum, or from a folded pulse profile (see for example \citealt{Muno02b} and \citealt{Watts05}). When using the power spectrum, the root mean square (rms)
fractional amplitude $r$ is given by
\begin{equation}
r = \left(\frac{P}{N_\gamma}\right)^\frac{1}{2}
\left(\frac{N_\gamma}{N_\gamma - N_b}\right)
\end{equation}
where $P$ is the signal power, $N_\gamma$ the total number
of photons, and $N_b$ the number of background photons (estimated from the period before or after the burst). Note that signal power is not
the same as the measured power. Once a power has been deemed
significant (i.e. unlikely to have occurred through noise alone),
one must then determine the signal power that would give rise to the
measured value, since measured power includes the effects of noise
\citep{Groth75}. For details of how to compute signal power from measured power, see \citet{Vaughan94}, \citet{Muno02b} or \citet{Watts05}. For sources
that also have accretion-powered pulsations, this value is often
corrected to remove the expected contribution due to
continuing pulsed accretion during the burst. Other related
quantities that can be computed are the amplitudes of any higher
harmonics of the burst oscillation frequency (often referred to as the
`harmonic content'), and the dependence of
the amplitude on photon energy.
There are several factors to be wary of when using values of amplitude
from the literature. Different groups use different definitions of amplitude: for a lightcurve that varies as $C + A \sin 2\pi \nu t$, different groups may quote, for the signal at frequency $\nu$, full fractional amplitude ($2A/C$), half fractional amplitude ($A/C$), or root mean square (rms) fractional amplitude ($A/C\sqrt{2}$). In addition, amplitude depends on the frequency model used. Fitting apparent
frequency drift will increase amplitude, but this must be taken into
account when testing models since the two are not independent. The
third factor, specific to the power
spectrum technique, is that amplitude is suppressed when frequency
drifts between Fourier bins \citep{vanderKlis89}. This effect can
sometimes be seen in published dynamical power spectra: compare for example the
dynamical power spectra in Figure 1 of \citet{Muno02b} with those in
Figure 1 of \citet{Watts09}, where an
interbin response function is used to reduce the effects of this
drop (although note that this does not affect the amplitudes in \citet{Muno02b}, which
were computed using folded pulse profiles).
Energy-dependent phase lags, which are of use when testing models, are computed by
comparing pulse profiles constructed from photons in different energy
bands. The degree to which the folded pulse profiles are offset is the phase lag.
This type of analysis can also be used to compare the phases of burst
oscillations to accretion-powered pulsations, for sources that show
both phenomena. Care must be taken when doing this to correct for
any contamination due to the continued presence of accretion-powered
pulsations during the burst. However the magnitude of the correction
is simple to estimate (see \citealt{Watts08b}).
Finally, one can analyse the dependence of the burst oscillation
characteristics on the properties of
the bursts and the accretion state of the star. Burst
properties may include the total integrated flux (fluence), peak flux, the presence or absence
of photospheric radius
expansion, duration, and rise and decay timescales. In terms of accretion
state, one can consider the overall accretion rate (which can be estimated in various ways) and, for the
pulsars, compare the properties of the burst oscillations to those of
the accretion-powered pulsations. Note that burst type also depends
strongly on accretion rate, so these two factors are not unrelated.
\subsection{Observational properties}
\subsubsection{Overview of burst oscillation sources}
Secure detections of burst oscillations have now been made in 17
sources (Table \ref{sources}), including 5 persistent
accretion-powered pulsars and 2 intermittent accretion-powered
pulsars. Note that all are transient accretors, and the use of the term persistent to describe the presence of accretion-powered pulsations
through accretion episodes - as used here -
should not be confused with its use to describe sources that are
persistently, as opposed to transiently, accreting. All of the burst oscillation sources are in Low Mass X-ray Binaries \citep{Tauris06} with orbital periods (where known) in the range 1.4 to 19 hours. There is one candidate ultracompact binary in this group, 4U 1738-34, but the orbital period for this source is not known, see \citealt{Galloway10b}. Ideally, for a detection to be secure, the same frequency
should have been seen in multiple bursts from the same source. For
one of the sources in Table \ref{sources} this is not the case. In
this case the high statistical significance of the detection is based
upon the fact
that the same frequency was seen in multiple independent time bins
within a single burst, very close to the spin frequency
observed from accretion-powered pulsations.
Table \ref{tentsources} gives details of several other sources for
which burst oscillation detections
have been claimed. Some of the detections are very marginal (below
the standard 3$\sigma$ threshold), and most
rely on a power in a
single independent time bin in a single burst. Also given in the
table is a summary of the analysis procedure used to estimate the
significance of the claimed detection. Comparing these procedures to
the ideal outlined in Section \ref{detection},
the significance of some of these results has probably been
over-estimated. As such, they should be considered tentative until
confirmed in a second burst.
\begin{deluxetable}{l c p{10cm}}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Sources with confirmed detections of burst oscillations \label{sources}}
\tablehead{\colhead{Source} & \colhead{Frequency (Hz)} &
\colhead{References\tablenotemark{\dag}} \\}
\tablecolumns{3}
\startdata
\sidehead{Persistent accretion-powered pulsars with burst
oscillations}
SAX J1808.4-3658 & 401 & \citet{Chakrabarty03} \\
IGR J17498-2921 & 401 & \citet{Linares11}, \citet{Chakraborty12} \\
XTE J1814-338 & 314 & \citet{Strohmayer03} \\
IGR J17511-3057 & 245 & \citet{Altamirano10a} \\
IGR J17480-2446 & \phantom{ }11 & \citet{Cavecchi11} \\
\sidehead{Intermittent accretion-powered pulsars with burst oscillations}
Aql X-1 & 550 & \citet{Zhang98} \\
HETE J1900.1-2455 & 377 & \citet{Watts09}\tablenotemark{\ddag} \\
\sidehead{Burst oscillation sources without detectable accretion-powered pulsations}
4U 1608-522 & 620 & \citet{Hartman03}, \citet{Galloway08} \\
SAX J1750.8-2900 & 601 & \citet{Kaaret02}, \citet{Galloway08} \\
GRS 1741.9-2853 & 589 & \citet{Strohmayer97b} \\
4U 1636-536 & 581 & \citet{Strohmayer98a}, \citet{Strohmayer02a} \\
X 1658-298 & 567 & \citet{Wijnands01} \\
EXO 0748-676 & 552 & \citet{Galloway10} \\
KS 1731-260 & 524 & \citet{Smith97}, \citet{Muno00} \\
4U 1728-34 & 363 & \citet{Strohmayer96b} \\
4U 1702-429 & 329 & \citet{Markwardt99} \\
IGR J17191-2821 & 294 & \citet{Altamirano10b} \\
\enddata
\tablenotetext{\dag}{Second (confirmation) references are given for
sources where the initial discovery rested on a single burst (but
see also note \ddag).}
\tablenotetext{\ddag}{Although burst oscillations from this source have only
been detected in a single burst, they were observed in multiple
independent time bins. We
therefore consider this detection to be secure.}
\end{deluxetable}
\begin{deluxetable}{l c p{11cm}}
\tabletypesize{\tiny}
\tablewidth{0pt}
\tablecaption{Sources with single burst or tentative burst
oscillation detections \label{tentsources}}
\tablehead{\colhead{Source} & \colhead{Frequency (Hz)} & \colhead{Remarks} \\ }
\tablecolumns{3}
\startdata
XTE J1739-285 & 1122 & \citet{Kaaret07} report a $\sim 4 \sigma$
signal in a 4s window in the tail of one burst
(of five observed). The significance was estimated using Monte Carlo
simulations, taking into account number of bursts, energy
bands, frequencies and time windows
searched. \citet{Galloway08}, however, report no significant signal in
the same data. The result seems very sensitive to the choice of
time windows (start point, overlapping vs. independent). \\
GS 1826-34 & \phantom{ }611 & \citet{Thompson05} report a $\sim 4\sigma$ signal after stacking power spectra from multiple 0.25 s time windows from 3 separate bursts, fitting the measured powers to obtain the distribution and taking into account the number of bursts, time windows and frequency bins used to create the power spectra. However the number of trials is underestimated: the authors made selections in energy, in addition to searching other time windows and fine-tuning both segment length and the number of segments included. \\
A 1744-361 & \phantom{ }530 & \citet{Bhattacharyya06d}
report a $\sim 6\sigma$ signal in a 4s window in the
rise of the one burst observed from this source. This
significance is estimated using $\chi^2$ with 2 d.o.f., taking into account the number of frequencies searched. \citet{Galloway08} confirm the detection in this burst,
but no other burst from this source has been observed with high time
resolution instruments.\\
4U 0614+09 & \phantom{ }415 & \citet{Strohmayer08} report
a $\sim 4 \sigma$ signal in a 5s window in the cooling tail of one
burst (of two observed). This significance is estimated using $\chi^2$ with 2 d.o.f., taking into account frequency and
time windows searched. \\
SAX J1748.9-2021 & \phantom{ }410 & \citet{Kaaret03} report a $4.4 \sigma$ oscillation in one time window of one burst from this source. Significance was estimated using the $\chi^2$ distribution with 2 d.o.f., taking into account the number of frequency bins searched. This source has since been discovered to be an intermittent accretion-powered pulsar with spin frequency 442 Hz \citep{Altamirano08}. Re-analysis by \citet{Altamirano08}, including the number of bursts and time windows searched in the number of trials, and not using overlapping windows to maximise power, found a revised significance for the burst oscillation candidate of $2.5\sigma$. \\
MXB 1730-335 & \phantom{ }306 & \citet{Fox01} report a
$\sim 2.5\sigma$ candidate signal after stacking power
spectra from
the 1s rising phase of 31 bursts. This significance is estimated
using
Monte Carlo simulations of the stacked spectra, taking into account
the number of frequency
bins searched.\\
4U 1916-053 & \phantom{ }270 & \citet{Galloway01} report a $4.6\sigma$ signal
during the $\sim 1$s rise of one burst. Significance is estimated using $\chi^2$ with 2 d.o.f., taking into account the number of frequency bins
searched. Weaker candidate signals are
present in adjacent frequency bins in the tail, but the signal cannot
be phase-connected between rise and tail. Burst oscillations are not
seen in the 13 other bursts from this source observed by
RXTE \citep{Galloway08}. \\
XB 1254-690 & \phantom{ }95 & \citet{Bhattacharyya07b} reports a $2\sigma$
candidate signal in the 1s rise of one burst (of five observed). This
significance is estimated using $\chi^2$ with 2 d.o.f., taking into account bursts and frequency bins
searched. \\
EXO 0748-676 & \phantom{ }45 & \citet{Villarreal04} report a $\sim 5.5\sigma$ signal after stacking power spectra from the tails of 38 bursts. Significance was estimated by fitting the distribution of measured powers, taking into account the number of frequency bins searched. Since this time, strong burst oscillations at 552 Hz have been discovered in this source (\citealt{Galloway10} and Table \ref{sources}). \citet{Galloway10} confirm the detection of the 45 Hz feature in the sample used by \citet{Villarreal04}, but do not find it when using a larger sample of bursts. Its nature remains unclear.
\enddata
\tablecomments{The table gives the significance quoted by the authors
for the candidate detection, and explains the methods used to derive
this significance. As explained in Section \ref{detection}, estimates
derived using $\chi^2$ with 2 d.o.f. rather than Monte
Carlo simulations tend to overestimate significance. The procedure
for estimating number of trials varies from paper to paper: ideally,
proper allowance should be made for the number of bursts,
energy bands, frequency bins {\bf and} time windows searched. The SAX J1748.9-2021 result is particularly informative in terms of illustrating the importance of using the correct number of trials. Whether to
give equal weight in terms of numbers of trials to every burst
searched remains a matter of debate. The number of RXTE proportional counter units active
during observations varies (with more photons maximising detection
chances), and in addition burst oscillations tend to occur
preferentially in certain accretion states (Section \ref{burstcond}). }
\end{deluxetable}
\subsubsection{Conditions in which burst oscillations are detected}
\label{burstcond}
For four of the five persistent pulsars, burst oscillations have been detected in
all observed bursts, irrespective of burst properties or accretion
state (see Table \ref{sources} for references) - although the persistent pulsars do not typically exhibit a wide range of accretion states. For the fifth persistent pulsar (IGR J17498-2921), oscillations have only been detected in the two brightest bursts. However the upper limits on the presence of oscillations in the weaker bursts are comparable to the amplitudes detected in the brighter bursts \citep{Chakraborty12}, so they could well be present at the same level in these other bursts. For the other sources (the non-pulsars and intermittent pulsars), burst oscillations are detected in only a subset of bursts.
A comprehensive analysis of burst and burst oscillation
properties by \citet{Galloway08} found that the properties of the
bursts (e.g. duration, recurrence
times) where oscillations are detected are not
unusual compared to the properties of bursts where oscillations
are not detected. There does however appear to be a correlation with
accretion state, as estimated from the color-color diagram, a plot of hard against soft X-ray colors. As accretion rate increases, Low Mass X-ray Binaries typically move from the top left (hard) to bottom right (soft), tracing out a Z-shaped pattern \citep{Hasinger89}. The persistent pulsars, including those with burst oscillations, tend to remain in the hard (low accretion rate) state. The sources that are not persistent pulsars tend to show burst oscillations when the source is in the soft (high accretion rate) state.
This leads, as previously noted by \citet{Muno01} and \citet{Muno04}, to
an apparent relationship between the occurrence of burst oscillations, spin frequency, burst type
and the presence of photospheric radius expansion (PRE). Sources with burst
oscillation frequency $< 400$ Hz tend to have short, most likely helium-dominated
bursts. Higher frequency sources tend to have longer bursts, that probably involve mixed
hydrogen/helium burning. In the soft state, short bursts are less likely to have PRE whilst long bursts are more likely. Since burst oscillations occur more often in the soft state, the low frequency sources are more likely to have oscillations in bursts without PRE, while high frequency sources are more likely to
have oscillations in bursts with PRE. The distinction is however not absolute and the relationship between these various factors is clearly complex. For a more in-depth discussion of the apparent link between burst type and rotation rate, see \citet{Galloway08}.
Given that burst oscillations (for sources that are not
persistent pulsars) seem to occur preferentially in certain accretion states, one
can ask whether the non-detection of burst oscillations in other
sources is at all surprising. \citet{Galloway08} characterized position on the Z-shaped track in the color-color diagram using a variable $S_z$, with low values of $S_z$ corresponding to harder states, and high
values to softer states. For 87\% of bursts with oscillations, they found $S_z>1.75$, cementing the link between burst oscillations and the soft, high accretion rate state. Table \ref{otherbursters} summarizes
the status for prolific bursters in their data set, for which burst oscillations have not been found. The
sample includes one intermittent pulsar, SAX J1748.9-2021. For most of
these sources the range of accretion states that were observed was
not sufficient to enable full characterization of the color-color
diagram. This means that the position variable $S_z$ could not be calculated for these
sources, and so it is difficult to judge whether the non-detection of burst oscillations
is unexpected. For the two sources that do have a
fully-characterized color-color diagram (4U 1705-44 and 4U 1746-37), however, there are bursts with $S_z>1.75$. Given that 57\% of the bursts with $S_z>1.75$ studied by \citet{Galloway08} showed burst
oscillations, it is perhaps rather surprising that no
oscillations have been found in these two sources (although the source with the most high $S_z$ bursts, 4U 1746-37, has very
unusual bursting behaviour in all respects). A more rigorous analysis of whether the non-detection of oscillations in the other bursters can be attributed to their being observed in harder accretion states would be desirable.
In terms of when burst oscillations occur during bursts, \citet{Galloway08} show
that although they can be observed at any point during the burst
(rise, peak or tail), they are most commonly detected in
the tails of bursts. Oscillation trains tend to be interrupted during the
peaks of bursts, particularly (although by no means exclusively) during episodes of
PRE (\citealt{Galloway08}, \citealt{Altamirano10a}).
Burst oscillations are clearly a common feature of many normal Type I X-ray
bursts. But what about the other types of thermonuclear burst? A number of systems (although none
of the confirmed burst oscillation sources in Table \ref{sources})
have shown intermediate
duration bursts that last several hundred seconds (\citealt{intZand07},
\citealt{Falanga08}, \citealt{Linares09}, \citealt{Kuulkers10}). At present there are two models to explain
the occurrence of such bursts, both involving the build-up and
ignition of a thick layer of helium. Either the system is
ultracompact, so that it accretes nearly pure helium (\citealt{intZand05}, \citealt{Cumming06})
or a thick layer of helium is built up from unstable hydrogen burning
at low accretion rates (\citealt{Peng07}, \citealt{Cooper07c}). Due to their
rarity, very few intermediate bursts have been observed with high time
resolution instruments. Where they have, no significant burst
oscillation signal has been found (\citealt{Linares09}, \citealt{Kuulkers10}).
Burst oscillations have however been detected in one superburst
\citep{Strohmayer02a}. The source in question, 4U 1636-536, also shows burst oscillations at the same
frequency in its Type I bursts. Superbursts, which last several
hours, are thought to be caused by unstable carbon ignition deeper
within the ocean than the Type I burst ignition point
(\citealt{Cumming01}, \citealt{Strohmayer02b}, \citealt{Kuulkers04}). The oscillations were
detected over about 800 s during the peak of the 4U 1636-536
superburst, but not during the long, decaying tail. High time resolution
data has only been obtained for one other superburst, from the
ultracompact source 4U 1820-30. Unlike 4U 1636-536, this source has
not shown burst oscillations in its Type I bursts. Unfortunately an antenna
malfunction on {\it RXTE} resulted in the loss of the high time
resolution data from the peak of this superburst \citep{Strohmayer02b}. However no oscillations
are detected during the superburst tail (unpublished analysis).
\begin{deluxetable}{lccp{8cm}}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Prolific bursters without
oscillations. \label{otherbursters}}
\tablehead{\colhead{Source} & \colhead{Number of bursts} & $S_z$ reported? & Bursts
with $S_z>1.75$ }
\tablecolumns{4}
\startdata
2E~1742.9$-$2929 & 84 & N & \nodata \\
Rapid~Burster & 66 & N & \nodata \\
GS~1826$-$24 & 65 & N & \nodata \\
Cyg~X-2 & 55 & N & \nodata \\
4U~1705$-$44 & 47 & Y & 8 \\
4U~1746$-$37 & 30 & Y & 20 \\
4U~1323$-$62 & 30 & N & \nodata \\
SAX~J1747.0$-$2853 & 23 & N & \nodata \\
EXO~1745$-$248 & 22 & N & \nodata \\
XTE~J1710$-$281 & 18 & N & \nodata \\
1M~0836$-$425 & 17 & N & \nodata \\
SAX~J1748.9$-$2021 & 16 & N & \nodata \\
4U~1916$-$053\tablenotemark{\dag} & 14 & N & \nodata \\
GX~17+2 & 12 & N & \nodata \\
4U~1735$-$44 & 11 & N & \nodata \\
\enddata
\tablenotetext{\dag}{There is a tentative detection of a burst
oscillation frequency for this source, see Table \ref{tentsources}.}
\tablecomments{Most burst oscillations are seen when sources are in a
particular accretion state with color-color diagram position
variable $S_z>1.75$, (\citealt{Galloway08} and see the text for more detail). This table lists all sources with more than 10 bursts in the RXTE Burst Catalogue (for bursts up to June 3 2007, \citealt{Galloway08}) for
which burst oscillations have not been detected. In most cases
$S_z$ could not be computed since the source had not been observed
over the full range of accretion states necessary to fully
characterize the color-color diagram. }
\end{deluxetable}
\subsubsection{Frequencies}
\label{frequencies}
The detection of burst oscillations in several persistent and
intermittent accretion-powered pulsars, sources for which the spin
frequency is known, has proven conclusively that burst oscillation
frequency is very close to the known spin frequency. How close,
however, depends on the source. Table \ref{spinfreqs} summarizes the
situation for the various persistent and intermittent
accretion-powered pulsars with burst oscillations. In some cases the
frequencies agree to within $10^{-8}$ Hz, while for others they are
separated by a few Hz.
\begin{deluxetable}{l p{4cm} p{8cm}}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Comparison of burst oscillation and spin frequency.\label{spinfreqs}}
\tablehead{\colhead{Source} & \colhead{References} &
\colhead{Frequency comparison} }
\tablecolumns{3}
\startdata
\sidehead{Persistent accretion-powered pulsars}
SAX J1808.4-3658 & \citet{Chakrabarty03} & Upwards drift of a few Hz
in the burst rise, starting below the spin frequency and
overshooting. In tail, frequency
exceeds spin by a few mHz. \\
IGR J17498-2921 & \citet{Chakraborty12} & The frequency of burst oscillations in the tail (they are not detected significantly in the rise) appears to be relatively stable, and within $\pm 0.25$ Hz of the spin frequency. However the precise offset, and limits on frequency drift, have yet to be quantified. \\
XTE J1814-338 & \citet{Strohmayer03}, \citet{Watts05}, \citet{Watts08b} & Frequency is
identical to the spin frequency within the errors ($\sim 10^{-8}$ Hz), with no drifts. The
exception is the brightest burst, which shows a 0.1 Hz downwards drift
in the burst rise. \\
IGR J17511-3057 & \citet{Altamirano10a} & Upwards drifts of $\sim$ 0.1
Hz in the burst rise, starting below the spin frequency. There appears
to be overshoot in some bursts, but whether this is significant has
yet to be quantified. The frequency stabilises very close to the spin frequency in
the tail (certainly within 0.05 Hz), but the precise offset has also yet to be quantified.\\
IGR J17480-2446 & \citet{Cavecchi11} & Frequency is identical to the
spin frequency within the errors ($\sim 10^{-4}$ Hz), with no drifts. \\
\sidehead{Intermittent accretion-powered pulsars}
Aql X-1 & \citet{Zhang98}, \citet{Muno02a}, \citet{Casella08} & Burst oscillations typically drift
upwards in frequency during the burst. The maximum asymptotic
frequency reached by the burst oscillations is 0.5 Hz below the spin
frequency measured in the one brief episode where the source
showed accretion-powered pulsations. \\
HETE J1900.1-2455 & \citet{Watts09} & Burst oscillations drift upwards
by $\sim 1$ Hz during the burst, with the maximum frequency being
$\sim 1$ Hz below the spin frequency. \\
\enddata
\end{deluxetable}
Most burst oscillation sources exhibit frequency drift, with the
frequency rising by a few Hz over the course of a burst. Although we do not have sufficient counts to resolve individual cycles, the drift can be modelled and the resulting signals can be highly coherent, with values of Q as high as 4000 \citep{Strohmayer99}. The most
comprehensive study of frequency evolution in burst oscillation trains
to date is that carried out by \citet{Muno02a}. In most cases
the frequency drifts upwards during the burst, by 1-3 Hz, to reach an asymptotic
maximum (although the signal cannot always be tracked through
the peak of the burst since the amplitude sometimes drops below the
detectability threshold). For the two intermittent pulsars, the
asympototic maximum is 0.5-1 Hz below the spin frequency (the drifts
for the regular pulsars are more unusual, see Table \ref{spinfreqs}). The asymptotic maximum for a given source
appears to be very
stable, with the fractional dispersion in asymptotic frequencies,
measured using bursts separated by several years, being $<
10^{-3}$. Although orbital motion might account for some of the
dispersion, it cannot account for all of the observed variation. In
terms of coherence, the majority of the oscillation trains
studied evolved smoothly in frequency, and hence appeared to be highly coherent. However in about 30\% of cases, evolution was not smooth, suggesting jumps in phase or frequency, or the simultaneous presence of two signals with very similar frequencies \citep{Muno02a}.
The largest frequency drifts reported in the literature are of order a few Hz. \citet{Wijnands01} reported an apparent drift of 5 Hz in a burst from X 1658-298, and \citet{Galloway01} reported a drift of 3.6 Hz in a burst from 4U 1916-053. In both cases, however, the burst oscillation signal drops below the detection threshold in the middle of the burst. Since frequency cannot be tracked continuously throughout the burst, it is not clear whether we are really seeing a single signal drifting, or artificially connecting separate signals or perhaps even noise peaks (since with many bursts a few outliers due to noise would not be unexpected). A rigorous analysis of the entire sample of burst oscillation observations, to determine limits on the size of the drifts that are compatible with the overall data set, has yet to be carried out.
In the one case where oscillations were observed in a superburst, the frequency drift was much smaller (0.04 Hz
over 800 s), and entirely consistent with the expected Doppler shifts
due to orbital motion \citep{Strohmayer02a}. The superburst oscillation frequency inferred from this
measurement was $\approx 0.4$ Hz higher than any of the
aysmptotic frequencies measured during regular Type I X-ray bursts
from this source (see also \citealt{Giles02}). If this source shows the
same behaviour as that seen in the intermittent pulsars
(Table \ref{spinfreqs}) it is likely that the superburst
oscillation frequency is closer to the true spin frequency of the star
than the frequency of the normal burst oscillations.
\subsubsection{Amplitudes}
The average amplitudes of burst oscillations (computed over the whole
burst) are highly variable, even for
single sources, but are mostly in
the range 2-20\% rms
\citep{Galloway08}. Higher average amplitudes have been measured for
some bursts where the signal in the rise dominates (\citealt{Strohmayer97a},\citealt{Strohmayer98a},\citealt{Bhattacharyya06a}, \citealt{Galloway08}), although the error bars are larger since fewer photons can be
accumulated in the short burst rise. For the oscillations detected during the superburst from 4U 1636-536, the amplitudes are $\sim 1$ \% rms, lower than that measured for oscillations detected in Type I bursts from this source \citep{Strohmayer02a}.
Amplitudes can also vary substantially during a burst. The
tendency for signals to disappear in the peaks of bursts has already
been mentioned. This occurs most often, but not exclusively, during
periods of photospheric radius expansion \citep{Galloway08}, and is seen in both pulsars and
non-pulsars. Significant variations in amplitude during bursts have
been found for a number of sources \citep{Muno02b} but there are
exceptions where the data are consistent with a constant amplitude
model \citep{Watts05}.
For the sources that are accretion-powered pulsars, one can
compare the amplitudes of the burst oscillations to those of the
accretion-powered pulsations (\citealt{Chakrabarty03}, \citealt{Strohmayer03}, \citealt{Watts09},
\citealt{Altamirano10a}, \citealt{Cavecchi11}, \citealt{Papitto11}, \citealt{Chakraborty12}). The burst oscillation amplitudes for the
persistent pulsars are very different. However for each
individual source the burst oscillation amplitude is within a few
percent
of (and mostly lower than) that the accretion-powered pulsations at
the time of the burst. The difference between the amplitudes of the
two types of pulsations (for at least some bursts) is statistically
significant \citep{Watts05}.
Harmonic content (a significant signal at the first
overtone of the burst oscillation frequency) has been found in most of the burst
oscillation trains from the persistent accretion-powered pulsars,
although not necessarily at the same level as that seen in the accretion-powered
pulsations (\citealt{Chakrabarty03}, \citealt{Strohmayer03}, \citealt{Watts05}, \citealt{Altamirano10a},
\citealt{Cavecchi11}). Burst oscillation only sources \citep{Muno02b}, and burst oscillations
from the intermittent pulsars (\citealt{Muno02b},\citealt{Watts09}), tend not to have significant harmonic
content, although see \citet{Bhattacharyya05b} for an
exception in the rising phases of some bursts.
Another property that has been studied in some detail is the
dependence of burst oscillation amplitude on photon energy. For burst
oscillation only sources, and burst oscillations from the intermittent
pulsars, amplitude rises with energy (\citealt{Muno03}, \citealt{Watts09}).
For persistent pulsars where this property has been analysed, however,
burst oscillation amplitude falls with energy
(\citealt{Chakrabarty03}, \citealt{Watts06}). For sources that show
accretion-powered pulsations (persistent or intermittent) burst
oscillation amplitudes have the same energy dependence as the
accretion-powered pulsations, irrespective of whether this is a rise
or a fall (references above and \citealt{Casella08}).
\subsubsection{Phase offsets}
\label{phases}
Searches for phase lags between pulse profiles constructed using
photons in different energy bands (energy dependent phase lags) have been carried out for
several burst oscillation sources. While in a few sources there are
marginal detections of hard lags (high energy pulse arriving later
than the low energy one), most burst oscillation profiles show no
statistically significant phase offsets (\citealt{Muno03}, \citealt{Watts06}, \citealt{Watts09}). This
differs markedly from the behaviour of accretion-powered
pulsations, which have significant soft lags (\citealt{Cui98},
\citealt{Gierlinski02}, \citealt{Galloway02}, \citealt{Kirsch04}, \citealt{Galloway05}, \citealt{Gierlinski05}, \citealt{Papitto10}).
For the persistent accretion-powered pulsars, one can also investigate
phase lags between the burst oscillation pulse profile and the
accretion-powered pulse profile. For XTE J1814-338, which has no detectable
frequency drifts in its burst oscillations (for all but the
final burst, which occurs at much lower accretion rates and has
quite different properties), the two sets of pulsations are completely
phase-locked, with constant phase offset (\citealt{Strohmayer03}, \citealt{Watts08b}). Indeed the burst oscillation profile is actually coincident (zero phase offset)
with the low energy accretion-powered pulse profile. Spectral
modelling indicates that the latter component originates from the neutron star
surface (\citealt{Gierlinski02}, \citealt{Gierlinski05}). Phase-locking of pulsations has also been reported
for IGR J17511-3057 \citep{Papitto10}, although frequency drifts in
this source complicate this type of analysis \citep{Altamirano10a}.
\section{Burst oscillation theories}
\subsection{Relevant physics}
Burst oscillations are associated with the surface
layers of the neutron star. As such there are several key pieces of
physics that must be considered when developing models for the
phenomenon. Before discussing specific models, it is worthwhile
reviewing these factors. Some of these intrinsic properties remain
unchanged during an X-ray burst, whilst others may evolve.
\subsubsection{Structure of the neutron star}
Figure \ref{oceanstructure} shows the stratified composition of the surface layers
of a neutron star, with ignition depths for different burst types
marked. When modelling how a burst develops, one needs
to take into account
the coupling between the various burning and non-burning ocean layers, the
solid crust, and the photosphere. Coupling
may be dynamical, chemical, or thermal. The layers will evolve and
expand due to heat generation during the burst, particularly during
episodes of strong photospheric radius expansion (\citealt{Paczynski83},
\citealt{Paczynski86}, \citealt{intZand10}).
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, clip]{WattsFig3.pdf}
\caption{The outer layers of an accreting neutron star, showing where thermonuclear
ignition takes place. Adapted from a figure in \citet{Lewin80}. }
\label{oceanstructure}
\end{center}
\end{figure}
\subsubsection{Heat generation and transport}
Nuclear burning is unstable when the heat released by a
themonuclear reaction causes an increase in reaction rate that cannot
be compensated for by cooling, resulting in a
thermonuclear runaway. Several such reactions are involved in X-ray
bursts. Hydrogen can burn unstably via the
cold CNO cycle: $^{12}\mathrm{C}(\mathrm{p}, \gamma)^{13}\mathrm{N}(\beta^+, \nu)^{13}\mathrm{C}(\mathrm{p},\gamma)^{14}\mathrm{N}(\mathrm{p}, \gamma)^{15}\mathrm{O}(\beta^+, \nu)^{15}\mathrm{N}(\mathrm{p}, \alpha)^{12}\mathrm{C}$, where the rate-controlling temperature dependent step is $^{14}\mathrm{N}(\mathrm{p}, \gamma)^{15}\mathrm{O}$. Helium burns unstably via the temperature-dependent triple alpha reaction: $3~^4\mathrm{He} \rightarrow ^{12}\mathrm{C} + \gamma$. For reviews of the unstable reactions, and their application to X-ray bursts, see \citet{Schwarzschild65}, \citet{Hansen75}, \citet{Fujimoto81}, \citet{Lewin93}, and \citet{Bildsten98b}. Superbursts are thought
to be caused by unstable burning of carbon (\citealt{Woosley76}, \citealt{Taam78},
\citealt{Brown98}, \citealt{Cumming01}, \citealt{Strohmayer02b}).
There are however uncertainties
inherent in modelling ignition and the progression of the
thermonuclear reactions that affect our understanding of the heat
generation process. Ignition conditions are known to depend on accretion
rate, composition of accreted material, and heat flux from the deep
crust (all quantities that are impossible to measure precisely).
Burning, sedimentation, and mixing between bursts are also likely to play a role
(\citealt{Heger07}, \citealt{Peng07}, \citealt{Piro07}, \citealt{Keek09}). Despite considerable theoretical
progress in this area, there still exists a substantial mismatch
between predicted and observed
recurrence times for most bursters, indicating that our understanding of ignition remains
poor (\citealt{vanParadijs88}, \citealt{Cornelisse03}, \citealt{Cumming05b}, \citealt{Strohmayer06}, \citealt{Galloway08}), although there are a few exceptions (\citealt{Galloway06}, \citealt{Heger07b}) The precise
details of the reaction chains are also not exactly known, despite excellent
experimental work aimed at pinning this down \citep{Schatz11}. These uncertainties will
affect not only ignition conditions but also
heat generation and compositional changes throughout the burst (see
for example \citealt{Fisker07}, \citealt{Cooper09}, \citealt{Davids11}, and \citealt{Schatz11}).
Heat transport is also important in models of burst
oscillation development. One-dimensional simulations show that radiative heat transport will be important
in all bursts, with convection also playing an important
role (\citealt{Woosley04}, \citealt{Weinberg06}). Heat
transport will also be important in determining how the thermonuclear
flame spreads around the star. Multi-dimensional calculations and
simulations have shown that convection, turbulence, conduction, and advection may
all play a role (\citealt{Fryxell82}, \citealt{Spitkovsky02}, \citealt{Malone11}).
\subsubsection{Rotation and flows}
\label{rotation}
If burst oscillation frequency is a good measure of spin frequency (as it appears to be, see Section \ref{frequencies}), then most burst oscillation sources rotate rapidly. Is the rotation sufficient, however, to affect the evolution of the burst?
The importance of rotation in burst dynamics can be estimated by calculating the Rossby number
$R_o$. This is the ratio of inertial to Coriolis
force terms in the Navier-Stokes equations, the hydrodynamical equations governing the
motions of the fluid layers where the burst takes place (see for
example \citealt{Pedlosky87}).
\begin{equation}
R_o = \frac{U}{Lf}
\end{equation}
where $U$ is the velocity of the motion, $L$ a characteristic length,
and $f = 4\pi\nu_s\cos\theta$ the Coriolis parameter ($\nu_s$ being
the spin frequency of the star, and $\theta$ the co-latitude).
Rotation starts to have important dynamical effects once $R_o \lesssim
1$ and the Coriolis force becomes the dominant force. This means that rotational effects must be taken into
consideration once lengthscales exceed $U/f$. As rotation rate
increases, the effects are felt on shorter lengthscales. We can now estimate whether rotation is likely to affect burst
oscillation mechanisms, given that oscillations have been found in stars
that rotate in the range 11-620 Hz (Table \ref{sources}).
Let us first
consider the effect on flame spread. One of the effects
of the Coriolis force is to balance pressure
gradients that develop as the hot fluid expands, hence slowing spreading \citep{Spitkovsky02}. The
appropriate speed for a flow driven by such pressure gradients is $U = \sqrt{gH}$ where $g$ is the
gravitational acceleration and $H$ the scale height of the hot fluid. As rotation rate
increases, the tendency of the Coriolis force to confine the spreading
flame operates over ever shorter lengthscales. Significant
confinement (slowing of flame spread) will occur once these are less
than the size of the star, $R$,
\citep{Cavecchi11}. This occurs for rotation rates
\begin{equation}
\nu_s > 25 ~\mathrm{Hz} \left(\frac{g}{10^{14}~ \mathrm{cm~s}^{-2}}
\right)^\frac{1}{2} \left(\frac{H}{10~\mathrm{m}}\right)^\frac{1}{2}
\left(\frac{10~\mathrm{km}}{R}\right)
\label{nusthresh}
\end{equation}
Clearly most of the burst oscillation sources are in the regime where
this will be relevant.
Rapid rotation will also affect oscillation modes (global wave patterns) that
might develop in the ocean and atmosphere, excited by the
thermonuclear burst. As above, simple estimates can be used to
determine whether rotation will have an effect. Shallow water gravity
waves (driven by buoyancy), for example, have a timescale $\tau = 1/N$, where $N =
\sqrt{g/H}$ is the Brunt-V\"ais\"al\"a frequency (see for example \citealt{Pringle07}). The characteristic
speed of such waves is $U = H/\tau = \sqrt{gH}$. This means that
modes with a wavelength $\lambda$ comparable to or larger than
$\sqrt{gH}/f$ will be sensitive to the
effects of rotation. Large-scale global modes (those
with the largest wavelengths,
$\lambda \sim R$) are
therefore likely to be affected by rotation for most of the burst
oscillation sources (compare to Equation \ref{nusthresh}).
Rotation can also affect ignition conditions. Rotation lowers the effective gravity $g_\mathrm{eff}$ at the
equator as compared to the poles, with the difference being given by
\begin{equation}
\frac{\delta g_\mathrm{eff}}{g_\mathrm{eff}} \sim
\left(\frac{\Omega_s}{\Omega_k}\right)^2
\sim 5\%
\left(\frac{\nu_s}{500~\mathrm{Hz}}\right)^2
\left(\frac{R}{10~\mathrm{km}}\right)^3 \left(\frac{1.4~
M_\odot}{M}\right),
\end{equation}
where $\Omega_k$ is the Keplerian angular velocity at the surface of the
neutron star. The reduced gravity leads to a higher rate of fuel accumulation at the equator: because of this ignition is likely to occur preferentially at equatorial latitudes for all but a small range of accretion rates (\citealt{Spitkovsky02}, \citealt{Cooper07d}).
The expansion associated with the X-ray burst could also in principle
lead to differential rotation or shearing flows. Calculations show
that the change in thickness of the burning layers as they expand during a
burst is $\Delta R \sim 20$ m \cite{Bildsten98b}. If angular momentum
were conserved, this would lead to a spin down of order
\begin{equation}
\Delta \nu_s \approx \frac{2\nu_s\Delta R}{R} = 2~\mathrm{Hz}
\left(\frac{\nu_s}{500~\mathrm{Hz}}\right) \left(\frac{\Delta
R}{20~\mathrm{m}}\right) \left(\frac{10~\mathrm{km}}{R}\right)
\end{equation}
Over the course of the burst, the hot layer could in principle achieve
multiple wraps
of the underlying star. Zonal flows (latitudinal differential
rotation involving east-west flow along latitude lines, as opposed to meridional flows, which involve north-south flow
along longitude lines) may also develop due to temperature
gradients as different
latitudes ignite at different times \citep{Spitkovsky02}. In practice any shear
flow (radial or latitudinal) would be attenuated by various
frictional processes or
shearing instabilities, for instance. However if shear flows do persist, they will
affect the hydrodynamics of the surface layers, modifying for example
the structure and stability of oscillation modes \citep{Pringle07}.
\subsubsection{Magnetic fields}
\label{mag}
Measuring magnetic field strength for the burst oscillation sources is very difficult. For some of the pulsars, it has been possible to put limits on field strength by observing spin-down between accretion episodes and calculating the field strength necessary for this to occur via magnetic dipole radiation. Using this method it has been determined that the accretion-powered pulsar SAX J1808.4-3658,
for example, has a field $\sim 10^8$ G (\citealt{Hartman08}, \citealt{Hartman09}). Most of the burst oscillation sources, however, are not pulsars. If we assume that this is because the magnetic field is too weak to cause channeled accretion at the observed accretion rates, then this sets an upper limit on field strength $\sim 10^{9}$ G (for a
discussion of the techniques used in making this type of estimate see \citealt{Psaltis99}). It is possible, however, that the magnetic field is simply aligned with the rotation axis \citep{Chen93} or that pulsations are obscured (\citealt{Titarchuk02}, \citealt{Gogus07}).
The importance of magnetic fields on the
dynamics of the burning
layers can be estimated by calculating the ratio of the Lorentz force
terms to the other force terms in the magnetohydrodynamic versions of
the Navier-Stokes equations. The magnetic contribution is often split
into two terms - a magnetic pressure and a magnetic tension (see for
example \citealt{Choudhuri98}). It should also be borne in mind that
the field is unlikely to remain static during the burst. Flows set up by
the burst (convection, spreading of fuel or flame front, shearing
due to differential rotation) may lead to amplification and
modification of the field (\citealt{Cumming00}, \citealt{Boutloukos10}). While magnetic pressure is unlikely
to be relevant compared to other pressure terms at the burning depth,
it will dominate in the very outermost layers of the atmosphere. Magnetic
tension, which can act to counterbalance pressure gradients
associated with accumulating fuel and flame spread, may be dynamically important (\citealt{Brown98}, \citealt{Cavecchi11}). Magnetic fields will also
introduce new types of oscillation and instability in the ocean layers
(see \citealt{Choudhuri98} for a general overview of magnetohydrodynamic oscillations, \citealt{Heng09}, and Section \ref{modes}). Magnetic fields will also affect conditions in the ocean layers even
prior to ignition. Channeling of accreting material onto the magnetic
poles of the neutron star may lead to a local over-density at the
magnetic polar caps, in addition to temperature and composition gradients \citep{Brown98}.
\subsection{Current models for burst oscillations}
Current models for burst oscillations fall into two categories, hotspot models and global mode models, illustrated in very general terms in Figure \ref{modelsum}. As will be discussed in detail below, it seems likely that burning spreads initially from an ignition point, forming a spreading hotspot in the rising phase of the burst. The burning front then either stalls, to leave a persistent hotspot, or spreads around the star, exciting large-scale waves in the ocean. Both could lead to a brightness asymmetry in the ocean during the tail of the burst. We will treat each class of model in turn, noting that they may not be mutually exclusive.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, clip]{WattsFig4.pdf}
\caption{Simplified schematic of current burst oscillation models. Burst oscillations require the presence of a brightness asymmetry that can be modulated by the star's rotation so that the observer sees pulsations at close to the spin frequency. Ignition is expected to start at a point, with a flame front then spreading across the star (left panel, red = hot, blue = cold). This is known as a spreading hotspot. What happens then is unclear. One possibility is that the flame front stalls, so that only part of the ocean burns, leaving a hotspot that persists in the tail of the burst (top right). The other possibility is that the flame front spreads around the star and excites large scale waves (global modes) in the ocean (bottom right). The peaks and troughs of the waves will differ in temperature so that the wave pattern gives rise to a brightness asymmetry.}
\label{modelsum}
\end{center}
\end{figure}
\subsubsection{Hotspot models}
\label{hotspot}
Hotspot models are based on the general principle that the thermonuclear burning is not occurring uniformly across the surface of the star, but is instead confined to a smaller region. This hotspot is then modulated by the star's rotation to give rise to the observed burst oscillations. Two types of hotspot model have been considered: spreading hotspots, generated temporarily in the rise as the burst ignites and flame spreads from the ignition site to engulf the star; and persistent hotspots caused by restriction of the burning to a small part of the stellar surface.
In the tails of most bursts, the blackbody radius obtained from fitting the overall burst spectrum is comparable to the radius of the star, implying that the flame has engulfed the entire surface (see for example \citealt{Galloway08}). However it takes hours to days to accumulate fuel between bursts, and less than a second for a thermonuclear runaway to develop \citep{Shara82}. For ignition conditions to be met simultaneously across the stellar surface, the thermal state would need to be consistent to better than one part in $\sim 10^4$ \citep{Cumming00}. The presence of slight asymmetries in accretion (due for example to magnetic channeling, or the presence of equatorial boundary layers), makes this unlikely - particularly for the accretion-powered pulsars. In the absence of a mechanism that could equalize the thermal state of the ocean to the required level, it is therefore assumed that ignition starts at a point and that a flame front then spreads across the stellar surface.
The presence of such a spreading hotspot would provide a simple explanation for the presence of burst oscillations in the rising phase of bursts. Results from a small sample of bursts appear to support this picture: in a few bursts that have strong oscillations in the rising phase, the amplitude of oscillations falls as the blackbody radius of the overall burst spectrum increases (\citealt{Strohmayer97a}, \citealt{Strohmayer98a}, \citealt{Bhattacharyya06a}). This is consistent with the idea of a flame front spreading to cover the star, since as the spot grows in size, the overall amplitude of pulsations will fall \citep{Muno02b}. There are however several problems and theoretical questions that then arise.
The first issue is that if point ignition and flame spread are occurring in all bursts, why do we not detect oscillations in the rising phase of all bursts? As discussed in Section \ref{burstcond}, oscillations are actually more common in the tails rather than the rising phase of bursts. One possibility is that this is a detection bias: it can be difficult to make detections in the rising phase of bursts given their short duration. Something that has yet to be done is a comprehensive study to check whether the current data, including upper limits, are consistent with the spreading hotspot model.
One factor that will also affect detectability is the latitude at which ignition occurs. On a rapidly rotating star, ignition is expected near the equator for most accretion rates due to the lower effective gravity, which makes it easier to build up the column depth of material necessary for ignition \citep{Spitkovsky02}. There are however small ranges of accretion rate close to the boundary between stable and unstable burning where ignition is expected to occur at higher latitudes \citep{Cooper07d}. Whether a burst ignites at the equator or at higher latitudes can have a major effect on the detectability of burst oscillations \citep{Maurer08}. For sources with a substantial amount of magnetic channeling, ignition may also occur preferentially at the magnetic poles due to local overdensities or extra heating \citep{Watts08b}.
How the flame then spreads is equally important to burst oscillation detectability, since the flame must be able to spread in such a way that an azimuthal asymmetry can persist. The processes controlling flame spread in X-ray bursts have been an open question for years, with various heat transfer mechanisms including conduction, turbulence, and convection all thought to play a role (\citealt{Fryxell82}, \citealt{Nozakura84}, \citealt{Bildsten95b}, \citealt{Zingale01}). \citet{Spitkovsky02} also pointed out the importance of hydrodynamical effects, in particular the interaction between the Coriolis force and expansion of the hot burning material. The Coriolis force will act to slow down flame spread, thereby preserving a hotspot. Indeed some degree of confinement may be essential to get the flame front established and propagating, since rapid expansion and lateral spread of the hot material might otherwise cause the flame to stall \citep{Zingale03}. Note however that Coriolis force confinement will not be effective for IGR J17480-2446, which has a rotation rate of only 11 Hz (see Section \ref{rotation} and \citealt{Cavecchi11}). The flame spread mechanism remains an unsolved problem with immense importance for burst oscillation models.
If the flame spreads across the whole star during the rising phase, the spreading hotspot will dissipate. The fact that different regions of the star ignite at different times will leave some residual temperature asymmetry in the tail. However the magnitude of the predicted difference is too low to explain the amplitudes of the burst oscillations observed in burst tails \citep{Cumming00}. Some additional mechanism, or modification to the model, is still required. One possibility is the excitation of large-scale waves (Section \ref{modes}). The other is that a hotspot survives because the burning front does not spread across the whole star.
One way in which this might occur is if fuel is confined, with the prime candidate for a confinement mechanism being the star's magnetic field. Matter channeled onto the magnetic polar caps of the star will be prevented from crossing field lines (and hence flowing across the star) until the overpressure is sufficiently large to deform the field lines. Estimates by \citet{Brown98} show that for matter to remain confined in the ignition depth for normal Type I bursts requires fields of at least $10^{10}$ G, much higher than the fields strengths estimated for the burst oscillation sources (Section \ref{mag}). Magnetohydrodynamical instabilities may also act to make magnetic confinement of fuel ineffective \citep{Litwin01}. Magnetic fuel confinement on burst oscillation sources is therefore expected to be relatively unlikely.
The alternative is that the flame front itself is confined: burning, once started, spreads some distance and then stalls. This requires heat transport to be impeded in some way. Several mechanisms have been explored in the literature. \citet{Payne06} suggested that magnetic field evolution on an accreting neutron star might result in the development of strong magnetic belts around the equator that would then impede north-south heat transport. On rapidly rotating stars, a reduction in Coriolis-mediated confinement as the flame front tries to cross the equator might result in stalling (\citealt{Spitkovsky02}, \citealt{Zingale03}). Both of these mechanisms would confine burning to one hemisphere of the star: it is unclear whether this would lead to an asymmetry of sufficient magnitude to explain burst oscillation amplitudes. Recently \citet{Cavecchi11} showed that the dynamical interaction between the expanding burning fuel and the radial magnetic field could in principle induce horizontal field components that might be large enough to restrict further spread. This would lead to more localized confinement. For this mechanism to work the initial field can in principle be somewhat lower than for fuel confinement, $\sim 10^9$ G (consistent with estimates for two sources, XTE J1814-338 and IGR J17480-2446). If ignition starts at the magnetic pole, this would result in burst oscillations that are phase-locked to the accretion-powered pulsations and have only minimal frequency drift, consistent with observations from these two sources (Sections \ref{frequencies} and \ref{phases}). This mechanism may not work, however, for sources that appear to have lower magnetic fields, unless something in the burst process (such as a convective dynamo) acts to increase the field temporarily \citep{Boutloukos10}.
A major question addressed even in the earliest papers on burst oscillations was how to explain the upwards frequency drift seen in most bursts (Section \ref{frequencies}). \citet{Strohmayer97a} pointed out that horizontal motion of the spreading flame front could lead to drifts in frequency of a few Hz. However in this case rotation would have to set a preferred direction for spread, to explain why the frequency always tends to rise. They therefore suggested an alternative possibility: that of angular momentum conservation during vertical expansion of the heated burning layers. Rapid expansion would cause a hotspot to rotate more slowly, giving an observed frequency below the spin rate, with frequency rising as the layers cooled and contracted again (see Section \ref{rotation}).
The expansion model depends on two key assumptions: that the burning layers can decouple from the underlying star; and that shearing within the burning layer itself is small enough to preserve the spot rather than causing it to be smeared out. The validity of these assumptions, and the magnitude of the expected frequency shift in first Newtonian gravity and then General Relativity, were examined by \citet{Cumming00} and \citet{Cumming02}. These authors examined a range of hydrodynamical coupling mechanisms, such as viscous or shearing instabilities, and found that it was in principle possible for the burning layer to remain decoupled for the timescale of the burst. Smearing out of the hotspot was more problematic, and keeping this coherent required convection or short wavelength baroclinic instabilities to operate inside the burning layer. Smearing was found to be particularly pronounced during episodes of photospheric radius expansion. The maximum frequency drifts predicted by these studies were $\sim 1-2$ Hz. Although larger drifts have been reported, as explained in Section \ref{frequencies} the largest drifts in the literature are not clear cut, and the data set as a whole may in fact be consistent with a smaller maximum drift. Magnetic wind-up and shearing effects may also be important in determining the coupling and angular momentum transfer between the expanding layers and the underlying star (\citealt{Cumming00}, \citealt{Lovelace07}). A role for the magnetic field may explain the different drift properties seen in burst oscillations from the accretion-powered millisecond pulsars, where the magnetic field is known to be dynamically important.
\subsubsection{Surface modes}
\label{modes}
Ignition of the burst and the subsequent spread of flame around the star may excite large-scale waves (global modes) in the neutron star ocean. Height differences associated with such oscillations would translate into hotter and cooler patches (the wave pattern), thus giving rise to differences in X-ray brightness. Non-axisymmetric modes could in principle be excited easily by the initial flame spread, and persist throughout the burst tail (depending on damping mechanisms).
The problem of finding global modes of oscillation in oceans is one with a long and venerable history, due to its applications to the Earth's oceans. As such, papers on this topic use a rich and sometimes confusing mix of terminology when referring to different mode types. The basic hydrodynamical equations of continuity (mass conservation), momentum and energy, applied to the oscillations of a shallow stably stratified ocean on a rotating sphere, reduce to what is known as Laplace's tidal equation (\citealt{Longuet68}, \citealt{Townsend05}, \citealt{Lou00}). The restoring forces in this simple system are buoyancy (due to temperature or composition gradients) and the Coriolis force.
Non-axisymmetric mode solutions to Laplace's tidal equation have dependence $\exp(im\varphi)$ on the azimuthal angle $\varphi$, where $m$ is an integer. Given a mode with an azimuthal number $m$ and frequency $\nu_r$ in the rotating frame of the star, the frequency $\nu_o$ seen by an inertial observer would be
\begin{equation}
\nu_o = m\nu_s + \nu_r
\label{obsfreq}
\end{equation}
where $\nu_s$ is the spin frequency of the star, and the sign of $\nu_r$ is positive or negative depending on whether the mode is prograde (eastbound) or retrograde (westbound) respectively. The angular structure of the modes, which depend on spherical harmonics, are characterized by the usual integers $l$ and $m$ (see for example \citealt{Mathews70}). The pattern has $m$ nodes in azimuth and $|l-m|$ nodes in latitude. For non-zero $\nu_r$, the brightness pattern moves around the star, permitting an offset from the spin-frequency. Frequency drift can be naturally explained if $\nu_r$ depends on the thermal state of the ocean, since this will change as the burst evolves and cools through the tail.
Laplace's tidal equation admits three families of mode solution:
\begin{itemize}
\item{Poincar\'{e} modes: propagate in the retrograde (westbound) direction and reduce to pure gravity waves (restored by buoyancy alone) in the non-rotating limit. For this reason they are often referred to as g-modes in the neutron star ocean oscillation literature (see for example \citealt{Bildsten96} and \citealt{Heyl04}). }
\item{Kelvin modes: propagate in the prograde (eastbound) direction and also reduce to pure gravity waves in the non-rotating limit. Unlike the Poincar\'e modes, they are in geostrophic balance. This means that the horizontal component of the Coriolis force balances the horizontal pressure gradient, see \citet{Pedlosky87}. As a result they involve purely azimuthal motions, and are confined to an equatorial waveguide.}
\item{Rossby modes: propagate in the retrograde (westbound) direction, and are also known as r-modes (\citealt{Papaloizou78a}, \citealt{Saio82}). In the limit of zero buoyancy, Rossby modes are driven purely by latitudinal variations in the Coriolis force. In the non-rotating limit they reduce to trivial zero-frequency solutions. If buoyancy is present then it will also act to restore these modes, and they are then often referred to as buoyant r-modes (see for example \citealt{Heyl04} and \citealt{Piro05b}).}
\end{itemize}
\citet{Heyl04} was the first to give serious consideration to whether such ocean modes might explain the properties of burst oscillations, particularly in the tails of bursts. He laid out a number of criteria that the modes would have to meet, including the fact that observed frequency should be a few Hz below the spin frequency of the star, and that low $l, m$ modes would be needed to ensure that a highly sinusoidal and high amplitude burst oscillation would result (the finer the pattern, the lower the contrast across the stellar surface). On the basis of these criteria he determined that the $l=2, m=1$ buoyant r-mode was the most likely candidate. This retrograde mode has $|\nu_r|$ of only a few Hz, giving an observed frequency slightly below the spin frequency. As the burning layers cool (changing the buoyancy), $|\nu_r|$ gets smaller, so that observed frequency would tend towards the spin frequency. Poincar\'e or g-modes were ruled out primarily on the grounds of their frequencies, since $|\nu_r|$ is expected to be much larger than a few Hz (\citealt{McDermott87}, \citealt{Strohmayer96a}, \citealt{Bildsten95a}, \citealt{Bildsten96}, \citealt{Bildsten98c}, \citealt{Heyl04}, \citealt{Maniopoulou04}). Kelvin modes were ruled out on the grounds that they are prograde, so that frequency would fall as the ocean cooled, in contrast to the majority of observed frequency drifts. Both Kelvin and Poincar\'e modes are also strongly confined to the equatorial regions on rapidly-rotating stars, which would reduce overall burst oscillation amplitudes (since there is no strong contrasting mode pattern across most of the surface). The buoyant r-mode, by contrast, occupies a wider band around the equator, giving greater contrast.
In addition to providing a natural explanation for the sinusoidal shape, overall frequency, and amplitude of the burst oscillations, mode models predict that the amplitude of the burst oscillation should increase with photon energy (\citealt{Heyl05}, \citealt{Lee05}, \citealt{Piro06}), in line with observations of the non-pulsars and the intermittent pulsars (although not those of the persistent pulsars). \citet{Heyl04} also argued that excitation of such a mode should be relatively easy, since the timescale of the mode matches the $\sim 1$ s rise time of the burst associated with the flame spread. \citet{Narayan07} and \citet{Cooper08} explored the issue of excitation in more detail, concluding that the mode would need to be unstable in order to grow to the required amplitudes in the tails of bursts. They considered various classical pumping mechanisms \citep{Saio93} and conclude that the main driver is likely to be the $\epsilon$ mechanism, which drives modes via nuclear energy generation. The role of the $\epsilon$ mechanism in neutron star envelopes was studied previously by \citet{McDermott87} and \citet{Strohmayer96a} for thermally unstable envelopes. \citet{Narayan07} and \citet{Cooper08}, by contrast, consider thermally stable envelopes, appropriate for the tails of bursts. For He-rich explosions, instability is more likely when the burst is radiation pressure dominated. Instability was much less likely to develop in H-rich bursts. Suppression of the instability occurs during episodes of convection, which may contribute to the observed disappearance of burst oscillations during the peaks of bright photospheric radius expansion bursts.
The major problem with the buoyant r-mode model is that the frequency drift predicted as the burning layer cools is $\sim 10$ Hz, an order of magnitude larger than the observed values. At $10^9$ K, a typical temperature in the peak of a burst, the buoyant r-mode frequency should be $|\nu_r| \gtrsim 10$ Hz, drifting to $|\nu_r| \sim 1$ Hz as the layer cools to $10^8$ K. Various solutions to this problem have been explored. \citet{Piro05a} pointed out the existence of another global mode of oscillation that is confined to the interface between the cool ocean below the burning layer and the underlying elastic crust. The observed frequency of this crust-interface mode would be $\sim 4$ Hz below the spin frequency of the star. If, as the buoyant r-mode drifted upwards in frequency, it could couple efficiently to this crustal-interface mode and transfer its energy via resonant conversion, this would naturally truncate the drift a few Hz short of the spin frequency \citep{Piro05b}. The frequency of the crustal interface mode is highly stable, so could account for the reported invariance in asymptotic frequencies. It could also explain why the burst oscillation frequency seen in a superburst is close to that seen in normal bursts, despite the differences in the burning layer, since the final frequency in both cases would be set by the crust-interface mode. Unfortunately detailed studies of the coupling between the buoyant r-mode and the crustal-interface wave indicate that resonant conversion is not effective \citep{Berkhout08}. The fact that the separation between spin frequency and burst oscillation frequency is much less than 4 Hz for the persistent and accretion-powered pulsars (Table \ref{spinfreqs}) is also problematic for the crust-interface mode model.
As a result, several alternative models have been explored. \citet{Heyl04} pointed out that if the r-mode were instead confined to the photospheric layers, the drift would be smaller. This idea has yet to be explored in detail, but as pointed out by \citet{Berkhout08} magnetic forces will become important in these low density layers, and could have a substantial effect on mode frequencies. \citet{Cumming05a} explored what might happen to the buoyant r-modes if strong zonal flows (latitudinal differential rotation) were to develop in the ocean layers in the immediate aftermath of flame spread. While shear-modified modes could in principle match both frequencies and drifts, his study showed that high $m$ modes were likely to grow much faster than the low $m$ modes necessary to explain burst oscillation properties. Whether zonal flows can develop in the first place is also still an open question \citep{Spitkovsky02}. Most recently \cite{Heng09} have considered the effects that magnetic fields might have on oceanic mode structure, by solving the shallow water magnetohydrodynamical equations for an incompressible ocean with a purely radial magnetic field. In addition to magnetically modified versions of the normal families of modes described above, they find new solutions dominated by magnetic and rotational effects that they call magnetostrophic modes. Unlike the previously known modes, which are confined to the equatorial regions on a rapidly rotating star, these new modes are stronger near the poles. Both prograde and retrograde modes are possible: however the model predicts a downwards rather than an upwards frequency drift for the retrograde modes as the ocean cools. More detailed calculations, however, are required to determine whether or not magnetically modified modes such as these might play a role in explaining burst oscillations.
Mode models remain an extremely promising mechanism to explain burst oscillations from the non-pulsars and the intermittent pulsars, although issues such as the frequency drift problem remain to be resolved. For the persistent accretion-powered pulsars, they have more difficulty: existing mode models cannot explain the extremely close agreement between spin and burst oscillation frequency (Table \ref{spinfreqs}), which would require $|\nu_r| \ll 1$ Hz for an $m=1$ mode; the unusual frequency drifts (or lack thereof); and the fact that burst oscillation amplitude decreases with photon energy. Magnetically modified modes may eventually account for these differences: alternatively sources with a dynamically important magnetic field may have a different burst oscillation mechanism (such as a magnetically confined hotspot, see Section \ref{hotspot}).
\section{Uses of burst oscillations}
Burst oscillations are an intrinsic part of the thermonuclear burst process. They can also, however, be used as tools to explore other areas of neutron star physics. In this Section I will expand briefly on some of these applications.
\subsection{Pulse profile modelling as a diagnostic of mass and radius}
The nature of the strong force, and the state of matter at extremes of temperature and density, is an extremely active field of research. Of particular interest is the transition from nucleons (neutrons and protons) to de-confined quarks and gluons or other more exotic states. At low densities this can be studied by experiments like the Large Hadron Collider (LHC) or in various Heavy Ion experiments such as the Relativistic Heavy Ion Collider (RHIC) and the Facility for Antiproton and Ion Research (FAIR). At higher densities, however, neutron stars are the only environment in the Universe where the transition can be explored \citep{Paerels10}.
Dense matter models, combined with the relativistic stellar structure equations, make predictions for mass and radii of neutron stars. By measuring these quantities, we can thus work back to the underlying nuclear physics \citep{Lattimer07}. Burst oscillations are expected to be powerful tools in this regard because the emission comes from the surface of the neutron star, deep within the gravitational potential well. As photons propagate towards us, they are affected by relativistic effects that depend on mass and radius, such as gravitational redshift and light-bending. The pulse profile associated with the surface emission pattern is modified by these processes. By modelling the shape and energy-dependence of the observed pulse profile using relativistic ray-tracing algorithms, researchers seek to deconvolve these effects and hence extract information about mass and radius. Typically one has to fit for the surface emission pattern and the geometry (such as observer inclination), as well as the parameters of interest.
The fact that stellar compactness will affect pulse profiles from a hotspot on the surface of the neutron star is long-established, with a series of papers exploring the effects of (for example) different metrics, rotation, and stellar oblateness (\citealt{Pechenick83}, \citealt{Chen89}, \citealt{Strohmayer92}, \citealt{Braje00}, \citealt{Cadeau05}, \citealt{Cadeau07}, \citealt{Morsink07}). The techniques developed in these papers are generic to all types of hotspot and have been applied extensively to accretion-powered pulsations as well as burst oscillations. Following the discovery of burst oscillations, several authors started to model the lightcurves and pulse profiles that might be expected from both static and spreading hotspots or mode patterns on the surface of bursting neutron stars (\citealt{Strohmayer97a}, \citealt{Miller98}, \citealt{Weinberg01}, \citealt{Muno02b}, \citealt{Muno03}, \citealt{Viironen04}, \citealt{Lee05}). Some attempts were also made at the inverse problem: that of fitting measured burst oscillation pulse profiles in order to obtain confidence contours on stellar compactness (\citealt{Nath02}, \citealt{Bhattacharyya05a}). Many more attempts have been made to obtain constraints by fitting accretion-powered pulsations, despite the complicating effects of the accretion funnel and shock on the emission, see \citet{Poutanen08} for a review. Burst oscillations offer in principle a somewhat cleaner system dominated by thermal emission. Although the constraints obtained using existing data were relatively weak, the potential of the technique to put tight bounds on mass and radius, given sufficient photons, was clear \citep{Strohmayer04}. As such pulse profile modelling of burst oscillations is now a major science driver for proposed future large area X-ray timing missions such as the {\it Large Observatory for X-ray Timing} (LOFT, \citealt{Feroci11}) and the {\it Advanced X-ray Timing Array} (AXTAR, \citealt{Ray10}). A concerted effort is now underway to address and resolve some of the outstanding issues that may introduce degeneracies into fits for mass and radius, such as the effect of changes in the surface emission pattern over the course of a burst.
\subsection{Neutron star spin}
The evidence outlined in Section \ref{frequencies} supports the identification of burst oscillation frequency as a good measure of neutron star spin, at least to within a few Hz. The main consequence of this has been to double the number of rapidly-rotating (above 10 Hz) accreting neutron stars with a known spin rate. Indeed the fastest spinning accreting neutron star currently known (4U 1608-522, with a spin of 620 Hz) is a burst oscillation source and not an accretion-powered pulsar.
Identifying the spin distribution of the various classes of neutron
star has been a longstanding research goal for many years, as they are an important element in our understanding of stellar and binary evolution. The discovery of rapidly rotating accreting neutron stars, in particular, was regarded as critical to confirming the recycling scenario for the formation of the millisecond radio pulsars (\citealt{Alpar82}, \citealt{Radhakrishnan82}, \citealt{Bhattacharya91}). With a larger number of stars in this class more detailed studies of population evolution are possible, revealing interesting discrepancies between models and observation (\citealt{Tauris06}, \citealt{Lorimer08}, \citealt{Hessels08}, \citealt{Kiziltan09}, \citealt{Tauris11}). These point to problems in our understanding of, for example, the mass transfer process, magnetic field decay, and accretion torques. Other formation routes for millisecond radio pulsars, such as accretion-induced collapse of white dwarves, may also need to be invoked to explain the observed populations.
The maximum spin rate that a neutron star can reach is also of great interest. A neutron star with a sub-millisecond spin period would place a tight and very clean constraint on the dense matter equation of state (\citealt{Lattimer07}, \citealt{Schaffner08}, \citealt{Haensel09}). This is because the break-up, or mass-shedding, frequency (the spin rate above which a neutron star would fly apart as rotational forces overwhelm gravitational attraction) depends strongly on the composition of the star. This in turn depends on the poorly constrained behavior of the strong force at supranuclear densities. Although no current burst oscillation source (and indeed not even the most rapidly rotating radio pulsar, \citealt{Hessels06}) rotates fast enough to rule out equations of state, the search for extremely rapid rotators does motivate burst oscillation searches. Whether burst oscillations could ever develop, or be detectable, on such rapidly rotating stars remains however an open question (see Section \ref{rotation}).
The current spin distribution of accreting neutron stars, as given by both the accretion-powered pulsars and the burst oscillation sources, hints at a maximum that is well below the spin rate suggested by simple estimates of accretion-induced spin-up over their lifetime (\citealt{Chakrabarty03}, \citealt{Chakrabarty08}). If the evolutionary estimates
are sound, there is a requirement for a braking mechanism to halt the
spin-up. Magnetic braking (due to interaction between the stellar
field and the accretion disk) is one possibility (\citealt{Ghosh78}, \citealt{White97}, \citealt{Andersson05}). Another, which
has generated a lot of excitement, is the
emission of gravitational waves (\citealt{Papaloizou78b}, \citealt{Wagoner84}, \citealt{Bildsten98a}). This has triggered major theoretical effort exploring the nature of possible gravitational wave generation mechanisms, such as crustal mountains (\citealt{Bildsten98a}, \citealt{Ushomirsky00}, \citealt{Melatos05}, \citealt{Haskell06}), or internal r-mode oscillations (\citealt{Bildsten98a}, \citealt{Andersson99}, \citealt{Levin99}, \citealt{Andersson00}, \citealt{Andersson02}, \citealt{Heyl02}, \citealt{Wagoner02}, \citealt{Nayyar06}, \citealt{Bondarescu07}, \citealt{Ho11}). Gravitational wave braking could also make accreting neutron stars promising continuous wave sources for future gravitational wave detectors, although
this is one application where knowing the spin to a very high
degree of precision (certainly better than a few Hz) would be important \citep{Watts08a}. Uncertainty in the spin rate has a very negative impact on gravitational wave detection threshold, since increasing the parameter space increases the numbers of trials (reducing detection sensitivity, see Section \ref{detection}) and increases the computational power needed to perform any search. Improving burst oscillation models so that we can make a more accurate diagnosis of spin rate would be of great benefit.
\section{Conclusions}
Burst oscillations are an intriguing part of the phenomenology of thermonuclear bursts on neutron stars, with wider uses in terms of measuring stellar spin and potentially constraining the dense matter equation of state. One of the legacies of the Rossi X-ray Timing Explorer has been a rich database of burst oscillation observations, giving us a clear understanding of their general properties and the conditions under which they develop. A satisfactory theoretical explanation of why and under what conditions burst oscillations develop, however, is still lacking. Burst evolution models that take full account of both the structure of the neutron star ocean and envelope, and the dynamical effects of magnetic fields, are needed to explore the mechanics of flame spread and the excitation of global oceanic modes. Such theoretical development is essential if we are take full advantage of the opportunities offered by the next generation of X-ray timing telescopes.
\section*{Acknowledgments}
ALW acknowledges support from a Netherlands Organisation for Scientific Research (NWO) Vidi Fellowship. She would like to thank Michiel van der Klis and Deepto Chakrabarty for their encouragement and feedback whilst writing this review. She would also like to thank Jean in 't Zand, Duncan Galloway, Diego Altamirano, Tullio Bagnoli, Yuri Levin and Yuri Cavecchi for their comments on the draft manuscript.
\bibliographystyle{Astronomy}
|
{
"timestamp": "2012-03-12T01:01:28",
"yymm": "1203",
"arxiv_id": "1203.2065",
"language": "en",
"url": "https://arxiv.org/abs/1203.2065"
}
|
\section*{\S0 Introduction}\
Let $(X,\B,m)$ be a standard, continuous, probability space (that is, $(X,\B)$ is a Polish space equipped with its Borel sets and a non-atomic $m\in\mathcal P(X)$ (the collection of probability measures on $(X,\B)$).
\
We'll denote by ${\tt MPT}={\tt MPT}(X,\B,m)$ the collection of invertible, probability preserving transformations of $(X,\B,m)$. This is a Polish space when equipped with the {\it coarse topology} with basic neighborhoods of form
\begin{align*}U(T_0,f_1,&\dots,f_N;\e):=\\ &\{T\in\text{\tt MPT}:\ \|f_j\circ T^s-f_j\circ T_0^s\|_{L^2(m)}<\e\ \forall\ 1\le j\le N,\ s=\pm 1\}\end{align*}
where $T_0\in\text{\tt MPT}\xbm$ and $f_1,\dots,f_N\in L^2(m)$.
\par Equipped with the coarse topology, {\tt MPT}$\xbm$ is a topological group under composition. It is embedded into the Polish, topological group $\mathcal U(L^2(m))$ of unitary operators on $L^2(m)$ equipped with the strong operator topology by the {\it Koopman representation} $U_Tf:=f\circ T\ \ (T\in\text{\tt MPT}\xbm,\ f\in L^2(m))$.
Accounts of the spectral theory of unitary operators can be found in \cite{KT} and \cite{N2}.
\subsection*{Recurrence and rigidity}
\
A sequence $q\in\Bbb N^\Bbb N(\uparrow):=\{q\in\Bbb N^\Bbb N:\ q_n<q_{n+1}\ \forall\ n\ge 1\}$ is called a sequence of {\em recurrence} for $T\in{\tt MPT}(X,\B,m)$ if $\limsup_{n\to\infty}\mu(A\cap T^{-q_n}A)>0$ for each $A\in\B$ of positive measure.
\
Rigidity is a stronger version of recurrence.
\
An sequence $q\in\Bbb N^\Bbb N(\uparrow)$ is called a {\em rigidity sequence} for $T\in{\tt MPT}\xbm$ if $\mu(T^{q_n}A\triangle A)\to 0$ for each $A\in\B$; equivalently
\begin{align*}&\tag{\dsmilitary}
T^{q_n}\xrightarrow[n\to\infty]{\tt MPT} \text{\tt Id}.\end{align*}
Using spectral theory one sees that (\dsmilitary) is equivalent to the {\tt restricted spectral type} $\s_T$ of $T$ (i.e. $U_T|_{L^2(m)_0}$) having the {\it Dirichlet property} along $q$, that is
\begin{align*}\tag{\ddag}
\chi_{q_n}\xrightarrow[n\to\infty]{L^2(\Bbb T,\sigma_T)}\,1.\end{align*} where $\chi_k(t):=e^{2\pi i kt}$.
\
Rigidity sequences for non-trivial transformations must be sparse. In particular, unless $T\in{\tt MPT}$ is purely periodic any rigidity sequence for $T$ has at most finite intersection with each of its translates whence has zero Banach density.
Additional properties of rigidity sequences are studied in \cite{BJLR} $\&$ \cite{EG} including the rigidity properties of {\tt lacunary} and {\tt super-lacunary} sequences, a sequence $q\in\incss$ being called
{\it lacunary} if $\tfrac{q_{n+1}}{q_n}\ge\l>1\ \forall\ n\ge 1$ and {\it super-lacunary} if $\tfrac{q_{n+1}}{q_n}\xyr[n\to\infty]{}\infty$.
\
\subsection*{Rigid factors, mild mixing and IP sets}\ \ \ \ Let $T\in{\tt MPT}\xbm$ and let $q\in\incss$. It is well known that the collection of sets
\begin{align*}\tag{\Wheelchair}\mathcal R(q):=\{A\in\B:\ m(A\D T^{q_n}A)\xyr[n\to\infty]{}0\}\end{align*}
is a $T$-invariant, $\s$-algebra. It corresponds to the maximal factor of $T$ which is rigid along $q$.
The transformation $T\in{\tt MPT}\xbm$ is called {\it mildly mixing} if it has no non-trivial, rigid factor along any $q\in\incss$ (as in \cite{FW}).
\
Since the spectral type $\s_S$ of a factor $S$ of $T$ is absolutely continuous with respect to $\s_T$,
it is evident that $T$ has some non-trivial rigid factor if and only if $\exists$ a {\it Dirichlet measure} $\mu\ll\s_T$,
(that is, one satisfying (\ddag) along some $q\in\incss$).
\
An {\it IP-set} is a collection of ``finite sum sets" of form
$$\text{\tt FS}\,(q):=\{q(F):\ F\in\mathcal F\}$$ where $q\in\incss$ and for
$$F\in\mathcal F:=\{\text{\tt finite, nonempty subsets of}\ \Bbb N\},\ q(F):=\sum_{j\in F}q_j.$$
This notion appears in combinatorics, ultrafilter theory, topological dynamics (see \cite{Fu} and \cite{HS}) and also in ergodic theory.
\
As shown in \cite{Fu}, $T\in{\tt MPT}\xbm$ is mildly mixing if and only if $\exists\ K\subset\Bbb N$ which intersects with every finite sum set so that
$$m(A\cap T^{-n}B)\xyr[n\to\infty,\ n\in K]{}m(A)m(B)\ \forall\ A,\ B\in\B;$$
equivalently (see \cite{HMP1}),
\
$T$ is not mildly mixing if and only if
$\exists\ q\in\incss$ so that
$$\underset{n\in\text{\tt\tiny FS(q)}}\inf\,|\widehat{\mu}(n)|\ >0.$$
The considerations involved give rise to the notion of
\subsection*{ IP convergence}
\
Let $q\in\incss$. We'll say that a sequence $a:\Bbb N\to Z$ (a metric space) {\it converges IP to $L\in Z$ along {\tt FS}$(q)$}
$$\text{\rm written}\ \ a(n)\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(q)} \ L,\ \text{\rm in}\ Z\ \ \ \text{\rm if}$$
$$a(q(F))\xrightarrow[F\in\mathcal F,\ \min\,F\to\infty]{Z} \ L.$$
This paper is about
\subsection*{IP-rigidity}
\
We'll say that $b\in\Bbb N^\Bbb N(\uparrow)$ is
an {\it IP-rigidity sequence for $T$} and that $T$ is {\it IP-rigid along $b$} if $$T^{n}\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)}\ \text{\tt Id\ {\rm in}\ MPT}.$$
\
Let
\begin{align*}&\text{\tt IPRWM}:=\{b\in\Bbb N^\Bbb N(\uparrow):\ \exists\ T\in\text{\tt MPT}\xbm, \text{\rm\small weakly mixing $\&$ IP-rigid along}\ b\}.\end{align*}
Any rigid transformation is IP-rigid on some subsequence (see \cite{Fu}). On the other hand if a transformation is
IP-rigid on $q\in\incss$, then it is rigid along much thicker subsequences (see \S5).
\
Similarly to (\Wheelchair), for $q\in\incss$, the collection
$$\mathcal R_{\tiny\tt IP}(q):=\{A\in\B:\ m(A\D T^{n}A)\xyr[n\to\infty]{{\tt FS}\,(q)}\ 0\}$$
is a $T$-invariant, $\s$-algebra. It corresponds to the maximal factor of $T$ which is IP-rigid along $q$.
As above, $T$ has a non-trivial factor, IP-rigid along $q$ if and only if
$\lim_{N\to\infty}\inf_{F\in\mathcal F,\ \min\,F\ge N}|\widehat{\s_T}(q(F))|>0.$
\
The existence of IP-Dirichlet measures along $b\in\Bbb N^\Bbb N(\uparrow)$ is related to the groups
\begin{align*}&G_p(b):=\{t\in\Bbb T:\ \sum_{n=1}^\infty\|b_nt\|^p<\infty\}\ (0<p<\infty)\ \&\\ & G_\infty(b):=\{t\in\Bbb T:\ \|b_nt\|\xyr[n\to\infty]{}0\}\end{align*}
where for $x\in\Bbb R,\ \|x\|:=\min_{n\in\Bbb Z}\,|x-n|$.
\
These groups are discussed in [AN] and [HMP2].
\
\subsection*{\large Results}
\
\proclaim{Proposition 1}\ \ \ \ Suppose that
$b\in \Bbb N^\Bbb N(\uparrow)$, then
\begin{align*}\ \ \ \ |G_1(b)|>\mathbf{\aleph_0}\ \ \Lra\ \ b\in \text{\tt IPRWM}.\end{align*} \endproclaim
Proposition 1 (which is folklore) can be proved using Propositions 1.1 and 1.2 (below).
\
\proclaim{Theorem 2} If $b\in\incss$, then
\begin{align*} b\in \text{\tt IPRWM}\ \Lra\ \ |G_2(b)|>\mathbf{\aleph_0}.\end{align*}
\endproclaim
Theorem 2 also provides an answer to a question in \cite{BJLR}:
{\Large\Smi} {\em if $b\in\text{\tt IPRWM}$, then some irrational rotation is rigid along $b$}
because if $|G_2(b)|>\aleph_0$ then $\exists\ \a\in G_2(b)\setminus\Bbb Q$.
It follows that rotation of $\Bbb T$ by $\a$ is rigid along $b$. \Checkedbox
The converse of theorem 2 holds for {\tt arithmetic} sequences and {\tt Erdos-Taylor} sequences for different reasons.
\
A sequence $q\in\incss$ is called {\it arithmetic} if it is either
\bul {\it multiplicative} in the sense that $q_n|q_{n+1}\ \forall\ n\ge 1$; or it is the
\bul {\it principal denominator sequence} of some $\a\in\Bbb T\setminus\Bbb Q$, being defined by
$q_0=1,\ q_1=a_1,\ q_{n+1}:=a_{n+1}q_n+q_{n-1}$ where $\a=[0;a_1,a_2,\dots]$ is the continued fraction expansion of $\a$.
\proclaim{Proposition 3} Let $b\in\incss$, be arithmetic. The following statements are equivalent.
\begin{align*}&\text{\rm (a)}\ \ \varlimsup_{n\to\infty}\frac{b_{n+1}}{b_n}=\infty.\ \ \text{\rm (b)}\ |G_1(b)|>\mathbf{\aleph_0}.\\ &
\ \text{\rm (c)}\ b\in \text{\tt IPRWM}.\ \ \text{\rm (d)}\ \ |G_2(b)|>\mathbf{\aleph_0}.\end{align*}
\endproclaim
\
The {\it Erdos-Taylor sequence associated to $(a_1,a_2,\dots)\in\Bbb N^\Bbb N$} is
$b=(b_1,b_2,\dots)\in\Bbb N^\Bbb N(\uparrow)$ defined by
$$b_1:=1.\ \ b_{n+1}:=a_{n}b_n+1.$$
Erdos-Taylor sequences were introduced in \cite{ET} and are considered to be ``extremely non-arithmetic''.
\proclaim{Proposition 4} If $b\in\incss$, is an Erdos-Taylor sequence, then
\begin{align*}\text{\rm (i)}\ \sum_{n\ge 1}\(\frac{b_n}{b_{n+1}}\)^2<\infty\ \iff\ \text{\rm (ii)}\ b\in \text{\tt IPRWM}\ \iff\ \text{\rm (iii)}\ \ |G_2(b)|>\mathbf{\aleph_0}.\end{align*}
\endproclaim
We'll see that there are super-lacunary Erdos-Taylor sequences $b\ \&\ q\in\incss$ satisfying
$|G_2(b)|>\mathbf{\aleph_0}\ \&\ G_1(b)=\{0\}$ and $G_2(q)=\{0\}$.
\subsection*{Eigenvalue Groups and theorem 2}
\
Groups of form $G_2$ appear as eigenvalue groups (see \cite{AN}). Eigenvalue groups and rigidity are related as follows:
\
An ergodic probability preserving transformation $S$ is not mildly mixing (i.e. has a rigid factor) if and only if there is a conservative, ergodic non-singular transformation $T$ so that $S\x T$ is not ergodic (see \cite{FW}). By the ergodic multiplier theorem of M. Keane (see e.g. \S2.7 of \cite{A}), this situation is characterized by $\s_S(e(T))>0$ where $\s_S$ is the restricted spectral type of $S$ and $e(T)$ is the group of eigenvalues of $T$.
\
We prove Theorem 2 in \S4 by considering a dyadic cocycle (see below) associated to $b\in\Bbb N^\Bbb N(\uparrow)$ over the dyadic adding machine.
The eigenvalue group of the Mackey range (as in p. 76-77 in \cite{Z}) of this cocycle is $G_2(b)$.
In case $b$ is a {\it growth sequence} as in \cite{A2}, that is $b(n)>\sum_{1\le k<n}b(k)$, then the Mackey range preserves a $\s$-finite measure and is isomorphic to the appropriate dyadic tower over the dyadic adding machine (defined in \cite{A2}).
\subsection*{Organization of the paper}
\
In \S1 we establish the basic results on Dirichlet sets and measures and begin to consider membership of {\tt IPRWM}.
\
In \S2 we consider the class of arithmetic sequences, and prove Proposition 3.
\
In \S3 we prove proposition 4 for Erdos-Taylor sequences and give our main examples.
\
The proofs of propositions 3 $\&$ 4 both use Theorem 2 which is established in \S4. In \S5 we make some quantitative remarks on the growth of rigid sequences for transformations IP-rigid along some
(particular) $b\in\incss$.
\
\section*{\S1 Dirichlet sets and measures}
\subsection*{Dirichlet sets}
A {\it Dirichlet set} is a subset $\G\subset\Bbb T$ of form
$$\G(b)=\{t\in\Bbb T:\ \chi_{b_n}(t)\xrightarrow[n\to\infty]{}\ 1\}$$
where $b\in\Bbb N^\Bbb N(\uparrow)\ \&\ \ \chi_n(t):=e^{2\pi int}$.\par An {\it IP-Dirichlet set} is a subset $\G\subset\Bbb T$ of form
$$\G(\text{\tt FS}\,(b))=\{t\in\Bbb T:\ \chi_{n}(t)\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)} \ \ 1\}$$ where $b\in\Bbb N^\Bbb N(\uparrow)$. Here, we have
\proclaim{Proposition 1.1}\ \ \ For $b\in\Bbb N^\Bbb N(\uparrow)$,
$$\G(\text{\tt FS}\,(b))=G_1(b).$$\endproclaim
\demo{Proof sketch of $\subseteq$}
\
It suffices to show that for $t\in\Bbb R$,
$$\|nt\|\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)} \ \ 0\ \Rightarrow\ \ \sum_{n\ge 1}\|b_nt\|<\infty.$$
For $x\in\Bbb R$, let $\lfl x\rcl$ be the nearest integer to $x$ (if there are two, take the lesser one), and let
$$\<x\>:=x-\lfl x\rcl,\ \text{then}\ \ \ |\<x\>|=\|x\|\le \frac12.$$
Fix $t\in\Bbb R$ so that $\|nt\|\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)} \ \ 0$, let $K>0$ be so that
\begin{align*}\|b(F)t\|<\frac1{16}\ \ \ \forall\ F\in\mathcal F,\ \min\,F\ge K.\end{align*}
If $F,\ G\subset [K,\infty)\cap\Bbb N$ are disjoint finite sets, then
$$\|b(F)t\|,\ \|b(G)t\|,\ \|b(F\cupdot G)t\|<\frac1{16}.$$
Since $\<b(F\cupdot G)t\>-\<b(F)t\>-\<b(G)t\>\in\Bbb Z$, this forces
\f$\<b(F\cupdot G)t\>=\<b(F)t\>+\<b(G)t\>.$
\
It follows that
$$\sum_{n\ge K,\ \<b_nt\>\ge 0}\|b_nt\|\le\frac1{16},\ \sum_{n\ge K,\ \ \<b_nt\><0}\|b_nt\|\le\frac1{16}\ \&\ \sum_{n\in\Bbb N,\ n\ge K}\|b_nt\|<\frac18.\ \ \ \text{\Checkedbox}.$$
\subsection*{Dirichlet measures}
\
A probability measure $\mu\in\mathcal P(\Bbb T)$ is
called
\bul a {\it Dirichlet} measure if
$$\|\chi_{b_n}-1\|_{L^2(\mu)}\xrightarrow[n\to\infty]{} \ \ 0$$ for some $b\in\Bbb N^\Bbb N(\uparrow)$ in which case $\mu$ is called {\it Dirichlet along $b$} and
\bul an {\it IP Dirichlet} measure if
$$\|\chi_n-1\|_{L^2(\mu)}\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)} \ \ 0$$ for some $b\in\Bbb N^\Bbb N(\uparrow)$ in which case $\mu$ is called {\it IP-Dirichlet along $b$}.
\
Evidently:
\sms $\chi_{n_k}\xrightarrow[k\to\infty]{L^2(\mu)} 1$ if and only if $\widehat\mu(n_k)\xrightarrow[k\to\infty]{} \mu(\Bbb T)$,
\sms if $\mu$ is IP-Dirichlet along $b$, then so is any $\nu\ll\mu$,
\sms if $\mu$ is IP-Dirichlet, then $\exists\ b\in\incss$ so that $\sum_{n\ge 1}\|\chi_{b_n}-1\|_{L^2(\mu)}<\infty$, whence $\mu(G_1(b))=1$ and $\mu$ is IP-rigid along $b$.
\
By Proposition 1.1, a totally atomic measure $\mu\in\mathcal P(\Bbb T)$ is IP-Dirichlet along $b$ if and only if $\mu(G_1(b))=1$.
Examples in \S4 (below) show that this is false for
continuous measures $\mu\in\mathcal P(\Bbb T)$.
\
\
\proclaim{Proposition 1.2}
\begin{align*}\text{\tt IPRWM}\ =\ \{b\in\Bbb N^\Bbb N(\uparrow):\ \exists\ \mu\in\mathcal P(\Bbb T)\ \ \text{\rm\small continuous $\&$ IP-Dirichlet along}\ b\}.\end{align*}\endproclaim\demo{Proof of $\subseteq$}
Suppose that $b\in\text{\tt IPRWM}$ and that $(X,\B,m,T)$ is a weakly mixing, probability preserving transformation so that
$T^{n}\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)}\ \text{\tt Id}$. Fix $f\in L^2(m),\ \int_Xfdm=0,\
\int_X|f|^2dm=1$. The spectral measure of $f$: $\mu\in\mathcal P(\Bbb T)$ is continuous and IP-Dirichlet along $b$.
\demo{Proof of $\supseteq$}\ \ Suppose that $\mu_0\in\mathcal P(\Bbb T)$ is continuous and IP-Dirichlet along $b$.
Let $\mu$ be the symmetrization of $\mu_0$ (also continuous and IP-Dirichlet along $b$) and let $(X,\B,m,T)$ be the shift of the Gaussian process
with spectral measure $\mu$. The spectral type of $(X,\B,m,T)$ is $\s_T=\sum_{n\ge 0}\mu^{n*}$ where $\mu^{n*}$ denotes the $n$-fold convolution
of $\mu$ with itself (see e.g. \cite{CFS}). Each $\mu^{K*}$ is continuous (whence $T$ is weakly mixing) and IP-Dirichlet along $b$
(since $\widehat{\mu^{K*}}(n)=\widehat{\mu}(n)^{K}\xrightarrow[n\to\infty]{\text{\tt\tiny FS}\,(b)}\ 1$). Every $\nu\ll\s_T$ is also IP-Dirichlet along $b$
and $T$ is IP-rigid along $b$. Thus $b\in\text{\tt IPRWM}$.\ \Checkedbox
\
It follows from proposition 1.2 that if $b\in\incss$ is an IP rigidity sequence for some $T\in\text{\tt MPT}$ not of discrete spectrum (i.e. $\s_T$ is not totally atomic) then $b\in\text{\tt IPRWM}$.
\
We complete this section with a ``mixed" multiplicative-finite sum condition for membership in {\tt IPRWM}.
\proclaim{Proposition 1.3}\ \ Suppose that $b\in\incss$ and that $\exists\ S\subset\Bbb N$ infinite, so that
$\sum_{n\in S}\frac{b_n}{b_{n+1}}<\infty$, and $b_n|b_{n+1}$ for $n\notin S$, then $b\in {\tt IPRWM}$.\endproclaim
If $\Bbb N\setminus S$ is finite, then $|G_1(b)|>\aleph_0$ by Theorem 5 in \cite{ET}, whence
$b\in {\tt IPRWM}$ by proposition 1.
\demo{Proof} \ \ Assume (without loss of generality) that $b_1=1$. We construct a weakly mixing
$T\in {\tt MPT}\xbm$ which is IP-rigid along $b$ by cutting and stacking as in Ch. 7 of \cite{N1}.
\
To this end, we construct a nested sequence of Rokhlin towers $(\tau_n)_{n\ge 1}$ of intervals where $\tau_n$ has height $b_n$.
Let $\tau_1$ be $[0,1]$. To construct $\tau_{n+1}$ from $\tau_n$:
\bul if $n\notin S\ \&\ b_{n+1}=a_n b_n,\ a_n\in\Bbb N,\ a_n\ge 2$ then we cut $\tau_n$ into $a_n$ columns and stack.
\bul If $n\in S\ \&\ b_{n+1}= a_nb_n+r_n$, $ a_n,\ r_n\in\Bbb N,\ 1\le r_n< b_n$, we cut $\tau_n$ into $ a_n$ columns and put one spacer interval above the $\lfl\frac{ a_n}2\rfl$'th column from the left and $r_n-1$ spacer intervals above the last column in the right and then we stack.
\
The tower $\tau_n$ is a stack of $b_n$ intervals of length $\prod_{j=1}^{n-1}\frac1{a_j}$ called {\it levels of $\tau_n$}.
\
It follows from \S7.31 in \cite{N1} that the transformation $T$ constructed preserves a finite measure $m$.
A standard argument as in the proof of Proposition 3.10 of \cite{BJLR} shows that $T$ is weakly mixing.
\
Next, we show that if $A$ is a union of levels in some $\tau_K$, then $$\sum_{n=1}^\infty m(A\D T^{b_n}A)<\infty.$$
\
To see this, we note first that $A$ is also a union of levels in every $\tau_n\ \ (n\ge K)$. Fix $n\ge K$ and write
$$S\cap [K,\infty)=\{s_1<s_2<s_3<\dots\}.$$
Since
$\frac1{a_{s_\ell}+1}\le\ \frac{b_{s_\ell}}{b_{s_\ell+1}}\le \frac1{a_{s_\ell}}$ our assumptions imply
$\sum_{\ell=1}^\infty \frac1{a_{s_\ell}}<\infty$.
\
To estimate $m(A\D T^{-b_n}A)$ for $n\in (s_{\ell-1},s_{\ell}]$ we consider the appearance of the tower $\tau_n$ as ``$\tau_n$-stalks" inside $\tau_{s_\ell+1}$.
A {\it $\tau_n$-stalk} in $\tau_{s_\ell+1}$ is a union $\frak s=\bigcupdot_{k=0}^{b_n-1}T^kB$ of levels of $\tau_{s_\ell+1}$ where $B$ is contained in the base of $\tau_n$.
Let $\frak s\in\B$ be a $\tau_n$-stalk in $\tau_{s_\ell+1}$, then
$$m(\frak s\cap A)=\prod_{j=n}^{s_\ell}\frac1{a_j}\cdot m(A).$$
\
By construction, $\tau_{s_\ell+1}$ consists entirely of $\tau_n$-stalks and spacer stalks added in $\tau_{s_\ell+1}$. Thus, all points of $A$ except those contained in the two $\tau_n$-stalks preceding the spacer stalks added in $\tau_{s_\ell+1}$, return to $A$ at time $b_n$, so
$$ m(A\D T^{-b_n}A)=2m(A\setminus T^{-b_n}A)=4\prod_{j=n}^{s_\ell}\frac1{a_j}\cdot m(A)\le \frac4{2^{s_\ell-n}}\cdot\frac1{a_{s_\ell}}\cdot m(A).$$
Thus, writing $s_0:=K-1$, we have
\begin{align*}\sum_{n=K}^\infty m(A\D T^{-b_n}A)&=
\sum_{\ell=1}^\infty\sum_{n\in (s_{\ell-1},s_\ell]}m(T^{b_n}(A)\Delta A)\\ &\le
\sum_{\ell=1}^\infty\sum_{n\in (s_{\ell-1},s_\ell]}\frac4{2^{s_\ell-n}}\cdot\frac1{a_{s_\ell}}\cdot m(A)\\ &\le 4m(A)
\sum_{\ell=1}^\infty \frac1{a_{s_\ell}}\\ &<\infty.\end{align*}
It follows from this that for $A$ a union of levels in some tower $\tau_n$ and $F=\{n_1<n_2<\dots<n_k\}\in\mathcal F$,
\begin{align*}m(A\D T^{b(F)}A)&=m(A\D T^{\sum_{j=1}^kb_{n_j}}A)\\ &\le
m(A\D T^{b_1}A)+m(T^{b_1}A\D T^{\sum_{j=1}^kb_{n_j}}A)\\ &=
m(A\D T^{b_1}A)+m(A\D T^{\sum_{j=2}^kb_{n_j}}A)\\ &\le\\ &\vdots\\ &\le\sum_{j=1}^k m(A\D T^{b_j}A)\\ &\le\sum_{n=\min\,F}^\infty m(T^{b_n}(A)\Delta A)\\ &
\xrightarrow[F\in\mathcal F,\ \min\,F\to\infty]{} 0.\end{align*}
The collection of measurable sets $\mathcal C$ satisfying this last convergence is a $\s$-algebra and
$$\mathcal C\supset\s\(\bigcup_{n\ge 1}\{\text{unions of levels in }\ \tau_n\}\)=\mathcal B.$$ Thus $T$ is IP-rigid along $b$.\ \ \Checkedbox
\subsection*{Remark 1.4}\ \ \ The proof of Proposition 1.3 establishes the following proposition:
\sms {\em \ Suppose that $b\in\incss,\ p>0$ and that $\exists\ S\subset\Bbb N$ infinite, so that
$\sum_{n\in S}(\frac{b_n}{b_{n+1}})^p<\infty$, and $b_n|b_{n+1}$ for $n\notin S$, then $\exists\ T\in{\tt MPT}\xbm$ weakly mixing and a dense collection $\mathcal A\subset\B$ so that \begin{align*}\tag{\dscommercial$(p)$}\ \ \ \ \ \ \ \ \ \sum_{n\ge 1}m(A\D T^{b_n}A)^p<\infty\ \ \forall\ A\in\mathcal A.\end{align*}}
As above, {\dscommercial$(1)$ $\Lra$ IP-rigidity along $b$ whence $\s_T(G_2(b)^c)=0$ by Theorem 2.
Using the spectral theorem, one sees that {\dscommercial$(\tfrac12)$ $\Lra$ $\s_T(G_1(b)^c)=0$.
\section*{\S2 Arithmetic sequences}
In this section, we prove proposition 3. The implications (b) $\Lra$ (c)\ $\Lra$ (d) $\Lra$ (a) follow from proposition 1, theorem 2 and theorem 16 in \cite{E} (respectively). None of these uses arithmeticity.
We turn to the remaining implication (a) $\Lra$ (b).
\proclaim{Lemma 2.1}
\
If $b\in\Bbb N^\Bbb N(\uparrow)$ is multiplicative and $\sup_{n\ge 1}\frac{b_{n+1}}{b_n}=\infty$, then
$|G_1(b)|>\mathbf{\aleph_0}$.\endproclaim C.f. theorem 3 in \cite{ET}.\demo{Proof}
Suppose that $b_{n+1}=a_{n+1}b_n$, where $a_n\ge 2\ \forall n\geq 1$.
\
Since $\sup_{n\ge 1}\,a_n=\infty$, $\exists$ a subsequence $(n_k)_{k\geq 1}$ such that $a_{n_{k+1}}/a_{n_k}\geq 3\ \forall\ k\ge 1$. Define $t:\Om:=\{0,1\}^\Bbb N\to [0,1]$ by
$$t(\om):=\sum_{k=1}^\infty\frac{\om_k}{b_{n_k}}.$$ Since
$$\sum_{k=L+1}^\infty\frac{\om_k}{b_{n_k}}\le \frac1{b_{n_L}}\sum_{j=1}^\infty\frac1{a_{n_L+1}a_{n_L+2}\cdots a_{n_j}}\le\frac1{b_{n_L}}\sum_{j=1}^\infty\frac1{3^j}< \frac1{b_{n_L}},$$
we have that $t:\Om:=\{0,1\}^\Bbb N\to [0,1]$ is strictly increasing (with respect to lexicographic order on $\Om$), whence injective and $|t(\Om)|>\mathbf{\aleph_0}$.
\
It suffices to show that $t(\Om)\subset G_1(b)$.
\
To see this, fix $\om\in\Om$. For $N\ge 1$, we have that
\begin{align*}
b_Nt(\om)=\sum_{k=1}^\infty\frac{b_N\om_k}{b_{n_k}}=
\sum_{k\ge 1,\ n_k\ge N}\frac{\om_k}{a_{N+1}a_{N+2}\cdots a_{n_k}}\ \ \mod\ 1,
\end{align*}
whence
\begin{align*}
\|b_Nt(\om)\|=
\sum_{k\ge 1,\ n_k\ge N}\frac{\om_k}{a_{N+1}a_{N+2}\cdots a_{n_k}}\le \sum_{k\ge 1,\ n_k\ge N}\frac1{a_{N+1}a_{N+2}\cdots a_{n_k}}=:\D_N.
\end{align*}
For $n_{K-1}<N\le n_K$,
\begin{align*}\D_N =
\sum_{k\ge K}\frac1{a_{N+1}a_{N+2}\cdots a_{n_k}}\le
\sum_{k\ge K}\frac1{a_{N+1}a_{N+2}\cdots a_{n_K}\cdot a_{n_k}}\le
\frac1{2^{n_K-N}a_{n_K}}.
\end{align*}
Thus
\begin{align*}\sum_{N=n_1}^\infty\|b_{N}t(\om)\|&\leq\sum_{N=n_1}^\infty\D_N=
\sum_{k=2}^\infty\sum_{N=n_{k-1}+1}^{n_k}\sum_{\nu=k}^\infty{1\over{a_{N+1}a_{N+2}\cdots a_{n_\nu}}}\\ &\le
\sum_{k=2}^\infty\sum_{N=n_{k-1}+1}^{n_k}\frac1{2^{n_k-N}a_{n_k}}\le
\sum_{k=2}^\infty\frac2{a_{n_k}}\le 4\end{align*}
and $t(\om)\in G_1(b)$.\ \ \ \Checkedbox
\proclaim{Lemma 2.2}
\
Let $q=q(\a)\in\Bbb N^\Bbb N(\uparrow)$ be the principal denominator sequence of $\a\in (0,1)\setminus\Bbb Q$. \
If $\sup_{n\ge 1}\frac{q_{n+1}}{q_n}=\infty$, then
$|G_1(q)|>\mathbf{\aleph_0}$.\endproclaim
\demo{Proof}
\ As shown in \cite{IN}, \par for any $t\in[0,1]$ there is a unique sequence $(\omega_n)_{n\geq 1}\in\prod_{n\geq 1}\{0,1,\cdots,a_n\}$ such that \sbul $\omega_k\leq a_k,\ \omega_k=a_k\ \Rightarrow\ \omega_{k+1}=0$ and \sbul $t=\sum_{n=1}^\infty \omega_n \<q_n\alpha\>\ \mod 1$.
\
Since $\sup_{n\ge 1}{a_n}=\sup_{n\ge 1}\frac{q_{n+1}}{q_n}=\infty$, we can choose a sub-sequence $a_{n_k}$ such that $\sum_{k\geq 1}{1/a_{n_k}}<\infty$ and define
$$t:\Om=\{0,1\}^\Bbb N\to \Bbb T\ \ \text{by}\ \ t(\om):=\sum_{k=1}^\infty \omega_{k} \<q_{n_k-1}\alpha\>\ \mod 1.$$
By the above, $t:\Om\to \Bbb T$ is injective. It follows that $|t(\Om)|>\mathbf{\aleph_0}$ and it suffices to show that
$t(\Om)\subset G_1(q)$.
\
We claim that $\sup_{\om\in\Omega}\sum_{n=1}^\infty \|q_nt(\om)\|<\infty$.
\
Fix $K\geq 1$ and consider $n_K-1\leq N\leq n_{K+1}-1$.
Then
\begin{align*}
\|q_Nt(\om)\|&\leq\sum_{k=1}^\infty \|\omega_{n_k-1}q_Nq_{n_k-1}\alpha\|\\
&\leq \sum_{k=1}^\infty\|q_Nq_{n_k-1}\alpha\|\\
&\leq \sum_{k=1}^K{q_{n_k-1}\over q_{N+1}}+\sum_{k=K+1}^\infty{q_N\over{q_{n_k}}}.\\
\end{align*}
Using the fact that $\frac{q_{j+n}}{q_j}\geq \sqrt{2}^{n-2}\ \forall\ j,\ n\ge 1$ we have for some absolute constant $C$,
$${q_{n_k-1}\over q_{N+1}}={q_{n_k-1}\over q_{n_K-1}}\cdot{q_{n_K-1}\over q_{n_K}}\cdot{q_{n_K}\over q_{N+1}}
\leq {C\over {\sqrt{2}^{n_K-n_k}}}\cdot{1\over a_{n_K}}\cdot {C\over{\sqrt{2}^{N-n_K}}}\ \ \ \text{for $k\leq K$}$$
and
$${q_N\over{q_{n_k}}}={q_N\over{q_{n_{K+1}-1}}}\cdot{q_{n_{K+1}-1}\over q_{n_{K+1}}}\cdot{q_{n_{K+1}}\over q_{n_k}}
\leq {C\over {\sqrt{2}^{n_{K+1}-N}}}\cdot{1\over{a_{n_{K+1}}}}\cdot{C\over \sqrt{2}^{n_k-n_{K+1}}}\ \ \ \text{for $k>K$.}$$
Therefore,
\begin{align*}
\sum_{N=n_K-1}^{n_{K+1}-2}\|q_Nt(\om)\|&\le \sum_{N=n_K-1}^{n_{K+1}-2}\left(\sum_{k=1}^K{q_{n_k-1}\over q_{N+1}}+\sum_{k=K+1}^\infty{q_N\over{q_{n_k}}}\right)\\
&\leq C^2\sum_{N=n_K-1}^{n_{K+1}-2}\left(\sum_{k=1}^K {1 \over {a_{n_{K}}\sqrt{2}^{N-n_k}}}+\sum_{k=K+1}^\infty {1\over a_{n_{K+1}}\sqrt{2}^{n_k-N}}\right)\\
&\leq C^3\sum_{N=n_K-1}^{n_{K+1}-2}(\frac1{a_{n_{K}}\sqrt{2}^{N-n_K}}+\frac1{a_{n_{K+1}}\sqrt{2}^{n_{K+1}-N}})\\
&\leq C^4({1\over a_{n_K}}+{1\over a_{n_{K+1}}})
\end{align*}
and
\begin{align*}
\sum_{N=1}^\infty \|q_Nt(\om)\|\le C^4\sum_{K=1}^\infty({1\over a_{n_K}}+{1\over a_{n_{K+1}}})<\infty.\ \ \ \text{\Checkedbox}
\end{align*}
Hence $q\in{\tt IPRWM}. \ \ \ \text{\Checkedbox}$
The proof of Proposition 3 is now complete.\ \ \Checkedbox
\section*{\S3 Super-lacunary sequences}
\
Suppose that $b=(b_1,b_2,\dots)\in\Bbb N^\Bbb N(\uparrow)$ is super-lacunary, i.e. $\frac{b_{n+1}}{b_n}\xyr[n\to\infty]{}\infty$.
\
As in theorem 17 in \cite{E},
we fix $N\ge 1$ with $\frac{b_{n+1}}{b_n}>10\ \forall\ n\ge N$ and let
$$E:=\bigcap_{n\ge N}E_n$$ where
$$E_n:=\{t\in [0,1]:\ \|b_nt\|\le\frac{4b_n}{b_{n+1}}\}.$$
Now
$$E_n\supseteq\bigcup_{k=1}^{b_n-1}I_{k,n}$$
where
$$I_{k,n}:=[\frac{k}{b_n}-\frac4{b_{n+1}},\frac{k}{b_n}+\frac4{b_{n+1}}].$$
For each $n\ge N$, the intervals $\{I_{k,n}:\ 1\le k<b_n\}$ are disjoint, and
each interval $I_{k,n}$ contains at least five disjoint intervals of form $I_{k',n+1}$. It follows that $E$ contains a Cantor set and $|E|>\aleph_0$. Thus, (c.f of theorem 5 in \cite{ET})
\proclaim{Proposition 3.1}\ \ \ Suppose that $b\in\incss\ \&\ p>0$, then
$$\sum_{n\ge 1}\(\frac{b_n}{b_{n+1}}\)^p<\infty\ \ \ \Lra\ \ |G_p(b)|>\aleph_0.$$\endproclaim
\demo{Proof}\ \ Let $N\ge 1\ \&\ E$ be as above, then for $t\in E$,
$$\sum_{n\ge 1}\|b_nt\|^p\le N+4^p\sum_{n\ge N}(\tfrac{b_n}{b_{n+1}})^p<\infty.\ \ \ \ \text{\Checkedbox}$$
\
\proclaim{Proposition 3.2}\ \ \ Suppose that $b\in\incss$ and that
$$\sum_{n\ge 1}\(\frac{b_n}{b_{n+1}}\)^2<\infty,\ \ \text{then}\ \ \ b\in{\tt IPRWM}.$$\endproclaim
\demo{Proof} \ \ By Proposition 1.2, it suffices to construct a continuous probability in $\Bbb T$ which is IP-Dirichlet along $b$.
\
To this end, let $N\ge 1,\ E$ and $\{I_{k,n}:\ 1\le k<b_n\}$ be as above. As above, for each $n\ge N$, the intervals $\{I_{k,n}:\ 1\le k<b_n\}$ are disjoint, and
each interval $I_{k,n}$ contains at least five disjoint intervals of form $I_{k',n+1}$.
\
Thus we may choose
$$I_n(\om)=I_{k_n(\om),n}\ \ \ (n\ge 1,\ \om\in\{0,1\}^n)$$ so that
$$I_{n+1}(\om,\e)\subset I_n(\om)\ \ \forall\ n\ge 1,\ \om\in\{0,1\}^n\ \&\ \e=0,1$$
and
$X_{n+1}(\om,0)<X_n(\om)<X_{n+1}(\om,1)$ where $X_n(\om):=\frac{k_n(\om)}{b_n}$.
\
Next, for $\om\in\Om:=\{0,1\}^\Bbb N$,
$$X_n(\om_1,\dots,\om_n)\xyr[n\to\infty]{}X(\om)\ \text{where}\ \ \bigcap_{n\ge 1}I_n(\om_1,\dots,\om_n)=\{X(\om)\}$$ and
\begin{align*}b_nX(\om)&=b_nX_n(\om)+b_n(X_{n+1}(\om)-X_n(\om))+b_n(X(\om)-X_{n+1}(\om))\\ &=k_n(\om)+\xi_n(\om)+\th_n(\om)
\end{align*}
where
$$\xi_n(\om):=b_n(X_{n+1}(\om)-X_n(\om))\ \&\ $$
$$\th_n(\om):=b_n(X(\om)-X_{n+1}(\om)).$$ Note that $|\th_n(\om)|\le\mathcal E_n:=\frac{4b_n}{b_{n+2}}$ and that by assumption, $\sum_{n\ge 1}\mathcal E_n<\infty$.
For $n\ge 1,\ \om\in\{0,1\}^n,\ \exists\ !\ p_{n,\om}:\{0,1\}\to (0,1)$ so that
$$p_{n,\om}(0)+p_{n,\om}(1)=1\ \&\ \ X_{n+1}(\om,0)p_{n,\om}(0)+X_{n+1}(\om,1)p_{n,\om}(1)=X_n(\om).$$
Define $P:\{\text{\tt cylinders}\}\to (0,1)$ by
$$P([a_1,a_2,\dots,a_n]):=\frac12\prod_{k=1}^{n-1}p_{k,(a_1,a_2,\dots,a_k)}(a_k).$$
It follows that $P$ is additive and by standard extension theory $\exists\ \Bbb P\in\mathcal P(\Om)$ extending $P$.
\
Denoting expectation with respect to $\Bbb P$ by $\Bbb E$ and writing $\om=(\om_1,\om_2,\dots,\om_n)\in\{0,1\}^n$, we have
$$\Bbb E(X_{n+1}\|\om_1,\om_2,\dots,\om_n)=X_{n+1}(\om,0)p_{n,\om}(0)+
X_{n+1}(\om,1)p_{n,\om}(1)=X_n(\om),$$ whence
$$\Bbb E(\xi_n\|\om_1,\om_2,\dots,\om_n)=b_n(\Bbb E(X_{n+1}\|\om_1,\om_2,\dots,\om_n)-X_n(\om))=0$$ and $\Bbb E(\xi_n)=0$.
\
For $n,\ k\ge 1$,
\begin{align*}\Bbb E(\xi_n\xi_{n+k})&=b_nb_{n+k}\Bbb E(\Bbb E(\xi_n\xi_{n+k}\|\om_1,\om_2,\dots,\om_{n+k})) \\ &=
b_nb_{n+k}\Bbb E((\xi_n(\om)\Bbb E\xi_{n+k}\|\om_1,\om_2,\dots,\om_{n+k}))=0\ \ \ \ \&\
\end{align*}
$$\Bbb E(\xi_n^2)=:\D_n\ \le\ \frac{16b_{n}^2}{b_{n+1}^2}.$$
By assumption $\sum_{n\ge 1}\D_n<\infty$, so
$\sum_{n\in K}\xi_n$ converges in $L^2(\Bbb P)$ for every $K\subset\Bbb N$ and
$$\Bbb E\((\sum_{n\in K}\xi_n)^2\)=\sum_{n\in K}E(\xi_n^2)=\sum_{n\in K}\D_n.$$
\
The measure $\mu:=\Bbb P\circ X^{-1}\in\mathcal P(\Bbb T)$ is continuous. We claim that it is IP-Dirichlet along $b$.
To check this, let $F\subset\Bbb N\cap [K,\infty)$ be finite and write $\Xi_F:=\sum_{n\in F}\xi_n$, then
\begin{align*}\|\chi_{b(F)}-1\|_{L^2(\mu)}& \le\|\<\tsum_{N\in F}b_Nt\>\|_{L^2(\mu)}\\ &=
\|\<\sum_{N\in F}(\xi_N+\th_N)\>\|_{L^2(\Bbb P)}\\ &
\le\|\<\Xi_F\>\|_{L^2(\Bbb P)}+\sum_{N\in F}\mathcal E_N.\end{align*}
Next,
\begin{align*}\|\<\Xi_F\>\|^2_{L^2(\Bbb P)}&=\Bbb E(\<\Xi_F\>^2)\\ &=
\Bbb E(1_{[|\Xi_F|\le\frac12]}\<\Xi_F\>^2)+\Bbb E(1_{[|\Xi_F|>\frac12]}\<\Xi_F\>^2)\\ &\le
\Bbb E(\Xi_F^2)+\frac14\Bbb P[|\Xi_F|>\frac12])\ \ \ \because\ \<\Xi_F\>^2\le\frac14,\\ &\le
2\Bbb E(\Xi_F^2)\ \ \ \text{\rm by Tchebychev's inequality}\\ &=2\sum_{N\in F}\Bbb E(\xi_N^2)\le 2\sum_{N=K}^\infty\D_N.\end{align*}
Thus,
\begin{align*}\|\chi_{n(F)}-1\|_{L^2(\mu)}& \le\|\<\Xi_F\>\|_{L^2(\Bbb P)}+\sum_{N=K}^\infty\mathcal E_N\\ &\le
\sqrt{2\sum_{N=K}^\infty\D_N}+\sum_{N=K}^\infty\mathcal E_N\\ &\xyr[K\to\infty]{}0\end{align*}
proving that $\mu$ is IP-Dirichlet along $b$.\ \ \ \Checkedbox
\subsection*{Remark}\ \ \ The converses to propositions 3.1 $\&$ 3.2 are false.
It is easy to construct $b\in\incss$ multiplicative, super-lacunary so that $\sum_{n\ge 1}(\tfrac{b_n}{b_{n+1}})^p=\infty \ \ \ \forall\ p>0$.
By proposition 3, $|G_1(b)|>\aleph_0\ \&\ \ T\in\text{\tt IPRWM}$.
\subsection*{\large Erdos-Taylor sequences $\&$ proposition 4}
\
We begin with a strong converse to Proposition 3.1 for Erdos-Taylor sequences:
\proclaim{Proposition 3.3}\ \ \ Suppose that $b\in\incss$ is an Erdos-Taylor sequence and let $p>0$, then
$$\sum_{n\ge 1}\(\frac{b_n}{b_{n+1}}\)^p=\infty\ \ \ \Lra\ \ G_p(b)=\{0\}.$$\endproclaim
This was stated in \cite{ET} for $p=1$ and $b$ the Erdos-Taylor sequence associated to $(2,3,\dots)$. See also Th\'eor\`eme 2 in \cite{P}.
\demo{Proof}\ \
Let $b\in\incss$ be the Erdos-Taylor sequence associated to $(a_1,a_2,\dots)\in\Bbb N^\Bbb N$ and let $t\in\Bbb R\setminus\Bbb Z$, then
$$\|b_nt\|<\frac{\|t\|}{2a_n}\ \Lra\ \|b_{n+1}t\|>\frac{\|t\|}{2}.$$
If $p>0$ and $\sum_{n\ge 1}(\frac{b_n}{b_{n+1}})^p=\infty$, then for $t\in\Bbb R\setminus\Bbb Z$,
\bul either $\|b_nt\|\ge\tfrac{\|t\|}{2a_n}$ eventually and $\sum_{n\ge 1}\|b_nt\|^p=\infty$, or
\bul $\|b_{n+1}t\|>\tfrac{\|t\|}{2}$ infinitely often and $\sum_{n\ge 1}\|b_nt\|^p=\infty$.
\
Either way, $t\notin G_p(b)$.\ \ \Checkedbox
\demo{Proof of Proposition 4} \ \ The implications (i) $\Lra$ (ii) $\Lra$ (iii) $\Lra$ (i) follow from proposition 3.2, theorem 2 and proposition 3.3 (respectively).\ \ \ \Checkedbox
\
\subsection*{Examples}
\
\bul If $b\in\incss$ is the Erdos-Taylor sequence associated to $(2,3,\dots)$, then
$$\text{\rm (a)}\ \ \ \ b\in\text{\tt IPRWM} \ \ \ \ \&\ \ \ \text{\rm (b)}\ \ \ \ G_1(b)=\{0\}.$$
\bul If $b\in\incss$ is the Erdos-Taylor sequence associated to $(a_2,a_3,\dots)$ where $a_n\to\infty$ and
$\sum_{n\ge 1}\frac1{a_n^2}=\infty$ (e.g. $a_n:=\lfl\sqrt n\rfl$), then $b$ is super-lacunary, $ G_2(b)=\{0\}$ and $b$ is not a sequence of IP-rigidity for any probability preserving transformation other than the identity.
\section*{\S4 Proof of Theorem 2}
\
We prove Theorem 2 using dyadic cocycles over the dyadic odometer.
\
Let $\Om:=\{0,1\}^\Bbb N,$ and let $P\in\mathcal P(\Om)$ be symmetric product measure: $P=\prod(\frac12,\tfrac12)$, and let $\tau:\Om\to\Om$ be the dyadic odometer
defined by
$$\tau(1,\dots,1,0,\om_{\ell+1},\dots)=(0,\dots,0,1,\om_{\ell+1},\dots)$$
where $\ell=\ell(\om):=\min\,\{n\ge 1:\ \om_n=0\}$.
\
The {\it dyadic cocycle $\varphi:\Om\to\Bbb Z$ associated to $b\in\incss$} is defined by
$$\varphi(\om):=b_{\ell(\om)}-\sum_{k=1}^{\ell(\om)-1}b_k$$ and its {\it skew product}
$\tau_\v:\Om\x\Bbb Z\to \Om\x\Bbb Z$ is defined by $\tau_\v(x,n)=(\tau(x),n+\v(x))$.
Define $Q:\Om\x\Bbb Z\to \Om\x\Bbb Z$ by $Q(x,n):=(x,n+1)$ and fix $p\in\mathcal P(\Om\x\Bbb Z),\ p\sim P\x\#$ where $\#$ is counting measure on $\Bbb Z$.
\
There is (see \cite{Z}, pp. 76--77)
an ergodic, non-singular transformation $(\frak X,\B(\frak X),\frak q,\frak T)$ of a standard probability space and a map $\pi:\Om\x\Bbb Z\to \frak X$ so that
$$p\circ\pi^{-1}=\frak q,\ \ \pi^{-1}\B(\frak X)=\{A\in\B(\Om\x\Bbb Z):\ \tau_\v A=A\}\ \ \&\ \ \pi\circ Q=\frak T\circ\pi.$$
The ergodic, non-singular transformation $(\frak X,\B(\frak X),\frak q,\frak T)$ is called the {\it Mackey range of $(\tau,\v)$.}
\
In case $b$ is a growth sequence, equivalently $\v:\Om\to\Bbb N$, there is a $\s$-finite, invariant $T$-invariant measure $\frak m\sim \frak q$ with respect to which the Mackey range $(\frak X,\B(\frak X),\frak m,\frak T)$ is isomorphic to the tower over $(\Om,\B(\Om),P,\tau)$ with height function $\v$ (aka the {\it dyadic tower with growth sequence $b$} in \cite{A2}).
The collection of eigenvalues of the Mackey range is
\begin{align*} e(\frak T):=\{t\in\Bbb T:\ \exists\ F\in L^\infty(\frak q),\ F\nequiv 0,\ F\circ\frak T=e^{2\pi it}F\}\end{align*} and it follows from the definitions that
\begin{align*} e(\frak T)=\mathcal T(\tau,\v):=\{s\in\Bbb T:\ \exists\ f\in L^\infty(P),\ f\nequiv 0,\ ,\ f\circ\tau=e^{2\pi is\v}f\}.\end{align*}
\
It it is shown in \S2 of \cite{AN} (see also Theorem 2.6.3 of \cite{A1}) that
\begin{align*}\tag{{\Large\Pointinghand}}\mathcal T(\tau,\v)&=G_2(b).
\end{align*}
.
\
Although formally, ({\Large\Pointinghand}) was only stated for growth sequences in \cite{AN} and \cite{A1}, the proofs do not use this condition and apply to arbitrary $b\in\incss$.
\
Consider the Polish group $\frak B(\mu):=\{f\in L^2(\mu):\ |f|\equiv 1\}$ equipped with $L^2(\mu)$-distance.
\
\proclaim{Lemma 4.1}
\
If the probability $\mu\in\mathcal P(\Bbb T)$ is IP Dirichlet along $b$, then $\exists\ \mathcal X:\Om\to\frak B(\mu)$ continuous so that
\begin{align*}\tag{\dsmedical}\sup_{\om\in\Om}\,\|\chi_{b(K(\om)\cap [1,n])}-\mathcal X(\om)\|_{L^2(\mu)}\xrightarrow[n\to\infty]{}\ 0\end{align*}where $K(\om):=\{n\ge 1:\ \om_n=1\}.$\endproclaim
\demo{Proof}
\sms Suppose that $\mu\in\mathcal P(\Bbb T)$ is IP Dirichlet along $b$. Fix $\om \in \Om$. We claim that the sequence $n\mapsto \chi_{b(K(\om)\cap [1,n])}$ is Cauchy in $L^2(\mu)$.
To see this, let
$$\mathcal E_n:=\sup_{F\in\mathcal F,\ \min\,F\ge n}\|\chi_{_{b(F)}}-1\|_{L^2(\mu)}$$
then by assumption $\mathcal E_n\xrightarrow[n\to\infty]{}\ 0$. Evidently
$$\|\chi_{b(K(\om)\cap [1,n])}-\chi_{b(K(\om)\cap [1,n+k])}\|_{L^2(\mu)}= \|\chi_{_{b(K(\om)\cap [n+1,n+k])}}-1\|_{L^2(\mu)}\le\mathcal E_n$$ whence $\exists\ \mathcal X:\Om\to \frak B(\mu)$ so that
$$\chi_{b(K(\om)\cap [1,n])}\xrightarrow[n\to\infty]{L^2(\mu)}\ \mathcal X(\om)\ \ \text{uniformly in}\ \om\in\Om.$$
For $\om\in\Om$,
$$\|\chi_{b(K(\om)\cap [1,n])}-\mathcal X(\om)\|_{L^2(\mu)}\xleftarrow[k\to\infty]{}\ \|\chi_{b(K(\om)\cap [1,n])}-\chi_{b(K(\om)\cap [1,n+k])}\|_{L^2(\mu)}\le 2\mathcal E_n$$ proving (\dsmedical). Clearly, for each $n\ge 1,\ \om\mapsto\chi_{b(K(\om)\cap [1,n])}$ is continuous ($\Om\to\frak B(\mu)$) and so continuity of $\mathcal X:\Om\to\frak B(\mu)$ follows from the
uniformity of the convergence. \ \ \Checkedbox
Note that the converse of Lemma 4.1 is also true.
\demo{Completion of the proof }
\
Now suppose that $\mu\in\mathcal P(\Bbb T)$ is IP-Dirichlet along $b$. By Lemma 4.1,
$\exists\ \mathcal X:\Om\to\frak B(\mu)$ satisfying
(\dsmedical).
We claim that
\begin{align*}\tag{\Radioactivity}\mathcal X(\tau\om)=\chi_{\varphi(\om)}\mathcal X(\om).\end{align*}
To see this, note that $\chi_{b(K(\om)\cap [1,n])}(t)=\prod_{k=1}^n\chi_{\om_kb_k}(t)=:\mathcal X_n(\om,t)$. For $n>\ell(\om)$,
$$\frac{\mathcal X_n(\tau\om)}{\mathcal X_n(\om)}=\frac{\chi_{b(K(\tau\om)\cap [1,n])}}{\chi_{b(K(\om)\cap [1,n])}}=
\prod_{k=1}^n\frac{\chi_{(\tau\om)_kb_k}}{\chi_{\om_kb_k}}=\chi_{\varphi(\om)}.$$
Since $\frac{\mathcal X_n(\tau\om)}{\mathcal X_n(\om)}\xyr[n\to\infty]{\frak B(\mu)}\frac{\mathcal X(\tau\om)}{\mathcal X(\om)}$, this proves (\Radioactivity).
\
By (\dsmedical), $\exists\ n_J\to\infty$ so that
$$\sum_{J\ge 1}\|\mathcal X_{n_J}-\mathcal X\|_{L^2(P\x\mu)}<\infty\le \sum_{J\ge 1}\sup_{\om\in\Om}\,\|\chi_{b(K(\om)\cap [1,n_J])}-\mathcal X(\om)\|_{L^2(\mu)}<\infty$$ (where $\mathcal X(\om,t):=\mathcal X(\om)(t)$). Hence $\mathcal X_{n_J}\to\mathcal X$
$P\x\mu$-a.e. and by Fubini's theorem,
$\exists\ \Lambda\in\B(\Bbb T),\ \mu(\Lambda)=1$ so that
$$\prod_{k=1}^{n_J}\chi_{\om_kb_k}(t)\xrightarrow[J\to\infty]{}\ \mathfrak X_t(\om)=\mathcal X(\om)(t)\ \forall\ t\in\Lambda\ \&\ P-\text{a.e.}\ \om\in\Om.$$
By (\Radioactivity), for $t\in\Lambda$,
$$\mathfrak X_t\circ\tau=e^{2\pi it\v}\mathfrak X_t\ \ P-\text{a.e.}$$
and $t\in \mathcal T(\tau,\v)$. By ({\Large\Pointinghand}), $t\in G_2(b)$.\ \ \ \ \Checkedbox
The following example shows that the converse to Theorem 2 is false.
\proclaim{Example 4.2}\ \ \ \ $\exists\ b\in\Bbb N^\Bbb N(\uparrow)\ \&\ \mu\in\mathcal P(\Bbb T)$ non-atomic, not IP-Dirichlet along $b$ but so that $\mu(G_2(b))=1$.\endproclaim
\demo{\tt Construction:}
\
Define $b\in\Bbb N^\Bbb N(\uparrow)$ by $b_n:=\prod_{k=1}^na_k$ with $a_k:=k+1$.
\
Consider the mapping
$t:\Om:=\prod_{k\ge 1}\{0,1\}\to [0,1]$ defined by
$$t(\om)=t(\om_1,\om_2,\dots):=\sum_{n\ge 1}\frac{\om_n}{b_n}.$$
This is injective and Borel measurable, so $t(\Om)$ is an uncountable, Borel set in $[0,1]$. We claim that
\begin{align*}\tag{\Biohazard}t(\Om)\subset G_2(b).\end{align*}
\demo{Proof of (\Biohazard)}
For $\om\in\Om$ and $N\ge 1$, we have that
$$b_Nt(\om)=\frac{\om_{N+1}}{a_{N+1}}+\frac{1}{a_{N+1}}\sum_{k\ge 2}\frac{\om_{N+k}}{a_{N+2}\dots a_{N+k}}\ \ \ \mod 1.$$
Now
$$|\sum_{k\ge 2}\frac{\om_{N+k}}{a_{N+2}\dots a_{N+k}}|\le\sum_{k\ge 2}\frac1{a_{N+2}\dots a_{N+k}}<\frac1{a_{N+1}}.$$
Thus, we have
\begin{align*}\tag{\Laserbeam}\<b_Nt(\om)\>=\frac{\om_{N+1}}{a_{N+1}}+\th_N(\om)\ \ \text{where}\ \ \th_N\le \tfrac{1}{a_{N+1}^2},\end{align*}
whence $|\<b_Nt(\om)\>|^2\le \tfrac2{N^2}$ and
$$\sum_{k=1}^\infty|\<b_kt(\om)\>|^2\le\frac{\pi}3.\ \ \ \text{\Checkedbox}\ \ \text{(\Biohazard)}$$
\
Now define $P\in\mathcal P(\Om)$ by $P:=\prod_{k\ge 1}(\tfrac12\d_0+\tfrac12\d_1)$ and set $\mu:=P\circ{t}^{-1}$, then $\mu\in\mathcal P(G_2(b))$. We now show that $\mu$ is not IP-Dirichlet along $b$.
\
Fix $1<\l<e^{\frac13}$, then by (\Laserbeam),
$$\big|\sum_{\l^N<k<\l^{N+1}}\<b_Nt(\om)\>-\sum_{\l^N<k<\l^{N+1}}\frac{\om_{N+1}}{a_{N+1}}\big|\le \sum_{\l^N<k<\l^{N+1}}\tfrac2{k^2}\xrightarrow[N\to\infty]{}0$$
and setting $s_n:=\sum_{j=1}^n\om_j, \kappa_n:=\lcl\l^n\rcl,\ \ell_n:=\lfl\l^{n+1}\rfl$, we have
\begin{align*}\sum_{\l^N<k<\l^{N+1}}\frac{\om_{k}}{a_{k}}&=\sum_{\kappa_N\le k\le \ell_N}\frac{s_k-s_{k-1}}{k+1}\\ &=\sum_{\kappa_N\le k\le \ell_N}\frac{s_k}{k+1}-\sum_{\kappa_N-1\le k\le \ell_N-1}\frac{s_k}{k+2}\\ &=
\frac{s_{\ell_N}}{\ell_N+1}-\frac{s_{\kappa_N}}{\kappa_N+2}+\sum_{\kappa_N\le k\le \ell_N-1}\frac{s_k}{(k+1)(k+2)}.\end{align*}
By the {\tt SLLN}, we have
$$\frac{s_N}{N}\xrightarrow[N\to\infty]{}\frac12\ \ \ \text{a.s.}$$
Writing $\G_N\approx\D_N$ as $N\to\infty$ to mean $\G_N\approx\D_N\xyr[N\to\infty]{}\ 0$, we have
for $P$-a.e. $\om\in\Om$, as $N\to\infty$:
\begin{align*}\sum_{\l^N<k<\l^{N+1}}\<b_Nt(\om)\> &\approx \sum_{\l^N<k<\l^{N+1}}\frac{\om_{k}}{a_{k}}\\ &=
\frac{s_{\ell_N}}{\ell_N+1}-\frac{s_{\kappa_N}}{\kappa_N+2}+\sum_{\kappa_N\le k\le \ell_N-1}\frac{s_k}{(k+1)(k+2)}\\ &\approx\sum_{\kappa_N\le k\le \ell_N-1}\frac1{2k}\\ &\lra \log\l.\end{align*}
Thus
\begin{align*}\sup_{F\in\mathcal F,\ \min\,F>\l^N}\|\chi_{b(F)}-1\|_{L^2(\mu)}^2 &\ge
\Bbb E(|\exp[2\pi i\sum_{\l^N<k<\l^{N+1}}b_Nt(\om)]-1|^2)\\ &\ge
4\Bbb E(|\sum_{\l^N<k<\l^{N+1}}\<b_Nt(\om)\>|^2)\\ &\lra 2\log\l.\ \ \ \boxtimes \end{align*}
\
\section*{\S5 Remarks on the thickness of rigidity sequences}
\
\subsection*{Remark 5.1} \ \ Rigidity sequences for weakly mixing transformations can be arbitrarily "large" within the limitation of density zero.
\
It follows from the definitions that for $b\in\incss$ a growth sequence,
$$|{\tt FS}\,(b)\cap [1,n]|\asymp 2^{c(n)}$$ where $c(n)=\min\,\{k\ge 1:\ b_k\ge n\}$ and if $T\in {\tt MPT}\xbm$ is IP-rigid along $b$, one might expect a rigid sequence at least of this thickness.
\
Indeed, in this case, by Theorem 2, $\s_T(e(\frak T)^c)=0$ where $\frak T$ is the dyadic tower with growth sequence $b$ (since by [AN] $e(\frak T)=G_2(b)$). By theorem 4 in \cite{A2} $\exists\ L\subset\Bbb N$ with
$$T^n\xyr[n\to\infty,\ n\in L]{{\tt MPT}}\ \text{\tt Id}\ \ \&\ \ \frac{|L\cap [1,n]|}{2^{c(n)}}\xyr[n\to\infty]{}\infty.$$
\
By the Corollary in \cite{A2}, $\forall\ a(n)>0,\ \tfrac{a(n)}n\xyr[n\to\infty]{}0$, there is a weakly mixing $T\in{\tt MPT}\xbm$ and
$L\subset\Bbb N$ such that
$$T^n\xyr[n\to\infty,\ n\in L]{{\tt MPT}}\ \text{\tt Id}\ \ \&\ \ \frac{|L\cap [1,n]|}{a(n)}\xyr[n\to\infty]{}\infty.$$
For more on this phenomenon, see \S3 in \cite{BJLR}.
\
\subsection*{Remark 5.2}\ \ Let $b\in\incss$ be a growth sequence and suppose that ${\tt H-dim}(G_1(b))>\a$ (where $\a\in (0,1)$ and {\tt H-dim} denotes Hausdorff dimension).
\
We claim that $\exists\ T\in{\tt MPT}$ weakly mixing $\&$ IP-rigid along $b$ with the property that for any sequence $L\subset\Bbb N$ along which $T$ is rigid:
\begin{align*}\tag{\Football}\sum_{n=1}^\infty\frac{|L\cap [1,n]|}{n^{2-\a}}<\infty.\end{align*} Note that it follows from this that
$\sum_{n=1}^\infty\frac{2^{c(n)}}{n^{2-\a}}<\infty.$\demo{Proof of (\Football)}
\
As in the proof of theorem 1 of \cite{A3}, it follows from Frostman's theorem (\cite{Fr}, see also \cite{KS}) that $\exists\ \mu\in\mathcal P(G_1(b))$ so that
\begin{align*} \sum_{n=1}^\infty\frac{|\widehat{\mu}(n)|}{n^{1-\a}}<\infty.\end{align*}
Let $T\in{\tt MPT}\xbm$ be the associated Gaussian automorphism. By Proposition 1, $T$ is IP-rigid along $b$.
\
Suppose that $T$ is rigid along $L\subset\Bbb N$. In particular $\exists\ N\ge 1$ so that $|\widehat{\mu}(n)|\ge\tfrac12\ \forall\ n\in L,\ n>N$.
\
It follows that
\begin{align*}\sum_{n=1}^\infty\frac{|L\cap [1,n]|}{n^{2-\a}}&=
\sum_{n=1}^\infty \sum_{k=1}^n1_L(k)\frac1{n^{2-\a}}\\ &=
\sum_{k=1}^\infty 1_L(k)\sum_{n=k}^\infty\frac1{n^{2-\a}}\\ &\le
\frac1{1-\a}\sum_{k=1}^\infty 1_L(k)\frac1{k^{1-\a}}\\ &\le\frac{N}{1-\a}+
\frac2{1-\a}\sum_{k=N+1}^\infty \frac{|\widehat{\mu}(k)|}{k^{1-\a}}\\ &<\infty.\ \ \ \ \text{\Checkedbox}\text{(\Football)}\end{align*}
\subsection*{Remark 5.3}\ \ The condition (\Football) is sharp. In \S5 of \cite{A3}, $\forall\ \a\in (0,1)$ a growth sequence $b^{(\a)}\in\incss$ is exhibited with
$$\text{\tt H-dim}\,(G_1(b^{(\a)}))=\a\ \&\ 2^{c^{(\a)}(n)}\gg n^{1-\a}$$
whence
if $T\in {\tt MPT}\xbm$ is so that $\s_T(G_2(b))=1$, then (again by theorem 4 in [A2]) $\exists\ L\subset\Bbb N$ rigid for $T$ with $\frac{|L\cap [1,n]|}{n^{1-\a}}\to\infty$ and therefore
\begin{align*}\sum_{n=1}^\infty\frac{|L\cap [1,n]|}{n^{2-\a}}=\infty.\end{align*}
|
{
"timestamp": "2012-05-28T02:01:01",
"yymm": "1203",
"arxiv_id": "1203.2257",
"language": "en",
"url": "https://arxiv.org/abs/1203.2257"
}
|
\section{Introduction\label{sec:intro}}
Since the pioneering effort by Eagles,\cite{Eagles} many researchers have
extensively and repeatedly addressed the transition of superconducting (SC)
properties from a BCS type to a Bose-Einstein condensation (BEC) type as the
strength of attractive potential between fermions is increased.
Following early studies of the BCS-BEC crossover\cite{Leggett,Miyake}
using continuum models with the superfluidity of $^3$He in mind, Nozi\`eres
and Schmitt-Rink\cite{N-SR} showed by approximation that the SC properties
smoothly evolve with the correlation strength in an attractive Hubbard model
(AHM).
Later, stimulated by the discovery of high-$T_{\rm c}$ cuprates with
a small coherence length, numerous researchers have tackled
AHM,\cite{Micnus} especially in two dimensions (2D); now, in connection
with the evolution of pseudogaps as the doping rate $\delta$ decreases
in the so-called underdoped regime,\cite{pseudogap} the problem of BCS-BEC
crossover as a function of $\delta$ is a subject of
urgency.\cite{cuprate,Levin}
Entering this century, we have become capable of directly observing phenomena
of crossover\cite{Regal,Zwierlein} and pseudogaps\cite{Gaebler,Feld}
in traps of ultracold dilute alkali gases,\cite{Bloch,Giorgini} for which
physical parameters can be artificially tuned.
Recent experimental advances have brought hope of obtaining similar observations
on optical lattices.
\par
In the above stream of research, AHM is one of the most important and basic
lattice models for studying the evolution of SC properties according to
the interaction strength $U/t$ ($U$: onsite interaction strength,
$t$: hopping integral between nearest-neighbor sites).
In early and later studies of AHM, mean-field-type\cite{Micnus} and
diagrammatic\cite{N-SR,Bickers,Deisz,TPSC} approaches were used;
although they successfully treated the weakly correlated regime,
where the original BCS theory is basically valid, and developed
a conceptual framework of the BCS-BEC crossover, more reliable methods
remain necessary to establish properties in the intermediately
(unitary) and strongly correlated (BEC) regimes.
First, as an unbiased way, quantum Monte Carlo (QMC) calculations
were implemented in the weakly and intermediately correlated regimes
($|U|\lsim W$, $W$: bandwidth), because QMC is free from the
negative-sign problem for AHM, but statistical fluctuation increases
with increasing $|U|/t$ and system size.
In 2D, SC transition of the Berezinskii-Kosterlitz-Thouless type was
confirmed and an $n$-dependent phase diagram was discussed
($n$: particle density).\cite{Moreo}
Then, it was shown in the normal state ($T>T_{\rm c}$) for intermediate
$|U|/t$'s that a thermal-activation-type behavior appears
in magnetic susceptibility, but that the charge compressibility
is almost $T$-independent.\cite{Randeria}
This spin-gap behavior was corroborated by a peak split in the density
of state.\cite{Singer}
The dynamical mean field theory (DMFT), which becomes exact in infinite
dimensions and is applicable to an arbitrary interaction strength, is
another important approach to AHM.
Early DMFT studies addressed normal branches without introducing
SC orders at low temperatures ($T<T_{\rm c}$\cite{Keller} and
$T=0$\cite{Capone}), and found that the normal state undergoes
a first-order transition from a Fermi liquid to a gapped state
at $|U|/W=1$-1.5, as $|U|/t$ increases.
Later, various properties of the SC phase were
calculated,\cite{Garg,Bauer,Koga}
and the crossover was characterized by the SC gap and superfluid
stiffness.\cite{Toschi}
\par
Another effective approach to AHM is a many-body variation theory,
which is applicable continuously in the entire range of correlation
strengths and particle densities.
In contrast to DMFT, the dimension and lattice form are realistically
specified, and one can treat wave-number-dependent properties
in low-lying states.
Furthermore, since wave functions are explicitly given, this approach
has advantages in forming a physical picture.
Because an AHM of a bipartite lattice is mapped to a repulsive Hubbard
model (RHM) by a canonical transformation,\cite{Dichtel,Shiba,Nagaoka}
one can develop a theory
relying on the knowledge of RHM.
Thus, the well-known Gutzwiller wave function (GWF)\cite{GWF} became
a primary trial function for the normal state; first, its properties
were studied\cite{Medina} using the so-called Gutzwiller approximation
(GA).\cite{GA}
As known for RHM, although GWF itself is always metallic,\cite{YS1}
additional GA induces a spurious metal-insulator (Brinkman-Rice)
transition\cite{BR} at finite $|U_{\rm BR}|/t$ in finite
lattice dimensions.
For $|U|>|U_{\rm BR}|$ in AHM, all the particles tightly form onsite
singlet pairs, and hopping completely ceases, so that the Brinkman-Rice
transition remains a metal-insulator (Mott) transition also in AHM.
Later, approximations similar to GA, which may be correct in infinite
dimensions, have also been applied to the SC state\cite{Suzuki,Saito,Bunemann}
to discuss the BCS-BEC crossover.
However, to avoid the ambiguity of GA in realistic dimensions and
to make use of the merits of the variation method, we need
to accurately estimate variational expectation values.
This claim is satisfied by a variational Monte Carlo (VMC)
method,\cite{McMillan,Ceperley,YS1,Umrigar} which treats local
correlation factors exactly without a minus sign problem.
A decade ago, a VMC method was applied to a normal state in AHM
to study a transition corresponding to the Mott transition in RHM
by introducing a binding factor between adjacent antiparallel
spinons.\cite{Y-PTP}
For simplicity, we call a singly occupied site a spinon.
However, the interpretation of the transition was incorrect on account
of the limitation of treated system sizes and an insufficient analysis.
Recently, VMC has been applied to solving problems with optical lattices
in a confinement potential.\cite{Fujiwara}
\par
In this study, on the basis of VMC calculations of high precision
for normal, SC and CDW states, we modify the previous results\cite{Y-PTP}
and make features of the BCS-BEC crossover in AHM on the square lattice
microscopically more clear.
We mainly discuss the following points:
(1) In both normal and SC states, a correlation between adjacent
antiparallel spinons, in addition to the Gutzwiller correlation, is
indispensable to qualitatively derive proper behavior.
(2) In the normal state, which underlies the SC state, a first-order
phase transition occurs at $|U_{\ma{c}}|\sim W$ from a Fermi-liquid
to a spin-gapped state.
This transition is caused by the competition between the size of
an antiparallel-spinon pair and the interpair distance, as in the case
of Mott transitions in RHM.\cite{Miyagawa,boson}
(3) The properties of SC noticeably change at approximately
$|U_{\ma{co}}|\sim|U_{\rm c}|$, which are compared with those derived
in a strongly correlated RHM for high-$T_{\rm c}$ cuprates.
Part of the present result has been published before.\cite{Tamura}
\par
The rest of this paper is organized as follows:
In \S\ref{sec:formulation}, we explain the model and method used
in this study.
In \S\ref{sec:normal}, we provide a discussion of the spin-gap
transition arising in the normal state, and of the features
in the spin-gapped regime.
In \S\ref{sec:SC}, we consider a BCS-BEC crossover from various
points of view.
In \S\ref{sec:summary}, we briefly summarize our main results.
\par
\section{Formulation\label{sec:formulation}}
In \S\ref{sec:model}, we introduce AHM and briefly mention its relation
to RHM.
In \S\ref{sec:wf}, we discuss trial wave functions for normal, SC,
and CDW phases.
In \S\ref{sec:VMC}, we briefly explain the setup of VMC calculations
in this work.
\par
\subsection{Attractive Hubbard model\label{sec:model}}
We consider a single-band attractive Hubbard model ($U\leq0$)
on a square lattice:
\begin{equation}
{\cal H}={\cal H}_t+{\cal H}_U
= \sum_{{\bf k}\sigma}\varepsilon_{\bf k}
c_{{\bf k}\sigma}^{\dag}c_{{\bf k}\sigma}
+U\sum_{j}n_{j\uparrow}n_{j\downarrow},
\label{eq:model}
\end{equation}
where $n_{j\sigma}=c_{j\sigma}^{\dag}c_{j\sigma}$, $c_{j\sigma}$, and
$c_{{\bf k}\sigma}$ are fermion annihilation operators in the Wannier
and Bloch representations, respectively, and
\begin{equation}
\varepsilon_{\bf k}=-2t(\cos k_x +\cos k_y).
\label{eq:dispersion}
\end{equation}
We use the hopping integral $t$ and lattice constant as the units
of energy and length, respectively.
Because the lattice has a particle-hole symmetry at $n=N/N_{\rm s}=1$
($N$: number of particles, $N_{\rm s}$: number of lattice sites),
and properties at half filling are deduced from the results of
RHM using corresponding wave functions,\cite{YTOT,YOT} as mentioned
below, we mostly treat cases of $n<1$.
The chemical potential term $-\zeta\sum_{j\sigma}n_{j\sigma}$ may be
added to eq.~(\ref{eq:model}) to adjust particle density, if necessary.
\par
In the following, we summarize the relation of AHM to RHM.
The attractive Hubbard Hamiltonian eq.~(\ref{eq:model}) on a bipartite
lattice satisfying the relation
$\varepsilon_{\bf k}=-\varepsilon_{-{\bf k}+{\bf Q}}$
[here ${\bf Q}=(\pi,\pi)$] is mapped by the canonical
transformation\cite{Dichtel,Shiba}
\begin{equation}
c_{{\bf k}\uparrow}=\tilde{c}_{{\bf k}\uparrow}, \quad
c_{{\bf k}\downarrow}=\tilde{c}^\dag_{-{\bf k}+{\bf Q}\downarrow}
\label{eq:canonical}
\end{equation}
to RHM with constant shifts:
\begin{equation}
\tilde{\cal H}=\sum_{{\bf k}\sigma}\varepsilon_{\bf k}
\tilde c_{{\bf k}\sigma}^\dag\tilde c_{{\bf k}\sigma}
+|U|\sum_j\tilde n_{j\uparrow}\tilde n_{j\downarrow}
+UN\tilde n_\uparrow
-h\sum_j\left(\tilde S^z_j+\frac{1}{2}\right),
\end{equation}
where $\tilde n_{j\sigma}=\tilde c_{j\sigma}^\dag\tilde c_{j\sigma}$,
$\tilde S^z_j=(\tilde n_{j\uparrow}-\tilde n_{j\downarrow})/2$ and
$\tilde n_\sigma=\tilde N_\sigma/N$.
A tilde denotes the representation transformed according
to eq.~(\ref{eq:canonical}).
The chemical potential $\zeta$ and $n$ in AHM are related to
the effective magnetic field as $h=2\zeta$ and to the magnetization
as $m=1-n$ in the $z$-direction in RHM, respectively.
Therefore, unless the original AHM has a spin polarization ($m=0$),
the particle density in the transformed RHM is always at half filling
($\tilde n=1$).
Also, the order parameters of CDW and onsite singlet pairing
defined as
\begin{eqnarray}
O_{\rm CDW}&=&\frac{1}{N}\left|\sum_j e^{i{\bf Q}\cdot{\bf r}_j}
\langle n_{j\uparrow}+n_{j\downarrow}-1\rangle\right|, \\
O_{\rm SC}&=&\frac{1}{N}\sum_j
\langle c_{j\uparrow}^\dag c_{j\downarrow}^\dag\rangle \quad\mbox{or}\quad
\frac{1}{N}\sum_j
\langle c_{j\downarrow}c_{j\uparrow}\rangle,
\end{eqnarray}
in AHM are transformed into the forms of the $z$- and $xy$-components
of the SDW order parameter:
\begin{eqnarray}
\tilde O_{\rm SDW}^z&=&\frac{1}{N}\left|\sum_j e^{i{\bf Q}\cdot{\bf r}_j}
\langle \tilde n_{j\uparrow}-\tilde n_{j\downarrow}\rangle\right|, \\
\tilde O_{\rm SDW}^\pm&=&\frac{1}{N}\sum_j
\langle\tilde c_{j\uparrow}^\dag\tilde c_{j\downarrow}\rangle
\quad\mbox{or}\quad \frac{1}{N}\sum_j
\langle\tilde c_{j\downarrow}^\dag\tilde c_{j\uparrow}\rangle,
\end{eqnarray}
respectively, in RHM.\cite{Nagaoka}
It is widely accepted that, at $T=0$, an antiferromagnetic (AFM) long-range
order with equal magnitudes of $O_{\rm SDW}^\alpha$ for $\alpha=x,y,z$
arises in the half-filled RHM on the square lattice for arbitrary $U$ ($>0$),
and that the AFM order in the $z$-direction is easily destroyed
by a field applied in the $z$-direction $h$, whereas the AFM orders
in the $xy$-plane survive.
This implies that the ground state of AHM possesses a singlet pairing
order for any $U$ and $n$ ($\zeta$), and simultaneously possesses a CDW
order of the same magnitude at half filling $n=1$ ($\zeta=0$).\cite{Nagaoka}
This argument was confirmed by direct calculations for AHM.\cite{Micnus}
Although the above mapping holds unconditionally in exact treatments,
when some approximation is applied, the validity of the mapping
has to be verified individually for each specific treatment.
\par
\subsection{Trial wave functions\label{sec:wf}}
As a development of our previous study,\cite{Y-PTP} we apply a many-body
variation theory to the Hamiltonian eq.~(\ref{eq:model}).
As a trial wave function, a two-body Jastrow-type
$\Psi={\cal P}\Phi_{\ma{MF}}$ was adopped,\cite{Jastrow}
where $\Phi_{\ma{MF}}$ is a one-body (mean-field) wave function and
${\cal P}$ is a many-body correlation (Jastrow) factor.
\par
As the many-body part, we use the form
${\cal P}={\cal P}_f{\cal P}_Q{\cal P}_{\rm G}$ in this work.
The onsite (Gutzwiller) projector
\begin{equation}
{\cal P}_{\rm G}=\prod_{j}
\left[1-\left(1-g\right)d_j\right],
\label{eq:gp}
\end{equation}
with $d_j=n_{j\uparrow}n_{j\downarrow}$, is the most important.
The variational parameter $g$ increases the number of doubly occupied
sites (doublons), and ranges over $1\le g<\infty$ for $U\le 0$;
in the limit of $g\rightarrow\infty$, singly occupied sites (spinons)
are not allowed in a nonmagnetic case.
If we put $\tilde g=1/g$, the properties of ${\cal P}_{\rm G}(\tilde g)$
for RHM are applicable to the present case ${\cal P}_{\rm G}(g)$.\cite{Y-PTP}
\par
To explain the importance of a binding factor between the up and down spinons
${\cal P}_Q$, it is convenient to refer to an effective Hamiltonian
in the strong-correlation limit ($t/|U|\rightarrow 0$):\cite{Emery}
\begin{equation}
{\cal H}_{\rm eff}=\frac{2t^2}{|U|}
\sum_{<i,j>}\left[\left(-b^{\dag}_ib_j+\rho_i\rho_j+\si_i\si_j\right)
+\mbox{H.c.}-\frac{1}{2} \right],
\label{eq:ste1}
\end{equation}
with
\begin{eqnarray}
b_i=c_{i\ua}c_{i\da},\;\;
\rho_i=\frac{1}{2}(n_{i\ua}+n_{i\da}-1),\;\;
\si_i=\frac{1}{2}(n_{i\ua}-n_{i\da}).
\label{eq:b}
\end{eqnarray}
The first term of eq.~(\ref{eq:ste1}) indicates the hopping of doublons.
The second is a repulsive interaction between doublons [or empty sites
(holons)] and an attractive interaction between a doublon and a holon
in nearest-neighbor (NN) sites.
The third works as an AFM-Ising interaction.
The expectation values of these terms can be reduced using antiparallel-spinon
configurations in NN sites.
To encourage such configurations, we introduce the attractive intersite
correlation,\cite{Y-PTP}
\begin{eqnarray}
{\cal P}_Q\=\prod_j\left(1-\mu Q_j\right)
\label{eq:PQ} \\
Q_j\=s^{\ua}_{j}\prod_{\tau}
(1-s^{\da}_{j+\tau})+s^{\da}_{j}\prod_{\tau}(1-s^{\ua}_{j+\tau})
\label{eq:Qj}
\end{eqnarray}
where $s_j^{\sigma}=n_{j\sigma}(1-n_{j-\sigma})$ (spinon projector),
and $\tau$ runs over NN sites of the site $j$.
In ${\cal P}_Q$, the parameter $\mu$ ($0\le\mu\le 1$) controls the strength
of binding between NN antiparallel spinons;
for $\mu=0$, spinons are free of binding, while in the limit
$\mu\rightarrow 1$, antiparallel spinons are necessarily paired as nearest
neighbors.
As we will see later, ${\cal P}_Q$ is indispensable for a spin-gap
transition\cite{Y-PTP} and a proper description of the SC state.
In fact, ${\cal P}_Q$ is the canonical transformation
through eq.~(\ref{eq:canonical}) of the doublon-holon binding factor
often used to describe Mott transitions in RHM.\cite{Kaplan,YS3}
A Mott transition in RHM corresponds to a spin-gap transition in AHM,
as we will see in \S\ref{sec:normal}.
Since $Q_j$ is a spin-dependent projector, the so-called spin
contamination\cite{contamination} arises in the wave function, namely,
$\Psi$ deviates from an eigenstate of ${\bf S}^2=(\sum_j{\bf S}_j)^2$.
However, in this case, the expectation values of $\langle{\bf S}^2\rangle$
estimated using a VMC method are as small as 0.15 (2) in the SC
(normal) state for $N=200$-$300$ at its maximum at $U\sim W$.
Because these values, particularly of the SC state, are much smaller than
those of the AFM state, the spin contamination is considered to have little
influence on the results.
\par
As a factor supplementary to ${\cal P}_Q$, a repulsive correlation
suited to eq.~(\ref{eq:ste1}) should be considered.
As a simple one, we check a repulsive factor between NN doublons:
\begin{eqnarray}
{\cal P}_f=\prod_j
\left[1-fd_j\left(1-\prod_{\tau}\bar d_{j+\tau}\right)\right],
\label{eq:pf}
\end{eqnarray}
where $f$ ($0\le f\le 1$) is a parameter,
$\bar d_j=1-d_j$, and $\tau$ runs over NN sites of the site $j$.
The projector ${\cal P}_f$ reduces the weight of configurations with
adjacent doublons by $1-f$; for $f\rightarrow 0$, the effect of
${\cal P}_f$ vanishes, and for $f\rightarrow 1$, a doublon cannot sit
in a NN site of another doublon.
\par
Now, we turn to the one-body part $\Phi$ of the wave function.
For a normal state, we adopt the Fermi sea $\Phi_{\rm F}$.
Since general features of $\Psi_{\rm N}={\cal P}\Phi_{\rm F}$ with $f=0$ were
studied in a previous paper,\cite{Y-PTP} here we focus on the properties
of the transition arising at $U\sim W$, which was regarded as a Mott
transition.\cite{Y-PTP}
\par
It is known that the BCS state $\Phi_{\rm BCS}$ can deal with
the BCS-BEC crossover in some degree;\cite{Leggett,N-SR} it is natural
to employ $\Phi_{\rm BCS}$ for a SC state:
\begin{eqnarray}
\Phi_{\rm BCS}=\left(\sum_{\m{k}}a_{\m{k}}
c_{\m{k}\ua}^{\dag}c_{-\m{k}\da}^{\dag} \right)^{N/2}|0\rangle.
\label{eq:BCS}
\end{eqnarray}
where the particle number is fixed and
\begin{eqnarray}
a_{\m{k}}=\frac{v_{\m{k}}}{u_{\m{k}}}=
\frac{\Delta_{\rm P}}{\varepsilon_{\m{k}}-\bar\zeta+
\sqrt{(\varepsilon_{\m{k}}-\bar\zeta)^2+\Delta_{\rm P}^2}},
\label{eq:ak}
\end{eqnarray}
with $\Delta_{\rm P}$ and $\bar\zeta$ being variational parameters
corresponding to the SC gap $\Delta_{\rm SC}$ and chemical potential
$\zeta$, respectively, in the weakly correlated limit, and
$$
u_{\bf k}^2\ (v_{\bf k}^2)=\frac{1}{2}\left[
1+(-)\frac{\varepsilon_{\rm k}-\bar\zeta}
{\sqrt{\left(\varepsilon_{\bf k}-\bar\zeta\right)^2
+\Delta_{\rm p}^2}}\right].
$$
For $\Delta_{\rm P}\rightarrow 0$, $\Phi_{\rm BCS}$ is reduced to
$\Phi_{\rm F}$.
Here, we assume $\Delta_{\rm P}$ to be a homogeneous $s$ wave on account
of the attractive contact potential.
For RHM, a form similar to eq.~(\ref{eq:ak}) with a $d_{x^2-y^2}$-wave pair
potential was studied,\cite{YTOT,Y12} where, as $\delta$ decreases, what
$\Delta_{\rm P}$ means deviates from the SC gap, and represents
a pseudogap.\cite{ZGRS,Paramekanti}
In contrast, in the present case, $\Delta_{\rm P}$ seems to reflect
the magnitude of $T_{\rm c}$ for any $|U|/t$, except for $n\rightarrow 1$,
for which $T_{\rm c}$ is considered to vanish owing to the CDW order.
The correlated SC function $\Psi_{\rm sc}={\cal P}\Phi_{\rm BCS}$ is
mapped using eq.~(\ref{eq:canonical}) to a projected AFM wave function
ordered in the $x$-$y$ plane at $n=1$.
\par
In addition, we check a CDW wave function for $n\sim 1$:
\begin{equation}
\Phi_{\rm CDW}=\prod_{{\bf k},\sigma}
\left(-\alpha_{\bf k}c_{{\bf k}\sigma}^\dag
+\beta_{\bf k}c_{{\bf k}+{\bf Q}\sigma}^\dag\right)|0\rangle,
\label{eq:CDW}
\end{equation}
where the ${\bf k}$ sum is taken in the Fermi sea, $\m{Q}=(\pi,\pi)$, and
\begin{equation}
\alpha_{\bf k}\ (\beta_{\bf k})=
\sqrt{\frac{1}{2}\left(1-(+)\frac{\varepsilon_{\bf k}}
{\sqrt{\varepsilon_{\bf k}^2+\Delta_{\rm c}^2}}\right)}\ ,
\label{eq:CDWcoeff}
\end{equation}
with $\Delta_{\rm c}$ being a parameter corresponding to the CDW gap.
$\Psi_{\rm CDW}={\cal P}\Phi_{\rm CDW}$ is mapped through
eq.~(\ref{eq:canonical}) to a projected AFM wave function ordered
in the $z$-direction at $n=1$.
Because the AFM order is isotropic in RHM, $\Psi_{\rm SC}$ and
$\Psi_{\rm CDW}$ should yield identical results at half filling.
\par
\subsection{Variational Monte Carlo method\label{sec:VMC}}
In estimating variational expectation values with respect to $\Psi$
discussed in \S\ref{sec:wf}, we use a VMC
method,\cite{McMillan,Ceperley,YS1,Umrigar} which gives virtually
exact values for finite but relatively large systems.
Since the number of variational parameters is not large, we execute
rounds of linear optimization for each parameter with the other
parameters fixed until the parameters as well as the energy converge
(typically 3-5 rounds) with $2.5\times 10^5$ particle configurations
generated through a Metropolis algorithm.
After the convergence, we continue to execute additional $20$ to $30$
rounds of iteration with successively renewed configuration sets.
We determine the optimized values by averaging the data obtained in the
additional rounds; in averaging, we exclude scattered data beyond the range
of twice the standard deviation.
Thus, the optimal value is an average of substantially more than several
million samples.
Physical quantities are computed with the optimized parameters thus
obtained with $2.5\times 10^5$ samples.
\par
We use systems of $L\times L$ sites of up to $L=32$ for $\Psi_{\rm N}$
and $L=24$ for $\Psi_{\rm SC}$ with the periodic-antiperiodic
boundary conditions to reduce level degeneracy.
We choose the particle densities to satisfy the closed-shell condition,
and mainly study $n=0.25$ (0.26), 0.5, and 0.75.
\par
\section{Spin-Gap Transition in Normal State\label{sec:normal}}
As mentioned in \S\ref{sec:model}, the ground state of AHM is SC
for any $U/t$ and $n$ (and CDW at $n=1$).
Therefore, the normal state, we address in this section, is not
the ground state of eq.~(\ref{eq:model}).
The significance to study $\Psi_{\rm N}$ is not only in a passive sense
that a normal state appears when the SC state is destroyed
by, e.g., magnetic field or impurities, but in that $\Psi_{\rm N}$
underlies $\Psi_{\rm SC}$, just as a SC transition is understood
by the instability of the Fermi sphere against an infinitesimal
attractive interaction in the BCS theory.
Namely, normal states are deeply involved in the mechanism of
SC transitions.
\par
In \S\ref{sec:energy}, we show the improvement in energy by introducing
${\cal P}_Q$ and ${\cal P}_f$, and the energy gains using the SC
and CDW states.
In \S\ref{sec:spingap}, we confirm the existence of a spin-gap transition
in the normal state.
In \S\ref{sec:picture}, we consider the mechanism of the spin-gap transition
and other properties.
\par
\subsection{Energy improvement\label{sec:energy}}
First, we briefly look at the energy improvement by the projection
factors ${\cal P}_Q$ and ${\cal P}_f$ in the normal, SC, and CDW states.
\par
\begin{table}[htbp]
\vspace{-0.0cm}
\caption{
Comparison of energy per site $E/t$ among three kinds of projection factors
${\cal P}_{\rm G}$, ${\cal P}_{\cal Q}\equiv{\cal P}_Q {\cal P}_{\rm G}$, and
${\cal P}_{\cal F}\equiv{\cal P}_f {\cal P}_Q {\cal P}_{\rm G}$ in normal
and SC states.
A system of $n=0.5$ and $L=12$ is used; the tendency is the same
for other $n$'s.
The last digit in each section includes some statistical errors.
}
\begin{center}
\begin{tabular}{r|l|l|l|l}
\hline
\multicolumn{1}{c|}{$|U|/t$} & \multicolumn{1}{c|}{1} &
\multicolumn{1}{c|}{3} & \multicolumn{1}{c|}{7} &
\multicolumn{1}{c}{10} \\
\hline \hline
${\cal P}_{\rm G}\ \Phi_{\ma{F}}$ &
$-1.37513$ & $-1.54791$ & $-2.0677$ & $-2.6215$ \\ \hline
${\cal P}_{\cal Q}\ \Phi_{\ma{F}}$ &
$-1.37536$ & $-1.55099$ & $-2.10435$& $-2.7417$ \\ \hline
${\cal P}_{\cal F}\ \Phi_{\ma{F}}$ &
$-1.37536$ & $-1.55097$ & $-2.10440$& $-2.7416$ \\ \hline\hline
${\cal P}_{\rm G}\ \Phi_{\ma{BCS}}$ &
$-1.37531$ & $-1.55686$ & $-2.17815$ & $-2.80500$ \\ \hline
${\cal P}_{\cal Q}\ \Phi_{\ma{BCS}}$ &
$-1.375535$& $-1.55908$ & $-2.18444$ & $-2.81162$ \\ \hline
${\cal P}_{\cal F}\ \Phi_{\ma{BCS}}$ &
$-1.375534$& $-1.55918$ & $-2.18681$ & $-2.81436$ \\ \hline
\end{tabular}
\label{table:energy}
\vspace{-0.4cm}
\end{center}
\end{table}
As studied in detail in ref.~\citen{Y-PTP}, the variational energy in
the normal state is considerably improved by ${\cal P}_Q$ on that of GWF,
especially for large $|U|/t$'s (Table \ref{table:energy}).
Moreover, it is known that the phase transition discussed in \S\ref{sec:spingap}
does not arise without ${\cal P}_Q$.\cite{YS1,Y-PTP}
Thus, the factor ${\cal P}_Q$ is indispensable to appropriately describe
the normal state.
These aspects of ${\cal P}_Q$ correspond to those of the doublon-holon
(D-H) binding factor in RHM.\cite{YS3,YOT}
On the other hand, the improvement by ${\cal P}_f$
on ${\cal P}_Q{\cal P}_{\rm G}\Phi$ is almost imperceptible for any $|U|/t$
as shown in Table \ref{table:energy}.
The optimized parameter $f$ is nearly zero, namely,
${\cal P}_f$ scarcely modifies the wave function.
\par
In the SC state, the improvement in $E/t$ by ${\cal P}_Q$
on ${\cal P}_{\rm G}\Phi_{\rm BCS}$ is not as large as that
for the normal state.
This is because the effect of binding between up and down spinons, i.e.,
the effect of singlet pair creation, is already included in
$\Phi_{\rm BCS}$ to some extent, as the one-body AFM state has some
D-H binding effect for RHM.\cite{YTOT}
Further energy reduction by ${\cal P}_f$ is again negligible for
small $|U|/t$'s and remains relatively small in magnitude for larger $|U|/t$'s,
as compared with the energy reduction by ${\cal P}_Q$.
The magnitude of energy reduction by ${\cal P}_f$ is similarly small
for $n=0.25$ and 0.75.
\par
Since we find that a short-range repulsive factor ${\cal P}_f$ produces
only negligible effects in all the cases we treat, we omit ${\cal P}_f$
and use the form
${\cal P}={\cal P}_{\cal Q}\equiv{\cal P}_Q{\cal P}_{\rm G}$
as the many-body factor in $\Psi$ in the rest of this paper, unless
otherwise specified.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm,clip]{65325fig1.eps}
\end{center}
\caption{(Color online)
Particle-density dependences of the energy gains by $\Psi_{\rm SC}$
and $\Psi_{\rm CDW}$ are plotted for five values of $|U|/t$.
For small $|U|/t$'s, the system size dependence (fluctuation) is large
owing to the large coherence length.
}
\vspace{-0.3cm}
\label{fig:cdw}
\end{figure}
Finally, we compare the energy gains using the SC and CDW states:
\begin{equation}
\Delta E_{\rm SC}\ (\Delta E_{\rm CDW})=
E_{\rm N}-E_{\rm SC}\ (E_{\rm CDW}),
\label{eq:Egain}
\end{equation}
where $E_{\rm N}$, $E_{\rm SC}$,
and $E_{\rm CDW}$ are the optimized energies per site for $\Psi_{\rm N}$,
$\Psi_{\rm SC}$ and $\Psi_{\rm CDW}$, respectively.
Figure \ref{fig:cdw} shows the $n$ dependences of $\Delta E_{\rm SC}$ and
$\Delta E_{\rm CDW}$ for large $n$'s.
At half filling ($n=1$), the SC and CDW states are degenerate, but this
degeneracy is immediately lifted for $n<1$.
$\Delta E_{\rm CDW}$ rapidly deteriorates and vanishes as $n$ decreases,
whereas $\Delta E_{\rm SC}$ preserves appreciable values for high densities
and gradually decays until $n=0$ (not shown).
This feature of $\Delta E$ coincides with what we discussed for
the canonical transformation in \S\ref{sec:model}.\cite{Micnus}
\par
In the remainder of this section, we will concentrate on $\Psi_{\rm N}$.
\par
\subsection{Spin-gap transition\label{sec:spingap}}
In a previous VMC study using ${\cal P}_{\cal Q}\Phi_{\rm F}$,\cite{Y-PTP}
a transition was detected with systems up to $L=12$
at $U=U_{\rm c}\sim 9t$.
However, this transition was misinterpreted as a continuous metal-insulator
transition.
In this subsection and the next, we study the features of this transition
more carefully.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.0cm,clip]{65325fig2.eps}
\end{center}
\caption{(Color online)
Momentum distribution function for some values of $|U|/t$ near transition
point $|U_{\ma{c}}|/t$ along path $(0,0)$-$(\pi,0)$-$(\pi,\pi)$-$(0,0)$.
The discontinuity at ${\bf k}_{\rm F}$ indicated by an arrow is used
to estimate $Z$ shown in Fig.~\ref{fig:Z}.
}
\vspace{-0.6cm}
\label{fig:nk}
\end{figure}
First, we confirm the existence of a transition.
In Fig.~\ref{fig:nk}, we plot the momentum distribution function
\begin{equation}
n(\m{k})=\frac{1}{2}\sum_\sigma
\left\langle c_{\m{k}\sigma}^{\dag}c_{\m{k}\sigma} \right\rangle
\label{eq:nk}
\end{equation}
for $|U|/t\sim 9$ and $n=0.25$ ($L=32$).
For $|U|/t\le 9.0$, $n({\bf k})$ has discontinuities on the $\Gamma$-X and
$\Gamma$-M segments, indicating that a Fermi surface exists and
the state is a Fermi liquid.
On the other hand, the discontinuity suddenly vanishes for $|U|/t\ge 9.05$,
and $n({\bf k})$ becomes a smooth function of ${\bf k}$.
It follows a certain gap opens and the state becomes a non-Fermi liquid
for $|U|>|U_{\rm c}|$, with $9.0<|U_{\rm c}|/t<9.05$ in this case.
Through similar analyses, we found $|U_{\rm c}|/t\sim 0.875$ (0.83)
for $n=0.195$ (0.121) for $L=32$; $|U_{\rm c}|/t$ tends to gradually
decrease with $n$.\cite{note-Uc}
Thus, a transition from a Fermi liquid to a non-Fermi liquid certainly
exists, as found in our previous study.\cite{Y-PTP}
According to similar analyses for $L=24$ and 28 and $n\sim 0.25$,
the system-size dependence of $|U_{\rm c}|/t$ is very small at these values
of $L$, but $|U_{\rm c}|/t$ tends to increase slightly as $L$ increases.
Such a feature is analogous to those of the Mott transitions
in RHM induced by D-H binding factors.\cite{YOT,Miyagawa,boson}
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.5cm,clip]{65325fig3.eps}
\end{center}
\caption{(Color online)
Quasiparticle renormalization factor as function of $|U|/t$ for three
particle densities, estimated from discontinuities of $n({\bf k})$
in Fig.~\ref{fig:nk} and other data.
The inset shows the optimized spinon-binding parameter as a function
of $|U|/t$ near $U=U_{\rm c}$ for the same systems as that in the main panel.
}
\label{fig:Z}
\end{figure}
Next, we check the order of this transition.
In Fig.~\ref{fig:Z}, the quasiparticle renormalization factor $Z$ is shown
vs $U/t$; $Z$ is obtained using $Z=n(k_\ma{F}-0)-n(k_\ma{F}+0)$
on the $\Gamma$-X segment, where the values of $n(k)$
at $k\rightarrow k_{\rm F}\pm 0$ are estimated using third-order
least-squares fits of the data for $k<k_{\rm F}$ and $k>k_{\rm F}$,
respectively.
There exist clear discontinuities in $Z$ at $U=U_{\rm c}$.
The optimized spinon-binding parameter $\mu$ plotted in the inset
of Fig.~\ref{fig:Z} also exhibits a large jump at $U=U_{\rm c}$.
In fact, other physical quantities show a similar discontinuous behavior.
Thus, we may safely conclude that this transition is not a continuous
transition but a first-order phase transition.
The reason why the previous study could not find the correct transition
order is that the discontinuous behavior of $n({\bf k})$ manifests itself
only for $L\gsim 18$.
The critical value $|U_{\rm c}|/t$ only slightly depends on $n$.
\par
Now, we consider the features of $\Psi_{\rm N}$ in the non-Fermi-liquid
regime $|U|>|U_{\rm c}|$.
As shown in the inset of Fig.~\ref{fig:Z}, the spinon-binding parameter
$\mu$ approaches unity, suggesting that almost all up and down spinons
are paired as singlets.
In discussing gap formation, the small-$|{\bf q}|$ behavior of the charge
(density) and spin structure factors
\begin{eqnarray}
N(\m{q})\=\frac{1}{N}
\sum_{j,l}e^{-i\m{q}\cdot\m{r}_l}\langle n_j n_{j+l}\rangle-n^2
\label{eq:Nq} \\
S(\m{q})\=\frac{1}{N}
\sum_{j,l}e^{-i\m{q}\cdot\m{r}_l}\langle S^z_j S^z_{j+l}\rangle
\label{eq:Sq}
\end{eqnarray}
provide us with useful information.
Assuming that the lowest excitation occurs at ${\bf q}={\bf 0}$, the energy
gap in the spin sector between the ground state $|\Psi_0\rangle$
and the first excited state $|\Psi_\m{q}\rangle$ is given
by the single-mode approximation (SMA)\cite{SMA} as
\begin{eqnarray}
\Delta_S
\=\frac{\langle\Psi_{\bf q}|({\cal H}-E_0)|\Psi_{\bf q}\rangle}
{\langle\Psi_{\bf q}|\Psi_{\bf q}\rangle}
=\frac{\langle\Psi_0|[S_{-{\bf q}},[{\cal H},S_{\bf q}]]|\Psi_0\rangle}
{\langle \Psi_{\bf q}|\Psi_{\bf q} \rangle} \nonumber \\
\=-\frac{1}{8}\lim_{q\ra0}\frac{q^2}{S(\m{q})}K
\label{eq:SMA}
\end{eqnarray}
where $K$ denotes the kinetic energy,
$|\Psi_\m{q}\rangle=S_\m{q}|\Psi_0\rangle$,
and
\begin{equation}
S_\m{q} = \frac{1}{\sqrt{N}}\sum_j e^{i\m{q}\cdot\m{r}_j}S^z_j\ .
\end{equation}
From eq.~(\ref{eq:SMA}), we find that $\Delta_S$ vanishes
if $S(\m{q})\propto q$ for $q\rightarrow 0$, whereas $\Delta_S$ becomes
finite, if $S(\m{q})\propto q^2$ for $q\rightarrow 0$.
The charge (density) gap $\Delta_N$ can be similarly treated.
\par
\begin{figure}[htbp]
\vspace{-0.1cm}
\begin{center}
\includegraphics[width=8.0cm,clip]{65325fig4.eps}
\end{center}
\caption{(Color online)
(a) Spin and (b) charge (density) structure factors along path
$(0,0)$-$(\pi,0)$-$(\pi,\pi)$-$(0,0)$ for some values of $|U|/t$
near $|U_{\rm c}|/t$ ($9.0<|U_{\rm c}|/t<9.05$).
The inset in (b) is a magnification of $S({\bf q})$ near
${\bf q}\sim{\bf 0}$ on the segment $(0,0)$-$(\pi,0)$.
The tendency is the same for other values of $n$.
}
\vspace{-0.1cm}
\label{fig:SqNq_normal}
\end{figure}
\begin{figure}[htbp]
\vspace{-0.0cm}
\begin{center}
\includegraphics[width=6.5cm,clip]{65325fig5.eps}
\end{center}
\caption{(Color online)
Spin gaps estimated by single-mode approximation for three particle
densities as functions of $|U|/t$.
The dash-dotted line is a visual guide for $n=0.5$.
The spin-gap transition occurs at $|U_\m{c}|/t\sim 9$.
}
\vspace{-0.3cm}
\label{fig:spingap}
\end{figure}
In Fig.~\ref{fig:SqNq_normal}, we show $S(\m{q})$ and $N(\m{q})$
for some values of $|U|/t$ near $U=U_{\rm c}$.
In the vicinity of $\m{q}=\m{0}$, as $|U|/t$ increases, $S(\m{q})$
abruptly changes its behavior from linear to quadratic at $U=U_{\rm c}$,
as shown in the inset of Fig.~\ref{fig:SqNq_normal}(b).
Thus, it is very likely that the spin gap is generated in the non-Fermi
liquid regime.
In Fig.~\ref{fig:spingap}, we plot the spin gap estimated using
eq.~(\ref{eq:SMA}) for the segment of $(0,0)$-$(\pi,0)$ of $S(\m{q})$;
the magnitude of $\Delta_S$ is proportional to $|U|$
($\Delta_S\sim 0.13|U|$ for $n=0.5$), and depends on $n$ only weakly.
In contrast, the behavior of $N(\m{q})$ shown
in Fig.~\ref{fig:SqNq_normal}(b) is almost unchanged including the
$2k_{\rm F}$ anomaly, if $|U|/t$ varies, and remains linear in $q$
for $q\rightarrow 0$ for $|U|>|U_{\rm c}|$.
Thus, low-energy properties with respect to the charge degree of
freedom are unlikely to be affected by this transition;
regarding the charge excitation, $\Psi_{\rm N}$ remains gapless
and conductivity is preserved in the spin-gapped regime.
This may be the first realization of a conductive spin-gapped normal
state in the variation theory.
\par
\subsection{Picture of transition and spin-gapped state
\label{sec:picture}}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig6.eps}
\end{center}
\caption{(Color online)
(a) Doublon density as function of $|U|/t$ for $\Psi_{\rm N}$ and
$\Psi_{\rm SC}$.
The dashed lines indicate the maximum $D$, i.e., $n/2$.
The inset in (b) shows the magnification of $D$ ($\Psi_{\rm N}$) for
$n=0.25$ near the transition point, at which $D$ displays a small jump.
(b) Average distances from up spinon to its nearest down spinon
for three particle densities as functions of $|U|/t$.
}
\vspace{-0.3cm}
\label{fig:dist_ud}
\end{figure}
To deepen our understanding of the above spin-gap transition, let us look at
some other quantities.
Figure \ref{fig:dist_ud}(a) shows the doublon density
\begin{equation}
D=\frac{1}{N_{\ma{s}}}\sum_j\langle b^\dag_jb_j\rangle.
\end{equation}
As $|U|/t$ increases, $D$ increases in the Fermi-liquid state
owing to the attractive correlation of ${\cal P}_{\rm G}$,
but it reaches almost its full value ($n/2$) at $U=U_{\rm c}$.
The main panel of Fig.~\ref{fig:dist_ud}(b) shows the average distance
from an up (down) spinon to its nearest down (up) spinon $r_{\rm ud}$.
Here, we measure distance $r$ by the stepwise (so-called
Manhattan) metric.
As $|U|/t$ increases in a small-$|U|/t$ regime, $r_{\rm ud}$ increases
because the densities of up and down spinons decrease owing to
doublon formation, and the binding correlation of ${\cal P}_Q$
is still weak, as in the inset of Fig.~\ref{fig:Z}.
However, $r_{\rm ud}$ abruptly drops when $U$ approaches $U_{\rm c}$,
and converges to unity for $|U|>|U_{\rm c}|$, because an up spinon and
a down spinon are tightly bound within NN sites ($\mu\rightarrow 1$).
Consequently, for $|U|>|U_{\rm c}|$, almost all particles form onsite
pairs, and even if a doublon resolves into spinons, they remain
an adjacent pair and do not itinerate as isolated spinons.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm,clip]{65325fig7.eps}
\end{center}
\caption{(Color online)
Illustration of mechanism of spin-gap transition.
The up and down arrows denote up and down spinons, respectively.
Typical configurations are shown for the two phases.
Although in each pair domain drawn with a pale circle, at least one up
spinon and one down spinon must exist, excess spinons can still move around
independently, slipping out of their original domain, as indicated
by the long dashed arrows in the Fermi-liquid phase.
In the spin-gapped phase, a spinon cannot itinerate independently of
the partner spinon out of the pair domain.
}
\vspace{-0.3cm}
\label{fig:up-down}
\end{figure}
Thus, we notice that this spin-gap transition can be understood
in parallel with a recently proposed picture of Mott transitions
owing to the D-H binding.\cite{Miyagawa,boson}
Here, we postulate that antiparallel spinon pairs with a pair domain
of size $\xi_{\rm ud}$ are created by the attractive correlation
of ${\cal P}_Q$.
We can appropriately define this binding length $\xi_{\rm ud}$ and
also the minimum distance from a spinon to its nearest parallel
spinon $\xi_{\rm uu}$ as
\begin{eqnarray}
\xi_{\rm ud}\=r_{\rm ud}+\sigma_{\rm ud},
\label{eq:xiud} \\
\xi_{\rm uu}\=r_{\rm uu}-\sigma_{\rm uu},
\label{eq:xiuu}
\end{eqnarray}
where $r_{\rm uu}$ is the average distance from an up (down) spinon
to its nearest up (down) spinon, and $\sigma_{\rm ud}$ and
$\sigma_{\rm uu}$ are the standard deviations of $r_{\rm ud}$ and
$r_{\rm uu}$, respectively.
In the spin-gapped phase, the relation $\xi_{\rm ud}<\xi_{\rm uu}$ holds,
indicating that the domains of pairs do not usually overlap, at least,
not in sequence.
Consequently, almost all pairs are isolated and an up spinon and a down spinon
are confined within $\xi_{\rm ud}$, resulting in singlet pairs
of small lengths with finite excitation gaps.
In contrast, in the Fermi-liquid phase, $\xi_{\rm ud}$ becomes longer
than $\xi_{\rm uu}$, indicating that the domains of spinon pairs overlap
with one another.
Then, an up spinon in a pair can exchange a partner down spinon
with a down spinon in an adjacent pair.
As a result, an up spinon and a down spinon can move independently by exchanging
their partner, as shown in long arrows in Fig.~\ref{fig:up-down},
and definite singlet pairs cannot be specified.
Thus, as $|U|/t$ is varied, a spin-gap transition takes place
when $\xi_{\rm ud}$ becomes equivalent to $\xi_{\rm uu}$, which is
expected to be a monotonically increasing function of $|U|/t$.
\par
\begin{figure}[htbp]
\vspace{-0.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig8.eps}
\end{center}
\vspace{-0.3cm}
\caption{(Color online)
The binding length of up and down spinon pairs, $\xi_{\rm ud}$, and
the minimum distance between spinon pairs, $\xi_{\rm uu}$, defined
by eqs.~(\ref{eq:xiud}) and (\ref{eq:xiuu}), respectively,
are plotted as functions of $U/t$ for three densities.
An arrow indicates the spin-gap transition point, the $n$ dependence of
which is small.
}
\label{fig:xi_normal}
\end{figure}
Figure \ref{fig:xi_normal} shows $\xi_{\rm ud}$ and $\xi_{\rm uu}$
estimated from the VMC results as functions of $|U|/t$ for three
particle densities.
As expected from Fig.~\ref{fig:dist_ud}(b), $\xi_{\rm ud}$ abruptly
drops at $U=U_{\rm c}$, whereas $\xi_{\rm ud}$ monotonically increases
as $|U|/t$ increases with a small jump at the transition point.
As a result, $\xi_{\rm ud}$ and $\xi_{\rm uu}$ intersect each other
at $U=U_{\rm c}$ for any $n$.
Thus, the scheme illustrated in Fig.~\ref{fig:up-down} is justified
to some extent.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig9a.eps}
\includegraphics[width=6.5cm,clip]{65325fig9b.eps}
\end{center}
\vspace{-0.3cm}
\caption{(Color online)
In the main panel, the kinetic energy $E_\ma{kin}$ and its components
$E_1$ and $E_2$ are depicted as functions of correlation strength
for two particle densities.
$E_1$ ($E_2$) is the contribution of hopping processes that do not
(do) change $D$, as shown in the illustration in the lower part.
The dashed line for $E_2/t$ is a guide proportional to $-t/|U|$
($|U|>|U_{\rm c}|$).
}
\label{fig:E1E2}
\end{figure}
Finally, we discuss the itinerancy of particles.
To this end, it is convenient to decompose the kinetic energy $E_{\ma{kin}}$
into two parts ($E_\ma{kin}=E_1+E_2$), namely, the contribution
of the hopping processes that do not (do) change the number
of doublons $E_1$ ($E_2$),\cite{Tocchio}
as shown in the lower part of Fig. \ref{fig:E1E2}.
In the main panel of Fig. \ref{fig:E1E2}, $E_1$, $E_2$, and $E_{\ma{kin}}$
are depicted as functions of $|U|/t$.
For $|U|<|U_{\rm c}|$, both $E_1$ and $E_2$ contribute to $E_{\ma{kin}}$
because isolated spinons are independently mobile, whereas
in the spin-gapped phase, $E_1$ almost vanishes ($E_{\ma{kin}}\sim E_2$)
for any particle density.
In this case, the independent motion of a spinon not accompanied
by an antiparallel spinon is strongly suppressed.
On the other hand, the contribution of the dissociation of a doublon
into a spinon pair and their reunion, $E_2$, remains appreciable
and is proportional to $-t^2/|U|$ for large $|U|/t$'s.
This point is in sharp contrast to a feature of the Brinkman-Rice
transition\cite{BR} derived using the Gutzwiller approximation;\cite{GA}
in this case, the motion of particles is completely prohibited
for $|U|>|U_{\rm BR}|=8|E(U=0)|$,\cite{Medina} so that the state becomes
insulating, and a charge (Mott) gap opens.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.5cm,clip]{65325fig10a.eps}
\end{center}
\vspace{0.0cm}
\begin{center}
\includegraphics[width=6.5cm,clip]{65325fig10b.eps}
\end{center}
\vspace{-0.3cm}
\caption{(Color online)
In the main panel, kinetic energy and two elements of
$\langle {\cal H}_{\rm eff}\rangle$ [eqs.~(\ref{eq:ED}) and
(\ref{eq:Erhosigma})]
normalized by $J=2t^2/U$ are plotted in the BEC regime as functions
of $|U|/t$.
Illustrated in the lower part is the relation between a single hopping
process in ${\cal H}$ and a doublon hopping process in ${\cal H}_{\rm eff}$.
The inset, discussed in \S\ref{sec:gain}, shows a comparison of the doublon
hopping $E_{\rm D}$ between the normal and SC states as a function of particle
density.
}
\label{fig:Dh}
\end{figure}
The above feature of $E_1=0$ ($|U|>|U_{\rm c}|$) for any $n$ is distinct
from that of the D-H binding state for strongly correlated RHM,
where $E_1$ is $n$-dependent ($\propto 1-n$) near half filling.\cite{Y12}
Thus, in doped Mott insulators in RHM, doped holes or particles play
a role of carriers, and bound D-H pairs are localized and not involved
in conduction.
On the other hand, in AHM, the motion of a doublon, which is composed of
two single particle-hopping processes as shown at the bottom
of Fig.~\ref{fig:Dh}, contributes to the kinetic energy.
To check this, we calculate the doublon hopping and diagonal terms in
eq.~(\ref{eq:ste1}), namely,
\begin{eqnarray}
\frac{E_{\ma{D}}}{J}\=
\frac{1}{N_{\rm s}}\sum_{j,\tau}\langle b_{j+\tau}^{\dag}b_j\rangle,
\label{eq:ED}
\\
\frac{E_{\rho\sigma}}{J}\=
\frac{1}{N_{\rm s}}\sum_{j,\tau}\langle \rho_{j+\tau}\rho_j+
\sigma_{j+\tau}\sigma_j\rangle,
\label{eq:Erhosigma}
\end{eqnarray}
with $J=2t^2/|U|$, in the original Hilbert space of eq.~(\ref{eq:model}).
In the main panel of Fig.~\ref{fig:Dh}, we compare them with
the single hopping contribution ($E_{\ma{kin}}\sim E_2$) for large $|U|/t$'s.
In addition to the large constant contribution of $E_{\rho\sigma}/J$,
the doublon hopping ($E_{\rm D}$) has an appreciable magnitude,
indicating the possibility of transport.
Thus, the normal state $\Psi_{\rm N}$ is conductive for any values
of $|U|/t$ and $n$ ($\ne 1$).\cite{note-kappa}
\par
\section{Crossover of Superconducting Properties\label{sec:SC}}
In \S\ref{sec:gain}, we discuss the BCS-BEC crossover in the light of
energy gain in the SC transition and of chemical potential, and show
that the SC transition in the BEC regime is induced by the kinetic-energy
gain.
In \S\ref{sec:Ps}, we discuss quantities that characterize the
BCS and BEC regimes.
In \S\ref{sec:coherence}, we roughly estimate the coherence length
and interpair distance, thereby giving an intuitive picture
of the crossover.
\par
\subsection{Energy gain and kinetic-energy-driven transition
\label{sec:gain}}
First, we discuss the BCS-BEC crossover from the point of view of
the energy difference per site between the normal ($\Psi_{\rm N}$)
and SC ($\Psi_{\rm SC}$) states $\Delta E$ $(\ge 0)$ defined
in eq.~(\ref{eq:Egain}).
In Fig.~\ref{fig:cond_ene}(a), the $|U|/t$ dependence of $\Delta E/t$ is
shown; $\Delta E/t$ increases as $\sim\exp(-t/U)$ corresponding
to the BCS theory for small $|U|/t$'s, reaches a maximum
at $|U|=|U_{\rm co}|\sim 8.7t$, and then decreases for $|U|>|U_{\rm co}|$
as $\sim t/|U|$.
As we will see later, various properties of SC actually exhibit qualitative
changes at approximately this $|U_{\rm co}|/t$ from a BCS type to a BEC type.
Note that normal-state properties are deeply involved in the
crossover;\cite{note-co}
$|U_{\rm co}|/t$ is affected by the spin-gap transition point
$|U_{\rm c}|/t$ in $\Psi_{\rm N}$, where $E_{\rm N}$ exhibits a cusp,
resulting in $U_{\rm co}\sim U_{\rm c}$.
Recall that the normal state $\Psi_{\rm N}$, underlying $\Psi_{\rm SC}$,
is a Fermi liquid for $|U|<|U_{\rm c}|$, but $\Psi_{\rm N}$ becomes
a spin-gapped state in the absence of a Fermi surface, as shown in
Fig.~\ref{fig:nk} for $|U|/t\ge 9.05$.
Namely, for $|U|>|U_{\rm c}|$, a SC transition cannot be interpreted
by the instability of the Fermi surface against an attractive interaction.
In this relation, $\Delta E$ means the condensation energy
for $|U|/t\sim 0$ according to the BCS theory, but $\Delta E$ probably
deviates from the condensation energy observed experimentally
for $|U|\gsim|U_{\rm co}|$, as in the case of high-$T_{\rm c}$
cuprates.\cite{Y12}
\par
\begin{figure}[htbp]
\vspace{0.3cm}
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig11.eps}
\end{center}
\caption{(Color online)
(a) Energy differences between normal and SC states as functions of $|U|/t$
for three particle densities.
The maximum point is $|U_\ma{co}|/t\sim 8.7$ for these $n$'s, as indicated
by a dashed line.
The dash-dotted lines for $n=0.5$ are visual guides for the two limit
$|U|/t\rightarrow 0$ [$\propto \exp(-\alpha t/|U|)$] and
$\rightarrow\infty$ [$\propto \beta t/|U|$] with $\alpha$ and $\beta$
being constants.
The kinetic and interaction components of $\Delta E$ are drawn in
(b) and (c), respectively.
}
\vspace{-0.5cm}
\label{fig:cond_ene}
\end{figure}
It is important to analyze $\Delta E$ into the kinetic part
$\Delta E_{\rm kin}$ and the interaction part $\Delta E_{\rm int}$
($\Delta E=\Delta E_{\rm kin}+\Delta E_{\rm int}$).
In the BCS theory, a SC transition is induced by lowering $E_{\rm int}$
at the cost of smaller loss in $E_{\rm kin}$.
On the other hand, it is known in a large-$U/t$ regime of RHM that
a SC transition occurs by reducing $E_{\rm kin}$ with a loss
in $E_{\rm int}$.\cite{YTOT,Y12}
In the latter case, the low-frequency sum rule of optical conductivity
$\sigma_1(\omega)$\cite{Tinkham} should be broken, namely, high-frequency
excitations in $\sigma_1(\omega)$ arise, because the sum of
$\sigma_1(\omega)$ is proportional to $-E_{\rm kin}$\cite{Maldague}
on a square lattice.
In Figs.~\ref{fig:cond_ene}(b) and \ref{fig:cond_ene}(c), we show
$\Delta E_{\rm kin}$ and $\Delta E_{\rm int}$, respectively,
for AHM.
For $|U|\lsim|U_{\rm co}|$, $E_{\rm int}$ ($E_{\rm kin}$) has a gain
(loss) by the SC transition in accordance with the BCS theory.
Meanwhile, for $|U|\gsim|U_{\rm co}|$, the situation is reverse;
the SC transition is driven by a gain in kinetic energy.
Correspondingly, the hopping of doublons (carriers) $|E_{\rm D}|$
becomes more enhanced in $\Psi_{\rm SC}$ than in $\Psi_{\rm N}$
in the BEC regime, as shown in the inset of Fig.~\ref{fig:Dh}.
Kinetic-energy-driven (SC or magnetic) transitions may be
rather general in strongly correlated systems.\cite{Nagaoka-Ferro,YTOT,Y12}
As mentioned previously, this reversal of driven force will be experimentally
found if $\sigma_1(\omega)$ is accurately measured in cold-atom
systems.
In fact, similar results were reached for AHM in infinite dimensions
by DMFT.\cite{Toschi,Kyung}
\par
\begin{figure}[htbp]
\vspace{0.0cm}
\begin{center}
\includegraphics[width=8.5cm,clip]{65325fig12.eps}
\end{center}
\caption{(Color online)
Optimized variational parameters,
(a) Gutzwiller (onsite) factor $g$, and
(b) NN antiparallel-spinon factor $\mu$ for $\Psi_{\rm N}$ and
$\Psi_{\rm SC}$ are shown as functions of $|U|/t$.
The symbols are common in (a) and (b).
The arrows indicate the spin-gap transition points in $\Psi_{\rm N}$.
}
\vspace{-0.5cm}
\label{fig:g_mu}
\end{figure}
Let us consider this SC transition in the light of the variational
parameters.
At the level of one-body wave function, $\Phi_{\rm BCS}$ improves
the energy over $\Phi_{\rm F}$ by creating onsite Cooper pairs
through the pair potential $\Delta_{\rm P}$ (Fig.~\ref{fig:delta_var}),
in accordance with the BCS theory.
Therefore, in the BCS regime, the number of doublons is expected to be more
in the SC state than in the normal state.
This is actually shown in Fig.~\ref{fig:dist_ud}(a).
Since the increase in $D$ hinders the motion of particles, the kinetic
energy is suppressed in the SC state, as in Fig.~\ref{fig:cond_ene}(b).
As $|U|$ increases, however, $D$ of $\Psi_{\rm N}$ increases more rapidly
and surpasses that of $\Psi_{\rm SC}$ at $|U_{\rm co}|$.
This reversal is brought about mainly by the correlation factor ${\cal P}$.
In Fig.~\ref{fig:g_mu}, we compare the optimized parameters in ${\cal P}$
between the normal and SC states.
The onsite attractive factor $g$ is certainly larger in $\Psi_{\rm N}$,
especially near $U=U_{\rm co}$.
The antiparallel-spinon binding factor $\mu$ mainly works for the
suppression of overgrown $D$ by $g$ in the BEC regime in order
to gain $E_{\rm kin}$.
\par
\begin{figure}[htbp]
\vspace{0.0cm}
\begin{center}
\includegraphics[width=6.5cm,clip]{65325fig13.eps}
\end{center}
\caption{(Color online)
The chemical potential estimated from $E/t$ at higher particle densities
($0.5\le n\le 1.0$) is shown by dots as a function of $|U|/t$.
The dash-dotted line denotes the value at half filling: $\zeta=U/2$.
The dashed line indicates the band bottom
$\varepsilon_{\rm L}$ ($-4t$).
}
\vspace{-0.5cm}
\label{fig:cp}
\end{figure}
In a one-body framework, when the chemical potential ($\zeta$) is
situated in an energy band, low-energy excitation in the SC phase is
described by Bogoliubov's quasiparticles, whereas when $\zeta$ becomes
lower than the band bottom $\varepsilon_{\rm L}$, the statistics
of the system becomes bosonic.
Thus, the BCS-BEC crossover point is roughly estimated using
$\zeta=\varepsilon_{\rm L}$.
We estimate $\zeta$ for $\Psi_{\rm SC}$ from
$\zeta=\partial E_{\rm SC}/\partial n$ (strictly finite differences).
Within statistical error, $E_{\rm SC}$ is almost a linear
function of $n$ for $0.5<n<1.0$, so that the $\zeta$ obtained
in this range becomes independent of $n$.
In Fig.~\ref{fig:cp}, we plot the thus-estimated $\zeta$ as a function of
$|U|/t$ with the value at half filling i.e., $\zeta=U/2t$.
$\zeta$ reaches the band bottom $\varepsilon_{\rm L}=-4t$
at $|U|/t\sim 7.9$.
The behavior of $\zeta$ here is consistent with those obtained
by DMFT.\cite{Garg,Bauer}
\par
\begin{figure}[htbp]
\vspace{0.0cm}
\begin{center}
\includegraphics[width=7.5cm,clip]{65325fig14.eps}
\end{center}
\caption{(Color online)
Evolution of momentum distribution function for $\Psi_{\rm SC}$
as $|U|/t$ varies.
The dashed line indicates $|U|/t=\infty$.
}
\vspace{-0.5cm}
\label{fig:nk_sc}
\end{figure}
Finally, let us look at the momentum distribution function
for $\Psi_{\rm SC}$ (Fig.~\ref{fig:nk_sc}).
As $|U|/t$ increases, a step-function form of $|U|/t=0$ changes
to a BCS type ($v_{\bf k}^2$) for small $|U|/t$ and is then smoothly
modified through $|U_{\rm co}|/t$ to a constant in the BEC limit
($|U|/t=\infty$).
Such evolution of $n({\bf k})$ has already been observed in an experiment
on an ultracold gas in a trap;\cite{Regal-nk} we hope for similar experiments
on optical lattices.
\par
\subsection{Pair correlation function and helicity modulus
\label{sec:Ps}}
As mentioned in \S\ref{sec:intro}, a previous study of Toschi
\etal\cite{Toschi}\
using DMFT
argued that appropriate quantities that trace the strength of SC
($T_{\rm c}$) in the BCS and BEC regimes are the gap
$\Delta_{\rm SC}\sim\langle c^\dag_\uparrow c^\dag_\downarrow\rangle$ and
the superfluid stiffness $D_{\rm s}$, respectively.
$\Delta_{\rm SC}$ indicates the cost of creating a Cooper pair, while
$D_{\rm s}$ characterizes the cost of realizeing phase coherence.
In this subsection, we start with the quantities corresponding to
$\Delta_{\rm SC}$ and $D_{\rm s}$.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.5cm,clip]{65325fig15.eps}
\end{center}
\caption{(Color online)
The onsite superconducting correlation function is drawn along
a path of ${\bf r}_\ell$, (0,0)-$(L/2,0)$-$(L/2,L/2)$-(0,0),
for various values of $|U|/t$.
For other values of $n$ and $L$, the behavior is basically the same.
}
\vspace{-0.1cm}
\label{fig:P(r)}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.0cm,clip]{65325fig16.eps}
\end{center}
\caption{(Color online)
System size dependence of $P_\infty$ for three values of $|U|/t$
at $n=0.5$.
In the cases of $L$ not satisfying the closed-shell condition at $n=0.5$,
we adopt the average of $n=0.5\pm\delta$ with the smallest $\delta$
satisfying the condition.
As visual guides, we show the first-order least-squares fits for $L\rightarrow\infty$
with dashed lines.
Solid (open) symbols represent the data of $\Psi_{\rm SC}$
($\Psi_{\rm N}$).
$\Psi_{\rm N}$ is bound to vanish for $L\rightarrow\infty$.
}
\vspace{-0.5cm}
\label{fig:P_infty_size}
\end{figure}
As an appropriate index of off-diagonal-long-range order (ODLRO)
in the present scheme, we consider a SC correlation function
of onsite pairing,\cite{noteP} defined as
\begin{eqnarray}
P(\m{r}_\ell)=\frac{1}{N_{\rm s}}\sum_{j}
\langle b_{j}^{\dag}b_{j+\ell} \rangle.
\label{eq:P}
\end{eqnarray}
The magnitude of ODLRO is given by the long-distance value
of $P(\m{r}_\ell)$, i.e.,
$P_\infty=\lim_{|\m{r}_\ell|\ra \infty}P(\m{r}_\ell)\sim\Delta_{\rm SC}^2$.
In the present VMC calculations with finite systems, we must check
the ${\bf r_\ell}$ dependence of $P(\m{r}_\ell)$.
In Fig.~\ref{fig:P(r)}, we plot $P(\m{r}_\ell)$ for various $|U|/t$'s
along a typical path on the lattice.
Since $P({\bf r})$ is substantially constant for $|{\bf r}|\ge 2$
for any value of $|U|/t$, consistently with a QMC study,\cite{Guerrero}
it is appropriate to put $P({\bf r})$ with the most distant
${\bf r}=(L/2,L/2)$ at $P_\infty$, and check its system-size dependence.
In Fig.~\ref{fig:P_infty_size}, we plot the thus-estimated $P_\infty$
as a function of $1/L$ for three values of $|U|/t$.
In the SC state, the system-size dependence of $P_{\infty}$ is very weak
for any $|U|/t$, and fitted well by a first-order least-squares method.
Thus, we may discuss $P_\infty$ with a finite but large $L$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig17.eps}
\end{center}
\vspace{-0.5cm}
\caption{(Color online)
Onsite superconducting correlation function of $\Psi_{\rm SC}$
plotted as a function of $|U|/t$ for three particle densities
and $L=20$.
The dash-dotted line is a visual guide of $\propto\exp(-t/|U|)$ for $n=0.75$.
}
\label{fig:P_infty}
\end{figure}
Figure \ref{fig:P_infty} shows the $|U|/t$ dependence of $P_\infty$
for $L=20$.
In the BCS regime ($|U|\lsim|U_{\rm co}|$), $P_\infty$ increases as
$\sim\exp(-t/U)$ as $|U|/t$ increases, as predicted by the BCS theory.
In this regime, the optimized $\Delta_{\rm P}$ is considered to
approximate the gap, as will be argued; we confirmed using the data
shown in Fig.~\ref{fig:delta_var} that the relation
$P_\infty\sim\Delta_{\rm P}^2$ holds.
On the other hand, in the BEC regime, $P_\infty$ tends to converge
at a finite value as $|U|/t$ increases, in accordance with the result
of $\Delta_{\rm SC}$ obtained using DMFT.\cite{Garg,Bauer}
Thus, the behavior of $P_\infty$ or $\Delta_{\rm SC}$ in this regime
does not coincide with the behavior of $T_{\rm c}$, $\sim t^2/|U|$,
which was naturally expected\cite{N-SR} and actually obtained by
DMFT.\cite{Keller,Toschi,Koga}
\par
Incidentally, a similar SC correlation function with the NN $d_{x^2-y^2}$-wave
pairing $P_d^\infty$ has been calculated for RHM in 2D by VMC with
the same class of trial functions.\cite{Y12}
In the strongly correlated regime (typically $U/t=12$), where the cuprates
are considered to be properly described, $P_d^\infty$ behaves
as the so-called dome shape as a function of doping rate $\delta$ ($=1-n$).
This dome shape closely agrees with the $\delta$ dependence of $T_{\rm c}$
experimentally observed for the cuprates.
If the framework of BSC-BEC crossover as a function of $\delta$ is
applicable to this case, $P_d^\infty$ decreases and scales with $T_{\rm c}$
as the parameter approaches the BEC limit ($\delta\rightarrow 0$).
Consequently, the behavior of $P_d^\infty$ in RHM does not fully
correspond to that of $P_\infty$ in this study.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig18.eps}
\end{center}
\caption{(Color online)
Helicity modulus as a function of $|U|/t$ for three particle densities.
The dash-dotted line indicates a visual guide of $\propto -t/|U|$ for $n=0.75$.
}
\vspace{-0.3cm}
\label{fig:helicity}
\end{figure}
Now, we turn to the helicity modulus $\rho_{\rm s}$, which is related to
superfluid stiffness $D_{\rm s}$ with $\rho_{\rm s}=D_{\rm s}/4\pi e^2$.
In the SC state, $D_{\rm s}$ is equivalent to the strength of the
delta-function component at $\omega=0$ in the optical conductivity
$\sigma_1(\omega)$\cite{Scalapino}
and represents the superfluid weight.
We calculate $\rho_{\rm s}$ as the coefficient of the quadratic term
in the increment in energy when the phase of order parameter
$\Delta_j$ is twisted by ${\bf q}$ as
$\Delta_j=|\Delta|e^{i{\bf q}\cdot{\bf r}_j}$:\cite{Fisher}
\begin{equation}
E({\bf q})-E(0)=2\rho_{\rm s}{\bf q}^2+{\cal O}({\bf q}^4),
\end{equation}
following the prescription of Denteneer \etal\cite{Denteneer}
In Fig.~\ref{fig:helicity}, the $|U|/t$ dependence of the $\rho_{\rm s}$ thus
obtained is plotted for three particle densities.
The resultant $\rho_{\rm s}$ here is almost independent of the {\bf q} used
($|{\bf q}|\sim 0.1$), and the system size dependence is negligible within
the symbols between $L=12$ and 20.
Regardless of $n$, $\rho_{\rm s}$ is a monotonically decreasing function
of $|U|/t$.
The behavior in the BCS regime is distinct from that of
$T_{\rm c}\sim \exp(-t/|U|)$, but $\rho_{\rm s}$ scales with
$T_{\rm c}\sim t^2/|U|$ in the BEC regime, indicating the strength
of SC in the BEC regime is determined not by the cost of creating
a pair but by the cost of realizing phase coherence.
The present result of $\rho_{\rm s}$ is consistent with the previous
results obtained by a VMC method with
${\cal P}_{\rm G}\Phi_{\rm BCS}$,\cite{Denteneer} QMC,\cite{Singer}
and DMFT.\cite{Garg,Toschi,Bauer}
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig19.eps}
\end{center}
\caption{(Color online)
Spin gap of SC state estimated similarly to that in Fig.~\ref{fig:spingap}
by single-mode approximation.
The dash-dotted straight line indicates an extrapolation from
large-$|U|/t$ values for $n=0.5$.
}
\vspace{-0.2cm}
\label{fig:spingap_sc}
\end{figure}
In the remainder of this subsection, we discuss some quantities related
to $P({\bf r})$ and $\rho_{\rm s}$.
First, we take up the small-$|{\bf q}|$ behavior of spin and charge
structure factors, eqs.~(\ref{eq:Nq}) and (\ref{eq:Sq}).
Like for the normal state, $N({\bf q})\propto |{\bf q}|$
for $|{\bf q}|\rightarrow 0$ for any $|U|/t$, showing that
$\Psi_{\rm SC}$ is conductive in particle density.
On the other hand, $S({\bf q})\propto |{\bf q}|^2$ for any $|U|/t$ ($>0$)
in contrast to the case of $\Psi_{\rm N}$, indicating that a spin gap opens
owing to pair formation.
We estimate the spin gap $\Delta_S$ for $\Psi_{\rm SC}$ using SMA
[eq.~(\ref{eq:SMA})], and show the $|U|/t$ dependence
in Fig.~\ref{fig:spingap_sc}.
Although we do not display the data for small $|U|/t$'s owing to the
relatively large errors due to the use of a finite system ($L=20$), $\Delta_S$
seems to be proportional to $\exp(-\alpha t/|U|)$ for small $|U|/t$'s.
On the other hand, $\Delta_S$ is proportional to $|U|/t$, and has
a magnitude similar to that of $\Psi_{\rm N}$.
$\Delta_S$ is almost independent of $n$.
Thus, $\Delta_S$ is a quantity that scales with $T_{\rm c}$
in the BCS regime.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig20.eps}
\end{center}
\caption{(Color online)
Condensate fractions of hard core bosons (or doublons) for three
particle densities as functions of $|U|/t$.
The dash-dotted line is a visual guide of $\propto \exp(-t/U)$ for n=0.26.
}
\vspace{-0.5cm}
\label{fig:BEC1}
\end{figure}
Next, let us consider the condensate fraction $\rho_0$.
We may regard $b^\dag_j$ as a creation operator of a hard-core
spinless boson at the site $j$; $b^\dag_j$ satisfies the Bose
commutation relation except for the same site.
Following Bose systems,\cite{boson} we define a quantity corresponding to
the condensate fraction or the ${\bf k}={\bf 0}$ element of momentum
distribution function $n_D({\bf k})$ for $b_j^\dag$:
\begin{equation}
\rho_0=\frac{1}{D}n_D({\bf 0})= \frac{1}{DN_{\rm s}}
\sum_{j,\ell}\langle b_{j}^{\dag}b_{j+\ell} \rangle.
\label{eq:condensate}
\end{equation}
In Fig.~\ref{fig:BEC1}, the $|U|/t$ dependence of $\rho_0$ is depicted
for three particle densities.
The behavior of $\rho_0$ for small $|U|/t$'s is $\rho_0\sim\exp(-t/|U|)$
and has a meaning similar to $P_\infty$.
In the BEC regime ($|U|>|U_{\rm co}|$), $\rho_0$ is almost constant,
indicating that a picture of hard-core bosons is justified in
the entire regime of BEC.
The suppression of $\rho_0$ with increasing $n$ is primarily because
a high density enhances the effect of onsite interaction.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm,clip]{65325fig21.eps}
\end{center}
\caption{(Color online)
Optimized gap parameter $\Delta_{\rm P}$ for several $n$'s as function of
$|U|/t$.
The dash-dotted line is a visual guide of $\propto \exp(-t/|U|)$.
The inset shows a comparison of the optimized values of $\Delta_{\rm P}$ between
${\cal P}_Q{\cal P}_{\rm G}\Phi_{\rm BCS}$ and
${\cal P}_{\rm G}\Phi_{\rm BCS}$ for $n=0.5$.
}
\vspace{-0.3cm}
\label{fig:delta_var}
\end{figure}
Finally, we consider the pairing gap parameter $\Delta_{\rm P}$ given
in eq.~(\ref{eq:ak}).
In Fig.~\ref{fig:delta_var}, we show the $|U|/t$ dependence of the
optimized $\Delta_{\rm P}$.
For $|U|<|U|_{\rm co}$, it is natural that $\Delta_{\rm P}$ represents
the SC gap of $\propto\exp(-t/|U|)$, as expected from the BCS theory.
In the BCS theory, $\Delta_{\rm SC}$ should continue to
linearly increase like $\Delta_S$ in Fig.~\ref{fig:spingap_sc}.
However, the $\Delta_{\rm P}$ of $\Psi_{\rm SC}$ exhibits a peak at
$U\sim U_{\rm co}$, and then decreases as $|U|/t$ increases,
similarly to $\Delta E$ [Fig.~\ref{fig:cond_ene}(a)].
In the BEC regime, it is probable that $\Delta_{\rm P}$ obeys
$\propto 2t^2/|U|=J$, because $J$ is the sole energy scale
for $|U|/t\rightarrow\infty$, according to eq.~(\ref{eq:ste1}).
It seems that $\Delta E$ and $\Delta_{\rm P}$ scale $T_{\rm c}$ in AHM.
\par
In the inset of Fig.~\ref{fig:delta_var}, we compare the optimized
$\Delta_{\rm P}$'s between $\Psi_{\rm SC}$ and
${\cal P}_{\rm G}\Phi_{\rm BCS}$.
Although the two $\Delta_{\rm P}$'s behave similarly in the BCS regime,
the $\Delta_{\rm P}$ of ${\cal P}_{\rm G}\Phi_{\rm BCS}$ monotonically
increases unlike the $\Delta_{\rm P}$ of $\Psi_{\rm SC}$ in the BEC regime.
It follows that the binding correlation between antiparallel spinons
is also significant for $\Psi_{\rm SC}$, especially in the BEC regime.
\par
In variational theories with $d_{x^2-y^2}$-wave SC states for
cuprates, the $d_{x^2-y^2}$-wave gap parameter $\Delta_d$, corresponding
to $\Delta_{\rm P}$ here, is considered to represent a
singlet-pairing gap (not necessarily SC gap).\cite{ZGRS,Paramekanti}
The optimized $\Delta_d$ monotonically increases as the doping rate,
the relevant parameter of the crossover, approaches the BEC limit
($\delta\rightarrow 0$), in contrast
to $T_{\rm c}$.\cite{ZGRS,Paramekanti,Y12}
The behavior of $\Delta_{\rm p}$ here is distinct from that of $\Delta_d$.
Again, we should be careful to consider the cuprate in the point of
view of the BCS-BEC crossover.
\par
\subsection{Coherence length and intuitive picture
\label{sec:coherence}}
The BCS-BEC crossover has been typically explained by whether
or not a domain of a Cooper pair overlaps with a domain of another
pair, as in Fig.~\ref{fig:up-down}.
To discuss this more quantitatively, we need to estimate a pair size
$\xi_{\rm pair}$ corresponding to the coherence length
and a distance between Cooper pairs $\tilde\xi_{\rm uu}$.
As for $\xi_{\rm pair}$, it is reasonable to refer to the BCS expression
of Pippard's coherence length,
\begin{equation}
\xi_0=\frac{\hbar v_{\rm F}}{\pi|\Delta_{\rm SC}|},
\label{eq:xi-BCS}
\end{equation}
in the BCS regime.
In the present study, $v_{\rm F}$ is a constant for any $|U|/t$,
because the renormalization of $k_{\rm F}$ by $|U|/t$ is not
introduced.
Thus, we assume $\xi_{\rm pair}=\alpha/\Delta_{\rm P}$, where $\alpha$ is
a constant determined so that $\xi_{\rm pair}$ can be smoothly connected
to the form on the BEC side.
In the BEC regime, eq.~(\ref{eq:xi-BCS}) does not work, because
$v_{\rm F}$ cannot be defined.
Thus, following eq.~(\ref{eq:xiud}), we naively assume
$\xi_{\rm pair}=\tilde\xi_{\rm ud}=\tilde r_{\rm ud}+\tilde\sigma_{\rm ud}$,
where $\tilde r_{\rm ud}$ and $\tilde\sigma_{\rm ud}$ denote
the average distance between a spin (not necessarily of a spinon)
and its nearest antiparallel spin and the standard deviation of
$\tilde r_{\rm ud}$, respectively.
Note that $\tilde r_{\rm ud}$ ($\tilde\sigma_{\rm ud}$) is different
from $r_{\rm ud}$ ($\sigma_{\rm ud}$) in eq.~(\ref{eq:xiud}) in that
spins that constitute doublons are taken into account.
Similarly, following eq.~(\ref{eq:xiuu}), we estimate an interpair
distance as the average minimum distance between a spin and
its nearest parallel spin,
$\tilde\xi_{\rm uu}=\tilde r_{\rm uu}-\tilde\sigma_{\rm uu}$.
\par
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.0cm,clip]{65325fig22.eps}
\end{center}
\caption{(Color online)
Two estimates ($\alpha/\Delta_{\rm P}$ and $\tilde\xi_{\rm ud}$) of
the pair size $\xi_{\rm pair}$, corresponding to the coherence length,
and the average minimum distance between Cooper pairs $\tilde\xi_{\rm uu}$
are compared as functions of $|U|/t$.
The dashed line indicates $|U|/t$ where $\xi_{\rm pair}$
intersects $\tilde\xi_{\rm uu}$.
}
\label{fig:xi_sc}
\end{figure}
In Fig.~\ref{fig:xi_sc}, the $|U|/t$ dependences of $\xi_{\rm pair}$ and
$\tilde\xi_{\rm uu}$ thus estimated are compared for $n=0.5$.
The pair size $\xi_{\rm pair}$ is a monotonically decreasing
function of $|U|/t$, whereas the interpair distance $\tilde\xi_{\rm uu}$
is almost independent of $|U|/t$.
Consequently, $\xi_{\rm pair}$ crosses $\tilde\xi_{\rm uu}$ at
$|U|=|U_\xi|\sim 6.2t$ at this particle concentration.
Thus, for $|U|<|U_\xi|$, Cooper pairs penetrate each other
($\xi_{\rm pair}>\tilde\xi_{\rm uu}$) as in the Fermi liquid phase
in Fig.~\ref{fig:up-down}.
On the other hand, for $|U|>|U_\xi|$, a pair becomes almost a point
hard-core boson ($\xi_{\rm pair}\sim 0$), and is isolated from other pairs
($\xi_{\rm pair}<\tilde\xi_{\rm uu}$).
Similarly, we estimated $|U_\xi|/t$ for other values of $n$, and
found $|U_\xi|\sim 6t$ irrespective of particle densities.\cite{note-xi}
Since the above estimation of $U_\xi$ is rather broad, we consider that
$U_\xi$ should be identical to $U_{\rm co}$.
\par
Finally, we point out the difference in pairing manner between the
spin-gap transition at $U_{\rm c}$ in the normal state (\S\ref{sec:picture})
and the crossover at $\sim U_{\rm co}$ in the SC state.
Bound spinon pairs in the normal state ($|U|>|U_{\rm c}|$) dissociate
into independent spinons immediately when the pair domains overlap
with each other at $U=U_{\rm c}$.
On the other hand, Cooper pairs in the SC state remain paired, even if
pair domains come to considerably overlap for $|U|\ll |U_{\rm co}|$
in the BCS regime; there is no critical change at $U=U_{\rm co}$.
Consequently, a phase transition (crossover) arises and a spin gap
closes (survives) on the weakly correlated side in the normal
(SC) state.
The stability of Cooper pairs against mutual overlap is a current
topic.\cite{Zhu}
\par
\section{Conclusions\label{sec:summary}}
Using a variational Monte Carlo (VMC) method, we studied the features of
a spin-gap transition in a normal state and of the BCS-BEC crossover
in a superconducting (SC) state in the attractive Hubbard model (AHM)
on the square lattice.
We summarize the main results below.
\par
(1)
In the normal state, we revealed that, unlike the simple Gutzwiller
wave function (GWF), a wave function with an antiparallel-spinon binding
correlation $P_Q$ [eqs.~(\ref{eq:PQ}) and (\ref{eq:Qj})] undergoes
a first-order transition from a Fermi liquid to a spin-gapped phase
at $|U_{\rm c}|/t\sim 9$.
In the spin-gapped phase, particle density current can flow through
the hopping of doublons.
The pseudogap phase above $T_{\rm c}$ for $|U|\gsim |U_{\rm c}|$ may be
deduced from the properties of this wave function.
The mechanism of this spin-gap transition is understood to be similar to that
of a Mott transition in a repulsive Hubbard model (RHM) induced by
a doublon-holon binding correlation.\cite{Miyagawa,boson}
We would also like to realize a variational normal state that is spin-gapped
and conductive in RHM.
\par
(2)
We first applied VMC to the SC state of AHM, and confirmed that,
as $|U|/t$ increases, the mechanism of superconductivity undergoes
a crossover at approximately $|U_{\rm co}|\sim|U_{\rm c}|$ from an BCS type
to a Bose-Einstein condensation (BEC) type.
$P_Q$ is again needed to suppress the gap, which is greatly overestimated
in GWF for $|U|\gsim |U_{\rm co}|$.
In the weak-correlation regime ($|U|<|U_{\rm co}|$), the strength of SC
($T_{\rm c}$) is scaled with quantities related to the SC gap
as $\sim\exp(-t/|U|)$, as expected from the BCS theory.
For $|U|>|U_{\rm co}|$, the superfluid stiffness, which is related to
the cost of phase coherence, scales with $T_{\rm c}$ as $t^2/|U|$.
Such typical features of this crossover are captured by the energy gain
in the SC transition $\Delta E$ in the whole range of $|U|/t$.
In the BEC regime, the SC transition is induced by a gain in kinetic
energy; this aspect is in contrast to the BCS theory, but is in accord with
the magnetic and SC transitions in strongly correlated RHM.\cite{YTOT,Y12}
Most features are consistent with the framework of BCS-BEC crossover
that previous studies provided.\par
(3)
The physics of a spin-gap transition in the normal state and the BCS-BEC
crossover in the SC state are explained in a semiquantitative manner
by a simple idea of the competition between the pair size $\xi_{\rm ud}$
and the interpair distance $\xi_{\rm uu}$, as shown
in Fig.~\ref{fig:up-down}.
This idea is equivalent to that of Mott transitions
in RHM,\cite{Miyagawa,boson} in which a doublon-holon pair corresponds
to the singlet pair here.
\par
(4)
In connection with that observed high-$T_{\rm c}$ cuprates, the $|U|/t$
dependence of the pair correlation function $P_\infty$ and the gap parameter
$\Delta_{\rm P}$ studied here qualitatively differ from the doping rate
($\delta$) dependence of the corresponding quantities ($P_d^\infty$ and
$\Delta_d$) in the strongly correlated RHM, when the relevant parameters
($|U|/t$ and $\delta$) are in the respective BEC regimes.
Furthermore, in a strongly correlated RHM, the $d_{x^2-y^2}$-wave SC
transition is always kinetic-energy-driven, regardless of $n$.\cite{Y12}
We will address this subject more carefully in upcoming publications.
\par
\bigskip
\begin{acknowledgments}
\noindent
{\bf Acknowledgments}
\par\medskip
The authors thank Tomoaki Miyagawa for helpful discussions.
This work is partially supported by Grant-in-Aids from the Ministry of
Education, Culture, Sports, Science, and Technology, Japan.
\end{acknowledgments}
|
{
"timestamp": "2012-04-25T02:03:25",
"yymm": "1203",
"arxiv_id": "1203.1719",
"language": "en",
"url": "https://arxiv.org/abs/1203.1719"
}
|
\section{Introduction}
\label{intro}
Ground- and space-based transit surveys are discovering a growing number of exoplanets in the super-Earth range, defined alternately as between 1 and 10 Earth masses or between 1 and 2 Earth radii. The largest such survey, being conducted by the {\it Kepler} spacecraft, has already announced hundreds of super-Earth candidates (Kepler Objects of Interest) and one confirmed super-Earth in the ``habitable zone'' \citep{BoruckiII}, although only a few of these candidates have been confirmed. Nevertheless, these surveys have yielded several exoplanets for which it is possible to begin characterizing their atmospheres via transit spectroscopy. This has been attempted by several groups for GJ 1214b \citep{Bean2011,Berta II,Croll,Crossfield,Desert,de Mooij}, an estimated 6.55 Earth-mass ($M_\Earth$) planet orbiting an M-dwarf star with an equilibrium temperature near 500 K \citep{Charbonneau}, with conflicting results. In particular, measured transit depths in the $K$ and $K_s$ bands are in direct conflict, and proposed atmospheric models have been unable to fit observations in other bands simultaneously.
GJ 1214's faintness ($m_V = 14.67$) makes it a challenge for spectroscopic observations. However, the recent discovery of transits of 55 Cnc e around a naked-eye star (Winn et al. 2011) holds promise for improved spectroscopic characterizations in the near future. These stars bring their own challenges of smaller transit depths and fewer nearby comparisons stars, but efforts are nonetheless underway to characterize them.
While transit and radial velocity observations can together constrain a planet's mass and radius (and thus its density), this leaves significant degeneracies in its bulk composition. Planets with similar masses and radii can take on different compositions, ranging from a mini-Neptune with a thick H/He layer to a true super-Earth composed mostly of silicates with a high-molecular-weight atmosphere to a ``water world'' whose bulk composition and atmosphere both have high water content \citep{Valencia,Fortney,Seager,Sotin,Rogers}. To further confuse matters, recent observations suggest that some giant exoplanets may have significantly higher C/O ratios than their parent stars, opening up the additional possibility of carbon-rich compositions \citep{Madhusudhan}.
GJ 1214b faces the additional complication that while its measured mass and density suggest a hydrogen-dominated atmosphere, the measured radius of the parent star, GJ 1214, is significantly larger than predicted by stellar models \citep{Charbonneau}. If the star is 15\% smaller than was measured, as these stellar models predict, GJ 1214b would be correspondingly smaller and denser (given the same transit depth), and a hydrogen-rich atmosphere would not be indicated. This uncertainty means that we cannot definitively distinguish between a hydrogen-rich and a hydrogen-poor atmosphere without spectroscopic data \citep{Miller-Ricci2010,Carter}.
Spectroscopic observations of both secondary eclipses (``emission'') and transits (``transmission'') of exoplanets have proven successful in characterizing the atmospheres of gas giants, e.g. \citep{Burrows2009,Charbonneau2002,Deming,Knutson,Swain}, and observations to characterize super-Earths continue. Transit spectra are particularly useful for measuring compositions. While secondary eclipses can be used to measure temperatures and compositions, the transit spectrum is most directly sensitive to the column depth at the limb of individual molecular and atomic species and is only weakly dependent on temperature (for a given scale height). In this paper, we present calculations of transit spectra for GJ 1214b for a variety of atmospheres in order to improve the current fits to the data. We also provide a library of models for future exploration of transit spectra of super-Earths.
Transit spectroscopy measures the apparent size of a planet in various wavelengths, which depends on the averaged properties of the planet's atmosphere. Because exoplanets cannot be resolved by current telescope technology, this transit spectrum is a measure of the total amount of starlight blocked in transit at each wavelength. Ignoring the effects of limb darkening, this includes a fixed component due to the solid disk of the planet, plus a wavelength-dependent component absorbed or scattered by the atmosphere. Within the atmosphere, different altitudes will contribute different amounts to this component because of the variations with pressure. However, molecular absorption features should still emerge from these spectra in a predictable way, enabling abundance characterization of planetary atmospheres based on transit observations. Conversely, the absence of these features in the observed spectrum might be indicative not only of their absence, but of a cloud or haze layer, as is inferred in the atmosphere of HD 189733b \citep{Pont}, but one should keep an open mind that a small scale height that reduces the cross section of the atmosphere could also cause this effect. The molecular features can be characterized by the relative increase in transit depth when compared with the continuum (or with a local minimum in transit depth if the continuum is obscured), a contrast that we hereafter refer to as the ``amplitude'' of the spectral features.
If the atmosphere is non-isothermal or not vertically mixed, this further complicates the results. Also, the limb of the planet as seen in transit represents the terminator, where there is likely to be circulation and a change in temperature and composition. Indeed, the physics at the terminator is more complicated than any model to date. However, one-dimensional theoretical spectra are not strongly dependent on these factors and provide valuable zeroth-order approximations to estimate atmospheric compositions \citep{Miller-Ricci2009}. In our models, hydrogen-rich atmospheres are taken to be in chemical equilibrium, while hydrogen-poor atmospheres are taken to be vertically mixed.
The physical parameters of an exoplanet should have predictable effects on its transit spectrum. These mainly act through the scale height of the atmosphere, which is given by
\begin{equation}
H = \frac{kT}{\mu g},
\end{equation}
where $k$ is Boltzmann's constant, $T$ is the atmospheric temperature, $\mu$ is the mean molecular weight, and $g$ is the surface gravity. If the atmosphere is approximately isothermal, $T$ is the ``equilibrium temperature.''
The amount of starlight absorbed in transit is proportional to the depth of the atmosphere and, thus, to its scale height. This should be visible in the transit spectra as proportional changes in the amplitudes of the spectral features and the slope of the Rayleigh continuum. Therefore, these amplitudes should be directly proportional to the temperature of the atmosphere and inversely proportional to its mean molecular weight. To first order, the amplitudes of spectral features should also be inversely proportional to surface gravity, but this relation is not exact since gravity changes slightly with altitude. In very thick atmospheres, the lower gravity at high altitudes should make the scale height larger, resulting in an even larger atmosphere and larger amplitudes than a simple estimate might suggest.
The remaining parameters that affect the transit spectrum are related to the composition structure of the atmosphere. In particular, for a hydrogen-rich atmosphere, metallicity is important. While for a hydrogen-rich atmosphere metallicity does not have a strong effect on mean molecular weight, the relatively large absorption opacities of metal compounds compared with molecular hydrogen mean that significantly larger-amplitude spectral features will be found at larger metallicities than at smaller ones. The addition of clouds or haze should have the opposite effect. The wavelength dependence of extinction opacity for haze is usually smaller than for molecular species, so the corresponding spectral features should be weaker. In the simplest case of a uniformly opaque cloud layer, suggested by Croll et al. (2011) and Bean et al. (2011) for GJ 1214b, only molecular absorption above the cloud tops contributes to the variation in the spectrum, and the molecular features are suppressed. All of these relations are seen in our calculated spectra.
We present new calculations of theoretical transit spectra of extrasolar super-Earths, including Mie scattering and a range of molecular compositions. Modeling of transit spectroscopy has been established for super-Earths previously, for example, in Miller-Ricci \& Fortney (2010), and it has been used to try to characterize the atmosphere of GJ 1214b. However, we present a number of new results and models including spectra for sub-solar metallicities, enhanced C-to-O ratios, and explicit calculations for a range of haze models. We also outline a library of models for super-Earths with different parameters. We present our methodology in Section \ref{method}. We present general conclusions and demonstrations of the systematics described above in Section \ref{general}. We apply our method to GJ 1214b in Section \ref{gj1214} in an attempt to produce a better fit to observations than current models. We then present some general conclusions in Section \ref{conclusion}.
\section{Methodology}
\label{method}
For each atmosphere model, we produce a theoretical spectrum by computing the transit depth at 2500 frequency points between 0.3 and 10 ${\rm \mu m}$. At each frequency point, we numerically integrate the optical depth along the line of sight of each impact parameter, then integrate the effective area over all impact parameters. For transiting planets, we use the initial observed transit depth as a fiducial value for the radius and then normalize the spectrum by computing the average transit depth over the observational bandpass, equating this to the observed depth. Without independent knowledge of the radius of the solid surface, a procedure like this is necessary for all models and modelers.
We use an isothermal temperature-pressure ({\it T-P}) profile at the equilibrium temperature. However, we set $T = 470$ K for GJ 1214b based on the work of Miller-Ricci \& Fortney (2010), who find with more ambitious calculations that this is approximately correct over the pressure range probed by transit spectroscopy, although their approach does not fully capture the conditions at the terminator and should not be considered more reliable than any other model. Additionally, transit spectra have been shown not to be strongly dependent on the exact T-P profile used \citep{Miller-Ricci2009}. We do account for the variation of gravity with altitude when computing the pressure profile, although this effect is only a few percent in transit depth, depending on the scale height of the atmosphere. We implement haze using the Mie approximation for a monodispersed, constant abundance haze between upper and lower pressure levels. We also include the option of completely opaque cloud decks by cutting off transmission at all frequencies at pressures greater than a specified level \citep{Crossfield}.
Our chemical equilibrium abundances for molecular species were taken from Burrows \& Sharp (1999) and molecular line and collisional opacities were taken from Sharp \& Burrows (2007). We compute Rayleigh cross sections on a per-molecule basis, based on each species’ electric polarizability and assuming a simple $\lambda^{-4}$ power-law dependence. For haze components, we use Mie scattering theory for several possible haze species. Polyacetylenes are polymers based on CH units produced by UV photolysis of simple hydrocarbons, the simplest being ${\rm (CH)_n}$, and are suggested to occur on Jupiter and giant exoplanets \citep{Sudarsky}. Tholins are more complex heteropolymers produced by UV photolysis of hydrocarbons, nitrogen, and oxygen and are thought to occur in the atmosphere of Titan and on the surfaces of other outer solar system bodies \citep{Khare}. They take the form of cross-linked chains of a wide variety of organic subunits, including aromatic hydrocarbons, other rings, alkanes, and polypeptides, with no definite order \citep{Sagan}. We also consider sulfuric acid because of its presence on Venus. Complex indices of refraction were taken from Khare et al. (1984) for tholins, from Bar-Nun et al. (1988) for polyacetylenes, and from Palmer \& Williams (1975) for sulfuric acid. Our absorption opacities for pure molecular species at a temperature and density representative of GJ 1214b are shown in Figure \ref{opacities}. The extinction opacities of hazes of various sizes and compositions are shown in Figure \ref{haze-xsec}.
We calculate transit spectra for a variety of atmospheric compositions. The hydrogen-rich compositions we compute include metal abundances varying from 0.01 to 30 times solar, solar abundances with methane set to zero, and otherwise-solar abundances with the abundances of carbon and oxygen reversed, resulting in a C/O ratio of $\sim$2. We consider an atmosphere with no methane because methane is easily destroyed by UV photolysis, and it has been suggested as a possibility to explain the appearance of GJ 1214b's spectrum \citep{Miller-Ricci2010,Bean2011,Desert}. We consider an elevated C/O ratio because such a possibility has been advanced for giant planets, including those with a low C/O ratio in the host star \citep{Madhusudhan} and for terrestrial planets \citep{Kuchner}. The C/O ratio is especially important because it strongly affects the abundances of common molecular species such as ${\rm H_2O, CO, CO_2, and \, CH_4}$.
For hydrogen-poor models, we use vertically mixed atmospheres that can include $\mathrm{H_2}$, $\mathrm{CO_2}$, $\mathrm{H_2O}$, $\mathrm{CH_4}$, $\mathrm{NH_3}$, $\mathrm{CO}$, and $\mathrm{N_2}$ in arbitrary proportions by volume. We assume nitrogen to have Rayleigh scattering opacity only.
To test the code, we compute transit spectra for atmospheres with pure molecular compositions and for solar metallicities over a range of metallicities, temperatures, and opaque clouds placed at varying altitudes. These calculations successfully reproduce the relations predicted by systematics, and they are discussed in detail in Section \ref{general}. The spectra for pure molecular atmospheres succeed in reproducing the molecular opacity features of each species (compare Figures \ref{opacities} and \ref{pure}). Smooth regions in the spectra represent the Rayleigh continuum (or, if the continuum is sufficiently transparent, the cloud tops or the planet's solid surface) at wavelengths where absorption is negligible.
We also compute spectra for GJ 1214b with compositions similar to those used in Miller-Ricci \& Fortney (2010) with a ``surface pressure'' of 1 bar. At this pressure level, the atmosphere becomes opaque at all wavelengths in most models, so a deeper atmosphere will not affect the spectrum. Our spectra successfully reproduce the molecular features of the Miller-Ricci \& Fortney (2010) computations. They have systematically larger transit depths than Miller-Ricci \& Fortney (2010) in the molecular features by 10\%-20\%, and they agree to better than 10\% with the slope of the Rayleigh continuum. The systematic offsets are most likely due to the use of different opacity tables.
\section{Generic Model Results}
\label{general}
For convenience, all generic spectra are computed using the physical parameters of GJ 1214b as given in Tables \ref{planets} and \ref{stars}, except as specified. They are also normalized to the initial measurement of GJ 1214b: 1.35\% (13,500 ppm) in the MEarth band of 0.7-1.0 ${\rm \mu m}$ \citep{Charbonneau}.
Generic spectra computed for hydrogen-rich atmospheres for which abundances are a fraction or multiple of solar are shown in Figure \ref{Metals}. These spectra show transit depths increasing with metallicity. This is expected since metals are the main source of extinction in a haze-free atmosphere. However, for this normalization, the transit depth of the Rayleigh scattering tail decreases with increasing metallicity. This is because of the normalization to the MEarth band, in which absorption dominates. The spectral features in that band increase in amplitude with increasing metallicity relative to the Rayleigh continuum, so the continuum has a smaller relative transit depth at large metallicities.
Two other hydrogen-rich atmosphere models are shown in Figure \ref{Other}. Swapping the abundances of carbon and oxygen from solar levels--in effect, raising the C/O ratio from 0.5 to 2.0--diminishes the spectral features of water and enhances those of methane. Some of these features overlap and, thus, those wavelengths are not strongly affected. The greater extinction opacity of methane at near-infrared wavelengths causes the Rayleigh continuum to be normalized to a lower transit depth. For the model with no methane, the spectrum is more strongly dominated by water features, and the normalization is not strongly affected.
The effects of temperature on the generic transit spectra are shown in Figure \ref{Temperature}. A higher temperature results in a larger scale height, and, as expected, a larger continuum slope at short wavelengths. As the temperature increases, two major chemical changes are evident as well--the suppression of methane and the appearance of strong alkali metal lines.
An opaque cloud layer can suppress the molecular spectral features if it lies at high enough altitudes. If light is not sufficiently blocked as it grazes the cloud tops, the observed radius should be very near the cloud tops in that wavelength. This effect is demonstrated in Figure \ref{Pressure}, which includes spectra with cloud layers at different pressure levels. Flat, horizontal sections of the spectra represent wavelengths where the atmosphere is transparent down to the cloud tops. If the clouds are deeper than the 1 bar level, they are usually obscured at all wavelengths, so higher pressures have no effect on the spectrum. However, for some atmospheres composed of pure molecular species, such as carbon dioxide, light can penetrate deeper at some infrared wavelengths, where both absorption and scattering are negligible, so these flat regions due to clouds can persist to very high pressures in such an atmosphere.
Models for pure atmospheric compositions are shown in Figure \ref{pure}, to be compared with Figure \ref{opacities}, which depicts the absorption opacities for the same species at a fixed temperature and density. Nitrogen is assumed to have Rayleigh scattering opacity only, as are the other species at wavelengths outside their respective absorption bands. All of these results are consistent with the expected dependencies on physical and atmospheric parameters and, thus, provide a further validation of our methodology.
We add haze to the atmosphere using the Mie approximation for spherical particles. We find that dense hazes ($n \gtrsim 10^{5} \, \mathrm{cm^{-3}}$ for most typical sizes) are essentially opaque in the wavelength range we study and result in flat regions in the spectrum. However, very thin hazes ($n \lesssim 10^2 \, \mathrm{cm^{-3}}$ for typical particle sizes) become transparent at some wavelengths. In general, these thin hazes tend to produce sloped regions of the spectrum similar to those produced by Rayleigh scattering. However, these regions do not span a wide range of transit depths because many molecular features persist to much lower pressures than typical hazes. Examples of spectra with haze included are shown in Figure \ref{Haze}.
\section{GJ 1214b}
\label{gj1214}
\subsection{Prior Observations}
\label{prior}
The photometric and spectroscopic data for GJ 1214b available at this writing are shown in Figure \ref{Data}. D\'{e}sert et al. (2011), working in the Warm Spitzer 3.6 and 4.5 ${\rm \mu m}$ bands, observed a flat transit spectrum of GJ 1214b and concluded that its atmosphere must have a small scale height and, thus, a high molecular weight, probably with a large proportion of water. Bean et al. (2010), working in the $I$ band, came to the same conclusion, although they also suggested that the planet could have a layer of clouds or hazes above the 300-mbar level.
This picture changed when Croll et al. (2011) observed a deep $K_s$-band transit, but shallow $J$ and $\mathrm{CH_4On \, (1.69 \, \mu m)}$ band transits, suggesting a hydrogen-rich, low-molecular-weight atmosphere, observable in the $K_s$ band, that included a haze layer to mute features at other wavelengths. The anomalous $K_s$-band feature was attributed to either water or methane.
However, Crossfield et al. (2011), based on near-IR spectroscopy, argue that a hydrogen-rich atmosphere is not favored, and that if a large-scale-height atmosphere were present, it must be out of chemical equilibrium (e.g., depleted in methane) or muted by a haze or cloud layer. Bean et al. (2011) revised their earlier assessment with data in the $R$, $J$, $H$, and $K$ bands. These data support a high-molecular-weight atmosphere, particularly with shallow K-band transits. However, their reported results in the R-band show some evidence for a Rayleigh scattering tail at short wavelengths, with a slope consistent with a hydrogen-rich atmosphere.
The most recent observations continue to give conflicting results. De Mooij et al. (2011), taking measurements in a wide range of visible and infrared bands, obtained results that reinforce both the conclusion of both apparent Rayleigh scattering at short wavelengths in Bean et al. (2011) and Croll's deep $K_s$-band measurement, providing further evidence for a hydrogen-rich atmosphere. However, the small relative size of the observed $K_s$-band feature, compared with theoretical predictions for a hydrogen-rich atmosphere, suggested an improbably low metallicity for the planet, despite orbiting a star with normal metallicity \citep{de Mooij}. Finally, Berta et al. (2011b) observed what appears to be a feature at $\sim$1.5 ${\rm \mu m}$ with the {\it Hubble Space Telescope's} WFC3, which could correspond to water, the depth of which is consistent with a high-molecular-weight atmosphere.
To explain these conflicting results, one of the most widely postulated models is a hydrogen-rich atmosphere with a layer of clouds or haze near or above the 100 mbar pressure level \citep{Croll,Bean2011}, or above the 10 mbar level in Berta et al. (2011b). Haze has also been suggested to explain the weak alkali metal features in transits of the giant planet HD 189733b \citep{Pont}. The other alternative is a high-molecular-weight atmosphere with a high water content \citep{Desert,Berta II}.
\subsection{Models without Haze}
\label{nohaze}
Our full list of atmosphere models for GJ 1214b is given in Table 3. Assuming that GJ 1214b has a low albedo and its heat flow is distributed uniformly over its surface, its equilibrium temperature is $\sim$555 K \citep{Miller-Ricci2010}. If we further assume that the characterization of a Rayleigh scattering tail is correct ($\alpha = 4$), then the complete data set suggests a mean molecular weight for GJ 1214b's atmosphere of around 2.7, based on the analytic model given in the Appendix. This mean molecular weight definitely represents a hydrogen-rich atmosphere, but it leaves the problem of the lack of expected spectral features unresolved.
Working from this initial prediction of a hydrogen-rich atmosphere, Figure \ref{solar-model} shows the spectrum of a solar-abundance atmosphere model for GJ 1214b compared with the observational data. It is normalized to the initial observation of GJ 1214b of a transit depth of 1.35\% (13500 ppm) in the ``MEarth'' bandpass of 0.7-1.0 ${\rm \mu m}$ \citep{Charbonneau}. This model spectrum bears almost no resemblance to the data, since the features are far too large and of different relative proportions than the observed features. Water features are most prominent in the solar model spectrum, with one that coincides with the suspected 1.5 ${\rm \mu m}$ feature, except in magnitude. The Rayleigh continuum is visible as a smooth rise in transit depth blueward of 0.7 ${\rm \mu m}$ with a similar slope to the observed rise at short wavelengths, but a far lower transit depth. Also, absorption dominates over scattering well into the optical range, so that the Rayleigh tail is predicted primarily at wavelengths shorter than the observations. Methane features can also be identified, such as those at 2.2 and 3.3 ${\rm \mu m}$, although the prominent methane features at shorter wavelengths overlap with water features.
By ignoring one or more observed features at a time, the small amplitude of the remaining features can be explained by one or more of four possibilities: (1) the atmosphere contains low abundances of molecules with large opacities; (2) the atmosphere contains an opaque cloud layer that mutes the molecular absorption features; (3) the atmosphere contains a translucent haze layer that mutes the molecular absorption features; and (4) the atmosphere has a high molecular weight, which results in a smaller scale height, a shallower atmosphere, and, thus, a smaller variation in transit depth.
Figure \ref{metals-model} shows the spectrum of two low-metallicity hydrogen-rich models. Hydrogen has a relatively low absorption opacity compared with metal compounds, so reducing the metal abundance makes most of the features much smaller in amplitude. The model spectrum begins to conform with the data at metallicity of ${\rm [Fe/H]} \lesssim -2$; the computed Rayleigh tail extends redward into the observed tail, and the spectrum is at least marginally consistent with most of the infrared observations, including the suggested 1.5${\rm \mu m}$ feature, the $K_s$-band, and the Spitzer bands. While this is a relatively good fit, given the difficulty of reconciling the various observations at different wavelengths, the star GJ 1214 has a metallicity of ${\rm [Fe/H]} = +0.39$ \citep{Charbonneau}, making the existence of an associated planet with such a low metallicity extremely unlikely. We therefore do not advance this solution.
Figure \ref{clouds-model} portrays spectra of model atmospheres with solar abundances with the addition at an opaque cloud layer at various pressure levels. An opaque cloud layer near or above the 100-mbar level has been suggested in several other models for GJ 1214b \citep{Croll,Bean2011}, and above the 10 mbar level in Berta et al. (2011b). We find that the spectrum becomes flat enough to be consistent with observations only if the cloud layer cuts off near or above the 1 mbar level. In this case, only very large features remain visible in transit in the rarefied regions above the cloud layer. If the cloud tops are between 1 mbar and 0.1 mbar, the spectrum is consistent with most of the measured infrared features (the ${\rm K_s}$ band being the primary exception) and would be a viable model if all of the high-amplitude features are rejected. Unfortunately, placing clouds at such a high altitude also suppresses the Rayleigh tail due to the molecular features, leaving the spectrum inconsistent with the observed short-wavelength rise. However, a translucent haze layer remains a viable alternative (see below).
Figure \ref{water-model} provides spectra of model atmospheres with high molecular weights, consisting of various proportions of water, with the remainder nitrogen, which we take to have only Rayleigh scattering opacity. With a higher molecular weight, the scale height of the atmosphere is smaller, as are the spectra features. Almost any proportion of water is consistent with most of the observed infrared features, again with the $K_s$ band being the primary exception. The best fit is an abundance of $\lesssim 10\%$, water, which reproduces the suggested 1.5 ${\rm \mu m}$ feature in Berta et al. (2011b) and the K-band measurements of Bean et al. (2011). However, these models fail to produce the short-wavelength rise, and the lower abundances fail to produce the slight observed rise in the mid-infrared \citep{Desert}. Furthermore, the molecular weights of water and nitrogen result in a scale height so small that the slope of the Rayleigh tail is much less than that of the observed rise. This can be confirmed using the analytic model in the Appendix. The slope of the Rayleigh tail in a pure water atmosphere (molecular weight 18) is expected to be less than one-sixth the slope of the observed rise.
As before, the addition of an opaque cloud layer only exacerbates the problem. This effect is illustrated in Figure \ref{water-clouds-model}, specifically, with a pure water atmosphere. As before, the addition of clouds improves the agreement with the observed molecular features, but obscures the Rayleigh tail, leaving it a flat continuum if the cloud tops are above 10 mbar.
Of the other molecules we consider, methane is the only one that might be consistent with the observations (Figure \ref{ch4}). A pure methane atmosphere fits the large transit depth observed in the $K_s$ band, but the overall fit is poor. A composition of $\lesssim 10\%$ methane does better, fitting the mid-infrared datapoints and the 1.5-${\rm \mu m}$ feature well, but it predicts additional near-infrared features that are not observed, and it does not fit either of the conflicting $K$-band and $K_s$-band measurements. Therefore, we conclude that the principal source of absorption opacity is most likely water.
\subsection{Models with Haze}
\label{haze}
On the problem of the short-wavelength rise, the high-molecular-weight model could possibly be salvaged with the addition of a haze layer. Some possible haze components in this temperature range are polyacetylene, tholin, and sulfuric acid. Sulfuric acid is known to be in the atmosphere of Venus, but is considered less probable than carbon-based haze species, most notably due to the uncertainty of producing it in a water- or hydrogen-rich atmosphere. For this reason, and because its extinction opacity has a similar wavelength dependence to tholins at most wavelengths and particle sizes (see Figure \ref{haze-xsec}), we do not consider it as a solution for GJ 1214b in this work. However, we do produce some general models including sulfuric acid haze in our online library. These similarities in behavior also mean that the haze composition is ambiguous without more data.
Figure \ref{haze-xsec} shows the extinction opacities of the all three of these haze components by particle size. At certain particle sizes, the opacities of these particles exhibit a slope significantly larger than that of the Rayleigh continuum, meaning that they could potentially be used to fit the observed rise, even in a hydrogen-poor atmosphere. Since we find that hydrogen-rich models also fail to recreate this rise without a haze layer, we reach the conclusion that if the short-wavelength rise is valid, then GJ 1214b possesses a haze layer, which necessarily mutes molecular absorption features, but produces a significant rise in transit depth at short wavelengths.
For a haze layer to recreate the observed rise in transit depth, it must be thick enough to avoid being nearly transparent at all wavelengths, which would produce a negligible effect on the spectrum. It must also be thin enough to avoid being nearly opaque at all wavelengths, since this would approach the limit of the opaque cloud layer, which also fails to agree with observations at short wavelengths. These limits vary with the particle density and the thickness of the haze layer.
For polyacetylene particles of any size, the wavelength dependence is weak at short wavelengths. A model representative of the effects of monodispersed polyacetylene haze in a hydrogen-poor atmosphere is shown in Figure \ref{poly-model}. The pressure range of the haze layer is chosen to be 0.1-0.001 mbar to maximize the contribution above Rayleigh scattering to total extinction at large radii, which should increase the slope of the scattering tail. This haze model has a vertical optical depth of $\tau = 0.0034$ at 0.85 ${\rm \mu m}$, the midpoint of the normalization band. The small slope of the opacity curve results in a similarly small slope of the scattering spectrum, only slightly larger than the pure Rayleigh tail with no haze. This has a very slight effect on the rest of the spectrum. The resulting slope remains too small to account for the observed tail, so we do not support a polyacetylene haze in a high-molecular weight atmosphere as a model of GJ 1214b.
The largest slope for haze extinction opacities occurs with very small ($\lesssim \, 0.03 \, {\rm \mu m}$) tholin particles, making this the best candidate to fit the observed rise in a hydrogen-poor atmosphere. Figure \ref{tholin-model} portrays several models with monodispersed tholin haze with the same geometric optical depth. As before, the haze layer is put between 0.1 mbar and 0.001 mbar. At 0.85 ${\rm \mu m}$, the middle of the normalization band, the vertical optical depth of the haze component in the 0.1 ${\rm \mu m}$ model is $\tau = 0.068$. The corresponding values for the 0.03 ${\rm \mu m}$ and 0.01 ${\rm \mu m}$ haze models are $\tau = 0.010$ and 0.021, respectively. Changes in transit depth in the 0.7-1.0 ${\rm \mu m}$ band affect the normalization, but all of the models are about equally consistent with the infrared observations. At short wavelengths, the slope of the rise in transit depth is greatest with the smallest particles, 0.01 ${\rm \mu m}$. This is consistent with observations if the data points with the largest transit depths are excluded. There is some evidence for this idea in the qualitative bifurcation of the data blueward of 0.8 ${\rm \mu m}$, although there are measured points from both Bean et al. (2011) and from de Mooij et al. (2011) at both large and small transit depths.
The problem of hydrogen-rich models with hazes presents similar constraints. The slope required to reproduce the observed short-wavelength rise is a less strict constraint because of the lower molecular weight of the atmosphere, a fact that justifies a renewed interest in polyacetylene haze. However, a hydrogen-rich atmosphere provides an additional constraint in that large opacities are needed in the near- and mid-infrared to mute the large absorption features, especially the suggested 1.5 ${\rm \mu m}$ feature.
From Figure \ref{haze-xsec}, the wavelength dependence of both haze species we consider is very similar at particle sizes of 1 ${\rm \mu m}$ and larger. Furthermore, the wavelength dependence is almost completely flat for 10 ${\rm \mu m}$ particles and is relatively flat with 1 ${\rm \mu m}$ particles, since the particles are similar in size or larger than the wavelengths of light of interest, so their effects are roughly proportional to their geometric cross sections. Figure \ref{tholin1} shows the effect of a solar-abundance atmosphere with various monodispersed tholin haze models with a particle size of 1 ${\rm \mu m}$. At 0.85 ${\rm \mu m}$, the vertical optical depths of the models are 0.20 for a particle density of $0.1 \, {\rm cm^{-3}}$ and 0.020 for a particle density of $0.01 \, {\rm cm^{-3}}$. Because of the flat wavelength dependence, especially in the visible, the resulting spectra are flattened uniformly to varying degrees, so that the short-wavelength rise is suppressed, and the model fails to achieve the goal of fitting the short-wavelength data. However, with a relatively large particle density of $0.1 \, {\rm cm^{-3}}$ and a high altitude of 0.1-0.001 mbar, the spectrum begins to conform with the near- and mid-infrared data. While it is easier to keep small particles aloft, we do not know the degree of turbulence in GJ 1214b's atmosphere, so we do not make any judgment concerning the plausibility of these conditions. Further research is needed to investigate this issue.
Figure \ref{tholin0.1} shows a set of models with hydrogen-rich atmospheres with monodispersed tholin hazes with a particle size of 0.1 ${\rm \mu m}$. At 0.85 ${\rm \mu m}$, the vertical optical depths of the models are 0.82 for a particle density of $1000 \, {\rm cm^{-3}}$ and 0.082 for a particle density of $100 \, {\rm cm^{-3}}$. This figure demonstrates a problem that occurs in a large portion of the parameter space of hazes. While the haze dominates the atmospheric opacity at visible wavelengths, its opacity becomes too small at infrared wavelengths, resulting in infrared features that remain too large in amplitude to fit the data. This problem often persists even if the haze opacity saturates in the visible, and the layer becomes uniformly opaque at short wavelengths. Depending upon the choice of parameters, these models can be made to fit the short-wavelength rise in the data and could do so even if some of these data were refuted. However, in each case, the infrared features either are far too large to fit the data or are normalized to an incorrect transit depth.
Figure \ref{tholin0.01} provides models with hydrogen-rich atmospheres with monodispersed tholin hazes with a particle size of 0.01 ${\rm \mu m}$. At 0.85 ${\rm \mu m}$, the vertical optical depths of the models are 0.26 for a particle density of $10^7 \, {\rm cm^{-3}}$ and 0.026 for a particle density of $10^6 \, {\rm cm^{-3}}$. The results are much the same as for the 0.1 ${\rm \mu m}$ models. The opacity saturates at short wavelengths, sometimes resulting in a flat spectrum in the visible, while lower infrared opacities result in near- and mid-infrared features that are too large to fit the data. In other cases, the opacity does not saturate at these wavelengths, but the slope of the predicted short-wavelength tail is much larger than is observed. Because of the difficultly of fitting the short-wavelength rise and the improbability of particles this small, we do not advance this model as an explanation for the observed features.
Polyacetylene has a spectral behavior similar to tholins at larger particle sizes ($\gtrsim$1 ${\rm \mu m}$), and at smaller particle sizes ($\lesssim$0.1 ${\rm \mu m}$), it suffers from even lower infrared opacities than tholins, so the same problems persist that occur for tholins. A haze model that might successfully fit both the observed short-wavelength rise in the data and the low amplitudes of the infrared features is one for which the opacity follows a Rayleigh-like $\lambda^{-4}$ behavior blueward of 1 ${\rm \mu m}$ and is roughly constant redward of 1 ${\rm \mu m}$. None of the hazes we test fits this description.
\subsection{Best-fit Models}
\label{best}
Our best fit models for GJ 1214b are given in Figure \ref{1214b-best}. We select five models by visual inspection. While this is not, in principle, the most accurate method, the systematic problems with the data set indicate that any quantitative assessment of the fits would not necessarily be any better. Moreover, many of our theoretical spectra are similar enough that a range of parameters of the haze layer (in some cases as large as a factor of two) would produce very similar fits. Thus, any quantitative fit would not produce a significant gain in precision over our "eyeball" technique.
Of our five selections, Model 1 is the only model we select that provides a good fit to the observed short-wavelength rise in transit depth. It utilizes a solar-abundance atmosphere with a tholin haze (although the choice of haze species is arbitrary) with a particle size of 0.1 ${\rm \mu m}$, a particle density of 100 ${\rm cm^{-3}}$, and a pressure range of 10 to 0.1 mbar. This model has a vertical optical depth at 0.85 ${\rm \mu m}$ of $\tau = 0.082$. It is consistent with many of the near- and mid- infrared data points, but the predicted spectral features have much larger amplitudes than are observed.
Models 2-4 are our best fits to the data points with low transit depths, with larger transit depths excluded. Model 2 does this with a solar-abundance atmosphere and a translucent tholin haze layer (the choice of haze species being arbitrary) with a particle size of 1 ${\rm \mu m}$, a particle density of 0.1 ${\rm cm^{-3}}$, and residing in a pressure range from 10-0.1 mbar; it has a vertical optical depth at 0.85 ${\rm \mu m}$ of $\tau = 0.20$. Model 3 instead uses solar abundances with a uniformly opaque cloud layer, with cloud tops at 0.3 mbar. This model provides a better fit in the near- and mid-infrared than the translucent haze.
Model 4 is our overall best fit using a high-molecular-weight atmosphere, specifically, 1\% ${\rm H_2O}$ and 99\% ${\rm N_2}$ with no clouds or haze. However, this model is not very sensitive to the proportion of water (within an order of magnitude) or to the addition of any haze that does not significantly obscure the absorption features.
Model 5 is our best fit to the short-wavelength rise with a high-molecular-weight atmosphere. It includes a 1\% ${\rm H_2O}$ and 99\% ${\rm N_2}$ atmosphere with a tholin haze with a particle size of 0.01 ${\rm \mu m}$, a particle density of $10^6 \, {\rm cm^{-3}}$, and residing in a pressure range from 0.1 to 0.001 mbar; it has a vertical optical depth at 0.85 ${\rm \mu m}$ of $\tau = 0.021$. Here, the choice of tholins is not arbitrary. Polyacetylene has a near-discontinuous wavelength dependence in the near-infrared at this particle size, which makes it unsuitable for this model. This haze model is relatively improbable because of its small particle size, and it is consistent only with the lower edge of the observed short-wavelength rise in transit depth. Therefore, we conclude that if this rise is valid, GJ 1214b should have a hydrogen-rich atmosphere.
A hydrogen-rich model provides a better numerical fit to the large-transit-depth data, and the systematics predict a low mean molecular weight for the atmosphere if the short-wavelength rise is valid. Furthermore, while it is possible to fit the low amplitudes of the observed near- and mid-infrared features with a hydrogen-rich model, it is not possible to fit the large-amplitude features with a hydrogen-poor model. Therefore, we conclude that a hydrogen-rich atmosphere is more probable overall for GJ 1214b, although we cannot rule out a hydrogen-poor model based on the current data. Because a hydrogen-rich model predicts larger-amplitude features, observations of the transit depth around 2.7 and 3.3 ${\rm \mu m}$, and to a lesser extent near 1.9 ${\rm \mu m}$, could be valuable for distinguishing hydrogen-rich from hydrogen-poor models.
We are unable to produce a model that fits all of the data, even after omitting either the ${\rm K_s}$-band or K-band measurements. Indeed, the best fit overall is the unacceptably improbable 0.01$\times$ solar model with no haze. If all of the non-conflicting data are valid, then we can say with confidence that there is a haze layer, but we do not make any predictions about its properties. Additionally, our methodology rules out any hazeless hydrogen-rich model in general and hydrogen-rich models with tholin hazes in the 0.01 ${\rm \mu m}$ range, as well as any model in which the largest source of absorption is not water. We look forward to refining our model in light of additional data.
\section{Discussion and Conclusions}
\label{conclusion}
We have developed a new capability to compute transit spectra of super-Earths that allows atmospheres with arbitrary proportions of common molecular species, along with hazes. Similar hazes have been used to characterize the atmospheres of giant planets, and they appear to be important in explaining the apparently conflicting observations of GJ 1214b. However, additional measurements are needed in a range of wavelengths to determine this object's transit spectrum more precisely.
If the observed short-wavelength rise is valid, then GJ 1214b most likely possesses a hydrogen-rich atmosphere with a haze of small ($\sim$0.1 ${\rm \mu m}$) particles, although there is also the possibility of an atmosphere rich in nitrogen and water, with very small ($\sim$0.01 ${\rm \mu m}$), very high altitude (up to 0.001 mbar) tholin haze particles. The latter model is a poorer fit in the visible and is physically less probable, but it cannot be ruled out by the current data. However, the hydrogen-rich model provides a very poor fit redward of 1 ${\rm \mu m}$.
Alternatively, the short-wavelength measurements may prove inaccurate, but the near-infrared measurements may be accurate, including the 1.5-${\rm \mu m}$ suspected water feature and the low-amplitude $K$-band measurements. This is a particularly plausible scenario because of the greater stellar variability and lower luminosity for M-dwarfs in the optical. In this case, in order of goodness of fit, GJ 1214b likely possesses (1) an ${\rm N_2-H_2O}$ atmosphere, (2) a solar-abundance atmosphere, with thick, opaque clouds at or above the 1 mbar level, or (3) a solar-abundance atmosphere with a translucent haze with medium-to-large ($\gtrsim$1 ${\rm \mu m}$) particles. For hydrogen-rich atmospheres, the models are not sensitive to the haze composition. High-molecular-weight models are not sensitive to the addition of hazes or clouds in general or to the abundance of water, within an order of magnitude. None of these models shows a short-wavelength rise in transit depth or a $K_s$-band feature as deep as the one suggested by the data.
In both cases, our calculations rule out a hazeless (and cloudless) hydrogen-rich atmosphere, which predicts a transit depth in the optical that is much smaller than the data for any reasonable metallicity. We also rule out a hydrogen-rich atmosphere with tholin particles in the $\sim$0.01 ${\rm \mu m}$ range and a high-molecular-weight atmosphere with significant amounts of absorbing species other than water, such as ${\rm NH_3, CO, or \, CO_2}$, which is inconsistent with the near-infrared data. Overall, the prospects for fitting the data appear better with a hydrogen-rich atmosphere, but we cannot rule out a hydrogen-poor composition.
Some of the data for GJ 1214b conflict, particularly those in the $K$ band and $K_s$ band. Even for the data that do not directly conflict, none of the models we test provides a good fit to all of the observed features simultaneously. However, we can rule out large classes of models that do not fit any of the observed features. In the case that all of the non-conflicting data are valid, we do not support any model without a haze layer.
We propose additional observational tests to distinguish hydrogen-rich from hydrogen-poor atmospheres. All of our hydrogen-rich models predict a strong water feature at 2.7 ${\rm \mu m}$ and a strong methane feature at 3.3 ${\rm \mu m}$, both of which should be unambiguously larger in amplitude than the features predicted by any reasonable hydrogen-poor model. The hydrogen-rich models also predict a water feature at 1.9 ${\rm \mu m}$ that could give an unambiguous result depending on the actual atmospheric parameters. Therefore, data at these wavelengths would provide strong evidence to distinguish among these models.
Our models correctly reproduce the expected systematics of transit spectra of planets with varying metallicities, temperatures, and cloud layers. They also reproduce the expected features from molecular absorption. Nevertheless, our models use an isothermal {\it T-P} profile. While this is a good first-order approximation, a more accurate {\it T-P} profile is desired to produce a more accurate transit spectrum, along with an atmospheric circulation model of the behavior at the terminator, which will not be in thermal or chemical equilibrium. A photochemical model is also desired to determine the abundances of molecular species created or destroyed by photolysis.
We have prepared an online library of theoretical transit spectra produced by our calculations, sampling the known and expected parameter space of super-Earths. This library includes models with equilibrium temperatures from 300 K to 1000 K and silicate-iron planets from 1 ${M_\Earth}$ to 10 ${M_\Earth}$. The library also includes models with atmospheric compositions of solar abundance, 0.3$\times$ solar, 3$\times$ solar, and pure ${\rm H_2O, CO_2, and \, CO}$ atmospheres, as well as a variety of haze models, including sulfuric acid hazes.
Very recently, we learned of the work of Murgas et al. (2012), who measured the transit depth of GJ 1214b at three additional wavelengths in the $R$ band. For completeness, these data points are included in Figure \ref{Data}. These points are consistent with the proposed rise in transit depth at short wavelengths; however, their uncertainties are too large to make any definite conclusions.
\acknowledgments
We thank Matteo Brogi and Zachory Berta for providing data in electronic form prior to publication, and Jacob Bean, Ian Crossfield, and Jean-Michel D\'{e}sert for helpful conversations, comments, and suggestions. The authors also acknowledge support in part under NASA ATP grant NNX07AG80G, {\it HST} grants HST-GO-12181.04-A, HST-GO-12314.03-A, and HST-GO-12550.02, and JPL/Spitzer Agreements 1417122, 1348668, 1371432, 1377197, and 1439064. We have prepared a large suite of models of transit spectra of super-Earths for a range of masses, radii, temperatures, and compositions. These models are made available at http://www.astro.princeton.edu/$\sim$arhowe and \\ http://www.astro.princeton.edu/$\sim$burrows.
|
{
"timestamp": "2012-08-31T02:00:24",
"yymm": "1203",
"arxiv_id": "1203.1921",
"language": "en",
"url": "https://arxiv.org/abs/1203.1921"
}
|
\section{Introduction}
Harmonically confined few-particle quantum systems and especially
their time-dependent properties
are an important subject of experimental
and theoretical research activities.
For example, correlated electrons in metal clusters \cite{baletto} or quantum
dots \cite{filinov1, filinov2, reimann}
and ultracold Bose and Fermi gases in traps or optical
lattices \cite{bloch1, giorgini} have been investigated
in recent years.
Particularly, Bose-Einstein condensation in low dimensions \cite{goerlitz} and
nonideality (interaction)
effects \cite{menotti, moritz, pedri, bloch2}, including
superfluidity and crystallization \cite{filinov3},
lately raised attention.
Among these properties the behaviour of the breathing mode (BM) attracts special
interest, as it is easily excited experimentally \cite{moritz} and turns out
\cite{bauch} to give information on a variety of system properties.
The BM describes a radial expansion and contraction of a finite system and
is characterized by two frequencies in the general quantum case.
In our previous work \cite{bauch} we have shown for a 2-particle
system that one of those frequencies changes with the
system dimensionality, the particle spin and the strength of the pair interaction.
These results were extended \cite{hochstuhl} to 4 and 6
particles and to different inverse power law
interaction potentials \( w(r) \propto r^{-d} \) with \(d=1,2,3 \).
In this paper we present new results for larger particle numbers
and we show how for different coupling strenghts the corresponding breathing
frequencies depend on the particle number.
For that purpose we present the results of time-dependent
Hartree-Fock simulations for up to 20 particles and compare them to
the results of exact CI results for up to 8 particles.
\section{Theory}
\subsection{Time-dependent Schr\"odinger equation}
We briefly recall the theoretical background of the BM \cite{bauch, hochstuhl}.
Generally, a system of \( N \) interacting particles can be described by the Hamiltonian
\begin{equation}
\label{eq:tidhamiltonian}
\hat H_0 = \sum _{i=1}^N \hat h_i + \sum_{i < j}^N w(|\mathbf{r}_i - \mathbf{r}_j|) \;,
\end{equation}
where
\begin{equation}
\hat h_i = \hat t_i + v(\mathbf r_i)
\end{equation}
is the single-particle Hamiltonian and \( w(|\mathbf{r}_i - \mathbf{r}_j|) \) is a
binary interaction potential. In this case,
the external single-particle potential \( v(\mathbf r_i) \) is
chosen to be harmonic, i.e.
\begin{equation}
v(\mathbf r _i) = \frac{1}{2} m \Omega^2 \mathbf r_i^2 \;.
\end{equation}
\(v\) serves as a trapping potential with the trapping
frequency \( \Omega \). In the following, we concentrate on
Coulomb-interacting particles with equal masses \(m\) and
equal charges \(e\), e.g. electrons.
Thus, the interaction potential has the form
\begin{equation}
w(|\mathbf{r}_i - \mathbf{r}_j|) = \frac{a}{|\mathbf{r}_i - \mathbf{r}_j|}
\end{equation}
with \( a \equiv e^2 / (4\pi \varepsilon_0 )\). Finally, using the notation \( \mathbf r \equiv (\mathbf r_1, \ldots, \mathbf r_N ) \), the \(N\)-particle
time-dependent Schr\"odinger equation (TDSE) reads
\begin{align}
\mathrm i \hbar \frac{\partial}{\partial t} \mathrm{\Psi}(\mathbf r, t) =& \Bigg[\sum_{i=1}^N
\bigg(\frac{-\hbar^2}{2m} \frac{\partial^2}{\partial \mathbf r_i^2}
+ \frac{1}{2} m \Omega^2 \mathbf r_i^2 \bigg) \nonumber \\
&
+\sum_{i < j}^N \frac{a}{|\mathbf{r}_i - \mathbf{r}_j|} \Bigg] \mathrm{\Psi}(\mathbf r, t) \;.
\end{align}
For convenience, we introduce the scaled
quantities \( \tilde{\mathbf r}_i = \mathbf r_i / l_0 \)
and \( \tilde t = \Omega t \), so that after omitting the
tilde symbol,
the TDSE can be written in the dimensionless form
\begin{align}
\label{eq:schroedrescaled}
\mathrm i \frac{\partial}{\partial t} \mathrm \Psi(\mathbf r, t) =& \Bigg[ \frac{1}{2}
\sum_{i=1}^N \bigg(-\frac{\partial^2}{\partial \mathbf r_i^2} +
\mathbf r_i^2 \bigg) \nonumber \\
&+ \lambda \sum_{i < j}^N \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|}
\Bigg] \mathrm \Psi(\mathbf r, t) \; ,
\end{align}
where \( l_0= \left( \hbar / \left( m\Omega\right)\right)^{-1/2} \)
is the well-known oscillator length and
\begin{equation}
\lambda = \frac{mal_0}{\hbar^2}
\end{equation}
is the so-called coupling parameter.
Due to the rescaling, there will only be dimensionless quantities
throughout this work. For example, lengths, times and energies
are given in units of \(l_0\), \(\Omega^{-1}\) and \( \hbar \Omega\),
respectively.
The meaning of \(\lambda\) can be interpretated as follows.
Defining the scale of the potential energy as
\begin{equation}
E_0 = \frac{1}{2}m\Omega^2 l_0^2
\end{equation}
and the mean interaction energy as
\begin{equation}
E_\mathrm{C} = \frac{a}{2l_0} \;,
\end{equation}
one finds
\begin{equation}
\lambda = \frac{E_\mathrm{C}}{E_0} \,.
\end{equation}
Hence, \( \lambda \) can be understood as the ratio of the interaction
energy and the confinement energy. The influence of the value of
\(\lambda\) is described later in Sec. \ref{sec:lambda}. The actual
excitation of the breathing mode is realized by a fast switch of
the trapping potential. For a short period of time \(t_\mathrm{off} \)
the single-particle potential \( v \) is completely turned off.
As a consequence, the time-dependent Hamiltonian takes the form
\begin{align}
\label{eq:tdhamiltonian}
\hat H(t) = &\sum _{i=1}^N \hat h_i +
\left[ \theta(t_0-t) + \theta(t - t_\mathrm{off}) \right] v(\mathbf r_i) \nonumber \\
&+ \sum_{i < j}^N w(|\mathbf{r}_i - \mathbf{r}_j|) \;.
\end{align}
During the off-time the particles are driven out of their initial state. When the potential is restored, the time-dependent expectation value of some quantities start to oscillate. In particular for the breathing mode, the oscillation of the single-particle potential energy is dominated by a beating of two harmonic oscillations. The corresponding frequencies will be called \( \omega_r \) and \( \omega_R \) from now on. Both a typical time series of the potential energy and the excitation process are demonstrated in Fig. \ref{fig:zeitserie}. In the next subsections we want to point out some algebraically accessible properties of the frequencies and their relations to the coupling parameter \(\lambda\).
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=1]{zeitserie.pdf}
\end{center}
\caption{Exemplary time series (\(N=2,\; \lambda=1 \)) of the potential energy \( \langle E_\mathrm{pot} \rangle \). The waveform shows a superposition of two harmonic oscillations with frequencies \( \omega_r \) and \( \omega_R \). The inset demonstrates the excitation of the breathing mode.}
\label{fig:zeitserie}
\end{figure}
\subsection{Separation of center-of-mass motion}
As it has already been shown\cite{bauch}, the system always possesses a universal breathing mode whose frequency has the value \(\omega_R=2\). In the following, we will derive this result and show that this value is independent of the coupling strength, the system dimensionality and the particle number.
The basic idea is the introduction of center-of-mass and relative coordinates for the solution of Eq. (\ref{eq:schroedrescaled}) with a product ansatz. The center-of-mass coordinate is given by
\begin{equation}
\mathbf{R} \equiv \frac{1}{N} \sum_{i=1}^N \mathbf{r}_i\;,
\end{equation}
and the set of relative coordinates is given by
\begin{equation}
\mathbf{x} \equiv \left( \mathbf{x}_{1},\; \mathbf{x}_{2},\dots, \mathbf{x}_{N-1} \right)\;,
\end{equation}
with the definition
\( \mathbf{x}_{i} \equiv \mathbf{r}_{i,i+1} \equiv \mathbf{r}_{i+1} -\mathbf{r}_{i} \).
Thus, \( \mathrm{\Psi}(\mathbf{r},t) \) in
Eq. (\ref{eq:schroedrescaled}) is replaced
by \(\mathrm{\Psi}(\mathbf{R},\mathbf{x} ,t)\).
Now the transformation is shown for each term in Eq. (\ref{eq:schroedrescaled}), starting with
\begin{align}
-\frac{1}{2}\sum_{i=1}^N \frac{\partial^2}{\partial \mathbf{r}_i^2}
&=-\frac{1}{2}\left( \sum_{i=1}^N \frac{\partial^2}{\partial \mathbf{R}^2} \frac{1}{N^2} +2\sum_{i=1}^{N-1} \frac{\partial^2}{\partial \mathbf{x}_{i}^2} \right) \nonumber \\
&=-\frac{1}{2N} \frac{\partial^2}{\partial \mathbf{R}^2} - \sum_{i=1}^{N-1} \frac{\partial^2}{\partial \mathbf{x}_{i}^2} \;.
\end{align}
For the second term we obtain
\begin{align}
\label{eq:potenergie_falle}
\frac{1}{2}\sum_{i=1}^N \mathbf{r}_i^2 = \frac{1}{2}N\mathbf{R}^2 + \frac{1}{4N} \sum_{i=1}^N\sum_{k=1}^N \mathbf{r}_{ik}^2 \;,
\end{align}
where \( \mathbf{r}_{ik}^2 \) still has to be expressed in relative coordinates \( \mathbf{x} \).
This can be be done as follows:
\begin{align}
\mathbf{r}_{ik}=
\begin{cases}
\displaystyle \sum_{l=i}^{k-1} \mathbf{x}_{l} & \text{for } i < k \\
\displaystyle \sum_{l=k}^{i-1} \mathbf{x}_{l} & \text{for } i > k \\
0 & \text{for } i = k \;.
\end{cases}
\end{align}
Finally, the third term takes the form
\begin{align}
\lambda \sum_{i<j}^N \frac{1}{|\mathbf{r}_i -\mathbf{r}_j|}
= \lambda \sum_{i<j}^N \frac{1}{\left| \sum_{l=i}^{j-1} \mathbf{x}_{l}\right|}\;.
\end{align}
As a result the Hamiltonian can be split in two independent contributions:
\begin{align}
\hat{H}_\mathbf{R} = -\frac{1}{2N} \frac{\partial^2}{\partial \mathbf{R}^2} + \frac{1}{2}N\mathbf{R}^2
\end{align}
and
\begin{align}
\hat{H}_{\mathbf{x}} = &-\sum_{i=1}^{N-1} \frac{\partial^2}{\partial \mathbf{x}_{i}^2} + \frac{1}{4N} \sum_{i=1}^N\sum_{k=1}^N \mathbf{r}_{ik}^2 \nonumber \\
&+ \lambda \sum_{i<j}^N \frac{1}{\left| \sum_{l=i}^{j-1} \mathbf{x}_{l}\right|} \;.
\end{align}
Hence, the TDSE takes the form
\begin{align}
\label{eq:tdse_trafo}
\mathrm{i}\frac{\partial}{\partial t} \mathrm{\Psi}(\mathbf{R},\mathbf{x},t) = \left( \hat{H}_\mathbf{R} + \hat{H}_{\mathbf{x}} \right) \mathrm{\Psi}(\mathbf{R},\mathbf{x},t) \;.
\end{align}
The product ansatz
\begin{equation}
\mathrm{\Psi}(\mathbf{R},\mathbf{x},t) = \phi(\mathbf{R},t) \, \varphi(\mathbf{x},t)
\end{equation}
yields the independent problems
\begin{equation}
\label{eq:tdse_sp}
\mathrm{i}\frac{\partial}{\partial t}\phi(\mathbf{R},t)= \hat{H}_\mathbf{R} \phi(\mathbf{R},t)
\end{equation}
and
\begin{equation}
\mathrm{i}\frac{\partial}{\partial t} \varphi(\mathbf{x},t) = \hat{H}_{\mathbf{x}} \varphi(\mathbf{x},t) \;.
\end{equation}
The center-of-mass problem can be transformed to the standard oscillator form
\begin{equation}
\label{eq:tdse_sp_scaled}
\mathrm{i}\frac{\partial}{\partial t} \phi(\tilde{\mathbf{R}},t)=\left(-\frac{1}{2} \frac{\partial^2}{\partial \tilde{\mathbf{R}}^2} + \frac{1}{2}\tilde{\mathbf{R}}^2 \right) \phi(\tilde{\mathbf{R}},t)\;,
\end{equation}
where the rescaling \( \tilde{\mathbf{R}} = \sqrt{N} \mathbf{R} \) has been used.
Now consider an initial state, which can be expressed by
\begin{equation}
\label{eq:tdse_anfangsbed}
\phi(\tilde{\mathbf{R}},t=0)=\sum_n c_n \phi_n(\tilde{\mathbf{R}}) \;,
\end{equation}
where \( \phi_n \) is a solution of
\begin{equation}
\left( -\frac{1}{2} \frac{\partial^2}{\partial \tilde{\mathbf{R}}^2} + \frac{1}{2}\tilde{\mathbf{R}}^2 \right) \phi_n(\tilde{\mathbf{R}}) = E_n \phi_n(\tilde{\mathbf{R}}) \;.
\end{equation}
The associated energy eigenvalues are well-known: \( E_n = n+ d/2 \) for all \( n \in \{ 0,1,2,\dots \} \), and the time evolution of the state in Eq. (\ref{eq:tdse_anfangsbed}) is given by
\begin{equation}
\phi(\tilde{\mathbf{R}},t)=\sum_n c_n \phi_n(\tilde{\mathbf{R}}) \mathrm{e}^{-\mathrm{i}E_nt}\;.
\end{equation}
The breathing mode manifests itself in the dynamics of the quantity \( \mathbf{r}^2 = \sum_{i=1}^N \mathbf{r}_i^2 \). Using center-of-mass and relative coordinates, this quantity can be expressed according to Eq. (\ref{eq:potenergie_falle}). For the first term in Eq. (\ref{eq:potenergie_falle}) a breathing frequency can be obtained by determining the expectation value
\begin{equation}
\label{eq:exvalue}
\langle \mathbf{R}^2 \rangle(t) = N^{-1/2}\langle \tilde{\mathbf{R}}^2 \rangle(t) \;.
\end{equation}
The result for this expression is
\begin{align}
\label{eq:ewbmqm}
\langle \tilde{\mathbf{R}}^2 \rangle(t)
&= \int \mathrm{d}\tilde{\mathbf{R}} \,\mathrm{d}\mathbf{x} \, \mathrm{\Psi}^*(\tilde{\mathbf{R}},\mathbf{x},t) \tilde{\mathbf{R}}^2 \mathrm{\Psi}(\tilde{\mathbf{R}},\mathbf{x},t) \nonumber \\
&= \sum_{i,j}c_i^*c_j\mathrm{e}^{-\mathrm{i}(j-i)t} ( \tilde{\mathbf{R}}^2)_{ij} \;,
\end{align}
with
\begin{align}
( \tilde{\mathbf{R}}^2)_{ij} = \int \mathrm{d}\tilde{\mathbf{R}} \, \phi_i^* (\tilde{\mathbf{R}}) \tilde{\mathbf{R}}^2 \phi_j(\tilde{\mathbf{R}}) \;.
\end{align}
It can be shown with a reduction to the matrix elements of a
one-dimensional harmonic oscillator that only the cases
\( i=j\) and \( i=j\pm2\) contribute to the last summation in Eq. (\ref{eq:ewbmqm})
(the case \( i=j\) does not correspond to an oscillation).
The only frequency appearing in the oscillation is thus
given by \( \omega_{{R}}=2 \). As the coupling parameter \(\lambda\)
does not appear in the center-of-mass problem, the center-of-mass mode with frequency \( \omega_{{R}}=2 \) is present for all couplings. In summary, we have shown that the frequency \( \omega_R \) is universal, but its amplitude tends to vanish for large particle numbers since it is proportional to \( N^{-1/2} \).
\subsection{Influence of coupling parameter and limiting cases}
\label{sec:lambda}
As we have seen in the last subsection, the introduction of relative and
center-of-mass coordinates has led to a separation ansatz.
The center-of-mass Hamiltonian \(\hat H_\mathbf{R} \) yields a breathing
frequency \( \omega_R = 2\). It has already been mentioned that the
breathing mode also exhibits another frequency \(\omega_r \). The properties
of this frequency are an interesting subject of numerical analysis. Only in two
limiting cases the values of \(\omega_r\) are known from algebraic calculations.
In the pure quantum limit, \( \lambda = 0\), the particles are completely uncorrelated.
As the interaction term in the Hamiltonian is cancelled out, a degenerate
frequency \(\omega_r=\omega_R=2\) occurs. On the contrary, in the classical
limit \( \lambda \to \infty\) the frequency \( \omega_r\) has
the value \(\sqrt{3} \) \cite{peeters, dubin, henning}. In both cases the
frequency does not depend on the particle number or the dimensionality of
the system. For arbitrary coupling parameters the values of \( \omega_r \)
are expected to be in the interval \(\left[\sqrt 3, 2 \right] \). To clarify
the influnce of the system parameters such as the coupling parameter and the
particle number on \( \omega_r\) is one of the main goals of our investigation.
\section{Simulation Methods}
Whereas the solution of the time-dependent Schr\"odinger equation could only be performed for 2 particles, the frequencies for up to 8 particles are still accessible through exact Configuration Interaction calculations.
These results are used to support the accuracy of time-dependent Hartree-Fock calculations, which have been performed for even higher particle numbers (up to 20).
Due to the high computational effort, we restrict ourselves to the solution of a 1-dimensional system. Nevertheless such a system can be regarded as a basic theoretical model which requires a deepened understanding.
In order to suppress spin effects, only spin-polarized systems are investigated. Before the breathing mode is excited, the system is in the energetically lowest anti-symmetric state.
In the following, we give a brief discussion of the employed methods and present their numerical results. It shall already be mentioned here that in order to avoid divergencies in the interaction potential all methods use a regularized Coulomb potential
\( \lambda / | \mathbf r_i - \mathbf r_j + \kappa ^2 | \), where \( \kappa \) is a small finite cut-off parameter.
\subsection{Time-dependent Schr\"odinger equation}
In our previous work \cite{bauch} we determined \(\omega_r(\lambda)\) for \(N=2 \) in the whole range \( \lambda = 0 \ldots \infty\). These values are the basis for the comparison with other methods. Our TDSE results were obtained by solving the time-dependent Schr\"odinger equation with two different methods. On the one hand, a standard grid-based Crank-Nicolson scheme was used, and on the other hand the wave function was expanded into a set of basis functions (oscillator eigenfunctions). The results confirm the values of the breathing frequencies in the limiting cases (\( \lambda = 0 \) and \( \lambda = \infty \)) and yield a continuous function \( \omega_r(\lambda) \) for all other couplings in between.
\subsection{Configuration Interaction}
Configuration Interaction (CI) is another method for obtaining numerically exact results. The basic idea of CI is to expand the wave function in a complete set of Slater determinants, which in turn are constructed with a complete single-particle basis.
It has already been stated in Eq. (\ref{eq:tdhamiltonian}) that the excitation of the breathing mode is realized by a fast Heaviside-type switch-off of the single-particle confinement potential. Assuming the excitation to be infinitely short, the Hamiltonian is given by
\begin{equation}
\hat{H}(t) = \hat{H}_0 + \eta \hat{H}_1 \delta(t) \;,
\end{equation}
where the time-independent part \(\hat{H}_0\) is that of Eq. (\ref{eq:tidhamiltonian}). If \( \hat{H}_0 \) is diagonalized by the eigenfunctions \( | \Psi_n \rangle\) with eigenvalues \( E_n \), the application of perturbation theory yields that the expectation value of an arbitrary observable can be calculated by
\begin{equation}
\langle \hat{A} \rangle (t) = \sum_{i,j} c^*_i c_j \: e^{\mathrm{i}(E_i - E_j) \,t } \: \langle \Psi_i | \hat{A} | \Psi_j \rangle \;,
\end{equation}
where \( c_{i,j} \) are time-independent coefficients. As a consequence of this relation, the oscillation of the expectation value is restricted to frequencies \( \omega_{ij} \equiv | E_i - E_j | \).
Instead of time-propagating the solution of the Schr\"odinger equation, we can use this result and extract the breathing frequencies from the eigenvalues of \( \hat{H}_0 \) with relatively little computational effort.
However, this method is only applicable for small particle numbers because of the strongly increasing size of the required basis sets for higher particle numbers.
All results were produced with a basis of oscillator functions. Especially for weak couplings this basis set is well adjusted to the physical problem and the number of basis functions can be kept low.
Just like the TDSE results the CI results can be used as a benchmark for the accuracy of the time-dependent Hartree-Fock results.
\subsection{Time-dependent Hartree-Fock}
For larger particle numbers (\(N \geq 7\)) approximation methods have to be employed, because the basis size for reasonable CI results dramatically increases. In the following, the interaction of the system is approximated on the mean-field level. The method of choice is the use of Nonequilibrium Green's functions $G^<(1,2)$, where $1\equiv(\mathbf{r}_1,t_1)$. Employing the Hartree-Fock (HF) approximation, the Green's functions obey the Keldysh/Kadanoff-Baym equations \cite{bonitzintro, kadanoffbaym}
\begin{align}
\label{MS1}
\left(i \partial_{t_1}-h^\mathrm{HF}_1 (1)\right)G^<(1,2) &= 0 \;,\\
\label{MS2}
h^\mathrm{HF}_1(1) &= h^0(1)+h^\mathrm{HF}[G^<](1)\;,
\end{align}
where $h^0(1)$ is the single-particle part of the Hamiltonian (\ref{eq:tidhamiltonian}).
In such a reduced quantum statistical description of the system all one-particle
quantities (and some \(N\)-particle quantities, e.g. the total energy) can be derived from the one-particle density matrix~\cite{bonitzteub}
\begin{equation}
F^1(\mathbf{r}_1,\mathbf{r}_2,t) = - \mathrm{i} \lim_{t' \to t^+ } G^<(\mathbf{r}_1,t,\mathbf{r}_2,t') \;,
\end{equation}
where the limit is taken from above. The advantage of the quantum statistical description is the fact that the particle number
is only a parameter and does not determine the size of a whole system of equations as in an equivalent wave-function-based method.
The Green's function is expanded in FE-DVR (finite-element discrete variable representation) basis functions
\cite{fedvr1, fedvr2} with up to $\sim$400 basis functions. The numerical procedure is to calculate the ground
state (with inverse temperature \(\beta\to\infty \)), switch off and on the confinement and continue propagating
the Green's function in time. In each step the expectation values $\langle E_\mathrm{pot} \rangle$ are calculated and saved.
Finally, one can analyze the spectrum of the time series and extract the breathing frequency out of it.
Compared to methods based on perturbation theory, which only involve the ground state energies, this method is quite time consuming.
Since the computation of just a single frequency can last more than one day on one CPU, the paramaters have to be chosen carefully in order to guarantee converged results
while keeping the durations of the computations acceptable.
\section{Results}
As mentioned before, we want to concentrate on presenting the results of our time-dependent Hartree-Fock (TDHF) calculations and show the dependency of \( \omega_r \) on the particle number. In this work we only investigate the cases \(\lambda~=~0.1\), \(0.3\) and \(1\). Larger values of \( \lambda \) would require to go beyond Hartree-Fock, which demands a very high computational effort. Before we show our results, we start by explaining some important aspects concerning the spectral determination of the breathing frequencies.
\begin{figure}[b]
\begin{center}
\includegraphics[scale=1]{plotspektra.pdf}
\end{center}
\caption{Comparison of the spectra for 2, 10 and 18 particles at different coupling strengths \( \lambda \). The left peaks represent the relative mode with frequency \( \omega_r \). Peaks with slightly higher frequencies than the center-of-mass frequency \( \omega_R = 2 \) can be found in case \( \lambda=1 \). For \( \lambda=0.1 \) these peaks and the center-of-mass peaks might overlap.}
\label{fig:spectra}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1]{spektrumges.pdf}
\end{center}
\caption{Spectrum corresponding to time series in Fig. \ref{fig:zeitserie} (TDHF) in comparison with the spectrum of the solution of the Schr\"odinger equation (TDSE) for the same configuration. The two breathing frequencies \( \omega_r \) and \( \omega_R \) can be identified in both spectra. However, the TDHF spectrum exhibits numerous additional peaks. One of those (unphysical) peaks is near \(\omega = 2\) and has a high amplitude. Aside from that both spectra show higher harmonics, which, however, do not agree with each other.}
\label{fig:spekges}
\end{figure}
\subsection{Spectral Analysis}
The spectra of all TDHF results were calculated from time series which have the length of at least \( t_\mathrm{prop} =2000\)
(in units \(\Omega^{-1}\)). Since the resolution in the frequency space is limited by the size of \( t_\mathrm{prop} \),
we applied spline interpolations to the spectra in order to achieve a higher accuracy for the frequency values.
Besides, each spectrum was calculated with a Blackman window in order to uncover peaks with small spectral weights.
In Fig. \ref{fig:spectra} the interesting part of the spectra around the breathing
frequencies is shown for different couplings and particle numbers. The peaks corresponding to \( \omega_r \) can be
clearly identified. However, the identification of the center-of-mass peaks corresponding to
\( \omega_R = 2 \) is problematic. Only in the case \( \lambda = 1 \) it is evidently possible
to find peaks at frequency \( \omega = 2 \), although they have a very small spectral weight and tend to vanish with
increasing particle number. Surprisingly, peaks with slightly higher frequencies than \( \omega = 2 \) and
a rather strong dependency on the particle number are observed. As their spectral weight is some orders of magnitude larger than that of the center-of-mass peaks,
one has to be careful not to confuse those peaks with the center-of-mass peaks. This is important when regarding the spectra in cases
\( \lambda = 0.1 \) and \( \lambda = 0.3 \), because for these couplings the small peaks
at \( \omega = 2 \) vanish. It is conceivable that these peaks and those slightly higher than \( \omega = 2 \) tend to overlap.
The question remains why the spectra show such strong additional peaks. In Fig. \ref{fig:spekges} the spectra
of both TDSE and TDHF calculations are shown. Both spectra were produced with the same parameters
(\( \lambda=1, N=2, \kappa^2 = 0.1\)). As it can be seen, the TDHF spectrum contains several additional peaks.
It turns out that the appearance of (unphysical) frequencies is a typical effect of the TDHF approximation.
Consequently, it can be a challenging task to distinguish between real physical frequencies and artefacts of
the approximation method. However, in both spectra the peaks of higher harmonics occur (Only the first higher
harmonics can be seen in the figure, but the full data even contain higher harmonics). They are caused by the finite duration
of the initial excitation. Unfortunately, the values of the higher harmonics are not in a good agreement with each other.
Taking into account that the spectral weight of the higher harmonics is quite small, it is presumable that the accuracy
of the calculations is not high enough to properly represent the higher harmonics. In the following, we want to focus on
the breathing frequency \( \omega_r \) and neglect all higher order spectral features.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1]{plot_spektN.pdf}
\end{center}
\caption{Part of the spectrum around the breathing frequencies for different particle numbers at coupling strength \( \lambda = 1 \). While the relative mode frequency \( \omega_r \) slightly changes with $N$ (black line), the center-of-mass mode frequency \( \omega_R\) (black rectangle) remains constant (apart from numerical errors). Note the strongly \(N\)-dependent additional peaks on the right, which must not be confused with the \( \omega_R\) peaks.}
\label{fig:spekN}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1]{tiling.pdf}
\end{center}
\caption{Breathing frequency $\omega_{r}$ vs. particle number $N$ for coupling parameters \(\lambda~=~0.1\), \(0.3\) and \(1\) with cut-off parameter \(\kappa^2=0.1\). For \( \lambda = 1 \) the frequencies obtained for different \(\kappa^2\) are also plotted. For comparison, the TDSE frequencies for two particles and the CI results are plotted as well.
For \(\kappa^2=0.1\), the \(N\)-dependencies are qualitatively the same for all \( \lambda\).}
\label{fig:omega_lambda}
\end{figure}
\subsection{$N$-dependency of the breathing frequency}
The TDHF calculations enabled us to obtain the breathing frequencies for up to 20 particles. Apart from the problem of clearly identifying the center-of-mass mode,
we found a typical behaviour of the breathing frequency \( \omega_r \), which is qualitatively the same for all couplings.
The change of the spectra with increasing particle number is illustrated in Fig. \ref{fig:spekN}. It is obvious that the breathing frequency \( \omega_r \)
decreases to a minimum (for 5 particles, see also Fig. \ref{fig:omega_lambda}) and afterwards monotonically increases.
The figure also shows the center-of-mass peaks, which, as expected from Eq. (\ref{eq:exvalue}), tend to vanish for high particle numbers.
Furthermore, the additional peaks near \( \omega = 2 \) exhibit a rather strong \( N\)-dependency, but appear to be converged.
After analyzing the spectra for the couplings \(\lambda~=~0.1\), \(0.3\) and \(1\), it turns out that the \( N\)-dependency of \( \omega_r\)
is qualitatively equal for all \( \lambda\).
It is typical that the frequencies attain a minimum for 5 particles, before they start steadily growing.
For a complete overview the explicit values of the breathing frequencies are summarized in the graphs of Fig. \ref{fig:omega_lambda}. For comparison,
we also show the TDSE values for \(N=2\) and the CI values for up to
8 particles as well as the breathing frequencies obtained with other cut-off parameters.
As expected, the TDHF and the CI values are almost the same (see Fig.
\ref{fig:omega_lambda}(a)) for small \(\lambda\).
With increasing \(\lambda\) a constant shift between both results arises.
The CI results confirm that the breathing frequency has a minimum, which, however, occurs for 6 particles instead of 5 particles in the case of TDHF.
Moreover, the variation of \( \kappa \) indicates that the results are converged in the
regime \( \kappa^2 = 0.01 \ldots 0.1 \), because the results just marginally differ from each other.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1]{gsproperties.pdf}
\end{center}
\caption{\(N\)-dependent ground state properties of the interacting system. The inverse coupling parameter \(\alpha\) has a minimum for 7 particles and afterwards increases with $N$ (a).
The difference of the addition energy continuously decreases and possibly reaches the ideal value 0.5 for large particle numbers (b). The pointwise deviation of
the densities $d$ has a maximum for 4 particles and tends to vanish for large particle numbers (c).}
\label{fig:indicators}
\end{figure}
For 20 particles the frequency \( \omega_r \) has not yet reached an asymptotic value. There are, however, several indicators that for \( N \to \infty\)
the frequency \( \omega_r \) converges to the value \( \omega = 2 \) characteristic for the ideal quantum limit. In the following, we just consider the ground state.
This assumption is justified by the fact that the breathing mode is computed in linear response, which should be derivable from the ground state properties.
Firstly, if the system converges to a pure quantum state, the interaction energy must become a negligible quantity.
Assuming that the Hartree-Fock part of the total energy represents the interaction energy,
it is possible to calculate the ratio of the one-particle energy \( E^1\) to the interaction energy \(E_\mathrm{HF}\):
\begin{align}
\alpha = \frac{\langle E^1 \rangle}{\langle E_\mathrm{HF}\rangle} = \frac{\langle E_\mathrm{kin}\rangle+\langle E_\mathrm{pot}\rangle }{\langle E_\mathrm{HF}\rangle} \;.
\end{align}
This quantity can be interpretated as the inverse of an effective coupling parameter.
The values of \(\alpha \) for up to 70 particles are shown in Fig. \ref{fig:indicators}(a). As expected, \(\alpha \)
increases with the particle number. Interestingly, \(\alpha \) has a minimum for 7 particles.
Secondly, one can easily show that for an ideal quantum system the total energy per particle increases by \(0.5\) each time a particle is added to the system, i.e.
\begin{align}
\frac{ \Delta E_\mathrm{tot}}{N} \equiv \frac{E^N_\mathrm{tot} - E^{N-1}_\mathrm{tot}}{N} = 0.5 \;.
\end{align}
Calculating this addition energy for the interacting system, we get a result which is presented in Fig. \ref{fig:indicators}(b). The values slowly converge against the ideal value \(0.5\),
although still larger values of $N$ would be required to prove this limit.
A third indicator is the particle density \( n(\mathbf r) \) (or in the one-dimensional system \( n(x) \)). We want to show that
for \(N \to \infty\) the density converges to the density of the ideal system. An appropriate quantity for that purpose is the pointwise
squared deviation of the normalized densities, i.e.
\begin{align}
d \equiv \frac{1}{N}\sqrt{ \sum_{i} \left( n_\mathrm{ideal}(x_i)-n_\mathrm{interacting}(x_i)\right)^2 } \;.
\end{align}
The factor \(1/N \) prevents this quantity from diverging and gives it a relative character. (Recall the normalization
\( \int n(x) \mathrm{d}x = N\).) The values for up to 500 particles are shown in Fig.
\ref{fig:indicators}(c). As expected, $d$ tends to vanish for large particle numbers and noticeably has a maximum for 4 particles. In order to demonstrate the change in the densities, the normalized densities of the ideal quantum system and the interacting system are plotted for 2 and 30 particles in Fig. \ref{fig:densities}.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=1]{densities.pdf}
\end{center}
\caption{Comparison of the normalized densities of an ideal quantum system and an interacting system ($\lambda=0.1$) for 2 and 30 particles. The deviation from the ideal system vanishes for large particle numbers.}
\label{fig:densities}
\end{figure}
Although all of the indicators are still not able to prove that for \(N \to \infty\) the system converges to an ideal quantum system,
they give an evident hint to this hypothesis.
An explanation for the transition to a pure quantum system might be that for large particle numbers the
system reaches an extension which is dominated by the trap potential (\(\propto \mathbf{r}_i^2 \)). This is a consequence of the Pauli exclusion principle for fermions. Moreover, it is apparent that
the quantities \(\alpha \) and \(d\) have an extremum near the minimum of the breathing frequency. Yet it is not possible to clearly reconstruct
that mimimum from the ground state.
\subsection{Conclusions}
We showed that the breathing frequency \( \omega_r \) bears an \(N\)-dependency which is similar for the
coupling strengths \(\lambda=~0.1\), \(0.3\) and \(1\). Presumably, the same behaviour can be observed for arbitrary
finite couplings. We found that the breathing frequencies decrease to a minimum
for 5 particles ($N=6$, for CI) and afterwards monotonically increase.
The origin of the minimum of $\omega_r$ is the competition between two
restoring forces: first, the Coulomb repulsion between
the particles which is strongest for $\lambda \to \infty$ (point charge)
and which favours $\omega_r = \sqrt{3}$. Second, the kinetic
energy of a quantum system that is maximal for $\lambda=0$. The minimum
appears where the ratio of the two, $1/\alpha$ has
its maximum. It may be expected that the occurence of this maximum is
related to particularly stable cluster configurations.
Although we did not investigate more than 20 particles, we found some hints that for \(N\to\infty\) the breathing frequency
converges to the frequency \( \omega = 2 \) of an ideal quantum system.
In contrast to the increase of $\omega_r$ with $N$ at finite $\lambda$,
for $\lambda \to \infty$, the frequency is independent
of $N$ and equals $\sqrt{3}$. Thus, presumably, there exists a critical
value of $\lambda$ where the crossover between the
two behaviours occurs.
It remains a subject of further research activities
to confirm these hypotheses. Furthermore, it would be interesting to extend this analysis to higher dimensions,
other interactions (e.g. dipole interaction) and other spin properties.
|
{
"timestamp": "2012-03-12T01:01:01",
"yymm": "1203",
"arxiv_id": "1203.2019",
"language": "en",
"url": "https://arxiv.org/abs/1203.2019"
}
|
\section{Introduction}
The recently proposed \emph{compressive binary search} (CBS) algorithm \cite{2012arXiv1202.0937D} aims to determine the location of a single non-zero entry in a vector $\bm{x} \in \mathbb{R}^n$ using $m$ adaptive linear projections of the form
\begin{eqnarray} \nonumber
y_i = \left\langle \bm{a}_i, \bm{x} \right\rangle + z_i \qquad i = 1,...,m
\end{eqnarray}
where $\bm{a}_i\in\mathbb{R}^n$ are sensing vectors with $||\bm{a}_i|| =1$ and $z_i$ are i.i.d. $\mathcal{N}(0,1)$. The CBS algorithm proposed in \cite{2012arXiv1202.0937D} finds the non-zero entry with vanishing probability of error as $n$ gets large provided
$\mu \geq C \sqrt{(n/m) \log \log_2 n}$
where $\mu$ is the amplitude of the non-zero entry.
A bit more precisely, Theorem 1 in \cite{2012arXiv1202.0937D} states that $\mathbb{P}_e \leq \delta$ provided
\begin{eqnarray} \label{eqn:CBSOrig}
\mu \geq \sqrt{ \frac{8n}{m}\left( \log \frac{1}{2\delta} + \log \log_2 n \right)}
\end{eqnarray}
where $\mathbb{P}_e$ is the probability the procedure fails to return an index corresponding to the non-zero entry. The dependence on $\delta$ is to be expected, but the authors of \cite{2012arXiv1202.0937D} rightly question whether the $\log\log_2 n$ term is needed. The main contribution of this paper is a simple modification of the algorithm proposed in \cite{2012arXiv1202.0937D} which eliminates this unnecessary and suboptimal dependence on $n$.
\section{Main Result}
The CBS algorithm proposed in \cite{2012arXiv1202.0937D} operates as follows.
The algorithm consists of $s_0 = \log_2 n$ steps (as in \cite{2012arXiv1202.0937D} we assume $n$ is dyadic). At each step of the algorithm, indexed by $s$, measurements are taken on progressively smaller dyadic subintervals of $\{1,\dots,n\}$. The sensing vectors are discrete Haar wavelets, and the sign of each measurement gives an indication of whether the non-zero entry is in the right or left half-interval of the wavelet's support. The CBS algorithm is outlined in Fig.~\ref{fig:ocbs}.
\begin{figure}[h] \small
\fbox{\parbox[b]{3.35in}{{\underline{\bf Compressive Binary Search (CBS)}} \\ \vspace{-.15in} \\
number of steps: $s_0 := \log_2 n$ \\
measurements per step:
$m_s$ , with $\sum_{s=1}^{s_0} m_s \leq m$ \\
initial support: $J_0^{(1)} := \{1,\dots,n\}$ \\ \vspace{-.1in} \\
for $s=1,\dots,s_0$ \\ \vspace{-.15in} \\
1) split: $J_1^{(s)}$ and $J_2^{(s)}$, left and right subinterval of $J_0^{(s)}$ \\
2) sensing vector: $\bm{u}^{(s)} = 2^{-(s_0-s+1)/2}$ on $J_1^{(s)}$, \\ \indent \ \ \ \ $\bm{u}^{(s)}= -2^{-(s_0-s+1)/2}$ on $J_2^{(s)}$, and $0$ otherwise \\
3) measure: $y_i^{(s)} = \langle \bm{u}^{(s)},\bm{x}\rangle + z_i^{(s)}, \\
\indent \ \ \ \ z_i^{(s)} \overset{iid}{\sim} {\cal N}(0,1), \ i=1,\dots,m_s$ \\
4) update support: $J_0^{(s+1)} = J_1^{(s)}$ if $\sum_{i=1}^{m_s}y_i^{(s)}>0$\\ \indent \ \ \ \ and $J_0^{(s+1)} = J_2^{(s)}$ otherwise \\
end \\ \vspace{-.1in} \\
output: $J_0^{(s_0+1)}$ (a single index)
}}
\caption{Compressive Binary Search algorithm.}
\label{fig:ocbs}
\end{figure}
Since the sensing vectors have unit norm, the magnitude of the non-zero entries of the sensing vectors grow as the algorithm proceeds from coarse-to-fine wavelets. Consequently, the SNR of the measurements grows exponentially as the procedure progresses. In \cite{2012arXiv1202.0937D}, the authors divide the $m$ measurements among the steps as follows. For $s=1,\dots,s_0$
\begin{eqnarray*}
m_s := \tilde{m}_s +1, \qquad \tilde{m}_s = \left \lfloor (m - s_0) \; 2^{-s} \right \rfloor
\end{eqnarray*}
It is easy to check that $\sum_{s=1}^{s_0} m_s \leq m$. This allocation roughly equalizes the SNR at each step. If $\mu$ satisfies (\ref{eqn:CBSOrig}), then the CBS algorithm is guaranteed to have a probability of error of at most $\delta/s_0$ at each step (and the union bound yields an overall probability of error of at most $\delta$).
Here, instead, we allocate the $m$ measurements as follows:
\begin{eqnarray*}
m_s := \tilde{m}_s +1, \qquad \tilde{m}_s = \left \lfloor (m - s_0) \; s\; 2^{-(s+1)} \right \rfloor.
\end{eqnarray*}
Again, it is easily verified that $\sum_{s=1}^{s_0} m_s \leq m$. With this allocation, the probability of error is at most a constant times $2^{-s}$ at each step; i.e.,
the probability of error decreases exponentially over the steps, rather than remaining constant as above. This simple modification is enough to eliminate the $\log\log_2 n$ term in the bound in \cite{2012arXiv1202.0937D}.
\begin{thm}
If $m \geq 2 \log_2 n$ and
\begin{eqnarray}
m_s := \tilde{m}_s +1, \qquad \tilde{m}_s = \left \lfloor (m - s_0) \; s\; 2^{-(s+1)} \right \rfloor
\end{eqnarray}
then $\sum_{s=1}^{s_0} m_s\leq m$ and the CBS algorithm succeeds with
$\mathbb{P}_e \leq \delta$
provided magnitude of the non-zero entry satisfies
\begin{eqnarray} \label{eqn:CBSimprov}
\mu \geq \sqrt{ \frac{16n}{m}\log\left( \frac{1}{2\delta} +1 \right)}.
\end{eqnarray}
\end{thm}
\begin{proof}
The total measurement budget satisfies
\begin{eqnarray*}
\sum_{s=1}^{s_0} m_s = s_0 + \sum_{s=1}^{s_0} \tilde{m}_s \leq s_0 + (m-s_0) \sum_{s=1}^{s_0} \; s \; 2^{-(s+1)} \leq m,
\end{eqnarray*}
since $\sum_{s=1}^{s_0} s 2^{-(s+1)} \leq 1$.
By the union bound and a Gaussian tail bound, the total error probability satisfies
\begin{eqnarray*}
\mathbb{P}_e &\leq &\frac{1}{2}\, \sum_{s=1}^{s_0} \exp \left( -\frac{m_s \mu^2 2^s}{4n}\right).
\end{eqnarray*}
Since $m_s \geq (m-s_0) s 2^{-(s+1)}$ and $m \geq 2 \log_2 n = 2 s_0$, we conclude $m_s \geq\frac{m}{2} s 2^{-(s+1)}$ and thus
\begin{eqnarray*}
\mathbb{P}_e &\leq &\frac{1}{2} \, \sum_{s=1}^{s_0} \exp \left( -\frac{m s\mu^2}{16n}\right).
\end{eqnarray*}
Letting $\mu \geq \sqrt{\frac{16n}{m} \log\left( \frac{1}{2\delta} +1 \right)}$ yields
\begin{eqnarray*}
\mathbb{P}_e & \leq & \frac{1}{2} \, \sum_{s=1}^{s_0} \left(\frac{1}{2\delta} +1\right)^{-s} \ \leq \ \delta \ .
\end{eqnarray*}
\end{proof}
\setcounter{thm}{0}
\begin{remark}
The authors of \cite{2012arXiv1202.0937D} also show that no procedure can succeed at the compressive binary search problem with probability greater than $1/2$ if $\mu \leq \sqrt{n/m}$.
Using the new allocation of measurements, the CBS algorithm succeeds with probability greater than $1/2$ if $\mu \geq \sqrt{16\log(2)n/m}$, within a small constant factor of the lower bound.
\end{remark}
\begin{remark}
An algorithm similar to CBS was proposed in \cite{4786013} as a special case of a more general approach to the adaptive sensing problem, although control of the SNR across measurement steps is not discussed. Additionally, the authors in \cite{5470144} suggest and analyze an adaptive group testing procedure with many of the main ideas of CBS. The adaptive group testing procedure also has a sub-optimal $\log \log n$ dependence.
\end{remark}
\begin{remark}
The CBS problem is closely related to the so-called {\em noisy binary search} problem \cite{horstein,bz,gbs,kk}. Noisy binary search addresses a version of the classic binary search problem with binary noise. If the SNR is equal in each step of CBS (as in
\cite{2012arXiv1202.0937D}), then it is equivalent to the ``naive'' noisy binary search algorithm discussed in \cite{kk}, which also suffers from the suboptimal $\log\log_2 n$ factor.
More sophisticated algorithms such as Horstein's algorithm \cite{horstein,bz,gbs} and binary search with backtracking \cite{kk} are optimal to within constants. The CBS problem, however, is different from noisy binary search in that more localized measurements (or ``queries'') are more reliable. \newpage \noindent Because of this unique feature, the optimal algorithms for noisy binary search do not yield optimal solutions for the CBS problem. Instead, all that is needed to eliminate the $\log \log_2 n$ factor is a measurement allocation that properly exploits the fact that more localized measurements have a larger SNR.
\end{remark}
\begin{figure}[htb]
\vspace{.3cm}
\centerline{\includegraphics[width=9.6cm]{CBScompare.eps}}
\caption{Numerical Simulation. Empirical performance of CBS (both original and modified) as a function of $\mu$ for $n= 4096$ and $m = 256$. 10,000 trials.
}
\vspace{.3cm}
\end{figure}
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2012-05-09T02:04:23",
"yymm": "1203",
"arxiv_id": "1203.1804",
"language": "en",
"url": "https://arxiv.org/abs/1203.1804"
}
|
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